Abstract parabolic evolution equations and their applications

Springer Monographs in Mathematics

For further volumes: www.springer.com/series/3733

Atsushi Yagi

Abstract Parabolic Evolution Equations and their Applications

Atsushi Yagi Department of Applied Physics Graduate School of Engineering Osaka University Suita, Osaka 565-0871 Japan [email protected]

ISSN 1439-7382 ISBN 978-3-642-04630-8 e-ISBN 978-3-642-04631-5 DOI 10.1007/978-3-642-04631-5 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2009939692 Mathematics Subject Classification (2000): 35K90, 35K57, 37L30, 92D25, 92D40 © Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Contents

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Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 1 Banach Spaces . . . . . . . . . . . . . . . . . . . . . 1.1 Distance of Subsets . . . . . . . . . . . . . . 1.2 Fractal Dimension . . . . . . . . . . . . . . . 2 Function Spaces with Values in a Banach Space . . . 2.1 Uniformly Bounded Function Spaces . . . . . 2.2 Continuously Differentiable Function Spaces . 2.3 Hölder Continuous Function Spaces . . . . . 2.4 Weighted Hölder Continuous Function Spaces 2.5 Space of Analytic Functions . . . . . . . . . . 3 Linear Operators . . . . . . . . . . . . . . . . . . . . 3.1 Bounded Linear Operators . . . . . . . . . . . 3.2 Part of Linear Operator . . . . . . . . . . . . 3.3 Multivalued Linear Operators . . . . . . . . . 3.4 Resolvent Equation and Pseudo-Resolvent . . 3.5 Spectra of Operators . . . . . . . . . . . . . . 4 Nonlinear Operators . . . . . . . . . . . . . . . . . . 4.1 Contraction Mappings . . . . . . . . . . . . . 4.2 Fréchet Differentiation . . . . . . . . . . . . . 5 Interpolation of Banach Spaces . . . . . . . . . . . . 5.1 Interpolation Spaces . . . . . . . . . . . . . . 5.2 Interpolation Theorem . . . . . . . . . . . . . 6 Adjoint Spaces and Operators . . . . . . . . . . . . . 6.1 Dual Spaces . . . . . . . . . . . . . . . . . . 6.2 Adjoint Spaces . . . . . . . . . . . . . . . . . 6.3 Adjoint Operators . . . . . . . . . . . . . . . 7 Extrapolation of Hilbert Spaces . . . . . . . . . . . . 7.1 Triplets of Spaces . . . . . . . . . . . . . . . 7.2 Extrapolation Theorem . . . . . . . . . . . . 8 Linear Operators Associated with Sesquilinear Forms 8.1 Sesquilinear Forms and Associated Operators 8.2 Adjoint Forms and Adjoint Operators . . . . .

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Integral Equations and Inequalities of Volterra Type . . . . . . 9.1 Generalized Inequality of Gronwall Type . . . . . . . . 9.2 Integral Inequality of Volterra Type . . . . . . . . . . . 9.3 Integral Equations of Volterra Type . . . . . . . . . . . 9.4 Dominate Convergence Theorem for Integral Equations Differential Inequalities . . . . . . . . . . . . . . . . . . . . . Sobolev–Lebesgue Spaces . . . . . . . . . . . . . . . . . . . . 11.1 Boundaries of Domains . . . . . . . . . . . . . . . . . 11.2 Sobolev Spaces of Integral Order . . . . . . . . . . . . 11.3 Sobolev–Lebesgue Spaces in Rn . . . . . . . . . . . . 11.4 Sobolev–Lebesgue Spaces in Rn+ or Bounded Domains 11.5 Interpolation Property . . . . . . . . . . . . . . . . . . 11.6 Embedding Theorems . . . . . . . . . . . . . . . . . . 11.7 Traces . . . . . . . . . . . . . . . . . . . . . . . . . . 11.8 Spaces H˚ ps (Ω) and Hp−s (Ω) . . . . . . . . . . . . . . 11.9 Product Spaces . . . . . . . . . . . . . . . . . . . . . . 11.10 Miscellaneous Results . . . . . . . . . . . . . . . . . . Notes and Further Researches . . . . . . . . . . . . . . . . . .

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Sectorial Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Sectorial Operators in Hilbert Spaces . . . . . . . . . . . . . . 1.1 Sectorial Operators Associated with Sesquilinear Form 1.2 Sectorial Operators in L2 Spaces . . . . . . . . . . . . 1.3 Shift Property in L2 . . . . . . . . . . . . . . . . . . . 1.4 Sectorial Operators in Product Spaces of L2 . . . . . . 2 Sectorial Operators in Banach Spaces . . . . . . . . . . . . . . 2.1 Sectorial Operators in Lp (1 < p < ∞) Spaces . . . . . 2.2 Extreme Cases p = 1 and p = ∞ . . . . . . . . . . . . 2.3 Adjoint Operators . . . . . . . . . . . . . . . . . . . . 2.4 Shift Property in Lp Spaces . . . . . . . . . . . . . . . 3 Sectorial Operators in Product Spaces . . . . . . . . . . . . . . 4 Yosida Approximation . . . . . . . . . . . . . . . . . . . . . . 4.1 Uniform Estimates . . . . . . . . . . . . . . . . . . . . 4.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . 5 Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 5.1 Analytic Semigroups . . . . . . . . . . . . . . . . . . 5.2 Contraction Semigroups . . . . . . . . . . . . . . . . . 5.3 Adjoint Semigroups . . . . . . . . . . . . . . . . . . . 5.4 Approximate Semigroups . . . . . . . . . . . . . . . . 6 Generation of Analytic Semigroups . . . . . . . . . . . . . . . 6.1 Generation Results in Hilbert Spaces . . . . . . . . . . 6.2 Generation Results in Lp Spaces . . . . . . . . . . . . 7 Fractional Powers of Linear Operators . . . . . . . . . . . . . 7.1 Integral Formula . . . . . . . . . . . . . . . . . . . . . 7.2 Law of Exponent . . . . . . . . . . . . . . . . . . . . . 7.3 Law of Exponent, II . . . . . . . . . . . . . . . . . . .

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7.4 Moment Inequality . . . . . . . . . . . . . . . . . . 7.5 Comparison of Domains of Fractional Powers . . . 7.6 Fractional Powers of Adjoint Operators . . . . . . . 7.7 Fractional Powers and Analytic Semigroups . . . . 7.8 Generation Theorem of Continuous Semigroups . . Purely Imaginary Powers and Square Root Problem . . . . 8.1 Bounded Purely Imaginary Powers . . . . . . . . . 8.2 Imaginary Powers for Maximal Accretive Operators 8.3 Heinz and Kato Inequality . . . . . . . . . . . . . . 8.4 Square Root Problem . . . . . . . . . . . . . . . . 8.5 Symmetric Forms and Square Root Problem . . . . Notes and Further Researches . . . . . . . . . . . . . . . .

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Linear Evolution Equations . . . . . . . . . . . . . . . . . . . Part I. Autonomous Equations . . . . . . . . . . . . . . . . 1 Analytic Semigroups . . . . . . . . . . . . . . . . . . . . . 1.1 Generation of Analytic Semigroups . . . . . . . . . 1.2 Generator of Analytic Semigroup . . . . . . . . . . 1.3 Perturbation for Generators of Analytic Semigroups 2 Cauchy Problems . . . . . . . . . . . . . . . . . . . . . . 2.1 Construction of Solutions . . . . . . . . . . . . . . 2.2 Maximal Regularity . . . . . . . . . . . . . . . . . 2.3 Higher-Order Maximal Regularity . . . . . . . . . 2.4 Spatial Regularity . . . . . . . . . . . . . . . . . . 3 Autonomous Parabolic Equations . . . . . . . . . . . . . . 3.1 Problems in Hilbert Spaces . . . . . . . . . . . . . 3.2 Problems in Lp Spaces . . . . . . . . . . . . . . . Part II. Nonautonomous Equations . . . . . . . . . . . . . 4 Nonautonomous Abstract Evolution Equations . . . . . . . 4.1 Structural Assumptions . . . . . . . . . . . . . . . 4.2 Some Consequences from (3.30) . . . . . . . . . . 4.3 Structural Assumptions and Yosida Approximation . 5 Evolution Operators . . . . . . . . . . . . . . . . . . . . . 5.1 Integral Equations . . . . . . . . . . . . . . . . . . 5.2 Convergence of Un (t, s) . . . . . . . . . . . . . . . 5.3 Basic Properties of U (t, s) . . . . . . . . . . . . . 6 Cauchy Problems . . . . . . . . . . . . . . . . . . . . . . 7 Conditions (3.29) and (3.30) . . . . . . . . . . . . . . . . . 7.1 Case of Hilbert Spaces . . . . . . . . . . . . . . . . 7.2 Case of Lp Spaces . . . . . . . . . . . . . . . . . . 8 Maximal Regularity . . . . . . . . . . . . . . . . . . . . . 8.1 Refined Properties of U (t, s) . . . . . . . . . . . . 8.2 Maximal Regularity . . . . . . . . . . . . . . . . . 8.3 Spatial Regularity . . . . . . . . . . . . . . . . . . 9 Variational Methods . . . . . . . . . . . . . . . . . . . . .

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Semilinear Evolution Equations . . . . . . . . . . . . . . . . 1 Semilinear Abstract Evolution Equations . . . . . . . . . 1.1 Cauchy Problems . . . . . . . . . . . . . . . . . 1.2 Proof of Theorem 4.1 . . . . . . . . . . . . . . . 1.3 Regularity for More Regular Initial Data . . . . . 1.4 Global Existence . . . . . . . . . . . . . . . . . . 1.5 Lipschitz Continuity of Solutions in Initial Data . 1.6 Case β = 0 . . . . . . . . . . . . . . . . . . . . . 2 Variational Methods . . . . . . . . . . . . . . . . . . . . 3 Semilinear Parabolic Equations . . . . . . . . . . . . . . 4 Competition System with Diffusions . . . . . . . . . . . 4.1 Construction of Local Solutions . . . . . . . . . . 4.2 Nonnegativity of Local Solutions . . . . . . . . . 4.3 Global Solutions . . . . . . . . . . . . . . . . . . 5 Some Model in Immunology . . . . . . . . . . . . . . . 5.1 Construction of Local Solutions . . . . . . . . . . 5.2 Nonnegativity of Local Solutions . . . . . . . . . 5.3 Global Solutions . . . . . . . . . . . . . . . . . . 6 Nonautonomous Semilinear Abstract Evolution Equations Notes and Further Researches . . . . . . . . . . . . . . .

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Quasilinear Evolution Equations . . . . . . . . . . 1 Quasilinear Abstract Evolution Equations . . . . 1.1 Structural Assumptions . . . . . . . . . 1.2 Proof of Theorem 5.1 . . . . . . . . . . 1.3 Stronger Results for More Regular Data 1.4 Case ν ≤ β . . . . . . . . . . . . . . . . 2 Lipschitz Continuity of Solutions in Initial Data 3 Equations Including Semilinear Terms . . . . . 3.1 Existence Theorem . . . . . . . . . . . 3.2 Proof of Theorem 5.5 . . . . . . . . . . 3.3 Stronger Results for More Regular Data 3.4 Case ν ≤ β . . . . . . . . . . . . . . . . 4 Lipschitz Continuity of Solutions in Initial Data 5 Variational Methods . . . . . . . . . . . . . . . 6 Quasilinear Parabolic Equations . . . . . . . . . 6.1 Abstract Formulation . . . . . . . . . . 6.2 Construction of Local Solutions . . . . .

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Nonautonomous Parabolic Equations . . . . . . 10.1 Problems in Hilbert Spaces . . . . . . . 10.2 Problems in Lp Spaces . . . . . . . . . 10.3 Some Example of the Critical Case ν = 0 Perturbed Problems . . . . . . . . . . . . . . . 11.1 Evolution Operator . . . . . . . . . . . 11.2 Cauchy Problem . . . . . . . . . . . . . Notes and Further Researches . . . . . . . . . .

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Some Model for Honeybee Colonies . . . . . . . . . . . . 7.1 Abstract Formulation . . . . . . . . . . . . . . . . 7.2 Construction of Local Solutions . . . . . . . . . . . 7.3 Nonnegativity of Local Solutions . . . . . . . . . . 7.4 A Priori Estimates . . . . . . . . . . . . . . . . . . 7.5 Global Solutions . . . . . . . . . . . . . . . . . . . Some Chemotaxis Model . . . . . . . . . . . . . . . . . . 8.1 Abstract Formulation . . . . . . . . . . . . . . . . 8.2 Basic Properties of Operators A(U ) . . . . . . . . . 8.3 Construction of Local Solutions . . . . . . . . . . . 8.4 Nonnegativity of Local Solutions . . . . . . . . . . 8.5 Global Solutions . . . . . . . . . . . . . . . . . . . Nonautonomous Quasilinear Abstract Evolution Equations Notes and Further Researches . . . . . . . . . . . . . . . .

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Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . 1 Basics of Dynamical Systems . . . . . . . . . . . . . . . . 1.1 Equilibria and Attractors . . . . . . . . . . . . . . . 1.2 ω-limit Sets and Global Attractor . . . . . . . . . . 1.3 Stable and Unstable Manifolds . . . . . . . . . . . 1.4 Exponential Attraction . . . . . . . . . . . . . . . . 2 Representation Theorem of Stable and Unstable Manifolds 2.1 Discrete Case . . . . . . . . . . . . . . . . . . . . 2.2 Continuous Case . . . . . . . . . . . . . . . . . . . 3 Exponentially Stable Equilibria . . . . . . . . . . . . . . . 4 Exponential Attractors . . . . . . . . . . . . . . . . . . . . 4.1 Contraction Semigroups . . . . . . . . . . . . . . . 4.2 Compact Perturbation of Contraction Operator, I . . 4.3 Compact Perturbation of Contraction Operator, II . 4.4 Squeezing Property . . . . . . . . . . . . . . . . . 4.5 Continuous Dynamical System . . . . . . . . . . . 4.6 Continuous Dependence on Parameter . . . . . . . 5 Dynamical Systems for Semilinear Evolution Equations . . 5.1 Dynamical System . . . . . . . . . . . . . . . . . . 5.2 Absorbing and Invariant Compact Set . . . . . . . . 5.3 Exponential Attractors . . . . . . . . . . . . . . . . 5.4 Squeezing Property of S(t) . . . . . . . . . . . . . 6 Stationary Solutions to Semilinear Equations . . . . . . . . 6.1 Equilibria of Dynamical System . . . . . . . . . . . 6.2 Exponential Stability of U . . . . . . . . . . . . . . 6.3 Unstable Manifold of U . . . . . . . . . . . . . . . 6.4 Reaction–Diffusion Systems . . . . . . . . . . . . . 7 Dynamical Systems for Quasilinear Evolution Equations . . 7.1 Structural Assumptions . . . . . . . . . . . . . . . 7.2 Dynamical System . . . . . . . . . . . . . . . . . .

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7.3 Absorbing and Invariant Compact Set . 7.4 Exponential Attractors . . . . . . . . . Stationary Solutions to Quasilinear Equations 8.1 Equilibria of Dynamical System . . . . 8.2 Exponential Stability of U . . . . . . . 8.3 Unstable Manifold of U . . . . . . . . 8.4 Proof of Proposition 6.10 . . . . . . . Notes and Further Researches . . . . . . . . .

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Numerical Analysis . . . . . . . . . . . . . . . . 1 Semilinear Evolution Equations . . . . . . . 1.1 Approximate Problems . . . . . . . 1.2 Local Estimates of Convergence . . 2 Quasilinear Evolution Equations . . . . . . 2.1 Approximate Problems . . . . . . . 2.2 Local Estimates of Convergence . . 2.3 Proof of Lemma 7.1 . . . . . . . . . 3 Two-Dimensional Finite Element Methods . 3.1 Approximate Operators Aξ . . . . . 3.2 Some properties of Cξ (Ω) and Πξ . 3.3 Projection Operator Pξ . . . . . . . 3.4 Ritz Operator Rξ . . . . . . . . . . . 3.5 Domains of Fractional Powers of Aξ 4 Convergence of Exponential Attractors . . . 4.1 Discrete Case . . . . . . . . . . . . 4.2 Continuous Case . . . . . . . . . . . Notes and Further Researches . . . . . . . .

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Semiconductor Models . . . . . . . . . . . . . . . 1 Prototype Semiconductor Equations . . . . . . 1.1 Model Equations . . . . . . . . . . . . 1.2 Some Preliminary Results . . . . . . . 2 Local Solutions . . . . . . . . . . . . . . . . 2.1 Construction of Local Solutions . . . . 2.2 Nonnegativity of Solutions . . . . . . 3 Global Solutions . . . . . . . . . . . . . . . . 3.1 A Priori Estimates for Local Solutions 3.2 Global Solutions . . . . . . . . . . . . 3.3 Global Norm Estimates . . . . . . . . 4 Dynamical System . . . . . . . . . . . . . . . 4.1 Construction of Dynamical System . . 4.2 Exponential Attractors . . . . . . . . . 5 Three-Dimensional Problem . . . . . . . . . . Notes and Further Researches . . . . . . . . .

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Activator–Inhibitor Models . . . . . . . . . . . . . 1 Prototype Reaction–Diffusion Model . . . . . . 2 Local Solutions . . . . . . . . . . . . . . . . . 2.1 Construction of Local Solutions . . . . . 2.2 Nonnegativity of Local Solutions . . . . 3 Global Solutions . . . . . . . . . . . . . . . . . 3.1 A Priori Estimates from Below . . . . . 3.2 A Priori Estimates for Sobolev Norms . 3.3 Global Solution . . . . . . . . . . . . . 3.4 Global Norm Estimates . . . . . . . . . 4 Dynamical System . . . . . . . . . . . . . . . . 4.1 Construction of Dynamical System . . . 4.2 Exponential Attractors . . . . . . . . . . 4.3 Squeezing Property of S(t) . . . . . . . 5 Instability of Homogeneous Stationary Solution 5.1 Localized Problem . . . . . . . . . . . . 5.2 Complexified Dynamical System . . . . 5.3 Unstable Manifold of U . . . . . . . . . Notes and Further Researches . . . . . . . . . .

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10 Belousov–Zhabotinskii Reaction Models . . . . . . 1 Field–Noyes Model . . . . . . . . . . . . . . . 2 Local Solutions . . . . . . . . . . . . . . . . . 2.1 Construction of Local Solutions . . . . . 2.2 Nonnegativity of Local Solutions . . . . 3 Global Solutions . . . . . . . . . . . . . . . . . 3.1 A Priori Estimates of Local Solutions . . 3.2 Global Existence . . . . . . . . . . . . . 3.3 Global Norm Estimate . . . . . . . . . . 4 Dynamical System . . . . . . . . . . . . . . . . 4.1 Construction of Dynamical System . . . 4.2 Exponential Attractors . . . . . . . . . . 5 Keener–Tyson Model . . . . . . . . . . . . . . 6 Local Solutions . . . . . . . . . . . . . . . . . 6.1 Construction of Local Solutions . . . . . 6.2 Nonnegativity of Local Solutions . . . . 7 Global Solutions . . . . . . . . . . . . . . . . . 7.1 A Priori Estimates of Local Solutions . . 7.2 Global Existence . . . . . . . . . . . . . 7.3 Global Norm Estimate . . . . . . . . . . 8 Dynamical System . . . . . . . . . . . . . . . . 8.1 Construction of Dynamical System . . . 8.2 Exponential Attractors . . . . . . . . . . 9 Instability of Homogeneous Stationary Solution 9.1 Localized Problem . . . . . . . . . . . . 9.2 Complexified Dynamical System . . . .

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373 374 374 374 376 377 377 379 379 379 379 379 380 382 382 383 383 383 384 385 385 385 386 386 386 387

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9.3 Case where fu (u, v) ≤ 0 . . . . . . . . . . . . . . . . . . . 387 9.4 Case where 0 < fu (u, v) < 1 . . . . . . . . . . . . . . . . 388 Notes and Further Researches . . . . . . . . . . . . . . . . . . . . 389 11 Forest Kinematic Model . . . . . . . . . . . . . . . . . . . . . . . 1 Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . 2 Local Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Abstract Formulation . . . . . . . . . . . . . . . . . . 2.2 Construction of Local Solutions . . . . . . . . . . . . . 2.3 Nonnegativity of Local Solutions . . . . . . . . . . . . 3 Global Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 A Priori Estimates for Local Solutions . . . . . . . . . 3.2 Global Solutions . . . . . . . . . . . . . . . . . . . . . 3.3 Global Norm Estimates . . . . . . . . . . . . . . . . . 4 Dynamical System . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Lyapunov Function . . . . . . . . . . . . . . . . . . . 4.2 ω-limit Sets . . . . . . . . . . . . . . . . . . . . . . . 4.3 Constituents of L2 -ω-limit Sets . . . . . . . . . . . . . f αδ < h < ∞ and where ab2 < 3(c + f ) . 4.4 Cases where c+f 4.5 Numerical Examples . . . . . . . . . . . . . . . . . . . 5 Homogeneous Stationary Solution . . . . . . . . . . . . . . . 5.1 Localized Problem in a Neighborhood of O . . . . . . ˜ O . . . . . . . . . . . . . . . . . . . 5.2 Spectrum of S(t) 5.3 Stability and Instability of O . . . . . . . . . . . . . . Notes and Further Researches . . . . . . . . . . . . . . . . . .

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391 391 392 392 393 394 395 395 398 398 400 400 403 405 405 408 409 410 411 412 414

12 Chemotaxis Models . . . . . . . . . . . . . . . . . . . . . . . 1 Chemotaxis Model Without Proliferation . . . . . . . . . 1.1 Abstract Formulation . . . . . . . . . . . . . . . 1.2 Construction of Local Solutions . . . . . . . . . . 1.3 Nonnegativity of Local Solutions . . . . . . . . . 2 Case where χ(ρ) = ρ . . . . . . . . . . . . . . . . . . . 2.1 Global Solutions for Small Initial Functions . . . 2.2 Lyapunov Function . . . . . . . . . . . . . . . . 2.3 Blowup of Solutions . . . . . . . . . . . . . . . . 3 Chemotaxis Model with Proliferation . . . . . . . . . . . 3.1 Local Solutions . . . . . . . . . . . . . . . . . . 3.2 A Priori Estimates of Local Solutions . . . . . . . 3.3 Global Solutions . . . . . . . . . . . . . . . . . . 3.4 Dynamical System . . . . . . . . . . . . . . . . . 3.5 Exponential Attractors . . . . . . . . . . . . . . . 3.6 Numerical Examples . . . . . . . . . . . . . . . . 4 Instability of Homogeneous Stationary Solution of (12.24) 4.1 Localized Problem . . . . . . . . . . . . . . . . . 4.2 Complexified Dynamical System . . . . . . . . . 4.3 Differentiability of F˜ (U ) . . . . . . . . . . . . .

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417 418 419 420 422 423 423 426 427 427 427 428 433 433 434 434 437 437 438 438

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4.4 Spectrum Separation Condition of A . . . . . . . . . . . . 438 4.5 Stability or Instability Conditions . . . . . . . . . . . . . . 441 Notes and Further Researches . . . . . . . . . . . . . . . . . . . . 442 13 Termite Mound Building Model . . . . . . . . . . . 1 Model equations . . . . . . . . . . . . . . . . . 2 Local Solutions . . . . . . . . . . . . . . . . . 2.1 Abstract Formulation . . . . . . . . . . 2.2 Construction of Local Solutions . . . . . 2.3 Nonnegativity of Local Solutions . . . . 3 Global Solutions . . . . . . . . . . . . . . . . . 3.1 A Priori Estimates for Local Solutions . 3.2 Construction of Global Solutions . . . . 3.3 Global Norm Estimates . . . . . . . . . 4 Dynamical System . . . . . . . . . . . . . . . . 4.1 Construction of Dynamical System . . . 4.2 Bounded absorbing set . . . . . . . . . . 4.3 Global Attractor . . . . . . . . . . . . . 4.4 Exponential Attractors . . . . . . . . . . 5 Instability of Homogeneous Stationary Solution 5.1 Localized Problem . . . . . . . . . . . . 5.2 Complexified Dynamical System . . . . 5.3 Differentiability of F˜ (U ) . . . . . . . . 5.4 Spectrum Separation Condition of A . . 5.5 Stability and Instability Conditions . . . Notes and Further Researches . . . . . . . . . .

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445 445 446 446 447 449 450 450 455 455 457 457 458 458 460 463 464 464 465 465 469 470

14 Adsorbate-Induced Phase Transition Model . . . . . . 1 Model Equations . . . . . . . . . . . . . . . . . . . 2 Local Solutions . . . . . . . . . . . . . . . . . . . 2.1 Abstract Formulation . . . . . . . . . . . . 2.2 Basic Properties of A(U ) . . . . . . . . . . 2.3 Construction of Local Solutions . . . . . . . 2.4 Lower and Upper Bound of Local Solutions 3 Global Solutions . . . . . . . . . . . . . . . . . . . 3.1 A Priori Estimates . . . . . . . . . . . . . . 3.2 Global Solutions . . . . . . . . . . . . . . . 3.3 Global Norm Estimates . . . . . . . . . . . 4 Dynamical System . . . . . . . . . . . . . . . . . . 4.1 Construction of Dynamical System . . . . . 4.2 Exponential Attractors . . . . . . . . . . . . 5 Homogeneous Stationary Solutions . . . . . . . . . Notes and Further Researches . . . . . . . . . . . .

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471 472 472 472 474 475 477 479 479 483 483 483 483 484 484 485

15 Lotka–Volterra Competition Model with Cross-Diffusion . . . . . . 487 1 Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 488

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489 489 490 492 493 494 495 495 505 511 511 512 513 513 514 514 514 517 521 521 524 524

16 Characterization of Domains of Fractional Powers . . . . . . . . . 1 Domains of Fractional Powers in Hilbert Spaces . . . . . . . . . 1.1 Case of Self-Adjoint Operators . . . . . . . . . . . . . . 1.2 Bounded H∞ Functional Calculus . . . . . . . . . . . . 1.3 Integrable Condition Along Contour . . . . . . . . . . . 1.4 Equivalence Theorem . . . . . . . . . . . . . . . . . . . 1.5 Counterexample . . . . . . . . . . . . . . . . . . . . . . 2 Some Result on Matrix Operators . . . . . . . . . . . . . . . . . 3 Domains of Fractional Powers in Banach Spaces . . . . . . . . . 3.1 Equivalence Theorem in Banach Spaces . . . . . . . . . 4 Domains of Fractional Powers of Elliptic Operators in L2 Spaces 4.1 Case of Domains with Lipschitz Boundary . . . . . . . . 4.2 Case of Domains with C2 Boundary . . . . . . . . . . . . 4.3 Case of Convex Domains . . . . . . . . . . . . . . . . . 5 Domains of Fractional Powers of Elliptic Operators in Lp Spaces 6 Domains of Fractional Powers Under Dirichlet Conditions . . . . 6.1 Domains in L2 Spaces . . . . . . . . . . . . . . . . . . . 6.2 Domains in Lp Spaces . . . . . . . . . . . . . . . . . . . Notes and Further Researches . . . . . . . . . . . . . . . . . . .

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527 527 527 529 532 532 537 540 542 542 545 545 546 552 552 558 558 560 561

3

4

5

6

Local Solutions . . . . . . . . . . . . . . . . . 2.1 Abstract Formulation . . . . . . . . . . 2.2 Basic Properties of Operators A(U ) . . . 2.3 Construction of Local Solutions . . . . . 2.4 Some Regularity Properties . . . . . . . 2.5 Nonnegativity of Local Solutions . . . . A Priori Estimates for Local Solutions . . . . . 3.1 Case where (15.16) is Valid . . . . . . . 3.2 Case where (15.17) is Valid . . . . . . . Global Solutions . . . . . . . . . . . . . . . . . 4.1 Global Solutions . . . . . . . . . . . . . 4.2 Global Norm Estimates . . . . . . . . . Dynamical System . . . . . . . . . . . . . . . . 5.1 Construction of Dynamical System . . . 5.2 Exponential Attractors . . . . . . . . . . Instability of Homogeneous Stationary Solution 6.1 Localized Problem . . . . . . . . . . . . 6.2 Dynamical System of Localized Problem ˜ ) . . . . . . . . 6.3 Differentiability of A(U 6.4 Spectrum Separation Condition of A . . 6.5 Instability of U . . . . . . . . . . . . . Notes and Further Researches . . . . . . . . . .

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Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579

Preface

This monograph is intended to present the fundamentals of the theory of abstract parabolic evolution equations and to show how to apply to various nonlinear diffusion equations and systems arising in science. The theory gives us a unified and systematic treatment for concrete nonlinear diffusion models. Three main approaches are known to the abstract parabolic evolution equations, namely, the semigroup methods, the variational methods, and the methods of using operational equations. In order to keep the volume of the monograph in reasonable length, we will focus on the semigroup methods. For other two approaches, see the related references in Bibliography. The semigroup methods, which go back to the invention of the analytic semigroups in the middle of the last century, are characterized by precise formulas representing the solutions of the Cauchy problem for evolution equations. The analytic semigroup e−tA generated by a linear operator −A provides directly a fundamental solution to the Cauchy problem for an autonomous linear evolution equation, dU dt + AU = F (t), 0 < t ≤ T , U (0) = U0 , and the solution is t given by the formula U (t) = e−tA U0 + 0 e−(t−s)A F (s) ds. The evolution opU (t, s) for linear operators A(t) gives the formula U (t) = U (t, 0)U0 + erator t U (t, s)F (s) ds representing a solution to a nonautonomous linear evolution 0 equation, dU dt + A(t)U = F (t), 0 < t ≤ T , U (0) = U0 , and the evolution operator U (t, s) can be obtained from the analytic semigroups e−sA(t) through an operator-valued integral equation of Volterra type. Furthermore, a solution to the Cauchy problem for semilinear evolution equation, dU dt + AU = F (U ), 0 < , can be found as a solution of the integral equation U (t) = t ≤ T , U (0) = U 0 t e−tA U0 + 0 e−(t−s)A F (U (s)) ds. Similarly, a solution to the quasilinear problem, dU = U , can be obtained as a solution of the dt + A(U )U = F (U ), 0 < t ≤ T , U (0) t 0 integral equation U (t) = UU (t, 0)U0 + 0 UU (t, s)F (U (s)) ds, where UU (t, s) denotes the evolution operator for linear operators AU (t) = A(U (t)). For these problems, the solution formulas provide us important information on solutions such as the uniqueness, maximal regularity, smoothing effect, and so forth. Especially, for nonlinear problems, one can derive the Lipschitz continuity of solutions with respect to the initial values, even their Fréchet differentiability. Such information on xv

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solutions then enable us develop the study to next stages. In fact, we can construct dynamical systems determined from the Cauchy problems, investigate asymptotic behavior of solutions, show the existence of attractors, investigate the stability and instability of stationary solutions, construct smooth stable and unstable manifolds, and even can analyze solutions numerically in a general framework. So far, a great number of nonlinear diffusion models have already been presented in order to describe diverse interesting phenomena in the real world. Our abstract results are available to these models. Among others, the second half of the present monograph is devoted to handling various diffusion systems in the science of self-organizations. The self-organizing process is usually described by a nonlinear dynamical system. When it is well abstractly formulated, the basic rules of selforganization, which are locally uniform in general, can be described by diffusions of constituents and suitable interactions or reactions among them, and consequently the process can be written down into a nonlinear diffusion system. We will pick up prototype diffusion systems arising in physics, chemistry, biology, and ecology. After constructing dynamical systems, we shall further study the asymptotic behavior of trajectories, existence of attractors, stability and instability of equilibria, construction of smooth stable and unstable manifolds. These results then give a clear meaning or a reasonable interpretation for many notions and ideas suggested in the science of self-organizations such as the emergent property, robustness of process, fluctuations, instabilization of homogeneous states, reduction of active degree of freedom, and so forth. They also seem to give us a certain standpoint for studying the self-organizations theoretically in the framework of infinitedimensional dynamical systems. Chapter 1 prepares all basic matters which are used as terminology or fundamental facts such as Banach and Hilbert spaces, interpolation of Banach spaces, extrapolation of Hilbert spaces, closed linear operators, contraction operators, linear operators associated with sesquilinear forms, integral equations and inequalities of Volterra type, Sobolev–Lebesgue spaces, etc. Chapter 2 reviews the basic properties and matters of sectorial operators in Banach spaces generally with their proof. Sectorial operators indeed play the principal role in the theory of abstract parabolic evolution equations. Especially, we shed light upon the two subject notions, the fractional powers of sectorial operators which are a complex interpolation of operator powers of integral order with the law of exponent and the analytic semigroups which are exponential functions generated by sectorial operators of angles less than π2 . In both we utilize the techniques for analytic functions in complex domains with values in a complex Banach space. We also present many concrete sectorial operators determined from strongly elliptic operators in Hilbert and Banach spaces. In applications, complete characterization or sharp estimation of the domains of fractional powers have critical importance. However, they require some more sophisticated techniques called H∞ functional calculus for sectorial operators. So, they are reviewed separately in Chap. 16. Chapter 3 is devoted to studying the Cauchy problems for linear evolution equations. We are concerned with construction of fundamental solutions which give us formulas representing the solutions to the Cauchy problem. For the autonomous problems, the fundamental solutions are given simply by analytic semi-

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groups. Meanwhile, for the nonautonomous problems, the fundamental solutions are given by evolution operators which are constructed by using operator-valued integral equations of Volterra type. Chapters 4 and 5 then handle the Cauchy problems for nonlinear evolution equations. In Chap. 4, we construct local solutions for the semilinear evolution equations and derive various properties of the local solutions from the solution formulas, uniqueness, maximal regularity, smoothing effect, Lipschitz continuity with respect to the initial values, etc. A sufficient condition for the global existence is also given. Chapter 5 is devoted to handling the quasilinear evolution equations. We similarly construct the local solutions and derive their significant properties from the solution formula as for the semilinear equations. Chapters 6 and 7 continue the study of the Cauchy problem for nonlinear evolution equations. In Chap. 6, first, we review basic matters and results of the theory of infinite-dimensional dynamical systems, generally with their proofs. Secondly, we consider applications of the theory to the dynamical systems determined from semilinear and quasilinear evolution equations; more precisely, we show general methods how to construct ω-limit sets, global attractors, exponential attractors, stable and unstable manifolds of equilibria, etc. In Chap. 7, we develop numerical analysis for semilinear and quasilinear evolution equations. Setting suitable approximate evolution equations in finite-dimensional spaces, we present general methods how to estimate the order of convergence of approximate solutions. All the subsequent Chaps. 8, 9, 10, 11, 12, 13, 14, and 15 are devoted to studying prototype nonlinear diffusion models which describe the kinetics of self-organizing processes. We are concerned with constructing dynamical systems, constructing attractors, investigating stability and instability of stationary solutions, and constructing the stable and unstable manifolds. Those chapters can be read independently. Chapter 8 handles Semiconductor models with drift-diffusions. The Semiconductor models seem to be the first nonlinear diffusion models presented for describing the kinetics of systems made of a number of constituents which interact each other from the macroscopic view. Chapters 9 and 10 handle Activator–Inhibitor models and Belousov–Zhabotinskii reaction models, respectively. They are both well known as classical examples of pattern formations. Their model equations are written down by reaction–diffusion systems. Chapter 11 treats some diffusion model describing the kinetics of a forest. The model equations are written down by a system of one linear diffusion equation and two ordinary differential equations, one is nonlinear, and the other linear. By the kinematic model, we can find a natural forest boundary. Chapters 12 and 13 are concerned with chemotaxis models. Chemotaxis is known as one of indispensable machinery which sustain the life. In Chap. 12, we handle a bacterium aggregating model by interaction between bacteria and chemoattractant in the manner of positive feedback. The model equations are written down by diffusion–advection–reaction equations. In Chap. 13, a termite mound building model is handled as well. Chapter 14 treats a phase transition model of a metallic surface adsorbing CO molecules. Remarkable adsorption patterns are self-organized on the surface by in-

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teraction between adsorbed molecules and a phase of the surface. The model equations are written down by diffusion–advection–reaction equations. Chapter 15 is finally devoted to handling a segregation model of competing biological species. We consider the species segregating process by using Lotka– Volterra competition model with self-diffusions and cross-diffusions. I would like to express my profound gratitude to the numerous collaborators and colleagues around the world who have made, for the long period, valuable comments and suggestions. In particular, I would like to thank Professors Alan McIntosh, Angelo Favini, Masayasu Mimura, Messoud Efendiev, Tohru Tsujikawa, and Shin-ichi Nakagiri. I would also like to express my hearty thanks to my past students, Drs. Etsushi Nakaguchi, Koichi Osaki, Sang-Uk Ryu, Masashi Aida, Yasuhiro Takei, Le Huy Chuan, Takanori Shirai, and Hideaki Fujimura, my present postgraduate student Doan Duy Hai and postdoctoral Gianluca Mola. The joint works with them were extremely important for presenting this monograph. They also kindly read the draft very carefully. Finally, I would like to express my appreciation to Springer for their efficient handling of the publication. Osaka, July 2009

Atsushi Yagi

Chapter 1

Preliminaries

1 Banach Spaces 1.1 Distance of Subsets Let X be a Banach space with norm  · . Throughout book, unless others are notified, Banach and Hilbert spaces are always defined over the complex field C. As usual, X is a metric space with the distance d(U, V ) = U − V  determined from  · . For U ∈ X and 0 < r < ∞, the set B(U ; r) = {U ∈ X; d(U , U ) < r} is called the ball with center U and radius r. Similarly, B(U ; r) = {U ∈ X; d(U , U ) ≤ r} is called the closed ball with center U and radius r. For a point U ∈ X and a subset A of X, their distance is defined by d(U, A) = inf d(U, V ). V ∈A

For a subset A ⊂ X and 0 < r < ∞, the r-neighborhood of A is defined by Wr (A) = {U ∈ X; d(U, A) < r}. For two subsets A and B of X, the Hausdorff pseudo-distance is defined by h(A, B) = sup d(U, B) = sup inf d(U, V ). U ∈A

U ∈A V ∈B

(1.1)

Unless A is a bounded subset, h(A, B) may possibly be infinity. It is seen by definition that h(A1 , A3 ) ≤ h(A1 , A2 ) + h(A2 , A3 ). A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_1, © Springer-Verlag Berlin Heidelberg 2010

(1.2) 1

2

1 Preliminaries

For two subsets A and B, their distance is defined by d(A, B) = max{h(A, B), h(B, A)}.

(1.3)

Unless A and B are bounded subsets, d(A, B) may possibly be infinity.

1.2 Fractal Dimension Let M be a compact subset of a Banach space X. For any ε > 0, M can be covered by a finite number of closed balls having a radius ε. Let N (ε) be the minimal number of balls with radius ε which cover the set M. We then define d(M) = lim sup ε→0

log N (ε) log 1ε

.

This superior limit, 0 ≤ d(M) ≤ ∞, is called the fractal dimension of M. In particular, when d(M) < ∞, M is said to have a finite fractal dimension. The following result is easily verified from the definition. Theorem 1.1 Let M be a compact subset of X with finite fractal dimension. Let Φ be a Lipschitz-continuous operator from M into another Banach space Y , namely, ΦU − ΦV Y ≤ LU − V X ,

U, V ∈ M,

with some constant L. Then, Φ(M) also has a finite fractal dimension with the relation dY (Φ(M)) ≤ dX (M). If M has a finite fractal dimension, then M can be embedded into a finitedimensional space by Mañé–Hölder-type projection as follows. Theorem 1.2 Let M be a compact subset of a real Banach space X with fractal dimension d < ∞. Let N > 2d be an integer, and let α0 = NN−2d (1+d) . Then, for any 0 < α < α0 , we have that for almost every bounded linear operator Π : X → RN , CΠ(U − V )α ≥ U − V ,

U, V ∈ M,

(1.4)

with some constant C > 0. For details, see [BEFN93, FO96, HY99]. Condition (1.4) shows in particular that Π is injective on the subset M and the inverse of Π on Π(M) is Hölder continuous with exponent α. As N → ∞, α0 → 1; so, α can also be taken close to 1; we get by Theorem 1.1 that, if (1.4) is valid with α = 1, then d(M) = d(Π(M)). The term “almost every” is in the sense of prevalence. Prevalence is a generalized notion of “almost every” for the points in Lebesgue measurable sets in Euclidean spaces for the vectors in infinite-dimensional spaces.

2 Function Spaces with Values in a Banach Space

3

Consider two Banach spaces X and Y such that Y ⊂ X continuously. Let M be a compact subset of Y with dY (M) < ∞. By Theorem 1.1, we verify that M has a finite fractal dimension in X, too, with dX (M) ≤ dY (M). However, the following theorem gives some converse estimation. Theorem 1.3 Let three Banach spaces X, Y , and Z be such that Z ⊂ Y ⊂ X with 1−θ continuous embeddings, and let  · Y ≤ D · θZ  · X on Z with some exponent 0 < θ < 1. If M is a compact set of Y with dY (M) < ∞ and is a bounded set of Z, then dX (M) ≤ dY (M) ≤ (1 − θ )−1 dX (M). Proof Let ε > 0, and let N X (ε) be the minimal number of balls of X with radius ε which cover M. There exist N X (ε) points Ui ∈ X such that M⊂

X (ε) N

X

B (Ui ; ε).

i=1 X

X

For each i, since M ∩ B (Ui ; ε) = ∅, there is a point Vi ∈ M ∩ B (Ui ; ε). Then, 1−θ provides the relation  · Y ≤ D · θZ  · X X

X

M ∩ B (Ui ; ε) ⊂ M ∩ B (Vi ; 2ε) ⊂ B

Y

 Vi ; D[δZ (M)]θ (2ε)1−θ ,

where δZ (M) denotes the diameter of M in Z. Therefore, M⊂

X (ε) N

B

Y

 Vi ; D[δZ (M)]θ (2ε)1−θ .

i=0

This shows that M can be covered by at least N X (ε)-balls of Y with radius ε˜ = D[δZ (M)]θ (2ε)1−θ , that is, N Y (˜ε ) ≤ N X (ε). We hence obtain that dY (M) = lim sup ε˜ →0

log N Y (˜ε ) log

1 ε˜

= (1 − θ )−1 lim sup ε→0

≤ lim sup ε→0

log N X (ε) log

1 ε

log N X (ε) log D[δ

1 θ 1−θ Z (M)] 2

1 + log ε1−θ

= (1 − θ )−1 dX (M).

It is already known that dX (M) ≤ dY (M).



2 Function Spaces with Values in a Banach Space Let X be a Banach space with norm  · . We introduce various function spaces consisting of functions defined on an interval of R or on a domain of C with values in X.

4

1 Preliminaries

2.1 Uniformly Bounded Function Spaces Let [a, b] be a bounded closed interval. By B([a, b]; X) we denote the space of uniformly bounded functions on [a, b] (not necessarily smooth or measurable). The space is a Banach space with the supremum norm F B = sup F (t). a≤t≤b

−η

By B{a} ((a, b]; X), where η > 0, we denote the space of X-valued functions which are uniformly bounded on (a, b] with weight function (t − a)η , namely, F ∈ −η B{a} ((a, b]; X) if and only if (t − a)η F ∈ B((a, b]; X). We equip the space with the norm   F B−η = sup (t − a)η F (t) = (t − a)η F B . {a}

a
2.2 Continuously Differentiable Function Spaces Let [a, b] be a bounded closed interval, and let m = 0, 1, 2, . . . , be an integer. Cm ([a, b]; X) denotes the space of m times continuously differentiable functions on [a, b]. When m = 0, C0 ([a, b]; X) is the space of continuous functions and is simply denoted by C([a, b]; X). The space is equipped with the maximum norm F Cm =

m  i=0

  max F (i) (t).

a≤t≤b

For any continuous function F ∈ C([a, b]; X), the integral of F (t) on the interval b [a, b] is defined in the usual way and is denoted by a F (t) dt. It holds that    

b

a

   F (t) dt  ≤

b

F (t) dt.

a

The following result is used very often. Theorem 1.4 Let A be a closed linear operator of X. If F ∈ C([a, b]; X) and AF ∈ C([a, b]; X), then  b  b F (t) dt = AF (t) dt. A a

a

Proof Let a = t0 < t1 < · · · < tN = b be a partition of [a, b] and take the sum N  n=1

F (τn )(tn − tn−1 )

with tn−1 ≤ τn ≤ tn .

2 Function Spaces with Values in a Banach Space

5

Clearly, A

N 

F (τn )(tn − tn−1 ) =

n=1

N 

AF (τn )(tn − tn−1 ).

n=1

Letting N → ∞ with the condition that max1≤n≤N (tn − tn−1 ) → 0, we conclude b that a F (t)dt ∈ D(A) and get the desired formula. 

2.3 Hölder Continuous Function Spaces For m = 0, 1, 2, . . . , and an exponent 0 < σ < 1, Cm+σ ([a, b]; X) denotes the space of m times continuously differentiable functions whose mth derivatives are Hölder continuous on [a, b] with exponent σ . The space is equipped with the norm F Cm+σ = F Cm +

F (m) (t) − F (m) (s) . |t − s|σ a≤s
By Cm,1 ([a, b]; X), we denote the space of m times continuously differentiable functions whose mth derivatives are Lipschitz continuous on [a, b]. The space is equipped with the norm F Cm,1 = F Cm +

F (m) (t) − F (m) (s) . |t − s| a≤s
By Cσ{a} ([a, b]; X), where 0 < σ < 1, we denote the space of continuous functions [a, b] which are Hölder continuous at least at t = a with exponent σ . We equip the space with the norm F (t) − F (a) . (t − a)σ a
F Cσ{a} = F C + sup

2.4 Weighted Hölder Continuous Function Spaces For two exponents 0 < σ < β ≤ 1, we define the function space Fβ,σ ((a, b]; X),

0 < σ < β ≤ 1,

(1.5)

as follows. The space Fβ,σ ((a, b]; X) consists of continuous functions on (a, b] (resp. [a, b]) when 0 < β < 1 (resp. β = 1) with the following three properties: (1) When β < 1, (t − a)1−β F (t) has a limit as t → a.

6

1 Preliminaries

(2) F is Hölder continuous with the exponent σ and with the weight (s − a)1−β+σ , i.e., (s − a)1−β+σ F (t) − F (s) (t − s)σ a≤s
= sup

sup

a≤t≤b a≤s
(s − a)1−β+σ F (t) − F (s) < ∞. (t − s)σ

(1.6)

(3) As t → a, ωF (t) = sup

a≤s
(s − a)1−β+σ F (t) − F (s) → 0. (t − s)σ

(1.7)

The space Fβ,σ ((a, b]; X) is equipped with the norm F Fβ,σ = sup (t − a)1−β F (t) + a≤t≤b

(s − a)1−β+σ F (t) − F (s) . (1.8) (t − s)σ a≤s
It is clear that  · Fβ,σ is a function norm. With this norm, the space Fβ,σ ((a, b]; X) becomes a Banach space. Let 0 < σ < β ≤ 1. If 0 < σ < σ , then

Fβ,σ ((a, b]; X) ⊂ Fβ,σ ((a, b]; X),

0 < σ < σ < β ≤ 1,

(1.9)

with continuous embedding. In fact, if F ∈ Fβ,σ ((a, b]; X), then F (t) − F (s) = F (t) − F (s)θ F (t) − F (s)1−θ θ  1−θ  ≤ (s − a)β−1−σ (t − s)σ (t − a)β−1 + (s − a)β−1 F Fβ,σ ≤ 21−θ (s − a)β−1−θσ (t − s)θσ F Fβ,σ with any 0 < θ < 1. From this estimate, we immediately verify that F ∈ Fβ,θσ ((a, b]; X), and hence (1.9) is valid. Remark 1.1 Let a = 0 and b > 0. When 0 < σ < β < 1, any function of the form F (t) = t β−1 G(t)

with G ∈ Cσ ([0, b]; X), G(0) = 0,

belongs to Fβ,σ ((0, b]; X). Indeed, this is easily verified by noticing the estimate

 t σ  t 1−σ  t β−2 β−1 β−1 β−2 1−σ |t −s | = (1 − β) τ dτ ≤ (1 − β) dτ τ dτ s

 ≤ (1 − β)

1−σ σ +1−β

1−σ

s

(t − s)σ s β−σ −1 ,

s

0 < s < t < ∞.

Similarly, when 0 < σ < β = 1, any function F ∈ Cσ ([0, b]; X) belongs to F1,σ ((0, b]; X).

3 Linear Operators

7

2.5 Space of Analytic Functions Let D denote a domain of the complex plane C. An X-valued function f (λ) defined for λ ∈ D is called an analytic function of D if f (λ) is differentiable in λ in the usual sense. All the properties of scalar analytic functions can be extended over the X-valued analytic functions. For example, for λ ∈ D, the Cauchy integral formula  1 f (λ) f (λ) = dμ (1.10) 2πi C μ − λ holds with any rectifiable Jordan curve C encircling λ in D. An analytic function is infinitely differentiable for λ ∈ D. In a neighborhood of each point λ ∈ D, f (μ) is developed in a Taylor series f (μ) =

∞  (μ − λ)n n=0

n!

f (n) (λ).

3 Linear Operators 3.1 Bounded Linear Operators Let X and Y be Banach spaces with norms  · X and  · Y , respectively. The space of all bounded linear operators from X into Y is denoted by L(X, Y ). The space is equipped with the operator norm AL(X,Y ) = sup AU Y . U X ≤1

With this norm, L(X, Y ) is a Banach space. When X = Y , L(X, X) is written as L(X) for brevity. The following theorem is called the uniform boundedness theorem or the resonance theorem. For the proof, see [Yos80, Corollary 1, p. 69]. Theorem 1.5 Let X and Y be Banach spaces. Let {Aα }α∈I be a family of bounded linear operators from X into Y with index set I . If supα∈I Aα U Y < ∞ for all U ∈ X, then supα∈I Aα L(X,Y ) < ∞. For each U ∈ X, the functional pU (·) defined by pU (A) = AU Y , A ∈ L(X, Y ), is a semi-norm on L(X, Y ). Obviously the family of semi-norms pU (·), U ∈ X satisfies the separation property, that is, pU (A) = 0 for all pU (·) implies A = 0. Then, L(X, Y ) becomes a locally convex linear topological space by the family of semi-norms (see [Yos80, Theorem, p. 26]), and the space is denoted by Ls (X, Y ). By Theorem 1.5, every fundamental directed set {Aα } of Ls (X, Y )

8

1 Preliminaries

is shown to converge to a bounded linear operator A, that is, Ls (X, Y ) is complete. Therefore, Ls (X, Y ) is a locally convex, complete linear topological space. The topology defined in this way is called the strong topology of L(X, Y ). By contrast, the topology defined by the operator norm is called the uniform topology of L(X, Y ). Consider a sequence {An } of L(X, Y ). We say that An is strongly convergent to a bounded linear operator A if An is convergent to A in the strong topology; in other words, An is strongly convergent to A if and only if An U → AU in Y as n → ∞ for every U ∈ X. Similarly, consider an L(X, Y )-valued function A(ω) defined in a set Ω ⊂ Rd . We say that A(ω) is strongly continuous at ω0 ∈ Ω if A(ω) is continuous at ω0 in the strong topology; in other words, A(ω) is strongly continuous at ω0 if and only if A(ω)U → A(ω0 )U in Y as ω → ω0 for every U ∈ X. One also says that A(ω) is strongly continuous for ω ∈ Ω if A(ω) is strongly continuous at any point ω ∈ Ω. Let Z be a third Banach space with norm  · Z . Then, the following results are verified. Theorem 1.6 If a sequence {An } (resp. {Bn }) of L(X, Y ) (resp. L(Y, Z)) converges strongly to a bounded linear operator A (resp. B), then {Bn An } is a sequence of L(X, Z) which converges strongly to BA. Proof Indeed, by Theorem 1.5, the strong convergence of Bn implies that supn Bn L(Y,Z) = C < ∞. Therefore, as n → ∞, Bn An U − BAU Z ≤ Bn (An − A)U Z + (Bn − B)AU Z ≤ C(An − A)U Y + (Bn − B)AU Z → 0 for every U ∈ X. Hence, Bn An is strongly convergent to BA.



Theorem 1.7 Let A(ω) (resp. B(ω)) be a strongly continuous function defined for ω ∈ Ω ⊂ Rd with values in L(X, Y ) (resp. L(Y, Z)). Then, B(ω)A(ω) is a strongly continuous function for ω ∈ Ω with values in L(X, Z). Proof The proof is quite analogous to that of Theorem 1.6.



3.2 Part of Linear Operator Let X be a Banach space, and let A be a linear operator of X into itself. The domain of A is denoted by D(A), and the range of A is denoted by R(A). Let Y be a linear subspace of X. An operator defined by D(A|Y ) = {U ∈ D(A) ∩ Y ; AU ∈ Y } and A|Y U = AU is called the part of A in Y . It is easily verified that A|Y is a linear operator of Y into Y . When D(A) ⊂ Y , D(A|Y ) = {U ∈ D(A); AU ∈ Y }.

3 Linear Operators

9

3.3 Multivalued Linear Operators A set-valued operator A : X → 2X is called a multivalued linear operator of X if the domain D(A) = {U ∈ X; AU = ∅} is a linear subspace of X and A enjoys the properties:  AU + AV ⊂ A(U + V ) for U, V ∈ D(A); zAU ⊂ A(zU ) for z ∈ C and U ∈ D(A). It is known that the inverse A−1 of any single-valued linear operator A is a multivalued linear operator. It is also known, see [FY99, Chap. I], that the multivalued linear operators have analogous properties to those of the single-valued linear operators. Especially, the resolvent set ρ(A) is defined as the set of λ’s such that the operators (λ − A)−1 are single-valued, bounded linear operators on X. The L(X)-valued function (λ − A)−1 for λ ∈ ρ(A) is called the resolvent of A.

3.4 Resolvent Equation and Pseudo-Resolvent Let A be a densely defined, closed linear operator in a Banach space X. Its resolvent set ρ(A) is defined by the set of all complex numbers λ such that (λ − A) has a bounded inverse (λ − A)−1 ∈ L(X). The L(X)-valued function (λ − A)−1 defined for λ ∈ ρ(A) is called the resolvent of A. It is known that ρ(A) is an open subset of C and (λ − A)−1 is an analytic function in ρ(A) with values in L(X). For each λ0 ∈ ρ(A), the resolvent is developed in the Taylor expansion (λ − A)−1 =

∞  (−1)n (λ − λ0 )n (λ0 − A)−(n+1) ,

−1  |λ − λ0 | < (λ0 − A)−1  .

n=0

(1.11) The complement of ρ(A) is called the spectrum of A and is denoted by σ (A), σ (A) being a closed subset of C. It is easily seen that (λ − A)−1 − (μ − A)−1 = −(λ − μ)(λ − A)−1 (μ − A)−1 ,

λ, μ ∈ ρ(A). (1.12)

This formula is called the resolvent equation of A. We can construct conversely from a family of bounded linear operators satisfying (1.12) a linear operator whose resolvent coincides with the given one. In fact, let R(λ) be an L(X)-valued function defined in a domain D ⊂ C and satisfying the relation R(λ) − R(μ) = −(λ − μ)R(λ)R(μ), Such a function R(λ) is called a pseudo-resolvent in D.

λ, μ ∈ D.

(1.13)

10

1 Preliminaries

Lemma 1.1 Let R(λ) be a pseudo-resolvent in D. (1.13) implies λ − R(λ)−1 = μ − R(μ)−1 for all pairs (λ, μ) ∈ D 2 in the sense of multivalued linear operators. Proof If U ∈ [λ − R(λ)−1 ]F , then U − λF ∈ −R(λ)−1 F ; therefore, it is equivalent to R(λ)(U − λF ) = −F . Operate (μ − λ)R(μ) to this identity and use relation (1.13). Then, [R(λ) − R(μ)](U − λF ) = (λ − μ)R(μ)F, −F − R(μ)(U − λF ) = (λ − μ)R(μ)F. Therefore, R(μ)(U − μF ) = −F ; hence, U ∈ [μ − R(μ)−1 ]F . This means that the assertion of lemma is valid.  This lemma clearly shows that A = λ − R(λ)−1 is defined in a multivalued sense independently of the choice of λ. In addition, (λ − A)−1 is a single-valued bounded linear operator and coincides with R(λ) for all λ ∈ G. Therefore, we obtain an extension of the well-known result on the pseudo-resolvent ([Yos80, Theorem 1, p. 216] or [Kat80, Theorem 1.3, p. 428]). Theorem 1.8 Let R(λ) be a pseudo-resolvent defined in a complex domain D. Then, there exists a multivalued linear operator A such that (λ − A)−1 = R(λ) for all λ ∈ D. If R(λ) is injective for some λ ∈ D, then A is single valued.

3.5 Spectra of Operators Let A be a densely defined, closed linear operator in a Banach space X, and let σ (A) be its spectrum. The spectral radius of A is defined by supλ∈σ (A) |λ| and is denoted by rA . It is known that A is a bounded linear operator of X if and only if A has a finite spectral radius. Furthermore, if A is bounded, it is shown that 1 rA = maxλ∈σ (A) |λ| = limn→∞ An  n (this limit always exists, see [Yos80, Theorem 3, p. 211]). In particular, rA ≤ A. The converse estimate is also true in some generalized sense. In fact, let r be any number such that rA < r < ∞. We define the new norm An U  U  = sup , U ∈ X. rn n≥0 Clearly,  ·  satisfies the norm axioms on X. Moreover, this norm is seen to be n equivalent to the original one because of limn→∞ Ar n  = 0. Let us denote by X the Banach space X but equipped with the norm  ·  . Then, since AU  ≤ rU  , U ∈ X, it follows that AL(X ) ≤ r. As the spectrum σ (A) is determined independently of choice of the norms of X, one deduces that rA ≤ AL(X ) ≤ r

with any rA < r < ∞.

(1.14)

3 Linear Operators

11

Let A be a bounded linear operator of X, and let σ (A) be its spectrum. For any function f (λ) which is analytic in a simply connected domain D containing σ (A), we define  1 f (λ)(λ − A)−1 dλ, f (A) = 2πi C where C is a rectifiable Jordan curve lying in D and encircling σ (A). This integral is defined in L(X), so f (A) is a bounded linear operator of X. Clearly, the definition of f (A) is independent of the choice of the Jordan curve C. The operator f (A) is called the functional calculus of A with respect to f (λ), and the integral is called the Dunford integral. The following result is well known as a spectral mapping theorem. Theorem 1.9 Let A ∈ L(X), let D ⊃ σ (A) be a simply connected domain, and let f (λ) be an analytic function in D. Then, the functional calculus f (A) ∈ L(X) with respect to f (λ) has its spectrum such that σ (f (A)) = f (σ (A)). For the proof, see [Dun43] (cf. also [Yos80, Corollary 1, p. 227]). Let us next consider spectrum separation for a closed linear operator A. We assume the following situation. The spectrum σ (A) is separated into two closed components Σ1 and Σ2 , Σ1 being bounded and Σ2 being possibly unbounded. Let C be a rectifiable Jordan curve such that Σ1 (resp. Σ2 ) is contained in the interior (resp. exterior) domain. By the formula  1 (λ − A)−1 dλ (1.15) P= 2πi C we introduce a bounded linear operator on X. Utilizing (1.12), we can verify that P is a projection operator on X, i.e., P 2 = P . Set X1 = P X = (1 − P )−1 0 and X2 = (1 − P )X = P −1 0; then, X1 and X2 are closed subspaces of X; furthermore, X is decomposed into the direct sum X1 ⊕ X2 whose projections are P and 1 − P . It is easy to see that R(P ) ⊂ D(A) (cf. Theorem 1.4) and AP is a bounded operator on X with AP ⊃ P A. Therefore, X1 ⊂ D(A); in addition, AP = AP 2 = P AP implies that X1 ⊂ D(A1 ) and AX1 ⊂ X1 . On the other hand, (1 − P )A = A(1 − P ) on D(A); therefore, if U ∈ D(A) ∩ X2 , then AU = (1 − P )AU ∈ X2 . In view of these properties, we are led to consider the parts A1 and A2 of A in X1 and X2 , respectively. Then, AU = A1 P U + A2 (1 − P )U for any U ∈ D(A) (note that (1 − P )U = U − P U ∈ D(A)). Therefore, A is decomposed into the direct sum A = A1 ⊕ A2 . It is easy to see that A1 is a bounded operator of X1 and A2 is a densely defined, closed linear operator of X2 . We have ρ(A) ⊂ ρ(A1 ) ∩ ρ(A2 ). And, for λ ∈ ρ(A), we have (λ − A)−1 = (λ − A1 )−1 ⊕ (λ − A2 )−1 . In the meantime, we consider  1 1 R1 (μ) = (λ − A)−1 dλ, 2πi C μ − λ where μ’s lie in the exterior domain of C. It is easy to see that R1 (μ)P = P R1 (μ) and R1 (μ)(μ − A) ⊂ (μ − A)R1 (μ) = P . These imply that (μ − A1 )R1 (μ) =

12

1 Preliminaries

R1 (μ)(μ − A1 ) on the space X1 ; hence, (μ − A1 )−1 = R1 (μ) on X1 . Since R1 (μ) is analytic in the exterior domain of C, we observe that (μ − A1 )−1 can be extended analytically even in the domain Σ2 . Similarly, we consider  1 1 R2 (μ) = (λ − A)−1 dλ, 2πi −C μ − λ where μ’s lie in the interior domain of C. Then, R2 (μ)(1 − P ) = (1 − P )R2 (μ) and R2 (μ)(μ − A) ⊂ (μ − A)R2 (μ) = 1 − P . These furthermore imply that (μ − A2 )−1 = R2 (μ) on the space X2 . Hence, (μ − A2 )−1 can be extended analytically in Σ1 . Ultimately, we can show that σ (A1 ) = Σ1 and σ (A2 ) = Σ2 . For the detailed proof, see [Kat80, Theorem 6.17, p. 178]. Theorem 1.10 Let the spectrum σ (A) of a densely defined, closed linear operator A be separated into two closed components Σk (k = 1, 2), at least Σ1 being bounded. Then, X can be decomposed into the continuous direct sum X = X1 ⊕ X2 of two closed subspaces Xk (k = 1, 2), and A is decomposed into A = A1 ⊕ A2 , where A1 is a bounded operator of X1 , and A2 is a densely defined, closed linear operator of X2 with spectra σ (Ak ) = Σk (k = 1, 2). Such a decomposition is unique. Proof It remains to prove the uniqueness of the decomposition. Assume that there exists another such decomposition X = X1 ⊕ X2 and A = A 1 ⊕ A 2 . Then,    1 1 (λ − A 1 )−1 ⊕ (λ − A 2 )−1 dλ (λ − A)−1 dλ = P= 2πi C 2πi C       1 1 (λ − A 1 )−1 dλ ⊕ (λ − A 2 )−1 dλ = 1X ⊕ 0X , = 1 2 2πi C 2πi C where 1X is the identity operator of X1 , and 0X is the zero operator of X2 . There1 2 fore, X1 = X1 and X2 = X2 . Then, it is clear that A1 = A 1 and A2 = A 2 . 

4 Nonlinear Operators 4.1 Contraction Mappings Let X be a Banach space with norm  · . Let X be a subset of X, X being a metric space with the induced distance d(U, V ) = U − V , U, V ∈ X. An operator S from X into X is called a contraction operator or a contraction mapping if S satisfies the Lipschitz condition SU − SV  ≤ δU − V , with constant 0 < δ < 1.

U, V ∈ X,

(1.16)

4 Nonlinear Operators

13

Concerning contraction mappings, we have the following fundamental result. For the proof, see, e.g., [Zei86, Theorem 1.A, p. 17]. Theorem 1.11 Let X be a closed subset of X. Let S : X → X be a contraction, and let S map X into itself. Then, S has a unique fixed point U ∈ X, i.e., SU = U .

4.2 Fréchet Differentiation Let X and Y be Banach spaces with norms  · X and  · Y , respectively. Let S be a continuous operator from an open set O of X into Y . Then, S is said to be Fréchet differentiable at a point U ∈ O if there exists a bounded linear operator A ∈ L(X, Y ) which satisfies o(h) = S(U + h) − SU − AhY ,

lim o(h)/hX = 0;

h→0

here o(h) is defined for h ∈ X such that U + h ∈ O. The linear operator A (if it exists) is uniquely determined and is called the Fréchet differential of S at U and is denoted by A = S (U ) = S U . When S is Fréchet differentiable at every point of O, S is said to be Fréchet differentiable in O. The mapping S : O → L(X, Y ), U → S U is called the Fréchet derivative of S. When the derivative S is continuous from O into L(X, Y ), we say that S is continuously Fréchet differentiable in O. The following theorems are all verified directly from the definition. Theorem 1.12 Let X, Y, Z be Banach spaces. Let S : O → Y be a continuous operator from an open set O ⊂ X which is Fréchet differentiable at U ∈ O. Let T : O → Z be a continuous operator from an open set O ⊂ Y containing SO, and let T be Fréchet differentiable at SU . Then, the operator T ◦ S : O → Z is Fréchet differentiable at U , and the differential is given by T (SU ) ◦ S (U ). Theorem 1.13 Let X, Y be Banach spaces. Let S : O → O be a continuous homeomorphism from an open set O ⊂ X onto an open set O ⊂ Y . If S is Fréchet differentiable at U ∈ O and if its differential S (U ) is a linear isomorphism from X onto Y , then S −1 is Fréchet differentiable at SU , and the differential is given by S (U )−1 . Theorem 1.14 Let O ⊂ X be an open set of a Banach space X. Let S : O → X be a contraction mapping satisfying (1.16). Then, the operator 1 − S is a homeomorphism from O onto the open set O = (1 − S)O, where 1 denotes the identity operator on X. Furthermore, if S is Fréchet differentiable in O, then the inverse operator (1 − S)−1 is also Fréchet differentiable in O with derivative [(1 − S)−1 ] = (1 − S )−1 . Proof Let us first verify that (1 − S)O is an open set. Let V˜ = (1 − S)U˜ ∈ (1 − S)O, and let B(U˜ ; r) ⊂ O with some r > 0. Consider the closed ball B(V˜ ; (1 − δ)r). For

14

1 Preliminaries

any V ∈ B(V˜ ; (1 − δ)r), put ΦU = V + SU . Then, it is clear that Φ is a contraction mapping defined on B(U˜ ; r) and maps B(U˜ ; r) into itself. Theorem 1.11 then yields that there exists a unique fixed point of Φ, i.e., U = ΦU = V + SU . This implies that V ∈ (1 − S)O, and therefore B(V˜ ; (1 − δ)r) ⊂ (I − S)O. Hence, O = (1 − S)O is an open set of X. Let us next verify that 1 − S is injective. If V = (1 − S)U and V˜ = (1 − S)U˜ for U, U˜ ∈ O, then U − U˜ = (V − V˜ ) + (SU − S U˜ ), and hence (1 − δ)U − U˜  ≤ V − V˜ . So, U − U˜  ≤ (1 − δ)−1 V − V˜ . This means that 1 − S is a one-to-one operator and that its inverse (1 − S)−1 is Lipschitz continuous. The second assertion of the theorem is now verified by Theorem 1.13. 

5 Interpolation of Banach Spaces 5.1 Interpolation Spaces Let X0 and X1 be two Banach spaces with norms  · X0 and  · X1 , respectively, X1 being embedded in X0 densely and continuously. Let S be the strip S = {z; 0 < Re z < 1}

(1.17)

in the complex plane C. By H(X0 , X1 ) we denote the space of analytic functions with the following properties: H(X0 , X1 ) = {F (z); F (z) is an analytic function for z ∈ S with values in X0 , is a bounded and continuous function for z ∈ S with values in X0 , and is a bounded and continuous function for z = 1 + iy with values in X1 }. For F ∈ H(X0 , X1 ), its norm is defined by  sup F (iy)X0 , F H = max −∞
sup

−∞
 F (1 + iy)X1 .

Then, H(X0 , X1 ) is a Banach space (see [Tri78, Theorem 1.9.1]). For each 0 ≤ θ ≤ 1, we define the space [X0 , X1 ]θ in the following way: [X0 , X1 ]θ = {U ∈ X0 ; there exists a function F ∈ H(X0 , X1 ) such that U = F (θ )}.

(1.18)

For U ∈ [X0 , X1 ]θ , its norm is defined by U θ =

inf

F ∈H,F (θ)=U

F H .

Then, [X0 , X1 ]θ is a Banach space with the norm  · θ (see [Tri78, Theorem 1.9.2]).

5 Interpolation of Banach Spaces

15

The following basic properties are verified: (1) (2) (3) (4)

[X0 , X1 ]0 = X0 and [X0 , X1 ]1 = X1 with isometries. For 0 < θ < 1, X1 ⊂ [X0 , X1 ]θ ⊂ X0 densely and continuously. θ For 0 < θ < 1, U θ ≤ U 1−θ X0 U X1 for all U ∈ X1 . For 0 ≤ θ < θ˜ ≤ 1, [X0 , X1 ]θ˜ ⊂ [X0 , X1 ]θ continuously.

Indeed, for any U ∈ X1 , consider the sequence of analytic functions Fn (z) = e−nz U . Obviously, Fn belongs to H(X0 , X1 ) with Fn (0) = U . Therefore, U ∈ [X0 , X1 ]0 with U 0 ≤ inf Fn H ≤ inf max{U X0 , e−n U X1 } = U X0 . n

n

Conversely, it is clear that U ∈ [X0 , X1 ]0 implies U ∈ X0 with U X0 ≤ U 0 . Hence, as we have seen, U X0 = U 0 for any U ∈ X1 . Since X1 is dense in X0 , we conclude that [X0 , X1 ]0 = X0 with isometry. Similarly, for any U ∈ X1 , consider the sequence of analytic functions Fn (z) = en(z−1) U . It is easy to see that Fn ∈ H(X0 , X1 ) with U = Fn (1). From this one verifies that U 1 ≤ U X1 . Conversely, it is clear that U 1 ≤ U X1 . Therefore, we have X1 = [X0 , X1 ]1 with isometry. Hence (1) is verified. Let U ∈ X1 be a nonzero vector. The function F (z) = (U X0 /U X1 )z−θ U is θ in H(X0 , X1 ) with U = F (θ ). Therefore, U θ ≤ F H ≤ U 1−θ X0 U X1 . Hence (3) is verified. The proofs of other properties are given in [Tri78, Theorem 1.9.3].

5.2 Interpolation Theorem Let X0 and X1 (resp. Y0 and Y1 ) be two Banach spaces with norms  · X0 and  · X1 (resp.  · Y0 and  · Y1 ), respectively, such that X1 ⊂ X0 (resp. Y1 ⊂ Y0 ) densely and continuously. Let [X0 , X1 ]θ (resp. [Y0 , Y1 ]θ ) be the interpolation space defined by (1.18). If T is a bounded linear operator from X0 into Y0 and from X1 into Y1 simultaneously, then T is automatically bounded from the interpolation space [X0 , X1 ]θ into [Y0 , Y1 ]θ . Theorem 1.15 Let X1 ⊂ X0 (resp. Y1 ⊂ Y0 ) densely and continuously, and let [X0 , X1 ]θ (resp. [Y0 , Y1 ]θ ) be the interpolation space. If T ∈ L(X0 , Y0 ) and T ∈ L(X1 , Y1 ), then T belongs to L([X0 , X1 ]θ , [Y0 , Y1 ]θ ) for any 0 < θ < 1 with the estimate θ T L([X0 ,X1 ]θ ,[Y0 ,Y1 ]θ ) ≤ T 1−θ L(X0 ,Y0 ) T L(X1 ,Y1 ) .

Proof If T is zero, the assertion is trivial. So, let T L(X0 ,Y0 ) = 0 and T L(X1 ,Y1 ) = 0. If U ∈ [X0 , X1 ]θ , then U = F (θ ) with some F ∈ H(X0 , X1 ).

16

1 Preliminaries

We then consider the analytic function G(z) = (T L(X0 ,Y0 ) /T L(X1 ,Y1 ) )z−θ T F (z). Clearly, G ∈ H(Y0 , Y1 ) with G(θ ) = T U . Therefore, T U ∈ [Y0 , Y1 ]θ with the estimate θ T U θ ≤ GH ≤ T 1−θ L(X0 ,Y0 ) T L(X1 ,Y1 ) F H .

Taking the infimum of F H for all possible F , we obtain the desired estimate. 

6 Adjoint Spaces and Operators 6.1 Dual Spaces Let X be a Banach space with norm  · . Regarding C as a Banach space, we consider the Banach space L(X, C), which is of course the space of all continuous linear operators from X into C. Such a linear operator is called a continuous linear functional on X. The space L(X, C) is denoted by X and is called the dual space of X. So, X is a Banach space of all continuous linear functionals on X equipped with the norm Φ = sup F ≤1 |Φ(F )| for Φ ∈ X . It is, however, convenient to change scalar multiplications from the usual one to the following one: (αΦ)(F ) = αΦ(F ),

α ∈ C, Φ ∈ X , F ∈ X.

(1.19)

Throughout the book, the dual space is equipped with this multiplication. It is not difficult to verify that X is again a Banach space with the same norm as before. As X is a Banach space, we can consider the second dual space X of X. Then, the operator ι from X into X is defined by (ιF )(Φ) = Φ(F ),

F ∈ X, Φ ∈ X .

Since |(ιF )(Φ)| = |Φ(F )| ≤ ΦF , ιF is certainly an element of X , and ιF  ≤ F  for any F ∈ X. Conversely, for any F ∈ X, the Hahn–Banach extension theorem below provides the existence of a functional Φ ∈ X such that Φ(F ) = F  and Φ = 1; therefore, F  = |(ιF )(Φ)| ≤ ιF . That is, ιF  = F  for any F ∈ X. In the meantime, it is verified that ι is a linear operator from X into X . Therefore, ι is an isometry from X onto the subspace ι(X) of X . The operator ι is called the canonical isometry of X. When ι is surjective, i.e., ι(X) = X , X is called a reflexive Banach space. When X is a Hilbert space with inner product (·, ·), the Riesz theorem yields that, for any Φ ∈ X , there exists an element F ∈ X such that Φ(G) = (G, F ), G ∈ X. On the basis of this fact, X is seen to be reflexive. Therefore, the Hilbert spaces are reflexive. The following theorem is called the Hahn–Banach extension theorem. For the proof, see [Yos80, Theorem 1, p. 106].

6 Adjoint Spaces and Operators

17

Theorem 1.16 Let Y be any subspace of a Banach space X. Let F : Y → C be a linear functional on Y satisfying the estimate |F (U )| ≤ MU , U ∈ Y , with some constant M ≥ 0. Then, F can be extended over the whole space X as a continuous linear functional F˜ ∈ X with estimate F˜  ≤ M.

6.2 Adjoint Spaces Let X and Y be Banach spaces with norms ·X and ·Y , respectively. A complexvalued function ·, · defined on the product space X × Y is called a sesquilinear form on X × Y if it satisfies  αF + β F˜ , G = αF, G + βF˜ , G, α, β ∈ C, F, F˜ ∈ X, G ∈ Y, ˜ ˜ ∈ Y. ˜ = αF, G + βF, G, α, β ∈ C, F ∈ X, G, G F, αG + β G Furthermore, a sesquilinear form on X × Y is called a duality product if it satisfies |F, G| ≤ F X GY ,

F ∈ X, G ∈ Y,

(1.20)

F X = sup |F, G|,

F ∈ X,

(1.21)

GY = sup |F, G|,

G ∈ Y.

(1.22)

GY ≤1

F X ≤1

When there exists such a duality product for two Banach spaces X and Y , the space Y is called an adjoint space of X with duality product ·, ·. Obviously, this relation is symmetric. That is, if Y is adjoint to X with ·, ·X×Y , then X is adjoint to Y with the duality product G, F Y ×X = F, GX×Y . This fact allows us to consider adjointness as a property for pairs of Banach spaces and to say that {X, Y } forms an adjoint pair with duality product ·, ·. If X is a Hilbert space, then X is clearly adjoint to itself with its inner product. In this sense, {X, X} is an adjoint pair. Let X be a Banach space, and X be its dual space defined above. We introduce the function on the product space X × X given by F, Φ = Φ(F ),

F ∈ X, Φ ∈ X .

By virtue of the definition (1.19), this function defines a sesquilinear form. Moreover, the function is shown to satisfy (1.20), (1.21), and (1.22). Especially, as mentioned above, Theorem 1.16 yields that, for any F ∈ X, there exists Φ ∈ X such that Φ(F ) = F X and ΦX = 1. Consequently, F X =

sup |Φ(F )| =

ΦX ≤1

sup |F, Φ|,

ΦX ≤1

F ∈ X.

Therefore, this function gives a duality product on X × X . In this way, {X, X } forms an adjoint pair.

18

1 Preliminaries

As verified above, {X, X } is an adjoint pair. So is {X , X }. Therefore, X has at least two adjoint spaces X and X . As is well known, if X is nonreflexive, then X and X are not canonically isomorphic. In order to denote an unspecified adjoint space of X, we shall use the notation X ∗ . Therefore, X ∗ may not be uniquely determined from X, quite differently from its dual space X . The norm of X ∗ will be denoted by  · ∗ . Let {X, X ∗ } be an adjoint pair with duality product ·, ·. For each G ∈ X ∗ , the linear functional ·, G is continuous on X due to (1.20); therefore, the correspondence J : G → J G from X∗ into X is defined by the relation (J G)(F ) = F, G,

F ∈ X, G ∈ X ∗ .

(1.23)

We observe from (1.20) that J GX = sup |(J G)(F )| = sup |F, G| ≤ GX∗ , F X ≤1

F X ≤1

G ∈ X∗ .

Conversely, by (1.22), GX∗ = sup |F, G| = sup |(J G)(F )| ≤ J GX , F X ≤1

F X ≤1

G ∈ X∗ .

Therefore, J preserves the norm. In view of (1.19), it is also easy to observe that J is a linear operator from X ∗ into X . Hence, J is an isometry from X ∗ onto the subspace J (X ∗ ) of X . When X is reflexive, i.e., ι(X) = X , we can show that J (X ∗ ) = X . Theorem 1.17 Let X be a reflexive Banach space, and let {X, X ∗ } be an adjoint pair with duality product ·, ·. Then, the isometry J : X ∗ → X defined by (1.23) is onto and hence an isomorphism. Consequently, for any Φ ∈ X , there exists a unique vector G ∈ X ∗ such that ΦX = GX∗ and Φ(F ) = F, G for all F ∈ X. Proof Suppose that J (X ∗ ) = X . Since J is a linear isometry, J (X ∗ ) is a closed subspace of X . Then, Theorem 1.16 provides the existence of a functional Ψ ∈ X such that Ψ (J (X ∗ )) = 0 and Ψ X = 1 (cf. [Yos80, Corollary, p. 109]). By the reflexivity of X, there exists F ∈ X such that ιF = Ψ , where ι is the canonical isometry. Then, F, G = (J G)(F ) = (ιF )(J G) = Ψ (J G) = 0, and, due to (1.21), F = 0. This contradicts F  = ιF  = 1.

G ∈ X∗ , 

This theorem shows that, when X is reflexive, every continuous linear functional on X can be represented by the duality product, namely, the Riesz theorem can be generalized to the Banach spaces. In addition, as X ∗ is isometric to X , the adjoint space X ∗ is also reflexive. When {X, X ∗ } is an adjoint pair, their product spaces also form an adjoint pair.

6 Adjoint Spaces and Operators

19

Proposition 1.1 Let {X, X ∗ } be an adjoint pair. Consider the product spaces ∗ ˜ ˜ ˜ X = {(F, F˜ ); F, F˜ ∈ X} and X∗ = {(G,   G); G, G ∈ X }) with norms (F, F )X = ˜ X∗ = G2∗ + G ˜ 2∗ , respectively. Then, {X, X∗ } F 2 + F˜ 2 and (G, G) also forms an adjoint pair with duality product ˜ = F, G + F˜ , G, ˜ (F, F˜ ), (G, G)

˜ ∈ X∗ . (F, F˜ ) ∈ X, (G, G)

Proof By Cauchy–Schwarz’s inequality, ˜ ≤ F G + F˜ G ˜ ≤ (F, F˜ )(G, G). ˜ |(F, F˜ ), (G, G)| Hence, inequality (1.20) is valid. For any F ∈ X, we easily verify from (1.21) that there exists a sequence Gn ∈ X ∗ such that Gn ∗ ≤ F  for all n ≥ 1 and that F, Gn  → F 2 as n → ∞. ˜ n  → F˜ 2 as n → ∞ with some sequence G ˜n ∈ Similarly, for any F˜ ∈ X, F˜ , G ∗ ˜ ˜ ˜ ˜ ˜ X satisfying Gn ∗ ≤ F . Here, put Vn = Gn /(F, F ) and Vn = Gn /(F, F˜ ). We then observe that (Vn , V˜n )X∗ ≤ 1 and ˜ n ]/(F, F˜ ) = (F, F˜ ), (Vn , V˜n ) [F, Gn  + F˜ , G ≤

sup (V ,V˜ )X∗ ≤1

(F, F˜ ), (V , V˜ ).

Letting n → ∞, we obtain that (F, F˜ ) ≤

sup (V ,V˜ )X∗ ≤1

|(F, F˜ ), (V , V˜ )|.

It is clear that (F, F˜ ) ≥ sup(V ,V˜ ) ∗ ≤1 |(F, F˜ ), (V , V˜ )|. Hence, (1.21) holds X true. Condition (1.22) is also verified in an analogous way. Hence, the sesquilinear form on X × X∗ is a duality product, and consequently {X, X∗ } is an adjoint pair.  Let {X, X ∗ } be an adjoint pair with duality product ·, ·. When F ∈ X and G ∈ satisfy F, G = 0, one says that F (resp. G) is orthogonal to G (resp. F ) or F and G are mutually orthogonal. For a subset H ⊂ X (resp. K ⊂ X ∗ ), H ⊥ (resp. K ⊥ ) denotes the set of all vectors G ∈ X ∗ (resp. F ∈ X) which are orthogonal to every vector of H (resp. K). Then, the following proposition is obvious. The proof is left to the reader. X∗

Proposition 1.2 Let {X, X ∗ } be an adjoint pair, and let H ⊂ X and K ⊂ X ∗ be subsets of X and X ∗ , respectively. Then, H ⊥ and K ⊥ are closed linear subspaces of X ∗ and X, respectively. We shall need the following result later.

20

1 Preliminaries

Proposition 1.3 Let X be a reflexive Banach space, and let {X, X ∗ } be an adjoint pair. If H is a closed linear subspace of X, then H ⊥⊥ = H . Proof By definition, it is clear that H ⊂ H ⊥⊥ . So, it suffices to prove the inverse inclusion. Let F ∈ H . By Theorem 1.16 (cf. also [Yos80, Corollary, p. 109]), we can construct a functional Φ ∈ X such that Φ(H ) = 0 and Φ(F ) = 0. Then, by Theorem 1.17, there exists a vector G ∈ X ∗ such that F˜ , G = 0 for every F˜ ∈ H and / H ⊥⊥ . F, G =

0; that is, G ∈ H ⊥ , and F is not orthogonal to G. Therefore, F ∈ ⊥⊥ ⊂ H.  This means that H We conclude this subsection by giving some examples of adjoint pairs. Let Ω be a domain in Rn . For 1 < p < ∞, the Lp space in Ω, which is denoted by Lp (Ω), is a reflexive Banach space with Lp (Ω) = Lp (Ω), where p1 + p1 = 1. When p = 2, L2 (Ω) has a Hilbert structure. Therefore, {Lp (Ω), Lp (Ω)} forms an adjoint pair with the duality product  f (x) g(x) dx, f ∈ Lp (Ω), g ∈ Lp (Ω). (1.24) f, gLp ,Lp = Ω

In the meantime, L1 (Ω) and L∞ (Ω) are nonreflexive Banach spaces. However, it is known that L1 (Ω) = L∞ (Ω). Therefore, {L1 (Ω), L∞ (Ω)} is also an adjoint pair with a similar duality product as (1.24) in which p = 1 and p = ∞.

6.3 Adjoint Operators Let {X, X ∗ } (resp. {Y, Y ∗ }) be an adjoint pair of Banach spaces with duality product ·, ·X×X∗ (resp. ·, ·Y ×Y ∗ ). Let A be a densely defined linear operator from a subspace D(A) ⊂ X into Y . Then, a linear operator A∗ with domain D(A∗ ) ⊂ Y ∗ into X ∗ is determined as follows. A vector Ψ ∈ Y ∗ is in D(A∗ ) if and only if there exists a vector Φ ∈ X ∗ such that AU, V Y ×Y ∗ = U, ΦX×X∗ for all U ∈ D(A). Since D(A) is dense in X, such a Φ is uniquely determined. For Ψ ∈ D(A∗ ), we define A∗ Ψ = Φ. Therefore, AU, Ψ Y ×Y ∗ = U, A∗ Ψ X×X∗

for all U ∈ D(A), Ψ ∈ D(A∗ ).

It is easily verified that D(A∗ ) is a linear subspace of Y ∗ and A∗ is a linear operator. This operator A∗ is then called the adjoint operator of A with respect to the adjoint pairs {X, X ∗ } and {Y, Y ∗ }. Similarly, if B is a densely defined linear operator from a subspace D(B) ⊂ Y ∗ into X ∗ , then its adjoint operator B ∗ with domain D(B ∗ ) is defined by U, BΨ X×X∗ = B ∗ U, Ψ Y ×Y ∗

for all Ψ ∈ D(B), U ∈ D(B ∗ ).

As before, B ∗ is a linear operator from the subspace D(B ∗ ) ⊂ X into Y . For the adjoint operators, we can show the following theorems.

6 Adjoint Spaces and Operators

21

Theorem 1.18 Let X and Y be reflexive Banach spaces, and let {X, X ∗ } and {Y, Y ∗ } be adjoint pairs. If T is a bounded linear operator from X into Y , then T ∗ is a bounded linear operator from Y ∗ into X ∗ . In addition, it holds that T  = T ∗  and T ∗∗ = T . Proof Since |T U, Ψ Y ×Y ∗ | ≤ T U X Ψ Y ∗ for U ∈ X and Ψ ∈ Y ∗ , it is seen that U → T U, Ψ Y ×Y ∗ is a continuous linear functional on X. Since X is reflexive, there exists a unique vector Φ ∈ X ∗ such that T U, Ψ Y ×Y ∗ = U, ΦX×X∗ for all U ∈ X. This means that Ψ ∈ D(T ∗ ) and T ∗ Ψ = Φ, i.e., T ∗ is everywhere defined. Furthermore, it is observed by (1.21) and (1.22) that T ∗ Ψ X∗ ≤ T Ψ Y ∗ ; therefore, T ∗ is a bounded linear operator with operator norm T ∗  ≤ T . Similarly, since T ∗ ∈ L(Y ∗ , X ∗ ), its adjoint operator T ∗∗ is defined. Since Y ∗ is reflexive (this follows from the reflexivity of Y ), it is verified that T ∗∗ is a bounded linear operator from X into Y with operator norm T ∗∗  ≤ T ∗ . But it is clear that T ∗∗ = T .  We next consider the case where X = Y and X ∗ = Y ∗ and {X, X ∗ } is an adjoint pair with duality product ·, ·. If A is a densely defined linear operator of X, then its adjoint operator A∗ is a linear operator acting in the adjoint space X ∗ . Theorem 1.19 Let X be a reflexive Banach space, and let {X, X ∗ } be an adjoint pair. If A is a densely defined, closed linear operator of X, then A∗ is also a densely defined, closed linear operator of X ∗ . In this case, A and A∗ enjoy the following properties: (1) A∗∗ = A. (2) λ ∈ ρ(A∗ ) if and only if λ ∈ ρ(A). (3) For λ ∈ ρ(A∗ ), it holds that (λ − A∗ )−1 = [(λ − A)−1 ]∗ . Proof For the proof, we introduce the product spaces X = {(F, F˜ ); F, F˜ ∈ X} and ˜ G, G ˜ ∈ X ∗ }. It is well known that X is also reflexive, and it is known X∗ = {(G, G); by Proposition 1.1 that X and X∗ form an adjoint pair. On X, we consider a linear operator given by R(F, F˜ ) = (−F˜ , F ). We can then show that G(A∗ ) = [R(G(A))]⊥ ,

(1.25)

where G(A) = {(U, AU ); U ∈ D(A)} ⊂ X denotes the graph of A and is a closed subspace of X because of the closedness of A. Indeed, (V , G) ∈ G(A∗ ) if and only if V ∈ D(A∗ ) and G = A∗ V ; so, this is equivalent to saying that AU, V  = U, G for all U ∈ D(A). Hence, (−AU, U ), (V , G) = 0 for all U ∈ D(A), and (V , G) is orthogonal to R(G(A)). In this way, (1.25) is shown. Suppose that D(A∗ ) is not dense. Then, there exists some nonzero functional Ψ ∈ (X ∗ ) such that Ψ (D(A∗ )) = 0. By Theorem 1.17, there exists a nonzero vector U ∈ X such that U, V  = 0 for all V ∈ D(A∗ ). This implies that (U, 0) is orthogonal to G(A∗ ). Consequently, by (1.25), (U, 0) ∈ [R(G(A))]⊥⊥ . By virtue of

22

1 Preliminaries

Proposition 1.3, it follows that (U, 0) ∈ R(G(A)) and consequently, U = A0. This contradicts U = 0. The closedness of G(A∗ ) in X∗ is concluded directly from (1.25) with the aid of Proposition 1.2. Similarly, F = A∗∗ U if and only if U, A∗ V  = F, V  for all V ∈ D(A∗ ); so, this is equivalent to saying that (−F, U ) is orthogonal to G(A∗ ). Hence, by virtue of Proposition 1.3, (−F, U ) ∈ R(G(A)); that is, (U, F ) ∈ G(A). Thus, we verify that A∗∗ = A. Let λ ∈ ρ(A), and put R(λ) = (λ − A)−1 ∈ L(X). By Theorem 1.18, R(λ)∗ ∈ L(X ∗ ). Then, we have (λ − A)U, R(λ)∗ G = U, G for all U ∈ D(A) and all G ∈ X ∗ . This implies that R(λ)∗ G ∈ D(A∗ ) and U, (λ − A∗ )R(λ)∗ G = U, G; hence, since D(A) is dense, (λ − A∗ )R(λ)∗ G = G for all G ∈ X ∗ . Similarly, we have R(λ)F, (λ − A∗ )V  = F, V  for all F ∈ X and all V ∈ D(A∗ ). Therefore, F, R(λ)∗ (λ − A∗ )V  = F, V ; hence, R(λ)∗ (λ − A∗ )V = V for all V ∈ D(A∗ ). We thus see that λ ∈ ρ(A∗ ) and (λ − A∗ )−1 = R(λ)∗ . We now know that λ ∈ ρ(A∗ ) implies λ ∈ ρ(A∗∗ ) = ρ(A). Hence, λ ∈ ρ(A∗ )  and λ ∈ ρ(A) are equivalent. Let X and Y be Hilbert spaces, {X, X} and {Y, Y } being adjoint pairs with their inner products. If A is a densely defined, closed linear operator from D(A) ⊂ X into Y , then its adjoint operator is from D(A∗ ) ⊂ Y into X. Similarly, if A is a densely defined, closed linear operator of X, then its adjoint A∗ is also a densely defined, closed linear operator of X with the property A∗∗ = A.

7 Extrapolation of Hilbert Spaces 7.1 Triplets of Spaces Consider two Hilbert spaces Z and X with inner products ((·, ·)) and (·, ·) and with norms  ·  and | · |, respectively. We assume that Z ⊂ X with dense and continuous embedding. Suppose that there exists a Banach space Z ∗ satisfying the following conditions: (1) Z ⊂ X ⊂ Z ∗ with dense and continuous embeddings. (2) {Z, Z ∗ } forms an adjoint pair with duality product ·, ·. (3) the duality product ·, · satisfies U, F  = (U, F )

for all U ∈ Z, F ∈ X.

(1.26)

Then the space Z ∗ is called an extrapolation space of Z ⊂ X, and the three spaces Z ⊂ X ⊂ Z ∗ are called a triplet of spaces. By the definition of duality product, ·, · must satisfy |U, Φ| ≤ U Φ∗ ,

U ∈ Z, Φ ∈ Z ∗ ,

(1.27)

7 Extrapolation of Hilbert Spaces

23

U  = sup |U, Φ|,

U ∈ Z,

(1.28)

Φ∗ = sup |U, Φ|,

Φ ∈ Z∗,

(1.29)

Φ∗ ≤1

U ≤1

where  · ∗ denotes the norm of Z ∗ . In addition, we see that, for U, V ∈ Z, U, V Z ∗ ×Z = V , U Z×Z ∗ = (V , U ) = (U, V ) = U, V Z×Z ∗ , i.e., U, V Z×Z ∗ = (U, V ) = U, V Z ∗ ×Z ,

U, V ∈ Z.

(1.30)

As a matter of fact, such an extrapolation space can be always constructed in a unique way. Existence. Let Z and X be the dual spaces of Z and X, respectively. By the continuity of embedding | · | ≤ D · , Ψ ∈ X implies Ψ ∈ Z with Ψ Z ≤ DΨ X . In addition, by the density of Z in X, if Ψ, Ψ˜ ∈ X satisfy Ψ (U ) = Ψ˜ (U ) for all U ∈ Z, then Ψ (F ) = Ψ˜ (F ) for all F ∈ X, namely, Ψ = Ψ˜ . These two facts mean that X is continuously embedded in Z . Furthermore, it is also shown that this embedding is dense. Indeed, suppose the contrary; then, the closure of X in Z is a closed subspace which is strictly contained in Z ; since ι(Z) = Z , where ι is the canonical isometry from Z onto Z , there exists a nonzero vector U ∈ Z such that (ιU )(Ψ ) = Ψ (U ) = 0 for all Ψ ∈ X ; as U ∈ Z ⊂ X, U must be the zero vector. Hence, X ⊂ Z with dense and continuous embedding. Let J : X → X be the correspondence defined by the relation (J G)(F ) = (F, G) for F, G ∈ X. By virtue of the Riesz theorem, J is a linear isometry from X onto X . By using this isometry, we introduce a new norm on X given by F ∗ = J F Z . By definition, F ∗ ≤ DJ F X = |F | for F ∈ X, that is,  · ∗ is a weaker norm than the original one. We in turn consider the completion of X with respect to the new norm. Let us define Z ∗ as the resulting Banach space. Since Z ∗ is a completion of X and since  · ∗ ≤ D| · |, X is densely and continuously embedded in Z ∗ . Since J F Z = F ∗ for F ∈ X and since X is dense in Z ∗ , we can extend the linear operator J over Z ∗ as a linear operator into Z , which will be denoted again by J . Moreover, it is verified that the extended J is onto Z and hence is an isometry from Z ∗ onto Z , i.e., J ΦZ = Φ∗ , Φ ∈ Z ∗ . On Z × Z ∗ , we define a sesquilinear form by U, Φ = (J Φ)(U ), U ∈ Z, Φ ∈ Z ∗ . Then, it is easily observed that (1.27), (1.28), and (1.29) are all fulfilled. Therefore, U, Φ is a duality product, and {Z, Z ∗ } becomes an adjoint pair. Finally, for any U ∈ Z and F ∈ X, U, F  = (J F )(U ) = (U, F ). Hence, (1.26) is valid. We have thus shown that Z ∗ is certainly an extrapolation space.  Uniqueness. Let Z ∗ be an extrapolation space. We have, for any F ∈ X, F ∗ = supU ≤1 |U, F | = supU ≤1 |(U, F )|. This means that the norm F ∗ is determined by Z and X only and is actually independent of Z ∗ . Furthermore, Z ∗ can be characterized as a completion of X with respect to the norm  · ∗ . These facts then  mean that the extrapolation space Z ∗ is determined uniquely.

24

1 Preliminaries

Let Z ⊂ X ⊂ Z ∗ be a triplet of spaces. Let J : Z ∗ → Z be the isometric isomorphism constructed above. By applying the Riesz theorem to Z, we have also a linear isometry JZ from Z onto Z such that (JZ U˜ )(U ) = ((U, U˜ )), U, U˜ ∈ Z. So, Z ∗ is isometric to Z with the isometric isomorphism J˜ = JZ−1 ◦ J . In particular, Z ∗ also enjoys Hilbert structure with the inner product ˜ ˜ ∗ = ((J˜Φ, J˜Φ)), ((Φ, Φ))

Φ, Φ˜ ∈ Z ∗ .

By definition,   ˜ ∗ = J −1 ◦ J Φ, J −1 ◦ J Φ˜ ((Φ, Φ)) Z Z   ˜ ˜ J −1 ◦ J Φ = J˜Φ, Φ, Φ, Φ˜ ∈ Z ∗ . = (J Φ) Z

(1.31)

Concerning the weak convergence in Z and Z ∗ , we easily observe the following results. Theorem 1.20 For a sequence {Un } of vectors in Z, Un converges weakly to a limit U in Z if and only if Un , Φ → U, Φ for all Φ ∈ Z ∗ . Conversely, for a sequence {Φn } of vectors in Z ∗ , Φn converges weakly to Φ in Z ∗ if and only if U, Φn  → U, Φ for all U ∈ Z. Proof Let {Un } be a sequence in Z. Then, Un → U weakly in Z if and only if Ψ (Un ) → Ψ (U ) for all Ψ ∈ Z . Since J is an isomorphism from Z ∗ onto Z , this is equivalent to saying that (J Φ)(Un ) → (J Φ)(U ) for all Φ ∈ Z ∗ . That is, Un , Φ → U, Φ for all Φ ∈ Z ∗ . ˜ Φn ))∗ = J˜Φ, ˜ Φn  for all Consider a sequence {Φn } in Z ∗ . By (1.31), ((Φ, ∗ ∗ ˜ ˜ Φ ∈ Z . Since J is an isomorphism from Z onto Z, Φn → Φ weakly in Z ∗ if  and only if U, Φn  → U, Φ for all U ∈ Z.

7.2 Extrapolation Theorem Concerning extrapolation of Hilbert spaces, we verify the following theorem. Theorem 1.21 Let Z ⊂ X ⊂ Z ∗ be a triplet of spaces. If T is a bounded self-adjoint operator of X and is a bounded linear operator on Z, then T can be extended on Z ∗ as a bounded linear operator with the estimate T L(Z ∗ ) ≤ T L(Z) . Proof For any F ∈ X, we have T F ∗ = sup |U, T F | = sup |(U, T F )| = sup |(T U, F )| ≤ T L(Z) F ∗ . U ≤1

U ≤1

U ≤1

Since X is dense in Z ∗ , T is uniquely extended on Z ∗ as a bounded operator.



We also remark an interpolation property which will be proved by (2.145) below.

8 Linear Operators Associated with Sesquilinear Forms

25

Theorem 1.22 Let Z ⊂ X ⊂ Z ∗ be a triplet. It is always true that [Z, Z ∗ ] 1 = X 2 with norm equivalence.

8 Linear Operators Associated with Sesquilinear Forms 8.1 Sesquilinear Forms and Associated Operators Let Z ⊂ X ⊂ Z ∗ be a triplet of spaces. By definition, {Z, Z ∗ } is an adjoint pair. In this subsection, we will use the duality product ·, ·Z ∗ ×Z rather than ·, ·Z×Z ∗ ; of course, ·, ·Z ∗ ×Z = ·, ·Z×Z ∗ . We consider a sesquilinear form a(U, V ) defined on Z ×Z, or shortly on Z. More precisely, let a(U, V ) be a complex-valued function defined for (U, V ) ∈ Z × Z satisfying  a(αU + β U˜ , V ) = αa(U, V ) + βa(U˜ , V ), α, β ∈ C, U, U˜ , V ∈ Z, a(U, αV + β V˜ ) = αa(U, V ) + βa(U, V˜ ), α, β ∈ C, U, V , V˜ ∈ Z. When a(U, V ) satisfies the condition |a(U, V )| ≤ MU  V ,

U, V ∈ Z,

(1.32)

with some constant M, we say that a(U, V ) is a continuous form. Obviously, (1.32) implies that a(Un , Vn ) → a(U, V ) if Un → U and Vn → V in Z, respectively. Consider a continuous sesquilinear form a(U, V ) on Z. For each U ∈ Z, a(U, ·) is a continuous linear functional on Z. Therefore, by Theorem 1.17, there exists a unique Φ ∈ Z ∗ such that a(U, V ) = V , ΦZ×Z ∗ for all V ∈ Z, i.e., a(U, V ) = Φ, V  for all V ∈ Z. The correspondence A : U → Φ is obviously a linear operator from Z into Z ∗ . This operator A is called the linear operator associated with the form a(U, V ). Therefore, a(U, V ) = AU, V 

for all U ∈ Z, V ∈ Z.

(1.33)

From this we observe that A is a bounded operator satisfying AU∗ = sup |AU, V | ≤ MU , V ≤1

U ∈ Z.

Hence, AL(Z,Z ∗ ) ≤ M. When a(U, V ) satisfies the condition Re a(U, U ) ≥ δU 2 ,

U ∈ Z,

(1.34)

with some positive constant δ > 0, we say that a(U, V ) is a coercive form. Clearly, (1.34) implies that U = 0 if a(U, U ) = 0. The following theorem is well known as the Lax–Milgram theorem. For details and its proof, see [Yos80, Theorem, p. 92] or [DL88, Theorem 7, p. 368].

26

1 Preliminaries

Theorem 1.23 Let a(U, V ) be a continuous and coercive sesquilinear form on Z (i.e., (1.32) and (1.34)). Then, for any Ψ ∈ Z , there exists a unique element V ∈ Z such that Ψ (U ) = a(U, V ) for U ∈ Z. Using this theorem, we can show that A is an isomorphism from Z onto Z ∗ . Theorem 1.24 Let a(U, V ) satisfy (1.32) and (1.34), and let A be a linear operator associated with a(U, V ). Then, A is an isomorphism from Z onto Z ∗ with δU  ≤ AU ∗ ≤ MU  as well and is a densely defined, closed linear operator in Z ∗ . Proof It is already known that A ∈ L(Z, Z ∗ ) with AL(Z,Z ∗ ) ≤ M. By (1.34), AU ∗ ≥ δU  for all U ∈ Z; this in particular shows that A is one-to-one. Furthermore, A is shown to be onto by virtue of the Lax–Milgram theorem applied to the sesquilinear form a ∗ (U, V ) = a(V , U ) (obviously, a ∗ (U, V ) also satisfies (1.32) and (1.34)). Indeed, for any Φ ∈ Z ∗ , J Φ ∈ Z ; so, there exists an element U ∈ Z such that (J Φ)(V ) = a ∗ (V , U ) = a(U, V ) for all V ∈ Z; then, by definition, Φ, V  = AU, V  for all V ∈ Z; (1.29) then implies that Φ = AU. Thus, A−1 is a bounded operator from Z ∗ into Z with the estimate A−1 L(Z ∗ ,Z) ≤ δ. Let us now consider A as a linear operator acting in Z ∗ . Since D(A) = Z is dense in Z ∗ , A is densely defined. In addition, since A−1 is a bounded operator on Z ∗ , A is a closed linear operator of Z ∗ .  In what follows, we are also concerned with the parts of A in X and Z. By definition, since D(A) ⊂ X, the part of A in X is given by  D(A|X ) = {U ∈ Z; AU ∈ X}, A|X U = AU. In view of (1.26), we verify that U ∈ D(A|X ) if and only if a(U, V ) is continuous in V with respect to the X norm. Moreover, if U ∈ D(A|X ), then a(U, V ) = (A|X U, V ) for all V ∈ Z. For simplicity, the part A|X is often denoted by A. Similarly, since Z = D(A), the part of A in Z is given by  D(A|Z ) = {U ∈ Z; AU ∈ Z}, A|Z U = AU. In view of (1.30), we verify that U ∈ D(A|Z ) if and only if a(U, V ) is continuous in V with respect to the Z ∗ norm. Also, for U ∈ D(A|Z ), we have a(U, V ) = U, V Z×Z ∗ for all V ∈ Z.

8.2 Adjoint Forms and Adjoint Operators When (1.34) is valid, the parts A|X and A|Z are densely defined, closed operators in X and Z, respectively. In order to prove these facts, we have to consider adjoint operators.

9 Integral Equations and Inequalities of Volterra Type

27

We first introduce a sesquilinear form given by a ∗ (U, V ) = a(V , U ),

(U, V ) ∈ Z × Z,

which is called the adjoint form of a(U, V ). Obviously, a ∗ (U, V ) is also continuous and coercive on Z. Let B be the linear operator associated with a ∗ (U, V ). As is shown above, under (1.32) and (1.34), B is a densely defined (as a matter of fact, D(B) = Z), closed linear operator of Z ∗ satisfying the relation a(U, V ) = a ∗ (V , U ) = BV, U  for U, V ∈ Z. Furthermore, AU, V  = a(U, V ) = U, BV for U, V ∈ Z. It is then possible to observe that A|Z is the adjoint operator B∗ of B with respect to {Z, Z ∗ }. Indeed, U = B∗ U˜ if and only if U, V Z×Z ∗ = U˜ , BV for all V ∈ Z; so, it is equivalent to that U, V Z ∗ ×Z = AU˜ , V  for all V ∈ Z; hence, U = B∗ U˜ if and only if U = AU˜ ∈ Z. Here, we utilized the property (1.30). We now obtained the following results. Theorem 1.25 Let A be the linear operator associated with a(U, V ). Under (1.32) and (1.34), A|X and A|Z are densely defined, closed linear operators in X and Z, respectively. The adjoint operators A∗ and (A|Z )∗ with respect to {Z, Z ∗ } are given by B|Z and B, respectively. Meanwhile, the adjoint operator (A|X )∗ with respect to {X, X} is given by B|X . Proof Since A|Z = B∗ , the density of D(A|Z ) in Z is obtained directly by applying Theorem 1.19. Since D(A|Z ) ⊂ D(A|X ) and since Z is dense in X, the density of D(A|X ) in X is also verified. Repeating the similar argument as for A|Z = B∗ , we can now verify that A|X is the adjoint operator (B|X )∗ of B|X with respect to the adjoint pair {X, X}. The second assertions are then obtained directly by (1) of Theorem 1.19. 

9 Integral Equations and Inequalities of Volterra Type 9.1 Generalized Inequality of Gronwall Type Let μ > 0 and ν > 0 be two parameters. We begin with introducing the function defined by the series Eμ,ν (t) =

∞  n=0

t nν , (μ + nν)

0 ≤ t < ∞,

(1.35)

where (ξ ) (0 < ξ < ∞) is the gamma function. We can estimate Eμ,ν (t) as follows. Lemma 1.2 Eμ,ν (t) ≤

2 (1 + t)2−μ et+1 , 0 ν

where 0 = min0<ξ <∞ (ξ ).

0 ≤ t < ∞,

(1.36)

28

1 Preliminaries

Proof We write Eμ,ν (t) =

∞  



=0 ≤μ+nν<+1

 t nν . (μ + nν)

In addition, notice that, since (ξ ) is monotonically decreasing for 0 < ξ ≤ 1, it holds that (ξ ) ≥ (1) = 1 for 0 < ξ < 1. Put 0 = min1≤ξ ≤2 (ξ ) > 0; then (ξ ) ≥ 0 for 1 ≤ ξ < 2. Finally, since (ξ ) is increasing for ξ > 2, we have, for  ≥ 2 and ξ such that  ≤ ξ <  + 1, (ξ ) ≥ () = ( − 1)!. Furthermore, we notice that the number of integers n such that  ≤ μ + nν <  + 1 does not exceed ν1 . First, let 1 ≤ t < ∞. Then, one sees that Eμ,ν (t) ≤

∞  t +1−μ t 1−μ t 2−μ + + . ν 0 ν ν( − 1)! =2

As a consequence, Eμ,ν (t) ≤

1 2−μ t t 1−μ 2 2−μ t + t t e ≤ e. ν 0 ν 0 ν

Secondly, let 0 ≤ t < 1. Since 0 ≤ t nν ≤ 1, we can argue in a similar way to conclude that ∞  1 1 1 1 1 2e Eμ,ν (t) ≤ + + ≤ + e≤ . ν 0 ν ν( − 1)! ν 0 ν 0 ν =2

Combining these estimates, we obtain the desired estimate (1.36).



We now present and prove a generalized inequality of Gronwall type. Theorem 1.26 Let 0 ≤ a ∈ C([0, T ]; R) be an increasing function, let b > 0 be a constant, and let ν > 0 be an exponent. If ϕ ∈ C([0, T ]; R) satisfies the integral inequality  t ϕ(t) ≤ a(t) + b (t − s)ν−1 ϕ(s) ds, 0 ≤ t ≤ T , 0

then

 1  ϕ(t) ≤ a(t)E1,ν [b(ν)] ν t ,

0 ≤ t ≤ T.

Proof We will show the estimate ϕ(t) ≤

n 

a(t)bk

k=0

+b

n+1

(ν)k kν t (1 + kν)

(ν)n+1 ((n + 1)ν)

 0

t

(t − s)(n+1)ν−1 ϕ(s) ds,

0 ≤ t ≤ T . (1.37)

9 Integral Equations and Inequalities of Volterra Type

29

Indeed, this is obvious for n = 0. Let us assume that this is true for n. Then, ϕ(t) ≤

n 

a(t)bk

k=0

(ν)k kν t (1 + kν)

 t (ν)n+1 (t − s)(n+1)ν−1 ((n + 1)ν) 0    s ν−1 × a(s) + b (s − σ ) ϕ(σ ) dσ ds. + bn+1

0

Now, since a(s) ≤ a(t) for 0 ≤ s ≤ t, it follows that  t a(t)t (n+1)ν ((n + 1)ν) a(t)t (n+1)ν = . (t − s)(n+1)ν−1 a(s) ds ≤ (n + 1)ν (1 + (n + 1)ν) 0 In addition, using the well-known formula  t  1 μ−1 ν−1 μ+ν−1 (t − σ ) (σ − s) dσ = (t − s) (1 − θ )μ−1 θ ν−1 dθ 0

s

(μ)(ν) , = (t − s)μ+ν−1 B(μ, ν) = (t − s)μ+ν−1 (μ + ν)

(1.38)

where B(μ, ν) (μ > 0, ν > 0) is the beta function, we observe that  t s (t − s)(n+1)ν−1 (s − σ )ν−1 ϕ(σ ) dσ ds 0

0

=

 t 0

=

t

(t − s)(n+1)ν−1 (s − σ )ν−1 ϕ(σ ) ds dσ

σ

((n + 1)ν)(ν) ((n + 2)ν)



t

(t − σ )(n+2)ν−1 ϕ(σ ) dσ.

0

Hence, (1.37) is obtained for n + 1, too. Letting n → ∞ in (1.37), we now conclude that ∞

 [b(ν)t ν ]n  1  ϕ(t) ≤ a(t) = a(t)E1,ν [b(ν)] ν t , (1 + nν) n=0

0 ≤ t ≤ T.



9.2 Integral Inequality of Volterra Type Let 0 < T < ∞ be fixed. Let ϕ(t, s) be a continuous function defined for 0 ≤ s < t ≤ T . The function ϕ(t, s) is said to have a weak singularity at t = s if there exists an exponent ε > 0 such that 0 ≤ ϕ(t, s) ≤ C(t − s)ε−1 ,

0 ≤ s < t ≤ T.

30

1 Preliminaries

Theorem 1.27 Let a ≥ 0 and b > 0 be constants, and let μ > 0 and ν > 0 be fixed exponents. If a continuous function ϕ(t, s) ≥ 0 defined for 0 ≤ s < t ≤ T has a weak singularity at t = s and satisfies the integral inequality  t (1.39) ϕ(t, s) ≤ a(t − s)μ−1 + b (t − τ )ν−1 ϕ(τ, s) dτ, 0 ≤ s < t ≤ T , s

then   1 ϕ(t, s) ≤ a(μ)(t − s)μ−1 Eμ,ν [b(ν)] ν (t − s) ,

0 ≤ s < t ≤ T,

(1.40)

where (·) is the gamma function, and Eμ,ν (·) is the function given by (1.35). Proof By induction, we will show the estimate ϕ(t, s) ≤

n 

abk

k=0

+ bn+1

(μ)(ν)k (t − s)μ+kν−1 (μ + kν) (ν)n+1 ((n + 1)ν)



t

(t − τ )(n+1)ν−1 ϕ(τ, s) ds.

(1.41)

s

Indeed, it is obvious for n = 0. Assume that this is true for n. Then, ϕ(t, s) ≤

n 

(μ)(ν)k (ν)n+1 (t − s)μ+kν−1 + bn+1 (μ + kν) ((n + 1)ν) k=0    t  τ (n+1)ν−1 μ−1 ν−1 a(τ − s) × (t − τ ) +b (τ − σ ) ϕ(σ, s) dσ dτ. abk

s

s

Hence, by (1.38),  t (μ)((n + 1)ν) . (t − τ )(n+1)ν−1 (τ − s)μ−1 dτ = (t − s)μ+(n+1)ν−1 (μ + (n + 1)ν) s In addition, by (1.38),  t τ s

(t − τ )(n+1)ν−1 (τ − σ )ν−1 ϕ(σ, s) dσ dτ

s

=

 t s

=

t

(t − τ )(n+1)ν−1 (τ − σ )ν−1 ϕ(σ, s) dτ dσ

σ

((n + 1)ν)(ν) ((n + 2)ν)



t

(t − σ )(n+2)ν−1 ϕ(σ, s) dσ.

s

Hence, (1.41) is obtained for n + 1, too. Letting n → ∞ in (1.41), we observe that ϕ(t, s) ≤

∞  n=0

abn

(μ)(ν)n (t − s)μ+nν−1 , (μ + nν)

0 ≤ s < t ≤ T.

9 Integral Equations and Inequalities of Volterra Type

31

By the definition (1.35), the desired estimate (1.40) is concluded.



In a quite analogous way, we can show the following result. The proof will be left to the reader. Theorem 1.28 Let a ≥ 0 and b > 0 be constants, and let μ > 0 and ν > 0 be fixed exponents. If a continuous function ψ(t, s) ≥ 0 defined for 0 ≤ s < t ≤ T has a weak singularity at t = s and satisfies the integral inequality  t ψ(t, τ )(τ − s)ν−1 dτ, 0 ≤ s < t ≤ T , (1.42) ψ(t, s) ≤ a(t − s)μ−1 + b s

then   1 ψ(t, s) ≤ a(μ)(t − s)μ−1 Eμ,ν [b(ν)] ν (t − s) ,

0 ≤ s < t ≤ T,

where (·) is the gamma function, and Eμ,ν (·) is the function given by (1.35).

9.3 Integral Equations of Volterra Type In this subsection, we consider integral equations of Volterra type. Let X be a Banach space, and let L(X) be the Banach space of all bounded linear operators on X equipped with the uniform operator norm  · . Let 0 < T < ∞ be a fixed number and put  = {(t, s) ∈ R2 ; 0 ≤ s < t ≤ T }. We consider the integral equation of Volterra type  t Φ(t, s) = P (t, s) + Q(t, τ )Φ(τ, s) dτ, 0 ≤ s < t ≤ T ,

(1.43)

s

in L(X). Here, P and Q are in C(; L(X)) and are assumed to satisfy the estimates P (t, s) ≤ a(t − s)μ−1 ,

0 ≤ s < t ≤ T,

(1.44)

Q(t, s) ≤ b(t − s)ν−1 ,

0 ≤ s < t ≤ T,

(1.45)

with some fixed exponents μ > 0 and ν > 0 and some constants a > 0 and b > 0, respectively. These mean that P and Q have weak singularities at t = s. We can construct a unique solution to (1.43). Theorem 1.29 Under (1.44) and (1.45), there exists a unique function Φ ∈ C(; L(X)) such that Φ has a weak singularity at t = s and satisfies (1.43). Moreover, Φ is estimated by   1 Φ(t, s) ≤ a(μ)(t − s)μ−1 Eμ.ν [b(ν)] ν (t − s) , 0 ≤ s < t ≤ T , (1.46) where (·) is the gamma function, and Eμ,ν (·) is the function given by (1.35).

32

1 Preliminaries

Proof Before proving the assertion of theorem, we present a lemma. ˜ ∈ C(; L(X)) satisfy Lemma 1.3 Let P˜ , Q ˜ P˜ (t, s) ≤ a(t ˜ − s)μ−1 ,

0 ≤ s < t ≤ T,

(1.47)

˜ s) ≤ b(t ˜ − s)ν˜ −1 , Q(t,

0 ≤ s < t ≤ T,

(1.48)

with some exponents μ˜ > 0 and ν˜ > 0 and some constants a˜ > 0 and b˜ > 0, respectively. Put  t ˜ τ )P˜ (τ, s) dτ, 0 ≤ s < t ≤ T . ˜ s) = Q(t, R(t, s

Then, R˜ belongs to C(; L(X)) and satisfies the estimate ˜ ˜ ν) ˜ ν −1 ˜ s) ≤ a˜ b(μ)(˜ , R(t, (t − s)μ+˜ (μ˜ + ν˜ )

0 ≤ s < t ≤ T.

(1.49)

Proof We observe in view of (1.38) that R˜ is well defined and satisfies (1.49). ˜ But this is not obvious because P˜ So, it suffices to verify the continuity of R. ˜ and Q have singularities. We here introduce a sequence of cutoff functions P˜k defined for all integers k > T1 such that P˜k ’s are defined for all 0 ≤ s ≤ t ≤ T and are continuous functions with values in L(X), that P˜k (t, s) = P˜ (t, s) for 0 ≤ s ≤ T − k1 , s + k1 ≤ t ≤ T , and that P˜k ’s satisfy the same estimate as (1.47). We also introduce a sequence Q˜ k satisfying (1.48), which is defined in a quite analogous way to P˜k . ˜ k ∈ C(; L(X)), it is easy to see that the function As P˜k , Q  t 1 R˜ k (t, s) = Q˜ k (t, τ )P˜k (τ, s) dτ, 0 ≤ s ≤ t ≤ T , k > , T s is also in C(; L(X)). Moreover, for 0 ≤ s ≤ T − k1 , s +   R(t, ˜ s) − R˜ k (t, s) ≤



≤t ≤T,

 Q(t, ˜ τ ) − Q˜ k (t, τ )P˜ (τ, s) dτ

t

s



  Q ˜ k (t, τ )P˜ (τ, s) − P˜k (τ, s) dτ

t

+ s

 ≤C

1 k

t t− k1

 +

s+ k1 

˜ (t − τ )μ−1 (τ − s)ν˜ −1 dτ

s

    ˜  1 ν˜ −1 1 μ−1 −1 −μ˜ −1 −˜ν ≤ C μ˜ k + ν˜ k t −s− t −s− . k k ˜ s) in L(X) as k → ∞ This estimate indeed shows that R˜ k (t, s) converges to R(t, and, for arbitrarily fixed 0 < δ < T , the convergence is uniform for t and s such that

9 Integral Equations and Inequalities of Volterra Type

33

˜ s) for 0 ≤ s ≤ T − δ, s + δ ≤ t ≤ T . As a result, we observe the continuity of R(t, 0≤s
n = 1, 2, 3, . . . .

It is then obvious that 

t

ΔΦ1 (t, s) =

Q(t, τ )P (τ, s) dτ, s

 ΔΦn (t, s) =

t

Q(t, τ )ΔΦn−1 (τ, s) dτ,

n = 2, 3, 4, . . . .

s

By (1.49),   ΔΦ1 (t, s) ≤ ab(μ)(ν) (t − s)μ+ν−1 . (μ + ν) In addition, it is clear that   ΔΦn (t, s) ≤ b



t

  (t − τ )ν−1 ΔΦn−1 (τ, s) dτ.

s

Then, by the same techniques as before, we can easily observe that n n   ΔΦn (t, s) ≤ ab (μ)(ν) (t − s)μ+nν−1 , (μ + nν)

Thanks to Lemma 1.2, we conclude that the series convergent in L(X), with the estimate

n = 1, 2, 3, . . . . ∞

n=1 ΔΦn (t, s)

(1.50)

is absolutely

∞        Φn (t, s) ≤ a(μ)(t − s)μ−1 Eμ,ν [b(ν)] ν1 (t − s) . Φ0 (t, s) + n=1

 Since Φn (t, s) = Φ0 (t, s) + nk=1  ΔΦk (t, s), we conclude also that Φn (t, s) is convergent to the limit Φ0 (t, s) + ∞ k=1 Φk (t, s) in L(X) for each 0 ≤ s < t ≤ T . It is also easy to see that, for arbitrarily fixed 0 < δ < T , the convergence is uniform in t and s such that 0 ≤ s ≤ T − δ, s + δ ≤ t ≤ T . This then yields the continuity of Φ(t, s) for 0 ≤ s < t ≤ T . It is not difficult to show that Φ(t, s) is a solution of (1.43).

34

1 Preliminaries

Let us finally verify the uniqueness of solution. Let Φ˜ ∈ C(; L(X)) be any other solution having a weak singularity at t = s. Then, we easily verify that ˜ s) satisfies the integral inequality (1.39) the function ϕ(t, s) = Φ(t, s) − Φ(t, in which a equals to zero. Consequently, ϕ(t, s) must be identically zero, i.e., ˜ s). Φ(t, s) ≡ Φ(t,  A quite analogous result can be also proved for the integral equation  t Ψ (t, s) = P (t, s) + Ψ (t, τ )Q(τ, s) dτ, 0 ≤ s < t ≤ T .

(1.51)

s

The proof of the following theorem is left to the reader. Theorem 1.30 Under (1.44) and (1.45), there exists a unique function Ψ ∈ C(; L(X)) such that Ψ has a weak singularity at t = s and satisfies (1.51). Moreover, Ψ is estimated by   1 Ψ (t, s) ≤ a(μ)(t − s)μ−1 Eμ.ν [b(ν)] ν (t − s) ,

0 ≤ s < t ≤ T,

(1.52)

where (·) is the gamma function, and Eμ,ν (·) is the function given by (1.35).

9.4 Dominate Convergence Theorem for Integral Equations We continue the review for integral equations of Volterra type and use the same notation as in the preceding subsection. For k = 1, 2, 3, . . . , consider the integral equation  t k k Φ (t, s) = P (t, s) + Qk (t, τ )Φ k (τ, s) dτ, 0 ≤ s < t ≤ T , (1.53) s

in L(X). Here, P k and Qk are all in C(; L(X)) and satisfy (1.44) and (1.45) uniformly for k. We assume that, as k → ∞, P k (t, s) and Qk (t, s) converge to P (t, s) and Q(t, s), respectively, in the strong topology of L(X) for each (t, s) ∈ . In addition, we assume that P , Q ∈ C(; L(X)). We then have the following convergence theorem. Theorem 1.31 Let P k , Qk ∈ C(; L(X)) satisfy (1.44) and (1.45) uniformly for k, and let, as k → ∞, P k and Qk converge to P , Q ∈ C(; L(X)), respectively, in the strong topology of L(X) for each (t, s) ∈ . Then, the solution Φ k of (1.53) converges to the solution Φ of (1.43) in the same strong topology for each (t, s) ∈ . Proof We recall that, for each k, the solution Φ k (t, s) is constructed by Φ k (t, s) = P k (t, s) +

∞  n=1

Φnk (t, s),

10 Differential Inequalities

35

where  Φ1k (t, s) =  ΔΦnk (t, s) =

t

Qk (t, τ )P k (τ, s) dτ,

s t

s

n = 2, 3, 4, . . . .

k Qk (t, τ )Φn−1 (τ, s) dτ,

For each fixed n, let k tend to infinity. Then, by similar arguments as in the proof of Theorem 1.6, we observe from assumptions that ΔΦnk (t, s) satisfies (1.50) uniformly for k and, as k → ∞, ΔΦnk (t, s) converges to ΔΦn (t, s) in the strong topology of L(X). We can now apply the dominate convergence theorem for series to conclude that Φ k (t, s) converges to Φ(t, s) strongly on X.  Such a convergence theorem of course can be proved for the integral equations 

t

Ψ (t, s) = P (t, s) + k

k

0 ≤ s < t ≤ T.

Ψ k (t, τ )Qk (τ, s) dτ,

(1.54)

s

Theorem 1.32 Let P k , Qk ∈ C(; L(X)) satisfy (1.44) and (1.45) uniformly for k, and let, as k → ∞, P k and Qk converge to P , Q ∈ C(; L(X)), respectively, in the strong topology of L(X) for each (t, s) ∈ . Then, the solution Ψ k of (1.54) converges to the solution Ψ of (1.51) in the same strong topology for each (t, s) ∈ .

10 Differential Inequalities Let a ∈ C([0, T ]; R) and f ∈ C([0, T ]; R), and let a function u ∈ C([0, T ]; R) ∩ C1 ((0, T ]; R) satisfy the differential inequality du + a(t)u ≤ f (t), dt

0 < t ≤ T.

(1.55)

Then, for any fixed 0 < t ≤ T , we have t t  t d  u(s)e− s a(τ )dτ = u (s) + a(s)u(s) e− s a(τ )dτ ≤ f (s)e− s a(τ )dτ . ds

Integrating this inequality for s on the interval [0, t], we obtain that u(t) − u(0)e

−

t 0

 a(τ )dτ

≤

t

f (s)e−

t s

a(τ )dτ

ds.

0

In this way, (1.55) implies that u(t) ≤ e

−

t 0

 a(τ )dτ

t

u(0) + 0

e−

t s

a(τ )dτ

f (s) ds,

0 < t ≤ T.

(1.56)

36

1 Preliminaries

In particular, if a(t) ≡ δ > 0, then u(t) ≤ e−δt u(0) +



t

e−δ(t−s) f (s) ds,

0 < t ≤ T.

(1.57)

0

In addition, if f (t) ≡ f > 0, then it follows that u(t) ≤ e−δt u(0) + f δ −1 ,

0 < t ≤ T.

(1.58)

We next consider the differential inequality du + δu ≤ f (t), dt

0 < t ≤ T,

(1.59)

with δ > 0 and f ∈ C([0, T ]; R). We assume that f (t) satisfies the condition  t f (τ ) dτ ≤ α(t − s) + β, 0 ≤ s < t ≤ T , (1.60) s

with some constants α ≥ 0 and β ≥ 0. Then, as seen above, if u ∈ C([0, T ]; R) ∩ C1 ((0, T ]; R) satisfies (1.59), then u is estimated by (1.57). By integration by parts,    t  t  t d −δ(t−s) −δ(t−s) − e f (s) ds = e f (τ ) dτ ds ds s 0 0  t   t  t = e−δt f (τ ) dτ + δ e−δ(t−s) f (τ ) dτ ds. 0

0

s

Therefore, (1.60) yields that  t  t e−δ(t−s) f (s) ds ≤ e−δt (αt + β) + δ e−δ(t−s) [α(t − s) + β] ds 0

0

= αδ

−1

(1 − e

−δt

) + β.

(1.61)

Hence, we have proved the following proposition. Proposition 1.4 Let δ > 0, and let f ∈ C([0, T ]; R) satisfy (1.60) with α ≥ 0 and β ≥ 0. If u ∈ C([0, T ]; R) ∩ C1 ((0, T ]; R) satisfies the differential inequality (1.59), then u is estimated by u(t) ≤ e−δt u(0) + αδ −1 + β,

0 < t ≤ T.

(1.62)

On the basis of these results, we can prove some other propositions which will be often utilized in showing a dissipative condition for nonlinear diffusion equations. Proposition 1.5 Let δ > 0, γ > 0, and let f ∈ C([0, T ]; R) satisfy (1.60) with α ≥ 0 and β ≥ 0. Assume that 0 ≤ u ∈ C([0, T ]; R) ∩ C1 ((0, T ]; R) and 0 ≤ v ∈

10 Differential Inequalities

37

C([0, T ]; R) satisfy the differential inequality du + δu + γ v ≤ f (t), dt

0 < t ≤ T.

Then, it holds for v that  t v(τ ) dτ ≤ αγ −1 (t − s) + [u(0) + αδ −1 + 2β]γ −1 ,

(1.63)

0 ≤ s < t ≤ T . (1.64)

s

Proof Since γ v ≥ 0, u satisfies (1.59), too. So, u is estimated by (1.62). Fix 0 ≤ s < t ≤ T . Integrate both hand sides of (1.63) for τ in the interval [s, t]. Then,  t  t  t u(t) − u(s) + δ u(τ ) dτ + γ v(τ ) dτ ≤ f (τ ) dτ. s

Since u ≥ 0,

 γ

t

s

s

 v(τ ) dτ ≤ u(s) +

s

t

f (τ ) dτ. s

By (1.60) and (1.65), we see that  t γ v(τ ) dτ ≤ e−δs u(0) + αδ −1 + β + α(t − s) + β. s



Hence, (1.64) is obtained.

Proposition 1.6 Let δ > 0, let f ∈ C([0, T ]; R) satisfy (1.60) with α ≥ 0 and β ≥ 0, and let 0 ≤ p ∈ C([0, T ]; R) satisfy (1.60) with α ≥ 0 and β ≥ 0, where δ > α ≥ 0. If 0 ≤ u ∈ C([0, T ]; R) ∩ C1 ((0, T ]; R) satisfies the differential inequality du + δu ≤ p(t)u + f (t), dt

0 < t ≤ T,

(1.65)

then u is estimated by  u(t) ≤ eβ e−(δ−α)t u(0) + eβ α (δ − α)−1 + β ,

0 < t ≤ T.

(1.66)

Proof Obviously, u satisfies (1.55) with a(t) = δ − p(t). So, the assumption on p yields that  t  t t − 0 [δ−p(τ )]dτ u(t) ≤ e u(0) + e− s [δ−p(τ )]dτ f (s) ds ≤ eβ e−(δ−α)t u(0) + eβ



0 t

e−(δ−α)(t−s) f (s) ds.

0

Therefore, (1.66) is verified by (1.61).



38

1 Preliminaries

Proposition 1.7 Let δ > 0, γ > 0, and let f ∈ C([0, T ]; R) (resp. g ∈ C([0, T ]; R)) satisfy (1.60) with α ≥ 0 and β ≥ 0 (resp. α ≥ 0 and β ≥ 0). Let 0 ≤ p ∈ C([0, T ]; R) satisfy (1.60) with α ≥ 0 and β ≥ 0. Assume that 0 ≤ u ∈ C([0, T ]; R) ∩ C1 ((0, T ]; R) and 0 ≤ v ∈ C([0, T ]; R) ∩ C1 ((0, T ]; R) satisfy the two differential inequalities  du 0 < t ≤ T, dt + δu + γ v ≤ f (t), (1.67) dv + δv ≤ p(t)v + g(t), 0 < t ≤ T. dt Then, v is estimated by v(t) ≤ eβ e−δt v(0) + eβ αγ −1 [u(0) + α γ −1 + α δ −1 + 2β ] + eβ (α δ −1 + β ),

0 ≤ t ≤ T.

(1.68)

Proof Let ξ > 0 be a parameter. From the second inequality of (1.67) we have dv + [δ + ξ − p(t)]v ≤ ξ v(t) + g(t). dt Therefore, using (1.56) with a(t) = δ + ξ − p(t), we obtain that  t   t − 0t [δ+ξ −p(τ )]dτ v(t) ≤ e v(0) + e− s [δ+ξ −p(τ )]dτ [ξ v(s) + g(s)] ds. 0

The assumption on p then yields that v(t) ≤ eβ e−(δ+ξ −α)t v(0) + eβ



t

e−(δ+ξ −α)(t−s) [ξ v(s) + g(s)] ds.

0

We here fix the ξ such as ξ = α. Then, since v satisfies (1.64), we observe by (1.61) that  t e−δ(t−s) αv(s) ds ≤ αγ −1 [u(0) + α γ −1 + α δ −1 + 2β ]. 0

Similarly, by (1.61), 

t

e−δ(t−s) g(s) ds ≤ α δ −1 + β .

0

Hence, the desired estimate has been shown.



11 Sobolev–Lebesgue Spaces An open and connected subset Ω of Rn is called a domain in Rn . Throughout the book, unless otherwise notified, the functions in Ω are always complex valued.

11 Sobolev–Lebesgue Spaces

39

Let Ω be a domain in Rn . For 1 ≤ p ≤ ∞, the set of all Lp functions in Ω is denoted by Lp (Ω). With the usual Lp norm  · Lp , Lp (Ω) is a Banach space. When Ω is a bounded domain, C(Ω) (resp. Ck (Ω), k = 1, 2, . . .) is the set of all continuous (resp. k-times continuously differentiable)functions on Ω. With the norm uC = maxx∈Ω |u(x)| (resp. uCk = uC + |α|≤k D α uC ), C(Ω) (resp. Ck (Ω)) becomes a Banach space. For 0 < σ < 1, Cσ (Ω) is the set of all Hölder continuous functions on Ω with exponent σ . Cσ (Ω) is a Banach space k+σ (Ω), k = with the norm uCσ = uC + supx,y∈Ω, x =y |u(x)−u(y)| |x−y|σ . Similarly, C 1, 2, . . . , are the sets of all Ck functions on Ω with kth-order derivatives in Cσ (Ω). Ck+σ (Ω)’s are also Banach spaces with the norms uCk+σ = uCk +  α |α|=k D uCσ . Meanwhile, the space of Lipschitz continuous functions on Ω is denoted by C0,1 (Ω). C0,1 (Ω) is a Banach space with the norm uC0,1 = uC + k,1 supx,y∈Ω, x =y |u(x)−u(y)| |x−y| . In an analogous way, the spaces C (Ω), k = 1, 2, . . . , and their norms are defined. By D(Ω) we denote the space of all infinitely differentiable functions in Ω with compact supports.

11.1 Boundaries of Domains Following [Gri85], we review basic things concerning the boundary of domain in Rn . In this subsection, we are concerned with a bounded domain Ω in Rn . We say that Ω has a continuous (resp. Lipschitz, Cm class (m = 1, 2, 3, . . .)) boundary ∂Ω if for every x ∈ ∂Ω, there exists a neighborhood V of x in Rn and a new orthogonal coordinate system (y1 , . . . , yn ) such that (1) V is a cube in the new coordinates: V = {(y1 , . . . , yn ); −ai < yi < ai , i = 1, . . . , n}. (2) There exists a continuous (resp. Lipschitz, Cm class) function ϕ which is defined in V = {(y1 , . . . , yn−1 ); −ai < yi < ai , i = 1, . . . , n − 1} and satisfies |ϕ(y )| ≤ an /2 for every y = (y1 , . . . , yn−1 ) ∈ V , Ω ∩ V = {y = (y , yn ) ∈ V ; yn > ϕ(y )}, ∂Ω ∩ V = {y = (y , yn ) ∈ V ; yn = ϕ(y )}. (3) ϕC(V ) ≤ c (resp. ϕLip(V ) ≤ c, ϕCm (V ) ≤ c) with some constant c > 0. If Ω is an n-dimensional manifold of Cm class (m = 1, 2, . . .) with boundary, then its boundary ∂Ω is automatically of Cm class. However, the same is not necessarily true for a Lipschitz manifold. Even if Φ(y1 , . . . , yn ) is a Lipschitz continuous function for which the equation Φ(y1 , . . . , yn ) = 0 defines a hypersurface

40

1 Preliminaries

yn = ϕ(y1 , . . . , yn−1 ), the function ϕ may be non-Lipschitz. A counterexample is given in [Gri85, Lemma 1.2.1.4]. Let Ω be a bounded domain with Lipschitz boundary. Then, on the boundary ∂Ω, a measure dS can be defined in a similar manner as in the case of smooth boundary, and an integral with respect to dS is defined for the functions on ∂Ω. If a function f has a support in ∂Ω ∩ V , where V is one of the neighborhoods mentioned above, then its integral is given in the form    f dS = f 1 + |∇ ϕ(y )|2 dy , V

∂Ω

where ∇ ϕ = (D1 ϕ, . . . , Dn−1 ϕ). Furthermore, for almost all boundary points x ∈ ∂Ω, the outer normal vectors ν(x) are defined. Indeed, if (y , yn ) ∈ ∂Ω ∩ V , then ν is given by  ν(y , yn ) = (D1 ϕ(y ), . . . , Dn−1 ϕ(y ), −1) 1 + |∇ ϕ(y )|2 . Then, the formula of integration by parts    Di u · v dx + u · Di v dx = [uv]νi (x) dS, Ω

Ω

i = 1, 2, . . . , n,

(1.69)

∂Ω

is valid for u, v ∈ C1 (Ω), where ν(x) = (ν1 (x), . . . , νn (x)), x ∈ ∂Ω.

11.2 Sobolev Spaces of Integral Order Let Ω be a domain in Rn . For 1 ≤ p ≤ ∞ and k = 0, 1, 2, . . . , the space Hpk (Ω) is defined as a set of all complex-valued Lp functions u whose derivatives D α u in the distribution sense up to order k belong to Lp (Ω), where α = (α1 , α2 , . . . , αn ) denotes a multiindex, and |α| = α1 + α2 + · · · + αn . The space Hpk (Ω) becomes a Banach space with the norm uHpk =

 |α|≤k

1 D

α

p uLp

p

,

u ∈ Hpk (Ω).

When p = 2, H2k (Ω) is obviously a Hilbert space with the inner product  (D α u, D α v)L2 , u, v ∈ H2k (Ω). (u, v)H k = 2

|α|≤k

Let Ω be the half-space Rn+ = {x = (x , xn ); x ∈ Rn−1 , xn > 0} or a bounded domain in Rn with Lipschitz boundary. According to [Ste70, Chap. VI, Theorems 5 and 5 ], we can construct an extension operator of functions in Ω to functions in Rn universally in 1 ≤ p ≤ ∞ and k = 0, 1, 2, . . . .

11 Sobolev–Lebesgue Spaces

41

Theorem 1.33 Let Ω be Rn+ or a bounded domain with Lipschitz boundary. There exists a linear operator E mapping functions in Ω to functions in Rn with the following properties: (1) (Eu)|Ω = u. (2) E is a continuous operator from Hpk (Ω) to Hpk (Rn ) for any 1 ≤ p ≤ ∞ and any k = 0, 1, 2, . . . , namely, EuHpk (Rn ) ≤ Ap,k uHpk (Ω) ,

u ∈ Hpk (Ω),

with some constant Ap,k > 0 depending on p and k.

11.3 Sobolev–Lebesgue Spaces in Rn When 1 < p < ∞, the Sobolev spaces Hpk (Ω) can be extended for nonintegral orders. In this subsection, we consider the case where Ω = Rn . Let s ≥ 0 be a nonnegative number. The space Hps (Rn ) is defined by  s   Hps (Rn ) = u ∈ S(Rn ) ; F−1 1 + |ξ |2 2 Fu ∈ Lp (Rn ) , where S(Rn ) denotes the set of tempered distributions in Rn , and F and F−1 denote the Fourier transform and the inverse Fourier transform on S(Rn ) , respectively. The space Hps (Rn ) is a Banach space with the norm    s  uHps = F−1 1 + |ξ |2 2 Fu L , p

u ∈ Hps (Rn ).

By the theory of Fourier multipliers, it is possible to verify that the two definitions of Hpk (Rn ) and Hps (Rn ) are equivalent for integral orders s = 0, 1, 2, . . . , that is, Hpk (Rn ) = Hps (Rn ) with norm equivalence for nonnegative integers s = k. When p = 2, the space H2s (Rn ) is seen to be a Hilbert space with the inner product (u, v)H2s =



1 + |ξ |2

s

2

s   Fu, 1 + |ξ |2 2 Fv L , 2

u, v ∈ H2s (Rn ).

Furthermore, for s = k + σ , where k = [s] is an integer and 0 < σ < 1, the norm of H2s (Rn ) is equivalent to u H s 2

= uL2 +

  |α|≤k

see [Tri78, 2.5.1, (15)].

Rn ×Rn

|D α u(x) − D α u(y)|2 dx dy |x − y|n+2σ

1 2

,

u ∈ H2s (Rn ),

42

1 Preliminaries

11.4 Sobolev–Lebesgue Spaces in Rn+ or Bounded Domains We intend to extend the definition of Hpk (Ω) for nonintegral orders. Let Ω denote the half-space Rn+ or a bounded domain in Rn with Lipschitz boundary. For 1 < p < ∞ and s ≥ 0, Hps (Ω) is defined as the set of all restrictions u of the functions in Hps (Rn ) to Ω. That is, a function u ∈ Lp (Ω) is in Hps (Ω) if and only if there exists a function U ∈ Hps (Rn ) such that U|Ω = u almost everywhere in Ω. For u ∈ Hps (Ω), its Hps norm is defined by uHps (Ω) =

inf

U ∈Hps (Rn ),U|Ω =u

U Hps (Rn ) .

With this norm, Hps (Ω) is a Banach space. Indeed, since the space K = {U ∈ Hps (Rn ); U = 0 in Ω} is a closed subspace of Hps (Rn ), Hps (Ω) is nothing more  than the quotient space Hps (Rn ) K. By virtue of Theorem 1.33, we can easily verify that such a definition is consistent with the former one when s is an integer. Let p = 2, and let s = k + σ , where k = [s] is an integer and 0 < σ < 1. Then, the norm of H2s (Ω) is as before equivalent to u H s 2

= uL2 +

  |α|≤k

Ω×Ω

|D α u(x) − D α u(y)|2 dx dy |x − y|n+2σ

1

u ∈ H2s (Ω),

2

, (1.70)

see [Tri78, Remark 4.4.2/2]. One sees that, for 0 < s0 < s1 < ∞, Hps1 (Ω) ⊂ Hps0 (Ω) ⊂ Lp (Ω) with continuous embeddings. Theorem 1.34 Let Ω be Rn , Rn+ , or a bounded domain with Lipschitz boundary. Let 1 < p < ∞ and s ≥ 0. Then, u ∈ Hps (Ω) if and only if Re u ∈ Hps (Ω) and Im u ∈ Hps (Ω). It holds that max{ Re uHps ,  Im uHps } ≤ uHps ≤ Re uHps +  Im uHps ,

u ∈ Hps (Ω).

Proof We will prove that u ∈ Hps (Ω) implies u ∈ Hps (Ω) with uHps = uHps , where u(x) = u(x) is the complex conjugate of u(x). Clearly, all the assertions of the theorem are easily verified by this property. Consider first the case where Ω = Rn . By definition, u ∈ Hps (Rn ) is equivalent to   s   ei(y−x)·ξ 1 + |ξ |2 2 u(x) dx dξ ∈ Lp Rny . Rn ×Rn

11 Sobolev–Lebesgue Spaces

43

Taking the complex conjugate of this function and changing integral variables by ξ = −ξ , we have  s    n (−1) ei(y−x)·ξ 1 + |ξ |2 2 u(x) dx dξ ∈ Lp Rny . Rn ×Rn

This then means that u ∈ Hps (Rn ) with uHps = uHps . Consider next the general case of Ω. By definition, u ∈ Hps (Ω) implies that there exists a sequence of functions Uk ∈ Hps (Rn ) such that u = Uk|Ω and Uk Hps (Rn ) → uHps (Ω) as k → ∞. Since u = U k|Ω , we conclude that u ∈ Hps (Ω) with uHps ≤  infk U k Hps = infk Uk Hps = uHps . Similarly, uHps = uHps ≤ uHps . These spaces Hps (Ω) are called by different names. When the order is integral, Hpk (Ω) is called the Sobolev space. When p = 2, H2s (Ω) is also called the Sobolev space. In the meantime, when 1 < p < ∞, p = 2, Hps (Ω) is called the Lebesgue space. When p = 2, H2s (Ω) is often abbreviated by H s (Ω). The Sobolev–Lebesgue spaces consisting of real-valued functions are denoted by Hps (Ω; R), 1 < p < ∞, s ≥ 0.

11.5 Interpolation Property In this subsection, we present the interpolation property of the Sobolev–Lebesgue spaces. This property is extremely useful for deriving various properties for the Sobolev–Lebesgue spaces. According to [Tri78, Theorems 2.4.2/1, 2.10.1 and 4.3.1/2], we have the following theorem. Theorem 1.35 Let Ω be Rn , Rn+ , or a bounded domain with Lipschitz boundary. Let 1 < p < ∞ and 0 ≤ s0 < s1 < ∞. Then  s (1.71) Hp0 (Ω), Hps1 (Ω) θ = Hps (Ω) with norm equivalence, where 0 ≤ θ ≤ 1 and s = (1 − θ )s0 + θ s1 . Let us use here this theorem to verify the boundedness of the extension operator E introduced above from Hps (Ω) to Hps (Rn ). Let s > 0 be nonintegral, and let Ω = Rn . Let k = [s] + 1 > s be an integer. Since E is a bounded operator from Lp (Ω) to Lp (Rn ) and from Hpk (Ω) to Hpk (Rn ) simultaneously, the interpolation theorem (Theorem 1.15) is applied to E to conclude that E is a bounded operator from Hps (Ω) to Hps (Rn ). In other words, if u ∈ Hps (Ω), then Eu ∈ Hps (Rn ) with the estimate EuHps (Rn ) ≤ Ap,s uHps (Ω) . We thus observe the following estimates: A−1 p,s EuHps (Rn ) ≤ uHps (Ω) ≤ EuHps (Rn ) ,

u ∈ Hps (Ω).

(1.72)

44

1 Preliminaries

As an immediate consequence, we get that a function u ∈ Lp (Ω) belongs to Hps (Ω) if and only if its extension Eu is in Hps (Rn ). Estimate (1.72) shows also that the norm of Hps (Ω) is equivalent to the graph norm E · Hps (Rn ) of E. When p = 2, since H2s (Rn ) is a Hilbert space, this fact yields that H s (Ω) = H2s (Ω) also enjoys a Hilbert structure with the inner product (u, v)H s (Ω) = (Eu, Ev)H s (Rn ) ,

u, v ∈ H s (Ω).

(1.73)

11.6 Embedding Theorems According to [Tri78, Theorem 2.8.1/Remark 2 and Theorem 4.6.1], we get the following embedding theorem. Theorem 1.36 Let Ω be Rn , Rn+ , or a bounded domain with Lipschitz boundary. Let 1 < p < ∞ and 0 ≤ s < ∞. (1) If 0 ≤ s < pn , then Hps (Ω) ⊂ Lr (Ω) where p ≤ r ≤ (2) If s = pn , then

with continuous embedding,

(1.74)

Hpp (Ω) ⊂ Lr (Ω) with continuous embedding,

(1.75)

pn n−ps .

n

where r is any finite number such that p ≤ r < ∞. (3) If s > pn , then  Hps (Ω) ⊂

C(Rn ) (resp. C(Rn+ )) when Ω = Rn (resp. Rn+ ), when Ω is bounded. C(Ω)

(1.76)

When Ω is bounded, the embedding is continuous. The following estimates are known as Gagliardo–Nirenberg’s inequality. When Ω = Rn , the proof is given in [Tan97, Theorems 3.3, 3.4, and 3.5]. Thanks to the extension theorem, Theorem 1.33, the proofs for other cases are immediately reduced to this case. Theorem 1.37 Let Ω be Rn , Rn+ , or a bounded domain with Lipschitz boundary, and let 1 ≤ q ≤ p ≤ ∞. (1) If 1 ≤ p < n, then Hp1 (Ω) ∩ Lq (Ω) ⊂ Lr (Ω)

11 Sobolev–Lebesgue Spaces

45

with the estimate uLr ≤ Cp,q,r uaH 1 u1−a Lq , p

where q ≤ r ≤

np n−p ,

u ∈ Hp1 (Ω) ∩ Lq (Ω),

and a is given by   1 1 1 1−a =a − + . r p n q

(1.77)

(2) If p = n, then Hn1 (Ω) ∩ Lq (Ω) ⊂ Lr (Ω) with the estimate uLr ≤ Cp,q,r uaH 1 u1−a Lq , n

u ∈ Hn1 (Ω) ∩ Lq (Ω),

where r is any finite number such that q ≤ r < ∞, and a is given by 1 1−a = . r q

(1.78)

(3) If n < p ≤ ∞, then Hp1 (Ω) ∩ Lq (Ω) ⊂ Lr (Ω) with the estimate uLr ≤ Cp,q,r uaH 1 u1−a Lq , p

u ∈ Hp1 (Ω) ∩ Lq (Ω),

where q ≤ r ≤ ∞, and a is given by   1 1−a 1 1 =a − + . r p n q

(1.79)

We finally present the compact embedding theorem. The proof is given in [Tri78, Theorem 4.10.1]. Theorem 1.38 Let Ω be a bounded domain in Rn with Lipschitz boundary. For 1 < p < ∞ and s > 0, the embedding i : Hps (Ω) → Lp (Ω) is compact.

11.7 Traces We are concerned with defining the values of f (x) on the boundary ∂Ω for the functions f (x) in Ω. Let us first consider the case where Ω = Rn+ . If 1 < p < ∞ and s > pn , we see from (1.76) that H s (Rn+ ) ⊂ C(Rn+ ). Therefore, the trace operator γ : f → f|∂Rn+ is

46

1 Preliminaries

defined from H s (Rn+ ) to C(∂Rn+ ), where ∂Rn+ = {x = (x , 0); x ∈ Rn−1 }. According to [Tri78, Theorem 2.9.3], if s > p1 , the operator γ can be extended as a bounded operator from Hps (Rn+ ) to Lp (∂Rn+ ). We want to extend this result to bounded domains with Lipschitz boundaries. In [Tri78], only the case where the boundary ∂Ω is sufficiently smooth is treated (see [Tri78, Theorem 4.7.1]). In [Gri85], the main interest is only in the Sobolev spaces Wps (Ω). So, it may be necessary to write up the full proof for Lebesgue spaces. However, we are able to carry out the proof in a quite analogous way to the case of Wps (Ω). So, we will omit the proof of the following theorem. It is the same for the proofs of Theorems 1.40, 1.41, and 1.42 below. Theorem 1.39 Let Ω be Rn+ or a bounded domain with Lipschitz boundary. Let 1 < p < ∞. If s > p1 , then the trace γ : f → f|∂Ω is a bounded operator from Hps (Ω) to Lp (∂Ω). On the contrary, the following two theorems show that the similar trace is no longer meaningful for s such that 0 ≤ s ≤ p1 . Theorem 1.40 Let Ω be as above, and let 1 < p < ∞. For 0 ≤ s ≤ p1 , the space D(Ω) is dense in Hps (Ω). Theorem 1.41 Let Ω be as above, and let 1 < p < ∞. For 0 ≤ s < p1 , the zero extension operator f → f ∼ , where f ∼ = f in Ω and f ∼ = 0 in Rn − Ω, is a bounded operator from Hps (Ω) to Hps (Rn ). For

1 p

< s ≤ 1, we can verify the following fact.

Theorem 1.42 Let Ω be as above, and let 1 < p < ∞. For p1 < s ≤ 1, a function u ∈ Hps (Ω) belongs to the closure of D(Ω) in Hps (Ω) if and only if u|∂Ω = 0.

11.8 Spaces H˚ ps (Ω) and Hp−s (Ω) For 1 < p < ∞ and s ≥ 0, the space H˚ ps (Ω) is defined as the closure of the set D(Ω) in the space Hps (Ω). Therefore, u ∈ H˚ ps (Ω) if and only if there exists a sequence un ∈ D(Ω) such that un → u in Hps (Ω). When Ω = Rn , H˚ ps (Rn ) = Hps (Rn ) for all 0 ≤ s < ∞. When Ω is Rn+ or a bounded domain with Lipschitz boundary, Theorem 1.40 implies that H˚ ps (Ω) = Hps (Ω)

for all 0 ≤ s ≤

1 , p

(1.80)

11 Sobolev–Lebesgue Spaces

47

but H˚ ps (Ω) = Hps (Ω)

for any

1 < s < ∞. p

In addition, Theorem 1.42 implies that   1 H˚ ps (Ω) = u ∈ Hps (Ω); u|∂Ω = 0 for < s ≤ 1. p

(1.81)

When p = 2, the space H˚ 2s (Ω) is abbreviated by H˚ s (Ω). For s ≥ 0, the space Hp−s (Ω) is defined as the dual space of H˚ ps (Ω), where 1 < p < ∞, 1 + 1 = 1. Therefore, {H˚ s (Ω), Hp−s (Ω)} is an adjoint pair with p

p

p

duality product ·, ·H˚ s

×Hp−s p

on H˚ ps (Ω) × Hp−s (Ω). Since D(Ω) ⊂ H˚ ps (Ω) with

dense embedding, we have Hp−s (Ω) = H˚ ps (Ω) ⊂ D(Ω) , namely, ϕ ∈ Hp−s (Ω) is characterized as a distribution in Ω which is continuous in D(Ω) with respect to the Hps (Ω) topology. As an immediate consequence, Lp (Ω) ⊂ Hp−s (Ω) with the relation u, f H˚ s

p

×Hp−s

= u, f Lp ×Lp ,

u ∈ H˚ ps (Ω), f ∈ Lp (Ω).

(1.82)

When p = p = 2, H2−s (Ω) is abbreviated as H −s (Ω). We easily verify that the three spaces H˚ s (Ω) ⊂ L2 (Ω) ⊂ H −s (Ω),

0 < s < ∞,

(1.83)

form a triplet. When Ω = Rn , Hp−s (Rn ) is given by  − s   Hp−s (Rn ) = f ∈ S(Rn ) ; F−1 1 + |ξ |2 2 Ff ∈ Lp (Rn ) . According to [Tri78, Theorem 2.6.1], for any −∞ < s < ∞,    s n  1 1 −s n Hp (R ) = Hp (R ) =1 . + p p This in particular shows that, for any s ≥ 0 and 1 < p < ∞, Hps (Rn ) is a reflexive Banach space. Estimate (1.72) then yields that Hps (Ω) is also a reflexive Banach space. In fact, (1.72) shows that Hps (Ω) is isomorphic to the subspace E(Hps (Ω)) of H s (Rn ). But it is clear that the image E(Hps (Ω)) is a closed subspace of H s (Rn ). It is then known that any closed subspace of a reflexive Banach space is reflexive. Therefore, Hps (Ω) must be reflexive. Furthermore, by the similar reason, H˚ ps (Ω) is also reflexive. We remember that H s (Ω), s ≥ 0, is a Hilbert space with the inner product given by (1.73). It is the same for H˚ s (Ω). We can finally show the following fundamental fact. Since the proof is quite analogous to the case of the Sobolev spaces Wps (Ω) (cf. [Gri85, Theorem 1.4.4.6]), we will omit the proof.

48

1 Preliminaries

Theorem 1.43 Let Ω be Rn+ or a bounded domain with Lipschitz boundary. Let 1 < p < ∞. For any −∞ < s < ∞, s = p1 , the partial derivation Di (i = 1, . . . , n) is a bounded operator from Hps (Ω) to Hps−1 (Ω).

11.9 Product Spaces Let Ω be Rn , Rn+ , or a bounded domain with Lipschitz boundary. For 1 ≤ p ≤ ∞, the product space Lp (Ω) is defined by ⎧ ⎫ ⎛ ⎞ f1 ⎪ ⎪ ⎨ ⎬ ⎜ .. ⎟ Lp (Ω) = F = ⎝ . ⎠ ; fj ∈ Lp (Ω) for j = 1, . . . ,  ⎪ ⎪ ⎩ ⎭ f

(1.84)

 p 1 and is equipped with the product norm F Lp = ( j =1 fj Lp ) p if 1 ≤ p < ∞ and F L∞ = max{f1 L∞ , . . . , f L∞ } if p = ∞. Similarly, for 1 < p < ∞ and s ≥ 0, the product space Hsp (Ω) is defined by ⎧ ⎫ ⎛ ⎞ u1 ⎪ ⎪ ⎨ ⎬ ⎜ ⎟ Hsp (Ω) = U = ⎝ ... ⎠ ; uj ∈ Hps (Ω) for j = 1, . . . ,  ⎪ ⎪ ⎩ ⎭ v

(1.85)

1  p and is equipped with the product norm U Hsp = ( j =1 uj H s ) p . p All the results concerning Lp (Ω) and Hps (Ω) can be naturally generalized to Lp (Ω) and Hsp (Ω). For example, when p = 2, L2 (Ω) and Hs (Ω) = Hs2 (Ω) are Hilbert spaces. We shall also use the notation t (f1 , . . . , f ) (resp. t (u1 , . . . , u )), in order to denote the vectors F ∈ Lp (Ω) (resp. U ∈ Hsp (Ω)). Here, t (· · · ) means the transposition of matrices.

11.10 Miscellaneous Results Throughout this subsection, Ω denotes a bounded domain with Lipschitz boundary. (1) For 1 ≤ p, q, r ≤ ∞ such that

1 p

+

1 q

uvLr ≤ uLp vLq ,

= 1r , it holds that u ∈ Lp (Ω), v ∈ Lq (Ω).

(1.86)

Indeed, this estimate is immediately observed by using the usual Hölder inequality for p1 + q1 = 1, where p = pr and q = qr .

11 Sobolev–Lebesgue Spaces

49

(2) Let k be a positive integer. If a ∈ Ck (Ω), then the multiplication u → au by a is a bounded operator on Hps (Ω) for any 1 < p < ∞ and any 0 ≤ s ≤ k with the estimate s

1− s

auHps ≤ Ck aCk k aC k uHps ,

u ∈ Hps (Ω).

(1.87)

It is clear that auLp ≤ aC uLp and that auHpk ≤ Ck aCk uHpk . Then, the estimate is immediately verified by applying the interpolation theorem and property, namely, Theorems 1.15 and 1.35. (3) Let n < p < ∞. Then, u, v ∈ Hp1 (Ω) implies uv ∈ Hp1 (Ω) with the estimate uvHp1 ≤ Cp uHp1 vHp1 ,

u, v ∈ Hp1 (Ω).

(1.88)

Indeed, due to (1.76), Hp1 (Ω) ⊂ L∞ (Ω). Then, the estimate is verified directly from ∇(uv) = v∇u + u∇v. (4) Let s ≥ 1 and s > n2 . Then, similarly, u, v ∈ H s (Ω) implies uv ∈ H s (Ω) with the estimate uvH s ≤ Cs uH s vH s ,

u, v ∈ H s (Ω).

(1.89)

In order to show this result, let us prove first that for any a ∈ H s (Ω), the multiplication u → au by a is a bounded operator from H s−1 (Ω) into itself and the estimate auH s−1 ≤ Cs aH s uH s−1 ,

u ∈ H s−1 (Ω)

(1.90)

holds. Indeed, let s = k + σ with an integer k ≥ 1 and 0 ≤ σ < 1. Then, u → au is a bounded operator from H k−1 (Ω) into itself, for we have D α (av) =







Cα ,α (D α a)(D α u),

|α |+|α |=k−1

where α denotes any multiindex such that |α| = k − 1. Here, due to (1.74), (1.75), and (1.76), we observe that (D α a)(D α u) ∈ L2 (Ω); hence, au ∈ H k−1 (Ω) with auH k−1 ≤ Ck−1 aH s uH k−1 . In a similar way, we can observe that u → au is a bounded operator from H k (Ω) into itself with auH k ≤ Ck aH s uH k . Then, the interpolation theorem and property (Theorems 1.15 and 1.35) provide that u → au is a bounded operator from H k−1+σ (Ω) into itself with auH k−1+σ ≤ Ck−1+σ aH s uH k−1−σ , namely, (1.90) is proved. Now, the proof of (1.89) is immediate. Let u, v ∈ H s (Ω); obviously, ∇(uv) = v∇u + u∇v and ∇u, ∇v ∈ H s−1 (Ω); (1.90) then ensures ∇(uv) ∈ H s−1 (Ω); consequently, for any multiindex α such that |α| = k, it holds that D α (uv) ∈ H σ (Ω). Therefore, on account of (1.70), we conclude that uv ∈ H k+σ (Ω). At the same time, estimate (1.90) is also verified.

50

1 Preliminaries

(5) Let F : R → R be a C1 function. Let n < p < ∞. Then, the correspondence u → F (u) is an operator from Hp1 (Ω; R) into itself. Moreover, there is an increasing function p(·) determined from F such that F (u)Hp1 ≤ p(uHp1 ),

u ∈ Hp1 (Ω; R).

(1.91)

We have ∇F (u) = F (u)∇u. So, (1.91) is verified directly from (1.76). Let, more strongly, F : R → R be a C2 function, and n < p < ∞. Then, the operator u → F (u) from Hp1 (Ω; R) into itself satisfies the Lipschitz condition F (u) − F (v)Hp1 ≤ sup p1 (θ u + (1 − θ )vHp1 )u − vHp1 , 0≤θ≤1

u, v ∈ Hp1 (Ω; R),

(1.92)

where p1 (·) is an increasing function determined from F . Indeed, since 1 F (u) − F (v) = 0 F (θ u + (1 − θ )v) dθ (u − v), this estimate is verified by (1.88) and (1.91). (6) Let s ≥ 1 and s > n2 . Let F : R → R be a Cm function, where m is an integer such that m ≥ s. Then, the correspondence u → F (u) is an operator from H s (Ω; R) into itself. Moreover, there is an increasing function p(·) determined from F such that F (u)H s ≤ p(uH s ),

u ∈ H s (Ω; R).

(1.93)

Let s = k + σ , where 0 ≤ σ < 1, and k ≥ 1 is an integer. For any u ∈ H s (Ω; R), consider the linear operator Tu w = F (u)∇w. Due to (1.74), (1.75), and (1.76), we can observe that Tu is a bounded operator from H k (Ω) into H k−1 (Ω) with Tu wH k−1 ≤ p(uH s )wH k and is a bounded operator from H k+1 (Ω) into H k (Ω) with Tu wH k ≤ p(uH s )uH k+1 . The interpolation theorem and property (Theorems 1.15 and 1.35) then provide that Tu is a bounded operator from H k+σ (Ω) into H k−1+σ (Ω) with Tu wH k−1+σ ≤ p(uH s )wH k−1+σ . We now apply Tu to u itself to verify that ∇F (u) = F (u)∇u = Tu u ∈ H k−1+σ (Ω). Then, for any multiindex α such that |α| = k, we have D α F (u) ∈ H σ (Ω). Hence, on account of (1.70), we conclude that F (u) ∈ H k+σ (Ω). Estimate (1.93) is also verified. Let s ≥ 1 and s > n2 . Let, more strongly, F : R → R be a Cm+1 function with m ≥ s. Then, the operator u → F (u) from H s (Ω; R) into itself satisfies the Lipschitz condition F (u) − F (v)H s ≤ sup p1 (θ u + (1 − θ )vH s )u − vH s , 0≤θ≤1

u, v ∈ H s (Ω; R),

(1.94)

where p1 (·) is an increasing function determined from F . This estimate is verified in an analogous way for (1.92) by using (1.89) and (1.93).

11 Sobolev–Lebesgue Spaces

51

(7) Let F : R → R be a piecewise smooth function with F ∈ L∞ (R) and with F (0) = 0 (hence, |F (u)| ≤ F L∞ |u| for u ∈ R). Then, according to [GT83, Theorem 7.8], the correspondence u → F (u) is an operator from Hp1 (Ω; R) into itself for 1 ≤ p ≤ ∞ with the estimate F (u)Hp1 ≤ CF L∞ uHp1 ,

u ∈ Hp1 (Ω; R).

(1.95)

i = 1, . . . , n,

(1.96)

In addition, F (u) has the property  / Fs , F (u)Di u if u(x) ∈ Di F (u) = 0 if u(x) ∈ Fs ,

where Fs denotes the set of singular points of F (u). (8) The correspondence f → ef is an operator from L∞ (Ω) into itself. The operator is Fréchet differentiable in L∞ (Ω). Its Fréchet differential at f ∈ L∞ (Ω) is the multiplication operator by the function ef , i.e., [Def ]h = ef h,

f, h ∈ L∞ (Ω).

(1.97)

This fact is easily verified by a direct calculation. (9) For 1 ≤ q < ∞, let us introduce the quantity  Nq,log (u) = |u|q log(|u| + 1) dx. Ω

All the results announced in Theorem 1.37 can be modified in the following form. For any ζ > 0, estimates in (1), (2), and (3) can be rewritten as q   uLr ≤ Cp,q,r ζ 1−a uaH 1 Nq,log (u)1−a + Cζ uLr q , p

u ∈ Hp1 (Ω; R) ∩ Lq (Ω),

(1.98)

with some constant Cζ > 0 dependent on ζ . The relations among p, q, and r np ; if p = n, then are the same as before, that is, if 1 ≤ p < n, then q ≤ r ≤ n−p q ≤ r < ∞; and if n < p ≤ ∞, then q ≤ r ≤ ∞. The exponent a is also given in each case as before, that is, by (1.77), (1.78), and (1.79), respectively. To prove the estimate, we use a cutoff function χR (u) such that χR (u) = −R for u ≤ −R, χR (u) = u for |u| ≤ R, and χR (u) = R for R ≤ u. Then, by Theorem 1.37 and (1.95), we observe in each case that u − χR (u)Lr ≤ Cp,q,r u − χR (u)aH 1 u − χR (u)1−a Lq p

1−a

 ≤

Cp,q,r uaH 1 p

|u| dx

≤

Cp,q,r uaH 1 Nq,log (u)1−a . p [log(R + 1)]1−a

q

|u|≥R

52

1 Preliminaries

Meanwhile, q

q

χR (u)Lr ≤ R 1− r uLr q . These two estimates readily yield the desired estimate (1.98). (10) Let H (u) be a C1,1 function defined by H (u) = 12 u2 for −∞ < u < 0 and H (u) ≡ 0 for 0 ≤ u < ∞. For any function u ∈ C1 ((a, b); L2 (Ω; R)), if we put  ψ(t) = H (u(t)) dx, a < t < b, (1.99) Ω

then ψ

∈ C1 (a, b)

with the derivative  H (u(t))u (t) dx, ψ (t) =

(1.100)

a < t < b.

Ω

In fact, we notice that, since H : R → R is Lipschitz continuous, the mapping u → H (u) is also Lipschitz continuous from L2 (Ω; R) into itself. From  1 H (θ u(t + h, x) + (1 − θ )u(t, x)) dθ H (u(t + h, x)) − H (u(t, x)) = 0

× [u(t + h, x) − u(t, x)],

a.e. x ∈ Ω,

we have  ψ(t + h) − ψ(t) =

1

 H (θ u(t + h) + (1 − θ )u(t)) dθ, u(t + h) − u(t) .

0

L2

Dividing this equality by h and letting h → 0, we obtain (1.100). (11) As a variation of (1.100), for any u ∈ C((a, b); H 1 (Ω; R)) ∩ C1 ((a, b); H 1 (Ω)∗ ), the function ψ(t) defined similarly by (1.99) is in C1 (a, b) with the derivative ψ (t) = H (u(t)), u (t)H 1 ×H 1∗ ,

(1.101)

a < t < b.

In fact, due to (1.95), H (u) ∈ H 1 (Ω; R) and H (u)H 1 ≤ uH 1 for any u ∈ H 1 (Ω; R). Furthermore, as v → 0 in H 1 (Ω; R), H (u + v)H 1 ≤ u + vH 1 and H (u + v) → H (u) in L2 (Ω; R); therefore, H (u + v) → H (u) in the weak topology of H 1 (Ω). Hence, u → H (u) is continuous in H 1 (Ω; R) from strong topology to weak topology. In the meantime, we have ψ(t + h) − ψ(t)  1  = H (θ u(t + h) + (1 − θ )u(t)) dθ, u(t + h) − u(t) (

0

= 0

) 1 H (θ u(t + h) + (1 − θ )u(t)) dθ, u(t + h) − u(t)

L2

. H 1 ×H 1∗

Notes and Further Researches

53

Hence, (1.101) is obtained. The continuity of ψ (t) is also verified by the continuity of the mapping u → H (u) from strong topology to weak topology. (12) Consider the Cauchy problem  du dt + p(t)u = q(t), 0 < t ≤ T , (1.102) u(0) = u0 , in the space L∞ (Ω). Here, p and q are given functions such that  p ∈ C((0, T ]; L∞ (Ω)) ∩ B((0, T ]; L∞ (Ω)), t 1−ν q ∈ B((0, T ]; L∞ (Ω)), q ∈ C((0, T ]; L∞ (Ω)), with some exponent 0 < ν ≤ 1. Then, for any initial value u0 ∈ L∞ (Ω), (1.102) has a unique solution u in the function space C([0, T ]; L∞ (Ω)) ∩ C1 ((0, T ]; L∞ (Ω)), which is given by the formula  t  t t e− s p(τ )dτ q(s) ds, u(t) = e− 0 p(s)ds u0 +

0 ≤ t ≤ T.

(1.103)

(1.104)

0

Indeed, by (1.97), the operator g → eg from L∞ (Ω) into itself is Fréchet differentiable with the derivative [Deg ]h = eg h. Therefore, we verify that the function u given by (1.104) certainly belongs to (1.103) and satisfies the equation in (1.102). Conversely, let u be any solution of (1.102). Then, for 0 < s ≤ t , we have t t d  −  t p(τ )dτ u(s) = e− s p(τ )dτ p(s)u(s) + e− s p(τ )dτ u (s) e s ds

= e−

t s

p(τ )dτ

q(s).

Integrating this equality in [0, t], we obtain (1.104).

Notes and Further Researches For the basic matters of the theory of functional analysis, we quote some of the classics, e.g., Hille–Phillips [HP57], Yosida [Yos80], and Brezis [Bre83]. For the basic matters of fractal dimension for the compact sets, see Eden–Foias– Nicolaenko–Temam [EFNT94]. Especially, for Mañé–Hölder-type projection, see [EFNT94, Chap. 10] and also Ben Artzi–Eden–Foias–Nicolaenko [BEFN93], Foias–Olson [FO96], and Hunt–Kaloshin [HY99]. Theorem 1.2 was obtained in [HY99]. For Dunford integrals, see Dunford–Schwartz [DS58, DS63]. Basics of multivalued linear operators are given in Favini–Yagi [FY99, Chap. I]. Basics of

54

1 Preliminaries

nonlinear operators are given in Deimling [Dei85] and Zeidler [Zei86]. Basics of interpolation spaces are given in Bergh–Löfström [BL76] and Tribel [Tri78, Chap. 1]. The triplets of spaces were presented and systematically used in Lions–Magenes [LM72], cf. also Dautray–Lions [DL88]. In these literatures, the triplet was defined by identifying X and its dual X , i.e., Z ⊂ X = X ⊂ Z . Such an identification is possible, since Z ⊂ X implies X ⊂ Z with dense and continuous embedding and since X and X are canonically isomorphic by the Riesz Theorem. By the uniqueness of triplet Z ⊂ X ⊂ Z ∗ , such a definition is of course essentially equivalent to ours presented in Sect. 7. For Sobolev–Lebesgue spaces, we quote Adams [Ada75], Grisvard [Gri85, Chap. 1], and Tribel [Tri78, Chap. 4]. The fundamental extension theorem, Theorem 1.33, was presented by Stein [Ste70].

Chapter 2

Sectorial Operators

Let X be a Banach space with norm  · . Let A be a densely defined, closed linear operator in X. We assume that the spectrum of A is contained in an open sectorial domain such that σ (A) ⊂ Σω = {λ ∈ C; |arg λ| < ω},

0 < ω ≤ π,

(2.1)

and its resolvent satisfies the estimate   (λ − A)−1  ≤ M , |λ|

λ∈ / Σω ,

(2.2)

with some constant M ≥ 1. We call such an operator A a sectorial operator of X. The condition (2.1) implicitly means that the origin is not in σ (A), namely, A has a bounded inverse A−1 on X. By (1.11) it then follows that λ ∈ ρ(A) if |λ| < A−1 −1 with the estimate   (λ − A)−1  ≤

A−1  , 1 − A−1 |λ|

|λ| < A−1 −1 .

(2.3)

By the similar reason, for λ0 = r0 e±iω , r0 > 0, we see that   r0 ⊂ ρ(A) λ ∈ C; |λ − λ0 | < M with the estimate   (λ − A)−1  ≤

M , r0 − M|λ − λ0 |

|λ − λ0 | <

r0 . M

r0 1 Since inf{arg λ; |λ − λ0 | < M } = sin−1 M , we observe that, for any angle ω such −1 1  that ω − sin M < ω < ω, the inclusion

σ (A) ⊂ Σω = {λ ∈ C; | arg λ| < ω } A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_2, © Springer-Verlag Berlin Heidelberg 2010

(2.4) 55

56

2 Sectorial Operators

is true, and the resolvent satisfies   (λ − A)−1  ≤ Mω , |λ|

λ∈ / Σω  ,

(2.5)

with some constant Mω ≥ M. For example, it is possible to take Mω =

M cos(ω − ω ) . 1 − M sin(ω − ω )

In this way, we verified that (2.1) and (2.2) always imply similar two conditions at least for some smaller angle than the ω. It is then natural to consider the infimum of angles ω for which the angle condition (2.4) and estimate (2.5) hold true. This infimum is called the angle of sectorial operator A and is denoted by ωA . Hence, for any ω , ωA < ω ≤ π , the angle condition (2.4) and estimate (2.5) take place. The first half of this chapter is devoted to explaining the systematic methods how to construct sectorial operators from sesquilinear forms on Hilbert spaces. Applying these methods, we shall realize various second-order strongly elliptic operators in domains equipped with boundary conditions as sectorial operators in Hilbert spaces and even Banach spaces. The second half is devoted to introducing basic materials concerning sectorial operators and showing their properties, e.g., the exponential functions and fractional powers. The specific properties of exponential functions then give us a fundamental motivation for defining in an abstract way the notion of analytic semigroups (see Chap. 3). The fractional powers Aθ , 0 ≤ θ ≤ 1, of a sectorial operator A of a Banach space X are a smooth (roughly speaking, analytic) interpolation of the identity operator 1 = A0 and the operator A = A1 itself. They are an interpolation not only for the spaces between the whole space X and the domain D(A) like [X, D(A)]θ but also the operations of 1 and A. Unfortunately, it is not always true that D(Aθ ) = [X, D(A)]θ for 0 < θ < 1 even if A is a sectorial operator of a Hilbert space (see Chap. 16, Sect. 1.5). However, this favorable property is certainly the case in many important examples in applications as will been explained in the last chapter, Chap. 16.

1 Sectorial Operators in Hilbert Spaces 1.1 Sectorial Operators Associated with Sesquilinear Form Let Z ⊂ X ⊂ Z ∗ be a triplet of spaces with norms  · , | · | and  · ∗ , respectively. Let (·, ·) denote the inner product of X, and ·, · the duality product on Z ∗ × Z. Consider a continuous and coercive sesquilinear form a(U, V ) defined on Z, i.e., |a(U, V )| ≤ MU V , Re a(U, U ) ≥ δU 2 ,

U, V ∈ Z, U ∈ Z,

(2.6) (2.7)

1 Sectorial Operators in Hilbert Spaces

57

with some constants M > δ > 0. Let A be the linear operator associated with the form, and let A|X and A|Z be its parts in the spaces X and Z, respectively. As already shown in Chap. 1, Sect. 8, A, A|X , and A|Z are densely defined, closed linear operators acting in Z ∗ , X, and Z, respectively. For each Re λ ≤ 0, consider the sesquilinear form a(U, V ) − λ(U, V ),

U, V ∈ Z.

Obviously this form is also continuous and coercive on Z. Therefore, Theorem 1.24 shows that the associated operator, which is clearly nothing more than A − λ, is an isomorphism from Z onto Z ∗ . In particular, λ ∈ ρ(A) if A is considered as an operator of Z ∗ . Let us next establish various estimates concerning the resolvent (λ − A)−1 for Re λ ≤ 0. Let U ∈ Z. Then, δU 2 ≤ Re a(U, U ) − Re λ|U |2 = Re (A − λ)U, U ≤ (A − λ)U ∗ U . Putting Φ = (λ − A)U , we obtain that   (λ − A)−1 Φ  ≤ δ −1 Φ∗ .

(2.8)

Since λ(λ − A)−1 Φ = A(λ − A)−1 Φ + Φ, the estimate then yields that     |λ|(λ − A)−1 Φ ∗ ≤ AL(Z,Z ∗ ) (λ − A)−1 Φ  + Φ∗ ≤ (Mδ −1 + 1)Φ∗ . Consequently,   |λ|(λ − A)−1 Φ ∗ ≤ (Mδ −1 + 1)Φ∗ ,

Φ ∈ Z∗.

Let now U ∈ D(A|X ). For Re λ ≤ 0, δU 2 ≤ Re a(U, U ) − Re λ|U |2 = Re((A − λ)U, U ) ≤ |(A − λ)U ||U |. From λ|U |2 = a(U, U ) + ((λ − A)U, U ) it follows that |λ||U |2 ≤ MU 2 + |(λ − A)U | |U | ≤ (Mδ −1 + 1)|(λ − A)U | |U |. Putting F = (λ − A)U , we obtain that   |λ|(λ − A)−1 F  ≤ (Mδ −1 + 1)|F |,

F ∈ X.

58

2 Sectorial Operators

Let again U ∈ Z. From λ(λ − A)−1 U − (λ − A)−1 AU = U it follows that     |λ|(λ − A)−1 U  ≤ (λ − A)−1 L(Z ∗ ,Z) AL(Z,Z ∗ ) U  + U . Therefore, by (2.8),   |λ|(λ − A)−1 U  ≤ (Mδ −1 + 1)U ,

U ∈ Z.

We have therefore shown for Re λ ≤ 0 the estimates   |λ|(λ − A)−1 Φ ∗ ≤ (Mδ −1 + 1)Φ∗ , Φ ∈ Z ∗ ,   |λ|(λ − A)−1 F  ≤ (Mδ −1 + 1)|F |, F ∈ X,   |λ|(λ − A)−1 U  ≤ (Mδ −1 + 1)U , U ∈ Z.

(2.9) (2.10) (2.11)

These estimates mean that all the operators A, A|X , and A|Z satisfy conditions (2.1) and (2.2) with angle ω = π2 and constant Mδ −1 + 1 in the spaces Z ∗ , X, and Z, respectively. In this way, we have proved that they are sectorial operators. Theorem 2.1 Let a(U, V ) be a sesquilinear form on Z satisfying (2.6) and (2.7), and let A, A|X , and A|Z be the linear operators of Z ∗ , X, and Z, respectively, which are determined from the form. Then, they satisfy (2.1) and (2.2) with angle ∗ ω = π2 and constant M+δ δ . Hence, they are sectorial operators of Z , X, and Z, π respectively, with angles < 2 . Remark 2.1 For λ ≤ 0 and U ∈ D(A|X ), we have δU 2 + |λ||U |2 ≤ Re a(U, U ) − λ|U |2 = Re((A − λ)U, U ) ≤ |(λ − A)U | |U |, (c−2 δ + |λ|)|U |2 ≤ |(λ − A)U ||U |, where c is the constant of embedding: |U | ≤ cU , U ∈ Z. Putting F = (λ − A)U , we obtain that   (|λ| + c−2 δ)(λ − A)−1 F  ≤ |F |. Hence, A|X satisfies the estimate   (λ − A|X )−1  ≤ 1/(|λ| + c−2 δ),

λ ≤ 0.

(2.12)

1.2 Sectorial Operators in L2 Spaces Let us utilize Theorem 2.1 for constructing various sectorial operators in L2 spaces.

1 Sectorial Operators in Hilbert Spaces

59

Consider the sesquilinear form a(u, v) =



n   n i,j =1 R

aij (x)Di uDj v dx +

Rn

c(x)uv dx,

u, v ∈ H 1 (Rn ), (2.13)

defined on H 1 (Rn ). Here, aij (x), 1 ≤ i, j ≤ n, are real-valued functions in Rn satisfying the conditions aij ∈ L∞ (Rn ), n 

1 ≤ i, j ≤ n,

aij (x)ξi ξj ≥ δ|ξ |2 ,

(2.14)

ξ = (ξ1 , . . . , ξn ) ∈ Rn , a.e. x ∈ Rn ,

(2.15)

i,j =1

with some constant δ > 0, and c(x) is a real-valued function satisfying c ∈ L∞ (Rn )

and c(x) ≥ c0 > 0,

Clearly, a(u, v) fulfills (2.6) with M = max{ tion, (2.15) implies that Re a(u, u) =

n   n i,j =1 R

 i,j

a.e. x ∈ Rn .

(2.16)

ai,j L∞ , cL∞ }. In addi-

aij (x)[(Re Di u)(Re Dj u) + (Im Di u)(Im Dj u)] dx



+

Rn

c(x)|u|2 dx ≥ δ∇u2L2 + c0 u2L2 .

Therefore, a(u, v) fulfills (2.7), too. We consider a triplet H 1 (Rn ) ⊂ L2 (Rn ) ⊂ H 1 (Rn )∗ = H −1 (Rn ) and an operator A associated with (2.13). Since

n  n   aij (x)Di uDj v dx = − Dj [aij (x)Di u], v n i,j =1 R

i,j =1

H −1 ×H 1

in the distribution sense, it naturally holds that Au = −

n 

Dj [aij (x)Di u] + c(x)u

in H −1 (Rn ).

i,j =1

Theorem 2.1 then yields the following results. Theorem 2.2 Let (2.14), (2.15), and (2.16) be satisfied. Then, the operator A associated with the form (2.13) and its parts satisfy (2.1) and (2.2) with ω = π2 and M determined by aij L∞ , cL∞ , δ, and c0 and are sectorial operators of H −1 (Rn ), L2 (Rn ), and H 1 (Rn ), respectively, with angles < π2 .

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2 Sectorial Operators

Let now Ω be any domain in Rn , and let Z be any closed subspace of H 1 (Ω) such that H˚ 1 (Ω) ⊂ Z ⊂ H 1 (Ω). We consider the sesquilinear form n  

a(u, v) =

 aij (x)Di uDj v dx +

i,j =1 Ω

c(x)uv dx,

u, v ∈ Z,

(2.17)

Ω

on Z. As before, aij (x), 1 ≤ i, j ≤ n, are real-valued functions in Ω satisfying the conditions aij ∈ L∞ (Ω), n 

1 ≤ i, j ≤ n,

aij (x)ξi ξj ≥ δ|ξ |2 ,

(2.18)

ξ = (ξ1 , . . . , ξn ) ∈ Rn , a.e. x ∈ Ω,

(2.19)

i,j =1

with some constant δ > 0, and c(x) is a real-valued function in Ω satisfying c ∈ L∞ (Ω)

and c(x) ≥ c0 > 0,

a.e. x ∈ Ω.

(2.20)

By the same reason as above, these conditions readily imply that a(u, v) fulfills (2.6) and (2.7) on Z. So, if we consider a triplet Z ⊂ L2 (Ω) ⊂ Z ∗ (note that, due to D(Ω) ⊂ H˚ 1 (Ω) ⊂ Z, Z is always dense in L2 (Ω)), then the operator A associated with the form (2.17) and its parts are all sectorial operators of Z ∗ , L2 (Ω), and Z, respectively, with angles < π2 . Let us investigate below more specific cases. Case where Z = H˚ 1 (Ω). In this case, Z ∗ coincides with H −1 (Ω). Since the space D(Ω) is dense in H˚ 1 (Ω), H −1 (Ω) is contained in the space of distributions D(Ω) . Therefore, it is possible to represent Au as Au = −

n 

Dj [aij (x)Di u] + c(x)u

in distribution sense.

(2.21)

i,j =1

This shows us that A and its parts are regarded as realizations of the differential operator (2.21) in H −1 (Ω), L2 (Ω), and H˚ 1 (Ω), respectively. We also know that, if Ω is a bounded domain with Lipschitz boundary, then D(A) = H˚ 1 (Ω) is given by (1.81), that is, u ∈ D(A) satisfies the Dirichlet boundary conditions γ u = 0 on ∂Ω.

(2.22)

Theorem 2.3 Let Ω be any domain in Rn . Let (2.18), (2.19), and (2.20) be satisfied. Then, the operator A associated with the form (2.17) and its parts satisfy (2.1) and (2.2) with ω = π2 and M determined by aij L∞ , cL∞ , δ, and c0 and are sectorial operators of H −1 (Ω), L2 (Ω), and H˚ 1 (Ω), respectively, with angles < π2 . In addition, if Ω is a bounded domain with Lipschitz boundary, then the functions in D(A) satisfy (2.22).

1 Sectorial Operators in Hilbert Spaces

61

Case where Z = H 1 (Ω). In this case, Z ∗ coincides no longer with any subspace of distributions. So, the associated operator A is not represented as a usual differential operator. But, let u ∈ D(A|L2 ), i.e., Au ∈ L2 (Ω), and assume that all are regular enough to justify the following arguments. Then, by Green’s formula, (Au, v)L2 = a(u, v) = −

+

n  

Dj [aij (x)Di u]v dx

i,j =1 Ω



n  

[νj (x)aij (x)Di u]v dS +

i,j =1 ∂Ω

c(x)uv dx,

v ∈ H 1 (Ω),

Ω

where ν(x) = (ν1 (x), . . . , νn (x)) denotes the outer normal vector at x ∈ ∂Ω. Since a(u, v) must be continuous in v with respect to the L2 (Ω) topology, the integral on ∂Ω must vanish. In other words, u must satisfy the boundary conditions n  ∂u ≡ νj (x)aij (x)Di u = 0 ∂νA

on ∂Ω.

(2.23)

i,j =1

Such conditions are called the Neumann-type boundary conditions on ∂Ω. We then arrive at  n  Dj [aij (x)Di u] + c(x)u, v , v ∈ H 1 (Ω); (Au, v)L2 = − i,j =1

L2

therefore, Au = −

n 

Dj [aij (x)Di u] + c(x)u

in L2 (Ω).

(2.24)

i,j =1

In this way, u satisfies the boundary conditions (2.23), and Au is given by (2.24). We also notice that, if Au = ϕ with u ∈ H 1 (Ω) and ϕ ∈ H 1 (Ω)∗ , then there exist sequences uk ∈ D(A|L2 ) and fk ∈ L2 (Ω) such that A|L2 uk = fk and that uk → u in H 1 (Ω) and fk → ϕ in H 1 (Ω)∗ . Indeed, it is sufficient to choose any sequence fk → ϕ in H 1 (Ω)∗ with fk ∈ L2 (Ω) and to put uk = (A|L2 )−1 fk . In this sense, A and its parts are regarded as realizations of the differential operator (2.21) under the Neumann-type boundary conditions (2.23) in H 1 (Ω)∗ , L2 (Ω), and H 1 (Ω), respectively. Theorem 2.4 Let Ω be any domain in Rn . Let (2.18), (2.19), and (2.20) be satisfied. Then, the operator A associated with the form (2.17) and its parts satisfy (2.1) and (2.2) with ω = π2 and M determined by aij L∞ , cL∞ , δ, and c0 and are sectorial operators of H 1 (Ω)∗ , L2 (Ω), and H 1 (Ω), respectively, with angles < π2 . Case where Z = H˚ D1 (Ω). Let Ω be a bounded domain with Lipschitz boundary. We now consider a splitting of the boundary ∂Ω into ΓD and ΓN , namely, ∂Ω =

62

2 Sectorial Operators

ΓD ∪ ΓN , ΓD ∩ ΓN = ∅, and ΓD is a nonempty open set of ∂Ω. We set Z = H˚ D1 (Ω) = {u ∈ H 1 (Ω); γ u = 0 on ΓD }.

(2.25)

The operator A associated with the form (2.17) and its parts are regarded as realizations of the differential operator (2.21) under the splitting boundary conditions γu=0

on ΓD

and

∂u = 0 on ΓN . ∂νA

(2.26)

Theorem 2.5 Let Ω be a bounded domain with Lipschitz boundary. Let (2.18), (2.19), and (2.20) be satisfied. Then, the operator A associated with the form (2.17) and its parts satisfy (2.1) and (2.2) with ω = π2 and M determined by aij L∞ , cL∞ , δ, and c0 and are sectorial operators of H˚ D1 (Ω)∗ , L2 (Ω), and H˚ D1 (Ω), respectively, with angles < π2 . Remark 2.2 When aij (x) ≡ δij (Kronecker’s delta), the differential operator n ∂u 2 Δ n= i=1 Di is called the Laplace operator and the boundary conditions ∂ν ≡ i=1 νi (x)Di u = 0 are called the Neumann boundary conditions. Following the procedure described above, we can define a realization of the Laplace operator in Ω under the Dirichlet, Neumann, or splitting boundary conditions on ∂Ω. Remark 2.3 Let Ω be a bounded domain with Lipschitz boundary. When Z = H˚ 1 (Ω) or H˚ D1 (Ω), it is possible to show the coercivity (2.7) without using the positivity of the function c(x). In fact, according to the Poincaré inequality (e.g., see [BF02, 1.1.3.b]), uL2 ≤ C∇uL2 for all u ∈ H˚ D1 (Ω), a fortiori for all u ∈ H˚ 1 (Ω). Therefore, (2.19), together with the condition c(x) ≥ 0, can imply (2.7).

1.3 Shift Property in L2 When condition (2.19) is satisfied, the differential operator A(D)u = −

n 

Dj [aij (x)Di u] + c(x)u

(2.27)

i,j =1

is said to be strongly elliptic. In general, strongly elliptic operators are expected to enjoy the regularity property that A(D)u ∈ L2 (Ω) implies u ∈ H 2 (Ω). Such a property is called the shift property. However, our operators introduced in the preceding subsection cannot enjoy this property, because aij (x) are merely measurable functions. In order to have the shift property, we assume that Ω is a bounded domain in Rn with C2 boundary and that aij (x) satisfy aij ∈ C1 (Ω),

1 ≤ i, j ≤ n.

(2.28)

1 Sectorial Operators in Hilbert Spaces

63

The shift property is verified by the widely developed theory of elliptic operators. Let A(D) be the elliptic operator in Ω given by (2.27). Let the conditions (2.19), (2.20), and (2.28) be satisfied. We will first consider the case where A(D) is equipped with the Dirichlet conditions. According to the theory of elliptic operators (see [Gri85, Theorem 2.2.2.3 and Corollary 2.2.2.4]), the mapping u → (A(D) + k)u is an isomorphism from the space {u ∈ H 2 (Ω); γ u = 0 on ∂Ω} onto L2 (Ω) if k is sufficiently large. In addition, it holds that uH 2 ≤ C[A(D) + k]uL2 ,

u ∈ H 2 (Ω), γ u = 0.

Let then A be the operator associated with the form (2.17) on H˚ 1 (Ω), and let A|L2 be its part in L2 (Ω). If A|L2 u = f ∈ L2 (Ω), then there exists a unique function u˜ ∈ ˜ v) = (A(D)u, ˜ v) = H 2 (Ω) such that [A(D) + k]u˜ = f + ku and γ u˜ = 0; since a(u, (k(u − u) ˜ + f, v), v ∈ H˚ 1 (Ω), it follows that A|L2 u˜ = k(u − u) ˜ + f ; therefore, (A|L2 +k)(u− ˜ u) = 0, i.e., u˜ = u. In this way, u ∈ D(A|L2 ) implies that u ∈ H 2 (Ω). We have therefore verified that

D(A|L2 ) = {u ∈ H 2 (Ω); γ u = 0 on ∂Ω}, (2.29) A|L2 u = A(D)u, as well as uH 2 ≤ C(A|L2 uL2 + uL2 ),

u ∈ D(A|L2 ).

(2.30)

In what follows, the domain will often be denoted by HD2 (Ω) = {u ∈ H 2 (Ω); γ u = 0 on ∂Ω}.

(2.31)

We next consider the case where A(D) is equipped with the Neumann-type boundary conditions. Similarly, according to the theory of elliptic operators (see [Gri85, Theorem 2.2.2.5 and Corollary 2.2.2.6]), the mapping u → (A(D) + k)u is ∂u an isomorphism from the space {u ∈ H 2 (Ω); ∂ν = 0 on ∂Ω} onto L2 (Ω) if k is A sufficiently large. In addition, it holds that uH 2 ≤ C[A(D) + k]uL2 ,

u ∈ H 2 (Ω),

∂u = 0. ∂νA

Let A be the operator associated with the form (2.17) on H 1 (Ω). Let A|L2 be its part in L2 (Ω). If A|L2 u = f ∈ L2 (Ω), then there exists a unique function u˜ ∈ H 2 (Ω) ∂ u˜ = 0 on ∂Ω. As before, it is easy to see such that [A(D) + k]u˜ = f + ku and ∂ν A that u˜ = u. Therefore, u ∈ D(A|L2 ) implies that u ∈ H 2 (Ω). We thus conclude that

∂u D(A|L2 ) = {u ∈ H 2 (Ω); ∂ν = 0 on ∂Ω}, A (2.32) A|L2 u = A(D)u as well as uH 2 ≤ C(A|L2 uL2 + uL2 ),

u ∈ D(A|L2 ).

(2.33)

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2 Sectorial Operators

In what follows, the domain will often be denoted by   ∂u HN2 (Ω) = u ∈ H 2 (Ω); = 0 on ∂Ω . ∂νA

(2.34)

The following theorem has therefore been verified. Theorem 2.6 Let Ω be a bounded domain with C2 boundary, and let (2.19), (2.20), and (2.28) be satisfied. Then, the parts A|L2 of the operators associated with the form (2.17) on H˚ 1 (Ω) and H 1 (Ω) are characterized by (2.29) and (2.32), respectively. In the case where Ω is convex and where aij (x) are symmetric in the sense that aij (x) = aj i (x),

1 ≤ i, j ≤ n,

(2.35)

we can weaken the C2 regularity of ∂Ω to Lipschitz. Indeed, according to Grisvard [Gri85, Theorems 3.1.3.1 and 3.1.3.3], the constants C appearing in the energy estimates (2.30) and (2.33) depend (roughly speaking) only on the negative part of the curvature of ∂Ω and are independent of the positive part. On the basis of this fact, the same shift properties are established for bounded convex domains. Note that any bounded convex domain has a Lipschitz boundary. Theorem 2.7 Let Ω be a bounded convex domain, and let (2.19), (2.20), (2.28), and (2.35) be satisfied. Then, the parts A|L2 of the operators associated with the form (2.17) on H˚ 1 (Ω) and H 1 (Ω) are again characterized by (2.29) and (2.32), respectively. Estimates (2.30) and (2.33) also hold true. For the proofs, see [Gri85, Theorems 3.2.1.2 and 3.2.1.3]. Let us now consider a very particular case where aij (x) = a(x)δij but a(x) is less regular than (2.28). Our elliptic operator is then written as A(D)u = −

n 

Di [a(x)Di u] + c(x)u.

(2.36)

i=1

We assume (2.19) for a(x)δij , which reads as a(x) ≥ δ > 0 in Ω. We assume also that a(x) belongs to either a ∈ Hp1 (Ω; R)

with n < p < ∞

(2.37)

n with s ≥ 1 and s > . 2

(2.38)

or a ∈ H s (Ω; R)

Note that, due to (1.76), Hp1 (Ω; R) ⊂ C(Ω) and H s (Ω; R) ⊂ C(Ω).

1 Sectorial Operators in Hilbert Spaces

65

Theorem 2.8 Let Ω be a C2 or convex, bounded domain in Rn . Let (2.19) and (2.20) be satisfied, and let a(x) belong to either (2.37) or (2.38). Let A(D) be the elliptic operator given by (2.36). Then, realizations A|L2 of A(D) in L2 (Ω) under the boundary conditions (2.22) and (2.23) (cf. Remark 2.2) have their domains given by (2.31) and (2.34), respectively. Estimates (2.30) and (2.33) also hold true. Proof Let us describe here the proof only for the Dirichlet boundary conditions. For the other case, it is quite similar. Let u ∈ D(A|L2 ). Then, Au = f ∈ L2 (Ω), and −

n 

Di [a(x)Di u] + c(x)u = f

in distribution sense.

(2.39)

i=1

We rewrite −a(x)Δu −

n  [Di a(x)]Di u + c(x)u = f i=1

and (−Δ + 1)u −

n 

  a(x)−1 [Di a(x)]Di u + a(x)−1 c(x) − 1 u = a(x)−1 f.

i=1

In view of this formula, let us introduce a realization Λ of elliptic operator −Δ + 1 under (2.22) in L2 (Ω). We know by Theorems 2.6 and 2.7 that Λ is a positive definite self-adjoint operator of L2 (Ω) with domain (2.31). In addition, by Theorems 16.12 and 16.13 in Chap. 16, we know also characterization of the domains of fractional powers Λθ for 0 < θ < 1, in particular D(Λθ ) ⊂ H 2θ (Ω). Keeping in mind this fact, we consider a realization B of the differential operator B(D)v = −

n 

  a(x)−1 [Di a(x)]Di v + a(x)−1 c(x) − 1 v

i=1 1+ n

in L2 (Ω) whose domain is given by D(B) = H p (Ω) (resp. D(B) = n H 2−s+ 2 (Ω)) if (2.37) (resp. (2.38)) is assumed. Since 1 + pn < 2 (resp. 2 − s + n 2 < 2), the operator B is dominated by Λ in the sense of (3.8). As a consequence, if β > 0 is a sufficiently large number, then −β ∈ ρ(Λ + B), see Theorem 3.3. We recall that Au = f implies that (Λ + B)u = a(x)−1 f . Then, let u˜ be the solution of (Λ + B + β)u˜ = βu + a(x)−1 f . As shown above, u˜ ∈ HD2 (Ω) with the estimate u ˜ H 2 ≤ Cβu + a(x)−1 f L2 . We can follow the calculations above inversely to obtain the equality −

n  i=1

Di [a(x)Di u˜ ] + c(x)u˜ + βa(x)u˜ = βa(x)u + f

in L2 (Ω).

66

2 Sectorial Operators

Subtract (2.39) from this. Then, −

n 

Di [a(x)Di (u˜ − u)] + c(x)(u˜ − u) + βa(x)(u˜ − u) = 0.

i=1

Hence, u˜ = u. In addition, uH 2 ≤ Cβu + a(x)−1 f L2 ≤ Cδ −1 f L2 .



We often need a higher shift property of strongly elliptic operators. For this, we assume that aij ∈ Cm+1 (Ω),

1 ≤ i, j ≤ n; m = 1, 2, 3, . . . .

(2.40)

Theorem 2.9 Let Ω be a bounded domain with Cm+2 boundary, and let (2.19), (2.20), and (2.40) be satisfied. Then, the parts A|L2 of the operators associated with the form (2.17) on H˚ 1 (Ω) and H 1 (Ω), respectively, have the property that A|L2 u ∈ H m (Ω) implies u ∈ H m+2 (Ω). The estimate uH m+2 ≤ C(A|L2 uH m + uH m ) also holds true. For the proof, see [Gri85, Theorem 2.5.1.1]. Let us consider finally the splitting boundary conditions. Let Ω be a bounded domain of Rn with Lipschitz boundary. Consider a splitting of ∂Ω, ∂Ω = ΓD ∪ ΓN and ΓD ∩ ΓN = ∅, ΓD being a nonempty open subset. We assume the two conditions |B(x0 ; R) ∩ ΓD | ≥ γ R n−1 |B(x0 ; R) ∩ Ω| ≥ γ R

n

for any x0 ∈ ΓD ,

for any x0 ∈ ΓN , B(x0 ; R) ∩ ΓD = ∅

(2.41) (2.42)

with some constant γ > 0 (cf. [BF02, (1.22)]), where B(x0 ; R) denotes the open ball with center x0 ∈ ∂Ω and radius R > 0. Theorem 2.10 Let Ω be a bounded domain with Lipschitz boundary which is split into two parts ΓD and ΓN as above with conditions (2.41) and (2.42). Let (2.18), (2.19), and (2.20) be satisfied. Then, the part A|L2 of the operator associated with the form (2.17) on H˚ D1 (Ω) satisfies D(A|L2 ) ⊂ Wp1 (Ω) and uWp1 ≤ C(A|L2 uL2 + uL2 ),

u ∈ D(A|L2 ),

(2.43)

for a certain exponent p > 2. This property was proved by Bensoussan–Frehse [BF02, Theorem 2.2].

1.4 Sectorial Operators in Product Spaces of L2 Let Ω be a bounded domain in Rn . We introduce two product Hilbert spaces L2 (Ω) and H1 (Ω) with  = 2 (see (1.84) and (1.85)). Consider the triplet of spaces

1 Sectorial Operators in Hilbert Spaces

67

H1 (Ω) ⊂ L2 (Ω) ⊂ H1 (Ω)∗ . By definition, it is easy to see that H1 (Ω)∗ is also given by the product space    ϕ H1 (Ω)∗ = ; ϕ ∈ H 1 (Ω)∗ and ψ ∈ H 1 (Ω)∗ (2.44) ψ with duality product     ϕ u , = ϕ, u + ψ, v , ψ v

    ϕ u 1 ∗ ∈ H (Ω) , ∈ H1 (Ω). ψ v

Consider the sesquilinear form  ˜ a(U, U ) = [a(x)∇u · ∇ u˜ + b(x)∇v · ∇ u˜ + c(x)∇u · ∇ v˜ Ω



˜ dx + k + d(x)∇v · ∇ v]     u u˜ ˜ U= ,U= ∈ H1 (Ω), v v˜

(uu˜ + v v) ˜ dx, Ω

(2.45)

defined on H1 (Ω). Here, the functions a(x), b(x), c(x), d(x) are real-valued continuous functions on Ω. These functions are assumed to satisfy inf a(x) = a0 > 0,

x∈Ω

inf d(x) = d0 > 0,

x∈Ω

(2.46)

and to satisfy either b(x)c(x) ≤ 0

on Ω

(2.47)

or b(x)c(x) ≥ 0 on Ω

and

inf [a(x)d(x) − b(x)c(x)] = η0 > 0.

x∈Ω

(2.48)

The constant k is a positive constant which will be fixed below. Our goal is to show that the associated operator A and its parts in L2 (Ω) and H1 (Ω) are sectorial operators in H1 (Ω)∗ , L2 (Ω), and H1 (Ω), respectively. It seems, however, difficult to apply Theorem 2.1 to the form (2.45) directly. Our way to arrive at the goal will be indirect. We begin with verifying the following lemma.   ϕ(x) 0  b(x)  Lemma 2.1 Let P (x) = a(x) c(x) d(x) . There exist a diagonal matrix D(x) = 0 1 with ϕ ∈ C1 (Ω) and infx∈Ω ϕ(x) = ϕ0 > 0 and a constant δ > 0 for which it holds that   D(x)P (x)D(x)−1 U, U ≥ δU 2 , x ∈ Ω, U ∈ R2 . (2.49) In addition, when (a, b, c, d) varies in a compact subset of C(Ω; R)4 satisfying (2.46) and either (2.47) or (2.48), the quantities ϕC1 , ϕ0 , and δ can be uniform.

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2 Sectorial Operators

Proof (I) Let us first consider a case where b(x) and c(x) are C1 functions on Ω. We have     D(x)P (x)D(x)−1 U, U = a(x)u2 + b(x)ϕ(x) + c(x)ϕ(x)−1 uv + d(x)v 2 for U = t (u, v) ∈ R2 . Let condition (2.47) be satisfied. In this case, we put  −b(x)c(x) + ε ϕε (x) = b(x)2 + ε 2 with parameter ε > 0. Then,   b(x)ϕε (x) + c(x)ϕε (x)−1  = 

|b(x)ε + c(x)ε 2 |

√ ≤ (1 + cC ) ε.

(b(x)2 + ε 2 )(−b(x)c(x) + ε) √ √ If ε > 0 is fixed in such a way that (1 + cC ) ε ≤ a0 d0 , then   a(x)u2 + b(x)ϕε (x) + c(x)ϕε (x)−1 uv + d(x)v 2 ≥ a0 u2 −



1 a0 d0 |u||v| + d0 v 2 ≥ (a0 u2 + d0 v 2 ). 2

Let next condition (2.48) be satisfied. In this case, we put  b(x)c(x) + ε ϕε (x) = , ε > 0. b(x)2 + ε 2 By (2.48), 2 c(x) + b(x)ε + c(x)ε 2 |   b(x)ϕε (x) + c(x)ϕε (x)−1  = |2b(x)  (b(x)2 + ε 2 )(b(x)c(x) + ε)  √ ≤ 2 b(x)c(x) + (1 + cC ) ε  √ ≤ 2 a(x)d(x) − η0 + (1 + cC ) ε.

Therefore,   a(x)u2 + b(x)ϕε (x) + c(x)ϕε (x)−1 uv + d(x)v 2   ≥ [a(x) − μ]u2 − 2 a(x)d(x) − η √  + (1 + cC ) ε |u||v| + [d(x) − μ]v 2 + μ(u2 + v 2 ). Let μ =

η0 a(x)+d(x) .

Then, since

 [a(x) − μ]u2 + [d(x) − μ]v 2 ≥ 2 (a(x) − μ)(d(x) − μ)|uv|  > 2 a(x)d(x) − η0 |uv|,

1 Sectorial Operators in Hilbert Spaces

69

it follows that   a(x)u2 + b(x)ϕε (x) + c(x)ϕε (x)−1 uv + d(x)v 2 √ η0 ≥ (u2 + v 2 ) − (1 + cC ) ε|uv|. a + dC Hence, if ε > 0 is sufficiently small, then the desired estimate (2.49) is satisfied. Since b(x) and c(x) are C1 functions, it is the same for ϕε (x). (II) Secondly, let us consider the general case. We notice that the C1 regularity of b(x) and c(x) is necessary only for ensuring that ϕε (x) is a C1 function and not for yielding estimate (2.49). So, if a new function ϕ(x), ˜ which is smooth on Ω, is taken sufficiently close to ϕε (x) in the topology of C(Ω), then (2.49) can be valid with ϕ(x). ˜ We also notice that, if (2.49) is satisfied for P (x) with some matrix D(x) = diag{ϕ(x), 1}, ϕ(x) being smooth, then it is satisfied for every P˜ (x), provided that ˜ ˜ a(x), ˜ b(x), c(x), ˜ and d(x) are all sufficiently close to a(x), b(x), c(x), and d(x), respectively, in the topology C(Ω). This shows that, when a(x), b(x), c(x), and d(x) vary in a compact subset of C(Ω), one can choose a finite number of matrices so that (2.49) is valid for every P (x) with at least one of these matrices.  We notice that multiplication of D(x) is an isomorphism of H1 (Ω). In view of this fact, we here introduce the auxiliary sesquilinear form   aD (U, V ) = a D(x)−1 U, D(x)V , U, V ∈ H1 (Ω), on H1 (Ω). On account of Lemma 2.1, aD (U, V ) is seen to satisfy the coercivity (2.7). In fact, we see that     a D(x)−1 U, D(x)U = a(x)|∇u|2 + b(x)ϕ(x)∇v · ∇u Ω

 + c(x)ϕ(x)−1 ∇u · ∇v + d(x)|∇v|2 dx   2  +k |u| + |v|2 dx + b(U, U ),   u U= ∈ H1 (Ω), v

Ω

where b(U, V ) denotes a sesquilinear form containing first derivatives of ϕ in its coefficients. Obviously, b(U, V ) satisfies |b(U, U )| ≤ Cϕ U  |U |. Therefore,     Re aD (U, U ) ≥ δ |∇u|2 + |∇v|2 + k |u|2 + |v|2 − Cϕ U  |U | δ ≥ U 2 + (k − Cϕ )|U |2 . 2

70

2 Sectorial Operators

Fix k large enough to satisfy k ≥ Cϕ . Then, aD (U, V ) satisfies (2.7) and (2.6). Hence, Theorem 2.1 is available to conclude that the operator AD associated with the form aD (U, V ) and its parts satisfy   (λ − AD )−1 

H1∗

    C , + (λ − AD|L2 )−1 L + (λ − AD|H1 )−1 H1 ≤ 2 |λ|

λ∈ / Σω ,

(2.50)

with some 0 ≤ ω < π2 . We now notice that     AU, V = a(U, V ) = aD D(x)U, D(x)−1 V = AD D(x)U, D(x)−1 V . In addition, we observe by the lemma below that multiplications of D(x) and D(x)−1 are isomorphisms of H1 (Ω)∗ as well as of L2 (Ω) and H1 (Ω) and that the duality product has the property     D(x)−1 Φ, V = Φ, D(x)−1 V , Φ ∈ H1 (Ω)∗ , V ∈ H1 (Ω). Then, it is deduced that A = D(x)−1 AD D(x) and the resolvent satisfies the relation (λ − A)−1 = D(x)(λ − AD )−1 D(x)−1 ,

λ∈ / Σω .

(2.51)

As a consequence of (2.50),   (λ − A)−1 

H1∗

    C , + (λ − A|L2 )−1 L + (λ − A|H1 )−1 H1 ≤ 2 |λ|

λ∈ / Σω . (2.52)

Lemma 2.2 The operation of multiplication by D(x) can be extended as a bounded operator on H1 (Ω)∗ with the property D(x)Φ, V = Φ, D(x)V ,

Φ ∈ H1 (Ω)∗ , V ∈ H1 (Ω).

(2.53)

It is the same for D(x)−1 . Proof For F ∈ X, we have D(x)F ∗ = sup |(D(x)F, V )| = sup |(F, D(x)V )| ≤ CϕC1 F ∗ . V ≤1

V ≤1

Since L2 (Ω) is dense in H1 (Ω)∗ , this shows that multiplication by D(x) can be extended to H1 (Ω)∗ uniquely as a bounded operator. Since D(x)Φ, V = Φ, D(x)V holds for Φ ∈ L2 (Ω) and V ∈ H1 (Ω), (2.53) is also verified by the usual argument.  We are now able to announce the following result.

2 Sectorial Operators in Banach Spaces

71

Theorem 2.11 Let Ω be any domain in Rn . Let a, b, c, d ∈ C(Ω; R) satisfy (2.46) and either (2.47) or (2.48). Then, for sufficiently large k, the operator A associated with the form (2.45) and its parts in L2 (Ω) and H1 (Ω) satisfy (2.1) and (2.2) in H1 (Ω)∗ , L2 (Ω), and H1 (Ω), respectively, with ω = π2 and some constant M. When (a, b, c, d) varies in a compact subset of C(Ω; R)4 satisfying (2.46) and either (2.47) or (2.48), the constants k and M can be uniform. We finally investigate a representation of A. Let us consider a case where a(x), b(x), c(x), and d(x) are all C1 functions. Let Λ be the operator associated with the sesquilinear form   ∇u · ∇v dx + uv dx, u, v ∈ H 1 (Ω), a(u, v) = Ω

Ω

H 1 (Ω)∗

i.e., Λ is a realization of −Δ + 1 in under the Neumann boundary conditions. Then the first integral in the right-hand side of (2.45) is written as    a(x)∇u · ∇ u˜ dx = ∇u · ∇(a u) ˜ dx − (∇u · ∇a)u˜ dx Ω

Ω

Ω

= a(x)Λu, u

˜ − ∇a · ∇u + a(x)u, u . ˜ It is the same for other integrals. Therefore, we have AU, V = a(U, V ) = a(x)Λu + b(x)Λv, u

˜ + c(x)Λu + d(x)Λv, v

˜ − ∇a · ∇u + ∇b · ∇v, u

˜ − ∇c · ∇u + ∇d · ∇v, v

˜ + (k − a(x))u − b(x)v, u

˜ + −c(x)u + (k − d(x))v, v . ˜ This then means that AU is given explicitly by the formula     a(x) b(x) ∇a · ∇ ∇b · ∇ AU = (Λ − 1)U − U + kU, c(x) d(x) ∇c · ∇ ∇d · ∇

U ∈ H1 (Ω). (2.54)

Furthermore, for A|L2 , we can see that   ∇ · (a∇·) ∇ · (b∇·) A|L2 U = − U + kU, ∇ · (c∇·) ∇ · (d∇·)

U ∈ H2N (Ω).

(2.55)

2 Sectorial Operators in Banach Spaces 2.1 Sectorial Operators in Lp (1 < p < ∞) Spaces Let Ω be a bounded domain in Rn . Consider the sesquilinear form a(u, v) =

n   i,j =1 Ω

 aij (x)Di uDj v dx +

c(x)uv dx, Ω

u, v ∈ Z,

(2.56)

72

2 Sectorial Operators

on Z, where Z = H˚ 1 (Ω) or H 1 (Ω). Here, aij (x), 1 ≤ i, j ≤ n, are real-valued functions in Ω satisfying conditions (2.18) and (2.19), and c(x) is a real-valued function in Ω satisfying (2.20). We consider the triplet Z ⊂ L2 (Ω) ⊂ Z ∗ and the operator A associated with the form (2.56). We already know that its part A|L2 in L2 (Ω) is a sectorial operator of L2 (Ω). In this section, we shall study A in the space Lp (Ω), where 1 < p < ∞. It is shown for 2 < p < ∞ that its part in Lp (Ω) becomes a sectorial operator of Lp (Ω). On the other hand, it is shown for 1 < p < 2 that a natural extension of A in Lp (Ω) defines a sectorial operator of Lp (Ω). Consider first the case where 2 < p < ∞. We begin with showing the following lemma. Lemma 2.3 Let 2 < p < ∞. For u ∈ Z (= H˚ 1 (Ω) or H 1 (Ω)), put u∗ε =  |u|2  p−2 2 u with a parameter ε > 0. Then, 1+ε|u|2     Re Au, u∗ε ≥ c0 u, u∗ε

(2.57)

and    Im Au, u∗  ≤

  Cnp Re Au, u∗ε √ 2δ p − 1

ε

(2.58)

with C = maxi,j aij L∞ , where δ and c0 are the constants in (2.19) and (2.20). Proof It is easily seen that u∗ε ∈ H 1 (Ω) with ∇u∗ε =



|u|2 1 + ε|u|2



|u|2 1 + ε|u|2

=

 p−2 2

∇u +

p − 2 |u|p−4 u 2 p ∇|u| 2 (1 + ε|u|2 ) 2

∇u +

 p − 2 |u|p−4  2 2 p |u| ∇u + u ∇u . 2 2 (1 + ε|u| ) 2

 p−2 2

We therefore have   Au, u∗ε =

  Ω

|u|2 1 + ε|u|2

p−2 + 2  Ω





n 

|u|p−4

Ω (1 + ε|u|2 )

(1 + ε|u|2 )

p−2 2

n 

|u|p−2 (1 + ε|u|2 ) 2

p 2

  aij (x)Di u u2 Dj u + |u|2 Dj u dx

i,j =1

dx

p

Ω

aij (x)Di uDj u dx

i,j =1

c(x)|u|p

+ =ε

 p−2  n 2

i,j =1

 aij (x)|u|2 Di uDj u dx + Ω

c(x)|u|p (1 + ε|u|2 )

p−2 2

dx

2 Sectorial Operators in Banach Spaces



1 + 2

73 n 

|u|p−4

Ω (1 + ε|u|2 )

p 2

 aij (x) p|u|2 Di uDj u

i,j =1

 + (p − 2)u2 Di uDj u dx.

Here, we put uDj u = αj + iβj , 1 ≤ j ≤ n. Then, it turns out that   Au, u∗ε = ε



n 

|u|p−2

aij (x)[(αi αj + βi βj ) + i(βi αj − αi βj )] dx

p

Ω

(1 + ε|u|2 ) 2



i,j =1 n 

|u|p−4

+

aij (x){[(p − 1)αi αj + βi βj ]

p

Ω

(1 + ε|u|2 ) 2

i,j =1



c(x)|u|p

+ i[(p − 1)βi αj − αi βj ]} dx + Ω

(1 + ε|u|2 )

p−2 2

dx.

Then, Re(Au, u∗ε ) can be estimated from below by using (2.19) and (2.20). In particular, we verify the desired estimate (2.57). In the meantime, | Im(Au, u∗ε )| is estimated as follows. We have n 

n 

aij |βi αj − αi βj | ≤ 2C

i,j =1

|αi βj | ≤ C

i,j =1

= Cn

n   2  αi + βj2 i,j =1

n n   2  Cn  αj + βj2 ≤ aij (αi αj + βi βj ). δ j =1

i,j =1

In a similar way, n 

aij |(p − 1)βi αj − αi βj | ≤ Cp

i,j =1

≤√

≤

Noting that

√p 2 p−1

n 

n Cp   |αi βj | = √ p − 1|αi βj | p − 1 i,j =1 i,j =1

n  Cp   (p − 1)αi2 + βj2 p − 1 i,j =1

n  Cnp aij [(p − 1)αi αj + βi βj ]. √ 2δ p − 1 i,j =1

≥ 1, we obtain from these the desired estimate (2.58).



Let Re λ ≤ 0. For the solution u to the equation (A − λ)u = f , where f ∈ Lp (Ω) ⊂ L2 (Ω), we verify the following fact.

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2 Sectorial Operators

Proposition 2.1 Let 2 < p < ∞, and let Re λ ≤ 0. If (A − λ)u = f with f ∈ Lp (Ω), then u belongs to Lp (Ω), and the estimates (| Re λ| + c0 )uLp ≤ f Lp ,   Cnp + 2 f Lp |λ|uLp ≤ √ 2δ p − 1

(2.59) (2.60)

hold with C = maxi,j aij L∞ , where δ and c0 are the constants in (2.19) and (2.20). Proof Taking the real part of the inner product of (A − λ)u = f and u∗ε in L2 (Ω), we have       Re Au, u∗ε − Re λ u, u∗ε = Re f, u∗ε . Since Re(Au, u∗ε ) ≥ 0 and (u, u∗ε ) ≥ 0, it follows that     Re Au, u∗ε + | Re λ|(u, u∗ ) = Re f, u∗ε ≤ f Lp u∗ε Lq , where

1 p

+

1 q

(2.61)

= 1. This, together with (2.57), yields that 

(c0 + | Re λ|) Ω



|u|p (1 + ε|u|2 )

p−2 2

dx ≤ f Lp

Ω

(1 + ε|u|2 )

 ≤ f Lp

1

|u|(p−1)q

(1 + ε|u|2 )

dx 1

|u|p

Ω

q

(p−2)q 2

q

p−2 2

dx

.

Hence,  (c0 + | Re λ|)

1

|u|p Ω

(1 + ε|u|2 )

p

p−2 2

dx

≤ f Lp .

Letting ε → 0, we deduce by Fatou’s lemma that u ∈ Lp (Ω) with (2.59). In order to prove (2.60), we notice that              |λ| u, u∗ε ≤  Au, u∗ε  +  f, u∗ε  ≤ Re Au, u∗ε + Im Au, u∗ε  +  f, u∗ε . Then, it follows from (2.58) and (2.61) that       Cnp |λ| u, u∗ε ≤ + 2 f Lp u∗ε L . √ q 2δ p − 1 Letting ε → 0, we obtain that  p |λ|uLp

≤

 Cnp p−1 + 2 f Lp uLp . √ 2δ p − 1

Hence, estimate (2.60) is deduced.



2 Sectorial Operators in Banach Spaces

75

Let us next consider the case where 1 < p < 2. We verify the following analogous lemma. Lemma 2.4 Let 1 < p < 2. For u ∈ Z, put u∗ε = (|u|2 + ε) ε > 0. Then,     Re Au, u∗ε ≥ c0 u, u∗ε

p−2 2

u with parameter (2.62)

and    Im Au, u∗  ≤ ε

  Cnp Re Au, u∗ε √ 2δ p − 1

(2.63)

with C = maxi,j aij L∞ , where δ and c0 are the constants in (2.19) and (2.20). Proof We can argue in an analogous way as before. It is observed that u∗ε ∈ H 1 (Ω) with   p−2  p−4   p − 2 2 |u| + ε 2 |u|2 ∇u + u2 ∇u . ∇u∗ε = |u|2 + ε 2 ∇u + 2 From this we can write   Au, u∗ε = ε



+

n  2  p−4 −2  2 |u| + ε |u| aij (x)|u|2 Di uDj u dx

Ω

1 2

i,j =1

 Ω

n  2  p−4   |u| + ε 2 aij (x) p|u|2 Di uDj u i,j =1

 + (p − 2)u2 Di uDj u dx +



  p−2 c(x) |u|2 + ε 2 |u|2 dx.

Ω

By putting uDj u = αj +iβj , 1 ≤ j ≤ n, we can repeat the same argument as before. Then, the desired estimates (2.62) and (2.63) are obtained.  Let Re λ ≤ 0. For the solution u to the equation (A−λ)u = f , where f ∈ L2 (Ω), we verify the following proposition. Proposition 2.2 Let 1 < p < 2, and let Re λ ≤ 0. If (A − λ)u = f with f ∈ L2 (Ω), then u ∈ H 1 (Ω) satisfies the estimates (| Re λ| + c0 )uLp ≤ f Lp ,   Cnp + 2 f Lp |λ|uLp ≤ √ 2δ p − 1

(2.64) (2.65)

with C = maxi,j aij L∞ , where δ and c0 are the constants in (2.19) and (2.20).

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2 Sectorial Operators

Proof We consider the real part of the inner product (A − λ)u = f and u∗ε in L2 (Ω). Then, since Re(Au, u∗ε ) ≥ 0 and (u, u∗ε ) ≥ 0, it follows that       Re Au, u∗ε + | Re λ| u, u∗ε ≤ f Lp u∗ε L . (2.66) q

By (2.62), 



|u|2 + ε

(c0 + | Re λ|)

 p−2 2

Ω

p−1

|u|2 dx ≤ f Lp uLp .

We here used the fact that u∗ε Lq ≤ uLp . Hence, letting ε → 0, we deduce by the Lebesgue dominate convergence theorem the desired estimate (2.64). Estimate (2.65) is obtained in a similar way from          |λ| u, u∗ε ≤ Re Au, u∗ε + Im Au, u∗ε  +  f, u∗ε . p−1

Indeed, by (2.63) and (2.66),    2  p−2 2 2 |λ| |u| + ε |u| dx ≤ Ω

   Cnp + 2 f Lp u∗ε L . √ q 2δ p − 1

Hence, letting ε → 0, (2.65) is also deduced.



We are now ready to define from A a linear operator Ap for each 1 < p < ∞. Let us first consider the case where 2 < p < ∞. In this case, the operator Ap is defined as the part of A in the space Lp (Ω). As Proposition 2.1 (λ = 0) shows that Au ∈ Lp (Ω) implies u ∈ Lp (Ω), we have in fact

D(Ap ) = {u ∈ Z; Au ∈ Lp (Ω)}, Ap u = Au. Proposition 2.1 shows also that, for every Re λ ≤ 0, the resolvent (λ − Ap )−1 exists and that the following estimates   (λ − Ap )−1  ≤ 1/(| Re λ| + c0 ), Re λ ≤ 0, (2.67) Lp      Cnp (λ − Ap )−1  ≤ + 2 |λ|, Re λ ≤ 0, λ = 0 (2.68) √ Lp 2δ p − 1 hold due to (2.59) and (2.60). Consider next the case where 1 < p < 2. Proposition 2.2 shows that, for every Re λ ≤ 0, the resolvent (λ − A)−1 acting on L2 (Ω) is continuous with respect to the Lp (Ω) norm. This means that the resolvent can be extended on Lp (Ω) as a bounded operator. Denote the extended operator by Rp (λ) for Re λ ≤ 0. Then the family of operators Rp (λ) defines a pseudo-resolvent in L(Lp (Ω)). According to Theorem 1.8, there exists a multivalued linear operator Ap of Lp (Ω) such that

2 Sectorial Operators in Banach Spaces

77

Rp (λ) = (λ − Ap )−1 for every Re λ ≤ 0. In this way, A−1 p is a continuous extension on L (Ω). Therefore, D(A ) ⊂ D(A ) and f = A2 u implies that f ∈ Ap u; of A−1 p 2 p 2 in particular, Ap is densely defined in Lp (Ω) (Indeed, D(A2 ) is dense in L2 (Ω), a fortiori in Lp (Ω)). Moreover, it is readily verified that u ∈ D(Ap ) and f ∈ Ap u if and only if there exist sequences un ∈ D(A2 ) and fn = A2 un such that un → u and fn → f in Lp (Ω). Finally, (2.64) and (2.65) yield that Ap also satisfies the same estimates as (2.67) and (2.68). To conclude for 1 < p < ∞ that Ap is a sectorial operator of Lp (Ω), it therefore remains to verify that Ap is densely defined in Lp (Ω) in the case where 2 < p < ∞ and that Ap is single valued in the case where 1 < p < 2. For these purposes, we notice the following proposition. Proposition 2.3 Let 1 < p < ∞ and f ∈ Lp (Ω). As λ → −∞, λ(λ − Ap )−1 f converges weakly to f in Lp (Ω). Proof When p = 2, the result is already known. Indeed, λ(λ − A2 )−1 f − f = A2 (λ − A2 )−1 f ; therefore, if f ∈ D(A2 ), then λ(λ − A2 )−1 f − f → 0 in L2 (Ω). By the denseness of D(A2 ) in L2 (Ω), λ(λ − A2 )−1 indeed converges strongly to the identity operator on L2 (Ω). Let 2 < p < ∞. Since the norms λ(λ − Ap )−1 f Lp are uniformly bounded for all λ ≤ 0, we can extract a sequence λn → −∞ such that λn (λn − Ap )−1 f → g weakly in Lp (Ω). However, since the sequence is strongly convergent to f in L2 (Ω), it must hold that g = f . Therefore, without extracting a sequence for λ, λ(λ − Ap )−1 f is weakly convergent to f in Lp (Ω). Let 1 < p < 2. If f ∈ L2 (Ω), then λ(λ − Ap )−1 f is strongly convergent to f in Lp (Ω). Then, by the uniform boundedness of λ(λ − Ap )−1 Lp for λ ≤ 0 and the denseness of L2 (Ω) in Lp (Ω), we conclude that λ(λ − Ap )−1 f → f in Lp (Ω) as λ → −∞.  We can now state the final result and complete its proof. Theorem 2.12 Let Ω be a bounded domain in Rn . Let (2.18), (2.19), and (2.20) be satisfied, and let A be an operator associated with the form (2.56) on Z = H˚ 1 (Ω) or H 1 (Ω). Then, for any 1 < p < ∞, the operator Ap defined from A in the way √ + 2, where specified above satisfies (2.1) and (2.2) with ω = π2 and constant 2δCnp p−1 C = maxi,j aij L∞ , δ, and c0 being the constants in (2.19) and (2.20). Therefore, Ap is a sectorial operator of Lp (Ω) with angle ωAp < π2 . Proof Let 2 < p < ∞. Since R((λ − Ap )−1 ) ⊂ D(Ap ), Proposition 2.3 shows that D(Ap ) is dense in the weak topology of Lp (Ω). But since D(Ap ) is a linear subspace, its denseness is valid even in the usual topology of Lp (Ω). Let 1 < p < 2. Assume that Ap 0  f ; then, (λ − Ap )0  f for any λ ≤ 0; that is, 0 = (λ − Ap )−1 f ; then Proposition 2.3 yields that f = 0. Therefore, Ap is a single-valued operator. 

78

2 Sectorial Operators

The operator Ap is called a realization of the elliptic operator (2.21) in Lp (Ω), 1 < p < ∞, under the Dirichlet- or Neumann-type boundary conditions ((2.22) or (2.23)) if Z = H˚ 1 (Ω) or H 1 (Ω), respectively. Remark 2.4 We notice that the resolvent of Ap satisfies the optimal estimate along the negative real axis (−∞, 0]. In fact, (2.67) shows that   (λ − Ap )−1  ≤ 1/(|λ| + c0 ), λ ≤ 0, (2.69) L p

holds for every 1 < p < ∞.

2.2 Extreme Cases p = 1 and p = ∞ In this subsection, we consider the extreme cases p = 1 and p = ∞. It is possible to define analogously sectorial operators A1 and A∞ acting in the spaces L1 (Ω) and L∞ (Ω), respectively. However, (2.60) and (2.65) have no more meaning in these spaces. Let Ω be a bounded domain in Rn . Let A be a linear operator associated with the form (2.56) on Z = H˚ 1 (Ω) or H 1 (Ω). Conditions (2.18), (2.19), and (2.20) are assumed as before. First, let us consider the case p = ∞. For Re λ ≤ 0, consider the equation (A − λ)u = f , where f ∈ L∞ (Ω), and let u ∈ H 1 (Ω) be its solution. By 1

(2.59), we see that c0 uLp ≤ f Lp ≤ |Ω| p f L∞ for any finite p > 2. Using the fact that uL∞ = limp→∞ uLp , we can conclude that u ∈ L∞ (Ω). Furthermore, due to (2.59), (| Re λ| + c0 )uL∞ ≤ f L∞ .

(2.70)

The operator A∞ is then defined as the part of A in the space L∞ (Ω). That is, D(A∞ ) = {u ∈ Z; Au ∈ L∞ (Ω)} and A∞ u = Au. Due to (2.70), λ − A∞ has a bounded inverse on L∞ (Ω) with the estimates   (λ − A∞ )−1  ≤ 1/(| Re λ| + c0 ), Re λ ≤ 0. (2.71) L ∞

In this way, we have observed that A∞ satisfies (2.1) and (2.2) with any ω > π2 . However, it is no longer possible to verify that D(A∞ ) is dense in L∞ (Ω), because Proposition 2.3 cannot be verified for p = ∞. We observe that λ(λ − A∞ )−1 f → f in L∞ (Ω) as λ → −∞ for f ∈ D(A∞ ) only. So, we have to consider again a part of A∞ in such a way that D(A˜ ∞ ) = {u ∈ D(A∞ ); A∞ u ∈ D(A∞ )} and A˜ ∞ u = A∞ u, where the closure is taken in L∞ (Ω). Then, the operator A˜ ∞ is densely defined in D(A∞ ). In fact, for any u ∈ D(A∞ ), uλ = λ(λ − A∞ )−1 u ∈ D(A˜ ∞ ) for λ < 0, and uλ → u in L∞ (Ω) as λ → −∞. Hence, D(A˜ ∞ ) is dense in D(A∞ ). Of course, A˜ ∞ also satisfies the same condition as (2.71). Secondly, let us consider the case p = 1. For Re λ ≤ 0, consider the equation (A − λ)u = f with f ∈ L2 (Ω). It is verified from (2.64) that (| Re λ| + c0 )uL1 ≤ f L1 .

(2.72)

2 Sectorial Operators in Banach Spaces

79

Here, we used the fact that uL1 = limp→1 uLp . This means that the resolvent (λ − A2 )−1 is continuous with respect to the L1 topology. As a consequence, each operator (λ − A2 )−1 is extended to a bounded operator R1 (λ) on L1 (Ω), and the family R1 (λ), Re λ ≤ 0, defines a pseudo-resolvent in L(L1 (Ω)). By a multivalued linear operator A1 of L1 (Ω), we have (λ − A1 )−1 = R1 (λ). Since D(A2 ) ⊂ D(A1 ), A1 is densely defined. Furthermore, as Proposition 2.3 is valid for p = 1, too, A1 is a single-valued linear operator. Finally, it is clear from (2.72) that   (λ − A1 )−1 

L1

≤ 1/(| Re λ| + c0 ),

Re λ ≤ 0.

(2.73)

In the cases where p = 1 and p = ∞, we have therefore obtained the following results. Theorem 2.13 Let Ω be a bounded domain in Rn . Let (2.18), (2.19), and (2.20) be satisfied, and let A be an operator associated with the form (2.56) on Z, where Z = H˚ 1 (Ω) or H 1 (Ω). Then, the linear operators A1 and A˜ ∞ defined from A as above satisfy (2.1) and (2.2) in L1 (Ω) and D(A∞ ) ⊂ L∞ (Ω), respectively, with any angle ω > π2 and constant M = | cos1 ω| , and are sectorial operators of L1 (Ω) and D(A∞ ), respectively, with angles ωA1 ≤

π 2

and ωA˜ ∞ ≤ π2 .

2.3 Adjoint Operators We return to the study for the case where 1 < p < ∞. In Sect. 2.1 above, we defined the realization Ap of strongly elliptic operator (2.27) in Lp (Ω) under the Dirichletor Neumann type-boundary conditions. We are then concerned with the adjoint operator A∗p with respect to the adjoint pair {Lp (Ω), Lp (Ω)}, where p1 + p1 = 1. Let Ω be a bounded domain in Rn . Under conditions (2.18), (2.19), and (2.20), consider the sesquilinear form (2.56) on Z, where Z = H˚ 1 (Ω) or H 1 (Ω). Let A be the operator associated with the from. Furthermore, let Ap be the sectorial operator of Lp (Ω) determined from A, where 1 < p < ∞. We consider an adjoint pair {Lp (Ω), Lp (Ω)}, where p1 + p1 = 1, and denote by A∗p the adjoint of A∗p with respect to this adjoint pair. In order to characterize A∗p , we use the adjoint form whose notion was already presented in Chap. 1, Sect. 8.2. Actually, the adjoint form to (2.56) is given by a ∗ (u, v) = a(v, u) =

n   i,j =1 Ω

 aj i (x)Di uDj v dx +

c(x)uv dx,

u, v ∈ Z,

Ω

(2.74) on Z. Let A∗ be the operator associated with this form. It is of course possible to determine a sectorial operator Bp of Lp (Ω) from A∗ , where 1 < p  < ∞. Our goal is to show that A∗p = Bp if p1 + p1 = 1.

80

2 Sectorial Operators

Theorem 2.14 Let Ω be a bounded domain in Rn , and let (2.18), (2.19), and (2.20) be satisfied. For 1 < p, p  < ∞, let Ap and Bp be sectorial operators of Lp (Ω) and Lp (Ω) determined from the operators associated with the form (2.56) and the adjoint form (2.74), respectively, on Z, where Z = H˚ 1 (Ω) or H 1 (Ω). Then, A∗p = Bp if p1 + p1 = 1. Proof When p = 2, L2 (Ω) is a Hilbert space. So, A∗2 = B2 is observed directly by applying Theorem 1.25 to the form (2.56) and its adjoint form (2.74). Let 2 < p < ∞ and p1 + p1 = 1. Let u ∈ D(Ap ), and let v ∈ D(Bp ). Since D(Ap ) ⊂ D(A2 ), we have Ap u = A2 u. In the meantime, we recall that Bp v = g if and only if there exist sequences vn ∈ D(B2 ) and gn = B2 vn such that vn → v and gn → g in Lp (Ω). Then, since Ap u, vn Lp ×Lp = (A2 u, vn )L2 = (u, B2 vn )L2 = u, Bp vn Lp ×Lp , it follows that Ap u, v Lp ×Lp = u, Bp v Lp ×Lp ,

u ∈ D(Ap ), v ∈ D(Bp ).

This then shows that, if v ∈ D(Bp ), then v ∈ D(A∗p ) with A∗p v = Bp v. On the other hand, we know that both Bp and A∗p have their bounded inverses on Lp (Ω). Hence, it must hold that D(A∗p ) = D(Bp ) and A∗p = Bp . Let next 1 < p < 2 and p1 + p1 = 1. Since 2 < p  < ∞, we can apply the result just obtained to Bp to conclude that Bp∗ = Ap . By virtue of Theorem 1.19, it then follows that A∗p = Bp∗∗ = Bp .  As a consequence of Theorem 2.14, we know that, if aij (x) are symmetric in the sense of (2.35), then A∗2 = B2 = A2 . Therefore, A2 = A is a positive definite selfadjoint operator of L2 (Ω). Furthermore, A has a domain D(A) ⊂ Z (= H˚ 1 (Ω) or H 1 (Ω)). By Theorem 1.38, A−1 is a compact operator of L2 (Ω). Then, the theory of spectral resolutions (see [Yos80, Sect. XI.9]) provides that A possesses an infinite number of positive eigenvalues 0 < λ1 < λ2 < · · · and is written by 

∞

A= 0

λ dE(λ) =

∞ 

λk Pk ,

k=1

where Pk = E(λk ) − E(λk − 0) denotes the orthogonal projection to the eigenspace Nk corresponding to the eigenvalue λk . Each Nk is a finite-dimensional subspace of L2 (Ω), and L2 (Ω) is the direct sum of Nk . Since ai,j (x) and c(x) are real-valued functions, we see, for each k, that u ∈ Nk implies u ∈ Nk , that is, u ∈ Nk implies Re u ∈ Nk and Im u ∈ Nk . This means that Nk can be generated by real functions alone. By the usual manner, we can then construct an orthonormal basis of Nk which is composed by real-valued functions alone. Corollary 2.1 Let Ω be a bounded domain in Rn . Let (2.18), (2.19), (2.20), and (2.35) be satisfied. Then, A2 is a positive definite self-adjoint operator of L2 (Ω).

2 Sectorial Operators in Banach Spaces

81

There exists an orthonormal basis of L2 (Ω) which is composed by real-valued eigenfunctions of A2 .

2.4 Shift Property in Lp Spaces For 1 < p < ∞, let Ap be the realization of strongly elliptic operator (2.27) in Lp (Ω) under the Dirichlet- or Neumann-type boundary conditions. In the case where p = 2, we already know that regularity of ∂Ω and aij (x) imply the shift property D(A2 ) ⊂ H 2 (Ω). This subsection is devoted to remarking that a similar property is valid for the general Ap , 1 < p < ∞, too. Let Ω be a bounded domain in Rn with C2 boundary. Let (2.19), (2.20), and (2.28) be satisfied. Let A(D) be the elliptic operator given by (2.27). Let us first consider the realization Ap which is equipped with the Dirichlet boundary conditions. According to Grisvard [Gri85, Theorem 2.4.1.3], the mapping u → (A(D) + k)u is an isomorphism from the space {u ∈ Hp2 (Ω); γ u = 0} onto Lp (Ω) if k is sufficiently large. For u ∈ D(Ap ), put Ap u = f ∈ Lp (Ω); then, there exists a function u˜ ∈ Hp2 (Ω) satisfying γ u˜ = 0 such that (A(D) + k)u˜ = f + ku, i.e., (Ap + k)u˜ = f + ku. Therefore, u˜ = u. In this way, we verify that

D(Ap ) = {u ∈ Hp2 (Ω); γ u = 0 on ∂Ω}, (2.75) Ap u = A(D)u, with uHp2 ≤ Cp (Ap uLp + uLp ),

u ∈ D(Ap ).

(2.76)

It is quite similar for the realization Ap equipped with the Neumann-type boundary conditions. According to Grisvard [Gri85, Theorem 2.4.1.3], the mapping u → ∂u = 0} onto Lp (Ω) (A(D) + k)u is an isomorphism from the space {u ∈ Hp2 (Ω); ∂ν A if k is sufficiently large. Then, by the same reason, we verify that

∂u = 0 on ∂Ω}, D(Ap ) = {u ∈ Hp2 (Ω); ∂ν A (2.77) Ap u = A(D)u, with uHp2 ≤ Cp (Ap uLp + uLp ),

u ∈ D(Ap ).

(2.78)

Theorem 2.15 Let Ω be a bounded domain with C2 boundary. Let (2.19), (2.20), and (2.28) be satisfied. Then, for each 1 < p < ∞, the realizations Ap of A(D) in Lp (Ω) under the Dirichlet- and Neumann-type boundary conditions are characterized by (2.75) and (2.77), respectively. Remark 2.5 A similar shift property is verified even in the extreme cases p = 1 and p = ∞. But the optimal shift property (i.e., D(A1 ) ⊂ H12 (Ω) and D(A∞ ) ⊂ 2 (Ω)) fails except the one-dimensional case. For characterization of these operH∞ ators, we refer the reader to Tanabe [Tan97, Chap. 5].

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2 Sectorial Operators

3 Sectorial Operators in Product Spaces Let X1 and X2 be Banach spaces with norms  · 1 and  · 2 , respectively, and let X be the product Banach space of X1 and X2 with norm F  = f1 1 + f2 2 , F = t (f , f ) ∈ X. 1 2 Let A1 and A2 be sectorial operators of X1 and X2 , respectively, with angles 0 ≤ ω1 < π and 0 ≤ ω2 < π . We consider a matrix operator of the form  A1 B , 0 A2    u1 ; u1 ∈ D(A1 ), u2 ∈ D(A2 ) ∩ D(B) , D(A) = u2 

A=

(2.79)

where B is a third linear operator from D(B) ⊂ X2 into X1 . If B is a bounded operator from D(A2 ) into X1 , then A is a sectorial operator of X. Theorem 2.16 Let Ak be two sectorial operators of Xk with angles 0 ≤ ωk < π , respectively, where k = 1, 2. Let B ∈ L(D(A2 ), X1 ). Then, the matrix operator A defined by (2.79) satisfies (2.1) and (2.2) for any ω such that ωA < ω ≤ π with constant Mω = max{M1,ω , M2,ω + M1,ω (M2,ω + 1)BA−1 2 }, where ωA = max{ω1 , ω2 }. Therefore, A is a sectorial operator of X with angle ≤ ωA . Proof The equality     f u (λ − A) 1 = 1 u2 f2 is obviously equivalent to

(λ − A1 )u1 − Bu2 = f1 , (λ − A2 )u2 = f2 .

/ Σω , then λ ∈ ρ(A), and the resolvent is written by Let ωA < ω < π . If λ ∈ −1

(λ − A)



(λ − A1 )−1 = 0

 −1 (λ − A1 )−1 BA−1 2 A2 (λ − A2 ) . (λ − A2 )−1

The resolvent is estimated by      (λ − A)−1  ≤ max M1,ω , M2,ω + M1,ω (M2,ω + 1)BA−1  |λ|−1 , 2

This shows that the assertion of theorem is true.

λ∈ / Σω . 

4 Yosida Approximation

83

4 Yosida Approximation 4.1 Uniform Estimates Let A be a sectorial operator of X with angle ωA . We define the sequence An of bounded linear operators on X by An = A(1 + n−1 A)−1 = n − n2 (n + A)−1 ,

n = 1, 2, 3, . . . .

(2.80)

This sequence {An }n=1,2,3,... is called the Yosida approximation for A. It is observed by some calculations that     nλ λ − An = n2 − (n − λ)(n + A) (n + A)−1 = (n − λ) − A (n + A)−1 . n−λ Therefore, if

nλ n−λ

∈ ρ(A), then λ ∈ ρ(An ), and the following representation is valid: −1

(λ − An )

 −1 nλ 1 (n + A) −A = n−λ n−λ  2  −1 n nλ 1 + −A = . λ−n n−λ n−λ

(2.81) (2.82)

Let ωA < ω < π . By definition, C − Σω ⊂ ρ(A). Note that λ ∈ C − Σω implies ∈ C − Σω . Indeed, consider the transform

nλ n−λ

λ →

nλ = n−λ

1 1 λ

−

1 n

(2.83)

in C which is composed with two elementary transforms λ → λ−1 and λ → λ − n1 . nλ These two transforms map C − Σω into itself. Consequently, the transform λ → n−λ also maps C − Σω into itself. In view of (2.81), we have C − Σω ⊂ ρ(An ), i.e., σ (An ) ⊂ Σω ,

n = 1, 2, 3, . . . .

(2.84)

Let us estimate next (λ − An )−1 . By the cosine theorem, we have |λ − n|2 = |λ|2 + n2 − 2|λ|n cos(arg λ) ≥ |λ|2 + n2 − 2|λ|n cos ω

(1 − cos ω)(|λ|2 + n2 ) if 0 < ω ≤ π2 , λ ∈ / Σω , ≥ π 2 2 if 2 < ω < π, λ ∈ / Σω . |λ| + n

(2.85)

We then obtain from (2.82) that ˜   (λ − An )−1  ≤ Mω , |λ|

λ∈ / Σω , n = 1, 2, 3, . . . ,

with some constant M˜ ω ≥ Mω independent of n.

(2.86)

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2 Sectorial Operators

We have thus verified that the Yosida approximation An satisfies (2.1) and (2.2) for any ω such that ωA < ω < π with some constant M˜ ω independent of n. In particular, An are sectorial operators of X with angles ωAn ≤ ωA .

4.2 Convergence As remarked in Chap. 1, Sect. 3.4, (λ − A)−1 is an L(X)-valued analytic function for λ ∈ ρ(A). It then follows from (2.82) that, for each λ ∈ / Σω , as n → ∞, (λ − An )−1 → (λ − A)−1

in L(X).

(2.87)

Let us see the convergence of An itself. To this end, we introduce the sequence of bounded operators Jn = (1 + n−1 A)−1 = 1 − A(n + A)−1 ,

n = 1, 2, 3, . . . .

(2.88)

It is clear that Jn  ≤ M. Meanwhile, (1 − Jn )u = (n + A)−1 Au → 0 for u ∈ D(A). The denseness of D(A) in X then yields that Jn converges to 1 strongly on X

(2.89)

(see Chap. 1, Sect. 3). We therefore observe for U ∈ D(A) that, as n → ∞, An U = Jn AU → AU

in X.

(2.90)

Let k be any positive integer. Then, as n → ∞, Jnk also converges to 1 strongly on X. On the other hand, R(Jnk ) is contained in D(Ak ). This then shows that D(Ak ) is dense in X for any integer k = 1, 2, 3, . . . .

(2.91)

5 Exponential Functions In this section, we consider a sectorial operator A of a Banach space X with angle 0 ≤ ωA < π2 . By ω, we denote an angle such that ωA < ω < π2 . By definition, σ (A) ⊂ Σω = {λ ∈ C; | arg λ| < ω},

ωA < ω <

π , 2

(2.92)

and   (λ − A)−1  ≤ Mω , |λ|

λ∈ / Σω , ωA < ω <

π . 2

(2.93)

5.1 Analytic Semigroups We define the family of bounded linear operators e−tA on X by the Dunford integral

5 Exponential Functions

85

e−tA =

1 2πi



e−tλ (λ − A)−1 dλ,

0 < t < ∞,

(2.94)

Γ

in the space L(X). The integral contour Γ is an infinite curve lying in ρ(A) which surrounds σ (A) counterclockwise. For example, we can take Γ = Γ− ∪ Γ+ , where Γ± : λ = re±iω , 0 ≤ r < ∞, which is oriented from ∞eiω to 0 and from 0 to ∞e−iω (note that 0 ∈ ρ(A)). Since |e−tλ | = e−t Re λ = e−t|λ| cos ω ,

λ ∈ Γ± ,

the integral along Γ is certainly convergent in L(X). Let 0 < t, t  < ∞. Then, e

−tA −t  A

e

 =

1 2πi

2  

 

Γ

Γ

e−tλ e−t λ (λ − A)−1 (λ − A)−1 dλ dλ,

where Γ  = Γ − 1 is another integral contour obtained by shifting Γ to the left. By virtue of the resolvent equation (1.12), e

−tA −t  A

e

 =

1 2πi

2   Γ

 

Γ

 e−tλ e−t λ  (λ − A)−1 − (λ − A)−1 dλ dλ. λ − λ

We here observe by the Cauchy integral formula that 1 2πi 1 2πi

 

 

Γ

e−t λ 1 dλ = λ − λ 2πi

Γ

e−tλ dλ = 0. λ − λ



 

|λ −λ|=1

e−t λ  dλ = e−t λ , λ − λ

Therefore, e

−tA −t  A

e

1 = 2πi







e−(t+t )λ (λ − A)−1 dλ = e−(t+t )A . Γ

We have thus shown the exponential law 





e−tA e−t A = e−t A e−tA = e−(t+t )A ,

0 < t, t  < ∞.

In this sense, the family of operators e−tA is called the exponential function generated by −A. Let us next observe that e−tA is extended as an L(X)-valued analytic function defined in some sectorial domain containing the real half-axis (0, ∞). In fact, if | arg z| < π2 − ω, then |e−zλ | = e− Re(zλ) ≤ e−|z||λ| cos(|arg z|+ω) ,

λ ∈ Γ± ,

86

2 Sectorial Operators

where Γ± is the same contour presented as above. Therefore, the integral  1 e−zλ (λ − A)−1 dλ, z ∈ Σ π2 −ω , e−zA = 2πi Γ

(2.95)

is convergent in L(X). In this way, e−tA is extended to the function e−zA in the sectorial domain Σ π2 −ω . In addition, it is verified that e−zA is differentiable in z and hence is an analytic function of z. We thus have the following result. Proposition 2.4 For any ωA < ω < π2 , e−zA is an L(X)-valued analytic function in Σ π2 −ω . In addition, its derivatives are given by 1 d k e−zA = (−A)k e−zA = k dz 2πi



(−λ)k e−zλ (λ − A)−1 dλ, Γ

z ∈ Σ π2 −ω , k = 0, 1, 2, . . . .

(2.96)

Proof Let us prove (2.96) by induction. When k = 0, (2.96) is trivial from definition. d k −zA Assume that (2.96) holds for k. By a direct calculation, it is verified that dx is ke differentiable in z, and the derivative is given by  d k+1 e−zA 1 = (−λ)k+1 e−zλ (λ − A)−1 dλ. 2πi Γ dzk+1 We here consider the product (−A)−1

d k+1 e−zA 1 =− k+1 2πi dz



(−λ)k+1 e−zλ A−1 (λ − A)−1 dλ. Γ

By (1.12), it follows that (−A)−1

d k+1 e−zA 1 = k+1 2πi dz





  d k e−zA (−λ)k e−zλ (λ − A)−1 + A−1 dλ = , dzk Γ

1 k −zλ dλ = 0 by the Cauchy integral theorem. This 2πi Γ (−λ) e k d −zA ) ⊂ D(A) and −A d k e−zA = d k+1 e−zA . Hence, (2.96) is R( dx ke dx k dx k+1

since

k + 1, too.

shows that proved for 

The norm of e−zA is estimated as follows. Proposition 2.5 For any φ such that 0 < φ < δφ > 0 and a constant Cφ > 0 such that e−zA  ≤ Cφ e−δφ |z| ,

π 2

− ω, there exist a positive exponent

z ∈ Σφ − {0}.

(2.97)

Proof First, let us consider variables such that 0 < |z| ≤ 1 and | arg z| ≤ φ. We shift the integral contour in (2.95) to C1 ∪ Γ1 , where C1 : λ = |z|−1 e−iϕ ,

ω ≤ ϕ ≤ 2π − ω,

and

Γ1 : λ = ρe±iω ,

|z|−1 ≤ ρ < ∞.

5 Exponential Functions

87

Then, since |e−zλ | ≤ e for λ ∈ C1 ,      −zλ −1  e (λ − A) dλ  ≤ C1

2π−ω

ω

eMω −1 |z| dϕ ≤ 2πeMω . |z|−1

Since |e−zλ | ≤ e−|z||λ| cos(φ+ω) for λ ∈ Γ1 ,    ∞  ∞   dρ −zλ −1 −ρ|z| cos(φ+ω) dρ   ≤ Mω = M . e (λ−A) dλ e e−ρ cos(φ+ω) ω   −1 ρ ρ |z| 1 Γ1 Therefore, e−zA  ≤ Cφ if 0 < |z| ≤ 1 and | arg z| ≤ φ. Next, let us consider variables such that |z| ≥ 1 and | arg z| ≤ φ. Let 0 < δ < A−1 −1 . On account of (2.3), we can shift the integral contour to C2 ∪ Γ2 , where C2 : λ = δeiϕ ,

and Γ2 : λ = ρe±iω ,

−ω ≤ ϕ ≤ ω,

δ ≤ ρ < ∞.

Then, since |e−zλ | ≤ e−δ|z| cos(φ+ω) for λ ∈ C2 ,    ω   −zλ −1   ≤ Mω e (λ − A) dλ e−δ|z| cos(φ+ω) dϕ ≤ 2ωMω e−δ|z| cos(φ+ω) .   C2

−ω

In addition,      −zλ −1   e (λ − A) dλ ≤ Mω  Γ2

∞

e−ρ|z| cos(φ+ω) dρ =

δ

Mω e−δ|z| cos(φ+ω) . |z| cos(φ + ω)

Therefore, e−zA  ≤ Cφ e−δ|z| cos(φ+ω) if |z| ≥ 1 and | arg z| ≤ φ. We have thus obtained the desired estimate (2.97).



We are now concerned with the limit of e−zA as z → 0. Proposition 2.6 For any φ such that 0 < φ < on X as z → 0 with z ∈ Σφ − {0}.

π 2

− ω, e−zA converges to 1 strongly

Proof First, let U ∈ D(A). Then, by the same argument as in the proof of Proposition 2.4,  1 −zA e U= e−zλ (λ − A)−1 (1 + A)−1 dλ(1 + A)U 2πi Γ   1 e−zλ  (λ − A)−1 + (1 + A)−1 dλ(1 + A)U = 2πi Γ λ + 1  1 e−zλ = (λ − A)−1 dλ(1 + A)U. 2πi Γ λ + 1 Then, by the Lebesgue dominated convergence theorem, we see that

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2 Sectorial Operators

e−zA U →

1 2πi



1 =− 2πi

Γ



1 (λ − A)−1 dλ(1 + A)U λ+1

|λ+1|=1

1 (λ − A)−1 dλ(1 + A)U = U λ+1

in X.

Next, let F ∈ X be a general vector. The denseness of D(A) in X and uniform boundedness of the norms e−zA  verified by (2.97) readily yield that e−zA F → F in X.  In view of this convergence, we are naturally led to define the value of e−zA at z = 0 by e−0A = 1 (the identity mapping on X). We finally show the differentiability of e−zA U at z = 0 for U ∈ D(A). Proposition 2.7 Let U ∈ D(A). For any φ such that 0 < φ < z−1 [e−zA − 1]U converges to −AU in X as z → 0 with z ∈ Σφ − {0}.

π 2

− ω,

Proof Let z ∈ Σφ − {0} and ε > 0. Then, [e−zA − e−εA ]U = (ε − z)



1

e−[θz+(1−θ)ε]A AU dθ.

0

Letting ε → 0, we have [e−zA − 1]U = −z z−1 [e−zA − 1]U → −AU .

1 0

e−θzA AU dθ . Therefore, as z → 0, 

In this subsection, we have proved for a sectorial operator A satisfying (2.92) and (2.93) that its exponential function e−zA is an analytic function in Σ π2 −ω enjoying the properties described in Propositions 2.4, 2.5, 2.6, and 2.7. It is also verified that e−zA satisfies the semigroup property 



e−zA e−z A = e−(z+z )A ,

z, z ∈ Σ π2 −ω .

(2.98)

Such an analytic function is called an analytic semigroup on X generated by −A. The precise definitions of analytic semigroups and their generators will be presented in Chap. 3.

5.2 Contraction Semigroups Let A be a sectorial operator of X satisfying (2.92) and (2.93). We assume that A satisfies the optimal estimate on the negative real axis, i.e.,   (λ − A)−1  ≤ 1/|λ|, λ < 0. (2.99) This condition implies that e−tA  ≤ 1 for all t ≥ 0. Such a semigroup is called a contraction semigroup.

5 Exponential Functions

89

Theorem 2.17 Let A be a sectorial operator of X satisfying (2.92), (2.93), and (2.99). Then, it holds that e−tA  ≤ 1 for any 0 ≤ t < ∞. Proof Let U ∈ D(A). By Proposition 2.7, limh0 h−1 [e−hA − 1]U = −AU . In the meantime, by (1.12), h−1 [(1+hA)−1 −1]U = −(1+hA)−1 AU → −AU as h  0. Therefore, we observe that limh0 h−1 [e−hA − (1 + hA)−1 ]U = 0. Furthermore, we verify from (2.99) that e−hA U  − U  [e−hA − (1 + hA)−1 ]U  ≤ , h h

h > 0.

Hence, lim suph0 h−1 [e−hA U  − U ] ≤ 0. By the same argument, we verify that D + e−tA U  ≤ 0 for every t ≥ 0, where D + denotes Dini’s derivative. We then have e−tA U  ≤ U . Indeed, suppose the contrary; then there exists some τ > 0 such that e−τ A U  > U . Let α = (e−τ A U  − U )/τ , and let the continuous function e−tA U  − αt on [0, τ ] hit its minimum at t0 . Since e−τ A U  − ατ = U , it is possible to assume that 0 ≤ t0 < τ . Then, e−tA U  − e−t0 A U  ≥ α(t − t0 ) for any t0 < t ≤ τ , and we conclude that (D + e−tA U )|t=t0 ≥ α > 0, which is a contradiction. Estimate (2.99) thus implies that, for U ∈ D(A), e−tA U  ≤ U  for every t ≥ 0. But, since D(A) is dense in X, the same estimate is valid for general F ∈ X, too. 

5.3 Adjoint Semigroups Let X be a reflexive Banach space, and let X ∗ be an adjoint space of X. Let {X, X ∗ } be an adjoint pair with duality product ·, · . Let A be a sectorial operator of X satisfying (2.92) and (2.93). So, −A generates an analytic semigroup e−zA on X which is given by (2.95). Let A∗ be the adjoint operator of A with respect to {X, X ∗ }. By Theorem 1.19, ∗ A is a sectorial operator of X ∗ with the same angle ωA as A. Therefore, −A∗ also generates an analytic semigroup. We observe that, for any F ∈ X and any G ∈ X ∗ ,   −zλ  1 e (λ − A)−1 F, G dλ e−zA F, G = 2πi Γ    1 F, e−zλ (λ − A∗ )−1 G dλ. = 2πi Γ Here, let us choose an integral contour as Γω : λ = ρe±iω , 0 ≤ ρ < ∞, where ωA < ω < π2 . If the integral variable λ varies on Γω counterclockwise, namely, from ∞eiω through 0 to ∞e−iω , then the conjugate λ varies on Γω in the opposite direction. Therefore,    1 ∗ e−zA F, G = F, − e−zλ (λ − A∗ )−1 G dλ = F, e−zA G , 2πi −Γω

90

2 Sectorial Operators

i.e., ∗

e−zA = [e−zA ]∗ .

(2.100)

∗

We call the exponential function e−zA the adjoint semigroup of e−zA .

5.4 Approximate Semigroups Let A be a sectorial operator of X satisfying (2.92) and (2.93). Let An be the Yosida approximation of A defined by (2.80). As verified by (2.84) and (2.86), An are also sectorial operators of X with angles ≤ ωA . So, each −An generates an analytic semigroup e−zAn given by the integral (2.95). By (2.97), for any 0 < φ < π2 − ω, e−zAn  ≤ Cφ e−δφ |z| ,

z ∈ Σφ ,

(2.101)

with some exponent δφ > 0 and constant Cφ which are independent of n. The sequence of analytic semigroups e−zAn is then called approximate semigroups of e−zA . As shown by (2.87), as n → ∞, (λ − An )−1 converges to (λ − A)−1 in L(X). Therefore, by the Lebesgue dominate convergence theorem, we verify for any z ∈ Σ π2 −ωA that e−zAn → e−zA

in L(X).

(2.102)

6 Generation of Analytic Semigroups 6.1 Generation Results in Hilbert Spaces Let Z ⊂ X ⊂ Z ∗ be a triplet of spaces. Consider a continuous and coercive sesquilinear form a(U, V ) defined on Z, that is, (2.6) and (2.7) are satisfied. Let A be a linear operator associated with a(U, V ), and let A|X and A|Z be its parts in X and Z, respectively. By Theorem 2.1, we already know that A, A|X , and A|Z are all sectorial operators with angles < π2 . Therefore, they are negative generators of analytic semigroups on Z ∗ , X, and Z, respectively. Summarizing known results for A and its parts, we can state the following generation theorem. Theorem 2.18 Let a(U, V ) satisfy (2.6) and (2.7). Let A be the associated operator, and let A|X and A|Z be its parts. Then, A, A|X , and A|Z satisfy (2.92) and (2.93) δ < ω < π2 with a in Z ∗ , X, and Z, respectively, for any ω such that π2 − sin−1 M+δ (M+δ) sin ω constant δ−(M+δ) cos ω , where M and δ are the constants in (2.6) and (2.7).

6 Generation of Analytic Semigroups

91

Proof By (2.8), (2.9), (2.10), and (2.11), A, A|X , and A|Z satisfy (2.1) and (2.2) in Z ∗ , X, and Z, respectively, with angle π2 and constant M+δ δ . Then the results are obtained by applying (2.4) and (2.5).  We know by (2.12) that the part A|X in X satisfies (2.99). Consequently, Theorem 2.17 yields that |e−tA|X | ≤ 1,

0 ≤ t < ∞.

We also remark that the operator A given by (2.54) and its parts are negative generators of analytic semigroups on the product Hilbert spaces. See Theorem 2.11.

6.2 Generation Results in Lp Spaces Let Ω be a bounded domain in Rn . For 1 < p < ∞, let Ap be the realization of the elliptic operator A(D) given by (2.27) in Lp (Ω) under the Dirichlet- or Neumanntype boundary conditions. By Theorem 2.12, we already know that Ap satisfies (2.1) and (2.2) with angle π2 and constant Mp given by (2.68). If we apply (2.4) and (2.5), then Ap is seen to satisfy (2.92) and (2.93). On the other hand, due to (2.69), Ap also satisfies (2.99). We thus obtain the following generation theorem. Theorem 2.19 Let Ω be a bounded domain in Rn . For 1 < p < ∞, let Ap be the realization of A(D) given by (2.27) in Lp (Ω) under the Dirichlet- and Neumanntype boundary conditions. Then, Ap satisfies (2.92) and (2.93) for any ω such M sin ω that π2 − sin−1 M1p < ω < π2 with constant 1−Mp p cos ω , where Mp is the constant in (2.68). The analytic semigroup e−tAp generated by −Ap satisfies the estimate e−tAp L(Lp ) ≤ 1 for 0 ≤ t < ∞.

For p = 1 and p = ∞, let A1 and A˜ ∞ be the realizations of A(D) in L1 (Ω) and L∞ (Ω), respectively. As seen by Theorem 2.13, there is no guarantee which ensures generation of analytic semigroup for A1 and A˜ ∞ . But (2.71) and (2.73) mean that they satisfy (2.99). As will be mentioned in Sect. 7.8, such a sectorial operator can generate a continuous semigroup. The following result is an immediate consequence of Theorem 2.28 announced below. Theorem 2.20 The realizations A1 and A˜ ∞ of A(D) are negative generators of continuous and contraction semigroups on L1 (Ω) and D(A∞ ), respectively. Remark 2.6 It is known in the regular case (see Sect. 2.4) that −A1 and −A˜ ∞ can generate analytic semigroups. Especially, let AC be a realization of A in the continuous function space C(Ω), then −AC generates an analytic semigroup in C(Ω). For details, see Tanabe [Tan97, Chap. 5] and Taira [Tai03].

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2 Sectorial Operators

7 Fractional Powers of Linear Operators Let X be a Banach space with norm  · . Let A be a sectorial operator of X with angle 0 ≤ ωA < π . We know that, for any integer n ∈ Z, the operator An is defined; indeed, when n > 0, An is a densely defined, closed operator of X (cf. (2.91)); when n < 0, An = (A−1 )−n = (A−n )−1 is a bounded operator of X; and, when n = 0, A0 = 1 (the identity on X). We are then concerned with extending the definition for all real exponents x ∈ R in a natural manner. By ω we denote an angle such that ωA < ω < π . By definition, σ (A) ⊂ Σω = {λ ∈ C; | arg λ| < ω},

ωA < ω < π,

(2.103)

and   (λ − A)−1  ≤ Mω , |λ|

λ∈ / Σω , ωA < ω < π,

(2.104)

with some constant Mω ≥ 1. As noticed in the beginning of the chapter, (2.103) implicitly means that 0 ∈ ρ(A). We may remark again that {λ ∈ C; |λ| ≤ δ} ⊂ ρ(A),

(2.105)

provided that 0 < δ < A−1 −1 .

7.1 Integral Formula We define, for each complex number z such that Re z > 0, the bounded linear operator  1 A−z = λ−z (λ − A)−1 dλ, (2.106) 2πi Γ using the Dunford integral in L(X). As a branch of the analytic function λ−z , we take the principal branch {λ ∈ C; | arg λ| < π} obtained by excluding the negative real axis from the complex plane. So, Γ is an integral contour surrounding σ (A) counterclockwise in C − (∞, 0] ∩ ρ(A). Here, we take Γ = Γ− ∪ Γ0 ∪ Γ+ such that Γ± : λ = ρe±iω ,

δ ≤ ρ < ∞,

and Γ0 : λ = δeiϕ ,

−ω ≤ ϕ ≤ ω,

(2.107)

where ω and δ are as in (2.103), (2.104), and (2.105). In addition, Γ is oriented from ∞eiω to δeiω , from δeiω to δe−iω , and from δe−iω to ∞e−iω . Since |λ−z | = |e−z log λ | = |e−z(log ρ±iω) | = e±(Im z)ω ρ − Re z ,

λ ∈ Γ± ,

we observe that the integral (2.106) is convergent in L(X). When z = n is a positive integer, we can shift the contour from Γ to a circle Cδ : |λ| = δ. Consequently, by

7 Fractional Powers of Linear Operators

93

the Cauchy integral formula,   1 1 λ−n (λ − A)−1 dλ = − λ−n (λ − A)−1 dλ 2πi Γ 2πi Cδ  n−1 d 1 =− (λ − A)−1 (n − 1)! dλn−1 λ=0   n −n −n = (−1) (λ − A) λ=0 = A . This formula shows that the new definition (2.106) is consistent with the former one for any positive integer. By a direct calculation, we can verify that A−z is an analytic function for Re z > 0 with values in L(X). Indeed, we have  1 d −z A =− (log λ)λ−z (λ − A)−1 dλ. dz 2πi Γ However, since the proof is direct, we may omit it. We are now concerned with convergence of A−z as z → 0. Theorem 2.21 For any 0 < φ < to 1 strongly on X.

π 2,

as z → 0 with z ∈ Σφ − {0}, A−z converges

Proof We shall first verify the uniform boundedness of A−z when z varies in the set {z ∈ C; | arg z| ≤ φ, 0 < |z| < 1}. We notice that for such a z, the contour can be shifted from Γ to the reduplicated half lines (∞eiπ , 0] ∪ [0, ∞e−iπ ) by letting ±ω → ±π and δ → 0. From (2.107) it indeed follows that  ∞  πi −z  1 A−z = (ρe ) − (ρe−πi )−z (ρ + A)−1 dρ 2πi 0  e−πiz − eπiz ∞ −z = ρ (ρ + A)−1 dρ 2πi 0  sin(πz) ∞ −z =− ρ (ρ + A)−1 dρ. (2.108) π 0 By (2.104) and (2.105), there exists some constant C ≥ 1 such that   (ρ + A)−1  ≤

C , ρ +1

0 ≤ ρ < ∞.

Therefore, C| sin(πz)| A  ≤ π −z

 0

∞

C| sin(πz)| ρ − Re z dρ = . ρ +1 sin(π Re z)

Noting that (cos φ)|z| ≤ Re z ≤ |z|, we conclude that sup

| arg z|≤φ, 0<|z|<1

A−z  < ∞.

(2.109)

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2 Sectorial Operators

We next verify that A−z converges to 1 strongly on the domain D(A) of A. As will be shown below, A−z satisfies the law of exponent (2.111). In particular, we have A−z A−1 = A−(z+1) . Therefore, since A−z is continuous at z = 1 in L(X), we see that, for any U ∈ D(A), as z → 0, [A−z − 1]U  = [A−z − 1]A−1 AU  ≤ A−z−1 − A−1  AU  → 0. We finally verify the convergence for general F ∈ X. The proof is immediate if we use the denseness of D(A) in X and uniform boundedness (2.109) of A−z just proved above.  Remark 2.7 When A is a positive definite self-adjoint operator in a Hilbert space X, we can define A−z by using alternatively the spectral resolution (see [Yos80, Chap. XI]). Of course, this definition gives us the same operator as (2.106). Indeed, let σ (A) ⊂ [δ, ∞), δ > 0, and let E(μ), μ ≥ δ, be the spectral resolution of A. Then, for Re z > 0,    ∞ 1 1 A−z = λ−x (λ − A)−1 dλ = λ−z (λ − μ)−1 dE(μ) dλ 2πi Γ 2πi Γ δ   ∞  ∞ 1 −z −1 λ (λ − μ) dλ dE(μ) = μ−z dE(μ). (2.110) = 2πi Γ δ δ

7.2 Law of Exponent We shall show that the fractional power A−z of sectorial operator satisfies the law of exponent, i.e., 



A−z A−z = A−(z+z ) ,

Re z > 0, Re z > 0.

(2.111)

By (1.12) we have −z

−z

A A

 =  =

1 2πi 1 2πi

2   Γ

2  



Γ



Γ

Γ

λ−z (λ )−z (λ − A)−1 (λ − A)−1 dλ dλ  λ−z (λ )−z  (λ − A)−1 − (λ − A)−1 dλ dλ,  λ −λ

where Γ is the integral contour given by (2.107), and Γ  is another integral contour obtained by substituting ω and δ  for ω and δ in such a way that ω < ω < π and 0 < δ  < δ, respectively. Then, by the Cauchy integral formula, 1 2πi





Γ

(λ )−z 1 dλ =  λ −λ 2πi





|λ −λ|= 2δ

(λ )−z  dλ = λ−z ,  λ −λ

7 Fractional Powers of Linear Operators

1 2πi

 Γ

95

λ−z dλ = 0. λ − λ

Therefore, we obtain the desired law  1    A−z A−z = λ−(z+z ) (λ − A)−1 dλ = A−(z+z ) . 2πi Γ We have thus verified the following fact. Theorem 2.22 The L(X)-valued function A−z is an analytic semigroup defined in the half-plane {z ∈ C; Re z > 0}. Using this fact, let us next observe that A−z is invertible for every Re z > 0. Indeed, if A−z0 F = 0 for some z0 , then, for any x > 0, A−(z0 +x) F = A−x A−z0 F = 0; since A−z F is analytic, A−z F must vanish identically. Then, by Theorem 2.21, F is necessarily equal to 0. Hence, for Re z > 0, A−z is one-to-one, namely, its inverse Az is a single-valued linear operator of X. The following definition is meaningful: Az = (A−z )−1

for Re z > 0.

If 0 < Re z1 < Re z2 , then A−z2 = A−z1 A−(z2 −z1 ) ; therefore, R(A−z2 ) ⊂ R(A−z1 ); this then means that D(Az2 ) ⊂ D(Az1 ). In particular, if 0 < Re z < k, where k is an integer, then D(Ak ) ⊂ D(Az ); therefore, D(Az ) is dense in X due to (2.91). Hence, Az is a densely defined, closed linear operator of X. In view of Theorem 2.21 it is natural to define A0 = 1. Then, for every real number −∞ < x < ∞, the fractional power Ax of A has been defined. The following properties are already established: (1) Ax are bounded operators on X for −∞ < x < 0, A0 = 1, and Ax are densely defined, closed linear operators of X for 0 < x < ∞. (2) D(Ax2 ) ⊂ D(Ax1 ) for 0 ≤ x1 < x2 < ∞.    (3) Ax Ax = Ax Ax = Ax+x for any −∞ < x, x  < ∞.

7.3 Law of Exponent, II Let α be a fixed exponent such that 0 < α < 1. We shall show that the fractional power Aα of a sectorial operator A is a sectorial operator with angle ≤ αωA and satisfies the second law of exponent, i.e., (Aα )z = Aαz ,

z ∈ C, Re z = 0.

We consider the family of bounded operators defined by  1 1 Rα (λ) = (μ − A)−1 dμ, 2πi Γ λ − μα

(2.112)

(2.113)

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2 Sectorial Operators

where Γ is the same integral contour given in (2.107). Let ω be another angle satisfying ω < ω < π . Then, if λ ∈ / Σαω , then λ = μα for any μ ∈ Γ . So, Rα (λ) is defined for λ ∈ / Σαω . By a direct calculation, Rα (λ) is seen to be an analytic function with values in L(X). Furthermore, by similar techniques as used to verify (2.111), we can verify that  1 1  / Σαω . (μ − A)−1 dμ, λ, λ ∈ Rα (λ)Rα (λ ) = 2πi Γ (λ − μα )(λ − μα )  1  1 1 1 Since (λ−μα )(λ  −μα ) = − λ−λ λ−μα − λ −μα , it follows that Rα (λ)Rα (λ ) = −

1 [Rα (λ) − Rα (λ )], λ − λ

λ, λ ∈ / Σαω ,

that is, Rα (λ) define a pseudo-resolvent. Since 0 − R(0)−1 = Aα , we see by Theorem 1.8 that this pseudo-resolvent is generated by Aα , i.e., Rα (λ) = (λ − Aα )−1 . Hence, we conclude that σ (Aα ) ⊂ Σαω . In order to verify (2.2), however, we need more careful calculations in which the integral contour in (2.113) must be shifted suitably depending on φ = | arg λ|, π−ωA (φ − αωA ) + ωA . Clearly, where αω ≤ φ ≤ π . We introduce the angle ω˜ = π−αω A ω˜ satisfies

π(1−α) (φ − αωA ), π > φ − α ω˜ = π−αω A (2.114) π(1−α) 2π(1 − α) > α(ω˜ − 2π) − (φ − 2π) > π−αω (φ − αωA ). A As a branch of the function μα , we retake the domain {μ; ω˜ − 2π < arg μ < ω}. ˜ Then, in view of the relation {arg μα ; ω˜ − 2π < arg μ < ω} ˜ ⊂ [α(ω˜ − 2π), α ω] ˜ ⊂ (φ − 2π, φ) (due to (2.114)), the equation λ − μα = 0 (λ being fixed) has no solution in this domain. Noting this fact, we then shift the integral contour Γ to the replicated ˜ ). After some calculations, it turns out that half-lines (∞ei ω˜ , 0] ∪ [0, ∞ei(ω−2π) ˜ ω−π)} ˜ sin πα ei{ω+α( π  ∞ ρα × (ρei ω˜ − A)−1 dρ. ˜ (λ − ρ α eiα ω˜ )(λ − ρ α eiα(ω−2π) ) 0

Rα (λ) = −

From this representation we have 

∞

Mω˜ ρ α−1 dρ ˜ |λ − ρ α eiα ω˜ ||λ − ρ α eiα(ω−2π) | 0  ∞ α−1 Mω˜ sin πα ρ dρ ≤ iφ α iα ω ˜ i(φ−2π) ˜ π|λ| |e − ρ e ||e − ρ α eiα(ω−2π) | 0  ∞ α−1 ρ Mω˜ sin πα  = 2α α π|λ| 0 ρ − 2ρ cos(φ − α ω) ˜ +1

Rα (λ) ≤

sin πα π

7 Fractional Powers of Linear Operators

×

97

dρ ρ 2α

− 2ρ α cos[α(ω˜ − 2π) − (φ

− 2π)] + 1

.

Due to (2.114), cos(φ − α ω) ˜ < 1 and cos[α(ω˜ − 2π) − (φ − 2π)] < 1. Therefore, C for λ ∈ / Σαω . we conclude that Rα (λ) ≤ |λ| Theorem 2.23 For 0 < α < 1, Aα is a sectorial operator of X with angle ≤ αωA . Proof To complete the proof, it suffices to notice that ω , together with ω, can be  taken arbitrarily close to ωA . We now know that Aα is a sectorial operator and its resolvent is given by (2.113). Then, verification of (2.112) is immediate. The proof may be left to the reader. Remark 2.8 For λ < 0, we can obtain more precise estimates. Indeed, if φ = ω˜ = π , then  ∞   ρ α−1 dρ (λ − Aα )−1  ≤ Mπ sin πα π|λ| ρ 2α − 2ρ α cos π(1 − α) + 1 0  ∞ dρ Mπ Mπ sin πα = = , λ < 0, 2 2 πα|λ| |λ| (ρ + cos πα) + sin πα 0 (2.115) where Mπ is the constant appearing in (2.2) with ω = π .

7.4 Moment Inequality This subsection is devoted to showing the moment inequality of the fractional powers. For 0 < θ < 1, let Aθ be the fractional powers of a sectorial operator A. Let Mπ π be the constant in (2.2) with ω = π , i.e., (λ − A)−1  ≤ M |λ| for λ < 0. Applying θ−1 (2.108) to A , we have for U ∈ D(A),  sin(πθ ) ∞ θ−1 Aθ U = Aθ−1 AU = − ρ (ρ + A)−1 AU dρ. (2.116) π 0 Here,    

0

R

   ρ θ−1 A(ρ + A)−1 U dρ  ≤

R

  ρ θ−1 A(ρ + A)−1  U  dρ

0



≤ (Mπ + 1) 0

R

ρ θ−1 dρ U  =

Mπ + 1 θ R U , θ

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2 Sectorial Operators

where 0 < R < ∞ is an arbitrary number. Similarly,  ∞   ∞     θ−1 −1  ρ (ρ + A) AU dρ  ρ θ−1 (ρ + A)−1  AU  dρ  ≤ R

R



≤ Mπ

∞

Mπ θ−1 R AU . 1−θ

ρ θ−2 dρAU  =

R

Therefore,

  sin(πθ ) U  θ AU  θ−1 A U  ≤ , (Mπ + 1) R + R π θ 1−θ θ

0 < R < ∞.

The right-hand side hits the minimum at R = AU /U . Hence, Aθ U  ≤

sin(πθ ) (Mπ + 1)AU θ U 1−θ , πθ(1 − θ )

U ∈ D(A).

(2.117)

This inequality is called a moment inequality for Aθ . If we put U = A−1 F , then we have another moment inequality of the form Aθ−1 F  ≤

sin(πθ ) (Mπ + 1)F θ A−1 F 1−θ , πθ(1 − θ )

F ∈ X.

(2.118)

Let 0 ≤ α < β < γ ≤ 1. We know that Aγ −α is also a sectorial operator. So, applying the moment inequality to Aγ −α with θ = γβ−α −α , we obtain that Aβ−α V  ≤

2 (sin π(β−α) γ −α )(γ − α)

π(γ − β)(β − α)

β−α

γ −β

(Mπ + 1)Aγ −α V  γ −α V  γ −α .

Putting V = Aα U , we conclude that Aβ U  ≤

2 (sin π(β−α) γ −α )(γ − α)

π(γ − β)(β − α) β−α

(Mπ + 1) γ −β

× Aγ U  γ −α Aα U  γ −α , Remark 2.9 We remark that the function θ=

1 2

and hence

sin(πθ) πθ(1−θ)

≤

sin(πθ) πθ(1−θ)

U ∈ D(Aγ ).

(2.119)

for 0 < θ < 1 hits its maximum at

4 π.

Remark 2.10 The generalized moment inequality (2.119) can be shown for any triplet of exponents 0 ≤ α < β < γ < ∞.

7.5 Comparison of Domains of Fractional Powers Let A be a sectorial operator of a Banach space X. Let Aθ , 0 < θ < 1, be the fractional powers of A. We are concerned with comparing D(Aθ ) with some other spaces.

7 Fractional Powers of Linear Operators

99

Let us introduce the spaces " !   Dθ (A) = U ∈ X; sup ρ θ A(ρ + A)−1 U  < ∞ ,

0 ≤ θ ≤ 1.

(2.120)

0<ρ<∞

Obviously, Dθ (A) are normed spaces with the norms   U Dθ (A) = sup ρ θ A(ρ + A)−1 U . 0<ρ<∞

Theorem 2.24 For any 0 < θ < 1, D(Aθ ) ⊂ Dθ (A), and the estimate U Dθ (A) ≤ C(Mπ + 1)2 Aθ U ,

U ∈ D(Aθ ),

(2.121)

holds true. Conversely, for any 0 < θ < θ  < 1, Dθ  (A) ⊂ D(Aθ ), and the estimate Aθ U  ≤ Cθ,θ  [U Dθ  (A) + (Mπ + 1)U ],

U ∈ Dθ  (A)

(2.122)

holds true. Proof The first assertion follows from the moment inequality (2.117). Indeed, if U ∈ D(Aθ ), then     A(ρ + A)−1 U  = A1−θ (ρ + A)−1 Aθ U   1−θ  θ ≤ C(Mπ + 1)A(ρ + A)−1 Aθ U  (ρ + A)−1 Aθ U  ≤ C(Mπ + 1)2 ρ −θ Aθ U . Hence, (2.121) is verified. The second assertion is verified from (2.116). For U ∈ Dθ  (A),  ∞ θ−1 −1 sin(πθ ) ρ θ−1 A(ρ + A)−1 U dρ. A U = −A π 0 The integral in the right-hand side is indeed convergent, since  ∞    θ−1 −1   ρ A(ρ + A) U dρ   0



≤ (Mπ + 1)

1



∞

ρ θ−1 dρU  +

0

1



ρ θ−θ −1 dρU Dθ  (A) . 

Hence, U ∈ D(Aθ ), and (2.122) is obtained.

Consider two sectorial operators A and B of X. We assume that D(A) ⊂ D(B) continuously, i.e., BU  ≤ DAU ,

U ∈ D(A).

(2.123)

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2 Sectorial Operators

Theorem 2.25 Let A and B be two sectorial operators of X satisfying (2.103) and  (2.104), and let (2.123) be satisfied. Then, for any 0 < θ < θ  < 1, D(Aθ ) ⊂ D(B θ ), and the estimate 

B θ U  ≤ Cθ  ,θ Aθ U ,



U ∈ D(Aθ ),

holds true, where Cθ,θ  > 0 is determined by θ, θ  , Mπ , and D. Proof Under (2.123), we will prove that Dθ  (A) ⊂ Dθ  (B). In fact, some calculations for resolvent yield the equality   A(ρ + A)−1 − B(ρ + B)−1 = ρ (ρ + B)−1 − (ρ + A)−1 = ρ(ρ + B)−1 [1 − BA−1 ]A(ρ + A)−1 ,

ρ ≥ 0. (2.124)

Therefore,     B(ρ + B)−1 U  ≤ [1 + Mπ (1 + D)]A(ρ + A)−1 U ,

U ∈ X.

Hence, Dθ  (A) ⊂ Dθ  (B) with U Dθ  (B) ≤ [1 + Mπ (1 + D)]U Dθ  (A) ,

U ∈ Dθ  (A). 

This, together with Theorem 2.24, then implies that D(Aθ ) ⊂ Dθ  (A) ⊂ Dθ  (B)  ⊂ D(B θ ). Theorem 2.26 Let A and B be two sectorial operators of X satisfying (2.103) and (2.104), and let (2.123) be satisfied. In addition, assume that there exists some exponent 0 ≤ ν < 1 such that (B − A)U  ≤ N Aν U ,

U ∈ D(A),

(2.125)

with constant N > 0. Then, for any 0 < θ < 1, D(Aθ ) ⊂ D(B θ ), and the estimate B θ U  ≤ C[Aθ U  + U ],

U ∈ D(Aθ ),

holds true, where C > 0 is determined by Mπ , D, N , and ν. Proof By (2.124) and (2.116), we have for U ∈ D(A), [Aθ − B θ ]A−θ U = −

sin(πθ ) π



∞

ρ θ (ρ + B)−1 [A − B](ρ + A)−1 A−θ U dρ.

0

The integral in the right-hand side is estimated as follows. For 0 ≤ ρ ≤ 1,

7 Fractional Powers of Linear Operators

   

0

1

101

  ρ θ (ρ + B)−1 [A − B](ρ + A)−1 A−θ U dρ   

≤ (Mπ + 1)2 (1 + D)

1

ρ θ−1 dρA−θ U .

0

Meanwhile, applying the moment inequality (2.117) (if ν > θ ), we observe that   ∞   θ −1 −ν ν−θ −1   ρ (ρ + B) [A − B]A A (ρ + A) U dρ   1  ∞ ρ θ−1 ρ max{ν−θ−1,−1} dρU . ≤ (Mπ + 1)2 (1 + N ) 1

Therefore, [Aθ − B θ ]A−θ U  ≤ C(A−θ U  + U ),

U ∈ D(A).

The denseness of D(A) in X and closedness of B θ yield that this estimate holds true for any F ∈ X. It then follows that D(B θ ) ⊃ R(A−θ ) = D(Aθ ). 

7.6 Fractional Powers of Adjoint Operators Let X be a reflexive Banach space, and let X ∗ be the adjoint space of X. Let {X, X ∗ } be an adjoint pair. Let A be a sectorial operator of X, and let A∗ be its adjoint operator. By Theorem 1.19, A∗ is seen to be a sectorial operator of X ∗ . So, their fractional powers A−z and A∗(−z) are defined for Re z > 0 by the integral formula (2.106). We can then easily prove that A∗(−z) = A(−z)∗ for Re z > 0 (by the same argument as for (2.100)). Considering these inverses, we prove that A∗z = Az∗ for Re z > 0. In particular, for real powers, we see that A∗x = Ax∗ ,

−∞ < x < ∞.

(2.126)

7.7 Fractional Powers and Analytic Semigroups Let A be a sectorial operator of X satisfying (2.92) and (2.93). As verified in Sect. 5, −A generates an analytic semigroup e−zA on X. By Proposition 2.4, R(e−tA ) ⊂ D(Aθ ) for any t > 0 and any 0 < θ < ∞. In this subsection, we shall investigate various properties concerning Aθ e−tA . We first estimate the norm of Aθ e−tA . Consider the operator defined by  1 θ E (t) = λθ e−tλ (λ − A)−1 dλ, 0 < t < ∞, 0 < θ < ∞, 2πi Γ

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2 Sectorial Operators

where Γ is the integral contour given by (2.107) but with ωA < ω < π2 . Then, by arguing in a similar way as for (2.111), it is easily verified that A−θ E θ (t) = E θ (t)A−θ = e−tA . Hence, E θ (t) = Aθ e−tA = e−tA Aθ , i.e.,  1 λθ e−tλ (λ − A)−1 dλ, 0 < t < ∞, 0 < θ < ∞. (2.127) Aθ e−tA = 2πi Γ We here shift Γ to Γω : λ = ρe±iω , 0 ≤ ρ < ∞ (i.e., δ → 0). Then, we obtain that   ∞ |λ|θ−1 e−t Re λ |dλ| = Mω ρ θ−1 e−tρ cos ω dρ Aθ e−tA  ≤ Mω 0

Γw

= Mω (cos ω)−θ (θ )t −θ ,

0 < t < ∞, 0 < θ < ∞,

where (·) denotes the gamma function. Let now 0 < θ ≤ 1. We have  t   t   1−θ −τ A  ≤ Cθ [e−tA − 1]A−θ  ≤  A e dτ τ θ−1 dτ ≤ Cθ t θ ,   0

0

(2.128)

0 < t < ∞. (2.129)

Meanwhile, we can verify for each 0 < θ ≤ 1 that, as t → 0, t θ Aθ e−tA converges to 0 strongly on X.

(2.130)

Indeed, if U ∈ D(Aθ ), it is clear that t θ e−tA Aθ U → 0 in X as t → 0. Then, for general F ∈ X, the convergence is verified by the uniform boundedness of t θ Aθ e−tA  due to (2.128) and the denseness of D(Aθ ) in X. Similarly, for each 0 ≤ θ < 1, as t → 0, t −θ [e−tA − 1]A−θ converges to 0 strongly on X.

(2.131)

Indeed, if U ∈ D(A1−θ ), then t −θ [e−tA − 1]A−θ U = t −θ [e−tA − 1]A−1 A1−θ U → 0 in X as t → 0 due to (2.129). Then, for general F ∈ X, the convergence is verified by the uniform boundedness of t −θ [e−tA − 1]A−θ  due to (2.129) and the denseness of D(A1−θ ) in X. Using the properties obtained above, we will show the following result. Theorem 2.27 Let 0 < β ≤ 1 and U0 ∈ D(Aβ ). Then, for any σ such that 0 < σ < β ≤ 1 and any 0 < T < ∞, Ae−tA U0 ∈ Fβ,σ ((0, T ]; X) with the estimate Ae−tA U0 Fβ,σ ((0,T ];X) ≤ CT Aβ U0 X . Proof In fact, since t 1−β Ae−tA U0 = t 1−β A1−β e−tA Aβ U0 ,

(2.132)

7 Fractional Powers of Linear Operators

103

(2.130) yields that lim t 1−β Ae−tA U0 = 0 when 0 < β < 1,

t→0

lim t 1−β Ae−tA U0 = AU0

t→0

when β = 1.

Therefore, t 1−β Ae−tA U0 has a limit as t → 0. By (2.128) and (2.129), s 1−β+σ A[e−tA − e−sA ]U0  [e−(t−s)A − 1]A−σ  ≤ (t − s)σ (t − s)σ × s 1−β+σ A1−β+σ e−sA (Aβ U0 ) ≤ CT Aβ U0 ,

0 < s < t ≤ T.

Therefore, (1.6) is fulfilled. In a similar way, s 1−β+σ A[e−tA − e−sA ]U0  ≤ CT s 1−β+σ A1−β+σ e−sA (Aβ U0 ). (t − s)σ Hence, by (2.130), lim sup

t→0 0<s
s 1−β+σ A[e−tA − e−sA ]U0  (t − s)σ

≤ CT lim sup s 1−β+σ A1−β+σ e−sA (Aβ U0 ) = 0. t→0 0<s
This shows that (1.7) is also fulfilled. We have thus verified (2.132).



Consider the Yosida approximation An of A. As proved in Sect. 4.1, (2.92) and (2.93) imply (2.84) and (2.86) with the same angle ω. Thanks to (2.128), it immediately follows that  θ −tA  A e n  ≤ M˜ ω (cos ω)−θ (θ )t −θ , n

0 < t < ∞, 0 < θ < ∞,

(2.133)

with the constant M˜ ω in (2.86). In addition, as verified by (2.87), (λ − An )−1 , λ ∈ / Σω , converges to (λ − A)−1 in L(X) as n → ∞. Then, we can use the Lebesgue dominated convergence theorem to the sequence of integrals given by (2.127) with A is replaced by An . As a result, we have, as n → ∞, Aθn e−tAn → Aθ e−tA for 0 < t < ∞ and 0 < θ < ∞.

in L(X)

(2.134)

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2 Sectorial Operators

7.8 Generation Theorem of Continuous Semigroups Consider in this subsection a sectorial operator A of X with angle just ωA = π2 . Theorem 2.23 provides that, for 0 < α < 1, Aα has an angle ≤ π2 α. This means that α −Aα generates an analytic semigroup e−tA on X. We are then interested in the α −tA limit of e as α  1. For this purpose, we will utilize the fact that (2.99) implies e−tA  ≤ 1 (see Theorem 2.17). We assume that (λ − A)−1  ≤ 1/|λ| for all λ < 0. α Then, by (2.115), it is the same for Aα . Hence, e−tA  ≤ 1 for any 0 < α < 1. α −tA It is now possible to consider the limit of e as α  1. For 0 < α < β < 1 and U ∈ D(A),  t  −tAβ  d  −(t−s)Aα −sAβ  −tAα e e U= −e e U ds ds 0  t α β e−(t−s)A (Aα − Aβ )e−sA U ds = 0



t

=

e−(t−s)A e−sA (Aα − Aβ )U ds. α

β

0

As a consequence,  −tAα β  e − e−tA U  ≤ t(Aα−1 − Aβ−1 )AU ,

0 ≤ t < ∞.

This then shows that e−tA U is convergent in X as α  1. Furthermore, for F ∈ X, we observe that   −tAα   β  α β α β  e − e−tA F  ≤  e−tA − e−tA (F − U ) +  e−tA − e−tA U  α

≤ 2F − U  + t(Aα−1 − Aβ−1 )AU . Since D(A) is dense in X, we conclude that e−tA F is also convergent in X. Denote α by e−tA F the limit of e−tA F for each 0 ≤ t < ∞. Obviously, e−tA is a bounded linear operator of X with the estimate e−tA  ≤ 1. The operators e−tA , 0 ≤ t < ∞, are then shown to enjoy the following properties: α

(1) e−(t+s)A = e−tA e−sA , 0 ≤ s, t < ∞, and e−0A = 1. (2) e−tA is strongly continuous for 0 ≤ t < ∞ on X. (2) For U ∈ D(A), e−tA U is a differentiable function with values in X with derivad −tA tive dt e U = −e−tA AU, 0 ≤ t < ∞. Property (1) is observed from that of e−tA by letting α  1. As verified above, the α convergence of e−tA F is uniform in t on any bounded interval [0, T ]; therefore, e−tA F is a continuous with values in X. For U ∈ D(A), letting α  1 in  t function α α [e−tA − 1]U = − 0 e−sA Aα U ds, we have  t −tA − 1]U = − e−sA AU ds, 0 < t < ∞. [e α

0

8 Purely Imaginary Powers and Square Root Problem

105

From this the property (3) is also verified. In general, a family of bounded operators having properties (1), (2), and (3) is called a continuous semigroup or a C0 semigroup on X. Thus, we have verified the following theorem. Theorem 2.28 Let A be a sectorial operator of X with angle ≤ π2 satisfying (2.99). Then, −A generates a continuous semigroup e−tA on X with the estimate e−tA  ≤ 1. It is possible to define the generator of a continuous semigroup on X. We are furthermore able to characterize the generator of a continuous semigroup by the Hille–Yosida theorem, which is the fundamental result in the theory of semigroups. For details, see Hille–Philips [HP57], Yosida [Yos80], and other references on semigroup theory (e.g., [EN00, Tan75, Paz83]).

8 Purely Imaginary Powers and Square Root Problem 8.1 Bounded Purely Imaginary Powers Let X be a Banach space with norm  · . Let Az be the fractional powers of a sectorial operator A of X. We are interested in the purely imaginary powers Aiy , y ∈ R. In view of Theorem 2.22, we define Aiy by

D(Aiy ) = {F ∈ X; limRe z<0, z→iy Az F exists in X}, Aiy F = limRe z<0, z→iy Az F. If U ∈ D(A), then U belongs to D(Aiy ) since Az U = Az−1 AU . So, Aiy is densely defined. It is obvious that Aiy is a linear operator. When the condition for every y ∈ R, D(Aiy ) = X, and Aiy is a bounded operator on X

(2.135)

is satisfied, A is said to have bounded purely imaginary powers. Assume that there exists an increasing function p(·) such that Aiy U  ≤ p(|y|)U ,

−∞ < y < ∞, U ∈ D(A).

(2.136)

Let R > 0 be an arbitrarily fixed number, and let −1 < Re z < 0 and |Im z| ≤ R. Then, we have Az U  = ARe z Ai Im z U  ≤ C1 p(R)U  for U ∈ D(A), where C1 = sup−1≤x<0 Ax . Note that C1 < ∞ due to (2.109). Since D(A) is dense in X, we have Az F  ≤ CR p(R)F  for F ∈ X, too, i.e., Az  ≤ CR p(R). This uniform boundedness indeed readily implies that the assertion of (2.135) is valid for every |y| < R. Therefore, (2.136) is a sufficient condition for (2.135). It is known that some favorable sectorial operators have bounded purely imaginary powers, but

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2 Sectorial Operators

others not. In this subsection, we shall remark some immediate consequences of (2.135). Let (2.135) be satisfied for A. From (2.111) it is easily verified that the   purely imaginary powers have the group property Ai(y+y ) = Aiy Aiy . For F ∈ X, Az F is an X-valued continuous function in {z; Re z ≤ 0}. In particular, sup|y|≤1 Aiy F  < ∞ for each F ∈ X. Then, the uniform boundedness theorem, Theorem 1.5, provides that sup|y|≤1 Aiy  = NA < ∞. For general y ∈ R, let n − 1 ≤ |y| < n with n = 1, 2, 3, . . . . Since | yn | < 1, it follows that Aiy  = y |y| Ai( n )n  ≤ NAn ≤ NA NA . Therefore, putting βA = log NA , we obtain that Aiy  ≤ NA eβA |y| ,

−∞ < y < ∞.

8.2 Imaginary Powers for Maximal Accretive Operators In this subsection, X is a Hilbert space with inner product (·, ·) and norm | · |. If A is a positive definite self-adjoint operator of X, then, as noticed in Remark 2.7, its fractional powers can alternatively be defined by using the spectral resolution. In fact, we have  ∞ |Aiy F |2 = |μiy |2 d|E(μ)F |2 = |F |2 , F ∈ X. δ

Therefore, A has bounded purely imaginary powers with Aiy  = 1, y ∈ R. This result was generalized by Kato to maximal accretive operators. A sectorial operator A of X is called a maximal accretive operator of X if the resolvent satisfies   (λ − A)−1  ≤ 1/(|λ| + δ), λ ≤ 0, (2.137) with some constant δ > 0. Before proving that the maximal accretive operators have bounded purely imaginary powers, we shall first present some their properties. Theorem 2.29 For a sectorial operator A of X, condition (2.137) is equivalent to the following one: Re(AU, U ) ≥ δ|U |2 ,

U ∈ D(A).

(2.138)

Proof Let (2.137) be satisfied. For λ ≤ 0 and U ∈ D(A), we have |(λ − A)U | ≥ (|λ| + δ)|U |. It therefore holds that ((λ − A)U, (λ − A)U ) ≥ (|λ| + δ)2 |U |2 , 2|λ| Re(AU, U ) + |AU |2 ≥ 2|λ|δ|U |2 + δ 2 |U |2 , 2 Re(AU, U ) + |λ|−1 |AU |2 ≥ 2δ|U |2 + |λ|−1 δ 2 |U |2 . Letting λ → −∞, we obtain (2.138).

8 Purely Imaginary Powers and Square Root Problem

107

Conversely, let (2.138) be satisfied. For λ ≤ 0 and U ∈ D(A), it holds that ((A − λ)U, U ) = (AU, U ) − λ|U |2 , Re((A − λ)U, U ) = Re(AU, U ) + |λ||U |2 , |(A − λ)U ||U | ≥ Re((A − λ)U, U ) ≥ (|λ| + δ)|U |2 , |(A − λ)U | ≥ (|λ| + δ)|U |. Putting U = (A − λ)−1 F , we obtain (2.137).



Proposition 2.8 Let A be a maximal accretive operator of X, i.e., (2.137) holds. Then, for each 0 < θ < 1, Re(Aθ U, U ) ≥ δ θ |U |2 ,

U ∈ D(Aθ ).

(2.139)

Proof Let U ∈ D(A). By (2.106), Re(Aθ U, U ) = Re(Aθ−1 AU, U ) =

sin(πθ ) π



∞

  ρ θ−1 Re A(ρ + A)−1 U, U dρ.

0

Here, we observe from (2.137) that     Re A(ρ + A)−1 U, U = |U |2 − ρ Re (ρ + A)−1 U, U ≥ |U |2 − ρ(ρ + δ)−1 |U |2 = δ(ρ + δ)−1 |U |2 . Therefore, Re(Aθ U, U ) ≥

sin(πθ ) π



∞

ρ θ−1 δ(ρ + δ)−1 |U |2 dρ = δ θ |U |2 .

0

For general U ∈ D(Aθ ), we consider the sequence Un = Jn U = (1 + n−1 A)−1 U, n = 1, 2, 3, . . . . Then, as Un → U and Aθ Un = Jn Aθ U → Aθ U in X, (2.139) is verified for this U , too.  Theorem 2.30 A maximal accretive operator A of X has bounded purely imaginary π powers, and its imaginary powers satisfy Aiy  ≤ e 2 |y| , −∞ < y < ∞. Proof We shall first prove the theorem for bounded maximal accretive operators. So, let A be a bounded operator satisfying (2.137). Let Az be the fractional powers of A. As A is bounded, Az are also bounded operators for all z ∈ C, and Az is an analytic function of C with values in L(X). It is the same for the adjoint operator A∗ . We here introduce a decomposition Az = H (z) + iK(z), where H (z) = (Az + A∗z )/2 and K(z) = (Az − A∗z )/2i. By Proposition 2.8, we see for 0 ≤ Re z ≤ 12 and U ∈ X, that |H (z)U |2 − |K(z)U |2 = Re(Az U, A∗z U ) = Re(A2 Re z U, U ) ≥ 0.

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2 Sectorial Operators

Similarly, for − 12 ≤ Re z < 0, |H (z)U |2 − |K(z)U |2 = Re(A2 Re z U, U ) = Re(A−2 Re z A2 Re z U, A2 Re z U ) ≥ 0. Therefore, |K(z)U | ≤ |H (z)U | for all | Re z| ≤ 12 and U ∈ X. In the meantime, for 0 ≤ Re z ≤ 12 , we have |H (z)U |2 ≥ (δ 2 Re z /2)|U |2 . Similarly, for − 12 ≤ Re z < 0,

1 2

Re(A2 Re z U, U ) ≥

1 1 Re(A2 Re z U, U ) = Re(A−2 Re z A2 Re z U, A2 Re z U ) 2 2 √ −2 ≥ (δ −2 Re z /2)|A2 Re z U |2 ≥ 2δ Re z A−2 Re z  |U |2 .

|H (z)U |2 ≥

These estimates show that H (z) is injective and its inverse is continuous from R(H (z)) onto X for each | Re z| ≤ 12 . We then consider the set ρ(H ) = {z; | Re z| ≤ 1 −1 ∈ L(X)}. Since, if z is real, then H (z) is self-adjoint, we observe 2 , H (z) 1 1 that [− 2 , 2 ] ⊂ ρ(H ). In addition, since H (z) is a continuous function with values in L(X), ρ(H ) is an open set of the strip {| Re z| ≤ 12 }. On the other hand, the set is shown to be closed. Indeed, if zn ∈ ρ(H ) and zn → z, then it follows from H (zn )−1 − H (zm )−1 = H (zn )−1 [H (zm ) − H (zn )]H (zm )−1 that H (zn )−1 is a Cauchy sequence of L(X) and has a limit (note that H (zn )−1  is uniformly bounded with respect to n), and therefore H (z)−1 ∈ L(X). Hence, we have verified that ρ(H ) coincides with the strip. Of course, H (z)−1 is also an analytic function. Our goal is then to show that, for | Re z| ≤ 12 ,            K(z)H (z)−1  ≤  e πz2 i − e− πz2 i e πz2 i + e− πz2 i −1  = tan πz . (2.140)  2  We already know that K(z)H (z)−1 is an analytic function satisfying K(z)H (z)−1  ≤ 1 for | Re z| ≤ 12 . Clearly, K(0)H (0)−1 = 0. On the other hand, tan( πz 2 ) is πz 1 also an analytic function of the strip satisfying | tan( 2 )| = 1 for | Re z| = 2 and limIm z→±∞ tan( πz 2 ) = ±i, and it has a unique zero at z = 0 with order 1. There−1 −1 fore, we can apply the maximum principle to the function (tan( πz 2 )) K(z)H (z) to obtain estimate (2.140). We now take purely imaginary numbers z = iy in (2.140). Then, it is seen for U ∈ X that |Aiy U | − |A∗iy U | ≤ |[Aiy − A∗iy ]U | ≤

eπ|y|/2 − e−π|y|/2 (|Aiy U | + |A∗iy U |), eπ|y|/2 + e−π|y|/2

e−π|y|/2 |Aiy U | ≤ eπ|y|/2 |A∗iy U |. Therefore, A∗(−iy) Aiy  ≤ eπ|y| . This implies that |Aiy U |2 = (A∗(−iy) Aiy U, U ) ≤ eπ|y| |U |2 . To complete the proof, let A be an unbounded maximal accretive operator. We use the Yosida approximation An . According to (2.82), condition (2.137) yields

8 Purely Imaginary Powers and Square Root Problem

109

that (λ − An )−1  ≤

1 nδ |λ|+δn for λ ≤ 0, where δn = n+δ . Therefore, An are maximal π iy iy An  ≤ e 2 |y| . For U ∈ D(A), Aiy U = limn→∞ An U . There-

accretive, and hence π fore, |Aiy U | ≤ e 2 |y| |U | for U ∈ D(A). As remarked, (2.136) implies (2.135).



Corollary 2.2 Let Z ⊂ X ⊂ Z ∗ be a triplet of spaces. Let a(U, V ) be a sesquilinear form on Z satisfying (2.6) and (2.7). Then, the part A = A|X of the operator A associated with the form has bounded purely imaginary powers on X. Proof By (2.12) we already know that the part A is a maximal accretive operator of X.  For a counterexample of sectorial operators of Hilbert spaces which have unbounded purely imaginary powers, see Chap. 16, Sect. 1.5.

8.3 Heinz and Kato Inequality We shall show some application of purely imaginary powers. Let X (resp. Y ) be a Hilbert space with inner product (·, ·)X (resp. (·, ·)Y ) and norm | · |X (resp. | · |Y ). Let A (resp. B) be a sectorial operator of X (resp. Y ). We assume that both A and B have bounded purely imaginary powers with the estimates Aiy L(X) ≤ NA eβA |y| ,

−∞ < y < ∞,

(2.141)

B iy L(Y ) ≤ NB eβB |y| ,

−∞ < y < ∞,

(2.142)

with some constants NA , NB ≥ 1 and exponents βA , βB ≥ 0. We prove the following theorem which is called the Heinz and Kato inequality. Theorem 2.31 Let A and B satisfy (2.141) and (2.142). Let T be a linear operator from X into Y such that T ∈ L(X, Y ) ∩ L(D(A), D(B)). Then, for every 0 < θ < 1, T is automatically bounded from D(Aθ ) into D(B θ ) with the norm estimate √

T L(D(Aθ ),D(B θ )) ≤ (NA NB )e(βA +βB )

θ(1−θ)

θ T 1−θ L(X,Y ) T L(D(A),D(B)) .

Proof Let B ∗ be the adjoint operator of B. For 0 < θ < 1, we consider an analytic function of the form ϕ(z) = eαz(z−1) (T L(X,Y ) /T L(D(A),D(B)) )z−θ × (B z−1 T A−z F, B ∗ V )Y ,

F ∈ X, V ∈ D(B ∗ ),

in the strip S = {z ∈ C; 0 < Re z < 1}, where α > 0 is a positive exponent to be fixed below. By the assumptions, ϕ(z) is an analytic function of S and is a continuous

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2 Sectorial Operators

function of S. Since |eαz(z−1) | = eα{x(x−1)−y } , where z = x + iy, ϕ(z) is seen to satisfy the condition lim| Im z|→∞ ϕ(z) = 0. Therefore, 2

|ϕ(θ )| ≤ max

sup

k=0,1 −∞
|ϕ(k + iy)|.

Meanwhile, we observe that sup

−∞
$ # 2 |ϕ(k + iy)| ≤ (NA NB ) sup e−αy +(βA +βB )y 0≤y<∞ 1−θ θ × T L(X,Y ) T L(D(A),D(B)) |F |X |V |Y ,

k = 0, 1.

Consequently, |ϕ(θ )| ≤ (NA NB )e(βA +βB )

2 /(4α)

1−θ θ T L(X,Y ) T L(D(A),D(B)) |F |X |V |Y .

That is, |(B θ−1 T A−θ F, B ∗ V )Y | ≤ (NA NB )eαθ(1−θ)+(βA +βB )

2 /(4α)

1−θ θ × T L(X,Y ) T L(D(A),D(B)) |F |X |V |Y .

Since V ∈ D(B ∗ ) is arbitrary, this implies that B θ−1 T A−θ F ∈ D(B) with the estimate |B θ T A−θ F |Y ≤ (NA NB )eαθ(1−θ)+(βA +βB )

2 /(4α)

1−θ θ T L(X,Y ) T L(D(A),D(B)) |F |X .

Choosing now α in such a way that αθ (1 − θ ) + (βA + βB )2 /(4α) hits its minimum, we obtain that |B θ T A−θ F |Y ≤ (NA NB )e(βA +βB )

√

θ(1−θ)

1−θ θ T L(X,Y ) T L(D(A),D(B)) |F |X .

Putting A−θ F = U , we obtain the desired result.



8.4 Square Root Problem Let Z ⊂ X ⊂ Z ∗ be a triplet of spaces. Consider a sectorial operator A associated with a sesquilinear form a(·, ·) defined on Z satisfying (2.6) and (2.7). Let A = A|X 1 1 be the part of A in X. We are concerned with the domains D(A 2 ) and D(A 2 ) of square root. In general, we can show the following results. Theorem 2.32 Let A be a sectorial operator associated with a sesquilinear form on Z satisfying (2.6) and (2.7), and let A be its part in X. For any θ and θ  such that

8 Purely Imaginary Powers and Square Root Problem

111





0 < θ < 12 < θ  < 1, it holds that D(Aθ ) ⊂ X ⊂ D(Aθ ) and D(Aθ ) ⊂ Z ⊂ D(Aθ ) with the estimates 

Cθ−1 Aθ · ∗ ≤ | · | ≤ Cθ  Aθ · ∗ , 

Cθ−1 |Aθ · | ≤  ·  ≤ Cθ  |Aθ · |, where the constants Cθ > 0 and Cθ  > 0 are determined by M, δ, and θ, θ  only. Let us begin with proving the following proposition. Proposition 2.9 For Re λ ≤ 0,  1 1 |λ| 2 (λ − A)−1 Φ  ≤ [(Mδ −1 + 1)δ −1 ] 2 Φ∗ , Φ ∈ Z ∗ ,  1 1 |λ| 2 (λ − A)−1 F  ≤ [(Mδ −1 + 1)δ −1 ] 2 |F |, F ∈ X.

(2.143) (2.144)

Proof Let U ∈ Z. From λ|U |2 = a(U, U ) + (λ − A)U, U it follows that |λ||U |2 ≤ MU 2 + (λ − A)U ∗ U . Putting (λ − A)U = Φ, we use (2.8) and (2.9). Then, 2  |λ|(λ − A)−1 Φ  ≤ (Mδ −2 + δ −1 )Φ2∗ . Let now U ∈ D(A|X ). Then, δU 2 ≤ Re a(U, U ) − Re λ|U |2 = Re(AU − λU, U ) ≤ |(A − λ)U ||U |. Putting (λ − A)U = F , we use (2.10). Then,  2 δ (λ − A)−1 F  ≤ (Mδ −1 + 1)|λ|−1 |F |2 .



Proof of Theorem 2.32 Let F ∈ X. For ρ > 0, it follows from (2.144) that    1 1 ρ 2 A(ρ + A)−1 F ∗ ≤ ρ 2 AL(Z,Z ∗ ) (ρ + A)−1 F  ≤ CAL(Z,Z ∗ ) |F |. This means that F ∈ D 1 (A) due to (2.120). Hence, X ⊂ D 1 (A). Similarly, for 2 2 U ∈ Z, it follows from (2.143) that  1 ρ 2 A(ρ + A)−1 U  ≤ CAU∗ ≤ CAL(Z,Z ∗ ) U . This means that U ∈ D 1 (A). Hence, Z ⊂ D 1 (A). Theorem 2.24 then provides that 2

2

X ⊂ D 1 (A) ⊂ D(Aθ ) and Z ⊂ D 1 (A) ⊂ D(Aθ ) for 0 < θ < 12 . 2



2





Let U ∈ Z. Since A−θ U = (A|Z )−θ U , we see that A−θ U ∈ Z. Furthermore,      Φ, A−θ U = Φ, A−1 A1−θ U = (A∗ )−1 Φ, A1−θ U ,

Φ ∈ Z∗.

112

2 Sectorial Operators

Since 0 < 1 − θ  < 12 , we have     | Φ, A−θ U | ≤ (A∗ )−1 Φ A1−θ U ∗ ≤ Cθ  δ −1 Φ∗ |U |. 

Therefore, A−θ U  ≤ Cθ  δ −1 |U |, U ∈ Z. Using the denseness of Z in X, we readily verify that this estimate holds for all F ∈ X. That is, for any F ∈ X,     A−θ F = A−θ F ∈ Z. Hence, D(Aθ ) = R(A−θ ) ⊂ Z.   Let F ∈ X. Then, A−θ F = A−θ F ∈ X. So,      (U, A−θ F ) = U, A1−θ A−1 F = (A∗ )1−θ U, A−1 F , U ∈ Z. 

Obviously, the embedding X ⊂ D(Aθ ) holds for A∗ , too; hence, X ⊂ D((A∗ )1−θ ). It then follows that     |(U, A−θ F )| ≤ (A∗ )1−θ U ∗ A−1 F  ≤ Cθ  δ|U |F ∗ . 

Consequently, |A−θ F | ≤ Cθ  δF ∗ , F ∈ X. Repeating the similar argument as be   fore, we conclude that D(Aθ ) = R(A−θ ) ⊂ X. 1

1

By Theorem 2.32, it is very natural to ask whether D(A 2 ) = X and D(A 2 ) = Z, or not. Such a problem is called the square root problem. McIntosh [McI72] succeeded in giving a counterexample for these properties. We can however show some partial results. Theorem 2.33 Under the same situation as in Theorem 2.32, it is true that 1 [D(A), Z ∗ ] 1 = X and [X, D(A)] 1 = D(A 2 ). 2

2

Proof As will be seen, we have [Z, Z ∗ ] 1 = X (see (2.145)). Then, as Z = D(A), 2 the first assertion is observed. As shown by Corollary 2.2, A has bounded purely imaginary powers on X. Then, according to Theorem 16.3, [X, D(A)]θ = D(Aθ ) for any 0 < θ < 1. Hence, the second assertion is also observed.  Remark 2.11 According to Kato [Kat61a, Kat62] and Lions [Lio62], it is known 1 1 1 1 in general that D(A 2 ) = D(A∗ 2 ) if and only if D(A 2 ) ⊂ Z and D(A∗ 2 ) ⊂ Z are 1 1 1 valid. In addition, D(A 2 ) ⊂ Z (resp. D(A∗ 2 ) ⊂ Z) is equivalent to Z ⊂ D(A∗ 2 ) 1 1 1 (resp. Z ⊂ D(A 2 )). Therefore, the coincidence of any two of D(A 2 ), D(A∗ 2 ) and Z implies the coincidence of the three.

8.5 Symmetric Forms and Square Root Problem Consider a triplet of spaces Z ⊂ X ⊂ Z ∗ . Let a(U, V ) be a sesquilinear form defined on Z. Assume that a(U, V ) satisfies, in addition to (2.6) and (2.7), the condition a(U, V ) = a(V , U ) = a ∗ (U, V ), Such a form is called a symmetric form.

U, V ∈ Z.

8 Purely Imaginary Powers and Square Root Problem

113

Let A be the linear operator associated with the form, and let A be its part in X. By (2.126), we have A∗ = A, namely, A is a self-adjoint operator of X. Since ˜ | ≤ |AU | δU 2 ≤ a(U, U ) = (AU, U ) ≤ |AU ||U |, U ∈ D(A), it follows that δ|U with some δ˜ > 0. Therefore, A is positive definite in X. Theorem 2.34 Let a(U, V ) be a symmetric sesquilinear form satisfying (2.6) and (2.7). Then, the part A of the associated operator A is a positive definite 1 self-adjoint operator of X. In addition, D(A 2 ) = Z with the norm equivalence 1 δU 2 ≤ |A 2 U |2 ≤ MU 2 . Proof The first assertion is already verified. For U ∈ D(A), we have   1 1 a(U, U ) = AU, U = (AU, U ) = A 2 U, A 2 U . 1

Hence, (2.6) and (2.7) imply that δU 2 ≤ |A 2 U |2 ≤ MU 2 for U ∈ D(A). Then, 1 the second assertion is verified by the denseness of D(A) both in D(A 2 ) and in Z.  This theorem gives an affirmative result for the square root problem for A. 1

Theorem 2.35 Under the same situation as in Theorem 2.34, D(A 2 ) = X with the 1 norm equivalence δ|F |2 ≤ A 2 F 2∗ ≤ M|F |2 . 1

1

1

Proof If U ∈ Z, then U ∈ D(A 2 ) with A 2 U = A 2 U . Therefore,  1      √ A 2 U  = sup  A 12 U, V  = sup  U, A 12 V  ≤ M|U |. ∗ V ≤1

V ≤1

1

1

On the other hand, since V = A− 2 U/A− 2 U  ∈ Z, we have  1     √    A 2 U  ≥  A 12 U, V  =  U, A 12 V  = |U |2 /A− 12 U  ≥ δ|U |. ∗ 1

Here, we used the estimate δA− 2 U 2 ≤ |U |2 . Therefore,  1 2 δ|U |2 ≤ A 2 U ∗ ≤ M|U |2 ,

U ∈ Z. 1

The desired result now follows from the denseness of Z both in D(A 2 ) and in X.  Corollary 2.3 Under the same situation as in Theorem 2.34, A has bounded purely imaginary powers on Z ∗ . Consequently, [D(A), Z ∗ ]θ = D(Aθ ) for any 0 ≤ θ ≤ 1. 1

1

Proof For y ∈ R, we have Aiy = A 2 Aiy A− 2 . Then the boundedness of Aiy is verified by Corollary 2.2 and Theorem 2.35. The second assertion is then verified by Theorem 16.3 given in Chap. 16. 

114

2 Sectorial Operators 1

Since D(A) = Z and D(A 2 ) = X, we observe in general that [Z, Z ∗ ] 1 = X.

(2.145)

2

Corollary 2.4 Under the same situation as in Theorem 2.34,  1  D(Aθ ) = D A 2 +θ = [X, Z]2θ

for any 0 ≤ θ ≤

1 2

with the norm equivalence δ θ U [X,Z]2θ ≤ |Aθ U | ≤ M θ U [X,Z]2θ ,  1  1 δ 2 +θ U [X,Z] ≤ A 2 +θ U  ∗

2θ

≤M

1 2 +θ

U [X,Z]2θ ,

U ∈ D(Aθ ),

(2.146)

 1  U ∈ D A 2 +θ .

(2.147)

Proof As seen by Theorem 16.1, if A˜ is a positive definite self-adjoint operator of X, ˜ θ for any 0 ≤ θ ≤ 1 with isometry. Let us apply this fact then D(A˜ θ ) = [X, D(A)] 1

1

1

to A 2 . Then, in view of Aθ = (A 2 )2θ , we have D(Aθ ) = [X, D(A 2 )]2θ = [X, Z]2θ for any 0 ≤ θ ≤ 12 . The equivalence estimates in (2.146) are observed by Theorem 1.15 and Theorem 2.1. 1 1 1 In view of Theorem 2.35, A 2 +θ = A 2 Aθ = A 2 Aθ for any 0 ≤ θ ≤ 12 . Therefore, 1

D(A 2 +θ ) = D(Aθ ) for any 0 ≤ θ ≤  − 1 −1 A 2 

L(Z ∗ ,X)

1 2

with

 1   1 |Aθ U | ≤ A 2 +θ U ∗ ≤ A 2 L(X,Z ∗ ) |Aθ U |,

 1  U ∈ D A 2 +θ .

Moreover, the estimates in (2.147) are observed by Theorem 2.35 and (2.146).



Notes and Further Researches The methods of determining sectorial operators from continuous and coercive sesquilinear forms provide us a very convenient and unified way for showing that realizations of various differential operators in many Hilbert spaces become actually sectorial operators. Theorem 2.1 was proved by using the techniques of variations which were devised by Lions and Magenes, see Lions–Magenes [LM72, Chap. 2, Sect. 9], Dautray–Lions [DL88, Chap. VI], and others. As seen by Theorem 2.11, the methods are available even for defining sectorial operators of matrix form. For obtaining Propositions 2.1 and 2.2, we used the same techniques as in Pazy [Paz83, Chap. 7] of energy estimates in Lp (Ω) (1 < p < ∞) for second-order strongly elliptic operators. For higher-order strongly elliptic operators, it is similarly possible to show that they determine sectorial operators in Lp (Ω), see Tanabe [Tan75, Tan97]. Strongly elliptic operators in L1 (Ω) were studied by Amann [Ama83], Guidetti

Notes and Further Researches

115

[Gui93], and Tanabe [Tan97]. Strongly elliptic operators in C(Ω) were studied by Stewart [Stew74, Stew80], Tanabe [Tan97], and Taira [Tai03]. As seen in these literatures, the strongly elliptic operators can generate analytic semigroups even on L1 (Ω) and C(Ω). By using only the energy estimates, as done in Theorem 2.12, we can verify that strongly elliptic operators define sectorial operators. However, in order to obtain the shift property, namely, the regularity of solutions for elliptic equations, we have to appeal to the theory of elliptic equations, e.g., see [LU68, LM72, GT83]. Especially, the optimal regularity of elliptic equations in nonsmooth domains was obtained by Grisvard [Gri85]. The Yosida approximation for sectorial operators has been introduced by Yosida (see [Yos48] and [Yos80, Theorem, p. 246]). Since (2.1) and (2.2) imply (2.84) and (2.86), it often suffices to describe the proof of favorable properties which hold true for all the sectorial operators only for bounded sectorial operators. The notion of analytic semigroups was introduced (see Hille–Phillips [HP57, Chap. XVII] and Yosida [Yos80, Chap. IX]) as a class of regular semigroups of general continuous semigroups, the theory of which has been originated independently by Hille [Hil48] and Yosida [Yos48]. So far, many classes of semigroups have been introduced other than the analytic semigroups. The n times or infinitely differentiable semigroups classify the continuous semigroups and interpolate the analytic semigroups and the mere continuous semigroups. On the other hand, the distribution semigroups introduced by Lions [Lio60] give a large class of semigroups which contains the continuous semigroups. The n times integrated semigroups introduced by Arendt [Are87] then give classification of distribution semigroups. For the continuous semigroups, including the analytic semigroups, we quote, besides the classics [HP57] and [Yos80], Pazy [Paz83], Engel–Nagel [EN00], and Dautray–Lions [DL91]. For other semigroups, we refer the reader to monographs Arendt–Batty– Hieber–Neubrander [ABHN01], Melnikova–Filinkov [MF01], Favini–Yagi [FY99], Yakubov–Yakubov [YY00], and others. Several equivalent ways are known for defining the fractional powers of sectorial operators in Banach spaces. In this chapter, we followed Kato’s one of utilizing formula (2.106) presented in [Kat60, Kat61a]. For systematic treatments of fractional powers, we quote a series of papers by Komatsu [Kom66, Kom67, Kom69a, Kom69b, Kom70, Kom72] and a monograph Carracedo–Alix [CA01]. Complete characterization of the domains of fractional powers of sectorial operators is a very interesting but hard problem. As will be seen in Chap. 16 (by Theorem 16.3 in Hilbert-space case and by Theorem 16.5 in reflexive Banach-space case), the boundedness of purely imaginary powers implies the coincidence of the domains of fractional powers and the interpolation spaces. Using these facts, we are able to characterize the domains of fractional powers for some favorable strongly elliptic operators. On the other hand, as will be seen through the structural assumptions made in the subsequent chapters, Chaps. 3, 4, and 5, we need not necessarily the complete characterization of fractional powers. Suitable estimates from interior or exterior (cf. Theorem 2.24) are often sufficient in order to apply the general results on abstract parabolic evolution equations to the initial-boundary-value problems of concrete parabolic differential equations.

116

2 Sectorial Operators

The Heinz and Kato inequality, Theorem 2.31, was first shown by Heinz [Hei51] for self-adjoint operators of Hilbert spaces and then was generalized together with Theorem 2.30 by Kato [Kat61b, Kat62] into the present form. Let A be the sectorial operator associated with a continuous and coercive sesquilinear form defined on a Hilbert space Z. As mentioned (see McIntosh 1 [McI72]), D(A 2 ) and Z do not coincide in general, but there are a number of papers which show that realizations of specific strongly elliptic operators give an affirmative answer to the Kato square root problem, see [McI85, McI90, ER97, Dia00, Dia03, AHMT01, AHLM02, AT03, AKM06, HMP08].

Chapter 3

Linear Evolution Equations

In Part I of this chapter, we handle the autonomous abstract evolution equation dU + AU = F (t), dt

0 < t ≤ T,

in a Banach space X, where A is a sectorial operator of X with angle ωA < π2 . Using the analytic semigroup e−tA generated by −A, we construct a unique solution together with a representation formula given by (3.13) for the Cauchy problem with initial condition U (0) = U0 . We also describe the precise definition of analytic semigroups introduced to classify regular semigroups from the general continuous semigroups, the generation theorem of which was proved independently by Hille and Yosida in 1948. From (3.13) we can derive the maximal regularity of solution in the sense that F ∈ Fβ,σ ((0, T ]; X), together with U0 ∈ D(Aβ ), implies that dU β,σ ((0, T ]; X). dt , AU ∈ F In Part II we handle the nonautonomous abstract evolution equation dU + A(t)U = F (t), dt

0 < t ≤ T,

in X, where A(t) are sectorial operators of X with angles < π2 satisfying D(A(t)) ⊂ D(A(s)ν ) with 0 < ν ≤ 1. (When ν = 1, this condition obviously means that D(A(t)) ≡ D(A(0)) are independent of t.) Under suitable structural assumptions on A(t), we construct an evolution operator U (t, s) for the family of operators A(t). The evolution operator (which coincides with e−(t−s)A in the particular case where A(t) ≡ A) yields the existence and uniqueness for the Cauchy problem with initial condition U (0) = U0 together with a representation formula given by (3.50). Construction of the evolution operator was first studied independently by Sobolevski˘ı and Tanabe around 1960 in a favorable case where D(A(t)) ≡ D(A(0)), i.e., ν = 1. Afterward (in 1964), Tanabe considered an unfavorable case where the domain condition mentioned above is invalid, i.e., ν = 0. As we are going to study the case where 0 < ν ≤ 1, the result presented in this chapter is regarded as an interpolation of these two classical results. Representation formula (3.50) gives again, as for the autonomous case, the maximal regularity of solution, that is, F ∈ Fβ,σ ((0, T ]; X), β,σ ((0, T ]; X). together with U0 ∈ D(A(0)β ), implies dU dt , AU ∈ F A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_3, © Springer-Verlag Berlin Heidelberg 2010

117

118

3 Linear Evolution Equations

Part I. Autonomous Equations 1 Analytic Semigroups 1.1 Generation of Analytic Semigroups Let X be a Banach space, and let L(X) be the space of all bounded linear operators on X. An L(X)-valued function U (z) defined in a sectorial domain z ∈ Σφ = {z ∈ C; |arg z| < φ},

0<φ<

π , 2

is called an analytic semigroup on X if U (z) satisfies: (1) U (z) is an analytic function in Σφ with values in L(X). (2) U (z) satisfies the semigroup property U (z)U (z ) = U (z + z ) for z, z ∈ Σφ . (3) For any φ  such that 0 < φ  < φ, U (z) converges to 1 strongly on X as z → 0 with z ∈ Σφ  − {0}. In view of the third property, we define the value of U (z) at z = 0 by U (0) = 1. Let U (z) be an analytic semigroup in Σφ . For 0 < φ  < φ, it holds that sup{z∈Σ  , |z|≤1} U (z)F  < ∞ for each F . The uniform boundedness theorem, Theφ orem 1.5, then provides that sup {z∈Σφ  , |z|≤1}

U (z) = Cφ  < ∞.

For general z ∈ Σφ  , let n − 1 < |z| ≤ n with n = 1, 2, 3, . . . . Then, it holds that |z| U (z) = U ( nz )n  ≤ Cφn ≤ Cφ  Cφ  . Therefore, U (z) ≤ Cφ  eβφ  |z| ,

z ∈ Σφ  , 0 < φ  < φ,

where βφ  = log Cφ  . An analytic semigroup U (z) in Σφ may be extended as analytic semigroup over some larger sectorial domains. So, the supremum of angles of sectors of all possible extensions is called the angle of the semigroup U (z) and is denoted by φU . For any angle 0 < φ < φU , the estimate U (z) ≤ Cφ e−βφ |z| ,

z ∈ Σφ , 0 < φ < φU ,

(3.1)

holds with some fixed exponent −∞ < βφ < ∞ and constant Cφ ≥ 1. In Chap. 2, Sect. 5, we have seen that any sectorial operator A having angle ωA < π2 generates an analytic semigroup e−zA in Σ π2 −ωA . In Chap. 2, Sect. 7, we have seen that any sectorial operator having angle ωA < π defines fractional powers A−z which are an analytic semigroup in the right half-plane. We intend to present general generation results for analytic semigroups.

1 Analytic Semigroups

119

Consider a densely defined, closed linear operator A whose spectrum σ (A) is contained in an open sectorial domain of the form σ (A) ⊂ β + Σω ,

−∞ < β < ∞, 0 < ω <

π , 2

(3.2)

and its resolvent satisfies   (λ − A)−1  ≤

Mω , |λ − β|

λ∈ / β + Σω ,

(3.3)

with some constant Mω ≥ 1. It is immediate to verify that A satisfies (3.2) and (3.3) if and only if A − β satisfies (2.92) and (2.93), namely, A − β is a sectorial operator with angle ωA−β < π2 . Then, the exponential function e−tA of A can be defined in an analogous way by the integral formula e−tA =

1 2πi



e−tλ (λ − A)−1 dλ,

0 < t < ∞,

(3.4)

Γβ

where Γβ = Γβ,+ ∪ Γβ,− , Γβ,± : λ = β + re±iω , 0 ≤ r < ∞, and is directed from β + ∞eiω to β and from β to β + ∞e−iω . Then, similar results to Propositions 2.4, 2.5, 2.6, and 2.7 can be verified for e−tA . More precisely, −A generates an analytic semigroup e−zA in Σ π2 −ω . In particular, for each 0 < φ < π2 − ω, it holds that e−zA  ≤ Cφ e−(β+δφ )|z| ,

z ∈ Σφ , 0 < φ <

π − ω, 2

(3.5)

with some exponent δφ > 0 and a constant Cφ ≥ 1. We summarize these results. Theorem 3.1 Let A be a densely defined, closed linear operator of X satisfying (3.2) and (3.3). Then, e−zA is an analytic semigroup defined in Σ π2 −ω and satisfies estimate (3.5).

1.2 Generator of Analytic Semigroup We want to show that any analytic semigroup U (z) can be written in the form U (z) = e−zAU using some linear operator AU which is called the generator of analytic semigroup. To this end, we shall first investigate a formula how to determine the generator −A from an analytic semigroup e−zA . Let e−zA be the analytic semigroup generated by an operator −A satisfying (3.2) and (3.3). For Re λ ≤ β, we consider the Laplace transform

120

3 Linear Evolution Equations



T

eλt e−tA dt =

ε



T

(λ − A)eλt e−tA dt (λ − A)−1

ε

 =

T

ε

d λt −tA (e e ) dt (λ − A)−1 dt

= (eλT e−T A − eλε e−εA )(λ − A)−1 , where 0 < ε < T < ∞. Since (3.5) holds, letting T → ∞ and ε → 0, we verify that  ∞ − eλt e−tA dt = (λ − A)−1 for Re λ ≤ β. (3.6) 0

Formula (3.6) then gives us a useful suggestion. Let U (z) be an analytic semigroup with angle φU . Since U (z) satisfies (3.1) for any 0 < φ < φU , we can consider the Laplace transform  ∞ R(λ) = − eλt U (t) dt for Re λ < βφ . 0

Let Re λ, Re μ < βφ . Then, 

∞ ∞

R(λ)R(μ) = 0



∞ ∞

eλt+μs U (t)U (s) dt ds =

0

0

eλt+μs U (t + s) dt ds.

0

Changing the variables by σ = t + s, τ = t − s, we have   1 ∞ σ ( λ+μ σ + λ−μ τ ) 2 R(λ)R(μ) = e 2 U (σ ) dτ dσ 2 0 −σ  ∞ 1 1 [R(λ) − R(μ)]. = (eλσ − eμσ )U (σ ) dσ = − λ−μ 0 λ−μ This shows that the family of bounded operators R(λ) defined for Re λ < βφ is a pseudo-resolvent. We here observe the following fact. Lemma 3.1 As λ → −∞, λR(λ) converges to 1 strongly on X. Proof For λ < 0, −

∞ 0

λeλt dt = 1. So, we have for λ < min{βφ , 0}, 

[λR(λ) − 1]F = −

∞

λeλt [U (t) − 1]F dt,

F ∈ X.

0

Let 0 < ε < 1. Then,  ε    λt  λe [U (t) − 1]F dt    ≤ sup [U (t) − 1]F . 0

0≤t≤ε

1 Analytic Semigroups

121

Meanwhile,  ∞      λt   ≤ (Cφ + 1)λ(λ + βφ )−1 e(λ+βφ )ε F . λe [U (t) − 1]F dt   ε

From these estimates we conclude that, as λ → −∞, [λR(λ) − 1]F → 0 in X.



This lemma proves that R(λ) is one-to-one, for, if R(λ)F = 0 for some F ∈ X, then it follows for any integer −n < βφ that R(−n)F = R(λ)F + (λ + n) × R(−n)R(λ)F = 0 and 0 = −nR(−n)F → F , i.e., F = 0. We now apply Theorem 1.8 to ensure the existence of a single-valued linear operator AU for which the formula  ∞ eλt U (t) dt, Re λ < βφ , (3.7) (λ − AU )−1 = R(λ) = − 0

is valid. Lemma 3.1 also implies that AU is densely defined in X. It is clear that AU is closed. We can state the theorem for characterizing the generators of analytic semigroups. Theorem 3.2 Let U (z) be an analytic semigroup on X with angle φU . Let AU be the linear operator determined by (3.7). Then, AU is a densely defined, closed linear operator satisfying (3.2) and (3.3) with any angle ω > π2 − φU , and −AU is the generator of U (z), i.e., U (z) = e−zAU . Proof Step 1. Let us prove that AU satisfies (3.2) and (3.3). So far, the resolvent of AU is defined only in the half-plane Re λ < βφ . By the analyticity of U (z) and estimate (3.1), we notice for λ < βφ that  (λ − AU ) = − e(λ−β)z eβz U (z) dz, where γ− : z = re−iφ , 0 ≤ r < ∞. γ−

Here, |e(λ−β)z | = e|λ−β||z| cos{arg(λ−β)−φ} ,

λ ∈ C, z ∈ γ− .

In the meantime, eβz U (z) ≤ Cφ eβ Re z−βφ |z| = Cφ e{(cos φ)β−βφ }|z| ,

z ∈ γ− .

So, if we set β = β˜φ < (cos φ)−1 βφ , then the integral is convergent for any λ outside of the half-plane {λ; |arg(λ − β˜φ ) − φ| < π2 }. As well, the estimate  ∞   ˜ ˜ ˜ (λ − AU )−1  ≤ Cφ e{|λ−βφ | cos[arg(λ−βφ )−φ]+δφ }r dr 0

≤

Cφ ˜ − cos[arg(λ − βφ ) − φ]|λ − β˜φ | + δ˜φ

is obtained, where δ˜φ = (cos φ)β˜φ − βφ > 0.

122

3 Linear Evolution Equations

Similarly, for any λ outside the half-plane {λ; |arg(λ − β˜φ ) + φ| < π2 }, the integral  ˜ ˜ −1 (λ − AU ) = − e(λ−βφ )z eβφ z U (z) dz, where γ+ : z = reiφ , 0 ≤ r < ∞, γ+

is convergent with the estimate   (λ − AU )−1  ≤

Cφ . ˜ − cos[arg(λ − βφ ) + φ]|λ − β˜φ | + δ˜φ

This indeed shows that the analytic function (λ − AU )−1 can be extended over the domain C − β + Σ π2 −φ . Since φ can be taken arbitrarily close to φU , ω can be arbitrarily close to π2 − φU . Step 2. We know that AU satisfies (3.2) and (3.3). By Theorem 3.1, the exponential function e−zAU is defined. So, let us verify that e−zAU = U (z). By the coincidence property of analytic functions, it is sufficient to prove that e−tAU = U (t) for t > 0. Fix φ and ω so that 0 < φ < φU and π2 − φ < ω < π2 , and take also β for which (3.2) and (3.3) are valid. Then, since ω > ωAU , from (3.4) we have    1 e−tλ (λ − AU )−1 dλ, − e−tAU = 2πi Γβ,+ Γβ,− where Γβ,± : λ = β + ρe±iω , 0 ≤ ρ < ∞. As −ω − φ < − π2 , we see for λ ∈ Γβ,− that (λ − AU )−1 = −

 eλz U (z) dz, γ−

where γ− : z = re−φ , 0 ≤ r < ∞. Therefore,    −tλ −1 − e (λ − AU ) dλ = eλ(z−t) U (z) dz dλ Γβ,−



Γβ,−



γ−

=

 e

γ−

λ(z−t)

U (z) dλ dz =

Γβ,−

γ−

1 U (z) dz. z−t

In a similar way, as ω + φ > π2 , we see that  Γβ,+

e−tλ (λ − A)−1 dλ = −





 eλ(z−t) U (z) dz dλ = −

Γβ,+

γ+

where γ+ : z = reiφ , 0 ≤ r < ∞. Hence,

γ+

1 U (z) dz, z−t

1 Analytic Semigroups

e−tAU =

123

     1 1 1 1 + − U (z) dz = U (z) dz 2πi 2πi |z−t|=ε z − t γ+ γ− z − t

with a sufficiently small radius ε > 0. Hence, by (1.10), e−tAU = U (t).



1.3 Perturbation for Generators of Analytic Semigroups We shall consider some perturbation of generators of analytic semigroup. Let A be a sectorial operator of a Banach space X with angle ωA < π2 . Let B a linear operator of X. For some exponent 0 < ν ≤ 1, we assume that B is dominated by A1−ν , where A1−ν is the fractional power of A, in the sense that D(A1−ν ) ⊂ D(B), and the estimate BAν−1  ≤ D

(3.8)

holds with some constant D > 0. Let ω > ωA . Since D(A) ⊂ D(B), we have

λ − (A + B) = 1 − B(λ − A)−1 (λ − A)

for λ ∈ / Σω .

Here,     B(λ − A)−1  ≤ BAν−1 A1−ν (λ − A)−1 . In addition, by the moment inequality (2.117) (with θ = 1 − ν),  1−ν      A (λ − A)−1  ≤ C A(λ − A)−1 1−ν (λ − A)−1 ν ≤ C|λ|−ν . Therefore, if λ ∈ / Σω and |λ| ≥ R and if R > 0 is fixed sufficiently large, then λ ∈ ρ(A + B), and the resolvent is given by

−1 (λ − (A + B))−1 = (λ − A)−1 1 − B(λ − A)−1 . This then means that A + B satisfies (3.2) and (3.3) with the ω fixed above and β = −(sin ω)−1 R. We thus obtain the following theorem. Theorem 3.3 Let A be a sectorial operator with angle ωA < π2 . If a linear operator B satisfies (3.8), then −(A + B) satisfies (3.2) and (3.3) with any angle ω > ωA . So, A + B generates an analytic semigroup e−z(A+B) .

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3 Linear Evolution Equations

2 Cauchy Problems 2.1 Construction of Solutions Let us consider the Cauchy problem for a linear evolution equation  dU 0 < t ≤ T, dt + AU = F (t), U (0) = U0 ,

(3.9)

in a Banach space X, where 0 < T < ∞ is fixed time. Here, A is a sectorial operator of X with angle ωA < π2 , that is, A satisfies (2.92) and (2.93). Throughout this section, the notation C will stand for a universal constant which is determined in each occurrence by the exponents and constants in (2.92)–(2.93) and by A−1  in a specific way. The function F is a given external force function defined in (0, T ] with values in X. We assume that F belongs to F ∈ Fβ,σ ((0, T ]; X),

0<σ <β ≤1

(3.10)

(see (1.5)). The initial value U0 is taken in X. We prove the following existence and uniqueness result. Theorem 3.4 Let A satisfy (2.92) and (2.93). For any function F satisfying (3.10) and any initial value U0 ∈ X, there exists a unique solution U to (3.9) in the function space: U ∈ C([0, T ]; X) ∩ C((0, T ]; D(A)) ∩ C1 ((0, T ]; X) with the estimate    dU   U (t) + t  (t)  + tAU (t) ≤ C(U0  + F Fβ,σ ), dt Moreover, U is necessarily given by the formula  t U (t) = e−tA U0 + e−(t−τ )A F (τ ) dτ,

0 < t ≤ T.

0 ≤ t ≤ T.

(3.11)

(3.12)

(3.13)

0

Proof For n = 1, 2, 3, . . . , let An be the Yosida approximation of A defined by (2.80). As shown in Chap. 2, Sect. 5.4, An also satisfies (2.92) and (2.93) uniformly and generates an analytic semigroups e−tAn . We then consider the function  t −tAn Un (t) = e U0 + e−(t−τ )An F (τ ) dτ, 0 ≤ t ≤ T . 0

Due to (2.101) and (2.102), as n → ∞, Un (t) is seen to converge to the function U (t) given by (3.13) in X pointwise. In addition, since An is a bounded operator,

2 Cauchy Problems

125

we verify directly that dUn = −An Un (t) + F (t), dt

0 < t ≤ T.

Therefore, with an arbitrary 0 < ε < T ,  t [F (τ ) − An Un (τ )] dτ, Un (t) = Un (ε) +

ε ≤ t ≤ T.

(3.14)

ε

Let us next observe the convergence of An Un (t). For this purpose, we write  t −tAn U0 + An e−(t−τ )An [F (τ ) − F (t)] dτ + (1 − e−tAn )F (t), An Un (t) = An e 0

where we used the formula  t

An e−(t−τ )An dτ = 1 − e−tAn .

(3.15)

0

By (1.8) and (2.133), we see that An Un (t) ≤ Ct −1 U0  + CF Fβ,σ



t

(t − τ )σ −1 τ β−σ −1 dτ

0

+ Ct β−1 F Fβ,σ ≤ C(t −1 U0  + t β−1 F Fβ,σ ),

0 < t ≤ T, (3.16)

C being independent of n. Furthermore, by (2.134), An Un (t) converges to  t Ae−(t−τ )A [F (τ ) − F (t)] dτ + [1 − e−tA ]F (t) W (t) = Ae−tA U0 + 0

in X pointwise for 0 < t ≤ T . In addition, W is a continuous function for 0 < t ≤ T (in fact, we can use an analogous argument to the proof of Lemma 1.3 in Chap. 1 for verifying the continuity of the second term). Letting n → ∞ in Un (t) = −1 A−1 n An Un (t), we obtain that U (t) = A W (t); this shows that U (t) ∈ D(A) and AU (t) = W (t) for every 0 < t ≤ T . In view of (3.16), we apply the Lebesgue dominate convergence theorem to (3.14) to obtain that  t [F (τ ) − AU (τ )] dτ, ε ≤ t ≤ T . U (t) = U (ε) + ε

This shows that U (t) is differentiable for ε ≤ t ≤ T . Since ε > 0 is arbitrary, U is a C1 function on (0, T ] and satisfies the equation there. It is easy from (3.13) to see that U is continuous even at t = 0 and satisfies the initial condition. It is also easy to verify that U satisfies (3.12). We have thus verified that U (t) is a solution in (3.11).

126

3 Linear Evolution Equations

We now show the uniqueness of solution. Let V be any solution in (3.11). Let 0 < ε < t ≤ T ; then ∂ −(t−τ )A V (τ ) = e−(t−τ )A F (τ ), e ∂τ

ε ≤ τ < t.

Integrating these functions in τ ∈ (ε, t), we have V (t) = e−(t−ε)A V (ε) +



t

e−(t−τ )A F (τ ) dτ,

ε < t ≤ T.

ε

Letting ε → 0, we conclude that V (t) = U (t) for every 0 < t ≤ T .



2.2 Maximal Regularity In addition to (3.10), let U0 be in D(Aβ ). Then, the solution U to (3.9) enjoys better regularity than (3.11). As a matter of fact, dU dt and AU belong to the same function space (3.10) as F and Ae−tA U0 do (recall (2.132)). In this sense, such a regularity is called the maximal regularity of solutions. Theorem 3.5 Let F ∈ Fβ,σ ((0, T ]; X) with 0 < σ < β ≤ 1, and let U0 ∈ D(Aβ ). Then, the solution U to (3.9) has the following regularity: Aβ U ∈ C([0, T ]; X),

(3.17)

dU , dt

(3.18)

AU ∈ Fβ,σ ((0, T ]; X),

with the estimates Aβ U C ≤ C[Aβ U0  + F Fβ,σ ],    dU  β    dt  β,σ + AU Fβ,σ ≤ C[A U0  + F Fβ,σ ]. F

(3.19) (3.20)

Proof Let us first prove (3.17). We already know that Aβ U (t) is continuous for t > 0. So, it suffices only to show that Aβ U (t) is continuous at t = 0. We use the expression A [U (t) − U0 ] = [e−tA − 1]Aβ U0 +



β

t

Aβ e−(t−τ )A [F (τ ) − F (t)] dτ

0

+ [1 − e

−tA

]A

β−1

F (t)

= J1 (t) + J2 (t) + J3 (t).

2 Cauchy Problems

127

Clearly, limt→0 J1 (t) = 0 in X. In addition, it is seen that 

t

J2 (t)X ≤ C

(t − τ )σ −β τ β−σ −1 dτ sup

0

0<τ
τ 1−β+σ F (t) − F (τ )X (t − τ )σ

≤ CB(1 − β + σ, β − σ )ωF (t). Therefore, from (1.7) it follows that limt→0 J2 (t) = 0 in X. Since J3 (t) = −t β−1 [e−tA − 1]Aβ−1 t 1−β F (t), it also follows from (2.131) that limt→0 J3 (t) = 0. In this way, Aβ [U (t) − U0 ] → 0 as t → 0. Estimate (3.19) has also been proved at the same time. Let us now prove (3.18). As above, we use the expression 

AU (t) = Ae−tA U0 +

t

Ae−(t−τ )A [F (τ ) − F (t)] dτ + [1 − e−tA ]F (t)

0

= I1 (t) + I2 (t) + I3 (t). By Theorem 2.27 we already know that I1 ∈ Fβ,σ ((0, T ]; X). So, it suffices to verify that Ii ∈ Fβ,σ ((0, T ]; X) for i = 2, 3. Proof for I2 The norm of I2 (t) is estimated by  I2 (t) ≤ C

t

(t − τ )σ −1 τ β−σ −1 dτ sup

0

≤ Ct

0<τ
β−1

τ 1−β+σ F (t) − F (τ )X (t − τ )σ

B(σ, β − σ )ωF (t).

Therefore, from (1.7) it follows that limt→0 t 1−β I2 (t) = 0 in X. We next observe that  t Ae−(t−τ )A [F (τ ) − F (t)] dτ I2 (t) − I2 (s) = s

+ [e−(t−s)A − 1]



s

Ae−(s−τ )A [F (τ ) − F (s)] dτ

0



t

+

Ae−(t−τ )A [F (s) − F (t)] dτ

s

= I21 (t, s) + I22 (t, s) + I23 (t, s). Then,  I21 (t, s) ≤ CωF (t) s

t

(t − τ )σ −1 τ β−σ −1 dτ ≤ C(t − s)σ s β−σ −1 ωF (t).

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3 Linear Evolution Equations

As for I22 (t, s), we estimate its norm as follows:  t−s   s   −ρA −(s−τ )A  Ae dρ Ae {F (τ ) − F (s)} dτ I22 (t, s) =    0

  = 

t−s

0



0

s 0



  A2 e−ρA e−(s−τ )A {F (τ ) − F (s)} dτ dρ   t−s

≤ CωF (s) 0



s

(ρ + s − τ )−2 (s − τ )σ τ β−σ −1 dτ dρ.

0

Furthermore, 

t−s



0

s

(ρ + s − τ )−2 (s − τ )σ τ β−σ −1 dτ dρ

0



s

= (t − s) 

(t − τ )−1 (s − τ )σ −1 τ β−σ −1 dτ

0 s

= (t − s)

(t − s + τ )−1 τ σ −1 (s − τ )β−σ −1 dτ.

0

Since  (t − s)

s s 2

(t − s + τ )−1 τ σ −1 (s − τ )β−σ −1 dτ σ −1



≤ 2(t − s) s

σ −1



s

(t − s)1−σ τ σ (t − s + τ )−1 (s − τ )β−σ −1 dτ

s 2

≤ 2(t − s) s

s

(s − τ )β−σ −1 dτ =

0

2 (t − s)σ s β−σ −1 β −σ

and 

s 2

(t − s)

(t − s + τ )−1 τ σ −1 (s − τ )β−σ −1 dτ

0

≤ (t − s)21−β+σ s β−σ −1



∞

(t − s + τ )−1 τ σ −1 dτ

0

=

21−β+σ π (t − s)σ s β−σ −1 , sin σ π

we get I22 (t, s) ≤ CωF (s)(t − s)σ s β−σ −1 ,

0 ≤ s < t ≤ T.

Finally, since I23 (t, s) = [e−(t−s)A − e−tA ][F (s) − F (t)],

(3.21)

2 Cauchy Problems

129

it is clear that I23 (t, s) ≤ CωF (t)(t − s)σ s β−σ −1 ,

0 ≤ s < t ≤ T.

In this way, we have concluded that I2 ∈ Fβ,σ ((0, T ]; X).



Proof for I3 Since t 1−β I3 (t) = [e−tA − 1]t 1−β F (t), it is easily seen that limt→0 t 1−β I3 (t) = 0. We next write I3 (t) − I3 (s) = [1 − e−tA ][F (t) − F (s)] − [e−(t−s)A − 1]A−σ Aσ e−sA F (s). Then, (1 − e−tA )[F (s) − F (t)] ≤ CωF (t)(t − s)σ s β−σ −1 ,

0 ≤ s < t ≤ T.

Similarly, by (2.129), [e−(t−s)A − 1]A−σ Aσ e−sA F (s) ≤ C(t − s)σ s β−σ −1 s σ Aσ e−sA s 1−β F (s), 0 ≤ s < t ≤ T. In addition, by (2.130), lims→0 s σ Aσ e−sA s 1−β F (s) = 0. In this way, I3 is verified to be in Fβ,σ ((0, T ]; X). We have thus proved that AU belongs to Fβ,σ ((0, T ]; X) and satisfies estidU mate (3.20). Since dU dt = −AU + F (t), dt also has the same property as AU and F .  Remark 3.1 As verified, limt→0 t 1−β I2 (t) = limt→0 t 1−β I3 (t) = 0 in X. Therefore, in view of (2.130), lim t 1−β AU (t) = 0 when 0 < β < 1,

t→0

lim t 1−β AU (t) = lim AU (t) = AU0

t→0

t→0

when β = 1.

2.3 Higher-Order Maximal Regularity If F is a continuously differentiable function with derivative F  ∈ Fβ,σ ((0, T ]; X) and if U0 and F (0) satisfy a compatibility condition, then we can show that the solution satisfies a maximal regularity of second order. Theorem 3.6 Let F ∈ Cβ ([0, T ]; X) ∩ C 1 ((0, T ]; X) and F  ∈ Fβ,σ ((0, T ]; X) with 0 < σ < β ≤ 1, and let U0 ∈ D(A) and F (0) satisfy the compatibility condition AU0 + F (0) ∈ D(Aβ ). Then, the solution U to (3.9) has the regularity dU d2U β 1 β,σ ((0, T ]; X). dt ∈ C ([0, T ]; X) ∩ C ((0, T ]; X) and dt 2 ∈ F

130

3 Linear Evolution Equations

Proof We notice that, since U0 ∈ D(A) and F ∈ F1,σ ((0, T ]; X), dU dt is also in F1,σ ((0, T ]; X). In particular, U is right differentiable at t = 0 with U+ (0) = AU0 + F (0). We then introduce the Cauchy problem for an equation of the form 

dU 1 1  dt + AU = F (t), U 1 (0) = U+ (0),

0 < t ≤ T,

for the unknown function U 1 = U 1 (t). Since U+ (0) ∈ D(Aβ ), Theorem 3.5 pro1

β,σ ((0, T ]; X). Let Δt > 0 vides the existence of a solution with derivative dU dt ∈ F be a small variable and put WΔt (t) = (Δt)−1 [U (t + Δt) − U (t)] − U 1 (t) for 0 ≤ t ≤ T − Δt. Then, it is easily seen that

 dW

+ AWΔt = GΔt (t),

Δt

dt

0 < t ≤ T − Δt,

W (0) = WΔt , where GΔt (t) = (Δt)−1 [F (t + Δt) − F (t)] − F  (t) and WΔt = (Δt)−1 [U (Δt) − U (0)] − U+ (0). We can now use Theorem 3.4 to obtain that WΔt (t) = e

−tA

 WΔt +

t

e−(t−s)A GΔt (s) ds,

0 ≤ t ≤ T − Δt.

0

Let Δt → 0. Then, WΔt → 0 in X, and GΔt (t) → 0 in X with the estimate 

1

GΔt (t) ≤ 0

F  (t + θ Δt) − F  (t) dθ ≤ F  Fβ,σ t β−σ −1 (Δt)σ .

Therefore, it follows that WΔt (t) → 0 in X, too. This means that the right derivative U+ of U is equal to U 1 for every 0 ≤ t < T . Of course, U  ≡ U+ ≡ U 1 . Hence, the desired result is proved. 

2.4 Spatial Regularity We remark that, if the external force function F has some spatial regularity, then the solution U to (3.9) does the same regularity. We assume that F ∈ Fβ,σ ((0, T ]; X) satisfies F (t) ∈ D(Aρ ) for all 0 < t ≤ T with the condition β−1−ρ

Aρ F ∈ B{0}

((0, T ]; X),

0 < ρ < σ.

(3.22)

Theorem 3.7 Let F ∈ Fβ,σ ((0, T ]; X), 0 < σ < β ≤ 1, satisfy (3.22) and let 1+ρ U also belong to Bβ−1−ρ ((0, T ]; X) with the U0 ∈ D(Aβ ). Then, Aρ dU {0} dt and A

3 Autonomous Parabolic Equations

131

estimate    ρ dU 

 A + A1+ρ U Bβ−1−ρ ≤ C Aβ U0  + F Fβ,σ + Aρ F Bβ−1−ρ .   {0} {0} dt Bβ−1−ρ {0}

Proof From the representation A1+ρ U (t) = A1+ρ−β e−tA Aβ U0  t + A1+ρ e−(t−τ )A [F (τ ) − F (t)] dτ + [1 − e−tA ]Aρ F (t) 0



we readily verify the desired result.

3 Autonomous Parabolic Equations We are concerned with the initial-boundary-value problem for the autonomous parabolic equation n ∂u Dj [aij (x)Di u] − c(x)u + f (x, t) = ∂t

in Ω × (0, T )

i,j =1

in a domain Ω ⊂ Rn , where 0 < T < ∞ is fixed time. The second-order differential n operator −Σi,j =1 Dj [aij (x)Di ·] + c(x) is a strongly elliptic operator. Such an equation is then rewritten abstractly as an evolution of the form (3.9) in which the linear operator is a realization of the elliptic operator under suitable boundary conditions. We can enjoy an advantage of the unified treatment of parabolic equations, for all abstract results in this chapter together with last two are available to these abstract parabolic equations. In this section, however, we will describe only a few examples. As for the concrete characterization of the domains of fractional powers of strongly elliptic operators, see Chap. 16.

3.1 Problems in Hilbert Spaces Case where Ω = Rn . Consider the initial value problem  ∂u n n i,j =1 Dj [aij (x)Di u] − c(x)u + f (x, t) in R × (0, T ), ∂t = u(x, 0) = u0 (x)

in Rn .

(3.23)

Here, real-valued functions aij (x) satisfy conditions (2.14) and (2.15), and c(x) condition (2.16). Let us handle this problem in the space X = H −1 (Rn ), for example. Then, Problem (3.23) is formulated as an abstract problem of form

132

3 Linear Evolution Equations

−1 n (3.9) linear operator is a realization A of

nin H (R ) in which the coefficient − i,j =1 Dj [aij (x)Di ·] + c(x) in H −1 (Rn ). According to Theorem 2.2, A is a sectorial operator of H −1 (Rn ) with domain D(A) = H 1 (Rn ) and with angle ωA < π2 , and consequently −A generates an analytic semigroup on H −1 (Rn ). Hence, all the results obtained above are available. For example, Theorem 3.5 yields that, for any f ∈ Fβ,σ ((0, T ]; H −1 (Rn )), where 0 < σ < β ≤ 1, and any u0 ∈ D(Aβ ), (3.23) possesses a unique solution in the function space:

 u ∈ C((0, T ]; H 1 (Rn )) ∩ C([0, T ]; D(Aβ )) ∩ C1 ((0, T ]; H −1 (Rn )), Au ∈ Fβ,σ ((0, T ]; H −1 (Rn )),

du dt ,

with the estimates    du  β  A uC +   dt  β,σ + AuFβ,σ ≤ C(A u0 H −1 + f Fβ,σ ). F β

Dirichlet conditions. Consider the initial-boundary-value problem ⎧

∂u ⎪ = ni,j =1 Dj [aij (x)Di u] − c(x)u + f (x, t) in Ω × (0, T ), ⎪ ⎨ ∂t u=0 on ∂Ω × (0, T ), ⎪ ⎪ ⎩ in Ω, u(x, 0) = u0 (x)

(3.24)

in a domain Ω ⊂ Rn . Here, real-valued functions aij (x) satisfy conditions (2.18) and (2.19), and c(x) condition (2.20). Let us handle this problem again in X = H −1 (Ω) = {H˚ 1 (Ω)}∗ . Then, Problem (3.24) is formulated as an abstract problem −1 of form linear operator is a realization A

n(3.9) in H (Ω) in which the coefficient of − i,j =1 Dj [aij (x)Di ·] + c(x) in H −1 (Ω) under the Dirichlet boundary conditions. According to Theorem 2.3, A is a sectorial operator of H −1 (Ω) with domain D(A) = H˚ 1 (Ω) and with angle ωA < π2 , and consequently −A generates an analytic semigroup on H −1 (Ω). We can state, for example, applying Theorem 3.5, that for any f ∈ Fβ,σ ((0, T ]; −1 H (Ω)), where 0 < σ < β ≤ 1, and any u0 ∈ D(Aβ ), (3.24) possesses a unique solution in the function space: 

u ∈ C((0, T ]; H˚ 1 (Ω)) ∩ C([0, T ]; D(Aβ )) ∩ C1 ((0, T ]; H −1 (Ω)), du dt ,

Au ∈ Fβ,σ ((0, T ]; H −1 (Ω)),

with the estimates    du  β  Aβ uC +   dt  β,σ + AuFβ,σ ≤ C(A u0 H −1 + f Fβ,σ ). F

3 Autonomous Parabolic Equations

Neumann-type conditions. Consider the initial-boundary-value problem ⎧

n ∂u ⎪ ⎪ i,j =1 Dj [aij (x)Di u] − c(x)u + f (x, t) in Ω × (0, T ), ∂t = ⎨ n on ∂Ω × (0, T ), i,j =1 νj (x)aij (x)Di u = 0 ⎪ ⎪ ⎩ in Ω, u(x, 0) = u0 (x)

133

(3.25)

in a domain Ω ⊂ Rn . Here, real-valued functions aij (x) satisfy conditions (2.18) and (2.19), and c(x) condition (2.20). We handle the problem in X = H 1 (Ω)∗ . Problem (3.25) is then written in the form (3.9) in the space H 1 (Ω)∗ in which the coefficient linear operator is a realization A of − ni,j =1 Dj [aij (x)Di ·] + c(x) in H 1 (Ω) under the Neumann-type boundary conditions. According to Theorem 2.4, A is a sectorial operator of H 1 (Ω)∗ with domain D(A) = H 1 (Ω) and with angle ωA < π2 . So, e−tA is an analytic semigroup on H 1 (Ω)∗ . By a direct application of Theorem 3.5, it follows that, for any f ∈ Fβ,σ ((0, T ]; 1 H (Ω)∗ ), where 0 < σ < β ≤ 1, and any u0 ∈ D(Aβ ), (3.25) possesses a unique solution in the function space:  u ∈ C((0, T ]; H 1 (Ω)) ∩ C([0, T ]; D(Aβ )) ∩ C1 ((0, T ]; H 1 (Ω)∗ ), Au ∈ Fβ,σ ((0, T ]; H 1 (Ω)∗ ),

du dt ,

with the estimate

   du  β  A uC +   dt  β,σ + AuFβ,σ ≤ C(A u0 H 1∗ + f Fβ,σ ). F β

3.2 Problems in Lp Spaces Consider the initial-boundary-value problem ⎧

n ∂u ⎪ ⎪ i,j =1 Dj [aij (x)Di u] − c(x)u + f (x, t) in Ω × (0, T ), ∂t = ⎨ n on ∂Ω × (0, T ), i,j =1 νj (x)aij (x)Di u = 0 ⎪ ⎪ ⎩ in Ω, u(x, 0) = u0 (x)

(3.26)

in a bounded domain Ω ⊂ Rn with C2 boundary ∂Ω. Here, real-valued functions aij (x) satisfy conditions (2.19) and (2.28), and c(x) condition (2.20). Let us handle this problem in X = Lp (Ω), 1 < p < ∞. For this purpose, we use the realization Ap of − i,j Dj [aij (x)Di ·] + c(x) in Lp (Ω) under the Neumann-type boundary conditions on ∂Ω which was defined in Chap. 2, Sect. 2.1. According to Theorem 2.12, the realization Ap is a sectorial operator of Lp (Ω) with angle ωAp < π2 and with domain D(Ap ) given by (2.77) (due to the shift property). So, −Ap generates an analytic semigroup on Lp (Ω). Taking this Ap as the coefficient linear operator, Problem (3.26) is written as an abstract problem of form (3.9)

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3 Linear Evolution Equations

in Lp (Ω). Then, by a direct application of Theorem 3.5, it follows that, for any β f ∈ Fβ,σ ((0, T ]; Lp (Ω)), 0 < σ < β ≤ 1, and any u0 ∈ D(Ap ), (3.26) possesses a unique solution u in the function space:  β u ∈ C((0, T ]; Hp2 (Ω)) ∩ C([0, T ]; D(Ap )) ∩ C1 ((0, T ]; Lp (Ω)), Ap u ∈ Fβ,σ ((0, T ]; Lp (Ω)),

du dt ,

with the estimate

   β      β  A u +  du    p  dt  β,σ + Ap uFβ,σ ≤ C Ap u0 Lp + f Fβ,σ . C F

Part II. Nonautonomous Equations 4 Nonautonomous Abstract Evolution Equations Let X be a Banach space with norm  · . We consider the Cauchy problem for a linear abstract evolution equation  dU dt + A(t)U = F (t), 0 < t ≤ T , U (0) = U0 , in X, where 0 < T < ∞ is fixed time. Here, A(t) is a sectorial operator of X with angle ωA(t) < π2 , that is, −A(t) generates an analytic semigroup e−τ A(t) on X. The function F is a given external force function on (0, T ]. The initial value U0 is taken in X.

4.1 Structural Assumptions In order to handle the Cauchy problem, we make the following assumptions for a family of the operators A(t) as structural assumptions. For 0 ≤ t ≤ T , the spectrum of A(t) is contained in a sectorial open domain σ (A(t)) ⊂ Σω = {λ ∈ C; |arg λ| < ω},

0 ≤ t ≤ T,

with some fixed angle 0 < ω < π2 , and the resolvent satisfies the estimate   (λ − A(t))−1  ≤ M/|λ|, λ ∈ / Σω , 0 ≤ t ≤ T ,

(3.27)

(3.28)

with some constant M ≥ 1. The domain D(A(t)) is allowed to vary with t, but there exists some fixed exponent 0 < ν ≤ 1 such that   D(A(s)) ⊂ D A(t)ν for all 0 ≤ s, t ≤ T . (3.29)

4 Nonautonomous Abstract Evolution Equations

135

(When ν = 1, this condition means that the domain D(A(t)) is independent of t .) In addition, A(t)−1 is Hölder continuous in t in the sense that 

 A(t)ν A(t)−1 − A(s)−1  ≤ N |t − s|μ , 0 ≤ s, t ≤ T , (3.30) with some fixed exponent 0 < μ ≤ 1 and some constant N > 0. The exponents μ and ν satisfy the relation 1 < μ + ν.

(3.31)

Remark 3.2 It is possible to handle the Cauchy problem also in the critical case where ν = 0 if we strengthen conditions (3.30) and (3.31) as (3.114) and (3.115) which are announced in Sect. 10.

4.2 Some Consequences from (3.30) Let a family of sectorial operators A(t) of X satisfy (3.27), (3.28), (3.29), (3.30), and (3.31). We shall collect some immediate consequences from these conditions, especially from (3.30). Condition (3.27) implies that A(t) has a bounded inverse on X. Since

A(t)−1 = A(s)−1 + A(s)−ν A(s)ν A(t)−1 − A(s)−1 , (3.30) (together with (2.118)) yields that     A(t)−1 − A(s)−1  ≤ 2(M + 1)N A(s)−1 ν |t − s|μ ,

0 ≤ s, t ≤ T .

(3.32)

This shows that A(·)−1 is in Cμ ([0, T ]; L(X)). In particular, the norm A(t)−1  is uniformly bounded, i.e.,       A(t)−1  ≤ A(0)−1  + 2(M + 1)N T μ A(0)−1 ν , 0 ≤ t ≤ T . Throughout Sects. 4, 5, 6, 7, and 8, the notation C will stand for a universal constant which is determined in each occurrence by the exponents and constants appearing in (3.27), (3.28), (3.29), (3.30), and (3.31) and by A(0)−1  in a specific way. Let e−τ A(t) be an analytic semigroup generated by −A(t). We can deduce from (3.30) regularity of e−τ A(t) with respect to the variable t . Proposition 3.1 Let 0 ≤ θ ≤ 2. Then, A(t)θ e−τ A(t) is an L(X)-valued continuous function for both τ > 0 and 0 ≤ t ≤ T . Proof First, fix t ∈ [0, T ]. Then, as shown in Chap. 2, Sect. 7.7, A(t)θ e−τ A(t) is continuous for τ > 0 as an L(X)-valued function. Hence, the proof of proposition will be completed if we verify the following lemma. 

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3 Linear Evolution Equations

Lemma 3.2 For 0 ≤ θ ≤ 2,   A(t)θ e−τ A(t) − A(s)θ e−τ A(s)  ≤ Cτ ν−θ−1 |t − s|μ ,

τ > 0, 0 ≤ s, t ≤ T .

Proof Notice the formula (λ − A(t))−1 − (λ − A(s))−1

= −A(t)(λ − A(t))−1 A(t)−1 − A(s)−1 A(s)(λ − A(s))−1 . Then, by the same techniques as for (3.32), we observe that   (λ − A(t))−1 − (λ − A(s))−1  ≤ C|λ|−ν |t − s|μ , λ ∈ / Σω , 0 ≤ s, t ≤ T . (3.33) We use (2.127) and write A(t)θ e−τ A(t) − A(s)θ e−τ A(s) =

1 2πi





λθ e−τ λ (λ − A(t))−1 − (λ − A(s))−1 dλ, Γω

where Γω : λ = ρe±iω , 0 ≤ ρ < ∞, is an integral contour. Then, using (3.33), we obtain that    A(t)θ e−τ A(t) − A(s)θ e−τ A(s)  ≤ C |λ|θ−ν e−τ Re λ |dλ||t − s|μ Γω

≤ Cτ ν−θ−1 |t − s|μ . When θ = 0 and ν = 1, this integral may not be convergent in a neighborhood λ = 0. But in this case, we are able to shift the integral contour Γω in the same way as for proving (2.128). Then, the assertion of lemma is proved without any change.  When θ = 0, the continuity of the semigroup is extended at τ = 0. In fact, we verify the following proposition. Proposition 3.2 e−τ A(t) is strongly continuous on X for τ ≥ 0 and 0 ≤ t ≤ T . Proof Obviously, it suffices to prove the result at τ = 0 only. We already know that   −τ A(t) [e − 1]A(t)−ν  ≤ Cτ ν , τ ≥ 0, 0 ≤ t ≤ T . Let U ∈ D(A(0)). Then,   [e−τ A(t) − 1]U  ≤ [e−τ A(t) − 1]A(t)−ν A(t)ν A(0)−1 A(0)U  ≤ CA(0)U τ ν ,

τ ≥ 0, 0 ≤ t ≤ T .

Furthermore, the denseness of D(A(0)) in X, together with the uniform boundedness of the norms e−τ A(t) − 1, yields that e−τ A(t) F → F in X as τ → 0 for every F ∈ X, the convergence being uniform in 0 ≤ t ≤ T . 

4 Nonautonomous Abstract Evolution Equations

137

4.3 Structural Assumptions and Yosida Approximation Let a family of sectorial operators A(t) satisfy (3.27), (3.28), (3.29), (3.30), and (3.31). For each n = 1, 2, 3, . . . , let An (t), 0 ≤ t ≤ T , be a family of Yosida approximations of A(t). The purpose of this subsection is to show that An (t), 0 ≤ t ≤ T , also satisfy these conditions uniformly in n; especially ω, μ, and ν can be the same as for A(t). As already verified by (2.84) and (2.86), σ (An (t)) ⊂ Σω , and the estimate   (λ − An (t))−1  ≤ M/|λ|, ˜ (3.34) λ∈ / Σω , 0 ≤ t ≤ T , holds with some constant M˜ ≥ 1 determined by M and ω only. In the meantime, it is possible to verify from (3.30) that 

 An (t)ν An (t)−1 − An (s)−1  ≤ N˜ |t − s|μ , 0 ≤ s, t ≤ T .

(3.35)

In fact, since An (t)−1 = n−1 + A(t)−1 due to (2.81), we have An (t)−1 − An (s)−1 = A(t)−1 − A(s)−1 . Then, (3.35) follows as a consequence of the two propositions presented below. Proposition 3.3 Let 0 < θ ≤ 1. For n = 1, 2, 3, . . . , it holds that An (t)θ A(t)−θ = (1 + n−1 A(t))−θ ,

0 ≤ t ≤ T.

(3.36)

Proof When θ = 1, the assertion is clear. So, we consider the case where 0 < θ < 1 only. Let us prove that A(t)−θ = An (t)−θ (1 + n−1 A(t))−θ . From (λ − (1 + n−1 A(t)))−1 = n(n(λ − 1) − A(t))−1 it is seen that σ (1 + −1 n A(t)) ⊂ Σω and the estimate   (λ − (1 + n−1 A(t)))−1  ≤

M M ≤ , |λ − 1| |λ| sin ω

λ∈ / Σω ,

(3.37)

holds. This means that 1 + n−1 A(t) are also sectorial operators with angles < ω. Their fractional powers are then given by  1 (1 + n−1 A(t))−θ = λ−θ (λ − (1 + n−1 A(t)))−1 dλ 2πi Γλ  1 λ−θ n(n(λ − 1) − A(t))−1 dλ. = 2πi Γλ Since 0 < θ < 1, we can take an integral contour which passes the origin. Here, we choose Γλ : λ = re±iω , 0 ≤ r < ∞. Then, change of variable η = n(λ − 1) yields that  1 (1 + n−1 η)−θ (η − A(t))−1 dη, (1 + n−1 A(t))−θ = 2πi Γη where Γη = Γλ − n, and the branch of the function (1 + n−1 η)−θ is C − (−∞, −n].

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3 Linear Evolution Equations

In the meantime, we observe by (2.81) that  −1  λ−θ 1 nλ dλ, An (t)−θ = (n + A(t)) − A(t) 2πi Γλ n − λ n−λ where Γλ is the same contour as above. We here use the integral formula  −1 −1   nλ 1 nλ  = (η − A(t))−1 dη , λ ∈ Γλ , − A(t) −η n−λ 2πi Γη n − λ 

choosing a contour Γη = Γω ∪ Γδ such that Γω : η = re±iω , δ ≤ r < ∞, and Γδ : η = δeiϕ , −ω ≤ ϕ ≤ ω , where ω is an angle such that ωA(t) < ω < ω, and δ is a radius such that 0 < δ < A(t)−1 −1 . Then, on D(A(t)), we obtain that  −1   λ−θ 1 nλ  (η − A(t))−1 dλ dη (n + A(t)) − η An (t)−θ = (2πi)2 Γλ Γη n − λ n − λ    1 1 nη −θ  = (η − A(t))−1 dη (n + A(t)). 2πi Γη n + η n + η Noting that

nη n+η

1 2πi

∈ Σω for η ∈ Γη , we used the formula  Γλ

λ−θ dλ =  λ − nη (n + η )−1



nη n + η

−θ ,

η  ∈ Γη  .

 nη −θ is given by C − [−n, 0]. We easily notice that the branch of the function n+η  We now consider the product    1 nη −θ  1 (η − A(t))−1 dη 2πi Γη n + η n + η  1 × (1 + n−1 η)−θ (η − A(t))−1 dη. 2πi Γη Then, it is deduced by the same techniques as for (2.98) and (2.111) (cf. also (16.4)) that this product is equal to    (1 + n−1 λ)−θ nλ −θ 1 (λ − A(t))−1 dλ. 2πi Γλ n+λ n+λ  nλ −θ  n −θ −θ  nλ  Furthermore, we observe that n+λ = = n+λ λ for λ ∈ Γλ , since arg n+λ arg(nλ) − arg(n + λ) and −π < arg(nλ) − arg(n + λ) < π . We therefore arrive at    (1 + n−1 λ)−θ nλ −θ 1 (λ − A(t))−1 dλ = A(t)−θ (n + A(t))−1 . 2πi Γλ n+λ n+λ Hence, we proved that A(t)−θ = An (t)−θ (1 + n−1 A(t))−θ .



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139

Proposition 3.4 Let 0 < θ ≤ 1. For n = 1, 2, 3, . . . , it holds that     An (t)θ A(t)−θ  ≤ 2 M + 1 M θ , 0 ≤ t ≤ T . sin ω

(3.38)

In addition, for each t , An (t)θ A(t)−θ converges to 1 strongly on X as n → ∞. Proof Applying the moment inequality (2.118) to 1 + n−1 A(t), we have         (1 + n−1 A(t))−θ  ≤ 2 M + 1 (1 + n−1 A(t))−1 θ ≤ 2 M + 1 M θ sin ω sin ω due to (3.37). Hence, (3.38) is an immediate consequence of (3.36). The strong convergence of An (t)θ A(t)−θ to 1 is first verified on the domain D(A(t)1−θ ). Indeed, it is seen that   An (t)θ A(t)−θ U = An (t)θ−1 (1 + n−1 A(t))−1 A(t)1−θ U → U, U ∈ D A(t)1−θ . Then, the convergence on X is verified by the uniform boundedness of An (t)θ A(t)−θ and the denseness of D(A(t)1−θ ) in X. 

5 Evolution Operators In this section, we shall construct under (3.27), (3.28), (3.29), (3.30), and (3.31) the evolution operator U (t, s) for the family of sectorial operators A(t). For each n = 1, 2, 3, . . . , let Un (t, s), 0 ≤ s ≤ t ≤ T , be the evolution operator for the family of Yosida approximations An (t), 0 ≤ t ≤ T , of A(t). In fact, Un (t, s) is obtained as a unique solution to the Cauchy problem of the abstract differential equation  dU n dt + An (t)Un = 0, s ≤ t ≤ T , Un (s) = 1, in the space L(X); note that An (·) ∈ C([0, T ]; L(X)) (due to (3.32) and (3.33)). Obviously, the problem is equivalent to the integral equation  t An (τ )Un (τ ) dτ, s ≤ t ≤ T . Un (t) = 1 + s

Then, the solution is constructed by the recurrence method as in the proof of Theorems 1.29 and 1.30. The evolution operator Un (t, s) is a continuous function defined for 0 ≤ s ≤ t ≤ T with values in L(X). By the definition, Un (t, s) enjoys the generalized semigroup property  Un (t, r)Un (r, s) = Un (t, s), 0 ≤ s ≤ r ≤ t ≤ T , (3.39) 0 ≤ s ≤ T. Un (s, s) = 1,

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3 Linear Evolution Equations

In addition, Un (t, s) is differentiable both in t and s with the derivatives ∂ Un (t, s) = −An (t)Un (t, s), ∂t ∂ Un (t, s) = Un (t, s)An (s). ∂s

(3.40) (3.41)

Verification of (3.41) is not direct. We have to consider another family of operators U˜ n (t, s), 0 ≤ s ≤ t ≤ T , where U˜ n (t, s) is a solution to the backward Cauchy problem  ˜ d Un ˜ ds − Un An (s) = 0, 0 ≤ s ≤ t, U˜ n (t) = 1, for 0 < t ≤ T . Clearly, U˜ n (t, s) satisfies (3.41). Moreover, Un (t, s) = U˜ n (t, s) for every 0 ≤ s ≤ t ≤ T . Indeed, since ∂ ˜ Un (t, τ )Un (τ, s) = 0 for all τ in (s, t), ∂τ it is deduced that U˜ n (t, τ )Un (τ, s) is constant in τ ; hence, taking τ = s and t, we conclude that Un (t, s) = U˜ n (t, s) for every 0 ≤ s ≤ t ≤ T .

5.1 Integral Equations We first derive two integral equations for Un (t, s). By using (3.41) we have  t ∂ [Un (t, τ )e−(τ −s)An (s) ] dτ Un (t, s) − e−(t−s)An (s) = − ∂τ s  t

Un (t, τ )An (τ ) An (τ )−1 − An (s)−1 = s

× An (s)e−(τ −s)An (s) dτ. Operating An (s)1−ν to this equality from the right-hand side and putting Vn (t, s) = Un (t, s)An (s)1−ν , 0 ≤ s ≤ t ≤ T ,

Dn (t, s) = An (t)ν An (t)−1 − An (s)−1 , 0 ≤ s ≤ t ≤ T , we obtain the first integral equation Vn (t, s) = An (s)1−ν e−(t−s)An (s)  t + Vn (t.τ )Dn (τ, s)An (s)2−ν e−(τ −s)An (s) dτ. s

(3.42)

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141

To obtain the other one, we use (3.40). Then, Un (t, s) − e−(t−s)An (t)  t

=− An (t)e−(t−τ )An (t) An (t)−1 − An (τ )−1 An (τ )Un (τ, s) dτ. s

Operating An (t) to this equality from the left-hand side and putting Wn (t, s) = An (t)Un (t, s) − An (t)e−(t−s)An (t) ,

0 ≤ s ≤ t ≤ T,

we obtain the integral equation  t Wn (t, s) = Rn (t, s) − An (t)2−ν e−(t−τ )An (t) Dn (t, τ )Wn (τ, s) dτ,

(3.43)

s

where

 Rn (t, s) = −

t

An (t)2 e−(t−τ )An (t)

s

× [An (t)−1 − An (τ )−1 ]An (τ )e−(τ −s)An (τ ) dτ.

(3.44)

Secondly, we shall derive some norm estimates for Un (t, s) which hold uniformly in n. From (2.133) and (3.35) it is readily seen that the integral kernels of (3.42) satisfy   An (s)1−ν e−(t−s)An (s)  ≤ C(t − s)ν−1 ,   Dn (t, s)An (s)2−ν e−(t−s)An (s)  ≤ C(t − s)μ+ν−2 , respectively. Then, we can apply Theorem 1.30 to (3.42). As a direct consequence, we have the following proposition. Proposition 3.5 The operator Un (t, s)An (s)1−ν satisfies   Un (t, s)An (s)1−ν  ≤ C(t − s)ν−1 , 0 ≤ s < t ≤ T ,

(3.45)

C > 0 being independent of n. Concerning the integral kernels of (3.43), we readily verify from (2.133) and (3.35) that   An (t)2−ν e−(t−s)An (t) Dn (t, s) ≤ C(t − s)μ+ν−2 . (3.46) As for Rn (t, s), however, we need more careful calculations. Lemma 3.3 Rn (t, s) satisfies the estimates Rn (t, s) ≤ C(t − s)μ+ν−2 , C > 0 being independent of n.

0 ≤ s < t ≤ T,

(3.47)

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3 Linear Evolution Equations

Proof Because of  

An (t)2 e−(t−τ )An (t) An (t)−1 − An (τ )−1 An (τ )e−(τ −s)An (τ )  ≤ C(t − τ )μ+ν−2 (τ − s)−1 , we cannot obtain the desired estimate by direct estimation of the integral (3.44). We have to decompose the integral suitably into several terms which admit uniform estimates. As a matter of fact, we write Rn (t, s) in the form  t

An (t)2 e−(t−τ )An (t) An (τ )−1 − An (t)−1 An (τ )e−(τ −s)An (τ ) dτ Rn (t, s) = r



r

+ 

An (t)2 e−(t−τ )An (t) [e−(τ −s)An (τ ) − e−(τ −s)An (s) ] dτ

s r

+

An (t)e−(t−τ )An (t) [An (s)e−(τ −s)An (s) − An (τ )e−(τ −s)An (τ ) ] dτ

s

+ An (t)e−(t−r)An (t) [e−(r−s)An (s) − e−(r−s)An (t) ], where r =

t+s 2 .

(3.48)

Here, we used the formula  r An (t)e−(t−τ )An (t) [An (t) − An (s)]e−(τ −s)An (s) dτ s

= An (t)e−(t−r)An (t) [e−(r−s)An (s) − e−(r−s)An (t) ]. As the family of Yosida approximations An (t) also satisfies the same structural assumptions, we can apply Lemma 3.2 to An (t). In other words, for 0 ≤ θ ≤ 2, it holds that   An (t)θ e−τ An (t) − An (s)θ e−τ An (s)  ≤ Cτ ν−θ−1 |t − s|μ , τ > 0, 0 ≤ s, t ≤ T . (3.49) This estimate is then available to estimate the decomposed terms in the right-hand side of (3.48). Indeed, by (2.133), (3.35), and (3.49) (with θ = 0, 1), we can obtain estimate (3.47). The details of the proof are however left to the reader.  It is now ready to apply Theorem 1.29 to (3.43). In fact, we obtain the following estimate. Proposition 3.6 The operator An (t)Un (t, s) satisfies An (t)Un (t, s) ≤ C(t − s)−1 ,

0 ≤ s < t ≤ T,

(3.50)

C > 0 being independent of n. Proof Applying Theorem 1.29 to (3.43), we immediately obtain that Wn (t, s) ≤ C(t − s)μ+ν−2 .

(3.51)

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143

Then, since An (t)e−(t−s)An (t)  ≤ C(t − s)−1 , we conclude the estimate of proposition. 

5.2 Convergence of Un (t, s) In this subsection, we intend to see convergence of Un (t, s) as n → ∞. To this end, we will utilize the dominated convergence theorem, Theorem 1.31, to the integral equation (3.43). From (2.102) and Proposition 3.4, we observe that, as n → ∞, An (t)2−ν e−(t−s)An (t) Dn (t, s) → A(t)2−ν e−(t−s)A(t) D(t, s) strongly on X, where

D(t, s) = A(t)ν A(t)−1 − A(s)−1 , As

0 ≤ s < t ≤ T.



A(t)2−ν e−(t−s)A(t) D(t, s) = A(t)2 e−(t−s)A(t) A(t)−1 − A(s)−1 ,

the limit function A(t)2−ν e−(t−s)A(t) D(t, s) is continuous for 0 ≤ s < t ≤ T in L(X). For verifying convergence of Rn (t, s), we return to the decomposition (3.48). Then, by (2.102), Rn (t, s) is seen to converge strongly to the bounded linear operator  t

R(t, s) = A(t)2 e−(t−τ )A(t) A(τ )−1 − A(t)−1 A(τ )e−(τ −s)A(τ ) dτ r



r

+ 

A(t)2 e−(t−τ )A(t) [e−(τ −s)A(τ ) − e−(τ −s)A(s) ] dτ

s r

+

A(t)e−(t−τ )A(t) [A(s)e−(τ −s)A(s) − A(τ )e−(τ −s)A(τ ) ] dτ

s

+ A(t)e−(t−r)A(t) [e−(r−s)A(s) − e−(r−s)A(t) ], where r = t+s 2 , for all 0 ≤ s < t ≤ T . Furthermore, by the same techniques as in the proof of Lemma 1.3, R(t, s) is verified to be continuous in L(X) for 0 ≤ s < t ≤ T . It is now ready to apply Theorem 1.31 to (3.43). Indeed, in view of (3.46) and (3.47), Theorem 1.31 immediately provides the following convergence result. Proposition 3.7 As n → ∞, Wn (t, s) converges to a bounded linear operator W (t, s)

(3.52)

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3 Linear Evolution Equations

strongly on X for all 0 ≤ s < t ≤ T . The limit W (t, s) is characterized as a solution to the integral equation  t W (t, s) = R(t, s) − A(t)2−ν e−(t−τ )A(t) D(t, τ )W (τ, s) dτ. (3.53) s

Since Un (t, s) = e−(t−s)An (t) + An (t)−1 Wn (t, s), as a consequence of the proposition it follows that Un (t, s) is also strongly convergent. We then define the operator U (t, s) as the limit of Un (t, s), namely, as n → ∞, Un (t, s) → U (t, s) = e−(t−s)A(t) + A(t)−1 W (t, s)

(3.54)

strongly on X for all 0 ≤ s < t ≤ T . We see from relation (3.54) that R(U (t, s)) ⊂ D(A(t)) and have A(t)U (t, s) = A(t)e−(t−s)A(t) + W (t, s). As a consequence, An (t)Un (t, s) → A(t)U (t, s) = A(t)e−(t−s)A(t) + W (t, s)

(3.55)

strongly on X for all 0 ≤ s < t ≤ T . If U ∈ D(A(s)1−ν ), then Un (t, s)An (s)1−ν U → U (t, s)A(s)1−ν U . But, since the norms Un (t, s)An (s)1−ν  are uniformly bounded in n (due to (3.45)), we verify that U (t, s)A(s)1−ν has a bounded extension on X. Moreover, as n → ∞, Un (t, s)An (s)1−ν → U (t, s)A(s)1−ν

(3.56)

strongly on X for all 0 ≤ s < t ≤ T .

5.3 Basic Properties of U (t, s) For 0 ≤ s ≤ T , we define U (s, s) = 1. Then, by (3.39), U (t, s) also enjoys the generalized semigroup property  U (t, s) = U (t, r)U (r, s), 0 ≤ s ≤ r ≤ t ≤ T , (3.57) U (s, s) = 1, 0 ≤ s ≤ T . The operator U (t, s) has the following properties. Proposition 3.8 A(t)U (t, s) is an L(X)-valued continuous function for 0 ≤ s < t ≤ T satisfying the estimate A(t)U (t, s) ≤ C(t − s)−1 , with some constant C > 0.

0 ≤ s < t ≤ T,

(3.58)

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145

Proof As W (t, s) has been obtained as a solution to the integral equation (3.53), W (t, s) is an L(X)-valued continuous function for 0 ≤ s < t ≤ T . This, together with Proposition 3.1 (θ = 1), then yields the continuity of A(t)U (t, s) in the uniform topology. Estimate (3.58) is an immediate consequence of (3.50) and (3.55).  Proposition 3.9 U (t, s) is an L(X)-valued strongly continuous function for 0 ≤ s ≤ t ≤ T with the estimate U (t, s) ≤ C,

0 ≤ s ≤ t ≤ T,

(3.59)

with some constant C > 0. Proof From (3.42) we have Un (t, s) = e

−(t−s)An (s)

 +

t

Vn (t, τ )Dn (τ, s)An (s)e−(τ −s)An (s) dτ.

s

Therefore, by (2.102), (3.35), and (3.45), Un (t, s) is estimated by  t

Un (t, s) ≤ C + C (t − τ )ν−1 (τ − s)μ−1 dτ ≤ C 1 + (t − s)μ+ν−1 ≤ C. s

This, together with (3.54), then yields estimate (3.59). As U (t, s) = A(t)−1 A(t)U (t, s), Proposition 3.8, together with (3.32), yields the continuity of U (t, s) in L(X) for 0 ≤ s < t ≤ T . So, it suffices to verify that U (t, s) → 1 strongly as (t, s) → (r, r) with s ≤ t , where 0 ≤ r ≤ T . By the same reason as above, we have Un (t, s) − e−(t−s)An (s)  ≤ C(t − s)μ+ν−1 . So, letting n → ∞, U (t, s) − e−(t−s)A(s)  ≤ C(t − s)μ+ν−1 . Therefore, as (t, s) → (r, r), U (t, s) − e−(t−s)A(s)  → 0. Then, the desired strong convergence of U (t, s) is deduced from Proposition 3.2.  We finally show the differentiability of U (t, s) in t . Let 0 ≤ s < t ≤ T . In view of (3.50), let n → ∞ in the formula  t+h −1 −1 h [Un (t + h, s) − Un (t, s)] = −h An (τ )Un (τ, s) dτ. t

By (3.55) we obtain that −1

h

−1



t+h

[U (t + h, s) − U (t, s)] = −h

A(τ )U (τ, s) dτ, t

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3 Linear Evolution Equations

which shows that U (t, s) is differentiable in t in the uniform topology with the derivative ∂ U (t, s) = −A(t)U (t, s), 0 ≤ s < t ≤ T . (3.60) ∂t We here sum up the obtained results concerning the evolution operator. Theorem 3.8 Let A(t) be a family of sectorial operators of X satisfying the structural assumptions (3.27), (3.28), (3.29), (3.30), and (3.31). Then, there exists a unique evolution operator U (t, s) satisfying (3.57) and having the following properties. A(t)U (t, s) is an L(X)-valued continuous function for 0 ≤ s < t ≤ T with estimate (3.58), and U (t, s) is an L(X)-valued strongly continuous function for 0 ≤ s ≤ t ≤ T with estimate (3.59). Furthermore, for t > s, U (t, s) is differentiable in t with the derivative (3.60). Proof It remains only to show the uniqueness of the evolution operator. Let U˜ (t, s) be any other evolution operator for the family A(t). For s < τ ≤ t , we have ∂ Un (t, τ )U˜ (τ, s) = Un (t, τ )[Jn (τ ) − 1]A(τ )U˜ (τ, s). ∂τ Integrating this equality in τ on the interval [s + ε, t], where ε > 0 is a small positive number, we obtain that  t U˜ (t, s) − Un (t, s + ε)U˜ (s + ε, s) = Un (t, τ )[Jn (τ ) − 1]A(τ )U˜ (τ, s) dτ. s+ε

Letting n → ∞, U˜ (t, s) − U (t, s + ε)U˜ (s + ε, s) = 0 (due to (2.89)). Finally, letting ε → 0, we conclude that U˜ (t, s) = U (t, s). Hence, the evolution operator is uniquely determined from A(t). 

6 Cauchy Problems We consider the Cauchy problem  dU dt + A(t)U = F (t),

0 < t ≤ T,

U (0) = U0 .

(3.61)

Here, A(t), 0 ≤ t ≤ T , is a family of sectorial operators of X which satisfies all the structural assumptions (3.27), (3.28), (3.29), (3.30), and (3.31). The function F is a given function on (0, T ] which belongs to the function space: F ∈ Fβ,σ ((0, T ]; X), The initial value U0 is taken in X.

0 < σ < β ≤ 1.

(3.62)

6 Cauchy Problems

147

Let U (t, s) be the evolution operator for the family of the operators A(t) constructed in the preceding section. We show the following existence and uniqueness result. Theorem 3.9 Under (3.27), (3.28), (3.29), (3.30), and (3.31), for any F belonging to (3.62) and any U0 ∈ X, there exists a unique solution U to (3.61) in the function space: U ∈ C([0, T ]; X) ∩ C1 ((0, T ]; X),

AU ∈ C((0, T ]; X),

with the estimate    dU   U (t) + t  (t)  + tA(t)U (t) ≤ C(U0  + F Fβ,σ ), dt Moreover, U is necessarily given by the formula  t U (t, τ )F (τ ) dτ, U (t) = U (t, 0)U0 +

(3.63)

0 < t ≤ T . (3.64)

0 ≤ t ≤ T.

(3.65)

0

Proof In order to prove this theorem, we will go back to the stage of constructing the evolution operator U (t, s). Let An (t) be the Yosida approximations for A(t) for each n = 1, 2, 3, . . . , and let Un (t, s) be the evolution operator for An (t). Using the operator Un (t, s), we define an X-valued function of the form  t Un (t) = Un (t, 0)U0 + Un (t, τ )F (τ ) dτ, 0 ≤ t ≤ T . 0

In view of (3.54), as n → ∞, this function converges to the function U given by (3.65) in X pointwise. It is easy to see that U is an X-valued continuous function on [0, T ]. By a direct calculation, we verify from (3.40) that Un is differentiable for t > 0 and the derivative is given by dUn (t) = −An (t)Un (t) + F (t), dt

0 < t ≤ T.

Therefore, for t > 0, −1

h

−1



[Un (t + h) − Un (t)] = h

t+h

[F (τ ) − An (τ )Un (τ )] dτ.

(3.66)

t

In the meantime, we consider the function An (t)Un (t). This function can be decomposed in the form  t An (t)Un (t) =An (t)Un (t, 0)U0 + An (t)Un (t, τ )[F (τ ) − F (t)] dτ  + 0

0 t

Wn (t, τ ) dτ F (t) + [1 − e−tAn (t) ]F (t).

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3 Linear Evolution Equations

Here, we used the formula 

t

An (t)e−(t−τ )An (t) dτ = 1 − e−tAn (t)

0

(which follows from (2.130)). By (1.8), (3.51), and (3.50), we then obtain the estimate An (t)Un (t) ≤ CF,U0 t −1 ,

0 < t ≤ T.

(3.67)

The constant CF,U0 > 0 depends on F and U0 . Furthermore, by (3.52) and (3.55), the function is seen to converge to the function  W (t) = A(t)U (t, 0)U0 +

t

A(t)U (t, τ )[F (τ ) − F (t)] dτ

0



t

+

W (t, τ ) dτ F (t) + [1 − e−tA(t) ]F (t)

0

in X pointwise. The limit function is also verified to be an X-valued continuous function for 0 < t ≤ T by arguing in a similar way as in the proof of Lemma 1.3. Since Un (t) = An (t)−1 An (t)Un (t), it follows that U (t) = A(t)−1 W (t); therefore, U (t) ∈ D(A(t)) and A(t)U (t) = W (t) for each 0 < t ≤ T . In view of (3.67), let n → ∞ in (3.66). Then, h−1 [U (t + h) − U (t)] = h−1



t+h

[F (τ ) − A(τ )U (τ )] dτ,

0 < t ≤ T.

t

Hence, dU (t) = −A(t)U (t) + F (t), dt

0 < t ≤ T,

which shows that U is a C1 function on (0, T ] and satisfies the equation of (3.61). It is easy to verify that U satisfies (3.64). It now remains only to show the uniqueness of solution. Let U˜ be any other solution in the space (3.63). Let 0 < ε < t ≤ T . By (3.41), ∂ Un (t, τ )U˜ (τ ) = Un (t, τ )[Jn (τ ) − 1]A(τ )U˜ (τ ) + Un (t, τ )F (τ ). ∂τ So, U˜ (t) = Un (t, ε)U˜ (ε) +



t

Un (t, τ )[Jn (τ ) − 1]A(τ )U˜ (τ ) dτ

ε



t

+

Un (t, τ )F (τ ) dτ. ε

7 Conditions (3.29) and (3.30)

149

Letting n → ∞, we obtain that U˜ (t) = U (t, ε)U˜ (ε) +



t

U (t, τ )F (τ ) dτ. ε

Finally, letting ε → 0, we conclude that U˜ (t) = U (t) for every 0 ≤ t ≤ T .



Remark 3.3 We needed the derivative of Un (t, s) in s in establishing the uniqueness of solution to problem (3.61). On the other hand, we used the derivative of Un (t, s) in t in constructing the solution.

7 Conditions (3.29) and (3.30) The objective of this section is to see how one can verify conditions (3.29) and (3.30) in concrete examples. The case where ν = 1 is the most favorable one. In this case, since

(3.68) A(t) A(t)−1 − A(s)−1 = −[A(t) − A(s)]A(s)−1 , we can calculate directly the difference A(t) − A(s) in order to verify (3.30). On the other hand, when ν < 1, there is no convenient representation for the term A(t)ν [A(t)−1 − A(s)−1 ], for we have to consider the fractional power A(t)ν .

7.1 Case of Hilbert Spaces Let Z ⊂ X ⊂ Z ∗ be a triplet of spaces. We consider a family of sesquilinear forms a(t; U, V ), 0 ≤ t ≤ T , which are all defined on Z and satisfy (2.6) and (2.7) with some uniform constants M > 0 and δ > 0, respectively. In addition, we assume the Hölder condition |a(t; U, V ) − a(s; U, V )| ≤ N |t − s|μ U V ,

0 ≤ s, t ≤ T ; U, V ∈ Z, (3.69)

with some fixed exponent 0 < μ ≤ 1 and some constant N > 0. We already know (see Chap. 1) that, from the sesquilinear forms, sectorial operators A(t), 0 ≤ t ≤ T , of Z ∗ and equally sectorial operators A(t), 0 ≤ t ≤ T , of X are defined. Consider first the family of A(t)’s. By Theorem 1.24, D(A(t)) coincides with Z, namely, (3.29) is fulfilled with ν = 1. In addition, using the formula (3.68) for A(t), we observe that

  A(t) A(t)−1 − A(s)−1 Φ, U    

= − a t; A(s)−1 Φ, U − a s; A(s)−1 Φ, U , Φ ∈ Z ∗ , U ∈ Z. (3.70)

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3 Linear Evolution Equations

So, (3.69) yields directly that 

 A(t) A(t)−1 − A(s)−1 Φ  ≤ N δ −1 |t − s|μ Φ∗ , ∗

Φ ∈ Z∗.

Thus, under (3.69), A(t)’s satisfy (3.29) and (3.30) with ν = 1 and the μ.

(3.71)

Secondly, consider the family of A(t)’s. According to Theorem 2.32, we have D(A(t)∗θ ) ⊂ Z for any 12 < θ ≤ 1. Therefore, for any ν such that 0 < ν < 12 , we observe from (3.70) that 

   A(t)ν A(t)−1 − A(s)−1 F, G = − a t; A(s)−1 F, A(t)∗(ν−1) G 

 (3.72) − a s; A(s)−1 F, A(t)∗(ν−1) G , F, G ∈ X. Hence, (3.69) implies that 

  A(t)ν A(t)−1 − A(s)−1 F, G  ≤ C|t − s|μ |F ||G|, namely,



 A(t)ν A(t)−1 − A(s)−1 F  ≤ N |t − s|μ |F |,

F ∈ X, G ∈ X,

F ∈ X.

Thus, under (3.69), A(t) satisfy (3.29) and (3.30) with any 0 < ν <

1 and the μ. 2

(3.73)

Instead of (3.69), if a Hölder condition of the form   |a(t; U, V ) − a(s; U, V )| ≤ N |t − s|μ |A(s)U |A(t)∗(1−ν) V , 0 ≤ s, t ≤ T ; U ∈ D(A(s)), V ∈ D(A(t)∗ ), is satisfied with some exponent these μ and ν.

1 2

≤ ν ≤ 1, then A(t) satisfy (3.29) and (3.30) with

7.2 Case of Lp Spaces We now see how conditions (3.29) and (3.30) are fulfilled in Banach-space cases. As an example, let us consider the family of sesquilinear forms a(t; u, v) =

n  i,j =1 Ω

 aij (x, t)Di uDj v dx +

c(x, t)uv dx,

u, v ∈ H 1 (Ω),

Ω

defined on H 1 (Ω), where Ω is a bounded domain in Rn with C2 boundary ∂Ω and 0 ≤ t ≤ T . We assume that aij satisfy condition (2.19) with some constant δ > 0

7 Conditions (3.29) and (3.30)

151

uniformly with respect to the variable t, that c(x, t) satisfies condition (2.20), and that  aij ∈ C([0, T ]; C1 (Ω)) ∩ Cμ ([0, T ]; Lp (Ω)), (3.74) c ∈ C([0, T ]; L∞ (Ω)) ∩ Cμ ([0, T ]; Lp (Ω)), with some exponents 0 < μ ≤ 1 and 1 < p < ∞. We already know that one can define, from the forms a(t; ·, ·), realizations

Ap (t) of the differential operators − ni,j =1 Dj [aij (x, t)Di ] + c(x, t) in the space Lp (Ω), 1 < p < ∞, under the Neumann-type boundary conditions ∂ν∂u = 0 on A(t) ∂Ω. We are concerned with the family of sectorial operators Ap (t), 0 ≤ t ≤ T , in the Banach space Lp (Ω). It is known by (2.77) and (2.78) that D(Ap (t)) = {u ∈ Hp2 (Ω); ∂ν∂u = 0 on ∂Ω} and A(t) uHp2 ≤ CAp (t)uLp ,

u ∈ D(Ap (t)).

(3.75)

Here, the constant C > 0 can be uniform with respect to t , because aij (·, t) vary only in some relatively compact subset of C1 (Ω). We consider also the adjoint operators Bp (t) = Ap (t)∗ (see Theorem 2.14) of Ap (t) with respect to the adjoint pair {Lp (Ω), Lp (Ω)}, where p1 + p1 = 1, Bp (t) being determined by the adjoint forms a ∗ (t; ·, ·). As will be shown by Theorems 16.7 and 16.10 in Chap. 16, we have, in favorable cases, that D(Bp (t)θ ) ⊂ Hp2θ (Ω) for any 0 < θ ≤ 1, and   uH 2θ ≤ C Bp (t)θ uL  , p

p

  u ∈ D Bp (t)θ ,

(3.76)

C > 0 being uniform for t . In the general case, according to (16.36), it is at least  true that D(Bp (t)θ ) ⊂ Hp2θ (Ω) for any 0 < θ  < θ ≤ 1 with the estimate   uH 2θ  ≤ Cθ  ,θ Bp (t)θ uL  , p

p

  u ∈ D Bp (t)θ ,

(3.77)

Cθ  ,θ > 0 being uniform for t . In what follows, let us consider the optimal case (3.76). For the general case, we shall have some weaker result, see Remark 3.4. ν Let 12 ≤ ν < p+1 2p . Then, by (16.36) again, we see that D(A(s)) ⊂ D(A(t) ) for all 0 ≤ s, t ≤ T ; hence, (3.29) is fulfilled. When p = 2, the formula (3.72) reads as 



 A2 (t)ν A2 (t)−1 − A2 (s)−1 f, g n  [ai,j (t, x) − aij (s, x)]Di A2 (s)−1 f · Dj A2 (t)∗(ν−1) g dx =− i,j =1 Ω



−

[c(x, t) − c(s, x)]A2 (s)−1 f · A2 (t)∗(ν−1) g dx,

Ω

f ∈ L2 (Ω), g ∈ D(A2 (t)∗ ).

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3 Linear Evolution Equations

For the general 1 < p < ∞, this formula yields that 

 Ap (t)ν Ap (t)−1 − Ap (s)−1 f, g L

p ×Lp 

n 

=−

[aij (t, x) − aij (s, x)]Di Ap (s)−1 f · Dj Bp (t)ν−1 g dx

i,j =1 Ω



[c(x, t) − c(s, x)]Ap (s)−1 f · Bp (t)ν−1 g dx,

− Ω

f ∈ Lp (Ω), g ∈ D(Bp (t)). Since 0 ≤ 2ν − 1 < p1 , we observe by (1.80) and (1.82) that      [aij (t, x) − aij (s, x)]Di Ap (s)−1 f · Dj Bp (t)ν−1 g dx    Ω     ≤ [aij (t) − aij (s)]Di Ap (s)−1 f H˚ 2ν−1 Dj Bp (t)ν−1 g H 1−2ν p

p

≤ aij (t) − aij (s)L(H 1 ,H 2ν−1 ) p

p

  Di Ap (s)−1 f 

  ν−1   g 1 Dj Bp  (t)

Hp

Hp1−2ν 

.

(3.78) In addition, by (3.75), Di Ap (s)−1 f Hp1 ≤ Cf Lp . By the boundedness of dif2(1−ν)

ferentiation Dj ∈ L(Hp (Ω), Hp1−2ν (Ω)) (due to Theorem 1.43) and estimate  ν−1 (3.76), we have Dj Bp (t) gH 1−2ν ≤ Cν gLp . Finally, Lemma 3.4 below, top

gether with assumption (3.74), provides that ⎧ 2pμ(1−ν)/n ⎪ ⎨C|t − s|  aij (t) − aij (s)L(H 1 ,Hp2ν−1 ) ≤ Cμ |t − s|2μ (1−ν) p ⎪ ⎩ C|t − s|2μ(1−ν)

if 1 < p < n, if p = n, if n < p < ∞,

where μ can be taken arbitrarily in such a way that 0 < μ < μ. In this way, under (3.74) and (3.76), we have shown that, for any condition (3.30) is fulfilled as ⎧ 2pμ(1−ν)/n ⎪ ⎨Cν |t − s| 

  ν −1 −1 Ap (t) Ap (t) − Ap (s)  ≤ Cν,μ |t − s|2μ (1−ν) L(Lp ) ⎪ ⎩ Cν |t − s|2μ(1−ν) μ being an arbitrary exponent such that 0 < μ < μ.

1 2

≤ν<

p+1 2p ,

if 1 < p < n, if p = n, if n < p < ∞, (3.79)

8 Maximal Regularity

153

Lemma 3.4 Let a ∈ C1 (Ω). Then, for 0 ≤ s ≤ 1,

auHps ≤

⎧ p(1−s) 1− p(1−s) ⎪ ⎪ CaC1 n aLpn uHp1 , ⎪ ⎨ 1−η(1−s)

η(1−s)

u ∈ Hp1 (Ω), if 1 < p < n,

aLp uHp1 , Cη aC1 ⎪ ⎪ ⎪ ⎩ 1−s uHp1 , CasC1 aL p

u ∈ Hp1 (Ω), if p = n, u ∈ Hp1 (Ω), if n < p < ∞,

where 0 < η < 1 is an arbitrary exponent smaller than 1. Proof We have 1−s 1−s auHps ≤ CausH 1 auL ≤ CasL(H 1 ,H 1 ) aL(H 1 ,L ) uHp1 , p p

p

p

p

p

u ∈ Hp1 (Ω).

It is easy to see that aL(Hp1 ,Hp1 ) ≤ CaC1 . In the meantime, if 1 < p < n, then 1− pn

aL(Hp1 ,Lp ) ≤ CaC

p

aLn p since 1− pn

auLp ≤ aLn uLpn/(n−p) ≤ aC

p

aLn p uHp1 ,

u ∈ Hp1 (Ω).

If n < p < ∞, then aL(Hp1 ,Lp ) ≤ CaLp because of auLp ≤ aLp uL∞ ≤ 1−η

CaLp uHp1 , u ∈ Hp1 (Ω). Similarly, if p = n, then aL(Hp1 ,Lp ) ≤ CaC η aLp

with an arbitrary exponent 0 < η < 1 less than 1.

× 

Remark 3.4 Even if we have only the weaker estimate (3.77), we can prove analoν−1 g  ≤ Cν,ν  gL  with  gous results. For 12 ≤ ν < p+1 2p , note that Dj Bp (t) p H 1−2ν p

any exponent ify that

ν

such that ν

< ν

p+1 2p .

<



 Ap (t)ν Ap (t)−1 − Ap (s)−1  L

p

Actually, under (3.74), it is possible to ver-

⎧ 2pμ(1−ν  )/n ⎪ ⎨Cν,ν  |t − s|   ≤ Cν,μ ,ν  |t − s|2μ (1−ν ) ⎪  ⎩ Cν,ν  |t − s|2μ(1−ν )

if 1 < p < n, if p = n, if n < p < ∞,

μ and ν  being arbitrary exponents such that 0 < μ < μ and ν < ν  < tively.

p+1 2p , respec-

8 Maximal Regularity In this section, we investigate refined properties of the evolution operator U (t, s). As an application, we shall establish the maximal regularity of the Cauchy problem. Another more important application will be made later for studying the quasilinear evolution equations.

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3 Linear Evolution Equations

8.1 Refined Properties of U (t, s) Let U (t, s) be the evolution operator for a family of A(t)’s constructed under assumptions (3.27), (3.28), (3.29), (3.30), and (3.31). In fact, U (t, s) satisfies the following estimates. For any 0 ≤ θ < μ + ν,   (3.80) R(U (t, s)) ⊂ D A(t)θ , 0 ≤ s < t ≤ T , with

  A(t)θ U (t, s) ≤ Cθ (t − s)−θ ,

0 ≤ s < t ≤ T.

(3.81)

For 0 ≤ θ < μ, U (t, s)A(s)θ is extended as a bounded operator on X with   U (t, s)A(s)θ  ≤ Cθ (t − s)−θ , 0 ≤ s < t ≤ T .

(3.82)

For 0 ≤ ϕ ≤ 1 and 0 ≤ ϕ ≤ θ < μ + ν,   A(t)θ U (t, s)A(s)ϕ−θ  ≤ Cθ (t − s)−ϕ ,

(3.83)

0 ≤ s < t ≤ T.

As for the differences of U (t, s) and e−(t−s)A(t) or e−(t−s)A(s) , we verify the following. For 0 ≤ θ < μ + ν and 0 ≤ ϕ ≤ 1,   A(t)θ [U (t, s) − e−(t−s)A(t) ]A(s)−ϕ  ≤ Cθ (t − s)ϕ−θ+μ+ν−1 , 0 ≤ s < t ≤ T . (3.84) For k = 0, 1,   A(t)k U (t, s)A(s)−k − e−(t−s)A(s)  ≤ C(t − s)μ+ν−1 , 0 ≤ s < t ≤ T . (3.85) For 0 < θ < 1,   A(t)θ U (t, s)A(s)−θ − e−(t−s)A(s)   C(1 − ν)(t − s)μ+ν−1 , 0 ≤ s ≤ t ≤ T , if ν < 1, ≤ −1 μ C log((t − s) + 1)|t − s| , 0 ≤ s ≤ t ≤ T , if ν = 1.

(3.86)

For 0 ≤ θ ≤ ν and 0 ≤ ϕ ≤ 1,   A(t)θ [U (t, s) − e−(t−s)A(s) ]A(s)−ϕ  ≤ C(t − s)ϕ−θ+μ+ν−1 ,

0 ≤ s < t ≤ T. (3.87) We shall describe the proof of all these properties below. For any 0 ≤ θ < μ + ν, it is seen from (3.43) that An (t)θ−1 Wn (t, s) = An (t)θ−1 Rn (t, s)  t − An (t)θ+1−ν e−(t−τ )An (t) Dn (t, τ )Wn (τ, s) dτ. s

8 Maximal Regularity

155

Here, we notice that   An (t)θ+1−ν e−(t−s)An (t) Dn (t, s) ≤ C(t − s)μ+ν−θ−1 .

(3.88)

In addition, by the same argument as in the proof of Proposition 2.1, we can verify that   An (t)θ−1 Rn (t, s) ≤ Cθ (t − s)μ+ν−θ−1 , 0 ≤ s < t ≤ T , L(X) and, as n → ∞, An (t)θ−1 Rn (t, s) converges strongly to the bounded linear operator A(t)θ−1 R(t, s) of X which satisfies the same estimate as above. As a consequence, An (t)θ−1 Wn (t, s) is also strongly convergent to a bounded operator of X. Since An (t)θ Un (t, s) = An (t)θ e−(t−s)An (t) + An (t)θ−1 Wn (t, s), this implies that (3.80) holds true and the limit of An (t)θ−1 Wn (t, s) is A(t)θ U (t, s) − A(t)θ e−(t−s)A(t) . Therefore, A(t)θ [U (t, s) − e−(t−s)A(t) ] = A(t)θ−1 R(t, s)  t − A(t)θ+1−ν e−(t−τ )A(t) D(t, τ )W (τ, s) dτ. s

Therefore, (3.81) is verified. As well, (3.84) is verified in the case of ϕ = 0. We observe from (3.42) that Un (t, s)An (s)θ = An (s)θ e−(t−s)An (s)  t + Vn (t, τ )Dn (τ, s)An (s)θ+1 e−(τ −s)An (s) dτ. s

Here, since   Dn (t, s)An (s)θ+1 e−(t−s)An (s)  ≤ C(t − s)μ−θ−1 L(X) for 0 ≤ θ < μ, this kernel has a weak singularity. Therefore, Un (t, s)An (s)θ is strongly convergent to a bounded linear operator of X which is a unique bounded extension of U (t, s)A(s)θ on X and which satisfies (3.82). The extension is also denoted by U (t, s)A(s)θ . To verify other properties, we have to use the following properties of the family of semigroups. Lemma 3.5 For 1 ≤ θ ≤ 2 and 0 ≤ ϕ ≤ 1,  

 A(t)θ e−τ A(t) − A(s)θ e−τ A(s) A(s)−ϕ  ≤ Cτ ϕ−θ+ν−1 |t − s|μ . For 0 ≤ ϕ ≤ θ < 1,  

 A(t)θ e−τ A(t) − A(s)θ e−τ A(s) A(s)−ϕ   C(θ − ϕ − ν + 1)τ ϕ−θ +ν−1 |t − s|μ ≤ C[log(τ −1 + 1) + 1]|t − s|μ

if ϕ < θ or ν < 1, if ϕ = θ and ν = 1.

(3.89)

(3.90)

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3 Linear Evolution Equations

For 0 ≤ θ ≤ ν and 0 ≤ ϕ ≤ 1,   A(t)θ [e−τ A(t) − e−τ A(s) ]A(s)−ϕ  ≤ Cτ ϕ−θ+ν−1 |t − s|μ . For 0 ≤ θ ≤ ν and 0 ≤ ϕ ≤ 1, 

 A(t)θ e−τ A(t) A(t)−ϕ − e−τ A(s) A(s)−ϕ  ⎧ ϕ−θ +ν−1 |t − s|μ ⎪ ⎨C(θ − ϕ − ν + 1)τ −1 ≤ C[log(τ + 1) + 1]|t − s|μ ⎪ ⎩ C(ϕ + ν − θ − 1)−1 |t − s|μ

if ϕ + ν < θ + 1, if ϕ + ν = θ + 1, if ϕ + ν > θ + 1.

(3.91)

(3.92)

Proof Consider first the case where 1 ≤ θ ≤ 2 and 0 ≤ ϕ ≤ 1. As in the proof of Lemma 3.2, we can write as

A(t)θ e−τ A(t) − A(s)θ e−τ A(s) A(s)−ϕ  1 = λθ e−τ λ A(t)1−ν (λ − A(t))−1 D(t, s)A(s)1−ϕ (λ − A(s))−1 dλ, 2πi Γ where Γ is a suitable integral contour. Indeed, we will take Γ as one of Γω : λ = ρe±iω , 0 ≤ ρ < ∞, or Γδ,ω : λ = ρe±iω , δ ≤ ρ < ∞; λ = δeiϑ , |ϑ| ≤ ω. By assumption, it holds that −1 ≤ θ − ν − ϕ < 2. So, except the case where θ − ν − ϕ = −1, we have   

 A(t)θ e−τ A(t) − A(s)θ e−τ A(s) A(s)−ϕ  ≤ C |λ|θ−ν−ϕ e−τ Re λ |dλ||t − s|μ Γω

≤ Cτ

ϕ−θ+ν−1

|t − s|μ .

The critical case θ − ν − ϕ = −1 takes place only when θ = ν = ϕ = 1; in this case, since θ = 1, the function λθ is analytic even on the negative axis; so, we can use the techniques of shifting integral contour (see (2.128)) to obtain (3.89). Let now 0 ≤ ϕ ≤ θ < 1. In the same way, we have   

 A(t)θ e−τ A(t) − A(s)θ e−τ A(s) A(s)−θ  ≤ C |λ|θ−ϕ−ν e−τ Re λ |dλ| |t − s|μ . Γ

Here, if ϕ < θ or ν < 1, then −1 < θ − ϕ − ν < 1. So, by the same reason as above,  |λ|θ−ϕ−ν e−τ Re λ |dλ| ≤ C(θ − ϕ − ν + 1)τ ϕ−θ+ν−1 . Γω

In the meantime, when ϕ = θ and ν = 1, 

|λ|−1 e−τ Re λ |dλ| ≤ Γδ,ω



ω

−ω

 dϑ + 2 δ

∞

e−τρ cos ω dρ ≤ C[log(τ −1 + 1) + 1]. ρ

8 Maximal Regularity

157

In this case, the shifting of integral contour is not available, for the function λθ is not analytic on (−∞, 0). We also remark that, as τ → 0, τ ε log(τ −1 + 1) tends to 0 with any small exponent ε > 0. Let 0 ≤ θ ≤ ν and 0 ≤ ϕ ≤ 1. We write A(t)θ [e−τ A(t) − e−τ A(s) ]A(s)−ϕ  1 = e−τ λ A(t)θ−ν+1 (λ − A(t))−1 D(t, s)A(s)1−ϕ (λ − A(s))−1 dλ. 2πi Γ By assumption, it holds that −2 ≤ θ − ϕ − ν ≤ 0. In the case where −1 < θ − ϕ − ν,   A(t)θ [e−τ A(t) − e−τ A(s) ]A(s)−ϕ   ≤C |λ|θ−ϕ−ν e−τ Re λ |dλ| |t − s|μ ≤ Cτ ϕ−θ+ν−1 |t − s|μ . Γω

In the case where −2 ≤ θ − ϕ − ν ≤ −1, we can always use the shifting techniques to obtain (3.91). For 0 ≤ θ ≤ ν and 0 ≤ ϕ ≤ 1, we write

A(t)θ e−τ A(t) A(t)−ϕ − e−τ A(s) A(s)−ϕ  1 λ−ϕ e−τ λ A(t)θ−ν+1 (λ − A(t))−1 D(t, s)A(s)(λ − A(s))−1 dλ. = 2πi Γ When −1 < θ − ϕ − ν, (3.92) is verified by the same arguments as before. In the critical case where θ − ϕ − ν = −1, the shifting of integral contour is not available except the special case θ = ν and ϕ = 1. When θ − ϕ − ν < −1, we have  |λ|θ−ϕ−ν e−τ Re λ |dλ| ≤ C(ϕ + ν − θ − 1)−1 .  Γδ,ω Let 0 < ϕ ≤ 1. To see (3.83), we notice that A(t)U (t, s)A(s)−ϕ is a solution to the integral equation A(t)U (t, s)A(s)−ϕ = A(t)e−(t−s)A(t) A(s)−ϕ  t − A(t)2−ν e−(t−τ )A(t) D(t, τ )A(τ )U (τ, s)A(s)−ϕ dτ. s

(3.93) Moreover, by (3.89),   A(t)e−(t−s)A(t) A(s)−ϕ  L(X)   −(t−s)A(t) ≤ [A(t)e − A(s)e−(t−s)A(s) ]A(s)−ϕ L(X)   + A(s)1−ϕ e−(t−s)A(s) L(X) ≤ C(t − s)ϕ−1 .

158

3 Linear Evolution Equations

Hence, (3.83) is verified for 0 < ϕ ≤ 1 and θ = 1. When ϕ = 0, (3.83) is already known. Let 0 ≤ θ < μ + ν and 0 < ϕ ≤ 1. From (3.93) it also follows that A(t)θ [U (t, s) − e−(t−s)A(t) ]A(s)−ϕ  t =− A(t)θ−ν+1 e−(t−τ )A(t) D(t, τ )A(τ )U (τ, s)A(s)−ϕ dτ. s

Therefore,   A(t)θ [U (t, s) − e−(t−s)A(t) ]A(s)−ϕ 

L(X)



t

≤C

(t − τ )−θ+μ+ν−1 (τ − s)ϕ−1 dτ ≤ C(t − s)ϕ−θ+μ+ν−1 ,

s

and hence (3.84) is verified for 0 < ϕ ≤ 1, too. Let 0 ≤ ϕ ≤ 1 and 0 ≤ ϕ ≤ θ < μ + ν. The estimate just established, together with Lemma 4.1, yields that   A(t)θ U (t, s)A(s)−ϕ − A(s)θ−ϕ e−(t−s)A(s)  L(X)   ≤ A(t)θ [U (t, s) − e−(t−s)A(t) ]A(s)−ϕ L(X)  

+  A(t)θ e−(t−s)A(t) − A(s)θ e−(t−s)A(s) A(s)−ϕ    ≤ Cθ log (t − s)−1 + 1 (t − s)ϕ−θ+μ+ν−1 .

L(X)

(3.94)

Therefore, (3.83) holds. Let θ = ϕ = 0 in (3.84). Then, U (t, s) − e−(t−s)A(t) L(X) ≤ C(t − s)μ+ν−1 . This, together with (3.91) (θ = ϕ = 0), therefore yields (3.85) in the case k = 0. Similarly, let θ = ϕ = 1 in (3.84); then (3.85) of the case k = 1 is verified by (3.89) (θ = ϕ = 1). Equation (3.86) is also verified in the same manner. Indeed, for 0 < θ = ϕ < 1,   A(t)θ [U (t, s) − e−(t−s)A(t) ]A(s)−θ  ≤ C(t − s)μ+ν−1 . L(X) This, together with (3.90) (θ = ϕ), then yields (3.86). For 0 ≤ θ ≤ ν and 0 ≤ ϕ ≤ 1, (3.84), together with (3.91), yields (3.87). We have thus established all the properties of U (t, s) to be shown. Remark 3.5 In some problems we may also need the property (3.82) for θ ≥ μ. In order to extend (3.82) to such an exponent θ , say θ = 1, we have to assume a dual condition of (3.30). That is, for some exponent 0 < ν  ≤ 1, the operators

8 Maximal Regularity

159 

[A(t)−1 − A(s)−1 ]A(s)ν have bounded extensions on X and satisfy the Hölder condition  

  A(t)−1 − A(s)−1 A(s)ν   ≤ N  |t − s|μ , 0 ≤ s, t ≤ T , (3.95) L(X) with another exponent 0 < μ ≤ 1 with the relation 1 < μ + ν  .

(3.96)

Furthermore, we assume that X is a reflexive Banach space. In the adjoint space X ∗ , let us consider the family of sectorial operators A∗ (t) ≡ A(T − t)∗ , 0 ≤ t ≤ T . By Theorem 2.32, the family of A∗ (t)’s is easily seen to satisfy (3.27) and (3.28). In addition, by (2.126), (3.95), together with (3.96), implies that A∗ (t) satisfies (3.29), (3.30), and (3.31) with μ = μ and ν = ν  . Consequently, there exists an evolution operator U ∗ (t, s), 0 ≤ s ≤ t ≤ T , for A∗ (t). Then, it is easily verified that U ∗ (t, s) = U (T − s, T − t)∗ for any 0 ≤ s ≤ t ≤ T . Indeed, for τ such that s ≤ τ ≤ t, we have ∂ [U (T − τ, T − t)∗ U ∗ (τ, s)] = [−A(T − τ )U (T − τ, T − t)]∗ U ∗ (τ, s) ∂τ + U (T − τ, T − t)∗ A∗ (τ )U ∗ (τ, s) = 0. Since A∗ (t)U ∗ (t, s) = A(T − t)∗ U (T − s, T − t)∗ = [U (T − s, T − t)A(T − t)]∗ , estimate (3.58) for A∗ (t) shows that U (T − s, T − t)A(T − t)θ L(X) ≤ C(t − s)−θ for 0 < θ < μ + ν  . We have thus seen that, when X is reflexive, (3.95), together with (3.96), yields property (3.82) for μ ≤ θ < μ + ν  , too.

8.2 Maximal Regularity We return to problem (3.61). Utilizing the refined properties of U (t, s), we shall show the maximal regularity of solutions. Theorem 3.10 Under (3.27), (3.28), (3.29), (3.30), and (3.31), let F ∈ Fβ,σ ((0, T ]; X), and let

0 < σ < min{β, μ + ν − 1},

  U0 ∈ D A(0)β ,

0 < β ≤ 1.

(3.97)

(3.98)

Then, the solution U to (3.61) given by (3.65) possesses the following regularity:

160

3 Linear Evolution Equations

A(t)β U ∈ C([0, T ]; X), dU , dt

(3.99)

A(t)U ∈ Fβ,σ ((0, T ]; X),

with the estimates   

 A(t)β U  ≤ C A(0)β U0  + F Fβ,σ , C     dU 

 β      dt  β,σ + A(t)U Fβ,σ ≤ C A(0) U0 + F Fβ,σ . F

(3.100)

(3.101) (3.102)

Proof Let us first verify (3.99). For 0 < t ≤ T , we notice that A(t)β U (t) = A(t)β−1 A(t)U (t). Since A(·)β−1 ∈ C([0, T ]; L(X)), (3.63) yields that A(t)β U (t) is continuous for t > 0. So, it suffices to show that A(t)β U (t) is continuous at t = 0. We use the expression

A(t)β U (t) = A(0)β U0 + A(t)β U (t, 0)A(0)−β − 1 A(0)β U0  t + A(t)β U (t, τ )[F (τ ) − F (t)] dτ 0

 +

t

A(t)β [U (t, τ ) − e−(t−τ )A(t) ] dτ F (t)

0

+t

β−1

[1 − e−tA(t) ]A(t)β−1 t 1−β F (t).

Then, by the same arguments as in the proof of Theorem 3.5, we directly see that A(t)β U (t) → A(0)β U0 as t → 0. Hence, (3.99) is proved. Estimate (3.101) is also verified in the same way. Let us now prove (3.100). We use the expression A(t)U (t) = A(0)e−tA(0) U0

+ A(t)U (t, 0)A(0)−β − A(0)1−β e−tA(0) A(0)β U0  t  t + A(t)U (t, τ )[F (τ ) − F (t)] dτ + W (t, τ ) dτ F (t) 0

0

+ [1 − e−tA(t) ]F (t) = W1 (t) + W2 (t) + W3 (t) + W4 (t) + W5 (t), where W (t, s) = A(t)[U (t, s) − e−(t−s)A(t) ]. By (2.132), it is already known that W1 ∈ Fβ,σ ((0, T ]; X). So, it remains to verify Wi ∈ Fβ,σ ((0, T ]; X) for 2 ≤ i ≤ 5. Proof for W2 We use (3.94) to observe that t 1−β W2 (t)X ≤ Ct μ+ν−1 log(t −1 + 1) → 0 as t → 0. In the meantime, W2 (t) − W2 (s) is seen to be decomposed as

8 Maximal Regularity

161

W2 (t) − W2 (s) = A(t)U (t, s)A(s)−1 − e−(t−s)A(s) A(s)U (s, 0)A(0)−β A(0)β U0 + [e−(t−s)A(s) − 1]A(s)−σ A(s)1+σ U (s, 0)A(0)−β A(0)β U0 − [e−(t−s)A(0) − 1]A(0)−σ A(0)1−β+σ e−sA(0) A(0)β U0 . Then, by (2.129), (3.83), and (3.85), 

 W2 (t) − W2 (s)X ≤ C s β−1 (t − s)μ+ν−1 + s β−σ −1 (t − s)σ A(0)β U0 X . Hence, since σ < μ + ν − 1, W2 satisfies (1.6). It is also readily verified by (2.130)  and (3.94) that W2 satisfies (1.7). Proof for W3 We shall argue as in the proof of Theorem 3.5. It is easy to see that limt→0 t 1−β W3 (t) = 0. In addition, W3 (t) − W3 (s) is decomposed as  W3 (t) − W3 (s) = s

t

A(t)U (t, τ )[F (τ ) − F (t)] dτ



s

+

 W (t, τ ) dτ + [e−(t−s)A(t) − e−tA(t) ] [F (s) − F (t)]

0



+ A(t)U (t, s)A(s)−1 − e−(t−s)A(s)  s × A(s)U (s, τ )[F (τ ) − F (s)] dτ 0



+ [e−(t−s)A(s) − 1]A(s)−σ + [e−(t−s)A(s) − 1]



s



A(s)σ W (s, τ )[F (τ ) − F (s)] dτ

0 s

A(s)e−(s−τ )A(s) [F (τ ) − F (s)] dτ.

0

Then, using (2.129), (3.81), (3.84), and (3.85) and especially using (3.21) for the last term, we can verify that

W3 (t) − W3 (s)X ≤ C s 1−β (t − s)μ+ν−1 + s 1−β+σ (t − s)σ F Fβ,σ . 

Hence, W3 satisfies (1.6). Similarly, (1.7) is also verified. Proof for W4 It is clear that limt→0 t 1−β W4 (t) = 0. We use the decomposition 

t

W4 (t) − W4 (s) =



s

W (t, τ ) dτ F (t) +

W (t, τ ) dτ [F (t) − F (s)]

0

s



+ A(t)U (t, s)A(s)−1 − e−(t−s)A(s) + [e

−(t−s)A(s)

−σ



− 1]A(s)

0



t

W (s, τ ) dτ F (s) s

s

A(s)σ W (s, τ ) dτ F (s)

162

3 Linear Evolution Equations

 +

s

[A(s)e−(t−τ )A(s) − A(t)e−(t−τ )A(t) ] dτ F (s)

0

+ A(t)U (t, s)A(s)−1 − e−(t−s)A(s) [1 − e−sA(s) ]F (s). Then, by (2.129), (3.49), (3.84), and (3.85),

W4 (t) − W4 (s)X ≤ C s β−1 (t − s)μ+ν−1 + t μ+ν−1 s β−σ −1 (t − s)σ F Fβ,σ . 

Hence, W4 satisfies (1.6) and (1.7). Proof for W5 We see that W5 is decomposed as W5 (t) − W5 (s) = [1 − e−tA(t) ][F (t) − F (s)] − [e−tA(t) − e−tA(s) ]F (s) − [e−(t−s)A(s) − 1]A(s)−σ A(s)σ e−sA(s) F (s). Then, by (2.129) and (3.49), W5 (t) − W5 (s)X ≤ Cs β−σ −1 (t − s)σ F Fβ,σ .

It is also verified that W5 satisfies (1.7). We have thus verified that AU belongs to Fβ,σ ((0, T ]; X) and satisfies estimate (3.102). Since dU dt = −A(t)U + F (t), we have accomplished the proof of (3.100) and (3.102).  Remark 3.6 When 0 < β < 1, t 1−β A(t)U (t) tends to 0 as t → 0. In the meantime, when β = 1, A(t)U (t) tends to A(0)U0 as t → 0.

8.3 Spatial Regularity We remark that, if the external force function F ∈ Fβ,σ ((0, T ]; X) has the spatial regularity β−1−ρ

A(t)ρ F ∈ B{0}

((0, T ]; X),

0 < ρ < σ,

(3.103)

the solution U to (3.61) has the same regularity. Theorem 3.11 Let U0 ∈ D(A(0)β ), 0 < β ≤ 1, and let F ∈ Fβ,σ ((0, T ]; X), 0 < 1+ρ U also belong to σ < min{β, μ + ν − 1}, satisfy (3.103). Then, A(t)ρ dU dt , A(t) β−1−ρ

B{0}

((0, T ]; X) with the estimate     A(t)ρ dU   dt 

β−1−ρ

B{0}

  + A(t)1+ρ U Bβ−1−ρ {0}

  

 ≤ C A(0)β U0  + F Fβ,σ + A(t)ρ F Bβ−1−ρ . {0}

9 Variational Methods

163

Proof The result is easily verified from the representation A(t)1+ρ U (t) = A(t)1+ρ U (t, 0)A(0)−β A(0)β U0  t + A(t)1+ρ U (t, τ )[F (τ ) − F (t)] dτ 0

 +

t

A(t)1+ρ [U (t, τ ) − e−(t−τ )A(t) ] dτ F (t)

0

+ [1 − e−tA(t) ]A(t)ρ F (t). Note that 1 + ρ < 1 + σ < μ + ν.



9 Variational Methods Let Z and X be two separable Hilbert spaces such that Z ⊂ X densely and continuously with norms  ·  and | · |, respectively. Let Z ⊂ X ⊂ Z ∗ be a triplet of spaces, the norm of Z ∗ being denoted by  · ∗ . Consider a family of sesquilinear forms a(t; U, V ), 0 ≤ t ≤ T , defined on Z and assume that the following conditions are satisfied: a(t; ·, ·) is a measurable function with respect to t,

(3.104)

|a(t; U, V )| ≤ MU V ,

(3.105)

Re a(t; U, U ) ≥ δU  , 2

U, V ∈ Z,

U ∈ Z,

(3.106)

with some constant M > 0 and positive constant δ > 0 independent of t . Let A(t) be a family of sectorial operators of Z ∗ which are associated with the forms a(t; U, V ). We then consider the Cauchy problem for an abstract evolution equation  dU dt + A(t)U = F (t), 0 < t ≤ T , (3.107) U (0) = U0 , in the space Z ∗ . The following theorem is known as the fundamental existence and uniqueness result. For the proof, see Theorems 1, 2, and 3 of Dautray–Lions [DL88, Chap. XVIII]. Theorem 3.12 Let A(t) be a family of sectorial operators of Z ∗ associated with sesquilinear forms satisfying (3.104), (3.105), and (3.106). For any F ∈ L2 (0, T ; Z ∗ ) and any U0 ∈ X, there exists a unique solution U to (3.107) in the function space: U ∈ L2 (0, T ; Z) ∩ C([0, T ]; X) ∩ H 1 (0, T ; Z ∗ ). In addition, U satisfies the estimates

164

3 Linear Evolution Equations

U L2 (0,T ;Z) + U C([0,T ];X) + U H 1 (0,T ;Z ∗ ) ≤ C(|U0 | + F L2 (0,T ;Z ∗ ) ), where C > 0 is a constant determined by M, δ, and T only.

10 Nonautonomous Parabolic Equations Let us consider a nonautonomous parabolic equation of the form n ∂u Dj [aij (x, t)Di u] − c(x, t)u + f (x, t) in Ω × (0, T ) = ∂t i,j =1

in a domain Ω ⊂ Rn , where 0 < T < ∞ is fixed time. It is possible to treat a number of different initial-boundary-value problems concerning this equation. In this section, however, we will describe only a few examples. As for the concrete characterization of the domains of fractional powers of strongly elliptic operators, see Chap. 16.

10.1 Problems in Hilbert Spaces Case where Ω = Rn . Consider the initial-value problem  ∂u n n i,j =1 Dj [aij (x, t)Di u] − c(x, t)u + f (x, t) in R × (0, T ), ∂t = u(x, 0) = u0 (x) Here, aij and c are such that  aij ∈ Cμ ([0, T ]; L∞ (Rn ; R)), c ∈ Cμ ([0, T ]; L∞ (Rn ; R)),

in Rn .

1 ≤ i, j ≤ n,

(3.108)

(3.109)

with some exponent 0 < μ ≤ 1. We assume further that, for each 0 ≤ t ≤ T , aij (x, t) satisfy (2.15) with some uniform constant δ > 0, and c(x, t) satisfies (2.16) with some uniform constant c0 > 0. Let us handle (3.108) in the space H −1 (Rn ) by formulating (3.108) as an abstract problem of the form (3.61) in which the coefficient linear operator is a realization A(t) of the differential operator − ni,j =1 Dj [aij (x, t)Di ] + c(x, t) in H −1 (Rn ). By Theorem 2.2, A(t) is a sectorial operator of H −1 (Rn ) with uniform domain D(A(t)) ≡ H 1 (Rn ) and with angle ωA(t) ≤ ωA < π2 , ωA being independent of t . Therefore, the family of operators A(t) fulfills conditions (3.27), (3.28), and (3.29) with ν = 1. In addition, it is clear that (3.109) implies (3.69); therefore, by virtue of (3.71), condition (3.30) is also fulfilled. Since ν = 1 in the present case, (3.31) is trivially fulfilled.

10 Nonautonomous Parabolic Equations

165

It is now ready to apply Theorem 3.10 to (3.108) to conclude the following result. For any f ∈ Fβ,σ ((0, T ]; H −1 (Rn )), where 0 < σ < β ≤ 1, and any u0 ∈ D(A(0)β ), (3.108) possesses a unique solution in the function space:  u ∈ C((0, T ]; H 1 (Rn )) ∩ C1 ((0, T ]; H −1 (Rn )), β,σ ((0, T ]; H −1 (Rn )), A(t)β u ∈ C([0, T ]; H −1 (Rn )); du dt , A(t)u ∈ F with the estimates

         β   A(t)β u +  du   dt  β,σ + A(t)uFβ,σ ≤ C A(0) u0 H −1 + f Fβ,σ . C F

Dirichlet conditions. Consider the initial-boundary-value problem ⎧ ∂u n ⎪ ⎨ ∂t = i,j =1 Dj [aij (x, t)Di u] − c(x, t)u + f (x, t) in Ω × (0, T ), u=0 on ∂Ω × (0, T ), (3.110) ⎪ ⎩ in Ω, u(x, 0) = u0 (x) in a domain Ω ⊂ Rn . Here, aij and c are such that  aij ∈ Cμ ([0, T ]; L∞ (Ω; R)), 1 ≤ i, j ≤ n, c ∈ Cμ ([0, T ]; L∞ (Ω; R)),

(3.111)

with some exponent 0 < μ ≤ 1. In addition, it is assumed that, for each 0 ≤ t ≤ T , aij (x, t) satisfy (2.19) with some uniform constant δ > 0, and c(x, t) satisfies (2.20) with some uniform constant c0 > 0. Let us handle (3.110) again in H −1 (Ω) = [H˚ 1 (Ω)]∗ . Indeed, (3.110) is formulated as an abstract problem of the form (3.61) in which the coefficient linear operator is a realization A(t) of the differential operator − ni,j =1 Dj [aij (x, t)Di ] + c(x, t) in H −1 (Ω) under the Dirichlet boundary conditions. By Theorem 2.3, A(t) is a sectorial operator of H −1 (Ω) with uniform domain D(A(t)) ≡ H˚ 1 (Ω) and with angles ωA(t) ≤ ωA < π2 , ωA being independent of t . Therefore, the family of operators A(t) fulfills conditions (3.27), (3.28), and (3.29) with ν = 1. In addition, (3.111) implies (3.69); therefore, by (3.71), condition (3.30) is also fulfilled. Clearly, (3.31) is also fulfilled. Theorem 3.5 readily provides that, for any f ∈ Fβ,σ ((0, T ]; H −1 (Ω)), where 0 < σ < β ≤ 1, and any u0 ∈ D(A(0)β ), (3.110) possesses a unique solution in the function space:  u ∈ C((0, T ]; H˚ 1 (Ω)) ∩ C1 ((0, T ]; H −1 (Ω)), β,σ ((0, T ]; H −1 (Ω)), A(t)β u ∈ C([0, T ]; H −1 (Ω)); du dt , A(t)u ∈ F with the estimates

         β   A(t)β u +  du   dt  β,σ + A(t)uFβ,σ ≤ C A(0) u0 H −1 + f Fβ,σ . C F

166

3 Linear Evolution Equations

Neumann-type conditions. Consider the initial-boundary-value problem ⎧

n ∂u ⎪ ⎪ i,j =1 Dj [aij (x, t)Di u] − c(x, t)u + f (x, t) in Ω × (0, T ), ∂t = ⎨ n on ∂Ω × (0, T ), (3.112) i,j =1 νj (x)aij (x, t)Di u = 0 ⎪ ⎪ ⎩ in Ω, u(x, 0) = u0 (x) in a domain Ω ⊂ Rn . Here, aij and c are as in (3.111). In addition, for each 0 ≤ t ≤ T , aij (x, t) satisfy (2.19) with some uniform constant δ > 0, and c(x, t) satisfies (2.20) with some uniform constant c0 > 0. We handle this problem in H 1 (Ω)∗ . Indeed, (3.112) is written as (3.61) in which

the coefficient linear operator is a realization A(t) of the differential operator − ni,j =1 Dj [aij (x, t)Di ] + c(x, t) in H 1 (Ω)∗ under the Neumann-type boundary conditions. By Theorem 2.4, A(t) is a sectorial operator of H 1 (Ω)∗ with uniform domain D(A(t)) ≡ H 1 (Ω) and with angle ωA(t) ≤ ωA < π2 , ωA being independent of t . Therefore, the family of operators A(t) fulfills conditions (3.27), (3.28), and (3.29) with ν = 1. In addition, (3.111) implies (3.69); therefore, by (3.71), condition (3.30) is also fulfilled. Clearly, (3.31) is also fulfilled. Theorem 3.5 readily provides that, for any f ∈ Fβ,σ ((0, T ]; H 1 (Ω)∗ ), where 0 < σ < β ≤ 1, and any u0 ∈ D(A(0)β ), (3.112) possesses a unique solution in the function space:  u ∈ C((0, T ]; H 1 (Ω)) ∩ C1 ((0, T ]; H 1 (Ω)∗ ), β,σ ((0, T ]; H 1 (Ω)∗ ), A(t)β u ∈ C([0, T ]; H 1 (Ω)∗ ); du dt , A(t)u ∈ F with the estimate

         β A(t)β u +  du  + A(t)u β,σ ≤ C A(0) u0  1∗ + f F β,σ . F   C H dt Fβ,σ

10.2 Problems in Lp Spaces Consider the initial-boundary-value problem ⎧

∂u ⎪ = ni,j =1 Dj [aij (x, t)Di u] − c(x, t)u + f (x, t) in Ω × (0, T ), ⎪ ∂t ⎨ n on ∂Ω × (0, T ), (3.113) i,j =1 νj (x)aij (x, t)Di u = 0 ⎪ ⎪ ⎩ in Ω, u(x, 0) = u0 (x) in a bounded domain Ω ⊂ Rn with C2 boundary ∂Ω. Here, aij and c are as in (3.74). In addition, for each 0 ≤ t ≤ T , aij (x, t) satisfy (2.19) with some uniform constant δ > 0, and c(x, t) satisfies (2.20) with some uniform constant c0 > 0. We now want to handle (3.113) in the space Lp (Ω), where n < p < ∞. Let Ap (t) be a realization of the differential operator − i,j Dj [aij (x, t)Di ] + c(x, t)

10 Nonautonomous Parabolic Equations

167

in Lp (Ω) under the Neumann-type boundary conditions ∂ν∂u = 0 on ∂Ω. Using A(t) Ap (t), we formulate (3.113) as an abstract problem of the form (3.61). Fix an exponent ν in such a way such that 12 ≤ ν < p+1 2p . As discussed in Sect. 7.2, a family of the operators Ap (t) can fulfill condition (3.29) with the ν. Similarly, by Remark 3.4, we know that the family can fulfill condition (3.30) with the Hölder exponent 2μ(1 − ν  ), where ν  is an arbitrarily fixed exponent such that ν < ν  < p+1 2p . Therefore, if μ in (3.74) is such that μ > 12 (and if ν  is taken sufficiently close to ν), then relation (3.31) can be fulfilled. In this way, we have verified that all the structural assumptions are fulfilled in Lp (Ω), too. Theorem 3.10 readily provides the following result. For any f ∈ Fβ,σ ((0, T ]; Lp (Ω)), where 0 < σ < β ≤ 1, and any u0 ∈ D(Ap (0)β ), (3.113) possesses a unique solution in the function space: 

u ∈ C((0, T ]; Hp2 (Ω)) ∩ C1 ((0, T ]; Lp (Ω)), β,σ ((0, T ]; L (Ω)), Ap (t)β u ∈ C([0, T ]; Lp (Ω)); du p dt , Ap (t)u ∈ F

with the estimate          β  Ap (t)β u +  du    dt  β,σ + Ap (t)uFβ,σ ≤ C Ap (0) u0 Lp + f Fβ,σ . C F

10.3 Some Example of the Critical Case ν = 0 Let us consider a very particular example such that A(t) is a multiplicative operator of the form A(t)f (x) = (x − t)−2 f (x) in Lp (0, 1) (1 < p < ∞) for 0 ≤ t ≤ 1 = T . It is easy to see that A(t) is a sectorial operator of Lp (Ω). The domain D(A(t)) of A(t) consists of the functions (x − t)2 f (x), where f ∈ Lp (0, 1). It is also verified that, for 0 < θ < 1, A(t)θ f (x) = |x − t|−2θ f (x) with D(A(t)θ ) = {|x − t|2θ f ; f ∈ Lp (0, 1)}. Therefore, condition (3.29) is not the case for any ν > 0. Nevertheless, there exists an evolution operator for A(t) which is given by

U (t, s) = e−

t s

(x−τ )−2 dτ

⎧ [(t−x)−1 +(x−s)−1 ] , ⎪ ⎨e = 0, ⎪ ⎩ [(t−x)−1 +(x−s)−1 ] e ,

t < x ≤ 1, s ≤ x ≤ t, 0 ≤ x < s,

for 0 ≤ s < t ≤ 1 and by U (s, s) = 1 for 0 ≤ s ≤ 1. We can in fact extend our abstract theory built in this chapter to the critical case where ν = 0. Let X be a Banach space. Consider a family of sectorial operators A(t), 0 ≤ t ≤ T , of X satisfying (3.27) and (3.28). We assume that A(·)−1 is strongly continuously differentiable on X for 0 ≤ t ≤ T

(3.114)

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3 Linear Evolution Equations

d and that there exists some fixed exponent 0 < ν  ≤ 1 such that R( dt A(t)−1 ) ⊂  ν D(A(t) ) for all t with the estimates     A(t)ν  d A(t)−1  ≤ N  , 0 ≤ t ≤ T , (3.115)   dt

N  > 0 being a constant. Under (3.27), (3.28), (3.114), and (3.115), we can construct a unique evolution operator U (t, s) for A(t) and can furthermore show similar results as Theorems 3.8, 3.9, and 3.10, see Tanabe [Tan64] (cf. also [Yag76] and [Wat77]).

11 Perturbed Problems We are concerned with the perturbed problem  dU dt + A(t)U + B(t)U = F (t),

0 < t ≤ T,

U (0) = U0 ,

(3.116)

in a Banach space X. Here, A(t), 0 ≤ t ≤ T , is a family of sectorial operators of X satisfying (3.27), (3.28), (3.29), (3.30), and (3.31). Meanwhile, B(t), 0 < t ≤ T , is a family of linear operators of X with singularity at t = 0. We assume that D(A(t)1−˜ν ) ⊂ D(B(t)) for all 0 < t ≤ T with the estimates   ˜ B(t)A(t)ν˜ −1  ≤ Dt ˜ μ−1 , 0 < t ≤ T, (3.117) D˜ > 0 being a constant. Here, μ˜ > 0 and ν˜ > 0 are some fixed exponents such that 0 < μ˜ ≤ 1,

0 < ν˜ ≤ 1 and 1 < μ˜ + ν˜ .

(3.118)

In addition, B(t)A(t)ν˜ −1 is assumed to satisfy a Hölder condition of the form   ˜ σ˜ −1 B(t)A(t)ν˜ −1 − B(s)A(s)ν˜ −1  ≤ Ls ˜ μ− (t − s)σ˜ , 0 < s ≤ t ≤ T , (3.119) with some fixed exponent 0 < σ˜ < min{μ + ν − 1, μ˜ + ν˜ − 1},

(3.120)

L˜ > 0 being a constant. The function F is a given external force function, and U0 is an initial value in X.

11.1 Evolution Operator The purpose of this subsection is to construct an evolution operator U˜ (t, s), 0 ≤ s ≤ t ≤ T , for the family of operators A(t) + B(t).

11 Perturbed Problems

169

We begin with introducing the approximation A˜ n (t) = An (t) + Bn (t),

n = 1, 2, 3, . . . , 0 < t ≤ T ,

where An (t) is the Yosida approximation of A(t), and Bn (t) = B(t)Jn (t)1−˜ν = B(t)A(t)ν˜ −1 An (t)1−˜ν ,

(3.121)

see Proposition 3.3. For any 0 < ε < T , A˜ n (t) is an L(X)-valued continuous function on [ε, T ]. So, we have an evolution operator U˜ n (t, s), ε ≤ s ≤ t ≤ T , for A˜ n (t) such that ∂ ˜ Un (t, s) = −A˜ n (t)U˜ n (t, s), ε ≤ s ≤ t ≤ T , ∂t ∂ ˜ Un (t, s) = U˜ n (t, s)A˜ n (s), ε ≤ s ≤ t ≤ T . ∂s Of course, U˜ n (t, s) does not depend on the choice of ε > 0. Therefore, the evolution operator is defined for any (t, s) such that 0 < s ≤ t ≤ T . The evolution operator U˜ n (t, s) satisfies integral equations. Due to (3.40),  t ∂ ˜ U˜ n (t, s) − Un (t, s) = − Un (t, τ )Un (τ, s) dτ s ∂τ  t =− U˜ n (t, τ )Bn (τ )Un (τ, s) dτ, s

where Un (t, s), 0 ≤ s ≤ t ≤ T , is the evolution operator for An (t). Therefore, U˜ n (t, s) is connected with Un (t, s) by  t ˜ Un (t, s) = Un (t, s) − U˜ n (t, τ )Bn (τ )Un (τ, s) dτ, 0 < s ≤ t ≤ T . (3.122) s

Operating An (t) to this equation, we obtain also an integral equation for W˜ n (t, s) = An (t)U˜ n (t, s) − An (t)Un (t, s) such that  t W˜ n (t, τ )Bn (τ )Un (τ, s) dτ, 0 < s ≤ t ≤ T , (3.123) W˜ n (t, s) = R˜ n (t, s) + s

where R˜ n (t, s) = −



t

An (t)Un (t, τ )Bn (τ )Un (τ, s) dτ. s

Let us now show the convergence of U˜ n (t, s) as n → ∞. From (3.121) we have Bn (τ )Un (τ, s) = B(τ )A(τ )ν˜ −1 An (τ )1−˜ν Un (τ, s). Therefore, by (3.81) and (3.117), ˜ ˜ ν −2 (τ − s)ν˜ −1 ≤ C(τ − s)μ+˜ , Bn (τ )Un (τ, s) ≤ Cτ μ−1

0 < s < τ ≤ T,

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3 Linear Evolution Equations

with some constant C > 0 independent of n. In addition, as n → ∞, Bn (τ )Un (τ, s) → B(τ )A(τ )ν˜ −1 A(τ )1−˜ν U (τ, s)

strongly on X.

We can then apply the dominated convergence theorems, Theorems 1.30 and 1.32, to (3.122) to conclude that U˜ n (t, s) satisfies the uniform estimate U˜ n (t, s) ≤ C0 and is strongly convergent on X to a bounded operator, which will be denoted by U˜ (t, s) for all 0 < s ≤ t ≤ T . Furthermore, U˜ (t, s) is characterized as a solution to the integral equation  t ˜ U (t, s) = U (t, s) − U˜ (t, τ )B(τ )U (τ, s) dτ, 0 < s ≤ t ≤ T . s

In the meantime, we notice for 0 < s < s  < T that U˜ (t, s  ) − U˜ (t, s) = U (t, s  ) − U (t, s) +  +



s

U˜ (t, τ )B(τ )U (τ, s) dτ

s t s

U˜ (t, τ )B(τ )U (τ, s  ) dτ [1 − U (s  , s)].

This implies that U˜ (t, s) is uniformly continuous in s ∈ (0, t] in the strong topology of L(X) (t > 0 being fixed). Therefore, as s → 0, U˜ (t, s) is strongly convergent on X, and the strong convergence is uniform in t . In this way, we conclude that U˜ (t, s) is defined even at s = 0, namely, for all 0 ≤ s ≤ t ≤ T , and is strongly continuous on X for 0 ≤ s ≤ t ≤ T . Let us next consider (3.123). We have the following proposition. Proposition 3.10 For any n = 1, 2, 3, . . . , R˜ n (t, s) satisfies ˜ ν −2 , R˜ n (t, s) ≤ C(t − s)μ+˜

0 ≤ s < t ≤ T,

with some uniform constant C > 0. As n → ∞, Rn (t, s) converges strongly to a ˜ s) on X. R(t, ˜ s) satisfies the same estimate and is bounded linear operator R(t, strongly continuous on X for 0 ≤ s < t ≤ T . Proof We can decompose R˜ n (t, s) in the form  t ˜ Rn (t, s) = − [An (t)Un (t, τ ) − An (t)e−(t−τ )An (t) ]Bn (τ )Un (τ, s) dτ s

 − 

t



An (t)e−(t−τ )An (t) B(τ )A(τ )ν˜ −1 − B(t)A(t)ν˜ −1 An (τ )1−˜ν Un (τ, s) dτ

t



An (t)e−(t−τ )An (t) B(t)A(t)ν˜ −1 1 − An (t)1−˜ν Un (t, τ )An (τ )ν˜ −1

s

− s

× An (τ )1−˜ν Un (τ, s) dτ + [e−(t−s)An (t) − 1]B(t)A(t)ν˜ −1 An (t)1−˜ν Un (t, s).

11 Perturbed Problems

171

We used the formula  −

t

An (t)e−(t−τ )An (t) dτ = e−(t−s)An (t) − 1.

s

We notice by (3.81), (3.85), and (3.120) that  

 1 − An (t)1−˜ν Un (t, τ )An (τ )ν˜ −1 An (τ )1−˜ν Un (τ, s)   

≤  1 − An (t)1−˜ν Un (t, τ )An (τ )ν˜ −1 An (τ )−σ˜ An (τ )1−˜ν +σ˜ Un (τ, s) ≤ C(t − τ )σ˜ (τ − s)ν˜ −σ˜ −1 ,

0 ≤ s < τ < t ≤ T.

Using this together with (3.81), (3.84), (3.117), and (3.119), we deduce that

˜ ν −3 ˜ ν −2 . + (t − s)μ+˜ R˜ n (t, s) ≤ C (t − s)μ+ν+μ+˜ Hence, the desired estimate is proved. The strong convergence of R˜ n (t, s) is also verified by using the decomposition presented above, since each term is easily seen to be strongly convergent on X as n → ∞ by (3.55) and Proposition 3.4.  It is now ready to apply Theorem 1.32 to (3.123) to deduce that W˜ n (t, s) is strongly convergent to a bounded linear operator W˜ (t, s) for each 0 ≤ s < t ≤ T , which is characterized as a solution to the integral equation ˜ s) + W˜ (t, s) = R(t,



t

W˜ (t, τ )B(τ )U (τ, s) dτ,

0 < s ≤ t ≤ T.

s

Consequently, An (t)U˜ n (t, s) = An (t)Un (t, s) + W˜ n (t, s) is strongly convergent to A(t)U (t, s) + W˜ (t, s). Since U˜ (t, s) = U (t, s) + A(t)−1 W˜ (t, s), it follows that R(U˜ (t, s)) ⊂ D(A(t)) and A(t)U˜ (t, s) = A(t)U (t, s) + W˜ (t, s). We can repeat the same arguments as in Sect. 5.3 for proving the analogous results to (3.57), (3.58), (3.59), and (3.60) for A(t) + B(t) and U˜ (t, s). In particular, since U˜ (t, s) ≤ C0 and A(t)U˜ (t, s) ≤ C1 (t − s)−1 hold, the moment inequality yields that for any 0 ≤ θ ≤ 1,   A(t)θ U˜ (t, s) ≤ Cθ (t − s)−θ , 0 ≤ s < t ≤ T . (3.124) It is also seen that for 0 ≤ θ ≤ μ, ˜   A(t)U˜ (t, s)A(s)−θ  ≤ Cθ (t − s)θ−1 ,

0 ≤ s < t ≤ T.

(3.125)

In fact, in the proof of Proposition 3.10, it is possible to utilize (3.83) with θ = 1 − ν˜ to obtain that for 0 ≤ θ ≤ 1 − ν˜ ,   ˜ ν +θ−2 R˜ n (t, s)An (s)−θ  ≤ C(t − s)μ+˜ , 0 ≤ s < t ≤ T.

172

3 Linear Evolution Equations

˜ s)A(s)−θ . Then, for 0 ≤ θ ≤ 1 − ν˜ , It is the same for R(t,   ˜ ν +θ−2 W˜ (t, s)A(s)−θ  ≤ C(t − s)μ+˜ , 0 ≤ s < t ≤ T. This means that (3.125) is valid for 0 ≤ θ ≤ 1 − ν˜ . For 1 − ν˜ < θ ≤ μ, ˜ it is observed that     ˜ W˜ (t, s)A(s)−θ  ≤ C W˜ (t, s)A(s)ν˜ −1  ≤ C(t − s)μ−1 , 0 ≤ s < t ≤ T. This then means that (3.125) is extensively valid for 1 − ν˜ < θ ≤ μ, ˜ too. We sum up the results obtained for the evolution operator U˜ (t, s). Theorem 3.13 Let (3.27), (3.28), (3.29), (3.30), and (3.31) be satisfied for A(t), and let (3.117), (3.118), (3.119), and (3.120) be satisfied for B(t). Then, there exists a unique evolution operator U˜ (t, s) for A(t) + B(t) satisfying (3.57), having the regularity that A(t)U˜ (t, s) is an L(X)-valued continuous function for 0 ≤ s < t ≤ T , and U˜ (t, s) is strongly continuous on X for 0 ≤ s ≤ t ≤ T , satisfies estimates (3.124) and (3.125), and is differentiable in t for t > s with ∂t∂ U˜ (t, s) = −[A(t) + B(t)]U˜ (t, s).

11.2 Cauchy Problem Let us consider problem (3.116) under conditions (3.27), (3.28), (3.29), (3.30), and (3.31) and (3.117), (3.118), (3.119), and (3.120). Let U˜ (t, s) be an evolution operator for A(t) + B(t) constructed above. As a matter of fact, we can repeat the same arguments as in Sect. 2 to prove the following theorem. So, the proof is left to the reader. Theorem 3.14 Let (3.27), (3.28), (3.29), (3.30), and (3.31) be satisfied, and let (3.117), (3.118), (3.119), and (3.120) be also satisfied. Then, for any function F satisfying (3.62) and any initial value U0 ∈ X, (3.116) possesses a unique solution U in the function space: U ∈ C([0, T ]; X) ∩ C1 ((0, T ]; X),

A(t)U ∈ C((0, T ]; X).

Moreover, U is given by the formula  t U˜ (t, s)F (s) ds, U (t) = U˜ (t, 0)U0 +

0 ≤ t ≤ T.

(3.126)

(3.127)

0

Concerning the regularity of B(t)U (t), we have the following result. Corollary 3.1 Let U be the solution obtained in Theorem 3.14. Then, B(t)U (t) satisfies the following estimate:  ˜ ν −2  B(t)U (t) ≤ C U0  + F Bβ−1 t μ+˜ , 0 < t ≤ T, (3.128) {0}

11 Perturbed Problems

173

  ˜ ν −σ˜ −2 B(t)U (t) − B(s)U (s) ≤ C U0  + F Bβ−1 (t − s)σ˜ s μ+˜ , {0}

0 < s < t ≤ T.

(3.129)

Proof The first estimate is verified directly from (3.127). Indeed,     t    ν˜ −1  1−˜ν ˜  ˜ U (t, s)F (s) ds  B(t)U (t) ≤ B(t)A(t) U (t, 0)U0 + A(t)  0

  ˜ ν −2 ≤ C U0  + F Bβ−1 t μ+˜ , {0}

due to (3.124). To prove the second one, we shall use rather a formula of the form  t U (t, s)[F (s) − B(s)U (s)] ds, 0 ≤ t ≤ T , U (t) = U (t, 0)U0 + 0

where U (t, s) denotes the evolution operator for A(t). Then, from

B(t)U (t, 0) − B(s)U (s, 0) = B(t)A(t)ν˜ −1 − B(s)A(s)ν˜ −1 A(s)1−˜ν U (s, 0)

+ B(t)A(t)ν˜ −1 A(t)1−˜ν U (t, s)A(s)ν˜ −1 − 1 A(s)−σ˜ × A(s)1−˜ν +σ˜ U (s, 0) we observe that ˜ ν −σ˜ −2 [B(t)U (t, 0) − B(s)U (s, 0)]U0  ≤ CU0 (t − s)σ˜ s μ+˜ .

Similarly, from  t  s B(t) U (t, τ )F (τ ) dτ − B(s) U (s, τ )F (τ ) dτ 0

= B(t)A(t)ν˜ −1



0 t

A(t)1−˜ν U (t, τ )F (τ ) dτ

s



+ B(t)A(t)ν˜ −1 A(t)1−˜ν U (t, s)A(s)ν˜ −1 − 1 A(s)−σ˜  s

× A(s)1−˜ν +σ˜ U (s, τ )F (τ ) dτ + B(t)A(t)ν˜ −1 − B(s)A(s)ν˜ −1 0

 ×

s

A(s)1−˜ν U (s, τ )F (τ ) dτ

0

we obtain that    t  s   B(t)  U (t, τ )F (τ ) dτ − B(s) U (s, τ )F (τ ) dτ   0

0

˜ ν −σ˜ −2 ≤ CF Bβ−1 (t − s)σ˜ s μ+˜ . {0}

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3 Linear Evolution Equations

It is the same for B(t) μ+˜ ˜ ν −2 ((0, T ]; X) B{0}

t 0

U (t, τ )B(τ )U (τ ) dτ , since (3.128) means that B(·)U (·) ∈

with   B(t)U Bμ+˜ ˜ ν −2 ≤ C U0  + F  β−1 . B {0}



{0}

˜

We notice from (3.128) and (3.129) that B(·)U (·) belongs to Fβ,σ˜ ((0, T ]; X) ˜ with any β˜ such that σ˜ < β˜ < μ+ ˜ ν˜ − 1. Therefore, if F ∈ Fβ,σ˜ ((0, T ]; X) and U0 ∈ ˜ D(A(0)β ), then Theorem 3.10 provides the maximal regularity that dU dt , A(t)U ∈ ˜

Fβ,σ˜ ((0, T ]; X) with the estimate     dU    β˜      dt  β,˜ σ˜ + A(t)U Fβ,˜ σ˜ ≤ C A(0) U0 + F Fβ,˜ σ˜ . F

(3.130)

Remark 3.7 Theorems 3.13 and 3.14 are applicable even for the autonomous case. Namely, A(t) ≡ A is a sectorial operator satisfying (2.92) and (2.93), and B(t) ≡ B is a linear operator satisfying (3.8). Then, it is easily verified that the evolution operator for A + B obtained by Theorem 3.13 is nothing more than the analytic semigroup e−t (A+B) given by Theorem 3.3. Therefore, in this case, we can read all the results in Theorem 3.14 by substituting e−(t−s)(A+B) for U˜ (t, s).

Notes and Further Researches There are three main approaches for linear abstract parabolic evolution equations, namely, the semigroup methods, the variational methods, and the methods of using operational equations. The semigroup methods have originated in the birth of the notion of analytic semigroups (see Notes of Chap. 2). The analytic semigroup was first used to study autonomous abstract parabolic evolution equations by Solomyak [Sol58], and those abstract results were then applied for solving concrete autonomous parabolic differential equations, see Hille [HP57], Yosida [Yos80], Krein [Kre67], and so on. Afterward, Sobolevskii [Sob61a] and Tanabe [Tan60] independently began to study nonautonomous abstract parabolic evolution equations. On the basis of analytic semigroups, they could construct an evolution operator for the Cauchy problem of the form (3.33). Sobolevskii and Tanabe considered first the case where the domains D(A(t)) of coefficient linear operators are independent of the variable t . Sobolevskii [Sob61b] and Kato [Kat61c] then considered the case where the domains D(A(t)) are variable with t but, for some exponent 0 < θ < 1, the domains D(A(t)θ ) of their fractional powers A(t)θ are independent of t . Tanabe [Tan64] then considered the case where the domains D(A(t)) vary completely with t in the sense that, for any exponent, D(A(t)θ ) are possibly variant (see also Kato–Tanabe [KT62]). The semigroup methods are the most constructive ones and provide us the solution formulas (3.13) and (3.51) which furthermore give us much information on the solutions of equations. For the details, see the books by Friedman

Notes and Further Researches

175

[Fri69] and Tanabe [Tan75, Tan97]. In particular, we can get the maximal regularity with respect to the initial values and the external force functions, i.e., Theorems 3.5 and 3.10. The variational methods have originated with Lions [Lio61]. Abstract equations are considered in Hilbert spaces. For L2 external force functions with values in a Hilbert space, unique H 1 solutions are immediately constructed together with the L2 maximal regularity, Theorem 3.12. For details, see Lions–Magenes [LM72] and Dautray–Lions [DL88]. Da Prato–Grisvard [DaPG75] (see also [DaPG84]) devised a method for solving operational equations of the form AU + BU = F given by the sum of two linear operators A and B. Acquistapace–Terreni [AcT86, AcT87] used this method for constructing C1 solutions for abstract parabolic evolution equations in Banach spaces. Lunardi [Lun95] studied the maximal regularity of C 1 solutions. In the meantime, Dore–Venni [DV87] used the method for constructing Wp1 (1 < p < ∞) solutions for Lp external force functions with values in a Banach space enjoying HT property (namely, a Banach space in which the Hilbert transform is bounded) under the condition that the coefficient linear operators have bounded purely imaginary powers. This result gives the Lp (1 < p < ∞) maximal regularity (see also Yamamoto [Yamm93]). Afterward, many authors have studied extensions of this result in the context of H∞ functional calculus in various directions. We refer the reader to the monograph due to Prüss [Pru02] and references therein. There is of course a great deal of important literature which deals with the parabolic differential equations directly by the functional analytical methods. However, we will here quote only some monographs by Ladyzhenskaya–Solonnikov– Uraltseva [LSU68], Amann [Ama95], and so forth. The main Theorems 3.8 and 3.9 were proved when D(A(t)) ≡ D(A(0)) (i.e., ν = 1) in [Sob61a] and [Tan60]. When the domain condition (3.29) fails for any ν > 0 (i.e., ν = 0, see Sect. 10.3), as shown by [Tan64], we have to assume condition (3.115). So, (3.29), (3.30), and (3.31) can be regarded as those of interpolating these conditions of the two limit cases that ν = 1 (with μ > 0) and ν = 0 (with μ > 1). In [Sob61b] and [Kat61c], it was assumed that D(A(t)θ ) are independent of t for some suitable exponent 0 < θ < 1, but we need not to assume such an independence condition. Theorems 3.8 and 3.9 were essentially presented by the paper [Yag88]. So far, maximal regularity was mainly investigated for the correspondence F → dU dt (from the external force function to the derivative of solution), and the initial value U0 was usually assumed to be such that U0 ∈ D(A(0)). Theorem 3.5 or 3.10 newly provides the maximal regularity for both initial value U0 ∈ D(A(0)β ) and external force function F ∈ Fβ,σ ((0, T ]; X). The results obtained in this chapter can be extended in various directions. For example, we are able to get rid of the density of domains D(A(t)) (cf. [Yag88]). Conditions (3.27) and (3.28) can be relaxed to   σ (A(t)) ⊂ Σα = λ ∈ C; c|Im λ|α < Re λ , (3.131)   (λ − A(t))−1  ≤ M , |λ|β

λ∈ / Σα ,

(3.132)

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3 Linear Evolution Equations

respectively, with some exponents 0 < β ≤ α ≤ 1, see [Yag89, Yag90]. We are able to generalize the results to the abstract parabolic evolution equations of the form dU + A(t)U  F (t), dt

0 < t ≤ T,

where A(t) is a family of sectorial multivalued linear operators (Chap. 1, Sect. 3.4), see Favini–Yagi [FY99]. Muramatsu–Tojima [MT01] have relaxed (3.30) to some integrability condition.

Chapter 4

Semilinear Evolution Equations

This chapter is devoted to the Cauchy problem for a semilinear evolution equation of the form (4.1) in a Banach space X. We want to present existence and uniqueness results in a sophisticated way enough to apply them to various models considered in the subsequent chapters (Chaps. 8, 9, 10, 11, 12, and 13). The equation in (4.1) contains two operators, a linear operator A and a nonlinear operator F . The linear operator A is a sectorial operator of X with angle ωA < π2 , and hence −A generates an analytic semigroup on X. Meanwhile, the nonlinear operator F is dominated by the fractional power Aη of A with some exponent 0 ≤ η < 1, and F satisfies a Lipschitz condition described by two fractional powers Aη and Aβ , where β is another exponent such that 0 ≤ β ≤ η < 1. In applications, we intend to take the β as small as possible in view of the fact that η is allowed to be arbitrarily close to 1 by the following two reasons. First, the initial values of (4.1) can be taken in D(Aβ ). Secondly, as Corollaries 4.1 and 4.3 show, the a priori estimates for local solutions of (4.1) with respect to the Aβ graph norm ensure extension of local solutions without limit so as to construct the global solutions. So, the case β = 0 is the optimal one.

1 Semilinear Abstract Evolution Equations 1.1 Cauchy Problems Let X be a Banach space with norm  · . We consider the Cauchy problem for a semilinear abstract evolution equation  dU 0 < t ≤ T, dt + AU = F (U ) + G(t), (4.1) U (0) = U0 , in X. Here, A is a sectorial operator of X satisfying (2.92) and (2.93). Meanwhile, F is a nonlinear operator from D(Aη ) into X, where 0 < η < 1. F is assumed to A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_4, © Springer-Verlag Berlin Heidelberg 2010

177

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satisfy a Lipschitz condition of the form F (U ) − F (V ) ≤ ϕ(Aβ U  + Aβ V )[Aη (U − V ) + (Aη U  + Aη V )Aβ (U − V )],

U, V ∈ D(Aη ), (4.2)

where β is a second exponent such that 0 < β ≤ η < 1,

(4.3)

ϕ(·) being some increasing continuous function. In particular, (4.2) implies the estimate F (U ) ≤ ψ(Aβ U )(Aη U  + 1),

U ∈ D(Aη ),

(4.4)

where ψ(ξ ) = F (0) + ϕ(ξ )(ξ + 1). The function G(t) is an external force function given in the space Fβ,σ ((0, T ]; X), 0 < σ < β. The initial value U0 is taken in D(Aβ ). We shall show the following result. Theorem 4.1 Let (2.92), (2.93), (4.2), and (4.3) be satisfied. Then, for any G ∈ Fβ,σ ((0, T ]; X), where 0 < σ < 1 − η, and any U0 ∈ D(Aβ ), (4.1) possesses a unique local solution U in the function space:  U ∈ C((0, TG,U0 ]; D(A)) ∩ C([0, TG,U0 ]; D(Aβ )) ∩ C1 ((0, TG,U0 ]; X), (4.5) dU AU ∈ Fβ,σ ((0, TG,U0 ]; X), dt , TG,U0 > 0 depending only on the norms GFβ,σ and Aβ U0 . In addition, U satisfies the estimates    dU   (4.6) Aβ U C +   dt  β,σ + AU Fβ,σ ≤ CG,U0 F with some constant CG,U0 > 0 depending on the norms GFβ,σ and Aβ U0 .

1.2 Proof of Theorem 4.1 For each S ∈ (0, T ], we set the following Banach space which consists of D(Aη ) valued continuous functions on (0, S]:   X(S) = U ∈ C((0, S]; D(Aη )) ∩ C([0, S]; D(Aβ )); sup t η−β Aη U (t) < ∞ , 0
U X(S) = sup t η−β Aη U (t) + sup Aβ U (t). 0
0≤t≤S

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In X(S), we consider the subset K(S) which consists of all functions such that t η−β Aη U (t) ≤ C1 , Aβ U (t) ≤ C2 ,

0 < t ≤ S,

(4.7)

0 ≤ t ≤ S,

(4.8)

where the two constants Ci > 0 (i = 1, 2) will be fixed below. Obviously, K(S) is a nonempty closed subset of X(S). For U ∈ K(S), we define the function {ΦU }(t) = e−tA U0 +



t

e−(t−s)A [F (U (s)) + G(s)] ds,

0 ≤ t ≤ S.

(4.9)

0

Our goal is then to show that Φ is a contraction mapping from K(S) into itself, provided that S is sufficiently small, and that the fixed point of Φ is the desired solution of (4.1). For this purpose, we shall divide the proof into several steps. Throughout the proof, CG,U0 denotes a universal constant which is determined in each occurrence by the constants appearing in (2.92), (2.93), (4.2), and (4.3) and by the function ϕ(·) and the norms GFβ,σ and Aβ U0 . Step 1. Let us first verify that Φ maps K(S) into itself. For U ∈ K(S), we have F (U (t)) ≤ ψ(C2 )(C1 t β−η + 1),

0 < t ≤ S,

(4.10)

due to (4.4). For any exponent θ such that β ≤ θ < 1, from Aθ {ΦU }(t) ≤ Aθ−β e−tA Aβ U0   t + Aθ e−(t−s)A [F (U (s) + G(s)] ds 0

we see that 

t

t θ−β Aθ {ΦU }(t) ≤ Aθ−β Aβ U0  + Aθ ψ(C2 )t θ−β  + Aθ Gβ,σ t θ−β

(t − s)−θ (C1 s β−η + 1) ds

0 t

(t − s)−θ s β−1 ds

0

≤ Aθ−β Aβ U0  + Aθ B(1 − θ, β)Gβ,σ + Aθ ψ(C2 )[C1 B(1 − θ, 1 + β − η)t 1−η + B(1 − θ, 1)t 1−β ].

(4.11)

Here, B(p, q) is the beta function, and Aξ is given by Aξ = sup t ξ Aξ e−tA  0
for 0 ≤ ξ ≤ 1

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4 Semilinear Evolution Equations

(note (2.128)). Apply these estimates with θ = η and θ = β. Then, it is observed that if C1 and C2 are fixed in such a way that C1 > Aη−β Aβ U0  + Aη B(1 − η, β)Gβ,σ , C2 > A0 Aβ U0  + Aβ B(1 − β, β)Gβ,σ , and if S is sufficiently small, then Φ(U ) satisfies (4.7) and (4.8). For 0 < s < t ≤ S, we see by the semigroup property that  t −(t−s)A −sA {ΦU }(t) = e e U0 + e−(t−τ )A [F (U (τ )) + G(τ )] dτ + e−(t−s)A



s s

e−(s−τ )A [F (U (τ ) + G(τ )] dτ.

0

From this it is verified that {ΦU }(t) − {ΦU }(s) = [e−(t−s)A − 1]{ΦU }(s)  t + e−(t−τ )A [F (U (τ )) + G(τ )] dτ. s

Then, by (4.10) and (4.11) (θ = η + σ < 1), we obtain that Aη [{ΦU }(t) − {ΦU }(s)] ≤ [e−(t−s)A − 1]A−σ Aη+σ {ΦU }(s)  t + Aη e−(t−τ )A [F (U (τ )) + G(τ )] dτ s

≤ CG,U0 A1−σ (t − s)σ s β−σ −η  t + CG,U0 Aη (t − τ )−η (τ β−1 + 1) dτ. s

Here, we also used property (2.129). Writing β − 1 as β − 1 = (η + σ − 1) + (β − σ − η), we have  t  t (t − τ )−η τ β−1 dτ ≤ (t − τ )−η (τ − s)η+σ −1 s β−σ −η dτ s

s

= B(1 − η, η + σ )(t − s)σ s β−σ −η . Hence, we have obtained that Aη [{ΦU }(t) − {ΦU }(s)] ≤ CG,U0 (t − s)σ s β−σ −η ,

0 < s < t ≤ S.

(4.12)

This in particular shows that ΦU ∈ C((0, S]; D(Aη )). It is also verified, by a similar calculation, that σ −σ

A [{ΦU }(t) − {ΦU }(s)] ≤ CG,U0 (t − s) s β

 + CG,U0

s

t

(t − τ )−β τ β−1 dτ.

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181

Using now the decomposition β − 1 = (β + σ − 1) − σ , we have  t  t (t − τ )−β τ β−1 dτ ≤ (t − τ )−β (τ − s)β+σ −1 s −σ dτ s

s

= B(1 − β, β + σ )(t − s)σ s −σ . Therefore, Aβ [{ΦU }(t) − {ΦU }(s)] ≤ CG,U0 (t − s)σ s −σ ,

0 < s < t ≤ S.

(4.13)

t We already know by Theorem 3.5 that e−tA U0 + 0 e−(t−s)A G(s)ds is continuous at t = 0 in the graph norm of D(Aβ ). In addition, it follows from (4.10) that, as t → 0,   t   t  β  −(t−s)A A  ≤ Aβ e F (U (s)) ds (t − s)−β (s β−η + 1) ds   0

0

≤ CAβ (t 1−η + t 1−β ) → 0. Therefore, ΦU ∈ C([0, S]; D(Aβ )). The mapping Φ has been shown to map K(S) into itself, provided that S > 0 is sufficiently small. Step 2. We next verify that Φ is a contraction mapping of X(S). Let U, V ∈ K(S) be any two functions. For θ such that β ≤ θ < 1, we can write  t θ A [{ΦU }(t) − {ΦV }(t)] = Aθ e−(t−s)A [F (U (s)) − F (V (s))] ds. 0

From (4.2), (4.7), and (4.8) it is seen that  t

θ−β

A [{ΦU }(t) − {ΦV }(t)] ≤ Aθ ϕ(2C2 )t θ

θ−β

t

(t − s)−θ

0

× {A [U (s) − V (s)] + (2C1 s + 1)A [U (s) − V (s)]} ds  t ≤ CAθ (C1 + 1)ϕ(2C2 )t θ−β (t − s)−θ (s β−η + 1) dsU − V X(S) . η

β−η

β

0

Applying these estimates with θ = η and θ = β, we easily conclude that ΦU − ΦV X(S) ≤ C(C1 + 1)ϕ(2C2 )S 1−η U − V X(S) , which shows that Φ is a contraction in X(S), provided that S > 0 is sufficiently small. Step 3. Let S > 0 be sufficiently small in such a way that Φ maps K(S) into itself and is a contraction with respect to the norm X(S). By the arguments of Steps 1 and 2, S = TG,U0 > 0 can be determined by the two norms Gβ,σ and Aβ U0 

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4 Semilinear Evolution Equations

only. Thanks to the fixed point theorem for contraction mappings, Theorem 1.11, there exists a unique function U ∈ K(TG,U0 ) such that U = ΦU . From now on, U stands for this function. Clearly,  t e−(t−s)A [F (U (s)) + G(s)] ds, 0 ≤ t ≤ TG,U0 . (4.14) U (t) = e−tA U0 + 0

We next verify that F (U ) ∈ Fβ,σ ((0, TG,U0 ]; X).

(4.15)

In fact, by(4.10), as t → 0, t 1−β F (U (t)) ≤ CG,U0 (t 1−η + t 1−β ) → 0. Furthermore, from (4.2), (4.12), and (4.13) it follows that F (U (t)) − F (U (s)) = F ({ΦU }(t)) − F ({ΦU }(s))  ≤ CG,U0 (t − s)σ s β−σ −η + (s β−η + 1)(t − s)σ s −σ ≤ CG,U0 (t − s)σ s β−σ −η . Consequently, s 1−β+σ F (U (t)) − F (U (s)) ≤ CG,U0 s 1−η , (t − s)σ

0 < s < t ≤ TG,U0 .

This shows that (1.6) and (1.7) are valid for F (U (t)). Hence, (4.15) is verified. We now know that F (U ) + G ∈ Fβ,σ ((0, T ]; X). In view of (4.14), we can apply Theorem 3.5 to conclude that U belongs to the space (4.5) with estimates (4.6) and is a solution to problem (4.1). Step 4. Let us finally verify the uniqueness of solutions. Let U˜ be any other solution to (4.1) on the interval [0, TG,U0 ] which belongs to the space (4.5). By the definition, t 1−β AU˜ (t) + Aβ U˜ (t) ≤ CU˜ ,

0 < t ≤ TG,U0 .

Moreover, we can verify the intermediate estimate t η−β Aη U˜ (t) ≤ CU˜ ,

0 < t ≤ TG,U0 .

Indeed, by the moment inequality (2.117) for the fractional power A1−β applied with θ = η−β 1−β , we have η−β

1−η

Aη−β V  ≤ CA1−β V  1−β V  1−β ,

V ∈ D(A1−β ).

Then, taking V = Aβ U˜ (t), we obtain the desired estimate.

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183

The formula U˜ (t) = e−tA U0 +



t

e−(t−s)A [F (U˜ (s)) + G(s)] ds,

0

0 ≤ t ≤ TG,U0 ,

together with (4.14), yields that U (t) − U˜ (t) =



t

e−(t−s)A [F (U (s)) − F (U˜ (s))] ds,

0

0 ≤ t ≤ TG,U0 .

We can then repeat the same arguments as in Step 2 to deduce that U − U˜ X(S) ≤ CU,U˜ S 1−η U − U˜ X(S) ,

0 < S ≤ TG,U0 .

This means that, if S > 0 is sufficiently small, then U (t) = U˜ (t) for every 0 ≤ t ≤ S. Let S˜ = sup{S; U (t) = U˜ (t) for all 0 ≤ t ≤ S}, and suppose that S˜ < TG,U0 . Then, ˜ = U˜ (S). ˜ We repeat the same procedure with the initial time S˜ and we have U (S) ˜ ˜ to derive that U (S˜ + t) = U˜ (S˜ + t) for every sufficiently initial value U (S) = U˜ (S) small t > 0, which is a contradiction. Hence, U (t) = U˜ (t) for every 0 < t ≤ TG,U0 .

1.3 Regularity for More Regular Initial Data For an external force function G ∈ Fγ ,σ ((0, T ]; X), β < γ ≤ 1, and an initial value U0 ∈ D(Aγ ), β < γ ≤ 1, we can verify a stronger regularity than (4.5) for the local solution of (4.1). Theorem 4.2 Let (2.92), (2.93), (4.2), and (4.3) be satisfied. Then, for any G ∈ Fγ ,σ ((0, T ]; X), 0 < σ < 1 − η, β < γ ≤ 1, and any U0 ∈ D(Aγ ), β < γ ≤ 1, the local solution U of (4.1) constructed in Theorem 4.1 has the following regularity:  U ∈ C((0, TG,U0 ]; D(A)) ∩ C([0, TG,U0 ]; D(Aγ )) ∩ C1 ((0, TG,U0 ]; X), (4.16) dU AU ∈ Fγ ,σ ((0, TG,U0 ]; X). dt , In addition, U satisfies the estimates    dU   Aγ U C +   dt  γ ,σ + AU Fγ ,σ ≤ CG,U0 F

(4.17)

with some constant CG,U0 > 0 depending on the norms GFγ ,σ and Aγ U0 . Proof From (4.14) we have A U (t) = A η

η−γ −tA

e

 A U0 + γ

0

t

Aη e−(t−s)A [F (U (s)) + G(s)] ds.

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4 Semilinear Evolution Equations

Therefore, in view of (4.4), Aη U (t) ≤ Aη−γ e−tA Aγ U0   t +C (t − s)−η [Aη U (s) + GFγ ,σ s γ −1 + 1] ds. 0

t When η ≤ γ ≤ 1, we see that Aη U (t) ≤ CG,U0 + C 0 (t − s)−η Aη U (s) ds. Theorem 1.27 then provides that Aη U (t) ≤ CG,U0 . When β < γ < η, however, we see that  t η γ −η +C (t − s)−η Aη U (s) ds. A U (t) ≤ CG,U0 t 0

In this case, the function

p(t) = t η−γ Aη U (t) 

p(t) ≤ CG,U0 + C

t

satisfies

t η−γ (t − s)−η s γ −η p(s) ds.

0

Let us solve the integral equation as follows. Let ε > 0 denote a small parameter. For 0 ≤ t ≤ ε,  t p(t) ≤ CG,U0 + C t η−γ (t − s)−η s γ −η ds sup p(s) 0

0≤s≤ε

≤ CG,U0 + Ct 1−η sup p(s) ≤ CG,U0 + Cε 1−η sup p(s). 0≤s≤ε

0≤s≤ε

Hence, (1 − Cε 1−η ) sup0≤s≤ε p(s) ≤ CG,U0 . If ε is taken sufficiently small so that Cε1−η ≤ 12 , then we conclude that p(t) ≤ CG,U0 , 0 ≤ t ≤ ε. Meanwhile, for ε < t ≤ TG,U0 , we have  t  ε p(t) ≤ CG,U0 + C t η−γ (t − s)−η s γ −η ds + CT η−γ ε γ −η (t − s)−η p(s) ds 0

 ≤ CG,U0 + C

ε

t

(t − s)−η p(s) ds.

ε

Then, Theorem 1.27 again provides that p(t) ≤ CU0 ,G , ε ≤ t ≤ TG,U0 . We have thus shown that Aη U (t) ≤ CG,U0 (t γ −η + 1),

0 < t ≤ TG,U0 .

(4.18)

Using (4.18) (instead of (4.7)) and (4.8), we will repeat the same argument as in Step 1 of the proof of Theorem 4.1. Then, it is concluded that Aη [U (t) − U (s)] ≤ CG,U0 (t − s)σ (s γ −η−σ + 1),

0 < s ≤ t ≤ TG,U0 ,

Aβ [U (t) − U (s)] ≤ CG,U0 (t − s)σ (s γ −β−σ + 1),

0 < s ≤ t ≤ TG,U0 .

and

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185

Furthermore, these estimates immediately imply that F (U ) ∈ Fγ ,σ ((0, TG,U0 ]; X) with the estimate F (U )Fγ ,σ ≤ CG,U0 . The desired regularity (4.16) is then an immediate consequence of Theorem 3.5. 

1.4 Global Existence We will return to the case where G ∈ Fβ,σ ((0, T ]; X) and U0 ∈ D(Aβ ). We recall that, according to Theorem 4.1, the interval [0, TG,U0 ] on which the local solution was constructed is determined by the norms GFβ,σ and Aβ U0  only. This fact immediately provides the following global existence of solutions which is utilized very often in applications. Corollary 4.1 Under the assumptions of Theorem 4.1, let G ∈ Fβ,σ ((0, T ]; X), 0 < σ < 1 − η, and U0 ∈ D(Aβ ). Assume that any local solution U of (4.1) in the function space C((0, TU ]; D(A)) ∩ C([0, TU ]; D(Aβ )) ∩ C1 ((0, TU ]; X) satisfies the estimate Aβ U (t) ≤ CG,U0 ,

0 ≤ t ≤ TU ,

(4.19)

with some constant CG,U0 > 0 independent of TU . Then, (4.1) possesses a unique global solution on the interval [0, T ]. Proof We extend the function G to a function G defined on the whole interval [0, ∞) by putting G(t) ≡ G(T ) for T < t < ∞. Clearly, we have GFβ,σ ((a,b];X) ≤ GFβ,σ ((0,T ];X) for any interval (a, b]. Let U1 be any vector in D(Aβ ) satisfying Aβ U1  ≤ CG,U0 . We here consider the Cauchy problem  dV dt + AV = F (V ) + G(t), t1 < t < ∞, (4.20) V (t1 ) = U1 , with initial time t1 such that 0 ≤ t1 < ∞. Theorem 4.1 then provides that there exists a length τ > 0 determined by GFβ,σ and CG,U0 only such that (4.20) always possesses a local solution on an interval [t1 , t1 + τ ]. On account of this fact, we can prove the assertion of the corollary. Let us start with the local solution to (4.1) on [0, TG,U0 ] obtained in Theorem 4.1. Clearly, TG,U0 ≥ τ . Setting t1 = TG,U0 − τ2 and U1 = U (t1 ), we consider problem (4.20) to obtain a local solution V on an interval [t1 , t1 + τ ]. By the uniqueness of solution, we have U (t) = V (t) for t1 ≤ t ≤ TG,U0 ; this means that we have constructed

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4 Semilinear Evolution Equations

a local solution to (4.1) on the interval [0, TG,U0 + τ2 ]. The a priori estimate (4.19) allows us to continue this procedure unlimitedly. Each time, the local solution is extended over the fixed length τ2 of interval. So, by finite times, the extended interval can cover the given interval [0, T ]. 

1.5 Lipschitz Continuity of Solutions in Initial Data Let G be a bounded ball G = {G ∈ Fβ,σ ((0, T ]; X); Gβ,σ ≤ R1 },

0 < R1 < ∞,

of the space Fβ,σ ((0, T ]; X), and let B be a bounded ball B = {U ∈ D(Aβ ); Aβ U  ≤ R2 },

0 < R2 < ∞,

of D(Aβ ). Then, for all G ∈ G and U0 ∈ B, there exists a local solution of (4.1) on some interval [0, TG,U0 ]. There is then time TG,B > 0 such that [0, TG,B ] ⊂ [0, TG,U0 ] for all pairs of (G, U0 ) ∈ G × B. We shall show the continuous dependence of solutions on (G, U0 ). Theorem 4.3 Under (2.92), (2.93), (4.2), and (4.3), let U and V be the solutions of (4.1) for the data (G, U0 ) and (H, V0 ) in G × B, respectively. Then, t η Aη [U (t) − V (t)] + t β Aβ [U (t) − V (t)] + U (t) − V (t) ≤ CG,B [U0 − V0  + t β G − H β,σ ],

0 < t ≤ TG,B .

Proof This theorem is proved by analogous arguments as in the proof of Theorem 4.1. Indeed, for 0 ≤ θ < 1, we see that  t Aθ [U (t) − V (t)] = Aθ e−tA (U0 − V0 ) + Aθ e−(t−s)A 0

× {[F (U (s)) − F (V (s))] + [G(s) − H (s)]} ds, 0 < t ≤ TB,G . As a consequence,  t θ Aθ [U (t) − V (t)] ≤ Aθ U0 − V0  + Aθ t θ  + CG,B Aθ t θ

t

(t − s)−θ G(s) − H (s) ds

0 t

(t − s)−θ {Aη [U (s) − V (s)]

0

+ (s β−η + 1)Aβ [U (s) − V (s)]} ds

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187

 ≤ Aθ U0 − V0  + Aθ G − H β,σ t θ  + CG,B Aθ t

θ

t

(t − s)−θ s β−1 ds

0 t

(t − s)−θ s −η p(s) ds,

0

where p(t) = t η Aη [U (t) − V (t)] + t β Aβ [U (t) − V (t)],

0 < t ≤ TG,B .

Using these estimates with θ = β and η, we obtain the integral inequality p(t) ≤ C[U0 − V0  + t β G − H β,σ ]  t η + CG,B t (t − s)−η + t β (t − s)−β s −η p(s) ds,

0 < t ≤ TG,B .

0

To solve this inequality, we use the same techniques as in the proof of Theorem 4.2. Arguing first in a small interval [0, ε] and then in the other interval [ε, TG,B ], we in fact conclude that p(t) ≤ CG,B [U0 − V0  + t β G − H β,σ ],

0 < t ≤ TG,B .

Finally, taking θ = 0, we conclude also that U (t) − V (t) ≤ CG,B [U0 − V0  + t β G − H β,σ ],

0 ≤ t ≤ TG,B .



Corollary 4.2 Under the conditions of Theorem 4.3, t η−β Aη [U (t) − V (t)] + Aβ [U (t) − V (t)] ≤ CG,B [Aβ (U0 − V0 ) + G − H β,σ ],

0 < t ≤ TG,B .

Proof The techniques used in the proof of Theorem 4.3 are available for verifying the assertion. It indeed suffices to substitute the estimate Aθ e−tA (U0 − V0 ) ≤ Aθ−β t β−θ Aβ (U0 − V0 ) with θ = β and η for Aθ e−tA (U0 − V0 ) ≤ Aθ t −θ U0 − V0 .



1.6 Case β = 0 It is possible to generalize some results of Theorem 4.1 to the critical case where β = 0. Let us assume that (4.2) is valid with β = 0, i.e., F (U ) − F (V ) ≤ ϕ(U  + V )[Aη (U − V ) + (Aη U  + Aη V )U − V ],

U, V ∈ D(Aη ),

(4.21)

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4 Semilinear Evolution Equations

with 0 ≤ η < 1.

(4.22)

We can then verify the following result. Theorem 4.4 Let (2.92), (2.93), (4.21), and (4.22) be satisfied. Then, for any G ∈ Fβ,σ ((0, T ]; X), 0 < σ < β ≤ 1 − η, and any U0 ∈ X, (4.1) possesses a unique local solution U in the function space: U ∈ C([0, TG,U0 ]; X) ∩ C1 ((0, TG,U0 ]; X), AU ∈ C((0, TG,U0 ]; X),

(4.23)

TG,U0 > 0 depending only on the norms GFβ,σ and U0 . In addition, U satisfies the estimates    dU   (4.24) (t) U (t) + t   + tAU (t) ≤ CG,U0 , 0 < t ≤ TG,U0 , dt with some constant CG,U0 > 0 depending on the norms GFβ,σ and U0 . Proof The method of proving this theorem is quite analogous to that of Theorem 4.1. We set for 0 < S ≤ T , the basic Banach space   X(S) = U ∈ C((0, S]; D(Aη )) ∩ C([0, S]; X); sup t η Aη U (t) < ∞ , 0
U X(S) = sup t A U (t) + sup U (t). η

0
η

0≤t≤S

We consider the subset K(S) which consists of functions such that t η Aη U (t) ≤ C1 , U (t) ≤ C2 ,

0 < t ≤ S,

(4.25)

0 ≤ t ≤ S,

(4.26)

with some constants Ci > 0 (i = 1, 2) which will be fixed appropriately. As for (4.12) and (4.13), we observe that (4.25) and (4.26) yield that Aη [{ΦU }(t) − {ΦU }(s)] ≤ C(t − s)σ s −σ −η , σ −σ

{ΦU }(t) − {ΦU }(s) ≤ C(t − s) s

,

0 < s < t ≤ S,

0 < s < t ≤ S,

where Φ is the mapping defined by (4.9). We can then show that, if Ci > 0 (i = 1, 2) are chosen appropriately and S > 0 is sufficiently small, then Φ maps the set K(S) into itself and is a contraction with respect to the norm X(S). Consequently, Φ possesses a unique fixed point U ∈ K(S). Since F (U ) = F (ΦU ) ∈ F β,σ ((0, S]; X),

(4.27)

we conclude that U is a unique solution of (4.1) in the function space (4.23). Estimate (4.24) is verified by Theorem 3.4. 

2 Variational Methods

189

By the same argument as in Corollary 4.1, we can similarly construct a global solution to (4.1). Corollary 4.3 Under the assumptions of Theorem 4.4, let G ∈ Fβ,σ ((0, T ]; X), 0 < σ < β ≤ 1 − η, and U0 ∈ X. Assume that any local solution U of (4.1) in the function space C((0, TU ]; D(A)) ∩ C([0, TU ]; X) ∩ C1 ((0, TU ]; X)

(4.28)

satisfies the estimate U (t) ≤ CG,U0 ,

0 ≤ t ≤ TU ,

(4.29)

with some constant CG,U0 > 0 independent of TU . Then, (4.1) possesses a unique global solution in the function space: U ∈ C((0, T ]; D(A)) ∩ C([0, T ]; X) ∩ C1 ((0, T ]; X). Let us finally verify the Lipschitz continuity of solutions in initial data. Let G be a bounded ball G = {G ∈ Fβ,σ ((0, T ]; X); Gβ,σ ≤ R1 },

0 < R1 < ∞,

of the space Fβ,σ ((0, T ]; X), 0 < σ < β ≤ 1 − η, and let B be a bounded ball B = {U ∈ X; U  ≤ R2 },

0 < R2 < ∞,

of X. Then, there is an interval [0, TG,B ] on which (4.1) has a unique local solution for any pair (G, U0 ) ∈ G × B. Then, we have the following result. Theorem 4.5 Under (2.92), (2.93), (4.21), and (4.22), let U and V be the solutions of (4.1) for the data (G, U0 ) and (H, V0 ) in G × B, respectively. Then, t η Aη [U (t) − V (t)] + U (t) − V (t) ≤ CG,B [U0 − V0  + t β G − H β,σ ],

0 < t ≤ TG,B .

As the proof of this theorem is quite analogous to that of Theorem 4.3, we may omit it.

2 Variational Methods We here present existence and uniqueness results for problem (4.1) by the variational methods. Let Z and X be two separable Hilbert spaces with dense and compact embedding Z ⊂ X. Let  ·  and | · | be the norms of Z and X, respectively. Let Z ⊂ X ⊂ Z ∗ be a triplet of spaces. Let  · ∗ be the norm of Z ∗ . Let a(U, V ) be a sesquilinear

190

4 Semilinear Evolution Equations

form defined on Z satisfying (2.6) and (2.7), and let A be a sectorial operator of Z ∗ associated with the form. We are concerned with the Cauchy problem for a semilinear evolution equation  dU dt + AU = F (U ) + G(t), 0 < t ≤ T , (4.30) U (0) = U0 , in Z ∗ . Here, F is a nonlinear operator from Z to Z ∗ and is assumed to satisfy the following conditions. For any positive number ζ > 0, there exist continuous increasing functions φζ (·) ≥ 0 and ψζ (·) ≥ 0 such that the following estimates hold: F (U )∗ ≤ ζ U  + φζ (|U |),

U ∈ Z;

(4.31)

F (U ) − F (V )∗ ≤ ζ U − V  + (U  + V  + 1)ψζ (|U | + |V |)|U − V |, U, V ∈ Z.

(4.32)

The external force function G is a Z ∗ -valued function belonging to L2 (0, T ; Z ∗ ), and the initial value U0 is taken in X. We then obtain the following theorem. For the proof, see Lions [Lio69] (see also [RY01]). Theorem 4.6 Let (4.31) and (4.32) be satisfied. Then, for any G ∈ L2 (0, T ; Z ∗ ) and any U0 ∈ X, there exists a unique local solution U to (4.30) in the function space: U ∈ L2 (0, TU0 ,G ; Z) ∩ C([0, TU0 ,G ]; X) ∩ H 1 (0, TU0 ,G ; Z ∗ ), where TU0 ,G > 0 is determined by the norms |U0 | and GL2 (0,T ;Z ∗ ) . In addition, U satisfies the estimates U L2 (0,TG,U0 ;Z) + U C([0,TG,U0 ];X) + U H 1 (0,TG,U

0

;Z ∗ )

≤ CG,U0 ,

where CG,U0 > 0 is a constant determined by the norms GL2 (0,T ;Z ∗ ) and |U0 |.

3 Semilinear Parabolic Equations We consider the initial-boundary-value problem for a semilinear parabolic equation ⎧ n ∂u ⎪ ⎪ i,j =1 Dj [aij (x)Di u] + f (x, u, ∇u) + g(x, t) in Ω × (0, T ), ∂t = ⎨ n on ∂Ω × (0, T ), (4.33) i,j =1 νj (x)aij (x)Di u = 0 ⎪ ⎪ ⎩ in Ω, u(x, 0) = u0 (x) in a bounded domain Ω ⊂ Rn with C2 boundary ∂Ω, where 0 < T < ∞ is fixed time. We assume that aij (x) satisfy (2.19) and (2.28). The function f (x, u, ζ ) is

4 Competition System with Diffusions

191

a complex-valued function for x ∈ Ω, u ∈ R + iR, and ζ ∈ (R + iR)n , which is a smooth function with respect to the real variables x ∈ Ω, Re u, Im u, Re ζ , and Im ζ . The function g(x, t) is an external force function, and u0 (x) is an initial function. We want to handle (4.33) in a Banach space X  = Lp (Ω), where n < p < ∞. Let Ap be a realization of the differential operator − i,j Dj [aij (x)Di ] + 1 in Lp (Ω) ∂u under the Neumann-type boundary conditions ∂ν = 0 on ∂Ω. It is known that the A domain D(Ap ) is characterized by (2.77). Using this operator, we obtain an abstract formulation of (4.33) of the form (4.1). Here, the nonlinear operator F (u) is given by F (u) = u + f (x, u, ∇u),

u ∈ Hps (Ω),

where the exponent s is taken as 1 + pn < s < 2. We notice by (1.76) that Hps (Ω) ⊂ C1 (Ω) with continuous embedding. We choose exponents β and η as 2s < β = η < 1. Then, according to (16.36) in η Chap. 16, it is observed that D(Ap ) ⊂ Hps (Ω). In addition, by a direct calculation, F (u) is shown to fulfill condition (4.2). So, the abstract results obtained above are available to (4.33). By Theorem 4.1, for any g ∈ Fβ,σ ((0, T ]; Lp (Ω)), 0 < σ < β 1 − η, and any u0 ∈ D(Ap ), there exists a unique local solution u to (4.33) in the function space:  β u ∈ C((0, Tg,u0 ]; D(Ap )) ∩ C([0, Tg,u0 ]; D(Ap )) ∩ C1 ((0, Tg,u0 ]; Lp (Ω)), du Ap u ∈ Fβ,σ ((0, Tg,u0 ]; Lp (Ω)), dt , β

Tg,u0 > 0 depending only on the norms gFβ,σ and Ap u0 . Furthermore, u satisfies the estimates      β  A U  +  du  p  dt  β,σ + Ap uFβ,σ ≤ Cg,u0 . C F

4 Competition System with Diffusions Consider the initial-boundary-value problem for the competition system ⎧ ∂u ⎪ = aΔu + cu − γ11 u2 − γ12 uv in Ω × (0, ∞), ⎪ ⎪ ∂t ⎪ ⎪ ⎨ ∂v = bΔv + dv − γ21 uv − γ22 v 2 in Ω × (0, ∞), ∂t ∂u ∂v ⎪ on ∂Ω × (0, ∞), ⎪ ⎪ ∂n = ∂n = 0 ⎪ ⎪ ⎩u(x, 0) = u (x), v(x, 0) = v (x) in Ω, 0

(4.34)

0

in a C2 or convex, bounded domain Ω ⊂ R3 . We consider two competing species of biological individuals in Ω. Their densities are denoted by u(x, t) and v(x, t), respectively. We consider diffusions of individuals, and the diffusion rates of species are denoted by a and b, respectively.

192

4 Semilinear Evolution Equations

We assume that a, b, c, d, and γij (i, j = 1, 2) are positive constants (> 0). Initial functions u0 , v0 are assumed to satisfy 0 ≤ u0 , v0 ∈ L2 (Ω).

4.1 Construction of Local Solutions We want to handle (4.34) in the product space X = L2 (Ω) (see (1.84)). The problem is then formulated as the Cauchy problem for an abstract equation  dU 0 < t < ∞, dt + AU = F (U ), (4.35) U (0) = U0 , in X. Here, A is a diagonal matrix operator of X given by A = diag {A1 , A2 } with domain D(A) = H2N (Ω), where A1 and A2 are realizations of operators −aΔ + 1 and −bΔ + 1, respectively, in L2 (Ω) under the Neumann boundary conditions on ∂Ω. It is known by (2.34) that A1 and A2 are positive definite self-adjoint operators of L2 (Ω) with domains D(A1 ) = D(A2 ) = HN2 (Ω). Furthermore, according to Theorems 16.7 and 16.9 in Chap. 16, D(Aθ1 ) = D(Aθ2 ) = H 2θ (Ω) if 0 ≤ θ < 34 and D(Aθ1 ) = D(Aθ2 ) = HN2θ (Ω) if 34 < θ ≤ 1. As a consequence, A is a positive definite self-adjoint operator of X, and the domains of fractional powers are given 3 by D(Aθ ) = H2θ (Ω) if 0 ≤ θ < 34 and D(Aθ ) = H2θ N (Ω) if 4 < θ ≤ 1. The nonlinear operator F : L4 (Ω) → L2 (Ω) is defined by     u (c + 1)u − γ11 u2 − γ12 uv , U= ∈ L4 (Ω). F (U ) = (d + 1)v − γ21 uv − γ22 v 2 v The space of initial functions is given by     u K = U0 = 0 ; 0 ≤ u0 ∈ L2 (Ω) and 0 ≤ v0 ∈ L2 (Ω) . v0 It is immediate to verify that the nonlinear operator F fulfills the condition (4.21) 2η with 34 < η < 1. Indeed, since D(Aη ) = HN (Ω) ⊂ L∞ (Ω), we have F (U ) − F (V )L2 ≤ C(U L∞ + V L∞ + 1)U − V L2 ≤ C(Aη U L2 + Aη V L2 + 1)U − V L2 ,

U, V ∈ D(Aη ).

Theorem 4.4 then provides the existence of local solutions. Indeed, for any U0 ∈ L2 (Ω), (4.35) possesses a unique local solution U in the function space:   U ∈ C (0, TU0 ]; H2N (Ω) ∩ C([0, TU0 ]; L2 (Ω)) ∩ C1 ((0, TU0 ]; L2 (Ω)). (4.36) Here, TU0 > 0 is determined by the norm U0 L2 only. Furthermore, the solution satisfies the estimates U (t)L2 + tAU (t)L2 ≤ CU0 .

(4.37)

4 Competition System with Diffusions

193

4.2 Nonnegativity of Local Solutions We shall show that the local solution constructed above is nonnegative. For U0 ∈ K, let U = t(u, v) be the local solution in the function space (4.36) with estimate (4.37), and let us prove that u(t) ≥ 0 and v(t) ≥ 0 for every 0 < t ≤ TU0 . We first notice that U (t) is real valued. Indeed, the complex conjugate U (t) of U (t) is also a local solution of (4.35) with the same initial value U0 . So, the uniqueness of solution implies that U (t) = U (t); hence, U (t) must be real valued. We will employ the truncation method. Let H (u) be a C1,1 cutoff function such that H (u) = 12 u2 for −∞ <u < 0 and H (u) ≡ 0 for 0 ≤ u < ∞. According to (1.100), the function ψ(t) = Ω H (u(t)) dx is continuously differentiable with the derivative  

ψ (t) = a H (u)Δu dx + H (u)(cu − γ11 u2 − γ12 uv) dx. Ω

Ω

By property (1.96), we have   

H (u)Δu dx = − ∇H (u) · ∇u dx = − |∇H (u)|2 dx ≤ 0. Ω

Ω

Ω

Meanwhile,  H (u)u(c − γ11 u − γ12 v) dx ≤ CH (u)uL1 (1 + uL∞ + vL∞ ) Ω

≤ CH (u)L1 (1 + uH 2η + vH 2η ), where η is the exponent fixed above. Therefore, it follows that ψ (t) ≤ C[1 + t η Aη U (t)L2 ]ψ(t); consequently, ψ(t) ≤ ψ(0)eC 0 (1+A U (τ )L2 )dτ . Estimates (4.37) imply that Aη U (τ )L2 ≤ CU0 τ −η , which means that Aη U (τ )L2 is integrable in (0, TU0 ). Hence, ψ(0) = 0 implies that ψ(t) ≡ 0, namely, u(t) ≥ 0 for 0 < t ≤ TU0 . It is the same for the proof of v(t) ≥ 0 for 0 < t ≤ TU0 .

4.3 Global Solutions For U0 ∈ K, let U (t) denote a nonnegative local solution of (4.35) on [0, TU ] in the function space:   0 ≤ u, v ∈ C (0, TU ]; HN2 (Ω) ∩ C([0, TU ]; L2 (Ω)) ∩ C1 ((0, TU ]; L2 (Ω)). (4.38) We shall obtain a priori estimates for local solutions. Proposition 4.1 There exists a constant C > 0 such that, for any local solution U in (4.38) with U0 ∈ K, it holds that U (t)L2 ≤ C(e−t U0 L2 + 1),

0 ≤ t ≤ TU .

(4.39)

194

4 Semilinear Evolution Equations

Proof Multiply the first equation of (4.34) by u and integrate the product in Ω. Then, 1 d 2 dt



 u2 dx + a Ω

 |∇u|2 dx =

Ω

(cu2 − γ11 u3 − γ u2 v) dx 

Ω

≤

(cu2 − γ11 u3 ) dx. Ω

Since 1 (2c + 1)3 cu2 − γ11 u3 ≤ − u2 + 2 2 27γ11

for all u ≥ 0,

it follows that 1 d 2 dt



1 u dx ≤ − 2 Ω

 u2 dx +

2

Ω

|Ω|(2c + 1)3 . 2 27γ11

Solving this differential inequality (cf. (1.58)), we obtain that 2|Ω|(2c + 1)3 , 2 27γ11

u(t)2L2 ≤ e−t u0 2L2 +

0 ≤ t ≤ TU .

We verify by the similar arguments that v(t)2L2 ≤ e−t v0 2L2 +

2|Ω|(2d + 1)3 , 2 27γ22

0 ≤ t ≤ TU .

This, together with the above, yields the desired estimate (4.39).



The global existence is now obtained immediately by Corollary 4.3. Indeed, (4.39) shows that estimate (4.29) holds for any nonnegative local solution with initial value U0 ∈ K. We can then repeat the same arguments as in Corollary 4.3 to conclude the global existence of a solution. We have thus deduced that, for any U0 ∈ K, there exists a unique global solution of (4.34) in the function space:   0 ≤ u, v ∈ C (0, ∞); HN2 (Ω) ∩ C([0, ∞); L2 (Ω)) ∩ C1 ((0, ∞); L2 (Ω)).

5 Some Model in Immunology We are concerned with the initial-boundary-value problem for some mathematical model describing the immune system. The problem is given by

5 Some Model in Immunology

195

⎧ ∂u ⎪ ⎪ ∂t = c − du − αuρ ⎪ ⎪ ⎪ ∂v ⎪ ⎪ ⎪ ∂t = αuρ − ev − βvw ⎪ ⎪ ⎪ ∂w ⎪ ⎪ ⎨ ∂t = aΔw − f w + γ (x) ∂ρ ∂t = bΔρ − gρ ⎪ ⎪ ⎪ ∂ρ ∂w ⎪ ⎪ = ∂n =0 ⎪ ⎪ ∂n

+ hv

in Ω × (0, ∞), in Ω × (0, ∞), in Ω × (0, ∞), in Ω × (0, ∞),

(4.40)

on ∂Ω × (0, ∞), ⎪ ⎪ ⎪ ⎪ v(x, 0) = v0 (x), u(x, 0) = u0 (x), ⎪ ⎪ ⎩ ρ(x, 0) = ρ0 (x) in Ω, w(x, 0) = w0 (x), in a C2 or convex, bounded domain Ω ⊂ R3 . Here, Ω denotes an organism or a body. The function u(x, t) denotes the density of uninfected target cells in Ω at time t, and v(x, t) denotes the density of infected cells by viruses. By w(x, t) the density of killer cells is denoted. The killer cells are produced at a rate γ (x) in Ω by the effect of the immune system and kill the infected cells. By ρ(x, t) the density of viruses is denoted. Viruses are produced by the infected cells and diffuses in Ω with diffusion rate b. We assume that 0 ≤ γ ∈ L2 (Ω)

(4.41)

and that a, b, c, d, e, f , and g are positive constants (>0) and α, β are also positive constants (>0). We will handle this problem in the product space   X = L2 (Ω) = t(u, v, w, ρ); u, v, w and ρ ∈ L2 (Ω) . The space of initial functions is given by   K = t(u0 , v0 , w0 , ρ0 ); 0 ≤ u0 , v0 , w0 and ρ0 ∈ L2 (Ω) .

(4.42)

5.1 Construction of Local Solutions Problem (4.40) is formulated in X as the Cauchy problem of the form (4.35). Here, the linear operator A is a diagonal matrix operator of X given by A = diag{d, e, A1 , A2 }, where A1 and A2 are realizations of differential operators −aΔ + f and −bΔ + g, respectively, in L2 (Ω) under the Neumann boundary conditions on ∂Ω. So, D(A) is given by   D(A) = t(u, v, w, ρ); u, v ∈ L2 (Ω) and w, ρ ∈ HN2 (Ω) (see (2.34)). Clearly, A is a positive definite self-adjoint operator of X. For 0 < θ < 1, its fractional powers Aθ are also diagonal operators diag{d θ , eθ , Aθ1 , Aθ2 } and, thanks to Theorems 16.7 and 16.9, their domains are characterized by  D(Aθ ) = {t(u, v, w, ρ); u, v ∈ L2 (Ω) and w, ρ ∈ H 2θ (Ω)}, 0 ≤ θ < 34 , D(Aθ ) = {t(u, v, w, ρ); u, v ∈ L2 (Ω) and w, ρ ∈ HN2θ (Ω)}, 34 < θ ≤ 1.

196

4 Semilinear Evolution Equations

The nonlinear operator F is given by F (U ) = t (c − αuρ, αuρ − βvw, γ (x), hv) with domain   D(F ) = t(u, v, w, ρ); u, v ∈ L2 (Ω) and w, ρ ∈ L∞ (Ω) . Fix an exponent η such that 34 < η < 1. Then, by virtue of (1.76), we have D(Aη ) ⊂ D(F ). Furthermore, it is immediate to see that F satisfies condition (4.21). Thanks to Theorem 4.4, we can state that, for any U0 = t(u0 , v0 , w0 , ρ0 ) ∈ K, there exists a unique local solution U = t(u, v, w, ρ) to (4.40) in the function space: U ∈ C((0, TU0 ]; D(A)) ∩ C([0, TU0 ]; X) ∩ C1 ((0, TU0 ]; X),

(4.43)

where TU0 > 0 is determined by the norm U0 X only. In addition, the local solution satisfies the estimates    dU   U (t)L2 + t  (4.44)  dt  + tAU (t)L2 ≤ CU0 , 0 < t ≤ TU0 , L2 CU0 > 0 being determined by U0 L2 .

5.2 Nonnegativity of Local Solutions For U0 ∈ K, let U be the local solution of (4.40) constructed above. We want to show that u(t) ≥ 0, v(t) ≥ 0, w(t) ≥ 0, and ρ(t) ≥ 0 for every 0 < t ≤ TU0 . For this purpose, however, we have to introduce the modified nonlinear operator F˜ (U ) = t (c − αuρ, αuρ − βvw, γ (x), hχ(Re v)),

U ∈ D(F˜ ) = D(F ),

where χ(v) denotes a cutoff function such that χ(v) ≡ 0 for −∞ < v < 0 and χ(v) = v for 0 ≤ v < ∞. We have to consider the auxiliary problem 

d U˜ dt

+ AU˜ = F˜ (U ),

U˜ (0) = U0 ,

0 < t < ∞,

(4.45)

in X. It is clear that the new nonlinear operator F˜ also satisfies (4.21) with the same ˜ L2 for v, v˜ ∈ L2 (Ω). So, exponent η because of χ(Re v) − χ(Re v) ˜ L2 ≤ v − v ˜ v, ˜ w, ˜ ρ) ˜ on an interval [0, T˜U0 ] in (4.45) possesses a unique local solution U˜ = t(u, the same function space (4.43). Then, it is seen that u(t) ˜ ≥ 0, v(t) ˜ ≥ 0, w(t) ˜ ≥ 0, ρ(t) ˜ ≥ 0 for every 0 < t ≤ T˜0 . In fact, let us first verify that U˜ (t) is real valued. It is clear that the complex conjugate U˜ (t) of U˜ (t) is also a local solution to (4.45) with the same initial value; so, by the uniqueness of solution, U˜ (t) = U˜ (t); hence, U˜ (t) is real valued.

5 Some Model in Immunology

197

Let us next use a cutoff function H (ρ) ˜ such that H (ρ) ˜ = 12 ρ˜ 2 for −∞ < ρ˜ < 0 and H (ρ) ˜ ≡ 0 for 0 ≤ ρ˜ < ∞. Since ρ˜ ∈ C([0, T˜U0 ];L2 (Ω)) ∩ C1 ((0, T˜U0 ]; L2 (Ω)) due to (4.43), we observe by (1.100) that ψ(t) = Ω H (ρ(t)) ˜ dx is continuously differentiable with the derivative   

ψ (t) = b H (ρ)Δ ˜ ρ˜ dx − g H (ρ) ˜ ρ˜ dx + h H (ρ)χ( ˜ v) ˜ dx. Ω

Ω

Ω

By property (1.96), we have    H (ρ)Δ ˜ ρ˜ dx = − ∇H (ρ) ˜ · ∇ ρ˜ dx = − |∇H (ρ)| ˜ 2 dx ≤ 0. Ω

Ω

Ω

˜ ≤ 0 and H (ρ) ˜ ρ˜ ≥ 0, it follows that ψ (t) ≤ 0; conseIn addition, since H (ρ) quently, ψ(t) ≤ ψ(0) for 0 < t ≤ T˜U0 . Thus, ψ(0) = 0 implies ψ(t) ≡ 0, namely, ρ(t) ˜ ≥ 0 for 0 ≤ t ≤ T˜U0 . We can repeat the same argument for u, ˜ v, ˜ w˜ also. In fact, knowing that ρ(t) ˜ ≥ 0, we are able to prove that u(t) ˜ ≥ 0 for 0 < t ≤ T˜U0 . Since ˜ ρ(t) ˜ ≥ 0 and w(t) ˜ ≥ 0, we γ (x) ≥ 0, w(t) ˜ ≥ 0 for 0 < t ≤ T˜U0 . Finally, since u(t) conclude that v(t) ˜ ≥ 0 for 0 < t ≤ T˜U0 . We now notice that v(t) ˜ ≥ 0 implies that χ(v(t)) ˜ = v(t); ˜ in other words, U˜ is a local solution of the original problem (4.40), too. The uniqueness of solution then implies that U (t) ≡ U˜ (t). Hence, u(t) ≥ 0, v(t) ≥ 0, w(t) ≥ 0, ρ(t) ≥ 0 for 0 < t ≤ T˜U0 . If T˜U0 ≥ TU0 , we finish the proof. If not, we put T0 = sup{0 < T  ≤ TU0 ; u(t) ≥ 0, v(t) ≥ 0, w(t) ≥ 0, ρ(t) ≥ 0 for every 0 < t ≤ T }. Since Ω H (u(T0 )) dx = limtT0 Ω H (u(t)) dx = 0, we see that u(T0 ) ≥ 0. By similar reasons, we have v(T0 ) ≥ 0, w(T0 ) ≥ 0, ρ(T0 ) ≥ 0. So, if T0 = TU0 , we finish the proof. If T0 < TU0 , we will consider again problem (4.45) but with the initial time T0 and the initial value U (T0 ). Repeating the same arguments as above, we conclude that there is τ > 0 such that u(t) ≥ 0, v(t) ≥ 0, w(t) ≥ 0, ρ(t) ≥ 0 for every t such that TU0 ≤ t ≤ TU0 + τ . This is a contradiction; hence, T0 = TU0 .

5.3 Global Solutions In order to construct a global solution, let us obtain a priori estimates for local solutions. For U0 ∈ K, let U = t(u, v, w, ρ) be a local solution of (4.40) on [0, TU ] in the function space: 0 ≤ U ∈ C((0, TU ]; D(A)) ∩ C([0, TU ]; X) ∩ C1 ((0, TU ]; X). Integrate the first equation of (4.40) in Ω. Then,   d u dx = (c − du − αuρ) dx. dt Ω Ω

198

4 Semilinear Evolution Equations

Similarly, by the second equation of (4.40),   d v dx = (αuρ − ev − βvw) dx. dt Ω Ω   d Therefore, dt Ω (u + v) dx ≤ c|Ω| − δ Ω (u + v) dx, where δ = min{d, e}. Consequently (cf. (1.58)), it follows that u(t) + v(t)L1 ≤ e−δt (u0 + v0 L1 ) + cδ −1 |Ω|,

0 ≤ t ≤ TU .

(4.46)

Multiply the first equation by u and integrate the product in Ω. Then,   1 d u2 dx = (cu − du2 − αu2 ρ) dx. 2 dt Ω Ω Meanwhile, multiplying the first equation by v and integrating the product in Ω, we have   du v dx = (cv − duv − αuvρ) dx. Ω dt Ω In a similar way, we can obtain from the second equation on v the equalities   dv u dx = (αu2 ρ − euv − βuvw) dx Ω dt Ω and 1 d 2 dt



 v 2 dx = Ω

(αuvρ − ev 2 − βv 2 w) dx. Ω

Therefore, summing up these four equalities, we have    1 d c(u + v) − δ(u + v)2 dx (u + v)2 dx ≤ 2 dt Ω Ω  ≤ C(u0 + v0 L1 + 1) − δ (u + v)2 dx Ω

(due to (4.46)). Hence, solving this differential inequality, we conclude that u(t) + v(t)2L2 ≤ e−2δt u0 + v0 2L2 + C(u0 + v0 L1 + 1),

0 ≤ t ≤ TU .

Since u(t) ≥ 0 and v(t) ≥ 0, it then follows that u(t)2L2 + v(t)2L2 ≤ e−2δt u0 + v0 2L2 + C(u0 + v0 L1 + 1), 0 ≤ t ≤ TU .

(4.47)

Multiply the fourth equation on ρ by ρ and integrate the product in Ω. Then, after some calculations,     1 d g h2 2 2 2 ρ dx + b |∇ρ| dx + g ρ dx ≤ ρ 2 dx + v(t)2L2 . 2 dt Ω 2 Ω 2g Ω Ω

6 Nonautonomous Semilinear Abstract Evolution Equations

199

Using (4.47), we obtain that     g 1 d ρ 2 dx + ρ 2 dx ≤ C u0 + v0 2L2 + 1 . 2 dt Ω 2 Ω Hence,   ρ(t)2L2 ≤ e−gt ρ0 2L2 + C u0 + v0 2L2 + 1 ,

0 ≤ t ≤ TU .

(4.48)

In view of (4.41), we can use the similar techniques to conclude that w(t)2L2 ≤ e−f w0 2L2 + C, 0 ≤ t ≤ TU . Hence, this, together with (4.47) and (4.48), yields that U (t)L2 ≤ C(U0 L2 + 1),

0 ≤ t ≤ TU .

(4.49)

Now, when we have the a priori estimates (4.29), the global existence is concluded by applying Corollary 4.3. Indeed, for any U0 ∈ K, (4.40) possesses a unique global solution in the function space: 0 ≤ U ∈ C((0, ∞); D(A)) ∩ C([0, ∞); X) ∩ C1 ((0, ∞); X).

6 Nonautonomous Semilinear Abstract Evolution Equations In this section, we will present some device for treating nonautonomous semilinear abstract evolution equations. Consider the Cauchy problem for a nonautonomous evolution equation  dU dt

+ AU = F (t, U ) + G(t),

0 < t ≤ T,

U (0) = U0 ,

(4.50)

in a Banach space X. Here, A is a sectorial operator of X satisfying (2.92) and (2.93). For each 0 ≤ t ≤ T , F (t, ·) is a nonlinear operator from D(Aη ) into X, where η is some fixed exponent. We assume the following Lipschitz condition: F (t, U ) − F (s, V ) ≤ ϕ(Aβ U  + Aβ V )   × Aη (U − V ) + (Aη U  + Aη V  + 1)[|t − s| + Aβ (U − V )] , (t, U ), (s, V ) ∈ [0, T ] × D(Aη ),

(4.51)

where β is a second fixed exponent satisfying the relation 0 ≤ β ≤ η < 1. The function G is a given external force function in Fβ,σ ((0, T ]; X). The initial value U0 is taken in D(Aβ ).

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4 Semilinear Evolution Equations

We consider the product Banach space X = C × X and rewrite problem (4.50) in the form  d  τ   1 0  τ     0  τ +1 dt U + 0 A U = F (χ(Re τ ),U ) + G(t) , 0 < t ≤ T , (4.52) τ  0 U (0) = U0 , where χ(t) is a C0,1 cutoff function for −∞ < t < ∞ such that χ(t) ≡ 0 for −∞ < t < 0, χ(t) = t for 0 ≤ t ≤ T , and χ(t) ≡ T for T < t < ∞. We can consider (4.52) as an autonomous abstract evolution equation in X for the unknown function t(τ, U ). Since τ (t) = t, problem (4.52) is certainly equivalent to (4.50). It is easy to see that the operator A = diag{1, A} is a sectorial operator of X satisfying (2.92) and (2.93). Furthermore, for 0 < θ < 1, Aθ = diag{1, Aθ } with domain D(Aθ ) = C × D(Aθ ). Therefore, condition (4.51) implies (4.2) if β > 0 and (4.21) if β = 0 for the nonlinear F(τ, U ) = t(τ + 1, F (χ(Re τ ), U )), which is an operator from D(Aη ) into X. In addition, U0 ∈ D(Aβ ) implies that t(0, U0 ) ∈ D(Aβ ). Hence, Theorems 4.1 if β > 0 and 4.4 if β = 0 are available to (4.52).

Notes and Further Researches The theory of semilinear abstract parabolic evolution equations has been developed in the study of the Navier–Stokes equations by the functional analytical methods. As pioneering works, we quote Fujita–Kato [FK64], Giga [Gig81, Gig85, Gig86], and Giga–Miyakawa [GM85]. For the variational methods, we quote Lions [Lio69] and Temam [Tem84]. When the Lipschitz condition (4.2) (or (4.21)) is valid with α = η, Theorem 4.1 (or 4.4) is quite standard (e.g., [Zei90, Theorem 19.I]). The generalization for the case where α < η was first done by von Wahl [vWa85] and Hoshino–Yamada [HY91]. The proof of Theorem 4.1 was presented by Osaki–Yagi [OY02]. Theorem 4.4 can be obtained by a slight modification of Theorem 4.1. In this chapter, we have described applications of the abstract theory only for the Lotka–Volterra competition model and some model in immunology. It is actually possible to apply our abstract results to enormous reaction–diffusion models presented from the real world. (We will indeed treat some of these in Chaps. 8, 9, 10, 11, 12, and 13). A historical study on the Lotka–Volterra model is seen in Israel– Gasca [IG02]. Stationary problem of (4.34) were studied by Dancer–Du [DD95a, DD95b], Kan-on [Kan02, Kan03, Kan06, Kan07], and others. Nakashima–Wakasa [NW07] and Dancer–Zhang [DZ02] studied the asymptotic behavior of solutions of the evolutional problem of (4.34). In addition, there is a great deal of literature which treats related problems to (4.34) including the homogeneous Dirichlet boundary conditions, variable coefficients, nonautonomous equations, and so forth. For mathematical models describing the immune system against the threat of invasion by bacteria and viruses, we refer the reader to Murray [Mur02, Chap. 10]. The reaction–diffusion model in (4.40) was obtained by incorporating spreading of killer cells and viruses to the ordinary differential equation model presented by Britton [Bri03, Sect. 6.5].

Chapter 5

Quasilinear Evolution Equations

This chapter is devoted to the Cauchy problem for quasilinear evolution equations of the forms (5.1) and (5.43) in Banach spaces. The linear operators in the equations are sectorial operators with angles < π2 but are allowed to depend on the unknown functions. Besides an underlying space X in which the problem is given, we shall introduce two more Banach spaces Y and Z such that Z ⊂ Y ⊂ X. The sectorial operators A(U ) are defined for U ∈ Z. Like (3.29), we make the assumption that D(A(V )) ⊂ D(A(U )ν ) for any U, V ∈ Z with some fixed exponent 0 < ν ≤ 1. In addition, instead of (3.30), A(U )ν [A(U )−1 − A(V )−1 ] is estimated by the Y norm of U − V , i.e., U − V Y . We shall use two more exponents α and β as graded scales to measure the spaces Y and Z, respectively. That means that we assume the inclusions D(A(U )α ) ⊂ Y and D(A(U )β ⊂ Z for all U ∈ Z with 0 ≤ α < β < 1. The exponents are also assumed to satisfy the relation 1 + α < β + ν. The optimal one is of course the case where ν = 1, which means that the domains D(A(U )) are independent of U . When ν ≤ β, we have to add the assumption of reflexibility of Z. The initial value U0 is taken in Z; more precisely, U0 must satisfy a compatibility condition of the form U0 ∈ D(A(U0 )β ) (⊂ Z). In applying the abstract results of this chapter, we have to set the three spaces Z ⊂ Y ⊂ X in a suitable way. The space X is a function space in which we want to handle a quasilinear equation to be considered. The space Z is chosen as large as possible with the condition that, for all U ∈ Z, A(U ) are well defined as sectorial operators of X. Recall that the initial values can be taken only in Z. The space Y is also chosen as close as possible to X in order that the exponent condition 1 + α < β + ν is valid. The optimal one is the case where Y = X, i.e., α = 0. The norm which we have to estimate a priori for obtaining global solutions is the Z norm of the local solutions, more precisely, the norm A(U (t))β U (t)X . The abstract results are actually applicable to various quasilinear problems. In Chaps. 14 and 15, we shall apply them to physical and biological models. A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_5, © Springer-Verlag Berlin Heidelberg 2010

201

202

5 Quasilinear Evolution Equations

1 Quasilinear Abstract Evolution Equations 1.1 Structural Assumptions We consider the Cauchy problem of an abstract quasilinear evolution equation  dU dt

+ A(U )U = F (t),

0 < t ≤ T,

(5.1)

U (0) = U0 ,

in a Banach space X. Let Z be a second Banach space continuously embedded in X, and let K be the open ball of Z, K = B Z (0; R) = {U ∈ Z; U Z < R},

0 < R < ∞.

For U ∈ K, A(U ) is a sectorial operator of X with angle ωA(U ) < π2 and with domain D(A(U )) which is allowed to depend on U . The function F is a given external force function on (0, T ]. The initial value U0 is taken in K(⊂ Z). We make the following assumptions. The spectrum σ (A(U )) is contained in a fixed open sectorial domain, i.e., σ (A(U )) ⊂ Σω = {λ ∈ C; |arg λ| < ω},

U ∈ K,

(5.2)

λ∈ / Σω , U ∈ K,

(5.3)

with some angle 0 < ω < π2 , and the resolvent satisfies   (λ − A(U ))−1  ≤ M/|λ|, L(X)

with a constant M ≥ 1. The domain D(A(U )) may vary with U ∈ K, but there is an exponent 0 < ν ≤ 1 such that   D(A(V )) ⊂ D A(U )ν for any pair of U, V ∈ K. (5.4) In addition, A(U ) satisfies a Lipschitz condition of the form    A(U )ν A(U )−1 − A(V )−1  ≤ N U − V Y , U, V ∈ K, L(X)

(5.5)

with a constant N > 0. Here, Y is a third Banach space such that Z ⊂ Y ⊂ X with continuous embeddings. There are two exponents 0 ≤ α < β < 1 such that, for any U ∈ K, D(A(U )α ) ⊂ Y and D(A(U )β ) ⊂ Z with the estimates  U˜ Y ≤ D1 A(U )α U˜ X , U˜ ∈ D(A(U )α ), U ∈ K, (5.6) U˜ ∈ D(A(U )β ), U ∈ K, U˜ Z ≤ D2 A(U )β U˜ X , Di > 0 (i = 1, 2) being some constants. The exponents satisfy the relations 0≤α<β <ν≤1

and 1 + α < β + ν.

(5.7)

1 Quasilinear Abstract Evolution Equations

203

From the relation   A(U )−1 = A(0)−1 + A(0)−ν A(0)ν A(U )−1 − A(0)−1 (5.5) (together with (2.118)) implies that    ν   A(U )−1  ≤ A(0)−1 L(X) + 2c(M + 1)N R A(0)−1 L(X) , L(X) where c > 0 is the embedding constant  · Y ≤ c · Z . The purpose of this section will be to construct a local solution to (5.1) for any F ∈ Fβ,σ ((0, T ]; X) and any U0 ∈ K satisfying the condition U0 ∈ D(A(U0 )β ). In this section, the notation C will stand for a universal constant which is determined in each occurrence by the angle ω, the exponents α, β, ν, σ , the initial constants T , R, M, N, D1 , D2 , c, and the norm A(0)−1 L(X) . When the constant depends also on some other factors, say F or U0 , we shall denote it by CF,U0 . Theorem 5.1 Let (5.2), (5.3), (5.4), (5.5), (5.6), and (5.7) be satisfied. Let F belong to F ∈ Fβ,σ ((0, T ]; X),

0 < σ < β − α + ν − 1,

(5.8)

and let U0 ∈ K satisfy the condition   U0 ∈ D A(U0 )β . Then, there exists a unique local solution U to (5.1) in the function space:  U ∈ C([0, TF,U0 ]; Z) ∩ Cβ−α ([0, TF,U0 ]; Y ) ∩ C1 ((0, TF,U0 ]; X), A(U )β U ∈ C([0, TF,U0 ]; X);

dU dt ,

A(U )U ∈ Fβ,σ ((0, TF,U0 ]; X),

(5.9)

(5.10)

where TF,U0 > 0 is determined by F and U0 . In addition, U satisfies the estimates       A(U )β U  +  dU  (5.11)  dt  β,σ + A(U )U Fβ,σ ≤ CF,U0 , C F where CF,U0 > 0 is a constant depending on F and U0 . Remark 5.1 It is very important to investigate the dependence of TF,U0 on the initial datum. We must notice by the proof of the theorem described below (see Steps 3 and 4) that TF,U0 depends not only on the norms   F Fβ,σ and A(U0 )β U0 X but also on the following moduli of continuity: ωF (t) = sup 0≤s
s 1−β+σ F (t) − F (s)X (t − s)σ

as t → 0,

(5.12)

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5 Quasilinear Evolution Equations

  ωU0 (t) = sup [e−sA(U0 ) − 1]A(U0 )β U0 X

as t → 0,

(5.13)

0≤s≤t

  ωF,U0 (t) = sup s β−1 [e−sA(U0 ) − 1]A(U0 )β−1 s 1−β F (s)X

as t → 0.

0≤s≤t

(5.14) In view of (2.131) and the existence of the limit lims→0 s 1−β F (s), we have ωF,U0 (t) → 0 as t → 0.

1.2 Proof of Theorem 5.1 The proof is divided into several steps. Step 1. For S such that 0 < S ≤ T , we set the Banach space μ

Z(S) = B([0, S]; Z) ∩ C{0} ([0, S]; Y ) with some fixed exponent μ such that 1 − ν + σ < μ < β − α. Recall that the space μ C{0} ([0, S]; Y ) (see Sect. 2.3, Chap. 1) is given by  U (t) − U (0)Y μ < ∞ C{0} ([0, S]; Y ) = U ∈ C([0, S]; Y ); sup tμ 0
(5.15)

with the norm U Cμ

{0} ([0,S]; Y )

= U C([0,S]; Y ) + sup

0
U (t) − U (0)Y . tμ

We set also the subset K(S) of Z(S) given by  K(S) = U ∈ Z(S); U (0) = U0 ; sup U (t)Z ≤ R1 and 0≤t≤S

sup 0≤s
U (t) − U (s)Y ≤ 1 . |t − s|μ

Here, R1 is a constant fixed as U0 Z < R1 < R.

(5.16)

Then, K(S) is clearly a nonempty closed subset of Z(S). Step 2. For each V ∈ K(S), let AV (t) be a family of sectorial operators AV (t) = A(V (t)), 0 ≤ t ≤ S. We then consider the Cauchy problem for a linear equation  dU dt + AV (t)U = F (t), 0 < t ≤ S, (5.17) U (0) = U0 .

1 Quasilinear Abstract Evolution Equations

205

It is quite easy to observe that the family of AV (t) satisfies all the structural assumptions (3.28), (3.29), (3.30), and (3.31) introduced in Chap. 3, Sect. 4.1 with the exponents μ and ν presented above and with some suitable constants independent of U0 satisfying U0 Z < R1 . As F and U0 satisfy (3.97) and (3.98), respectively, there exists a unique solution U to (5.17) in the space: U ∈ C([0, S]; X) ∩ C1 ((0, S]; X),

AV U ∈ Fβ,σ ((0, S]; X).

Indeed, the solution is given by

t

U (t) = UV (t, 0)U0 +

UV (t, s)F (s) ds,

0 ≤ t ≤ S,

(5.18)

0

where UV (t, s) denotes the evolution operator for the family of AV (t). We can then define the mapping Φ from K(S) to Z(S) by {Φ(V )}(t) = U (t),

0 ≤ t ≤ S, for V ∈ K(S).

Step 3. If S > 0 is sufficiently small, then Φ maps the set K(S) into itself. Indeed, for U = Φ(V ), we write U (t) as U (t) = U0 + [e−tA(U0 ) − 1]U0 + [UV (t, 0) − e−tAV (0) ]AV (0)−β A(U0 )β U0

t

t + UV (t, s)[F (s) − F (t)] ds + [UV (t, s) − e−(t−s)AV (s) ] ds F (t) 0

+

0 t

[e−(t−s)AV (s) − e−(t−s)AV (0) ] ds F (t) − [e−tA(U0 ) − 1]A(U0 )−1 F (t),

0

using (3.15). Then, by (5.6) and (5.13),   [e−tA(U0 ) − 1]U0 Z ≤ D2 [e−tA(U0 ) − 1]A(U0 )β U0 X ≤ D2 ωU0 (t). Utilizing (3.87) and (5.6), we verify that   [UV (t, 0) − e−tAV (0) ]AV (0)−β A(U0 )β U0  Z   β −tA (0) V ≤ D2 AV (t) [UV (t, 0) − e ]AV (0)−β L(X) U0 β ≤ Ct μ+ν−1 U0 β ; here and in what follows, U0 β denotes the quantity A(U0 )β U0 X . By (3.80), (5.6), (5.8), and (5.12),   t    UV (t, s)[F (s) − F (t)] ds    0



≤C 0

Z

t

(t − s)σ −β s β−σ −1 ds sup 0<s
s 1−β+σ F (t) − F (s)X ≤ CωF (t). (t − s)σ

206

5 Quasilinear Evolution Equations

Similarly, by (3.87) and (5.6),   t    [UV (t, s) − e−(t−s)AV (s) ] ds F (t)   0

Z



t

≤C 0

(t − s)−β+μ+ν−1 ds t β−1 F Fβ,σ ≤ Ct μ+ν−1 F Fβ,σ .

By (3.90) and (5.6),  t     [e−(t−s)AV (s) − e−(t−s)AV (0) ] ds F (t)   0



≤C 0

t

Z

(t − s)ν−β−1 s μ ds t β−1 F Fβ,σ ≤ Ct μ+ν−1 F Fβ,σ .

Finally, by (5.6) and (5.14),  −tA(U )  0 − 1]A(U )−1 F (t) ≤ D ω [e 0 2 F,U0 (t). Z We have therefore verified that U (t) − U0 Z ≤ C[ωF (t) + ωU0 (t) + ωF,U0 (t)] + Ct μ+ν−1 (F Fβ,σ + U0 β ),

0 < t ≤ S.

(5.19)

Hence, if S > 0 is sufficiently small, then (5.16) implies that sup U (t)Z ≤ R1 .

(5.20)

0≤t≤S

Furthermore, by (3.101), we have   AV (t)β U (t) ≤ C(F Fβ,σ + U0 β ). X

(5.21)

To verify the Hölder condition, let us notice from (5.18) that

s U (t) = UV (t, s)UV (s, 0)U0 + UV (t, s) UV (s, τ )F (τ ) dτ 0



t

+

UV (t, τ )F (τ ) dτ. s

Therefore,

U (t) − U (s) = [UV (t, s) − 1]U (s) +

t

UV (t, τ )F (τ ) dτ s

= {[UV (t, s) − e−(t−s)AV (s) ] + [e−(t−s)AV (s) − 1]}AV (s)−β

t β × AV (s) U (s) + UV (t, τ )F (τ ) dτ, 0 ≤ s < t ≤ S. s

1 Quasilinear Abstract Evolution Equations

207

Then, by (3.87) and (2.129),   AV (t)α [UV (t, s) − e−(t−s)AV (s) ]AV (s)−β  ≤ C(t − s)β−α+μ+ν−1 , L(X)   AV (s)α [e−(t−s)AV (s) − 1]AV (s)−β  ≤ C(t − s)β−α , L(X) respectively. By (3.80),   t    AV (t)α UV (t, τ )F (τ ) dτ    s X

t ≤ CF β,σ (t − τ )−α (τ − s)β−1 dτ s

≤ CF β,σ (t − s)β−α . In view of (5.6) and (5.21), we therefore verified that U (t) − U (s)Y ≤ C(F β,σ + U0 β )(t − s)β−α .

(5.22)

Hence, if S > 0 is sufficiently small, then U (t) − U (s)Y ≤ (t − s)μ ,

0 ≤ s ≤ t ≤ S.

Step 4. If S > 0 is sufficiently small, then the mapping Φ : K(S) → Z(S) is a contraction with respect to the  · Z(S) norm. Indeed, for Ui = Φ(Vi ), Vi ∈ K(S), i = 1, 2, we have

t U1 (t) − U2 (t) = [UV1 (t, 0) − UV2 (t, 0)]U0 + [UV1 (t, s) − UV2 (t, s)]F (s) ds. 0

Here, we will prove the following proposition. Proposition 5.1 For 0 ≤ θ < ν,   AV (t)θ [UV (t, 0) − UV (t, 0)]U0  1 1 2 X ≤ Cθ t β−θ+μ+ν−1 U0 β V1 − V2 Cμ

{0} ([0,S];Y )

,

0 ≤ t ≤ S,

(5.23)

,

0 ≤ t ≤ S.

(5.24)

and  

t   AV (t)θ [UV1 (t, s) − UV2 (t, s)]F (s) ds    1 0

≤ Cθ t

β−θ+μ+ν−1

F β,σ V1 − V2 Cμ

X

{0} ([0,S];Y )

Proof In order to show these results, which play a fundamental role in the proof, we have to employ the evolution operators UVi ,n (t, s) (i = 1, 2) for the families of Yosida approximations AVi ,n (t) (i = 1, 2) for AVi (t). Indeed, we observe that

208

5 Quasilinear Evolution Equations

AV1 ,n (t)θ [UV1 ,n (t, 0) − UV2 ,n (t, 0)]AV2 ,n (0)−β

t   = AV1 ,n (t)θ UV1 ,n (t, s)AV1 ,n (s) AV1 ,n (s)−1 − AV2 ,n (s)−1 0

× AV2 ,n (s)UV2 ,n (s, 0)AV2 ,n (0)−β ds. Letting n → ∞, we obtain that AV1 (t)θ [UV1 (t, 0) − UV2 (t, 0)]AV2 (0)−β

t   = AV1 (t)θ UV1 (t, s)AV1 (s)1−ν AV1 (s)ν AV1 (s)−1 − AV2 (s)−1 0

×AV2 (s)UV2 (s, 0)AV2 (0)−β ds. Therefore,   AV (t)θ [UV (t, 0) − UV (t, 0)]AV (0)−β  1 1 2 2 L(X)

t ≤C (t − s)ν−θ−1 s β−1 V1 (s) − V2 (s)Y ds

0 t

≤C 0

(t − s)ν−θ−1 s μ+β−1 ds V1 − V2 Cμ

{0} ([0,S];Y )

.

From this the first assertion is verified. We next write

t AV1 ,n (t)θ [UV1 ,n (t, s) − UV2 ,n (t, s)]F (s) ds =

0 t

t

0

s

  AV1 ,n (t)θ UV1 ,n (t, τ )AV1 ,n (τ ) AV1 ,n (τ )−1 − AV2 ,n (τ )−1

× AV2 ,n (τ )UV2 ,n (τ, s)F (s) dτ ds

t   = AV1 ,n (t)θ UV1 ,n (t, τ )AV1 ,n (τ ) AV1 ,n (τ )−1 − AV2 ,n (τ )−1 0



× AV2 ,n (τ )

τ 0

By Theorem 3.10, AV2 ,n (τ ) 

 AV ,n (τ )  2

τ 0

UV2 ,n (τ, s)F (s) ds dτ. τ 0

UV2 ,n (τ, s)F (s) ds satisfies the uniform estimate

  β−1 UV2 ,n (τ, s)F (s) ds  F Fβ,σ ((0,S]; X)  ≤ Cτ X

and converges, as n → ∞, to a continuous function g(τ ) on (0, S]. Then, letting n → ∞, we obtain that

1 Quasilinear Abstract Evolution Equations

AV1 (t)θ

0 t

= 0

t

209

[UV1 (t, s) − UV2 (t, s)]F (s) ds

  AV1 (t)θ UV1 (t, τ )AV1 (τ )1−ν AV1 (τ )ν AV1 (τ )−1 − AV2 (τ )−1 g(τ ) dτ.

Hence,  

t   AV (t)θ  [U (t, s) − U (t, s)]F (s) ds V1 V2  1  0 X

t ≤C (t − τ )ν−θ−1 τ β−1 V1 (τ ) − V2 (τ )Y dτ F Fβ,σ ((0,S]; X) 0

≤C

0

t

(t − τ )ν−θ−1 τ μ+β−1 dτ V1 − V2 Cμ

{0} ((0,S];Y )

F β,σ . 

Thus, we have verified the second estimate of the proposition. Using this proposition with θ = β, we see that   AV (t)β [U1 (t) − U2 (t)] 1 X ≤ CS μ+ν−1 (F β,σ + U0 β )V1 − V2 Cμ

{0} ([0,S];Y )

,

0 ≤ t ≤ S. (5.25)

Hence, U1 − U2 B([0,S];Z) ≤ C(F β,σ + U0 β )S μ+ν−1 V1 − V2 Cμ

{0} ([0,S];Y )

μ

. (5.26)

The C{0} norm of U1 − U2 is also estimated in a quite similar way by applying Proposition 5.1 with θ = α. Indeed,   AV (t)α [UV (t, 0) − UV (t, 0)]U0  1 1 2 X ≤ Ct β−α+μ+ν−1 U0 β V1 − V2 Cμ

{0} ([0,S];Y )

≤ Ct μ S β+ν−α−1 U0 β V1 − V2 Cμ

{0} ([0,S];Y )

,

0 < t ≤ S.

,

0 < t ≤ S.

Similarly,  

t    AV (t)α [U (t, s) − U (t, s)]F (s) ds V1 V2   1 0

≤ Ct S

μ β+ν−α−1

Therefore,

F β,σ V1 − V2 Cμ

X

{0} ([0,S];Y )

210

5 Quasilinear Evolution Equations

sup t −μ [U1 (t) − U2 (t)] − [U1 (0) − U2 (0)]Y 0
≤ CS β+ν−α−1 (F β,σ + U0 β )V1 − V2 Cμ

{0} ([0,S];Y )

,

0 < S ≤ T. (5.27)

This, together with (5.26), then yields that U1 − U2 Z(S) ≤ CS μ+ν−1 (F β,σ + U0 β )V1 − V2 Z(S) ,

V1 , V2 ∈ K(S).

Hence, Φ is a contraction mapping from K(S) to Z(S), provided that S > 0 is sufficiently small. Step 5. Let TF,U0 = S > 0 be small enough to make the mapping Φ : K(S) → K(S) a contraction. The fixed point theorem, Theorem 1.11, ensures the existence of a unique fixed point U ∈ K(S). Since U satisfies the formula

t UU (t, s)F (s) ds, 0 ≤ t ≤ TF,U0 , (5.28) U (t) = UU (t, 0)U0 + 0

U is shown to be a local solution to (5.1) on the interval [0, TF,U0 ]. Since 0 < σ < μ + ν − 1, U must satisfy all the conditions of (5.10), except that U ∈ Cβ−α ([0, TF,U0 ]; Y ) and U ∈ C([0, TF,U0 ]; Z); see Theorem 3.10. The Hölder condition was however verified by (5.22). Similarly, by (5.19) it is already seen that U (t) is continuous at t = 0 in the Z norm. Meanwhile, for 0 < t ≤ TU0 ,F , we have AU (t + h)β [U (t + h) − U (t)] = AU (t + h)β U (t + h) − AU (t)β U (t)   + AU (t + h)β AU (t + h)−1 − AU (t)−1 AU (t)U (t)   − AU (t + h)β−1 − AU (t)β−1 AU (t)U (t). Because of AU (·)β−1 ∈ C([0, TF,U0 ]; L(X)), we verify that U (t + h) − U (t)Z → 0

as h → 0.

Therefore, U (t) is also continuous in the topology of Z for t > 0. We already know that the family of operators AU (t), 0 ≤ t ≤ TF,U0 , satisfies (3.27)–(3.31) and that U (t) is represented by (5.28). Therefore, (5.11) is an immediate consequence of Theorem 3.10. Step 6. Let us finally verify the uniqueness of solutions of (5.1) in the space (5.10). Under (5.8) and (5.9), let U˜ be any other local solution belonging to the function space (5.10) on an interval [0, TU˜ ]. We assume that TU˜ ≤ TU0 ,F and prove that U˜ (t) = U (t) for every 0 ≤ t ≤ TU˜ . Since there exists an evolution operator UU˜ (t, s) for the family of operators AU˜ (t) = A(U˜ (t)), 0 ≤ t ≤ TU˜ , U˜ is written by the formula

t ˜ UU˜ (t, s)F (s) ds, 0 ≤ t ≤ TU˜ . U (t) = UU˜ (t, 0)U0 + 0

1 Quasilinear Abstract Evolution Equations

211

Then, we can repeat a quite similar argument as in Step 4 to obtain the estimate U˜ − U Z(S) ≤ CU˜ S μ+ν−1 U˜ − U Z(S) ,

0 < S ≤ TU˜ .

This then shows that U˜ (t) = U (t) for every t ∈ [0, S] if S > 0 is sufficiently small. That is, the uniqueness is true in a neighborhood of the initial time. We here consider the interval I = {S ∈ (0, TU˜ ]; U˜ (t) = U (t) for every t ∈ [0, S]}. By the discussion above, I is nonempty. In addition, I is open in (0, TU˜ ]. Indeed, if S ∈ I and S < TU˜ , then VS = U˜ (S) = U (S) ∈ K satisfies the condition VS ∈ D(A(VS )). It is clear that F ∈ F1,σ ([S, TU˜ ]). Then, we can construct a local solution to the problem  dV dt + A(V )V = F (t), S < t ≤ T , V (S) = VS . As the solution is unique in a neighborhood of the initial time S, we conclude that V (t) = U˜ (t) = U (t) for every t > S which is sufficiently close to S. Therefore, I is open. In the meantime, it is clear that I is closed in (0, TU˜ ]. Hence, I = (0, TU˜ ]. We have thus accomplished the proof of Theorem 5.1. 

1.3 Stronger Results for More Regular Data When F and U0 in problem (5.1) satisfy stronger conditions than (5.8) and (5.9), respectively, we can strengthen the results of Theorem 5.1. Let β < γ ≤ 1. Let F belong to F ∈ Fγ ,σ ((0, T ]; X),

0 < σ < β − α + ν − 1.

Let U0 satisfy the stronger compatibility condition   U0 ∈ D A(U0 )γ , β < γ ≤ 1.

(5.29)

(5.30)

Theorem 5.2 Let (5.2), (5.3), (5.4), (5.5), (5.6), and (5.7) be satisfied. Let F and U0 ∈ K satisfy (5.29) and (5.30), respectively. Then, there exists a unique local solution to (5.1) in the function space:  U ∈ C([0, TF,U0 ]; Z) ∩ Cγ −α ([0, TF,U0 ]; Y ) ∩ C1 ((0, TF,U0 ]; X), (5.31) γ ,σ ((0, T A(U )γ U ∈ C([0, TF,U0 ]; X); dU F,U0 ]; X), dt , A(U )U ∈ F where TF,U0 > 0 is determined by the norms F Fγ ,σ ((0,T ];X)) and A(U0 )γ U0 X only. Furthermore, U satisfies the estimates       A(U )γ U  +  dU  (5.32)  dt  γ ,σ + A(U )U Fγ ,σ ≤ CF,U0 , C F

212

5 Quasilinear Evolution Equations

where CF,U0 > 0 is a constant determined by the norms F Fγ ,σ and A(U0 )γ U0 X . Proof By the definition (5.13), we have     ωU0 (t) ≤ sup [e−sA(U0 ) − 1]A(U0 )β−γ A(U0 )γ U0 X ≤ Ct γ −β A(U0 )γ U0 X . 0≤s≤t

Similarly, by the definitions (5.12) and (5.14), ωF (t) + ωU0 ,F (t) ≤ Ct γ −β F Fγ ,σ . These estimates mean that (5.20) is valid if S γ −β (F Fγ ,σ + A(U0 )γ U0 X ) is sufficiently small. Hence, TF,U0 depends only on these quantities. By the same arguments as in Step 3 of the proof of Theorem 5.1, (5.29) and (5.30) yield that, for γ ≤ θ ≤ 1,   A(U (t))θ U (t) ≤ CU ,F t γ −θ , 0 < t ≤ TF,U . 0 0 X Then, the other properties of U can also be verified by the same techniques as in Steps 3 and 5 of the proof of Theorem 5.1.  As an immediate consequence, we are able to state the global existence of solutions. Corollary 5.1 Under the assumptions of Theorem 5.2, let F and U0 ∈ K satisfy (5.29) and (5.30), respectively. Assume that any local solution U of (5.1) in the function space (5.31), with TF,U0 replaced by TU , satisfies the estimates  U (t)Z ≤ R1 < R, 0 ≤ t ≤ TU , (5.33) A(U (t))γ U (t)X ≤ CF,U0 , 0 ≤ t ≤ TU , with some constants R1 and CF,U0 independent of TU . Then, (5.1) possesses a unique global solution on the interval [0, T ]. Proof We extend the function F to a function F defined on the whole interval [0, ∞) by putting F (t) ≡ F (T ) for T < t < ∞. Taking an initial value U1 which satisfies U1 Z ≤ R1 and A(U1 )γ U1 X ≤ CF,U0 and setting the initial time 0 ≤ T1 < ∞ arbitrarily, we consider the following problem  dV dt + A(V )V = F (t), T1 < t < ∞, (5.34) V (T1 ) = U1 . Theorem 5.2 then ensures the existence of a unique local solution on an interval [T1 , T1 + τ ], where τ > 0 is uniform with respect to U1 and T1 . Note that F Fγ ,σ ((a,b];X) ≤ F Fγ ,σ ((0,T ];X) for any finite interval (a, b].

1 Quasilinear Abstract Evolution Equations

213

This fact proves the assertion of the corollary. Consider first the local solution U constructed in Theorem 5.2 on the interval [0, TF,U0 ], where TF,U0 ≥ τ . Set T1 = TF,U0 − τ2 and U1 = U (T1 ). Then, since U1 satisfies condition (5.33), there exists a local solution V to (5.34) on the interval [T1 , T1 +τ ]. By the uniqueness of solutions, U (t) = V (t) for every T1 ≤ t ≤ TF,U0 . This then means that U has been extended to the interval [0, TF,U0 + τ2 ] as a local solution. Due to (5.33), we can repeat this procedure unlimitedly. Each time, the local solution is extended over the length τ2 of interval. Hence, we can arrive at the full interval [0, T ] by finite times. 

1.4 Case ν ≤ β We assumed the relation β < ν in (5.7) as one of the structural conditions of our abstract theory. But, this is not an indispensable one. We shall show that the lack of this relation can be recovered by the reflexivity of Z. Instead of (5.7), let the relations 0 ≤ α < ν ≤ β < 1 and 1 + α < β + ν

(5.35)

be satisfied. We assume the following condition: Z is a reflexive Banach space.

(5.36)

Theorem 5.3 Let (5.3), (5.4), (5.5), (5.6), (5.35), and (5.36) be satisfied. Let F be as in (5.8), and let U0 satisfy condition (5.9) with   A(U0 )β U0  < R/D2 . X Then, there exists a unique local solution to (5.1) on an interval [0, TF,U0 ] in the function space:  U ∈ B([0, TF,U0 ]; Z) ∩ Cβ−α ([0, TF,U0 ]; Y ) ∩ C1 ((0, TF,U0 ]; X), (5.37) β,σ ((0, T A(U )β U ∈ C([0, TF,U0 ]; X); dU F,U0 ]; X). dt , A(U )U ∈ F Here, TF,U0 > 0 is determined by F and U0 . Proof In the present case, Proposition 5.1 is not applicable with θ = β in showing (5.26). This fact makes us change the underlying Banach spaces from Z(S) to the space μ

Y(S) = C{0} ([0, S]; Y )

with 1 − ν + σ < μ < β − α.

By the new condition (5.36), the closed ball {U ∈ Z; U Z ≤ R1 }, where R1 is a fixed number such that   D2 A(U0 )β U0 X < R1 < R,

214

5 Quasilinear Evolution Equations

is weakly compact. As a consequence, this ball is a closed subset of Y . (In fact, let Un Z ≤ R1 and Un → U in Y ; then, we can assume that Un → V weakly in Z with V Z ≤ R1 . Since Z ⊂ Y with continuous embedding, Φ ∈ Y  implies Φ|Z ∈ Z  ; therefore, Φ(Un ) = Φ|Z (Un ) → Φ(V ), i.e., Un → V weakly in Y , too. Hence, U = V . Therefore, the set K(S) introduced in the proof of Theorem 5.1 is a closed subset of Y(S). In showing that Φ maps K(S) into itself, we will utilize the following formula. For U = Φ(V ), by (5.18) it holds that AV (t)β U (t) = A(U0 )β U0 + [e−tA(U0 ) − 1]A(U0 )β U0   + AV (t)β UV (t, 0)AV (0)−β − e−tAV (0) A(U0 )β U0

t + AV (t)β UV (t, s)[F (s) − F (t)] ds 0

+

t

AV (t)β [UV (t, s) − e−(t−s)AV (t) ] ds F (t)

0

− [e−tAV (t) − 1]AV (t)β−1 F (t). Then, it is verified by (2.129), (3.80), (3.84), and (3.86) that     AV (t)β U (t) ≤ A(U0 )β U0  + C[ωU (t) + ωF (t)] 0 X X   μ+ν−1 β + Ct A(U0 ) U0 X + F Fβ,σ + ω˜ F,U0 (t), where

  ω˜ F,U0 (t) = t β−1 [e−tAV (t) − 1]AV (t)β−1 t 1−β F (t)X .

Therefore, by (5.6),   U (t)Z ≤ D2 A(U0 )β U0 X + C[ωU0 (t) + ωF (t)]    + Ct μ+ν−1 A(U0 )β U0 X + F Fβ,σ + ω˜ F,U0 (t).

(5.38)

As will be seen by the lemma below, we have limt→0 ω˜ F,U0 (t) = 0. Hence, if S > 0 is sufficiently small, then sup U (t)Z ≤ R1 . 0≤t≤S

Lemma 5.1 As t → 0, ω˜ F,U0 (t) → 0. Proof We have   ω˜ F,U0 (t) ≤ t β−1 [e−tAV (t) − 1]AV (t)β−1 [t 1−β F (t) − F0 ]X   + t β−1 [e−tAV (t) − 1]AV (t)β−1 F0 X ,

2 Lipschitz Continuity of Solutions in Initial Data

215

where F0 = limt→0 t 1−β F (t). The first term in the right-hand side is seen to converge to 0. For the second term, we use the similar argument as for (2.131). When F0 ∈ D(A(0)), it follows that     t β−1 [e−tAV (t) − 1]AV (t)β−1 F0  ≤ t β−1 [e−tAV (t) − 1]AV (t)β−ν−1  X

  × AV (t)ν A(0)−1 

L(X)

L(X)

A(0)F0 X → 0.

For general F0 ∈ X, the desired convergence is deduced by the uniform boundedness of t β−1 [e−tAV (t) − 1]AV (t)β−1 L(X) and the density of D(A(0)) in X.  The same Lipschitz condition as (5.22) is verified. Hence, we conclude that Φ is a mapping from K(S) into itself if S > 0 is sufficiently small. As estimate (5.27) was proved without using the relation β < ν, Φ is a contraction with respect to the Y(S) norm if S > 0 sufficiently small. We can apply again the fixed point theorem to Φ : K(S) → K(S) to obtain the desired solution U . By analogous arguments as before, U is shown to belong to the space (5.37). In the present case, U may not be a continuous function with values in Z. We know only that A(U )β U is an X-valued continuous function on [0, TF,U0 ] by Theorem 3.10. 

2 Lipschitz Continuity of Solutions in Initial Data Under (5.2)–(5.7), we shall show the Lipschitz continuity of the local solutions with respect to initial data (F, U0 ). We introduce the set of functions F by E(T ) = {F ∈ Fγ ,σ (T ); F Fγ ,σ ((0,T ];X) ≤ L1 },

0 < σ < ν + β − α − 1,

with some constant L1 > 0, and also introduce the set of initial values  

B = U0 ∈ Z; U0 Z ≤ R1 and A(U0 )γ U0 X ≤ R2 , β < γ ≤ 1, with some constants 0 < R1 < R and 0 < R2 < ∞. By Theorem 5.1, there exists a unique local solution of (5.1) for each pair (F, U0 ) ∈ E(T ) × B on some interval [0, TF,U0 ]. Let TE,B = inf(F,U0 )∈E(T )×B TF,U0 ; then TE,B is positive. Our goal is to prove the following theorem. Theorem 5.4 Let (5.2), (5.3), (5.4), (5.5), (5.6), and (5.7) be satisfied. Let U (resp. V ) be the local solution of (5.1) with initial data (F, U0 ) ∈ E(T ) × B (resp. (G, V0 ) ∈ E(T ) × B). Then, there exist some constants CE,B > 0 depending only on E(T ) and B such that t β U (t) − V (t)Z + t α U (t) − V (t)Y + U (t) − V (t)X ≤ CB,E (U0 − V0 X + t β F − GFβ,σ ),

0 < t ≤ TE,B .

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5 Quasilinear Evolution Equations

Proof Let UU (t, s) (resp. UV (t, s)) denote the evolution operator for a family of AU (t) = A(U (t)) (resp. AV (t) = A(V (t))). It follows from (5.28) that U (t) − V (t) = UU (t, 0)(U0 − V0 ) + [UU (t, 0) − UV (t, 0)]V0

t + [UU (t, s) − UV (t, s)]G(s) ds 0



t

+

UU (t, s)[F (s) − G(s)] ds.

0

Then, by (3.80) and (5.6) we have   UU (t, 0)(U0 − V0 )Y ≤ D1 AU (t)α UU (t, 0)(U0 − V0 )X ≤ Ct −α U0 − V0 X . Similarly,    t  t     α    UU (t, s)[F (s) − G(s)] ds  ≤ D1  AU (t) UU (t, s)[F (s) − G(s)] ds    0 0 Y X

t ≤ CF − GFβ,σ (t − s)−α s β−1 ds 0

≤ Ct

β−α

F − GFβ,σ .

Repeating the same arguments as in the proof of Proposition 5.1, we can verify for 0 ≤ θ < ν that   AU (t)θ [UU (t, 0) − UV (t, 0)]V0  X

t ≤ CV0 β (t − s)ν−θ−1 s β−1 U (s) − V (s)Y ds

(5.39)

0

and  

t   AU (t)θ [UU (t, s) − UV (t, s)]G(s) ds    0 X

t ≤ CGFβ,σ (t − s)ν−θ−1 s β−1 U (s) − V (s)Y ds. 0

Therefore, applying these estimates with θ = α, we observe that  t     [UU (t, 0) − UV (t, 0)]V0 Y +  [UU (t, s) − UV (t, s)]G(s) ds   0 Y

t ≤ CE,B (t − s)ν−α−1 s β−1 U (s) − V (s)Y ds. 0

(5.40)

2 Lipschitz Continuity of Solutions in Initial Data

217

In this way, we have obtained the integral inequality

t ϕ(t) ≤ CE(t) + CE,B t α (t − s)ν−α−1 s β−α−1 ϕ(s) ds

(5.41)

0

for the function ϕ(t) which is defined by ϕ(t) = t α U (t) − V (t)Y ,

0 ≤ t ≤ TE,B ,

where E(t) = U0 − V0 X + t β F − GFβ,σ ,

0 ≤ t ≤ TE,B .

To solve this integral inequality, we first consider small variable t . For any s such that 0 ≤ s ≤ t , we have

s ϕ(s) ≤ CE(s) + CE,B s α (s − σ )ν−α−1 σ β−α−1 dσ sup ϕ(σ ) 0≤σ ≤s

0

≤ CE(t) + CE,B t ν+β−α−1 sup ϕ(s). 0≤s≤t

Therefore, it follows that   1 − CE,B t ν+β−α−1 sup ϕ(s) ≤ CE(t). 0≤s≤t

Fix an ε > 0 small enough to satisfy CE,B ε ν+β−α−1 < 1. Then, for 0 ≤ t ≤ ε, ϕ(t) ≤ sup ϕ(s) ≤ CE(t),

0 ≤ t ≤ ε.

0≤s≤t

Secondly, let us consider variable t such that t ≥ ε. We observe that

ε (t − s)ν−α−1 s β−α−1 ds ϕ(t) ≤ CE(t) + CE,B t α E(ε) 0

+ CE,B t α ε β−α−1 



≤ CE,B E(t) +

t

t

(t − s)ν−α−1 ϕ(s) ds

ε



(t − s)

ν−α−1

ϕ(s) ds .

ε

Applying Theorem 1.27, we conclude that ϕ(t) ≤ CE,B E(t),

ε ≤ t ≤ TE,B .

Thus, for all 0 ≤ t ≤ TE,B , ϕ(t) ≤ CE,B E(t),

i.e.,

t α U (t) − V (t)Y ≤ CE,B E(t),

0 ≤ t ≤ TE,B . (5.42)

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5 Quasilinear Evolution Equations

Estimation for Z and X norms of U (t) − V (t) is now immediate. By (3.80) and (5.6), UU (t, 0)(U0 − V0 )Z ≤ Ct −β U0 − V0 X . Similarly,   t    UU (t, s)[F (s) − G(s)] ds    0 Z   t   β  ≤ D2  AU (t) UU (t, s)[F (s) − G(s)] ds   0



≤ CF − GFβ,σ

0

t

X

(t − s)−β s β−1 ds ≤ CF − GFβ,σ .

In addition, applying (5.39) and (5.40) with θ = β, we have  t     [UU (t, 0) − UV (t, 0)]V0 Z +  [UU (t, s) − UV (t, s)]G(s) ds  

≤ CE,B

0

t

Z

(t − s)ν−β−1 s β−1 U (s) − V (s)Y ds

0

and, therefore, by (5.42),

t ≤ CE,B U0 − V0 X (t − s)ν−β−1 s β−α−1 ds ≤ CE,B t ν−α−1 U0 − V0 X . 0

Summing up these estimates, we conclude that t β U (t) − V (t)Z ≤ CE,B E(t),

0 ≤ t ≤ TE,B .

It is similar for the estimation of U (t) − V (t)X . It suffices to apply (5.39) and (5.40) with θ = 0. 

3 Equations Including Semilinear Terms In this section, we consider the Cauchy problem for a quasilinear equation of the form  dU dt + A(U )U = F (U ) + G(t), 0 < t ≤ T , (5.43) U (0) = U0 , in a Banach space X. For U ∈ K, A(U ) is a sectorial operator of X with angle ωA(U ) < π2 , where K is an open ball K = {U ∈ Z; U Z < R}, 0 < R < ∞, of a second Banach space Z continuously embedded in X. The operator F is a nonlinear operator from another Banach space W into X, W being continuously embedded in Z. The function G is a given external force function. The initial value U0 is taken in K.

3 Equations Including Semilinear Terms

219

3.1 Existence Theorem For the linear operators A(U ), we make the same structural assumptions as in Sect. 1.1. That is, we use the three Banach spaces Z ⊂ Y ⊂ X. The operator A(U ), U ∈ K, is a sectorial operator of X satisfying conditions (5.2) and (5.3) uniformly. The domain D(A(U )) is allowed to vary with U ∈ K, but A(U ) is assumed to satisfy (5.4) and the Lipschitz condition (5.5). Furthermore, the domains of fractional powers D(A(U )α ) and D(A(U )β ) are contained in Y and Z with (5.6), respectively, with the exponents α and β satisfying relations (5.7). For the nonlinear operator F , we assume the Lipschitz condition F (U ) − F (V )X ≤ ϕ(U Z + V Z )[U − V W + (U W + V W ) × U − V Z ],

U, V ∈ W,

(5.44)

where ϕ(·) is some continuous increasing function. This condition obviously implies that F (U )X ≤ F (0)X + ϕ(U Z )(1 + U Z )U W ,

U ∈ W.

(5.45)

For the space W , we assume an analogous condition to that for Y and Z. There is an exponent 0 < η < 1 such that D(A(U )η ) ⊂ W with the estimate     U˜ W ≤ D3 A(U )η U˜ X , U˜ ∈ D A(U )η for U ∈ K, (5.46) D3 > 0 being some constant. Finally, the exponents satisfy the relation β < η < ν.

(5.47)

As in the preceding section, the notation C will stand for a universal constant which is determined by the angle, the exponents, the norm A(0)−1 , and the constants appearing in (5.2)–(5.7). In this section, C may depend also on the function ϕ(·), the norm F (0)X , the exponent η, and the constant D3 . We shall prove the following theorem. Theorem 5.5 Let (5.2), (5.3), (5.4), (5.5), (5.6), and (5.7) be satisfied, and let (5.44), (5.46), and (5.47) be satisfied. Let G belong to (5.8) with 0 < σ < min{β − α + ν − 1, 1 − η}. Let U0 ∈ K satisfy (5.9). Then, there exists a unique local solution U to (5.43) on an interval [0, TG,U0 ] in the function space: ⎧ β−η ⎪ ⎨U ∈ B{0} ((0, TG,U0 ]; W ) ∩ C([0, TG,U0 ]; Z) ∩Cβ−α ([0, TG,U0 ]; Y ) ∩ C1 ((0, TG,U0 ]; X), ⎪ ⎩ β,σ ((0, T A(U )β U ∈ C([0, TG,U0 ]; X); dU G,U0 ]; X); dt , A(U )U ∈ F (5.48) here TG,U0 > 0 is determined by G and U0 . In addition, U satisfies the estimates

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5 Quasilinear Evolution Equations

      A(U )β U  +  dU   dt  C

+ A(U )U Fβ,σ ≤ CG,U0 ,

F β,σ

(5.49)

where CG,U0 > 0 is a constant depending on G and U0 . Remark 5.2 The dependence of TG,U0 on G and U0 is quite similar to that in Theorem 5.1. As will be seen in the proof below (see Steps 3 and 4), TG,U0 is determined by GFβ,σ and A(U0 )β U0 X and by the moduli of continuity ωG (t), ωU0 (t), and ωG,U0 (t) as t → 0, which are already defined by (5.12), (5.13), and (5.14), respectively.

3.2 Proof of Theorem 5.5 The proof consists of several steps. Step 1. For each 0 < S ≤ T , we set the underlying Banach space −ρ

μ

W(S) = B{0} ((0, S]; W ) ∩ B([0, S]; Z) ∩ C{0} ([0, S]; Y ) with some fixed exponents μ and ρ such that 1 − ν + σ < μ < β − α (cf. the proof of Theorem 5.1) and 1 − σ − β < ρ < 1 − β. Therefore, we have the relations η − β < 1 − σ − β < ρ < 1 − β < 1 − μ − α. −ρ

(5.50)

μ

For the function spaces B{0} ((0, S]; W ) and C{0} ([0, S]; Y ), see Chap 1, Sect. 2. We also set a subset F(S) of W(S) as follows:  F(S) = U ∈ W(S); U (0) = U0 , sup U (t)Z ≤ R1 , sup t ρ U (t)W ≤ 1, 0≤t≤S

sup 0≤s
sup 0≤s
s κ U (t) − U (s)

Z

(t − s)σ

0
≤ 1 and

s ρ+σ U (t) − U (s)W ≤ 1 . (t − s)σ

Here, κ is some fixed exponent such that σ < κ < σ + 1 − β − ρ,

(5.51)

and, R1 is a fixed radius such that U0 Z < R1 < R. It is clear that F(S) is a closed subset of W(S), and it is easy to see that F(S) is a nonempty set for any small S > 0, since e−tA(U0 ) U0 ∈ F(S). Step 2. For each V ∈ F(S), we denote by FV (t) the function given by FV (t) = F (V (t)), 0 < t ≤ S. From (5.45) it follows that t 1−β FV (t)X ≤ t 1−β F (0)X + ϕ(R1 )(1 + R1 )t 1−β−ρ ,

0 < t ≤ S.

3 Equations Including Semilinear Terms

221

Hence, t 1−β FV is an X-valued continuous function with limt→0 t 1−β FV (t) = 0. Similarly, ωFV (t) = sup

0≤s
s 1−β+σ FV (t) − FV (s)X (t − s)σ

≤ ϕ(2R1 ) sup 0≤s
s 1−β+σ [V (t) − V (s)W + s −ρ V (t) − V (s)Z ] (t − s)σ

≤ 2ϕ(2R1 )t 1+σ −κ−β−ρ . Due to (5.51), this shows that limt→0 ωFV (t) = 0. In this way, we have verified that FV ∈ Fβ,σ ((0, S]; X) with the estimate FV Fβ,σ ≤ CS 1+σ −κ−β−ρ ,

V ∈ F(S).

(5.52)

We now consider for each FV ∈ Fβ,σ ((0, S]; X), where V ∈ F(S), the Cauchy problem for a quasilinear equation  dU dt + A(U )U = FV (t) + G(t), 0 < t ≤ S, (5.53) U (0) = U0 . According to Theorem 5.1, this problem possesses a unique local solution U on some interval [0, TFV +G,U0 ], and as noticed by Remark 5.1, TFV +G,U0 > 0 is determined by the norms FV + GFβ,σ and A(U0 )β U0 X and by the moduli of continuity ωFV +G (t), ωU0 (t) and ωFV +G,U0 (t) which are defined by (5.12), (5.13), and (5.14), respectively. By (5.52), FV + GFβ,σ ≤ CS 1+σ −κ−β−ρ + GFβ,σ ,

V ∈ F(S).

(5.54)

Similarly, we easily verify that ωFV +G (t) ≤ ωFV (t) + ωG (t) ≤ ϕ(2R1 )t 1+σ −κ−β−ρ + ωG (t), ωFV +G,U0 (t) ≤ ωFV ,U0 (t) + ωG,U0 (t) ≤ Ct 1+σ −κ−β−ρ + ωG,U0 (t). These then mean that the infimum of TFV +G,U0 , when V varies in F(S), is positive, i.e., infV ∈F(S) TFV +G,U0 = T˜G,U0 > 0. According to the proof of Theorem 5.1, the solution U belongs to the function space (5.10). In particular, we see that  U (t)Z ≤ R1 , 0 ≤ t ≤ T˜G,U0 , (5.55) μ U (t) − U (s)Y ≤ (t − s) , 0 ≤ s ≤ t ≤ T˜G,U0 . For the solution U , we denote by UU (t, s) the evolution operator for the family of A(U (t)). Then, U itself is represented by

t U (t) = UU (t, 0)U0 + UU (t, s)[FV (s) + G(s)] ds, 0 ≤ t ≤ S. (5.56) 0

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5 Quasilinear Evolution Equations

We are thus led to define a mapping U = Ψ (V ) for V ∈ F(S), where U is the solution of (5.53). Step 3. We shall prove that Ψ maps the subset F(S) into itself if S > 0 is sufficiently small. Let U = Ψ (V ), V ∈ F(S). As (5.55) is the case, the family of AU (t), 0 ≤ t ≤ S (≤ T˜G,U0 ) satisfies all the structural assumptions (3.27)–(3.31). Then, we can argue as in Step 3 of the proof of Theorem 5.1 by substituting UU (t, s) for UV (t, s). By repeating the same arguments as for (5.19), it is deduced that U (t) − U0 Z ≤ C[t 1+σ −κ−β−ρ + ωG (t) + ωU0 (t) + ωG,U0 (t)] + Ct μ+ν−1 (FV + GFβ,σ + U0 β ),

0 ≤ t ≤ S.

Therefore, if S > 0 is sufficiently small, then sup

sup {Ψ (V )}(t)Z ≤ R1 .

V ∈F(S) 0≤t≤S

We next apply the moment inequality (2.119) for estimating AU (t)θ U (t)X . Indeed, due to (3.101) and (3.102), for β ≤ θ ≤ 1,   AU (t)θ U (t) ≤ Ct β−θ (FV + GFβ,σ + U0 β ), 0 ≤ t ≤ S. (5.57) X Therefore, in view of (5.46), t ρ U (t)W ≤ D3 t ρ AU (t)η U (t)X ≤ Ct ρ+β−η (U0 β + FV + GFβ,σ ) ≤ 1,

0 ≤ t ≤ S,

if S > 0 is sufficiently small. For proving the Hölder condition of U , we write

t

U (t) − U (s) =[UU (t, s) − 1]U (s) +

UU (t, τ )[FV (τ ) + G(τ )] dτ

s

={[UU (t, s) − e−(t−s)AU (s) ] + [e−(t−s)AU (s) − 1]}AU (s)−(η+σ )

t η+σ ×AU (s) U (s) + UU (t, τ )[FV (τ ) + G(τ )] dτ, s

0 ≤ s < t ≤ S. Then, by (2.129) and (3.87) (note that η < ν),   AU (t)η [UU (t, s) − e−(t−s)AU (s) ]AU (s)−(η+σ )  ≤ C(t − s)σ +μ+ν−1 , L(X)   AU (s)η [e−(t−s)AU (s) − 1]AU (s)−(η+σ )  ≤ C(t − s)σ . L(X) By (3.81),

3 Equations Including Semilinear Terms



 AU (t)η UU (t, τ )

t

s

≤ C(t − s)σ

t

223

τ L(X)

β−1

dτ ≤ C

t

(t − τ )−η τ β−1 dτ

s

(t − τ )−η−σ τ β−1 dτ ≤ C(t − s)σ



t

(t − τ )−η−σ τ β−1 dτ

0

s

≤ C(t − s) t

σ β−η−σ

.

In view of (5.46) and (5.57) (θ = η + σ ), we therefore obtain that U (t) − U (s)W ≤ C(FV + GFβ,σ + U0 β )(t − s)σ s β−η−σ .

(5.58)

Hence, if S > 0 is sufficiently small, then sup 0<s
s ρ+σ U (t) − U (s)W ≤ CS ρ+β−η (FV + GFβ,σ + U0 β ) ≤ 1. (t − s)σ

In a quite analogous way, if S > 0 is sufficiently small, then sup 0<s
s κ U (t) − U (s)Z ≤ CS κ−σ (FV + GFβ,σ + U0 β ) ≤ 1. (t − s)σ

We thus verified that Ψ maps F(S) into itself, provided that S > 0 is sufficiently small. Step 4. We shall prove that Ψ : F(S) → W(S) is a contraction with respect to the W(S) norm if S > 0 is sufficiently small. In fact, let Vi ∈ F(S) and Ui = Ψ (Vi ), i = 1, 2. From (5.56) it is seen that U1 (t) − U2 (t) = [UU1 (t, 0) − UU2 (t, 0)]U0

t + [UU1 (t, s) − UU2 (t, s)][FV1 (s) + G(s)] ds 0



t

+ 0

UU2 (t, s)[FV1 (s) − FV2 (s)] ds.

For 0 ≤ θ < ν, using (5.23) of Proposition 5.1, we have   AU (t)θ [UU (t, 0) − UU (t, 0)]U0  1

1

≤ Cθ t

2

β−θ+μ+ν−1

X

U0 β U1 − U2 Cμ

{0} ([0,S];Y )

Similarly, using (5.24),  

t   AU (t)θ [UU1 (t, s) − UU2 (t, s)][FV1 (s) + G(s)] ds    1 0

≤ Cθ (GFβ,σ + 1)t

β−θ+μ+ν−1

U1 − U2 Cμ

(5.59)

.

X

{0} ([0,S];Y )

.

(5.60)

224

5 Quasilinear Evolution Equations

By (5.44), we have  

t   AU (t)θ UU2 (t, s)[FV1 (s) − FV2 (s)] ds   2 

≤C

0

t

X

(t − s)−θ s −ρ [s ρ V1 (s) − V2 (s)W + V1 (s) − V2 (s)Z ] ds

0

≤ Cθ t −θ−ρ+1 (V1 − V2 B−ρ ((0,S];W ) + V1 − V2 B([0,S];Z) ).

(5.61)

{0}

Applying these estimates with θ = α, we obtain that t −μ U1 (t) − U2 (t)Y ≤ C(GFβ,σ + U0 β + 1)S β−α+ν−1 U1 − U2 Cμ

{0} ([0,S];Y )

+ CS 1−μ−α−ρ (V1 − V2 B−ρ ((0,S];W ) + V1 − V2 B([0,S];Z) ), {0}

0 < t ≤ S.

(5.62)

Similarly, applying the estimates with θ = β, U1 (t) − U2 (t)Z ≤ C(GFβ,σ + U0 β + 1)S μ+ν−1 U1 − U2 Cμ

{0} ([0,S];Y )

+ CS 1−β−ρ (V1 − V2 B−ρ ((0,S];W ) {0}

+ V1 − V2 B([0,S];Z) ),

0 < t ≤ S.

(5.63)

Finally, applying the estimates with θ = η, we have t ρ U1 (t) − U2 (t)W ≤ C(GFβ,σ + U0 β + 1)S ρ+β−η+μ+ν−1 × U1 − U2 Cμ

{0} ([0,S];Y )

+ V1 − V2 B([0,S];Z) ),

+ CS 1−η (V1 − V2 B−ρ ((0,S];W ) {0}

0 < t ≤ S.

(5.64)

Therefore, summing up (5.62), (5.63), and (5.64), we have U1 − U2 W(S) ≤ C(GFβ,σ + U0 β + 1)S μ+ν−1 U1 − U2 Cμ

{0} ([0,S];Y )

+ CS 1−β−ρ (V1 − V2 B−ρ ((0,S];W ) + V1 − V2 B([0,S];Z) ) {0}

(due to (5.50)). In this way, we conclude that Ψ is a contraction with respect to the W(S) norm, provided that S > 0 is sufficiently small. Step 5. Let TG,U0 = S > 0 be small enough to make Ψ : F(S) → F(S) to be a contraction. Then, there exists a unique fixed point U ∈ F(S) of Ψ which satisfies the formula

t U (t) = UU (t, 0)U0 + UU (t, s)[FU (s) + G(s)] ds, 0 ≤ t ≤ TG,U0 . (5.65) 0

3 Equations Including Semilinear Terms

225

Then, U is a local solution to (5.43) on [0, TG,U0 ] belonging to the space (5.10). In addition, applying (5.57) with θ = η, we verify by (5.46) that U (t)W ≤ Ct β−η (FU + GFβ,σ + U0 β ). Hence, U belongs to the space (5.48). In view of (5.55), the family of operators AU (t), 0 ≤ t ≤ TG,U0 , satisfies (3.27)–(3.31). Therefore, (5.49) is readily verified by Theorem 3.10 and (5.54). The uniqueness of a local solution is also verified in an analogous way as in the proof of Theorem 5.1. We have thus accomplished the proof of Theorem 5.5. 

3.3 Stronger Results for More Regular Data Let G belong to the function space (5.29). Let U0 satisfy the compatibility condition (5.30), which is stronger than (5.8). Then, Theorem 5.5 can be strengthened as follows. Theorem 5.6 Let (5.2), (5.3), (5.4), (5.5), (5.6), and (5.7) be satisfied, and let (5.44), (5.46), and (5.47) be satisfied. Let G and U0 ∈ K satisfy (5.29) and (5.30), respectively, with 0 < σ < min{β − α + ν − 1, 1 − η}. Then, there exists a unique local solution U to (5.43) on an interval [0, TG,U0 ] in the function space (5.31). Here, TG,U0 > 0 is determined by the norms GFγ ,σ and A(U0 )γ U0 X alone. Furthermore, U satisfies the estimates       A(U )γ U  +  dU  (5.66)  dt  γ ,σ + A(U )U Fγ ,σ ≤ CG,U0 C F with a constant CG,U0 > 0 determined by the norms GFγ ,σ and A(U0 )γ U0 X . As noticed in Remark 5.2, the time TG,U0 > 0 is determined by ωG (t), ωU0 , and ωG,U0 (t) in addition to GFβ,σ and A(U0 )β U0 X . Then, the proof of this theorem is quite analogous to that of Theorem 5.2, and we may leave it to the reader. By the same argument as in Corollary 5.1, we verify the following global existence. The proof may now be obvious. Corollary 5.2 Under the assumptions of Theorem 5.6, let G and U0 ∈ K satisfy (5.29) and (5.30), respectively. Assume that any local solution U of (5.43) in the function space (5.31), with TG,U0 replaced by TU , satisfies the estimates  0 ≤ t ≤ TU , U (t)Z ≤ R1 < R, (5.67) A(U (t))γ U (t)X ≤ CG,U0 , 0 ≤ t ≤ TU , with some constants R1 and CG,U0 independent of TU . Then, (5.43) possesses a unique global solution on the interval [0, T ].

3.4 Case ν ≤ β The condition β < ν assumed above is not an indispensable one for the local existence. As in Sect. 1.4, the lack of this condition can be recovered by reflexivity of Z.

226

5 Quasilinear Evolution Equations

Instead of (5.44), we assume that F satisfies F (U ) − F (V )X ≤ ϕ(U Z + V Z )U − V W ,

U, V ∈ Z,

(5.68)

with some continuous increasing function ϕ(·), and that the space W satisfies the relation Z ⊂ W (⊂ Y ⊂ X) with continuous embedding. We also assume relations (5.35) and, instead of (5.47), (5.69)

α < η < ν.

Theorem 5.7 Let (5.2), (5.3), (5.4), (5.5), (5.6), (5.35), and (5.36) be satisfied, and let (5.46), (5.68), and (5.69) be satisfied. Let G belong to (5.8) with 0 < σ < min{β − α + ν − 1, 1 − η}. Let U0 satisfy condition (5.9) with    A(U0 )β U0  < R D2 . X Then, there exists a unique local solution to (5.43) on an interval [0, TG,U0 ] in the function space: ⎧ β−η ([0, T ⎪ G,U0 ]; W ) ⎨U ∈ B([0, TG,U0 ]; Z) ∩ C ∩ Cβ−α ([0, TG,U0 ]; Y ) ∩ C1 ((0, TG,U0 ]; X), ⎪ ⎩ β,σ ((0, T A(U )β U ∈ C([0, TG,U0 ]; X); dU G,U0 ]; X). dt , A(U )U ∈ F (5.70) Here, TG,U0 > 0 is determined by G and U0 . Proof In the present case, (5.63) is no longer available. So, we have to change the underlying spaces from W(S) to ζ μ ˜ W(S) = C{0} ((0, S]; W ) ∩ C{0} ([0, S]; Y ),

0 < S ≤ T,

with some fixed exponents μ and ζ such that 1 − ν + σ < μ < β − α and 0 < ζ < ˜ β − η, respectively. Accordingly, we set the subset of W(S) by  ˜ ˜ F(S) = U ∈ W(S); U (0) = U0 , sup U (t)Z ≤ R1 0≤t≤S

and

sup 0≤s
s σ U (t) − U (s) (t − s)σ

W

≤1 ,

where R1 is a fixed number such that D2 A(U0 )β U0 X < R1 < R. Owing to (5.36), the ball {U ∈ Z; U Z ≤ R1 } is a closed subset of Y . (Confer the proof of Theo˜ ˜ rem 5.3.) Consequently, F(S) is a nonempty closed subset of W(S). ˜ Then, it is easy to see We denote FV (t) = F (V (t)), 0 ≤ t ≤ T , for V ∈ F(S). that FV ∈ F1,σ ((0, T ]; X), and therefore FV + G ∈ Fβ,σ ((0, T ]; X). Theorem 5.3 ˜ then provides that, for each V ∈ F(S), the quasilinear problem (5.53) possesses a unique local solution U on an interval [0, TFV +G,U0 ] in the function space (5.37). Furthermore, as verified by its proof, we have infV ∈F(S) TFV +G,U0 = T˜G,U0 > 0. ˜

3 Equations Including Semilinear Terms

227

˜ Hence, by formula (5.56), we define a mapping U = Ψ V for V ∈ F(S) (0 < S ≤ ˜ TG,U0 ). ˜ If S > 0 is sufficiently small, then Ψ maps F(S) into itself. In fact, by (5.38),   {Ψ V }(t)Z ≤ D2 A(U0 )β U0 X + C[ωU0 (t) + ωFV +G (t)]     + Ct μ+ν−1 FV + GFβ,σ + A(U0 )β U0 X + ω˜ FV +G,U0 (t). This shows that, if S > 0 is sufficiently small, then sup

sup (Ψ V )(t)Z ≤ R1 .

0≤t≤S ˜ V ∈F(S)

For verifying the Hölder condition, we can use the same argument as for (5.58). In view of the relations η < ν and σ < 1 − η, estimate (5.58) is verified without any change. Therefore, if S > 0 is sufficiently small, then sup

sup

0≤s
s σ (Ψ V )(t) − (Ψ V )(s)W ≤ 1. (t − s)σ

˜ Hence, Ψ maps F(S) into itself, provided that S > 0 is sufficiently small. Let us verify that, if S > 0 is sufficiently small, then Ψ is a contraction in the ˜ W(S) norm. This is also verified in the same way as before. In fact, let Ui = Ψ Vi ˜ with Vi ∈ F(S) (i = 1, 2). For 0 ≤ θ < ν, estimates (5.59) and (5.60) are valid for these Ui and Vi . Instead of (5.61), we verify that  

t   AU (t)θ  U (t, s)[F (s) − F (s)] ds U2 V1 V2  2  0 X

t ≤C (t − s)−θ s ζ s −ζ V1 (s) − V2 (s)W ds 0

≤ Cθ t ζ −θ+1 V1 − V2 Cζ

{0} ((0,S];W )

.

Then, using these estimates with θ = η and α, we easily obtain that t −ζ U1 (t) − U2 (t)W + t −α U1 (t) − U2 (t)Y ≤ CG,U0 S μ+ν−1 U1 − U2 W(S) + C(S 1−η + S ζ +1−α−μ )V1 − V2 W(S) , ˜ ˜ 0 ≤ t ≤ S. ˜ This shows that Ψ is a contraction with respect to the W(S) norm, provided that S > 0 is sufficiently small (note that 1 − α − μ > 1 − β > 0). ˜ We thus conclude that Ψ has a unique fixed point U ∈ F(S) which satisfies formula (5.65). Then, U is the solution to (5.43) in the space (5.70) which enjoys all the desired properties. 

228

5 Quasilinear Evolution Equations

4 Lipschitz Continuity of Solutions in Initial Data We are concerned with the continuous dependence of the local solutions of (5.43) with respect to the initial data (G, U0 ). Let (5.2), (5.3), (5.4), (5.5), (5.6), and (5.7) be satisfied, and let (5.44), (5.46), and (5.47) be satisfied, namely, we return to the case where β < ν. We consider the set of functions G given by E(T ) = {G ∈ Fγ ,σ ((0, T ]; X); GFγ ,σ ≤ L1 }, 0 < σ < min{β − α + ν − 1, 1 − η} with some constant L1 > 0. We consider also the set of initial values  

B = U0 ∈ Z; U0 Z ≤ R1 and A(U0 )γ U0 X ≤ R2 , β < γ ≤ 1, with some constants 0 < R1 < R and 0 < R2 < ∞. By Theorem 5.6, there exists a unique local solution to (5.43) for each pair (G, U0 ) ∈ E(T ) × B on some interval [0, TG,U0 ]. Let TE,B = inf(G,U0 )∈E(T )×B TG,U0 > 0. Before proving the Lipschitz continuity, we shall show estimates of solutions by the initial data. Theorem 5.8 Let (5.2), (5.3), (5.4), (5.5), (5.6), and (5.7) be satisfied, and let (5.44), (5.46), and (5.47) be satisfied. Let U be the local solution of (5.43) with (G, U0 ) ∈ E(T ) × B. Then, there exist some constants CE,B > 0 depending on E(T ) and B only such that t η U (t)W + t β U (t)Z + t α U (t)Y + U (t)X ≤ CE,B (U0 X + tF (0)X + t β GFβ,σ ),

0 ≤ t ≤ TE,B .

Proof Let FU (t) = F (U (t)), and let UU (t, s) be the evolution operator for the family of AU (t) = A(U (t)). Operating A(U (t))θ (0 ≤ θ < 1) to the solution formula (5.65), we have AU (t)θ U (t) = AU (t)θ UU (t, 0)U0

t + AU (t)θ UU (t, s)[FU (s) + G(s)] ds,

0 < t ≤ TB,E .

0

Here, by (5.45), FU (t)X ≤ F (0)X + D3 ϕ(R)(1 + R)AU (t)η U (t)X . Therefore, 

t      AU (t)θ U (t) ≤Cθ t −θ U0 X + (t − s)−θ F (0)X + AU (s)η U (s)X X 0

+ s β−1 GFβ,σ ds . 

4 Lipschitz Continuity of Solutions in Initial Data

229

Consequently,    t θ AU (t)θ U (t)X ≤ Cθ U0  + tF (0)X + t β GFβ,σ

t

+ 0

  t θ (t − s)−θ s −η s η AU (s)η U (s)X ds .

If we put θ = η, then an integral inequality with respect to AU (s)η U (s)X is obtained, and the inequality can be easily solved by the same techniques used for (5.41). As a result, we conclude that   t η AU (t)η U (t)X ≤ CE,B [U0 X + tF (0)X + t β GFβ,σ ], 0 < t ≤ TE,B . Hence, for 0 ≤ θ < 1, we have 

t   t θ AU (t)θ U (t)X ≤ Cθ U0  + tF (0)X + t β GFβ,σ + t θ (t − s)−θ s −η 0

× [U0 X + sF (0)X + s β GFβ,σ ] ds . Putting θ = β, α, and 0, we can obtain the desired estimates.



Corollary 5.3 Under the conditions of Theorem 5.8, we have t η−β U (t)W + U (t)Z    ≤ CE,B A(U0 )β U0 X + t 1−β F (0)X + GFβ,σ ,

0 ≤ t ≤ TE,B .

Proof For β ≤ θ < 1, we use the estimate   AU (t)θ UU (t, 0)U0  ≤ CE,B t β−θ U0 β . X Then, it follows that 

t    AU (t)θ U (t) ≤ Cθ CB,E t β−θ U0 β + (t − s)−θ F (0)X X 0

   η β−1   + AU (s) U (s) X + s Gβ,σ ds . Putting θ = η, we have t

   AU (t) U (t) X ≤ CB,E U0 β + t 1−β F (0)X + GFβ,σ



η−β 

η



t

+ 0

  t η−β (t − s)−η s β−η s η−β AU (s)η U (s)X ds .

230

5 Quasilinear Evolution Equations

So, solving this integral inequality, we conclude that   t η−β AU (t)η U (t)X ≤ CB,E [U0 β + t 1−β F (0)X + Gβ,σ ], By putting θ = β, we also verify that   AU (t)β U (t) ≤ CE,B [U0 β + t 1−β F (0)X + Gβ,σ ], X

0 < t ≤ TE,B .

0 < t ≤ TE,B . 

Theorem 5.9 Let (5.2), (5.3), (5.4), (5.5), (5.6), and (5.7) be satisfied, and let (5.44), (5.46), and (5.47) be satisfied. Let U (resp. V ) be the local solution of (5.43) with (G1 , U0 ) ∈ E(T ) × B (resp. (G2 , V0 ) ∈ E(T ) × B). Then, there exist some constants CE,B > 0 depending on E(T ) and B only such that t η U (t) − V (t)W + t β U (t) − V (t)Z + t α U (t) − V (t)Y + U (t) − V (t)X ≤ CE,B (U0 − V0 X + t β G1 − G2 Fβ,σ ),

0 ≤ t ≤ TE,B .

Proof Let FU (t) = F (U (t)) (resp. FV (t) = F (V (t))), and let UU (t, s) (resp. UV (t, s)) be the evolution operator for the family of AU (t) = A(U (t)) (resp. AV (t) = A(V (t))). By (5.65) we can write U − V as U (t) − V (t) = UU (t, 0)(U0 − V0 ) + [UU (t, 0) − UV (t, 0)]V0

t + [UU (t, s) − UV (t, s)][FV (s) + G2 (s)] ds

0 t

+

UU (t, s)[FU (s) − FV (s)] ds

0 t

+

UU (t, s)[G1 (s) − G2 (s)] ds.

(5.71)

0

For 0 ≤ θ < ν, operating AU (t)θ , we estimate the X norm of the right-hand side. By (5.44),  

t   AU (t)θ UU (t, s)[FU (s) − FV (s)] ds   

≤C

0

t

X

(t − s)−θ [U (s) − V (s)W + s −ρ U (s) − V (s)Z ] ds.

0

The norms of other terms are estimated in the same way as in the proof of Theorem 5.4. Indeed, by (3.80),   AU (t)θ UU (t, 0)(U0 − V0 ) ≤ Ct −θ U0 − V0 X (5.72) X and

  t    AU (t)θ UU (t, s)[G1 (s) − G2 (s)] ds  ≤ Ct β−θ G1 − G2 Fβ,σ .   0

X

4 Lipschitz Continuity of Solutions in Initial Data

231

By the same arguments as for (5.39) and (5.40), we obtain that   AU (t)θ [UU (t, 0) − UV (t, 0)]V0  X

t   ≤ C A(V0 )β V0 X (t − s)ν−θ−1 s β−1 U (s) − V (s)Y ds 0

and

 

t   AU (t)θ  [U (t, s) − U (t, s)][F (s) + G (s)] ds U V V 2   0 X

t ≤ CFV + G2 Fβ,σ (t − s)ν−θ−1 s β−1 U (s) − V (s)Y ds. 0

Applying these estimates with θ = η, β, and α and using (5.6) and (5.46), we immediately observe that the function ϕ(t) = t η U (t) − V (t)W + t β U (t) − V (t)Z + t α U (t) − V (t)Y , 0 < t ≤ TE,B , satisfies the integral inequality ϕ(t) ≤ CE,B (U0 − V0 X + t β G1 − G2 Fβ,σ )

t    θ −θ −η −ρ−β ν−1 β−α−1 ϕ(s) ds. + CE,B t (t − s) s + s + (t − s) s 0

θ=η,β,α

We can then solve this integral inequality by the same technique as for (5.41) (see (5.50)) and can in fact conclude that ϕ(t) ≤ CE,B (U0 − V0 X + t β G1 − G2 Fβ,σ ),

0 ≤ t ≤ TE,B .

As in the proof of Theorem 5.4, we can estimate the norm U (t) − V (t)X , too, directly from (5.71). Hence, we have accomplished the proof.  Corollary 5.4 Under the conditions of Theorem 5.9, it holds that t η−α U (t) − V (t)W + t β−α U (t) − V (t)Z + U (t) − V (t)Y    ≤ CB,E A(U0 )α (U0 − V0 )X + t β−α G1 − G2 Fβ,σ , 0 ≤ t ≤ TB,E . Proof For α ≤ θ ≤ 1, it holds that   AU (t)θ UU (t, 0)(U0 − V0 ) X     θ −α ≤ AU (t) UU (t, 0)AU (0) L(X) AU (0)α (U0 − V0 )X   ≤ CE,B t α−θ AU (0)α (U0 − V0 )X , 0 < t ≤ TE,B .

232

5 Quasilinear Evolution Equations

We will use this instead of (5.72). Then, by a similar argument as in the proof Theorem 5.9, it is deduced that     ϕ(t) ≤ CE,B t α A(U0 )α (U0 − V0 )X + t β G1 − G2 Fβ,σ , 0 ≤ t ≤ TE,B . 

Hence, the desired estimate is obtained.

5 Variational Methods In this section, we will remark on the variational methods for studying problem (5.1). Let X and Z be two Hilbert spaces such that Z ⊂ X with dense and compact embedding. Let | · | (resp.  · ) be the norm of X (resp. Z). Let Z ⊂ X ⊂ Z ∗ be a triplet of spaces. Let  · ∗ be the norm of Z ∗ . We consider the Cauchy problem for a quasilinear parabolic evolution equation  dU dt + A(U )U = F (U ) + G(t), 0 < t ≤ T , (5.73) U (0) = U0 , in Z ∗ . Here, the sectorial operators A(U ) are given as follows. Let K be an open ball of X, K = {E ∈ X; |E| < R},

0 < R < ∞.

For each E ∈ K, let a(E; ·, ·) be a sectorial operator on Z and assume the following conditions: |a(E; U, V )| ≤ MU V , Re a(E; U, U ) ≥ δU  , 2

E ∈ K, U, V ∈ Z,

E ∈ K, U ∈ Z,

(5.74) (5.75)

|a(E1 ; U, V ) − a(E2 ; U, V )| ≤ N |E1 − E2 |U V , Ei ∈ K (i = 1, 2), U, V ∈ Z,

(5.76)

with constants δ > 0 and M, N independent of E ∈ K. By Theorem 2.1, we know that the associated operator A(E) is defined and is a sectorial operator of Z ∗ with angle ωA(E) < π2 . The operator F : K ∩ Z → Z ∗ is a nonlinear operator satisfying the following conditions. For any ζ > 0, there exists a constant Lζ such that F (U )∗ ≤ ζ U  + Lζ ,

U ∈ K ∩ Z,

(5.77)

F (U ) − F (V )∗ ≤ ζ U − V  + Lζ (U  + V  + 1)|U − V |, U, V ∈ K ∩ Z.

(5.78)

The function G(t) is a given external force function belonging to L2 (0, T ; Z ∗ ). The initial value U0 is taken in K.

6 Quasilinear Parabolic Equations

233

We can then prove the following results in a quite analogous way to the proof of Theorem 4.6. Theorem 5.10 Let (5.74), (5.75), (5.76), (5.77), and (5.78) be satisfied. Then, for any G ∈ L2 (0, T ; Z ∗ ) and any U0 ∈ K, there exists a unique local solution U to (5.73) in the function space: U ∈ L2 ((0, TG,U0 ); Z) ∩ C([0, TG,U0 ]; X) ∩ H 1 ((0, TG,U0 ); Z ∗ ), where TG,U0 > 0 is determined by the norms GL2 (0,T ;Z ∗ ) and |U0 | only. In addition, U satisfies the estimates U L2 ((0,TG,U0 );Z) + U C([0,TG,U0 ];X) + U H 1 ((0,TG,U

0

);Z ∗ )

≤ CG,U0 ,

where CG,U0 > 0 depends on the norms |U0 | and GL2 ((0,T );Z ∗ ) .

6 Quasilinear Parabolic Equations We consider the initial-boundary-value problem for a quasilinear parabolic equation ⎧ ∂u n ⎪ ⎨ ∂t = i,j =1 Dj [aij (x, u)Di u] + f (x, u, ∇u) + g(x, t) n νj (x)aij (x, u)Di u = 0 ⎪ ⎩ i,j =1 u(x, 0) = u0 (x)

in Ω × (0, T ),

on ∂Ω × (0, T ), in Ω, (5.79) in a bounded domain Ω ⊂ Rn with C2 boundary ∂Ω, where 0 < T < ∞ is fixed time. Here, aij (x, u) are real-valued functions for (x, u) ∈ Ω × (R + iR) which are smooth functions with respect to the real variables x ∈ Ω, Re u and Im u. We assume that aij (x, u) satisfy conditions (2.19) and (2.28) for every u ∈ R + iR with some uniform constant δ > 0. The function f (x, u, ζ ) is a complex-valued function for (x, u, ζ ) ∈ Ω × (R + iR) × (R + iR)n which is a smooth function with respect to the real variables x ∈ Ω, Re u, Im u, Re ζ , and Im ζ . The function g(x, t) is an external force function, and u0 (x) is an initial function.

6.1 Abstract Formulation Let us formulate problem (5.79) as the Cauchy problem of the form (5.43) in X = Lp (Ω), where n < p < ∞. We set the Sobolev space Hps (Ω) as the space Z where s s 1 s is taken as p+n 2p < 2 < 1, Hp (Ω) being embedded in C (Ω) continuously due to (1.76). For each u ∈ K = {u ∈ Hps (Ω); uHps < R}, 0 < R < ∞, we consider a real ization Ap (u) of the differential operator − i,j Dj [aij (x, u(x))Di ·] + 1 in Lp (Ω)

234

5 Quasilinear Evolution Equations

u˜ under the Neumann-type boundary conditions ∂ν∂A(u) = 0 on ∂Ω. The domain of Ap (u) is given by (2.77) and depends on u. The nonlinear operator F (u) is given by

F (u) = u + f (x, u, ∇u),

u ∈ Hps (Ω).

We set Hp1 (Ω) as the space W . In this way, we rewrite (5.79) in the form (5.43).

6.2 Construction of Local Solutions We already know by Theorem 2.12 that Ap (u) satisfy (5.2) and (5.3). We fix an exponent ν as 12 < ν < p+1 2p . Then, according to the fact (16.36) presented in Chap. 16, 

condition (5.4) is fulfilled. As for the third space Y , we set Hp2ν −1 (Ω) = Y , where

the exponent ν  is fixed as ν < ν  <

p+1 2p .

In order to verify the Lipschitz condition

(5.5), we argue as in Chap 3, Sect. 7.2. For u, v ∈ K, we can verify as for (3.78) that        Ap (u)ν Ap (u)−1 − Ap (v)−1 f, g L ×L  p

p

  ≤ C sup aij (u) − aij (v)L(H 1 ,H 2ν  −1 ) Di Ap (v)−1 f H 1 p

i,j

  × Dj Bp (u)ν−1 g 

Hp1−2ν 



,

p

p

f ∈ Lp (Ω), g ∈ Lp (Ω).

Here, Bp (u) denotes the adjoint operator of Ap (u). Since D(Ap (v)) ⊂ Hp2 (Ω), it follows that Di Ap (v)−1 f Hp1 ≤ Cf Lp . In the meantime, since D(Bp1−ν  (u)) ⊂ 2(1−ν  )

2(1−ν  )

(Ω) due to (16.36) and since Dj ∈ L(Hp Hp that Dj Bp (u)ν−1 gH 1−2ν  ≤ CgLp . Therefore,



(Ω), Hp1−2ν (Ω)), it follows 

p

   Ap (u)ν Ap (u)−1 − Ap (v)−1  L(L

p)

≤ C sup aij (u) − aij (v)L(H 1 ,H 2ν  −1 ) , i,j

p

p

u, v ∈ K.

Then, condition (5.5) is verified by the following two Lemmas 5.2 and 5.3. Lemma 5.2 Let a ∈ Hps (Ω), where n < p < ∞ and 0 ≤ s ≤ 1. Then, auHps ≤ CaHps uHp1 for u ∈ Hp1 (Ω). Proof Taking any function u ∈ Hp1 (Ω), we consider the linear operator a → au, which is a bounded operator from Lp (Ω) into itself with auLp ≤ CaLp uHp1 and at the same time is a bounded operator from Hp1 (Ω) into itself with auHp1 ≤ CaHp1 uHp1 due to (1.88). Then, by Theorem 1.15, the linear operator is a

7 Some Model for Honeybee Colonies

235

bounded operator from Hps (Ω) into itself for every 0 < s < 1 with auHps ≤  CaHps uHp1 , the constant C > 0 being independent of u. Lemma 5.3 Let a(x, u) be a function for (x, u) ∈ Ω × (R + iR) which is a smooth function with respect to the real variables Re u and Im u. Then, for 0 ≤ s ≤ 1, a(x, u) − a(x, v)Hps ≤ ϕ(uC1 + vC1 )u − vHps ,

u, v ∈ C1 (Ω),

where ϕ(·) is some continuous increasing function determined by a(x, u). Proof Indeed,

1

a(x, u) − a(x, v) =

+ 0

0 1

∂a (x, θ Re u + (1 − θ ) Re v + i Im v) dθ · Re(u − v) ∂ Re u

∂a (x, Re v + iθ Im u + i(1 − θ ) Im v) dθ · i Im(u − v). ∂ Im u

It then suffices to apply (1.87) to the function in the right-hand side with m = 1.  Let us now fix the exponents α and β so that (5.6) is verified. By (16.36), if we  1 take these exponents in such a way that 2ν 2−1 < α < 2p and 2s < β < 1, then the two space conditions in (5.6) are verified. In addition, it holds that 1+α<1+

p+n 1 1 ≤ + < β + ν, 2p 2p 2

which means that (5.35) is also fulfilled. Note, however, that ν < β is the case bep+n cause of ν < p+1 2p ≤ 2p < β. But, as investigated in Chap. 1, Sect. 11.8, we know that Z = Hps (Ω) is a reflexive space, i.e., (5.36) is valid. Let us finally verify the conditions on F and W . By the definition of F , condition (5.68) is verified directly. If we take η in such a way that 12 < η < ν, then (5.46) and (5.69) are fulfilled. We are thus ready to apply Theorem 5.7 to (5.79). For any function g ∈ Fβ,σ ((0, T ]; Lp (Ω)), 0 < σ < min{β − α + ν − 1, 1 − η}, and for any u0 ∈ K satisfying Ap (u0 )β u0 Lp < R/D2 , (5.79) possesses a unique local solution in the function space given by (5.70).

7 Some Model for Honeybee Colonies Let us consider the initial-boundary-value problem for a system of quasilinear parabolic and ordinary equations which is given by

236

5 Quasilinear Evolution Equations

⎧  u  ∂u ⎪ ⎪ ∂t = a1 s(u)θ1 (v)θ2 (w) − a2 k1 +u [1 + c(w)] ⎪ ⎪  v  ⎪ ∂v ⎪ ⎪ ⎪ ∂t = a3 s(v)θ1 (u)θ2 (w) − a4 k1 +v [1 + c(w)] ⎪ ⎪ ⎪ ∂w ⎪ ⎪ ⎨ ∂t = a5 θ1 (u)θ1 (v)(1 − w)ρ  k2   ∂ρ ⎪ ∂t = d∇ · k2 +u+v ∇ρ ⎪ ⎪ ⎪ ∂ρ ⎪ ⎪ ⎪ ∂n = 0 ⎪ ⎪ ⎪ u(x, 0) = u0 (x), v(x, 0) = v0 (x), ⎪ ⎪ ⎪ ⎩ w(x, 0) = w (x), ρ(x, 0) = ρ (x) 0 0

in Ω × (0, ∞), in Ω × (0, ∞), in Ω × (0, ∞), in Ω × (0, ∞),

(5.80)

on ∂Ω × (0, ∞), in Ω,

in a C2 or convex, bounded domain Ω ⊂ R2 . This model was presented for describing the process of a comb pattern formation observed in honeybee colony. The function u(x, t) denotes the rate of honey in Ω at time t, and v(x, t) denotes the rate of pollen. These rates are expected to be 0 ≤ u(x, t) ≤ 1 and 0 ≤ v(x, t) ≤ 1 for every (x, t) ∈ Ω × (0, ∞). The third function w(x, t) denotes the rate of brood in Ω at time t, w(x, t) being also expected to be 0 ≤ w(x, t) ≤ 1 for every (x, t) ∈ Ω × (0, ∞). Meanwhile, the function ρ(x, t) denotes the movement of the egg-laying queen. The queen moves by random walkk2 depends on the densities of honey and pollen. ing, but the diffusion rate k2 +u+v The unknown function ρ(x, t) is assumed to satisfy the homogeneous Neumann boundary conditions on ∂Ω. Initial functions u0 (x), v0 (x), w0 (x), and ρ0 (x) are given functions in Ω such that 0 ≤ u0 ≤ 1, 0 ≤ v0 ≤ 1, 0 ≤ w0 ≤ 1, and ρ0 ≥ 0, respectively. We assume that s(u), θi (u) (i = 1, 2) are all smooth decreasing functions for 0 ≤ u ≤ 1 such that s(0) = θi (0) = 1 and s(1) = θi (1) = 0 with s  (0) = θi (0) = 0 and s  (1) = θi (1) = 0. While, c(w) is a smooth increasing function for 0 ≤ w ≤ 1 such that c(0) ≥ 0. We assume also that ai (i = 1, 2, . . . , 5), ki (i = 1, 2), and d are positive constants (> 0).

7.1 Abstract Formulation We want to treat (5.80) in the product space ⎧⎛ ⎞ ⎫ u ⎪ ⎪ ⎪ ⎪ ⎨⎜ ⎟ ⎬ v 1 1 1 ⎟ ; u ∈ H (Ω), v ∈ H (Ω), w ∈ H (Ω), ρ ∈ L2 (Ω) , X= ⎜ p p p ⎝w ⎠ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ρ p being a fixed number such that 2 < p ≤ 4. Let 0 < ε < 1 be a fixed exponent. The space K of initial values is defined as follows. A vector U0 = t (u0 , v0 , w0 , ρ0 ) ∈ X belongs to K if and only if u0 , v0 , w0 ∈ Hp1 (Ω) and ρ0 ∈ H ε (Ω) with the conditions 0 ≤ u0 ≤ 1,

0 ≤ v0 ≤ 1,

0 ≤ w0 ≤ 1,

and 0 ≤ ρ0 , ρ0 L1 = 1.

(5.81)

7 Some Model for Honeybee Colonies

237

We then formulate (5.80) as the Cauchy problem for an abstract equation  dU dt

+ A(U )U = F (U ),

0 < t < ∞,

(5.82)

U (0) = U0 ,

in X. Here, for U = t (u, v, w, ρ) ∈ X, A(U ) is a linear operator given by A(U ) = diag{1, 1, 1, A(u, v)}, where A(u, v) is a realization of the differential operator # $  k2 −d∇ · ∇· + 1 k2 + χ2 (Re u) + χ2 (Re v) in the space L2 (Ω) under the Neumann boundary conditions on ∂Ω, χ2 (u) being a smooth cutoff function for −∞ < u < ∞ such that χ2 (u) ≡ − k22 for −∞ < u ≤ − k22 , χ2 (u) = u for 0 ≤ u ≤ 1, and χ2 (u) ≡ 1 + k22 for 1 + k22 ≤ u < ∞. Actually, A(u, v) is a positive definite self-adjoint operator of L2 (Ω). Note that Theorem 2.8 is available to verify that the domain D(A(u, v)) is independent of (u, v) and is given by D(A(u, v)) = HN2 (Ω) (see (2.34)). Consequently, we observe that D(A(U )) ≡ D = Hp1 (Ω) × Hp1 (Ω) × Hp1 (Ω) × HN2 (Ω),

U ∈ X.

(5.83)

In particular, D(A(U )) is independent of U ∈ X. In addition, for 0 ≤ θ ≤ 1, we have A(U )θ = diag{1, 1, 1, A(u, v)θ }. Since A(u, v) is a self-adjoint operator, it follows that D(A(u, v)θ ) coincides with the interpolation space [L2 (Ω), HN2 (Ω)]θ characterized by Theorems 16.7 and 16.9. Therefore, for every U ∈ X, 

D A(U )

θ



 ≡ Dθ =

[Hp1 (Ω)]3 × H 2θ (Ω) for 0 ≤ θ < 34 , [Hp1 (Ω)]3 × HN2θ (Ω) for

3 4

< θ ≤ 1.

(5.84)

The nonlinear operator F is defined by ⎛ ⎞  1 (Re u)  [1 + c(Re ˜ w)] u + a1 s˜ (Re u)θ˜1 (Re v)θ˜2 (Re w) − a2 k1χ+χ 1 (Re u) ⎜ ⎟ ⎜v + a3 s˜ (Re v)θ˜1 (Re u)θ˜2 (Re w) − a4  χ1 (Re v) [1 + c(Re ⎟ ˜ w)] ⎜ ⎟. k +χ (Re v) 1 1 F (U ) = ⎜ ⎟ ⎝ ⎠ ˜ ˜ w + a5 θ1 (Re u)θ1 (Re v)(1 − w)ρ ρ Here, s˜ (u), θ˜i (u) (i = 1, 2) denote C1,1 extensions of s(u), θi (u), respectively, to the whole real line (−∞, ∞) in such a way that s˜ (u) = s(u) for 0 ≤ u ≤ 1, s˜ (u) ≡ 1 for −∞ < u ≤ 0, and s˜ (u) ≡ 0 for 1 ≤ u < ∞. It is the same for θ˜i (u) (i = 1, 2). Similarly, c(w) ˜ is a C1,1 extension of c(w) for −∞ < w < ∞ such that c(w) ˜ = c(w) for 0 ≤ w ≤ 1 and − 12 ≤ c(w) ˜ ≤ c(1) + 12 for −∞ < w < ∞. While, χ1 (u) is a smooth cutoff function for −∞ < u < ∞ such that χ1 (u) ≡ − k21 for −∞ < u ≤ − k21 , χ1 (u) = u for 0 ≤ u ≤ 1, and χ1 (u) ≡ 1 + k21 for 1 + k21 ≤ u < ∞. Let us take 1 2θ 1 the exponent θ = p−1 p (> 2 ). Then, H (Ω) ⊂ Hp (Ω). Note also property (1.95)

238

5 Quasilinear Evolution Equations

and the fact that Hp1 (Ω) is a Banach algebra (see (1.88)). As a result, we conclude that F : W ≡ Hp1 (Ω) × Hp1 (Ω) × Hp1 (Ω) × H 2θ (Ω) → X.

(5.85)

The initial values U0 are taken in K. In this way, problem (5.80) has been formulated in the form (5.82).

7.2 Construction of Local Solutions It is immediate to apply Theorem 5.6 to (5.82) for constructing local solutions. By the definition, it is clear that A(U ) is a sectorial operator of X with angle ωA(U ) ≤ ω < π2 , namely, (5.2) and (5.3) are fulfilled with some angle ω and constant M ≥ 1. In view of (5.83), (5.4) is also fulfilled with ν = 1. For verifying (5.5), we will use (3.68). Then, it is sufficient to verify that [A(U ) − A(V )]U˜ X ≤ CU − V Y U˜ D ,

U˜ ∈ D.

But, since ˜ H2, ∇ · [χ∇ u] ˜ L2 ≤ CχHp1 u

χ ∈ Hp1 (Ω), u˜ ∈ HN2 (Ω),

the estimate is observed directly by setting Y = X as the space Y . In addition, since X = Y = Z, (5.6) is also fulfilled with α = 0 and an arbitrarily fixed β such that 0 < β < 2ε , as well as (5.7). As verified by (5.85), F is an operator from W into X. Furthermore, F is seen to satisfy (5.44). Indeed, ˜s (u1 ) − s˜ (u2 )Hp1 ≤ C[∇[˜s (u1 ) − s˜ (u2 )]Lp + ˜s (u1 ) − s˜ (u2 )Lp ] ≤ C[˜s  (u1 )∇u1 − s˜  (u2 )∇u2 Lp + u1 − u2 Lp ] ≤ C(u1 Hp1 + u2 Hp1 + 1)u1 − u2 Hp1 ,

u1 , u2 ∈ Hp1 (Ω).

It is the same for other functions contained in the components of F . On account of (5.84), (5.46) is fulfilled with η = θ = p−1 p , as well as (5.47) because of 0 < β < < 12 < p−1 p = η < 1. We finally verify that U0 satisfies (5.30) if γ is fixed as γ = 2ε . Thus, for any U0 ∈ K, (5.82) possesses a unique local solution in the function space: ε 2

U ∈ C((0, TU0 ]; D) ∩ C([0, TU0 ]; Dγ ) ∩ C1 ((0, TU0 ]; X), where TU0 > 0 is determined by the norm U0 D ε . By (5.66), 2 belong to dU , dt

with any 0 < σ < min 2ε , p1 .

ε

A(U )U ∈ F 2 ,σ ((0, TU0 ]; X)

dU dt

(5.86) and A(U )U

(5.87)

7 Some Model for Honeybee Colonies

239

7.3 Nonnegativity of Local Solutions For U0 ∈ K, let U (t) be the local solution of (5.82) constructed above in (5.86) and (5.87). We shall prove that the solution inherits all the properties described in (5.81). Let us first verify that U (t) is real valued. Indeed, the complex conjugate U (t) of U (t) is also a local solution of (5.82) with the same initial value U0 . So, the uniqueness of solution implies that U (t) = U (t); hence, U (t) is real valued. Let H (u) be a C1,1 cutoff function given by H (u) = 12 u2 for −∞  < u < 0 and H (u) ≡ 0 for 0 ≤ u < ∞. According to (1.100), the function ψ(t) = Ω H (u(t)) dx is continuously differentiable with the derivative  $ #

χ1 (u)   ˜ ˜ H (u) a1 s˜ (u)θ1 (v)θ2 (w) − a2 [1 + c(w)] ˜ dx. ψ (t) = k1 + χ1 (u) Ω ˜ and χ1 (u), we observe that By the definitions of s˜ (u), θ˜i (v) (i = 1, 2), c(w), # $

χ1 (u)   H (u) [1 + c(w)] ˜ dx ψ (t) ≤ −a2 k1 + χ1 (u) Ω



≤C |H  (u)χ1 (u)| dx ≤ C H (u) dx. Ω

Ω

Since ψ  (t) ≤ Cψ(t), it follows that ψ(t) ≤ eCt ψ(0) = 0, i.e., ψ(t) ≡ 0. This shows that u(t) ≥ 0 for every 0 ≤ t ≤ TU0 .  We next consider the function ψ1 (t) = Ω H (1 − u(t)) dx. By similar arguments, we observe that ψ1 (t) ≤ 0. Hence, ψ1 (t) ≤ ψ1 (0) = 0, i.e., u(t) ≤ 1 for every 0 ≤ t ≤ T U0 . We can repeat quite similar arguments as those above to prove that 0 ≤ v(t) ≤ 1 for 0 ≤ t ≤ TU0 . In the meantime, the nonnegativity of ρ(t) is easily verified by the same techniques as in Chap. 4, Sect. 4.2. From the differential equation of ρ, we  d ρ(t) dx ≡ 0. Therefore, ρ(t)L1 = Ω ρ(t) dx ≡ Ω ρ0 dx = 1. Now, have dt Ω since ρ(t) ≥ 0, we can carry out the proof of 0 ≤ w(t) ≤ 1. We will in fact repeat the same arguments as for u and v in view of the known estimate 1

1−ε

1

ρ(t)L∞ ≤ Cρ(t)H 1+ε ≤ Cρ(t)H2−ε2 ρ(t)H2−εε ≤ CU t − 2 ,

0 < t ≤ TU 0 .

7.4 A Priori Estimates In this subsection, let U0 ∈ K satisfy a supplement condition that ρ0 ∈ L∞ (Ω) ∩ H 1 (Ω). Let U denote a local solution of (5.82) on [0, TU ] with U (t) ∈ K, 0 ≤ t ≤ TU , which belongs to the function space: U ∈ C((0, TU ]; D) ∩ C([0, TU ]; Dγ ) ∩ C1 ((0, TU ]; X). We shall establish a priori estimates for local solutions.

(5.88)

240

5 Quasilinear Evolution Equations

Proposition 5.2 There exists a continuous increasing function p(·) such that the estimate U (t)Dγ ≤ p(U0 X + ρ0 L∞ + ρ0 H 1 + t),

0 ≤ t ≤ TU ,

(5.89)

holds for any local solution of (5.82) in the function space (5.88). Proof The proof will consist of a several steps. Step 1. Let 2 < q < ∞ be any number. Multiply the equation of ρ by qρ q−1 and integrate the product in Ω. Then,



d k2 ρ q−2 q |∇ρ|2 dx ≤ 0. ρ dx = −dq(q − 1) dt Ω Ω k2 + u + v Therefore, ρ(t)Lq ≤ ρ0 Lq , 0 ≤ t ≤ TU . Letting q → ∞, it follows that ρ(t)L∞ ≤ ρ0 L∞ ,

0 ≤ t ≤ TU .

(5.90)

k2 Step 2. Multiply now the equation of ρ by ∇ · [ k2 +u+v ∇ρ] and integrate the product in Ω. Then,   2

 

  k2 k2 ∂ρ ∇· ∇ρ dx = ∇ · ∇ρ  dx. k2 + u + v k2 + u + v Ω ∂t Ω

By a direct calculation, the left-hand side is equal to  



k2 d ∂ρ k2 ∇· ∇ρ dx = − |∇ρ|2 dx k2 + u + v dt Ω 2(k2 + u + v) Ω ∂t

k2 (ut + vt ) + |∇ρ|2 dx. 2 2(k + u + v) 2 Ω Here, from the equations of u and v we have ut ≤ a1 and vt ≤ a3 . Therefore, 2

 

  k2 k2 d 2  |∇ρ| dx + ∇ · ∇ρ  dx dt Ω 2(k2 + u + v) k + u + v 2 Ω

k2 a1 + a3 |∇ρ|2 dx. ≤ k2 Ω 2(k2 + u + v) Consequently,



k2 2 Ω 2(k2 +u+v) |∇ρ| dx

∇ρ(t)2L2 ≤

≤ 12 e

a1 +a3 k2 t

ρ0 2H 1 , i.e.,

k2 + 2 a1k+a3 t e 2 ρ0 2H 1 , k2

0 ≤ t ≤ TU .

As well, we obtain that 2

t      a1 +a3  k2  dx ≤ k2 + 2 e k2 t − 1 ρ0 2 1 , ∇ · ∇ρ   H k2 + u + v 2k2 0 Ω

(5.91)

0 ≤ t ≤ TU . (5.92)

7 Some Model for Honeybee Colonies

241

Step 3. We shall use the notation p(ρ0 ) = p(ρ0 L∞ + ρ0 H 1 ), where p(·) in the right-hand side denotes a suitable continuous increasing function. Differentiating the equation of u with respect to the variable xi (i = 1, 2), we observe that ∂ Di u = f1 (u, v, w)Di u + f2 (u, v, w)Di v + f3 (u, v, w)Di w, ∂t where fj (u, v, w) (j = 1, 2, 3) are real smooth functions defined for 0 ≤ u ≤ 1, 0 ≤ v ≤ 1, 0 ≤ w ≤ 1. Multiply this equation by p|Di u|p−2 Di u and integrate the product in Ω. After some calculations, we have

 d p p p  |Di u|p dx ≤ C Di uLp + Di vLp + Di wLp . dt Ω d Note that du |u|p = p|u|p−2 u for −∞ < u < ∞. By using the equations of v and w (in view of (5.90)), we similarly obtain that

 d p p p  |Di v|p dx ≤ C Di uLp + Di vLp + Di wLp , dt Ω

 d p p p p  |Di w|p dx ≤ Cp(ρ0 ) Di uLp + Di vLp + Di wLp + Di ρLp . dt Ω

Therefore,

  d |Di u|p + |Di v|p + |Di w|p dx dt Ω  p p p p  ≤ Cp(ρ0 ) Di u(t)Lp + Di v(t)Lp + Di w(t)Lp + Di ρ(t)Lp . Here, by (5.90) and (5.91), p

p−2

p−2

Di ρLp ≤ CDi ρH 1 Di ρ2L2 ≤ Cp(ρ0 )eμt ρ(t)H 2 , where μ =

a1 +a3 k2 .

In addition, from

$ k2 k2 k2 ∇ρ = Δρ − (∇u + ∇v) · ∇ρ ∇· k2 + u + v k2 + u + v (k2 + u + v)2 #

we see that ΔρL2

 # $  k2 k2 + 2  1  ∇· ∇ρ  ≤ + (∇uLp + ∇vLp )∇ρL 2p   k2 k2 + u + v k2 p−2 L2  # $  k2 k2 + 2  ∇ · ∇ρ  ≤   k2 k2 + u + v L2

242

5 Quasilinear Evolution Equations 2

p−2

+ C(∇uLp + ∇vLp )ρHp 2 ρHp1  # $  k2 k2 + 2   ∇· ≤ ∇ρ   + ζ ρH 2 k2  k2 + u + v L2 p p & % μ p−2 + Cζ p(ρ0 )e 2 t ∇uLp−2 + ∇v Lp p with any number ζ > 0 and some constant Cζ > 0 depending on ζ . Since ρH 2 ≤ C(ΔρL2 + ρL2 ) for all ρ ∈ HN2 (Ω) (see (2.33)), it follows that  # $ p p & ' μ % (   k2 p−2 p−2  2 t ∇u + Cp(ρ ) e + ∇v ∇ρ  + 1 ρH 2 ≤ C ∇ · 0 Lp Lp  k2 + u + v L2 and that p−2 ρH 2

 # $p−2   k2  ≤ C ∇ · ∇ρ   k2 + u + v L2   μ(p−2) t  p p  ∇uLp + ∇vLp + 1 . + Cp(ρ0 ) e 2

In this way, we arrive at the differential inequality

  d |∇u|p + |∇v|p + |∇w|p dx dt Ω  pμ p p p t 2 ∇uLp + ∇vLp + ∇wLp ≤ Cp(ρ0 )e  #  $p−2   k2   + ∇ · +1 . ∇ρ  k2 + u + v L2 By virtue of (5.92) (recall that p was taken as 2 < p ≤ 4), we can solve this differential inequality to conclude that ∇u(t)Lp + ∇v(t)Lp + ∇w(t)Lp ≤ p(U0 X + ρ0 L∞ + ρ0 H 1 + t),

0 ≤ t ≤ TU .

This, together with (5.91), then yields the desired estimate (5.89).



7.5 Global Solutions Utilizing the a priori estimates (5.89), we shall construct a global solution to (5.82). For U0 ∈ K, we know that there exists a local solution having the properties (5.81) at least on an interval [0, TU0 ]. Let 0 < t1 < TU0 . Then, U1 = U (t1 ) ∈ K

8 Some Chemotaxis Model

243

and ρ1 = ρ(t1 ) ∈ HN2 (Ω). We next consider problem (5.82) with the initial value U1 on an interval [t1 , T ], where the end time T > 0 is any finite time. Proposition 5.2 ensures estimate (5.89) for any local solution V , i.e., V (t)Dγ ≤ p(U1 X + ρ1 L∞ + ρ1 H 1 + T ), t1 ≤ t ≤ TV . Then, the local solution V can always be extended over an interval [t1 , TV + τ ] as local solution, τ > 0 being dependent only on p(U1 X + ρ1 L∞ + ρ1 H 1 + T ) and hence being independent of the extreme time TV (cf. Corollary 5.2). This means that our Cauchy problem possesses a global solution on the interval [t1 , T ]. This argument is meaningful for any finite time T > 0. So, we conclude the global existence of solution. For any initial value U0 ∈ K, there exists a unique global solution to (5.82) with U (t) ∈ K, 0 ≤ t < ∞, in the function space: U ∈ C((0, ∞); D) ∩ C([0, ∞); Dγ ) ∩ C1 ((0, ∞); X).

8 Some Chemotaxis Model Let us consider the initial-boundary-value problem for a chemotaxis model in biology which is given by ⎧ ∂u 2 in Ω × (0, ∞), ⎪ ∂t = Δ(au + αu ) − μ∇ · (u∇ρ) ⎪ ⎪ ⎪ ⎨ ∂ρ = bΔρ − dρ + νu in Ω × (0, ∞), ∂t (5.93) ∂ρ ⎪ ∂u ⎪ = = 0 on ∂Ω × (0, ∞), ⎪ ∂n ∂n ⎪ ⎩ ρ(x, 0) = ρ0 (x) in Ω, u(x, 0) = u0 (x), in a C2 or convex, bounded domain Ω ⊂ R2 . In the domain Ω, biological individuals (bacteria) exist. The density of bacteria at time t is denoted by u(x, t). The function ρ(x, t) denotes the concentration of chemical substance at time t . Bacteria diffuse not only by random walking with diffusion rate a but also by self-diffusion with rate α. They also have a directed movement which is called the chemotaxis in the sense that they have a tendency to move toward higher concentration of the chemical substance. The flow by this effect is given by μu∇ρ with some mobility constant μ. Bacteria produce the chemical substance by themselves at a rate ν. The chemical substance diffuses with diffusion rate b. For the details, see Chap. 12. We assume that a, b, d, α, μ, and ν are positive constants (>0). Let 0 < ε < 12 be arbitrarily fixed. The initial functions are taken as 0 ≤ u0 ∈ H 1+ε (Ω) and 0 ≤ ρ0 ∈ H 1+ε (Ω).

(5.94)

8.1 Abstract Formulation Let us set X = L2 (Ω) (see (1.84)) as the underlying space X. We want to formulate (5.93) as the Cauchy problem for an abstract equation

244

5 Quasilinear Evolution Equations



+ A(U )U = F (U ), U (0) = U0 , dU dt

0 < t < ∞,

(5.95)

in X. Here, A(U ) is a linear operator of X defined for U ∈ Z = H1+ε1 (Ω), where 0 < ε1 < ε, which is given by # $ $ # u A1 (u) B(u) ∈ Z. with D(A(U )) = H2N (Ω), U = A(U ) = ρ 0 A2 The operator A1 (u) is a realization of the differential operator −{∇ · [a + 2αχ(Re u)]∇·} + 1 in L2 (Ω) under the Neumann boundary conditions on ∂Ω, where χ(u) is a smooth cutoff function such that χ(u) = u for u ≥ 0 and χ(u) ≡ −δ for u ≤ −δ, δ > 0 being some small positive constant such that 4αδ ≤ a. By Theorem 2.4, A1 (u) is a positive definite self-adjoint operator of L2 (Ω), and by Theorem 2.8, their domain is given by D(A1 (u)) = HN2 (Ω) (see (2.34)). The operator A2 is a realization of the operator −bΔ + d in L2 (Ω) under the same boundary conditions on ∂Ω and is a positive definite self-adjoint operator of L2 (Ω) with domain D(A2 ) = HN2 (Ω); and the operator B(u) is given by μ∇ · (u∇·). Since ∇ · (u∇v) = ∇u · ∇v + uΔu, we observe from (1.74), (1.75), and (1.76) that ∇ · (u∇v)L2 ≤ CuH 1+ε1 vH 2 , This means that D(B(u)) ⊃ H 2 (Ω). The operator F : X → X is given by # $ u F (U ) = , νu

u ∈ H 1+ε1 (Ω), v ∈ H 2 (Ω).

U=

# $ u ∈ X. ρ

The initial value U0 is given by U0 = t (u0 , ρ0 ), u0 and ρ0 satisfying (5.94).

8.2 Basic Properties of Operators A(U ) Let 0 < R < ∞, and let KR = {U ∈ Z; U Z < R} be an open ball of Z. Proposition 5.3 For any U ∈ Z, we have D(A(U )) = H2N (Ω). For each 0 < R < ∞, there is a constant CR > 0 such that CR−1 U˜ H2 ≤ A(U )U˜ X ≤ CR U˜ H2 , U˜ ∈ D(A(U )), holds for every U ∈ KR . Proof This result is easily verified by (2.33) which is ensured by Theorem 2.8.  Proposition 5.4 For U ∈ KR , A(U ) is a sectorial operator of X with angle ωA(U ) ≤ ωR < π2 and satisfies (5.3) with MR ≥ 1 uniformly. Proof As A1 (u) and A2 are positive definite self-adjoint operators of L2 (Ω), this result is obtained by applying Theorem 1.8 to each operator A(U ). 

8 Some Chemotaxis Model

245

For s > 32 , we put  HsN (Ω) =

∂U U ∈ H (Ω); = 0 on ∂Ω . ∂n s

Proposition 5.5 For all U ∈ Z, we have   D A(U )θ = H2θ (Ω),   D A(U )θ = H2θ N (Ω),

3 0≤θ < , 4 3 < θ ≤ 1. 4

Let 0 ≤ θ ≤ 1, θ = 34 . For 0 < R < ∞, there is a constant CR,θ > 0 such that   −1 ˜ U H2θ ≤ A(U )θ U˜ X ≤ CR,θ U˜ H2θ , CR,θ

  U˜ ∈ D A(U )θ ; U ∈ KR .

Proof In order to prove the assertion of theorem, we will appeal to Theorem 16.4 presented in Chap. 16. For applying Theorem 16.4, it is in fact sufficient to verify that for each u, the adjoint operator B(u)∗ : D(B(u)∗ ) ⊂ L2 (Ω) → L2 (Ω) of B(u) has a domain such that D(B(u)∗ ) ⊃ HN2 (Ω) and is a bounded operator from HN2 (Ω) into L2 (Ω). However, this is easily verified. It is clear that HN2 (Ω) ⊂ D(B(u)∗ ) and B(u)∗ v = μ∇ · (u∇v) for v ∈ HN2 (Ω). Hence, B(u)∗ is bounded from HN2 (Ω) into L2 (Ω). We thus know that D(A(U )θ ) = [X, H2N (Ω)]θ for 0 ≤ θ ≤ 1. Then, the desired  characterization of D(A(U )θ ) is verified by Theorem 16.7.

8.3 Construction of Local Solutions We are ready to apply Theorem 5.6 to construct local solutions to (5.95). Let U0 be any initial value satisfying (5.94). We can then choose a radius R > 0 in such a way that U0 ∈ KR . It is already known that, for U ∈ KR , A(U ) fulfills (5.2) and (5.3). Since D(A(U )) = H2N (Ω) for every U ∈ KR , (5.4) is also fulfilled with ν = 1. In view of (3.68), (5.5) is equivalent to showing that [A(U ) − A(V )]U˜ X ≤ CR U − V Y U˜ H2 ,

U˜ ∈ H2N (Ω),

with a suitable space Y . We here notice that ˜ ∇ · [χ(Re u)∇ u] ˜ = χ(Re u)Δu˜ + χ  (Re u)∇(Re u) · ∇ u. From this we observe that ˜ H2 ∇ · [χ(Re u)∇ u] ˜ L2 ≤ (χ(Re u)L∞ + χ  (Re u)L∞ uH θ+1 )u ≤ (χ(Re u)H θ+1 + χ  (Re u)H θ+1 uH θ+1 )u ˜ H2, (5.96)

246

5 Quasilinear Evolution Equations

whatever 0 < θ < 1 is. In view of this fact, we set Y = H1+ε0 (Ω), where ε0 is a third exponent such that 0 < ε0 < ε1 < ε. Then, (5.5) is verified immediately with the aid of (1.89) and (1.94). By virtue of Proposition 5.5, (5.6) is then fulfilled with 1+ε1 0 α = 1+ε 2 and β = 2 . Relation (5.7) is now obvious. The nonlinear operator F is defined on W = H1+ε2 (Ω), where ε < ε2 < 12 . It is trivial that (5.44) is fulfilled. Conditions (5.46) and (5.47) are also verified, since 2 Proposition 5.5 provides that D(A(U )η ) = W with η = 1+ε 2 for every U ∈ KR . The 1+ε initial value U0 satisfies (5.30) if γ is taken as γ = 2 . Theorem 5.6 then provides that, for any U0 ∈ KR satisfying (5.94), (5.95) possesses a unique local solution in the function space:   U ∈ C (0, TU0 ]; H2N (Ω) ∩ C([0, TU0 ]; H1+ε (Ω)) ∩ C1 ((0, TU0 ]; L2 (Ω)), (5.97) where TU0 > 0 is determined by the norm U0 H1+ε . By (5.66), A(U )U belongs to dU , dt

A(U )U ∈ F

1+ε 2 ,σ

((0, TU0 ]; L2 (Ω))

(5.98)

with any 0 < σ < min{ε1 − ε0 , 1−ε 2 }.

8.4 Nonnegativity of Local Solutions For U0 satisfying (5.94), let U (t) be the local solution of (5.95) constructed above in the function space (5.97) and (5.98). We want to prove that u(t) ≥ 0 and v(t) ≥ 0 for every 0 < t ≤ TU0 . Let us notice first that U (t) is real valued. The complex conjugate U (t) of U (t) is also a local solution of (5.94) with the same initial value U0 . So, the uniqueness of solution implies that U (t) = U (t); hence, U (t) is real valued. We then utilize the truncation method as in Sect. 7.3. Let H (u) be a C1,1 cutoff function given by H (u) = 12 u2 for −∞ < u < 0 and H (u) ≡ 0 for 0 ≤ u < ∞. According to (1.100), the function ψ(t) = Ω H (u(t)) dx is continuously differentiable with the derivative



  H (u)∇ · {[a + 2αχ(u)]∇u} dx − μ H  (u)∇ · (u∇ρ) dx. ψ (t) = Ω

Ω

Using property (1.96), we have



H  (u)∇ · {[a + 2αχ(u)]∇u} dx = − [a + 2αχ(u)]∇H  (u) · ∇u dx Ω



=−

Ω

[a + 2αχ(u)]|∇H  (u)|2 dx ≤ −a

Ω

By a similar reason,



|∇H  (u)|2 dx. Ω

(5.99)

9 Nonautonomous Quasilinear Abstract Evolution Equations

−μ

H  (u)∇ · (u∇ρ) dx = μ

Ω



247

u∇H  (u) · ∇ρ dx

Ω

=μ

H  (u)∇H  (u) · ∇ρ dx

Ω

≤ μH  (u)Lp ∇H  (u)L2 ∇ρLq 2(1− p1 )

≤ CH  (u)H 1 ≤

2

H  (u)Lp 2 ∇ρLq

a p H  (u)2H 1 + CH  (u)2L2 ∇ρLq , 2

where 2 < p, q < ∞ are any numbers satisfying

1 p

+

1 q

= 12 . Therefore, it follows C

t

p

[∇ρ(s)

+1]ds

Lq that ψ  (t) ≤ C[∇v(t)Lq +1]ψ(t). Consequently, ψ(t) ≤ ψ(0)e 0 for 0 < t < TU0 . p p From the regularity (5.98), we notice that ∇ρ(s)Lq ≤ Cq ρ(s)H 2 ≤ Cq CU0 ×

p

s p(ε−1)/2 . It is possible to choose p > 2 arbitrarily close to 2, which shows that p ∇ρ(·)Lq is integrable in (0, TU0 ). Then, ψ(0) = 0 implies ψ(t) ≡ 0. Thus, u(t) ≥ 0 for every 0 < t ≤ TU0 . It is similar (actually easier) for the proof of ρ(t) ≥ 0 for 0 < t ≤ TU0 . Hence, we have accomplished the proof of nonnegativity of U (t). Now, since u(t) ≥ 0, it holds that χ(Re u(t)) = u(t); this then means that the local solution of (5.95) is regarded as a local solution to the original problem (5.93).

8.5 Global Solutions We are able to build up a priori estimates for local solutions of (5.95) with initial values satisfying (5.94) and an additional condition U0 ∈ H2 (Ω). In fact, there exists a continuous increasing function p(·) such that, for any local solution of (5.95) in (5.97) (in which TU0 must be replaced by TU ) with initial value satisfying (5.94) and U0 ∈ H2 (Ω), it holds that U (t))H2 ≤ p(U0 H2 ),

0 < t ≤ TU .

(5.100)

For showing these estimates step-by-step, we need more complicated calculations for the norms of U (t) in each step. We will not describe the proof here, cf. Chap. 12, Sect. 3.2, Chap. 15, Sects. 3.1 and 3.2. It is of course immediate to conclude global existence of solutions for (5.95) utilizing (5.100) (cf. Corollary 5.2). Note also that, for any initial value satisfying (5.94), its local solution always satisfies U (t) ∈ H2 (Ω) for any t > 0.

9 Nonautonomous Quasilinear Abstract Evolution Equations In this section, we will show how to reduce nonautonomous problems to autonomous ones as in Chap. 4, Sect. 5. Consider the Cauchy problem for a quasilinear

248

5 Quasilinear Evolution Equations

abstract evolution equation  dU dt + A(t, U )U = F (t, U ) + G(t),

0 < t ≤ T,

U (0) = U0 ,

(5.101)

in a Banach X. Let Z be a second Banach space continuously embedded in X, and let K = {U ∈ Z; U Z < R}, 0 < R < ∞, be an open ball of Z. For (t, U ) ∈ [0, T ] × K, A(t, U ) is a sectorial operator of X with angle ωA(t,U ) < π2 . For each 0 ≤ t ≤ T , F (t, ·) is a nonlinear operator from W into X, where W is another Banach space continuously embedded in X. The function G is a given X-valued function. The initial value U0 is taken in K. For the sectorial operators A(t, U ), we make the same structural assumptions as in Sect. 1.1. We use three Banach spaces Z ⊂ Y ⊂ X. The operator A(t, U ) is a sectorial operator of X satisfying (5.2) and (5.3). The domain D(A(t, U )) is allowed to vary with (t, U ) ∈ [0, T ] × K, but it is assumed that there exists a fixed exponent 0 < ν ≤ 1 such that   D(A(s, V )) ⊂ D A(t, U )ν for any pair (t, U ), (s, V ) ∈ [0, T ] × K (5.102) (analogously to (5.4)). The inverse A(t, U )−1 satisfies the Lipschitz condition    A(t, U )ν A(t, U )−1 − A(s, V )−1  L(X) ≤ N (|t − s| + U − V Y ),

(t, U ), (s, V ) ∈ [0, T ] × K,

(5.103)

with a constant N > 0 (instead of (5.5)). Furthermore, the domains of fractional powers D(A(t, U )α ) and D(A(t, U )β ) are contained in Y and Z with estimate (5.6), respectively, and with exponents 0 ≤ α < β < 1 satisfying relations (5.7). Similarly, D(A(t, U )η ) is contained in W with estimate (5.46). When (5.47) is the case, i.e., β < ν, we assume the Lipschitz condition F (t, U ) − F (s, V )X ≤ ϕ(U Z + V Z )[U − V W + (U W + V W + 1)(|t − s| + U − V Z )], (t, U ), (s, V ) ∈ [0, T ] × W,

(5.104)

where ϕ(·) is some continuous increasing function. On the other hand, when (5.69) is the case, i.e., ν ≤ β, we have to assume not only the Lipschitz condition F (t, U ) − F (s, V )X ≤ ϕ(U Z + V Z )(|t − s| + U − V W ), (t, U ), (s, V ) ∈ [0, T ] × Z, but also the reflexivity of Z (see (5.36)). We now rewrite problem (5.101) as the following one:  d  τ  1  τ     0  0 τ +1 dt U + 0 A(χ(Re τ ),U ) U = F (χ(Re τ ),U ) + G(t) , 0 τ  U (0) = U0 ,

(5.105)

(5.106)

Notes and Further Researches

249

in the product space X = C × X for the unknown function t (τ, U ), where χ(t) is a C0,1 cutoff function for −∞ < t < ∞ such that χ(t) ≡ 0 for −∞ < t < 0, χ(t) = t for 0 ≤ t ≤ T , and χ(t) ≡ T for T < t < ∞. Since τ (t) = t , (5.106) is certainly equivalent to (5.101). It is not difficult to verify that the conditions assumed above, including (5.102), (5.103), (5.104), and (5.105), imply all the structural conditions for the linear operators A(τ, U ) = diag{1, A(χ(Re τ ), U )},

(τ, U ) ∈ C × K,

of X and for the nonlinear operator F(τ, U ) = t (τ + 1, F (χ(Re τ ), U )) from C × W into X. Hence, Theorems 5.5, 5.6, 5.7, and others are available to (5.106).

Notes and Further Researches The abstract problems of the form (5.1) or (5.43) were first studied by Sobolevskii [Sob61b], where the independence of the domains D(A(t, U )ν ) is assumed for some 0 < ν ≤ 1. The analyticity of solutions was studied by Furuya [Fur81, Fur83], and the global existence of solutions was studied by Lunardi [Lun84, Lun85]. Under a similar assumption as (5.5), especially in the case where β ≤ ν (together with (5.36)), the problems were handled by Yagi [Yag91, Yag02]. For the nonautonomous problem (5.101), see [Yag04]. In Favini–Yagi [FY04], the Cauchy problem for the multivalued quasilinear equation dU + A(U )U  F (U ), dt

0 < t ≤ T,

was handled. For the study of (5.1) or (5.43) by the variational methods, we quote Lions [Lio69, DL88]. Theorem 5.10 can be proved by similar arguments as in Ryu–Yagi [RY01]. By the methods of operational equations (see [DaPG75]), the abstract problems can also be studied; we quote Prüss [Pru02]. There is of course a great deal of important literature which deals with the quasilinear parabolic differential equations directly by the functional analytic methods. But we will here quote only Ladyzhenskaya–Solonnikov–Uraltseva [LSU68], Amann [Ama86, Ama93, Ama95], Quittner–Souplet [QS07], and so forth. The abstract results obtained in this chapter are available for various model equations presented from the real world. In Sects. 7 and 8, some of them were represented. The honeybee colony model (5.80) was introduced in Camazine–Sneyd– Jenkins–Murray [CSJM90] for describing pattern formation created spontaneously by the egg-laying queen. For more detailed observations and arguments for modeling, see Camazine et al. [CDFSTB, Chap. 16]. The chemotaxis system with selfdiffusion (5.93) is one of specific forms of the chemotaxis model presented by Keller–Segel [KS70]. Moreover, these abstract results will be utilized in Chaps. 14 and 15.

Chapter 6

Dynamical Systems

In this chapter, we study dynamical systems determined from the Cauchy problems for abstract semilinear and quasilinear evolution equations in Banach spaces. The theory of infinite-dimensional dynamical systems was developed by Ladyzhenskaya, Vishik, Temam, and many other mathematicians. The first half of the chapter is mainly devoted to studying two basic matters of the dynamical systems, namely, stable and unstable manifolds of the equilibria and attractors. We shall use the notation (S(t), X, X) to denote a dynamical system. Here, X is a Banach space and denotes the universal space in which a dynamical system is considered. The notation X denotes a subset of X and is a metric space. This is the space consisting of all possible states the system can take and is called the phase space. Meanwhile, S(t) (0 ≤ t < ∞) is a family of nonlinear operators acting on X with the semigroup property S(t)S(s) = S(t + s) and with S(0) = 1 (the identity operator on X). It is assumed that S(t) is continuous in both space and time variables in the sense that the correspondence (t, U0 ) → S(t)U0 is continuous from [0, ∞) × X into X. The operators S(t) give a propagation law of states of the system. For U0 ∈ X, the continuous function S(·)U0 for 0 ≤ t < ∞ with values in X denotes a transition of states starting from U0 at t = 0. The trace of points S(·)U0 which vary in the phase space X as time t goes on from 0 to ∞ is called the trajectory starting from U0 . The semigroup property implies S(t + s)U0 = S(t)S(s)U0 for any 0 ≤ s, t < ∞; this means that, s being fixed, a trajectory passing by S(s)U0 at time s coincides with that starting from S(s)U0 at time 0. The totality of such trajectories is called a dynamical system and is denoted by the triplet (S(t), X, X). A point U ∈ X which satisfies S(t)U = U is called a fixed point of S(t). A point U which is a fixed point for every S(t), that is, S(t)U = U for every 0 ≤ t < ∞, is called an equilibrium of dynamical system. For each equilibrium U , one can define a stable manifold M− (U ) and an unstable manifold M+ (U ) by (6.11) and (6.12), respectively. We then want to construct stable and unstable manifolds which are smooth in a neighborhood of U . If M− (U ) is a neighborhood of U , then U is stable. In the meantime, if M+ (U ) = {U }, then U is unstable. We shall indeed show that, if S(t) is Fréchet differentiable in a neighborhood of U with Hölder continuous derivative (6.19) and if the differential S(t) U satisfies a spectrum separation condition (6.20), then one can construct the desired smooth stable and unstable manifolds in a neighborhood of U . A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_6, © Springer-Verlag Berlin Heidelberg 2010

251

252

6 Dynamical Systems

For (S(t), X, X), a subset A of X is called an attractor if A is invariant (see (6.4)) and if A attracts, as t → ∞, all the trajectories starting in a neighborhood of A. Among others, an attractor A is called a global attractor of dynamical system if A is a compact set of X, if A is strictly invariant (see (6.5)), and if A attracts all trajectories in X. We know however that the global attractor is not necessarily robusts. Consider a dynamical system (Sξ (t), X, X) containing a parameter ξ and assume that (Sξ (t), X, X) possesses a global attractor Aξ , even if the semigroup Sξ (t) depends on ξ smoothly, Aξ may be not continuous in ξ . It is only upper semicontinuous in general. In addition, the global attractor may not be rich enough to describe the nature of the system, for, since A is strictly invariant, A can easily miss some important trajectories which correspond to a specific transition of states like pattern formations. As a more robust attractor, Foias–Sell–Temam introduced the notion of inertial manifolds which are hyperbolic (and thus robust), compact, invariant, finite-dimensional C1 manifolds that attract all the trajectories at exponential rates. But one can ensure the existence of inertial manifolds only under a very restrictive assumption called the spectral gap condition. Consequently, we do not know whether the dynamical systems constructed from many important models have inertial manifolds or not. In 1994, Eden–Foias–Nicolaenko–Temam have founded the new concept of exponential attractors (sometimes called inertial sets) to overcome this difficulty. An attractor M containing the global attractor is called an exponential attractor if M is a compact set of X with finite fractal dimension and if M attracts all the trajectories at exponential rates. (So, M need not to be a smooth manifold.) We notice that the exponential attractor is not uniquely determined and usually exists as a family. But the exponential attractor Mξ can depend continuously on a parameter ξ in a certain specific sense if the semigroup Sξ (t) depends on ξ smoothly, see Theorem 6.17. Two kinds of a sufficient condition on S(t) are known that ensure the existence of exponential attractors for (S(t), X, X). One is a squeezing property presented by the founders of the notion themselves. The squeezing property is meaningful only when X is a Hilbert space. When X is a Hilbert space, the operator S(t) is said to enjoy a squeezing property if, for any pair (U, V ) of points U, V ∈ X, it holds that either S(t)U − S(t)V ≤ δ U − V with some fixed exponent 0 ≤ δ < 14 or (1 − P )[S(t)U − S(t)V ] ≤ P [S(t)U − S(t)V ] with some fixed finite-rank orthogonal projection P (see (6.45)–(6.46)). The other is a compact perturbation of a contraction operator presented first by Efendiev–Miranville–Zelik. This property is meaningful in general Banach spaces, and there are two types. The operator S(t) is said to be a compact perturbation of a contraction operator of type I if the difference S(t)U − S(t)V is divided into two parts S0 (U, V ) and K(U, V ) such that S0 (U, V ) X ≤ δ U − V X with some fixed exponent 0 ≤ δ < 12 and that K(U, V ) Z ≤ L U − V X with some constant L > 0 and some fixed subspace Z which is compactly embedded in X (see (6.28)–(6.29)). Similarly, S(t) is said to be a compact perturbation of a contraction operator of type II if the norm S(t)U − S(t)V X is estimated by the sum of norms δ U − V X and KU −KV X with some fixed exponent 0 ≤ δ < 12 and with some Lipschitz continuous nonlinear operator K from X into a subspace Z which is compactly embedded in X (see (6.38)–(6.39)). As will be seen, the squeezing property provides sharper

1 Basics of Dynamical Systems

253

dimension estimates for exponential attractors than those of the compact perturbation condition. On the other hand, we have to notice that the property can easily be verified only when (S(t), X, X) are constructed from semilinear problems in Hilbert spaces of the form (6.53). Unfortunately, no convenient method is known so far for dynamical systems from quasilinear problems. In contrast, the method of using the compact perturbation condition on S(t) can easily be available to many concrete models. In the second half of the chapter, we shall consider dynamical systems which are determined by the Cauchy problems of semilinear and quasilinear abstract evolution equations. We are concerned with describing general methods for constructing stable and unstable manifolds for equilibria, i.e., stationary solutions. Similarly, we are concerned with general methods for constructing exponential attractors. For semilinear problems, stable–unstable manifolds are constructed under conditions (6.68), (6.69), and (6.76) and exponential attractors under (6.59) and (6.60), respectively. For quasilinear problems, stable–unstable manifolds are constructed under (6.115), (6.116) and (6.127) and exponential attractors under (6.103) and (6.104), respectively. We remark also that all these conditions are described mainly in terms of linear and nonlinear operators in the equations.

1 Basics of Dynamical Systems In this section, we shall prepare the basic matters of dynamical systems in Banach spaces. For details, see [Lad91, BV92, Tem97, Rob01] and others.

1.1 Equilibria and Attractors Let X be a Banach space with norm · . Let X be a subset of X, X being a metric space with the distance d(U, V ) = U − V induced by · . A family of nonlinear operators S(t), 0 ≤ t < ∞, from X into itself is called a (nonlinear) semigroup on X if S(t) satisfies (1) S(0) = 1 (identity mapping on X). (2) S(t)S(s) = S(t + s), 0 ≤ t, s < ∞ (semigroup property). When a semigroup S(t) on X has the property that the mapping G(t, U ) = S(t)U is continuous from [0, ∞) × X to X,

(6.1)

S(t) is called a continuous (nonlinear) semigroup on X. Let {Un }n=0,1,2,... be any sequence of X which converges to U in X. Then, for any fixed 0 < T < ∞, the set [0, T ] × {U0 , U1 , U2 , . . . , U } is a compact set of [0, ∞) × X. Therefore, if S(t) is a continuous semigroup, G is uniformly continuous on [0, T ] × {U0 , U1 , U2 , . . . , U }.

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Thus, for any fixed 0 < T < ∞, Un → U

sup S(t)Un − S(t)U → 0.

implies

(6.2)

0≤t≤T

Let S(t) be a continuous semigroup on X. For each U0 ∈ X, S(·)U0 defines an X-valued continuous function for 0 ≤ t < ∞. The trace of the function S(·)U0 in X is called a trajectory starting from U0 . The totality of trajectories starting from the points in X are denoted by the triplet (S(t), X, X) and is called a dynamical system. The set X is called a phase space, and X a universal space. Let (S(t), X, X) be a dynamical system. A point U ∈ X is called a fixed point of S(t) if S(t)U = U . If U is a fixed point for all operators S(t), i.e., S(t)U = U

for every t ≥ 0,

(6.3)

then U is called an equilibrium of (S(t), X, X). More generally, a set A ⊂ X is said to be invariant under S(t) if S(t)A ⊂ A. If A is invariant under all operators S(t), i.e., S(t)A ⊂ A

for every t ≥ 0,

(6.4)

then A is called an invariant set of (S(t), X, X). Similarly, a set A ⊂ X is said to be strictly invariant under S(t) if S(t)A = A. If S(t)A = A

for every t ≥ 0,

(6.5)

then A is called a strictly invariant set of (S(t), X, X). Two important notions of absorbing and attraction are defined as follows. A set A is said to absorb a set B if there is a certain time t0 ≥ 0 such that S(t)B is contained in A for every t ≥ t0 , i.e., S(t)B ⊂ A

for every t ≥ t0 .

We call a subset A ⊂ X an absorbing set of (S(t), X, X) if A ⊂ X absorbs every bounded set of X, namely, if, for any bounded set B ⊂ X, there is a time tB ≥ 0 (depending on B) such that S(t)B ⊂ A

for every t ≥ tB .

In the meantime, a set A is said to attract a set B if for any ε > 0, the εneighborhood Wε (A) of A absorbs B. Using the Hausdorff pseudo-distance h(·, ·) (see (1.1)), this definition can be described as lim h(S(t)B, A) = 0.

t→∞

(6.6)

Indeed, if (6.6) holds true, then for any ε > 0, there exists some tε > 0 such that h(S(t)B, A) ≤ 2ε for t ≥ tε ; therefore, for any U ∈ S(t)B, d(U, A) ≤ 2ε ; hence, U ∈ Wε (A), i.e., S(t)B ⊂ Wε (A)

for every t ≥ tε .

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Conversely, this condition implies h(S(t)B, A) ≤ ε for every t ≥ tε . So, if A attracts B, then (6.6) is valid. An equilibrium U of (S(t), X, X) is said to be asymptotically stable if some neighborhood W of U is attracted by U , i.e., lim h(S(t)W, U ) = 0.

t→∞

(6.7)

We simply say that U is stable if for any positive number ε > 0, there exists a neighborhood W of U which is attracted by an ε-neighborhood of U in such a way that lim sup h(S(t)W, U ) ≤ ε. t→∞

Clearly, if U is asymptotically stable, it is stable. Asymptotic stability is therefore stronger than mere stability. When U is not stable, U is said to be unstable. In other words, U is unstable if and only if there exists a positive number ε0 > 0 such that, for any neighborhood W of U , it holds that lim sup h(S(t)W, U ) > ε0 .

(6.8)

t→∞

That is, for any neighborhood W , there are a time sequence tn → ∞ and a sequence of points Un ∈ W such that d(S(tn )Un , U ) ≥ ε0 for all n. An invariant set A of (S(t), X, X) is called an attractor if some open neighborhood W of A is attracted by A, namely, A is an attractor if S(t)A ⊂ A for every t and lim h(S(t)W, A) = 0

t→∞

for some neighborhood W of A. An attractor A is called a global attractor of (S(t), X, X) if (1) A is a compact subset of X. (2) S(t)A = A for every t > 0. (3) A attracts every bounded subset of X, i.e., for every bounded B ⊂ X, lim h(S(t)B, A) = 0.

t→∞

(6.9)

Let B be a bounded subset of X which contains the global attractor A. Then, A = S(t)A ⊂ S(t)B, i.e., h(A, S(t)B) = 0 for any t > 0; hence, limt→∞ d(S(t)B, A) = 0, where d(·, ·) is the distance of two sets. By definition, if it exists, the global attractor A is always unique.

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1.2 ω-limit Sets and Global Attractor Consider a dynamical system (S(t), X, X). For each U0 ∈ X, we put  {S(τ )U0 ; t ≤ τ < ∞} (closure in X). ω(U0 ) = 0≤t<∞

This set is called the ω-limit set of the trajectory S(t)U0 . It is easy to see that U ∈ ω(U0 ) if and only if there exists a sequence of time tn → ∞ such that S(tn )U0 → U in X. Of course, ω(U0 ) may be an empty set. But, if ω(U0 ) = ∅, then it is a strictly invariant set of (S(t), X, X). The set ω(U0 ) consists of a single point U if and only if S(t)U0 → U as t → ∞; in this case, U is an equilibrium. Theorem 6.1 Let a trajectory {S(t)U0 ; 0 ≤ t < ∞} be relatively compact in X, that is, {S(t)U0 ; 0 ≤ t < ∞} is a compact set of X. Then, ω(U0 ) is a nonempty strictly invariant set and is a connected set. Proof By the compactness, the sequence S(n)U0 contains a convergent subsequence in X; hence, ω(U0 ) = ∅. It is always true that S(t)ω(U0 ) ⊂ ω(U0 ) for every t ≥ 0. Conversely, let U ∈ ω(U0 ) and t > 0. There exists a sequence tn → ∞ such that S(tn )U0 → U ; for tn ≥ t, S(tn )U0 = S(t)S(tn − t)U0 , and S(tn − t)U0 contains a convergent subsequence with a limit U  ∈ ω(U0 ); hence, U = S(t)U  . In this way, S(t)ω(U0 ) = ω(U0 ) for every t ≥ 0. Consider two open sets Oi (i = 1, 2) of X such that ω(U0 ) ⊂ O1 ∪ O2 and O1 ∩ O2 = ∅. Suppose that there exist points Ui ∈ ω(U0 ) ∩ Oi (i = 1, 2); then, there exist sequences sm → ∞ and tn → ∞ such that S(sm )U0 → U1 and S(tn )U0 → U2 , respectively; here, it is possible to assume that S(sm )U0 ∈ O1 and S(tn )U0 ∈ O2 . If sm < tn , then there exists a time τ ∈ (sm , tn ) such that S(τ )U0 ∈ / O1 ∪ O2 because of the connectedness of the arc {S(t)U0 ; sm ≤ t ≤ tn } of the trajectory S(t)U0 ; similarly, if tn < sm , then there is a time τ  ∈ (tn , sm ) such that S(τ  )U0 ∈ / O1 ∪ O2 ; therefore, we can take a sequence of times τk → ∞ for which S(τk )U0 ∈ / O1 ∪ O2 holds. Furthermore, this implies that there exists a point U ∈ ω(U0 ) such that U∈ / O1 ∪ O2 , which is a contradiction.  Assume that (S(t), X, X) possesses an global attractor A. Then, for any U0 ∈ X, ω(U0 ) = ∅ and ω(U0 ) ⊂ A. In fact, since {U0 } is a bounded set, the trajectory S(t)U0 is attracted by A; therefore, for each integer n ≥ 1, there exists a time tn ≥ n such that d(S(tn )U0 , A) ≤ n1 and that d(S(tn )U0 , Vn ) ≤ n2 with some Vn ∈ A. Since the sequence Vn contains a subsequence which converges to a limit V ∈ A, it follows that S(tn )U0 → V and hence V ∈ ω(U0 ). Moreover, it is easy to see that ω(U0 ) ⊂ A. The following theorem is very convenient for constructing global attractors. Theorem 6.2 Let (S(t), X, X) be a dynamical system. Assume that there exists a compact absorbing set B.

(6.10)

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257

Then, (S(t), X, X) has a global attractor A. In addition, if B is a connected set, then it is the same for A. Proof Since B itself is absorbed by B, there exists a time tB ≥ 0 such that S(t)B ⊂ B for every t ≥ tB . We then consider the set ˜ = X



S(t)B ⊂ B (closure in X).

tB ≤t<∞

For any bounded subset B of X, if t ≥ tB + tB , then S(t)B = S(t − tB )S(tB )B ⊂ ˜ This means that X ˜ is also an absorbing set. Since S(t) is a continS(t − tB )B ⊂ X. uous operator, we have     ˜ S(t) S(τ )B ⊂ S(t)S(τ )B ⊂ X, tB ≤τ <∞

tB ≤t<∞

˜ is invariant under S(t) for every t > 0. Finally, since X ˜ is a which shows that X ˜ ˜ is a closed subset of X and is contained in B, X is also a compact set of X. Thus, X compact, absorbing, and invariant set. ˜ X) is also a dynamical system. As X ˜ is an absorbing We know that (S(t), X, set, the original dynamical system (S(t), X, X) is reduced to the smaller dynamical ˜ X) in which the phase space is a compact subset of X. So, it now system (S(t), X, ˜ X). suffices to construct a global attractor for (S(t), X, We set  ˜ A= S(t)X. 0≤t<∞

˜ = S(t)S(t  − t)X ˜ ⊂ ˜ is a compact set; in addition, since S(t  )X For each t, S(t)X  ˜ ˜ S(t)X for t > t, S(t)X is a decreasing family of compact subsets. Therefore, A is a ˜ which is a compact set of X. nonempty subset of X By definition, U ∈ A if and only if there exist a time sequence tn tending ˜ such that S(tn )Un converges to U . Into ∞ and a sequence of points Un ∈ X ˜ such deed, if U ∈ A, then for each integer n ≥ 0, there exists an element Un ∈ X that U = S(n)Un ; hence, the assertion is clear. Conversely, if S(tn )Un → U with ˜ then for each fixed t ≥ 0, the sequence S(tn − t)Un (tn > t) tn → ∞ and Un ∈ X, ˜ therefore, from contains some subsequence which is convergent to a point V ∈ X; S(tn )Un = S(t)S(tn − t)Un it follows that U = S(t)V ; this means that U ∈ A. Using this fact, A can be shown to be a strictly invariant set. Let U ∈ A, and let tn → ∞ and Un ∈ B be such that S(tn )Un → U ; then, for any t ≥ 0, S(t)S(tn )Un = S(tn + t)Un , and this sequence converges to S(t)U ; therefore, S(t)U ∈ A, that is, S(t)A ⊂ A. Conversely, let U ∈ A and fix t > 0. There exist ˜ such that S(tn )Un → U . a time sequence tn → ∞ and a sequence Un of points of X We write S(tn )Un = S(t)S(tn − t)Un for tn ≥ t; then, there exists a subsequence ˜ i.e., U = S(t)V ; by the of S(tn − t)Un which is convergent to some point V ∈ X, verified fact, V must be in A; therefore, U ∈ S(t)A and hence A ⊂ S(t)A. Thus, we verify that S(t)A = A for every t > 0.

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˜ A) = 0. Indeed, suppose the contrary; Let us verify now that limt→∞ h(S(t)X, then, there would exist some α > 0 such that, for any T > 0, we can find some t ≥ T ˜ A) > α. This then means that there exist a time sequence tn → ∞ such that h(S(t)X, ˜ such that d(S(tn )Un , A) ≥ α. By the compactness of X, ˜ we and a sequence Un ∈ X ˜ can assume that S(tn )Un is convergent to some point V ∈ X. Clearly, d(V , A) ≥ α. Meanwhile, as noticed above, V must be a point of A; hence, d(V , A) = 0, a contradiction. Let us finally show that A is connected under the condition that B is. Suppose the contrary. Then, there would exist two open sets Oi (i = 1, 2) of X such that A ⊂ O1 ∪ O2 , O1 ∩ O2 = ∅, and A ∩ Oi = ∅ for i = 1, 2. Since A is compact, there exists an ε-neighborhood of A such that Wε (A) ⊂ O1 ∪ O2 . Applying (6.9) to B, we have that S(t)B ⊂ Wε (A) ⊂ O1 ∪ O2 for sufficiently large t. Since A ⊂ S(t)B, S(t)B ∩ Oi = ∅ for i = 1, 2. Therefore, for such t, S(t)B is neither connected. But this contradicts the fact that any image of a connected set by a continuous mapping is connected.  In the proof of this theorem, we have already obtained the following result. Corollary 6.1 Let (S(t), X, X) possess a compact absorbing set B. Then, there ˜ ⊂ B. The asymptotic behavior of exists a compact, absorbing, and invariant set X the original dynamical system (S(t), X, X) is therefore reduced to that of a smaller ˜ X) having a compact phase space. one (S(t), X, Remark 6.1 In the proof of Theorem 6.2, it is possible to construct the global at˜ In fact, A can be set directly by tractor A directly without using the set X. A = ω(B) =





S(s)B

(closure in X).

0≤t<∞ t≤s<∞

The proof of the theorem is quite analogous as before.

1.3 Stable and Unstable Manifolds Let (S(t), X, X) be a dynamical system. If there exists a function U : (−∞, 0] → X satisfying the relation S(t)U (−τ ) = U (t − τ )

for any 0 ≤ t ≤ τ < ∞

with U (0) = U0 , then the trace of the function U (−t) for 0 ≤ t < ∞ is called a negative trajectory ending at U0 . On the other hand, the trace of the function U (t) = S(t)U0 for 0 ≤ t < ∞ is called a positive trajectory starting from U0 . Of course, the positive trajectory exists for any starting point in X. Let U be an equilibrium of (S(t), X, X). The set   (6.11) M− (U ) = U0 ∈ X; lim S(t)U0 = U t→∞

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259

is called a stable manifold of U . In the meantime, the set  M+ (U ) = U0 ∈ X; ∃ a negative trajectory U (−t) ending at U0  such that lim U (−t) = U t→∞

(6.12)

is called an unstable manifold of U . From these definitions it is easily verified that M− (U ) (resp. M+ (U )) is an invariant (resp. strictly invariant) set, i.e., S(t)(M− (U )) ⊂ M− (U )

and S(t)(M+ (U )) = M+ (U )

for every t ≥ 0.

The proof is left to the reader. If U is asymptotically stable, say a neighborhood W of U satisfies (6.7), then obviously the stable manifold M− (U ) contains the neighborhood W . At the same time, M+ (U ) must consist of the unique element U . To the contrary, if M+ (U ) = {U }, then d(U0 , U ) = ε0 > 0 with some point U0 ∈ M+ (U ). Let U (−t) be a negative trajectory ending at U0 ; then, for any open neighborhood W of U , there exists an integer N such that U (−n) ∈ W for any n ≥ N ; consequently, d(S(n)U (−n), U ) = d(U0 , U ) = ε0 ; hence, (6.8) holds. In this way, the existence of elements of M+ (U ) different from U means that U is unstable. Fix time 0 < t ∗ < ∞ arbitrarily and put S = S(t ∗ ). Then, S is a continuous operator on X, and a discrete dynamical system (S n , X, X) is defined. Clearly, U is an equilibrium of (S n , X, X), too. Analogously, stable and unstable manifolds at U are defined for (S n , X, X). The set   W− (U ) = U0 ∈ X; lim S n U0 = U n→∞

is called a stable manifold of U in (S n , X, X). In the meantime, the set  W+ (U ) = U0 ∈ X; ∃{U−n }n=1,2,3,... ⊂ X, SU−n = U−n+1 for n ≥ 1 and  lim U−n = U n→∞

is called an unstable manifold of U in (S n , X, X). By these definitions, it is easy to see that S(W− (U )) ⊂ W− (U ) and S(W+ (U )) = W+ (U ). We can then verify the following coincidence theorem. Theorem 6.3 M− (U ) = W− (U )

and M+ (U ) = W+ (U ).

In particular, W∓ (U ) are independent of the choice of t ∗ . Proof It is clear that U0 ∈ M− (U ) implies U0 ∈ W− (U ). Hence, M− (U ) ⊂ W− (U ). Conversely, let U0 ∈ W− (U ). For any t > 0, let t = nt ∗ + r with

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integer n ≥ 0 and 0 ≤ r < t ∗ . Then, S(t)U0 − U = S(r)S n U0 − U ≤ sup0≤r≤t ∗ S(r)S n U0 − S(r)U . As t → ∞, n also tends to ∞; therefore, (6.4) yields limt→∞ S(t)U0 − U = 0, which means that U0 ∈ M− (U ). Hence, W− (U ) ⊂ M− (U ). The coincidence of unstable manifolds is also verified in a similar way. By definition, it is obvious that M+ (U ) ⊂ W+ (U ). Let next U0 ∈ W+ (U ). From a sequence {U−n }n=1,2,3,... such that SU−n = U−n+1 , we define the function U : (−∞, 0] → X as U (−t) = S((n + 1)t ∗ − t)U−(n+1) for t ∈ (nt ∗ , (n + 1)t ∗ ], where n = 0, 1, 2, . . . , and as U (0) = U0 . Then, it is seen that the relation S(t)U (−τ ) = U (t − τ ) holds for any 0 ≤ t ≤ τ . Indeed, if τ = 0, then t = 0; so, the relation is clear. Let nt ∗ < τ ≤ (n + 1)t ∗ with some integer n ≥ 0. If τ − t = mt ∗ with some integer m ≥ 0, then S(t)U (−τ ) = S(t)S((n + 1)t ∗ − τ )U−(n+1) = S((n + 1 − m)t ∗ )U−(n+1) = U−m = U (t − τ ). Otherwise, mt ∗ < τ − t < (m + 1)t ∗ with some integer m ≥ 0 (it is clear that n ≥ m); then, S(t)U (−τ ) = S(t + (n + 1)t ∗ − τ )U−(n+1) = S((m + 1)t ∗ + t − τ )S n−m U−(n+1) = S((m + 1)t ∗ + t − τ )U−(m+1) = U (t − τ ). Therefore, in both cases, the function U (−t) satisfies the relation S(t)U (−τ ) = U (t − τ ). We now verify the convergence limt→∞ U (−t) = U . Let nt ∗ < t ≤ (n + 1)t ∗ . Then, U (−t) − U = S((n + 1)t ∗ − t)U−(n+1) − S((n + 1)t ∗ − t)U ≤ sup0≤r≤t ∗ S(r)U−(n+1) − S(r)U . Since n → ∞ as t → ∞, it is observed by (6.4) that limt→∞ U (−t) − U = 0. Therefore, U0 ∈ M+ (U ). Hence, W+ (U ) ⊂  M+ (U ). Let O be any neighborhood of U in X. Localizing the stable and unstable manifolds W∓ (U ) in this neighborhood, we will consider the sets   W− (U ; O) = U0 ∈ O; S n U0 ∈ O for n ≥ 1 and lim S n U0 = U (6.13) n→∞

and

 W+ (U ; O) = U0 ∈ O; ∃{U−n }n=1,2,3,... ⊂ O, SU−n = U−n+1 for n ≥ 1 and  lim U−n = U , n→∞ (6.14) respectively. The set W− (U ; O) (resp. W+ (U ; O)) is called a local stable manifold (resp. a local unstable manifold) of U in a neighborhood O. By definition, it is clear that S(W− (U , O)) ⊂ W− (U , O))

and S(W+ (U ; O)) ⊃ W+ (U , O))

and that W− (U ; O) ⊂ W− (U )

and W+ (U ; O) ⊂ W+ (U ).

Moreover, it is verified without difficulty that W− (U ) =

∞  n=0

S −n (W− (U ; O)) and W+ (U ) =

∞  n=0

S n (W+ (U ; O)). (6.15)

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These facts, together with Theorem 6.3, then yield the following results. Theorem 6.4 We have M− (U ) =

∞ 

S −n (W− (U ; O)) and M+ (U ) =

n=0

∞ 

S n (W+ (U ; O)).

n=0

The global attractor, if it exists, always contains the unstable manifold. Theorem 6.5 Assume that (S(t), X, X) has a global attractor A. Then, M+ (U ) ⊂ A. Proof Let U0 ∈ M+ (U ). Then, there exists a negative trajectory U (−t) in X ending at U0 . The set B = {U (−t); 0 ≤ t < ∞} is a bounded set. Since A attracts B, we have limt→∞ h(S(t)B, A) = 0. In view of S(t)U (−t) = U (0) = U0 , we have d(U0 , A) = d(S(t)U (−t), A) ≤ h(S(t)B, A) → 0 as t → ∞. Hence, d(U0 , A) = 0, i.e., U0 ∈ A. 

1.4 Exponential Attraction An equilibrium U of (S(t), X, X) is said to be exponentially stable if there exists some neighborhood W of U which U attracts exponentially in the sense that h(S(t)W, U ) ≤ C0 e−kt ,

0 ≤ t < ∞,

(6.16)

with some constant C0 > 0 and some exponent k > 0. In a neighborhood of an exponentially stable equilibrium, one can find an invariant set to define a small dynamical system. Theorem 6.6 Let U be an exponentially stable equilibrium of (S(t), X, X). Then, there exists some neighborhood W of U such that W is an invariant set (therefore, (S(t), W, X) defines a new dynamical system) and W is attracted exponentially by U in the sense (6.16). Proof There is some open ball W = B(U ; r), r > 0, for which (6.16) holds. Then r −kT take a time T > 0 such that C0 e  = 2 . Then, for every t ≥ T , S(t)W ⊂ W . So, if we set W = 0≤t<∞ S(t)W = 0≤t≤T S(t)W , then S(t) maps W into itself for every 0 ≤ t < ∞; namely, W is invariant.  When 0 ≤ τ ≤ T , S(τ )W ⊂ W = 0≤t≤T S(t)W ; therefore, it holds that h(S(τ )W, U ) ≤ sup0≤t≤T C0 e−kt = C0 . Meanwhile, when τ ≥ T , S(τ )W =  0≤t≤T S(t + τ )W ⊂ S(τ − T ) 0≤t≤T S(t + T )W ⊂ S(τ − T )W ; therefore,  h(S(τ )W, U ) ≤ C0 ekT e−kτ . These show that U attracts W exponentially.

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We now introduce the notion of exponential attractor. Let a dynamical system (S(t), X, X) possess a global attractor A. A subset M ⊂ X is called an exponential attractor of (S(t), X, X) if M has the following properties: (1) M is a compact set of X such that A ⊂ M ⊂ X with finite fractal dimension dF (M). (2) M is an invariant set, i.e., S(t)M ⊂ M for every t > 0. (3) There exists some exponent k > 0 such that, for any bounded set B ⊂ X, it holds that h(S(t)B, M) ≤ CB e−kt

for 0 ≤ t < ∞

(6.17)

with a suitable constant CB > 0 depending on B. We must notice that, even if it exists, the exponential attractor is not determined uniquely in general. Usually, they exist as a family. To see this, assume that a dynamical system (S(t), X, X) possesses an exponential attractor M and that, for each t > 0, S(t) satisfies the Lipschitz condition S(t)U − S(t)V ≤ Lt U − V ,

U, V ∈ X.

(6.18)

For 0 < T < ∞, set MT = S(T )M. Then, MT is seen to be an exponential attractor of (S(t), X, X). In fact, since M is a compact set, its image MT by the continuous operator S(T ) is also a compact set of X. Since MT ⊂ M, MT has also a finite fractal dimension with dF (MT ) ≤ dF (M). By the semigroup property, MT is also an invariant set. Let us finally verify (6.17) for MT . For t ≥ T , it follows from (6.18) that d(S(t)U, S(T )V ) = d(S(T )S(t − T )U, S(T )V ) ≤ LT d(S(t − T )U, V ),

U ∈ X, V ∈ M.

Taking the infimum in the right-hand side with respect to V ∈ M, we have d(S(t)U, MT ) = d(S(t)U, S(T )M) ≤ LT d(S(t − T )U, M),

U ∈ X.

Let B be any bounded subset of X. Taking now the supremum in the left-hand side with respect to U ∈ B, we have h(S(t)B, MT ) ≤ LT h(S(t − T )B, M). Therefore, (6.17) implies that h(S(t)B, MT ) ≤ LT CB e−k(t−T ) = CB LT ekT e−kt ,

T ≤ t < ∞.

Meanwhile, since MT ⊂ M, h(S(t)B, MT ) ≤ h(S(t)B, M) + δ(M) ≤ CB + δ(M),

0 ≤ t ≤ T,

where δ(M) is the diameter of M. Therefore, MT also attracts B at an exponential rate. Hence, MT becomes a smaller exponential attractor than M.

2 Representation Theorem of Stable and Unstable Manifolds

263

Theorem 6.7 If (S(t), X, X) possesses an exponential attractor M and if S(t) satisfies (6.18), then in any ε neighborhood Wε (A) of the global attractor A, there exists another exponential attractor Mε , i.e., A ⊂ Mε ⊂ Wε (A). Proof Since M is a bounded set, limT →∞ h(S(T )M, A) = 0. So, for any ε > 0, there is a time T > 0 such that h(S(T )M, A) < ε, namely, MT = S(T )M ⊂ Wε (A). In addition, A = S(T )A ⊂ S(T )M = MT . 

2 Representation Theorem of Stable and Unstable Manifolds Throughout this section, we handle a dynamical system whose phase space is a whole space X, namely, a dynamical system of the form (S(t), X, X).

2.1 Discrete Case Let (S n , X, X) be a discrete dynamical system in a Banach space X, where S is a continuous nonlinear operator acting on X. We assume that S has a fixed point U ∈ X, so U is an equilibrium of (S n , X, X). We assume also that S is Fréchet ˜ of U and the derivative satisfies the Hölder differentiable in an open neighborhood O condition S  U − S  V ≤ D U − V α ,

˜ U, V ∈ O,

(6.19)

with some exponent 0 < α ≤ 1 and constant D > 0. When the spectrum σ (S  U ) of S  U does not intersect with the unit circle, i.e., σ (S  U ) ∩ {λ ∈ C; |λ| = 1} = ∅,

(6.20)

U is called a hyperbolic equilibrium. Let U be hyperbolic. According to Theorem 1.10, the space X and the operator S  U are decomposed in a unique way into a direct sum X = Xi ⊕ Xe of two closed subspaces and into a direct sum S  U = Si ⊕ Se in such a way that (1) Si and Se are bounded operators of Xi and Xe , respectively. (2) The spectra of Si and Se are given by σ (Si ) = σ (S  U ) ∩ {λ ∈ C; |λ| < 1} and σ (Se ) = σ (S  U ) ∩ {λ ∈ C; |λ| > 1}, respectively. The spectral radius of Si then satisfies r(Si ) = 1 − 2δi with some small δi > 0, and Se is invertible on Xe with r(Se −1 ) = 1 − 2δe with some small δe > 0. Using these numbers, we introduce the new norm on X by n Ui 0 = sup Si Ui /(1 − δi )n , Ui ∈ Xi , (6.21) n≥0

264

6 Dynamical Systems

−n Ue 0 = sup Se Ue /(1 − δe )n ,

Ue ∈ X e ,

(6.22)

n≥0

Ui + Ue 0 = max{ Ui 0 , Ue 0 },

U = Ui + Ue ∈ X.

(6.23)

It is immediate to verify that · 0 defines a norm on X which is equivalent to the original one · . Furthermore, as seen in (1.14), it is true that  S Ui ≤ (1 − δi ) Ui 0 , Ui ∈ Xi , i 0  −1 S Ue ≤ (1 − δe ) Ue 0 , Ue ∈ Xe . e 0 Under such a situation, we consider the local stable and unstable manifolds ˜ W∓ (U , O) defined by (6.13) and (6.14), respectively, in a neighborhood of O ⊂ O. When O is sufficiently small, we can establish the following representation theorem. Theorem 6.8 Let U be an equilibrium of (S n , X, X). Let S be Fréchet differentiable ˜ of U with derivative satisfying (6.19). Assume that U is in an open neighborhood O hyperbolic (see (6.20)) and let X be given by the continuous direct sum of two closed subspaces Xi and Xe with the properties described above. Then, for a sufficiently ˜ of U , there exists a one-to-one correspondence small open neighborhood O ⊂ O between the points of W− (U ; O) and the points of Oi ⊂ Xi such that W− (U ; O)  U0 = U + Ui + Ge (Ui ), Xi ⊃ Oi  Ui → Ge (Ui ) ∈ Xe , Ge ∈ C1+α

Ge (0) = 0, Ge (0) = 0,

when 0 < α < 1 and

Ge ∈ C1,1

when α = 1.

Similarly, there exists a one-to-one correspondence between the points of W+ (U ; O) and the points of Oe ⊂ Xe such that W+ (U ; O)  U0 = U + Gi (Ue ) + Ue , Xe ⊃ Oe  Ue → Gi (Ue ) ∈ Xi , Gi ∈ C1+α

Gi (0) = 0, Gi (0) = 0,

when 0 < α < 1 and

Gi ∈ C1,1

when α = 1.

Here, Oi and Oe are some suitable open neighborhoods of the origin of Xi and Xe , respectively. Proof Let us first prove the representation result for W+ (U ; O). For simplicity, we assume that U = 0, so that S0 = 0. Pi and Pe denote the projections from X onto Xi and Xe , respectively. By (6.23), we have Pi 0 ≤ 1 and Pe 0 ≤ 1. Let us consider the Banach space   X = U = {Un }n=0,1,2,... ⊂ X; lim Un = 0 n→∞

2 Representation Theorem of Stable and Unstable Manifolds

265

with norm U X = sup Un 0 . n

The zero vector of X is denoted by 0. We define the linear operator L : X → X by (LU )0 = Si Pi U1 + 0e , (LU )n = Si Pi Un+1 + Se

−1

Pe Un−1 ,

n ≥ 1.

In addition, the nonlinear operator R : X → X is defined by (RU )0 = Pi (S − S  0)U1 + 0e , (RU )n = Pi (S − S  0)Un+1 − Se

−1

Pe (S − S  0)Un ,

n ≥ 1.

Obviously, R is an operator of class C1+α (resp. C1,1 ) when 0 < α < 1 (resp. α = 1) with R0 = 0. We finally put Φ = L + R. Contractibility of Φ. Let O denote any neighborhood of 0 in X. From O we can always define a neighborhood O of 0 = (0, 0, 0, . . .) in X in such a way that O = {U = {Un }n ∈ X; Un ∈ O}. Then, O is said to be a corresponding neighborhood of O. Let us show that, if O is sufficiently small, then Φ is a contraction mapping from O into X. By (6.21) and (6.22), L satisfies LU X ≤ max{1 − δi , 1 − δe } U X ,

U ∈ X.

In the meantime, R can satisfy RU − RV X ≤ C( U X + V X )α U − V X ,

U , V ∈ O ,

in a suitable neighborhood O  , C > 0 being some constant. Indeed, this is readily verified by the following estimate: 1     (S − S 0)U − (S − S 0)V = {S (θ U + (1 − θ )V ) − S 0}(U − V ) dθ 0

≤ C sup θ U + (1 − θ )V α U − V ,

U, V ∈ O ;

0≤θ≤1

here O is a sufficiently small neighborhood of 0 in which any segment connecting ˜ appearing in (6.19). Theretwo points of O is contained in the neighborhood of O fore, we conclude that, if O is sufficiently small, then R satisfies RU − RV X ≤

1 min{δi , δe } U − V X , 2

in the corresponding neighborhood O.

U , V ∈ O,

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6 Dynamical Systems

In this way, Φ is shown to be a contraction on such a neighborhood. One-to-one correspondence. Let O be a sufficiently small open neighborhood of 0, and let O be the corresponding neighborhood on which Φ is a contraction. We can then apply the inverse mapping theorem, Theorem 1.14, to I − Φ, where I is the identity operator of X, to conclude that I − Φ is a homeomorphism from O onto some open neighborhood O  of 0, both I − Φ and (I − Φ)−1 being of class C1+α (resp. C1,1 ) when 0 < α < 1 (resp. α = 1). Let U0 ∈ W+ (0; O), and let {Un }n=0,1,2,... be a sequence in O such that SUn = Un−1 for n ≥ 1 and limn→∞ Un = 0. Then, U = {Un }n is an element of O. So, let (I − Φ)U = U  = {Un }n . Then, by definition, Pi (U0 − SU1 ) + Pe U0 = U0 Pi (Un − SUn+1 ) − Se

−1

for n = 0,

(6.24)

Pe (Un−1 − SUn ) = Un

for n ≥ 1.

(6.25)

Since SUn = Un−1 , it follows that U0 = Pe U0 ∈ Xe

and Un = 0 for n ≥ 1.

Conversely, let Ue ∈ Xe and U  = (Ue , 0, 0, . . .) ∈ O  . Then, there exists an element U = {Un }n ∈ O such that (I − Φ)U = U  . As relations (6.24) (U0 = Ue ) and (6.25) (Un = 0) are still valid, it follows that Pi (U0 − SU1 ) = 0 and Pe U0 = Ue , Pi (Un − SUn+1 ) = 0

and Pe (Un−1 − SUn ) = 0 for n ≥ 1.

This indeed shows that U0 ∈ W+ (0; O) and Pe U0 = Ue . Set Oe = {Ue ∈ Xe ; (Ue , 0, 0, . . .) ∈ O  }. Obviously, Oe is an open neighborhood of the origin 0e of Xe . As shown above, there is one-to-one correspondence between the points in W+ (0; O) and the points in Oe . Mapping Gi . We observe more precisely that the point Ue ∈ Oe corresponds to the point U0 = Ue + Gi (Ue ) ∈ W+ (0; O) in such a way that Gi (Ue ) = Pi Π0 (I − Φ)−1 (Ue , 0, 0, . . .),

Ue ∈ O e ,

where Π0 is the projection from X to X such that Π0 ({Un }n ) = U0 . Clearly, Gi is a C1+α (resp. C1,1 ) mapping when 0 < α < 1 (resp. α = 1) from Oe into Xi with Gi (0) = 0. Since (I − Φ) 0 = I − L, we have {(I − Φ)−1 } 0 = (I − L)−1 . Therefore, (Gi 0)Ue = Pi Π0 (I − L)−1 (Ue , 0, 0, . . .),

Ue ∈ X e .

2 Representation Theorem of Stable and Unstable Manifolds

267

We easily verify that (I − L)−1 (Ue , 0, 0, . . .) = {Un }n implies Pi Un = Si Pi Un+1 for n ≥ 0, and hence Pi Un = 0 for every n. This then shows that Gi 0 = 0. The representation result for W− (0; O) is also deduced in a quite analogous way. We shall use a contraction mapping Ψ instead of Φ which is given by Ψ = M + Q. Here, M : X → X is the linear operator given by (MU )0 = 0i + Se

−1

Pe U1 ,

(MU )n = Si Pi Un−1 + Se

−1

Pe Un+1 ,

n ≥ 1,

and Q : X → X is the nonlinear operator given by (QU )0 = 0i − Se

−1

Pe (S − S  0)U0 ,

(QU )n = Pi (S − S  0)Un−1 − Se

−1

Pe (S − S  0)Un ,

n ≥ 1.

The equivalent condition to (I − Ψ )U = U  is written as Pi U0 + Se

−1

Pe (SU0 − U1 ) = U0

−Pi (SUn−1 − Un ) + Se

−1

The rest of proof is left to the reader.

for n = 0,

Pe (SUn − Un+1 ) = Un

for n ≥ 1. 

Remark 6.2 Since X is the continuous direct sum of Xi and Xe , we can identify X as the product space Xi × Xe by the natural isomorphism U = Ui + Ue → (Ui , Ue ). If we use this identification, then the points Ui + Ge (Ui ) correspond to (Ui , Ge (Ui )). The theorem obtained above then shows that the local stable manifold W− (U ; O) can be represented as the sum of U and the graph of the mapping Ge : Oi → Xe . Therefore, W− (U ; O) is tangent to U + Xi × {0} at the equilibrium U . It is the same for the local unstable manifold W+ (U ; O). That is, W+ (U ; O) ⊂ Xi × Xe can be represented as the sum of U and the graph of the mapping Gi : Oe → Xi . Therefore, W+ (U ; O) is tangent to U + {0} × Xe at the U . ˜ This theorem shows that, for sufficiently small open neighborhoods O ⊂ O, W− (U ; O) and W+ (U ; O) are C1+α (resp. C1,1 ) manifolds, when 0 < α < 1 (resp. α = 1), with dimensions dim Xi and dim Xe , respectively. We also remark that the whole stable and unstable manifolds W− (U ) and W+ (U ) are given by the images of these localized manifolds by the operators S −n and S n , respectively, due to (6.15).

2.2 Continuous Case We shall now consider a continuous dynamical system (S(t), X, X), where S(t) is a continuous semigroup acting on the whole space X. We assume that, for some

268

6 Dynamical Systems

fixed 0 < t ∗ < ∞, S = S(t ∗ ) is Fréchet differentiable in an open neighborhood ˜ of U and its derivative S  U satisfies the same condition as in the discrete case O (6.19). In addition, we assume that U is a hyperbolic equilibrium of the discrete dynamical system (S n , X, X) (S = S(t ∗ )) in the sense described above. Let Xi = Xi (U ) and Xe = Xe (U ) be invariant subspaces of S  U such that X = Xi + Xe and the parts (S  U )|Xi and (S  U )|Xe of S  U have their spectra in {λ ∈ C; |λ| < 1} and in {λ ∈ C; |λ| > 1}, respectively. The following result is then a direct consequence of Theorem 6.8. Theorem 6.9 Let U be an equilibrium of (S(t), X, X). Let, for some fixed ˜ of U with 0 < t ∗ < ∞, S(t ∗ ) be Fréchet differentiable in an open neighborhood O derivative satisfying (6.19), and let U be hyperbolic with invariants subspaces Xi and Xe of the derivative [S(t ∗ )] U . Then, for sufficiently small open neighborhoods ˜ the local stable and unstable manifolds in O for (S(t ∗ )n , X, X) can be O ⊂ O, represented in such a way that W− (U ; O)  U0 = U + Ui + Ge (Ui ), Xi ⊃ Oi  Ui → Ge (Ui ) ∈ Xe , Ge ∈ C1+α

Ge (0) = 0, Ge (0) = 0,

when 0 < α < 1 and Ge ∈ C1,1

when α = 1

and W+ (U ; O)  U0 = U + Gi (Ue ) + Ue , Xe ⊃ Oe  Ue → Gi (Ue ) ∈ Xi , Gi ∈ C1+α

Gi (0) = 0, Gi (0) = 0,

when 0 < α < 1 and Gi ∈ C1,1

when α = 1,

respectively. Here, Oi and Oe are some suitable open neighborhoods of the origin of Xi and Xe , respectively. ˜ Theorem 6.9 shows that, for sufficiently small open neighborhoods O ⊂ O, 1+α 1,1 (resp. C ) manifolds when 0 < α < 1 (resp. W− (U ; O) and W+ (U ; O) are C α = 1), with dimensions dim Xi and dim Xe , respectively. In addition, as shown by Theorem 6.4, the whole stable and unstable manifolds M− (U ) and M+ (U ) are given by the images of these localized manifolds by the operators S(t ∗ )−n and S(t ∗ )n .

3 Exponentially Stable Equilibria Consider a dynamical system (S(t), X, X) in a Banach space X. Let U ∈ X be an equilibrium. We assume that S(t) is a contraction at the equilibrium in the following sense. There exist a time t ∗ > 0 and a constant 0 < δ < 1 such that S(t ∗ )U − U = S(t ∗ )U − S(t ∗ )U ≤ δ U − U , for some R > 0.

U ∈ B(U ; R) ∩ X, (6.26)

4 Exponential Attractors

269

Theorem 6.10 For an equilibrium U , let (6.26) be satisfied with some t ∗ > 0 and 0 < δ < 1. Then, there exists a radius r > 0 such that 1

S(t)U − U ≤ Rδ −1 e−( t ∗ log δ

−1 )t

,

U ∈ B(U ; r) ∩ X.

Proof We first notice that there exists an r > 0 such that {S(τ )U ; 0 ≤ τ ≤ t ∗ , U − U ≤ r} ⊂ B(U ; R). Indeed, suppose the contrary. Then, for any integer n ≥ 1, there exist some τn , 0 ≤ τn ≤ t ∗ , and some Un ∈ X such that Un − U ≤ n1 and S(τn )Un − U > R. We can then assume that τn → τ and Un → U as n → ∞. From (6.1) it follows that S(τ )U − U ≥ R, which clearly contradicts S(τ )U = U . Let next U ∈ B(U ; r) ∩ X. For any t > 0, there exists an integer n ≥ 0 such that t = nt ∗ + τ with some 0 ≤ τ < t ∗ . Then, S(τ )U − U ≤ R. Furthermore, using (6.26) repeatedly, we verify that S(t ∗ )n S(τ )U − U ≤ δ n R. Since S(t)U = S(t ∗ )n S(τ )U , we conclude that 1

S(t)U − U ≤ Rδ − t ∗ δ t ∗ ≤ Rδ −1 e−( t ∗ log δ τ

t

−1 )t



.

4 Exponential Attractors 4.1 Contraction Semigroups Consider a dynamical system (S(t), X, X) in a Banach space X, where the phase X is a closed bounded subset of X. We assume for some t ∗ > 0 that S(t ∗ ) is a contraction operator from X into itself, i.e., S(t ∗ )U − S(t ∗ )V ≤ δ U − V ,

U, V ∈ X,

(6.27)

with some exponent 0 < δ < 1. Let us consider under (6.27) a discrete dynamical system (S n , X, X), where S = S(t ∗ ). By virtue of the fixed point theorem for contraction mappings, Theorem 1.11, there exists a unique equilibrium U of (S n , X, X). Furthermore, for any integer n ≥ 0, S n U − U = S n U − S n U ≤ δ n U − U ,

U ∈ X.

We then verify that U is an equilibrium of (S(t), X, X), too. Indeed, for any t > 0, SS(t)U = S(t ∗ + t)U = S(t)SU = S(t)U ; that is, S(t)U is also a fixed point of S; by the uniqueness of fixed point, S(t)U = U . Furthermore, let t = nt ∗ + τ with integer n ≥ 0 and 0 ≤ τ < t ∗ . Then, S(t)U − U = S n S(τ )U − S n U ≤ Rδ

t−τ t∗

1

= Rδ −1 e−( t ∗ log δ

where R = δ(X) = supU,V ∈X U − V is the diameter of X. We have thus obtained the following result.

−1 )t

,

U ∈ X,

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6 Dynamical Systems

Theorem 6.11 Let X be a bounded closed subset of X, and let (6.27) be satisfied. Then, (S(t), X, X) has a unique equilibrium U which attracts every trajectory at the 1 −1 exponential rate d(S(t)X, U ) ≤ Rδ −1 e−( t ∗ log δ )t , where R is the diameter of X.

4.2 Compact Perturbation of Contraction Operator, I Let (S(t), X, X) be a dynamical system in a Banach space X, where X is a closed bounded subset of X. In the subsequent two subsections, we shall consider compact perturbations of condition (6.27) by two ways. In the first way, the semigroup S(t ∗ ) is divided into two parts in which the first part is a contraction like (6.27), and the second one is a Lipschitz continuous compact operator (see (6.29)). In the second way, we divide the norm S(t ∗ )U − S(t ∗ )V X into two parts in which the first part is again a contraction and the second one is estimated by the norm of a Lipschitz continuous compact operator (see (6.38)). In this subsection, we assume for some fixed t ∗ > 0 that S(t ∗ ) is divided into two parts as follows. There are two operators S0 : X × X → X and K : X × X → Z, where Z is a second Banach space compactly embedded in X and such that for each pair (U, V ) ∈ X2 , it holds that S(t ∗ )U − S(t ∗ )V = S0 (U, V ) + K(U, V ). In addition, S0 and K satisfy S0 (U, V ) X ≤ δ U − V X ,

U, V ∈ X,

(6.28)

K(U, V ) Z ≤ L U − V X ,

U, V ∈ X,

(6.29)

with some constants 0 ≤ δ < 12 and L > 0, respectively. When conditions (6.28)– (6.29) are satisfied, S(t ∗ ) is said to be a compact perturbation of contraction operator of type I. Putting S = S(t ∗ ), let us consider the discrete dynamical system (S n , X, X). We can prove the following theorem. Theorem 6.12 Let (6.28) and (6.29) be satisfied with 0 ≤ δ < 12 . Let θ be any ∗ exponent such that 0 < θ < 1−2δ 2L . Then, there exists an exponential attractor Mθ n for (S , X, X) with the following properties: (1) M∗θ contains a global attractor A∗ of (S n , X, X). (2) M∗θ is a compact set of X with finite fractal dimension dF (M∗θ ) ≤ log Nθ / log

1 . aθ

(6.30)

(3) M∗θ is invariant under S, i.e., SM∗θ ⊂ M∗θ . (4) h(S n X, M∗θ ) ≤ Raθn for all integers n ≥ 0. Here, R = δ(X) is the diameter of X, 0 < aθ < 1 is the exponent given by aθ = 2(δ + θ L), and Nθ is the minimal number of balls of X with radius θ which cover Z the closed unit ball B (0; 1) of Z.

4 Exponential Attractors

271

Proof The proof is divided into three steps. Step 1. For n = 0, 1, 2, . . . , let us construct inductively a finite covering of S n X such that n

SnX ⊂

Nθ 

B(Wn,i ; Ra n ),

where a = aθ ,

(6.31)

i=1

with centers Wn,i ∈ S n X, 1 ≤ i ≤ Nθn . For n = 0, it is clear that S 0 X = X ⊂ B(W0 ; R) with an arbitrarily fixed W0 ∈ X. So, (6.31) is trivial. Assume that the covering (6.31) is defined for n − 1. Then, we have Nθn−1

S X = S(S n

n−1

X) ⊂



S(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X).

i=1

So, it suffices to cover each set S(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X) by Nθ closed balls with centers in S n X and radius Ra n . We consider the image K(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X, Wn−1,i ). From (6.29) we see that Z

K(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X, Wn−1,i ) ⊂ B (0; LRa n−1 ). Then, by the compactness of closed bounded balls of Z in X, the last ball can be covered by Nθ closed balls B(V˜n−1,j ; θ LRa n−1 ), 1 ≤ j ≤ Nθ , of X with centers V˜n−1,j ∈ X and radius θ LRa n−1 . Therefore, we obtain that K(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X, Wn−1,i ) ⊂

Nθ 

B(V˜n−1,j ; θ LRa n−1 ).

j =1

Furthermore, K(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X, Wn−1,i ) + SWn−1,i ⊂

Nθ 

B(V˜n−1,j + SWn−1,i ; θ LRa n−1 ).

(6.32)

j =1

Let U ∈ B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X, and let SU − SWn−1,i = S0 (U, Wn−1,i ) + K(U, Wn−1,i ). By (6.32), there is some j such that K(U, Wn−1,i ) + SWn−1,i ∈ B(V˜n−1,j + SWn−1,i ; θ LRa n−1 ).

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6 Dynamical Systems

Since SU = S0 (U, Wn−1,i )+K(U, Wn−1,i )+SWn−1,i and since S0 (U, Wn−1,i ) X ≤ δ U − Wn−1,i X ≤ δRa n−1 due to (6.28), it follows that SU ∈ B(V˜n−1,j + SWn−1,i ; (δ + θ L)Ra n−1 ). Hence, S(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X) ⊂

Nθ 

B(V˜n−1,j + SWn−1,i ; (δ + θ L)Ra n−1 ).

j =1

We are here allowed to assume that S(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X) ∩ B(V˜n−1,j + SWn−1,i ; (δ + θ L)Ra n−1 ) = ∅ for every j , since, if not for some j , we can exclude such balls from the covering. So, we can choose for each j , a point Vn−1,i,j such that Vn−1,i,j ∈ B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X, SVn−1,i,j ∈ B(V˜n−1,j + SWn−1,i ; (δ + θ L)Ra n−1 ).

(6.33)

Substituting for each j , the central point SVn−1,i for V˜n−1,j +SWn−1,i and doubling the radius, we obtain the covering S(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X) ⊂

Nθ 

B(SVn−1,i,j ; aθ Ra n−1 ).

(6.34)

j =1

From (6.33) we notice that SVn−1,i,j ∈ S n X. It is of course possible to construct the same covering as (6.34) for all other balls B(Wn−1,i ; Ra n−1 ), 1 ≤ i ≤ Nθn−1 . The desired covering (6.31) for n is thus obtained just locating central points as   Wn,i ; 1 ≤ i ≤ Nθn = SVn−1,i,j ; 1 ≤ i ≤ Nθn−1 , 1 ≤ j ≤ Nθ ⊂ S n X.

Step 2. Let

 P = Wn,i ; 0 ≤ n < ∞, 1 ≤ i ≤ Nθn

be a collection of all central points of the covering (6.31) and set M∗θ in such a way that M∗θ =

∞ 

S n P.

n=0

Then, as will be shown, M∗θ is an exponential attractor for (S n , X, X). Let us verify in this step that Mθ has a finite fractal dimension.

4 Exponential Attractors

273

We will first cover M∗θ by closed balls with radius Ra n . We know by (6.31) that is covered by Nθn balls with radius Ra n . We then decompose M∗θ as 

n−1   ∗ p m S Wm,i ; 0 ≤ m ≤ n − 1, 1 ≤ i ≤ Nθ Mθ =

SnX

p=0



n−1 

Sp



Wm,i ; n ≤ m < ∞, 1 ≤ i

≤ Nθm

 ∞  

 SpP

.

p=n

p=0

Then,

n−1 

 ∞     S p Wm,i ; n ≤ m < ∞, 1 ≤ i ≤ Nθm S p P ⊂ S n X. p=n

p=0

In the meantime, the number of points of Nθm } is calculated by

n−1

p=0 S

p {W

m,i ;

0 ≤ m ≤ n − 1, 1 ≤ i ≤

 n(Nθn − 1)  nNθn < . n 1 + Nθ + Nθ2 + · · · + Nθn−1 = Nθ − 1 Nθ − 1 N n+1 +(n−1)N n

Therefore, M∗θ is covered at least by θ Nθ −1 θ balls of radius Ra n . Let next 0 < ε < R be any number. By N (ε) we denote the minimal number of balls with radius ε which cover M∗θ . Let Ra n+1 ≤ ε ≤ Ra n . Then, we have N (ε) ≤ N (Ra n+1 ) ≤

Nθn+2 + nNθn+1 . Nθ − 1

(6.35)

In particular, this means that M∗θ is a precompact set of X. As M∗θ is closed, M∗θ is a compact set of X. Furthermore, it follows from (6.35) and the estimate log Ra1 n ≤ log 1ε that log N (ε) log 1ε

≤

log(Nθn+2 + nNθn ) − log(Nθ − 1) n log a1 − log R1

.

Consequently, we obtain that dF (M∗θ ) = lim sup

log N (ε)

ε→0

log 1ε

≤

log Nθ log a1

.

Step 3. Let us verify that M∗θ enjoys the other properties as an exponential attractor. Since S is a continuous mapping, we have S

∞  n=0

SnP ⊂ S

∞  n=0

SnP ⊂

∞  n=0

S n P.

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6 Dynamical Systems

Hence, M∗θ is an invariant set of S. If U ∈ S n X, we see from (6.31) that   d U, M∗θ ≤ d(U, P) ≤ Ra n . Consequently,

  h S n X, M∗θ ≤ Ra n .

(6.36)

∗ attracts the phase space X at an exponential rate. This shows that M θ ∗ n X. Since X ⊃ M∗ and since M∗ is an invariant comWe set A = ∞ n=0 S  θ ∞θ n ∗ n pact  set, it follows that ∞ n=0 S X ⊃ n=0 S Mθ = ∅. On the other hand, let n n+m U U∈ ∞ n+m = n=0 S X. Then, for any integers n, m ≥ 1, we can write U = S ∗ n m m S (S Un+m ) with Un+m ∈ X; since d(S Un+m , Mθ ) → 0 as m → ∞, we can assume that S m Un+m → V ∈ M∗θ as m → used the compactness of M∗θ ); ∞∞ (we ∞ ∗ n n n therefore, U =S V ∈ S Mθ . Hence, n=0 S X ⊂ n=0 S n M∗θ . So that A∗ = ∞ ∞ n n ∗ ∗ n=0 S X = n=0 S Mθ . In particular, by Theorem 6.2, A is a strictly invari-

ant compact set. Let us finally verify that A∗ attracts the phase space X. Let · X ≤ c · Z . Then, (6.28) and (6.29) give SU − SV X ≤ (δ + cL) U − V X , By (1.2),

U, V ∈ X.

(6.37)

    h(S m+n X, A∗ ) ≤ h S m+n X, S m M∗θ + h S m M∗θ , A∗ .

Here, (6.36) and (6.37) yield that     h S m+n X, S m M∗θ ≤ (δ + cL)m h S n X, M∗θ ≤ (δ + cL)m Ra n . Therefore,

  h(S m+n X, A∗ ) ≤ R(δ + cL)m a n + h S m M∗θ , A∗ .

Letting n → ∞, we have lim supn→∞ h(S n X, A∗ ) ≤ h(S m M∗θ , A∗ ). By Theorem 6.2, we already know that limm→∞ h(S m M∗θ , A∗ ) = 0. Hence, it follows that limn→∞ h(S n X, A∗ ) = 0. Thus, we have shown that A∗ is a global attractor of (S n , X, X). 

4.3 Compact Perturbation of Contraction Operator, II Let (S(t), X, X) be a dynamical system in a Banach space X, where X is a closed bounded subset of X. In this subsection, we assume for some fixed t ∗ > 0 that S = S(t ∗ ) is a compact perturbation of a contraction operator in the sense that SU − SV X ≤ δ U − V X + KU − KV X ,

U, V ∈ X.

(6.38)

4 Exponential Attractors

275

Here, δ is a constant such that 0 ≤ δ < 12 , and K is an operator from X into a second Banach Z which is compactly embedded in X and satisfies the Lipschitz condition KU − KV Z ≤ L U − V X ,

U, V ∈ X,

(6.39)

with some L > 0. When conditions (6.38)–(6.39) are satisfied, S(t ∗ ) is said to be a compact perturbation of contraction operator of type II. We are then able to prove a similar result as Theorem 6.12 for the discrete dynamical system (S n , X, X). Theorem 6.13 Let (6.38) and (6.39) be satisfied with 0 ≤ δ < 12 . Let θ be any ∗ exponent such that 0 < θ < 1−2δ 2L . Then, there exists an exponential attractor Mθ n for (S , X, X) having the same properties as those announced in Theorem 6.12. Proof We can follow the similar procedure as in the proof of Theorem 6.12. Let us construct under (6.38)–(6.39) the same covering as before. For n = 0, 1, 2, . . . , there is a covering of S n X such that n

Nθ 

S X⊂ n

B(Wn,i ; Ra n ),

where a = aθ ,

(6.40)

i=1

with centers Wn,i ∈ S n X, 1 ≤ i ≤ Nθn . Here, as before, R = δ(X) is the diameter of X, 0 < aθ < 1 is the exponent given by aθ = 2(δ + θ L), and Nθ is the minimal Z number of balls of X with radius θ which cover the closed unit ball B (0; 1) of Z. For n = 0, it is clear that S 0 X = X ⊂ B(W0 ; R) with arbitrarily fixed W0 ∈ X. Assume that the covering (6.40) is defined for n − 1. Then, we have Nθn−1

S X = S(S n

n−1

X) ⊂



S(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X).

i=1

So, it suffices to cover each set S(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X) by Nθ closed balls with centers in S n X and the radius Ra n . From (6.39) we see that Z

K(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X) ⊂ B (KWn−1,i ; LRa n−1 ). Then, by the compactness of closed bounded balls of Z in X, the last ball can be covered by Nθ closed balls of X in such a way that Z

B (KWn−1,i ; LRa n−1 ) ⊂

Nθ  j =1

B(V˜n−1,i,j ; θ LRa n−1 )

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6 Dynamical Systems

with centers V˜n−1,i,j ∈ X, 1 ≤ j ≤ Nθ , and radius θ LRa n−1 . Therefore, K(B(Wn−1,i ; Ra

n−1

)∩S

n−1

X) ⊂

Nθ 

B(V˜n−1,i,j ; θ LRa n−1 ).

(6.41)

j =1

We are here allowed to assume that K(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X) ∩ B(V˜n−1,i,j ; θ LRa n−1 ) = ∅ for every j , since, if not for some j , we can exclude these balls from the covering. So, we can choose for each j , a point Vn−1,i,j such that Vn−1,i,j ∈ B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X, KVn−1,i,j ∈ B(V˜n−1,i,j ; θ LRa n−1 ).

(6.42)

Therefore, from (6.41) it is deduced that K(B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X) ⊂

Nθ 

B(KVn−1,i,j ; 2θ LRa n−1 ).

j =1

Let now U ∈ B(Wn−1,i ; Ra n−1 ) ∩ S n−1 X. Then, there is some j such that KU ∈ B(KVn−1,i,j ; 2θ LRa n−1 ). As a consequence, it follows from (6.38) that SU − SVn−1,i,j X ≤ δ U − Vn−1,i,j X + KU − KVn−1,i,j X ≤ δ U − Vn−1,i,j X + 2θ LRa n−1 . In addition, by (6.42), U − Vn−1,i,j X ≤ U − Wn−1,i X + Wn−1,i,j − Vn−1,i,j X ≤ 2Ra n−1 , so that SU − SVn−1,i,j X ≤ 2δRa n−1 + 2Lθ Ra n−1 = Ra n . Hence, it holds that S(B(Wn−1,i ; Ra

n−1

)∩S

n−1

X) ⊂

Nθ 

B(SVn−1,i,j ; Ra n ).

(6.43)

j =1

We observe that SVn−1,i,j ∈ S n X due to (6.42). Covering of the form (6.43) can of course be constructed for all other balls B(Wn−1,i ; Ra n−1 ), 1 ≤ i ≤ Nθn−1 . Therefore, the desired covering (6.40) for n is obtained by locating central points as   Wn,i ; 1 ≤ i ≤ Nθn = SVn−1,i,j ; 1 ≤ i ≤ Nθn−1 , 1 ≤ j ≤ Nθ ⊂ S n X. We here recall that in Steps 2 and 3, the proof of Theorem 6.12 was carried out only on the basis of the covering (6.31) and conditions (6.28)–(6.29) were never used. As (6.40) has been established, we shall repeat the same arguments as in Steps 2 and 3 of the proof of the previous theorem to complete our proof. 

4 Exponential Attractors

277

4.4 Squeezing Property In this subsection, let X be a Hilbert space. We consider a dynamical system (S(t), X, X) in X, where X is a compact subset of X. For some fixed 0 < t ∗ < ∞, assume that S = S(t ∗ ) satisfies the Lipschitz condition SU − SV ≤ L U − V ,

U, V ∈ X,

(6.44)

and the following condition: either SU − SV ≤ δ U − V

(6.45)

or (1 − P )(SU − SV ) ≤ P (SU − SV )

(6.46)

holds for any pair (U, V ) in X × X with some fixed constant 0 ≤ δ < and some fixed orthogonal projection P from X onto a finite-dimensional subspace Z in X, say dim Z = N < ∞. Especially, the second condition that either (6.45) or (6.46) is valid is called a squeezing property for S. Since 1 4

SU − SV 2 = P (SU − SV ) 2 + (1 − P )(SU − SV ) 2 , it is easy to observe from the squeezing property that √ SU − SV ≤ δ U − V + 2 P (SU − SV ) for any pair (U, V ) ∈ X × X. In addition, (6.44) implies that P (SU − SV ) ≤ L U − V ,

U, V ∈ X.

As the finite-dimensional space Z is compactly embedded in X, we notice that the squeezing property (6.45)–(6.46), together with (6.44), √ implies that S fulfills the compact perturbation condition (6.38)–(6.39) with K = 2P S. As a consequence √ , there exists an exponential attractor M∗ for of Theorem 6.13, for 0 < θ < 1−2δ θ 2 2L the discrete dynamical system (S n , X, X). It is, however, known that we can acquire a more precise dimension estimate for M∗θ than (6.30) without using the covering number Nθ . Theorem 6.14 Let X be a Hilbert space, and let X be a compact phase space. Let (6.44) and (6.45)–(6.46) be satisfied with 0 ≤ δ < 14 . Then, for any 0 < θ < 1 − 2δ, there exists an exponential attractor M∗θ for (S n , X, X) with fractal dimension     3L 1 ∗ +1 log , 1 dF (Mθ ) ≤ N max log θ aθ and the same properties as (1), (3), and (4) announced in Theorem 6.12, where aθ = 2δ + θ .

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6 Dynamical Systems

The proof of this theorem requires us more refined techniques than those for proving Theorem 6.13. We refer the reader to Eden, Foias, Nicolaenko, and Temam [EFNT94, Theorem 2.1]. We here remark only the following fact. Remark 6.3 The squeezing property implies a sharper covering than (6.31) or (6.40). Let W ∈ X and r > 0. We are concerned with the image S(B(W ; r) ∩ X) of a closed ball B(W ; r) by S. So, let U ∈ B(W ; r) ∩ X. If (6.45) takes place for the pair (U, W ), then SU ∈ B(SW ; δr). On the other hand, if (6.46) takes place, then it follows that SU − SW 2 ≤ 2 Re(SU − SW, P (SU − SW )). Therefore, SU − SW P (SU − SW ) ≤ 2 Re(SU − SW, P (SU − SW )).

(6.47)

In view of this, we consider the cone of X defined by  C = U˜ ∈ X − {0};

max

V ∈Z, V Z =1

Re

 1 (U˜ , V ) ≥ . U˜ V 2

Then, (6.47) shows that SU − SW ∈ C. Hence, (6.46), together with (6.44), implies that SU ∈ (SW + C) ∩ B(SW ; Lr). We thus observed that (6.44) and (6.45)–(6.46) imply that S(B(W ; r) ∩ X) ⊂ B(SW ; δr) ∪ [(SW + C) ∩ B(SW ; Lr)].

4.5 Continuous Dynamical System Let us return to the continuous dynamical system (S(t), X, X) in a Banach space X, where X is a closed bounded subset of X. We assume for some fixed t ∗ > 0 that S(t ∗ ) satisfies the compact perturbation condition either (6.28)–(6.29) or (6.38)– (6.39). In addition, we assume that G(t, U0 ) = S(t)U0 satisfies a Lipschitz condition on [0, t ∗ ] × X, i.e., G(t, U ) − G(s, V ) ≤ L1 (|t − s| + U − V ),

t, s ∈ [0, t ∗ ], U, V ∈ X. (6.48)

Then, on the basis of the exponential attractors M∗θ for (S(t ∗ )n , X, X), we can readily construct exponential attractors for (S(t), X, X). Theorem 6.15 Let (S(t), X, X) be a dynamical system in a Banach space X with closed bounded phase space X. For some fixed t ∗ > 0, let S(t ∗ ) satisfy either (6.28)– (6.29) or (6.38)–(6.39), and let (6.48) be satisfied. Then, for any 0 < θ < 1−2δ 2L , there exists an exponential attractor Mθ for (S(t), X, X) with the following properties: (1) Mθ contains a global attractor A of (S(t), X, X).

4 Exponential Attractors

279

(2) Mθ is a compact set of X with finite fractal dimension   dF (Mθ ) ≤ dF M∗θ + 1. (3) Mθ is an invariant set, i.e., S(t)Mθ ⊂ Mθ for every 0 < t < ∞. −1 1 (4) h(S(t)X, Mθ ) ≤ Raθ−1 e−( t ∗ log aθ )t for every 0 < t < ∞. Here, M∗θ is the exponential attractor of (S(t ∗ )n , X, X) constructed in Theorem 6.12 or Theorem 6.13, R = δ(X) is the diameter of X, and aθ = 2(δ + θ L). Proof Let M∗θ be the exponential attractor of (S(t ∗ )n , X, X) constructed by Theorem 6.12 or 6.13. We consider the image G([0, t ∗ ] × M∗θ ) of a set [0, t ∗ ] × M∗θ by G and denote it by Mθ , i.e.,    S(τ )M∗θ . Mθ = G [0, t ∗ ] × M∗θ = 0≤τ ≤t ∗

Our goal is then to show that Mθ is the desired exponential attractor. Since [0, t ∗ ]×M∗θ is a compact set of C×X and since G is a continuous mapping on [0, t ∗ ] × M∗θ , its image Mθ is also a compact set of X. In addition, since G is Lipschitz continuous, we obtain by Theorem 1.1 that     dF (Mθ ) ≤ dF [0, t ∗ ] × M∗θ = dF M∗θ + 1. Let 0 < t < ∞ and write t = nt ∗ + τ with integer n ≥ 0 and 0 ≤ τ < t ∗ . Then, S(t)M∗θ = S(τ )S(nt ∗ )M∗θ = S(τ )S(t ∗ )n M∗θ ⊂ S(τ )M∗θ .   ∗ S(τ )M∗θ = Mθ . Conversely, it is clear that Therefore, 0≤t<∞ S(t)Mθ ⊂ 0≤τ ≤t ∗   ∗ Mθ ⊂ 0≤t<∞ S(t)Mθ . Hence, Mθ = 0≤t<∞ S(t)M∗θ . This then shows the invariance of Mθ under S(t) for every 0 < t < ∞. For 0 < t < ∞, writing t = nt ∗ + τ , we see that       h S(t)X, M∗θ = h S(t ∗ )n S(τ )X, M∗θ ≤ h S(t ∗ )n X, M∗θ ≤ Raθn due to (4) of Theorem 6.12 or 6.13. Furthermore, since Mθ ⊃ M∗θ ,   −1 1 (t−τ )/t ∗ h(S(t)X, Mθ ) ≤ h S(t)X, M∗θ ≤ Raθ ≤ Raθ−1 e−( t ∗ log aθ )t .

(6.49)

Hence, Mθ also attracts X at an exponential rate.  We set A = is compact, we have 0≤t<∞ S(t)X ⊃ 0≤t<∞ S(t)X. Since Mθ   0≤t<∞ S(t)Mθ = ∅. Meanwhile, let U ∈ 0≤t<∞ S(t)X. Then, for 0 ≤ s, t < ∞, U = S(t + s)Ut+s = S(t)S(s)Ut+s with some suitable Ut+s ∈ X. By (6.49), lims→∞ d(S(s)Ut+s , Mθ ) = 0; therefore, by the compactness of Mθ , we can assume that, as s →∞, S(s)Ut+s → V ∈ Mθ ; hence, U = S(t)V  ∈ S(t)Mθ , i.e.,   S(t)X ⊂ S(t)M . Thus, A = S(t)X = θ 0≤t<∞ 0≤t<∞ 0≤t<∞ 0≤t<∞ S(t) × Mθ . By Theorem 6.2, we conclude that A is a global attractor of (S(t), Mθ , X).

280

6 Dynamical Systems

In particular, A is a compact set of X and is strictly invariant under S(t) for every 0 < t < ∞. We can then show that A is a global attractor for (S(t), X, X), too. The proof is quite similar to that of Theorem 6.12, and so, we may omit it.  Let X be a Hilbert space. Consider a dynamical system (S(t), X, X) in X with compact phase space X. We assume for some fixed t ∗ > 0 that S(t ∗ ) satisfies (6.44) and (6.45)–(6.46). Then, by the same arguments as for Theorem 6.15, we are able to construct exponential attractors Mθ for (S(t), X, X). Theorem 6.16 Let (S(t), X, X) be a dynamical system in a Hilbert space X with compact phase space X. For some fixed t ∗ > 0, let S(t ∗ ) satisfy (6.44) and (6.45)– (6.46), and let (6.48) be satisfied. Then, for any exponent 0 < θ < 1 − 2δ, there exists an exponential attractor Mθ for (S(t), X, X) with finite fractal dimension   dF (Mθ ) ≤ dF M∗θ + 1 (6.50) and with the same properties as (1), (3), and (4) announced in Theorem 6.15. Here, M∗θ is the exponential attractor of (S(t ∗ )n , X, X) constructed in Theorem 6.14, and aθ = 2δ + θ .

4.6 Continuous Dependence on Parameter In this subsection, we want to show that exponential attractors can depend continuously on a parameter, provided that their corresponding semigroups change smoothly. Consider a family of dynamical systems (Sξ (t), Xξ , X) in a Banach space X parameterized by 0 ≤ ξ ≤ 1. The phase spaces Xξ are all compact subsets of X, and Sξ (t) are continuous semigroups acting on Xξ , respectively.  We assume that there exists a compact set B of X such that B ⊂ 0≤ξ ≤1 Xξ and that B absorbs the phase space Xξ uniformly for ξ . That is, there exists some fixed time t ∗ > 0 such that Sξ (t)Xξ ⊂ B

for any t ∗ ≤ t < ∞ and every 0 ≤ ξ ≤ 1.

(6.51)

We then consider a family of discrete dynamical systems (Sξn , B, X), where Sξ = Sξ (t ∗ ). We assume that, for each ξ , Sξ satisfies the compact perturbation condition (6.38)–(6.39) with some fixed number 0 ≤ δ < 12 and some constant L > 0 and with some fixed Banach space Z compactly embedded in X. We assume also that the semigroups Sξ (t) satisfy (6.48) with some uniform constant L1 > 0. By Theorem 6.15, we see for each ξ that (Sξ (t), Xξ , X) has exponential attractors. As ξ → 0, let Sξ (t) tend to S0 (t) on B for 0 ≤ t ≤ t ∗ at the rate sup

sup Sξ (τ )U − S0 (τ )U X ≤ Oξ ,

0≤τ ≤t ∗ U ∈B

(6.52)

where Oξ stands for the convergence rate of Sξ (t) to S0 (t) such that limξ →0 Oξ = 0.

5 Dynamical Systems for Semilinear Evolution Equations

281

Under this situation, we can prove the following theorem. Theorem 6.17 For a family of dynamical systems (Sξ (t), Xξ , X), 0 ≤ ξ ≤ 1, assume that there exists a uniform absorbing set B satisfying (6.51) and that (Sξ (t), Xξ , X) satisfy (6.38)–(6.39) and (6.48) uniformly for ξ . If Sξ (t) converges to S0 (t) locally at the rate (6.52), then there exist exponential attractors Mξ of (Sξ (t), Xξ , X), 0 ≤ ξ ≤ 1, for which the estimate d(Mξ , M0 ) ≤ COξκ ,

0 < ξ ≤ 1,

holds with some exponent 0 < κ < 1 and constant C > 0. Such a theorem can be proved under a more general situation. As a matter of fact, an existence of uniform absorbing set like (6.51) is not indispensable. In Chap. 7, we shall consider approximation problems in an abstract framework. There the phase spaces Xξ will be approximate spaces of X0 with finite dimensions such that Xξ ⊂  X0 for 0 < ξ ≤ 1 but 0<ξ ≤1 Xξ = ∅. A similar result, Theorem 7.4, will be proved under such a situation. So, the proof of Theorem 6.17 will be given in Chap. 7 as a special case of Theorem 7.4. We are able to prove a similar result under the compact perturbation condition (6.28)–(6.29), too.

5 Dynamical Systems for Semilinear Evolution Equations 5.1 Dynamical System We consider the Cauchy problem for a semilinear evolution equation  dU dt + AU = F (U ), 0 < t < ∞, U (0) = U0 ,

(6.53)

in a Banach space X. Here, A is a sectorial operator of X with angle ωA < π2 (see (2.92) and (2.93)). The nonlinear operator F is defined from D(Aη ) into X, where η is some exponent such that 0 < η < 1, and is assumed with another exponent β such that 0 ≤ β < η to satisfy the Lipschitz condition either (4.2) (when β > 0) or (4.21) (when β = 0). We will set D(Aβ ) which is regarded as a Banach space with the graph norm Aβ · as the space of initial values and will denote it by Dβ . Of course, when β = 0, we have simply Dβ = X. According to Theorem 4.1 when β > 0 and Theorem 4.4 when β = 0, for each U0 ∈ Dβ , there exists a unique local solution to (6.53) in the function space: U ∈ C((0, TU0 ]; D(A)) ∩ C([0, TU0 ]; Dβ ) ∩ C1 ((0, TU0 ]; X), TU0 > 0 depending only on the norm U0 Dβ .

(6.54)

282

6 Dynamical Systems

To obtain the global existence of a solution, we have to establish a priori estimate for any local solution with the initial value U0 . We here assume such an estimate. More precisely, assume that there exists an increasing continuous function p(·) > 0 such that the estimate U (t; U0 ) Dβ ≤ p( U0 Dβ ),

0 ≤ t ≤ T U , U0 ∈ D β ,

(6.55)

holds for any local solution U (t; U0 ) on an interval [0, TU ] with the initial value U0 . Then, by Corollary 4.1 (when β > 0) and Corollary 4.3 (when β = 0), we can deduce that (6.53) possesses a global solution on [0, ∞) in the function space (6.54) (in which TU0 must be any finite time). Clearly, (6.55) yields that U (t; U0 ) Dβ ≤ p( U0 Dβ ),

0 ≤ t < ∞, U0 ∈ Dβ .

(6.56)

Moreover, we can show a norm estimate stronger than (6.56). Proposition 6.1 For U (t; U0 ), it holds that AU (t; U0 ) X ≤ (1 + t β−1 )p1 ( U0 Dβ ),

0 < t < ∞, U0 ∈ Dβ ,

(6.57)

with another continuous increasing function p1 (·). Proof Let 0 ≤ s < ∞ and consider a similar problem as (6.53) but with initial time s and with an initial value U (s; U0 ). We then apply Theorems 4.1 and 4.4 to this problem to conclude that there exist τ > 0 (depending only on U (s; U0 ) Dβ and hence only on p( U0 Dβ )) and a continuous increasing function p1 (·) such that AU (t) X ≤ (t − s)β−1 p1 ( U (s; U0 ) Dβ ),

s < t ≤ s + τ.

First, applying this with s = 0, we have AU (t) X ≤ t β−1 p1 ( U0 Dβ ),

0 < t ≤ τ.

Secondly, taking s = t − τ , we have AU (t) X ≤ τ β−1 p1 ( U (t − τ ) Dβ ) ≤ τ β−1 p1 (p0 ( U0 Dβ )),

τ < t < ∞. 

Combining these estimates, we obtain the desired one.

Under (6.55), (6.53) possesses a unique global solution U (t; U0 ) in the function space: U (·; U0 ) ∈ C((0, ∞); D(A)) ∩ C([0, ∞); Dβ ) ∩ C1 ((0, ∞); X),

U0 ∈ D β ,

and (6.56) and (6.57) hold true. We next set S(t)U0 = U (t; U0 ), 0 ≤ t < ∞. Then, the uniqueness of solutions readily implies that S(t) is a nonlinear semigroup acting on Dβ . Let us establish

5 Dynamical Systems for Semilinear Evolution Equations

283

its continuity on Dβ . For any 0 < R < ∞, let KR = {U ∈ Dβ ; U Dβ < R} be an open ball. By Corollary 4.2 (when β > 0) and Theorem 4.5 (when β = 0), we know that there exists tR > 0 such that S(t)U0 − S(t)V0 Dβ ≤ LR U0 − V0 Dβ , 0 ≤ t ≤ tR ; U0 , V0 ∈ KR .  In view of 0≤t<∞ S(t)KR ⊂ Kp(R) (due to (6.56)), we will apply this estimate on the larger ball Kp(R) ⊃ KR . Then, we can prove the following result. Proposition 6.2 For any 0 < R < ∞, it holds that S(t)U0 − S(t)V0 Dβ ≤ Ln+1 p(R) U0 − V0 Dβ , t ∈ [ntp(R) , (n + 1)tp(R) ]; n = 0, 1, 2, . . . ; U0 , V0 ∈ KR .

(6.58)

Proof In fact, this is true when n = 0 (note that R ≤ p(R)). Assume that the estimate is true for n − 1. For ntp(R) < t ≤ (n + 1)tp(R) , we have S(t) = S(t − ntp(R) )S(ntp(R) ). Since S(ntp(R) )U0 and S(ntp(R) )V0 are in Kp(R) , we easily observe that the estimate is true for n, too.  Estimate (6.58) means that S(t) is Lipschitz continuous in the ball KR and the Lipschitz constant is uniformly bounded in t on any finite time interval. In the meantime, for each U0 ∈ KR , S(·)U0 is continuous from [0, ∞) to Dβ . Hence, the mapping G : [0, ∞) × KR → Dβ , where G(t, U0 ) = S(t)U0 , is seen to be continuous. As R is arbitrary, we conclude that S(t) is continuous on the whole space Dβ . In this way, under (2.92), (2.93), either (4.2) or (4.21), and (6.55), it has been proved that problem (6.53) determines a dynamical system (S(t), Dβ , Dβ ). When β > 0, it is often very important to construct a dynamical system in the universal space X (instead of Dβ ). To this end, we introduce for any 0 < R < ∞, the phase space  {S(t)U0 ; 0 ≤ t < ∞} ⊃ KR . XR = U0 ∈KR

Clearly, S(t) maps XR into itself, namely, XR is invariant under S(t). In addition, since XR ⊂ Kp(R) due to (6.56), XR is a bounded set of Dβ . With the aid of Theorem 4.3, we can argue in a similar way as for the proof of (6.58) to deduce the following proposition. Proposition 6.3 For any 0 < R < ∞, it holds that S(t)U0 − S(t)V0 X ≤ Ln+1 p(R) U0 − V0 X , t ∈ [ntp(R) , (n + 1)tp(R) ]; n = 0, 1, 2, . . . ; U0 , V0 ∈ XR ; with some constant Lp(R) > 0. This ultimately implies that G : [0, ∞) × XR → X is continuous, where XR is equipped with the induced distance from X. Thus, problem (6.53) determines a dynamical system (S(t), XR , X) even in the universal space X, too.

284

6 Dynamical Systems

5.2 Absorbing and Invariant Compact Set We shall now make (in addition to the above) the assumptions that  X is a reflexive Banach space; D(A) is compactly embedded in X.

(6.59)

As a consequence, we observe the following fact. Proposition 6.4 Let D = D(A) be a Banach space with the graph norm · D = D A · . Then, any bounded closed ball B (0; R), 0 < R < ∞, of D is a compact subset of X. D

Proof Indeed, it suffices to prove that the ball is closed in X. Let Un ∈ B (0; R) and Un → U in X. Since D is reflexive (note that D and X are isomorphic), D B (0; R) is weakly compact; therefore, we can assume that Un → V weakly in D with V D ≤ R. Since D ⊂ X densely and continuously, we have X  ⊂ D ; D hence, also Un → V weakly in X. Consequently, U = V ∈ B (0; R).  We assume also a dissipative condition, that is, there exists a number C˜ > 0 for which the following assertion is valid. For any bounded subset B of Dβ , there is a time tB such that ˜ sup sup S(t)U0 Dβ ≤ C.

U0 ∈B t≥tB

(6.60)

A stronger dissipative condition is then deduced by the same argument as for (6.57). In fact, there exists a number C˜ 1 > 0 such that, for any bounded subset B of Dβ , it holds that sup sup S(t)U0 D ≤ C˜ 1

U0 ∈B t≥tB

(6.61)

with a suitable time tB depending on B. Using this constant C˜ 1 , we introduce the closed ball D B = B (0; C˜ 1 ).

Estimate (6.61) obviously means that, for any bounded subset B of Dβ , there exists a time tB such that S(t)B ⊂ B for all t ≥ tB , that is, B is an absorbing set. As noticed above, B is a compact set of X. We furthermore consider the set X=



S(t)B ⊂ B

(closure in X).

tB ≤t<∞

Note that, as B is a bounded set of Dβ , there exists tB > 0 such that the images S(t)B are also absorbed by itself for all t ≥ tB . As B is an absorbing set, it is the

5 Dynamical Systems for Semilinear Evolution Equations

285

same for X. In addition, since S(t) is continuous with respect to the X norm (recall Proposition 6.3), it holds that      S(t) S(τ )B ⊂ S(t)S(τ )B ⊂ S(τ )B. tB ≤τ <∞

tB ≤τ <∞

tB ≤τ <∞

This means that X is an invariant set. Under (6.59) and (6.60), we have in this way observed that the asymptotic behavior of the dynamical system (S(t), XR , X) for any 0 < R < ∞ is reduced to that of a dynamical system (S(t), X, X) in which the phase space X is a compact set of X and is a bounded set of D.

5.3 Exponential Attractors We are ready to apply Theorem 6.15 to the dynamical system (S(t), X, X). Theorem 4.3 (when β > 0) and Theorem 4.5 (when β = 0) immediately provide that, for a sufficiently small time t ∗ > 0, the operator S(t ∗ ) from X into Dβ fulfills the compact perturbation condition (6.38)–(6.39) with Z = D(Aη ) and δ = 0. Note that the second condition of (6.59) implies that Z is compactly embedded in X. Let us next verify (6.48). Proposition 6.3 obviously provides that S(t)U0 − S(t)V0 X ≤ CX U0 − V0 X ,

0 ≤ t ≤ t ∗ , U0 , V0 ∈ X,

with some constant CX > 0. As X is a bounded set of D, t S(t)V0 − S(s)V0 = [F (S(τ )V0 ) − AS(τ )V0 ] dτ s

≤ CX (t − s),

0 ≤ s < t ≤ t ∗ , V0 ∈ X.

These two estimates obviously yield (6.48). We have thus deduced, under (2.92), (2.93), either (4.2) or (4.21), (6.55), (6.59), and (6.60), that the dynamical system (S(t), X, X) possesses a family of exponential attractors M. Let us finally verify that each M is an exponential attractor of the dynamical system (S(t), X, Dβ ), too, where X is equipped with the induced distance from Dβ . By the moment inequality (2.117), it is readily seen that hDβ (S(t)X, M) ≤ CX h(S(t)X, M)1−β ≤ CX e−(1−β)kt ,

0 ≤ t < ∞,

where hDβ (·, ·) denotes the Hausdorff pseudo-distance in Dβ . This shows that M attracts S(t)X at an exponential rate even in Dβ . Meanwhile, by Theorem 1.3, the fractal dimension dDβ (M) of M in Dβ is estimated by (1 − β)−1 dX (M). Hence, M keeps its properties as an exponential attractor in (S(t), X, Dβ ).

286

6 Dynamical Systems

5.4 Squeezing Property of S(t) Let X be a Hilbert space. For the operator A in (6.53), we now assume that  A is a positive definite self-adjoint operator of X; (6.62) D(A) is compactly embedded in X. For the nonlinear operator F , we assume (4.2) (when β > 0) or (4.21) (when β = 0). We assume also that problem (6.53) determines a dynamical system (S(t), X, X) in which X is a compact set of X and is a bounded set of D(A). Consequently, F satisfies the Lipschitz condition F (U ) − F (V ) ≤ D Aη (U − V ) ,

U, V ∈ X, 0 ≤ η < 1.

(6.63)

On account of (6.62), there exist positive eigenvalues 0 < λ1 ≤ λ2 ≤ λ3 ≤ · · · → ∞ of A, and the corresponding eigenvectors U1 , U2 , U3 , . . . compose an orthonormal basis of X. We consider a finite-dimensional subspace Z = Span{U1 , U2 , U3 , . . . , UN }. Let P : X → Z be the orthogonal projection. We can show that the squeezing property (6.45)–(6.46) is fulfilled on X if N is sufficiently large. Let t ∗ > 0 be a time such that the estimate t η Aη [S(t)U − S(t)V ] + S(t)U − S(t)V ≤ C U − V ,

U, V ∈ X,

holds for any 0 < t ≤ t ∗ (see Theorem 4.3 or 4.5) and set S = S(t ∗ ). In view of (6.45) and (6.46), it suffices to prove that, if (1 − P )[SU − SV ] > P [SU − SV ] for U, V ∈ X, then (6.45) is valid with a suitable constant 0 ≤ δ < 14 . So, let (1 − P )[SU − SV ] > SU − SV be satisfied. Then, SU − SV 2 = P [SU − SV ] 2 + (1 − P )[SU − SV ] 2 < 2 (1 − P )[SU − SV ] 2 . In the meantime, we have (1 − P )SU = e

−t ∗ A

(1 − P )U +

t∗

e−(t

∗ −s)A

(1 − P )F (S(s)U ) ds.

0

It is the same for (1 − P )SV . We then observe that e−t

∗A

(1 − P )(U − V ) ≤ e−t

∗λ N+1

1 U − V ≤ √ U − V 2 34

if N is sufficiently large. Similarly, due to (6.63),

6 Stationary Solutions to Semilinear Equations





t∗

e−(t

0

∗ −s)A



≤D

t∗

287

(1 − P )[F (S(s)U ) − F (S(s)V )] ds

e−(t

∗ −s)λ N+1

F (S(s)U ) − F (S(s)V ) ds

0



t∗

1 s −η ds U − V ≤ √ U − V 2 34 0  t∗ ∗ if N is sufficiently large. We here used the fact that limλ→∞ 0 e−(t −s)λ s −η ds = 0. 1 Hence, it follows that SU − SV 2 ≤ 17 U − V 2 , namely, (6.45) is fulfilled. We have thus shown under (6.62) that S(t) enjoys the squeezing property. ≤D

e−(t

∗ −s)λ N+1

6 Stationary Solutions to Semilinear Equations 6.1 Equilibria of Dynamical System Consider problem (6.53) in a Banach space X. We assume that the linear operator A and the nonlinear operators F satisfy (2.92), (2.93), and either (4.2) (when β > 0) or (4.21) (when β = 0). In addition, we assume that the a priori estimate (6.55) holds for any U0 ∈ Dβ . As shown, a dynamical system (S(t), Dβ , Dβ ) is constructed, where Dβ = D(Aβ ) (and D0 = X when β = 0). Let (6.53) have a stationary solution U (t) ≡ U , i.e., AU = F (U ). Clearly, U is an equilibrium of (S(t), Dβ , Dβ ). Let R > 0 be such that U Dβ < R. In this section, we shall investigate the stability and instability of U by applying Theorems 6.9 and 6.10, respectively, in which the Fréchet differential of S(t) at U played an essential role.

6.2 Exponential Stability of U Let us assume that F : D(Aη ) → X is Fréchet differentiable at U , i.e.,  ΔF (h) = F (U + h) − F (U ) − F  (U )h, F  (U ) ∈ L(D(Aη ), X), limh→0 oF (h)/ Aη h = 0. oF (h) = ΔF (h) ,

(6.64)

Here, oF (h) is defined for all h ∈ D(Aη ). In view of (4.2) or (4.21), we observe that oF (h) ≤ C Aη h for all h such that Aη h ≤ 1. Let now h ∈ Dβ denote a small variable satisfying h Dβ < R − U Dβ . For such an h, let Uh (t) = S(t)(U + h). By Theorem 4.3 or 4.5, there exists T > 0 such that t η Aη [Uh (t) − U ] + t β Aβ [Uh (t) − U ] ≤ C h ,

0 < t ≤ T.

(6.65)

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6 Dynamical Systems

We here introduce the linear problem  dV  h dt + AVh = F (U )Vh ,

0 < t ≤ T,

Vh (0) = h, in X. Since F  (U ) ∈ L(D(Aη ), X), we know by Theorem 3.3 that A − F  (U ) generates an analytic semigroup on X (cf. Remark 3.7). By Theorem 3.14 the solution of the problem is given by Vh (t) = e−tA h, where A = A − F  (U ). Put ΔU (t) = Uh (t) − U − Vh (t). Then, since F (Uh (t)) − F (U ) − F  (U )Vh (t) = F  (U )ΔU (t) + ΔF (Uh (t) − U ), it is observed that  dΔ  U dt + AΔU = F (U )ΔU + ΔF (Uh (t) − U ),

0 < t ≤ T,

ΔU (0) = 0. Using the semigroup e−tA , ΔU (t) is written by

t e−(t−s)A ΔF (Uh (s) − U ) ds, ΔU (t) =

0 < t ≤ T.

0

Note that ΔF (Uh − U ) ∈ Fβ,σ ((0, T ]; X) due to (4.15) (when β > 0) and ΔF (Uh − U ) ∈ F1−η,σ ((0, T ]; X) due to (4.27) (when β = 0). Therefore, for any 0 ≤ θ < 1,

t (t − s)−θ oF (Uh (s) − U ) ds. Aθ ΔU (t) ≤ C 0

Furthermore, by (6.65),

t θ A ΔU (t) ≤ C (t − s)−θ s −η oF (Uh (s) − U )/ Aη (Uh (s) − U ) ds h . 0

Since Aη (Uh (t) − U ) → 0 as h → 0, we conclude by the dominate convergence theorem that, as h → 0 (a fortiori as h Dβ → 0), 0 ≤ Aθ ΔU (t) / h Dβ ≤ Aθ ΔU (t) / h → 0.

(6.66)

Thus, under (6.64), the operator S(t) : Dβ → D(Aθ ), where 0 < t ≤ T and 0 ≤ θ < 1, is Fréchet differentiable at U with differential S(t) U = e−tA ,

where A = A − F  (U ).

Moreover, since S(2t)U = S(t)S(t)U and S(t)U = U , we verify that S(2t) is also Fréchet differentiable at U . Repeating this argument, S(t) : Dβ → D(Aθ ) is Fréchet differentiable at U for every time 0 < t < ∞.

6 Stationary Solutions to Semilinear Equations

289

We can now give a sufficient condition for exponential stability of U . Let us assume that the spectrum of A is contained in the right half-plane, i.e., σ (A) = σ (A − F  (U )) ⊂ {λ ∈ C; Re λ > 0}.

(6.67)

This condition, together with Theorem 3.3, yields that A is a sectorial operator, namely, A satisfies (2.92) and (2.93). Proposition 2.5 then provides that e−tA ≤ Ce−dt , 0 ≤ t < ∞, with some positive exponent d > 0. As a consequence, e−tA U Dβ ≤ Aβ e−A e−(t−1)A U ≤ Ce−d(t−1) U Dβ . ∗

This implies that e−t A L(Dβ ) ≤ δ < 1 for any sufficiently large t ∗ > 0. Let K˜ r = B Dβ (0; r) be an open ball of Dβ . Then, (6.66) (θ = β) implies that S(t ∗ )(U + h) − S(t ∗ )U − S(t ∗ ) h Dβ = ΔU (t ∗ ) Dβ ≤

1−δ h Dβ , 2

if r > 0 is sufficiently small. Hence,   1−δ δ+1 S(t ∗ )(U + h) − U Dβ ≤ δ + h Dβ = h Dβ , 2 2

h ∈ K˜ r ,

h ∈ K˜ r .

This shows that (6.26) holds true, that is, S(t ∗ ) is a contraction at U . We have thus proved by Theorem 6.10 that, under (6.64) and (6.67), U is an exponentially stable equilibrium of (S(t), Dβ , Dβ ).

6.3 Unstable Manifold of U Let U ∈ D be again a stationary solution of (6.53) and be an equilibrium of (S(t), Dβ , Dβ ). In this subsection, we assume that F : D(Aη ) → X is of class C1,1 in a neighborhood of U . More precisely, there is a number r > 0 such that, for any U ∈ D(Aη ) ∩ B Dβ (U ; r), F is Fréchet differentiable and the derivative satisfies F  (U )h ≤ L1 Aη h ,

U ∈ D(Aη ) ∩ B Dβ (U ; r), h ∈ D(Aη ),

(6.68)

with some constant L1 > 0. In addition, [F  (U1 ) − F  (U2 )]h ≤ L2 Aβ (U1 − U2 ) Aη h , U1 , U2 ∈ D(Aη ) ∩ B Dβ (U ; r), h ∈ D(Aη ),

(6.69)

with some constant L2 > 0. Put ΔF (U ; h) = F (U + h) − F (U ) − F  (U )h. Then, (6.69) implies that ΔF (U ; h) ≤ L2 Aβ h Aη h ,

U + h, U ∈ D(Aη ) ∩ B Dβ (U ; r).

(6.70)

290

6 Dynamical Systems

Consider an open neighborhood B Dβ (U ; r1 ) of U . Let U0 ∈ B Dβ (U ; r1 ) be an initial value, and let U (t) = S(t)U0 . By Corollary 4.2 (when β > 0) or Theorem 4.5 (when β = 0), we have t η−β Aη (U (t) − U ) + Aβ (U (t) − U ) ≤ C Aβ (U0 − U ) , 0 ≤ t ≤ T1 , U0 ∈ B Dβ (U ; r1 ), for sufficiently small time T1 > 0. So, if r1 > 0 and T1 > 0 are sufficiently small, then U (t) ∈ B Dβ (U ; r) for 0 ≤ t ≤ T1 . Consequently, F  (U (t)) is defined. Consider also a small vector variable h ∈ B Dβ (0; r2 ) and let Uh (t) = S(t)(U0 + h). By Theorem 4.3 or 4.5, we have t η Aη [Uh (t) − U (t)] + t β Aβ [Uh (t) − U (t)] ≤ C h , 0 < t ≤ T2 , U0 ∈ B Dβ (U ; r1 ), h ∈ B Dβ (0; r2 ),

(6.71)

for sufficiently small time T2 > 0. We here introduce the linear problem  dV

h

dt

+ AVh = F  (U (t))Vh ,

0 < t ≤ T,

Vh (0) = h,

(6.72)

in X, where T = min{T1 , T2 }. To solve this problem, we will employ Theorem 3.14. Let B(t) = −F  (U (t)), 0 < t ≤ T . Then, it is obvious from (6.68) that B(t)A−η ≤ L1 ,

0 < t ≤ T.

(6.73)

Furthermore, noticing (4.13), we have [B(t) − B(s)]A−η ≤ L2 Aβ [U (t) − U (s)] ≤ Cs −σ˜ (t − s)σ˜ ,

0 < s ≤ t ≤ T,

(6.74) (6.75)

with any exponent σ˜ such that 0 < σ˜ < 1 − η. These estimates (6.73) and (6.74) show that (3.117), (3.118), and (3.119) are satisfied with μ˜ = 1 and ν˜ = 1 − η. Therefore, Theorems 3.13 and 3.14 provide the existence of the evolution operator U˜ (t, s) for the family of operators A − F  (U (t)) which describes the solution of (6.72) in the form Vh (t) = U˜ (t, 0)h. Let ΔU (t) = Uh (t) − U (t) − Vh (t). Since F (Uh (t))−F (U (t))−F  (U (t))Vh (t) = F  (U (t))ΔU (t)+ΔF (U (t); Uh (t)−U (t)), we have  dΔ

U

dt

+ AΔU = F  (U (t))ΔU + ΔF (U (t); Uh (t) − U (t)),

ΔU (0) = 0.

6 Stationary Solutions to Semilinear Equations

291

Then, ΔU is also described by U˜ (t, s) in the form

t ΔU (t) = U˜ (t, s)ΔF (U (s); Uh (s) − U (s)) ds. 0

Note that ΔF (U ; Uh − U ) ∈ Fβ,σ ((0, T ]; X) (when β > 0) due to (4.15) and ΔF (U ; Uh − U ) ∈ F1−η,σ ((0, T ]; X) (when β = 0) due to (4.27). Therefore, on account of (3.124) and (6.70), for any 0 ≤ θ < 1,

t θ A ΔU (t) ≤ C (t − s)−θ Aβ [Uh (s) − U (s)] Aη [Uh (s) − U (s)] ds. 0

Hence, by (6.71),

t

Aθ ΔU (t) ≤ C

(t − s)−θ s −η Aβ [Uh (s) − U (s)] ds h .

0

Aβ [U

Since h (s) − U (s)] → 0 as h → 0, it follows that, as h → 0 (a fortiori as h Dβ → 0), 0 ≤ Aθ ΔU (t) / h Dβ ≤ Aθ ΔU (t) / h → 0. Thus, under (6.68) and (6.69), S(t) : Dβ → D(Aθ ) is Fréchet differentiable in the neighborhood B Dβ (U ; r1 ) for 0 ≤ t ≤ T , where r1 > 0 and T > 0 are sufficiently small. Furthermore, its derivative is given by S(t) U0 = U˜ (t, 0),

0 < t ≤ T , U0 ∈ B Dβ (U ; r1 ),

where U˜ (t, s) is the evolution operator for the family of operators A − F  (S(t)U0 ). Let U1 (t, 0) = S(t) U1 and U2 (t, 0) = S(t) U2 . Then,

t Aθ U2 (t, s)[F  (S(s)U1 ) − F  (S(s)U2 )]A−η Aθ [U1 (t, 0) − U2 (t, 0)]h = 0

× Aη U1 (s, 0)A−β Aβ h ds. By (3.124), (3.125), and (6.69), it follows that

t θ (t − s)−θ s β−η ds h Dβ A [U1 (t, 0) − U2 (t, 0)]h ≤ C 0

× sup Aβ [S(s)U1 − S(s)U2 ] . 0≤s≤T

Therefore, on account of (6.58), S(t) U1 − S(t) U2 L(Dβ ,D(Aθ )) = U1 (t, 0) − U2 (t, 0) L(Dβ ,D(Aθ )) ≤ Ct 1+β−θ−η U1 − U2 Dβ ,

0 ≤ t ≤ T ; U1 , U2 ∈ B Dβ (U ; r1 ).

292

6 Dynamical Systems

Let us assume now that the spectrum of A = A − F  (U ) is separated by the imaginary axis, i.e., σ (A) ∩ {λ ∈ C; Re λ = 0} = ∅.

(6.76)

Denote σ− (A) = σ (A) ∩ {λ ∈ C; Re λ < 0} and σ+ (A) = σ (A) ∩ {λ ∈ C; Re λ > 0}. Since F  (U ) satisfies (6.68), A satisfies (3.2). This necessarily implies that σ− (A) is a bounded closed set. We can then decompose the space X into X = X− ⊕ X+ and the operator A into A = A− ⊕ A+ by applying Theorem 1.10. Here, X∓ are closed subspaces of X with continuous projection, and A∓ are the parts of A in X∓ , respectively. The operator A− is a bounded operator of X− with σ (A− ) = σ− (A); in particular, X− ⊂ D(A). In the meantime, A+ is a densely defined, closed linear operator of X+ with D(A+ ) = D(A) ∩ X+ and σ (A+ ) = σ+ (A). For λ ∈ ρ(A), the resolvent (λ − A)−1 is also decomposed into (λ − A)−1 = (λ − A− )−1 ⊕ (λ − A+ )−1 ; in particular, we verify that A+ is a sectorial operator of X+ . As a consequence, we obtain a decomposition of semigroup e−tA of the form e−tA = e−tA− ⊕ e−tA+ . Since A− is a bounded operator, we can use the spectral mapping theorem, Theorem 1.9, to conclude that σ (e−tA− ) = e−tσ− (A) ⊂ {λ ∈ C; |λ| > 1} for any 0 < t < ∞. In the meantime, on account of Proposition 2.5, we have e−tA+ ≤ Ce−δt with some δ > 0. Therefore, 1 the spectral radius of e−tA+ is estimated by limn→∞ e−ntA+ n ≤ e−δt ; hence, σ (e−tA+ ) ⊂ {λ ∈ C; |λ| < 1}. In this way, we have verified from (6.76) that σ (e−tA ) ∩ {λ ∈ C; |λ| = 1} = ∅,

0 < t < ∞,

and σ (e−tA ) ∩ {λ ∈ C; |λ| > 1} = e−tσ− (A) ,

0 < t < ∞.

Moreover, the subspace Xe of X corresponding to the exterior spectrum σ (e−tA ) ∩ {λ ∈ C; |λ| > 1} is the space X− . When β > 0, let us consider e−tA as a bounded operator from Dβ into itself. We observe the following fact. Lemma 6.1 Let t > 0. For λ = 0, λ ∈ σ (e−tA ) if and only if λ ∈ σ Dβ (e−tA ), where σ Dβ (·) denotes the spectrum of an operator in the space Dβ . Proof Let λ ∈ ρ(e−tA ). Then,   (λ − e−tA )−1 = λ−1 e−tA (λ − e−tA )−1 + 1 . From this it is deduced that (λ − e−tA )−1 is a bounded operator from Dβ into itself. Therefore, (λ − e−tA )−1 is a bounded inverse of λ − e−tA in the space Dβ , too, i.e., λ ∈ ρ Dβ (e−tA ). Conversely, let λ ∈ ρ Dβ (e−tA ). Taking a λ0 ∈ ρ(A), we have   (λ − e−tA )−1 = λ−1 A−β Aβ (λ − e−tA )−1 A−β Aβ (λ0 − A)−1 (λ0 − A)e−tA + 1 .

6 Stationary Solutions to Semilinear Equations

293

Since Aβ (λ − e−tA )−1 A−β ∈ L(X), this shows that (λ − e−tA )−1 is continuous in the norm of X, too. Since Dβ is dense in X, it follows that (λ − e−tA )−1 is extended over X as a bounded operator of X, and hence λ ∈ ρ(e−tA ). Thus, the assertion of lemma is verified.  In view of this lemma, we conclude that (6.76) equally implies σ Dβ (e−tA ) ∩ {λ ∈ C; |λ| = 1} = ∅,

0 < t < ∞,

and σ Dβ (e−tA ) ∩ {λ ∈ C; |λ| > 1} = e−tσ− (A) ,

0 < t < ∞.

Moreover, the subspace Dβ,e of Dβ which corresponds to the exterior spectrum σ Dβ (e−tA ) ∩ {λ ∈ C; |λ| > 1} is nothing more than X− . Indeed, the last assertion is verified as follows. Let X = Xe ⊕ Xi and Dβ = Dβ,e ⊕ Dβ,i be the decompositions of X and Dβ , respectively, corresponding to the spectrum separation of the semigroup e−tA by the unit circle. Let P and Pβ be the projections from X to Xi and from Dβ to Dβ,i . By (1.15), if U ∈ Dβ , then obviously Pβ U = P U ; consequently, (1 − Pβ )U = (1 − P )U for U ∈ Dβ ; in particular, Dβ,e ⊂ Xe . While, as shown above, Xe = X− ⊂ D(A) ⊂ Dβ ; hence, Xe ⊂ Dβ,e , i.e., Xe = Dβ,e . In this way, under (6.68), (6.69), and (6.76), we have shown that S(t) U satisfies (6.20), that is, U is a hyperbolic equilibrium of (S(t), Dβ , Dβ ). The exterior spec trum of S(t) U is given by e−tσ− (A−F (U )) . By virtue of Theorem 6.9, there exists a smooth local unstable manifold W+ (U ; O) of U which has a dimension dim X− and is tangent to the subspace U + X− of Dβ at U .

6.4 Reaction–Diffusion Systems Let us apply our general results obtained above to the initial-boundary-value problem ⎧ ∂u ⎪ in Ω × (0, ∞), ⎪ ∂t = aΔu + f (u, v) ⎪ ⎪ ⎪ ⎨ ∂v = bΔv + g(u, v) in Ω × (0, ∞), ∂t (6.77) ∂u ∂v ⎪ = ∂n = 0 on ∂Ω × (0, ∞), ⎪ ⎪ ∂n ⎪ ⎪ ⎩u(x, 0) = u (x), v(x, 0) = v (x) in Ω, 0

0

in a C2 or convex, bounded domain Ω ⊂ Rd (d = 1, 2, 3). Here, a > 0 and b > 0 are diffusion constants. The kinetic functions f (u, v) and g(u, v) are real smooth functions defined for (u, v) ∈ R2 . Let (u, v) ≡ (u, v) be a stationary solution to (6.77), that is, (u, v) ∈ R2 satisfies f (u, v) = g(u, v) = 0.

(6.78)

294

6 Dynamical Systems

We assume that f (u, v) and g(u, v) can be extended over a complex neighborhood D ⊂ C2 of (u, v) and the extended functions, still denoted by f (u, v) and g(u, v), are both differentiable as complex variable functions from D into C: f (u, v) and g(u, v) are complex differentiable functions in D.

(6.79)

We set the underlying space by    f X= ; f ∈ L2 (Ω) and g ∈ L2 (Ω) , g and formulate problem (6.77) as the Cauchy problem for an abstract equation of the form (6.53). The linear operator A of X is defined by      u A1 0 2 2 , D(A) = ; u ∈ HN (Ω) and v ∈ HN (Ω) , A= v 0 A2 where A1 and A2 are realizations of the differential operators −aΔ+1 and −bΔ+ 1 in L2 (Ω), respectively, under the homogeneous Neumann boundary conditions on ∂Ω. Then, A is a positive definite self-adjoint operator of X. By virtue of Theorems 16.7 and 16.9, the domains of its fractional powers are given by    3 u θ 2θ 2θ ; u ∈ H (Ω) and v ∈ H (Ω) for 0 ≤ θ < D(A ) = v 4 and D(Aθ ) =

   u ; u ∈ HN2θ (Ω) and v ∈ HN2θ (Ω) v

for

3 < θ ≤ 1. 4

The nonlinear operator F from D(Aη ) into X is defined by     u + f (u, v) u F (U ) = , U= ∈ D(Aη ), v + g(u, v) v where the exponent η is taken in such a way that d4 < η < 1. Since H s (Ω) ⊂ C(Ω) for s > d2 (due to (1.76)), we observe that D(Aη ) ⊂ C(Ω). We take another exponent β so that d4 < β < η < 1. Then, D(Aη ) ⊂ D(Aβ ) ⊂ C(Ω). By direct calculations, it is easily verified from (6.79) that F is Fréchet differentiable in a neighborhood of U = t (u, v) with derivative   1 + fu (u, v) fv (u, v)  h, F (U )h = 1 + gv (u, v) gu (u, v)   u U= ∈ B Dβ (U ; r), h ∈ D(Aη ), v where Dβ = D(Aβ ) and r > 0. Furthermore, the derivative fulfills conditions (6.68) and (6.69). Let us then investigate how the spectral condition (6.67) or the separation condition (6.76) is fulfilled by A = A − F  (U ).

6 Stationary Solutions to Semilinear Equations

295

We here consider the self-adjoint operator Λ of L2 (Ω) which is a realization of −Δ in L2 (Ω) under the homogeneous Neumann boundary conditions on ∂Ω. As is well known, Λ possesses denumerable nonnegative eigenvalues, and the corresponding real eigenfunctions can constitute an orthonormal basis of L2 (Ω), see Corollary 2.1. So, let 0 = μ 0 < μ1 ≤ μ 2 ≤ · · · → ∞ 1

be eigenvalues of Λ, and let φ0 = |Ω|− 2 , φ1 , φ2 , . . . , be the corresponding real eigenfunctions which constitute an orthonormal basis. We notice that X is decomposed into an infinite orthogonal direct sum of subspaces Xk , where each Xk is a two-dimensional subspace given by       φk 0 + ηk ; ξk , ηk ∈ C , k = 0, 1, 2, . . . , Xk = U = ξ k 0 φk   0  φk  X= ∞ k=0 Xk . Obviously, Φk = 0 and Ψk = φk are orthonormal bases of Xk with components of real functions. Since     aΛ + 1 0 1+fu fv A= and F  (U ) = , 0 bΛ + 1 gu 1 + gv where f u = fu (u, v), f v = fv (u, v), g u = gu (u, v), and g v = gv (u, v), we notice each Xk is invariant also that A maps the subspace Xk into itself. In other words,  under A. In this way, A is also decomposed into the sum A = ∞ k=0 Ak , where Ak is the part of A in Xk . Let us consider the part Ak in Xk . Obviously, Ak Φk = (aμk − f u )Φk − g u Ψk ,

Ak Ψk = −f v Φk + (bμk − g v )Ψk .

The transformation matrix M k of Ak with respect to the bases Φk , Ψk is given by   aμk − f u −g u Mk = . −f v bμk − g v So, the characteristic equation of M k is given by |λ − M k | ≡ λ2 − [(a + b)μk − (f u + g v )]λ + abμ2k − (ag v + bf u )μk + (f u g v − f v g u ) = 0. If the discriminant Δk is nonnegative, i.e., Δk = (a − b)2 μ2k + 2(a − b)(g v − f u )μk + (f u − g v )2 + 4f v g u ≥ 0, then the characteristic equation has two real solutions λk ≤ λk satisfying 2[abμ2k − (ag v + bf u )μk + (f u g v − f v g u )] ≤ λk ≤ λk . √ (a + b)μk − (f u + g v ) + Δk

(6.80)

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6 Dynamical Systems

On the other hand, if Δk < 0, then the solutions λk and λk are complex and satisfy Re λk = Re λk = [(a + b)μk − (f u + g v )]/2, √ |Im λk | = |Im λk | = |a − b|μk + O( μk ). Both cases, as k → ∞, λk and λk lie in Σω + 1 with some fixed angle ω such   π  that arctan |a−b| a+b < ω < 2 ; in addition, |λk | = O(μk ) and |λk | = O(μk ). Let ψk : Xk → C2 be a natural isomorphism such that ψk (ξk Φk + ηk Ψk ) = t (ξk , ηk ) ( ψk = ψk−1 = 1); then, Ak and its transformation matrix M k satisfy the relation Ak = ψk−1 M k ψk . Therefore, (λ − Ak )−1 = ψk−1 (λ − M k )−1 ψk

for λ = λk , λk .

(6.81)

Hence, σ (Ak ) = {λk , λk } for each k = 0, 1, 2, . . . . Let us now consider (λ − A)−1 . Take an angle ω such that ω < ω < π2 , and let λ∈ / Σω + 1. For all sufficiently large k such that λk , λk ∈ Σω + 1, we have   1 λ − bμk + g v −g u (λ − M k )−1 = . −f v λ − aμk + f u (λ − λk )(λ − λk ) In addition, |λ − λk |2 ≥ |λ|2 + |λk |2 − 2|λ||λk | cos(ω − ω), |λ − λk |2 ≥ |λ|2 + |λk |2 − 2|λ||λk | cos(ω − ω). Therefore, in view of (6.81), we obtain that (λ − Ak )−1 L(Xk ) ≤ Cλ for all such k, where Cλ > 0 are independent of k. This then means that λ ∈ ρ(A) if and only if λ ∈ ρ(Ak ) for each k = 0, 1, 2, . . . . By these considerations, we can now state the following. The operator A fulfills (6.67) or (6.76) if and only if the part Ak fulfills the same for each k = 0, 1, 2, . . . . Condition (6.67). A necessary and sufficient condition that A0 (μ0 = 0) satisfies (6.67) is given by f u + g v < 0 and f u g v > f v g u . Therefore, a necessary and sufficient condition that every Ak satisfies (6.67) is that f u + g v < 0, f u g v > f v g u , and that (ag v + bf u )μk < abμ2k + (f u g v − f v g u ) for every μk . As (ag v + bf u )μ < abμ2 + (f u g v − f v g u ) for all μ ≥ 0,  provided that ag v + bf u < 2 ab(f u g v − f v g u ), we find that the triplet of inequalities  f u + g v < 0, f u g v > f v g u , and ag v + bf u < 2 ab(f u g v − f v g u ) (6.82) is a sufficient condition for (6.67).

7 Dynamical Systems for Quasilinear Evolution Equations

297

Condition (6.76). A necessary and sufficient condition that equation (6.80) has a purely imaginary solution is given by abμ2k − (ag v + bf u )μk + (f u g v − f v g u ) = 0, or by (a + b)μk = f u + g v and abμ2k − (ag v + bf u )μk + (f u g v − f v g u ) > 0. So, it is equivalent to the condition that abμ2k − (ag v + bf u )μk + (f u g v − f v g u ) = 0, or μk = (f u + g v )/(a + b) and −(a + b)2 f u g v > (ag v − bf u )2 . Therefore, a necessary and sufficient condition in order that (6.76) is fulfilled (in other words, (6.80) has no purely imaginary solution) is that  abμ2k − (ag v + bf u )μk + (f u g v − f v g u ) = 0 for all μk , and μk = (f u + g v )/(a + b) for all μk ’s or − (a + b)2 f v g u ≤ (ag v − bf u )2 . (6.83)

Condition σ (A) ∩ {λ ∈ C; Re λ < 0} = ∅. Assume that, for some μk , abμ2k − (ag v + bf u )μk + (f u g v − f v g u ) < 0.

(6.84)

For such a μk , the characteristic equation (6.80) has solutions λk < 0 < λk . There ξ  fore, λk ∈ σ (A) ∩ {λ ∈ C; Re λ < 0}. Let ηk be an eigenvector of M k correspondk

ing to the negative eigenvalue λk . Since M k is a real matrix, ξk and ηk are real num ξ  bers. It then follows that ψk−1 ηk ∈ Xk is an eigenfunction of Ak corresponding to k ξ   ξ  φ  the eigenvalue λk . Since the eigenfunction is given by ψk−1 ηk = ηk φk , we observe k

k k

that the eigenfunction of Ak corresponding to the negative eigenvalue λk is a realvalued function. Thus, under (6.84), we deduce that A has a negative eigenvalue and the corresponding eigenfunction is a real-valued function componentwise.

7 Dynamical Systems for Quasilinear Evolution Equations 7.1 Structural Assumptions We consider the Cauchy problem for a quasilinear abstract parabolic evolution equation  dU dt + A(U )U = F (U ), 0 < t < ∞, (6.85) U (0) = U0 , in a Banach space X. Let Z be a second Banach space continuously embedded in X. For U ∈ Z, A(U ) is a sectorial operator of X with angle ωA(U ) < π2 . The domain D(A(U ) is uniform, i.e., D(A(U )) ≡ D,

U ∈ Z.

(6.86)

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6 Dynamical Systems

The domain D is a Banach space equipped with the graph norm A(0) · X of the operator A(0). The operator F is a nonlinear operator from a third Banach space W into X, where W ⊂ Z ⊂ X with continuous embeddings. We make the following structural assumptions. For each 0 < R < ∞, let KR = {U ∈ Z; U Z < R} denote an open ball of Z. For U ∈ KR , A(U ) satisfies σ (A(U )) ⊂ ΣR = {λ ∈ C; | arg λ| < ωR },

U ∈ KR ,

(6.87)

with some uniform angle 0 < ωR < π2 , and the resolvent satisfies  (λ − A(U ))−1 L(X) ≤ MR |λ|,

λ∈ / ΣR , U ∈ KR ,

(6.88)

with some uniform constant MR ≥ 1. In addition, A(U ) satisfies the Lipschitz condition   A(U ) A(U )−1 − A(V )−1 ≤ NR U − V Y , U, V ∈ KR , (6.89) L(X) with some constant NR > 0, where Y is another Banach space such that Z ⊂ Y ⊂ X with continuous embeddings. It is verified from (6.89) that there is a constant CR > 0 such that CR−1 A(0)U˜ X ≤ A(U )U˜ X ≤ CR A(0)U˜ X ,

U ∈ KR , U˜ ∈ D.

(6.90)

For the nonlinear operator F : W → X, we assume the Lipschitz condition F (U ) − F (V ) X ≤ ϕ( U Z + V Z )[ U − V W + ( U W + V W ) × U − V Z ],

U, V ∈ W,

(6.91)

with some continuous increasing function ϕ(·). We may recall that (6.91) implies (5.45). For the spaces W ⊂ Z ⊂ Y ⊂ X, we assume that there exist exponents 0 ≤ α < β < η < 1 such that, for every U ∈ Z, D(A(U )α ) ⊂ Y, D(A(U )β ) ⊂ Z, D(A(U )η ) ⊂ W with continuous embeddings such that ⎧ α α ⎪ ⎨ U˜ Y ≤ D1,R A(U ) U˜ X , U˜ ∈ D(A(U ) ), U ∈ KR , (6.92) U˜ Z ≤ D2,R A(U )β U˜ X , U˜ ∈ D(A(U )β ), U ∈ KR , ⎪ ⎩ ˜ U W ≤ D3,R A(U )η U˜ X , U˜ ∈ D(A(U )η ), U ∈ KR , with some constants Di,R > 0 (i = 1, 2, 3). We furthermore introduce another exponent γ such that β < γ < η and assume that the domain D(A(U )γ ) of the fractional power is independent of U ∈ Z with the equivalence estimate CR−1 A(0)γ U˜ X ≤ A(U )γ U˜ X ≤ CR A(0)γ U˜ X , U˜ ∈ D(A(0)γ ), U ∈ KR , (6.93)

7 Dynamical Systems for Quasilinear Evolution Equations

299

with some constant CR > 0. The domain D(A(U )γ ) is denoted by Dγ , which is a Banach space with the graph norm · Dγ = A(0)γ · X . This space is then used as a space of initial values for problem (6.85). Since D(A(0)γ ) ⊂ D(A(0)β ) ⊂ Z due to (6.92), Dγ is continuously embedded in Z; so, let · Z ≤ c · Dγ . For ˜ be an open ball of Dγ . Then, 0 < R˜ < ∞, let K˜ R˜ = {U ∈ Dγ ; U Dγ < R} K˜ R˜ ⊂ KcR˜ ,

0 < R˜ < ∞.

(6.94)

7.2 Dynamical System Applying the results obtained in Chap. 5, we can see the local existence of solution ˜ to (6.85) for each initial value U0 ∈ Dγ . In fact, let U0 ∈ Dγ with U0 Dγ < R; ˜ then, U0 ∈ Z with U0 Z < cR = R due to (6.94). The family of operators A(U ), U ∈ KR , and the nonlinear operator F satisfy all the structural assumptions (5.2), (5.3), (5.4), (5.5), (5.6), (5.7), (5.46), and (5.47) with ν = 1. In view of U0 ∈ D(A(0)γ ) = D(A(U0 )γ ), we get by Theorem 5.6 the existence of a unique local solution U (t) on some interval [0, TU0 ] in the function space: U ∈ C((0, TU0 ]; D) ∩ C([0, TU0 ]; Dγ ) ∩ C1 ((0, TU0 ]; X),

(6.95)

where TU0 > 0 is determined by the norm A(0)γ U0 X . In order to obtain the global existence, we have to establish a priori estimates for any local solution with the initial value U0 . We here assume that there exists some fixed continuous increasing function p(·) > 0 such that the estimate U (t) Dγ ≤ p( U0 Dγ ),

0 ≤ t ≤ T U , U0 ∈ D γ ,

(6.96)

holds for any local solution U on [0, TU ] to (6.85). This means that any local solution lies in a fixed open ball Kcp( U0 Dγ ) of Z. Since the family of operators A(U ), U ∈ Kcp( U0 Dγ ) , also satisfies all the structural assumptions and since A(U (t))γ U (t) ≤ Ccp( U0 Dγ ) p( U0 Dγ ) due to (6.93), we can use Corollary 5.2 to conclude the existence of a global solution U (t; U0 ) in the function space (6.95) in which TU0 is an arbitrary finite time. Clearly, (6.96) implies that U (t; U0 ) Dγ ≤ p( U0 Dγ ),

0 ≤ t < ∞, U0 ∈ Dγ .

(6.97)

Furthermore, by the same arguments as in Proposition 6.1, the following stronger estimate is verified. Proposition 6.5 For U (t; U0 ), it holds that U (t; U0 ) D ≤ (t γ −1 + 1)p1 ( U0 Dγ ), with some continuous increasing function p1 (·).

0 < t < ∞, U0 ∈ Dγ ,

(6.98)

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6 Dynamical Systems

In view of γ < η < 1, we can use the moment inequality (2.119). Then, (6.97) and (6.98) yield that U (t; U0 ) W ≤ (t γ −η + 1)pη ( U0 Dγ ),

0 ≤ t < ∞, U0 ∈ Dγ ,

(6.99)

with some continuous increasing function pη (·). For U0 ∈ Dγ , we set S(t)U0 = U (t; U0 ). The uniqueness of solutions implies that S(t) is a nonlinear semigroup acting on Dγ . Let us show that S(t) is continuous on Dγ . Let 0 < R˜ < ∞. For U0 ∈ K˜ R˜ , it holds that U0 Z < cR˜ and A(U0 )γ U0 X ≤ CcR˜ . So, Corollary 5.4 is applicable for the solutions S(t)U0 , U0 ∈ K˜ R˜ , to see that t η−α S(t)U0 − S(t)V0 W + t β−α S(t)U0 − S(t)V0 Z + S(t)U0 − S(t)V0 Y 0 < t ≤ tR˜ , U0 , V0 ∈ K˜ R˜ .

≤ CR˜ U0 − V0 Dγ ,

(6.100)

On the basis of this estimate, we can deduce a Hölder condition of S(t). Proposition 6.6 With the exponent δ = min{ γ1−η −α , 1}, S(t) satisfies S(t)U0 − S(t)V0 Dγ ≤ LR˜ U0 − V0 δDγ ,

0 ≤ t ≤ tR˜ , U0 , V0 ∈ K˜ R˜ .

Proof For simplicity, let us use the notation U (t) = S(t)U0 and V (t) = S(t)V0 , and let UU (t, s) and UV (t, s) denote the evolution operators for the families of operators AU (t) = A(U (t)) and AV (t) = A(V (t)), respectively. We repeat the similar argument as in the proof of Theorem 5.9. Indeed, as shown in (5.71), U (t) − V (t) = UU (t, 0)(U0 − V0 ) + [UU (t, 0) − UV (t, 0)]V0

t + [UU (t, s) − UV (t, s)]F (V (s)) ds 0

+

t

UU (t, s)[F (U (s)) − F (V (s))] ds.

0

Here, by (6.93), AU (t)γ UU (t, 0)(U0 − V0 ) ≤ C ˜ A(U0 )γ (U0 − V0 ) ≤ C ˜ U0 − V0 D . γ R R X X By the same arguments as for (5.39) but substituting γ for β, we verify that AU (t)γ [UU (t, 0) − UV (t, 0)]V0 X

t ≤ CR˜ V0 Dγ (t − s)−γ s γ −1 U (s) − V (s) Y ds. 0

Hence, by (3.83), AU (t)γ [UU (t, 0) − UV (t, 0)]V0 ≤ C ˜ U0 − V0 D . γ R X

7 Dynamical Systems for Quasilinear Evolution Equations

301

Similarly, by the same arguments as for (5.40), we verify that

t AU (t)γ [UU (t, s) − UV (t, s)]F (V (s)) ds 0



≤ CR˜ F (V (·)) Fγ ,σ

X

t

(t − s)−γ s γ −1 U (s) − V (s) Y ds.

0

Hence, by (6.100),

t AU (t)γ [UU (t, s) − UV (t, s)]F (V (s)) ds ≤ CR˜ U0 − V0 Dγ . 0

X

In the meantime, by (6.91),

t AU (t)γ U (t, s)[F (U (s)) − F (V (s))] ds U 0

≤ CR˜

t

X

(t − s)−γ [ U (s) − V (s) W

0

+ ( U (s) W + V (s) W ) U (s) − V (s) Z ] ds. We notice from (6.99) and (6.100) that U (s) − V (s) W ≤ ( U (s) W + V (s) W )1−δ U (s) − V (s) δW ≤ CR˜ s γ (1−δ)+αδ−η U0 − V0 δDγ with any 0 < δ ≤ 1. So, taking δ = min{ γ1−η −α , 1}, we have

t

(t − s)−γ s γ (1−δ)+αδ−η ds ≤ C.

0

Similarly, ( U (s) W + V (s) W ) U (s) − V (s) Z 



≤ CR˜ ( U (s) W + V (s) W )( U (s) Z + V (s) Z )1−δ U (s) − V (s) δZ 



≤ CR˜ s γ −η+(α−β)δ U0 − V0 δDγ 1−η with any 0 < δ  ≤ 1. So, taking δ  = min{ β−α , 1}, we have



t



(t − s)−γ s γ −η+(α−β)δ ds ≤ C.

0

Hence, we conclude that

t AU (t)γ ≤ C ˜ U0 − V0 δ U (t, s)[F (U (s)) − F (V (s))] ds U Dγ R 0

X

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with the exponent δ = min{δ, δ  }. Summing up these estimates, we obtain the desired one.



The a priori estimate (6.96) implies that 

{S(t)U0 ; 0 ≤ t < ∞} ⊂ K˜ p(R) ˜ .

U0 ∈K˜ R˜

We now apply Proposition 6.6 on the ball K˜ p(R) ˜ . Proposition 6.7 For any 0 < R˜ < ∞, it holds that n+1

S(t)U0 − S(t)V0 Dγ ≤ Ln+1˜ U0 − V0 δDγ , p(R)

˜ t ∈ [ntp(R) ˜ , (n + 1)tp(R) ˜ ]; n = 0, 1, 2, . . . ; U0 , V0 ∈ KR˜ .

(6.101)

Proof In fact, when n = 0, this is already verified by Proposition 6.6 because of ˜ Assume that this is true for n − 1. For nt ˜ < t ≤ (n + 1)t ˜ , we R˜ ≤ p(R). p(R) p(R) have S(t) = S(t − ntp(R) ˜ )S(ntR˜ ); then (6.101) is easily obtained since S(ntp(R) ˜ )U0 and S(nt ˜ )V0 are in K˜ ˜ .  p(R)

p(R)

Estimate (6.101) means that S(t) is Hölder continuous on the ball K˜ R˜ and the Hölder exponent is uniform in t on any finite time interval. Meanwhile, for each U0 ∈ K˜ R˜ , S(·)U0 is continuous from [0, ∞) to Dγ . Then, the mapping G : [0, ∞) × K˜ R˜ → Dγ , where G(t, U0 ) = S(t)U0 , is continuous. As R˜ can be taken arbitrarily, we conclude that S(t) is a continuous semigroup on Dγ . Thus, under (6.87), (6.88), (6.89), (6.91), (6.92), (6.93), and (6.96), problem (6.85) determines a dynamical system (S(t), Dγ , Dγ ), where Dγ = D(A(0)γ ). It is often important to construct a dynamical system in the universal space X instead of Dγ . To define such a dynamical system, we introduce the phase space XR˜ =



{S(t)U0 ; 0 ≤ t < ∞},

0 < R˜ < ∞.

(6.102)

U0 ∈K˜ R˜

Clearly, XR˜ is invariant under S(t) for every 0 < t < ∞, and XR˜ ⊂ K˜ p(R) ˜ due to (6.96). With the aid of Theorem 5.9, we can then repeat the same arguments as in Proposition 6.3 to deduce the Lipschitz continuity of S(t) in the X norm. Proposition 6.8 For any 0 < R˜ < ∞, it holds that S(t)U0 − S(t)V0 X ≤ Ln+1˜ U0 − V0 X , p(R)

t ∈ [ntp(R) ˜ , (n + 1)tp(R) ˜ ]; n = 0, 1, 2, . . . ; U0 , V0 ∈ XR˜ .

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303

Consequently, G : [0, ∞) × XR˜ → X is continuous, where XR˜ is equipped with the induced distance from X. Thus, problem (6.85) defines a dynamical system (S(t), XR˜ , X) in the space X, too. Clearly, K˜ R˜ ⊂ XR˜ for 0 < R˜ < ∞.

7.3 Absorbing and Invariant Compact Set We now add two assumptions that  X is a reflexive Banach space, D is compactly embedded in X.

(6.103)

As for Proposition 6.4, these assumptions immediately imply the following result. D

Proposition 6.9 Any closed bounded ball B (0; R), 0 < R < ∞, of D is a compact set of X. We assume also the dissipative condition that there exists a number C˜ > 0 for which the following estimate is valid. For any bounded subset B of Dγ , there is a time tB such that ˜ sup sup S(t)U0 Dγ ≤ C.

U0 ∈B t≥tB

(6.104)

As for Proposition 6.5 (actually Proposition 6.1), this condition implies a stronger one. There exists a number C˜ 1 > 0 such that, for any bounded subset B of Dγ , it holds that sup sup S(t)U0 D ≤ C˜ 1

U0 ∈B t≥tB

(6.105)

with a suitable time tB depending on B. Using this constant C˜ 1 , we introduce the closed ball D B = B (0; C˜ 1 ).

Then (6.105) obviously means that B is an absorbing set. As noticed above, B is a compact set of X. We next introduce the set X=



S(t)B ⊂ B

(closure in X).

(6.106)

tB ≤t<∞

Note that, as B is a bounded subset of Dγ , the images S(t)B of B are absorbed by itself for all t ≥ tB . Then, by the same argument as in the proof of Theorem 6.2, X is shown to be an absorbing and invariant set. Under (6.103) and (6.104), we have observed that the asymptotic behavior of any dynamical system (S(t), XR , X) is reduced to that of a dynamical system (S(t), X, X) whose phase space is a compact set of X and is a bounded set of D.

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6 Dynamical Systems

7.4 Exponential Attractors We are ready to apply Theorem 6.15 to the system (S(t), X, X). We use Theorem 5.9 to conclude for a sufficiently small time t ∗ > 0 that the operator S(t ∗ ) from X into Dγ satisfies the compact perturbation condition (6.38)–(6.39) (with δ = 0 and Z = Dγ ). Similarly, by Proposition 6.8, S(t)U0 − S(t)V0 X ≤ CX U0 − V0 X ,

0 ≤ t ≤ t ∗ , U0 , V0 ∈ X,

with some constant CX > 0. Meanwhile, by (5.45) and (6.90), t [F (S(τ )V ) − A(S(τ )V )S(τ )V ] dτ S(t)V0 − S(s)V0 X = 0 0 0 s

≤ CX (t − s),

X

∗

0 ≤ s < t ≤ t , V0 ∈ X.

Hence, condition (6.48) is fulfilled. Under (6.87), (6.88), (6.89), (6.91), (6.92), (6.93), (6.96), (6.103), and (6.104), we have proved that (S(t), X, X) possesses a family of exponential attractors M. Let us equally verify that M are exponential attractors of the dynamical system (S(t), X, Dγ ), too, where X is equipped with the induced distance from Dγ . Let M be an exponential attractor of (S(t), X, X) satisfying (6.17). By the moment inequality, we have hDγ (S(t)X, M) ≤ CX h(S(t)X, M)1−γ ≤ CX e−(1−γ )kt ,

0 ≤ t < ∞,

where hDγ (·, ·) denotes the Hausdorff pseudo-distance in Dγ . In addition, by Theorem 1.3, the fractal dimension dDγ (M) of M in Dγ is estimated by (1 − γ )−1 dX (M), where dX (M) denotes that of M in X. These facts show that M is an exponential attractor of (S(t), X, Dγ ), too.

8 Stationary Solutions to Quasilinear Equations 8.1 Equilibria of Dynamical System Let U be a stationary solution of (6.85), i.e., U ∈ D and A(U )U = F (U ). We assume that (6.85) determines a dynamical system (S(t), Dγ , Dγ ). More precisely, all the structural conditions (6.87), (6.88), (6.89), (6.91), (6.92), and (6.93) are satisfied and the a priori estimates (6.96) hold for all initial values in Dγ . So, U is an equi˜ Then, U Z < R, librium of (S(t), Dγ , Dγ ). Let R˜ > 0 be such that U Dγ < R. Z where R = cR˜ (see (6.94)), and let KR be the open ball B (0; R) of Z. In this section, we shall investigate the stability and instability of U by applying Theorems 6.9 and 6.10, respectively.

8 Stationary Solutions to Quasilinear Equations

305

8.2 Exponential Stability of U Let us assume the following conditions. The operator A : Z → L(D, X) is Fréchet differentiable at U with respect to the Y norm. Its differential A (U ) is a bounded operator from Y to L(D, X) and, equivalently, is a bilinear operator from Y × D into X. So, the derivative is written by A (U )[U, U˜ ] ∈ X for [U, U˜ ] ∈ Y × D. Therefore,  ΔA (h) = A(U + h) − A(U ) − A (U )[h, ·], A (U ) ∈ L(Y, L(D, X)), (6.107) limh→0 oA (h)/ h Y = 0. oA (h) = ΔA (h) L(D,X) , Here, oA (h) is defined for h ∈ Z such that U + h ∈ KR , and oA (h) is uniformly bounded for h such that h ∈ KR − U . The nonlinear operator F : W → X is also Fréchet differentiable at U , i.e.,  ΔF (h) = F (U + h) − F (U ) − F  (U )h, F  (U ) ∈ L(W, X), (6.108) limh→0 oF (h)/ h W = 0. oF (h) = ΔF (h) X , Here, oF (h) is defined for all h ∈ W . In view of (6.91), oF (h) ≤ C h W for all h ∈ W such that h W < 1. From now on, let h denote a small variable in Dγ such that h Dγ < R˜ − U Dγ , and let Uh (t) = S(t)(U + h) be the solution to (6.85) with initial value U + h. According to Theorem 5.9, if T > 0 is sufficiently small, then t η Uh (t) − U W + t β Uh (t) − U Z + t α Uh (t) − U Y ≤ C h X ,

0 ≤ t ≤ T. (6.109)

We consider the linear problem  dV

h

dt

+ A(U )Vh + A (U )[Vh , U ] = F  (U )Vh ,

0 < t ≤ T,

Vh (0) = h,

(6.110)

in X with the initial value h. The linear operator of this problem can be regarded as a perturbation of A(U ) in a sense that condition (3.8) is fulfilled with ν˜ = 1 − η. Therefore, by Theorem 3.3, A(U ) + A (U )[·, U ] − F  (U ) generates an analytic semigroup e−tA on X, where A = A(U ) + A (U )[·, U ] − F  (U ). In addition, by Theorem 3.14 (cf. Remark 3.7), the solution of the problem is given by Vh (t) = e−tA h. In particular, we notice by maximal regularity (3.130) that ˜

A(U )Vh ∈ Fβ,σ˜ ((0, T ]; X) ˜ σ˜ such that 0 < σ˜ < β˜ < min{1 − η, γ } and that with some exponents β, Vh (t) D ≤ Ct β−1 A(U )β h , 0 < t ≤ T .

(6.111)

(6.112)

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6 Dynamical Systems

We set ΔU (t) = Uh (t) − U − Vh (t), 0 < t ≤ T . It is observed that A(Uh )Uh − A(U )U − A(U )Vh − A (U )[Vh , U ] = A(Uh )ΔU + A (U )[ΔU , U ] + A (U )[Uh − U , Vh ] + ΔA (Uh − U )(U + Vh ) and F (Uh ) − F (U ) − F  (U )Vh = F  (U )ΔU + ΔF (Uh − U ). This then means that ΔU (t) is regarded as a solution to the linear problem  dΔ   U dt + A(Uh (t))ΔU + A (U )[ΔU , U ] − F (U )ΔU = G(t), ΔU (0) = 0, where the external force function is given by G(t) = −A (U )[Uh (t) − U , Vh (t)] − ΔA (Uh (t) − U )(U + Vh (t)) + ΔF (Uh (t) − U ). The linear operators in the equation are regarded as perturbations of the operators A(t) = A(Uh (t)), 0 ≤ t ≤ T . More precisely, we can show that A(t) = A(Uh (t)) and B(t) ≡ B = A (U )[·, U ] − F  (U ) satisfy the assumptions in Theorems 3.13 and 3.14. In fact, it is already known that A(t) satisfy assumptions (3.27), (3.28), (3.29), (3.30), and (3.31) with ν = 1 and with some exponent μ = β − α > 0 (due to (6.89) and (5.48)). In the meantime, B is a bounded operator from W into X. Therefore, it satisfies (3.117) and (3.118) with μ˜ = 1 and with some exponent 0 < ν˜ < 1 − η. Moreover, by using (3.92) with θ = η and ϕ = 1 − ν˜ , we observe that     B A(t)ν˜ −1 − A(s)ν˜ −1 ≤ B L(W,X) A(t)η A(t)ν˜ −1 − A(s)ν˜ −1 ≤ C|t − s|η . Hence, (3.119) and (3.120) are satisfied. Consequently, there exists an evolution operator U˜ (t, s) for the operators A(t) + B. In addition, ΔU is written by

t ΔU (t) = U˜ (t, s)G(s) ds, 0 < t ≤ T . 0 ˜

Here we used a fact that G ∈ Fβ,σ˜ ((0, T ]; X) due to (6.111). Utilizing this expression, we want to estimate the norm ΔU (t) Dγ . By (3.124) and (6.93),

t ΔU (t) Dγ ≤ C A(t)γ ΔU (t) X ≤ C (t − s)−γ G(s) X ds. 0

Here, it follows from (6.109) and (6.112) that A (U )[Uh (s) − U , Vh (s)] X ≤ C Uh (s) − U Y Vh (s) D ≤ Cs −α h X s β−1 A(U )β h X ,

8 Stationary Solutions to Quasilinear Equations

307

ΔA (Uh (s) − U )(U + Vh (s)) X ≤ oA (Uh (s) − U ) U + Vh (s) D −α ≤ oA (Uh (s) − U ) Uh (s) − U −1 h X Y s  β−1  A(U )β h + 1 , × s X

and −η ΔF (Uh (s) − U ) X ≤ oF (Uh (s) − U ) Uh (s) − U −1 W s h X .

Hence,  G(s) X ≤ C(s β−α−1 + s −η ) A(U )β h X + oA (Uh (s) − U ) Uh (s) − U −1 Y

 + oF (Uh (s) − U ) Uh (s) − U −1 W h X . As a consequence,

ΔU (t) Dγ / h X ≤ C

0

t

 (t − s)−γ (s β−α−1 + s −η ) A(U )β h X

+ oA (Uh (s) − U ) Uh (s) − U −1 Y

 + oF (Uh (s) − U ) Uh (s) − U −1 W ds. β/γ

1−β/γ

, h X → 0 implies Since A(U )β h X ≤ C A(U )γ h X h X A(U )β h X → 0. Similarly, h X → 0 implies Uh (s) − U W → 0 and Uh (s) − U Y → 0 due to (6.109). Therefore, by virtue of the dominate convergence theorem, we conclude that, as h X → 0 (a fortiori as h Dγ → 0), 0 ≤ ΔU (t) Dγ / h Dγ ≤ ΔU (t) Dγ / h X → 0.

(6.113)

Thus, under (6.107) and (6.108), the operator S(t) : Dγ → Dγ is Fréchet differentiable at U for any time t such that 0 < t ≤ T . Its differential is given by S(t) U = e−tA

where A = A(U ) + A (U )[·, U ] − F  (U ).

As S(t)U = U , we verify that S(2t) = S(t)S(t) is also Fréchet differentiable at U . Repeating this argument, we conclude that S(t) is Fréchet differentiable at U for every time 0 < t < ∞. Let us now assume that U satisfies the spectral condition σ (A) = σ (A(U ) + A (U )[·, U ] − F  (U )) ⊂ {λ ∈ C; Re λ > 0}.

(6.114)

This condition, together with Theorem 3.3, implies that A is a sectorial operator, i.e., A satisfies (2.92) and (2.93). Proposition 2.5 then provides the exponentially decaying estimate e−tA L(X) ≤ Ce−dt , 0 ≤ t < ∞. Then, we can repeat the same

308

6 Dynamical Systems

arguments as in Sect. 6.2 for verifying that S(t ∗ ) is a contraction at U . Hence, by Theorem 6.10, we conclude that U is exponentially stable. We have thus proved under (6.107), (6.108), and (6.114) that U is an exponentially stable equilibrium of (S(t), Dγ , Dγ ).

8.3 Unstable Manifold of U For simplicity, we will consider only the case where the semilinear term is given by F (U ) = kU with some complex number k. (For the general case, see Remark 6.4 below.) So, U is a stationary solution such that A(U )U = kU . We assume the following conditions. The operator A : Z → L(D, X) is Fréchet differentiable in a neighborhood B Z (U ; r) of U with respect to the Y norm, and the derivative satisfies the following estimates: A (U )[h, U˜ ] X ≤ N1 h Y U˜ D ,

U ∈ B Z (U ; r), h ∈ Z, U˜ ∈ D, (6.115)

{A (U1 ) − A (U2 )}[h, U˜ ] X ≤ N2 U1 − U2 Z h Y U˜ D , 



U1 , U2 ∈ B Z (U ; r), h ∈ Z, U˜ ∈ D,

(6.116)

with some constants Ni > 0 (i = 1, 2). Put ΔA (U ; h) = A(U + h) − A(U ) − A (U )[h, ·]. Then, (6.116) implies that ΔA (U ; h) L(D,X) ≤ C h Z h Y ,

U + h, U ∈ B Z (U ; r).

(6.117)

Consider an open neighborhood B Dγ (U ; r1 ) of U . Let U0 ∈ B Dγ (U ; r1 ) be an initial value, and let U (t) = S(t)U0 . By Proposition 6.6 and (6.92), we have U (t) − U Z ≤ C U0 − U δDγ ,

0 ≤ t ≤ T1 , U0 ∈ B Dγ (U ; r1 ),

for sufficiently small time T1 > 0. So, if r1 > 0 and T1 > 0 are sufficiently small, then U (t) ∈ B Z (U ; r) for 0 ≤ t ≤ T1 . Consequently, A (U (t)) is defined. Consider also a small vector variable h ∈ B Dγ (0; r2 ) and let Uh (t) = S(t)(U0 + h). By Theorem 5.9, we have t α Uh (t) − U (t) Y ≤ C h X ,

0 < t ≤ T2 , U0 ∈ B Dγ (U ; r1 ), h ∈ B Dγ (0; r2 ), (6.118)

for sufficiently small T2 > 0. We introduce the linear problem  dV  h dt + A(U (t))Vh + A (U (t))[Vh , U (t)] = kVh , Vh (0) = h,

0 < t ≤ T,

(6.119)

8 Stationary Solutions to Quasilinear Equations

309

in X with T = min{T1 , T2 }. The linear operators in the equation can then be treated as perturbations of operators A(t) = A(U (t)). In fact, we verify the following lemma. Lemma 6.2 Set B(t) = A (U (t))[·, U (t)] − k. Then, B(t)A(t)−β ≤ Ct γ −1 , 0 < t ≤ T , L(X) B(t)A(t)−β − B(s)A(s)−β ≤ Cs γ −1−σ (t − s)σ , L(X)

(6.120) 0 < s < t ≤ T, (6.121)

with any σ such that 0 < σ < β − α. Proof From (5.66), (6.92), and (6.115) it follows that B(t)A(t)−β ≤ B(t)A(t)−α ≤ C U (t) D ≤ Ct γ −1 . Hence, (6.120) is verified. In the meantime, B(t)A(t)−β − B(s)A(s)−β ≤ [B(t) − B(s)]A(t)−β   + B(s) A(t)−β − A(s)−β . By (6.116), [B(t) − B(s)]A(t)−β ≤ C[ U (t) − U (s) Z U (t) D + U (t) − U (s) D ]. Here, by (6.92), U (t) − U (s) Z ≤ C A(t)β [U (t) − U (s)] X β

1−β

≤ C U (t) − U (s) D U (t) − U (s) X . From (5.66), in particular from A(t)U (t) ∈ Fγ ,σ ((0, T ]; X), it follows that  U (t) − U (s) D ≤ C A(t)U (t) − A(s)U (s) X   + A(t) A(t)−1 − A(s)−1 A(s)U (s) X

≤ Cs

γ −1−σ

with any 0 < σ < β − α. In addition,

(t − s)

dU dt

σ

∈ Fγ ,σ ((0, T ]; X) implies that

U (t) − U (s) ≤ Cs γ −1 (t − s). Therefore,

[B(t) − B(s)]A(t)−β ≤ Cs γ −1−σ (t − s)σ .

(6.122)

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6 Dynamical Systems

We obtain from Theorem 2.25, (5.66), and (6.115) that     B(s) A(t)−β − A(s)−β ≤ C B(s)A(0)−α A(0)α A(t)−β − A(s)−β ≤ C U (s) D U (t) − U (s) Y ≤ Cs γ −1 (t − s)σ . 

This, together with (6.122), then yields (6.121).

We have just verified that A(t) = A(U (t)) and B(t) = A (U (t))[·, U (t)] − k satisfy (3.117), (3.118), and (3.119) with μ˜ = γ and ν˜ = 1 − β. Consequently, there exists an evolution operator U˜ (t, s) for the family of operators A(t) + B(t) by which the solution of (6.119) is written as Vh (t) = U˜ (t, 0)h. By the maximal regularity (3.130), ˜

A(U )Vh ∈ Fβ,σ˜ ((0, T ]; X)

(6.123)

˜ σ˜ such that 0 < σ˜ < β˜ < γ − β, and with some exponents β, Vh (t) D ≤ Ct β−1 A(U0 )β h , 0 < t ≤ T .

(6.124)

We now put ΔU (t) = Uh (t) − U (t) − Vh (t),

0 < t ≤ T.

It is observed that A(Uh )Uh − A(U )U − A(U )Vh − A (U )[Vh , U ] = A(Uh )ΔU + A (U )[ΔU , U ] + A (U )[Uh − U, Vh ] + ΔA (U ; Uh − U )(U + Vh ). This then means that ΔU (t) is a solution to the linear problem  dΔU  dt + A(Uh (t))ΔU + A (U )[ΔU , U ] − kΔU = G(t), ΔU (0) = 0,

0 < t ≤ T,

where G(t) = −A (U (t))[Uh (t) − U (t), Vh (t)] − ΔA (U (t); Uh (t) − U (t))[U (t) + Vh (t)]. By the same arguments as before, the linear operators are seen to be perturbations of the operators A(Uh (t)) with μ˜ = γ and ν˜ = 1 − β. Therefore, by using an evolution operator U˜ h (t, s) for the linear operators A(Uh (t)) + A (U (t))[·, U (t)] − k, we obtain the expression

t

ΔU (t) = 0

U˜ h (t, s)G(s) ds,

0 < t ≤ T.

8 Stationary Solutions to Quasilinear Equations

311

Then, in view of (6.117), (6.118), (6.123), and (6.124), we have

t  ΔU (t) Dγ / h X ≤ C (t − s)−γ s β−α−1 A(U0 )β h X 0

 + ΔA (U (s); Uh (s) − U (s)) L(D,X) Uh (t) − U (t) −1 Y ds. We can now repeat the same arguments as in the preceding subsection to verify that, as h X → 0 (a fortiori as h Dγ → 0), 0 ≤ ΔU (t) Dγ / h Dγ ≤ ΔU (t) Dγ / h X → 0. Thus, under (6.115) and (6.116), S(t) : Dγ → Dγ is Fréchet differentiable in a neighborhood B Dγ (U ; r1 ) of U for any time t such that 0 < t ≤ T , T > 0 being sufficiently small, with the derivative S(t) U0 = U˜ (t, 0), where U˜ (t, s) is an evolution operator for a family of the linear operators A(S(t)U0 ) + A (S(t)U0 )[·, S(t)U0 ] − k,

0 ≤ t ≤ T , U0 ∈ B Dγ (U ; r1 ).

In particular, S(t) U = e−tA , where A = A(U ) + A (U )[·, U ] − k. Let us investigate next the regularity of the derivative S(t) U0 with respect to U0 . For this purpose, let Ui ∈ B Dγ (U ; r1 ), Ui (t) = S(t)Ui , and Ai (t) ≡ A(Ui (t)) for i = 1, 2. We denote also by Ui (t, s) the evolution operator for the operators Ai (t). Similarly, let A˜ i (t) ≡ A(Ui (t)) + A (Ui (t))[·, Ui (t)] − k and denote by U˜ i (t, s) the evolution operator for the operators A˜ i (t). Then, it is derived that U˜ 2 (t, 0) − U˜ 1 (t, 0) L(Dγ ) ≤ C U2 − U1 δDγ ,

0 ≤ t ≤ T,

(6.125)

where δ = min{ γ1−η −α , 1} (cf. Proposition 6.6). That is, S(t) U2 − S(t) U1 L(Dγ ) ≤ C U2 − U1 δDγ ,

0 ≤ t ≤ T ; U1 , U2 ∈ B Dγ (U ; r1 ).

In order to prove (6.125), we write A1 (t)γ [U˜ 2 (t, 0) − U˜ 1 (t, 0)]h

t = A1 (t)γ U˜ 1 (t, τ )[A(U1 (τ )) − A(U2 (τ ))] 0

× A(U2 (τ ))−1 A(U2 (τ ))U˜ 2 (τ, s)h dτ

t + A1 (t)γ U˜ 1 (t, τ ){A (U1 (τ )) − A (U2 (τ ))}[U˜ 2 (τ, 0)h, U1 (τ )] dτ 0

+

t

A1 (t)γ U˜ 1 (t, τ )A (U2 (τ ))[U˜ 2 (τ, 0)h, U1 (τ ) − U2 (τ )] dτ.

0

Then, by (6.89), (6.116), and Proposition 6.6, we immediately verify that the first and second integrals in the right-hand side are estimated by C U2 − U1 δDγ h Dγ .

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6 Dynamical Systems

Meanwhile, the estimate for the third one is more profound (it is necessary to extend the known estimate (5.39) or (5.23) to the critical case θ = ν = 1). We can in fact show the following proposition. The proof of this proposition will be described in the next subsection. Proposition 6.10 We have U2 (t) − U1 (t) D ≤ Ct γ −1 U2 − U1 δDγ ,

0 < t ≤ T.

(6.126)

Clearly, (6.126) implies that the third integral is also estimated by C U2 − U1 δDγ h Dγ ; hence, (6.125) has been proved. Let us now make the assumption that the operator A = A(U ) + A (U )[·, U ] − k as an operator of X satisfies the spectrum separation condition σ (A) ∩ {λ ∈ C; Re λ = 0} = ∅.

(6.127)

Let σ− (A) = σ (A) ∩ {λ ∈ C; Re λ < 0} and σ+ (A) = σ (A) ∩ {λ ∈ C; Re λ > 0}, and let X = X− ⊕ X+ be the decomposition of X corresponding to the spectrum separation. By the same arguments as in Sect. 6.3, we can verify that the spectrum σ Dγ (S(t) U ) is separated by the unit circle as (6.20) and the exterior spectrum is given by e−tσ− (A) . We have thus shown that, under (6.115) and (6.116), if U satisfies (6.127), then U is a hyperbolic equilibrium of (S(t), Dγ , Dγ ). The exterior spectrum σ Dγ (S(t) U ) ∩ {λ ∈ C; |λ| > 1} is given by e−tσ− (A) . Therefore, by virtue of Theorem 6.9, there exists a smooth local unstable manifold W+ (U ; O) of U with dimension dim X− which is tangent to the subspace U + X− of Dγ at U . Remark 6.4 It is possible to derive analogous results for the general case where the term F (U ) is not necessarily of the form F (U ) = kU under the conditions F  (U )h X ≤ L1 h W ,

U ∈ W ∩ B Z (U ; r), h ∈ W ;

[F  (U1 ) − F  (U2 )]h X ≤ L2 U1 − U2 Z h W , U1 , U2 ∈ W ∩ B Z (U ; r), h ∈ W, where Li > 0 (i = 1, 2) are some constants. Instead of (6.119), however, we have to handle the more complicated problem  dV   h dt + A(U (t))Vh + A (U (t))[Vh , U (t)] = F (U (t))Vh , 0 < t ≤ T , Vh (0) = h. For constructing an evolution operator for this problem, we must go back to the study of perturbed problems in Chap. 3, Sect. 10 and in fact must generalize the main theorems for the operators B(t) ≡ B1 (t) + B2 (t), each family Bi (t) satisfying conditions (3.117), (3.118), and (3.119) with some exponents 0 < μ˜ i , ν˜ i ≤ 1 depending on i = 1, 2.

8 Stationary Solutions to Quasilinear Equations

313

8.4 Proof of Proposition 6.10 We easily observe that Ui (t) = ekt Ui (t, 0)Ui for i = 1, 2. So, A1 (t)[U2 (t) − U1 (t)] = ekt A1 (t){U2 (t, 0)[U2 − U1 ] + [U2 (t, 0) − U1 (t, 0)]U1 }. Clearly, the first term in the right-hand side is estimated by A1 (t)U2 (t, 0)A2 (0)−γ A2 (0)γ [U2 − U1 ] ≤ Ct γ −1 U2 − U1 D . γ X In the same way as in the proof of Proposition 5.1, the second term is written by

t A1 (t)[U2 (t, 0) − U1 (t, 0)]U1 = A1 (t)U1 (t, τ )[A1 (t) − A2 (t)]U2 (τ, 0)U1 dτ 0



+

t

A1 (t)U1 (t, τ )[A1 (τ ) − A2 (τ )

0

− A1 (t) + A2 (t)]U2 (τ, 0)U1 dτ = J1 (t) + J2 (t). Furthermore, J1 (t) is decomposed into the form

t J1 (t) = [A1 (t)U1 (t, τ ) − A1 (t)e−(t−τ )A1 (t) ][A2 (t) − A1 (t)]U2 (τ, 0)U1 dτ 0



t

+

A1 (t)e−(t−τ )A1 (t) [A2 (t) − A1 (t)][1 − e−(t−τ )A2 (τ ) ]U2 (τ, 0)U1 dτ

0 t

+

A1 (t)e−(t−τ )A1 (t) [A2 (t) − A1 (t)][e−(t−τ )A2 (τ ) − U2 (t, τ )]

0

× U2 (τ, 0)U1 dτ + [1 − e−tA1 (t) ][A2 (t) − A1 (t)]U2 (t, 0)U1 . t Here, we used the formula 0 A1 (t)e−(t−τ )A1 (t) dτ = 1 − e−tA1 (t) . Then, by applying the estimates for the evolution operator and (6.100), we deduce the estimate J1 (t) X ≤ Ct γ −1 U2 − U1 Dγ . For estimating J2 (t), we write [Ai (t) − Ai (τ )]U˜ =



1

A (θ Ui (t) + (1 − θ )Ui (τ ))[Ui (t) − Ui (τ ), U˜ ] dθ

0

for i = 1, 2. Then, {[A2 (t) − A2 (τ )] − [A1 (t) − A1 (τ )]}U˜

1 = A (θ U1 (t) + (1 − θ )U1 (τ ))[(U2 (t) − U2 (τ )) − (U1 (t) − U1 (τ )), U˜ ] dθ 0

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6 Dynamical Systems

+

1

{A (θ U2 (t) + (1 − θ )U2 (τ ))

0

− A (θ U1 (t) + (1 − θ )U1 (τ ))}[U2 (t) − U2 (τ ), U˜ ] dθ. Then, by (6.115) and (6.116), there exists a positive exponent ε > 0 such that {[A2 (t) − A2 (τ )] − [A1 (t) − A1 (τ )]}U˜ X ≤ C(t − τ )ε U2 − U1 δDγ U˜ D . This estimate furthermore implies that J2 (t) X ≤ Ct γ −1 U2 − U1 δDγ . We have thus accomplished the proof of Proposition 6.10.

Notes and Further Researches We refer the reader for the general matters of dynamical systems in infinitedimensional spaces to the books by Ladyzhenskaya [Lad91], Babin–Vishik [BV92], Temam [Tem97], and Robinson [Rob01]. Especially, for the attractors, we cite the literature Cholewa–Doltko [CD00], Chepyzhov–Vishik [CV02], and Miranville– Zelik [MZ08]. For the inertial manifolds, see Foias–Sell–Temam [FST88] and [Tem97, Chap. 8]. In constructing smooth stable and unstable manifolds in Theorem 6.8, we followed the methods due to Wells [Wel76], cf. also [BV92, Chap. 5] and [Tem97, Chap. VII: Part B]. The notion of exponential attractors was founded by Eden–Foias–Nicolaenko– Temam [EFNT94]. The squeezing property (6.45)–(6.46) was also introduced by the founders. Efendiev–Miranville–Zelik [EMZ00] introduced the condition of compact perturbation of a contraction operator for constructing exponential attractors. (Actually in [EMZ00], δ was taken as δ = 0. For the general case, see Takei and Yagi [TY06].) The two types of conditions (6.28)–(6.29) and (6.38)–(6.39) may seem to be very close. We see, however, that the squeezing property (6.45)–(6.46) implies (6.28)–(6.29) but not (6.38)–(6.39). In the meantime, there exists a semigroup S(t) which fulfills (6.28)–(6.29) but does not (6.38)–(6.39) (see Chap. 13). It is in fact possible to verify the existence of exponential attractors in various mathematical models. After the notion had been introduced by Eden et al., almost hundred papers were already devoted to constructing exponential attractors for dissipative nonlinear systems. We here quote only some results concerning the reaction– diffusion equations [BN95, EMZ04, GP07], the damped wave equations [FGMZ04, MPZ07], Navier–Stokes equations [EFN93, EFNS93, KB06], Ginzburg–Landau equation [Gao95, LL08], Cahn–Hilliard equations [Gal08, MZ04], the phase field systems [BHC93, MZ02, GP05], and others [Du03, EM93, FaN96, Gal98]. In Chaps. 8, 9, 10, 12, 13, 14, and 15, we will also utilize the squeezing property and the two types of compact perturbation of a contraction operator in order to construct exponential attractors for the dynamical systems determined from the model equations under consideration.

Notes and Further Researches

315

The continuous dependence of exponential attractors on a parameter was also investigated by the founders Eden–Foias–Nicolaenko–Temam in [EFNT94, Chap. 4]. Theorem 6.17 then provides us a very general result on this parameter dependence. Indeed, the theorem indicates that, if the semigroup Sξ (t) satisfies the compact perturbation condition of type I uniformly for the parameter ξ and if Sξ (t) converges to S0 (t) only on a finite time interval [0, t ∗ ], then the corresponding exponential attractor Mξ converges to M0 . This result was obtained by Efendiev–Yagi [EY05]. A similar result can be proved under the compact perturbation condition of type II, too.

Chapter 7

Numerical Analysis

This chapter is devoted to numerical analysis for nonlinear abstract parabolic evolution equations. We consider approximate problems by discretizing the spatial variable in the Cauchy problems of the forms (4.1) and (5.43) in Banach spaces. We investigate the order of convergence for the approximation problems. In the first half, we show local estimates of convergence. We introduce the approximate equations ξ dU ξ ), ξ = Fξ (U + Aξ U dt

0 < t < ∞,

and ξ dU ξ )U ξ = Fξ (U ξ ), 0 < t < ∞, + Aξ (U dt in Banach spaces Xξ , where 0 ≤ ξ ≤ 1, for the semilinear and quasilinear equations in (4.1) and (5.43), respectively. Here, Xξ denote a family of approximate spaces such that X0 = X is an underlying Banach space in which (4.1) and (5.43) are considered, and for 0 < ξ ≤ 1, Xξ are finite-dimensional subspaces of X. Under suitable structural assumptions on Xξ , Aξ , Aξ (U ), and Fξ (U ), we derive order estimates of convergence of approximate solutions in a finite interval [0, T ]. In the second half, we show some global estimates. We assume that the Cauchy problems for the approximate equations mentioned above have global solutions and determine dynamical systems (Sξ (t), Xξ , Xξ ) for 0 ≤ ξ ≤ 1. Let us assume furthermore that Sξ (t) satisfy conditions (6.38)–(6.39) uniformly for 0 ≤ ξ ≤ 1. Consequently, each dynamical system (Sξ (t), Xξ , Xξ ) has exponential attractors. We then establish that the convergence of semigroups Sξ (t) → S0 (t) as ξ → 0, together with that of phase spaces Xξ → X0 , implies the convergence of exponential attractors with respect to the distance (1.3).

1 Semilinear Evolution Equations We consider the Cauchy problem for a semilinear equation A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_7, © Springer-Verlag Berlin Heidelberg 2010

317

318

7 Numerical Analysis

 dU dt

+ AU = F (U ),

0 < t < ∞,

(7.1)

U (0) = U0 ,

in a Banach space X. Here, A is a sectorial operator of X with angle ωA < π2 , that is, satisfies (2.92) and (2.93). The operator F is a nonlinear operator from D(Aβ ) into X, where 0 < β < 1. We assume that F satisfies the Lipschitz condition F (U ) − F (V ) ≤ ϕ(Aβ U  + Aβ V )Aβ (U − V ),

U, V ∈ D(Aβ ), (7.2)

where ϕ(·) is some continuous increasing function. This means that (4.2) is satisfied with β = η. The initial value U0 is taken in D(Aγ ), where γ is such that β ≤ γ ≤ 1. By Theorems 4.1 and 4.2, there exists a unique solution to (7.1) on an interval [0, T ] in the function space:  U ∈ C((0, T ]; D(A)) ∩ C([0, T ]; D(Aγ )) ∩ C1 ((0, T ]; X), (7.3) AU ∈ Fγ ,σ ((0, T ]; X), with 0 < σ < 1 − β.

1.1 Approximate Problems Let {Xξ }0<ξ ≤1 be a family of finite-dimensional subspaces of X such that Xξ ⊂ D(Aβ ). For each 0 < ξ ≤ 1, the subspace Xξ is equipped with the induced norm of X. The space Xξ is an approximate subspace to X in the sense that there exists a projection Pξ : X → Xξ such that sup0<ξ ≤1 Pξ L(X,Xξ ) < ∞ and that, as ξ → 0,  (1 − Pξ )A−1 L(X) ≤ C0 Oξ , (7.4) 1−β Aβ (1 − Pξ )A−1 L(X) ≤ Cβ Oξ . 1−β

Here, Oξ denotes a quantity which tends to 0 as ξ → 0, namely, Oξ and Oξ denote the approximation rates of the space Xξ to the spaces X and D(Aβ ), respectively. We consider a family {Aξ }0<ξ ≤1 of approximate operators to A and make the following structural assumptions. For 0 < ξ ≤ 1, Aξ is a bounded sectorial operator ω < π2 ,  ω being some fixed angle, and satisfies (2.93) of Xξ with angle ωAξ ≤   ≥ 1. The domain D(Aβ ) is embedded in D(Aβ ) with some uniform constant M ξ  > 0 such that uniformly in ξ , that is, there exists a uniform constant D  β    A U Aβ U ξ  ≤ D ξ , ξ

 ∈ Xξ , 0 < ξ ≤ 1. U

(7.5)

In addition, as ξ → 0, Aξ converges to A in the sense that (1 − Rξ )A−1 L(X) ≤ D1 Oξ ,

(7.6)

where Rξ stands for the Ritz operator from D(A) into Xξ which is defined by Rξ = A−1 ξ Pξ A. Finally, we assume that

1 Semilinear Evolution Equations

319

Aξ L(Xξ ) ≤ D2 Oξ−1 .

(7.7)

We define the nonlinear operator Fξ (U ) = Pξ F (U ). So, Fξ is an operator from Xξ into itself. Using Aξ and Fξ , we here formulate the approximate problem ⎧ ⎨ d Uξ + A U ξ ), 0 < t < ∞, ξ ξ = Fξ (U dt (7.8) ⎩U  ,  (0) = U 0,ξ

ξ

0,ξ is given by U 0,ξ = Pξ U0 . For the in Xξ to problem (7.1). The initial value U initial value, we assume that  β  (7.9) sup Aξ Pξ U0  < ∞. 0<ξ ≤1

β

In view of (7.5), Fξ and Aξ are seen to fulfill a similar condition to (7.2) in the space Xξ . Again, by Theorem 4.1, (7.8) possesses a unique solution on an interval [0, T ] in the function space: ξ ∈ Fβ,σ ((0, T ]; Xξ ). Aξ U

ξ ∈ C((0, T ]; Xξ ) ∩ C1 ((0, T ]; Xξ ), U

(7.10)

 D,  and others appearing in (7.2), (7.5), and Since the angle  ω and the constants M, (7.9) are independent of ξ , it is deduced that the interval [0, T ] is uniform for ξ ; moreover, it coincides with that of (7.3). In particular, we have  β  ξ (t) < ∞. (7.11) sup sup Aξ U 0<ξ ≤1 0≤t≤T

 will denote a universal constant which is determined by the exIn what follows, C ponents and constants appearing in the structural assumptions on Aξ . In particular,  is independent of 0 < ξ ≤ 1. C

1.2 Local Estimates of Convergence ξ (t). We can give error estimates between U (t) and U ξ be solutions Theorem 7.1 Under (2.92), (2.93), (7.2), (7.5), and (7.9), let U and U to (7.1) and (7.8) in the function spaces (7.3) and (7.10), respectively. Then, (7.4), (7.6), and (7.7) provide that ξ (t)] ≤ CU t γ −1 O , 0 < ξ ≤ 1, 0 < t ≤ T , (7.12) Aβ [U (t) − U ξ  ξ (t) ≤ CU t γ −1 Oξ + tO 1−β , 0 < ξ ≤ 1, 0 < t ≤ T , (7.13) U (t) − U ξ 1−β

CU > 0 being some constant depending on U .

320

7 Numerical Analysis

Proof We consider the projection of problem (7.1) on Xξ given by  dPξ U 0 < t ≤ T, dt + Pξ AU = Pξ F (U ), Pξ U (0) = Pξ U0 , and combine this with (7.8) to obtain that ⎧d    ⎪ ⎨ dt (Pξ U − Uξ ) + Aξ (Pξ U − Uξ ) = [Aξ Pξ U − Pξ AU ] + Pξ [F (U ) − F (Uξ )], 0 < t ≤ T, ⎪ ⎩ ξ (0) = 0. Pξ U (0) − U Then, by using (3.13) we have  t ξ (t) = ξ (s))] ds Pξ U (t) − U e−(t−s)Aξ Pξ [F (U (s)) − F (U 0



+

t

e−(t−s)Aξ [Aξ Pξ − Pξ A]U (s) ds.

0

ξ (t) can be represented by Therefore, the difference U (t) − U  t ξ (s))] ds ξ (t) = (1 − Pξ )U (t) + e−(t−s)Aξ Pξ [F (U (s)) − F (U U (t) − U 0



t

+

e−(t−s)Aξ [Aξ Pξ − Pξ A]U (s) ds.

(7.14)

0 β

Let us estimate the Aβ or Aξ norms (recall (7.5)) for the terms in the right-hand side. By (7.3) and (7.4), Aβ (1 − Pξ )U (t) ≤ Aβ (1 − Pξ )A−1 L(X) AU (t) ≤ CU t γ −1 Oξ

1−β

.

Similarly, by (7.2) and (7.11),  t     Aβ e−(t−s)Aξ Pξ [F (U (s)) − F (U ξ (s))] ds  ξ   0

 ≤C



t

ξ (s)] ds. (t − s)−β Aβ [U (s) − U

0

Meanwhile, the third term is decomposed into  t β Aξ e−(t−s)Aξ [Aξ Pξ − Pξ A]U (s) ds 0  t β Aξ e−(t−s)Aξ Aξ [(Pξ − 1)A−1 + (1 − Rξ )A−1 ][AU (s) − AU (t)] ds = 0  t β Aξ e−(t−s)Aξ Aξ [(Pξ − 1)A−1 + (1 − Rξ )A−1 ]AU (t) ds = I + II. + 0

2 Quasilinear Evolution Equations

321

By (2.128) (θ = 1), (7.3), (7.4), and (7.6),  t 1−β 1−β I  ≤ CU Oξ (t − s)σ −1 s γ −σ −1 ds ≤ CU t γ −1 Oξ . 0

Here, in view of (7.7), we used the moment inequality  β A 

ξ L(Xξ )

 0 1−β    ξ β  −β ≤ CA L(Xξ ) Aξ (X ) ≤ COξ ξ

(see (2.117)). The term II is rewritten as II = Aξ (1 − e−tAξ )[(Pξ − 1)A−1 + (1 − Rξ )A−1 ]AU (t) β

 γ −1 O 1−β . The function p(t) = t 1−γ Aβ [U (t) − and is estimated by II ≤ Ct ξ ξ (t)], 0 ≤ t ≤ T , has been shown in this way to satisfy the integral inequality U

 t 1−β 1−γ −β γ −1 p(t) ≤ CU Oξ + t (t − s) s p(s) ds . 0

It is possible to solve this inequality easily by the same techniques as in the proof 1−β of Theorem 4.2. As a result, we obtain that p(t) ≤ CU Oξ , 0 ≤ t ≤ T . Hence, estimate (7.12) is proved. To obtain estimate (7.13), it is now sufficient to estimate the X norm for the right-hand side of (7.14) by using (7.2), (7.3), (7.4), (7.6), and (7.12). 

2 Quasilinear Evolution Equations In this section, we consider the quasilinear problem  dU dt + A(U )U = F (U ), 0 < t < ∞, U (0) = U0 ,

(7.15)

in a Banach space X. Here, for U ∈ K, A(U ) is a sectorial operator of X with angle ωA(U ) < π2 and with uniform domain D(A(U )) ≡ D, where K is an open ball K = {U ∈ Z; U Z < R}, 0 < R < ∞, of a second Banach space Z continuously embedded in X. We assume that the spectrum σ (A(U )) is contained in some fixed open sectorial domain σ (A(U )) ⊂ Σω = {λ ∈ C; |arg λ| < ω},

0<ω<

π , U ∈ K, 2

(7.16)

and the resolvent satisfies   (λ − A(U ))−1 

L(X)

≤

M , |λ|

λ∈ / Σω , U ∈ K,

(7.17)

322

7 Numerical Analysis

with a uniform constant M ≥ 1. Furthermore, it is assumed that A(U ) satisfies a Lipschitz condition of the form    A(U ) A(U )−1 − A(V )−1 

L(X)

≤ NU − V Y ,

U, V ∈ K,

(7.18)

with some constant N > 0, where Y is a third Banach space such that Z ⊂ Y ⊂ X with continuous embeddings. The following space conditions are also assumed. There exist two exponents 0 < α < β < 1 such that D(A(U )α ) ⊂ Y and D(A(U )β ) ⊂ Z for every U ∈ K with the estimates  U˜ Y ≤ D1 A(U )α U˜ X , U˜ ∈ D(A(U )α ), U ∈ K, (7.19) U˜ Z ≤ D2 A(U )β U˜ X , U˜ ∈ D(A(U )β ), U ∈ K, with some constants Di > 0 (i = 1, 2). The nonlinear operator F is defined in the space Z and maps Z into X. Furthermore, it satisfies the Lipschitz condition F (U ) − F (V )X ≤ ϕ(U Z + V Z ) U − V Y ,

U, V ∈ Z,

(7.20)

ϕ(·) being a continuous increasing function. The initial value U0 is taken in K and is assumed to satisfy the compatibility condition     1 γ U0 ∈ D A(U0 ) with some max β, < γ ≤ 1. (7.21) 2 Under (7.16), (7.17), (7.18), (7.19), (7.20), and (7.21), Theorems 5.5 and 5.6 (ν = 1 and W = Z) ensure the existence of a unique local solution U to (7.15) on an interval [0, T ] in the function space: 

U ∈ C((0, T ]; D) ∩ Cγ −α ([0, T ]; Y ) ∩ C1 ((0, T ]; X), A(U )U ∈ Fγ ,σ ((0, T ]; X), A(U )γ U ∈ C([0, T ]; X),

(7.22)

with 0 < σ < min{β − α, 1 − β}.

2.1 Approximate Problems Let {Xξ }0<ξ ≤1 be a family of finite-dimension al subspaces of X such that Xξ ⊂ Z ⊂ Y . Normally, the subspaces Xξ are equipped with the induced norm of X, but when Xξ are equipped with the induced norm of Y or Z, these spaces will be denoted by Yξ or Zξ , respectively. The spaces Xξ are a family of approximate spaces to X in the following sense. For 0 < ξ ≤ 1, there exists a projection Pξ : X → Xξ

2 Quasilinear Evolution Equations

323

such that sup0<ξ ≤1 Pξ L(X,Xξ ) < ∞ and, as ξ → 0, ⎧ −1 ⎪ ⎨[1 − Pξ ]A(U ) L(X) ≤ C0 Oξ , α A(U ) [1 − Pξ ]A(U )−1 L(X) ≤ Cα Oξ1−α , ⎪ ⎩ 1−β A(U )β [1 − Pξ ]A(U )−1 L(X) ≤ Cβ Oξ ,

U ∈ K, U ∈ K, U ∈ K,

(7.23)

where Oξ denotes a quantity which tends to 0 as ξ → 0, namely, Oξ , Oξ1−α , and 1−β

denote the approximation rates of the space Xξ to the spaces X, D(A(U )α ) Oξ and D(A(U )β ), respectively. For 0 < ξ ≤ 1, we consider the family of approximate operators Aξ (U ), U ∈ K. The operators Aξ (U ) are bounded sectorial operators of Xξ satisfying the spectral condition σ (Aξ (U )) ⊂ Σ ω}, ω = {λ ∈ C; |arg λ| < 

0< ω<

π , U ∈ K, 2

(7.24)

 ω being some fixed angle, and satisfying the estimate   (λ − Aξ (U ))−1 

L(Xξ )

≤

 M , |λ|

λ∈ / Σωˆ , U ∈ K,

(7.25)

 ≥ 1 being a uniform constant. Furthermore, they are assumed to satisfy two LipM schitz conditions of the forms    Aξ (U ) Aξ (U )−1 − Aξ (V )−1  U − V Y , U, V ∈ K, (7.26) ≤N L(Xξ )    Aξ (U )−1 − Aξ (V )−1 Aξ (U ) U − V Y , U, V ∈ K, (7.27) ≤N L(X ) ξ

 > 0. In addition, for every U ∈ K, D(Aξ (U )α ) ⊂ with some uniform constant N D(A(U )α ) and D(Aξ (U )β ) ⊂ D(A(U )β ) with the estimates 

1 Aξ (U )α U ξ ∈ Xξ , U ∈ K, ξ X ≤ D ξ Xξ , U A(U )α U β β 2 Aξ (U ) U ξ ∈ Xξ , U ∈ K, ξ X ≤ D ξ Xξ , U A(U ) U

(7.28)

i > 0 (i = 1, 2) being some uniform constants. For every U ∈ K, as ξ → 0, Aξ (U ) D converges to A(U ) in the sense that   [1 − Rξ (U )]A(U )−1  ≤ C1 Oξ , U ∈ K, (7.29) L(X)   [Rξ (U ) − Rξ (V )]A(U )−1  ≤ C2 Oξ U − V Y , U, V ∈ K, (7.30) L(X) where Rξ (U ) stands for the Ritz operator from D into Xξ defined by Rξ (U ) = Aξ (U )−1 Pξ A(U ). Finally, we assume that Aξ (U )L(Xξ ) ≤ C3 Oξ−1 ,

U ∈ K.

(7.31)

324

7 Numerical Analysis

We also define the nonlinear operator Fξ (U ) = Pξ F (U ) for U ∈ Zξ ⊂ Z. Then, Fξ is an operator from Zξ into Xξ . Using Aξ (U ) and Fξ (U ), we introduce the approximate problem ⎧ ⎨ d Uξ + A (U ξ = Fξ (U ξ ), 0 < t < ∞, ξ ξ )U dt (7.32) ⎩U  ,  (0) = U 0,ξ

ξ

0,ξ is taken as U 0,ξ = Pξ U0 . We in Xξ to problem (7.15). Here, the initial value U assume the condition   2 sup Aξ (Pξ U0 )β Pξ U0 X < R/D (7.33) ξ

0<ξ ≤1

(cf. (7.21)). Again by Theorem 5.6, we can verify the local existence of solutions ξ on an to (7.32) which is uniform for ξ . Indeed, there exists a unique solution U interval [0, T ] in the function space:  ξ ∈ C([0, T ]; Zξ ) ∩ Cβ−α ([0, T ]; Yξ ) ∩ C1 ((0, T ]; Xξ ), U (7.34) ξ )β U ξ )U ξ ∈ Fβ,σ ((0, T ]; Xξ ). ξ ∈ C([0, T ]; Xξ ), Aξ (U Aξ (U As all the exponents and constants in the structural assumptions (7.24), (7.25), (7.26), and (7.28) are uniform in ξ and as condition (7.33) is assumed, we can naturally assume that the local solution to (7.32) exists on an interval [0, T ], which is uniform for ξ and even coincides with that appearing in (7.22). In particular, we have sup

sup

0<ξ ≤ξ0 0≤s
sup

ξ (t) − U ξ (s)Yξ U < ∞, (t − s)β−α

ξ (t)Zξ < R. sup U

0<ξ ≤ξ0 0≤t≤T

(7.35) (7.36)

 will denote a universal constant which is determined by the expoIn what follows, C nents and constants appearing in the structural assumptions on Aξ (U ). In particular,  is independent of 0 < ξ ≤ 1. C

2.2 Local Estimates of Convergence Under the situations described above, we will establish error estimates between U ξ . and U Theorem 7.2 Under (7.16), (7.17), (7.18), (7.19), (7.20), and (7.21), let U be the local solution of (7.15) in (7.22). Moreover, under (7.24), (7.25), (7.26), (7.28), and

2 Quasilinear Evolution Equations

325

ξ be a local solution of (7.32) in (7.34). Then, (7.23), (7.27), (7.29), (7.33), let U (7.30), and (7.31) provide that ξ (t)Y ≤ CU t γ −1 O 1−α , U (t) − U ξ

0 < ξ ≤ ξ0 , 0 < t ≤ T ,

(7.37)

CU > 0 being some constant depending on U . ξ (t)), 0 ≤ t ≤ T . By (7.26) Proof Consider the family of operators Aξ (t) ≡ Aξ (U and (7.35), we observe that    Aξ (t) Aξ (t)−1 − Aξ (s)−1   − s|β−α , 0 ≤ s, t ≤ T . ≤ C|t L(X ) ξ

This then shows that Aξ (t) satisfy the structural conditions (3.29), (3.30), and (3.31); as a consequence, there exists an evolution operator Uξ (t, s) on Xξ for the operators Aξ (t). By virtue of (3.81), for 0 ≤ θ ≤ 1,   Aξ (t)θ Uξ (t, s)  − s)−θ , 0 ≤ s < t ≤ T . ≤ C(t (7.38) L(X ) ξ

Furthermore, according to Remark 3.5, (7.27) provides that  − s)−1 , Uξ (t, s)Aξ (s)L(Xξ ) ≤ C(t

0 ≤ s < t ≤ T.

(7.39)

Introducing a projection of problem (7.15) on Xξ , we consider the problem  dPξ U dt + Pξ A(U )U = Pξ F (U ), 0 < t ≤ T , Pξ U (0) = Pξ U0 . Combining this with (7.32), we have ⎧d   ⎪ ⎨ dt (Pξ U − Uξ ) + Aξ (t)(Pξ U − Uξ ) ξ )Pξ U − Pξ A(U )U ] + Pξ [F (U ) − F (U ξ )], = [Aξ (U ⎪ ⎩  Pξ U (0) − Uξ (0) = 0.

0 < t ≤ T,

Let Uξ (t, s) be the evolution operator for the operators Aξ (t). Then,  t ξ (t) = ξ (s))] ds Pξ U (t) − U Uξ (t, s)Pξ [F (U (s)) − F (U 0



t

+

ξ (s))Pξ − Pξ A(U (s))]U (s) ds. Uξ (t, s)[Aξ (U

0

Therefore, ξ (t) = (1 − Pξ )U (t) + U (t) − U  + 0



t

ξ (s))] ds Uξ (t, s)Pξ [F (U (s)) − F (U

0 t

ξ (s))Pξ − Pξ A(U (s))]U (s) ds. Uξ (t, s)[Aξ (U

(7.40)

326

7 Numerical Analysis

Let us estimate the Y norms of the terms in the right-hand side. By (7.22) and (7.23),   (1 − Pξ )U (t)Y ≤ (1 − Pξ )A(U (t))−1 L(X,Y ) A(U (t))U (t)X ≤ CU t γ −1 Oξ1−α . Similarly, by (7.20),   t  t    Uξ (t, s)Pξ [F (U (s)) − F (U ξ (s))] ds  ≤ C  (t − s)−α U (s) − U ξ (s)Y ds.   0

0

Y

By the Lemma 7.1 (θ = α) announced and proved below, the third term is estimated by

 t γ −1 1−α −α γ −1  Oξ + (t − s) s U (s) − Uξ (s)Y ds . CU t 0

Hence, the function integral inequality

ξ (t)Y , p(t) = t 1−γ U (t) − U



p(t) ≤ CU Oξ1−α

t

+

t

1−γ

0 ≤ t ≤ T , is seen to satisfy the

−α 2(γ −1)

(t − s)

s

p(s) ds .

0

Since γ > α and γ > 12 , it is possible to solve this inequality by the same techniques as in the proof of Theorem 5.9 and deduce that p(t) ≤ CU Oξ1−α , 0 ≤ t ≤ T . We have thus obtained the desired estimate (7.37).  Lemma 7.1 For 0 ≤ θ < 1,  t     Aξ (t)θ Uξ (t, s)[Aξ (U ξ (s))Pξ − Pξ A(U (s))]U (s) ds    0



≤ CU t

γ −1

 Oξ1−θ

t

+

−θ γ −1

(t − s)

s

X

 U (s) − Uξ (s)Y ds ,

0 < t ≤ T.

0

Before proving Lemma 7.1, we will state a corollary of Theorem 7.2 concerning the error estimates by the X norm and the Z norm. Corollary 7.1 Under the conditions of Theorem 7.2, (7.23), (7.29), (7.30), and (7.31) provide that  ξ (t)X ≤ CU t γ −1 Oξ + t γ O 1−α , 0 < ξ ≤ 1, 0 < t ≤ T , (7.41) U (t) − U ξ  ξ (t)Z ≤ CU t γ −1 t γ −β O 1−α + O 1−β , 0 < ξ ≤ 1, 0 < t ≤ T , U (t) − U ξ ξ (7.42) CU > 0 being some constant depending on U .

2 Quasilinear Evolution Equations

327

Proof To prove (7.41), it suffices to estimate the X norm of the right-hand side of (7.40) by using (7.23), (7.37), and Lemma 7.1 (θ = 0). Similarly, to prove (7.42), it suffices to estimate the Z norm of the right-hand side of (7.40) by using (7.23), (7.37), and Lemma 7.1 (θ = β). 

2.3 Proof of Lemma 7.1 We can write  t ξ (s))Pξ − Pξ A(U (s))]U (s) ds Aξ (t)θ Uξ (t, s)[Aξ (U 0



t

=

 ξ (s)) Aξ (U (s))−1 − Aξ (U ξ (s))−1 Aξ (t)θ Uξ (t, s)Aξ (U

0

× Pξ A(U (s))U (s) ds  t  ξ (s)) Pξ − Aξ (U (s))−1 Pξ A(U (s)) U (s) ds Aξ (t)θ Uξ (t, s)Aξ (U + 0

= I + II. By (7.22), (7.26), and (7.38), we observe that  t  (t − s)−θ s γ −1 U (s) − U ξ (s)Y ds. I X ≤ C 0

To estimate II, we utilize the formula U (s) = U (t) + A(U (s))−1 G(t, s),

0 ≤ s < t ≤ T,

where  G(t, s) = A(U (s)) A(U (s))−1 − A(U (t))−1 A(U (t))U (t) + [A(U (s))U (s) − A(U (t))U (t)]. It follows from (7.18) and (7.22) that  G(t, s)X ≤ CU (t − s)β−α t γ −1 + (t − s)σ s γ −σ −1 .

(7.43)

Then, II is decomposed into  t ξ (s))[Pξ − Rξ (U (s))]A(U (s))−1 G(t, s) ds II = Aξ (t)θ Uξ (t, s)Aξ (U 0



+ 0

t

ξ (s))[Rξ (U (t)) − Rξ (U (s))]U (t) ds Aξ (t)θ Uξ (t, s)Aξ (U

328

7 Numerical Analysis



t

+

ξ (s))[Pξ − Rξ (U (t))]U (t) ds Aξ (t)θ Uξ (t, s)Aξ (U

0

= II 1 + II 2 + II 3 . By (7.29), (7.39), and (7.43), we have  t  γ −1 II 1 X ≤ CU Oξ1−θ (t − s)β−α−1 + (t − s)σ −1 s γ −σ −1 ds ≤ CU t γ −1 Oξ1−θ . t 0

Here, we used, in view of (7.31), the moment inequality     0 1−θ Aξ (U )θ    ξ (U )θ  −θ ≤ CA L(Xξ ) Aξ (U ) L(X ) ≤ COξ , L(X ) ξ

ξ

U ∈K

(7.44)

(see (2.117)). Similarly, by (7.22), (7.30), (7.39), and (7.44),  t II 2 X ≤ CU Oξ1−θ (t − s)β−α−1 ds t γ −1 ≤ CU t γ −1 Oξ1−θ . 0

Finally, II 3 is rewritten as II 3 = Aξ (t)θ [Uξ (t, 0) − 1][Rξ (U (t)) − Pξ ]A(U (t))−1 A(U (t))U (t). Therefore, by (7.22), (7.29), (7.39), and (7.44), II 3 X ≤ CU t γ −1 Oξ1−θ . We have thus accomplished the proof of lemma.

3 Two-Dimensional Finite Element Methods 3.1 Approximate Operators Aξ We are interested in the question of how to verify the various structural assumptions made in the preceding sections. As a simple example, we consider discretization of spatial variables by the finite element method for a strongly elliptic operator in a convex polygonal domain in R2 . Let Ω be a two-dimensional convex polygonal domain. Consider the sesquilinear form a(u, v) =

2   i,j =1 Ω

 aij (x)Di uDj v dx +

c(x)uv dx,

u, v ∈ H 1 (Ω).

Ω

Here, aij (x), 1 ≤ i, j ≤ 2, are real-valued functions in Ω satisfying aij ∈ C1 (Ω) (1 ≤ i, j ≤ 2) and a12 (x) = a21 (x),

(7.45)

3 Two-Dimensional Finite Element Methods 2 

329

aij (x)ζi ζj ≥ δ|ζ |2 ,

ζ = (ζ1 , ζ2 ) ∈ R2 , x ∈ Ω,

(7.46)

i,j =1

with some positive constant δ > 0, and c(x) is a real-valued function satisfying c ∈ L∞ (Ω) and

c(x) ≥ c0 > 0 a.e. x ∈ Ω.

(7.47)

Then, a(u, v) satisfies (2.6) and (2.7). Consider a triplet of spaces H 1 (Ω) ⊂ L2 (Ω) ⊂ H 1 (Ω)∗ , and let A be the linear operator  associated with this form which is a realization of the differential operator − 2i,j =1 Dj [aij (x)Di ·] + c(x) under ∂u the Neumann-type boundary conditions ∂ν = 0 on ∂Ω. As shown by Theorem 2.1, A A is a positive definite self-adjoint operator of L2 (Ω). By Theorem 16.9, the domains of fractional powers Aθ are characterized by  H 2θ (Ω) if 0 ≤ θ < 34 , θ (7.48) D(A ) = HN2θ (Ω) if 34 < θ ≤ 1. Consider a family {τξ }0<ξ ≤ξ0 of triangulations to Ω with size parameter ξ = max{dσ ; σ ∈ τξ } > 0, where σ is a triangle appearing in τξ , and dσ is its diameter. It is assumed that {τξ }0<ξ ≤ξ0 is regular in the sense that c−1 ξ ≤ 2ρσ ≤ dσ ≤ 2rσ ≤ cξ,

σ ∈ τ ξ , 0 < ξ ≤ ξ0 ,

(7.49)

with some positive constant c > 0, where ρσ and rσ are the radii of incircle and circumcircle of σ , respectively. Put, for each 0 < ξ ≤ ξ0 , Cξ (Ω) = { u ∈ C(Ω);  u|σ is linear in each σ ∈ τξ }. By Πξ we denote the interpolation operator Πξ : C(Ω) → Cξ (Ω). When Cξ (Ω) is equipped with the L2 (resp. H 1 ) inner product, the space is denoted by Xξ (resp. Zξ ). Let Zξ ⊂ Xξ ⊂ Zξ∗ be a triplet of spaces. Since Cξ (Ω) is a finitedimensional space, the three spaces coincide as Banach spaces, but they furnish different norms. Consider the given form a(u, v) on a restricted space Zξ . It is easily observed that the sesquilinear form on Zξ also satisfies (2.6) and (2.7) with some uniform  and   and  constants, say, M δ , respectively. (The constants M δ are in independent of the size parameter ξ .) Denote by Aξ the associated linear operator. By definition, Aξ is a linear operator from Cξ (Ω) into itself. However, by Theorem 2.1, Aξ is a sectorial operator with respect to the different norms of Zξ∗ , Xξ , and Zξ simultaneously. The family {Aξ }0<ξ ≤ξ0 is called the approximate operators to A.

3.2 Some properties of Cξ (Ω) and Πξ Let us first see the following property of Cξ (Ω).

330

7 Numerical Analysis

Proposition 7.1 For 0 ≤ s < 32 , Cξ (Ω) is continuously embedded in H s (Ω) with the estimate uξ L2 ,  uξ H s ≤ Cs ξ −s 

 uξ ∈ Cξ (Ω),

(7.50)

Cs > 0 being independent of ξ . Proof When s = 0, (7.50) is trivial and, when s = 1, is very standard (e.g., see [Cia78, Theorem 3.2.6]). Then the result follows immediately for 0 ≤ s ≤ 1 by interpolation property (1.71) and the estimate  uξ H s ≤ C uξ sH 1  uξ 1−s L2 . It therefore suffices to consider the case where s = 1 + ε, 0 < ε < 12 . We can estimate directly the H 1+ε norm of  uξ ∈ Cξ (Ω) using the fact that the H 1+ε norm is given by a sum of integrals in Ω × Ω. According to (1.70), we have  uξ 2H 1+ε

=  uξ 2L2

+

2   j =1

Ω×Ω

|∂j  uξ (x) − ∂j  uξ (y)|2 dx dy. |x − y|2+2ε

(7.51)

uξ )|σ is constant in each σ ∈ τξ , it follows that Since  uσj = (∂j   |∂j  uξ (x) − ∂j  uξ (y)|2 dx dy |x − y|2+2ε Ω×Ω   dx dy | uσj −  uσ j |2 = 2+2ε σ ×σ |x − y| σ,σ ∈τξ σ =σ



≤



2 | uσj | + | u

σ,σ ∈τξ σ =σ

≤4



2

σ ×(Ω\σ )

σ ∈τξ

σ ×(Ω\σ )

|



| uσj |2

It is then sufficient to verify 

σ j

2



 σ ×σ

dx dy |x − y|2+2ε

dx dy . |x − y|2+2ε

dx dy ≤ Cε (rσ )2−2ε |x − y|2+2ε

(7.52)

with some constant Cε > 0 independent of σ , since this, together with (7.49), implies that  uξ 2L2 + Cε | uσj |2 rσ2−2ε  uξ 2H 1+ε ≤  j

≤  uξ 2L2 + Cε ξ −2ε

σ

 j

| uσj |2 |σ | ≤ Cε ξ −2ε  uξ 2H 1 ,

σ

where |σ | denotes the area of σ , and hence implies the desired estimate (7.50).

3 Two-Dimensional Finite Element Methods

331

Let us now show estimate (7.52). Let Bσ denote the circumcircular domain of σ with radius rσ , and Uσ the circular domain with the same center as Bσ but with radius 2rσ . Let li , i = 1, 2, 3, be the three lines in R2 obtained by prolonging the sides of σ . Each li partitions Uσ into two subsets, so denote by Si the one disjoint with σ . Then, we obviously see that  σ ×(Ω\σ )

dx dy ≤ |x − y|2+2ε

 Bσ

 dx dy + |x − y|2+2ε 3

×(R2 \U

σ)

i=1

 σ ×Si

dx dy . |x − y|2+2ε

For the first integral, we have   rσ  ∞  2π  2π dx dy rρ dθ dϕ dρ dr = 2+2ε 2 2 [r + ρ − 2rρ cos(ϕ − θ )]1+ε Bσ ×(R2 \Uσ ) |x − y| 0 2rσ 0 0  1 ∞ rρ dρ dr ≤ (2π)2 rσ2−2ε ≤ Cε rσ2−2ε . 2+2ε (r − ρ) 0 2 For the second integrals, we can assume, without loss of generality, that σ ⊂ [0, 4rσ ] × [0, 4rσ ], Si ⊂ [−4rσ , 0] × [0, 4rσ ], and ∂σ ∩ ∂Si ⊂ {0} × [0, 4rσ ]. Then, changing variables (x2 , y2 ) → (z, w) = (x2 − y2 , x2 + y2 ), we have   0  4rσ  4rσ  4rσ dx dy dx2 dy2 dx1 dy1 ≤ 2+2ε (|x1 − y1 |2 + |x2 − y2 |2 )1+ε σ ×Si |x − y| 0 0 −4rσ 0

 0  1  2  1 2 dz dw ≤ (4rσ )2−2ε dx1 dy1 2 2 1+ε −1 0 0 −1 (|x1 − y1 | + |z| )

 0  1  |x1 −y1 |−1 4 dζ dx1 dy1 2−2ε ≤ (4rσ ) 2 )1+ε |x − y |1+2ε −1 (1 + ζ 1 1 −|x1 −y1 | −1 0  ∞  1 1 dζ dx1 dy1 ≤ Cε rσ2−2ε × 2 1+ε 1+2ε −∞ (1 + ζ ) 0 0 (x1 + y1 ) ≤ Cε rσ2−2ε . 

Hence, we have verified (7.52). Secondly, let us show some estimate for 1 − Πξ . Proposition 7.2 For 0 ≤ s < 32 , we have (1 − Πξ )uH s ≤ Cs ξ 2−s uH 2 ,

u ∈ H 2 (Ω),

(7.53)

Cs > 0 being independent of ξ . Proof When s = 0 or 1, (7.53) is very standard (e.g., see [Cia78, Theorem 3.2.1]). Then, for 0 ≤ s ≤ 1, (7.53) follows by the interpolation property as before. So, let us prove the result for s = 1 + ε, 0 < ε < 12 .

332

7 Numerical Analysis

Let u ∈ H 2 (Ω) and  uξ = Πξ u. By definition, we see that 

|[∂j  uξ (x) − ∂j u(x)] − [∂j  uξ (y) − ∂j u(y)]|2 dx dy |x − y|2+2ε Ω×Ω   |∂j u(x) − ∂j u(y)|2 dx dy ≤ |x − y|2+2ε σ ×σ σ ∈τ ξ

 

+4

σ ×(Ω\σ )

σ ∈τξ

|∂j  uξ (x) − ∂j u(x)|2 dx dy. |x − y|2+2ε

The first integrals are estimated by   σ ∈τξ

≤

σ ×σ

|∂j u(x) − ∂j u(y)|2

  σ ×σ ×(0,1)

σ

dx dy |x − y|2+2ε

|∇(∂j u)[(1 − ω)y + ωx]|2

dx dy dω . |x − y|2ε

If we change the variables (x, y, ω) → (z, y, ω) = ((1 − ω)y + ωx, y, ω), the domain of integration is given by {(z, y, ω); z, y ∈ σ, ω(z, y) < ω < 1}, where ω(z, y) = inf{ω; y + ω−1 (z − y) ∈ σ } ≥ |z−y| 2rσ . Then, ≤

   σ

σ

σ

1 |z−y| 2rσ

|∇(∂j u)(z)|2 ω2ε−2

dω dy dz |z − y|2ε

  (2rσ )1−2ε 1  dy dz |∇(∂j u)(z)|2 1 − 2ε σ σ σ |z − y|  ≤ Cε ∇(∂j u)2L2 (σ ) rσ2−2ε ≤ Cε u2H 2 ξ 2−2ε .

≤

σ

For the second integrals, we utilize Hölder’s inequality:  σ ×(Ω\σ )

|∂j  uξ (x) − ∂j u(x)|2 dx dy |x − y|2+2ε  1  

 ≤

|∂j  uξ (x) − ∂j u(x)|2p dx σ

where 1 < q <  Ω\σ

p

σ 1 2ε ,

and

1 p

+

dy ≤ |x − y|2+2ε

1 q

Ω\σ

dy |x − y|2+2ε

q

1 q

dx

= 1. Since

 |x−y|>dist(x,∂σ )

dy π = dist(x, ∂σ )−2ε , 2+2ε ε |x − y|

,

3 Two-Dimensional Finite Element Methods

we see that   σ

Ω\σ

dy |x − y|2+2ε

q

333

 dx ≤ σ

≤ Cε

Cε dist(x, ∂σ )−2εq dx 

rσ



−rσ

rσ

η−2εq dη dξ = Cε,q rσ2−2εq .

0

uξ − ∂j uL2p (σ ) , we notice by Lemma 7.2 below and (7.49) In order to estimate ∂j  that   1 1 1− p1 p p −1 ∂j  uξ − ∂j uL2p (σ ) ≤ Cp uH 2 (σ  u − u + r  u − u 1 ξ ξ σ H (σ ) ) H 1 (σ )   1 1 1− p1 −1 p +rp (r u ) · r u ≤ Cp uH 2 (σ 2 2 σ σ σ H (σ ) H (σ ) ) 1

= Cp rσp uH 2 (σ ) . Hence, we have   σ ×(Ω\σ )

σ ∈τξ

≤



|∂j  uξ (x) − ∂j u(x)|2 dx dy |x − y|2+2ε

2

2

Cε rσp u2H 2 (σ ) · rσq

−2ε

≤ Cε ξ 2−2ε u2H 2 .

σ



In view of (1.70), we have thus proved the proposition. Lemma 7.2 For 1 ≤ p < ∞,  wL2p (σ ) ≤ Cp

rσ ρσ

1− 1

p

1− 1

1

1

∇wL2 (σp ) wLp 2 (σ ) + ρσp

−1

wL2 (σ ) ,

w ∈ H 1 (σ ), σ ∈ τξ ,

(7.54)

Cp > 0 being a constant independent of σ and ξ . Proof Let σ0 = {(z1 , z2 ) ∈ R2 ; z1 > 0, z2 > 0, z1 + z2 < 1} be the reference triangle, and φσ : σ0 → σ an affine mapping. Then, we have 1

1

Cq−1 |σ | q w ◦ φσ Lq (σ0 ) ≤ wLq (σ ) ≤ Cq |σ | q w ◦ φσ Lq (σ0 ) and − q1

∇(w ◦ φσ )Lq (σ0 ) ≤ Cq rσ |σ |

∇wLq (σ )

for any 1 ≤ q ≤ ∞. Now, (7.54) is readily verified by applying Theorem 1.37 (with p = n = 2) to w ◦ φσ . 

334

7 Numerical Analysis

3.3 Projection Operator Pξ Consider Xξ = (Cξ (Ω),  · L2 ) as a closed subspace of L2 (Ω). Let Pξ : L2 (Ω) → Xξ be the orthogonal projection. By the definition, if u ∈ C(Ω), then (1 − Pξ )uL2 = min u − vL2 ≤ u − Πξ uL2 . v∈Xξ

Therefore, by (7.53) (s = 0), if u ∈ H 2 (Ω), then (1 − Pξ )uL2 ≤ Cξ 2 uH 2 . More generally, for 0 ≤ s < 32 , we obtain by (7.50) and (7.53) that (1 − Pξ )uH s ≤ (1 − Πξ )uH s + (Πξ − Pξ )uH s ≤ Cs [ξ 2−s uH 2 + ξ −s (Πξ − Pξ )uL2 ] ≤ Cs ξ 2−s uH 2 , u ∈ H 2 (Ω).

(7.55)

As is seen, we have (1 − Pξ )uL2 ≤ Cξ 2 AuL2 for u ∈ D(A). We then use the Heinz and Kato inequality, Theorem 2.31, with X = Y = L2 (Ω) and T = 1 − Pξ , B = 1. Then, 1 − Pξ is a bounded operator from D(Aθ ) into L2 (Ω) whose operator norm is estimated by θ θ (1 − Pξ )uL2 ≤ C1 − Pξ 1−θ L(L2 ,L2 ) 1 − Pξ L(D(A),L2 ) A uL2 ,

u ∈ D(Aθ ).

Therefore, for 0 ≤ θ ≤ 1, (1 − Pξ )uL2 ≤ Cξ 2θ Aθ uL2 ,

u ∈ D(Aθ ).

(7.56)

In view of (7.48), (7.55) shows that assumptions (7.4) and (7.23) are actually fulfilled for any 0 < β < 34 with Oξ = O(ξ 2 ).

3.4 Ritz Operator Rξ Let Zξ = (Cξ (Ω),  · H 1 ). Since a(u, v) is symmetric, a(u, v) can be regarded as an inner product of Zξ . Hence, for each u ∈ H 1 (Ω), there exists a unique vector Rξ u ∈ Zξ such that a(Rξ u, vξ ) = a(u, vξ ) for all  v ξ ∈ Zξ .   Then, Rξ uH 1 ≤ (M/ δ )uH 1 , that is, Rξ is a bounded operator from H 1 (Ω) into Zξ . Operator Rξ is called the Ritz operator. If u ∈ D(A), then     a(u, vξ ) = (Au, vξ ) = (Pξ Au, vξ ) = Aξ A−1 vξ = a A−1 vξ . ξ Pξ Au, ξ Pξ Au, Therefore, Rξ u = A−1 ξ Pξ Au.

3 Two-Dimensional Finite Element Methods

335

For u ∈ H 1 (Ω), since a((1 − Rξ )u, (Πξ − Rξ )u) = 0, we have a((1 − Rξ )u, (1 − Rξ )u) = a((1 − Rξ )u, (1 − Πξ )u). By (2.6) and (2.7),   − Rξ )uH 1 (1 − Πξ )uH 1 . δ (1 − Rξ )u2H 1 ≤ a((1 − Rξ )u, (1 − Rξ )u) ≤ M(1 Therefore, by (7.53) (s = 1), (1 − Rξ )uH 1 ≤ C(1 − Πξ )uH 1 ≤ Cξ uH 2 ,

u ∈ H 2 (Ω).

Furthermore, it is seen (Aubin–Nitsche’s trick) that (1 − Rξ )u2L2 = a(A−1 (1 − Rξ )u, (1 − Rξ )u) = a((1 − Rξ )A−1 (1 − Rξ )u, (1 − Rξ )u) ≤ C1 − Rξ L(H 2 ,H 1 ) A−1 L(L2 ,H 2 ) × (1 − Rξ )uL2 (1 − Rξ )uH 1 . Hence, (1 − Rξ )uL2 ≤ Cξ (1 − Rξ )uH 1 ≤ Cξ 2 uH 2 ,

u ∈ H 2 (Ω).

More generally, for 0 ≤ s < 32 , (1 − Rξ )uH s ≤ (1 − Πξ )uH s + (Πξ − Rξ )uH s ≤ Cs [ξ 2−s uH 2 + ξ −s (Πξ − Rξ )uL2 ] ≤ Cs ξ 2−s uH 2 , In addition, for 0 ≤ θ <

3 4

u ∈ H 2 (Ω).

(7.57)

and for u ∈ D(A),

(1 − Rξ )u2L2 = a((1 − Rξ )A−1 (1 − Rξ )u, (1 − Rξ )u) = a((1 − Rξ )A−1 (1 − Rξ )u, u) = (Aθ (1 − Rξ )A−1 (1 − Rξ )u, A1−θ u) ≤ C1 − Rξ L(H 2 ,H 2θ ) A−1 L(L2 ,H 2 ) (1 − Rξ )uL2 A1−θ uL2 due to (7.48), and therefore (1 − Rξ )uL2 ≤ Cξ 2(1−θ) A1−θ uL2 , Hence, it is deduced that, for

1 4

u ∈ D(A).

< θ ≤ 1,

(1 − Rξ )uL2 ≤ Cθ ξ 2θ Aθ uL2 , Cθ > 0 being a constant independent of ξ .

u ∈ D(Aθ ),

(7.58)

336

7 Numerical Analysis

We have thus verified that assumptions (7.6) and (7.29) are fulfilled with Oξ = O(ξ 2 ).

3.5 Domains of Fractional Powers of Aξ 1

1

Since a(u, v) is symmetric, Theorem 2.34 yields that D(Aξ2 ) = Zξ and Aξ2  uξ L2 ≤ C uξ H 1 . Therefore, in view of (7.50), we have  1   1  uξ L2 ≤ C Aξ2  uξ H 1 ≤ Cξ −1 Aξ2  uξ L ≤ Cξ −1  uξ H 1 ≤ Cξ −2  uξ L2 . Aξ  2

Therefore, the moment inequality (2.117) yields that, for 0 ≤ θ ≤ 1,  θ  −2θ  A   uξ L2 ,  uξ ∈ Xξ . ξ uξ L ≤ Cξ 2

(7.59)

For u ∈ D(A), we have Aξ Pξ u = Aξ (Pξ − Rξ )u + Aξ Rξ u = [Aξ (Pξ − Rξ )A−1 + Pξ ]Au. Hence, by (7.56), (7.58), and (7.59), we have Aξ Pξ uL2 ≤ CAuL2 , u ∈ D(A). The Heinz and Kato inequality then provides that, for 0 ≤ θ ≤ 1,  θ  A Pξ u ≤ CAθ uL , u ∈ D(Aθ ). (7.60) 2 ξ L 2

Conversely, for 0 ≤ θ < 34 , we observe that 

       θ−1 uξ , v = a  uξ , Aθ−1 v = a  uξ , Rξ Aθ−1 v = Aθξ  uξ , A1−θ v Aθ  ξ Rξ A     θ−1 θ−1 = Aθξ  uξ , A1−θ v + Aθξ  uξ , A1−θ v . ξ (Rξ − Pξ )A ξ Pξ A

Therefore, by (7.56), (7.58), (7.59), and (7.60),       θ   A uξ , v  ≤ C Aθξ  uξ L ξ 2(θ−1) (Rξ − Pξ )Aθ−1 vL2 + A1−θ Pξ Aθ−1 v L ξ 2 2  θ    ≤ Cθ Aξ  uξ L vL2 . 2

uξ L2 ≤ Cθ Aθξ  uξ L2 . Thus, for 0 ≤ θ < 34 , This implies that Aθ   θ  A  uξ 

L2

  ≤ Cθ Aθξ  uξ L , 2

 uξ ∈ Xξ .

(7.61)

Estimate (7.61) then shows that assumptions (7.5) and (7.28) are fulfilled for α < β < 34 . Similarly, (7.59) shows that assumptions (7.7) and (7.31) are fulfilled with Oξ−1 = O(ξ −2 ). Finally, (7.60) means that, if u0 ∈ D(Aβ ), then β

sup0<ξ ≤ξ0 Aξ Pξ u0  ≤ CAβ u0 L2 ; hence, assumptions (7.9) and (7.33) are also fulfilled.

4 Convergence of Exponential Attractors

337

4 Convergence of Exponential Attractors The purpose of this section is to establish some global estimate of convergence of the approximate solutions. We shall begin with discrete dynamical systems.

4.1 Discrete Case Let (Sξn , Xξ , X) be a family of discrete dynamical systems parameterized by a number 0 ≤ ξ ≤ 1. Here, the universal space X is a Banach space with norm  · X . The phase spaces Xξ are all compact subsets of X which are metric spaces equipped with the induced distance of X. We assume that for every 0 < ξ ≤ 1, Xξ ⊂ X0 and that, as ξ → 0, Xξ converges to X0 in the sense that d(Xξ , X0 ) ≤ D1 Oξ ,

(7.62)

D1 > 0 being some constant. Here, d(X1 , X2 ) is the distance defined by (1.3), and Oξ stands for the rate of approximations and tends to 0 as ξ → 0. For 0 ≤ ξ ≤ 1, the operator Sξ is a nonlinear operator from Xξ into itself which satisfies uniformly the compact perturbation conditions (6.38)–(6.39) of type II, i.e., Sξ U − Sξ V X ≤ δU − V X + Kξ U − Kξ V X , Kξ U − Kξ V Z ≤ LU − V X ,

U, V ∈ Xξ , (7.63)

U, V ∈ Xξ ,

(7.64)

with some fixed 0 ≤ δ < 12 and some uniform constant L > 0. In addition, Kξ is an operator from Xξ into a second Banach space Z with norm  · Z which is compactly embedded in X. Clearly, (7.63) and (7.64) imply the uniform Lipschitz condition Sξ U − Sξ V X ≤ (δ + cL)U − V X ,

U, V ∈ Xξ ,

(7.65)

where c > 0 is an embedding constant such that  · X ≤ c · Z . We assume also that, as ξ → 0, Sξ converges to S0 at a rate sup Sξ U − S0 U X ≤ D2 Oξ ,

U ∈Xξ

0 < ξ ≤ 1,

(7.66)

D2 > 0 being some constant. We already know by Theorem 6.13 that, under (7.63) and (7.64), each system (Sξn , Xξ , X) has exponential attractors. We will then show that the convergence (7.62) and (7.66) provide continuous dependence of the exponential attractors as ξ → 0. Theorem 7.3 Under (7.63) and (7.64), let (7.62) and (7.66) be satisfied. Then, there exist exponential attractors M∗ξ for (Sξn , Xξ , X), 0 ≤ ξ ≤ 1, respectively, for which the estimate   (7.67) d M∗ξ , M∗0 ≤ C ∗ Oξκ , 0 < ξ ≤ 1, holds with some exponent 0 < κ < 1 and some constant C ∗ > 0.

338

7 Numerical Analysis

Proof We will remember the method employed for proving Theorem 6.13 in constructing exponential attractors of the dynamical system (S0n , X0 , X). Let 0 < θ < 1−2δ 2L , and let a = 2(δ +Lθ ) < 1. Let us introduce for n = 0, 1, 2, . . . , an Ra n -radius covering of S0n X0 by a finite number of closed balls of X with centers in S0n X0 , where R is the diameter of X0 . More precisely, let n

S0n X0

⊂

Nθ 

X

B (Wn,i ; Ra n ),

Wn,i ∈ S0n X0 ,

(7.68)

i=1

where Nθ is the minimal number of balls of X with radii θ which can cover the unit Z closed ball B (0; 1) of Z centered at the origin. Then, the set defined by M∗0 =

∞ 

S0n P0 ,

  where P0 = Wn,i ; 0 ≤ n < ∞, 1 ≤ i ≤ Nθn ,

n=0

gives an exponential attractor for (S0n , X0 , X). We are then going to construct an exponential attractor for (Sξn , Xξ , X), 0 < ξ ≤ 1, on the basis of the set P0 . Since Wn,i ∈ S0n X0 , there exists Vn,i ∈ X0 such that Wn,i = S0n Vn,i . Furthermore, by (7.62), there exists Vn,i,ξ ∈ Xξ such that Vn,i − Vn,i,ξ X ≤ D1 Oξ . Setting Wn,i,ξ = Sξn Vn,i,ξ , we use Lemma 7.3 be presented below. Then, in view of (7.66), Wn,i − Wn,i,ξ X ≤ L˜ n Oξ with some constant L˜ > 1. Therefore, we obtain from (7.68) the new covering n

S0n X0 ⊂

Nθ 

X B (Wn,i,ξ ; Ra n + L˜ n Oξ ),

Wn,i,ξ ∈ Sξn Xξ .

(7.69)

i=1

In the meantime, Lemma 7.3, together with (7.62) and (7.66), yields also that   Sξn Xξ ⊂ W(L˜ n Oξ ) S0n X0 , (7.70) where W(L˜ n Oξ ) (S0n X0 ) denotes the L˜ n Oξ -neighborhood of S0n X0 in X. Therefore, it follows that n

Sξn Xξ ⊂

Nθ 

X B (Wn,i,ξ ; Ra n + 2L˜ n Oξ ),

Wn,i,ξ ∈ Sξn Xξ .

i=1

Lemma 7.3 Let L0 = δ + cL. Then, for every n = 0, 1, 2, . . . ,   n    S Uξ − S n U0  ≤ Ln+1 − 1 /(L0 − 1) sup Sξ V − S0 V X ξ 0 0 X 

+ Uξ − U0 X ,

V ∈Xξ

Uξ ∈ Xξ , U0 ∈ X0 .

(7.71)

4 Convergence of Exponential Attractors

339

Proof We have Sξn Uξ − S0n U0 = Sξn Uξ − S0n Uξ + S0n Uξ − S0n U0 =

n  

   S0n−i Sξi Uξ − S0n−i+1 Sξi−1 Uξ + S0n Uξ − S0n U0 .

i=1

By the Lipschitz condition (7.65) (ξ = 0), n   n    S U ξ − S n U 0  ≤ Sξ S i−1 Uξ − S0 S i−1 Uξ  + Ln Uξ − U0 X Ln−i ξ 0 0 ξ ξ 0 X X i=1

 ≤

n 

Ln−i 0

i=0

 

 sup Sξ V − S0 V X + Uξ − U0 X .

V ∈Xξ



Hence, the desired estimate is obtained.

ξ be the largest Let us accomplish the proof of theorem. For each 0 < ξ ≤ 1, let N n n n n ˜ ˜ ξ and Ra n < integer satisfying 2L Oξ ≤ Ra , i.e., 2L Oξ ≤ Ra for all 0 ≤ n ≤ N n ˜   2L Oξ for all Nξ < n < ∞. If we use (7.71) for 0 ≤ n ≤ Nξ , then n

Sξn Xξ

⊂

Nθ 

X

B (Wn,i,ξ ; 2Ra n ),

Wn,i,ξ ∈ Sξn Xξ .

i=1 

ξ with central points WNξ ,i,ξ , 1 ≤ i ≤ N Nξ , and following the Starting from n = N θ same procedure as in Step 1 of the proof of Theorem 6.12, we can construct inducξ < n < ∞, a 2Ra n -radius covering of S n Xξ such that tively for every N ξ n

Sξn Xξ

⊂

Nθ 

X B (W˜ n,i,ξ ; 2Ra n ),

W˜ n,i,ξ ∈ Sξn Xξ .

i=1

Put     ξ , 1 ≤ i ≤ Nθn ∪ W˜ n,i,ξ ; N ξ < n < ∞, 1 ≤ i ≤ Nθn . Pξ = Wn,i,ξ ; 0 ≤ n ≤ N Then, for all n = 0, 1, 2, . . . ,

  h Sξn Xξ , Pξ ≤ 2Ra n .

(7.72)

 n Moreover, by the same reason as for M∗0 , the set M∗ξ = ∞ n=0 Sξ Pξ gives an expon nential attractor for (Sξ , Xξ , X). It now remains to estimate the distance d(M∗ξ , M∗0 ). By definition, M∗ξ

⊂

 Nξ  n=0

Sξn

    Nξ +1 k ξ − n, 1 ≤ i ≤ Nθ Sξ Wk,i,ξ ; 0 ≤ k ≤ N Xξ .

340

7 Numerical Analysis

ξ + 1), together with (7.68) (n = N ξ + 1), yields that The inclusion (7.70) (n = N  N +1   h Sξ ξ Xξ , M∗0 ≤ h W  ≤h W



 Nξ +1  ∗  X 0 , M0 S0



 Nξ +1     S0 X0 , P0 ≤ L˜ Nξ +1 Oξ + Ra Nξ +1

(L˜ Nξ +1 Oξ ) (L˜ Nξ +1 Oξ )



≤ 3L˜ Nξ +1 Oξ . Meanwhile, due to Lemma 7.3,  n    S Wk,i,ξ − S n Wk,i  ≤ Ln+1 − 1 /(L0 − 1) ξ 0 0 X    × D2 Oξ + L˜ k Oξ ≤ C L˜ Nξ +1 Oξ ; therefore, it follows that    d Sξn Wk,i,ξ , M∗0 ≤ C L˜ Nξ +1 Oξ ,

(7.73)

ξ − n, 1 ≤ i ≤ Nθk . 0≤k≤N 

Thus, we have proved that h(M∗ξ , M∗0 ) ≤ C L˜ Nξ +1 Oξ .  ξ = log(R/2)−log Oξ , we easily observe that L˜ Nξ +1 Oξ ≤ CO κ with the As N ξ ˜ exponent κ =

log L−log a − log a . ˜ log L−log a

The converse estimate h(M∗0 , M∗ξ ) ≤ COξκ is also verified in a similar way. Indeed, by definition,  M∗0

⊂



Nξ 

S0n

    Nξ +1 k ξ − n, 1 ≤ i ≤ Nθ S0 Wk,i ; 0 ≤ k ≤ N X0 .

n=0

ξ ) to obtain that We use (7.69) (n = N  N +1   N   N  h S0 ξ X0 , M∗ξ ≤ h S0 ξ X0 , M∗ξ ≤ h S0 ξ X0 , Pξ 





≤ L˜ Nξ Oξ + Ra Nξ ≤ (1 + 2a −1 )L˜ Nξ +1 Oξ . Meanwhile, from (7.73) we conclude that    ξ − n, 1 ≤ i ≤ Nθk . d S0n Wk,i , M∗ξ ≤ C L˜ Nξ +1 Oξ , 0 ≤ k ≤ N Hence, h(M∗0 , M∗ξ ) ≤ COξκ with the same exponent κ as above.



4.2 Continuous Case Let (Sξ (t), Xξ , X) be a family of continuous dynamical systems parameterized by 0 ≤ ξ ≤ 1 in a Banach space X. The phase spaces Xξ are all closed subsets of X with

4 Convergence of Exponential Attractors

341

the relation Xξ ⊂ X0 for 0 < ξ ≤ 1, and Sξ (t) are continuous nonlinear semigroups acting on Xξ , respectively. We assume that there exists a compact set B of X for which the set Xξ ∩ B is an absorbing set of (Sξ (t), Xξ , X) for every 0 ≤ ξ ≤ 1. Furthermore, we assume that Xξ ∩ B absorbs itself uniformly for ξ , that is, there exists a uniform time t ∗ > 0 such that Sξ (t)(Xξ ∩ B) ⊂ Xξ ∩ B

for every t ∗ ≤ t < ∞ and 0 ≤ ξ ≤ 1.

(7.74)

We then set Bξ = Xξ ∩ B. Clearly, Bξ is a compact set of X with the relation Bξ ⊂ B0 . Furthermore, if Sξ is set by Sξ = Sξ (t ∗ ), then (Sξn , Bξ , X) defines a discrete dynamical system for each 0 ≤ ξ ≤ 1. Let us assume that conditions (7.62), (7.63), and (7.64) are satisfied for (Sξn , Bξ , X). In addition, let Sξ (t) satisfy the Lipschitz condition Sξ (t)U − Sξ (s)V X ≤ L1 (|t − s| + U − V X ),

0 ≤ s, t ≤ t ∗ , U, V ∈ Bξ , (7.75)

with some uniform constant L1 > 0. Finally, instead of (7.66), let the convergence condition sup

sup Sξ (τ )U − S0 (τ )U X ≤ D3 Oξ ,

0≤τ ≤t ∗ U ∈Bξ

0 < ξ ≤ 1,

(7.76)

be satisfied with some constant D3 > 0. We can then prove a convergence theorem of exponential attractors for the continuous case, namely, Theorem 7.4 below. In fact, by applying Theorem 7.3 to (Sξn , Bξ , X), we deduce that there exist exponential attractors M∗ξ , 0 ≤ ξ ≤ 1, enjoying property (7.67). For each 0 ≤ ξ ≤ 1, we then set Mξ =



Sξ (τ )M∗ξ .

(7.77)

0≤τ ≤t ∗

It is seen that this Mξ gives an exponential attractor for the continuous system (Sξ (t), Xξ , X). Indeed, by the same arguments as in the proof of Theorem 6.15, Mξ is a compact set of X with finite fractal dimension less than dF (M∗ξ ) + 1 and contains the global attractor Aξ of (Sξ (t), Xξ , X), i.e., Aξ ⊂ Mξ ⊂ Xξ . Similarly, Mξ is observed to be invariant under Sξ (t) for every 0 < t < ∞. Finally, the exponential attraction is verified by the following proposition. Proposition 7.3 There exist an exponent k > 0 such that, for any bounded set B of Xξ , h(Sξ (t)B, Mξ ) ≤ CB e−kt ,

tB ≤ t < ∞,

with suitable time tB ≥ 0 and a constant CB > 0 depending on B.

(7.78)

342

7 Numerical Analysis

Proof First, let us consider the case where B = Bξ . Let 0 < t < ∞, and let t = nt ∗ + τ , where n ≥ 0 is an integer and 0 ≤ τ < t ∗ . Since Mξ ⊃ Sξ (τ )M∗ξ , we have   h(Sξ (t)Bξ , Mξ ) ≤ h Sξ (t)Bξ , S(τ )M∗ξ . Using (7.75) (t = s = τ ) and noting (7.72), we see that   h(Sξ (t)Bξ , Mξ ) ≤ h Sξ (τ )Sξ (t ∗ )n Bξ , S(τ )M∗ξ   ≤ L1 h Sξ (t ∗ )n Bξ , M∗ξ ≤ 2L1 Ra n ≤ 2L1 Ra −1 e−(log a

−1 /t ∗ )t

,

0 ≤ t < ∞.

This shows that (7.78) is valid for Bξ with k = log a −1 /t ∗ , tBξ = 0 and CBξ = 2L1 R/a. Secondly, let us consider the general case. Let B be any bounded set of Xξ . By definition, Sξ (t)B ⊂ Bξ for any t such that t ≥ tB , tB ≥ 0 being a suitable time. Then, h(Sξ (t)B, Mξ ) = h(Sξ (t − tB )Sξ (tB )B, Mξ ) ≤ h(Sξ (t − tB )Bξ , Mξ ) ≤ CBξ e−k(t−tB ) = CBξ ektB e−kt ,

tB ≤ t < ∞. 

Hence, the assertion of proposition is verified.

In this way, the set Mξ given by (7.77) has been shown to be an exponential attractor of (Sξ (t), Xξ , X) for each ξ . We can then verify that Mξ also enjoys the same property as (7.67). Theorem 7.4 Under (7.63), (7.64), (7.74), and (7.75), assume (7.62) and (7.76) for the phase spaces Bξ and the semigroups Sξ (t), respectively. Then, there exists an exponential attractor Mξ for (Sξ (t), Xξ , X) for which the estimate d(Mξ , M0 ) ≤ COξκ ,

0 < ξ ≤ 1,

(7.79)

holds with some exponent 0 < κ < 1 and constant C > 0. Proof It now suffices to verify under (7.67) that the set Mξ given by (7.77) has property (7.79). Let Uξ ∈ Mξ be any element. Then, Uξ = Sξ (τ )Uξ∗ with some 0 ≤ τ ≤ t ∗ and some Uξ∗ ∈ M∗ξ . By (7.67), there exists an element U0∗ ∈ M∗0 such that d(Uξ∗ , U0∗ ) ≤ C ∗ Oξκ . Put U0 = S0 (τ )U0∗ ∈ M0 . Then, by (7.76) and (7.75),   d(Uξ , M0 ) ≤ d(Uξ , U0 ) = d Sξ (τ )Uξ∗ , S0 (τ )U0∗     ≤ d Sξ (τ )Uξ∗ , S0 (τ )Uξ∗ + d S0 (τ )Uξ∗ , S0 (τ )U0∗

Notes and Further Researches

343

≤ D3 Oξ + L1 C ∗ Oξκ . Thus, we obtain that h(Mξ , M0 ) ≤ COξκ . By the same argument, we can observe also h(M0 , Mξ ) ≤ COξκ . Hence, (7.79) has been verified. 

Notes and Further Researches The local convergence theorems, Theorems 7.1 and 7.2, for semilinear and quasilinear equations, respectively, have been obtained by the semigroup methods. The methods are based on the fact that the sectorial operators with angles < π2 acting in infinite-dimensional spaces can be approximated by those with the similar angles < π2 in finite-dimensional spaces. Those were founded by Fujita–Mizutani [FM76], Baker–Bramble–Thomée [BBT77], Brenner–Thomée [BT79], and Piskarev [Pis79], cf. also Ushijima [Ush75, Ush79], and were developed by Crouzeix–Larsson– Piskarev–Thomée [CLPT93], Palencia [Pal93, Pal95], and others. In the meanwhile, the methods were applied to nonautonomous linear parabolic equations by Suzuki [Suz78, Suz82] and Sammon [Sam82, Sam83] and to semilinear parabolic equations by Keeling [Kee90]. Theorems 7.1 and 7.2 were obtained by Nakaguchi–Yagi [NY99b, NY01, NY02]. The alternative way for constructing approximate solutions for parabolic equations is the variational methods, see the books by Dautray–Lions [DL91]. For the linear problems, we quote Helfrich [Hel74], Bramble–Schatz–Thomée–Wahlbin [BSTW77] and Savaré [Sav93], and for the nonlinear problems, Lubich–Ostermann [LO93, LO95] and Nakaguchi–Yagi [NY03]. In Sects. 1 and 2, we handled only semidiscretization on the spatial variables. However, it is known that the usual discretization schemes for ordinary differential equations in finite-dimensional spaces can be properly extended for ordinary evolution equations in infinite-dimensional spaces, too, cf. Le Roux [LeR79] and Ostermann–Roche [OR92]. In the papers Lubich–Ostermann [LO95] and Nakaguchi–Yagi [NY99a], the Runge–Kutta methods (cf. Butcher [But87] and Iserles [Ise96]) were applied to the Cauchy problem for quasilinear abstract evolution equations in Banach spaces, and some convergence theorems were obtained. These results show that the Runge–Kutta methods are available for the approximate problems (7.32) in order to build full-discretization schemes for the continuous problem (7.15) as done in [NY99b, NY01, NY02]. The structural assumptions made in Sects. 1 and 2 can actually be verified by concrete discretization schemes. Section 3 was devoted to verifying some of them for the finite element methods in two-dimensional domains. Propositions 7.1 and 7.2 were first verified in [NY02]. In higher-dimensional domains, however, the verification is left to be studied. We may have to use higher-order finite element methods for obtaining a similar result as Proposition 7.1. For the finite element methods, we quote the monographs Ciarlet [Cia78], Fujita–Suzuki [FS91], and Thomée [Tho97].

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7 Numerical Analysis

Theorems 7.3 and 7.4 were proved by modifying the arguments in Efendiev–Yagi [EY05] (cf. also [AY04b]). As seen in the subsequent chapters, the positivity of solutions of the model equations is extremely important in constructing exponential attractors. Nevertheless, the approximate equations do not admit positivity in general (cf. Aida–Yagi [AY03]). Recently, Saito [Sai04, Sai07] and Efendiev–Nakaguchi– Wendland [ENW09] have presented discretization schemes which conserve positivity. Using them, we can show that the approximate problems generate dynamical systems in which exponential attractors equally exist. Theorems 7.3 and 7.4 are then applicable to the family of dynamical systems.

Chapter 8

Semiconductor Models

In 1950, Shockley presented the drift-diffusion model ⎧ ∂u ⎪ ⎨ ∂t = aΔu − ∇ · [μu∇χ] + f (1 − uv) + g(x) ∂v = bΔv + ∇ · [νv∇χ] + f (1 − uv) + g(x) ⎪ ⎩ ∂t 0 = cΔχ − u + v + h(x)

in Ω × (0, ∞), in Ω × (0, ∞), in Ω × (0, ∞),

for describing the flows of electrons and holes in a semiconductor system. Here, the unknown functions u(x, t) and v(x, t) denote the densities of electrons and holes, respectively, in a semiconductor device Ω at time t. Electrons and holes diffuse with positive diffusion rates a > 0 and b > 0. The terms −∇ · [μu∇χ] and ∇ · [νv∇χ] denote the drift-diffusions of electrons and holes, where μ and ν denote their mobilities, respectively. The function χ(x, t) denotes electrostatic potential and is determined by the Poisson equation, where c > 0 is the dielectric constant. Meanwhile, the reaction term f (1 − uv) denotes the effects of generation and recombination of electrons and holes. The functions g(x) ≥ 0 and h(x) are given, and both denote external constraints to the system. In this chapter, we show the global existence of solutions and construct a dynamical system and exponential attractors.

1 Prototype Semiconductor Equations 1.1 Model Equations

We consider the initial-boundary-value problem for a semiconductor model A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_8, © Springer-Verlag Berlin Heidelberg 2010

345

346

8 Semiconductor Models

⎧ ∂u ⎪ ⎪ ∂t = aΔu − μ∇ · [u∇χ] + f (1 − uv) + g(x) ⎪ ⎪ ⎪ ∂v ⎪ ⎪ ⎪ ∂t = bΔv + ν∇ · [v∇χ] + f (1 − uv) + g(x) ⎪ ⎪ ⎪ ⎨0 = cΔχ − u + v + h(x) ⎪ u=v=χ =0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂u = ∂v = ∂χ = 0 ⎪ ∂n ∂n ∂n ⎪ ⎪ ⎪ ⎩ u(x, 0) = u0 (x), v(x, 0) = v0 (x)

in Ω × (0, ∞), in Ω × (0, ∞), in Ω × (0, ∞), on ΓD × (0, ∞),

(8.1)

on ΓN × (0, ∞), in Ω,

in a two-dimensional bounded domain Ω with Lipschitz boundary ∂Ω. We make the following assumptions. The boundary ∂Ω of Ω is split into two parts ΓD and ΓN in such a way that (2.41) and (2.42) are satisfied. In particular, ΓD is a nonempty open set of ∂Ω. On ΓD , the Dirichlet boundary conditions are imposed on u, v, and χ ; on the other hand, on ΓN , the Neumann boundary conditions are imposed. At x ∈ ΓN , n(x) denotes the outer normal vector of the boundary. The mobilities μ and ν of drift-diffusions are both positive constants (> 0). Physically these mobilities are determined by the scattering process of electrons and holes by the crystal lattices, the impurities, and so on, and may depend on u, v and ∇u, ∇v. In this chapter, however, we are concerned with the simplest case that μ and ν are constants. From a physical point of view, this means that the densities of electrons and holes are not so high. The rates of generation and recombination of electrons and holes are nonnegative constants, and the two rates coincide. These rates are denoted by f ≥ 0. If the two rates are positive constants, then it is always possible to make them equal by suitable normalization. The function g(x) is a given nonnegative function such that 0 ≤ g ∈ L2 (Ω),

(8.2)

and h(x) is a given real function such that h ∈ L∞ (Ω; R).

(8.3)

1.2 Some Preliminary Results Let H˚ D1 (Ω) be the space given by (2.25). We consider the sesquilinear form  ∇u · ∇v dx, u, v ∈ H˚ D1 (Ω). a(u, v) = Ω

It is easy to see that this form satisfies (2.6) and (2.7) on H˚ D1 (Ω). For (2.7), we may recall Remark 2.3. Let H˚ D1 (Ω) ⊂ L2 (Ω) ⊂ H˚ D1 (Ω)∗ = HD−1 (Ω) be a triplet of spaces, and let Λ be the associated sectorial operator of HD−1 (Ω) which is an isomorphism from H˚ D1 (Ω) onto HD−1 (Ω) and whose part in L2 (Ω) is a positive

1 Prototype Semiconductor Equations

347

definite self-adjoint operator. As noticed by Theorem 2.5, Λ is considered as a realization of the Laplace operator −Δ in HD−1 (Ω) under the Dirichlet boundary conditions on ΓD and the Neumann boundary conditions on ΓN . 1 By Theorem 2.35, the domain of the square root Λ 2 coincides with L2 (Ω), i.e.,  1 D Λ 2 = L2 (Ω). Moreover, for

1 2

(8.4)

≤ θ ≤ 1, it holds that



D(Λθ ) = L2 (Ω), H˚ D1 (Ω) 2θ −1 ⊂ [L2 (Ω), H 1 (Ω)]2θ−1 = H 2θ−1 (Ω).

(8.5)

The part Λ|L2 may not have the optimal shift property that Λu ∈ L2 (Ω) always implies u ∈ H 2 (Ω). According to Theorem 2.10, however, it holds that D(Λ|L2 ) ⊂ Wp10 (Ω) with the estimate u Wp1 ≤ C Λu L2 , 0

u ∈ D(Λ|L2 ),

(8.6)

where p0 is a certain fixed number such that p0 > 2.

2

We finally extend the definition of the drift term ∇ · [u∇χ] for all u ∈ H p (Ω) and all χ ∈ D(Λ|L2 ) ∩ Wp1 (Ω), provided that 2 < p < ∞. Proposition 8.1 Let 2 < p < ∞ be arbitrarily fixed. The correspondence (u, χ) → ∇ · [u∇χ] is continuous from H 2 (Ω) × [D(Λ|L2 ) ∩ Wp1 (Ω)] into L2 (Ω). It is possi2

ble to extend this correspondence continuously from H p (Ω) × [D(Λ|L2 ) ∩ Wp1 (Ω)] into HD−1 (Ω). Proof Let u ∈ H 2 (Ω) and χ ∈ D(Λ|L2 ) ∩ Wp1 (Ω). Then, we observe directly that ∇ · [u∇χ] = ∇u · ∇χ − uΛχ ∈ L2 (Ω) in the sense of distribution. In addition, by the definition of Λ,   ∇χ · ∇(uv) dx = ∇χ · (v∇u + u∇v) dx, (Λχ, uv)L2 = Ω

Ω

v ∈ H˚ D1 (Ω),

since u ∈ H 2 (Ω) and v ∈ H˚ D1 (Ω) imply that uv ∈ H˚ D1 (Ω). Noting (1.26), we obtain that  u∇χ · ∇v dx, v ∈ H˚ D1 (Ω).

∇ · [u∇χ], vH −1 ×H˚ 1 = (∇ · [u∇χ], v)L2 = − D

D

Ω

2

For u ∈ H p (Ω) and χ ∈ D(ΛL2 ) ∩ Wp1 (Ω), it is observed (due to (1.86)) that  |u∇χ∇v| dx ≤ u L Ω

2p p−2

∇χ Lp ∇v L2 ≤ Cp u

2

Hp

χ Wp1 v H˚ 1 . D

348

8 Semiconductor Models 2

Since H 2 (Ω) is a dense subspace of H p (Ω), we are led to extend ∇ · [u∇χ] ∈ 2

HD−1 (Ω) continuously for all u ∈ H p (Ω) and χ ∈ D(Λ|L2 ) ∩ Wp1 (Ω) by the formula 

∇ · [u∇χ], vH −1 ×H˚ 1 = − u∇χ · ∇v dx, v ∈ H˚ D1 (Ω). (8.7) D

D

Ω

We also have the same estimate ∇ · [u∇χ] H −1 ≤ C u D

2

Hp

χ Wp1 ,

2

u ∈ H p (Ω), χ ∈ D(Λ|L2 ) ∩ Wp1 (Ω).

(8.8) 

Of course, this proposition is available with p = p0 . However, in this case it holds that D(Λ|L2 ) ∩ Wp10 (Ω) = D(Λ|L2 ) due to (8.6). We choose as an underlying space for dealing with (8.1) the product space of HD−1 (Ω), i.e.,   ϕ X= ; ϕ ∈ HD−1 (Ω) and ψ ∈ HD−1 (Ω) . (8.9) ψ For electrons and holes, the initial densities are taken in the space   u0 ; 0 ≤ u0 ∈ L2 (Ω) and 0 ≤ v0 ∈ L2 (Ω) . K= v0

(8.10)

2 Local Solutions 2.1 Construction of Local Solutions We formulate problem (8.1) as the Cauchy problem for an abstract equation  dU 0 < t < ∞, dt + AU = F (U ), (8.11) U (0) = U0 , in the space X. Here, A is a diagonal matrix operator A = diag{aΛ, bΛ} in X with domain   u 1 1 ˚ ˚ D(A) = ; u ∈ HD (Ω) and v ∈ HD (Ω) , v where Λ is the operator introduced above. From (8.4) and (8.5) we have  1 D(A 2 ) = L2 (Ω), D(Aθ ) ⊂ H2θ−1 (Ω), 12 < θ ≤ 1

(8.12)

2 Local Solutions

349

(see (1.84) and (1.85)). The nonlinear operator F is given by F (U ) =

−μ∇ · [u∇(cΛ)−1 (−u + v + h(x))] + f (1 − uv) + g(x) ν∇ · [v∇(cΛ)−1 (−u + v + h(x))] + f (1 − uv) + g(x)

with domain D(F ) =

  2 2 u ; u ∈ H p0 (Ω) and v ∈ H p0 (Ω) , v

where p0 > 2 is the number for which (8.6) holds. In view of (8.7), F (U ) is well defined. Let η = 12 + p10 < 1. Then, by (8.12) we have D(Aη ) ⊂ D(F ). The initial value U0 is taken in K. We are going to apply Theorem 4.1 with β = 12 and η = 12 + p10 . By virtue of (8.6) and (8.8), 

∇ · [u∇Λ−1 (−u + v + h(x))] H −1 ≤ C u H 2/p0 − u + v + h L2 , D

∇ · [v∇Λ−1 (−u + v + h(x))] H −1 ≤ C v H 2/p0 − u + v + h L2 . D

Similarly, uv H −1 = D

     sup  uvw dx  ≤ C

w H˚ 1 ≤1

Ω

D

sup

w H˚ 1 ≤1 D

u L

2p0 p0 −2

v L2 w Lp0

≤ C u H 2/p0 v L2 . Therefore, we verify by (8.12) that  1   1   F (U ) − F (V ) X ≤C A 2 U X + A 2 V X + 1 Aη (U − V ) X  1 

+ ( Aη U X + Aη V X + 1)A 2 (U − V )X , U, V ∈ D(Aη ). This shows that (4.2) is fulfilled with β = 12 . Consequently, Theorem 4.1 is available to (8.11). We thus conclude that, for any U0 ∈ K, there exists a unique local solution U to (8.11) in the function space: U ∈ C((0, TU0 ]; D(A)) ∩ C([0, TU0 ]; L2 (Ω)) ∩ C1 ((0, TU0 ]; X), where TU0 > 0 is determined by the norm U0 L2 only. Moreover, √ t AU (t) X + U (t) L2 ≤ CU0 ,

0 < t ≤ TU 0 ,

where CU0 > 0 is also determined by the norm U0 L2 only.

(8.13)

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8 Semiconductor Models

2.2 Nonnegativity of Solutions For U0 ∈ K, let U = t (u, v) be the local solution obtained above in the function space (8.13). We will prove that u(t) ≥ 0 and v(t) ≥ 0 for every 0 < t ≤ TU0 . Let us first verify that U (t) is real valued. Indeed, it is clear that the complex conjugate U (t) of U (t) is also a local solution of (8.11) with the same initial value; so, by the uniqueness of solution, U (t) = U (t); hence, U (t) is real valued. Let us next introduce auxiliary linear problems. It is not difficult to construct sequences {uk }, {vk } of functions such that uk , vk ∈ C0,1 ([0, TU0 ]; L∞ (Ω))

(8.14)

and that, as k → ∞, uk → u and vk → v in C([0, TU0 ]; L2 (Ω)). (For example, we cut off the values of |u(t)| ≥ k and |v(t)| ≥ k and use the mollifier concerning the variable t.) Using uk and vk , we formulate the linear problem ⎧ ⎨ d U˜ k + AU˜ = B (t)U˜ + F, 0 < t ≤ T , k k k U0 dt (8.15) ⎩U˜ k (0) = U0 , in X. Here, Bk (t) is the family of closed linear operators given by

−μ∇ · [u˜ k ∇χk (t)] − u˜ k vk (t) u˜ Bk (t)U˜ k = , U˜ k = k ∈ D(Bk (t)), v˜k ν∇ · [v˜k ∇χk (t)] − uk (t)v˜k where χk (t) = (cΛ)−1 [−uk (t) + vk (t) + h], with domains D(Bk (t)) ≡ D(Aη ), and F is the vector of X given by F = [f + g(x)]e, where e = t (1, 1). The operators Bk (t) are weaker linear operators than A in the sense that D(A) ⊂ D(Aη ) ≡ D(Bk (t)) for all 0 < t ≤ TU0 . In addition, they are seen from (8.7) to have regularity [Bk (t) − Bk (s)]A−η L(X) ≤ Ck |t − s|,

0 ≤ s, t ≤ TU0 ,

due to (8.14). Then, we can apply Theorem 3.14 with μ˜ = 1 − η and ν˜ = 1 to this problem. It is then deduced that problem (8.15) possesses a unique solution U˜ k in the function space: U˜ k ∈ C((0, TU0 ]; D(A)) ∩ C([0, TU0 ]; L2 (Ω)) ∩ C1 ((0, TU0 ]; X). Furthermore, by direct calculations (cf. Theorem 4.3), it is possible to verify that t η Aη [U˜ k (t) − U (t)] X ≤ C

 1  sup A 2 [Uk (t) − U (t)]X ,

0≤t≤TU0

that is, as k → ∞, U˜ k (t) converges to U (t) for every t ∈ (0, TU0 ] in D(Aη ). Therefore, if we show that u˜ k (t) ≥ 0 and v˜k (t) ≥ 0, then it follows that u(t) ≥ 0 and v(t) ≥ 0.

2 Local Solutions

351

Our goal is therefore to prove that u˜ k (t) ≥ 0 and v˜k (t) ≥ 0 for every 0 < t ≤ TU0 . To this end, we use the truncation method. Let H (u) be a C1,1 cutoff function such that H (u) = 12 u2 for −∞ < u < 0 and H (u) ≡ 0 for 0 ≤ u < ∞. Since     u˜ k ∈ C (0, TU0 ]; H˚ D1 (Ω) ∩ C1 (0, TU0 ]; HD−1 (Ω) ,  we get by (1.101) that the function ψk (t) = Ω H (u˜ k (x, t)) dx is continuously differentiable with the derivative ψk (t) = H  (u˜ k ), −aΛu˜ k − μ∇ · [u˜ k ∇χk ]H˚ 1 ×H −1 D D dt + (H  (u˜ k ), −u˜ k vk + f + g(x))L2 . By property (1.96), we have 

H  (u˜ k ), −aΛu˜ k H˚ 1 ×H −1 = −a D

D



∇H  (u˜ k ) · ∇ u˜ k dx

Ω

|∇H  (u˜ k )|2 dx ≤ 0.

= −a Ω

Similarly, by (8.7) and Lemma 8.1 presented below,   u˜ k ∇H  (u˜ k ) · ∇χk dx

H (u˜ k ), −μ∇ · [u˜ k ∇χk ]H˚ 1 ×H −1 = μ D

D

Ω



H  (u˜ k )∇H  (u˜ k ) · ∇χk dx

=μ Ω

=

μ 2c



H  (u˜ k )2 (−uk + vk + h) dx.

Ω

Finally, due to (8.2) and f ≥ 0,   H  (u˜ k )[−u˜ k vk + f + g(x)] dx ≤ − vk H  (u˜ k )2 dx. Ω

Ω

Thus, in view of (8.3) and (8.14), ψk (t) ≤ Ck ψk (t); consequently, ψk (t) ≤ ψk (0)eCk t . Therefore, ψk (0) = 0 implies ψk (t) ≡ 0, that is, u˜ k (t) ≥ 0 for every 0 < t ≤ TU0 . It is the same for v˜k (t) ≥ 0 for every 0 < t ≤ TU0 . Lemma 8.1 For u ∈ H˚ D1 (Ω) and χ ∈ D(Λ|L2 ) ∩ Wp1 (Ω), where 2 < p < ∞, we have   1 u∇u · ∇χ dx = u2 Λχ dx. 2 Ω Ω Proof If u ∈ H˚ D1 (Ω) ∩ L∞ (Ω), then u2 ∈ H˚ D1 (Ω). Therefore, the result is verified directly noting that u∇u = 12 ∇u2 . For general u ∈ H˚ D1 (Ω), it is sufficient to approximate u by cutoff functions Ψk (u) ∈ H˚ D1 (Ω) ∩ L∞ (Ω), where Ψk (u) ≡ −k for

352

8 Semiconductor Models

u ≤ −k, Ψk (u) = u for |u| ≤ kd, and Ψk (u) ≡ k for u ≥ k. Owing to (1.96), we verify that Ψ (uk ) → u in H˚ D1 (Ω) as k → ∞. Meanwhile, note by (1.79) that it follows  from u ∈ H˚ D1 (Ω) ⊂ H 1 (Ω) that u ∈ L 2p (Ω). p−2

3 Global Solutions 3.1 A Priori Estimates for Local Solutions For U0 ∈ K, let U = t (u, v) denote a local solution of (8.11) on [0, TU ] in the function space:  0 ≤ u ∈ C((0, TU ]; H˚ D1 (Ω)) ∩ C([0, TU ]; L2 (Ω)) ∩ C1 ((0, TU ]; HD−1 (Ω)), 0 ≤ v ∈ C((0, TU ]; H˚ D1 (Ω)) ∩ C([0, TU ]; L2 (Ω)) ∩ C1 ((0, TU ]; HD−1 (Ω)). (8.16) Proposition 8.2 There exists a constant C > 0 such that, for any U0 ∈ K and any local solution of (8.11) in the function space (8.16), it holds that U (t) L2 ≤ C( U0 L2 + 1),

0 ≤ t ≤ TU , U0 ∈ K.

(8.17)

Proof Consider the duality product of HD−1 (Ω) × H˚ D1 (Ω) between the first equation of (8.11) and u. Then,     1 d 2 2 u dx + a |∇u| dx − μ u∇u · ∇χ dx + f u2 v dx 2 dt Ω Ω Ω Ω  = [f + g(x)]u dx, Ω

where χ = (cΛ)−1 [−u + v + h(x)]. Similarly, by the second equation of (8.11),     1 d 2 2 v dx + b |∇v| dx + ν v∇v · ∇χ dx + f uv 2 dx 2 dt Ω Ω Ω Ω  = [f + g(x)]v dx. Ω

We sum up these two equalities after multiplying the first one by ν and the second one by μ, respectively. Then, since Lemma 8.1 yields that   μν μν (−u∇u · ∇χ + v∇v · ∇χ) dx = (v 2 − u2 )Λχ dx 2 Ω Ω  μν = (u − v)2 (u + v) dx 2c Ω  μν + h(x)(v 2 − u2 ) dx, 2c Ω

3 Global Solutions

353

we have 1 d 2 dt



 (νu + μv ) dx + 2

2

Ω



 aν|∇u|2 + bμ|∇v|2 dx

Ω

  μν + (u − v)2 (u + v) dx + f (νu + μv)uv dx 2c Ω Ω   μν = [f + g(x)](νu + μv) dx + h(x)(u2 − v 2 ) dx. 2c Ω Ω

Furthermore, by (8.2),  [f + g(x)](νu + μv) dx ≤ f + g L2 μu + νv L2 Ω

 

≤ f + g L2 ζ u 2L2 + v 2L2 + Cζ

with an arbitrarily small parameter ζ > 0. Similarly, by (8.3),  

ζ (u − v)2 (u + v) + Cζ (u + v) dx h(x)(u2 − v 2 ) dx ≤ h L∞ Ω

Ω

 ≤ h L∞



ζ (u − v)2 (u + v) + ζ (u2 + v 2 ) + Cζ dx.

Ω

Meanwhile, by Poincaré’s inequality (see Remark 2.3),     2 2 aν|∇u| + bμ|∇v| dx ≥ α (u2 + v 2 ) dx Ω

Ω

with some positive number α > 0. Therefore, taking ζ > 0 sufficiently small, we obtain that   α 1 d 2 2 (νu + μv ) dx + (u2 + v 2 ) dx ≤ C 2 dt Ω 2 Ω with some constant C > 0. Hence (cf. (1.58)),  

u(t) 2L2 + v(t) 2L2 ≤ C e−δt u0 2L2 + v0 2L2 + 1 , with some positive exponent δ > 0 and constant C > 0.

0 ≤ t ≤ TU ,

(8.18) 

3.2 Global Solutions As the a priori estimates (8.17) are verified for local solutions, we can apply Corollary 4.1 to conclude the global existence of solutions. Thus, for any U0 ∈ K, (8.11)

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8 Semiconductor Models

possesses a unique global solution in the function space:  0 ≤ u ∈ C((0, ∞); H˚ D1 (Ω)) ∩ C([0, ∞); L2 (Ω)) ∩ C1 ((0, ∞); HD−1 (Ω)), 0 ≤ v ∈ C((0, ∞); H˚ D1 (Ω)) ∩ C([0, ∞); L2 (Ω)) ∩ C1 ((0, ∞); HD−1 (Ω)). (8.19)

3.3 Global Norm Estimates For U0 ∈ K, let U (t; U0 ) be the global solution of (8.11) with the initial value U0 in the function space (8.19). By (8.18), there exist an exponent δ > 0 and a constant C˜ > 0 such that, for any U0 ∈ K, it holds that ˜ −δt U0 L2 + 1), U (t; U0 ) L2 ≤ C(e

0 ≤ t < ∞, U0 ∈ K.

(8.20)

4 Dynamical System 4.1 Construction of Dynamical System We are now in a position to construct a dynamical system determined from problem (8.11) in the universal space X by following the general methods described in Chap. 6, Sect. 5, where the exponent β will be taken as β = 12 . We already know that, for any U0 ∈ K, (8.11) and hence (8.1) possesses a unique global solution U (t; U0 ) in the function space (8.19). By S(t)U0 = U (t; U0 ), we define a nonlinear semigroup S(t) acting on K. Estimate (8.20) shows that (6.56) is fulfilled. So, (8.11) defines a dynamical system (S(t), K, L2 (Ω)). Recall that 1 D(A 2 ) = L2 (Ω).

4.2 Exponential Attractors We shall construct exponential attractors for the dynamical system (S(t), K, L2 (Ω)). To this end, however, it suffices to verify (6.59) and (6.60). The space condition (6.59) is clear. Meanwhile, (8.20) shows that the dissipative condition (6.60) is also fulfilled. Hence, (S(t), K, L2 (Ω)) enjoys a family of exponential attractors.

5 Three-Dimensional Problem In this section, we want to discuss the three-dimensional problem. Let Ω ⊂ R3 be a bounded domain with Lipschitz boundary ∂Ω which is split into two parts ΓD and ΓN satisfying (2.41) and (2.42).

Notes and Further Researches

355

As a matter of fact, we can argue in a similar way as in the two-dimensional case. However, in order to extend definition of the drift term ∇ · [u∇χ] as (8.7), we have to assume the stronger shift property that Λu ∈ L2 (Ω) implies u ∈ Wp11 (Ω) with some p1 > 3 and the estimate u Wp1 ≤ C Λu L2 , 1

u ∈ D(Λ|L2 ).

(8.21)

We set the same underlying space X as (8.9). Three-dimensional problem (8.1) is then written as an abstract problem of the form (8.8), where A and F are the same linear and nonlinear operators, respectively, as before. The only difference is that F has the domain   3 3 u D(F ) = ; u ∈ H p1 (Ω) and v ∈ H p1 (Ω) , v where p1 > 3 is the number for which (8.21) is valid. Then, D(Aη ) ⊂ D(F ) with η = 12 + 2p3 1 < 1 due to (8.12) (which is independent of the dimensions), and F is

shown to fulfill (4.2) with β = 12 and η = 12 + 2p3 1 . We can then repeat the same arguments as in the preceding sections to obtain the local existence of solutions, nonnegativity of local solutions, and the a priori estimates of local solutions. Hence, under (8.21), we can deduce the uniqueness and global existence of solutions for any pair (u0 , v0 ) of initial functions in K (see (8.10)).

Notes and Further Researches The semiconductor model (8.1) was presented by Shockley to describe the flows of electrons and holes in a semiconductor in the middle of the last century. In his historic book [Sho50] published in 1950, we may find the origin of the macroscopic studies of physical systems which consist of many individuals possessing some uniform properties by the analytical methods. For the physical backgrounds of the semiconductor model and the details of modeling, we refer to the references Sze [Sze81], Selberherr [Sel83], and Gajewski–Gröger [GG90]. Many authors have already contributed to the study of semiconductor equations. The evolutional problems have been studied by many papers including [FI95a, FI95b, FI95c, Gaj85, GG86, Jer87, Moc74, Moc75, ST85]. In particular, the asymptotic behavior of solutions was studied in [FI95c, Gaj85, GG86, Moc74]. Mock [Moc74] first proved, in a simple case, that every solution converges to the stationary solution at an exponential rate. Gajewski [Gaj85] and Gajewski–Gröger [GG86] generalized this result, but they still assumed some conditions which guarantee the unique solvability of the stationary problem. In the general case where the stationary solutions are not unique, Fang–Ito [FI95c] constructed a global attractor for the dynamical system determined from (8.1). More precisely, they constructed a global attractor with finite Hausdorff dimension under the spectral gap condition for the operator Λ. The stationary problems have been studied, e.g., by [FN94, FN96, Jer85,

356

8 Semiconductor Models

Moc72, Mar86] and [BF02, Chap. 6]. The existence of stationary solutions to (8.1) was shown in various situations. On the contrary, Bensoussan–Frehse [BF02, Theorem 6.2] gave some special conditions which imply the uniqueness of stationary solutions. The same results as in this chapter was already obtained in Favini–Lorenzi– Yagi [FLY06]. However, in [FLY06], the abstract result of [OY02] was used to construct local solutions. In addition, the result due to [EMZ00] was used to construct exponential attractors. As shown in Sect. 4.2, we were able to construct a finite-dimensional global attractor without any spectral gap condition as assumed by Fang–Ito [FI95c]. For constructing (local) solutions to (8.11), we indeed made an essential use of the shift property (8.6) which was established in Bensoussan– Frehse [BF02] (cf. also Lorenzi [Lor78]). In the three-dimensional case, however, no general conditions are known for ensuring such a shift property (8.21). Recently, mathematical models incorporating quantum effects are attracting interest of researchers. Ancona–Tiersten [AnT87] and Ancona–Iafrate [AI89] presented the following quantum-drift-diffusion model: √ ⎧ Δ u

⎨ ∂u √ + F, = aΔu − ∇ · [μu∇χ] − ∇ · αu∇ ∂t u √

⎩ ∂v = bΔv + ∇ · [νv∇χ] − ∇ · βv∇ Δ√ v + F ∂t v (cf. also Gardner [Gar94]). Here, u = u(x, t) and v = v(x, t) denote as before the densities√of electrons and holes, respectively, in a semiconductor device. The terms √ Δ u Δ v √ √ −αu∇ u and −βv v denote quantum corrections for the flows of u and v, respectively, α > 0 and β > 0 being some positive coefficients, and F denotes the rate of generation and recombination of electrons and holes. Problem of constructing the dynamical system determined from such a quantum model remains entirely to be studied.

Chapter 9

Activator–Inhibitor Models

Let us consider the activator–inhibitor system. The general model equation is given by the reaction–diffusion system  ∂u ∂t = aΔu + γf (u, v) in Ω × (0, ∞), (9.1) ∂v ∂t = bΔv + γ g(u, v) in Ω × (0, ∞), in a domain Ω ⊂ Rd . Here, u denotes the concentration of the activator, and v the concentration of the inhibitor in Ω, respectively. The functions f (u, v) and g(u, v) denote the kinetics and are real smooth functions defined for u ≥ 0 and v ≥ 0. The activator and the inhibitor diffuse in Ω with diffusion rates a > 0 and b > 0, respectively. The constant γ > 0 denotes the reaction rate. Species of the activator or inhibitor must be specified in each activator–inhibitor system under consideration. The form of kinetic functions f (u, v) and g(u, v) must also be fixed in an adequate manner. Let the system possess a positive homogeneous stationary solution (u, v), i.e., f (u, v) = 0 and g(u, v) = 0. The partial derivatives of the kinetics at (u, v) are denoted by f u = fu (u, v), f v = fv (u, v), g u = gu (u, v), and g v = gv (u, v). We assume that f u > 0, that is, the activator is created autocatalystically, and that g v < 0 with the relation f u + g v < 0, that is, the inhibitor degrades faster than the activator grows. We assume also that f v < 0 and g u > 0; therefore, the inhibitor restrains growth of the activator, and oppositely, the activator produces the inhibitor, too. The relation f u g v > f v g u is also assumed. Hence, the partial derivatives are assumed to satisfy the following inequalities: f u > 0,

f u + g v < 0,

It then holds that f u + g v < 0,

f ugv > f v gu,

and f u g v > f v g u .

(9.2)

 and ag v + bf u < 2 ab(f u g v − f v g u ) (9.3)

A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_9, © Springer-Verlag Berlin Heidelberg 2010

357

358

9 Activator–Inhibitor Models

if the diffusion constants a and b coincide, i.e., a = b. According to the arguments in Chap. 6, Sect. 6.4 (since (6.82) is valid), this condition indicates that the homogeneous solution (u, v) is a stable stationary solution. Thus, under (9.2), the coincidence a = b provides the stability of the homogeneous solution (u, v). On the contrary, if the diffusion rate b of the inhibitor is greater than the a of the activator and their difference is large enough, then the last inequality in (9.3) becomes false, that is, (6.82) in Chap. 6, Sect. 6.4 is no longer valid. This means that the stationary solution (u, v) loses stability. Thus, under the same condition as (9.2), the relation b > a with sufficiently large difference can provide the instability of the homogeneous stationary solution. Such a principle is called the diffusion-driven instability and was originally proposed by Turing in 1952. This chapter is devoted to handling a typical model, choosing suitable functions f (u, v) and g(u, v), and to showing rigorously that the Turing instability principle actually takes place.

1 Prototype Reaction–Diffusion Model We take the kinetic functions f (u, v) = k1 − k2 u +

k3 u2 , v

g(u, v) = k4 u2 − k5 v,

presented by Gierer–Meinhardt [GM72] and Meinhardt [Mei82] (cf. also Meinhardt [Mei92] and Murray [Mur03, Chap. 2]), where ki are positive constants for i = 1, . . . , 5. After nondimensionalization (see [Mur03, p. 78]), the equations are written by  ∂u u2 ∂t = aΔu + γ (d − cu + v ) in Ω × (0, ∞), ∂v ∂t

= bΔv + γ (u2 − v)

in Ω × (0, ∞).

So, we consider the initial-boundary-value problem ⎧ ∂u 2 = aΔu + γ (d − cu + uv ) ⎪ ⎪ ∂t ⎪ ⎪ ⎨ ∂v 2 ∂t = bΔv + γ (u − v) ⎪ ∂u ⎪ = ∂v = 0 ⎪ ⎪ ⎩ ∂n ∂n v(x, 0) = v0 (x) u(x, 0) = u0 (x),

in Ω × (0, ∞), in Ω × (0, ∞), on ∂Ω × (0, ∞), in Ω,

(9.4)

where 0 < a ≤ b are diffusion constants of the activator and inhibitor, respectively, and where c, d, and γ are positive constants (> 0). We consider this problem in a three-dimensional C2 or convex, bounded domain Ω. Let from now on f (u, v) and g(u, v) denote the functions f (u, v) = d − cu +

u2 , v

g(u, v) = u2 − v,

(9.5)

2 Local Solutions

359

respectively. Then, fu (u, v) = −c + gu (u, v) = 2u,

2u u2 , fv (u, v) = − 2 , v v gv (u, v) − 1.

It is easy to verify that (9.4) possesses a unique homogeneous stationary solution (u, v), which is given by d +1 u= c

 and v =

d +1 c

2 (9.6)

.

Obviously, f u = fu (u, v) =

c(1 − d) , d +1

f v = fv (u, v) = −

g u = gu (u, v) =

2(d + 1) , c

g v = gv (u, v) = −1.

c2 , (d + 1)2

We then easily verify that, at least under the assumptions that 0 < d < 1 and 0 < c <

d +1 , 1−d

(9.7)

the partial differential coefficients f u , f v , g u , and g v satisfy all the inequalities in (9.2). That is, under (9.7), a = b implies the stability of the stationary solution (u, v). We shall see below that, if b is large enough with respect to a, then even under (9.7), (u, v) becomes unstable.

2 Local Solutions 2.1 Construction of Local Solutions We want to handle problem (9.4) in the L2 space. We then set the underlying space X as the product space of L2 (Ω), i.e., X = L2 (Ω). In addition, we set the space of initial functions by

  3 3 u0 K= ; 0 ≤ u0 ∈ H 4 (Ω) and v0 ∈ H 4 (Ω), ess inf v0 > 0 . (9.8) Ω v0 As the equation of (9.4) on u has a singularity at v = 0, we will impose the condition ess infΩ v0 > 0 on the initial functions v0 . We introduce a diagonal matrix operator A = diag{A1 , A2 } of X with domain D(A) = H2N (Ω), where A1 and A2 are realizations of the operators −aΔ + γ c

360

9 Activator–Inhibitor Models

and −bΔ + γ in L2 (Ω), respectively, under the homogeneous Neumann boundary conditions on ∂Ω. On account of Theorem 1.25, A1 and A2 are positive definite self-adjoint operators of L2 (Ω) with domains HN2 (Ω) (see (2.34)), H2N (Ω) being the product space of HN2 (Ω). Consequently, A is a positive definite self-adjoint operator of X. Furthermore, on account of Theorems 16.7 and 16.9 in Chap. 16, it is verified that 3 D(Aθ ) = H2θ (Ω) if 0 ≤ θ < , 4 3 < θ ≤ 1. D(Aθ ) = H2θ N (Ω) if 4

(9.9) (9.10)

Let ε > 0 be an arbitrarily fixed positive number. We introduce a nonlinear operator defined as follows. Let η be a fixed exponent such that 34 < η < 1. Let Fε : D(Aη ) → X be given by   γ d + γ u2 /χε (Re v) u (9.11) , U = ∈ D(Aη ). Fε (U ) = v γ u2 Here, χε (ξ ) is a cutoff function such that χε (ξ ) = ξ for ε ≤ ξ < ∞ and χε (ξ ) ≡ ε for −∞ < ξ ≤ ε; obviously, χε (ξ ) is a Lipschitz continuous function defined for ξ ∈ R. In view of (1.76) and (9.10), it holds true that D(Aη ) ⊂ H2η (Ω) ⊂ L∞ (Ω), which shows that Fε certainly maps D(Aη ) into X. Problem (9.4), at least for U0 ∈ K satisfying ess infΩ v0 > ε, is now formulated as the Cauchy problem for a semilinear abstract equation  dU dt + AU = Fε (U ), 0 < t < ∞, (9.12) U (0) = U0 , in X. Let us apply the general results in Chap. 4 to construct local solutions. Our nonlinear operator Fε fulfills (4.2). In fact, we have γ u2

 2   γ u˜ 2 ˜ 2 |χε (Re v) − χε (Re v)| ˜ + |u2 − u˜ 2 | ≤ Cε |u| + |u| χ (Re v) − χ (Re v) ˜ ε ε   ≤ Cε |u|2 + |u| ˜ 2 + 1 (|v − v| ˜ + |u − u|). ˜ Therefore, ˜ L∞ + u − u ˜ L∞ ) Fε (U ) − Fε (U˜ ) X ≤ Cε ( u2 L2 + u˜ 2 L2 + 1)( v − v   ≤ Cε u 2L4 + u ˜ 2L4 + 1 ( v − v ˜ L∞ + u − u ˜ L∞ ). 3

3

Noting that D(A 8 ) = H 4 (Ω) ⊂ L4 (Ω), we obtain that  3 2  3 2  Fε (U ) − Fε (U˜ ) X ≤ Cε A 8 U X + A 8 U˜ X + 1 × Aη (U − U˜ ) X ,

U, U˜ ∈ D(Aη ).

(9.13)

2 Local Solutions

361

This means that (4.2) is satisfied by Fε with β = 38 and the η fixed above. We can now apply Theorem 4.1. As a consequence, for any U0 ∈ K, (9.12) possesses a unique local solution in the function space:   3  U ∈ C((0, TU0 ]; D(A)) ∩ C [0, TU0 ]; D A 8 ∩ C1 ((0, TU0 ]; X),

(9.14)

3

where TU0 > 0 is determined by the norm A 8 U0 X only.

2.2 Nonnegativity of Local Solutions For U0 ∈ K, let U (t) = t (u(t), v(t)) be the local solution constructed above. We want to prove that u(t) ≥ 0 and v(t) > 0 for every 0 < t ≤ TU0 by using the truncation method. Let us first verify that U (t) is real valued. Indeed, the complex conjugate U (t) of U (t) is also a local solution of (9.12) with the same initial value U0 . So, the uniqueness of solution implies that U (t) = U (t); hence, U (t) is real valued. Let H (u) be a C1,1 cutoff function given by H (u) = 12 u2 for −∞  < u < 0 and H (u) ≡ 0 for 0 ≤ u < ∞. According to (1.100), the function ϕ(t) = Ω H (u(t)) dx is continuously differentiable with the derivative ϕ  (t) = a



H  (u)Δu dx + γ Ω

 u2 H  (u) d − cu + dx. χε (v) Ω



Property (1.96) provides that    H  (u)Δu dx = − ∇H  (u) · ∇u dx = − |∇H  (u)|2 dx ≤ 0. Ω

Ω

Ω

In addition, since H  (u) ≤ 0 and H  (u)u ≥ 0, it is seen that   u2 H  (u) d − cu + dx ≤ 0. χε (v) Ω Hence, ϕ  (t) ≤ 0; consequently, ϕ(t) ≤ ϕ(0) for every 0 < t ≤ TU0 . Thus, ϕ(0) = 0 implies ϕ(t) ≡ 0, that is, u(t) ≥ 0 for 0 < t ≤ TU0 . As for v(t), we consider ψ(t) = Ω H (v(t) − e−γ t ε0 ) dx, where ε0 = ess infΩ v0 > 0. Then, from   ψ (t) = b H  (v − e−γ t ε0 )Δ[v − e−γ t ε0 ] dx Ω



−γ Ω

H  (v − e−γ t ε0 )[v − e−γ t ε0 ] dx + γ



H  (v − e−γ t ε0 )u2 dx

Ω

it follows that ψ  (t) ≤ 0. Hence, v(t) ≥ e−γ t ε0 for 0 < t ≤ TU0 .

362

9 Activator–Inhibitor Models

This shows that, if ess infΩ v0 = ε0 > ε, then the local solution of (9.12) constructed above is a solution to (9.4), too, at least for t such that (v(t) ≥)e−γ t ε0 ≥ ε. The lower estimate for v(t) obtained in this subsection will be improved below by a uniform one.

3 Global Solutions 3.1 A Priori Estimates from Below Let U0 = t (u0 , v0 ) be an initial value in K with ess infx∈Ω v0 (x) = ε0 > 0. Let U = t (u, v) be any local solution of (9.12) on [0, T ] in the function space: U 

3

0 ≤ u ∈ C((0, TU ]; HN2 (Ω)) ∩ C([0, TU ]; H 4 (Ω)) ∩ C1 ((0, TU ]; L2 (Ω)), 3

0 < v ∈ C((0, TU ]; HN2 (Ω)) ∩ C([0, TU ]; H 4 (Ω)) ∩ C1 ((0, TU ]; L2 (Ω)). (9.15) Then, it holds that u(t) ≥ (d/c)(1 − e−γ ct ), v(t) ≥ ε0 e

−γ t

+ e(t),

0 ≤ t ≤ TU ,

0 ≤ t ≤ TU .

(9.16) (9.17)

t Here, e(t) denotes the function e(t) = γ (d/c)2 e−γ t 0 eγ s (1 − e−γ cs )2 ds; clearly, e(t) > 0 for t > 0 and limt→∞ e(t) = (d/c)2 . Note that these lower estimates are independent of the fixed number ε > 0 in (9.12). In fact, these estimates are verified by a quite analogous method to that used in Sect. 2.2. We consider the function u(t) ˜ = u(t) − (d/c)(1 − e−γ ct ), By (1.100), ϕ(t) ˜ = 

 Ω

H (u(t)) ˜ dx is continuously differentiable with the derivative

d u˜ H (u) ˜ dx = ϕ˜ (t) = dt Ω 



0 ≤ t ≤ TU .



  γ u2 −γ ct H (u) ˜ aΔu + γ (d − cu) + − γ de dx. χε (v) Ω 

By the same arguments as before (note that ∇ u˜ = ∇u), we conclude that ϕ˜  (t) ≤ 0. Hence, ϕ(t) ˜ ≡ 0 for 0 ≤ t ≤ TU . This shows that (9.16) is valid. Consider next the function v(t) ˜ = v(t) − [ε0 e−γ t + e(t)], ˜ We put ψ(t) =

 Ω

H (v(t)) ˜ dx. Then,

0 ≤ t ≤ TU .

3 Global Solutions

363



d v˜ dx dt Ω 

= H  (v) ˜ bΔv + γ (u2 − v) + ε0 γ e−γ t

ψ˜  (t) =

H  (v) ˜

Ω

 + γ e(t) − γ (d/c)2 (1 − e−γ ct )2 dx. Since (9.16) is already known, it follows that     2 ˜ ψ (t) ≤ −b H (v)|∇v| ˜ dx − γ H  (v) ˜ v˜ dx ≤ 0. Ω

Ω

˜ Hence, ψ(t) ≡ 0 for 0 ≤ t ≤ TU . We have thus verified (9.17).

3.2 A Priori Estimates for Sobolev Norms Let U0 ∈ K be an initial value satisfying ess infΩ v0 (x) = ε0 > 0. Let ε be such that 0 < ε ≤ min {e−γ t ε0 + e(t)}. 0≤t<∞

Let U = t (u, v) be any local solution of (9.12) on [0, TU ] in the function space (9.15) with the lower estimate v(t) ≥ e−γ t ε0 + e(t) ≥ ε,

0 ≤ t ≤ TU .

(9.18)

By using the same techniques as in Masuda–Takahashi [MaT87] and Jian [Jia06], 2 we are able to estimate L2 norm of the term uv . Proposition 9.1 Let U be any local solution of (9.12) in the space (9.15) which satisfies (9.18). Then,   2    u   u(t)2   ≤ Cε e−γ ct  0  + 1 , 0 ≤ t ≤ TU ,  (9.19) v   v(t)  0 L2 L2 Cε > 0 being a constant independent of U0 and TU . Proof Let 2 < p < ∞ and 0 < q < ∞ be exponents to be fixed below. We put p w(t) = u(t) v(t)q . Then, d dt



qup ∂v pup−1 ∂u − dx v q ∂t v q+1 ∂t Ω   p−1 qup pu aΔu − bΔv dx = vq v q+1 Ω  

w dx = Ω

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9 Activator–Inhibitor Models

  +γ Ω

  pup−1 u2 qup 2 d − cu + − (u − v) dx vq v v q+1

=I + II. By integration by parts, the first integral I is rewritten in the form   up−2 up−1 ap(p − 1) q |∇u|2 − (a + b)pq q+1 ∇u · ∇v I =− v v Ω  p u + bq(q + 1) q+2 |∇v|2 dx v  up−2 =− ap(p − 1)|v∇u|2 − (a + b)pq(v∇u) · (u∇v) q+2 Ω v  + bq(q + 1)|u∇v|2 dx. We here assume that p and q satisfy √ √ pq 2 ab . ≤ √ (p − 1)(q + 1) a + b

(9.20)

Then, it is easily observed that I ≤ 0. In the meantime, the second integral is equal to   −q p−1   II = γ v du − (pc − q)up + v −(q+1) (pup+1 − qup+2 ) dx. Ω

Here, we notice that dup−1 ≤ ζ up + Cp,ζ p+1

pu

 p+2

≤ζ u

for all u ≥ 0,

+ Cp,ζ 

for all u ≥ 0,

with arbitrarily small numbers ζ > 0 and ζ  > 0, provided that Cp,ζ > 0 and Cp,ζ  > 0 are chosen in a suitable way. We assume in addition to (9.20) that q<

cp . 2

(9.21)

Then, fixing ζ > 0 and ζ  > 0 sufficiently small and noting that v(t) ≥ ε (due to (9.18)), we obtain that  γ cp w dx + Cε,p,q . II ≤ − 2 Ω  γ cp  d In this way, we have dt Ω w dx ≤ − 2 Ω w dx + Cε,p,q and conclude that w(t) L1 ≤ e−

γ cp 2 t

w(0) L1 + Cε,p,q ,

0 ≤ t ≤ TU .

(9.22)

3 Global Solutions

365

In view of (9.20) and (9.21), we can take p = 4, provided that q > 0 is taken sufficiently small. Let us indeed fix p = 4 and q such that 0 < q ≤ 2. Then, since 4 4 uv 2 L1 ≤ ε q−2 vuq L1 , (9.22) yields the desired estimate (9.19).  The equation on u is written as the abstract equation du u(t)2 + A1 u = d + , dt v(t)

0 < t ≤ TU ,

in L2 (Ω). Therefore, u(t) = e−tA1 u0 +

 0

t

 u(s)2 ds, e−(t−s)A1 d + v(s)

where e−tA1 is the analytic semigroup on L2 (Ω) generated by −A1 and satisfies the estimate e−tA1 L2 ≤ e−γ ct . Furthermore,   t 3 3 3 u(s)2 ds. A18 u(t) = e−tA1 A18 u0 + A18 e−(t−s)A1 d + v(s) 0 Using this expression, we immediately derive from (9.19) that  38  A u(t) 1

L2

 3 

 ≤ Cε e−δt A18 u0 L + 1 , 2

0 ≤ t ≤ TU ,

(9.23)

with some positive exponent δ > 0 and a constant Cε > 0 independent of U0 and TU . As well,  3 

 u(t) L4 ≤ Cε e−δt A18 u0 L + 1 , 0 ≤ t ≤ TU . 2

We can repeat the same argument for v(t) to conclude that  38  A v(t) 2

L2

 3  

 3   ≤ Cε e−δt A18 u0 L + A28 v0 L + 1 , 2

2

0 ≤ t ≤ TU .

Hence, this, together with (9.23), provides the final estimate  3   

 A 8 U (t) ≤ Cε e−δt A 38 U0  + 1 , X X

0 ≤ t ≤ TU .

(9.24)

3.3 Global Solution Let U0 ∈ K be an initial value with ess infΩ v0 (x) = ε0 > 0. Taking ε > 0 small enough to satisfy min {ε0 e−γ t + e(t)} ≥ ε,

0≤t<∞

we consider problem (9.12) with the initial value U0 . As shown by (9.17), any local solution satisfies (9.18), i.e., v(t) ≥ ε. In the meantime, thanks to the estimate (9.24),

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9 Activator–Inhibitor Models

we can extend any local solution of (9.12) with a fixed time interval determined by 3 the norm A 8 U0 X only (see Corollary 4.1). Hence, (9.12) has a unique global solution on the whole interval [0, ∞) in the function space:  3 0 ≤ u ∈ C((0, ∞); HN2 (Ω)) ∩ C([0, ∞); H 4 (Ω)) ∩ C1 ((0, ∞); L2 (Ω)), 3

0 < v ∈ C((0, ∞); HN2 (Ω)) ∩ C([0, ∞); H 4 (Ω)) ∩ C1 ((0, ∞); L2 (Ω)). (9.25) Since v satisfies (9.18) for every t > 0, this function U is not only a solution of (9.12) but also that of the original equation (9.4).

3.4 Global Norm Estimates For U0 ∈ K, let U (t) = U (t; U0 ) be the global of (9.12) with the initial value U0 in the function space (9.25). Of course, estimate (9.24) holds for the global solution, i.e.,  3   

 A 8 U (t; U0 ) ≤ Cε e−δt A 38 U0  + 1 , 0 ≤ t < ∞. (9.26) X X

4 Dynamical System 4.1 Construction of Dynamical System For U0 ∈ K, let U (t; U0 ) denote the global solution of (9.4) with the initial value U0 . From (9.17) we notice that, if U (t; U0 ) = t (u(t), v(t)), then v(t) ≥ e(t) for every 0 ≤ t < ∞. This lower estimate is independent of the initial value. As we are concerned with asymptotic behavior of U (t; U0 ) for U0 ∈ K, without loss of generality we assume that all the initial functions v0 uniformly satisfy the estimate v0 (x) ≥ ε0 = e(1). Furthermore, v(t) ≥ ε for every 0 ≤ t < ∞, where ε > 0 is given by ε = min {e(1)e−γ t + e(t)}. 0≤t<∞

(9.27)

In this way, we are naturally led to restrict our attention to problem (9.12) (not the original one) in which the cutoff positive constant ε is given by (9.27). So, we are concerned with constructing a dynamical system for problem (9.12) with the phase 3 space K in the universal space D 3 = D(A 8 ). 8 For this purpose, we can entirely follow the general methods described in Chap. 6, Sect. 5, where β = 38 . We already know that, for any U0 ∈ K, (9.12) possesses a unique global solution U (t; U0 ) in the function space (9.25). So, a nonlinear semigroup S(t) acting on K is defined by S(t)U0 = U (t; U0 ). By (9.26), estimate (6.56) is fulfilled. Hence, (S(t), K, D 3 ) is a dynamical system determined from 8 (9.12).

5 Instability of Homogeneous Stationary Solution

367

For any exponent 38 < θ < 1, we can construct a similar dynamical system in a universal space Dθ = D(Aθ ) with phase space Kθ = K ∩ Dθ . The nonlinear operator Fε defined by (9.11) fulfills (4.21) and a fortiori (4.2) with β = θ . Let D Kθ,R = K ∩ B θ (0; R). Applying Theorem 4.1 to (9.12), we conclude that there exist τ˜R > 0 and a constant Cθ,R > 0 such that Aθ S(t)U0 X ≤ Cθ,R ,

0 ≤ t ≤ τ˜R ; U0 ∈ Kθ,R .

Meanwhile, by Proposition 6.1,  3  Aθ S(t)U0 X ≤ Cθ,R 1 + t −(θ− 8 ) ,

0 < t < ∞; U0 ∈ Kθ,R .

These two estimates then yield that Aθ S(t)U0 X ≤ Cθ,R ,

0 ≤ t < ∞; U0 ∈ Kθ,R .

Hence, (6.56) is fulfilled with the exponent dynamical system in Dθ .

3 8

< θ < 1. So, (S(t), Kθ , Dθ ) defines a

4.2 Exponential Attractors Let us construct exponential attractors for (S(t), K, D 3 ). To this end, we can fol8 low the methods established in Chap. 6, Sect. 5, namely, it suffices to verify (6.59) and (6.60). It is clear that the space condition (6.59) is valid. Meanwhile, (9.26) shows that the dissipative condition (6.60) is also valid. So, it is concluded that (S(t), K, D 3 ) has exponential attractors. 8

For any exponent 38 ≤ θ < 1, we can equally construct exponential attractors of the dynamical system (S(t), Kθ , Dθ ), too.

4.3 Squeezing Property of S(t) As in Chap. 6, Sect. 5.2, we can construct a subset X of K which is a compact set of X, is a bounded set of D(A), and is an absorbing and invariant set of S(t). Then, (S(t), X, X) is a dynamical system in the space X. It is clear that (6.62) is valid. Then, S(t) enjoys the squeezing property (6.45)–(6.46). Hence, by Theorem 6.16, (S(t), X, X) has exponential attractors whose fractal dimensions are precisely estimated by (6.50).

5 Instability of Homogeneous Stationary Solution As mentioned in the beginning of this chapter, our next objective is to show that the homogeneous stationary solution U of (9.4) becomes unstable, provided that b is

368

9 Activator–Inhibitor Models

sufficiently large. For this purpose, we intend to construct a local unstable manifold of U employing the method described in Chap. 6, Sect. 6. Since we are concerned with only the local unstable manifold, we introduce a localized problem of (9.4) in a neighborhood of U .

5.1 Localized Problem We introduce the following initial-boundary-value problem ⎧ 2 ∂u ⎪ = aΔu + γ [d − cu + ϕ(u) ] in Ω × (0, ∞), ⎪ ⎪ ∂t ψ(v) ⎪ ⎪ ⎨ ∂v 2 in Ω × (0, ∞), ∂t = bΔv + γ [ϕ(u) − v] ⎪ ∂u ∂v ⎪ on ∂Ω × (0, ∞), ⎪ ∂n = ∂n = 0 ⎪ ⎪ ⎩u(x, 0) = u (x), v(x, 0) = v0 (x) in Ω. 0

(9.28)

Here, ϕ(u) is a cutoff function of u in a complex neighborhood of u such that u−u ϕ(u) ≡ u for |u − u| < 1 and ϕ(u) = |u−u| + u for |u − u| ≥ 1. Clearly, ϕ(u) is a bounded Lipschitz continuous function in C. Meanwhile, ψ(v) ≡ v for |v − v| < r v−v r + v for |v − v| ≥ r, where r > 0 is taken in such a way that and ψ(v) = |v−v|

r < v2 . Then, ψ(v) is a bounded Lipschitz continuous function in C with the estimate |ψ(v)| ≥ v2 for all v ∈ C. We can then repeat the same arguments as in Sects. 3 and 5 to construct local solutions and global solutions to (9.28) for all initial values U0 ∈ D(Aθ ) in the function space: U ∈ C((0, ∞); D(A)) ∩ C([0, ∞); D(Aθ )) ∩ C1 ((0, ∞); X)

for any exponent θ such that 38 ≤ θ < 1. The homogeneous stationary solution U of (9.4) or of (9.12) is clearly a stationary solution to (9.28), too.

5.2 Complexified Dynamical System ˜ acting on The localized problem (9.28) introduced above defines a semigroup S(t) 3 θ ˜ Dθ = D(A ), 8 ≤ θ < 1, and defines a dynamical system (S(t), Dθ , Dθ ) which we call a comlexified dynamical system. We are concerned with the case 34 < θ < 1 because of the embedding Dθ = 2θ HN (Ω) ⊂ C(Ω). In this case, any solution of problem (9.12) is a solution of (9.28) in a small neighborhood of U in Dθ . Naturally, in such a neighborhood, any trajec˜ Dθ , Dθ ). In particular, U is an equilibrium tory of (S(t), Kθ , Dθ ) is that of (S(t), ˜ ˜ Dθ , Dθ ), then it is the of (S(t), Dθ , Dθ ). If U is stable as an equilibrium of (S(t),

5 Instability of Homogeneous Stationary Solution

369

same as that of (S(t), Kθ , Dθ ). On the other hand, even if U is unstable as an equi˜ librium of (S(t), Dθ , Dθ ), U is not automatically unstable as that of (S(t), Kθ , Dθ ). ˜ Nevertheless, as will be explained below, the instability of U in (S(t), Dθ , Dθ ) provides us an information which indicates its instability even in the original dynamical system (S(t), Kθ , Dθ ).

5.3 Unstable Manifold of U Problem (9.28) is now formulated in X as the Cauchy problem of the form (9.12) in which the nonlinear operator F˜ : D(Aη ) → X is given by   u γ d + γ ϕ(u)2 /ψ(v) , U = ∈ D(Aη ), F˜ (U ) = v γ ϕ(u)2 where 34 < θ < η < 1. In the complex neighborhoods |u − u| < 1 of u and |v − v| < r of v, we have ϕ(u) ≡ u and ψ(v) ≡ v. So, in a small neighborhood of U in D(Aη ) ⊂ D(Aθ ) ⊂ C(Ω), F˜ is Fréchet differentiable with the derivative   2γ u γ u2 h1 −  2 ˜ v v F (U ) h = , h2 2γ u 0   η h u U= ∈ B D(A ) (U ; r), h = 1 ∈ D(Aη ). h2 v Then, it is possible to apply the results obtained in Chap. 6, Sect. 6. The derivative obviously fulfills conditions (6.68) and (6.69). In order to verify the spectrum separation condition (6.76), it suffices to check condition (6.83). To this end, we have to make the assumption that abμ2k − γ (ag v + bf u )μk + γ 2 (f u g v − f v g u ) = 0 for all μk ,

(9.29)

where 0 = μ0 < μ1 ≤ μ2 ≤ · · · are the eigenvalues of the Laplace operator −Δ under the homogeneous Neumann boundary conditions on ∂Ω. Since f u + g v < 0 (due to (9.2)), we have μk = (f u + g v )/(a + b) for all μk . Therefore, under (9.29), (6.83) and hence (6.76) is fulfilled. Under (9.7) (and hence under (9.2)), if a and b are such that  ag v + bf u > 2 ab(f u g v − f v g u ), (9.30) then there may exist eigenvalues μk such that abμ2k − γ (ag v + bf u )μk + γ 2 (f u g v − f v g u ) < 0. Note that (9.30) is actually satisfied if a is fixed and b is sufficiently large because of ˜ + (U ; O) which f u > 0. In such a case, U has a nontrivial local unstable manifold W

370

9 Activator–Inhibitor Models

is tangential to the space U + X− at U , where X− is the sum of the eigenspaces of the linearized operator A = A − F˜  (U ) corresponding to the negative eigenvalues. ˜ + (U ; O) = dim X− = N (U ), where We already know that dim W   (9.31) N(U ) = # μk ; abμ2k − γ (ag v + bf u )μk + γ 2 (f u g v − f v g u ) < 0 . ˜ + (U ; O) with dimenAs a consequence, U possesses a local unstable manifold W ˜ sion N(U ) in the dynamical system (S(t), Dθ , Dθ ). ˜ + (U ; O) by using (9.31). For Let us furthermore estimate the dimension of M simplicity, let Ω be a smooth bounded domain in R3 . According to [Tri78, Theorem 5.6.2], we have N (μ) ≡



0≤μk ≤μ

 3 1 = O μ2

as μ → ∞.

In the meantime, let 0 < μ1 < μ2 < ∞ be the solution to the equation abμ2 − γ (ag v + bf u )μ + γ 2 (f u g v − f v g u ) = 0 for μ. Clearly,  γ (ag v + bf u ) ∓ γ (ag v + bf u )2 − 4ab(f u g v − f v g u ) μ 1 , μ2 = . 2ab Then, we have N(U ) = N (μ2 ) − N (μ1 ). Remark 9.1 Let us consider some specific case where the diffusion constants are such that a = γ −1 a  and b = γ b with a  > 0 and b > 0. Fixing all other constants in (9.4), we let γ tend to infinity. It is easy to see that μ1 (γ ) ≤ C1 and μ2 (γ ) ≥ c2 γ 2 as γ → ∞ with some two positive constants C1 and c2 (> 0). Consequently, N(μ2 (γ )) − N (μ1 (γ )) ≥ Cγ 3 ; this means that, as γ → ∞, ˜ + (U ; O) = N (U ) ≥ Cγ 3 → ∞. dim W Let us consider finally the instability of U as an equilibrium of the original dynamical system (S(t), Kθ , Dθ ). We already know that U is an unstable equilib˜ + (U ; O) with dimen˜ rium of (S(t), Dθ , Dθ ) having a local unstable manifold W sion dim X− which is tangential to U + X− at U . Furthermore, the sum X− of the eigenspaces of A corresponding to the negative eigenvalues possesses an orthogonal basis which consists of real-valued functions (cf. Corollary 2.1). These facts then indicate that the local unstable manifold W+ (U ; O) in (S(t), Kθ , Dθ ) is neither trivial, that is, U is an unstable equilibrium of (S(t), Kθ , Dθ ).

Notes and Further Researches After the Turing’s pioneering work [Tur52], his instabilty principle was applied extensively. It is known that various pattern formations exhibited by physical, chemical, or biological systems can be reproduced by making use of activator–inhibitor

Notes and Further Researches

371

models. Accordingly, there are a certain number of forms for the kinetic functions on the activator and inhibitor, see [Mei92] and [Mur02, Chap. 6]. Gierer–Meinhardt [GM72] and Meihardt [Mei82] presented the functions k3 u2 , v

f (u, v) = k1 − k2 u +

g(u, v) = k4 u2 − k5 v.

(9.32)

The system (9.1) having these kinetic functions is called the Gierer–Meinhardt system. More generally, when the system has the kinetic functions f (u, v) = k1 − k2 u +

k3 up , vq

g(u, v) =

k4 ur − k5 v, vs

(9.33)

it is often called the generalized Gierer–Meinhardt system. When the kinetic functions are k2 up k3 ur g(u, v) = s − k4 v (9.34) f (u, v) = −k1 u + q , v v and when one imposes the Dirichlet boundary conditions on u and v, the problem is called the singular Gierer–Meinhardt system, see [CM00, CM03]. A system consisting of the diffusion equation on u and an ordinary differential equation  on v which is obtained by letting b → ∞ and replacing g(u, v) by the integral Ω g(u, v) dx is called a shadow system of the Gierer–Meinhardt system. Murray [Mur88] (cf. also [Mur03, Chap. 3]) used the functions f (u, v) = k1 − k2 u − h(u, v), h(u, v) =

g(u, v) = k3 − k4 v − h(u, v),

k5 uv k6 + k7 u + k8 u2

in order to reproduce the animal coat patterns. A large number of papers have already been published for the Gierer–Meinhardt system. For the case of (9.32), the global existence was first obtained by Rothe [Rot84]. Masuda–Takahashi [MaT87] proved the global existence of solutions for the general case of (9.33), but they had to assume some restrictive condition on the exponents p, q, r, and s which excludes the case of (9.32). Afterward, Li–Chen– Qin [LCQ95] and Jiang [Jia06] got rid of such a restriction and proved the global q existence in the case where 0 < p − 1 < r and p−1 r < s+1 (which coves the case where p = 2, q = 1, r = 2, and s = 0). Wu–Huang [WH00] studied the asymptotic behavior of the global solutions. Evolutional problems for the shadow system were studied by Miyamoto [Miy06]. Exponential attractors are newly constructed for the Gierer–Meinhardt system. The results in Sect. 5 are also new. More articles have been published to contribute to the stationary problems of the Gierer–Meinhardt system than those to the evolutional problems. However, we here quote only pioneering papers Haken–Olbrich [HO78] and Takagi [Tak79] and recent papers Jiang–Ni [JN07] and Iron–Rumsey [IR07].

Chapter 10

Belousov–Zhabotinskii Reaction Models

This chapter is devoted to studying reaction–diffusion models for the Belousov– Zhabotinskii reaction. The reaction which consists of more than ten elementary chemical reactions arising simultaneously and which does not tend to any chemical equilibrium is known as one of typical self-organization phenomena in chemistry. In 1974, Field–Noyes presented a simple mathematical model for describing the complicated mechanism from a global point of view. Their reaction–diffusion model of dimensionless form is given by ⎧ ∂u −1 2 ⎪ ⎨ ∂t = aΔu + ε (qw − uw + u − u ) in Ω × (0, ∞), ∂v in Ω × (0, ∞), ∂t = bΔv + u − v ⎪ ⎩ ∂w −1 in Ω × (0, ∞). ∂t = dΔw + δ (−qw − uw + cv) Here, u denotes the concentration of HBrO2 , v the concentration of Ce4+ , and w the concentration of Br− , respectively, in a vessel represented by Ω. The positive constants δ, ε, q, and c are chemical parameters, where especially δ, ε, and q are considered to be small. In 1986, the model was simplified more by Keener–Tyson by assuming that δ cv is sufficiently small so that w is in quasi-equilibrium, i.e., w = u+q , and by using some scaling in the spatial variable so that the diffusions of u and v are εa and εb, respectively. Their model is then given by  ∂u ∂t ∂v ∂t

= εaΔu +

1 ε

 

u(1 − u) − cv u−q in Ω × (0, ∞), u+q

= εbΔv + u − v

in Ω × (0, ∞).

(10.1)

In this chapter, we consider the initial-boundary-value problems of these equations under the Neumann boundary conditions. We show the global existence of solutions, construct a dynamical system, and construct exponential attractors. Especially, for the Keener–Tyson model, we show that, if ε and q are sufficiently small in a specific sense, then the problem has no stable homogeneous stationary solutions. In addition, in such a case, the fractal dimension of exponential attractor is estimated from below by some quantity which tends to infinity as ε → 0. A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_10, © Springer-Verlag Berlin Heidelberg 2010

373

374

10 Belousov–Zhabotinskii Reaction Models

1 Field–Noyes Model Let us first consider the Field–Noyes model. We consider the initial-boundary-value problem ⎧ ∂u −1 2 ⎪ in Ω × (0, ∞), ⎪ ∂t = aΔu + ε (qw − uw + u − u ) ⎪ ⎪ ⎪ ∂v ⎪ ⎪ = bΔv + u − v in Ω × (0, ∞), ⎪ ⎨ ∂t ∂w −1 in Ω × (0, ∞), ∂t = dΔw + δ (−qw − uw + cv) ⎪ ⎪ ⎪ ∂u ∂v ∂w ⎪ on ∂Ω × (0, ∞), ⎪ ∂n = ∂n = ∂n = 0 ⎪ ⎪ ⎪ ⎩u(x, 0) = u (x), v(x, 0) = v (x), w(x, 0) = w (x) in Ω, 0 0 0 (10.2) in a three-dimensional C2 or convex, bounded domain Ω. Here, δ, ε, q, and c are positive constants (>0), and a, b, and d are positive diffusion rates. The three unknown functions u, v and w are assumed to satisfy the homogeneous Neumann boundary conditions on ∂Ω.

2 Local Solutions 2.1 Construction of Local Solutions We construct local solutions by formulating (10.2) as the Cauchy problem for a semilinear abstract equation. We set the underlying space X as the product ⎧⎛ ⎞ ⎫ ⎨ f ⎬ (10.3) X = L2 (Ω) = ⎝ g ⎠ ; f ∈ L2 (Ω), g ∈ L2 (Ω), h ∈ L2 (Ω) . ⎩ ⎭ h We introduce a diagonal matrix operator A = diag {A1 , A2 , A3 } of X, where A1 , A2 and A3 are realizations of −aΔ + ε −1 , −bΔ + 1 and −dΔ + δ −1 q, respectively, in L2 (Ω) under the homogeneous Neumann boundary conditions on ∂Ω. They are positive definite self-adjoint operators of L2 (Ω) with domains HN2 (Ω), see Theorems 2.6 and 2.7. Therefore, A is a positive definite self-adjoint operator of X. Furthermore, by Theorems 16.7 and 16.9 in Chap. 16, the domains of its fractional powers are given by 3 D(Aθ ) = H2θ (Ω) if 0 ≤ θ < , 4 3 < θ ≤ 1. D(Aθ ) = H2θ N (Ω) if 4

2 Local Solutions

375

We also define the nonlinear operator F : D(Aη ) → X by ⎛ −1 ⎞ ε (qw − uw + 2u − u2 ) ⎠ , U ∈ D(Aη ), u F (U ) = ⎝ δ −1 (−uw + cv)

(10.4) 2η

where the exponent η is fixed in such a way that 34 < η < 1. Since HN (Ω) ⊂ H 2η (Ω) ⊂ C(Ω) due to (1.76), it follows that D(Aη ) ⊂ C(Ω). We also set the space of initial values by ⎧⎛ ⎞ ⎫ ⎨ u0 ⎬ K = ⎝ v0 ⎠ ; 0 ≤ u0 ∈ L2 (Ω), 0 ≤ v0 ∈ L2 (Ω), 0 ≤ w0 ∈ L2 (Ω) . ⎩ ⎭ w0 We can then formulate (10.2) as the Cauchy problem  dU 0 < t < ∞, dt + AU = F (U ),

(10.5)

U (0) = U0 ,

in X. Applications of the general results in Chap. 4 to (10.5) are quite easy. The structural assumption (4.21) on F is immediately verified. Indeed, from |u2 − u˜ 2 | ≤ (|u| + |u|)|u ˜ − u| ˜ it follows that ˜ L∞ )u − u ˜ L2 u2 − u˜ 2 L2 ≤ (uL∞ + u ≤ (uH 2η + u ˜ H 2η )u − u ˜ L2 ,

2η

u, u˜ ∈ HN (Ω).

Similarly, ˜ L2 + u ˜ H 2η w − w ˜ L2 ), uw − u˜ w ˜ L2 ≤ C(wH 2η u − u 2η

u, w, u, ˜ w˜ ∈ HN (Ω). η

η

η

2η

Therefore, in view of D(A1 ) = D(A2 ) = D(A3 ) = HN (Ω), we deduce that F (U ) − F (U˜ )X ≤ Cε (Aη U X + Aη U˜ X + 1)U − U˜ X ,

U, U˜ ∈ D(Aη ). (10.6)

Hence, F fulfills (4.21). Theorem 4.4 (with G(t) ≡ 0) thus provides that, for any initial value U0 ∈ X(⊃ K), (10.5) possesses a unique local solution in the function space: U ∈ C((0, TU0 ]; D(A)) ∩ C([0, TU0 ]; X) ∩ C1 ((0, TU0 ]; X),

(10.7)

where TU0 > 0 depends only on the norm U0 X . It also holds that tAU (t)X + U (t)X ≤ CU0 ,

0 < t ≤ TU 0 ,

(10.8)

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10 Belousov–Zhabotinskii Reaction Models

where CU0 > 0 also depends only on the norm U0 X .

2.2 Nonnegativity of Local Solutions For U0 ∈ K, let U = t (u, v, w) be the local solution of (10.5) constructed above. We want to show that u(t) ≥ 0, v(t) ≥ 0, and w(t) ≥ 0 for every 0 < t ≤ TU0 . To this end, we have to use the modified nonlinear operator ⎞ ⎛ ⎞ ⎛ −1 u ε (qw − uw + 2u − u2 ) ⎠ , U = ⎝ v ⎠ ∈ D(Aη ), |u| F˜ (U ) = ⎝ w δ −1 (−uw + cv) to (10.4). And we consider an auxiliary problem  ˜ dU ˜ ˜ ˜ dt + AU = F (U ), 0 ≤ t < ∞, U˜ (0) = U0 .

(10.9)

We can repeat the same arguments as in the preceding subsection to construct a unique local solution U˜ = t (u, ˜ v, ˜ w) ˜ to (10.9) on an interval [0, T˜U0 ] in the same ˜ solution space as (10.7), U also satisfying (10.8). The local solution U˜ (t) is then real valued, since the complex conjugate U˜ (t) of U˜ (t) is also a local solution of (10.9) with the same initial value, and the uniqueness of solution implies that U˜ (t) = U˜ (t). Furthermore, as is shown below, it holds that u(t) ˜ ≥ 0, v(t) ˜ ≥ 0 and w(t) ˜ ≥ 0 for 0 < t ≤ T˜U0 . Let us verify first that v(t) ˜ ≥ 0 by the truncation method. Let H (v) ˜ be a C1,1 cut1 2 ˜ ≡0 off function for −∞ < v˜ < ∞ given by H (v) ˜ = 2v˜ for −∞ < v˜ < 0 and H (v) ˜ dx is continuously diffor 0 ≤ v˜ < ∞. By (1.100), the function ϕ(t) = Ω H (v(t)) ferentiable with the derivative   H (v)Δ ˜ v˜ dx + H (v)(| ˜ u| ˜ − v) ˜ dx. ϕ (t) = b Ω

Ω

By property (1.96),    H (v)Δ ˜ v˜ dx = − ∇H (v) ˜ · ∇ v˜ dx = − |∇H (v)| ˜ 2 dx ≤ 0. Ω

Ω

Ω

In addition, since H (v) ˜ ≤ 0 and H (v) ˜ v˜ ≥ 0, it follows that ϕ (t) ≤ 0. Conse˜ quently, ϕ(t) ≤ ϕ(0) for 0 < t ≤ TU0 . Thus, ϕ(0) = 0 implies ϕ(t) ≡ 0, that is, v(t) ˜ ≥ 0 for 0 ≤ t ≤ T˜U0 .  ˜ By Let us next verify that w(t) ˜ ≥ 0. We use the function ψ(t) = Ω H (w(t))dx. the same reason as for ϕ(t), we have   −1 H (w)Δ ˜ w˜ dx + δ H (w)(−q ˜ w˜ − u˜ w˜ + cv) ˜ dx. ψ (t) = d Ω

Ω

3 Global Solutions

377



Similarly, by property (1.96), Ω H (w)Δ ˜ w˜ dx ≤ 0. Since H (w) ˜ ≤ 0, 0 ≤ H (w) ˜ w˜ ≤ 2H (w) ˜ and v(t) ˜ ≥ 0, it is seen that   H (w)(−q ˜ w˜ − u˜ w˜ + cv) ˜ dx ≤ 2u(t) ˜ H (w) ˜ dx. L∞ Ω

Ω

t ˜ ˜ Hence, ψ (t) ≤ 2u(t) L∞ ψ(t); consequently, ψ(t) ≤ ψ(0) exp[2 0 u(s) L∞ ds] η ˜ ˜ ˜ for 0 < t ≤ TU0 . We here know that u(s) L∞ ≤ CA U (s)X and, in view of (10.8), that Aη U (s)X ≤ CU0 s −η . Thus, ψ(0) = 0 implies ψ(t) ≡ 0, that is, w(t) ˜ ≥ 0 for 0 < t ≤ T˜U0 .  ˜ dx. Using the known reLet us finally consider the function φ(t) = Ω H (u(t)) −η , we can repeat the same ˜ sults that w(t) ˜ ≥ 0 and u(t) ˜ L∞ + w(t) L∞ ≤ CU0 t arguments as for w(t) ˜ to verify that u(t) ˜ ≥ 0. In this way, we have shown that u(t) ˜ ≥ 0, v(t) ˜ ≥ 0, and w(t) ˜ ≥ 0 for 0 ≤ t ≤ T˜U0 . The componentwise nonnegativity of U˜ (t) now implies that F˜ (U˜ (t)) = F (U˜ (t)) for 0 ≤ t ≤ T˜U0 ; this then means that U˜ (t) is a local solution of (10.5), too. By the uniqueness of a solution of (10.5), we must have U˜ (t) = U (t) for 0 ≤ t ≤ T˜U0 . Hence, if T˜U0 ≥ TU0 , we finish the proof. Otherwise, we put T0 = sup{0 < T ≤ TU0 ; u(t)  ≥ 0, v(t) ≥ 0, w(t) ≥ 0 for every 0 < t ≤ T }. Since Ω H (u(T0 )) dx = limtT0 Ω H (u(t)) dx = 0, we see that u(T0 ) ≥ 0. By a similar reason, we have v(T0 ) ≥ 0 and w(T0 ) ≥ 0. So, if T0 = TU0 , we finish the proof. If T0 < TU0 , we will again consider problem (10.9) but with initial time T0 and initial value U (T0 ). Then, we conclude that there is τ > 0 such that u(t) ≥ 0, v(t) ≥ 0, and w(t) ≥ 0 for every t such that T0 ≤ t ≤ T0 + τ . This is a contradiction; hence, T0 = TU0 .

3 Global Solutions 3.1 A Priori Estimates of Local Solutions For U0 ∈ K, let U = t (u, v, w) denote a local solution of (10.5) on [0, TU ] in the function space:  0 ≤ U ∈ C (0, TU ]; H2N (Ω) ∩ C([0, TU ]; L2 (Ω)) ∩ C1 ((0, TU ]; L2 (Ω)). (10.10) Consider the inner product of the equation of (10.5) and U in X. From the equation on u we have    1 d 2 2 −1 u dx + a |∇u| dx + ε u2 dx 2 dt Ω Ω Ω  = ε −1 (quw − u2 w + 2u2 − u3 ) dx 

Ω

≤ Ω

ε −1 (4−1 q 2 w + 2u2 − u3 ) dx,

(10.11)

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10 Belousov–Zhabotinskii Reaction Models

where we used the inequality qu ≤ 1 d 2 dt



q2 4

+ u2 . By the equation on v,



1 v dx + b |∇v| dx + 2 Ω Ω 2







(uv − 2−1 v 2 ) dx

v dx =

2

2

Ω

Ω

(u2 − 4−1 v 2 ) dx.

≤

(10.12)

Ω

Finally, by the equation on w, 1 d 2 dt



 w 2 dx + d



Ω

= 

|∇w|2 dx + Ω

δ −1 q 2

 w 2 dx Ω

δ −1 (−2−1 qw 2 − uw 2 + cvw) dx

Ω

≤

δ −1 (−4−1 qw 2 + c2 q −1 v 2 ) dx.

(10.13)

Ω

After multiplying (10.12) by a parameter ξ > 0, we sum up (10.11), (10.12), and (10.13) to obtain that   1 1 d (u2 + ξ v 2 + w 2 ) dx + (2ε −1 u2 + ξ v 2 + δ −1 qw 2 ) dx 2 dt Ω 2 Ω   −1 ≤ (2ε + ξ − ε−1 u)u2 + (c2 q −1 δ −1 − 4−1 ξ )v 2 Ω

+ 4−1 q(ε −1 q − δ −1 w)w dx. Let us fix the parameter ξ so that ξ = 4c2 q −1 δ −1 and let us notice the following two estimates (2ε−1 + ξ )u2 ≤ ε −1 u3 +

4 (2ε −1 + ξ )3 ε 2 27

ε −1 qw ≤ δ −1 w 2 + 4−1 δ 2 ε −2 q 2 Then, d dt



for all 0 ≤ u < ∞,

for all 0 ≤ w < ∞. 

(u2 + ξ v 2 + w 2 ) dx + μ Ω

(u2 + ξ v 2 + w 2 ) dx ≤ C Ω

with μ = min{2ε −1 , 1, δ −1 q} > 0 and some constant C > 0. Solving this differential inequality (cf. (1.58)), we obtain that u(t)2L2 + ξ v(t)2L2 + w(t)2L2  ≤ e−μt u0 2L2 + ξ v0 2L2 + w0 2L2 + Cμ−1 ,

0 ≤ t ≤ TU . (10.14)

4 Dynamical System

379

3.2 Global Existence Let U0 ∈ K. For any local solution U of (10.5) in the function space (10.10), from (10.14) it follows that  U (t)X ≤ C U0 X + 1 , 0 ≤ t ≤ TU . (10.15) Then, we can apply Corollary 4.3 to conclude the global existence of solution. Indeed, for any U0 ∈ K, (10.5) possesses a unique global solution in the function space:  0 ≤ U ∈ C (0, ∞); H2N (Ω) ∩ C([0, ∞); L2 (Ω)) ∩ C1 ((0, ∞); L2 (Ω)). (10.16)

3.3 Global Norm Estimate For U0 ∈ K, the global solution U (t; U0 ) of (10.5) in (10.16) with initial value U0 satisfies the estimate ˜ U (t; U0 )X ≤ C[e−μt U0 X + 1],

0 ≤ t < ∞, U0 ∈ K,

(10.17)

with some exponent μ˜ > 0 and constant C > 0. Indeed, this is verified directly from (10.14).

4 Dynamical System 4.1 Construction of Dynamical System We can now construct a dynamical system determined from (10.5). We take the universal space X given by (10.3) and follow the general methods described in Chap. 6, Sect. 5. We already know that, for any U0 ∈ K, (10.5) possesses a unique global solution U (t; U0 ) in the function space (10.16). By S(t)U0 = U (t; U0 ) we define a nonlinear semigroup S(t) acting on K. Estimate (10.17) shows that (6.56) is fulfilled. So, (10.5) determines a dynamical system (S(t), K, X).

4.2 Exponential Attractors Let us construct exponential attractors for the dynamical system (S(t), K, X). Thanks to the general results, it suffices to verify (6.59) and (6.60). But, the space condition (6.59) is clear. Meanwhile, (10.17) shows that the dissipative condition (6.60) is also fulfilled. Hence, there exists an absorbing and invariant set X of

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10 Belousov–Zhabotinskii Reaction Models

(S(t), K, X) such that X is a compact subset of X and is a bounded subset of D(A). Furthermore, the dynamical system (S(t), X, X) enjoys a family of exponential attractors M. It is clear in the present case that (6.62) is valid. Therefore, S(t) enjoys the squeezing property (6.45)–(6.46). Hence, (S(t), X, X) has exponential attractors whose fractal dimensions are estimated precisely by (6.50).

5 Keener–Tyson Model We now consider the initial-boundary-value problem ⎧   u−q

∂u 1 ⎪ in Ω × (0, ∞), ⎪ ∂t = aΔu + ε 2 u(1 − u) − cv u+q ⎪ ⎪ ⎪ ∂v 1 ⎨ = bΔv + (u − v) in Ω × (0, ∞), ∂t ε ∂v ⎪ on ∂Ω × (0, ∞), ⎪ ∂u ⎪ ∂n = ∂n = 0 ⎪ ⎪ ⎩ v(x, 0) = v0 (x) in Ω, u(x, 0) = u0 (x),

(10.18)

in a three-dimensional C2 or convex, bounded domain Ω. Note that we have rewritten the equations for u and v in the original Keener–Tyson model (10.1) by scaling the temporal variable as τ = ε −1 t . The new temporal variable τ is again denoted by t as in (10.18). We fix the coefficients a > 0,

b > 0,

c > 0,

0 < q < 1.

(10.19)

In the meantime, 0 < ε ≤ 1 is considered as a control parameter of the system. We use the notation   f (u, v) = u(1 − u) − cv u−q u+q , (10.20) g(u, v) = u − v. For u ≥ 0 and v ≥ 0, the equations f (u, v) = 0 and g(u, v) = 0 obviously have two solutions (0, 0) and (u, v). Here, u is a unique positive solution to the quadratic equation (u + q)(1 − u) = c(u − q),

(10.21)

and v = u. More precisely,  

u = v = 1 − c − q + (c + q − 1)2 + 4q(c + 1) /2.

(10.22)

It is easy to see from (10.21) that q < u < 1. Thus, problem (10.18) has two homogeneous stationary solutions (u, v) ≡ (0, 0) and (u, v).

5 Keener–Tyson Model

381

We are then concerned with the stability and instability of these homogeneous stationary solutions. Clearly, we have fu (u, v) = 1 − 2u − gu (u, v) = 1,



2cqv , (u + q)2

fv (u, v) = −c

 u−q , u+q

gv (u, v) = −1.

In particular, fu (0, 0) = 1,

fv (0, 0) = c,

gu (0, 0) = 1,

gv (0, 0) = −1.

These imply that the relation fu (0, 0)gv (0, 0) − fv (0, 0)gu (0, 0) = −(1 + c) < 0 always takes place, which shows on account of (6.84) that (0, 0) is always unstable. We denote the partial differential coefficients as f u = fu (u, v) = 1 − 2u − g u = gu (u, v) = 1,

2cqu , (u + q)2

f v = fv (u, v) = u − 1,

g v = gv (u, v) = −1.

It is clear that f u < 1. This estimate is actually optimal when the coefficients of equations are taken as in (10.19); indeed, when c = 1, it is seen from (10.22) that √ √ u = O( q), and hence f u = 1 − O( q) when q is small. Since f u − (1 − u) = 2cqu −u − (u+q) 2 < 0, it is seen that the differential coefficients always satisfy the relation f u g v > f v g u . According to (6.82), a sufficient condition in order that (u, v) satisfies the stability condition (6.67) is read as 1 f + g v < 0, ε u

f ugv > f v gu

 and εag v + bf u < 2 εab(f u g v − f v g u ).

(10.23) When f u ≤ 0, all these inequalities are fulfilled automatically whatever ε > 0 is; hence, in this case, (u, v) is stable. On the other hand, when 0 < f u < 1, these inequalities may fail depending on the control parameter ε > 0. Indeed, if  ε > max

2

b2 f u



  , fu , ( ab(f u g v − f v g u ) + −abf v g u )2

then the inequalities in (10.23) are still valid; therefore, (u, v) is stable. But, if ε > 0 is small enough, then some of inequalities in (10.23) fail; so, (u, v) is no longer stable.

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10 Belousov–Zhabotinskii Reaction Models

6 Local Solutions 6.1 Construction of Local Solutions It is carried out in an analogous way to the Field–Noyes model by formulating (10.18) as the Cauchy problem of the form (10.5) in the product space X = L2 (Ω). We introduce a diagonal matrix operator A = diag{A1 , A2 } of X, where A1 and A2 are realizations of −aΔ + 1 and −bΔ + 1, respectively, in L2 (Ω) under the homogeneous Neumann boundary conditions on ∂Ω. They are positive definite selfadjoint operators of L2 (Ω) with domains HN2 (Ω). Therefore, A is a positive definite self-adjoint operator of X. Furthermore, the domains of its fractional powers are 3 given by D(Aθ ) = H2θ (Ω) if 0 ≤ θ < 34 and D(Aθ ) = H2θ N (Ω) if 4 < θ ≤ 1. We η introduce also a nonlinear operator F : D(A ) → X given by   u + ε −2 [u(1 − u) − cv(u − q)(|u| + q)−1 ] F (U ) = , U ∈ D(Aη ), (10.24) ε −1 u + (1 − ε−1 )v 2η

where the exponent η is fixed as 34 < η < 1. Since HN (Ω) ⊂ H 2η (Ω) ⊂ C(Ω) due to (1.76), it follows that D(Aη ) ⊂ C(Ω). The space of initial values is given by    u0 K= (10.25) ; 0 ≤ u0 ∈ L2 (Ω) and 0 ≤ v0 ∈ L2 (Ω) . v0 Problem (10.18) is then formulated as the Cauchy problem for an abstract evolution equation  dU 0 < t < ∞, dt + AU = F (U ), (10.26) U (0) = U0 , of the same form as (10.5) in X. Applications of the general results in Chap. 4 are quite direct as before. The Lipschitz condition (4.21) on F is easily verified. Indeed, from    uv u˜ v˜   ≤ C[|v − v| ˜ + (|v| + |v|)|u ˜ − u|] ˜  |u| + q − |u| ˜ +q it follows that    uv u˜ v˜    ≤ C[v − v ˜ L2 + (vH 2η + v ˜ H 2η )u − u ˜ L2 ],  |u| + q − |u| ˜ + q L2 2η

2η

u, v ∈ HN (Ω); u, ˜ v˜ ∈ HN (Ω). η

η

2η

Therefore, in view of D(A1 ) = D(A2 ) = HN (Ω), we deduce that F (U ) − F (U˜ )X ≤ Cε (Aη U X + Aη U˜ X + 1)U − U˜ X , U, U˜ ∈ D(Aη ),

(10.27)

7 Global Solutions

383

hence, F fulfills (4.21). We thus conclude that, for any initial value U0 ∈ K, (10.26) possesses a unique local solution in the function space: U ∈ C((0, TU0 ]; D(A)) ∩ C([0, TU0 ]; X) ∩ C1 ((0, TU0 ]; X),

(10.28)

where TU0 > 0 depends only on the norm U0 X .

6.2 Nonnegativity of Local Solutions For U0 ∈ K, let U = t (u, v) be the local solution of (10.26) constructed above. It is then shown by the same methods as in Sect. 2.2 that u(t) ≥ 0 and v(t) ≥ 0 for every 0 < t ≤ T U0 . We introduce the modified nonlinear operator     u u + ε−1 [|u|(1 − u) − c|v|(u − q)(|u| + q)−1 ] ˜ , U= ∈ D(Aη ), F (U ) = v ε −1 |u| + (1 − ε−1 )v to (10.24) and consider the same auxiliary problem as (10.9). Let U˜ (t) denote the local solution of the auxiliary problem with initial value U0 ∈ K on an interval [0, T˜U0 ]. Employing the truncation method as before, we conclude that U˜ (t) ≥ 0 componentwise for every 0 < t ≤ T˜U0 . Consequently, F˜ (U˜ (t)) = F (U˜ (t)) for 0 ≤ t ≤ T˜U0 . This means that U˜ (t) is a local solution of (10.26), too. By the uniqueness of local solution, we deduce that U˜ (t) = U (t) for 0 ≤ t ≤ T˜U0 . Therefore, U (t) ≥ 0 for 0 ≤ t ≤ T˜U0 . We can repeat the similar arguments until we obtain that U (t) ≥ 0 for every 0 < t ≤ TU0 . Now, since u(t) ≥ 0, we have |u(t)| = u(t). This means that the local solution U of (10.26) is a local solution of the original problem (10.18), too.

7 Global Solutions 7.1 A Priori Estimates of Local Solutions For U0 ∈ K, let U = t (u, v) denote a local solution of (10.26) on [0, TU ] in the function space:  0 ≤ U ∈ C (0, TU ]; H2N (Ω) ∩ C([0, TU ]; L2 (Ω)) ∩ C1 ((0, TU ]; L2 (Ω)). (10.29) Consider the inner product of the equation of (10.26) and U in X. From the equation on u we have    1 1 d 2 2 u dx + a |∇u| dx + u2 dx 2 dt Ω 2 Ω Ω

384

10 Belousov–Zhabotinskii Reaction Models

 =

[(2

−1

+ε

−2

)u − ε 2

−2 3

u ] dx − ε

Ω

−2

 c

uv Ω

u−q dx. u+q

u−q u−q If u > q, then −ε−2 cuv u+q ≤ 0. And, if 0 ≤ u ≤ q, then −ε −2 cuv u+q ≤ ε −2 cqv. So,    1 d 1 u2 dx + a |∇u|2 dx + u2 dx 2 dt Ω 2 Ω Ω   −1 −2 2 −2 3 −2 ≤ [(2 + ε )u − ε u ] dx + ε cq v dx. Ω

Ω

In the meantime, from the equation on v, we have     1 d 1 v 2 dx + b |∇v|2 dx + v 2 dx = [ε −1 uv + (2−1 − ε −1 )v 2 ] dx 2 dt Ω 2 Ω Ω Ω      3 −2 2 −1 −ε ε u + v 2 dx. ≤ 4 Ω Therefore, it follows that   1 1 d (u2 + v 2 ) dx + (u2 + v 2 ) dx 2 dt Ω 2 Ω       3 −1 −2 2 −2 3 −2 −1 −ε ≤ [(2 + 2ε )u − ε u ] + ε cqv + v 2 dx. 4 Ω Here, we notice that 4 (2−1 + 2ε −2 )u2 ≤ ε −2 u3 + (2 + 2−1 ε 2 )3 ε −2 for all 0 ≤ u < ∞, 27   3 2 −2 −1 v + (4 − 3ε)−1 c2 q 2 ε −3 for all 0 ≤ v < ∞. ε cqv ≤ ε − 4 Hence, d dt



 (u2 + v 2 ) dx + Ω

(u2 + v 2 ) dx ≤ Cε−3 .

Ω

Solving this differential inequality (cf. (1.58)), we obtain that  u(t)2L2 + v(t)2L2 ≤ e−t u0 2L2 + v0 2L2 + Cε −3 , 0 ≤ t ≤ TU .

(10.30)

7.2 Global Existence Let U0 ∈ K. For any local solution U of (10.26) in (10.29), from (10.30) it follows that  U (t)X ≤ U0 2X + Cε −3 , 0 ≤ t ≤ TU . (10.31)

8 Dynamical System

385

Then, we can use Corollary 4.3 to conclude the global existence of solution. Indeed, for any U0 ∈ K, (10.26) possesses a unique global solution in the function space:  0 ≤ U ∈ C (0, ∞); H2N (Ω) ∩ C([0, ∞); L2 (Ω)) ∩ C1 ((0, ∞); L2 (Ω)). (10.32)

7.3 Global Norm Estimate For U0 ∈ K, let U (t; U0 ) be the global solution of (10.26) in (10.32) with the initial value U0 . Then, (10.30) yields that U (t; U0 )2X ≤ e−t U0 2X + Cε −3 ,

0 ≤ t < ∞, U0 ∈ K.

(10.33)

8 Dynamical System 8.1 Construction of Dynamical System Let us now construct a dynamical system determined from (10.26). As before, we can follow the general methods described in Chap. 6, Sect. 5. We already know that, for any U0 ∈ K, (10.26) and hence (10.18) possesses a unique global solution U (t; U0 ) in the function space (10.32). By S(t)U0 = U (t; U0 ), we define a nonlinear semigroup S(t) acting on K. Estimate (10.33) shows that (6.56) is fulfilled. So, (10.26) determines a dynamical system (S(t), K, X). For any exponent 0 < θ < 1, (10.26) equally determines a dynamical system in the universal space Dθ = D(Aθ ) with phase space Kθ = K ∩ Dθ . Indeed, the nonlinear operator F defined by (10.24) fulfills (4.21) and a fortiori (4.2) with β = θ . D Let Kθ,R = K ∩ B θ (0; R). Applying Theorem 4.1, we conclude that there exist τ˜R > 0 and a constant Cθ,R > 0 such that Aθ S(t)U0 X ≤ Cθ,R ,

0 ≤ t ≤ τ˜R , U0 ∈ Kθ,R .

Meanwhile, by Proposition 6.1, Aθ S(t)U0 X ≤ CAS(t)U0 θX S(t)U0 1−θ X ≤ Cθ,R (1 + t −θ ),

0 < t < ∞, U0 ∈ Kθ,R .

These two estimates then yield that Aθ S(t)U0 X ≤ Cθ,R ,

0 ≤ t < ∞, U0 ∈ Kθ,R .

This means that the global norm estimate (6.56) is valid in the space Dθ , too. Then, by the similar arguments as in the proof of Proposition 6.2, S(t)U0 is seen to be continuous from [0, ∞) × Kθ to Kθ . So, (S(t), Kθ , Dθ ) becomes a dynamical system in Dθ .

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10 Belousov–Zhabotinskii Reaction Models

8.2 Exponential Attractors We construct exponential attractors for (S(t), K, X). Thanks to the general results, it suffices to verify (6.59) and (6.60). However, the space condition (6.59) is clear. Meanwhile, (10.33) shows that the dissipative condition (6.60) is also fulfilled. Hence, there exists an absorbing and invariant set X of (S(t), K, X) such that X is a compact subset of X and is a bounded subset of D(A). In addition, (S(t), X, X) has a family of exponential attractors M. For any exponent 0 < θ < 1, these exponential attractors M keep their properties as exponential attractors in the universal space Dθ , too. Hence, the exponential attractors M of (S(t), K, X) are exponential attractors of (S(t), Kθ , Dθ ). As before, we can verify that S(t) enjoys the squeezing property (6.45)–(6.46). Hence, by Theorem 6.16, (S(t), X, X) has exponential attractors whose fractal dimensions are estimated by (6.50).

9 Instability of Homogeneous Stationary Solution 9.1 Localized Problem Let U = t (u, v) (actually, u = v) be the positive homogeneous stationary solution to (10.18) given by (10.22). We intend to investigate its stability and instability. For this purpose, we begin with introducing the localized problem of (10.18) in a neighborhood of U ⎧ ∂u ⎪ ∂t = aΔu − u + χ(u) ⎪ ⎪  

⎪ ⎪ ⎪ + ε−2 χ(u)(1 − χ(u)) − cχ(v) χ(u)−q ⎪ χ(u)+q ⎨ ∂v −1 −1 ∂t = bΔv − v + ε χ(u) + (1 − ε )χ(v) ⎪ ⎪ ⎪ ⎪ ∂u = ∂v = 0 ⎪ ⎪ ⎪ ⎩ ∂n ∂n v(x, 0) = v0 (x) u(x, 0) = u0 (x),

in Ω × (0, ∞), in Ω × (0, ∞),

(10.34)

on ∂Ω × (0, ∞), in Ω.

Here, χ(u) is a cutoff function of u in a complex neighborhood of u such that u−u u + u for |u − u| ≥ u. It is clear χ(u) ≡ u for |u − u| < u and χ(u) = |u−u| that |χ(u) + q| ≥ q for u ∈ C, |χ(u)| ≤ 2u for u ∈ C, and |χ(u1 ) − χ(u2 )| ≤ |u1 − u2 | for u1 , u2 ∈ C. Since u = v, χ(v) also is a cutoff function of v in the same neighborhood of v. We can then repeat the same arguments as in Sects. 6 and 7 to construct local solutions and global solutions to (10.34) for any initial value U0 ∈ D(Aθ ), where 0 ≤ θ < 1, in the function space: U ∈ C((0, ∞); D(A)) ∩ C([0, ∞); D(Aθ )) ∩ C1 ((0, ∞); X).

9 Instability of Homogeneous Stationary Solution

387

9.2 Complexified Dynamical System ˜ The localized problem (10.34) defines a semigroup S(t) acting on Dθ = D(Aθ ). ˜ Therefore, (10.34) defines a complexified dynamical system (S(t), Dθ , Dθ ) for any 0 ≤ θ < 1. Let us fix 34 < θ < 1. Then, Dθ ⊂ C(Ω). In a suitable neighborhood of U in Dθ , any solution to the original problem (10.18) is a solution of (10.34), too. In such ˜ a neighborhood, any trajectory of (S(t), Kθ , Dθ ) is that of (S(t), Dθ , Dθ ). In par˜ ticular, U is an equilibrium of (S(t), Dθ , Dθ ). If U is stable as an equilibrium of ˜ (S(t), Dθ , Dθ ), then it is equally stable as that of (S(t), Kθ , Dθ ). However, even if ˜ U is unstable as an equilibrium of (S(t), Dθ , Dθ ), we cannot say that U is automatically unstable as that of (S(t), Kθ , Dθ ). The situation is quite similar as for the ˜ activator–inhibitor model handled in Chap. 9. The instability of U in (S(t), Dθ , Dθ ) provides us an information which indicates the instability in the original dynamical system (S(t), Kθ , Dθ ), too.

9.3 Case where fu (u, v) ≤ 0 Problem (10.34) is formulated as the Cauchy problem of an abstract equation in X of the form (10.26) in which the nonlinear operator F˜ : D(Aη ) → X, where 34 < θ < η < 1, is given by   χ(u) + ε −2 [χ(u)(1 − χ(u)) − cχ(v)(χ(u) − q)(χ(u) + q)−1 ] ˜ F (U ) = . ε −1 χ(u) + (1 − ε−1 )χ(v) In the complex neighborhood |u − u| < u and |v − v| < v of (u, v), we have χ(u) ≡ u and χ(v) ≡ v. So, in a suitable neighborhood of U in D(Aη ) ⊂ D(Aβ ) ⊂ C(Ω), F˜ is Fréchet differentiable with the derivative   1 + ε−2 [(1 − 2u) − 2cqv(u + q)−2 ] −ε −2 c(u − q)(u + q)−1 F˜ (U ) h = 1 1 − ε −1       η h h u × 1 , U= ∈ B D(A ) (U ; r), h = 1 ∈ D(Aη ). h2 h2 v It is now possible to apply the general results obtained in Chap. 6, Sect. 6. It is easy to check that F˜ fulfills conditions (6.68) and (6.69), and a fortiori (6.64). Let us consider in the present subsection the case where fu (u, v) ≤ 0.

(10.35)

Since the inequalities in (10.23) are all satisfied (whatever ε is), condition (6.82) and hence condition (6.67) is fulfilled. We already know that (6.64) and (6.67) provide the exponential stability. Hence, under (10.35), U is an exponentially stable ˜ equilibrium of (S(t), Dθ , Dθ ).

388

10 Belousov–Zhabotinskii Reaction Models

9.4 Case where 0 < fu (u, v) < 1 Let us next consider the case where 0 < fu (u, v) < 1. We can still follow the arguments of Chap. 6, Sect. 6. If ε > 0 is large enough to satisfy  ε > max



2

b2 f u

  , fu , ( ab(f u g v − f v g u ) + −abf v g u )2

then the inequalities in (10.23) are all valid. Hence, by the same reason as before, ˜ U is an exponentially stable equilibrium of (S(t), Dθ , Dθ ). For the remaining ε, we make the assumption that 

abμ2k − ε −2 (εag v + bf u )μk + ε −3 (f u g v − f v g u ) = 0  −εag v + bf u > (a + b) −εf v g u .

for all μk , and

(10.36) Here, 0 = μ0 < μ1 ≤ μ2 ≤ · · · are the eigenvalues of the Laplace operator −Δ in L2 (Ω) under the homogeneous Neumann boundary conditions on ∂Ω. This assumption means that condition (6.83) is fulfilled. We already know that (6.83) implies the spectrum separation condition (6.76). Hence, (6.68), (6.69), and (6.76) are fulfilled. We have thus shown that, under (10.36), U has a smooth local un˜ + (U ; O) which is tangential to the space U + X− at U , where stable manifold W X− is the eigenspace of A = A − F˜ (U ) for the negative eigenvalues. Furthermore, ˜ + (U ; O) = dim X− = N (U ), where dim W   N (U ) = # μk ; abμ2k − ε −2 (εag v + bf u )μk + ε −3 (f u g v − f v g u ) < 0 . (10.37) We remark that if ε is sufficiently small, then U has a nontrivial local unstable manifold. Indeed, if ε is sufficiently small, then we have  εag v + bf u > 2 εab(f u g v − f v g u ), which actually implies N (U ) ≥ 1. We remark also that, if ε → 0 (the other parameters in (10.34) are fixed), then ˜ + (U , O) tends to infinity. For simplicity, assume that Ω is a smooth bounded dim W domain in R3 . Then, according to [Tri78, Theorem 5.6.2], N (μ) ≡ 0≤μk ≤μ 1 = 3

3

3

O(μ 2 ). Therefore, N (U ) = N (μ2 (ε)) − N (μ1 (ε)) = O(μ2 (ε) 2 ) − O(μ1 (ε) 2 ), where 0 < μ1 (ε) < μ2 (ε) < ∞ are the solutions to the equation abμ2 −ε −2 (εag v + bf u )μ + ε −3 (f u g v − f v g u ) = 0 on μ. From μ1 (ε), μ2 (ε) =

(εag v + bf u )2 ∓



(εag v + bf u )2 − 4εab(f u g v − f v g u ) 2ε 2 ab

Notes and Further Researches

389

we find two positive constants C1 and c2 such that μ1 (ε) ≤ C1 ε −1 and μ2 (ε) ≥ c2 ε −2 , respectively. Hence, as ε → 0, ˜ + (U ; O) = N (U ) ≥ Cε −3 → ∞. dim W Let us finally consider the instability of U as an equilibrium of (S(t), Kθ , Dθ ). ˜ As is proved above, U has in (S(t), Dθ , Dθ ) a smooth local unstable manifold ˜ W+ (U ; O) with dimension N (U ) which is tangential to U + X− at U . Furthermore, X− is the eigenspace which corresponds to negative eigenvalues of A and possesses an orthogonal basis which consists of real-valued functions (cf. Corollary 2.1). These facts indicate that U is an unstable equilibrium of (S(t), Kθ , Dθ ), too.

Notes and Further Researches The Field–Noyes model was introduced in the papers [FN74a, FN74b], cf. also [TF80]. The model has already attracted interest of many mathematicians. Traveling wave solutions for one-dimensional problem were first constructed by Murray [Mur76] and were succeeded by the papers [FT79, Gib80, GC95, LS96, Tro77, WXY94, YW87]. The stability of traveling wave solutions was discussed by KL90]. In the meantime, there are only a few papers on the asymptotic behavior of solutions of the Field and Noyes model. Wang–Wu WW95] constructed for the d-dimensional problem under the Dirichlet boundary conditions a finite-dimensional global attractor. (But they handle the case where the parameter q vanishes.) They succeeded also in constructing an inertial manifold when Ω is a square domain. Pao Pao88] proved in some specific cases that solutions satisfying the Robin boundary conditions converge to the homogeneous stationary solutions as t → ∞. The stationary problems were studied by BK93, PW04]. For the numerical study, we quote MM82]. The results obtained in Sect. 4 are new. The Keener–Tyson model (10.18) was presented in the paper KT86]. The results on the Keener–Tyson model in this chapter were announced in YOS] without full proof. In YOS], the continuous dependence of exponential attractors M with respect to the parameter ε is also proved by employing Theorem 6.17.

Chapter 11

Forest Kinematic Model

In this chapter, we handle a forest kinematic model. In 1994, Kuznetsov, Antonovsky, Biktashev, and Aponina presented a system of parabolic-ordinary differential equations for three unknown functions u, v, and w for describing kinetics of a simplified forest ecosystem. The unknown function w (density of seeds) satisfies a linear diffusion equation in a domain Ω ⊂ R2 , v (density of old age class trees) satisfies a linear ordinary differential equation at each point of Ω, and u (density of young age class trees) does a nonlinear ordinary differential equation at each point of Ω with a cubic kinetic function on u and v. We treat this system as a degenerate parabolic system; “degenerate” means that the ordinary differential equations for u and v are regarded as a diffusion equation with vanishing diffusion rate. We may lose of course spatial smoothing of solutions for u and v. What is worse, at least in the level of numerical simulation (see Sect. 4.5), u and v can tend to discontinuous functions of Ω as t → ∞ even if they start from continuous functions at t = 0. These evidences make us be careful for choosing underlying function spaces X1 and X2 for treating the equations on u and v, respectively. First, Xi (i = 1, 2) must be closed in the usual product of functions, because the equations contain a cubic kinetic function on u and v and the solution u, v has no smoothing. Secondly, Xi must contain discontinuous functions in Ω. We are thus led to choose L∞ (Ω) as the spaces Xi (i = 1, 2).

1 Model Equations Consider the initial-boundary-value problem ⎧ ∂u ⎪ in Ω × (0, ∞), ⎪ ∂t = βδw − γ (v)u − f u ⎪ ⎪ ⎪ ∂v ⎪ ⎪ = f u − hv in Ω × (0, ∞), ⎪ ⎨ ∂t ∂w in Ω × (0, ∞), ∂t = dΔw − βw + αv ⎪ ⎪ ⎪ ⎪ w=0 on ∂Ω × (0, ∞), ⎪ ⎪ ⎪ ⎪ ⎩u(x, 0) = u (x), v(x, 0) = v (x), w(x, 0) = w (x) in Ω, 0 0 0 (11.1) A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_11, © Springer-Verlag Berlin Heidelberg 2010

391

392

11 Forest Kinematic Model

in a two-dimensional C2 or convex, bounded domain Ω. In Ω a prototype forest ecosystem which simply consists of mono-species is considered. We suppose that age generations of trees are divided only into two classes, namely, the young age class and old age class. The unknown functions u(x, t) and v(x, t) denote the tree densities of the young and old age classes in Ω at time t, respectively. The third unknown function w(x, t) denotes the density of seeds in the air over Ω at time t. The third equation describes the kinetics of seeds; d is a diffusion rate in the air, and α and β are the seed production and seed deposition rates, respectively. While the first and second equations describe the growth of the young and old age class trees, respectively, δ is the seed establishment rate (0 < δ ≤ 1), γ (v) is the mortality of young trees, which is allowed to depend on the old-tree density v, f is the aging rate, and h is the mortality of old generation trees. On w, we impose the homogeneous Dirichlet conditions on ∂Ω. The initial functions u0 (x), v0 (x), and w0 (x) are given nonnegative functions in Ω. We assume that the mortality γ (v) is given by a square function of the form γ (v) = a(v − b)2 + c,

(11.2)

which means that the mortality hits its minimum when the old-tree density is a specific optimal value b. We assume also that a, b, c, d, f, h, α, and β are positive constants (> 0) and that δ is a constant such that 0 < δ ≤ 1. We shall first prove the global existence of solutions and then construct a dynamical system determined from (11.1). Fortunately, our dynamical system has a Lyapunov function given by (11.31). Thanks to this Lyapunov function, we can investigate the asymptotic behavior of solutions, and indeed they are shown to tend to some stationary solutions in a certain weak sense. Furthermore, we see that the structure of stationary solutions changes drastically depending on the parameters in the system, especially on the parameter h.

2 Local Solutions 2.1 Abstract Formulation We will handle equations on u and v in L∞ (Ω) and that on w in L2 (Ω). The underlying space is therefore set by ⎛ u ⎞  X = ⎝ v ⎠ ; u, v ∈ L∞ (Ω), w ∈ L2 (Ω) . (11.3) w In addition, the space of initial functions is taken as ⎛ u ⎞  ⎝ ⎠ v ; 0 ≤ u, v ∈ L∞ (Ω), 0 ≤ w ∈ L2 (Ω) . K= w

(11.4)

2 Local Solutions

393

We formulate problem (11.1) as the Cauchy problem for an abstract semilinear equation  dU dt + AU = F (U ), 0 < t < ∞, (11.5) U (0) = U0 , in X. Here, A = diag{f, h, Λ} is a diagonal matrix operator of X with domain ⎛ u ⎞  2 D(A) = ⎝ v ⎠ ; u, v ∈ L∞ (Ω), w ∈ HD (Ω) , w where Λ is a realization of the operator −dΔ + β in L2 (Ω) under the homogeneous Dirichlet boundary conditions and is a positive definite self-adjoint operator of L2 (Ω) with D(Λ) = HD2 (Ω) (see (2.31)). By Theorems 16.12 and 16.13 in Chap. 16, the domains of Λθ , 0 < θ < 1, are characterized by D(Λθ ) = H 2θ (Ω) for 0 ≤ θ < 14 and D(Λθ ) = HD2θ (Ω) for 14 < θ ≤ 1, θ = 34 . Then, A is a sectorial operator of X. For 0 < θ < 1, its fractional powers are given by Aθ = diag{f θ , hθ , Λθ } with ⎛ u ⎞  D(Aθ ) = ⎝ v ⎠ ; u, v ∈ L∞ (Ω), w ∈ D(Λθ ) . w

The nonlinear operator F is given by ⎛ ⎞ βδw − γ (v)u ⎠, fu F (U ) = ⎝ αv

⎛ ⎞ u U = ⎝ v ⎠ ∈ D(Aη ), w

(11.6)

where η is an arbitrarily fixed exponent such that 12 < η < 1. As D(Λη ) ⊂ L∞ (Ω) due to (1.76) and consequently as D(Aη ) ⊂ L∞ (Ω), F is certainly an operator from D(Aη ) into X.

2.2 Construction of Local Solutions It is possible to apply Theorem 4.4 to (11.5). Indeed, after some direct calculations, we get

F (U ) − F (U˜ )X ≤ C U 2X + U˜ 2X + 1 Aη (U − U˜ )X , U, U˜ ∈ D(Aη ), (11.7) with some constant C > 0. This shows that (4.21) is fulfilled. Therefore, for any U0 ∈ K, (11.5) possesses a unique local solution in the function space:  u, v ∈ C([0, TU0 ]; L∞ (Ω)) ∩ C1 ((0, TU0 ]; L∞ (Ω)), w ∈ C((0, TU0 ]; HD2 (Ω)) ∩ C([0, TU0 ]; L2 (Ω)) ∩ C1 ((0, TU0 ]; L2 (Ω)). (11.8)

394

11 Forest Kinematic Model

Here, TU0 > 0 is determined by the norm U0 X only. Moreover, the local solution satisfies the estimate tAU (t)X + U (t)X ≤ CU0 ,

0 < t ≤ TU 0 ,

(11.9)

with some constant CU0 > 0 depending only on U0 X . We observe also that w(t)L∞ ≤ Cw(t)H 2η ≤ CU0 t −η ,

0 < t ≤ TU 0 .

(11.10)

Then, thanks to (1.104), we can verify the representation formulas u(t) = e

−

t

0 [γ (v(s))+f ] ds

v(t) = e−ht v0 + f

 u0 + βδ

t

e−

t s

[γ (v(τ ))+f ] dτ

w(s) ds,

(11.11)

0



t

e−h(t−s) u(s) ds.

(11.12)

0

In verifying (11.11), (11.10) was used.

2.3 Nonnegativity of Local Solutions For U0 ∈ K, let U = t (u, v, w) be the local solution of (11.5) constructed above. Our goal is to show that u(t) ≥ 0, v(t) ≥ 0, and w(t) ≥ 0 for every 0 < t ≤ TU0 . For this purpose, however, we have to introduce the modified nonlinear operator F˜ (U ) = t (βδw − γ (v)u, f u, αχ(Re v)),

U ∈ D(Aη ),

where χ(v) is a cutoff function given by χ(v) ≡ 0 for −∞ < v < 0 and χ(v) = v for 0 ≤ v < ∞. We also have to consider the auxiliary problem 

d U˜ dt

+ AU˜ = F˜ (U˜ ), ˜ U (0) = U0 .

0 < t < ∞,

(11.13)

We can in fact repeat the same arguments as in the preceding subsection to de˜ v, ˜ w) ˜ duce that, for any U0 ∈ K, (11.13) possesses a unique local solution U˜ = t (u, on an interval [0, T˜U0 ] in the function space (11.8)–(11.9) (note that χ(Re v1 ) − χ(Re v2 )L2 ≤ v1 − v2 L2 for v1 , v2 ∈ L2 (Ω)). Before showing the nonnegativity of u, v, w, we will show that of u, ˜ v, ˜ w. ˜ Let us first notice that U˜ is real valued. Indeed, if U˜ is a local solution of (11.13), then its complex conjugate is also a local solution with the same initial value; so, by the uniqueness of solution, they must coincide; this implies that U˜ is real valued. Let us secondly verify the nonnegativity of w. ˜ We use another cutoff function H (w) ˜ which is the C1,1 function given by H (w) ˜ = 12 w˜ 2 for −∞ < w˜ < 0

3 Global Solutions

395

and H (w) ˜ ≡ 0 for 0 ≤ w˜ < ∞. According to (1.100), the function ψ(t) =  H ( w(t)) ˜ dx is continuously differentiable with the derivative Ω   ψ (t) = d H (w)Δ ˜ w˜ dx + H (w)[−β ˜ w˜ + αχ(v)] ˜ dx. Ω

Ω

By property (1.96),    H (w)Δ ˜ w˜ dx = − ∇H (w) ˜ · ∇ w˜ dx = − |∇H (w)| ˜ 2 dx ≤ 0. Ω



Ω

Ω

H (w)[−β ˜ w˜

In addition, it is clear that Ω + αχ(v)] ˜ dx ≤ 0. Hence, ψ (t) ≤ 0; consequently, ψ(t) ≤ ψ(0) for 0 < t ≤ T˜U0 . Thus, ψ(0) = 0 implies ψ(t) ≡ 0, i.e., w(t) ˜ ≥ 0 for 0 < t ≤ T˜U0 . As for u˜ and v, ˜ we use representations (11.11) and (11.12), which hold equally for U˜ . Therefore, we conclude that u(t) ˜ ≥ 0 and v(t) ˜ ≥ 0 for 0 < t ≤ T˜U0 . We now notice that v(t) ˜ ≥ 0 implies that χ(v(t)) ˜ = v(t); ˜ in other words, U˜ is a local solution of the original problem (11.5), too. The uniqueness of solution then yields that U (t) ≡ U˜ (t). Hence, u(t) ≥ 0, v(t) ≥ 0, w(t) ≥ 0 for 0 < t ≤ T˜U0 . If T˜U0 ≥ TU0 , we finish the proof. If not, we put T  0 = sup{0 < T ≤ ≥ 0, v(t) ≥ 0, w(t) ≥ 0 for every 0 < t ≤ T }. Since TU0 ; u(t) Ω H (u(T0 )) dx =  limtT0 Ω H (u(t)) dx = 0, we see that u(T0 ) ≥ 0. By similar reasons, we have v(T0 ) ≥ 0, w(T0 ) ≥ 0. So, if T0 = TU0 , we finish the proof. If T0 < TU0 , we will consider again problem (11.13) but with the initial time T0 and the initial value U (T0 ). Repeating the same arguments above, we conclude that there is τ > 0 such that u(t) ≥ 0, v(t) ≥ 0, w(t) ≥ 0 for every t such that TU0 ≤ t ≤ TU0 + τ . This is a contradiction; hence, T0 = TU0 .

3 Global Solutions 3.1 A Priori Estimates for Local Solutions For U0 ∈ K, let U = t (u, v, w) be a local solution of (11.5) on [0, TU ] in the function space:  0 ≤ u, v ∈ C([0, TU ]; L∞ (Ω)) ∩ C1 ((0, TU ]; L∞ (Ω)), 0 ≤ w ∈ C([0, TU ]; L2 (Ω)) ∩ C((0, TU ]; HD2 (Ω)) ∩ C1 ((0, TU ]; L2 (Ω)). (11.14) We establish the following a priori estimates for local solutions. Proposition 11.1 There exist an exponent ρ > 0 and a constant C > 0 such that the estimate U (t)X ≤ C[e−ρt U0 X + 1],

0 ≤ t ≤ TU ,

holds for any local solution U in the function space (11.14).

(11.15)

396

11 Forest Kinematic Model

Proof In the proof, C, C1 , C2 , . . . , stand for some constants which are determined only by Ω and by the initial constants in the equations of (11.1). Similarly, ρ stands for some positive exponent determined in the same manner. Step 1. Let us first estimate the norm U (t)L2 . Multiply the first equation of (11.1) by u and integrate the product in Ω. Then,     1 d 2 2 u dx + f u dx = βδ wu dx − γ (v)u2 dx 2 dt Ω Ω Ω Ω    f u2 dx + C1 w 2 dx − γ (v)u2 dx. ≤ 2 Ω Ω Ω (11.16) Multiply the third equation of (11.1) by w and integrate the product in Ω. Then,    1 d 2 2 w dx + d |∇w| dx + β w 2 dx 2 dt Ω Ω Ω    β vw dx ≤ w 2 dx + C2 v 2 dx. =α 2 Ω Ω Ω Let C3 > 0 be a constant such that C1 C3 ≤ β4 . Multiply (11.16) by C3 and add the product to the above inequality. Then, we obtain that     C3 d 1 d C3 f β 2 2 2 u dx + w dx + u dx + w 2 dx 2 dt Ω 2 dt Ω 2 Ω 4 Ω   v 2 dx − C3 γ (v)u2 dx. (11.17) ≤ C2 Ω

Ω

Multiply next the second equation of (11.1) by v and integrate the product in Ω. Then,    1 d 2 2 v dx + h v dx = f uv dx. 2 dt Ω Ω Ω Let C4 > 0 be a constant such that C4 h ≥ 2C2 . Multiply the present equality by C4 and add the product to inequality (11.17) to obtain that     C3 d C4 d 1 d C3 f u2 dx + v 2 dx + w 2 dx + u2 dx 2 dt Ω 2 dt Ω 2 dt Ω 2 Ω     β 2 2 v dx + w dx ≤ C4 f uv dx − C3 γ (v)u2 dx. + C2 4 Ω Ω Ω Ω We here notice that   C2f 2 C4 f uv − C3 γ (v)u2 = − C3 a(v − b)2 u2 − C4 f (v − b)u + 4 4C3 a

3 Global Solutions

397

    C 2 f 2 b2 C 2 f 2 1 b2 − C3 cu2 − C4 f bu + 4 + 4 + 4C3 c 4C3 a c   C 2 f 2 1 b2 ≤ 4 + . 4C3 a c Therefore, d dt



 (C3 u + C4 v + w ) dx + ρ 2

2

(C3 u2 + C4 v 2 + w 2 ) dx ≤ C.

2

Ω

Ω

Hence (cf. (1.58)), we conclude that   U (t)2L2 ≤ C e−ρt U0 2L2 + 1 ,

0 ≤ t ≤ TU .

(11.18)

Step 2. We will estimate the norm w(t)L∞ . We have  t   η − t−s Λ  − t−s Λ  βt  βt Λη w(t) = Λη e− 2 Λ e− 2 Λ w0 + α Λ e 2 e 2 v(s) ds. 0

Then, βt

w(t)H 2η ≤ C(1 + t −η )e− 2 w0 L2 + C e−tΛ 



t

0

 β 1 + (t − s)−η e− 2 (t−s) v(s)L2 ds,

≤ e−βt

where we used the estimate for t ≥ 0 (recall that Λ ≥ β). FurtherL2 more, by (11.18),  t   β 1 + (t − s)−η e− 2 (t−s) v(s)L2 ds 0



t

 β 1 + (t − s)−η e− 2 (t−s) ds

≤C 0

+ Ce

−ρ t

 0

≤ C[e

−ρ t

t

 β 1 + (t − s)−η e−( 2 −ρ )(t−s) e−(ρ−ρ )s dsU0 L2

U0 L2 + 1],

where 0 < ρ < min{ β2 , ρ}. Replacing ρ by new one, we thus conclude that w(t)L∞ ≤ Cw(t)H 2η ≤ C[(1 + t −η )e−ρt U0 L2 + 1],

0 < t ≤ TU . (11.19)

Step 3. Let us finally estimate the norms u(t)L∞ and v(t)L∞ by using (11.11) and (11.12). In view of (11.19), we have  t −f t e−f (t−s) [(1 + s −η )e−ρs U0 L2 + 1] ds. u(t)L∞ ≤ e u0 L∞ + C 0

398

11 Forest Kinematic Model

Therefore, replacing ρ by new one, we conclude that u(t)L∞ ≤ C[e−ρt U0 X + 1],

0 ≤ t ≤ TU .

In a similarly way, by (11.12), v(t)L∞ ≤ C[e−ρt U0 X + 1],

0 ≤ t ≤ TU .

These, together with (11.18), finally yield the desired estimates (11.15).



3.2 Global Solutions As the a priori estimates have been established for local solutions of (11.5) with U0 ∈ K, we can apply Corollary 4.3 to conclude the global existence of solutions. Indeed, for any U0 ∈ K, there exists a unique global solution to (11.5) in the function space:  0 ≤ u, v ∈ C([0, ∞); L∞ (Ω)) ∩ C1 ((0, ∞); L∞ (Ω)), 0 ≤ w ∈ C((0, ∞); HD2 (Ω)) ∩ C([0, ∞); L2 (Ω)) ∩ C1 ((0, ∞); L2 (Ω)). (11.20)

3.3 Global Norm Estimates For U0 ∈ K, let U (t) = U (t; U0 ) be the global solution of (11.5) with the initial value U0 in the function space (11.20). From (11.15) it immediately follows that U (t; U0 )X ≤ C[e−ρt U0 X + 1],

0 ≤ t < ∞, U0 ∈ K.

(11.21)

In particular, U (t; U0 )X ≤ C[U0 X + 1],

0 ≤ t < ∞, U0 ∈ K.

This, together with (11.9), yields that AU (t; U0 ) ≤ (1 + t −1 )p0 (U0 X ),

0 < t < ∞, U0 ∈ K,

(11.22)

with a suitable continuous increasing function p0 (·). In particular, w(t)H 2 ≤ (1 + t −1 )p0 (U0 X ),

0 < t < ∞, U0 ∈ K.

As a consequence, w(t)L∞ ≤ (1 + t −η )p0 (U0 X ),

0 < t < ∞, U0 ∈ K.

We can further prove the following norm estimates.

(11.23)

3 Global Solutions

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Proposition 11.2 For the derivative U (t), it holds that u (t)L∞ ≤ (1 + t −η )p1 (U0 X ),

v (t)L∞ ≤ p1 (U0 X ),

0 < t < ∞, U0 ∈ K,

0 < t < ∞, U0 ∈ K,

w (t)L2 ≤ (1 + t −1 )p1 (U0 X ),

(11.24) (11.25)

0 < t < ∞, U0 ∈ K,

(11.26)

where p1 (·) is an appropriate continuous increasing function. Proof In view of (11.23), (11.24) is verified directly from the equation of u. In the meantime, (11.25) is obvious by (11.21). The last estimate (11.26) is an immediate consequence of (11.22).  Proposition 11.3 For the second order derivative U (t), it holds that u (t)L∞ ≤ (1 + t −1−η )p2 (U0 X ),

v (t)L∞ ≤ (1 + t

−η

0 < t < ∞, U0 ∈ K,

(11.27)

)p2 (U0 X ),

0 < t < ∞, U0 ∈ K,

(11.28)

w (t)L2 ≤ (1 + t −2 )p2 (U0 X ),

0 < t < ∞, U0 ∈ K,

(11.29)

where p2 (·) is an appropriate continuous increasing function. Proof By the equation of v in (11.5), v (t) = f u (t) − hv (t),

0 < t < ∞.

Then, v ∈ C2 ((0, ∞); L∞ (Ω)), and estimate (11.28) is verified by (11.24) and (11.25). We can next use the regularity property of solutions of linear evolution equations, regarding the equation of w as a linear equation with the unknown function w and given force function v. Since v belongs to C2 ([τ, ∞); L2 (Ω)), we conclude by Theorem 3.6 that w ∈ C2 ((τ, ∞); L2 (Ω)) with the estimate   w (t)L2 + w (t)H 2 ≤ C 1 + (t − τ )−1 w (τ )L2 ,

τ < t < ∞,

where τ > 0 is any positive time. So, taking τ = 2t , we verify (11.29) and the estimate w (t)H 2 ≤ (1 + t −2 )p2 (U0 X ),

0 < t < ∞.

As a consequence, we verify that w (t)L∞ ≤ Cw (t)H 2η ≤ (1 + t −1−η )p2 (U0 X ),

0 < t < ∞.

Then, (11.27) is observed by differentiating the equation of u in t .



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4 Dynamical System We now construct a dynamical system determined from (11.5). It is possible to follow the general strategy described in Chap. 6, Sect. 5.1 by taking β = 0. Estimate (11.21) indeed shows that (6.56) is fulfilled; therefore, it is immediate to conclude that (11.5) defines a dynamical system (S(t), K, X). For any exponent 0 < θ < 1, (11.5) equally defines a dynamical system in the universal space Dθ = D(Aθ ) with the phase space Kθ = K ∩ Dθ . Indeed, the nonlinear operator F defined by (11.6) fulfills (4.21) and a fortiori (4.2) with β = θ . D Let Kθ,R = K ∩ B θ (0; R). Applying Theorem 3.12, we conclude that there exist τ˜R > 0 and a constant Cθ,R > 0 such that Aθ S(t)U0 X ≤ Cθ,R ,

0 ≤ t ≤ τ˜R , U0 ∈ Kθ,R .

This, together with (11.22), then yields that Aθ S(t)U0 X ≤ Cθ,R ,

0 ≤ t < ∞, U0 ∈ Kθ,R .

This means that (6.56) is valid in the space Dθ , too. Therefore, for any 0 < θ < 1, (S(t), Kθ , Dθ ) defines a dynamical system. X Estimate (11.15) means that, for any KR = K ∩ B (0; R), there exists tR > 0 such that ˜ =K ˜ S(t)KR ⊂ K ∩ B X (0; 2C) 2C

for any tR ≤ t < ∞.

˜ Thus, we Moreover, let B = S(1)K2C˜ . Then, by (11.22), B ⊂ B D(A) (0; 2p(2C)). have observed that (S(t), K, X) possesses an absorbing set B (⊂ K) which is a bounded set of D(A).

4.1 Lyapunov Function We can build up a Lyapunov function Ψ (U ) for (S(t), K, X). Let U0 ∈ K, and let S(t)U0 = U (t). Put ϕ(t) = f u(t) − hv(t), 0 ≤ t < ∞. From the equations of u and v in (11.5) it is easily obtained that ∂ϕ = f {βδw − [γ (v) + f ]u} − h(f u − hv) ∂t = fβδw − [γ (v) + f + h]ϕ − h[γ (v) + f ]v. Multiply this by ϕ(t) = ∂v ∂t and integrate the product in Ω. Then,    ∂v d 1 d ϕ 2 dx + h Γ (v) dx − fβδ w dx 2 dt Ω dt Ω Ω ∂t  2  ∂v = − [γ (v) + f + h] dx, ∂t Ω v where Γ (v) = 0 [γ (v)v + f v] dv.

4 Dynamical System

401

Meanwhile, multiply the equation of w in (11.5) by ∂w ∂t and integrate the product in Ω. Then,       d d ∂w 2 β d ∂w 2 2 |∇w| dx + w dx − α v dx. dx = − 2 dt Ω 2 dt Ω Ω ∂t Ω ∂t From these two energy equalities we obtain that    α 2 dfβδ d fβ 2 δ 2 2 ϕ + |∇w| + hαΓ (v) + w − (f αβδ)vw dx dt Ω 2 2 2  2      ∂v ∂w 2 α[γ (v) + f + h] dx ≤ 0. (11.30) =− + fβδ ∂t ∂t Ω Note that hαΓ (v) +

fβ 2 δ 2 w − (f αβδ)vw ≥ −C 2

for all v, w ≥ 0,

with some constant C > 0. So, the functional given by   α dfβδ (f u − hv)2 + |∇w|2 + hαΓ (v) Ψ (U ) = 2 2 Ω  1 fβ 2 δ 2 w − (f αβδ)vw dx, U ∈ D A 2 , + 2

(11.31)

becomes a Lyapunov function for the dynamical system (S(t), K, X). The Lyapunov function provides the following two propositions. Proposition 11.4 For any trajectory S(t)U0 = U (t), it holds that   ∞  dU 2    dt (t) dt < ∞. 1 L2 Proof Integrate both sides of (11.30) on an interval [1, T ]. Then,  2     T  ∂v ∂w 2 α[γ (v) + f + h] dx dt + fβδ ∂t ∂t Ω 1   α dfβδ ≤ ϕ(1)2 + |∇w(1)|2 + hαΓ (v(1)) 2 Ω 2  fβ 2 δ 2 w(1) + f αβδv(T )w(T ) dx. + 2 By (11.21) it is observed that  2     ∞  ∂v ∂w 2 α[γ (v) + f + h] dx dt < ∞. + fβδ ∂t ∂t 1 Ω

(11.32)

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Differentiating the equation of u in (11.1) with respect to t, we have ∂u ∂w ∂v ∂ 2u − γ (v)u − [γ (v) + f ] . = βδ ∂t ∂t ∂t ∂t 2 ∂u ∂t

Multiply this by 1 d 2 dt

  Ω

∂u ∂t

and integrate the product in Ω. Then,  

2

  2  ∂w ∂v ∂u ∂u − γ (v)u dx − (γ (v) + f ) dx ∂t ∂t ∂t ∂t Ω Ω     2  ∂v ∂w 2 dx ≤ C(U0 X + 1) + ∂t ∂t Ω   2 f ∂u − dx. 2 Ω ∂t

dx =

βδ

Integrating this inequality on [1, T ], we obtain that f 2



T 1

  Ω

∂u ∂t

2 dx dt ≤

 

2 ∂u (1) dx Ω ∂t  + C(U0 X + 1)

1 2

T

1

On account of (11.32), we conclude that sired integrability has been verified.

f 2

  Ω

∞ 1

∂v ∂t

2

∂u 2 Ω ( ∂t ) dx dt

 +

∂w ∂t

2  dx dt.

< ∞. Hence, the de

Proposition 11.5 For any trajectory S(t)U0 = U (t), the derivative 0 as t → ∞ in the L2 norm.

dU dt (t)

tends to

d dU  dt (t)2L2 | ≤ CU0 for 1 ≤ t < ∞ Proof Propositions 11.2 and 11.3 yield that | dt with some constant CU0 > 0 depending on U0 X . This, together with integrability dU 2 of  dU  dt (t)L2 in (1, ∞), implies the desired convergence  dt (t)L2 → 0.

We finally verify the following fact. 1

Theorem 11.1 The Lyapunov function Ψ : D(A 2 ) → C is Fréchet differentiable, and U ∈ D(A) is a stationary solution of (11.1) if and only if Ψ (U ) = 0. Proof By direct calculations it is observed that Ψ is Fréchet differentiable, and its derivative is given by   Ψ (U )U = α(f u − hv)(f u − hv) + dfβδ∇w · ∇w + hαΓ (v)v Ω

 + fβ 2 δww − f αβδ(vw + wv) dx,

4 Dynamical System

403

⎛ ⎞ ⎛ ⎞ u u 1 U = ⎝v ⎠, U = ⎝v ⎠ ∈ D A2 . w w Consequently, Ψ (U )U =





f α(f u − hv)u + α[h2 v + hγ (v)v − fβδw]v

Ω

 + fβδ[d∇w · ∇w + (βw − αv)w] dx. Therefore, Ψ (U ) = 0 if and only if f u − hv = 0, h2 v + hγ (v)v − fβδw = 0 and  [d∇w · ∇w + (βw − αv)w] dx = 0 for all w ∈ H˚ 1 (Ω). Ω

Furthermore, Ψ (U ) = 0 if and only if f u − hv = 0, hu + γ (v)u − βδw = 0, and −dΔw + βw − αv = 0 in H −1 (Ω). Hence, Ψ (U ) = 0 if and only if U is a stationary solution of (11.1). 

4.2 ω-limit Sets For each U0 ∈ K, the ω-limit set of the trajectory S(t)U0 is defined by ω(U0 ) =



{S(τ )U0 ; t ≤ τ < ∞} (closure in X),

t≥0

see Chap. 6, Sect. 1.2. However, we have the following observation. Remark 11.1 As will be seen in Sect. 4.5, numerical examples suggest that there exists a trajectory which starts from an initial value U0 ∈ K consisting of a continuous function of Ω componentwise but, as t → ∞, converges to a discontinuous stationary solution U = t (u, v, w). If this is the case, then any sequence of the form S(tn )U0 cannot converge to U in the topology of X. (Indeed, on account of the topology of X, u and v must be continuous functions; in the meantime, since w ∈ H 2 (Ω), w is also a continuous function.) Therefore, for U0 ∈ K, it must hold that ω(U0 ) = ∅. This suggests also that the dynamical system (S(t), K, X) cannot have a global attractor. In view of this observation, we are led to introduce other types of ω-limit sets. A sequence Un = t (un , vn , wn ) in X is said to be L2 convergent to U = t (u, v, w) ∈ X as n → ∞ if u → u in L (Ω), v → v in L (Ω), and w → w n 2 n 2 n in L2 (Ω), respectively. Then, using this topology, we define the L2 -ω-limit set of

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11 Forest Kinematic Model

S(t)U0 , U0 ∈ K, by L2 -ω(U0 ) =



{S(τ )U0 ; t ≤ τ < ∞}

(closure in the L2 topology of X).

t≥0

In the meantime, a sequence Un = t (un , vn , wn ) in X is said to be weak* convergent to U = t (u, v, w) ∈ X as n → ∞ if un → u weak* in L∞ (Ω), vn → v weak* in L∞ (Ω), and wn → w in L2 (Ω), respectively. Using this topology, we define the w*-ω-limit set of S(t)U0 , U0 ∈ K, by w*-ω(U0 ) =



{S(τ )U0 ; t ≤ τ < ∞} (closure in the weak* topology of X).

t≥0

Concerning these ω-limit sets, we have the following results. Theorem 11.2 For each U0 ∈ K, w*-ω(U0 ) is a nonempty set. Proof Let U0 ∈ K and S(t)U0 = U (t). On account of (11.22), we notice that the Banach–Alaoglu theorem of sequentially weak* compact version (see Brezis [Bre83, IV. 3. C]) and the compact embedding theorem, i.e., H 2 (Ω) ⊂ L2 (Ω), are available to conclude that there exists a time sequence {tn } tending to ∞ such that u(tn ) → u weak* in L∞ (Ω), v(tn ) → v weak* in L∞ (Ω), and w(tn ) → w in  L2 (Ω), that is, U (tn ) → U in the weak* topology of X. Theorem 11.3 For each U0 ∈ K, ω(U0 ) ⊂ L2 -ω(U0 ) ⊂ w*-ω(U0 ). Proof The first relation ω(U0 ) ⊂ L2 -ω(U0 ) is obvious by the definition. Let U ∈ L2 -ω(U0 ). Then, there exists a sequence {tn }  ∞ such that S(tn )U0 = Un (t) → U in the L2 topology of X. Let ϕ ∈ L1 (Ω). For any f ∈ L2 (Ω),          ϕ[u(tn ) − u] dx  ≤ ϕ − f L u(tn ) − uL +  f [u(tn ) − u] dx . ∞ 1     Ω

Ω

Since L2 (Ω) is dense in L1 (Ω) and since (11.21) is true, we verify that, as tn → ∞,  | Ω ϕ[u(tn ) − u] dx| → 0. Hence, u(tn ) → u in the weak* topology of L∞ (Ω). It is the same for the weak* convergence of v(tn ) to v. Thus, we have U ∈ w*-ω(U0 ).  Theorem 11.4 For U0 ∈ K, let there exist a sequence {tn } tending to ∞ such that the sequence of functions S(tn )U0 converges to U ∈ X componentwise almost everywhere in Ω. Then, U ∈ L2 -ω(U0 ). Proof By (11.21) and (11.23), the almost everywhere convergence implies the L2 convergence of the sequences of u(tn ), v(tn ), and w(tn ). Hence, U ∈ L2 -ω(U0 ). 

4 Dynamical System

405

4.3 Constituents of L2 -ω-limit Sets Generally speaking, if there exists a Lyapunov function, then one can prove that the ω-limit set consists of equilibria of a dynamical system. However, in the present case, we do not know that S(t) is continuous with respect to the weak* topology of X. We can merely prove the assertion only for the L2 -ω-limit set. Theorem 11.5 If L2 -ω(U0 ) is nonempty, then the set consists of equilibria of (S(t), K, X). Proof We first notice that S(t) : K → X is continuous with respect to the L2 topology of X. Indeed, by similar techniques as in Step 1 of the proof of Proposition 11.1, we can establish the local Lipschitz condition S(t)U0 − S(t)U˜ 0 L2 ≤ CT ,R U0 − U˜ 0 L2 , 0 ≤ t ≤ T ; U0 , U˜ 0 ∈ K with U0 L2 + U˜ 0 L2 ≤ R. The detailed proof is however left to the reader. Let U ∈ L2 -ω(U0 ). Then, there exists a sequence tn → ∞ such that S(tn )U0 = U (tn ) → U in the L2 topology. By (11.22), we can take a subsequence {w(tn )} of {w(tn )} such that w(tn ) → w in H 1 (Ω). It is then easy to see that w = w . Meanwhile, u(tn ) → u and v(tn ) → v in any Lp topology with any finite 2 ≤ p < ∞. By these facts we conclude that the value Ψ (U (tn )) of the Lyapunov function is convergent to Ψ (U ) as tn → ∞. That is, Ψ (U ) = lim Ψ (U (tn )) = inf Ψ (S(t)U0 ) ≡ Ψ∞ . n →∞

0≤t<∞

We have therefore seen that, for every U ∈ L2 -ω(U0 ), it holds that Ψ (U ) ≡ Ψ∞ . In the meantime, we see that S(t)U ∈ L2 -ω(U0 ) for any vector U ∈ L2 -ω(U0 ) and any time 0 < t < ∞. In fact, S(tn )U0 → U in the L2 topology implies that S(t + tn )U0 = S(t)S(tn )U0 → S(t)U in the L2 topology; therefore, by definition, S(t)U ∈ L2 -ω(U0 ). Let us now complete the proof. If U ∈ L2 -ω(U0 ), then U (t) = S(t)U ∈ d L2 -ω(U0 ) for every t ≥ 0; therefore, Ψ (U (t)) ≡ Ψ∞ and dt Ψ (U (t)) ≡ 0. Hence, ∂v ∂w by (11.30), ∂t (t) ≡ ∂t (t) ≡ 0. Furthermore, from the equation of v it follows that f u(t) ≡ hv(t); hence, be an equilibrium.

4.4 Cases where

∂u ∂t (t)

f αδ c+f

≡ 0. Thus,

dU dt (t)

≡ 0 for 0 < t < ∞, that is, U must 

< h < ∞ and where ab2 < 3(c + f )

In some particular cases, we can prove stronger results than Theorem 11.2. We first f αδ show that, when c+f < h < ∞, every trajectory converges to the zero stationary solution.

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11 Forest Kinematic Model

f αδ Theorem 11.6 Let c+f < h < ∞. Then, w*-ω(U0 ) = L2 -ω(U0 ) = ω(U0 ) = {O} for every U0 ∈ K, where O = t (0, 0, 0).

Proof Let U0 ∈ K, and let S(t)U0 = U (t) be the global solution. Multiply the equation of u in (11.5) by 2(c + f )u and integrate the product in Ω. Then, (c + f )

d dt





 u2 dx + 2(c + f )2 Ω

u2 dx − 2(c + f )βδ Ω



wu dx Ω

(v − b)2 u2 dx ≤ 0.

= −2a(c + f )

(11.33)

Ω

Similarly, multiply the equation of v by Then,

2(c+f )αδ v f







(c + f )αδ d f dt

and integrate the product in Ω.

v 2 dx + 2(αδ)2

v 2 dx − 2(c + f )αδ

Ω

Ω

uv dx Ω

  f αδ 2(c + f )αδ h− + v 2 dx = 0. f c+f Ω

(11.34)

Multiply the equation of w by 2βδ 2 w and integrate the product in Ω. Then, d βδ dt



 w dx + 2(βδ)

2

2

Ω

 w dx − 2αβδ

2

2

Ω



|∇w|2 dx ≤ 0.

= −2dβδ 2

2

vw dx Ω

(11.35)

Ω

Summing up (11.33), (11.34), and (11.35), we obtain that d dt

      (c + f )αδv 2 (c + f )u2 + + βδ 2 w 2 dx + 2 (c + f )2 u2 + (αδv)2 f Ω Ω   2 2 + (βδw) − 2[(c + f )αδuv + αβδ vw + (c + f )βδwu] + 3εv 2 dx ≤ 0,

where ε =

2(c+f )αδ f αδ (h − c+f ) > 0. 3f

We here notice that

  2 (c + f )2 u2 + (αδv)2 + (βδw)2 − (c + f )αδuv − αβδ 2 vw − (c + f )βδwu + 3εv 2     

2

(c + f )αδu  2 2 αβδ 2 w 2 − α δ +ε v + = √ α2 δ2 + ε v − √ α2δ2 + ε α2δ2 + ε   (c + f )2 u2 (βδw)2 + v2 + 2 2 . + [βδw − (c + f )u]2 + ε 2 2 α δ +ε α δ +ε

4 Dynamical System

407

Therefore, with an appropriate exponent ρ > 0,   d (C1 u2 + C2 v 2 + C3 w 2 ) dx + ρ (C1 u2 + C2 v 2 + C3 w 2 ) dx ≤ 0, dt Ω Ω where Ci > 0 (i = 1, 2, 3) are some positive constants. Thus, U (t)L2 ≤ Ce−ρt U0 L2 ,

0 < t < ∞, U0 ∈ K.

This shows that, as t → ∞, S(t)U0 converges to O exponentially in the L2 topology. We can then repeat similar calculations as in Steps 2 and 3 of the proof of Proposition 11.1 to derive the exponential decay of S(t)U0 in the X norm. In particular, the assertion of theorem is verified.  We secondly consider the case where ab2 < 3(c + f ). We notice in this case that the cubic function Q(v) ≡

h [γ (v) + f ]v, fβδ

−∞ < v < ∞,

(11.36)

has the following properties. Lemma 11.1 When ab2 < 3(c + f ), Q (v) > 0 for all −∞ < v < ∞. Therefore, w = Q(v) is an increasing function, and its inverse v = Q−1 (w) is a single-valued smooth function for −∞ < w < ∞; moreover, the inverse function is uniformly Lipschitz continuous, i.e., |Q−1 (w) − Q−1 (w)| ˜ ≤ L|w − w|, ˜ −∞ < w, w˜ < ∞. On the contrary, when ab2 ≥ 3(c + f ), Q (v) has real roots 0 < v1 ≤ v2 < ∞ (when ab2 = 3(c + f ), v1 = v2 ). Proof We obviously see from (11.2) that h (3av 2 − 4abv + ab2 + c + f ) fβδ     2b 2 ab2 − 3(c + f ) h 3a v − . − = fβδ 3 3

Q (v) =

Therefore, all the assertions of lemma are clear.



Using these properties of Q(v), we prove the following theorem. Theorem 11.7 Let ab2 < 3(c + f ). Then, L2 -ω(U0 ) = w*-ω(U0 ) for every U0 ∈ K. Proof Let S(t)U0 = t (u(t), v(t), w(t)). Consider any time sequence {tn } which tends to ∞ as n → ∞. By (11.22), w(tn )H 2 is a bounded sequence; so, we can choose a subsequence {tn } for which {w(tn )} is convergent to w in H 1+ε (Ω) and

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11 Forest Kinematic Model

hence in L∞ (Ω). From the first and second equations of (11.1), it is easily observed that 1 du γ (v(tn )) + f dv Q(v(tn )) = w(tn ) − (tn ) − (tn ). βδ dt fβδ dt So, by (11.36),   1 du γ (v(tn )) + f dv v(tn ) = Q−1 w(tn ) − (tn ) − (tn ) . βδ dt fβδ dt Since w(tn ) → w in L∞ (Ω) and since Proposition 11.5 is valid, we conclude that v(tn ) converges to v = Q−1 (w) in L2 (Ω). Since Proposition 11.5 provides in particular that, as t → ∞, f u(t)−hv(t) → 0 in L2 (Ω), we conclude also that u(tn ) converges to fh v in L2 (Ω). Thus, we have shown that t (u(tn ), v(tn ), w(tn )) → t (u, v, w) in L (Ω). 2 We now know that any sequence t (u(tn ), v(tn ), w(tn )) has a subsequence which converges to some vector of X in the L2 topology. In particular, the relation w*-ω(U0 ) ⊂ L2 -ω(U0 ) must be true. 

4.5 Numerical Examples We here present some numerical example. Let Ω = [0, 1] × [0, 1] be a quadratic domain. Set the parameters as α = β = δ = 1.0, f = g = 1.0, a = 1.0, b = 3.0, c = 0.2, and d = 0.05. We treat h as a control parameter and put h = 0.0488 in order that the parameters satisfy the relation 0
f αβδ , (ab2 + c + f )(dμ0 + β)

(11.37)

where μ0 > 0 denotes the minimal eigenvalue of the realization of −Δ in L2 (Ω) under the homogeneous Dirichlet boundary conditions and is given by 2.0π 2 in the present case. As will be shown by Theorem 11.9 in the subsequent section, when (11.37) takes place, the zero solution O, which is the only homogeneous stationary solution of (11.1), is unstable. We take continuous initial functions t (u0 , v0 , w0 ) ∈ K which are of the shape of circular truncated cone and are put at the center of Ω. In the initial stage, the numerical solution U = t (u, v, w), and especially u, changes very rapidly. Gradually, the solution stabilizes and tends (at least in the numerical sense) to a nonhomogeneous stationary solution t (u, v, w), see Figs. 1 and 2. The value of the Lyapunov function also stabilizes as t → ∞ along the trajectory. As we can see, the graphs of u and v have a clear curve in Ω near the boundary ∂Ω on which the densities of the young age class trees and of old age class trees are discontinuous. From an ecological point of view, such a curve is regarded as the canonical forest boundary and is very important.

5 Homogeneous Stationary Solution

Fig. 1 Sections of graphs v, w by the plane x =

409

1 2

Fig. 2 Sections of graphs v, w by the plane y = x

5 Homogeneous Stationary Solution In this section, we investigate homogeneous stationary solutions of (11.1). Let (u, v, w) be any homogeneous stationary solution of (11.1). Because of the Dirichlet boundary conditions on w, we must have w = 0. Consequently, it follows that u = v = 0. Hence, O = t (0, 0, 0) is a unique homogeneous solution of (11.1). We are going to prove by Theorems 11.9, 11.10, and 11.11 below that f αβδ , O is unstable. (ab2 +c+f )(dμ0 +β) f αβδ f αδ < h ≤ c+f , O is stable. (ab2 +c+f )(dμ0 +β) f αδ c+f < h < ∞, O is globally stable.

(1) When 0 < h ≤ (2) When (3) When

Here, μ0 > 0 is the minimal eigenvalue of the operator −Δ in L2 (Ω) equipped with the homogeneous Dirichlet boundary conditions.

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11 Forest Kinematic Model

5.1 Localized Problem in a Neighborhood of O For investigating the stability and instability of O, we can follow the general strategy in Chap. 6, Sect. 6. Let χ(u) be a cutoff function defined on the complex plane C such that χ(u) = u for u : |u| < 1 and χ(u) = u/|u| for u : |u| ≥ 1. The localized problem in a neighborhood of O is given by ⎧ ∂u ⎪ in Ω × (0, ∞), ⎪ ∂t = βδχ(w) − γ (χ(v))χ(u) − f u ⎪ ⎪ ⎪ ∂v ⎪ ⎪ in Ω × (0, ∞), ⎨ ∂t = f χ(u) − hv ∂w ⎪ ∂t = dΔw − βw + αχ(v) ⎪ ⎪ ⎪ w =0 ⎪ ⎪ ⎪ ⎩u(x, 0) = u (x), v(x, 0) = v (x), 0 0

in Ω × (0, ∞), on ∂Ω × (0, ∞), w(x, 0) = w0 (x) in Ω. (11.38) We can then repeat the same arguments as in Sect. 1 to construct local solutions and global solutions for all initial values from X in the function space: U ∈ C((0, ∞); D(A)) ∩ C([0, ∞); X) ∩ C1 ((0, ∞); X). ˜ acting Therefore, the localized problem (11.38) defines a continuous semigroup S(t) ˜ on X. Furthermore, it is verified that (S(t), X, X) defines a dynamical system which is a complexified dynamical system of (S(t), K, X). It is also verified that, if U0 ∈ D(Aθ ) with 0 < θ < 1, then U belongs to: U ∈ C((0, ∞); D(A)) ∩ C([0, X); D(Aθ )) ∩ C1 ((0, ∞); D(Aθ )). ˜ From this we can equally verify that, for any 0 < θ < 1, (S(t), Dθ , Dθ ) is a dynamθ ical system in Dθ , where Dθ = D(A ). We now fix 12 < θ < 1 in order to have Dθ ⊂ L∞ (Ω). Then, in a suitable neighborhood of O in the space Dθ , any trajectory of (S(t), Kθ , Dθ ) is that of ˜ ˜ (S(t), Dθ , Dθ ). In particular, O is an equilibrium of (S(t), Dθ , Dθ ), too. Therefore, ˜ if O is stable as an equilibrium of (S(t), Dθ , Dθ ), then it is the same as that of (S(t), Kθ , Dθ ). However, we cannot say automatically that, even if O is unstable ˜ in (S(t), Dθ , Dθ ), it is the same as that of (S(t), Kθ , Dθ ). Nevertheless, the insta˜ bility of O in (S(t), Dθ , Dθ ) provides an information which indicates its instability in the original system (S(t), Kθ , Dθ ). In fact, that shows existence of a smooth lo˜ cal unstable manifold W˜ + (O; O) in (S(t), Dθ , Dθ ), which is tangential to the space X− at O, where X− is a finite- or infinite-dimensional eigenspace corresponding to the negative eigenvalues of the linearized operator A − F˜ (O) given by (11.39). Furthermore, X− has a basis consisting of vectors which are componentwise realvalued functions in Ω. ˜ Let us next verify the Fréchet differentiability of S(t) in a neighborhood of O. ˜ is determined by the Cauchy problem for a semilinear equation of We know that S(t)

5 Homogeneous Stationary Solution

411

the form (11.5) in which the nonlinear operator is replaced by the following operator F˜ (U ) : D(Aη ) → X, where 12 < θ < η < 1: ⎛ ⎞ ⎛ ⎞ βδχ(w) − γ (χ(v))χ(u) u ⎠ , U = ⎝ v ⎠ ∈ D(Aη ). f χ(u) F˜ (U ) = ⎝ αχ(v) w In a neighborhood of O in D(Aη ) ⊂ D(Aθ ) ⊂ L∞ (Ω), F˜ is Fréchet differentiable with the derivative ⎛ ⎞ βδr − γ (v)p − γ (v)uq ⎠, fp F˜ (U ) · H = ⎝ αq ⎛ ⎞ p η U ∈ B D(A ) (O; r), H = ⎝ q ⎠ ∈ D(Aη ). r Therefore, F˜ is of class C1,1 and satisfies conditions (6.68) and (6.69). So, the gen˜ : Dθ → eral results obtained in Chap. 6, Sect. 6 are available to conclude that S(t) Dθ is of class C1,1 in a neighborhood of O in Dθ for 0 ≤ t ≤ T , where T > 0. In ˜ is Fréchet differentiable at O for any 0 ≤ t < ∞ with the differential particular, S(t) ˜ O = e−tA , where e−tA is the analytic semigroup on X generated by S(t) ⎛ ⎞ m 0 −βδ h 0 ⎠ (11.39) −A = −(A − F˜ (O)) = ⎝−f 0 −α Λ with m = γ (0) + f = ab2 + c + f .

˜ O 5.2 Spectrum of S(t) Let us characterize in this subsection the spectrum of A in X, i.e., σ X (A). To this end, let us find λ ∈ C such that, for any H = t (p, q, r) ∈ X, the vector equation (λ − A)U = H has a unique solution U = t (u, v, w) ∈ D(A). Obviously, this problem is equivalent to ⎧ ⎪ ⎨(λ − m)u + βδw = p, f u + (λ − h)v = q, ⎪ ⎩ αv + (λ − Λ)w = r. It then follows that [(λ − m)(λ − h)(λ − Λ) + f αβδ]w = f αp − α(λ − m)q + (λ − m)(λ − h)r.

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11 Forest Kinematic Model

Therefore, if λ is a solution of the equation (λ − m)(λ − h) = 0, i.e., λ = m or λ = h, then w cannot belong to H 2 (Ω) in general. This means that m and h do not belong to ρ X (A), i.e., m, h ∈ σ X (A). So, let λ be neither m nor h. Then, we verify that λ ∈ ρ X (A)

⇔

λ+

f αβδ ∈ ρ L2 (Λ). (λ − m)(λ − h)

λ ∈ σ X (A)

⇔

λ+

f αβδ = dμn + β (λ − m)(λ − h)

In other words,

for some n = 0, 1, 2, . . . , where (0 <)μ0 < μ1 ≤ μ2 ≤ · · · denote the eigenvalues of the self-adjoint operator −Δ in L2 (Ω) equipped with the homogeneous Dirichlet boundary conditions. These equations are then rewritten as (λ − dμn − β)(λ − m)(λ − h) + f αβδ = 0,

n = 0, 1, 2, . . . .

(11.40)

We have thus seen that the elements of σ X (A) are given by m and h and by the totality of solutions of infinite number of cubic equations (11.40).

5.3 Stability and Instability of O Let us investigate the stability and instability of O by applying the general results obtained in Chap. 6, Sect. 6.3. We begin with verifying the hyperbolicity of O. Theorem 11.8 The equilibrium O is hyperbolic if and only if mh(dμn + β) − f αβδ = 0 for every n = 0, 1, 2, . . . , where m = ab2 + c + f . Proof The necessity is trivial because if mh(dμn + β) − f αβδ = 0 with some μn , then λ = 0 is a solution of the nth equation in (11.40), and so 0 ∈ σ X (A). Now, let mh(dμn + β) − f αβδ = 0 for every n = 0, 1, 2, . . . . It is easy to see that (11.40) has no purely imaginary solutions. Indeed, assume that λ = iy, y ∈ R, is a solution of (11.40) for some n = n0 . Then, by direct calculation, we observe that y = 0 and  y 2 = (m + h)(dμn0 + β) + mh, y 2 (m + h + dμn0 + β) = mh(dμn0 + β) − f αβδ. However, these two equalities are inconsistent.



On the basis of this theorem, we can see stability and instability conditions for O as follows.

5 Homogeneous Stationary Solution

Theorem 11.9 Let 0 < h < over, let the condition μn =

413

f αβδ , (ab2 +c+f )(dμ0 +β)

i.e., μ0 <

β[f αδ − (ab2 + c + f )h] , (ab2 + c + f )dh

β[f αδ−(ab2 +c+f )h] . (ab2 +c+f )hd

n = 0, 1, 2, . . . ,

More-

(11.41)

˜ be satisfied. Then, O is an unstable equilibrium of (S(t), Dθ , Dθ ). Proof By Theorem 11.8, we see that O is hyperbolic. So, it suffices to verify that A has a spectrum such that σ X (A) ∩ {λ ∈ C; Re λ < 0} = ∅. The equations in (11.40) are rewritten by λ3 − [m + h + (dμn + β)]λ2 + [mh + (m + h)(dμn + β)]λ + f αβδ − mh(dμn + β) = 0.

(11.42)

By virtue of the Routh–Hurwitz theorem, we verify that, for μn satisfying β(f αδ − mh) < μn , mhd the equations in (11.42) have all their solutions in the right half-plane {λ ∈ C; Re λ > 0}. On the other hand, for μn satisfying μ0 ≤ μn <

β(f αδ − mh) , mhd

the equations of (11.42) have negative real solutions λn , and these λn are in σ− (A) = ˜ + (O; O) σ X (A) ∩ {λ ∈ C; Re λ < 0}. Consequently, the local unstable manifold W is nontrivial and has the same dimension as dim X− , where X− is the subspace of  Dθ corresponding the spectrum σ− (A). αδ−(ab +c+f )h] < h < ∞, i.e., β[f (ab < μ0 . 2 +c+f )hd ˜ Then, O is an exponentially stable equilibrium of (S(t), Dθ , Dθ ).

Theorem 11.10 Let

f αβδ (ab2 +c+f )(dμ0 +β)

2

Proof As seen in the proof of Theorem 11.9, in the present case, we verify that all the equations in (11.42) have their solutions in the right half-plane ˜  {λ ∈ C; Re λ > 0}. Therefore, O is exponentially stable in (S(t), Dθ , Dθ ). f αδ Theorem 11.11 Let c+f < h < ∞. Then, O is a globally stable equilibrium of (S(t), Kθ , Dθ ). That is, every trajectory tends to O as t → ∞.

Proof For U0 ∈ K, let U (t) = t (u(t), v(t), w(t)) = S(t)U0 . By Theorem 11.6, we already know that w(t)L2 → 0 as t → ∞. Using that 1−θ Λθ w(t)L2 ≤ CΛw(t)θL2 w(t)L , 2

0 < t < ∞,

we observe that Λθ w(t)L2 → 0, too. Hence, S(t)U0 → O as t → ∞ in the topol ogy of Dθ .

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11 Forest Kinematic Model

Notes and Further Researches In theoretical forestry, many kinds of mathematical models have been introduced for describing the growing or decaying process of individual trees, trees in a plot of forest, and even of all the trees which interact one another in a forest. Botkin et al. [BJW72] (cf. also [Bot93]) may be the first and most basic mathematical model. The authors presented the Individual-Based Model for a plot of a forest (100 m2 ∼ 300 m2 ) which gives a growth equation of individual trees in the plot per year on the basis of a field work [BSLW70]. This individual model was generalized by Pacala et al. [PCS96] and [PD95] as the Individual-Based ContinuousSpace Model on the basis of works [CFPB94, PCSK95, RPS94]. In the meantime, macroscopic forest models concerning with the age-dependent tree relationship have also been introduced by Antonovsky [Ant75], Antonovsky et al. [AK83], and so forth. Such a model is called the Age-Structured Model. The model (11.1) treated in this chapter has been proposed by Kuznetsov–Antonovsky–Biktashev– Aponina [KABA94], which is classified as the Age-Structured Continuous-Space Model incorporating the seed dispersion in the space whose process has been studied by Okubo–Levin [OL89]. In modeling, the authors incorporated also the effect that there exists some optimal value of the density of old age class trees under which the recruitment of young age class trees is maximal, according to the works by Antonovsky–Korzukhin [AK83] and Korzukhin [Kor80] (cf. also the Janzen– Connell hypothesis [Jan70]). Indeed, the mortality γ (v) given by (11.2) is the only nonlinear term in the model equations (11.1) but provides a specific discontinuity in the densities of u and v which can be considered as a canonical boundary of the forest ecosystem (cf. Figs. 1 and 2 and also [Yag08c]). In the original paper [KABA94], the authors studied stationary solutions and traveling wave solutions reducing equations in (11.1) into the system of crossdiffusion type  ∂u ∂2v ∂t = ρv − γ (v)u − f u + K ∂x 2 , (11.43) ∂v ∂t = f u − hv, in the one-dimensional case. Wu [Wu95, Wu97] proved analytically the existence of traveling wave solutions to (11.43) and their stability. Wu–Lin [WL99] studied the stability and instability for homogeneous stationary solutions to the modified system of (11.1) in which the production term αv in the equation of w is replaced by the spatial average α Ω v dx. Lin–Liu [LL00] studied the further modified system in which (in addition to the term αv) the establishment term δβw in the equation of u is also replaced by δβw(t − τ ) with time delay τ > 0. The construction of a dynamical system for (11.1) in the underlying space X was done by Chuan–Yagi [CY06] and Shirai–Chuan–Yagi [SCY07a]. The structure of the ω-limit set of trajectories was studied by Chuan–Tsujikawa–Yagi [CTY06] and Shirai–Chuan–Yagi [SCY07b]. Although we know the existence of the Lyapunov function given by (11.31), there remain many essential questions unanswered for the ω-limit sets. For example, does the limit set w*-ω(U0 ) consist only of stationary solutions of (11.1)? In any case, is the limit set L2 -ω(U0 ) nonempty? When do the

Notes and Further Researches

415

two limit sets w*-ω(U0 ) and L2 -ω(U0 ) coincide? Can we utilize the Simon and Łojasiewicz method (cf. [Sim83] and [Loj63, Loj65]) in order to show that the ω-limit sets are singletons? The stability and instability of the homogeneous stationary solution O were studied by Shirai–Chuan–Yagi [SCY08]. In Chuan–Tsujikawa–Yagi [CTY09], the existence of an infinite number of discontinuous stationary solutions was indicated but under the homogeneous Neumann boundary conditions on w. In the Dirichlet case, however, there is no such a rigorous result. Mola–Yagi [MY09] treated a forest kinematic model with memory.

Chapter 12

Chemotaxis Models

Budrene–Berg discovered [BB91, BB95] a remarkable aggregation pattern generated by chemotactic bacteria. In 1995, Woodward–Tyson–Myerscough–Murray– Budrene–Berg presented the mathematical model ⎧  u    ⎨ ∂u = aΔu − μ∇ · ∇ρ + cu 1 − us in Ω × (0, ∞), ∂t (1+αρ)2 (12.1) rρ p ∂ρ u su ⎩ = bΔρ + in Ω × (0, ∞), ∂t 1+βuq − 1+γρ for describing the process of pattern formation. Here, Ω is a two-dimensional domain where bacteria are bred. The function u(x, t) denotes the population density of bacteria in Ω at time t . The function ρ(x, t) denotes the concentration of chemical attractant in Ω at time t . The movement of biological individuals consists of two effects: one is a random walk, and the other is a directed movement in a sense that they have a tendency to move toward a higher concentration direction of the chemical substance. The latter directed movement is called chemotaxis. In 1970, Keller–Segel introduced a mathematical formulation for chemotaxis by the diffusion–advection equation ∂u = aΔu − μ∇ · [u∇χ(ρ)], ∂t where χ(ρ) is a sensitivity function of bacteria toward the chemoattractant, and μ > 0 is an attraction rate. Several forms are proposed as sensitivity functions. The ρ typical ones are such that χ(ρ) = ρ, ρ 2 , log(ρ + 1), log ρ, ρ+1 , etc. In (12.1), the ρ authors took the function χ(ρ) = 1+αρ . In (12.1), the term cu(1 − us ) denotes the proliferation of bacteria by the logistic law, where s denotes the concentration of the substrate. In this model, s is assumed sup to be spatially homogeneous and constant. The term 1+βu q denotes the production of chemoattractant by bacteria, where p, q are some nonnegative exponents and ur ρ denotes the consumption of chemoattractant β ≥ 0 is a constant. The term − 1+γρ by bacteria, where r is some nonnegative exponent, and γ ≥ 0 is a constant. The bacteria and the chemical substance diffuse with rates a > 0 and b > 0, respectively. A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_12, © Springer-Verlag Berlin Heidelberg 2010

417

418

12 Chemotaxis Models

We will consider in this chapter the following simplified model: ∂u 2 in Ω × (0, ∞), ∂t = aΔu − μ∇ · [u∇χ(ρ)] + cu − γ u ∂ρ ∂t

= bΔρ − dρ + νu

in Ω × (0, ∞).

(12.2)

The sensitivity function will not be specified but will be assumed to satisfy condiρ tion (12.4). Actually, χ(ρ) = ρ, log(ρ +1), and ρ+1 are allowed. For the production term, we take p = 1 and β = 0. We ignore the consumption of chemical substance by bacteria, i.e., r = 0; instead we consider the unconstrained resolution which is denoted by −dρ with rate d > 0. First, we construct a dynamical system for the initial-boundary-value problem of (12.2) imposing the homogeneous Neumann boundary conditions for u and ρ and show that the dynamical system possesses exponential attractors. Secondly, we investigate  stability and instability of the homogeneous stationary solution  cνthe (u, ρ) = γc , dγ . In fact, we show that, when the product μν is small, (u, ρ) is stable and, on the other hand, that, when μν is large enough, (u, ρ) is no longer stable. Moreover, we show that the degree of instability tends to infinity as μν → ∞ and consequently the fractal dimensions of exponential attractors also tend to infinity.

1 Chemotaxis Model Without Proliferation Before studying (12.2), let us consider a chemotaxis model without proliferation. We are concerned with the following problem: ⎧ ∂u ⎪ in Ω × (0, ∞), ⎪ ∂t = aΔu − μ∇ · [u∇χ(ρ)] ⎪ ⎪ ⎪ ⎨ ∂ρ = bΔρ − dρ + νu in Ω × (0, ∞), ∂t (12.3) ∂ρ ∂u ⎪ = ∂n = 0 on ∂Ω × (0, ∞), ⎪ ⎪ ∂n ⎪ ⎪ ⎩u(x, 0) = u (x), ρ(x, 0) = ρ (x) in Ω. 0 0 As mentioned above, such a chemotaxis model was founded by Keller–Segel [KS70]. The form of the equations of (12.3) is of some particular case of the general ones presented in the paper. Here, Ω is a two-dimensional C2 or convex, bounded domain. We assume that the sensitivity function χ(ρ) is a real smooth function for 0 ≤ ρ < ∞ satisfying the condition   i  d χ  sup  i (ρ) < ∞ for i = 1, 2. (12.4) dρ 0≤ρ<∞

We assume also that a, b, d, μ, and ν are positive constants (>0). First, we construct local solutions for problem (12.3). Secondly, in the case where χ(ρ) is linear (= ρ), we establish a priori estimates of local solutions for the initial functions u0 (x) ≥ 0 and ρ0 (x) ≥ 0 with sufficiently small norm u0 L1 . Consequently, for such initial functions, we can show the global existence of solutions.

1 Chemotaxis Model Without Proliferation

419

However, we must notice the fact due to Herrero–Velázquez [HV96] that the local solutions to (12.3) blow up in finite time, provided that the norms u0 L1 of their initial functions are large.

1.1 Abstract Formulation Let us formulate problem (12.3) as the Cauchy problem for an abstract evolution equation. It will be spontaneous to try to handle the first equation of (12.3) on u in the space L2 (Ω). Then, we are naturally led to handle the second one in the Sobolev space H 2 (Ω). As will be observed in establishing a priori estimates for (12.3) (see Propositions 12.2 and 12.3), we have to couple the norms u(t)2L2 and ρ(t)2H 2 . This means that it is natural to handle (12.3) in the product space of u ∈ L2 (Ω) and ρ ∈ H 2 (Ω). By this reason, our underlying space is taken as   u X= ; u ∈ L2 (Ω) and ρ ∈ HN2 (Ω) . (12.5) ρ Since ρ satisfies the homogeneous Neumann boundary conditions, the space for ρ is taken as HN2 (Ω). In the underlying space X, the chemotactic term ∇ · [u∇χ(ρ)] can be treated as a lower term. In other words, we are able to formulate problem (12.3) as a semilinear problem of the form (4.1), although the first diffusion equation of (12.3) on u is quasilinear in form. This is the merit of taking an underlying space of the form (12.5). In this way, we rewrite (12.3) by dU 0 < t < ∞, dt + AU = F (U ), (12.6) U (0) = U0 , in X, where U = t (u, ρ). Here, the linear operator A is given by 

A1 0 . A= −ν A2

(12.7)

The two operators A1 = −aΔ + 1 and A2 = −bΔ + d are positive definite selfadjoint operators of L2 (Ω) with domains D(A1 ) = D(A2 ) = HN2 (Ω). However, A2 4 (Ω) = D(A2 ) = {ρ ∈ H 2 (Ω); Δρ ∈ H 2 (Ω)} is regarded as an operator from HN 2 N N 2 2 into D(A2 ) = HN (Ω). (Note that, as Ω is not assumed to be of C4 class, we cannot 4 (Ω) ⊂ H 4 (Ω).) So, the domain of A is given by expect that HN 2   u 2 4 D(A) = ; u ∈ HN (Ω) and ρ ∈ HN 2 (Ω) . ρ The nonlinear operator F is given by

 −μ∇ · [u∇χ(Re ρ)] + u F (U ) = . 0

(12.8)

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12 Chemotaxis Models

Since ∇ · [u∇χ] = ∇u · ∇χ + uΔχ , it is possible to see that ∇ · [u∇χ]L2 ≤ CuH 1+ε χH 2 ,

u ∈ H 1+ε (Ω), χ ∈ HN2 (Ω)

(12.9)

with an arbitrary fixed exponent 0 < ε < 12 . Indeed, note that H 1+ε (Ω) ⊂ L∞ (Ω) due to (1.76) and H 1+ε (Ω) ⊂ Hp1 (Ω) for p such that 2 < p < 2/(1 − ε) due to (1.74). We then observe that F (U ) is an operator from D(Aη ) into X, where the exponent η is given by η = 1+ε 2 . Note also by (1.93) that ρ → χ(Re ρ) is a bounded operator from HN2 (Ω) into itself.

1.2 Construction of Local Solutions In order to show the existence of local solutions to (12.6), let us verify that A and F fulfill the structural assumptions made in Chap. 4, Sect. 1. Denote by B the multiplicative operator by the number −ν. Then, A is obviously written by

 A1 0 A= . (12.10) B A2 We regard B as a linear operator from D(B) ⊂ L2 (Ω) to HN2 (Ω); clearly, D(B) = HN2 (Ω), and B is a bounded operator from D(A1 ) onto HN2 (Ω). Hence, it is an immediate consequence of Theorem 2.16 that the operator A is a sectorial operator of the product space X. Let us next investigate the domains of fractional powers Aθ for 0 ≤ θ ≤ 1. According to Theorems 16.7 and 16.7 in Chap. 16, the domains D(Aθ1 ) and D(Aθ2 ) are characterized by     D Aθ1 = D Aθ2 = H 2θ (Ω),     D Aθ1 = D Aθ2 = HN2θ (Ω),

3 0≤θ < , 4 3 < θ ≤ 1. 4

Consequently, we have     2θ (Ω) = ρ ∈ HN2 (Ω); Δρ ∈ H 2(θ−1) (Ω) , D Aθ2 = HN   2(θ−1) 2θ 2 (Ω) D(Aθ2 ) = HN 2 (Ω) = ρ ∈ HN (Ω); Δρ ∈ HN

7 1≤θ < , 4 7 < θ ≤ 2. 4

Consider a diagonal operator AD = diag {A1 , A2 } acting in X. Then, AD is a selfadjoint operator of X; therefore, AθD = diag {Aθ1 , Aθ2 }. More precisely,    θ u 3 2(θ+1) 2θ ; u ∈ H (Ω) and ρ ∈ HN (Ω) , 0 < θ < , D AD = ρ 4     u 3 2(θ+1) D AθD = ; u ∈ HN2θ (Ω) and ρ ∈ HN 2 (Ω) , < θ ≤ 1. ρ 4

1 Chemotaxis Model Without Proliferation

421

We then verify the following property. Proposition 12.1 The domains of fractional powers of A are given by   D(Aθ ) = D AθD , 0 ≤ θ ≤ 1, with the norm equivalence     C −1 AθD ·X ≤ Aθ · X ≤ C AθD ·X ,

(12.11)

0 ≤ θ ≤ 1.

Proof We will apply Theorem 16.4 to A. It is clear that the analogous statement to Theorem 16.4 is true for matrix operators of the form (12.10). The multiplicative operator B is clearly an operator from L2 (Ω) to HN2 (Ω) = D(A2 ) with the domain D(B) = HN2 (Ω). Therefore, B is a bounded operator from D(A1 ) into HN2 (Ω). In the meantime, u ∈ HN2 (Ω) belongs to D(B ∗ ) if and only if the functional (u, B·)H 2 is continuous with respect to the L2 topology. Since N

(u, Bv)H 2 = (A2 u, −A2 νv)L2 = −ν(A2 u, A2 v)L2 , u ∈ D(A22 ) implies u ∈ D(B ∗ ). N

This indeed means that D(A22 ) ⊂ D(B ∗ ) and B ∗ is a bounded operator from D(A22 ) into L2 (Ω). Hence, we have verified that Theorem 16.4 is available to A. We now know that D(Aθ ) = [D(A), X]θ . From this we deduce that D(Aθ ) = [D(A), X]θ = [D(AD ), X]θ = D(AθD ) because of the coincidence D(A) = D(AD ). The norm equivalence is also verified by Theorem 16.4.  Note that ρ → χ(Re ρ) is a continuous and bounded operator from HN2 (Ω) into itself due to (1.93). Furthermore, by (12.9) and (12.11) (θ = η), we observe that F satisfies the Lipschitz condition (4.21). We therefore know that Theorem 4.4 is available to problem (12.6). As (4.21) is valid in the present case, we are accordingly led to set the space of initial values by   u0 (12.12) ; 0 ≤ u0 ∈ L2 (Ω) and 0 ≤ ρ0 ∈ HN2 (Ω) . K= ρ0 We have thus proved that, for each U0 ∈ K, (12.6) (and hence (12.3)) possesses a unique local solution U = t (u, ρ) in the function space: u ∈ C((0, TU0 ]; HN2 (Ω)) ∩ C([0, TU0 ]; L2 (Ω)) ∩ C1 ((0, TU0 ]; L2 (Ω)), 4 (Ω)) ∩ C([0, T ]; H 2 (Ω)) ∩ C1 ((0, T ]; H 2 (Ω)). ρ ∈ C((0, TU0 ]; HN U0 U0 2 N N (12.13) In addition, it holds for U that    dU   + tAU (t)X + U (t)X ≤ CU , 0 < t ≤ TU . t (12.14) (t) 0 0  dt  X Here, the positive time TU0 > 0 and the constant CU0 > 0 are determined by the norm U0 X only.

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12 Chemotaxis Models

1.3 Nonnegativity of Local Solutions For U0 ∈ K, let U = t (u, ρ) be the local solution of (12.6) constructed above. Our goal is to prove that u(t) ≥ 0 and ρ(t) ≥ 0 for every 0 < t ≤ TU0 by the truncation method. Let us first verify that U (t) is real valued. Indeed, the complex conjugate U (t) of U (t) is also a local solution of (12.6) with the same initial value. Therefore, the uniqueness of solution implies that U (t) = U (t); hence, U (t) is real valued. Let H (u) be a C1,1 cutoff function such that H (u) = 12 u2 for −∞  < u < 0 and H (u) ≡ 0 for 0 ≤ u < ∞. According to (1.100), the function ψ(t) = Ω H (u(t)) dx is continuously differentiable with the derivative ψ  (t) = (H  (u), aΔu − μ∇ · [u∇χ(ρ)])L2 . Applying (1.96) to H  (u), we observe that 



(H (u), aΔu)L2 = −a





∇H (u) · ∇u dx = −a Ω

|∇H  (u)|2 dx.

Ω

In the meantime, by (1.96) again, 



−(H (u), μ∇ · [u∇χ(ρ)])L2 = μ

u∇H  (u) · ∇χ(ρ) dx

Ω



H  (u)∇H  (u) · ∇χ(ρ) dx

=μ Ω

=−

μ 2



H  (u)2 Δχ(ρ) dx. Ω

Since Δχ(ρ) = χ  (ρ)Δρ + χ  (ρ)|∇ρ|2 ,

(12.15)

(12.13) shows that Δχ(ρ)L2 ≤ CU for 0 ≤ t ≤ TU0 . Therefore, −(H  (u), μ∇ · [u∇χ(ρ)])L2 ≤

μ H  (u)2L4 Δχ(ρ)L2 2

≤ CU H  (u)H 1 H  (u)L2 a ≤ H  (u)2H 1 + CU H  (u)2L2 . 2 Hence, ψ  (t) ≤ CU ψ(t); consequently, ψ(t) ≤ ψ(0)eCU t . Then, ψ(0) = 0 implies ψ(t) ≡ 0, i.e., u(t) ≥ 0 for 0 < t ≤ TU0 . It is similar for the proof of ρ(t) ≥ 0 for 0 < t ≤ TU 0 .

2 Case where χ (ρ) = ρ

423

2 Case where χ(ρ) = ρ In this section, let us consider problem (12.6) in a particular case where the sensitivity function is given by χ(ρ) = ρ,

0 ≤ ρ < ∞.

(12.16)

We can establish a priori estimates for sufficiently small initial functions u0 , and consequently we obtain the global existence of solutions.

2.1 Global Solutions for Small Initial Functions Let U0 ∈ K, and let U be any local solution of (12.6) on [0, TU ] in the function space: 0 ≤ u ∈ C((0, TU ]; HN2 (Ω)) ∩ C([0, TU ]; L2 (Ω)) ∩ C1 ((0, TU ]; L2 (Ω)), 4 (Ω)) ∩ C([0, T ]; H 2 (Ω)) ∩ C1 ((0, T ]; H 2 (Ω)). 0 ≤ ρ ∈ C((0, TU ]; HN U U 2 N N (12.17) Proposition 12.2 There exists a positive number r > 0 and a continuous increasing function p(·) such that the estimate u(t)L2 + ρ(t)H 2 ≤ p(u0 L2 + ρ0 H 2 ),

0 ≤ t ≤ TU ,

(12.18)

holds for any local solution U = t (u, ρ) of (12.6) in (12.17) with initial value U0 ∈ K satisfying the condition u0 L1 ≤ r. Proof In the proof, the notation p(·) stands for some continuous increasing functions which are determined only by Ω and by the initial constants in the equations of (12.3). Similarly, the notation C stands for some constants, which are determined in the same way as p(·). Step 1. Integrate the first equation of (12.6) in Ω. Then, obviously,  d u dx = 0. dt Ω Since u(t) ≥ 0, it follows that u(t)L1 = u0 L1 ,

0 ≤ t ≤ TU .

Step 2. We here introduce the quantity  u log(u + 1) dx, N1,log (u) =

0 ≤ u ∈ L2 (Ω).

Ω

The purpose of this step is to estimate N1,log (u(t)) for the local solution U .

(12.19)

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Multiply the first equation of (12.6) by log(u + 1) and integrate the product in Ω. Then, since 

 log(u + 1)Δu dx = − Ω

Ω

1 |∇(u + 1)|2 dx = −4 u+1



√ |∇ u + 1|2 dx

Ω

and 

 log(u + 1)∇ · [u∇ρ] dx = Ω

∇[(u + 1) log(u + 1) − u] · ∇ρ dx Ω



 u log(u + 1)Δρ dx =

+

[u − log(u + 1)]Δρ dx,

Ω

Ω

it follows that d dt



 [(u + 1) log(u + 1) − u] dx + 4a Ω

√ |∇ u + 1|2 dx

Ω



=μ Ω

[log(u + 1) − u]Δρ dx ≤ ζ Δρ2L2 + Cζ u2L2

(12.20)

with any ζ > 0. In the meantime, from the second equation of (12.6) we obtain that 1 d 2 dt







ρ 2 dx + b Ω



|∇ρ|2 dx + d Ω

ρ 2 dx = ν Ω

uρ dx ≤ Ω

d ρ2L2 + Cu2L2 2

and 1 d 2 dt





=ν

|∇ρ| dx + b Ω



 |Δρ| dx + d

2

|∇ρ|2 dx

2

Ω

Ω

b uΔρ dx ≤ Δρ2L2 + Cu2L2 . 4 Ω

Fixing the parameter ζ as ζ = b4 , we sum up these three differential inequalities to obtain that   √   d ψ(t) + dψ(t) + 4a |∇ u + 1|2 dx + b |Δρ|2 dx ≤ C u2L2 + 1 , dt Ω Ω where ψ(t) = (u + 1) log(u + 1) − uL1 + ρ2H 1 . We notice the fact that 1/3 2/3 1/3 √ 4/3 uL2 ≤ uL1 uL4 ≤ uL1  u + 1L8 √ √ 1/3 √ 1/3 ≤ CuL1  u + 1L2 (∇ u + 1L2 +  u + 1L2 ),

0 ≤ u ∈ H 1 (Ω).

2 Case where χ (ρ) = ρ

425

Hence, on account of (12.19), d ψ(t) + dψ(t) + 4a dt



√



|∇ u + 1| dx + b

|Δρ|2 dx

2

Ω

Ω

2  √  ≤ u0 L3 1 p(u0 L1 ) ∇ u + 12L2 + 1 . 2

We now fix a radius r > 0 small enough to satisfy that r 3 p(r) ≤ 3a. Then, if u0 L1 ≤ r, it holds that  √   d a|∇ u + 1|2 + b|Δρ|2 dx ≤ C. ψ(t) + dψ(t) + dt Ω As a consequence (cf. (1.58)), ψ(t) ≤ e−dt ψ(0) + C,

0 ≤ t ≤ TU .

In particular,     N1,log (u(t)) + ρ(t)2H 1 ≤ C e−dt N1,log (u0 ) + ρ0 2H 1 + 1 ,

0 ≤ t ≤ TU . (12.21)

Proposition 1.5 is also available to conclude that  t     u(τ ) + 12H 1 + Δρ(τ )2L2 dτ s

  ≤ C (t − s) + N1,log (u0 ) + ρ0 2H 1 + 1 ,

0 ≤ s < t ≤ TU .

Step 3. The present step is devoted to showing the estimate u(t)2L2 + ρ(t)2H 2 ≤ p(N1,log (u0 ) + ρ0 H 1 )     × e−δt u0 2L2 + ρ0 2H 2 + 1 ,

0 ≤ t ≤ TU ,

with some exponent δ > 0. Now, when estimate (12.21) is established, the techniques of its proof are quite analogous to that of the proof (Step 4) of Proposition 12.3 below, which establishes a similar a priori estimate for a general sensitivity function including the case of (12.16). So, the continued proof will be given there.  It is now possible to apply Corollary 4.3 to conclude the global existence of solutions. In fact, there is a number r > 0 such that, for any U0 ∈ K satisfying u0 L1 ≤ r, (12.3) possesses a unique global solution in the function space: 0 ≤ u ∈ C((0, ∞); HN2 (Ω)) ∩ C([0, ∞); L2 (Ω)) ∩ C1 ((0, ∞); L2 (Ω)), 4 (Ω)) ∩ C([0, ∞); H 2 (Ω)) ∩ C1 ((0, ∞); H 2 (Ω)). 0 ≤ ρ ∈ C((0, ∞); HN 2 N N (12.22)

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12 Chemotaxis Models

2.2 Lyapunov Function When (12.16) holds, we can construct a Lyapunov function. Let U = t (u, ρ) denote a local or global solution of (12.6) in the function space (12.17). Let ε > 0 be a ∂uε positive parameter and put uε = u + ε. It is clear that ∂u ∂t = ∂t and that aΔu − μ∇ · [u∇ρ] = aΔuε − μ∇ · [uε ∇ρ] + εμΔρ. Therefore, the first equation of (12.6) is written as ∂uε = ∇ · [uε ∇(a log uε − μρ)] + εμΔρ. ∂t Multiply this equation by (a log uε − μρ) and integrate the product in Ω. Then,    ∂uε ∂u uε |∇(a log uε − μρ)|2 dx + Rε , a log uε dx − μ ρ dx = − Ω ∂t Ω ∂t Ω  where Rε = εμ Ω (a log uε − μρ)Δρ dx. In the meantime, multiplying the equation of ρ by ∂ρ ∂t and integrating the product in Ω, we have  Ω

∂ρ ∂t

2

b d dx = − 2 dt



d d |∇ρ| dx − 2 dt Ω



 ρ dx + ν

2

2

Ω

u Ω

∂ρ dx. ∂t

Therefore, d dt

 

 bμ dμ 2 |∇ρ|2 + ρ − μνuρ dx 2 2 Ω

2    ∂ρ νuε |∇(a log uε − μρ)|2 + μ dx + νRε ≤ νRε . =− ∂t Ω aν(uε log uε − uε ) +

Let 0 < s < t ≤ TU . Integrating this inequality in [s, t], we obtain that   bμ dμ aν(uε (τ ) log uε (τ ) − uε (τ )) + |∇ρ(τ )|2 + ρ(τ )2 2 2 Ω  τ =t  t − μνu(τ )ρ(τ ) dx ≤ νRε (τ ) dτ. τ =s

It is not difficult to verify that limε→0

s

t s

Rε (τ ) dτ = 0. Hence, if we set

   bμ 2 dμ 2 aν(u log u−u)+ |∇ρ| + ρ −μνuρ dx, Φ(U ) = 2 2 Ω

U ∈ X, (12.23)

then Φ(U (t)) ≤ Φ(U (s)). This means that Φ(U ) plays a role of the Lyapunov function for (12.3). Note that the function u log u can be extended for all 0 ≤ u < ∞ as a continuous function.

3 Chemotaxis Model with Proliferation

427

2.3 Blowup of Solutions Herrero–Velázquez [HV96, HV97] proved that, for suitable initial functions (at least, when u0 L1 are sufficiently large), the local solutions to (12.6) blow up in finite time. Theorem 12.1 There are initial values U0 ∈ K for which the local solutions to (12.6) satisfy lim supt→T U (t)X = ∞ with finite time T < ∞. Hence, (12.6) with the initial values U0 have no global solutions.

3 Chemotaxis Model with Proliferation Let us consider the initial-boundary-value problem for the chemotaxis model with proliferation ⎧ ∂u 2 in Ω × (0, ∞), ⎪ ⎪ ∂t = aΔu − μ∇ · [u∇χ(ρ)] + cu − γ u ⎪ ⎪ ⎪ ⎨ ∂ρ = bΔρ − dρ + νu in Ω × (0, ∞), ∂t (12.24) ∂ρ ∂u ⎪ = =0 on ∂Ω × (0, ∞), ⎪ ⎪ ∂n ∂n ⎪ ⎪ ⎩u(x, 0) = u (x), ρ(x, 0) = ρ (x) in Ω, 0

0

in a two-dimensional C2 or convex, bounded domain Ω. We still assume that χ(ρ) satisfies (12.4) and assume that γ > 0 is a positive constant. As before, we handle problem (12.24) in the product space X given by (12.5), and, as a space of initial functions, we take the space K given by (12.12).

3.1 Local Solutions In order to construct local solutions to (12.24), we can argue in a quite analogous way as for problem (12.3). In fact, we rewrite (12.24) as an abstract problem of the form (12.6) in X. The linear operator A is given in a similar way by (12.7). However, now, the nonlinear operator F is given (instead of (12.8)) by 

−μ∇ · [u∇χ(Re ρ)] + (c + 1)u − γ u2 . (12.25) F (U ) = 0 Let 0 < ε < 12 be arbitrarily fixed and put η = 1+ε 2 . Then, as before (by (12.9) and (12.11)), F is an operator from D(Aη ) into X. It is clear that u2 L2 ≤ CuL2 uH 1+ε ,

u ∈ H 1+ε (Ω).

Then, this, together with (12.9), implies that F fulfills the Lipschitz condition (4.21).

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12 Chemotaxis Models

For each U0 ∈ K, we can therefore construct a unique local solution to (12.24). Furthermore, by the truncation method, we can verify the nonnegativity of the local solution. In this way, for any initial value U0 ∈ K, problem (12.24) possesses a unique local solution in the function space: 0 ≤ u ∈ C((0, TU0 ]; HN2 (Ω)) ∩ C([0, TU0 ]; L2 (Ω)) ∩ C1 ((0, TU0 ]; L2 (Ω)), 4 (Ω)) ∩ C([0, T ]; H 2 (Ω)) ∩ C1 ((0, T ]; H 2 (Ω)), 0 ≤ ρ ∈ C((0, TU0 ]; HN U0 U0 2 N N where the positive time TU0 > 0 is determined by the norm U0 X only. In addition, estimate (12.14) holds for the present local solution, too.

3.2 A Priori Estimates of Local Solutions For U0 ∈ K, let U = t (u, ρ) denote a local solution of (12.24) on [0, TU ] in the function space: 0 ≤ u ∈ C((0, TU ]; HN2 (Ω)) ∩ C([0, TU ]; L2 (Ω)) ∩ C1 ((0, TU ]; L2 (Ω)), 4 (Ω)) ∩ C([0, T ]; H 2 (Ω)) ∩ C1 ((0, T ]; H 2 (Ω)). 0 ≤ ρ ∈ C((0, TU ]; HN U U 2 N N (12.26) We can establish the following a priori estimates for local solutions. Proposition 12.3 There exists a continuous increasing function p(·) such that the estimate U (t)X ≤ p(U0 X ),

0 ≤ t ≤ TU ,

(12.27)

holds for any local solution U of (12.24) in the function space (12.26). Proof We put f (u) = cu − γ u2 ,

0 ≤ u < ∞.

The notation p(·) stands for some continuous increasing functions which are determined only by χ(·) and Ω and by the initial constants in the equations of (12.24). Similarly, the notation C stands for some constants which are determined in the same way as p(·). Step 1. Integration of the first equation of (12.24) in Ω gives     (c + 1)2 d 2 u dx = (cu − γ u ) dx ≤ dx. −u + dt Ω 4γ Ω Ω Solving this differential inequality (cf. (1.58)), we see that  Ω

u dx ≤ e−t u0 L1 +

(c + 1)2 |Ω|. 4γ

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429

Since u(t) ≥ 0, we conclude that u(t)L1 ≤ e−t u0 L1 +

(c + 1)2 |Ω|, 4γ

0 ≤ t < TU .

(12.28)

Furthermore, for any 0 ≤ s < t ≤ TU , we have u(t)L1 − u(s)L1 = So,

 γ s

t

 u(τ )2L2

 t s

t

dτ ≤ c s

(cu − γ u2 ) dx dτ.

Ω

u(τ )L1 dτ + u(s)L1 ;

hence, on account of (12.28), 

t s



(c + 1)2 u(τ )2L2 dτ ≤ γ −1 u0 L1 + |Ω| [c(t − s) + 1], 4γ

0 ≤ s < t ≤ TU . (12.29)

Step 2. Multiply the second equation of (12.24) by Δρ and integrate the product in Ω. Then,    1 d 2 2 |∇ρ| dx + b |Δρ| dx + d |∇ρ|2 dx 2 dt Ω Ω Ω  b ν2 uΔρ dx ≤ Δρ2L2 + u2L2 . = −ν 2 2b Ω Similarly, multiplying the second equation of (12.24) by ρ, we obtain that     1 d 2 2 2 ρ dx + b |∇ρ| dx + d ρ dx = −ν uρ dx 2 dt Ω Ω Ω Ω ≤

d ν2 ρ2L2 + u2L2 . 2 2d

Summing up these differential inequalities, we obtain that       d |∇ρ|2 + ρ 2 dx + b|Δρ|2 + 2(d + b)|∇ρ|2 + dρ 2 dx dt Ω Ω ≤ ν 2 (b + d)(bd)−1 u(t)2L2 .

(12.30)

As (12.29) holds for u(t), it is possible to solve this by applying Proposition 1.4. Consequently, we deduce that ρ(t)2H 1 ≤ e−dt ρ0 2H 1 + C(u0 L1 + 1),

0 ≤ t ≤ TU .

(12.31)

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12 Chemotaxis Models

At the same time,  t Δρ(τ )2L2 dτ ≤ C(u0 L1 + 1)[(t − s) + ρ0 H 1 + 1],

0 ≤ s < t ≤ TU .

s

Furthermore, since it holds that ρH 2 ≤ C(ΔρL2 + ρL2 ),

ρ ∈ HN2 (Ω),

due to (2.33), it is also verified that  t ρ(τ )2H 2 dτ ≤ C(u0 L1 + 1)[(t − s) + ρ0 H 1 + 1],

(12.32)

0 ≤ s < t ≤ TU .

s

(12.33)

Step 3. In this step, we use the notation p1 (U0 ) = p(u0 L1 + ρ0 H 1 ), where p(·) is some continuous increasing function. Multiply the first equation of (12.24) by log(u + 1) and integrate the product in Ω. Then, by the same calculation as for (12.20), we have   √ d [(u + 1) log(u + 1) − u] dx + 4a |∇ u + 1|2 dx dt Ω Ω   = μ [log(u + 1) − u]Δχ(ρ) dx + f (u) log(u + 1) dx. Ω

Ω

In addition, we observe that Δχ(ρ) = χ  (ρ)Δρ + χ  (ρ)|∇ρ|2 . Then, the first integral in the right-hand side can be estimated as follows. By (12.4),    [log(u + 1) − u]χ  (ρ)Δρ dx ≤ CuL2 ΔρL2 ≤ C u2L2 + Δρ2L2 , Ω

while, by (12.4), (12.31), and (12.32),  [log(u + 1) − u]χ  (ρ)|∇ρ|2 dx ≤ CuL2 ∇ρ2L4 ≤ CuL2 ρH 2 ρH 1 Ω

  ≤ CρH 1 u2L2 + ρ2H 2   ≤ p1 (U0 ) u2L2 + ρ2H 2 .

As for the second integral, it is easy to see that f (u) log(u + 1) ≤ cu2

for all 0 ≤ u < ∞.

3 Chemotaxis Model with Proliferation

431

Similarly, 1 (u + 1) log(u + 1) ≤ u2 + u for all 0 ≤ u < ∞. 2 These estimates thus yield that

  1 [f (u) + u + 1] log(u + 1) dx ≤ c + u2L2 + uL1 . 2 Ω Our inequality is then reduced to   √ d [(u + 1) log(u + 1) − u] dx + 4a |∇ u + 1|2 dx dt Ω Ω    + [(u + 1) log(u + 1) − u] ≤ p1 (U0 ) u(t)2L2 + ρ(t)2H 2 + 1 . Ω

As we already know that (12.29) and (12.33) are valid, it is possible to solve this inequality by using Proposition 1.4. We then deduce that N1,log (u(t)) ≤ e−t N1,log (u0 ) + p1 (U0 ),

0 ≤ t ≤ TU .

(12.34)

Step 4. In this step, we use the notation p2 (U0 ) = p(N1,log (u0 ) + ρ0 H 1 ). Multiply the first equation of (12.24) by u and integrate the product in Ω. Then,     1 d u2 dx + a |∇u|2 dx = μ u∇u · ∇χ(ρ) dx + f (u)u dx. 2 dt Ω Ω Ω Ω Here,  u∇u · ∇χ(ρ) dx = − Ω 

1 2

 u2 Δχ(ρ) dx, Ω



Δχ(ρ) = χ (ρ)Δρ + χ (ρ)|∇ρ|2 . Furthermore, with the aid of (1.98) (p = 2, q = 1, and r = 3), we verify that 2

1

uL3 ≤ ζ uH3 1 N1,log (u) 3 + Cζ uL1 with any ζ > 0, and, by the moment inequality (2.117), we observe that 1 2 1 2  3 2  1 1 ΔρL3 ≤ CΔρH3 1 ΔρL3 2 ≤ CA2 ρH3 1 A2 ρL3 2 ≤ C A22 ρ L3 2 A22 ρ L3 2 .

Therefore, on account of (12.4), (12.31), and (12.34),      3 2  χ  (ρ)u2 Δρ dx ≤ Cu2L3 ΔρL3 ≤ p2 (U0 ) ζ u2H 1 + A22 ρ L + Cζ . − Ω

2

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12 Chemotaxis Models

Similarly,  − Ω

4

2

χ  (ρ)u2 |∇ρ|2 dx ≤ Cu2L3 ∇ρ2L6 ≤ Cu2L3 ρH3 2 ρH3 1     3 2  ≤ p2 (U0 ) ζ u2H 1 + A22 ρ L + Cζ 2

with any ζ > 0. Thus, we obtain that    1 d 2 2 u dx + a |∇u| dx + γ u3 dx 2 dt Ω Ω Ω     3 2  ≤ p2 (U0 ) ζ u2H 1 + A22 ρ L + Cζ . 2

(12.35)

In this step, it is necessary to combine two inequalities for u and ρ. Multiply the second equation of (12.24) by A22 ρ and integrate the product in Ω. Then,    1 3  3 2 1 d |A2 ρ|2 dx + A22 ρ  dx = ν A22 u · A22 ρ dx 2 dt Ω Ω Ω 3  1 2 ≤ A22 ρ L + Cu2H 1 , 2 2 i.e., d dt



 |A2 ρ|2 dx + Ω

Ω

 32 2 A ρ  dx ≤ Cu(t)2 1 . 2 H

After multiplying a parameter 2ξ > 0 to (12.35), we add the product to this inequality to obtain that     3 2  2    d ξ u + |A2 ρ|2 dx + (2aξ − C)|∇u|2 + 2γ ξ u3 dx + A22 ρ  dx dt Ω Ω Ω     3   2 ≤ p2 (U0 )ξ ζ u2H 1 + A22 ρ L + Cζ . 2

Now, fix the parameters ξ and ζ so that 2aξ − C ≥ 1 and p2 (U0 )ξ ζ ≤ 12 . Then, we arrive at the differential inequality  3 2  1 dψ + δψ + u(t)2H 1 + A22 ρ(t)L ≤ p2 (U0 ) 2 dt 2 for ψ(t) = ξ u(t)2L2 +A2 ρ(t)2L2 with some exponent δ > 0. We can now employ Proposition 1.4 to conclude that   ξ u(t)2L2 + A2 ρ(t)2L2 ≤ e−δt ξ u0 2L2 + A2 ρ0 2H 2 + p2 (U0 ), 0 ≤ t ≤ TU , i.e.,     u(t)2L2 + ρ(t)2H 2 ≤ p2 (U0 ) e−δt u0 2L2 + ρ0 2H 2 + 1 ,

0 ≤ t ≤ TU . (12.36)

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433

As well,  t  3 2    u(τ )2H 1 + A22 ρ(τ )L dτ ≤ p2 (U0 ) (t − s) + u0 2L2 + ρ0 2H 2 + 1 , 2

s

0 ≤ s < t ≤ TU . We have in this way established the desired a priori estimate (12.27).



3.3 Global Solutions We can now apply Corollary 4.3 to conclude the global existence of solutions. As a result, for any U0 ∈ K, (12.24) possesses a unique global solution in the function space: 0 ≤ u ∈ C((0, ∞); HN2 (Ω)) ∩ C([0, ∞); L2 (Ω)) ∩ C1 ((0, ∞); L2 (Ω)), 4 (Ω)) ∩ C([0, ∞); H 2 (Ω)) ∩ C1 ((0, ∞); H 2 (Ω)). 0 ≤ ρ ∈ C((0, ∞); HN 2 N N (12.37) For U0 ∈ K, let U (t; U0 ) be the global solution of (12.24) with the initial value U0 in the function space (12.37). Recalling that Proposition 12.3 was established by four steps, let us apply (12.28) to U (t; U0 ) in the interval 0, 4t , (12.31) in the       interval 4t , 2t4 , (12.34) in the interval 2t4 , 3t4 , and (12.36) in the interval 3t4 , t , respectively. Then, we obtain the (more refined than (12.27)) estimate U (t; U0 )X ≤ p(e−δt p(U0 X ) + 1),

0 ≤ t < ∞, U0 ∈ K,

(12.38)

with some suitable exponent δ > 0.

3.4 Dynamical System Let us construct a dynamical system determined from problem (12.24) in the universal space X by following the general methods in Chap. 6, Sect. 5. Indeed, we can take β = 0. Furthermore, (12.38) shows that (6.56) is fulfilled. Consequently, it is deduced that problem (12.24) determines a dynamical system (S(t), K, X). For any exponent 0 < θ < 1, (12.24) equally determines a dynamical system in a universal space Dθ = D(Aθ ) with phase space Kθ = K ∩ Dθ . Indeed, the nonlinear operator F defined by (12.25) fulfills (4.21) and a fortiori (4.2) with β = θ . Let D Kθ,R = K ∩ B θ (0; R). Applying Theorem 4.1, we conclude that there exist τ˜R > 0 and a constant Cθ,R > 0 such that Aθ S(t)U0 X ≤ Cθ,R ,

0 ≤ t ≤ τ˜R , U0 ∈ Kθ,R .

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12 Chemotaxis Models

Meanwhile, by Proposition 6.1, Aθ S(t)U0 X ≤ Cθ,R t −θ ,

0 < t < ∞, U0 ∈ Kθ,R .

These two estimates then yield that Aθ S(t)U0 X ≤ Cθ,R ,

0 ≤ t < ∞, U0 ∈ Kθ,R .

This means that (6.56) is valid in the space Dθ , too. So, (S(t), Kθ , Dθ ) defines a dynamical system.

3.5 Exponential Attractors Let us next construct exponential attractors for (S(t), K, X). To this end, it suffices to verify (6.59) and (6.60). However, the space condition (6.59) is fulfilled in the present case. Meanwhile, (12.38) shows that the dissipative condition (6.60) is also fulfilled. Hence, (S(t), K, X) possesses a family of exponential attractors. For any exponent 0 < θ < 1, these exponential attractors are seen to keep their properties as exponential attractors in the universal space Dθ , too. Remark 12.1 As the linear operator A given by (12.7) is not symmetric, A does not satisfy (6.62). So, it is unclear whether S(t) enjoys the squeezing property (6.45)– (6.46) or not. However, if Ω is of C3 class, then one can use the shift property (Δu ∈ H 1 (Ω), together with u ∈ HN2 (Ω), implies u ∈ H 3 (Ω)) to rewrite (12.24) into the form (12.6) in an underlying space different from (12.5) in which the linear operator is a self-adjoint operator satisfying (6.62). For details, see Osaki– Tsujikawa–Yagi–Mimura [OTYM02]. In such an abstract formulation, the semigroup enjoys the squeezing property, and therefore the dimension of exponential attractor is estimated precisely by (6.50).

3.6 Numerical Examples In this subsection, we present some numerical examples. Let Ω = [−8, 8] × [−8, 8] be a quadratic domain. We consider the linear sensitivity function χ(ρ) = ρ and the growth function f (u) = u2 (1 − u). More precisely, we consider the problem ⎧ ∂u 2 ⎪ ∂t = 0.0625Δu − μ∇ · [u∇ρ] + u (1 − u) in Ω × (0, ∞), ⎪ ⎪ ⎪ ⎨ ∂ρ = Δρ − 32ρ + u in Ω × (0, ∞), ∂t ∂ρ ⎪ ∂u ⎪ = =0 on ∂Ω × (0, ∞), ⎪ ⎪ ⎩ ∂n ∂n ρ(x, 0) = ρ0 (x) in Ω. u(x, 0) = u0 (x),

(12.39)

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435

Fig. 1 Stationary patterns for μ = 6.2, μ = 7.2, and μ = 8.5

We treat μ > 0 as a control parameter. As the growth function in (12.39) is a cubic function, we cannot apply the results obtained in this section to (12.39) directly, but, as a matter of fact, we can repeat the same arguments in order to conclude for any 0 < μ < ∞ that (12.39) equally defines a dynamical system (S(t), K, X) which possesses exponential attractors, cf. Aida–Efendiev–Yagi [AEY05]. Note 1 ) whatever also that (12.39) has a homogeneous stationary solution (u, ρ) = (1, 32 0 < μ < ∞ is. 1 ), where ε(x) is a The initial functions (u0 (x), ρ0 (x)) are taken as (1 + ε(x), 32 small perturbation which vanishes outside of a small disk centered at the origin. So, (u0 , ρ0 ) is taken in a neighborhood of the stationary solution (u, ρ). When μ is small enough, the homogeneous stationary solution (u, ρ) is asymptotically stable (cf. (12.48)). The numerical solution is also observed to tend to the stationary solution (u, ρ). When μ = 6.2, the homogeneous stationary solution is no longer stable (cf. (12.49)), and the numerical solution is observed to tend to an inhomogeneous stationary solution with honeycomb structure as in Fig. 1(a). (In the figure, white indicates high concentration of biological individuals, and black oppositely shows the graph of u.) As μ increases from 6.2 to 7.2 and 8.5, the inhomogeneous stationary solution changes its types from honeycomb to stripe and perforated stripe as Fig. 1 shows. When μ still increases, such an inhomogeneous and ordered stationary solution is no longer found. That may lose its stability or completely vanish. Instead, the numerical solution is observed to tend to some moving pattern. When μ = 9.0, a moving perforated labyrinthine pattern is observed as in Fig. 2. When μ = 11.0, a chaotic spot pattern is observed as in Fig. 3. Each spot continues to move in a chaotic manner, a few spots are combined here and there in Ω; on the other hand, some new spots are generated to conserve the total number of spots at every moment. When μ becomes larger, moving patterns as in Figs. 2 and 3 disappear. Instead, the numerical solution is observed to tend to a stationary pattern again. A number of steady spots are regularly located in Ω as in Fig. 4, but these stationary patterns seem to be quite different from those in Fig. 1.

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12 Chemotaxis Models

Fig. 2 Moving perforated labyrinthine pattern for μ = 9.0

Fig. 3 Chaotic spot pattern for μ = 11.0

Fig. 4 Stationary patterns for μ = 25.0, μ = 40.0, and μ = 70.0

These numerical results seem to be very interesting when we interpret them in accordance with the analytical result that the dimension of exponential attractors of the dynamical system for (12.39) increases as the parameter μ increases (cf. (12.50)).

4 Instability of Homogeneous Stationary Solution of (12.24)

437

4 Instability of Homogeneous Stationary Solution of (12.24) We are concerned with homogeneous stationary solutions to (12.24). The homogeneous solution is given as a solution to cu − γ u2 = 0, −dρ + νu = 0,  cν  . We want to investigate the stability and instanamely, (u, ρ) = (0, 0) and γc , dγ  cν  . bility of the nonzero homogeneous stationary solution (u, ρ) = γc , dγ In this section, we make the new assumption on χ(ρ) that χ(ρ) is real analytic in a neighborhood of ρ.

(12.40)

4.1 Localized Problem From (12.40) we can extend the sensitivity function as an analytic function in a complex neighborhood |ρ − ρ| < R; furthermore, this analytic function can be extended over the whole complex plane in such a way that the extended function, denoted by χ˜ (ρ), is a C2 function with respect to the real variables (ρ  , ρ  ) ∈ R2 such that ρ = ρ  + iρ  ∈ C with uniformly bounded partial derivatives in C up to the second order. Similarly, let ϕ(u) be an extension of the function u such that ϕ(u) ≡ u in a complex neighborhood |u − u| < R and ϕ(u) is a C1 function with respect to the real variable (u , u ) ∈ R2 such that u = u + iu ∈ C with uniformly bounded partial derivatives up to the first order. Using these functions, we introduce the localized problem ⎧ ∂u 2 ⎪ ∂t = aΔu − u − μ∇ · [ϕ(u)∇ χ˜ (ρ)] + (c + 1)ϕ(u) − γ ϕ(u) ⎪ ⎪ ⎪ ⎨ ∂ρ = bΔρ − dρ + νu ∂t ∂ρ ⎪ ∂u ⎪ ∂n = ∂n =0 ⎪ ⎪ ⎩ ρ(x, 0) = ρ0 (x) u(x, 0) = u0 (x),

in Ω × (0, ∞), in Ω × (0, ∞),

on ∂Ω × (0, ∞), in Ω. (12.41) We can handle this localized problem in a quite analogous way as for (12.24). In fact, the problem is formulated as the Cauchy problem for an abstract equation in X of the form (12.6) in which the nonlinear operator F˜ : D(Aη ) → X, where 1+ε 1 2 ≤ η < 1 and 0 < ε < 2 , is given by 

−μ∇ · [ϕ(u)∇ χ˜ (ρ)] + (c + 1)ϕ(u) − γ ϕ(u)2 ˜ , F (U ) = 0

 u U= ∈ D(Aη ). ρ

Obviously, F˜ is a nonlinear operator from D(Aη ) into X. Let 0 ≤ θ < η. It is then possible to construct local solutions and global solutions to (12.41) for any initial

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12 Chemotaxis Models

value U0 ∈ D(Aθ ) in the function space: U ∈ C([0, ∞); D(Aθ )) ∩ C((0, ∞); D(A)) ∩ C 1 ((0, ∞); X).

4.2 Complexified Dynamical System ˜ Problem (12.41) then defines a nonlinear semigroup S(t) acting on Dθ = D(Aθ ), ˜ 0 ≤ θ < η. Furthermore, (12.41) defines a complexified dynamical system (S(t), Dθ , Dθ ) in Dθ . We are concerned with the case where 12 < θ < η. Since Dθ ⊂ C(Ω) for θ > 12 , any solution to (12.24) is a solution of (12.41) in a suitable neighborhood of U in Dθ . Naturally, in such a neighborhood, any trajectory of (S(t), Kθ , Dθ ) is that of ˜ ˜ Dθ , Dθ ). If U is stable (S(t), Dθ , Dθ ). In particular, U is an equilibrium of (S(t), ˜ as an equilibrium of (S(t), Dθ , Dθ ), then it is the same as that of (S(t), Kθ , Dθ ). ˜ Dθ , Dθ ), we On the other hand, even if U is unstable as an equilibrium of (S(t), cannot say that U is automatically unstable as that of (S(t), Kθ , Dθ ). However, as ˜ Dθ , Dθ ) provides us an important information before, the instability of U in (S(t), which indicates the instability even in the original system (S(t), Kθ , Dθ ), too. We can follow the general methods described in Chap. 6, Sect. 6. It is indeed sufficient to verify the differentiability conditions (6.68) and (6.69) for the nonlinear operator F˜ (U ) and the spectrum separation condition (6.76) for the linearized operator A − F˜  (U ).

4.3 Differentiability of F˜ (U ) If r > 0 is sufficiently small so that Aθ (U − U ) < r implies U − U C < R, then, θ in the open ball B D(A ) (U ; r), F˜ : D(Aη ) → X is seen to be Fréchet differentiable with the derivative

 −μ∇ · [h∇χ(ρ)] − μ∇ · {u∇[χ  (ρ)]} + (c + 1)h − 2γ uh  ˜ , F (U )H = 0



 u h η D(Aβ ) ∈ D(A ) ∩ B U= (U ; r), H = ∈ D(Aη ). (12.42) ρ  Furthermore, it is immediate to check that F˜ (U ) fulfills conditions (6.68) and (6.69).

4.4 Spectrum Separation Condition of A For A = A − F˜  (U ), let us verify condition (6.76).

4 Instability of Homogeneous Stationary Solution of (12.24)

439

Let Λ be the realization of −Δ in L2 (Ω) under the homogeneous Neumann boundary conditions. Let 0 = μ 0 < μ1 ≤ μ 2 ≤ · · · → ∞ 1

be the eigenvalues of Λ in L2 (Ω), and let φ0 = |Ω|− 2 , φ1 , φ2 , . . . , be the corresponding real-valued eigenfunctions which constitute an orthonormal basis of L2 (Ω) (cf. Corollary 2.1). Using Λ, we can characterize the space HN2 (Ω) by  ∞ ∞   2 2 2 HN (Ω) = D(Λ) = ρ = ηk φk ; (μk + 1) |ηk | < ∞ k=0

k=0

with the inner product (ρ, ρ  )H 2 = N

∞  (μk + 1)2 ηk ηk ,

ρ=

k=0

∞ 

ηk φk , ρ  =

k=0

∞ 

ηk φk .

k=0

Then, we notice that our underlying space X given by (12.5) is obtained as the infinite sum of two-dimensional orthogonal subspaces 



 φk 0 + ηk ; ξk , ηk ∈ C , k = 0, 1, 2, . . . , Xk = U = ξ k 0 φk equipped with the norm U 2Xk = |ξk |2 + (μk + 1)2 |ηk |2 . It is then obvious that Φk =

 φk 0

and Ψk = (μk + 1)−1

0 φk



are orthonormal bases of Xk . According to (12.42), we have

 c + 1 − 2γ u −μuχ  (ρ)Δ . F˜  (U ) = 0 0 Note that u =

c γ

and put χ1 = χ  (ρ) > 0. Since

 aΛ + 1 0 A= −ν bΛ + d

−c + 1  ˜ and F (U ) = 0

 μuχ1 Λ , 0

we observe that A = A − F˜  (U ) maps the subspace Xk into itself, that is, Xk is invariantunder A for every k. Consequently, the operator A can also be decomposed as A = ∞ k=0 Ak , where Ak is the part of A in Xk . Obviously, Ak Φk = (aμk + c)Φk − ν(μk + 1)Ψk ,

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12 Chemotaxis Models

Ak Ψk = −μuχ1 μk (μk + 1)−1 Φk + (bμk + d)Ψk . The transformation matrix M k of Ak with respect to the basis Φk , Ψk is then given by 

aμk + c −ν(μk + 1) . Mk = bμk + d −μuχ1 μk (μk + 1)−1 Since |λ − M k | = [λ − (aμk + c)][λ − (bμk + d)] − μνuχ1 μk , the characteristic equation of M k is written by λ2 − [(a + b)μk + c + d]λ + (aμk + c)(bμk + d) − μνuχ1 μk = 0. So, the discriminant satisfies Dk = [(a − b)μk + c − d]2 + 4μνuχ1 μk ≥ 0. That is, for every k = 0, 1, 2, . . . , the matrix M k has two real eigenvalues λk ≤ λk . Furthermore, for λ = λk , λk , (λ − M k )−1 =

1 λ − (bμk + d) (λ − λk )(λ − λk ) −μuχ1 μk (μk + 1)−1

 −ν(μk + 1) . λ − (aμk + c) (12.43)

For verifying (6.76), we here make the assumption that abμ2k + (ad + bc − μνuχ1 )μk + cd = 0 for all k = 0, 1, 2, . . . .

(12.44)

If (aμk + c)(bμk + d) − μνuχ1 μk = abμ2k + (ad + bc − μνuχ1 )μk + cd > 0, (12.45) then the eigenvalues of M k are both positive, i.e., 0 < λk ≤ λk . Furthermore, the following estimate abμ2k + (ad + bc − μνuχ1 )μk + cd < λk ≤ λk (a + b)μk + c + d < (a + b)μk + c + d −

abμ2k + (ad + bc − μνuχ1 )μk + cd (12.46) (a + b)μk + c + d

holds for the eigenvalues. In the meantime, if abμ2k + (ad + bc − μνuχ1 )μk + cd < 0,

(12.47)

then one of the eigenvalues is negative, and the other is positive, i.e., λk < 0 < λk .

4 Instability of Homogeneous Stationary Solution of (12.24)

441

Let λ ∈ iR. Consider the parts Ak on Xk . Let ψk : Xk → C2 be a natural isomorphism such that ψk (ξk Φk + ηk Ψk ) = t (ξk , ηk ) (ψk  = ψk−1  = 1); then Ak is written as Ak = ψk−1 M k ψk . Consequently, (λ − Ak )−1 = ψk−1 (λ − M k )−1 ψk . Therefore, if k are sufficiently large so that (12.45) take place, then (12.43) provides, with the aid of (12.46) and the obvious inequalities |λ − λk | ≥ λk and |λ − λk | ≥ λk , that   (λ − Ak )−1  ≤ Cλ , L(X ) k

Cλ > 0 being independent of k. This fact means that λ ∈ iR belongs to ρ(A) if and / σ (A) if and only only if λ ∈ ρ(Ak ) for every k = 0, 1, 2, . . . . In other words, λ ∈ λ∈ / σ (Ak ) = {λk , λk } for every k. We already know that, under (12.44), iR ∩ σ (Ak ) = ∅ for every k = 0, 1, 2, . . . . Hence, iR ∩ σ (A) = ∅, that is, the separation condition (6.76) is fulfilled.

4.5 Stability or Instability Conditions Assume the condition



μνuχ1 <

√ √ ad + bc.

(12.48)

Then, since √ √ abμ2 + (ad + bc)μ + cd ≥ ( ad + bc)2 μ

for all μ ≥ 0,

it follows from (12.48) that (12.45) takes place, that is, A fulfills (6.67), and U is a ˜ stable equilibrium of (S(t), Dθ , Dθ ). On the other hand, under (12.44), if the condition   N(U ) = # μk ; abμ2k + (ad + bc − μνuχ1 )μk + cd < 0 ≥ 1 (12.49) takes place, then σ (A) ∩ {λ; Re λ < 0} =  ∅; therefore, U has a nontrivial local un˜ + (U ; O) and is an unstable equilibrium of (S(t), ˜ Dθ , Dθ ). stable manifold W   For μk satisfying (12.47), let λk < 0 < λk . Since M k is a real matrix, there exists a real eigenvector uk ∈ R2 corresponding to λk . Meanwhile, ψk−1 uk is an eigenfunction of Ak corresponding to λk . Since φk is a real eigenfunction of Λ, Φk and Ψk are real valued. Therefore, A has an eigenvector for λk < 0 whose components are real-valued functions. Let us consider the instability of U as an equilibrium of the original dynamical system (S(t), Kθ , Dθ ). We already know that U is an unsta˜ + (U ; O) with ˜ ble equilibrium of (S(t), Dθ , Dθ ) having a local unstable manifold W dimension N(U ) which is tangential to U + X− at U . Furthermore, the sum X− of eigenspaces of A corresponding to the negative eigenvalues contain an orthogonal

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12 Chemotaxis Models

basis which consists of real-valued functions. These facts indicate that the local unstable manifold W+ (U ; O) for the original dynamical system is neither trivial, that is, U is an unstable equilibrium of (S(t), Kθ , Dθ ). ˜ + (U , O) = dim X− is equal to the number N (U ) It is already known that dim W given by (12.49). We easily verify that, if μν → ∞, the other parameters being all fixed, then N(U ) → ∞. In this sense, as μν → ∞, ˜ + (U ; O) = N (U ) → ∞. dim W

(12.50)

Notes and Further Researches The diffusion–advection model for chemotactic phenomenon was first introduced by Keller–Segel [KS70]. Afterward, this modeling was developed further by Nanjundiah [Nan73] and others [AC98, AL85, Den77, LK83, MM92]. More detailed considerations on the sensitivity functions were made by [FL91]. On the basis of these works, several models were presented for understanding theoretically the pattern formation by chemotactic bacteria discovered by Budrene–Berg [BB91, BB95]. The model (12.1) was introduced by Woodward–Tyson–Myerscough–Murray– Budrene–Berg [WTMB] (cf. also [Mur03, Chap. 5]). Meanwhile, the model (12.2) was presented by Mimura–Tsujikawa [MT96]. We want to quote also [KS93]. The evolutional problem for the model (12.3) (without proliferation) was first studied by Childress–Percus [CP81]. When χ(ρ) = ρ, the global existence for initial functions having small L1 norm u0 L1 was obtained by Ryu–Yagi [Yag97b, RY01]. The Lyapunov function (12.23) was constructed by Nagai–Senba–Yosida [NSY97] using techniques due to Biler–Hebish–Nadzieja [BHN94]. Recently, Feireisl–Laurençot–Petzeltová [FLP07] showed the convergence of global solutions to equilibria as t → ∞ by using the Simon and Łojasiewicz method, see [Sim83] and [Loj63, Loj65]. The blowup of solutions of (12.3) was proved by Herrero–Velázquez [HV96, HV97]. Some related results were shown by Nagai– Senba–Suzuki [NSS00] and Horstmann–Wang [HW01]. The stationary problem for (12.3) was studied by [LNT88, MW06, Sch85, SS00]. For one-dimensional problem, we cite Osaki–Yagi [OY01] and Kang–Kolokolnikov–Ward [KKW07]. Finally, we quote reviewing papers due to Horstmann [Hor03, Hor04] and the references therein. For the model (12.24) (with proliferation), a dynamical system with not only global solutions was constructed by Osaki–Tsujikawa–Yagi–Mimura [OTYM02]. Under C3 regularity of Ω, they proved the existence of exponential attractors by showing the squeezing property of the semigroup. When Ω is a convex domain, the similar results were obtained by Aida–Efendiev–Yagi [AEY05] by using the compact perturbation. The construction of a smooth unstable manifold for the homogeneous stationary solution U was done by Aida–Tsujikawa–Efendiev–Yagi– Mimura [ATEYM]. In these papers, χ(ρ) is assumed to satisfy (12.4). Aida–Osaki– Tsujikawa–Yagi–Mimura [AOTYM] constructed global solutions for (12.24) in the case where χ(ρ) has a singularity at ρ = 0. But, in such a case, it still remains to

Notes and Further Researches

443

study the asymptotic behavior of solutions. In the case where χ(ρ) = ρ 2 , even the existence of global solutions is unknown. Although various stationary solutions to (12.39) were found numerically in Sect. 3.6, little is known about the stationary problem for (12.39) and (12.24). The numerical results in Sect. 3.6 were obtained by Aida [Aid03], Aida–Yagi [AY04a], and Hai–Yagi [HY09]. The techniques described in this chapter are available to other similar chemotaxis models. For example, we can treat the more complicated model ⎧  u   s2  ∂u ⎪ in Ω × (0, ∞), ⎪ ∂t = aΔu − μ∇ · (ρ+1)2 ∇ρ + f u δ 1+s 2 − u ⎪ ⎨ ∂ρ u2 in Ω × (0, ∞), ∂t = bΔρ − uρ + νs g+u2 ⎪ ⎪ ⎪ ⎩ ∂s = cΔs − hu s 2 in Ω × (0, ∞), ∂t 1+s 2 which was presented by Murray in [Mur03, Chap. 5]. Here, u and ρ are the same as in (12.1), and s denotes the concentration of stimulant in Ω. We shall consider in the subsequent chapter a chemotaxis model arising in ecology.

Chapter 13

Termite Mound Building Model

In 1977, Deneubourg introduced a diffusion–advection model for describing the initial stage that termites build spontaneously their mound. The model consisting of three variables is written in the form ⎧ ∂u ⎪ ⎪ ∂t = aΔu − μ∇ · [u∇ρ] − cu + f ⎨ ∂v v ∂t = −dv + ν(1 − K )u ⎪ ⎪ ⎩ ∂ρ ∂t = bΔρ − gρ + ζ v

in Ω × (0, ∞), in Ω × (0, ∞), in Ω × (0, ∞).

Here, Ω ⊂ R3 is a domain where termites walk around. The function u = u(x, t) denotes the density of termites loading soils in their mouthes in Ω at time t , v = v(x, t) is the density of deposited material still active in Ω, and ρ = ρ(x, t) is the concentration of pheromone. Termites deposit the loading soils at rate ν(1 − Kv ), where K is the capacity. Pheromone is emitted from the active deposited material at rate ζ . Pheromone acts as chemoattractant for laden termites. The attraction process is described by Keller–Segel model as in Chap. 12 with sensitivity function χ(ρ) = ρ. This model was brought up by Nicolis–Prigogine in their book Self-Organization in Nonequilibrium System. In this chapter, we show that the dynamical system determined from the model possesses finite-dimensional attractors and that its unique homogeneous stationary solution, which is given by (13.2), becomes unstable if the chemotactic parameters μ and ζ are sufficiently large.

1 Model equations We consider the initial-boundary-value problem A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_13, © Springer-Verlag Berlin Heidelberg 2010

445

446

13 Termite Mound Building Model

⎧ ∂u ⎪ ⎪ ∂t = aΔu − μ∇ · [u∇ρ] − cu + f ⎪ ⎪ ⎪ ∂v v ⎪ ⎪ ⎪ ∂t = −dv + ν(1 − K )u ⎨ ∂ρ ∂t = bΔρ − gρ + ζ v ⎪ ⎪ ⎪ ∂ρ ∂u ⎪ ⎪ ∂n = ∂n = 0 ⎪ ⎪ ⎪ ⎩u(x, 0) = u (x), v(x, 0) = v (x), 0 0

in Ω × (0, ∞), in Ω × (0, ∞), in Ω × (0, ∞), on ∂Ω × (0, ∞), ρ(x, 0) = ρ0 (x)

in Ω,

(13.1) in a three-dimensional bounded domain Ω. We assume that the boundary ∂Ω of Ω is of class C3 . For u and ρ, we impose the homogeneous Neumann boundary conditions on ∂Ω. We also assume that the diffusion rates a and b of u and ρ are positive (> 0) and that the parameters c, d, f, g, μ, ν, ζ , and K are positive (> 0) constants. It is easy to see that (13.1) has the unique homogeneous stationary solution   Kf ν Kf νζ f , , . (13.2) U= c Kcd + f ν (Kcd + f ν)g We shall first prove the global existence of solutions to (13.1) and construct a dynamical system. We shall next show that the dynamical system possesses exponential attractors and that the homogeneous stationary solution U becomes unstable if μ and ζ are sufficiently large.

2 Local Solutions 2.1 Abstract Formulation Let us formulate problem (13.1) as the Cauchy problem for an abstract evolution equation. We want to handle the first equation of (13.1) on u in the L2 space. In order to rewrite (13.1) into a semilinear abstract equation, we will choose the H 1 Sobolev space for handling the third equation on ρ. It is then natural to handle the second equation on v in the same H 1 space, because the equation has no smoothing effect. So, our underlying space is set by ⎧⎛ ⎞ ⎫ ⎨ u ⎬ X = ⎝ v ⎠ ; u ∈ L2 (Ω), v ∈ H 1 (Ω), ρ ∈ H 1 (Ω) . (13.3) ⎩ ⎭ ρ We are then led to formulate (13.1) as the problem  dU 0 < t < ∞, dt + AU = F (U ), U (0) = U0 ,

(13.4)

in X. Here, A = diag {A1 , d, A2 } is a diagonal matrix operator of X, where A1 = −aΔ + c and A2 = −bΔ + g are positive definite self-adjoint operators of L2 (Ω)

2 Local Solutions

447

with domains D(A1 ) = D(A2 ) = HN2 (Ω) (see (2.34)). But, A2 must be regarded 3

1

as an operator from D(A22 ) = {ρ ∈ HN2 (Ω); Δρ ∈ H 1 (Ω)} into D(A22 ) = H 1 (Ω). Note that, as Ω is of C3 class, Theorem 2.9 ensures the shift property that Δρ ∈ ∂ρ = 0 implies that ρ ∈ H 3 (Ω). Therefore, H 1 (Ω) with ∂n    32  ∂ρ 3 3 D A2 = HN (Ω) = ρ ∈ H (Ω); = 0 on ∂Ω . ∂n Consequently, the domain of A is given by ⎧⎛ ⎞ ⎫ ⎨ u ⎬ D(A) = ⎝ v ⎠ ; u ∈ HN2 (Ω), v ∈ H 1 (Ω), ρ ∈ HN3 (Ω) . ⎩ ⎭ ρ

(13.5)

Obviously, A is a positive definite self-adjoint operator of X. The nonlinear operator F is given by ⎛ ⎞ ⎞ ⎛ u −μ∇ · [u∇ρ] + f (13.6) F (U ) = ⎝ ν(1 − Kv )u ⎠ , U = ⎝ v ⎠ ∈ D(Aη ), ζv ρ 3 4

where η is a fixed exponent such that

< η < 1.

2.2 Construction of Local Solutions Let us first characterize the domains of fractional powers Aθ for 0 < θ < 1. Since A is diagonal, it is clear that Aθ = diag {Aθ1 , d θ , Aθ2 } for 0 < θ < 1, where Aθ2 is θ+ 1

1

an operator from D(A2 2 ) to D(A22 ). According to Theorems 16.7 and 16.9 in Chap. 16, it is known that  D(Aθ1 ) = D(Aθ2 ) = H 2θ (Ω), 0 ≤ θ < 34 , D(Aθ1 ) = D(Aθ2 ) = HN2θ (Ω), 34 < θ ≤ 1. In addition, as Ω is of C3 class, it is seen that   ∂ρ D(Aθ2 ) = HN2θ (Ω) = ρ ∈ H 2θ (Ω); = 0 on ∂Ω , ∂n

3 1≤θ ≤ . 2

As a consequence, it is deduced that ⎛u⎞  D(Aθ ) = ⎝ v ⎠ ; u ∈ H 2θ (Ω), v ∈ H 1 (Ω), ρ ∈ H 2θ+1 (Ω) , ρ

1 0≤θ < , 4 (13.7)

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13 Termite Mound Building Model

⎛u⎞



D(Aθ ) = ⎝ v ⎠ ; u ∈ H 2θ (Ω), v ∈ H 1 (Ω), ρ ρ

∈ HN2θ+1 (Ω)

,

⎛u⎞  D(Aθ ) = ⎝ v ⎠ ; u ∈ HN2θ (Ω), v ∈ H 1 (Ω), ρ ∈ HN2θ+1 (Ω) , ρ

1 3 <θ < , 4 4 (13.8) 3 < θ ≤ 1. 4 (13.9)

Let us verify that the operator F given by (13.6) fulfills condition (4.2) with β = 12 and the η fixed above. Clearly, ∇ · [u∇ρ] = ∇u · ∇ρ + uΔρ. Then, by (1.76) and Theorem 1.43, we have ∇u · ∇ρL2 ≤ ∇uL2 ∇ρL∞ ≤ CuH 1 ∇ρH 2η ≤ CuH 1 ρH 2η+1 . Similarly, by (1.74), (1.86), and Theorem 1.43, uΔρL2 ≤ uL6 ΔρL3 ≤ CuH 1 Δρ

1

H2

≤ CuH 1 ρH 2η+1 .

Therefore, it holds that ∇ · [u∇ρ] − ∇ · [u∇ ˜ ρ] ˜ L2 ≤ C(uH 1 ρ − ρ ˜ H 2η+1 + ρ ˜ H 2η+1 u − u ˜ H 1 ). In the meantime, similar arguments yield that uvH 1 ≤ CuH 2η vH 1 and uv − u˜ v ˜ H 1 ≤ C(uH 2η v − v ˜ H 1 + v ˜ H 1 u − u ˜ H 2η ). In view of (13.9), this means that (4.2) is fulfilled with β = 12 . We introduce, as (4.2) is valid with β = 12 , the space of initial values K by  ⎛u ⎞ 0 K = ⎝ v0 ⎠ ; 0 ≤ u0 ∈ H 1 (Ω), v0 ∈ H 1 (Ω) with 0 ≤ v0 ≤ K, 0 ≤ ρ0 ∈ HN2 (Ω) . ρ0 (13.10) Then, by Theorem 4.1, for each U0 ∈ K, there exists a unique local solution U to (13.4) in the function space:    1   U ∈ C((0, TU0 ]; D(A)) ∩ C [0, TU0 ]; D A 2 ∩ C1 (0, TU0 ]; X ,

(13.11)

1

where TU0 > 0 is determined by the norm A 2 U0 X only. In addition,  1  1 t 2 AU (t)X + A 2 U (t)X ≤ CU0 , 1

where CU0 > 0 is also determined by A 2 U0 X only.

0 < t ≤ TU 0 ,

(13.12)

2 Local Solutions

449

2.3 Nonnegativity of Local Solutions Let U (t) be a unique local solution of (13.4) in (13.11) for U0 ∈ K. We want to show that U (t) ∈ K for every 0 < t ≤ TU0 . The local solution U is real valued. Indeed, the complex conjugate U of U is clearly a solution of (13.4) with the same initial value; then, by the uniqueness of solution, it must hold that U (t) = U (t) for every 0 < t ≤ TU0 ; hence, U (t) is a real-valued function. Let us denote U = t (u, v, ρ). We first prove that u ≥ 0 by the truncation method. 2 Let H (u) be a C1,1 cutoff function such that H (u) = u2 for −∞ < u < 0 and H (u) ≡ 0 for 0 ≤ u < ∞. Consider the function ϕ(t) = Ω H (u(t)) dx. By (1.100), ϕ is a continuously differentiable function with the derivative   ϕ (t) = H  (u){aΔu − μ∇ · [u∇ρ] − cu + f } dx. Ω

Applying (1.96) to H  (u), it is observed that    H  (u)aΔu = −a ∇H  (u) · ∇u dx = −a |∇H  (u)|2 dx. Ω

Ω

Ω

In the meantime,      − H (u)μ∇ · [u∇ρ] dx = μ u∇H (u) · ∇ρ dx = μ H  (u)∇H  (u) · ∇ρ dx Ω

Ω

μ = 2



Ω

μ ∇H  (u)2 · ∇ρ dx = − 2 Ω



H  (u)2 Δρ dx.

Ω

Since Δρ(t)L2 ≤ CU for 0 ≤ t ≤ TU0 , we have    μ H  (u)2 Δρ dx ≤ C H  (u)2 L ΔρL2 ≤ CH  (u)2L4 ΔρL2 − 2 2 Ω 3

1

≤ CU H  (u)H2 1 H  (u)L2 2 a ≤ H  (u)2H 1 + CU H  (u)2L2 . 2

 Finally, it is clear that Ω H  (u)[−cu + f ] dx ≤ 0. Hence, we proved that ϕ  (t) ≤ CU ϕ(t) and that ϕ(t) ≤ eCU t ϕ(0). Consequently, ϕ(0) = 0 implies ϕ(t) ≡ 0, i.e., u(t) ≥ 0 for 0 < t ≤ TU0 . We next verify that 0 ≤ v(t) ≤ K for 0 < t ≤ TU0 . Since u ≥ 0 is already seen, the proof for v(t) ≥ 0 is similar or even easier. In order to prove that v(t) ≤ K, we rewrite the equation of v in the form   ∂w νu =− d+ w + dK, ∂t K

450

13 Termite Mound Building Model

where w = K − v. Then, the property w0 = K − v0 ≥ 0 at t = 0 implies w(t) ≥ 0 for every 0 < t ≤ TU0 . In this way, we observe that 0 ≤ v(t) ≤ K for 0 < t ≤ TU0 . We finally have to prove that ρ(t) ≥ 0 for 0 < t ≤ TU0 . Since v ≥ 0 is already known, this is also seen by the truncation method employed above. We have thus proved that U0 ∈ K implies U (t) ∈ K for each 0 < t ≤ TU0 .

3 Global Solutions 3.1 A Priori Estimates for Local Solutions For U0 ∈ K ∩ D(A), let U (t) denote any local solution of (13.4) on [0, TU ] in the function space:  1 U ∈ C((0, TU ]; D(A)) ∩ C([0, TU ]; D(A 2 ) ∩ C1 ((0, TU ]; X), (13.13) U (t) ∈ K, 0 ≤ t ≤ TU . We intend to establish the a priori estimate for such a local solution. For this purpose, we have to employ a realization of −bΔ + g under the homogeneous Neumann boundary conditions in the space L6 (Ω). We know by Theorem 2.12 that the realization, denoted by A2,6 , is a sectorial operator of L6 (Ω) with angle < π2 and generates an analytic semigroup e−tA2,6 on L6 (Ω) which satisfies the estimate e−tA2,6 L6 ≤ Ce−δ1 t , 0 ≤ t < ∞, with some exponent δ1 > 0. Moreover, ∂ρ = 0 on ∂Ω}. Since we know by Theorem 2.15 that D(A2,6 ) = {ρ ∈ H62 (Ω); ∂n 3

HN3 (Ω) ⊂ D(A2,6 ), we see that ρ0 ∈ D(A22 ) implies ρ0 ∈ D(A2,6 ). Furthermore, for 0 < θ < 1, consider the fractional power Aθ2,6 . On account of Remark 16.6 in 

Chap. 16, we verify that D(Aθ2,6 ) ⊂ H62θ (Ω) for any 0 < θ  < θ . In particular, we 9

8

9 8 10 1 (Ω) due to (1.76). and θ  = 10 ; then, D(A2,6 ) ⊂ H65 (Ω) ⊂ H∞ take θ = 10 We can state what we want to prove now.

Proposition 13.1 There exists a continuous increasing function p(·) such that the estimate   9   9     1  A 2 U (t) + A 10 ρ(t) ≤ p A 12 U0  + A 10 ρ0  , 2,6 2,6 X L X L

0 ≤ t ≤ TU , (13.14) holds for any local solution U of (13.4) with U0 ∈ K ∩ D(A) in the function space (13.13). 6

6

Proof Throughout the proof, the notation p(·) stands for some continuous increasing functions which are determined only by Ω and by the initial constants in the equations of (13.1). Similarly, the notation C stands for some constants which are determined only by Ω and by the initial constants in (13.1).

3 Global Solutions

451

Step 1. Integrate the first equation of (13.4) in Ω. Then,   d u dx = −c u dx + f |Ω|. dt Ω Ω So,

 Ω

u dx = e−ct u0 L1 + f |Ω|



t

e−c(t−s) ds.

0

Since u(t) ≥ 0, it follows that u(t)L1 ≤ e−ct u0 L1 +

f |Ω|, c

0 ≤ t ≤ TU .

(13.15)

Step 2. In this step, let us treat the third equation of (13.4) in L6 (Ω). From (13.13) v we see that  dv dt L2 =  − dv + ν(1 − K )uL2 ≤ C, so that v(t) − v(s)L2 ≤ 1

C|t − s|. This then implies v ∈ C 3 ([0, TU ]; L6 (Ω)) because of 2

1

v(t) − v(s)L6 ≤ v(t) − v(s)L3 ∞ v(t) − v(s)L3 2 ,

0 ≤ s < t ≤ TU .

Then, thanks to Theorem 3.4, ρ(t) is written by  t e−(t−s)A2,6 ζ v(s) ds, ρ(t) = e−tA2,6 ρ0 + 0 9

10 where A2,6 is the realization of −bΔ + g in L6 (Ω) mentioned above. Operate A2,6 to the equality. Then,  t 9 9 9 10 10 10 −(t−s)A2,6 A2,6 ρ(t) = e−A2,6 t A2,6 ρ0 + A2,6 e ζ v(s) ds.

0

On account of 0 ≤ v(s) ≤ K, we have  9   109  A ρ(t) ≤ Ce−δ1 t A 10 ρ0  + C 2,6 2,6 L L 6

6



t 0

    ≤ C e−δ1 t A2,6 ρ0 L + 1 , 9 10

9

δ1

(t − s)− 10 e− 2 (t−s) v(s)L6 ds 0 ≤ t ≤ TU .

6

In particular,  −δ t  109   1 A  ρ(t)H∞ 1 ≤C e 2,6 ρ0 L + 1 , 6

0 ≤ t ≤ TU ,

with some exponent δ1 > 0. Step 3. In this step, let p1 (U0 ) denote the quantity  109    p1 (U0 ) = p u0 L1 + A2,6 ρ0 L . 6

(13.16)

452

13 Termite Mound Building Model

Multiply the first equation of (13.4) by u and integrate the product in Ω. Then,    1 d u2 dx + a |∇u|2 dx + c u2 dx 2 dt Ω Ω Ω   =μ u∇u · ∇ρ dx + f u dx Ω

Ω

  7 a ≤ ∇u2L2 + C uL3 7 + ∇ρ14 L14 + f uL1 . 4 3 Here, by (1.77), we have 28

7

uL 7 ≤ CuH351 uL351 . 3

So, 7

28

7

uL3 7 ≤ CuH151 uL151 ≤ ζ˜ u2H 1 + Cζ˜ u7L1 3

with an arbitrary number ζ˜ > 0. In the meantime, it is clear that ∇ρL14 ≤ ρH∞ 1 . We take ζ˜ > 0 small enough to ensure that    a c 1 d u2 dx + |∇u|2 dx + u2 dx ≤ p1 (U0 ). 2 dt Ω 2 Ω 2 Ω Hence, solving this inequality (see (1.57)), we conclude that u(t)2L2 ≤ e−ct u0 2L2 + p1 (U0 ),

0 < t ≤ TU .

(13.17)

In addition, by Proposition 1.5, we conclude that 

t s

  u(τ )2H 1 dτ ≤ C p1 (U0 )(t − s) + u0 2L2 ,

0 ≤ s < t ≤ TU .

(13.18)

Step 4. Differentiate the second equation of (13.4) by xi (i = 1, 2, 3). Then,     νu v ∂ Di v = − d + Di v + ν 1 − Di u. ∂t K K Multiply this equation by Di v and integrate the product in Ω. Then,        1 d νu v 2 2 |Di v| dx + |Di v| dx = (Di u)(Di v) dx d+ ν− 2 dt Ω K K Ω Ω ≤

d Di v2L2 + CDi u2L2 . 2

3 Global Solutions

453

So, it follows that d dt



 |Di v|2 dx + d Ω

Ω

|Di v|2 dx ≤ CDi u(t)2L2 .

Employing Proposition 1.4, in view of (13.18), we conclude that Di v(t)2L2 ≤ e−dt Di v0 2L2 + p1 (U0 ) + u0 2L2 ,

0 ≤ t ≤ TU .

We sum up these estimates for i = 1, 2, 3 to conclude that   v(t)2H 1 ≤ C e−dt v0 2H 1 + p1 (U0 ) + u0 2L2 , 0 ≤ t ≤ TU .

(13.19)

Step 5. In view of (13.19), let p2 (U0 ) denote p2 (U0 ) = p(v0 H 1 + u0 L2 + p1 (U0 )). Let us use the triplet H 1 (Ω) ⊂ L2 (Ω) ⊂ H 1 (Ω)∗ . We remember that A2 can be extended as an operator from H 1 (Ω) to H 1 (Ω)∗ , see Theorem 2.4. On the other hand, A2 is an operator from HN3 (Ω) to H 1 (Ω). So, A22 maps HN3 (Ω) to H 1 (Ω)∗ . We consider the duality product of the third equation of (13.4), which has a meaning in H 1 (Ω), and the vector A22 ρ(t) ∈ H 1 (Ω)∗ . Then, since   ρ1 , A22 ρ2 H 1 ×H 1∗ = A2 ρ1 , A2 ρ2 H 1 ×H 1∗ = (A2 ρ1 , A2 ρ2 )L2 we have 1 d 2 dt





 32 2 A ρ  dx = ζ

|A2 ρ|2 dx + Ω

for ρ1 ∈ HN2 (Ω), ρ2 ∈ HN3 (Ω),

Ω

2



1

Ω

3

A22 v · A22 ρ dx

1  3 2 ≤ A22 ρ L + Cv2H 1 . 2 2 On account of (13.19), we obtain that    3 2 d 2 |A2 ρ| dx + A22 ρ  dx ≤ p2 (U0 ). dt Ω Ω −1

(13.20)

3

Since A2 ρL2 ≤ A2 2 L(L2 ) A22 ρL2 , it follows that    2 d 2 |A2 ρ| dx + δ2 A2 ρ  dx ≤ p2 (U0 ), dt Ω Ω −1

where δ2 = A2 2 −1 L(L2 ) > 0. Solving this (cf. (1.58)), we conclude that ρ(t)2H 2 ≤ Ce−δ2 t ρ0 2H 2 + p2 (U0 ),

0 ≤ t ≤ TU .

(13.21)

454

13 Termite Mound Building Model

As well, by Proposition 1.5, we conclude that  t 3  2  A ρ(τ )2 dt ≤ C[p2 (U0 )(t − s) + ρ0  2 ], H 2 L

0 ≤ s < t ≤ TU .

2

s

In other words,  t   ρ(τ )2 3 dτ ≤ C[p2 (U0 )(t − s) + ρ0  2 ], H H

0 ≤ s < t ≤ TU .

s

Step 6. On account of (13.21), let us denote p3 (U0 ) = p(ρ0 H 2 + p2 (U0 )). Multiply the first equation of (13.4) by Δu and integrate the product in Ω. Then,     1 d |∇u|2 dx + a |Δu|2 dx + c |∇u|2 dx = μ ∇ · [u∇ρ]Δu dx 2 dt Ω Ω Ω Ω   a 2 ≤ |Δu| dx + C (|∇u · ∇ρ| + u|Δρ|)2 dx. 4 Ω Ω Here,  Ω

|∇u|2 |∇ρ|2 dx ≤ C∇u2L2 ∇ρ2L∞ ≤ CuH 2 uL2 ρ2H 1

∞

≤ ζ˜ u2H 2 + Cζ˜ u2L2 ρ4H 1 ∞   2 2 ≤ ζ˜ ΔuL2 + Cζ˜ uL2 ρ4H 1 + 1 , ∞

where ζ˜ > 0 is an arbitrary number. Similarly, using H (Ω) ⊂ L∞ (Ω) (due to (1.76)), we have  5 1 u2 |Δρ|2 dx ≤ u2L∞ Δρ2L2 ≤ u2 5 ρ2H 2 ≤ uH3 2 uL3 2 ρ2H 2 5 3

H3

Ω

≤ ζ˜ u2H 2 + Cζ˜ u2L2 ρ12 H2   ≤ ζ˜ Δu2L2 + Cζ˜ u2L2 ρ12 +1 , H2 where ζ˜ > 0 is an arbitrary number. Therefore, by (13.17) and (13.21), we obtain that    d |∇u|2 dx + a |Δu|2 dx + c |∇u|2 dx ≤ p3 (U0 ). (13.22) dt Ω Ω Ω Solving this (cf. (1.58)), we conclude that ∇u(t)2L2 ≤ e−ct ∇u0 2L2 + p3 (U0 ),

0 ≤ t ≤ TU .

3 Global Solutions

455

This, together with (13.17), yields that u(t)2H 1 ≤ e−ct u0 2H 1 + p3 (U0 ),

0 ≤ t ≤ TU .

As well, by Proposition 1.5,  t   Δu(τ )2L2 dτ ≤ C p3 (U0 )(t − s) + u0 2H 1 ,

(13.23)

0 ≤ s ≤ t ≤ TU .

s

By (13.16), (13.19), (13.21), and (13.23), we have obtained the desired estimate. 

3.2 Construction of Global Solutions Let U0 ∈ K. As shown in Sect. 2.2, there exists a unique local solution to (13.4) with the initial value U0 on an interval [0, TU0 ]. Let 0 < t1 < TU0 . Then, U1 = U (t1 ) ∈ D(A) ∩ K. We consider problem (13.4) but substituting t1 for the initial time 0 and U1 for the initial value U0 . Of course, the problem also possesses a unique local solution in (13.11). Since U0 ∈ D(A) ∩ K, we can apply Proposition 13.1 to any local solution of the problem with initial time t1 and initial value U1 . Then, Corollary 4.1 provides the existence of the global solution. In this way, we verify that, for any U0 ∈ K, (13.4) possesses a unique global solution in the function space:  1 U ∈ C((0, ∞); D(A)) ∩ C([0, ∞); D(A 2 )) ∩ C1 ((0, ∞); X), (13.24) U (t) ∈ K, 0 < t < ∞.

3.3 Global Norm Estimates For U0 ∈ K, let U (t; U0 ) be the global solution of (13.4) with the initial value U0 in (13.24). We observe that  1       A 2 U (t; U0 ) ≤ p e−δt p A 12 U0  + 1 , 0 ≤ t < ∞, U0 ∈ K, (13.25) X X where p(·)’s are some continuous increasing functions, and δ is a positive exponent. In fact, let us first assume that U0 ∈ D(A) ∩ K. We will apply estimates (13.15), (13.16), (13.17), (13.19), (13.21), and (13.23) established above. Applying (13.23) on [ 4t5 , t] and using (13.14), we have u(t)2H 1

  2     4t  4t  ≤ e u + p3 U 5 H 1 5    c 4t ≤ e− 5 t p(AU0 X ) + p3 U . 5 − 5c t 

456

13 Termite Mound Building Model

By the definition of p3 (·), we observe that         2  4t  4t 4t   = p ρ . p3 U + p2 U 5 5 H 2 5 Applying (13.21) on

 3t 5

 , 4t5 and using (13.14), we have

     2      4t  δ   ≤ Ce− 52 t ρ 3t  + p2 U 3t ρ   5 H 2 5 H 2 5    δ2 3t . ≤ e− 5 t p(AU0 X ) + p2 U 5 Let us estimate p2 (U ( 3t5 )). It is the same for p2 (U ( 4t5 )). By the definition of p2 (·), we observe that   2         2  3t   3t  3t    + p1 U 3t v p2 U = p u . + 5 5 L2  5 H 1 5 Applying (13.19) on

 2t 5

 , 3t5 and using (13.14), we have

  2   2   2     3t      v  ≤ e− d5 t v 2t  + u 2t  + p1 U 2t   5  1 5  1  5  5 H

H

L2

  2     2t  2t − d5 t   . p(AU0 X ) + u + p1 U ≤e 5  5 L2

In addition, applying (13.17) on

t

 2t

5, 5

, we have

  2   2     2t    u  ≤ e− 5c t u t  + p1 U t   5  5 L2 5 L2    c t ≤ e− 5 t p(AU0 X ) + p1 U . 5 (It is the same for the estimate u( 3t5 )2L2 .) Let us next estimate p1 (U ( 5t )). (It is the same for p1 (U ( 2t5 )).) By the definition of p1 (·), we observe that  9           t   10 t  t    A2,6 ρ p1 U + . = p u   5 5 L1 5 L6 Finally, we apply (13.15) and (13.16) on [0, 5t ] to obtain that     t   −ct   u  5  ≤ C e 5 AU0 X + 1 , L1

4 Dynamical System

457

 9    10  A ρ t   2,6 5 

 δ1  ≤ C e− 5 t AU0 X + 1 .

L6

We have thus verified that u(t)H 1 ≤ p(e−δt p(AU0 X ) + 1),

0 ≤ t < ∞,

with some exponent δ > 0. It is possible to estimate v(t)H 1 and ρ(t)H 2 in an analogous way. We therefore observe that   1 A 2 U (t) ≤ p(e−δt p(AU0 X ) + 1), 0 ≤ t < ∞, (13.26) X with some exponent δ > 0. Let now U0 ∈ K, and let us verify (13.25). By (13.12), there exists a time τ = 1 1 τ (A 2 U0 X ) > 0 depending only on the norm A 2 U0 X such that  1   1   1 t 2 AU (t)X + A 2 U (t)X ≤ p A 2 U0 X ,

0 < t ≤ τ.

(13.27)

For τ ≤ t < ∞, (13.26) yields that   1 A 2 U (t) ≤ p(e−δ(t−τ ) p(AU (τ )X ) + 1), X

τ ≤ t < ∞.

Hence,  1        A 2 U (t) ≤ p e−δt eδτ p τ − 12 p A 12 U0  +1 , X X

τ ≤ t < ∞.

This, together with (13.27), immediately implies (13.25).

4 Dynamical System 4.1 Construction of Dynamical System Let us construct a dynamical system determined from problem (13.4). We can entirely follow the general methods in Chap. 6, Sect. 5 by taking β = 12 . Indeed, (13.25) shows that (6.56) is fulfilled. Hence, problem (13.4) determines a dynamical 1 system (S(t), K, D 1 ), where D 1 = D(A 2 ). 2

2

For any exponent 12 < θ < 1, (13.4) equally determines a dynamical system in the universal space Dθ = D(Aθ ) with phase space Kθ = K ∩ Dθ . Indeed, the nonlinear operator F defined by (13.6) fulfills (4.2) with β = 12 and a fortiori with β = θ . D

Let Kθ,R = K ∩ B θ (0; R). Applying Theorem 4.1, we conclude that there exist τ˜R > 0 and a constant Cθ,R > 0 such that Aθ S(t)U0 X ≤ Cθ,R ,

0 ≤ t ≤ τ˜R , U0 ∈ Kθ,R .

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13 Termite Mound Building Model

Meanwhile, by Proposition 6.1, 1

Aθ S(t)U0 X ≤ Cθ,R t 2 −θ ,

0 < t < ∞, U0 ∈ Kθ,R .

These two estimates then yield that Aθ S(t)U0 X ≤ Cθ,R ,

0 ≤ t < ∞, U0 ∈ Kθ,R .

This means that (6.56) is valid in the space Dθ , too. So, (S(t), Kθ , Dθ ) is a dynamical system. Let us finally construct a dynamicalsystem in the universal space X. For any 0 < R < ∞, let KR = K 1 ,R . Let XR = 0≤t<∞ S(t)KR be an invariant set of S(t) 2 containing KR which we equip with the induced distance from X. Proposition 6.3 then provides that G : [0, ∞) × XR → X is continuous, where G(t, U0 ) = S(t)U0 . This means that (S(t), XR , X) defines a dynamical system in X.

4.2 Bounded absorbing set Consider the dynamical system (S(t), K, D 1 ) constructed above. We observe from 2 (13.25) that there exists a number C˜ > 0 for which the following assertion is valid. For any bounded subset B of K, there exists a time tB > 0 such that ˜ sup sup S(t)U0 D 1 ≤ C.

U0 ∈B t≥tB

(13.28)

2

D1

˜ is an absorbing set. This means that the closed ball B 2 (0; C) We can then follow the same arguments as in Chap. 6, Sect. 5.2 to construct an absorbing set X ⊂ K which is invariant under S(t), is a closed subset of X, and is a bounded subset of D(A). Let R > 0 be a radius large enough to satisfy X ⊂ KR ⊂ XR . As (S(t), XR , X) is already known to be a dynamical system in X, we conclude that (S(t), X, X) also defines a dynamical system in X. Furthermore, since X is an absorbing set of (S(t), K, D 1 ), the asymptotic behavior of trajectories 2 of (S(t), K, D 1 ) is reduced to that of (S(t), X, X). 2

4.3 Global Attractor We are concerned with constructing a global attractor for (S(t), X, X). We must however notice that the general methods described in Chap. 6, Sect. 5.3 are not available, because the space condition (6.59) fails now. The domain D(A) given by (13.3) is not compactly embedded in X. So, we have to construct it in a direct way.

4 Dynamical System

459

We set !

A=

(closure in X).

S(s)X

(13.29)

0≤t<∞ t≤s<∞

Since (13.4) always has a homogeneous stationary solution U given by (13.2), U belongs to A; hence, A = ∅. Let us begin with proving that A is a compact set of X. For U0 ∈ X, denote S(t)U0 = t (u(t), v(t), ρ(t)) for 0 ≤ t < ∞. Since X is a bounded subset of D(A), there is a constant C˜  > 0 such that sup

sup [u(t)H 2 + v(t)H 1 + ρ(t)H 3 ] ≤ C˜  .

U0 ∈X 0≤t<∞

(13.30)

In the meantime, by the equation on v, v must be described by v(t) = e−

t

νu(s) 0 (d+ K ) ds



t

v0 + ν

e−

t s

) (d+ νu(τ K ) dτ

u(s) ds

(13.31)

0

(cf. (1.104)). By direct calculations, it can be seen that  t  νu(s) lim e− 0 (d+ K ) ds v0 H 1 = 0

t→∞

and

 t     t ) − s (d+ νu(τ ) dτ  K u(s) ds  sup  e 

0≤t<∞

0

< ∞. H2

More strongly,  t  νu(s) lim sup e− 0 (d+ K ) ds v0 H 1 = 0,

(13.32)

t→∞ U ∈X 0

and there exists a number r˜ > 0 such that   t    ) − st (d+ νu(τ K ) dτ u(s) ds  e sup sup    U0 ∈X 0≤t<∞

0

H2

≤ r˜ .

In view of these facts, we consider the subset  ⎛ u ⎞ B = ⎝ v ⎠ ; uH 2 + vH 1 + ρH 3 ≤ C˜  and vH 2 ≤ r˜ . ρ Then, B is a compact set of X (cf. Theorem 1.38 and Proposition 6.4). In addition, (13.32) implies that " # ! lim h S(s)X, B = 0, t→∞

t≤s<∞

460

13 Termite Mound Building Model

 where h(·, ·) is the pseudo-distance of X defined by (1.1). Since A ⊂ t≤s<∞ S(s)X for every t, it follows that h(A, B) = 0. Since A is a closed subset of X and B is a compact set of X, we must have A ⊂ B. Hence, A is a compact set of X. Let us prove next that S(t)A = A for any 0 ≤ t < ∞. By definition, U ∈ A if and only if there exists a time sequence tn tending to ∞ and a sequence Un ∈ X such that S(tn )Un → U in X. So, if U ∈ A, then there exists a sequence such that S(tn )Un → U in X; then, S(t + tn )Un → S(t)U in X; hence, S(t)U ∈ A. Conversely, for any 0 < t < ∞, consider tn such that tn > t and write S(tn )Un = S(t)S(tn − t)Un . Let S(tn − t)Un = t (un (tn − t), vn (tn − t), ρn (tn − t)). By (13.31), we have the decomposition t

(un (tn − t), vn (tn − t), ρ(tn − t))     = t 0, vn1 (tn − t), 0 + t un (tn − t), vn2 (tn − t), ρ(tn − t)

such that limn→∞ t (0, vn1 (tn − t), 0) = 0 in X (due to (13.32)) and t (un (tn − t), vn2 (tn − t), ρn (tn − t)) ∈ B. This yields that there exists a subsequence S(tn − t)Un of S(tn − t)Un such that S(tn − t)Un is convergent in X; by definition, its limit V is in A; therefore, S(tn )Un = S(t)S(tn − t)Un implies U = S(t)V , i.e., U ∈ S(t)A. Let us finally prove that limt→∞ h(S(t)X, A) = 0. Suppose the contrary; then, there would exist a number δ > 0 and a time sequence tn tending to ∞ such that h(S(tn )X, A) ≥ δ. This then means that there also exists a sequence Un ∈ X such that d(S(tn )Un , A) ≥ δ. By the similar argument as above, we can show that there exists a subsequence S(tn )Un which converges to a limit V ∈ A. On the other hand, it must hold for V that d(V , A) ≥ δ. The set A has thus been shown to be a global attractor of (S(t), X, X).

4.4 Exponential Attractors This subsection is devoted to constructing exponential attractors for (S(t), X, X). As mentioned above, the general methods described in Chap. 6, Sect. 5.3 are not available. So, we have to prove that S(t) satisfies the compact perturbation condition (6.28)–(6.29) of type I in a direct way. For this purpose, we reset the space of initial functions (instead of (13.10)) by $ ˜ = t (u0 , v0 , ρ0 ); 0 ≤ u0 ∈ H 1 (Ω), v0 ∈ H 2 (Ω) with K % 0 ≤ v0 ≤ K, 0 ≤ ρ0 ∈ HN2 (Ω) . The difference from (13.10) obviously is that v0 are taken in H 2 (Ω). We then ver˜ then S(t)U0 ∈ K ˜ for every 0 ≤ t < ∞. In fact, let S(t)U0 = ify that, if U0 ∈ K, t (u(t), v(t), ρ(t)). It suffices to verify v(t) ∈ H 2 (Ω). We already know that v(t) is represented by (13.31). In addition, by Proposition 6.1, it is observed that   1  u(t)H 2 ≤ 1 + t − 2 p U0 D 1 , 2

0 < t < ∞.

(13.33)

4 Dynamical System

This implies that e−

t

461

t 0

u(s) ds ∈ H 2 (Ω). Therefore, in view of (1.93), we see that

∈ H 2 (Ω); as a consequence, v(t) ∈ H 2 (Ω) for every 0 ≤ t < ∞. Let us estimate the H 2 norm of v(t) precisely. By direct calculations, 0

u(s) ds

 t  ν Di Dj e− 0 (d+ K u) ds v0 & t '  t  ν ν − 0t (d+ Kν u) ds =e (Di u) ds (Dj u) ds v0 − Dj v0 0 K 0 K &  t '  ν − 0t (d+ Kν u) ds +e {(Di Dj u)v0 + (Dj u)(Di v0 )} ds + Di Dj v0 . − 0 K Similarly,  t  t ν e− s (d+ K u) dτ u ds Di Dj 0 t

& t ' ν ν e (Di u) dτ (Dj u) dτ u − Dj u ds = 0 s K s K ' &  t  t  t ν ν + e− s (d+ K u) dτ − {(Di Dj u)u + (Dj u)(Di u)} dτ + Di Dj u ds. 0 s K 

−

t s

(d+ Kν u) dτ



t

Hence,   t  Di Dj e− 0 (d+ Kν u(s)) ds v0 

L2

and

≤e

−dt



t 0

2 u(s)H 2 ds + 1

   t    t − s (d+ Kν u(τ )) dτ D i D j e u(s) ds    0



t

≤

e−d(t−s)

0



t

s

L2

v0 H 2 (13.34)

2

u(τ )H 2 dτ + 1

u(s)H 2 ds.

(13.35)

Hence, in view of (13.33), v(t)H 2 ≤ p(U0 D 1 + v0 H 2 ),

0 ≤ t < ∞.

2

We next construct a new phase space which is a compact set of X. Recall for (S(t), X, X) that X is a bounded subset of D(A) and that the strong estimate (13.30) is valid. On account of these facts, we can deduce from (13.34) and (13.35) that a similar dissipative condition to (13.28) is also true for H 2 norm of v(t; U0 ), pro˜ More precisely, there exists a constant C˜  > 0 such that, for any vided that U0 ∈ K. ˜ of the form U0 D + v0  2 ≤ R, ˜ it holds that bounded subset B of K H 1 2

sup sup v(t; U0 )H 2 ≤ C˜ 

U0 ∈B t≥tB

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13 Termite Mound Building Model

with a suitable time tB depending on B. In other words, the mentioned bounded ˜ v0  2 ≤ C˜  }. This fact naturally subset B is absorbed by the set X ∩ {U0 ∈ K; H leads us to introduce the phase space ˜= X

!

˜ v0  2 ≤ C˜  }) (closure in X). S(t)(X ∩ {U0 ∈ K; H

0≤t<∞

˜ v0  2 ≤ C˜  } ⊂ X, ˜ and, on the other hand, X ˜ is contained Obviously, X ∩ {U0 ∈ K; H  ˜ ˜ ˜ in X ∩ {U0 ∈ K; v0 H 2 ≤ p(C )}. Consequently, X is a compact set of X and is ˜ is invariant under S(t). In this way, we have a bounded set of D(A). Clearly, X ˜ X) whose phase space is a compact set constructed a dynamical system (S(t), X, of X. ˜ X) by showing that Our goal is to construct exponential attractors for (S(t), X, S(t) fulfills, for a sufficiently large time t ∗ > 0, condition (6.28)–(6.29). As the second Banach space Z, we take a produced space of the form % $ Z = U = t (u, v, ρ) ∈ D 1 ; v ∈ H 2η (Ω) , 2

where 34 < η < 1 is the exponent fixed in Sect. 3.1. Obviously, Z is compactly embedded in X (cf. Theorem 1.38). ˜ (resp. U˜ 0 ∈ X), ˜ and let S(t)U0 = t (u(t), v(t), ρ(t)) (resp. S(t)U˜ 0 = Let U0 ∈ X t (u(t), ˜ v(t), ˜ ρ(t))). ˜ In view of (13.31), we have ⎛ S(t)U0 = ⎝e−

t

νu(s) 0 (d+ K ) ds

⎞ u(t)  t −  t (d+ νu(τ ) ) dτ K v0 + ν 0 e s u(s) ds ⎠ , ρ(t)

˜ U0 ∈ X.

We then decompose S(t)U0 − S(t)U˜ 0 into the form ⎛ S(t)U0 − S(t)U˜ 0 = ⎝e−

t

⎞

⎛

⎞ u(t) − u(t) ˜ (v0 − v˜0 )⎠ + ⎝k1 (U0 , U˜ 0 ) + k2 (U0 , U˜ 0 )⎠ ρ(t) − ρ(t) ˜ 0

0

νu(s) 0 (d+ K ) ds

≡ S0 (U0 , U˜ 0 ) + K(U0 , U˜ 0 ). Here, t   t νu(s) ν u(s) ˜ k1 (U0 , U˜ 0 ) = e− 0 (d+ K ) ds − e− 0 (d+ K ) ds v˜0 ,  t t  −  t (d+ νu(τ ) ) dτ  ν u(τ ˜ ) K k2 (U0 , U˜ 0 ) = ν e s u(s) − e− s (d+ K ) dτ u(s) ˜ ds. 0

We then observe by using (1.90) with s = 2 that   −  t (d+ νu(s) ) ds e 0 K (v0 − v˜0 )

H1

 t ≤ Ce−dt e− 0

νu(s) K ds

 

H2

v0 − v˜0 H 1 .

5 Instability of Homogeneous Stationary Solution

463

On account of (13.30) and (13.34), we obtain that e− for a sufficiently large time t ∗ > 0, it is deduced that 1 S0 (U0 , U˜ 0 )X ≤ U0 − U˜ 0 X , 4

t 0

νu(s) K ds

H 2 ≤ Ct 2 . Hence,

˜ U0 , U˜ 0 ∈ X.

From now on, we will fix time t = t ∗ . We similarly observe by using (1.89) with s = 2η that  t∗  −  t ∗ (d+ νu(s) ) ds   ν u(s) ˜ e 0 K − e− 0 (d+ K ) ds v˜0 

H 2η

  t∗ ≤ C e − 0

νu(s) K ds

− e−

 t∗ 0

ν u(s) ˜ K ds

 

H 2η

v˜0 H 2η .

In addition, applying (1.94), we obtain that  −  t∗ e 0

νu(s) K ds

−e

−

 t∗ 0

ν u(s) ˜ K ds

 

H 2η

  ≤ C 

0

t∗

  [u(s) − u(s)] ˜ ds  

H 2η

(due to (13.30)). Finally, we use the general result, Theorem 4.3, to deduce that  t ∗   t∗     [u(s) − u(s)] ˜ ds ≤ C s −η ds U0 − U˜ 0 X ≤ CU0 − U˜ 0 X .   H 2η

0

0

We have thus shown that k1 (U0 , U˜ 0 )H 2η ≤ CU0 − U˜ 0 X .

(13.36)

It is verified in a similar way that k2 (U0 , U˜ 0 )H 2η ≤ CU0 − U˜ 0 X .

(13.37)

In the meantime, we know by Theorem 4.3 that ˜ ∗ )H 1 + ρ(t ∗ ) − ρ(t ˜ ∗ )H 2 ] ≤ CU0 − U˜ 0 X . t ∗ 2 [u(t ∗ ) − u(t 1

(13.38)

Hence, (13.36), (13.37), and (13.38) yield that K(U0 , U˜ 0 )Z ≤ CU0 − U˜ 0 X ,

˜ U0 , U˜ 0 ∈ X.

˜ is a We thus proved that S = S(t ∗ ) satisfies (6.28)–(6.29). Furthermore, since X ˜ bounded set of D(A), S(t) fulfills the Lipschitz condition (6.48) on [0, t ∗ ] × X. ˜ X) possesses exponenTherefore, by Theorem 6.15, we can conclude that (S(t), X, tial attractors.

5 Instability of Homogeneous Stationary Solution We are concerned with a homogeneous stationary solution to (13.1). Obviously, (13.1) has a unique homogeneous solution U = t (u, v, ρ) given by (13.2). We investigate the stability and instability of this equilibrium U .

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13 Termite Mound Building Model

5.1 Localized Problem We introduce cutoff functions of functions u and ρ in a complex neighborhood of u and ρ, respectively. Let ϕ(u) be a function of u = u + iu such that ϕ(u) ≡ u in a complex neighborhood of u, satisfies |ϕ(u) − u| ≤ Kd 2ν for all u ∈ C, and is a 1   2 C function with respect to the real variable (u , u ) ∈ R with uniformly bounded partial derivatives. Let χ(ρ) be a function of ρ = ρ  + iρ  such that χ(ρ) ≡ ρ in a complex neighborhood of ρ, satisfies |χ(ρ) − ρ| ≤ 1 for all ρ ∈ C, and is a C2 function with respect to the real variables (ρ  , ρ  ) ∈ R2 with uniformly bounded partial derivatives up to the second order. Consider the localized problem ⎧ ∂u in Ω × (0, ∞), ⎪ ⎪ ∂t = aΔu − μ∇ · [ϕ(u)∇χ(ρ)] − cu + f ⎪ ⎪ ⎪ ∂v v ⎪ in Ω × (0, ∞), ⎪ ⎨ ∂t = −dv + ν(1 − K )ϕ(u) ∂ρ

∂t = bΔρ − gρ + ζ v ⎪ ⎪ ⎪ ∂u ⎪ ⎪ = ∂ρ = 0 ⎪ ⎪ ⎩ ∂n ∂n u(x, 0) = u0 (x), v(x, 0) = v0 (x),

in Ω × (0, ∞),

on ∂Ω × (0, ∞), in Ω. (13.39) We can handle this localized problem in a quite analogous way as for (13.1). In fact, the problem is formulated as the Cauchy problem for an abstract equation in X of the form (13.4) in which the nonlinear operator F˜ : D(Aη ) → X, where 34 < η < 1, is given by ⎛ ⎞ ⎛ ⎞ −μ∇ · [ϕ(u)∇χ(ρ)] + f u ⎠ , U = ⎝ v ⎠ ∈ D(Aη ). ν(1 − Kv )ϕ(u) F˜ (U ) = ⎝ ζv ρ ρ(x, 0) = ρ0 (x)

Obviously, F˜ is a nonlinear operator from D(Aη ) into X. Let 12 ≤ θ < η. We can then construct local solutions and global solutions to (13.39) for all initial values U0 ∈ D(Aθ ) in the function space: U ∈ C([0, ∞); D(Aθ )) ∩ C((0, ∞); D(A)) ∩ C1 ((0, ∞); X).

5.2 Complexified Dynamical System ˜ The localized problem (13.39) then defines a nonlinear semigroup S(t) acting on 1 θ Dθ = D(A ), 2 ≤ θ < η. Furthermore, the problem defines a complexified dynam˜ ical system (S(t), Dθ , Dθ ). We are concerned with the case where 34 < θ < η. Since Dθ ⊂ C(Ω) for θ > 34 , any solution to (13.39) is a solution of (13.1) in a suitable neighborhood of U in Dθ . Naturally, in such a neighborhood, any trajectory of (S(t), Kθ , Dθ ) is that of ˜ ˜ Dθ , Dθ ). If U is stable (S(t), Dθ , Dθ ). In particular, U is an equilibrium of (S(t), ˜ as an equilibrium of (S(t), Dθ , Dθ ), then it is the same as that of (S(t), Kθ , Dθ ). On

5 Instability of Homogeneous Stationary Solution

465

˜ the other hand, even if U is unstable as an equilibrium of (S(t), Dθ , Dθ ), we cannot say that U is equally unstable as that of (S(t), Kθ , Dθ ). Nevertheless, the instabil˜ ity of U in (S(t), Dθ , Dθ ) provides us an important information which indicates its instability in the original system (S(t), Kθ , Dθ ), too. We follow the general methods described in Chap. 6, Sect. 6. It is indeed sufficient to verify the differentiability conditions (6.68) and (6.69) for the nonlinear operator F˜ (U ) and the spectrum separation condition (6.76) for the linearized operator A − F˜  (U ).

5.3 Differentiability of F˜ (U ) If r > 0 is sufficiently small in such a way that Aθ (U − U ) < r implies ϕ(u) = θ u and χ(ρ) = ρ, then, in the open ball B D(A ) (U ; r), F˜ : D(Aη ) → X is Fréchet differentiable, and the derivative is given by ⎛ ⎞ −μ{∇ · [h∇ρ] + ∇ · [u∇]} ⎠, ν(h − Kv h − Ku k) F˜  (U )H = ⎝ ζk ⎛ ⎞ ⎛ ⎞ h u θ U = ⎝ v ⎠ ∈ D(Aη ) ∩ B D(A ) (U ; r), H = ⎝ k ⎠ ∈ D(Aη ). (13.40)  ρ Furthermore, we easily verify that F˜ (U ) fulfills (6.68) and (6.69).

5.4 Spectrum Separation Condition of A For A = A − F˜  (U ), let us verify the spectrum condition (6.76). Let Λ denote the realization of −Δ in L2 (Ω) under the homogeneous Neumann boundary conditions. The operator Λ possesses denumerable nonnegative eigenvalues, and the corresponding real-valued eigenfunctions can constitute an orthonormal basis of L2 (Ω). So, let 0 = μ 0 < μ1 ≤ μ 2 ≤ · · · → ∞ 1

be eigenvalues of Λ, and let φ0 = |Ω|− 2 , φ1 , φ2 , . . . , be corresponding real eigenfunctions which constitute an orthonormal basis. Using Λ, we can characterize the space H 1 (Ω) by   ∞ ∞ ( (  1 1 2 ηk φk ; (μk + 1)|ηk | < ∞ H (Ω) = D Λ 2 = ρ = k=0

k=0

466

13 Termite Mound Building Model

with the inner product (ρ, ρ  )H 1 =

∞ ( (μk + 1)ηk ηk ,

ρ=

k=0

∞ (

ηk φk , ρ  =

k=0

∞ (

ηk φk .

k=0

Then, we notice ) that our underlying space X given by (13.3) is obtained as the infinite sum X = ∞ k=0 Xk of three-dimensional orthogonal subspaces ⎛ ⎞ ⎛ ⎞ ⎛ ⎞   φk 0 0 Xk = U = ξk ⎝ 0 ⎠ + ηk ⎝φk ⎠ + ζk ⎝ 0 ⎠ ; ξk , ηk , ζk ∈ C φk 0 0 equipped with the norm U 2Xk = |ξk |2 + (μk + 1)|ηk |2 + (μk + 1)|ζk |2 . Therefore, ⎛

⎞ φk Φk = ⎝ 0 ⎠ , 0

⎛ − 12

Ψk = (μk + 1)

⎞ 0 ⎝φk ⎠ , 0

form an orthonormal basis of Xk . According to (13.40), we have ⎛

0 F˜  (U ) = ⎝ν(1 − 0

0 −ν Ku ζ

v K)

⎛ − 12

Θk = (μk + 1)

⎞ 0 ⎝0⎠ φk

⎞ μuΛ 0 ⎠. 0

Therefore, since A = diag {aΛ + c, d, bΛ + g}, the operator A = A − F˜  (U ) maps the subspace Xk into itself, namely, Xk is invariant under ) A for every k. Consequently, the operator A can also be decomposed as A = ∞ k=0 Ak , where Ak is the part of A in Xk . Hence,   1 v Ak Φk = (aμk + c)Φk − ν(μk + 1) 2 1 − Ψk , K   νu Ψk − ζ Θk , Ak Ψk = d + K 1

Ak Θk = −μuμk (μk + 1)− 2 Φk + (bμk + g)Θk . The transformation matrix M k of Ak with respect to the basis Φk , Ψk , Θk is then given by ⎞ ⎛ 1 −ν(1 − Kv )(μk + 1) 2 0 aμk + c ⎟ ⎜ Mk = ⎝ 0 d + νu −ζ ⎠ . K 1 −μuμk (μk + 1)− 2 0 bμk + g

5 Instability of Homogeneous Stationary Solution

467

The characteristic equation of M k is given by &   '  v νu [λ − (bμk + g)] + μνζ u 1 − μk = 0. [λ − (aμk + c)] λ − d + K K We here make the assumption     v νu (aμk + c)(bμk + g) − μνζ u 1 − μk = 0 for all k = 0, 1, 2, . . . . d+ K K (13.41) According to Proposition 13.2 announced below, if     v νu d+ (13.42) (aμk + c)(bμk + g) − μνζ u 1 − μk > 0, K K then M k has three positive eigenvalues 0 < λk ≤ λk ≤ λ k . In the meantime, if     v νu (aμk + c)(bμk + g) − μνζ u 1 − μk < 0, d+ K K

(13.43)

then M k has one negative eigenvalue λk and has either two positive eigenvalues   0 < λk ≤ λ k or two complex eigenvalues λk , λk with positive real parts. Let λ ∈ iR. Let k be sufficiently large for which (13.42) holds and min{aμk + c, bμk + g} ≥ d +

νu . K

Then, by Proposition 13.2, we obtain that min {aμk + c, bμk + g} ≤ λk ≤ λ k . Therefore, applying (13.44) to M k and noting the obvious inequalities |λ − λk | ≥  λk , |λ − λk | ≥ λk and |λ − λ k | ≥ λk , we obtain that   (λ − Ak )−1 

L(Xk )

≤ Cλ ,

where Cλ > 0 is independent of all such k. This means that λ ∈ iR belongs to ρ(A) if and only if λ ∈ ρ(Ak ) for every k = 0, 1, 2, . . . . In other words, λ ∈ σ (A) if and only if λ ∈ σ (Ak ) = {λk , λk , λ k } for some k. We already know that, under (13.41), iR ∩ σ (Ak ) = ∅ for every k = 0, 1, 2, . . . . Hence, iR ∩ σ (A) = ∅, that is, the separation condition (6.76) is fulfilled. Proposition 13.2 Consider a 3 × 3 matrix ⎛ α −ν β M =⎝ 0 −μ 0

⎞ 0 −ζ ⎠ , γ

468

13 Termite Mound Building Model

where 0 < α ≤ β ≤ γ , and μ, ν, and ζ are three positive constants. Let αβγ = μνζ . Then, M has no purely imaginary eigenvalues. When μνζ < αβγ , M has three positive eigenvalues 0 < λ ≤ λ ≤ λ such that 0 < λ < α ≤ β < λ ≤ λ < γ . When μνζ > αβγ , M has one negative eigenvalue λ < 0 and has either two positive eigenvalues 0 < λ ≤ λ or two complex eigenvalues λ , λ with positive real part. For λ ∈ / σ (M), (λ − M)−1 is given by (λ − M)−1 =

1 (λ − λ )(λ − λ )(λ − λ ) ⎛ (λ − β)(λ − γ ) −ν(λ − γ ) μζ (λ − α)(λ − γ ) ×⎝ −μ(λ − β) μν

⎞ νζ −ζ (λ − α) ⎠ . (λ − α)(λ − β) (13.44)

Proof We have det (λ − M) = (λ − α)(λ − β)(λ − γ ) + μνζ = λ3 − (α + β + γ )λ2 + (αβ + βγ + γ α)λ − αβγ + μνζ. Suppose that the equation det (λ − M) = 0 has a purely imaginary solution, say, λ = iy (y ∈ R − {0}). Then, it must hold for y that  y 2 = αβ + βγ + γ α, (α + β + γ )y 2 + μνζ = αβγ . However, this is impossible by the assumptions. We rewrite the equation det (λ − M) = 0 in the form (λ − α)(λ − β)(λ − γ ) = −μνζ.

(13.45)

The function in the left-hand side is a cubic function with roots 0 < α ≤ β ≤ γ and, when λ = 0, takes the value −αβζ . Therefore, if −μνζ > −αβγ , i.e., μνζ < αβγ , then (13.45) has three positive solutions 0 < λ ≤ λ ≤ λ . In addition, it must hold that 0 < λ < α ≤ β < λ ≤ λ < γ . In the meantime, if −μνζ < −αβζ , i.e., μνζ > αβγ , (13.45) has one negative solution λ < 0 and two other solutions λ and λ . Assume that λ and λ are complex solutions x ± iy (y = 0). Then, we readily observe that  x 3 − 3xy 2 − (α + β + γ )(x 2 − y 2 ) + (αβ + βγ + γ α)x − αβγ + μνζ = 0, y 2 = 3x 2 − 2(α + β + γ )x + (αβ + βγ + γ α). Therefore, x must be a solution to the cubic equation   8x 3 − 8(α + β + γ )x 2 + 2 (α + β + γ )2 + (αβ + βγ + γ α) x − (α + β + γ )(αβ + βγ + γ α) + αβγ − μνζ = 0,

5 Instability of Homogeneous Stationary Solution

469

whence x=

8(α + β + γ )x 2 + (α + β + γ )(αβ + βγ + γ α) − αβγ + μνζ > 0. 8x 2 + 2[(α + β + γ )2 + (αβ + βγ + γ α)]

The formula (13.44) for λ ∈ σ (M) is finally verified by a direct calculation.



5.5 Stability and Instability Conditions Assume the condition



v μνζ u 1 − K





 √ νu √ < d+ ( ag + bc)2 K

(13.46)

or equivalently (due to (13.2))   , √ √ νf √ μνζf < cd 1 + ( ag + bc). Kcd Then, since

√ √ (a μ˜ + c)(bμ˜ + g) ≥ ( ag + bc)2 μ˜ for all μ˜ ≥ 0,

(13.46) shows that (13.42) holds, that is, A fulfills (6.67), and U is a stable equilib˜ rium of (S(t), Dβ , Dβ ). On the other hand, under (13.41), if the condition       v νu (aμk + c)(bμk + g) − μνζ u 1 − μk < 0 ≥ 1 N (U ) = # μk ; d + K K holds, then σ (A) ∩ {λ; Re λ < 0} =  ∅; therefore, U has a nontrivial local unstable ˜ + (U ; O) and is an unstable equilibrium of (S(t), ˜ Dθ , Dθ ). manifold W  For μk satisfying (13.43), let λk < 0. Since M k is a real matrix, there exists a real eigenvector uk ∈ R3 corresponding to λk . Meanwhile, ψk−1 uk , where ψk : Xk → C3 is a natural isomorphism such that ψk (ξk Φk + ηk Ψk + ζk Θk ) = t (ξk , ηk , ζk ) is an eigenfunction of Ak corresponding to λk . Since φk is a real eigenfunction of Λ, we have that Φk , Ψk , and Θk are real vectors. Therefore, A has an eigenfunction for λk < 0 whose components are real-valued functions. Thus, since the local unsta˜ + (U ; O) with dimension N (U ) is tangential to U + X− at U and ble manifold W since the eigenspace X− of A corresponding to negative eigenvalues contains an orthogonal basis which consist of real-valued functions, the local unstable manifold W+ (U ; O) in (S(t), Kθ , Dθ ) must be nontrivial, that is, U must be an unstable equilibrium of (S(t), Kθ , Dθ ), too. If μζ is small enough to satisfy (13.46), then U is stable. On the other hand, if μζ increases, other parameters being fixed, then N (U ) is positive and even does tend to ∞; in particular, U becomes unstable. Note that u and v are independent of μ and ζ .

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13 Termite Mound Building Model

Notes and Further Researches The diffusion–advection model (13.1) was introduced by Deneubourg [Den77]. The model was brought up by Nicoli–Prigogine in their book [NP77, Chap. 18] and also by Camazine–Deneubourg–Franks–Sneyd–Theraulaz–Bonabeau [CDFSTB, Chap. 18]. In the original model in [Den77], there is no limit on the capacity of cements in Ω, i.e., K = ∞. The condition K < ∞ obviously indicates saturation of the density of deposited soils and plays a role of negative feedback. As a matter of fact, we incorporated this limit mainly by the technical reasons in analysis. Note that v0 ≤ K implies automatically v(t) ≤ K for every t > 0, and this fact was essentially utilized in the proof of Proposition 13.1. It is unknown whether one can establish similar results without K < ∞ or not.

Chapter 14

Adsorbate-Induced Phase Transition Model

In 1990, Jakubith et al. [JREOE] and Ertl [Ert91] found out that, in the catalytic oxidation of CO molecules on a Pt(110) surface, adsorbed CO molecules and O atoms form various types of spatio-temporal patterns, such as propagating wave, spiral, target, stripe, and chaotic patterns. To understand the mechanism of these phenomena from macroscopic point of view, Hildebrand–Kuperman–Wio–Mikhailov–Ertl and Hildebrand–Ipsen–Mikhailov–Ertl presented a simple kinematic model of the surface reaction coupled to a structural phase transition of the surface, i.e., ⎧ ∂u −αχ(ρ) u − gu + h(1 − u) ⎪ ⎨ ∂t = aΔu − μ∇ · [u(1 − u)∇χ(ρ)] − f e in Ω × (0, ∞), (14.1) ⎪ ⎩ ∂ρ 1 ∂t = bΔρ + dρ(ρ + u − 1)(1 − ρ) − ν(ρ − 2 ) in Ω × (0, ∞). The two-dimensional domain Ω denotes a Pt surface on which the patterns are performed. The unknown functions u(x, t) and ρ(x, t) denote the adsorbate coverage rate of the surface by CO molecules and the structural state of surface at a position x ∈ Ω and time t , respectively. Both of them take values in the interval 0 ≤ u, ρ ≤ 1. The term dρ(ρ + u − 1)(1 − ρ) indicates that the surface structure denoted by ρ has two stable states at ρ = 0 and ρ = 1 and one unstable state at ρ = 1 − u. The nonlinear advection −μ∇ · [u(1 − u)∇χ(ρ)] shows a flow of u on Ω induced by the gradient of the local chemical potential χ(ρ) with mobility 1 − u. A typical form of χ(ρ) is such that χ(ρ) = ρ 2 (3 − 2ρ). In the growth function of u, f eα(ρ) denotes the desorption rate of the molecules depending on χ(ρ), g the desorption rate by the effect of a chemical reaction, and h denotes the adsorption rate. However, the nature of the term ν(ρ − 12 ) is rather different from others. As a matter of fact, this term has been incorporated from the view point of thermodynamics. In fact, if ν = 0, h and ρ ≡ 1, u ≡ f e−αh+g+h are homogeneous stationary sothen ρ ≡ 0, u ≡ f +g+h lutions, and both of them are stable. On the contrary, if ν > 0, ρ ≡ 0 or ρ ≡ 1 is no longer a stationary solution. Furthermore, for a suitable case, there exists no stable homogeneous stationary solution. In this chapter, we formulate the diffusion system (14.1) as a quasilinear abstract evolution equation in a suitable product space of Sobolev spaces. We construct a dynamical system determined from (14.1) and exponential attractors. A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_14, © Springer-Verlag Berlin Heidelberg 2010

471

472

14 Adsorbate-Induced Phase Transition Model

1 Model Equations We consider the initial-boundary-value problem for a diffusion system ⎧ ∂u −αχ(ρ) u − gu + h(1 − u) ⎪ ⎪ ∂t = aΔu − μ∇ · [u(1 − u)∇χ(ρ)] − f e ⎪ ⎪ ⎪ ⎪ ⎪ in Ω × (0, ∞), ⎪ ⎨ ∂ρ 1 ∂t = bΔρ + dρ(ρ + u − 1)(1 − ρ) − ν(ρ − 2 ) in Ω × (0, ∞), ⎪ ⎪ ⎪ ∂ρ ∂u ⎪ ⎪ ∂n = ∂n = 0 on ∂Ω × (0, ∞), ⎪ ⎪ ⎪ ⎩u(x, 0) = u (x), ρ(x, 0) = ρ (x) in Ω, 0 0

(14.2)

in a two-dimensional C2 or convex, bounded domain Ω. The coefficients a, b, d, f, g, h, α, and μ are positive constants (> 0), and ν is a nonnegative one (≥ 0). On u and ρ, we impose the homogeneous Neumann boundary conditions. We take initial functions which satisfy 0 ≤ u0 ≤ 1 and 0 ≤ ρ0 ≤ 1. As shown below, these conditions imply the same ones for the solution. The function χ(ρ) is given by the cubic function χ(ρ) = ρ 2 (3 − 2ρ).

(14.3)

2 Local Solutions 2.1 Abstract Formulation Let us formulate problem (14.2) as the Cauchy problem for an abstract evolution equation. It may be spontaneous to handle the first equation of (14.2) in the space L2 (Ω). Then, we are led to handle the second one in the Sobolev space H 2 (Ω). In fact, as seen in Sect. 3.1, the norms u(t)2L2 and ρ(t)H 2 are naturally combined for establishing a priori estimates of local solutions. This means that the product space   u 2 (14.4) X= ; u ∈ L2 (Ω) and ρ ∈ HN (Ω) ρ is one of suitable spaces as an underlying space. Since ρ satisfies the homogeneous Neumann boundary conditions, the space for ρ is actually taken as HN2 (Ω). In the space X, the nonlinear term ∇ · [u(1 − u)∇χ(ρ)] and others can been treated as a lower term except uρ(1 − ρ). The term uρ(1 − ρ) is a principal term, because uρ(1 − ρ) must be in HN2 (Ω), but the optimal regularity of u that one can expect as a solution of the diffusion equation is the same. By this reason, (14.2) is formulated as a quasilinear problem of the form (5.1). As for Y and Z, we set   5 1 u Y =Z= ; u ∈ H 2 (Ω) and ρ ∈ HN2 (Ω) , (14.5) ρ

2 Local Solutions

473 5

respectively, HN2 (Ω) being specified below. For each U = t (u, ρ) ∈ Z, A(U ) is defined by      A1 0 u˜ u˜ A(U )U˜ = , U˜ = ∈ D(A(U )). B(U ) A2 ρ˜ ρ˜

(14.6)

Here, A1 = −aΔ + g and A2 = −bΔ + ν are self-adjoint operators of L2 (Ω) with D(A1 ) = D(A2 ) = HN2 (Ω), but A2 is at present considered as an operator from 4 (Ω) = D(A2 ) = {ρ˜ ∈ H 2 (Ω); Δρ˜ ∈ H 2 (Ω)} to H 2 (Ω) = D(A ). (Note HN 2 2 N N N 2 4 (Ω) ⊂ that, as Ω is not assumed to be of class C4 , we cannot expect that HN 2 H 4 (Ω).) Moreover, for each U = t (u, ρ) ∈ Z, B(U ) is the linear operator given by B(U )u˜ = dρ(ρ − 1)u, ˜

u˜ ∈ HN2 (Ω).

On account of (1.89), we observe that B(U )u ˜ H 2 ≤ CρH 2 (ρH 2 + 1)u ˜ H 2 ≤ CU Z (U Z + 1)u ˜ H2. Hence, B(U ) is a bounded linear operator from HN2 (Ω) into itself, i.e., 

B(U ) ∈ L HN2 (Ω) , U ∈ Z. Then, A(U ) is a linear operator of X with the constant domain   u˜ 2 4 D(A(U )) ≡ D = ; u˜ ∈ HN (Ω) and ρ˜ ∈ HN 2 (Ω) , ρ˜

U ∈ Z.

The nonlinear operator F is given by   −μ∇ · [u(1 − u)∇χ(ρ)] − f e−αχ(ρ) u + h(1 − u) F (U ) = , −dρ(1 − ρ)2 + ν2 Here, W is the fourth space given by   u 2−ε 4−ε W= ; u ∈ HN (Ω) and ρ ∈ HN 2 (Ω) ρ

(14.7)

(14.8)

  u ∈ W. U= ρ

(14.9)

4−ε with an arbitrary fixed exponent 0 < ε < 12 , the space HN 2 (Ω) being specified below. We finally set the space of initial values by   u0 1 3 ; u0 ∈ H (Ω) and ρ0 ∈ HN (Ω) with 0 ≤ u0 ≤ 1 and 0 ≤ ρ0 ≤ 1 . K= ρ0 (14.10) In this way, problem (14.2) is rewritten as the Cauchy problem for a quasilinear abstract evolution equation of the form dU dt + A(U )U = F (U ), 0 < t < ∞, (14.11) U (0) = U0 .

474

14 Adsorbate-Induced Phase Transition Model

2.2 Basic Properties of A(U ) Let 0 < R < ∞ and put KR = {U ∈ Z; U Z < R}. We will investigate basic properties of A(U ) for U ∈ KR . For λ ∈ / (0, ∞), we see that       u˜ u f˜ ∈ D, F˜ = ˜ ∈ X, (λ − A(U ))U˜ = F˜ , U = ∈ KR , U˜ = ρ˜ ρ ζ if and only if



u˜ = (λ − A1 )−1 f˜, ρ˜ = (λ − A2 )−1 [ζ˜ + B(U )(λ − A1 )−1 f˜].

In view of (14.7), we verify that, for each U ∈ KR , A(U ) is a sectorial operator with angle ωA(U ) = 0. We next characterize the domains of fractional powers of A(U ), U ∈ KR . By Theorems 16.7 and 16.9 in Chap. 16, we know that, for 0 ≤ θ ≤ 1, D(Aθ1 ) = D(Aθ2 ) = H 2θ (Ω), 0 ≤ θ < 34 , 3 D(Aθ1 ) = D(Aθ2 ) = HN2θ (Ω) = {u ∈ H 2θ (Ω); ∂u ∂n = 0 on ∂Ω}, 4 < θ ≤ 1. Consequently, for 1 ≤ θ ≤ 2, 2θ (Ω) = {ρ ∈ H 2 (Ω); Δρ ∈ H 2(θ−1) (Ω)}, D(Aθ2 ) = HN N 2θ (Ω) = {ρ ∈ H 2 (Ω); Δρ ∈ H 2(θ−1) (Ω)}, D(Aθ2 ) = HN 2 N N We notice that



A1 A(0) = 0

0 A2

1 ≤ θ < 74 , 7 4 < θ ≤ 2.



is a positive definite self-adjoint operator of X. Then, it is deduced that, for 0 ≤ θ ≤ 1, D(A(0)θ ) = {t (u, ρ); u ∈ D(Aθ1 ) and ρ ∈ D(Aθ+1 2 )}. As a consequence,   

3 u 2(θ+1) (Ω) , 0 ≤ θ < , ; u ∈ H 2θ (Ω) and ρ ∈ HN D A(0)θ = ρ 4 

D A(0)θ =

  u 2(θ+1) ; u ∈ HN2θ (Ω) and ρ ∈ HN (Ω) , 2 ρ

(14.12) 3 < θ ≤ 1. 4 (14.13)

We in fact see that the domains of A(U )θ coincide with those of A(0)θ . Proposition 14.1 For every U ∈ Z, it holds that D(A(U )θ ) ≡ D(A(0)θ ), 0 ≤ θ ≤ 1. For each 0 < R < ∞, there exists a constant CR > 0 such that



CR−1 A(0)θ · X ≤ A(U )θ · X ≤ CR A(0)θ · X , 0 ≤ θ ≤ 1, U ∈ KR .

2 Local Solutions

475

Proof Let us apply Theorem 16.4. Of course, we can easily read the statement

A1 0  of theorem for matrix operators of the form B(U ) A2 . We already know due to (14.7) that B(U ) is a closed linear operator of L2 (Ω) into HN2 (Ω) with the domain D(B(U )) = HN2 (Ω) = D(A1 ) and is a bounded operator from D(A1 ) into HN2 (Ω) with the estimate B(U )L(D(A1 ),H 2 ) ≤ CρH 2 (ρH 2 + 1), N

U ∈ Z.

(14.14)

Let us verify that B(U )∗ is a bounded operator from D(A22 ) into L2 (Ω). Clearly, u˜ ∈ D(B(U )∗ ) if and only if the functional (u, ˜ B(U )·)H 2 is continuous with respect N

to the L2 topology. By the way, for u˜ ∈ D(A22 ), it is seen that

 ˜ A2 B(U )v)L2 = A22 u, ˜ B(U )v L , (u, ˜ B(U )v)H 2 = (A2 u, 2

N

and hence   (u, ˜ B(U )v)H 2  ≤ dρL∞ (ρL∞ + 1)A22 u ˜ L2 vL2 . N

This means that D(A22 ) ⊂ D(B(U )∗ ) and B(U )∗ is a bounded operator from D(A22 ) into L2 (Ω) with the estimate B(U )∗ L(D(A2 ),L2 ) ≤ CρL∞ (ρL∞ + 1), 2

U ∈ Z.

(14.15)

Theorem 16.4 is therefore available to conclude that D(A(U )θ ) = [X, D]θ =  D(A(0)θ ). Finally, the norm equivalence is verified by (14.14) and (14.15).

2.3 Construction of Local Solutions Let us apply Theorem 5.6 to construct local solutions to (14.11). We have to verify for any 0 < R < ∞ that all structural assumptions (5.2), (5.3), (5.4), (5.5), (5.6), (5.7) and (5.44), (5.45), (5.46), (5.47) are fulfilled by A(U ), where U ∈ KR = {U ∈ Z; U Z < R}, and by F . We already know that the angles of all A(U ) are zero. So, (5.2) is fulfilled with an arbitrarily fixed angle ω such that 0 < ω < π2 . Let U, V ∈ KR with U = t (u, ρ) and V = t (v, ζ ). We know that (5.4) is valid with ν = 1 (cf. (14.8)). Condition (5.5) is verified with Y = X. Indeed, since   AR (U ) AR (U )−1 − AR (V )−1 F˜ = [AR (V ) − AR (U )]AR (V )−1 F˜   0 0 = AR (V )−1 F˜ , B(U ) − B(V ) 0

F˜ ∈ X,

(5.5) is nothing more than ˜ H2, [ρ(ρ − 1) − ζ (ζ − 1)]u ˜ H 2 ≤ CR ρ − ζ H 2 u

u˜ ∈ HN2 (Ω).

476

14 Adsorbate-Induced Phase Transition Model

But this is also verified immediately by (1.89). We set α = 0 and β = 14 . Then, on account of (14.12), (5.6) and (5.7) are seen to hold true. Let us next verify the conditions on F . For U = t (u, ρ) ∈ W , we have ∇ · [u(1 − u)∇χ(ρ)] = ∇[u(1 − u)] · ∇χ(ρ) + u(1 − u)Δχ(ρ), (14.16) ∇χ(ρ) = χ  (ρ)∇ρ and Δχ(ρ) = χ  (ρ)|∇ρ|2 + χ  (ρ)Δρ. As χ(ρ) is given by (14.3), it is seen that ∇ · [u(1 − u)∇χ(ρ)] = 6(1 − 2u)ρ(1 − ρ)∇u · ∇ρ + 6u(1 − u)(1 − 2ρ)|∇ρ|2 + 6u(1 − u)ρ(1 − ρ)Δρ.

(14.17)

We recall the definitions (14.5) and (14.9). Then, by (1.74), (1.76), and (1.86), (1 − 2u)ρ(1 − ρ)∇u · ∇ρL2 ≤ C(uL4 + 1)ρL∞ (ρL∞ + 1)∇uL 2 ∇ρL ε

≤ C(u

1

H2

4 1−2ε

+ 1)ρH 1+ε (ρH 1+ε + 1)uH 2−ε ρ

H

3+2ε 2

≤ CU 2X (U X + 1)(U Z + 1)U W . Similarly,

u(1 − u)(1 − 2ρ)|∇ρ|2 ≤ CuL (uL + 1)(ρL + 1)∇ρ2 ∞ ∞ 4 L8 L2

 2 ≤ CuH 1+ε u 1 + 1 (ρH 1+ε + 1)ρ 7 H2

≤ CU 2X (U X

H4

+ 1)(U Z + 1)U W .

Finally, u(1 − u)ρ(1 − ρ)ΔρL2 ≤ CuL4 (uL4 + 1)ρL∞ (ρL∞ + 1)ΔρL∞ ≤ Cu

1

H2

(u

1

H2

+ 1)ρH 1+ε (ρH 1+ε + 1)ρH3+ε

≤ CU X (U X + 1)U Z (U Z + 1)U W . We have thus obtained that ∇ · [u(1 − u)∇χ(ρ)]L2 ≤ CU X (U X + 1)U Z (U Z + 1)U W . For t (u, ρ), t (v, ζ ) ∈ W , we can repeat similar calculations to estimate the L2 norm of ∇ ·[u(1−u)∇χ(ρ)]−∇ ·[v(1−v)∇χ(ζ )] using the representation (14.17). As a result, we then conclude that ∇ · [u(1 − u)∇χ(ρ)] − ∇ · [v(1 − v)∇χ(ζ )]L2

 ≤ C U 4Z + V 4Z + 1 [U − V W + (U W + V W )U − V Z ].

2 Local Solutions

477

For t (u, ρ), t (v, ζ ) ∈ W , e−αχ(ρ) u − e−αχ(ζ ) vL2 ≤ eC(ρL∞ +ζ L∞ +1) (uL2 + vL2 + 1)[u − vL2 + ρ − ζ L∞ ] 3

3

≤ eC(U X +V X +1) (U X + V X + 1)U − V X . 3

3

Similarly, by (1.88),

 ρ 2 (2 − ρ) − ζ 2 (2 − ζ )H 2 ≤ C ρ2H 2 + ζ 2H 2 + 1 ρ − ζ H 2

 ≤ C U 2X + V 2X + 1 U − V X . In this way, we verified that F fulfills (5.44) with Z and W given by (14.5) and (14.9). Thanks to Proposition 14.1, it suffices to set η = 1 − 2ε to verify (5.46) for each KR . Finally, (5.47) is clear. 1 Let us take the initial value U0 in D 1 = D(A(0) 2 ). Obviously, K is a subset of 2

D 1 . Then, U0 ∈ D 1 satisfies (5.30) with γ = 12 . By Theorem 5.6, we obtain the 2

2

following local existence of solutions. For any initial value t (u0 , ρ0 ) ∈ D 1 (⊃ K), 2 there exists a unique local solution to (14.11) in the function space: ⎧ 1 1 1 1 ⎪ 4 2 ⎪ ⎨u ∈ C([0, TU0 ]; H (Ω)) ∩ C ([0, TU0 ]; H 5(Ω)) ∩ C ((0, TU0 ]; L2 (Ω)), 3 (Ω)) ∩ C 14 ([0, T ]; H 2 (Ω)) ∩ C1 ((0, T ]; H 2 (Ω)), ρ ∈ C([0, TU0 ]; HN U0 U0 N N ⎪ ⎪ 1 ⎩t 2 u ∈ C([0, T ]; H 2 (Ω)), t 12 ρ ∈ C([0, T ]; H4 (Ω)). U0

N

U0

N2

(14.18) Here, TU0 > 0 is determined by the norm U0 D 1 = u0 H 1 + ρ0 H3 only. 2

N

2.4 Lower and Upper Bound of Local Solutions We want to prove that, if the initial functions in D 1 satisfy the conditions 0 ≤ u0 , 2 ρ0 ≤ 1, then the local solution constructed above also satisfies the same conditions. Let t (u0 , ρ0 ) ∈ D 1 satisfy 0 ≤ u0 ≤ 1 and 0 ≤ ρ0 ≤ 1. Let U = t (u, ρ) be the 2 local solution of (14.11) in (14.18). The local solution U is real valued. Indeed, the complex conjugate U of U is also a solution of (14.11) with the same initial value; then, by the uniqueness of solution, it must hold that U (t) = U (t) for every 0 < t ≤ TU0 ; hence, U (t) is real valued. Let us first prove that u ≥ 0 and ρ ≥ 0 for 0 < t ≤ TU0 by the truncation method. 2 Let H (u) be a C1,1 cutoff function such that H (u) = u2 for −∞ < u < 0 and H (u) ≡ 0 for 0 ≤ u < ∞. Consider the function ϕ(t) = Ω H (u(t)) dx. As seen by (1.100), ϕ is continuously differentiable, and its derivative is given by ϕ  (t) = (H  (u), aΔu − μ∇ · [u(1 − u)∇χ(ρ)] − f eαχ(ρ) u − gu + h(1 − u))L2 .

478

14 Adsorbate-Induced Phase Transition Model

Applying (1.96) to H  (u), it is observed that   (H  (u), aΔu)L2 = −a ∇H  (u) · ∇u dx = −a |∇H  (u)|2 dx ≤ 0. Ω

Ω

In the meantime, by (1.96) again, 



−(H (u), μ∇ · [u(1 − u)∇χ(ρ)])L2 = μ  =μ Ω

u(1 − u)∇H  (u) · ∇χ(ρ) dx

Ω

H  (u)(1 − H  (u))∇H  (u) · ∇χ(ρ) dx

  1 1   2 ∇ =μ − H (u) H (u) · ∇χ(ρ) dx 2 3 Ω    1 1  = −μ − H (u) H  (u)2 Δχ(ρ) dx. 3 Ω 2 



Since Δχ(ρ) = χ  (ρ)|∇ρ|2 + χ  (ρ)Δρ, (14.18) yields that Δχ(ρ)L2 ≤ CU for 0 ≤ t ≤ TU0 . By (1.96), (14.18) yields also that ∇H  (u)L2 ≤ ∇uL2 ≤ CU . Therefore,

 −(H  (u), μ∇ · [u(1 − u)∇χ(ρ)]) ≤ C H  (u)2L4 + H  (u)3L6 ΔρL2

 ≤ CU H  (u)H 1 H  (u)L2 + H  (u)2H 1 H  (u)L2 a ≤ CU H  (u)H 1 H  (u)L2 ≤ H  (u)2H 1 + CU H  (u)2L2 . 2 Finally, (H  (u), −f eαχ(ρ) u − gu + h(1 − u))L2 ≤ (H  (u), −[f eαχ(ρ) + g + h]u)L2 ≤ CU H  (u)2L2 . Hence, we have proved that ϕ  (t) ≤ CU H  (u(t))2L2 ≤ CU ϕ(t). Consequently, ϕ(t) ≤ eCU t ϕ(0) and ϕ(0) = 0 implies ϕ(t) ≡ 0, i.e., u(t) ≥ 0 for 0 < t ≤ TU0 . Consider next the function ψ(t) = Ω H (ρ(t)) dx. The derivative of this function is given by    1  . ψ (t) = H (ρ), bΔρ + dρ(ρ + u − 1)(1 − ρ) − ν ρ − 2 L2 

By the same reason, 

(H (ρ), bΔρ)L2 = −b

 Ω

|∇H  (ρ)|2 dx ≤ 0.

3 Global Solutions

479

In the meantime, (14.18) yields that    1  H (ρ), dρ(ρ + u − 1)(1 − ρ) − ν ρ − 2 L2 ≤ (H  (ρ), [d(ρ + u − 1)(1 − ρ) − ν]ρ)L2

 b ≤ CU uL2 H  (ρ)2L4 + H  (ρ)2L2 ≤ H  (ρ)2H 1 + CU H  (ρ)2L2 . 2 Therefore, ψ  (t) ≤ CU H  (ρ)2L2 ≤ CU ψ(t). Hence, ρ(t) ≥ 0 for 0 < t ≤ TU0 . Let us prove now that u(t) ≤ 1 and ρ(t) ≤ 1 for every 0 < t ≤ TU0 . For this purpose, we introduce two functions w(t) = 1 − u(t) and ξ(t) = 1 − ρ(t) defined on the interval [0, TU0 ]. The equations of (14.2) are then rewritten in terms of w and ξ by ⎧ ∂w αχ(1−ξ ) (1 − w) ⎪ ⎨ ∂t = aΔw + μ∇ · [w(1 − w)∇χ(1 − ξ )] + f e ⎪ ⎩ ∂ξ ∂t

+ g(1 − w) − hw,

= bΔξ + dξ(ξ + w − 1)(1 − ξ ) − ν(ξ − 12 ).

We can repeat the same arguments as above to deduce from w0 ≥ 0 and ξ0 ≥ 0 that w(t) ≥ 0 and ξ(t) ≥ 0 for every 0 < t ≤ TU0 . We have thus shown that, for any U0 = t (u0 , ρ0 ) ∈ K, (14.2) possesses a unique local solution on [0, TU0 ] in (14.18) such that t (u(t), ρ(t)) ∈ K for every 0 < t ≤ TU 0 .

3 Global Solutions 3.1 A Priori Estimates Let U0 = t (u0 , ρ0 ) be an initial value in K. Let U = t (u, ρ) be any local solution to (14.11) on an interval [0, TU ] in the function space: u ∈ C((0, TU ]; HN2 (Ω)) ∩ C([0, TU ]; H 1 (Ω)) ∩ C1 ((0, TU ]; L2 (Ω)), 4 (Ω)) ∩ C([0, T ]; H3 (Ω)) ∩ C1 ((0, T ]; H 2 (Ω)), ρ ∈ C((0, TU ]; HN U U 2 N N (14.19) and let u and ρ satisfy 0 ≤ u(t) ≤ 1 and 0 ≤ ρ(t) ≤ 1,

0 ≤ t ≤ TU .

(14.20)

We then show the following a priori estimates. Proposition 14.2 There exists a continuous increasing function p(·) such that, for any local solution U of (14.2) in (14.19) with (14.20), the estimate u(t)H 1 + ρ(t)H3 ≤ p(u0 H 1 + ρ0 H3 ), holds.

0 ≤ t ≤ TU ,

(14.21)

480

14 Adsorbate-Induced Phase Transition Model

Proof This estimate will be established by a few steps. Throughout the proof, we shall use universal notation δ and p(·) to denote positive exponents and continuous increasing functions determined by the constants a, b, d, f, g, h, α, μ, and ν and by Ω. They may therefore change from occurrence to occurrence. We use also the notation P (u, ρ) = −f eαχ(ρ) + h(1 − u), ν Q(u, ρ) = dρ(u + ρ − 1)(1 − ρ) + . 2 Step 1. Multiply the second equation of (14.2) by Δρ and integrate the product in Ω. Then, noting (14.20), we have     1 d 2 2 2 |∇ρ| dx + b |Δρ| dx + ν |∇ρ| dx = − Q(u, ρ)Δρ dx 2 dt Ω Ω Ω Ω b ≤ Δρ2L2 + C, 2 i.e., d dt



 |∇ρ| dx + b Ω

 |Δρ| dx + 2ν

2

|∇ρ|2 dx ≤ C.

2

Ω

Ω

Therefore (cf. (1.58)), we conclude that ∇ρ(t)2L2 ≤ e−2νt ρ0 2H 1 + C. Moreover, in view of (14.20),   ρ(t)2H 1 ≤ C e−2νt ρ0 2H 1 + 1 ,

0 ≤ t ≤ TU .

(14.22)

Step 2. Multiply the first equation of (14.2) by u and integrate the product in Ω. Then,    1 d 2 2 u dx + a |∇u| dx + g u2 dx 2 dt Ω Ω Ω   =μ u(1 − u)∇χ(ρ) · ∇u dx + P (u, ρ)u dx Ω

 ≤C

Ω

  a (|∇u||∇ρ| + 1) dx ≤ ∇u2L2 + C ∇ρ2L2 + 1 . 2 Ω

On account of (14.22), we obtain that      d 2 2 u dx + a |∇u| dx + 2g u2 dx ≤ C ρ0 2H 1 + 1 . dt Ω Ω Ω

(14.23)

3 Global Solutions

481

In the meantime, multiply the second equation of (14.2) by Δ2 ρ and integrate the product in Ω. Then, 1 d 2 dt







|Δρ|2 dx + b Ω

Ω



=−

|∇Δρ|2 dx + ν

|Δρ|2 dx Ω

  b ∇Q(u, ρ) · ∇Δρ dx ≤ ∇Δρ2L2 + C ∇u2L2 + ∇ρ2L2 . 2 Ω

On account of (14.22), we obtain that d dt



 |Δρ| dx +b

 |∇Δρ| dx +2ν

2

Ω

2

Ω

Ω

  |Δρ|2 dx ≤ C∇u2L2 +C ρ0 2H 1 +1 .

Multiply a parameter ξ > 0 to the differential inequality (14.23) and add the product to this inequality to obtain that d dt





 ξ u2 + |Δρ|2 dx +

Ω



+ Ω





 (aξ − C)|∇u|2 + b|∇Δρ|2 dx

Ω



   2gξ u2 + 2ν|Δρ|2 dx ≤ Cξ ρ0 2H 1 + 1 .

Fix the parameter ξ such that aξ ≥ C + 1. Solving this differential inequality (cf. (1.58)), we conclude that  

 u(t)2L2 + ρ(t)2H 2 ≤ C e−δt u0 2L2 + ρ0 2H 2 + ρ0 2H 1 + 1   ≤ C e−δt ρ0 2H 2 + ρ0 2H 1 + 1 , 0 ≤ t ≤ TU . (14.24) Step 3. Multiply the first equation of (14.2) by Δu and integrate the product in Ω. Then, after some estimation,    1 d 2 2 |∇u| dx + a |Δu| dx + g |∇u|2 dx 2 dt Ω Ω Ω     a ≤ |∇ · [u(1 − u)∇χ(ρ)]|2 + P (u, ρ)2 dx. |Δu|2 dx + C 2 Ω Ω As (14.20) is the case, we observe from (14.16) that 

|∇ · [u(1 − u)∇χ(ρ)]|2 ≤ C |Δρ|2 + |∇ρ|4 + |∇u|2 |∇ρ|2 . In addition, by (14.24), 



 Δρ2L2 + ∇ρ4L4 ≤ C ρ2H 2 + ρ2H 2 ρ2H 1 ≤ C ρ0 4H 2 + 1 ,

(14.25)

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14 Adsorbate-Induced Phase Transition Model

and by (2.33), 4

2

(|∇u||∇ρ|)2L2 ≤ C∇u2L3 ∇ρ2L6 ≤ CuH3 2 uL3 2 ∇ρ2H 1

 ≤ ζ Δu2L2 + Cζ ρ0 6H 2 + 1

(14.26)

with any ζ > 0. Therefore, we obtain that    d a |∇u|2 dx + |Δu|2 dx + 2g |∇u|2 dx ≤ p(ρ0 H 2 ). dt Ω 2 Ω Ω

(14.27)

In the meantime, we observe in view of (5.4) that     ∂ d 2 |∇Δρ| dx = −2 Δρ Δ2 ρ dx. dt Ω Ω ∂t After operating Δ to the second equation of (14.2), take the product of the second equation and Δ2 ρ and integrate it in Ω. Then, it follows that    1 d 2 2 2 |∇Δρ| dx + b |Δ ρ| dx + ν |∇Δρ|2 dx 2 dt Ω Ω Ω    b =− ΔQ(u, ρ)Δ2 ρ dx ≤ |Δ2 ρ|2 dx + C |ΔQ(u, ρ)|2 dx. 2 Ω Ω Ω By a direct calculation, we observe that ΔQ(u, ρ) = 2Quρ ∇u · ∇ρ + Qρρ |∇ρ|2 + Qu Δu + Qρ Δρ. Therefore, on account of (5.71), it follows that 

|ΔQ(u, ρ)|2 ≤ C |Δu|2 + |Δρ|2 + |∇u|2 |∇ρ|2 + |∇ρ|4 . Here we can use estimates (14.25) and (14.26) to arrive at    d |∇Δρ|2 dx +b |Δ2 ρ|2 dx +2ν |∇Δρ|2 dx ≤ ζ Δu2L2 +Cζ p(ρ0 H 2 ) dt Ω Ω Ω with any ζ > 0. Fixing the parameter ζ as ζ = a4 , we sum up this differential inequality and (14.27). Then,    

 a d 2 2 2 2 2 |∇u| + |∇Δρ| dx + |Δu| + b|Δ ρ| dx dt Ω Ω 4 

 2g|∇u|2 + 2ν|∇Δρ|2 dx ≤ p(ρ0 H 2 ). + Ω

Solving this (cf. (1.58)), we conclude that

 u(t)2H 1 + ρ(t)2H3 ≤ Ce−δt u0 2H 1 + ρ0 2H3 + p(ρ0 H 2 ),

0 ≤ t ≤ TU , (14.28)

4 Dynamical System

483

with some exponent δ > 0. We have thus established the desired estimate (14.21).



3.2 Global Solutions We can now appeal to Corollary 5.2 in order to conclude the global existence of the solution. In fact, for any U0 ∈ K, problem (14.2) possesses a unique global solution in the function space: u ∈ C((0, ∞); HN2 (Ω)) ∩ C([0, ∞); H 1 (Ω)) ∩ C1 ((0, ∞); L2 (Ω)), 4 (Ω)) ∩ C([0, ∞); H3 (Ω)) ∩ C1 ((0, ∞); H 2 (Ω)), ρ ∈ C((0, ∞); HN 2 N N (14.29) with the properties 0 ≤ u(t) ≤ 1 and 0 ≤ ρ(t) ≤ 1 for every 0 ≤ t < ∞.

(14.30)

3.3 Global Norm Estimates For U0 ∈ K, let U (t; U0 ) be the global solution of (14.2) in (14.29). Then, U (·; U0 ) satisfies the global norm estimate  



U (t; U0 )2D 1 ≤ p e−δt p U0 D 1 + 1 , 0 ≤ t < ∞, U0 ∈ K, (14.31) 2

2

with a suitable exponent δ > 0 and some continuous increasing functions p(·).   This estimate is indeed obtained by applying (14.22) to U (·; U0 ) on the interval 0, 3t ,     applying (14.24) on the interval 3t , 2t3 , and applying (14.28) on the interval 2t3 , t .

4 Dynamical System 4.1 Construction of Dynamical System Let us verify that problem (14.2) defines a dynamical system. We can entirely follow the general methods described in Chap. 6, Sect. 7. We already know that, for any U0 ∈ K, (14.2) possesses a unique global solution U (t; U0 ) in (14.29) with (14.30). By S(t)U0 = U (t; U0 ) we define a nonlinear semigroup S(t) acting on K. We set the underlying space X = Y by (14.4), the space Z by (14.5), and the space W by (14.9). We also set γ = 12 , i.e., Dγ = D 1 . The space of initial values K is given by (14.10), 2 and this K becomes a phase space of a dynamical system. The basic assumptions (6.87), (6.88), (6.89), (6.91), (6.92), and (6.93) have already been verified in Sect. 1. The global estimate (6.97) is verified by (14.31). We hence deduce that problem (14.2) defines a dynamical system (S(t), K, D 1 ). 2

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14 Adsorbate-Induced Phase Transition Model

4.2 Exponential Attractors For constructing a family of exponential attractors, it is sufficient to verify conditions (6.103) and (6.104). In the present case, (6.103) is obvious. In addition, (14.31) shows that the dissipative condition (6.104) is also fulfilled. Hence, (S(t), K, D 1 ) 2 possesses exponential attractors. Remark 14.1 As problem (14.2) is rewritten as a quasilinear equation of the form (14.11), it is unclear whether S(t) enjoys the squeezing property (6.45)–(6.46) or not. However, if Ω is of C3 class, then one can use the shift property that Δu ∈ 3 H 1 (Ω) with ∂u ∂n = 0 on ∂Ω implies u ∈ H (Ω) in order to formulate (14.2) as a semilinear abstract equation of the form (6.53) in which A is a self-adjoint operator satisfying (6.62). As shown in Tsujikawa–Yagi [TY02], such a formulation ensures the squeezing property of S(t), and therefore the dimension of exponential attractor can be estimated precisely by (6.50).

5 Homogeneous Stationary Solutions h When ν = 0, (u, ρ) = ( f +g+h , 0) and (u, ρ) = (hf e−α + g + h, 1) are homogeneous stationary solutions of (14.11). It is possible by the methods presented in Chap. 6, Sect. 8.2 to verify that they are always stable equilibria of (S(t), K, D 1 ). 2 When ν > 0, however, neither ρ = 0 nor ρ = 1 is a stationary solution of the diffusion equation for ρ in (14.2). In this case, the homogeneous stationary solution (u, ρ) is characterized as a solution to the system of equations ⎧ h ⎨u = −αχ(ρ) , fe +g+h (14.32) ⎩u = 1 − ρ + ν(ρ− 12 ) , dρ(1−ρ)

lying in the region 0 < u < 1 and 0 < ρ < 1. In view of this fact, we consider two functions ⎧ h ⎨u = U1 (ρ) ≡ −αχ(ρ) , fe +g+h ⎩u = U (ρ) ≡ 1 − ρ + 2

ν(ρ− 12 )

(14.33)

0 < ρ < 1,

dρ(1−ρ) ,

0 < ρ < 1.

It is easy to see that limρ→0 U2 (ρ) = −∞ and limρ→1 U2 (ρ) = ∞. As U1 (ρ) is continuous on the closed interval [0, 1] with 0 < U2 (ρ) < 1, this means that (14.32) has always at least one solution in the region (14.33). We can observe the following fact. Proposition 14.3 Let

ν d

≥

12 5e

+ 14 . Then, (14.32) has exactly one solution in (14.33).

Notes and Further Researches

485

Proof We have −ρ 2 (1 − ρ)2 − dν ρ(1 − ρ) + dU2 (ρ) = dρ ρ 2 (1 − ρ)2

ν 2d

≥

ν 1 d − 4 , 4ρ 2 (1 − ρ)2

0 < ρ < 1.

(14.34) Therefore, U2 (ρ) is increasing for 0 < ρ < 1. Denote by ρ 0 the unique value of ρ such that U2 (ρ) = 0. Similarly, denote by ρ 1 the unique value of ρ such that U2 (ρ) = 1. In order that (14.32) has a solution in (14.33), ρ must therefore lie in the interval [ρ 0 , ρ 1 ]. In the meantime, since dν > 14 + 12 > 9 , we see that 14 < ρ 0 < 12

5e1 3 16 1 3 and 2 < ρ 1 < 4 . Hence, ρ must lie in the interval 4 , 4 . In the meantime, we notice that f hαχ  (ρ)e−αχ(ρ) 3f hαe−αχ(ρ) 3αe−α(ρ) dU1 ≤ < (ρ) = , 2 −αχ(ρ) 2 dρ 2 2(f + g + h) [f e + g + h] Since 32 αe−αχ(ρ) ≤

48 5e

for

1 4

0 < ρ < 1.

< ρ < 34 , it is seen that

48 dU2 (ρ) < dρ 5e

for

3 1 <ρ< . 4 4

Therefore, from (14.34) it follows that   ν 1 1 3 48 dU1 dU2 (ρ) ≥ 4 − ≥ > (ρ) for < ρ < . dρ d 4 5e dρ 4 4 This means that two functions u = U1 (ρ) and u = U2 (ρ) have at most one intersecting point for 14 < ρ < 34 . As we already know that the two functions have at least one intersecting point, we have proved the proposition.  1 In the case where 0 < dν < 12 5e + 4 , the number of solutions of (14.32) in (14.33) can change in various ways depending on the other parameters f, g, h, and α. As shown by Proposition 14.3, if dν is sufficiently large, then (14.2) has a unique homogeneous stationary solution. Furthermore, at least by numerical computations, see Takei [Take04] and Takei–Tsujikawa–Yagi [TTY05], we can verify that the unique homogeneous stationary solution becomes unstable if the advection rate μ is sufficiently large.

Notes and Further Researches The model (14.1) was presented by the papers Hildebrand–Kuperman–Wio– Mikhailov–Ertl [HKWME] and Hildebrand–Ipsen–Mikhailov–Ertl [HIME03]. Especially, for the term ν(ρ − 12 ), we refer to Hildebrand [Hil99]. When Ω ⊂ R2 is a bounded domain of class C3 , one can formulate (14.2) as a semilinear abstract problem of the form (4.1). In this case, Tsujikawa–Yagi [TY02] constructed a dynamical system and also exponential attractors by verifying the

486

14 Adsorbate-Induced Phase Transition Model

squeezing property of a semigroup. On the contrary, when Ω is merely of class C2 , it is impossible to formulate (14.2) as a semilinear problem. We have to rewrite it rather as a quasilinear problem. When Ω ⊂ R2 is of class C2 or convex, Takei– Efendiev–Tsujikawa–Yagi [TETY06] constructed a dynamical system and a family of exponential attractors by showing that the semigroup is a compact perturbation of a contraction operator. The results announced in Sect. 4 were obtained in [TETY06]. So far, there is no literature on the analytical study for stationary problems of (14.2). In the papers Takei [Take04] and Takei–Tsujikawa–Yagi [TTY05], the authors presented numerical results on (14.2) which announce that various types of pattern solutions are found out and that some of them indicate very good accordance with the experimental results due to Jakubith–Rotermund–Engel–Oertzen–Ertl [JREOE] and Ertl [Ert91].

Chapter 15

Lotka–Volterra Competition Model with Cross-Diffusion

Consider the Lotka–Volterra competition model  du 2 dt = cu − γ11 u − γ12 uv for 0 < t < ∞, dv dt

= dv − γ21 uv − γ22 v 2

for 0 < t < ∞,

(15.1)

for two species, say A and B, where u = u(t) denotes the total population of A, and v = v(t) the total population of B, respectively, at time t. Let the conditions dγ12 > cγ22

and cγ21 > dγ11

(15.2)

be satisfied. We note that these conditions imply γ11 γ22 < γ12 γ21 , that is, interspecies competition is stronger than intraspecies competition. Then the equations  u(c − γ11 u − γ12 v) = 0, v(d − γ21 u − γ22 v) = 0 have four nonnegative solutions O = (0, 0),

U u = (c/γ11 , 0),

U v = (0, d/γ22 ),

U = (u, v), (15.3)

where u = (cγ22 − dγ12 )/D and v = (dγ11 − cγ21 )/D with D = γ11 γ22 − γ12 γ21 < 0 (due to (15.2)). The solutions O, U u , and U v are extinct solutions; meanwhile, U is a coexistent solution. By the theory ordinary equations, it is easy under (15.2) to see that the coexistent solution U is always unstable. This means that in the case where interspecies competition dominates intraspecies one, it is impossible to explain the coexistence of the species A and B by the ordinary system (15.1). Consider then the competition model with diffusions  ∂u 2 ∂t = aΔu + cu − γ11 u − γ12 uv in Ω × (0, ∞), (15.4) ∂v 2 in Ω × (0, ∞), ∂t = bΔv + dv − γ21 uv − γ22 v where u = u(x, t) denotes the density of A in a bounded domain Ω ⊂ R2 and v = v(x, t) that of B at time t, respectively. Concerning this system, Kishimoto and A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_15, © Springer-Verlag Berlin Heidelberg 2010

487

488

15 Lotka–Volterra Competition Model with Cross-Diffusion

Weinberger have showed that, when Ω is convex, any nonhomogeneous stationary solutions are unstable. However, as explained above, the homogeneous stationary solution U cannot be stable under (15.2). This means that in such a domain Ω the simple Lotka–Volterra system (15.4) with diffusions on species can neither explain the coexistence of A and B. We have to pay attention, however, that in some nonconvex domain, Matano–Mimura succeeded in constructing a coexistent stable stationary solution to (15.4) even under (15.2). In this chapter, we intend to consider the Lotka–Volterra competition model with cross-diffusions which was presented by Shigesada–Kawasaki–Teramoto in 1979. The model equations are given by  ∂u 2 2 ∂t = Δ(au + α11 u + α12 uv) + cu − γ11 u − γ12 uv in Ω × (0, ∞), (15.5) ∂v 2 2 in Ω × (0, ∞), ∂t = Δ(bv + α21 uv + α22 v ) + dv − γ21 uv − γ22 v where α11 and α22 are self-diffusion rates of A and B, respectively, and α12 and α21 are cross-diffusion rates acting between A and B. Under suitable conditions on αij (cf. (15.15)), the global existence for (15.5) will be shown, and a dynamical system will be constructed from (15.5). Moreover, the existence of exponential attractors will be shown for the dynamical system. However, proving analytically the existence of coexistent stable solutions still remains as an open problem.

1 Model Equations Consider the initial-boundary-value problem ⎧ ∂u 2 2 ⎪ ⎪ ∂t = Δ(au + α11 u + α12 uv) + cu − γ11 u − γ12 uv ⎪ ⎪ ⎪ ⎨ ∂v = Δ(bv + α21 uv + α22 v 2 ) + dv − γ21 uv − γ22 v 2 ∂t ∂u ∂v ⎪ ⎪ ⎪ ∂n = ∂n = 0 ⎪ ⎪ ⎩u(x, 0) = u (x), v(x, 0) = v (x) 0

0

in Ω × (0, ∞), in Ω × (0, ∞), on ∂Ω × (0, ∞), in Ω,

(15.6) in a two-dimensional bounded C3 domain Ω. Here, u = u(x, t) and v = v(x, t) denote the densities of two competing species A and B, respectively, in Ω at time t. The coefficients γii > 0 (i = 1, 2) denote the intraspecies competition rates in A and B, respectively, and γij > 0 (i = j ) denote the interspecies competition rates between A and B. The terms Δ(α11 u2 ) and Δ(α22 v 2 ) denote the self-diffusions of A and B, respectively, with rates αii ≥ 0 (i = 1, 2), and Δ(α12 uv) and Δ(α21 uv) denote the cross-diffusions between A and B with rates αij ≥ 0 (i = j ). We assume also that a, b, c, and d are positive constants (>0). The space of initial functions is set by

 u0 1+ε 1+ε (Ω) and 0 ≤ v0 ∈ H (Ω) , (15.7) K= ; 0 ≤ u0 ∈ H v0 where ε denotes an arbitrary fixed exponent such that 0 < ε < 12 .

2 Local Solutions

489

Remark 15.1 The C3 regularity of the domain Ω will be used only in building up a priori estimates for local solutions of (15.6). Indeed, we are able to obtain the local existence of solutions under C2 regularity or convexity of Ω only.

2 Local Solutions 2.1 Abstract Formulation Let us treat (15.6) in the product space X = L2 (Ω) (see (1.84)). Problem (15.6) is written as the Cauchy problem for an abstract equation  dU dt

+ A(U )U = F (U ),

0 < t < ∞,

U (0) = U0 ,

(15.8)

in X. Here, A(U ) is a linear operator defined for U ∈ Z = H1+ε1 (Ω) (see (1.85)), where 0 < ε1 < ε. For U = t (u, v) ∈ Z, let us consider the sesquilinear form  a(U ; U1 , U2 ) =

{[a + 2α11 χ(Re u) + α12 χ(Re v)]∇u1 · ∇u2 Ω

+ α12 χ(Re u)∇v1 · ∇u2 + α21 χ(Re v)∇u1 · ∇v 2 + [b + α21 χ(Re u) + 2α22 χ(Re v)]∇v1 · ∇v 2 + k( U Z )(u1 u2 + v1 v 2 )} dx on H1 (Ω), where U1 = t (u1 , v1 ), U2 = t (u2 , v2 ) ∈ H1 (Ω). Here, χ(u) is a smooth cutoff function such that χ(u) = u for u ≥ 0 and χ(u) = −δ for u ≤ −δ, where δ > 0 is a small positive constant satisfying (2α11 + α12 )δ ≤

a 4

b and (α21 + 2α22 )δ ≤ , 4

(15.9)

and k(r) > 0 is some smooth increasing function for r ≥ 0. We then observe that the coefficients in the sesquilinear form satisfy (2.46) and (2.48), for (due to (15.9)) it is observed that [a + 2α11 χ(Re u)][b + 2α22 χ(Re v)] + [a + 2α11 χ(Re u)]α21 χ(Re u) + [b + 2α22 χ(Re v)]α12 χ(Re v) ≥

32 ab ab − (aα21 + bα12 )δ ≥ . 2 16 4

By virtue of Theorem 2.11, the sesquilinear form a(U ; ·, ·) defines sectorial operators A(U )|H1 , A(U )|L2 , and A(U ) with respect to the triplet H1 (Ω) ⊂ L2 (Ω) ⊂

490

15 Lotka–Volterra Competition Model with Cross-Diffusion

H1 (Ω)∗ . We then define A(U ) = A(U )|L2 . In view of (2.55), A(U )U1 = k( U Z )U1  ∇ · [a + 2α11 χ(Re u) + α12 χ(Re v)]∇u1 + ∇ · [α12 χ(Re u)]∇v1 − . ∇ · [α21 χ(Re v)]∇u1 + ∇ · [b + α21 χ(Re u) + 2α22 χ(Re v)]∇v1 The nonlinear operator F (U ) : Z → X is given by   u cu − γ11 u2 − γ12 uv F (U ) = k( U Z ) + , v dv − γ21 uv − γ22 v 2

U=

 u ∈ Z. v

The initial value U0 is taken in K which was set by (15.7) or more generally in the product space H1+ε (Ω) (⊃ K).

2.2 Basic Properties of Operators A(U ) Let 0 < R < ∞, and let KR = {U ∈ Z; U Z < R} be an open ball of Z. Proposition 15.1 For each 0 < R < ∞, A(U ), where U ∈ KR , is a sectorial operator of X with angle ωA(U ) ≤ ωR < π2 and satisfies (2.93) with some constant MR , provided that k(·) is chosen appropriately. Proof We notice that KR is a relatively compact subset of C(Ω). So, when U varies in KR , the quartet a + 2α11 χ(Re u) + α12 χ(Re v), α12 χ(Re u),

α21 χ(Re v), b + α21 χ(Re u) + 2α22 χ(Re v) varies in a compact subset of C(Ω)4 satisfying conditions (2.46) and (2.48). Therefore, the desired result is obtained by Theorem 2.11.  For s > 32 , we put



HsN (Ω) =

t

(u, v) ∈ Hs (Ω);

∂u ∂v = = 0 on ∂Ω . ∂n ∂n

Proposition 15.2 For any U ∈ Z, we have D(A(U )) = H2N (Ω). For each 0 < R < ∞, there is a constant CR > 0 such that CR−1 U˜ H2 ≤ A(U )U˜ X ≤ CR U˜ H2 , U˜ ∈ D(A(U )), hold for all U ∈ KR . Proof On account of (1.93), χ(Re u) and χ(Re v) vary in a bounded subset of H 1+ε1 (Ω). Then, it is possible to argue as for the proof of Theorem 2.8. On account of (2.54), A(U ) is written in the form A(U ) = P (U )[(Λ − 1)I + B(U )],

2 Local Solutions

491

where Λ is a realization of −Δ + 1 in L2 (Ω), I is the identity matrix, P (U ) is the matrix  a + 2α11 χ(Re u) + α12 χ(Re v) α12 χ(Re u) , P (U ) = b + α21 χ(Re u) + 2α22 χ(Re v) α21 χ(Re v) and B(U ) is the matrix operator given by B(U ) = k( U Z )P (U )−1 − P (U )−1  [2α11 ∇χ(Re u) + α12 ∇χ(Re v)] · ∇ α12 ∇χ(Re u) · ∇ × . [α21 ∇χ(Re u) + 2α22 ∇χ(Re v)] · ∇ α21 ∇χ(Re v) · ∇ It is clear that ΛI is a positive definite self-adjoint operator of L2 (Ω) with domain H2N (Ω). In the meantime, we see by (1.93) that ˜ L2 ≤ C ∇u L2/(1−ε1 ) ∇ u ˜ L2/ε1 ∇χ(Re u) · ∇ u ˜ L2 = χ (Re u)∇(Re u) · ∇ u ≤ C ∇u H ε1 ∇ u ˜ H 1−ε1 ≤ C u H 1+ε1 Λ1−ε1 /2 u ˜ L2 , u˜ ∈ HN2 (Ω). This means that −I + B(U ) is dominated by Λ1−ε1 /2 I in the sense of (3.8) (ν = ε1 /2), and Theorem 3.3 yields that (Λ − 1)I + B(U ) is the generator of an analytic semigroup on X with domain D((Λ− 1)I + B(U )) = D(ΛI ) = H2N (Ω). Therefore, if U˜ ∈ D(A(U )), then there exists a vector U˜ 1 ∈ H2N (Ω) such that A(U )U˜ = P (U )[(Λ − 1)I + B(U )]U˜ 1 = A(U )U˜ 1 . Hence, U˜ = U˜ 1 . In this way, we conclude that D(A(U )) = H2N (Ω). The second assertion of proposition is verified from the relative compactness of KR in the space C(Ω) (cf. Theorem 2.11).  Proposition 15.3 For any U ∈ Z, we have

3 D A(U )θ = H2θ (Ω) for 0 ≤ θ < , 4

3 < θ ≤ 1. D A(U )θ = H2θ N (Ω) for 4 Let 0 ≤ θ ≤ 1, θ = 34 . For each 0 < R < ∞, there is a constant CR,θ > 0 such that   −1 ˜ CR,θ U H2θ ≤ A(U )θ U˜ X ≤ CR,θ U˜ H2θ ,

U˜ ∈ D A(U )θ ,

hold for all U ∈ KR . Proof We recall Lemma 2.1. Then, for each U ∈ Z, there exists a C1 diagonal matrix D(U ) for which the diagonal elements of D(U )P (U )D(U )−1 dominate others in

492

15 Lotka–Volterra Competition Model with Cross-Diffusion

the sense of (2.49). Then, by (2.51), we have (λ − A(U ))−1 = D(U )(λ − A(U )D )−1 D(U )−1 , where A(U )D is a sectorial operator of L2 (Ω) associated with a suitable sesquilinear form on H1 (Ω). We apply Theorem 2.30 to A(U )D π to verify that A(U )D has bounded purely imaginary powers [A(U )D ]iy L2 ≤ e 2 |y| , y ∈ R, on L2 (Ω). Then, since A(U )iy = D(U )[A(U )Dπ]iy D(U )−1 , A(U ) also has bounded purely imaginary powers A(U )iy L2 ≤ Me 2 |y| on L2 (Ω). We next use Theorem 16.3 in Chap. 16 to conclude that D(A(U )θ ) = [L2 (Ω), H2N (Ω)]θ for any 0 ≤ θ ≤ 1. Furthermore, by Theorems 16.7 and 16.9, we verify that [L2 (Ω), H2N (Ω)]θ = H2θ (Ω) 3 for 0 ≤ θ < 34 and [L2 (Ω), H2N (Ω)]θ = H2θ N (Ω) for 4 < θ ≤ 1. Hence, the first assertion has been verified. The second assertion is verified by the relative compactness of KR in the space C(Ω) (cf. Lemma 2.1, Theorem 2.11, and Proposition 15.2). 

2.3 Construction of Local Solutions Let us construct local solutions to problem (15.8) by applying the general results obtained in Chap. 5, Sect. 3. Let 0 < R < ∞, and let A(U ) be defined for U ∈ KR . It is already known that A(U ) is a sectorial operator of X with angle ωA(U ) ≤ ωR < π2 , that is, (5.2) and (5.3) are fulfilled. Since D(A(U )) = H2N (Ω) for U ∈ KR , (5.4) is also fulfilled with ν = 1. In view of (3.68) it is sufficient for verifying (5.5) to show that [A(U ) − A(V )]U˜ X ≤ CR U − V Y U˜ H2 ,

U˜ ∈ H2N (Ω).

(15.10)

But, this can be done by a direct calculation with space Y = H1+ε0 (Ω), where ε0 is a third exponent chosen so that 0 < ε0 < ε1 < ε (cf. (5.96)). By Proposition 15.3, 1+ε1 0 (5.6) is also fulfilled with α = 1+ε 2 and β = 2 . Finally, (5.7) is obvious. The 1+ε nonlinear operator F is defined on W = H 2 (Ω), where ε < ε2 < 12 . It is easy to see that (5.44) is fulfilled. Conditions (5.46) and (5.47) are also verified, since 2 D(A(U )η ) = W for U ∈ KR with η = 1+ε 2 . We are now ready to apply Theorem 5.6 to (15.8) (with G ≡ 0). In fact, U0 ∈ KR satisfies (5.30) if γ is taken as γ = 1+ε 2 . Thus, for any U0 ∈ KR , problem (15.8) restricted in KR possesses a unique local solution in the function space:

U ∈ C (0, TU0 ]; H2N (Ω) ∩ C([0, TU0 ]; H1+ε (Ω)) ∩ C1 ((0, TU0 ]; L2 (Ω)), (15.11) where TU0 > 0 is determined by the norm U0 H1+ε . By (5.66), A(U )U belongs to the space dU , dt

A(U )U ∈ F

with any 0 < σ < min{ε1 − ε0 , 1−ε 2 }.

1+ε 2 ,σ

((0, TU0 ]; L2 (Ω))

(15.12)

2 Local Solutions

493

2.4 Some Regularity Properties We shall verify some regularity property of local solutions in the variable t which will be necessary for establishing their a priori estimates. Our goal is to prove that, for U0 ∈ KR , the local solution constructed above enjoys the regularity U ∈ C1 ((0, TU0 ]; H2θ (Ω)) ∩ C2 ((0, TU0 ]; H1 (Ω)∗ ),

0 ≤ θ < 1.

To prove these, we regard U (t) as a solution of the linear problem  dU τ < t ≤ TU 0 , dt + A(t)U = F (t), U (τ ) = Uτ ,

(15.13)

(15.14)

in X, where τ > 0 is arbitrarily fixed time, and Uτ is the initial value Uτ = U (τ ) ∈ H2N (Ω). Here, the linear operator A(t) is given by A(t) = A(U (t)), and the function F (t) is given by F (t) = F (U (t)). Then, it is clear from (15.11) that Uτ ∈ D(A(τ )) and F (t) ∈ H2N (Ω) for all t ∈ [τ, TU0 ] with AF ∈ C([τ, TU0 ]; L2 (Ω))

and

F ∈ C0,1 ([τ, TU0 ]; L2 (Ω)).

In addition, from (3.68) and (15.10) we know that    A(t) A(t)−1 − A(s)−1  ≤ CU U (t) − U (s) Y , 0 L 2

τ ≤ s, t ≤ TU0 .

We notice that U (t) − U (s) L2 ≤ CU0 |t − s|,

τ ≤ s, t ≤ TU0 .

This, together with the property Y = [L2 (Ω), H2 (Ω)] 1+ε0 , yields that 2

1+ε0

1−ε0

U (t) − U (s) Y ≤ C U (t) − U (s) H22 U (t) − U (s) L22 ≤ CU0 |t − s|

1−ε0 2

.

0 Let us here apply Theorem 3.11 to problem (15.14) with μ = 1−ε 2 , ν = 1, an ex0 ponent σ such that 1−ε 2 < σ < β = 1, and any ρ1 such that 0 < ρ1 < min{μ + ν − 1−ε0 1, σ } = 2 . We then conclude that A(t)ρ1 dU dt ∈ B([τ, TU0 ]; X), and hence

U (t) − U (s) H2ρ1 ≤ CU0 |t − s|,

τ ≤ s, t ≤ TU0 .

Assume that 0 = ρ0 < ρ1 < · · · < ρk are defined such that U (t) − U (s) H2ρk ≤ CU0 |t − s|,

τ ≤ s, t ≤ TU0 .

2ρk (Ω) ⊂ H1+ε0 (Ω) ⊂ H2 (Ω), 0 In addition, assume that ρk < 1+ε 2 . Then, since H we can repeat a similar argument to conclude that the same estimate holds for any

494

15 Lotka–Volterra Competition Model with Cross-Diffusion

ρk+1 such that ρk < ρk+1 < integer K, that is,

1−ε0 2(1−ρk ) . It is then possible to have ρK

U (t) − U (s) Y ≤ C U (t) − U (s) H2ρK ≤ CU0 |t − s|,

≥

1+ε0 2

by a finite

τ ≤ s, t ≤ TU0 .

We can apply again Theorem 3.11 to conclude that, for any 0 ≤ θ < 1, A(t)θ dU dt ∈ B([τ, TU0 ]; X); hence, the first assertion of (15.13) has been established. To prove the second assertion of (15.13), we consider the function A(U (t)) × U (t) − F (U (t)) defined on (0, TU0 ] with values in H1 (Ω)∗ . It is now immediate to observe that the function is continuously differentiable in t .

2.5 Nonnegativity of Local Solutions Let U0 ∈ KR ∩ K. For the local solution U = t (u, v) constructed above in (15.11) and (15.12), we shall prove that u(t) ≥ 0 and v(t) ≥ 0 for every 0 < t ≤ TU0 . Let us first verify that U (t) is real valued. Indeed, the complex conjugate U (t) of U (t) is also a local solution of (15.8) with the same initial value. So, the uniqueness of solution implies that U (t) = U (t); hence, U (t) is real valued. Let H (u) be a C1,1 cutoff function such that H (u) = 12 u2 for −∞ < u < 0 and H (u) ≡ 0 for 0 ≤ u < ∞. By (1.100), we know that the function ψ(t) = Ω H (u(t)) dx is continuously differentiable with the derivative 

ψ (t) = H (u)∇ · {[a + 2α11 χ(u) + α12 χ(v)]∇u} dx Ω



+

H (u)∇ · {α12 χ(u)∇v} dx +

Ω



H (u)(cu − γ11 u2 − γ12 uv) dx.

Ω

By property (1.96), we have  H (u)∇ · {[a + 2α11 χ(u) + α12 χ(v)]∇u} dx Ω



=−

[a + 2α11 χ(u) + α12 χ(v)]|∇H (u)|2 dx ≤ −

Ω

Similarly,  

H (u)∇ · {α12 χ(u)∇v}dx = −

Ω,u=0

Ω

3a 4



|∇H (u)|2 dx. Ω

α12 χ(u)∇H (u) · ∇v dx

≤ C H (u) Lp ∇H (u) L2 ∇v Lq 2(1− p1 )

≤ C H (u) H 1 ≤

2

H (u) Lp 2 ∇v Lq

a p H (u) 2H 1 + C H (u) 2L2 ∇v Lq , 2

3 A Priori Estimates for Local Solutions

495

where 2 < p, q < ∞ are any numbers satisfying p1 + q1 = 12 . In addition, 

|H (u)| |u|2 + |uv| dx ≤ ( u L∞ + v L∞ ) H (u)u L1 ≤ CU0 H (u) 2L2 . Ω

Therefore, we conclude that ψ (t) ≤ CU0 [ ∇v(t) Lq + 1]ψ(t); consequently, p

C

t

p

[ ∇v(s)

+1] ds

Lq ψ(t) ≤ ψ(0)e U0 0 for 0 < t ≤ TU0 . In the meantime, we observe p p from (15.12) that ∇v(s) Lq ≤ Cq v(s) H 2 ≤ Cq CU0 s p(ε−1)/2 . As p > 2 can be

p

arbitrarily close to 2, it is possible to take a suitable p so that ∇v(·) Lq is integrable in (0, TU0 ). Hence, ψ(0) = 0 implies ψ(t) ≡ 0, i.e., u(t) ≥ 0 for every 0 < t ≤ TU0 . It is the same for the proof of v(t) ≥ 0 for every 0 < t ≤ TU0 .

3 A Priori Estimates for Local Solutions We want to establish a priori estimates for local solutions of (15.8), but, so far, we can do this only in some particular cases. In [Yag08b], a priori estimates were shown under the condition 0 ≤ α12 α21 ≤ 64 α11 α22 .

(15.15)

In this chapter, we are concerned with the case that the two self-diffusions vanish, i.e., α11 = α22 = 0. In the viewpoint of segregation of two competing species, such a case may be more interesting. Condition (15.15) then implies α12 α21 = 0. So, we will first establish a priori estimates in the case where α11 = α22 = α21 = 0 and α12 = α > 0.

(15.16)

When α11 = α22 = 0, the restriction α12 α21 = 0 is not indispensable. Indeed, we will secondly establish a priori estimates in the following particular case where a = b,

α12 = α21 = α > 0,

c = d,

γ12 = γ21 = γ ,

α11 = α22 = 0

and

γ11 = γ22 = μ.

(15.17)

For U0 ∈ K ∩ H2 (Ω), let U denote a local solution of (15.8) on [0, TU ] in the function space:

0 ≤ U ∈ C (0, TU ]; H2N (Ω) ∩ C([0, TU ]; H1+ε (Ω)) ∩ C1 ((0, TU ]; H1 (Ω)) ∩ C2 ((0, TU ]; H1 (Ω)∗ ).

(15.18)

3.1 Case where (15.16) is Valid Proposition 15.4 Let (15.16) be satisfied. There exists a continuous increasing function p(·) such that, for any local solution U of (15.8) in (15.18) with initial

496

15 Lotka–Volterra Competition Model with Cross-Diffusion

value U0 ∈ K ∩ H2 (Ω), it holds that U (t) H2 ≤ p( U0 H2 ),

0 ≤ t ≤ TU .

(15.19)

Proof In this proof, we shall use the notation P (u, v) = au + αuv,

Q(u, v) = bv,

f (u, v) = cu − γ11 u − γ12 uv,

g(u, v) = dv − γ21 uv − γ22 v 2 .

2

Step 1. Integrate the first equation of (15.6) in Ω. Then,    d u dx = f (u, v) dx ≤ (cu − γ11 u2 ) dx. dt Ω Ω Ω

(15.20)

Since cu − γ11 u2 ≤ −u +

(c + 1)2 4γ11

for all u ≥ 0,

we conclude (cf. (1.58)) that u(t) L1 ≤ e−t u0 L1 +

(c + 1)2 |Ω|, 4γ11

0 ≤ t ≤ TU .

(15.21)

We obtain from (15.20) and (15.21) that d u(t) L1 ≤ c u(t) L1 − γ11 u(t) 2L2 ≤ C( u0 L1 + 1) − γ11 u(t) 2L2 . dt Therefore, for 0 ≤ s < t ≤ TU ,  u(t) L1 − u(s) L1 ≤ C( u0 L1 + 1)(t − s) − γ11

s

t

u(τ ) 2L2 dτ.

In this way, it follows that  γ11 s

t

u(τ ) 2L2 dτ ≤ C( u0 L1 + 1)(t − s) + u(s) L1 ,

i.e.,  s

t

u(τ ) 2L2 dτ ≤ C( u0 L1 + 1)[(t − s) + 1],

0 ≤ s < t ≤ TU .

(15.22)

Step 2. Let 2 < q < ∞. Multiply the second equation of (15.6) by qv q−1 and integrate the product in Ω. Then,

3 A Priori Estimates for Local Solutions

d dt

497







v q dx = −q(q − 1)b Ω

 ≤q

v q−2 |∇v|2 dx + q Ω

gv q−1 dx Ω

(d − γ22 v)v q dx. Ω

Noting that, for example, (d − γ22 v)v q ≤ −dv q + γ22 (2d/γ22 )q+1

for all v ≥ 0

(divide the proof into two cases 0 ≤ v ≤ 2d/γ22 and 2d/γ22 < v < ∞), we have   d v q dx ≤ −dq v q dx + γ22 q(2d/γ22 )q+1 |Ω|. dt Ω Ω Therefore (cf. (1.58)), v(t) Lq ≤ e−dqt v0 Lq + 2(2d/γ22 )q |Ω|, q

so that

q

1 1 v(t) Lq ≤ 2 q e−dt v0 Lq + (2d/γ22 )(2|Ω|) q .

On account of limq→∞ v Lq = v L∞ , we conclude that v(t) L∞ ≤ e−dt v0 L∞ +

2d , γ22

0 ≤ t ≤ TU .

(15.23)

Step 3. In this step, multiply the second equation of (15.6) by Δv and integrate the product in Ω. Then, in view of (15.23),    1 d |∇v|2 dx + b |Δv|2 dx = gΔv dx 2 dt Ω Ω Ω  ≤ C v0 L∞ (u + v + 1)|Δv| dx b ≤ 2

 Ω

Ω



|Δv|2 dx + C v0 2L∞

(u + v + 1)2 dx. Ω

Therefore,  

d |∇v|2 dx + b |Δv|2 dx ≤ C v0 2L∞ u(t) 2L2 + v0 2L∞ + 1 . (15.24) dt Ω Ω Similarly, multiply the second equation by v. Then, due to (15.21),     d 2 2 2 v dx + b |∇v| dx = gv dx ≤ C v0 L∞ (u + v + 1) dx dt Ω Ω Ω Ω ≤ C v0 2L∞ ( u0 L1 + v0 L∞ + 1).

498

15 Lotka–Volterra Competition Model with Cross-Diffusion

This inequality, together with (15.24), then yields that   2

2

d v + |∇v|2 dx + b v + |∇v|2 + |Δv|2 dx dt Ω Ω

2 2 ≤ C v0 L∞ u(t) L2 + u0 L1 + v0 2L∞ + 1 . On account of (15.22), Proposition 1.4 is now available to solve this differential inequality and to conclude that

v(t) 2H 1 ≤ e−bt v0 2H 1 + C v0 2L∞ u0 L1 + v0 2L∞ + 1 , 0 ≤ t ≤ TU . (15.25) It also holds that  t

Δv(τ ) 2L2 dτ ≤ C v0 2L∞ u0 L1 + v0 2L∞ + 1 [(t − s) + v0 H 1 + 1], s

0 ≤ s < t ≤ TU

(15.26)

(cf. (1.64)). Step 4. In this step, we shall use the abbreviated notation p1 (U0 ) = p( u0 L1 + v0 L∞ + v0 H 1 ), where p(·) is a continuous increasing functions which varies in each occurrence. Introducing the quantity  N1,log (u) = u log(u + 1) dx, 0 ≤ u ∈ L2 (Ω), Ω

we intend to estimate N1,log (u(t)) for the local solution. Multiply the first equation of (15.6) by log(u + 1) and integrate the product in Ω. Then, by the same calculations as for (12.20), we observe that   √ d [(u + 1) log(u + 1) − u] dx + 4 (a + αv)|∇ u + 1|2 dx dt Ω Ω   = −α [log(u + 1) − u]Δv dx + f log(u + 1) dx. Ω

Here,

 −α Ω

Ω

[log(u + 1) − u]Δv dx ≤ C u 2L2 + Δv 2L2 .

In addition, since (u + 1) log(u + 1) ≤

u2 + u for all u ≥ 0, 2

3 A Priori Estimates for Local Solutions

499

it is clear that 

 (u + 1) log(u + 1) dx + Ω

Ω

f log(u + 1) dx ≤ C u 2L2 .

Consequently, we observe that   d [(u + 1) log(u + 1) − u] dx + [(u + 1) log(u + 1) − u] dx dt Ω Ω

2 2 ≤ C u(t) L2 + Δv(t) L2 . On account of (15.22) and (15.26), we can then solve this differential inequality by using Proposition 1.4 and conclude that N1,log (u(t)) ≤ e−t N1,log (u0 ) + p1 (U0 ),

0 ≤ t ≤ TU .

(15.27)

Step 5. In this step, we shall use the abbreviated notation p2 (U0 ) = p(N1,log (u0 ) + v0 L∞ + v0 H 1 ), where p(·) is a continuous increasing function which varies in each occurrence. Multiply the first equation of (15.8) by u and integrate the product in Ω. Then,     1 d u2 dx + Pu |∇u|2 dx = − Pv ∇u · ∇v dx + f u dx. 2 dt Ω Ω Ω Ω Therefore,     1 d 2 2 u dx + (a + αv)|∇u| dx = −α u∇u · ∇v dx + f u dx. 2 dt Ω Ω Ω Ω Here,    α α 2 u∇u · ∇v dx = − ∇u · ∇v dx = u2 Δv dx ≤ C u 2L3 Δv L3 . −α 2 Ω 2 Ω Ω With the aid of (1.98) (p = 2, q = 1 and r = 3) and the interpolation (2.145), we verify that 2

1

u L3 ≤ ζ u H3 1 N1,log (u) 3 + Cζ u L1 with any ζ > 0 and 1

2

1

1

1

2

1

Δv L3 ≤ C Δv H3 1 Δv L3 2 ≤ C Δv H3 1 Δv H3 1 Δv H3 1∗ ≤ C v H3 3 v H3 1 , respectively. By (15.21), (15.25), and (15.27), it therefore follows that

  u 2L3 Δv L3 ≤ p2 (U0 ) ζ u 2H 1 + v 2H 3 + Cζ .

500

15 Lotka–Volterra Competition Model with Cross-Diffusion

We can thus deduce that    d u2 dx + 2a |∇u|2 dx + γ22 u3 dx dt Ω Ω Ω

  2 2 ≤ p2 (U0 ) ζ ∇u L2 + ∇Δv L2 + Cζ , utilizing the shift property that Δv ∈ H 1 (Ω) with with the estimate v H 3 ≤ C( ∇Δv L2 + v L2 ),

∂v ∂n

(15.28)

= 0 on ∂Ω implies v ∈ H 3 (Ω)

v ∈ H 3 (Ω) ∩ HN2 (Ω),

(15.29)

and the elementary estimate u 2L2 ≤ ζ ∇u 2L2 + Cζ u 2L1 ,

u ∈ H 1 (Ω),

(15.30)

with any ζ > 0. The latter estimate (15.30) is verified from u 2L2 ≤ C u H 1 u L1 ≤ C( ∇u L2 + u L2 ) u L1

≤ ζ˜ ∇u 2L2 + u 2L2 + Cζ˜ u 2L1 , ζ˜ > 0 being an arbitrary positive number. In this step, we have to combine two differential inequalities on u and v. Considering the duality product of the second equation of (15.8) with Δ2 v in H 1 (Ω) × H 1 (Ω)∗ , we get    1 d |Δv|2 dx + b |∇Δv|2 dx = − ∇g · ∇Δv dx 2 dt Ω Ω Ω ≤

b 2 ∇Δv 2L2 + ∇g 2L2 . 4 b

Here, ∇g = −γ21 v∇u + (d − γ12 u − 2γ22 v)∇v. Moreover, 2 2γ21 b

 Ω

|∇u|2 v 2 dx ≤ C ∇u 2L2 v 2L∞ ≤ p2 (U0 ) u 2H 1 .

Meanwhile, due to (15.21) and (15.27), 2 2γ21 b

 Ω

4

2

2

4

u2 |∇v|2 dx ≤ C u 2L3 ∇v 2L6 ≤ C u H3 1 u L3 1 v H3 3 v H3 1

b ≤ ∇Δv 2L2 + p2 (U0 ) ∇u 2L2 + 1 . 4

Here, we utilized again (15.29) and (15.30). In this way, we obtain the inequality  

d 2 |Δv| dx + b |∇Δv|2 dx ≤ p2 (U0 ) ∇u 2L2 + 1 . (15.31) dt Ω Ω

3 A Priori Estimates for Local Solutions

501

Multiply a parameter ξ > 0 to (15.28) and add the product to (15.31). Then,    2



d 2 2 2 ξ u + |Δv| dx + 2aξ |∇u| + b|∇Δv| dx + γ22 ξ u3 dx dt Ω Ω Ω

  (15.32) ≤ p2 (U0 ) ζ ξ ∇u 2L2 + ∇Δv 2L2 + ∇u 2L2 + Cζ ξ . We here fix the two parameters ξ > 0 and ζ > 0 such that 2aξ > P2 (U0 )(ζ ξ +1) and b > P2 (U0 )ξ ζ (namely, first fix ξ sufficiently large and then ζ sufficiently small). As a result, we arrive at 

d |∇u|2 + |∇Δv|2 dx ≤ p2 (U0 ) ψ1 + δ1 ψ1 + γ1 dt Ω with suitable δ1 > 0 and γ1 > 0, where we put ψ1 (t) = ξ u(t) 2L2 + Δv(t) 2L2 . Proposition 1.4 then yields that

  u(t) 2L2 + Δv(t) 2L2 ≤ p2 (U0 ) e−δ1 t u0 2L2 + v0 2H 2 + 1 , 0 ≤ t ≤ TU . (15.33) Obviously v(t) H 2 can substitute for Δv(t) L2 due to (12.32). As well,  t     u(τ ) 2H 1 + v(τ ) 2H 3 dτ ≤ p2 (U0 ) (t − s) + u0 2L2 + v0 2H 2 + 1 , s

0 ≤ s < t ≤ TU .

(15.34)

Step 6. We shall use the notation p3 (U0 ) = p( u0 L2 + v0 H 2 ). Our goal in this step is to estimate the norm u(t) H 1 . d P (u, v) = Pu ut + Pv vt and integrate Multiply the first equation of (15.8) by dt the product in Ω. Then,     1 d 2 2 |∇P | dx + Pu ut dx = − Pv ut vt dx + f (Pu ut + Pv vt ) dx 2 dt Ω Ω Ω Ω  ≤ C [u|ut ||vt | + (u3 + v 3 + 1)(|ut | + |vt |)] dx. Ω

Here, on account of (15.33),  1 1 1 1 u|ut ||vt | dx ≤ u L4 ut L2 vt L4 ≤ C u H2 1 u L2 2 ut L2 vt H2 1 vt L2 2 Ω

1

1

≤ p3 (U0 ) u H2 1 ut L2 Δv + g H2 1 ≤ p3 (U0 ) ut L2 ( v H 3 + u H 1 + 1)

a ≤ ut 2L2 + p3 (U0 ) u 2H 1 + v 2H 3 + 1 . 8

502

15 Lotka–Volterra Competition Model with Cross-Diffusion

Similarly,  u3 |ut | dx ≤ u 3L6 ut L2 ≤ u 2H 1 u L2 ut L2 Ω

a ≤ p3 (U0 ) u 2H 1 ut L2 ≤ ut 2L2 + p3 (U0 ) u 4H 1 8 and

 Ω

u3 |vt | dx ≤ u 3L6 vt L2 ≤ p3 (U0 ) u 2H 1 .

Furthermore, it is immediate from ∇P (u, v) = Pu ∇u + Pv ∇v to see that 1 1   ∇u L2 ≤ C[ ∇P L2 + u L4 v H 2 ] ≤ C ∇P L2 + u H2 1 u L2 2 v H 2 ,

0 ≤ u, v ∈ H 2 (Ω).

(15.35)

In this way, we verify the inequality   d 2 |∇P | dx + a|ut |2 dx dt Ω Ω

≤ p3 (U0 ) ∇P (t) 2L2 u(t) 2H 1 + u(t) 2H 1 + v(t) 2H 3 + 1 . We now want to solve this differential inequality by using Proposition 1.6. To this ∂v end, we have to introduce an estimate of ∇P L2 by ut L2 . Since ∂u ∂n = ∂n = 0 2 implies P ∈ HN (Ω), we have ∇P 2L2 ≤ ζ ΔP 2L2 + Cζ P 2L2 ,

u, v ∈ HN2 (Ω),

with any ζ > 0 (cf. (12.32)). Furthermore,





 ΔP 2L2 ≤ 2 ut 2L2 + f 2L2 ≤ 2 ut 2L2 + C v 4L4 + 1 u 4L4 + 1 

 ≤ 2 ut 2L2 + p3 (U0 ) u 2H 1 + 1 . (15.36) Therefore,

∇P 2L2 ≤ 2ζ ut 2L2 + Cζ p3 (U0 ) u 2H 1 + 1 . Multiplying a parameter ξ > 0 to this inequality, we add the product to the differential inequality above to obtain that   

d 2 2 2 |∇P | dx + ξ − p3 (U0 ) u(t) H1 |∇P | dx + (a − 2ξ ζ ) |ut |2 dx dt Ω Ω Ω

2 2 (15.37) ≤ p3 (U0 ) u(t) H 1 + v(t) H 3 + 1 .

3 A Priori Estimates for Local Solutions

503

We then fix the parameter ξ > 0 in such a way that u 2H 1 satisfies  p3 (U0 ) s

t

u(τ ) 2H 1 dτ ≤ p3 (U0 )[(t − s) + 1] ≤ (ξ + 1)(t − s) + p3 (U0 ),

0 ≤ s < t ≤ TU . Note that on account of (15.34) it is possible to take ξ in such a way. Then we fix ζ such that 2ξ ζ ≤ a2 . These indeed means that Proposition 1.6 is now available to (15.37). As a result, we conclude that   P (t) 2H 1 ≤ p3 (U0 ) e−t P (U0 ) 2H 1 + 1 ,

0 ≤ t ≤ TU .

Moreover, in view of (15.35),   u(t) 2H 1 ≤ p3 (U0 ) e−t P (U0 ) 2H 1 + 1 ,

0 ≤ t ≤ TU .

(15.38)

Once (15.38) is verified, Proposition 1.5 is also available to (15.37) to conclude that  t   ut (τ ) 2L2 dτ ≤ p3 (U0 ) (t − s) + P (U0 ) 2H 1 + 1 , 0 ≤ s < t ≤ TU , s

and, in view of (15.36), 

t s

  P (U (τ )) 2H 2 ds ≤ p3 (U0 ) (t − s) + P (U0 ) 2H 1 + 1 ,

0 ≤ s < t ≤ TU . (15.39)

Step 7. We shall use the notation p4 (U0 ) = p( P (U0 ) H 1 + v0 H 2 ). Differentiate the first equation of (15.8) in t and consider the duality product with ut in H 1 (Ω)∗ × H 1 (Ω). Then,  

1 d Pu |∇ut |2 + Pv ∇ut · ∇vt dx u2t dx + 2 dt Ω Ω   = − (ut ∇Pu + vt ∇Pv ) · ∇ut dx + (fu ut + fv vt )ut dx. Ω

Ω

After some obvious estimations,    1 d u2t dx + (a + αv)|∇ut |2 ≤ α u|∇ut ||∇vt | dx 2 dt Ω Ω Ω  

+ C (|∇u| + |∇v|)(|ut | + |vt |)|∇ut | dx + C (u + v + 1) u2t + vt2 dx. Ω

Ω

504

15 Lotka–Volterra Competition Model with Cross-Diffusion

Here,  u|∇ut ||∇vt |dx ≤ u L∞ ∇ut L2 ∇vt L2 ≤ ζ ∇ut 2L2 + Cζ u 2H 2 ∇vt 2L2 Ω

with any ζ > 0. In addition, by Lemma 15.1 presented below, u H 2 ≤ p4 (U0 )( ΔP L2 + 1) ≤ p4 (U0 )( ut L2 + 1) (recall (15.36)) and, by a direct calculation, ∇vt L2 ≤ ∇(Δv + g) L2 ≤ p4 (U0 )( v H 3 + 1). Similarly, by Lemma 15.1,  (|∇u| + |∇v|)(|ut | + |vt |)|∇ut | dx Ω

≤ |∇u| + |∇v| L4 |ut | + |vt | L4 ∇ut L2 1 1 1 1 1 1 1 1 ≤ u H2 2 u H2 1 + v H2 2 v H2 1 ut H2 1 ut L2 2 + vt H2 1 vt L2 2 ∇ut L2 1 1 1 1



≤ p4 (U0 ) ut L2 2 + 1 ∇ut L2 2 ut L2 2 + ut L2 + v H2 3 + 1 ∇ut L2

≤ ζ ∇ut 2L2 + Cζ p4 (U0 ) ut 2L2 ut 2L2 + v 2H 3 + 1

with any ζ > 0. Finally, 



(u + v + 1) u2t + vt2 dx ≤ u + v + 1 L∞ ut 2L2 + vt 2L2 Ω



≤ p4 (U0 ) ut 3L2 + 1 .

Hence, fixing ζ > 0 small enough in a suitable way, it is verified that  



d u2t dx + a|∇ut |2 dx ≤ p4 (U0 ) ut 2L2 + 1 P 2H 2 + v 2H 3 + 1 . dt Ω Ω As we already know that (15.34) and (15.39) are valid, we can employ Proposition 1.7 to solve this inequality by combing this with (15.37). As a result, it follows that   ut (t) 2L2 ≤ p4 (U0 ) e−at ΔP (U0 ) + f (U0 ) 2L2 + 1 , 0 ≤ t ≤ TU . By Lemma 15.1 again,   u(t) 2H 2 ≤ p4 (U0 ) e−at u0 2H 2 + 1 ,

0 ≤ t ≤ TU .

As well, by Proposition 1.5,  t ∇ut (τ ) 2L2 dτ ≤ p4 (U0 )[(t − s) + u0 H 2 + 1], s

(15.40)

0 ≤ s < t ≤ TU .

3 A Priori Estimates for Local Solutions

505



We have thus established the desired estimate (15.19). Lemma 15.1 For 0 ≤ u, v ∈ HN2 (Ω), it holds that

u H 2 ≤ C P 2H 1 + v 2H 2 + 1 ( ΔP L2 + P L2 ).

(15.41)

Proof We know that u H 2 ≤ C( Δu L2 + u L2 ). By a direct calculation, we see that   Δu = Pu−1 ΔP − 2Pu−1 Puv ∇P · ∇v + 2Pv Pu−1 Puv |∇v|2 . Then, the desired estimate (15.41) is verified by ∇P 2L4 ≤ C P H 2 P H 1 ≤ C( ΔP L2 + P L2 ) P H 1 . Recall that the boundary conditions

∂u ∂n

=

∂v ∂n

= 0 imply

∂P ∂n

= 0 on ∂Ω.



3.2 Case where (15.17) is Valid Let us put w = v − u. Then, in view of (15.17), problem (15.6) is rewritten in the form ⎧ ∂u 2 2 ⎪ ⎪ ∂t = Δ(au + αu + αuw) + cu − (γ + μ)u − γ uw in Ω × (0, ∞), ⎪ ⎪ ⎪ ⎨ ∂w = aΔw + cw − 2μuw − μw 2 in Ω × (0, ∞), ∂t ∂u ∂w ⎪ = on ∂Ω × (0, ∞), ⎪ ⎪ ∂n ∂n = 0 ⎪ ⎪ ⎩u(x, 0) = u (x), w(x, 0) = w (x) ≡ v (x) − u (x) in Ω. 0

0

0

0

(15.42) Utilizing this form, we establish the following proposition. Proposition 15.5 Let (15.17) be satisfied. There exists a continuous increasing function p(·) such that, for any local solution U in the space (15.18) with initial value U0 ∈ K ∩ H2 (Ω), it holds that U (t) H2 ≤ p( U0 H2 ),

0 ≤ t ≤ TU .

Proof Throughout the proof, we shall use the notation P (u, w) = au + αu2 + αuw,

Q(u, w) = aw,

f (u, w) = cu − (γ + μ)u2 − γ uw,

g(u, w) = cw − 2μuw − μw 2 .

Step 1. As in Step 1 of the proof of Proposition 15.4, integration of the first and second equations of (15.6) in Ω provides that     d d 2 u dx ≤ (cu − μu ) dx and v dx ≤ (cv − μv 2 ) dx, dt Ω dt Ω Ω Ω

506

15 Lotka–Volterra Competition Model with Cross-Diffusion

respectively. Since cu − μu2 ≤ −u +

(c + 1)2 4μ

for all u ≥ 0,

we conclude (cf. (1.58)) that u(t) L1 + v(t) L1 ≤ e−t ( u0 L1 + v0 L1 ) +

(c + 1)2 |Ω|, 2μ

0 ≤ t ≤ TU . (15.43)

Consequently, w(t) L1 ≤ e−t ( u0 L1 + v0 L1 ) +

(c + 1)2 |Ω|, 2μ

0 ≤ t ≤ TU .

(15.44)

0 ≤ t ≤ TU ,

(15.45)

Step 2. For any m = 1, 2, 3, . . . , it holds that

−t 2m 2m w(t) 2m L2m ≤ e w0 L2m + Cm w0 L2(m−1) + 1 ,

where Cm > 0 is some constant depending on m. When m = 1, · L0 is replaced by · L1 . Let us first verify (15.45) for m = 1. Multiply the second equation of (15.42) by w and integrate the product in Ω. Then, 1 d 2 dt







w 2 dx + a Ω



|∇w|2 dx = c

w 2 dx − μ

Ω

Ω

(2u + w)w 2 dx dx Ω

≤ c w 2L2 ; note that u + w = v ≥ 0. We have

w 2L2 ≤ C w H 1 w L1 ≤ η ∇w 2L2 + w 2L2 + Cη w 2L1 ,

w ∈ H 1 (Ω),

with any η > 0 and some constant Cη > 0. Hence, w 2L2 ≤ η(1 − η)−1 ∇w 2L2 + Cη w 2L1 ,

w ∈ H 1 (Ω).

Therefore, in view of (15.44), d dt



 w 2 dx + a Ω

 |∇w|2 dx +

Ω

Ω



w 2 dx ≤ C w0 2L1 + 1 .

Consequently,

w(t) 2L2 ≤ e−t w0 2L2 + C w0 2L1 + 1 ,

0 ≤ t ≤ TU .

(15.46)

3 A Priori Estimates for Local Solutions

507

Now let us assume that (15.45) holds for m − 1 (m ≥ 2). Multiply the second equation of (15.42) by w 2m−1 and integrate the product in Ω. Then,    1 d a(2m − 1) m 2 w 2m dx + |∇w | dx + c w 2m dx 2m dt Ω m2 Ω Ω  − μ (2u + w)w 2m dx ≤ c w m 2L2 . Ω

We notice that w m 2L2 ≤ η ∇w m 2L2 + Cη w m 2L1 ,

w m ∈ H 1 (Ω),

with any η > 0. Since 2m w m 2L1 = w 2m Lm ≤ Cm w L2(m−1) ,

we conclude by the assumption of induction that  

d 2m w dx + w 2m dx ≤ Cm w0 2m L2(m−1) + 1 . dt Ω Ω Consequently, we verify (15.45). As a particular result of (15.45), we can verify that t

w(t) 6L6 ≤ C e− 3 w0 6L6 + w0 6L1 + 1 , 0 ≤ t ≤ TU .

(15.47)

In fact, for 0 < t ≤ TU , let us apply (15.45) repeatedly on [ 2t3 , t] with m = 3, on  t 2t  t 3 , 3 with m = 2, and on [0, 3 ] with m = 1. Then,   6   6   t  2t   + C w 2t  + 1 w w(t) 6L6 ≤ e− 3    3 L6 3 L4    6   6   

    2t  t  t  − 3t       + w +1 ≤C e w 3  + w 3  3 L2 L6 L4       6

   2t  t  − 3t  6    ≤C e w 3  + w 3  + w0 L2 + w0 L1 + 1 , L6 L4 0 ≤ t ≤ TU . Therefore, (15.47) is verified by (15.45) again. Step 3. In this step, we shall use the notation p1 (U0 ) = p( u0 L1 + w0 L6 ). Multiply the first equation of (15.42) by u and integrate the product in Ω. Then,     1 d 2 2 u dx + Pu |∇u| dx = − Pw ∇u · ∇w dx + f u dx. 2 dt Ω Ω Ω Ω

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15 Lotka–Volterra Competition Model with Cross-Diffusion

Since Pu = a + 2αu + αw ≥ a + αu, it follows that     μ 1 d 2 2 3 u dx + (a + αu)|∇u| dx + u dx ≤ C (u|∇u||∇w| + 1) dx. 2 dt Ω 2 Ω Ω Ω Here,  u|∇u||∇w| dx ≤

C Ω

α 2



μ 4

u|∇u|2 dx + Ω



 u3 dx + C Ω

|∇w|3 dx, Ω

and ∇w 3L3 ≤ C w H 2 w 2H 1 ≤ C w 2H 2 w L2 . Therefore, d dt





μ u dx + (2a + αu)|∇u| dx + 2 Ω Ω 2



2

Ω

u3 dx ≤ p1 (U0 ) w 2H2 .

(15.48)

Next, multiply the second equation of (15.42) by Δw and integrate the product in Ω. Then,    1 d a 2 2 |∇w| dx + a |Δw| dx = − gΔw dx ≤ Δw 2L2 + C g 2L2 . 2 dt Ω 4 Ω Ω Therefore,   

d |∇w|2 dx + |∇w|2 dx + a |Δw|2 dx ≤ ζ u 3L3 + Cζ w 6L6 + 1 dt Ω Ω Ω with any ζ > 0. Here, we used (12.32). After multiplying this inequality by a parameter ξ > 0, we add the product to the inequality (15.48). Then, in view of (15.47),    d 2a|∇u|2 + (aξ − p1 (U0 ))|Δw|2 dx ψ 1 + ψ1 + dt Ω   μ u3 dx ≤ ξ Cζ p1 (U0 ), (15.49) − ξζ + 2 Ω where ψ1 (t) = u(t) 2L2 + ξ ∇w(t) 2L2 . Fix the parameters ξ > 0 and ζ > 0 such that aξ > p1 (U0 ) + 1 and ξ ζ < μ4 . By Proposition 1.4, we conclude that

u(t) 2L2 + w(t) 2H 1 ≤ e−t u0 2L2 + w0 2H 1 + p1 (U0 ),

0 ≤ t ≤ TU . (15.50)

As well,  t     u(τ ) 2H 1 + w(τ ) 2H 2 dτ ≤ p1 (U0 ) (t − s) + u0 2L2 + w0 2H 1 + 1 , s

0 ≤ s < t ≤ TU .

(15.51)

Step 4. In this step, we shall use the notation p2 (U0 ) = p( u0 L2 + w0 H 1 ).

3 A Priori Estimates for Local Solutions

509

d Multiply the first equation of (15.42) by dt P (u, w) = Pu ut + Pw wt and integrate the product in Ω. Then,     1 d |∇P |2 dx + Pu u2t dx = − Pw ut wt dx + f (Pu ut + Pw wt ) dx 2 dt Ω Ω Ω Ω  

 u|ut ||wt | + u3 + |w|3 + 1 (|ut | + |wt |) dx ≤C Ω

α ≤ 2





Ω

uu2t dx + C



Ω





 uwt2 + u5 + |w|3 + 1 |ut | + u3 + |w|3 + 1 |wt | dx.

Here, in view of (15.50),  1 2

uwt2 dx ≤ u L3 wt 2L3 ≤ C u H3 1 u L3 2 Δw 2L3 + g 2L3 Ω

1 2 4

≤ p2 (U0 ) u H3 1 w H3 3 w H3 2 + uw 2L3 + 1 1 4

≤ p2 (U0 ) u H3 1 w H3 3 + u H 1 + 1

≤ ζ u 2H 1 + w 2H 3 + Cζ p2 (U0 )

with any ζ > 0. Similarly,  u5 dx ≤ C u 3H 1 u 2L2 ≤ ζ u 4H 1 + Cζ p2 (U0 ), Ω

and

 Ω

5 4

u3 wt dx ≤ C u 3L 9 wt L3 ≤ C u H3 1 u L3 2 Δw 2L3 + g 2L3 2

5 2

≤ p2 (U0 ) u H3 1 w H3 3 + u H 1 + 1 ≤ ζ u 4H 1 + w 2H 1

+ Cζ p2 (U0 ) with any ζ > 0. Furthermore, from ∇P = Pu ∇u + Pw ∇w it follows that ∇u L2 ≤ C[ ∇P L2 + ∇w L2 ]. In addition, since

∂u ∂n

=

∂w ∂n

(15.52)

= 0 implies P ∈ HN2 (Ω),

∇P 2L2 ≤ ζ ΔP 2L2 + Cζ P 2L2 with any ζ > 0. In this way, it is obtained (due to (15.29)) that    d |∇P |2 dx + |∇P |2 dx + a|ΔP |2 dx dt Ω Ω Ω

2 2 2 ≤ ζ ∇P L2 u H 1 + ∇Δw L2 + Cζ p2 (U0 ).

(15.53)

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15 Lotka–Volterra Competition Model with Cross-Diffusion

We next consider the duality product of the second equation of (15.42) with Δ2 w in H 1 (Ω) × H 1 (Ω)∗ in view of the shift property (15.29) applied to w. Then,    1 d |Δw|2 dx + a |∇Δw|2 dx = − ∇g · ∇Δw dx 2 dt Ω Ω Ω ≤

a 2 ∇Δw 2L2 + ∇g 2L2 . 4 a

Here,  1 3 2 |∇u|2 w 2 dx ≤ C ∇u 2L2 w 2 3 ≤ C u 2H 1 w H2 3 w H2 1 a Ω H2 1

≤ p2 (U0 ) u 2H 1 w H2 3 ≤ ζ u 4H 1 + w 2H 3 + Cζ p2 (U0 ), and 2 a



1 3 2 u2 |∇w|2 dx ≤ u 2L4 ∇w 2L4 ≤ C u H 1 u L2 w H2 3 v H2 1 a Ω 1

≤ p2 (U0 ) u H 1 w H2 3 ≤ ζ u 2H 1 + w 2H 3 + Cζ p2 (U0 )

with any ζ > 0. Therefore,    d |Δw|2 dx + |Δw|2 dx + a |∇Δw|2 dx dt Ω Ω Ω

≤ ζ ∇P 2L2 u 2H 1 + ∇Δw 2L2 + Cζ p2 (U0 ). We add the differential inequality (15.53) to this one to obtain that  d ψ2 + ψ2 + (a − 2ζ ) (|ΔP |2 + |∇Δw|2 ) dx dt Ω ≤ ζp2 (U0 ) u(t) 2H 1 ψ2 + Cζ p2 (U0 ),

(15.54)

where ψ2 (t) = ∇P (t) 2H 1 + Δw(t) 2L2 . As we can choose the ζ > 0 arbitrarily small, Proposition 1.6 is available to (15.54). Note on account of (15.51) that u 2H 1 is shown to satisfy (1.60). In this way, we conclude that

  t P (U (t)) 2H 1 + Δw(t) 2L2 ≤ p2 (U0 ) e− 2 P (U0 ) 2H 1 + w0 2H 2 + 1 , 0 ≤ t ≤ TU , and, by (15.52),

  t u(t) 2H 1 + w(t) 2H 2 ≤ p2 (U0 ) e− 2 P (U0 ) 2H 1 + w0 2H 2 + 1 , 0 ≤ t ≤ TU .

(15.55)

4 Global Solutions

511

Proposition 1.5 is now available to (15.54) to conclude that  t

P (U (τ )) 2H 2 + w(τ ) 2H 3 dτ s

  ≤ p2 (U0 ) (t − s) + P (U0 ) 2H 1 + w0 2H 2 + 1 ,

0 ≤ s < t ≤ TU . (15.56)

Step 5. Let us use the notation p3 (U0 ) = p( P (U0 ) H 1 + w0 H 2 ). We want to establish in this step the estimate   u(t) 2H 2 ≤ p3 (U0 ) e−δt u0 2H 2 + 1 ,

0 ≤ t ≤ TU ,

with some exponent δ > 0 and the estimate  t   ∇ut (τ ) 2L2 dτ ≤ p3 (U0 ) (t − s) + u0 2H 2 + 1 ,

(15.57)

0 ≤ s < t ≤ TU .

s

Its proof is however carried out by quite analogous arguments as in Step 7 of the proof of Proposition 15.4 using (15.55), (15.54), and (15.56). The only difference may be replacement of Lemma 15.1 used before by Lemma 15.2 below. We have thus accomplished the proof of proposition.  Lemma 15.2 For u, w ∈ HN2 (Ω) with u ≥ 0 and u + w ≥ 0, it holds that

u H 2 ≤ C P 2H 1 + w 2H 2 + 1 ( ΔP L2 + P L2 ).

(15.58)

Proof We know that u H 2 ≤ C( Δu L2 + u L2 ). By a direct calculation, we see that    Δu = Pu−1 ΔP − Puu Pu−2 |∇P |2 + 2Pu−1 Puu Pw Pu−1 − Puw ∇P · ∇w    + Pw Pu−1 2Puv − Puu Pw Pu−1 |∇w|2 . Then, the desired estimate (15.58) is verified by ∇P 2L4 ≤ C P H 2 P H 1 ≤ C( ΔP L2 + P L2 ) P H 1 .



4 Global Solutions 4.1 Global Solutions By Propositions 15.4 and 15.5, the global existence of solutions can be deduced. Indeed, we know that, for any initial value U0 ∈ K, there exists a nonnegative local solution at least on an interval [0, TU0 ]. Let 0 < t1 < TU0 and U1 = U (t1 ). Then, U1 ∈ K ∩ H2 (Ω). We next consider problem (15.8) but with the initial time t1 and

512

15 Lotka–Volterra Competition Model with Cross-Diffusion

with the initial value U1 in the ball KR of Z, where R = p( U1 H2 ). Proposition 15.4 (when (15.16) is valid) and Proposition 15.5 (when (15.17) is valid) ensure that any local solution starting from U1 stays at any time in KR . In addition, any local solution V on [t1 , TV ] starting from U1 can be extended over an interval [t1 , TV + τ ] as local solution, τ > 0 being dependent only on sup0≤t≤TV V (t) H1+ε and hence being independent of the extreme time TV (cf. Corollary 5.2). This means that problem (15.8) on [t1 , ∞) with the initial value U1 possesses a unique global solution. Under (15.16) or (15.17), we have thus deduced that, for any initial value U0 ∈ K, there exists a unique global solution to (15.8) in the function space:

0 ≤ U ∈ C (0, ∞); H2N (Ω) ∩ C([0, ∞); H1+ε (Ω)) ∩ C1 ((0, ∞); H1 (Ω)) ∩ C2 ((0, ∞); H1 (Ω)∗ ).

(15.59)

4.2 Global Norm Estimates For U0 ∈ K, let U (t; U0 ) be the global solution of (15.8) with the initial value U0 in (15.59). We can then show a global norm estimate. Proposition 15.6 Let (15.16) or (15.17) be valid. Let U0 ∈ K ∩ H2 (Ω). Then, U (t; U0 ) H2 ≤ p(e−δt p( U0 H2 ) + 1),

0 ≤ t < ∞; U0 ∈ K ∩ H2 (Ω), (15.60)

where p(·) are some continuous increasing functions, and δ > 0 is a positive exponent, all independent of initial values. Proof We recall that, in the case where (15.16) is valid, Proposition 15.4   essen  was tially proved in six steps. Consider an interval [0, t] and divide it as 0, 6t , 6t , 2t6 ,  5t   2t 3t   3t 4t   4t 5t  6 , 6 , 6, 6 , 6 , 6 , and 6 , t . Let us first apply (15.40) to U (t; U0 ) in the subinterval 5t6 , t . We then observe that            5t   5t  5t  −δt      + + p u U (t) H2 ≤ e p U v 6  2   6 6 H2 H1 H           5t   5t  ≤ e−δt p( U0 H2 ) + p  u 6  1 + v 6  2 . H H  4t 5t  Let us next apply (15.38) in 6 , 6 . Then,          5t   + v 5t  u    6 6 H 2 H1              4t   4t   4t   4t  −δt        ≤ e p u + v + + p u v 6  2    6 6 6 H1 H2 L2 H

5 Dynamical System

513

    4t   ≤ e−δt p( U0 H2 ) + p  u 6 

L2

    4t   + v 6 

.

H2

    In a similar manner, we apply repeatedly (15.33) in 3t6 , 4t6 , (15.27) in 2t6 , 3t6 ,  t  t 2t  (15.21) in 6 , 6 , and (15.23) and (15.25) in 0, 6 . Then, we finally observe that        t    u  + v t  ≤ e−δt ( u0 L + v0 L ) + C ≤ e−δt U0 2 + C. H ∞ 1  6   6  L1 L∞ In this way, (15.60) has been verified. Analogous arguments are equally available in the proof of (15.60) under (15.17). Noticing that Proposition 15.5  established    in five   steps,we consider   the divi was sion of the interval [0, t] into 0, 5t , 5t , 2t5 , 2t5 , 3t5 , 3t5 , 4t5 and 4t5 , t . Then, we       apply repeatedly (15.57) in 4t5 , t , (15.55) in 3t5 , 4t5 , (15.50) in 2t5 , 3t5 , (15.47)      in 5t , 2t5 , and (15.43) and (15.44) in 0, 5t . In this way, (15.60) is verified. Proposition 15.7 Let (15.16) or (15.17) be valid. Let U0 ∈ K. Then, U (t; U0 ) H1+ε ≤ p(e−δt p( U0 H1+ε ) + 1),

0 < t < ∞, U0 ∈ K,

(15.61)

where p(·) are some continuous increasing functions, and δ > 0 is a positive exponent, all independent of initial values. Proof Let us apply Theorem 5.6 to (15.8). Then, by (5.66), t

1−ε 2

U (t) H2 + U (t) H1+ε ≤ p( U0 H1+ε ),

0 < t ≤ τ,

with some τ > 0 depending on the norm U0 H1+ε . Meanwhile, for τ ≤ t < ∞, (15.60) yields that ε−1

U (t) H2 ≤ p(e−δ(t−τ ) p( U (τ ) H2 ) + 1) ≤ p e−δt eδτ p τ 2 p( U0 H1+ε ) + 1 ≤ p(e−δt p( U0 H1+ε ) + 1),

τ ≤ t < ∞.

These two estimates obviously provide the desired one.



5 Dynamical System 5.1 Construction of Dynamical System Under (15.16) or (15.17), let us verify that problem (15.8) defines a dynamical system in H1+ε (Ω) with phase space K. We already know that, for any U0 ∈ K, (15.8) possesses a unique global solution U (t; U0 ) in (15.59). By S(t)U0 = U (t; U0 ) we define a nonlinear semigroup S(t) acting on K.

514

15 Lotka–Volterra Competition Model with Cross-Diffusion

We can entirely follow the general methods described in Chap. 6, Sect. 7 setting the spaces X = L2 (Ω), Y = H1+ε0 (Ω), Z = H1+ε1 (Ω), W = H1+ε2 (Ω), and Dγ = H1+ε (Ω), where 0 < ε0 < ε1 < ε < ε2 < 12 . The space of initial values K is given by (15.7). Actually, the basic assumptions (6.87), (6.88), (6.89), (6.91), (6.92), and (6.93) have already been verified in Chap. 12, Sect. 3. In addition, the global estimate (6.97) is certainly shown by (15.61). It is hence deduced that problem (15.8) defines a dynamical system (S(t), K, H1+ε (Ω)).

5.2 Exponential Attractors For constructing a family of exponential attractors, it is sufficient to verify conditions (6.103) and (6.104). In the present case, (6.103) is obvious. In addition, (15.61) shows that the dissipative condition (6.104) is fulfilled, too. Hence, (S(t), K, H1+ε (Ω)) has a family of exponential attractors.

6 Instability of Homogeneous Stationary Solution Under (15.16) or (15.17), we have constructed a dynamical system determined from (15.8) in H1+ε (Ω). In this section, we investigate the stability and instability of homogeneous stationary solutions of (15.8). We are concerned with the case where relation (15.2) takes place. Then, (15.8) has four nonnegative homogeneous stationary solutions O = (0, 0), U u = (c/γ11 , 0), U v = (0, d/γ22 ), and U = (u, v), where u = (cγ22 − dγ12 )/D and v = (dγ11 − cγ21 )/D with D = γ11 γ22 − γ12 γ21 < 0. The purpose of this section is to construct stable and unstable local manifolds for the stationary solution U . It is of course possible to construct those for other homogeneous stationary solutions in an analogous way. For simplicity, we will consider only the case where (15.16) is valid.

6.1 Localized Problem We begin with introducing the localized problem of (15.8) in a neighborhood of U given by ⎧ ∂u ⎪ ⎪ ∂t = ∇ · {[a + αχ2 (v)]∇u + αχ1 (u)∇v} − u + ϕ(u, v)(c + 1 − γ11 u − γ12 v)u ⎪ ⎪ ⎪ in Ω × (0, ∞), ⎪ ⎪ ⎨ ∂v ∂t = bΔv − v + ϕ(u, v)(d + 1 − γ21 u − γ22 v)v in Ω × (0, ∞), ⎪ ⎪ ∂u ∂v ⎪ ⎪ ⎪ ∂n = ∂n = 0 on ∂Ω × (0, ∞), ⎪ ⎪ ⎩ u(x, 0) = u0 (x), v(x, 0) = v0 (x) in Ω. (15.62)

6 Instability of Homogeneous Stationary Solution

515

Here, χ1 (u) (resp. χ2 (v)) denotes a cutoff function defined for u ∈ C (resp. v ∈ C) such that χ1 (u) = u for |u − u| < r (resp. χ2 (v) = v for |v − v| < r) and χ1 (u) = 2r(u−u) 2r(v−v) |u−u| − u for |u − u| ≥ 2r (resp. χ2 (v) = |v−v| − v for |v − v| ≥ 2r). Moreover, χ1 (u) (resp. χ2 (v)) is sufficiently smooth with respect to the real variables Re u + i Im u ∈ R + iR (resp. Re v + i Im v ∈ R + iR). In addition, the radius r > 0 is sufficiently small so that v 0
(15.63)

Meanwhile, ϕ(u, v) is another cutoff function defined for (u, v) ∈ C2 such that ϕ(u, v) ≡ 1 for |u| + |v| ≤ u + v + 1, ϕ(u, v) ≡ 0 for |u| + |v| ≥ u + v + 2, and 0 ≤ ϕ(u, v) ≤ 1 which is sufficiently smooth with respect to the four real variables (Re u + i Im u, Re v + i Im v) ∈ (R + iR)2 . We formulate problem (15.62) in the space X = L2 (Ω) as for (15.6). We set H1+ε (Ω), where 0 < ε < 12 , as the space of initial functions. Then, (15.62) can be rewritten by  dU ˜ dt + A(U )U = kU, 0 < t < ∞, (15.64) U (0) = U0 . Here, for U ∈ Z = H1+ε1 (Ω), where 0 < ε1 < ε, A(U ) is the linear operator given by  −∇ · {[a + αχ2 (v)]∇ u˜ + αχ1 (u)∇ v} ˜ + u˜ ˜ ˜ A(U )U = −bΔv˜ + v˜  (c + 1 − γ11 u − γ12 v)u˜ + k U˜ , − ϕ(u, v) (d + 1 − γ21 u − γ22 v)v˜   u u˜ ˜ )), U= , U˜ = ∈ D(A(U v v˜ ˜ )) ≡ H2 (Ω), where k is a sufficiently large constant to be specified with D(A(U N ˜ ) is a sectorial operator of X below. By the following proposition, we see that A(U π with ωA(U ˜ )≤ω< 2. ˜ ), U ∈ Z, is a sectorial operaProposition 15.8 If k is sufficiently large, then A(U π tor of X with ωA(U ≤ ω < , ω being some fixed angle. ˜ ) 2 Proof First, we notice from (15.63) that Re χ2 (v) = Re[χ2 (v) − v] + v ≥ −2r + v ≥ 0,

v ∈ C.

As a consequence, Re χ2 (v) ≥ 0 for all v ∈ H 1+ε1 (Ω). Second, consider for U ∈ Z, the sesquilinear form   u˜ 1 u˜ 2 dx a(U ; u˜ 1 , u˜ 2 ) = [a + αχ2 (v)]∇ u˜ 1 · ∇ u˜ 2 dx + Ω

Ω

(15.65)

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15 Lotka–Volterra Competition Model with Cross-Diffusion

defined on H 1 (Ω). We verify that the form fulfills (2.6) and (2.7) uniformly with δ = min{a, 1}. By Theorem 2.1, it is deduced that the realization A1 (U ) = −∇ · {[a + αχ2 (v)]∇·} + 1 is a sectorial operator of L2 (Ω), and its angle is uniformly less than some fixed angle ω < π2 . Since [a + αχ2 (v)]−1 ∇·{[a + αχ2 (v)]∇·} = Δ· + α[a + αχ2 (v)]−1 ∇[χ2 (v)] · ∇· and since Theorem 3.3 is available to the operator in the right-hand side, we observe that A1 (U ) has the uniform domain D(A1 (U )) ≡ HN2 (Ω). It is clear that A2 = −Δ + 1 is a positive definite self-adjoint operator of L2 (Ω) with D(A2 ) = HN2 (Ω). Then, Theorem 2.16 provides that the operator  −∇·{[a + αχ2 (v)]∇·} + 1 −∇ · {[αχ1 (u)]∇·} 0 −bΔ + 1 is a sectorial operator of L2 (Ω) with angle less than the angle ω. Finally, the desired result is deduced by Theorem 3.3 because of ϕ(u, v)(c + 1 − γ11 u − γ12 v) L∞ ≤ (c + 1 + γ11 + γ12 )(u + v + 2),

U ∈ Z,

ϕ(u, v)(d + 1 − γ21 u − γ12 v) L∞ ≤ (d + 1 + γ21 + γ22 )(u + v + 2),

U ∈ Z. 

We furthermore verify by virtue of Theorem 16.4 that 

˜ )θ ) = H2θ (Ω) D(A(U

for 0 ≤ θ < 34 ,

˜ )θ ) = H2θ (Ω) D(A(U N

for

3 4

< θ ≤ 1,

with the estimates   ˜ )θ U˜  ≤ Cθ U˜ H2θ , Cθ−1 U˜ H2θ ≤ A(U X

˜ )θ , U ∈ Z, θ = 3 , U˜ ∈ D A(U 4

Cθ > 0 being some constant depending on θ . ˜ ) satisfy the Lipschitz Let us set Y = H1+ε0 (Ω) with 0 < ε0 < ε1 < ε. Then, A(U condition (5.5) with ν = 1. We can now apply Theorem 5.6 to construct a local solution to (15.64). For any U0 ∈ H1+ε (Ω), there exists a local solution to (15.64) in the function space:

U ∈ C (0, TU0 ]; H2N (Ω) ∩ C([0, TU0 ]; H1+ε (Ω)) ∩ C1 ((0, TU0 ]; L2 (Ω)), where TU0 > 0 is determined by the norm U0 H1+ε . It is also verified that U ∈ C1 ((0, TU0 ]; H1 (Ω)) ∩ C2 ((0, TU0 ]; H1 (Ω)∗ ) by following the same procedure as for the proof of (15.13).

6 Instability of Homogeneous Stationary Solution

517

6.2 Dynamical System of Localized Problem Let us construct a dynamical system determined from (15.64) in the universal space H1+ε (Ω). To this end, we begin with showing a priori estimates for local solutions. For U0 ∈ H2N (Ω), let U be any local solution of (15.64) in the function space:

U ∈ C (0, TU ]; H2N (Ω) ∩ C([0, TU ]; H1+ε (Ω)) ∩ C1 ((0, TU ]; H1 (Ω)) ∩ C2 ((0, TU0 ]; H1 (Ω)∗ ),

(15.66)

where [0, TU ] denotes the interval on which U is defined. Proposition 15.9 There exists a continuous increasing function p(·) such that, for any local solution U in the space (15.66) with initial value U0 ∈ H2 (Ω), it holds that U (t) H2 ≤ p( U0 H2 ),

0 ≤ t ≤ TU .

(15.67)

Proof The proof consists of several steps. Step 1. Multiply the second equation of (15.64) by Δv and integrate the product in Ω. Noting that     ∂ ∂v 1 d ∇v · ∇v dx = − Re Δv dx, |∇v|2 dx = Re 2 dt Ω Ω ∂t Ω ∂t we have 1 d 2 dt



 |∇v|2 dx + b Ω

 |Δv|2 dx +

Ω



|∇v|2 dx Ω

b = − Re ϕ(u, v)(d + 1 − γ21 u − γ22 v)v Δv dx ≤ 2 Ω

 |Δv|2 dx + C. Ω

Therefore, d dt



 |∇v|2 dx + b Ω

 |Δv|2 dx + 2

Ω

|∇v|2 dx ≤ C. Ω

Hence, we obtain that ∇v(t) 2L2 ≤ e−2t ∇v0 2L2 + C,

0 ≤ t ≤ TU .

In a similar way, multiplying the second equation of (15.64) by v and integrating the product in Ω, we obtain that v(t) 2L2 ≤ e−2t v0 2L2 + C,

0 ≤ t ≤ TU .

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15 Lotka–Volterra Competition Model with Cross-Diffusion

Hence, v(t) 2H 1 ≤ e−2t v0 2H 1 + C,

0 ≤ t ≤ TU .

(15.68)

As well, 

t s

  v(s) 2H 2 ds ≤ C (t − s) + v0 2H 1 ,

0 ≤ s ≤ t ≤ TU .

Step 2. Multiplying the first equation of (15.64) by u and integrating the product in Ω, we have    1 d 2 2 |u| dx + Re [a + αχ2 (v)]|∇u| dx + |u|2 dx 2 dt Ω Ω Ω   αχ1 (u)∇v · ∇u dx + Re ϕ(u, v)(c + 1 − γ11 u − γ12 v)|u|2 dx = − Re a 2

≤

Ω

 Ω

Ω



|∇u|2 dx + C ∇v 2L2 + 1 .

Since Re χ2 (v) ≥ 0 due to (15.65), it follows that d dt



 |u|2 dx + a Ω

 |∇u|2 dx + 2

Ω

Ω



|u|2 dx ≤ C ∇v 2L2 + 1 .

(15.69)

In the meantime, consider the duality product of the second equation of (15.64) with Δ2 v in H 1 (Ω) × H 1 (Ω)∗ in view of the shift property (15.29). Since 1 d 2 dt





∂ Δv, Δv |Δv| dx = Re ∂t Ω





2

H 1∗ ×H 1

∂v 2 ,Δ v = Re ∂t

 , H 1 ×H 1∗

we have 1 d 2 dt



 |Δv| dx + b Ω

 |∇Δv| dx +

2

|Δv|2 dx

2

Ω

Ω

= Reϕ(u, v)(d + 1 − γ21 u − γ22 v)v, Δ2 vH 1 ×H 1∗ 

b ≤ |∇Δv|2 dx + C ∇u 2L2 + ∇v 2L2 . 2 Ω Therefore,   

d |Δv|2 dx + b |∇Δv|2 dx + 2 |Δv|2 dx ≤ C ∇u 2L2 + ∇v 2L2 . dt Ω Ω Ω After multiplying (15.69) by a parameter ξ > 0, we add the product to this inequality to obtain due to (15.68) that

6 Instability of Homogeneous Stationary Solution

d dt





ξ |u|2 + |Δv|2 dx +

Ω



519

  (aξ − C)|∇u|2 + b|∇Δv|2 dx

Ω



+ Ω



2 ξ |u|2 + |Δv|2 dx ≤ C v0 2H 1 + 1 .

Fixing the parameter ξ such that ξ > Ca , we conclude that



ξ u(t) 2L2 + v(t) 2H 2 ≤ e−2t ξ u0 2L2 + v0 2H 2 + C v0 2H 1 + 1 , As well,  t s

0 ≤ t ≤ TU . (15.70)



 u(τ ) 2H 1 + v(τ ) 2H 3 dτ ≤ p1 (v0 ) (t − s) + u0 2L2 + v0 2H 2 ,

0 ≤ s ≤ t ≤ TU , where p1 (v0 ) = p( v0 H 1 ), p(·) being some continuous increasing function. Step 3. In this step, we use the notation p2 (U0 ) = p( u0 L2 + v0 H 2 ). Multiply the first equation of (15.64) by Δu and integrate the product in Ω. Then, after some calculations,    1 d 2 2 |∇u| dx + Re [a + αχ2 (v)]|Δu| dx + |∇u|2 dx 2 dt Ω Ω Ω  ≤ (|Δv| + |∇u · ∇v| + 1)|Δu| dx. Ω

Since Re χ2 (v) ≥ 0 due to (15.65), we have   

d 2 2 |∇u| dx +a |Δu| dx +2 |∇u|2 dx ≤ C Δv 2L2 + ∇u·∇v 2L2 +1 . dt Ω Ω Ω We already know by (15.70) that v(t) H 2 ≤ C( v0 2H 2 + u0 2L2 + 1). In addition, 2

4

∇u · ∇v 2L2 ≤ ∇u 2L3 ∇v 2L6 ≤ C u H3 2 u H3 1 v 2H 2 4

2

≤ C u H3 2 u L3 2 v 2H 2 ≤ ζ u 2H 2 + Cζ u 2L2 v 6H 2 with any number ζ > 0. Therefore, in view of (15.70),    d a |∇u|2 dx + |Δu|2 dx + |∇u|2 dx ≤ p2 (U0 ). dt Ω 2 Ω Ω We thus conclude that u(t) 2H 1 ≤ e−t u0 2H 1 + p2 (U0 ),

0 ≤ t ≤ TU ,

(15.71)

520

and

15 Lotka–Volterra Competition Model with Cross-Diffusion



t s

  u(τ ) 2H 2 dτ ≤ p2 (U0 ) (t − s) + u0 2H 1 ,

0 ≤ s ≤ t ≤ TU .

Step 4. Differentiate the first equation of (15.62) in t . Then, ∂ut = ∇·{[a + αχ2 (v)]∇ut + αχ1 (u)∇vt } − ut ∂t + α∇·[(χ2,Re v Re vt + χ2,Im v Im vt )∇u + (χ1,Re u Re ut + χ1,Im u Im ut )∇v] + (ϕRe u Re ut + ϕIm u Im ut + ϕRe v Re vt + ϕIm v Im vt )(c + 1 − γ11 u − γ12 v)u + ϕ(c + 1 − 2γ11 u − γ12 v)ut + ϕγ12 uvt . Regarding this as an equation for ut = ∂u ∂t , we take the duality product with ut in H 1 (Ω)∗ × H 1 (Ω). Then, by analogous arguments as in Step 7 of the proof of Proposition 15.4, we can conclude that u(t) 2H 2 ≤ Ce−δt u0 2H 2 + p3 (U0 ),

0 ≤ t ≤ TU ,

(15.72)

with some positive exponent δ > 0 and p3 (U0 ) = p( u0 H 1 + v0 H 2 ), and that  t   ∇ut (τ ) 2L2 dτ ≤ p3 (U0 ) (t − s) + u0 2H 2 , 0 ≤ s ≤ t ≤ TU . s

The desired estimate (15.67) has then been established.



Now that the a priori estimates are verified, following the same procedure used in Chap. 12, Sect. 4, we can construct a unique global solution U (t; U0 ) of (15.64) for each initial value U0 ∈ H1+ε (Ω) in the function space:

U ∈ C (0, ∞); H2N (Ω) ∩ C([0, ∞); H1+ε (Ω)) ∩ C1 ((0, ∞); H1 (Ω)) ∩ C2 ((0, ∞); H1 (Ω)∗ ).

(15.73)

As a consequence of (15.70), (15.71), and (15.72), the global solution satisfies the estimate U (t; U0 ) H1+ε ≤ p(e−δt p( U0 H1+ε ) + 1),

0 < t < ∞, U0 ∈ H1+ε (Ω), (15.74) with some positive exponent δ > 0 and some continuous increasing functions p(·). Furthermore, we can determine a dynamical system from (15.64) in the universal space H1+ε (Ω) having the whole phase space H1+ε (Ω). In fact, if ˜ we set S(t)U 0 = U (t; U0 ), where U (t; U0 ) is the global solution of (15.64) in ˜ (15.73) with U (0; U0 ) = U0 ∈ H1+ε (Ω), then S(t) is shown to be a continu˜ H1+ε (Ω), H1+ε (Ω)) defines a dyous semigroup on H1+ε (Ω). Therefore, (S(t), namical system which is a complexified dynamical system of the original system

6 Instability of Homogeneous Stationary Solution

521

(S(t), K, H1+ε (Ω)) in a neighborhood of U . Moreover, the dissipative condition ˜ (6.104) is verified by (15.74). Therefore, the system (S(t), H1+ε (Ω), H1+ε (Ω)) has a family of exponential attractors too.

˜ ) 6.3 Differentiability of A(U ˜ Of course, the point U is an equilibrium of (S(t), H1+ε (Ω), H1+ε (Ω)). We then ˜ H1+ε (Ω), intend to construct stable and unstable local manifolds of U in (S(t), 1+ε H (Ω)). As a matter of fact, we can follow the general methods described in Chap. 6, Sect. 8. It is in fact sufficient to verify the differentiability conditions ˜ ) and the spectrum separation con(6.115) and (6.116) for the linear operators A(U ˜ dition (6.127) for the linearized operator A(U ) + A˜ (U )[·, U ] − k. We recall that the spaces Y, Z, and Dγ were set as Y = H1+ε0 (Ω), Z = 1+ε H 1 (Ω), and Dγ = H1+ε (Ω), respectively, where 0 < ε0 < ε1 < ε < 12 . If ρ > 0 is sufficiently small so that U − U H1+ε1 < ρ implies U − U C < r, then, in the 1+ε ˜ ) is given by open ball B H 1 (U ; ρ), the linear operator A(U  ˜ + u˜ ˜ )U˜ = −∇ · [(a + αv)∇ u˜ + αu∇ v] A(U −bΔv˜ + v˜  (c + 1 − γ11 u − γ12 v)u˜ − + k U˜ , (d + 1 − γ21 u − γ22 v)v˜ U ∈ BH

1+ε1

(U ; ρ), U˜ ∈ H2N (Ω).

(15.75)

We observe that the correspondence A˜ : B H 1 (U ; r) → L(H2N (Ω), L2 (Ω)) is lin˜ is Fréchet differentiable for U ∈ B H1+ε1 (U ; ρ) with the derivative ear. So, A(·)   ∇ · (∇ u˜ + h∇ v) ˜ (γ11 h + γ12 )u˜ , + A˜ (U )[H, U˜ ] = −α 0 (γ21 h + γ22 )v˜  h H1+ε1 2 ˜ U ∈B (U ; ρ), U ∈ HN (Ω), H = ∈ H1+ε1 (Ω). (15.76)  1+ε

This derivative obviously fulfills (6.115). It is clear that it also fulfills (6.116), be1+ε cause A˜ (U ) is independent of U ∈ B H 1 (U ; ρ).

6.4 Spectrum Separation Condition of A ˜ ) + A˜ (U )[·, U ] − k, let us verify (6.127). Let Λ be a realization of For A = A(U −Δ in L2 (Ω) under the homogeneous Neumann boundary conditions. We use the

522

15 Lotka–Volterra Competition Model with Cross-Diffusion

fact that Λ possesses denumerable nonnegative eigenvalues and the corresponding real eigenfunctions can constitute an orthonormal basis of L2 (Ω). So, let 0 = μ0 < μ1 ≤ μ2 ≤ · · · → ∞ 1

be the eigenvalues of Λ, and let φ0 = |Ω|− 2 , φ1 , φ2 , . . . , be the corresponding real eigenfunctions which constitute an orthonormal basis. Then, the underlying space L2 (Ω) is obtained as the infinite sum of two-dimensional orthogonal subspaces   

φj 0 ; ξ j , ηj ∈ C , X j = U = ξj + ηj 0 φj

j = 0, 1, 2, . . . ,

equipped with the norm U 2Xj = |ξj |2 + |ηj |2 . Obviously,

 Φj =

φj 0



and Ψj =

0 φj



form an orthonormal basis of Xj for each j . From (15.75) and (15.76) we observe that   (a + αv)Λ αuΛ ˜ γ11 u γ12 u ˜ ˜ AU = U. U+ 0 bΛ γ21 v γ22 v This shows that the operator A maps the subspace Xj into itself, that is, Xj are invariant under A. As a consequence, the operator A is also decomposed as A = ∞ j =0 Aj , where Aj is the part of A in Xj . Clearly, Aj Φj = [(a + αv)μj + γ11 u]Φj + γ21 vΨj , Aj Ψj = (αuμj + γ12 u)Φj + (bμj + γ22 v)Ψj . The transformation matrix M j of Aj with respect to the basis {Φj , Ψj } is given by  (a + αv)μj + γ11 u Mj = αuμj + γ12 u

γ21 v . bμj + γ22 v

Since |λ − M j | = [λ − (a + αv)μj − γ11 u][λ − bμj − γ22 v] − (αuμj + γ12 u)γ21 v, the characteristic equation of M j is written as λ2 − [(a + αv + b)μj + γ11 u + γ22 v]λ + Q(μj ) = 0,

6 Instability of Homogeneous Stationary Solution

523

where Q(μ) is the square function Q(μ) = (a +αv)bμ2 +[αv(γ22 v −γ21 u)+aγ22 v +bγ11 u]μ+(γ11 γ22 −γ12 γ21 )u v. (15.77) Furthermore, the discriminant is written as Dj = [(a + αv − b)μj + γ11 u − γ22 v]2 + 4(αγ21 μj + γ12 γ21 )u v > 0. Therefore, the matrix M j has two real eigenvalues λj < λ j . If the relation Q(μj ) > 0

(15.78)

takes place, then the eigenvalues are both positive, i.e., 0 < λj < λ j . Furthermore, the estimate Q(μj ) < λj < λ j < (a + αv + b)μj + γ11 u + γ22 v (a + αv + b)μj + γ11 u + γ22 v −

Q(μj ) (a + αv + b)μj + γ11 u + γ22 v

(15.79)

holds for the eigenvalues. On the other hand, if the opposite relation Q(μj ) < 0

(15.80)

takes place, then one of the eigenvalues must be negative and the other positive, i.e., λj < 0 < λ j . For verifying (6.127), we here make the assumption that Q(μj ) = 0 for all j = 0, 1, 2, . . . .

(15.81)

Consider the parts Aj on Xj . Let ψj : Xj → C2 be a natural isomorphism such that ψj (ξj Φj + ηj Ψj ) = t (ξj , ηj ) ( ψj = ψj−1 = 1); then Aj is written as Aj =

ψj−1 M j ψj . Therefore,

(λ − Aj )−1 = ψj−1 (λ − M j )−1 ψj

for λ = λj , λ j .

(15.82)

Hence, σ (Aj ) = {λj , λ j }. Let λ ∈ iR. For all j large enough to satisfy (15.78), we have  1 λ − bμj − γ22 v γ21 v −1 (λ − M j ) = . λ − (a + αv)μj − γ11 u (λ − λj )(λ − λ j ) αuμj + γ12 u Then, by (15.79) and (15.82) and by the obvious inequalities |λ − λj | ≥ λj and |λ − λ j | ≥ λ j , we obtain that   (λ − Aj )−1 

L(Xj )

≤ Cλ ,

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15 Lotka–Volterra Competition Model with Cross-Diffusion

where Cλ > 0 may depend on λ but is independent of j . This means that λ ∈ ρ(A) if and only if λ ∈ ρ(Aj ) for each j = 0, 1, 2, . . . . In other words, λ ∈ σ (A) if and only λ ∈ σ (Aj ) = {λj , λ j } for some j . We can now verify (6.127). Indeed, if λ ∈ iR−{0}, then λ = λj , λ j for every j , because λj and λ j are both real. Meanwhile, if (15.81) is valid, any λj , λ j cannot be zero. Therefore, under (15.81), iR∩σ (A) = ∅, that is, (6.127) is fulfilled.

6.5 Instability of U Put N (U ) = #{μj ; Q(μj ) < 0}. From (15.2) we have γ11 γ22 < γ12 γ21 . This then implies Q(0) < 0, which means that (15.80) holds for μ0 = 0, i.e., N (U ) ≥ 1. It is always the case that σ (A) ∩ {λ ∈ C; Re λ < 0} =  ∅, and hence, under (15.81), U is an unstable equilibrium of ˜ + (U ; O). ˜ (S(t), H1+ε (Ω), H1+ε (Ω)) and has a nontrivial unstable local manifold W

For μj satisfying (15.80), let λj < 0 < λj . Since M j is a real matrix, there exists

a real eigenvector uj ∈ R2 corresponding to λj . Meanwhile, ψj−1 uj is an eigenfunction of Aj corresponding to λj . Since φj is a real eigenfunction of Λ, Φj and Ψj are real valued. Therefore, A has an eigenfunction for λj < 0 whose components ˜ + (U ; O) of U with dimenare real-valued functions. The local unstable manifold W sion N(U ) is tangent to U + X− at U , where X− is the sum of the eigenspaces of A corresponding to the negative eigenvalues. These facts indicate that U is an unstable equilibrium of (S(t), K, H1+ε (Ω)), too. It is also easy to compute the number N (U ) from the parameters of equations appearing in (15.6).

Notes and Further Researches For the mathematical analysis of problem (15.1), we refer the reader to [Mur02, Chap. 3]. We have already handled the model (15.4) in Chap. 4, Sect. 4. We indeed constructed nonnegative global solutions. It is then immediate to construct a dynamical system and exponential attractors following the general procedure described in Chap. 6, Sect. 5. The fact that, when Ω is a convex domain, the diffusion model (15.4) possesses no nonhomogeneous stationary solutions was proved by Kishimoto [Kis81] and Kishimoto–Weinberger [KW85]. On the contrary, the existence of nonhomogeneous stationary solutions in a nonconvex domain was shown by Matano–Mimura [MM83]. The competition model (15.5) was presented by Shigesada–Kawasaki–Teramoto [SKT79]. They also introduced the diffusion term of form Δ[(a + α11 u + α12 v)u].

Notes and Further Researches

525

This term is obviously decomposed into Δ[(a + α11 u + α12 v)u] = ∇ · [(a + 2α11 u + α12 v)∇u] + ∇ · [α12 u∇v]. The first one then denotes a density-dependent Fickian diffusion of u, and the second one denotes an advection of u in the direction of decreasing v. Such densitydependent diffusions were studied systematically by Okubo–Levin [OL01]. They call the diffusion of this class repulsive translation of u with respect to the density (a + α11 u + α12 v). We want also to quote [Hua05]. Problem (15.6) has in fact attracted interest of many mathematicians. For the two-dimensional problem, the general global existence is only known so far under the condition (15.15), i.e., 0 ≤ α12 α21 ≤ 64 α11 α22 , cf. [Yag08b]. First, in the case where 0 < α21 < 8α11 and 0 < α12 < 8α22 , the global existence was proved by Yagi [Yag93a] (cf. also [Yag93b]); afterward, the result was extended by Ichikawa– Yamada [IY00] to the case where 0 < α12 α21 < 64α11 α22 . Next, for the critical case where α12 α21 = 0 (note that in such a case it is allowed that α11 = α22 = 0), the global existence was first obtained by Masuda–Mimura (cf. [Mim81]) for the one-dimensional problem. For the two-dimensional case, this was obtained by Yagi [Yag97a]; afterward, the similar result was shown by Lou–Ni–Wu [LNW98] but in a different framework. It is then very natural to ask whether condition (15.15) is necessary for the global existence of problem (15.6) or not. But, as the result of Sect. 4 shows, we know that (15.6) can have global solutions for initial values in K in the particular case where (15.17) holds and hence (15.15) does not. Such a result was first obtained for the one-dimensional problem by Kim [Kim84]; afterward, a similar result was shown for the two-dimensional one by [Yag93a]. In the meantime, no result which shows blowup of some solution is known when (15.15) does not hold. So, the role of condition (15.15) is still unclear. It seems that we have no satisfactory answer to the global existence yet. For the N -dimensional problem (N ≥ 3), Deuring [Deu87] first considered the global existence in the case where α12 and α21 are sufficiently small. Afterward, Pozio–Tesei [PT90] got rid of such smallness (under the Dirichlet boundary conditions) but assuming a higher order of decaying in the growth function of u or v. Yamada [Yam95] also studied in a similar spirit. Wiegner [Wie92] applied the Amann’s theory [Ama93] on abstract parabolic equations to the N -dimensional problem of (15.6). More recently, Choi–Lui–Yamada [CLY03, CLY04] tried to extend the result [LNW98] to the N -dimensional case, but they still needed the smallness of α12 > 0 (α21 = 0) or positivity of α11 > 0. In the meantime, little is known on the asymptotic behavior of solutions. Redlinger [Red95] constructed the global attractor for the one-dimensional problem containing growth functions at a higher order of decaying. Shim [Shi02, Shi03, Shi03b] established uniformly bounded estimates and the convergence of solutions in some suitable cases. Construction of exponential attractors under (15.15) was done in [Yag08b]. Construction of those under (15.17) is new. Construction of the smooth unstable manifold for the homogeneous stationary solution (u, v) is also new.

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15 Lotka–Volterra Competition Model with Cross-Diffusion

A number of papers on the stationary problem for (15.6) have already been published. We will here quote only some of them [Mim81, LN96, LN99, Wu05]. For the full references, we may refer the reader to the references therein. There are also many heuristic or numerical results suggesting the existence of stable (stationary or periodic) solutions which indicate the coexistence of two species. We here cite some of them [IMN06, Kan93, LNY04, MK80, MNTT84]. It seems however that there are no results showing definitively the coexistence in a rigorous way yet.

Chapter 16

Characterization of Domains of Fractional Powers

This chapter is devoted to characterizing the domains of fractional powers Aθ for a sectorial operator A in a Hilbert or Banach space. However, the situations are quite different in Hilbert and Banach space cases. Anyway, in both cases, the boundedness of the H∞ functional calculus for A plays an important role for the complete characterization of D(Aθ ). In the Hilbert space case, the H∞ functional calculus for sectorial operators was invented by McIntosh in 1986 and was generalized in the Banach space case by Cowling–Doust–McIntosh–Yagi in 1996. In the Hilbert space case, Theorem 2.30 due to Kato ensures that any sectorial operator associated with a sesquilinear form has a bounded H∞ functional calculus. To the contrary, in the Banach space case, there is no convenient sufficient condition ensuring the boundedness of the functional calculus. Therefore, for elliptic differential operators in Lp (Ω), 1 < p < ∞, p = 2, equipped with suitable boundary conditions, we have to appeal to the theory of harmonic analysis or the theory of integral operators in order to prove the boundedness by some specific methods. On the other hand, we may notice that estimation of the domain D(Aθ ) from interior or exterior is considerably easy as we have already done in Theorems 2.24 and 2.25 and will do in Remark 16.6. In our abstract results, we have repeatedly used the fractional powers of coefficient linear operators in abstract evolution equations for describing our structural assumptions such as (3.29), (3.30), (4.2), (4.21), (5.4), (5.5), and (5.6). For verifying these assumptions, however, their complete characterization is not necessarily required, but their appropriate estimation is often sufficient.

1 Domains of Fractional Powers in Hilbert Spaces 1.1 Case of Self-Adjoint Operators Let X be a Hilbert space. When A is a positive definite self-adjoint operator of X, the domains of its fractional powers coincide with the interpolation spaces. A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5_16, © Springer-Verlag Berlin Heidelberg 2010

527

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16 Characterization of Domains of Fractional Powers

Theorem 16.1 Let A be a positive definite self-adjoint operator of X. For any 0 < θ < 1, [X, D(A)]θ = D(Aθ ) with isometry. Proof We first recall that A−z F is an X-valued analytic function for Re z > 0 and is a continuous function for Re z ≥ 0 for each F ∈ X with the estimate A−z  ≤ 1 for Re z ≥ 0, see Remark 2.7. Let U ∈ D(Aθ ). We consider the analytic function F (z) = A−z Aθ U . Then, F ∈ H(X, D(A)) with U = F (θ ), where H(X, D(A)) denotes the function space defined in Chap. 1, Sect. 5.1. Therefore, U ∈ [X, D(A)]θ , and   sup Aiy  Aθ U  = Aθ U . U [X,D(A)]θ ≤ F H ≤ −∞
Conversely, let U ∈ [X, D(A)]θ . By definition, there exists a function F ∈ H(X, D(A)) such that U = F (θ ). We then introduce a function of the form 2

Φ(z) = eε(z−θ) (Az−1 F (z), AV ),

ε > 0, V ∈ D(A).

Obviously, Φ(z) is holomorphic in the strip S = {z ∈ C; 0 < Re z < 1} and is continuous in the closure S. Furthermore, it is observed that lim|Im z|→∞ Φ(z) = 0 because 2 2 of |eε(z−θ) | ≤ eε(1−|Im z| ) . Therefore,   sup |Φ(iy)|, sup |Φ(1 + iy)| sup |Φ(z)| ≤ max z∈S

≤

−∞


sup

−∞


−∞
Aiy  eε F H V .

Taking z = θ , we obtain that |(Aθ−1 U, AV )| ≤ eε F H V ,

ε > 0, V ∈ D(A).

This means that U ∈ D(Aθ ) with the estimate Aθ U  ≤ eε F H . Taking the infimum of F H for all possible F and afterwards tending ε → 0, we observe that Aθ U  ≤ U [X,D(A)]θ . We hence verified that D(Aθ ) = [X, D(A)]θ with the isometry Aθ U  = U [X,D(A)]θ ,

U ∈ D(Aθ ).



We can in addition verify that A satisfies the following square function estimate. Let A ≥ δ and let {E(μ)}δ≤μ<∞ be the spectral resolution of A. Let Γω : λ = ρe±iω , 0 ≤ ρ < ∞, where 0 < ω ≤ π , and let 0 < θ < 1. Then, for F ∈ X,    ∞ 2θ−1 2(1−θ) 2  |λ| μ λ2θ−1 A1−θ (λ − A)−1 F  |dλ| = dE(μ)F 2 |dλ| |λ − μ|2 Γω Γω δ   ∞  ∞ ρ 2θ −1 μ2(1−θ) dρ dE(μ)F 2 = ikω − μ|2 |ρe δ 0 k=±1

1 Domains of Fractional Powers in Hilbert Spaces

=

 

∞

k=±1 0

ρ 2θ−1 dρ |ρeikω − 1|2



∞

529

dE(μ)F 2 .

δ

Meanwhile,  0

∞

ρ 2θ−1 dρ = |ρeikω − 1|2 =

 0

∞

ρ 2θ−1 (ρ − cos ω)2 + sin2 ω

dρ

π sin[(π − ω)(2θ − 1)] . sin(π − ω) sin[π(2θ − 1)]

Hence, it follows that  2  2π sin[(π − ω)(2θ − 1)] F 2 , λ2θ−1 A1−θ (λ − A)−1 F  |dλ| = sin(π − ω) sin[π(2θ − 1)] Γω

F ∈ X. (16.1)

1.2 Bounded H∞ Functional Calculus When A is not a self-adjoint operator, it is known that the domains of the fractional powers and the interpolation spaces do not necessarily coincide, see Sect. 1.5. In order that D(Aθ ) and [X, D(A)]θ coincide, A must have a bounded H∞ functional calculus, and also A must satisfy an integrable condition on some half-lines. We here introduce the notions of bounded H∞ functional calculus and integrable condition on half-lines. Let X be a Banach space with norm  · . Let A be a sectorial operator of X with angle 0 ≤ ωA < π . For any angle ω such that ωA < ω ≤ π , H∞ (Σω ) denotes the Banach algebra constituted by all bounded holomorphic functions defined in the sectorial domain Σω equipped with the norm f ∞ = sup |f (λ)|, λ∈Σω

f ∈ H∞ (Σω ).

Let H˚ ∞ (Σω ) be the subspace of H∞ (Σω ) defined by   H˚ ∞ (Σω ) = f ∈ H∞ (Σω ); sup |λ|νf |f (λ)| < ∞ with some νf > 0 . λ∈Σω

Then, for any function in H˚ ∞ (Σω ), a functional calculus of A can be defined by the integral  1 f (λ)(λ − A)−1 dλ, f ∈ H˚ ∞ (Σω ), (16.2) f (A) = 2πi Γ in the space L(X). Here, Γ is an integral contour such that Γ = Γ− ∪ Γ0 ∪ Γ+ ,

Γ0 : λ = δeiϕ (−ω ≤ ϕ ≤ ω ) and Γ± : λ = ρe±iω (δ ≤ ρ < ∞),

(16.3)

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16 Characterization of Domains of Fractional Powers

ω being any angle such that ωA < ω < ω, and δ being any constant such that 0 < δ < A−1 −1 . Since f (λ)(λ − A)−1 is integrable on Γ , f (A) is a bounded operator of X, and its definition is of course independent of the choice of the integral contour Γ lying in Σω and surrounding σ (A). Furthermore, for functions in H˚ ∞ (Σω ), it is easily observed that (f + g)(A) = f (A) + g(A) and

(f g)(A) = f (A)g(A),

f, g ∈ H˚ ∞ (Σω ). (16.4)

For the proof, recall that of the Law of Exponent (2.111). If the correspondence f → f (A) is extended from H∞ (Σω ) to L(X) as a bounded homomorphism of Banach algebra, then A is said to have a bounded H∞ functional calculus in Σω . By the definition, if A has a bounded H∞ functional calculus, then the estimate f (A) ≤ Cω f ∞ ,

f ∈ H∞ (Σω ),

(16.5)

must hold with some constant Cω > 0 as well as the formulas (f + g)(A) = f (A) + g(A) and If A has a bounded H∞ from (16.6) that

(f g)(A) = f (A)g(A),

f, g ∈ H∞ (Σω ). (16.6) functional calculus, then for any f ∈ H∞ (Σω ), it is seen



f (A)(1 + A)−1 F = f · (1 + λ)−1 (A)F,

F ∈ X.

In the meantime, [f · (1 + λ)−1 ](A) is determined by formula (16.2). These two things then mean that the value of f (A)U is uniquely determined for U ∈ D(A + 1) = D(A) from an integral of the form (16.2). Since f (A) is a bounded operator and since D(A) is dense, f (A) must also be uniquely determined. In other words, the bounded H∞ functional calculus of A, if it exists, is uniquely characterized by the three conditions (16.2), (16.5), and (16.6). The following convergence theorem is also observed. Theorem 16.2 Let a sectorial operator A with angle ωA have a bounded H∞ functional calculus in a domain Σω , ωA < ω ≤ π . If fn → f in H∞ (Σω ) as n → ∞, then fn (A) → f (A) in L(X). If fn ∞ are uniformly bounded for n and if fn → f pointwise in Σω as n → ∞, then fn (A) converges to f (A) strongly on X. Proof The first assertion is obvious from (16.5) and (16.6), since fn (A)−f (A) ≤ Cω fn − f ∞ . So, it suffices to prove the second assertion. Since fn · (1 + λ)−1 ∈ H˚ ∞ (Γω ) for every n, the Lebesgue dominated convergence theorem is applicable to the sequence fn · (1 + λ)−1 to conclude that fn (A)(1 + A)−1 is strongly convergent to f (A)(1 + A)−1 on X, that is, fn (A) converges to f (A) strongly on D(1 + A) = D(A). Then, the desired result is obtained by the uniform boundedness of fn (A) for n and by the density of D(A) in X. 

1 Domains of Fractional Powers in Hilbert Spaces

531

Let us finally present a sufficient condition in order that A has a bounded functional calculus. Assume that there exists a constant Cω > 0 for which the functional calculus f → f (A) satisfies on H˚ ∞ (Σω ) the estimate f (A)F  ≤ Cω f ∞ F ,

F ∈ X, f ∈ H˚ ∞ (Σω ).

(16.7)

For general f ∈ H∞ (Σω ), it is very natural to define the values of f (A) as follows: 

f (A)U = f · (1 + λ)−1 (A)(1 + A)U, U ∈ D(A2 ). Here, due to (16.4),

 f · (1 + λ)−1 (A)(1 + A)U 

= f · (1 + λ)−1 (1 + n−1 λ)−1 (A)(1 + n−1 A)(1 + A)U 

= f · (1 + n−1 λ)−1 (A)(1 + n−1 A)U. Condition (16.7) implies that  

  f · (1 + n−1 λ)−1 (A)(1 + n−1 A)U    ≤ Cω f ∞ sup (1 + n−1 λ)−1 (1 + n−1 A)U . λ∈Σω

Therefore,

  −1 π f (A)U  ≤ Cω sin max ,ω f ∞ (1 + n−1 A)U . 2 Letting n → ∞, we conclude that

f (A)U  ≤ Cω

  −1 π sin max ,ω f ∞ U , 2

U ∈ D(A2 ).

Since D(A2 ) is dense in X, f (A) is extended uniquely up to X as a bounded operator with the estimate

f (A) ≤ Cω

−1   π ,ω sin max f ∞ , 2

f ∈ H∞ (Σω ).

For f, g ∈ H∞ (Σω ), it is not difficult to verify that [f (A) + g(A)]U = (f + g) × (A)U and f (A)g(A)U = (fg)(A)U for all U ∈ D(A2 ). Therefore, (16.6) is also fulfilled. In this way, under (16.7), we have constructed a bounded H∞ functional calculus of A in Σω .

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16 Characterization of Domains of Fractional Powers

1.3 Integrable Condition Along Contour Let X be a Hilbert space with inner product (·, ·) and norm  · . Let A be a sectorial operator of X with angle 0 ≤ ωA < π . We say that A satisfies an integrable condition along the V-shaped contour Γω : λ = ρe±iω (0 ≤ ρ < ∞), where ωA < ω ≤ π , if the estimate    |λ|2θ−1 A2(1−θ) (λ − A)−2 F, G |dλ| ≤ Cω,θ F G, F, G ∈ X, (16.8) Γω

is valid for an exponent 0 < θ < 1 with some constant Cω,θ > 0.

1.4 Equivalence Theorem Let X be a Hilbert space with inner product (·, ·) and norm  · . Let A be a sectorial operator of X with angle ωA . In this subsection, we prove that the integrable condition along contours, the boundedness of H∞ functional calculus, and the boundedness of purely imaginary powers (see (2.135)) are all equivalent conditions in order that the domains of fractional powers and the complex interpolation spaces coincide. Theorem 16.3 Let A be a sectorial operator of X with angle ωA , 0 ≤ ωA < π , and let A∗ be its adjoint operator. Then, the following four conditions are mutually equivalent: (1) Along any V-shaped contour Γω , ωA < ω ≤ π , the integrable condition (16.8) holds for every exponent 0 < θ < 1. (2) In any domain Σω , ωA < ω ≤ π , A has a bounded H∞ functional calculus. (3) The purely imaginary powers Aiy are bounded on X and satisfy the estimates Aiy  ≤ N eβ|y| (y ∈ R) with some exponent β ≥ 0 and constant N ≥ 1. (4) For every 0 < θ < 1, D(Aθ ) = [X, D(A)]θ and D(A∗ θ ) = [X, D(A∗ )]θ with norm equivalence. Proof (1) ⇒ (2). Let (16.8) hold with an angle ω , ωA < ω < π , and an exponent 1

˚ 2 < θ < 1, and let ω < ω ≤ π . For f ∈ H∞ (Σω ), f (A) is given by (16.2) with an integral contour Γ given by (16.3). For F ∈ X and V ∈ D(A∗2(1−θ) ), we have (f (A)F, V ) = (f (A)A2(θ−1) F, A∗2(1−θ) V )    1 = f (λ)λ2(θ−1) (λ − A)−1 F, A∗2(1−θ) V dλ. 2πi Γ λ Let F (λ) = 0 f (μ)μ2(θ−1) dμ be a primitive of f (λ)λ2(θ−1) in Σω . Clearly,  |F (λ)| ≤ f ∞

0

1

|tλ|2(θ−1) |λ| dt = (2θ − 1)−1 f ∞ |λ|2θ−1 ,

λ ∈ Σω .

1 Domains of Fractional Powers in Hilbert Spaces

533

Then, by integration by parts, we have    −1 (f (A)F, V ) = F (λ) (λ − A)−2 F, A∗2(1−θ) V dλ. 2πi Γω

Here, we shifted the integral contour Γ to Γω by letting δ → 0. Then, (16.8) implies that    f ∞ |λ|2θ −1 A2(1−θ) (λ − A)−2 F, V |dλ| |(f (A)F, V )| ≤ 2π(2θ − 1) Γω

≤ [2π(2θ − 1)]−1 Cω ,θ f ∞ F V . Since D(A∗2(1−θ) ) is dense in X, it follows that 

f (A)F  ≤ 2π(2θ − 1)−1 Cω ,θ f ∞ F ,

F ∈ X.

This shows that f (A), f ∈ H˚ ∞ (Σω ), satisfies the sufficient condition (16.7). Thus, condition (16.8) along the contour Γω implies the boundedness of H∞ functional calculus for any angle ω > ω . In the case where we take an exponent 0 < θ < 12 , we can argue in a quite analogous way. The only difference is that we may use a primitive F (λ) = ∞ 2(θ−1) dμ of f (λ)λ2(θ−1) . This primitive obviously satisfies the estimate f (μ)μ λ  |F (λ)| ≤ f ∞

1

∞

 |tλ|2(θ−1) |λ| dt ≤ f ∞ |λ|2θ−1

= (1 − 2θ )−1 f ∞ |λ|2θ −1 ,

∞

t 2(θ−1) dt

1

λ ∈ Σω .

In the case where we take θ = 12 , however, it is impossible to derive directly the boundedness of H∞ functional calculus. We have to follow an indirect way of proof (see Remark 16.1). (2) ⇒ (3). For any angle ω, ωA < ω ≤ π , the holomorphic function λiy (−∞ < y < ∞) belongs to H∞ (Σω ). Therefore, from estimate (16.5) of bounded functional calculus it follows that Aiy  ≤ Cω λiy ∞ ≤ Cω eω|y| (−∞ < y < ∞). (3) ⇒ (4). The purely imaginary powers satisfy Aiy  ≤ N eβ|y| with some N ≥ 1 and some β ≥ 0. It is then immediate to verify that the analytic semigroup A−z defined for Re z > 0 is strongly continuous on X for Re z ≥ 0, that is, A−z satisfies the bounded extension (2.135). We can then repeat the same arguments as in the proof of Theorem 16.1. In fact, 2 for U ∈ D(Aθ ), the analytic function F (z) = e(z−θ) A−z Aθ U is in H(X, D(A)) with F (θ ) = U . Therefore, U ∈ [X, D(A)]θ , and U [X,D(A)]θ ≤ N eβ

2 /4

emax{θ

2 ,(1−θ)2 }

Aθ U .

(16.9)

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16 Characterization of Domains of Fractional Powers

Conversely, if U ∈ [X, D(A)]θ , then there exists an analytic function F ∈ H(X, D(A)) such that U = F (θ ). Then, we consider a function of the form Φ(z) = e(z−θ) (Az−1 F (z), A∗ V ), 2

V ∈ D(A∗ ),

which is a holomorphic function in the strip S and is a continuous function on the closure S. By the maximum principle, it is obtained that   sup |Φ(iy)|, sup |Φ(1 + iy)| sup |Φ(z)| ≤ max −∞
z∈S

≤e

max{θ 2 ,(1−θ)2 }

−∞
 sup

−∞
 2 e−y Aiy  F H V .

Therefore, |(Aθ−1 U, A∗ V )| ≤ N eβ

2 /4

emax{θ

2 ,(1−θ)2 }

F H V .

Consequently, U ∈ D(Aθ ), and Aθ U  ≤ N eβ

2 /4

emax{θ

2 ,(1−θ)2 }

F H .

Furthermore, taking the infimum of F H for all possible F , we have Aθ U  ≤ N eβ

2 /4

emax{θ

2 ,(1−θ)2 }

U [X,D(A)]θ .

(4) ⇒ (1). We first prove that the inclusion D(Aθ ) ⊂ [X, D(A)]θ implies a square function estimate of A of the form (16.1). Then, the desired estimate will be shown as an immediate consequence of these estimates. Lemma 16.1 For 0 < θ < 1, the inclusion D(Aθ ) ⊂ [X, D(A)]θ with estimate U [X,D(A)]θ ≤ Dθ Aθ U ,

U ∈ D(Aθ ),

implies the square function estimate  2  |λ|2θ−1 A1−θ (λ − A)−1 F  |dλ| ≤ D˜ ω,θ F 2 ,

F ∈ X,

(16.10)

(16.11)

Γω

along any contour Γω , ωA < ω ≤ π . Here, D˜ ω,θ is given by 2π(2Mω + 1)Dθ sin[(π − ω)(2θ − 1)] D˜ ω,θ = . sin(π − ω) sin[π(2θ − 1)] Proof Let us regard D(A) as a Hilbert space with the inner product (A·, A·). Then, since D(A) ⊂ X, we can consider the triplet D(A) ⊂ X ⊂ D(A)∗ . On D(A), h(U, V ) = (AU, AV ) is obviously a symmetric sesquilinear form; consequently,

1 Domains of Fractional Powers in Hilbert Spaces

535

the associated operator H of h(·, ·) is defined, and its part H|X is a positive defi1 nite self-adjoint operator of X. Put H = {H|X } 2 ; then, according to Theorem 2.34, D(H ) = D(A) with isometry H U  = AU  for U ∈ D(A). In the meantime, by Theorem 16.1, D(H θ ) = [X, D(H )]θ = [X, D(A)]θ with isometry. Therefore, (16.10) implies H θ U  ≤ Dθ Aθ U  for U ∈ D(Aθ ). Since A1−θ (λ − A)−1 = A(λ − A)−1 A−θ 

= λ(λ − A)−1 AH −1 − A(λ − A)−1 H (λ − H )−1 A−θ , it follows that  1−θ    A (λ − A)−1 F  ≤ (2Mω + 1)H 1−θ (λ − H )−1 H θ A−θ F , Hence, 

F ∈ X.

 2 |λ|2θ−1 A1−θ (λ − A)−1 F  |dλ|

Γω



2  |λ|2θ −1 H 1−θ (λ − H )−1 H θ A−θ F  |dλ|,

≤ (2Mω + 1)

F ∈ X.

Γω

Since H is self-adjoint, we can now utilize (16.1) to conclude (16.11).



For 0 < θ < 1, let D(Aθ ) ⊂ [X, D(A)]θ and D(A∗θ ) ⊂ [X, D(A∗ )]θ with continuous embedding U [X,D(A)]θ ≤ Dθ U , U ∈ D(Aθ ), (16.12) U [X,D(A∗ )]θ ≤ Dθ∗ U , U ∈ D(A∗θ ). Then, by Lemma 16.1, we have not only (16.11) but also   2 ∗ |λ|2θ−1 A∗(1−θ) (λ − A∗ )−1 F  |dλ| ≤ D˜ ω,θ F 2 ,

F ∈ X,

Γω

∗ > 0. Consequently, with a suitable constant D˜ ω,θ    |λ|2θ −1 A2(1−θ) (λ − A)−2 F, G |dλ| Γω



1 2  2 |λ|2θ −1 A1−θ (λ − A)−1 F  |dλ|

≤ Γω

 ×

1 2  2 |λ|2θ −1 A∗(1−θ) (λ − A∗ )−1 G |dλ|

Γω ∗ ≤ [D˜ ω,θ D˜ ω,θ ] 2 F  G, 1

F, G ∈ X.

Thus, (16.12) implies (16.8) for any V-shaped contour Γω , ωA < ω ≤ π .

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16 Characterization of Domains of Fractional Powers



We have thus accomplished the proof of theorem.

Remark 16.1 If the integrable condition (16.8) holds for an angle ω, ωA < ω < π , and for an exponent 0 < θ < 1, then actually it holds for every ω, ωA < ω ≤ π and for every 0 < θ < 1. In fact, in the proof of (1) ⇒ (2), we have shown that the integrable condition (16.8) with an ω , ωA < ω < π , and with a θ such that 0 < θ < 1, θ = 12 , implies the boundedness of H∞ functional calculus for any angle ω, ω < ω ≤ π . And we have shown that this implies (3). Since (1) and (3) are equivalent, (16.8) must actually hold for every ω, ωA < ω ≤ π , and every 0 < θ < 1. It therefore remains to show that (16.8) with an angle ω , ωA < ω < π , and with θ = 12 also implies the boundedness of H∞ functional calculus. Let F ∈ X and V ∈ D(A∗ ). For y = 0, we have    1 λiy−1 (λ − A)−1 F, A∗ V dλ, (Aiy F, V ) = (Aiy−1 F, A∗ V ) = 2πi Γ where Γ is an integral contour given by (16.3). By integration by parts,    1 (Aiy F, V ) = λiy (λ − A)−2 F, A∗ V dλ. 2πy Γω

Here, we shifted the integral contour Γ to Γω by letting δ → 0. Therefore, (16.8) with θ = 12 implies that 1 |(A F, V )| ≤ 2π|y| iy





eω |y| Cω , 1   2 U V . |λ | A(λ − A)−2 F, V |dλ| ≤ 2π|y| Γω

iy



This shows that Aiy  ≤ (2π)−1 Cω , 1 eω |y| for |y| ≥ 1. For |y| < 1, we observe by 2



the law of exponent that Aiy  ≤ Ai(y+2) A−2i  ≤ [(2π)−1 e2ω Cω , 1 ]2 . Hence, condition (16.8) (θ = 12 ) certainly implies (3), and of course (2) too.

2

Remark 16.2 If an H∞ functional calculus of A is bounded in a sectorial domain Σω , ωA < ω ≤ π , then so is in every domain Σω , ωA < ω ≤ π . Indeed, as shown in the proof of (2) ⇒ (3), the boundedness for an angle ω implies (3). Remark 16.3 If the purely imaginary powers Aiy are bounded, then they satisfy the estimates Aiy  ≤ Nω eω|y| (−∞ < y < ∞) with an arbitrary exponent ω > ωA and some constant Nω > 0 dependent on ω. Indeed, under this assumption, A has a bounded H∞ functional calculus in any sectorial domain Σω , ωA < ω ≤ π . So, apply (16.5) to the holomorphic function λiy . Remark 16.4 As shown in the proof of (4) ⇒ (1), the inclusions D(Aθ ) ⊂ [X, D(A)]θ and D(A∗θ ) ⊂ [X, D(A∗θ) ]θ for an exponent 0 < θ < 1 imply (1). Therefore, these inclusions imply D(Aθ ) = [X, D(A)]θ and D(A∗θ ) = [X, D(A∗ )]θ for every 0 < θ < 1.

1 Domains of Fractional Powers in Hilbert Spaces

537

1.5 Counterexample In this subsection, we present an example of sectorial operator which does not sat1 isfy the condition D(A 2 ) = [X, D(A)] 1 and hence neither conditions (1)–(4) in 2 Theorem 16.3. For nonnegative integers n = 0, 1, 2, . . . , let Dn = diag{1, 4, . . . , 4n } be an (n + 1) × (n + 1) diagonal matrix, and let Hn = {hj k ; 0 ≤ j, k ≤ n} be an (n + 1) × (n + 1) Hilbert matrix such that hj k = 0 for j = k and hj k = 1/(j − k) for j = k. We consider Dn and Hn as linear operators of the Hilbert space Cn+1 . Clearly, Dn is a self-adjoint operator satisfying   Dn (λ − Dn )−1  ≤

1 , sin ω

λ ∈ Σω = {λ ∈ C; | arg λ| < ω},

for an arbitrarily fixed 0 < ω ≤ π2 . As will be verified by the lemma below, the operator norm of Hn is uniformly estimated by Hn  ≤ π. We next define the matrix-valued holomorphic function An (z) = (1 + zHn )Dn , where r is fixed as 0 < r <

sin ω π .

|z| ≤ r,

For λ ∈ Σω , we have

−1 (λ − An (z))−1 = (λ − Dn )−1 1 − zHn (λ − Dn )−1 =

∞ 

m (λ − Dn )−1 Hn Dn (λ − Dn )−1 zm ,

|z| ≤ r,

m=0

because of Hn Dn (λ − Dn )−1  ≤   (λ − An (z))−1  ≤



π sin ω .

Furthermore,

 sin ω |λ|, sin ω − πr

λ∈ / Σω , |z| ≤ r,

(16.13)

and   An (z)−1  ≤

1 , 1 − πr

|z| ≤ r.

(16.14)

It is also clear that (1 − πr)Dn ξ  ≤ An (z)ξ  ≤ (1 + πr)Dn ξ ,

ξ ∈ Cn+1 , |z| ≤ r. 1

(16.15)

We are now concerned with the square root of An (z). In fact, An (z)− 2 is a holomorphic function on the closed disk |z| ≤ r which can be written by

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16 Characterization of Domains of Fractional Powers 1

An (z)− 2 = =

1 2πi ∞  m=0



1

λ− 2 (λ − An (z))−1 dλ

Γω



1 2πi

m 1 λ− 2 (λ − Dn )−1 Hn Dn (λ − Dn )−1 zm dλ,

Γω

(16.16) where Γω : λ = ρe±ωi , 0 ≤ ρ < ∞. Then, it holds that  1 1 maxDn2 An (z)− 2  ≥



|z|=r

 1 n+1 r, √ log 2 3 2

n = 0, 1, 2, . . . .

(16.17)

To verify this underestimate, we introduce the matrix Qn =

1 2πi



1

1

Dn2 An (z)− 2 z−2 dz,

n = 0, 1, 2, . . . ,

Cr

where Cr : |z| = r. Obviously,  1 1 Qn  ≤ r −1 maxDn2 An (z)− 2 .

(16.18)

|z|=r

While, from (16.16), Qn is calculated as   1

m 1 − 12 2 Qn = λ D (λ − Dn )−1 Hn Dn (λ − Dn )−1 zm−2 dλ dz n (2πi)2 Cr Γω m=0  1 1 1 λ− 2 Dn2 (λ − Dn )−1 Hn Dn (λ − Dn )−1 dλ = 2πi Γω    1 λ−1/2 j k dλ · 2 = 4 h ; 0 ≤ j, k ≤ n jk 2πi Γω (λ − 4j )(λ − 4k )   2k hj k ; 0 ≤ j, k ≤ n . = − j 2 + 2k ∞ 

√ Then, for e = (1, 1, 1, . . .)/ n + 1 ∈ Cn+1 , 1

Qn e = √ n+1 1

=√ n+1



n  k=0

n−j  k=1

2k − j ; 0≤j ≤n (2 + 2k )(j − k)



 j  2k 1 − ; 0≤j ≤n . (2k + 1)k (2k + 1)k k=1

1 Domains of Fractional Powers in Hilbert Spaces

539

The first half components, 0 ≤ j ≤ n2 , of Qn e are estimated from below by √

n−j 1 1 2k − 1 ≥ √ n + 1 k=1 (2k + 1)k 3 n + 1 k=1 k  n+1−j log(n + 1 − j ) 1 1 dx > > √ . √ x 3 n+1 1 3 n+1

1

n−j 

Therefore, Qn 2 ≥ Qn e2 >

 1 log2 (n + 1 − j ) 9(n + 1) n 0≤j ≤ 2

>

n+1 1 n+1 n+1 1 · log2 = log2 . 9(n + 1) 2 2 18 2

This, together with (16.18), then yields (16.17). We can construct, under these preparations, our counterexample. Let X = ∞ n+1 be the sum of Hilbert spaces Cn+1 . Choose, for each n, a point n=0 C zn , |zn | = r, such that   1   21 Dn An (zn )− 12  = maxDn2 An (z)− 12 ,

(16.19)

|z|=r

 and let A = ∞ By (16.13) and n=0 An (zn ) be the sum of linear operators An (zn ). (16.14), A is a sectorial operator of X with angle ωA ≤ ω. Let Λ = ∞ n=0 Dn be the sum of operators Dn . Then, Λ is a self-adjoint operator of X. In view of (16.15), D(A) coincides with D(Λ) with norm equivalence. In addition, [X, D(A)]θ = [X, D(Λ)]θ = D(Λθ ) for every 0 < θ < 1. 1 The operator A cannot satisfy [X, D(A)] 1 = D(A 2 ), for the coincidence im1

2

1

1

1

plies D(A 2 ) = D(Λ 2 ) with norm equivalence and consequently Λ 2 A− 2 ∈ L(X). However, the last assertion is impossible because of (16.17) and (16.19). Lemma 16.2 For every n, it holds that Hn  ≤ π . Proof Let 2 (Z) denote the 2 space of all sequences ξ = (. . . , ξ−1 , ξ0 , ξ1 , . . .) de 1 fined on Z with ξ  = ( n∈Z |ξn |2 ) 2 < ∞, and let L2 (−π, π) denote the L2 space on the interval (−π, π). The Fourier transform from 2 (Z) to L2 (−π, π) is defined by  √ F[ξ ](x) = ( 2π)−1 ξn einx , ξ ∈ 2 (Z). n∈Z

√ As is well known, the family {( 2π)−1 einx }n∈Z is an orthonormal basis of the space L2 (−π, π), and the Fourier transform is an isometry, i.e., ξ 2 = F[ξ ]L2 , ξ ∈ 2 (Z).

540

16 Characterization of Domains of Fractional Powers

Let H be the Hilbert transform in 2 (Z) defined by the matrix H = {hj k ; j, k ∈ Z} such that hj k = 0 for j = k and hj k = 1/(j − k) for j = k. We then observe that   √ F[H ξ ](x) = ( 2π)−1 hj k ξk eij x j ∈Z k∈Z

  √ = ( 2π)−1 k∈Z j ∈Z−{k}

= 2ih(x)F[ξ ](x), where h(x) =

∞  j =1

 1 i(j −k)x ξk eikx e j −k

ξ ∈ 2 (Z),

⎧ ⎪−(π + x)/2, sin j x ⎨ = 0, ⎪ j ⎩ (π − x)/2,

−π < x < 0, x = 0, 0 < x < π.

Since hL∞ ≤ π2 , it is deduced that F[H ξ ]L2 ≤ πF[ξ ]L2 . Therefore, H is a bounded operator of 2 (Z) with norm H  ≤ π . For each n = 0, 1, 2, . . . , let Rn be the injection from the (n + 1)-dimensional space Cn+1 into 2 (Z) defined by Rn (ξ0 , ξ1 , . . . , ξn ) = (. . . , 0, ξ0 , ξ1 , . . . , ξn , 0, . . .). Conversely, let Pn be the projection from 2 (Z) onto Cn+1 such that Pn Rn = 1 on Cn+1 . Then, it is obvious that the matrix operator Hn is given by Hn = Pn H Rn . Hence, it follows that Hn  ≤ Pn H Rn  ≤ π . 

2 Some Result on Matrix Operators Let Xk , k = 1, 2, be two Hilbert spaces with inner products (·, ·)k and norms  · k , respectively, and let X be the product Hilbert space of X1 and X2 . Let Ak , k = 1, 2, be two sectorial operators of Xk with angles 0 ≤ ωk < π , respectively. Consider a matrix operator of the form  A=

A1 0

 B , A2

D(A) =

   u1 ; u1 ∈ D(A1 ) and u2 ∈ D(A2 ) , u2

(16.20)

where B is a third linear operator from D(B) ⊂ X2 into X1 . We assume that D(B) ⊃ D(A2 ) and B is a bounded operator from D(A2 ) to X1 and, for its adjoint B ∗ : D(B ∗ ) ⊂ X1 → X2 , assume that D(B ∗ ) ⊃ D(A∗1 ) and B ∗ is a bounded operator from D(A∗1 ) to X2 .

2 Some Result on Matrix Operators

541

It is already known by Theorem 2.16 that A becomes a sectorial operator of X with angle ωA = max{ω1 , ω2 }. It is also known that, for ωA < ω ≤ π ,   (λ − A1 )−1 (λ − A1 )−1 B(λ − A2 )−1 −1 , λ∈ / Σω . (λ − A) = 0 (λ − A2 )−1 We verify the following theorem. Theorem 16.4 Let A1 (resp. A2 ) be a sectorial operator of a Hilbert space X1 (resp. X2 ) with angle ω1 (resp. ω2 ). Let B : D(B) ⊂ X2 → X1 be a linear operator such that D(B) ⊃ D(A2 ) and B ∈ L(D(A2 ), X1 ) and such that D(B ∗ ) ⊃ D(A∗1 ) and B ∗ ∈ L(D(A∗1 ), X2 ). If both A1 and A2 satisfy the equivalent conditions in Theorem 16.3, then the matrix operator A defined by (16.20) also satisfies the similar conditions in the product space X. In particular, D(Aθ ) = [X, D(A)]θ and D(A∗θ ) = [X, D(A∗ )]θ for every 0 < θ < 1. Proof Our goal is to show that A has a bounded H∞ functional calculus. We beiy gin however with noticing the lemma below. By assumption, we have Ak  ≤ Nk eβk |y| (−∞ < y < ∞), k = 1, 2, with exponents βk ≥ 0 and constants Nk ≥ 1, respectively. Then, we observe the following result. Lemma 16.3 For any 0 ≤ θ ≤ 1, √  θ−1      A BA−θ u ≤ (N1 N2 )e(β1 +β2 ) θ(1−θ) B ∗ A∗−1 1−θ BA−1 θ u2 , 1 2 1 2 1   u ∈ D A1−θ . (16.21) 2 Proof When θ = 1, (16.21) is clear from the assumption. Meanwhile, when θ = 0, from  −1    A1 Bu, g 1 = u, B ∗ A∗−1 1 g 2 , u ∈ D(B), g ∈ X1 , it follows that

   −1  A Bu ≤ B ∗ A∗−1 u2 , 1 1 1

u ∈ D(B).

Hence, (16.21) is clear in this case, too. Moreover, this means that A−1 1 B has a

−1 ∗ ∗−1 unique bounded extension A−1 1 B from X2 into X1 with A1 B ≤ B A1 . For 0 < θ < 1, we can apply the Heinz–Kato inequality. Indeed, we know that −1 A1 B ∈ L(X2 , X1 ), and it is clear that A−1 1 B ∈ L(D(A2 ), D(A1 )) with

 −1  A B  1 L(D(A

2 ),D(A1 ))

  . ≤ BA−1 2

θ θ Therefore, Theorem 2.31 provides that A−1 1 B ∈ L(D(A2 ), D(A1 )) with

 −1  A B  1

L(D(Aθ2 ),D(Aθ1 ))

√  θ(1−θ)  ∗

≤ (N1 N2 )e(β1 +β2 )

This then shows that (16.21) is valid for any 0 < θ < 1.

1−θ  −1 θ  BA  . B A∗−1 1 2 

542

16 Characterization of Domains of Fractional Powers

Let f ∈ H˚ ∞ (Σω ), where ωA < ω ≤ π . For F = (f (A)F, G) =

1 2πi

1 = 2πi



 g1   f1  f2 , G = g2 ∈ X,

  f (λ) (λ − A)−1 F, G dλ

Γω



Γω

1 + 2πi

  

  f (λ) (λ − A1 )−1 f1 , g1 1 + (λ − A2 )−1 f2 , g2 2 dλ

 Γω

  f (λ) (λ − A1 )−1 B(λ − A2 )−1 f2 , g1 1 dλ = I + II.

Since A1 and A2 have a bounded H∞ functional calculus, the first term is estimated by |I | ≤ C1,ω f ∞ f1 1 g1 1 + C2,ω f ∞ f2 2 g2 2 . Using (16.21) with θ = 12 , we estimate the second term by  1 1 1  − 12   f ∞ A BA− 2 A 2 (λ − A2 )−1 f2 , A∗ 2 λ − A∗ −1 g1 |dλ| |II| ≤ 1 1 2 2 1 1 2π Γω   12   1   C˜ B f ∞ A (λ − A2 )−1 f2  A∗ 2 λ − A∗ −1 g1  |dλ|, ≤ 1 2 1 2 1 2π Γω −1 2 2 where C˜ B = (N1 N2 )e(β1 +β2 )/2 B ∗ A∗−1 1  BA2  . Here, we utilize Lemma 16.1 to conclude that 1

1

∗ |II| ≤ (2π)−1 C˜ B D˜ 2,ω, 1 D˜ 1,ω, 1 f ∞ f2 2 g1 1 . 2

2

Therefore,

 ∗ |(f (A)F, G)| ≤ C1,ω + C2,ω + (2π)−1 C˜ B D˜ 2,ω, 1 D˜ 1,ω, 1 f ∞ F G 2

2

and

 ∗ f (A)F  ≤ C1,ω + C2,ω + (2π)−1 C˜ B D˜ 2,ω, 1 D˜ 1,ω, 1 f ∞ F . 2

2

In this way, we have proved that estimate (16.7) is valid for A. Consequently, the  H∞ functional calculus for A is bounded in H∞ (Σω ).

3 Domains of Fractional Powers in Banach Spaces 3.1 Equivalence Theorem in Banach Spaces Let X be a reflexive Banach space with norm  · . Let X ∗ be an adjoint space of X with norm  · ∗ , their duality product on X × X ∗ being denoted by ·, ·. We want

3 Domains of Fractional Powers in Banach Spaces

543

to generalize Theorem 16.3 to the sectorial operators of Banach spaces. However, this is possible only partially. Theorem 16.5 Let X be a reflexive Banach space, and let A be a sectorial operator of X with angle 0 ≤ ωA < π . For A and its adjoint operator A∗ , consider the following four conditions: (1) Along any V-shaped contour Γω , ωA < ω ≤ π , the similar integrable condition 

  |λ|2θ −1 A2(1−θ) (λ − A)−2 F, G |dλ| Γω

≤ Cω,θ F G∗ ,

F ∈ X, G ∈ X ∗ ,

(16.22)

as (16.8) holds for every exponent 0 < θ < 1 with some constant Cω,θ > 0. (2) In any domain Σω , ωA < ω ≤ π , A has a bounded H∞ functional calculus. (3) The purely imaginary powers Aiy are bounded on X and satisfy the estimates Aiy  ≤ Neβ|y| (y ∈ R) with some exponent β ≥ 0 and constant N ≥ 1. (4) For every 0 < θ < 1, D(Aθ ) = [X, D(A)]θ and D(A∗θ ) = [X ∗ , D(A∗ )]θ with norm equivalence. Then, (1) and (2) are mutually equivalent. Either (1) or (2) implies (3). (3) implies (4). Proof (1) ⇒ (2). Thanks to the Theorem 1.19 on the adjoint operators in reflexive Banach spaces, we can follow quite similar arguments to those used in the proof of (1) ⇒ (2) of Theorem 16.3. Let ωA < ω < ω < π . For 12 < θ < 1, (16.22) with angle ω yields that |f (A)F, V | ≤ [2π(2θ − 1)]−1 Cω ,θ f ∞ F V ∗ , F ∈ X, V ∈ D(A∗2(1−θ) ), for any f ∈ H˚ ∞ (Σω ). Similarly, for 0 < θ < 12 , (16.22) with angle ω yields that |f (A)F, V | ≤ [2π(1 − 2θ )]−1 Cω ,θ f ∞ F V ∗ , F ∈ X, V ∈ D(A∗2(1−θ) ), for any f ∈ H˚ ∞ (Σω ). Anyway, since (16.22) holds for every 0 < θ < 1, any f ∈ H˚ ∞ (Σω ) satisfies (16.7). Hence, A possesses a bounded H∞ functional calculus in H∞ (Σω ). (2) ⇒ (1). Let F ∈ X and G ∈ X ∗ , and let 0 < ε < R < ∞. It is then easy to see that there exist Borel functions hk (ρ) (k = ±1) with |hk | = 1 such that

544

16 Characterization of Domains of Fractional Powers



  |λ|2θ−1 A2(1−θ) (λ − A)−2 F, G | dλ|

Γω,ε,R



= ε

R



  ρ 2θ−1 A2(1−θ) (ρekωi − A)−2 F, G hk (ρ) dρ,

k=±1

where the contour Γω,ε,R : λ ∈ Γω , ε ≤ |λ| ≤ R, is a part of Γω . Here, hk (ρ) may depend on F and G. Define holomorphic functions bk (z) in Σω , ωA < ω < ω, by the formulas  R ρ 2θ−1 z2(1−θ) (ρekωi − z)−2 hk (ρ) dρ. bk (z) = ε

Then, it is verified that  |bk (z)| ≤ |z|2(1−θ)

R

ρ 2θ−1 |ρekωi − z|−2 dρ

ε

≤



∞

sup

|ζ |=1,| arg ζ |<ω

ρ 2θ−1 |ρekωi − ζ |−2 dρ = Cω,ω ,θ ,

z ∈ Σω ,

0

that is, bk ∈ H∞ (Σω ) with bk ∞ ≤ Cω,ω ,θ . We notice that Cω,ω ,θ is independent of F and G, although bk (z) is not. Furthermore, it is clear that  bk (A) =

R

ρ 2(θ−1) A2(1−θ) (ρekωi − A)−2 hk (ρ) dρ.

ε

We therefore have    |λ|2θ−1 A2(1−θ) (λ − A)−2 F, G |dλ| Γω,ε,R

=



bk (A)F, G ≤

k=±1



bk (A)F G∗ ≤ 2Cω,ω ,θ F G∗ .

k=±1

Taking limits as ε → 0 and R → ∞, we deduce estimate (16.22). (2) ⇒ (3). In the proof of (2) ⇒ (3) of Theorem 16.3, no Hilbert structure of X was used. So, the same argument can be repeated in the present case. (3) ⇒ (4). In this part we are also able to repeat the same argument as in the proof of (3) ⇒ (4) of Theorem 16.3. The only difference is that, as the function Φ(z), we have to use Φ(z) = e(z−θ) Az−1 F (z), A∗ V , 2

We have thus accomplished the proof of theorem.

V ∈ D(A∗ ). 

4 Domains of Fractional Powers of Elliptic Operators in L2 Spaces

545

4 Domains of Fractional Powers of Elliptic Operators in L2 Spaces Let Ω be a bounded domain in Rn with Lipschitz boundary. Consider the sesquilinear form  n   aij (x)Di uDj v dx + c(x)uv dx, u, v ∈ H 1 (Ω), (16.23) a(u, v) = i,j =1 Ω

Ω

on the space H 1 (Ω). As in Chap. 2, Sect. 1, aij (x), 1 ≤ i, j ≤ n, are real-valued functions in Ω satisfying the conditions aij ∈ L∞ (Ω), n 

1 ≤ i, j ≤ n,

aij (x)ξi ξj ≥ δ|ξ |2 ,

(16.24)

ξ = (ξ1 , . . . , ξn ) ∈ Rn , a.e. x ∈ Ω,

(16.25)

i,j =1

with some constant δ > 0, and c(x) is a real-valued function in Ω satisfying c ∈ L∞ (Ω) and

c(x) ≥ c0 > 0, a.e. x ∈ Ω.

(16.26)

These conditions imply that a(·, ·) fulfills (2.6) and (2.7) on H 1 (Ω). We consider the triplet H 1 (Ω) ⊂ L2 (Ω) ⊂ H 1 (Ω)∗ . The operator associated with a(·, ·) and its part in L2 (Ω) are denoted by A and by A = A|L2 , respectively. The operator A (resp. A) is a sectorial operator of H 1 (Ω)∗ (resp. L2 (Ω)) with angle < π2 . The operator A (resp. A) is regarded as a realization of the strongly  elliptic operator − ni,j =1 Dj [aij (x)Di ·] + c(x) in H 1 (Ω)∗ (resp. L2 (Ω)) under the Neumann-type boundary conditions n  ∂u ≡ νj (x)aij (x)Di u = 0 ∂νA

on ∂Ω.

(16.27)

i,j =1

In this section, we are concerned with characterizing the domains of fractional powers Aθ and Aθ for 0 ≤ θ ≤ 1.

4.1 Case of Domains with Lipschitz Boundary We first consider the case where Ω is a bounded domain of Rn with Lipschitz boundary ∂Ω. We assume that aij (x) are symmetric, i.e., aij (x) = aj i (x)

a.e. x ∈ Ω, 1 ≤ i, j ≤ n.

(16.28)

According to Theorem 2.14, this condition implies that A∗ = A|H 1 with respect to the adjoint pair {H 1 (Ω), H 1 (Ω)∗ } and A = A∗ in L2 (Ω). In particular, A is a

546

16 Characterization of Domains of Fractional Powers

positive definite self-adjoint operator of L2 (Ω). We can then use Corollary 2.4 to conclude that  1  D(Aθ ) = D A 2 +θ = [L2 (Ω), H 1 (Ω)]2θ

for 0 ≤ θ ≤

1 2

with norm equivalence C −1 u[L2 ,H 1 ]2θ ≤ Aθ uL2 ≤ Cu[L2 ,H 1 ]2θ , u ∈ D(Aθ ),  1   1  C −1 u[L2 ,H 1 ]2θ ≤ A 2 +θ uH 1∗ ≤ Cu[L2 ,H 1 ]2θ , u ∈ D A 2 +θ , where the constant C > 0 is determined by δ, c0 and the norms aij L∞ , cL∞ only. Since [L2 (Ω), H 1 (Ω)]θ = H θ (Ω) for 0 ≤ θ ≤ 1 with norm equivalence due to (1.71), we obtain the following results. Theorem 16.6 Let Ω be a bounded domain in Rn with Lipschitz boundary. Let (16.24), (16.25), (16.26), and (16.28) be satisfied. Then, the operator A of H 1 (Ω)∗ and its part A in L2 (Ω) satisfy  1  1 D(Aθ ) = D A 2 +θ = H 2θ (Ω) for all 0 ≤ θ ≤ 2 with norm equivalence C −1 uH 2θ ≤ Aθ uL2 ≤ CuH 2θ , u ∈ D(Aθ ),  1   1  C −1 uH 2θ ≤ A 2 +θ uH 1∗ ≤ CuH 2θ , u ∈ D A 2 +θ , C > 0 being determined by Ω and by δ, c0 , and the norms aij L∞ , cL∞ .

4.2 Case of Domains with C2 Boundary We want to characterize the domains D(Aθ ) when 12 ≤ θ ≤ 1. For this purpose, we assume that Ω is a bounded domain with C2 boundary and the functions aij (x) satisfy aij (x) ∈ C1 (Ω),

1 ≤ i, j ≤ n.

(16.29)

By the trace theorem, Theorem 1.39, the Neumann-type boundary conditions (16.27) have a meaning for any function u ∈ H s (Ω), s > 32 . In view of this fact, we define, for 32 < s ≤ 2, the closed subspace of H s (Ω) by   ∂u HNs (Ω) = u ∈ H s (Ω); = 0 on ∂Ω . ∂νA

4 Domains of Fractional Powers of Elliptic Operators in L2 Spaces

547

By Theorem 2.6, we already know that D(A) = HN2 (Ω) with the estimate ˜ δu H 2 ≤ AuL2 ,

u ∈ D(A),

(16.30)

δ˜ > 0 being a positive constant. In the present case, the adjoint operator A∗ of A in L2 (Ω) is utilized in an essential way. Since A∗ is determined from the adjoint sesquilinear form a ∗ (u, v) = a(v, u) of a(u, v), it is seen that D(A∗ ) = {v ∈ H 2 (Ω); ∂ν∂v∗ = 0 on ∂Ω}, where A n ∂ ∂ i,j =1 νi (x)aij (x)Dj is the adjoint boundary operator of ∂νA . By the same ∂νA∗ = reason as for (16.30), it holds that δ ∗ vH 2 ≤ A∗ vL2 ,

v ∈ D(A∗ ),

with a positive constant δ ∗ > 0. We then verify the following characterization theorem. Theorem 16.7 Let Ω be a bounded domain in Rn with C2 boundary. Let (16.25), (16.26), and (16.29) be satisfied. Then, H 2θ (Ω) if 0 ≤ θ < 34 ,

 D(Aθ ) = L2 (Ω), HN2 (Ω) θ = HN2θ (Ω) if 34 < θ ≤ 1, with norm equivalence C −1 uH 2θ ≤ Aθ uL2 ≤ CuH 2θ ,

u ∈ D(Aθ ),

˜ δ ∗ , and the norms aij C1 , cL∞ . C > 0 being determined by Ω and by δ, c0 , δ, Proof The proof is divided into several parts. (I) D(Aθ ) ⊂ H 2θ (Ω). By Corollary 2.1, the purely imaginary powers of A satπ isfy Aiy L(L2 ) ≤ e 2 |y| , −∞ < y < ∞. Therefore, this fact and Theorem 16.3 yield that D(Aθ ) = [L2 (Ω), D(A)]θ for every 0 ≤ θ ≤ 1 with the estimate u[L2 ,D(A)]θ ≤ eπ

2 /16

emax{θ

2 ,(1−θ)2 }

Aθ uL2 ,

u ∈ D(Aθ )

(due to (16.9)). In the meantime, by applying the interpolation theorem, Theorem 1.15, to the embedding j : D(A) → H 2 (Ω), we see that [L2 (Ω), D(A)]θ ⊂ [L2 (Ω), H 2 (Ω)]θ ,

0 ≤ θ ≤ 1,

with δ˜θ u[L2 ,H 2 ]θ ≤ u[L2 ,D(A)]θ ,

u ∈ [L2 (Ω), D(A)]θ .

Furthermore, by Theorem 1.35, we have [L2 (Ω), H 2 (Ω)]θ = H 2θ (Ω) with norm equivalence. Therefore, we conclude that D(Aθ ) ⊂ H 2θ (Ω) for every 0 ≤ θ ≤ 1 with the estimate uH 2θ ≤ C δ˜−θ Aθ uL2 ,

u ∈ D(Aθ ).

548

16 Characterization of Domains of Fractional Powers

It is known that D(A) is dense in D(Aθ ), namely, if u ∈ D(Aθ ), then there is a sequence uk ∈ D(A) such that uk → u in D(Aθ ) and a fortiori in H 2θ (Ω). This implies that, if u ∈ D(Aθ ), then u satisfies (16.27) if 32 < 2θ ≤ 2. We have therefore shown that D(Aθ ) ⊂ HN2θ (Ω) for every

3 4

< θ ≤ 1.

(II) D(Aθ ) ⊃ H 2θ (Ω) for 12 ≤ θ < 34 . We first notice that the argument in the first part is available for the adjoint operator A∗ , too. So, we have D(A∗θ ) ⊂ H 2θ (Ω) for every 0 ≤ θ ≤ 1 with vH 2θ ≤ C(δ ∗ )−θ A∗θ vL2 , Let

v ∈ D(A∗θ ).

(16.31)

≤ θ < 34 , and let u ∈ H 2θ (Ω) and v ∈ D(A∗ ). Using that   1 −1 ∗(θ−1) θ−1 ∗ −1 A = λ (λ − A ) dλ = λθ (λ − A∗ )−2 dλ, 2πi Γ 2πθi Γ 1 2

we have (u, A∗θ v) = (u, A∗(θ−1) A∗ v) = −1 = 2πθi

 λ

θ

 

−1 2πθi



  λθ u, A∗ (λ − A∗ )−2 v dλ Γ ∗ −2

aij Di u, Dj (λ − A )

Γ





∗ −2

v + u, c(λ − A )



v



dλ.

i,j

(16.32) Furthermore, (u, A∗θ v) =

−1 2πθi

 λθ Γ

 

aij Di u, Dj A∗(θ−1) A∗(1−θ) (λ − A∗ )−2 v



i,j

   + u, c(λ − A∗ )−2 v dλ. We here note the following lemma. Lemma 16.4 We have |(aij Di u, Dj A∗(θ−1) g)| ≤ CuH 2θ gL2

for all g ∈ L2 (Ω).

Proof By Theorem 1.43 and (1.87), we observe that aij Di u ∈ H 2θ−1 (Ω). In addition, due to (1.80), H 2θ −1 (Ω) = H˚ 2θ −1 (Ω). In the meantime, by Theorem 1.43 and (16.31), Dj A∗(θ−1) is a bounded linear operator from L2 (Ω) into H 1−2θ (Ω). Then, since H 1−2θ (Ω) = {H˚ 2θ −1 (Ω)}∗ , we observe that |(aij Di u, Dj A∗(θ−1) g)| = aij Di u, Dj A∗(θ−1) gH˚ 2θ−1 ×H 1−2θ ≤ CuH 2θ gL2 . 

4 Domains of Fractional Powers of Elliptic Operators in L2 Spaces

549

This lemma provides, for each 1 ≤ i, j ≤ n, the existence of a function u˜ ij ∈ L2 (Ω) such that (aij Di u, Dj A∗(θ−1) g) = (u˜ ij , g)

for all g ∈ L2 (Ω)

with the estimate u˜ ij L2 ≤ CuH 2θ .

(16.33)

Hence, it follows that (u, A∗θ v) =

−1 2πθi

 λθ Γ

    u˜ ij , A∗(1−θ) (λ − A∗ )−2 v + u, c(λ − A∗ )−2 v dλ.

  i,j

As noticed above, A satisfies (3) and therefore (1) of Theorem 16.3. So, we see that |(u, A∗θ v)| ≤ C

 |λ|θ



  1−θ    A (λ − A)−2 u˜ ij , v + (λ − A)−2 (cu), v |dλ|

Γ

≤C



ij

 u˜ ij L2 + uL2 vL2 .

ij

Thus, by (16.33), |(u, A∗θ v)| ≤ CuH 2θ vL2 . By Theorem 1.19, this then implies that u ∈ D((A∗θ )∗ ) = D(Aθ ) with Aθ uL2 ≤ CuH 2θ . (III) D(Aθ ) ⊃ HN2θ (Ω) for 34 < θ ≤ 1. Let v ∈ D(A∗ ). By the same reason as before, −1 (u, A v) = 2πθi ∗θ

 λ Γ

θ

3 4

 

< θ ≤ 1, and let u ∈ HN2θ (Ω) and

∗ −2

aij Di u, Dj (λ − A )





∗ −2

v + u, c(λ − A )



v

i,j

The boundary conditions on u imply that 

∗ −2

aij Di u, Dj (λ − A )

i,j







v =

νj aij Di u · (λ − A∗ )−2 v dS

∂Ω i,j

−



Dj [aij Di u], (λ − A∗ )−2 v



i,j

=−



 Dj [aij Di u], (λ − A∗ )−2 v .

i,j



dλ.

550

16 Characterization of Domains of Fractional Powers

Therefore, we obtain that     1 Dj [aij Di u], A∗(θ−1) A∗(1−θ) (λ − A∗ )−2 v λθ (u, A∗θ v) = 2πθi Γ i,j

   − u, c(λ − A∗ )−2 v dλ. Here, we verify the following lemma. Lemma 16.5 We have |(Dj [aij Di u], A∗(θ−1) g)| ≤ CuH 2θ gL2

for all g ∈ L2 (Ω).

Proof We know that aij Di u ∈ H 2θ −1 (Ω) (due to Theorem 1.43 and (1.87)). In addition, since 2θ − 1 = 12 , Theorem 1.43 provides that Dj is a bounded operator from H 2θ−1 (Ω) into H 2(θ−1) (Ω). In the meantime, A∗(θ−1) is a bounded operator from L2 (Ω) into H 2(1−θ) (Ω). In addition, H 2(1−θ) (Ω) = H˚ 2(1−θ) (Ω) (by (1.80)). Then, since H 2(θ−1) (Ω) = {H˚ 2(1−θ) (Ω)}∗ , it follows that |(Dj [aij Di u], A∗(θ−1) g)| = Dj [aij Di u], A∗(θ−1) gH 2(θ−1) ×H˚ 2(1−θ) ≤ CuH 2θ gL2 .



This lemma implies, for each 1 ≤ i, j ≤ n, the existence of a function u˜ ij ∈ L2 (Ω) such that (Dj [aij Di u], A∗(θ−1) g) = (u˜ ij , g)

for all g ∈ L2 (Ω)

with the estimate u˜ ij L2 ≤ CuH 2θ .

(16.34)

Thus, we obtain that     1−θ    ∗θ θ −2 −2 A (λ − A) u˜ ij , v + (λ − A) (cu), v |dλ| |λ| |(u, A v)| ≤ C Γ

≤C



i,j



u˜ ij L2 + uL2 vL2 .

i,j

Hence, by (16.34), |(u, A∗θ v)| ≤ CuH 2θ vL2 . Consequently, u ∈ D(Aθ ) with Aθ uL2 ≤ CuH 2θ . (IV) D(Aθ ) ⊃ H 2θ (Ω) for 0 ≤ θ ≤ 12 . We know that Aiy  ≤ e 2 |y| for the purely π

1

y

imaginary power. Then, by Law of Exponent II (i.e., (2.112)), (A 2 )iy  = Ai 2  ≤

4 Domains of Fractional Powers of Elliptic Operators in L2 Spaces

551

1

1

e 4 |y| . Therefore, by Theorem 16.3, D(Aθ ) = D((A 2 )2θ ) = [L2 (Ω), D(A 2 )]2θ with π

 1 2θ  π2 2 2 Aθ uL2 =  A 2 uL ≤ e 64 emax{(2θ) ,(1−2θ) } u[L2 ,D(A1/2 )]2θ . 2

1 2

We already know that D(A ) = H 1 (Ω). So, the desired result is immediately obtained by the interpolation property (1.71) of the Sobolev spaces. We have thus accomplished the proof of theorem.  Remark 16.5 The case where θ = 3 4

3 2

3 2 3 4

3 4

is critical. As is shown in the proof, we have 3

D(A ) ⊂ H (Ω). However, H (Ω) ⊂ D(A 4 ) is not true in general. In a favorable case, actually, the domain D(A ) can be characterized as the space of functions in 3 H 2 (Ω) which satisfy the Neumann-type boundary conditions (16.27) in a certain weak sense, see Seeley [See71a, See71b]. As for Aθ , we verify the following theorem. Theorem 16.8 Let Ω be a bounded C2 domain in Rn . Let (16.25), (16.26), and (16.29) be satisfied. Then,  1  D A 2 +θ = H 2θ (Ω)

for 0 ≤ θ ≤

1 2

with norm equivalence  1  C −1 uH 2θ ≤ A 2 +θ uH 1 (Ω)∗ ≤ CuH 2θ ,

 1  u ∈ D A 2 +θ ,

˜ δ ∗ , and the norms aij C1 , cL∞ . C > 0 being determined by Ω and by δ, c0 , δ, Proof From Theorem 16.7 we have  1  C −1 uH 1 ≤ A 2 uL ≤ CuH 1 , 2

 1 u ∈ D A2 .

Similarly, by Theorem 1.24, C −1 uH 1 ≤ AuH 1 (Ω)∗ ≤ CuH 1 , 1

1

u ∈ H 1 (Ω).

1

1

Since A− 2 = A 2 A−1 = A 2 A−1 , these two estimates imply that D(A 2 ) = L2 (Ω) with  1  C −1 f L2 ≤ A 2 f H 1 (Ω)∗ ≤ Cf L2 , f ∈ L2 (Ω). 1

1

1

1

For 0 ≤ θ ≤ 12 , we notice that A 2 +θ = A 2 Aθ = A 2 Aθ . Hence, D(A 2 +θ ) = 1

D(Aθ ) with C −1 Aθ uL2 ≤ A 2 +θ uH 1 (Ω)∗ ≤ CAθ uL2 . Then, the desired result is verified directly from Theorem 16.7. 

552

16 Characterization of Domains of Fractional Powers

4.3 Case of Convex Domains The characterization proved in Theorem 16.7 can be generalized to a case of nonsmooth domain. In this subsection, we assume that Ω is a bounded convex domain in Rn and that aij (x) are symmetric, that is, (16.28) is satisfied. As observed by Theorem 2.7, D(A) coincides with HN2 (Ω), and estimate (16.30) holds as for the C2 domains. We can then repeat the same arguments as in the proof of Theorem 16.7 to characterize the domains of Aθ , 0 < θ < 1. In fact, the following results are similarly verified. Since the proof is quite analogous, we may omit it. Theorem 16.9 Let Ω be a bounded convex domain in Rn . Let (16.25), (16.26), (16.28), and (16.29) be satisfied. Then, D(A ) = θ





L2 (Ω), HN2 (Ω) θ

=

H 2θ (Ω) if 0 ≤ θ < 34 , HN2θ (Ω) if

3 4

< θ ≤ 1,

with norm equivalence C −1 uH 2θ ≤ Aθ uL2 ≤ CuH 2θ ,

u ∈ D(Aθ ),

˜ and the norms aij C1 , cL∞ . C > 0 being determined by Ω and by δ, c0 , δ,

5 Domains of Fractional Powers of Elliptic Operators in Lp Spaces In thissection, we consider a realization Ap of the strongly elliptic operator − nij =1 Dj [aij (x)Di ·] + c(x) in the space Lp (Ω), 1 < p < ∞, under the Neumann-type boundary conditions (16.27). As in the case of L2 (Ω), we can characterize the domains of the fractional powers Aθp for 0 ≤ θ ≤ 1. For this purpose, we assume that Ω is a bounded domain in Rn with C2 boundary ∂Ω. The functions aij (x), 1 ≤ i, j ≤ n, are assumed to satisfy conditions (16.25) and (16.29), and c(x) is assumed to satisfy (16.26). Under these conditions, we already know that, for 1 < p < ∞, the realization Ap is a sectorial operator of Lp (Ω), and its domain is given by  D(Ap ) =

u ∈ Hp2 (Ω);

∂u = 0 on ∂Ω ∂νA



with the estimate δ˜p uHp2 ≤ Ap uLp ,

u ∈ D(Ap ),

δ˜p > 0 being a positive constant. Indeed, recall Theorem 2.15.

5 Domains of Fractional Powers of Elliptic Operators in Lp Spaces

553

We introduce an adjoint pair {Lp (Ω), Lp (Ω)}, where p1 + p1 = 1. Then, the adjoint operator of Ap with respect to the pair {Lp (Ω), Lp (Ω)} is defined. Its adjoint operator is denoted by Bp = A∗p and is a sectorial operator of Lp (Ω) with domain    ∂v ≡ νi (x)aij (x)Dj v = 0 on ∂Ω . D(Bp ) = v ∈ Hp2 (Ω); ∂νA∗ i,j

The following estimate is again true: δp∗ vH 2 ≤ Bp vLp , p

v ∈ D(Bp ),

δp∗ > 0 being a positive constant. By definition, Ap u, vLp ×Lp = u, Bp vLp ×Lp , For 1 +

1 p

u ∈ D(Ap ), v ∈ D(Bp ).

< s ≤ 2, we consider the closed subspace of Hps (Ω) given by   ∂u s (Ω) = u ∈ Hps (Ω); = 0 on ∂Ω . Hp,N ∂νA

Note that, by the trace theorem, Theorem 1.39, if u ∈ Hps (Ω) with s > 1 + p1 , then ∈ Lp (∂Ω), and the boundary conditions (16.27) have a meaning. We can then extend Theorem 16.7 to the operator Ap by the following two theorems. ∂u ∂νA

Theorem 16.10 Let Ω be a bounded domain in Rn with C2 boundary. Let (16.25), (16.26), and (16.29) be satisfied. In addition, we assume that Ap has bounded purely iy imaginary powers in Lp (Ω), namely, Ap L(Lp ) ≤ Np eβp |y| , −∞ < y < ∞. Then, ⎧ p+1  θ  ⎨Hp2θ (Ω) if 0 ≤ θ < 2p , D Ap ⊂ ⎩H 2θ (Ω) if p+1 < θ ≤ 1, p,N 2p with continuous embedding   uHp2θ ≤ Cp Aθp uL , 2

  u ∈ D Aθp ,

Cp > 0 being determined by Ω and by δ˜p , Np , and βp . Proof The proof is carried out in a quite analogous way to the first part of proof of Theorem 16.7. In fact, by Theorem 16.5, we have D(Aθp ) = [Lp (Ω), D(Ap )]θ with the estimate    2 2 2  u[Lp ,D(Ap )]θ ≤ Np eβp /4 emax{θ ,(1−θ) } Aθp uL , u ∈ D Aθp p

554

16 Characterization of Domains of Fractional Powers

(cf. (16.9)). In the meantime, the interpolation theorem, Theorem 1.15, applied to the embedding j : D(Ap ) → Hp2 (Ω) yields that

 [Lp (Ω), D(Ap )]θ ⊂ Lp (Ω), Hp2 (Ω) θ ,

0 ≤ θ ≤ 1,

with δ˜pθ u[Lp ,Hp2 ]θ ≤ u[Lp ,D(Ap )]θ ,

u ∈ [Lp (Ω), D(Ap )]θ .

Furthermore, Theorem 1.35 yields that [Lp (Ω), Hp2 (Ω)]θ = Hp2θ (Ω) with norm equivalence. Hence, for every 0 ≤ θ ≤ 1, D(Aθp ) ⊂ Hp2θ (Ω) with   uHp2θ ≤ Cp Aθp uL ,

  u ∈ D Aθp .

p

As D(Ap ) is dense in D(Aθp ), for any u ∈ D(Aθp ), there is a sequence uk ∈ D(Ap ) such that uk → u in D(Aθp ). This implies that every function u ∈ p+1 2p < θ ≤ 1, satisfies the boundary 2θ (Ω) for p+1 < θ ≤ 1. θ D(Ap ) ⊂ Hp,N 2p

D(Aθp ),

conditions

∂u ∂νA

= 0 on ∂Ω, that is, 

Theorem 16.11 Let Ω be a bounded domain in Rn with C2 boundary. Let (16.25), (16.26), and (16.29) be satisfied. Now, assume that Ap has a bounded H∞ functional calculus in Lp (Ω), namely, f (Ap )L(Lp ) ≤ Cω f ∞ , f ∈ H∞ (Σω ), for some angle ω such that ωAp < ω ≤ π , where ωAp denotes the angle of Ap . Then, ⎧  θ  ⎨Hp2θ (Ω) D Ap = ⎩H 2θ (Ω) p,N

if 0 ≤ θ < if

p+1 2p

p+1 2p ,

< θ ≤ 1,

with norm equivalence   Cp−1 uHp2θ ≤ Aθp uL ≤ Cp uHp2θ , p

  u ∈ D Aθp ,

Cp > 0 being determined by Ω and by δ˜p , δp∗ , Cω , and the norms aij C1 , cL∞ . Proof According to Theorem 16.5, the bounded H∞ functional calculus for Ap implies the boundedness of the purely imaginary powers of Ap . Therefore, it has θ already been deduced that D(Aθp ) ⊂ Hp2θ (Ω) for 0 ≤ θ < p+1 2p and D(Ap ) ⊂ 2θ (Ω) for p+1 < θ ≤ 1 with continuous embeddings. So, it suffices to prove Hp,N 2p the converse relations of inclusion. However, this is carried out in a quite analogous way as in the proof of Theorem 16.7, Steps (II)–(IV). −iy

∗ (I) D(Aθp ) ⊃ Hp2θ (Ω) for 12 ≤ θ < p+1 2p . We first notice that, since Bp = (Ap ) with respect to the adjoint pair {Lp (Ω), Lp (Ω)}, Theorem 16.10 is available for the iy

5 Domains of Fractional Powers of Elliptic Operators in Lp Spaces

555

adjoint operator Bp . Therefore, for every 0 ≤ θ ≤ 1,   vH 2θ ≤ Cp (δ ∗ )−θ Bpθ v L , p

Let

1 2

≤θ <

p+1 2p ,

  u, Bpθ v L

p ×Lp

=

  v ∈ D Bpθ .

p

(16.35)

and let u ∈ Hp2θ (Ω) and v ∈ D(Bp ). As for (16.32), we have −1 2πθi

 λθ

 

Γ

1−θ −2 aij Di u, Dj Bpθ−1 v

Bp (λ − Bp )

i,j

 Lp ×Lp



  + u, c(λ − Bp )−2 v L

p ×Lp

dλ.

Here, we note the following lemma. Lemma 16.6 We have   aij Di u, Dj B θ−1

g p

Lp ×Lp

≤ Cu

Hp2θ gLp

for all g ∈ Lp (Ω).

Proof By Theorem 1.43 and (1.87), we observe that aij Di u ∈ Hp2θ−1 (Ω). In addition, due to (1.80), Hp2θ −1 (Ω) = H˚ p2θ −1 (Ω). In the meantime, (16.35) means that 2(1−θ)

is a bounded operator from Lp (Ω) into Hp

Bpθ−1

2(1−θ)

Dj is a bounded linear operator from Hp

(Ω) = {H˚ p2θ−1 (Ω)}∗ , it follows that H 1−2θ

(Ω). Since 2(1 − θ ) =

1 p ,

(Ω) into Hp1−2θ (Ω). Then, since

p

  aij Di u, Dj B θ−1

g p

Lp ×Lp

  = aij Di u, Dj B θ−1

g p

H˚ p2θ−1 ×Hp1−2θ



≤ CuHp2θ gLp .



This lemma provides, for each 1 ≤ i, j ≤ n, the existence of a function u˜ ij ∈ Lp (Ω) such that   aij Di u, Dj Bpθ−1

g L

p ×Lp

= u˜ ij , gLp ×Lp

for all g ∈ Lp (Ω)

with the estimate u˜ ij Lp ≤ CuH 2θ . By assumption, Ap has a bounded H∞ functional calculus, and, according to Theorem 16.5, this is equivalent to the integrable condition (16.22) along the integral

556

16 Characterization of Domains of Fractional Powers

contour Γ . Therefore, we conclude that

   1−θ    ≤ Cp A (λ − Ap )−2 u˜ ij , v u, B θ v |λ|θ p

Lp ×Lp

p

Γ

i,j

  + (λ − Ap )−2 (cu), v L

p ×Lp

≤ Cp



Lp ×Lp



 |dλ|

 u˜ ij Lp + uLp vLp .

i,j

Hence, |u, Bpθ vLp ×Lp | ≤ Cp uHp2θ vLp .

By virtue of Theorem 1.19, we then deduce that u ∈ D((Bpθ )∗ ) = D(Aθp ) with Aθp uLp ≤ Cp uHp2θ . 2θ (Ω) for p+1 < θ ≤ 1. Let p+1 < θ ≤ 1, and let u ∈ H 2θ (Ω) (II) D(Aθp ) ⊃ Hp,N p,N 2p 2p and v ∈ D(Bp ). By the same reason as before,

      −1 aij Di u, Dj (λ − Bp )−2 v L ×L

λθ u, Bpθ v L ×L = p p p p 2πθi Γ i,j



  + u, c(λ − Bp )−2 v L

p ×Lp

dλ.

The boundary conditions on u yield that   aij Di u, Dj (λ − Bp )−2 v L

p ×Lp

i,j





=

νj aij Di u · (λ − Bp )−2 v dS

∂Ω i,j

−



Dj [aij Di u], (λ − Bp )−2 v

 Lp ×Lp

i,j

=−



Dj [aij Di u], (λ − Bp )−2 v

i,j

 Lp ×Lp

.

Hence, 



u, Bpθ v Lp ×L

p

1 = 2πθi 

 λ Γ

θ

 

1−θ −2 v Dj [aij Di u], Bpθ−1

Bp (λ − Bp )

i,j −2

− u, c(λ − B ) p

Here, we note the following lemma.

v

 Lp ×Lp

 dλ.

 Lp ×Lp

5 Domains of Fractional Powers of Elliptic Operators in Lp Spaces

Lemma 16.7 We have   Dj [aij Di u], B θ−1 p g L

p ×Lp

557

≤ Cu

Hp2θ gLp

for all g ∈ Lp (Ω).

Proof By Theorem 1.43 and (1.87), aij u ∈ Hp2θ −1 (Ω). Since 2θ − 1 = a bounded operator from Hp2θ −1 (Ω) into shows that

2(θ−1) Hp (Ω).

is a bounded operator from Lp (Ω) into Hp

(Ω). In addition,

2(1−θ) since 2(1 − θ ) < p1 , it is the case by (1.80) that Hp2(1−θ) (Ω) = H˚ p

(Ω).

2(θ−1) 2(1−θ) in view of Hp (Ω) = {H˚ p

(Ω)}∗ , we verify that

  Dj [aij Di u], B θ−1

g p

Lp ×Lp

Dj is

In the meantime, (16.35) 2(1−θ)

Bpθ−1

1 p,

  = Dj [aij Di u], B θ−1

g p

2(θ−1) 2(1−θ) Hp ×H˚ p

Then,



≤ CuHp2θ gLp .



This lemma yields, for each 1 ≤ i, j ≤ n, the existence of a function u˜ ij ∈ Lp (Ω) such that   ˜ ij , gLp ×Lp for all g ∈ Lp (Ω) Dj [aij Di u], Bpθ−1

g L ×L = u p

p

with the estimate u˜ ij Lp ≤ CuH 2θ . Hence, we conclude that   u, B θ v p

Lp ×Lp



≤ Cp

  1−θ  A (λ − Ap )−2 u˜ ij , v |λ| p L θ

Γ

p ×Lp

i,j

  + (λ − Ap )−2 (cu), v ≤ Cp



Lp ×Lp



 |dλ|

 u˜ ij Lp + uLp vLp .

i,j

Moreover, |u, Bpθ vLp ×Lp | ≤ CuHp2θ vLp . Consequently, we conclude that u ∈ D(Aθp ) with Aθp uLp ≤ Cp uHp2θ . (III) D(Aθp ) ⊃ Hp2θ (Ω) for 0 ≤ θ ≤ 12 . We know that Ap L(Lp ) ≤ Np eβp |y| , −∞ < y < ∞, with some constant Np ≥ 1 and exponent βp > 0. Then, by Law iy

1

iy

of Exponent II (i.e., (2.112)), (Ap2 )iy L(Lp ) = Ap2 L(Lp ) ≤ Np e(βp /2)|y| . There1

1

fore, it follows by Theorem 16.5 that D(Aθp ) = D((Ap2 )2θ ) = [Lp (Ω), D(Ap2 )]2θ with  θ   1   A u =  Ap2 2θ u ≤ Np eβp2 /16 emax{(2θ)2 ,(1−2θ)2 } u . 1/2 p L L [L ,D(A )] p

p

p

p

2θ

558

16 Characterization of Domains of Fractional Powers 1

We already know that D(Ap2 ) = Hp1 (Ω). So, the desired inclusion is readily verified by Theorem 1.35.  Remark 16.6 Even if the operator Ap may enjoy neither bounded purely imaginary powers nor a bounded H∞ functional calculus in Lp (Ω), one can prove some analogous weak results as Theorems 16.10 and 16.11. In fact, under (16.25), (16.26), and (16.29), it is possible to verified that ⎧ ⎨H 2θ

(Ω) ⊂ D(Aθ ) ⊂ H 2θ (Ω) if 0 ≤ θ < θ < θ

< p+1 , p p p 2p (16.36) ⎩H 2θ

(Ω) ⊂ D(Aθ ) ⊂ H 2θ (Ω) if p+1 < θ < θ < θ

≤ 1, p p,N p,N 2p with continuous embeddings. −δ+iy , −∞ < y < ∞, are In the proof of these results, we can use the facts that Ap bounded operators of Lp (Ω) with any positive exponent δ > 0 and that an integral condition    |λ|2θ−1 A2(1−θ−δ) (λ − Ap )−1 f, g L ×L |dλ| ≤ Cω,θ,δ f Lp gLp , p p

Γω

p

f ∈ Lp (Ω), g ∈ Lp (Ω), is always valid for 0 < θ < 1 and any positive exponent 0 < δ < 1 − θ , where  ωAp < ω < π2 . The detailed proof is however left to the reader.

6 Domains of Fractional Powers Under Dirichlet Conditions 6.1 Domains in L2 Spaces Let Ω be a bounded domain in Rn with Lipschitz boundary ∂Ω. Consider the sesquilinear form a(u, v) =

n   i,j =1 Ω

 aij (x)Di uDj v dx +

c(x)uv dx,

u, v ∈ H˚ 1 (Ω), (16.37)

Ω

on the space H˚ 1 (Ω). As before, aij (x), 1 ≤ i, j ≤ n, are assumed to be real-valued functions and to satisfy (16.25) and (16.29), and, c(x) is a real-valued function in Ω satisfying (16.26). Then a(·, ·) fulfills (2.6) and (2.7) on H˚ 1 (Ω). We consider the triplet H˚ 1 (Ω) ⊂ L2 (Ω) ⊂ H −1 (Ω). The operator A associated with the form and its part A in L2 (Ω) are defined and are sectorial operators of H −1 (Ω) and L2 (Ω), respectively.  These operators A and A are regarded as realizations of the elliptic operator − ni,j =1 Dj [aij (x)Di ·]+c(x) in H −1 (Ω) and L2 (Ω), respectively, under the Dirichlet boundary conditions γ u = 0 on ∂Ω, where γ is the trace operator defined by γ u = u|∂Ω .

6 Domains of Fractional Powers Under Dirichlet Conditions

559

One can then verify analogous characterization results as Theorems 16.7 and 16.9 for the domains of fractional powers of A. Case of C2 boundary. Let ∂Ω be of C2 class. For subspace of H s (Ω) by

1 2

< s ≤ 2, we define a closed

HDs (Ω) = {u ∈ H s (Ω); γ u = 0}.

(16.38)

Note that, by Theorem 1.39, the Dirichlet conditions γ u = 0 on ∂Ω are meaningful for any function u ∈ H s (Ω), provided that s > 12 . By the shift property (2.29), we already know that D(A) = HD2 (Ω) with the estimate ˜ δu H 2 ≤ AuL2 ,

u ∈ D(A),

(16.39)

δ˜ > 0 being a positive constant. The adjoint operator A∗ of A in L2 (Ω) is obtained from the adjoint sesquilinear form a ∗ (u, v) = a(v, u) of a(u, v). So, it follows that D(A∗ ) = HD2 (Ω). By the same reason as for (16.39), it holds that δ ∗ vH 2 ≤ A∗ vL2 ,

v ∈ D(A∗ ),

with some positive constant δ ∗ > 0. The following characterization is then shown. For the proof, see [Yag08a]. Theorem 16.12 Let Ω be a bounded domain in Rn with C2 boundary. Let (16.25), (16.26), and (16.29) be satisfied. Then, H 2θ (Ω) if 0 ≤ θ < 14 ,

 θ 2 D(A ) = L2 (Ω), HD (Ω) θ = HD2θ (Ω) if 14 < θ ≤ 1, with norm equivalence C −1 uH 2θ ≤ Aθ uL2 ≤ CuH 2θ ,

u ∈ D(Aθ ),

˜ δ ∗ , and the norms aij C1 , cL∞ . C > 0 being determined by Ω and by δ, c0 , δ, Case of convex domains. Let Ω be a bounded convex domain in Rn . In addition to  (16.25) and (16.29), let aij (x) satisfy (16.28). In this case, the realization A of − ni,j =1 Dj [aij (x)Di ·] + c(x) in L2 (Ω) under the Dirichlet boundary conditions is a positive definite self-adjoint operator of L2 (Ω). By the shift property (2.29), its domain is still given by D(A) = HD2 (Ω), and estimate (16.39) is still valid. The following characterization is shown. For the proof, see [Yag08a]. Theorem 16.13 Let Ω be a bounded convex domain in Rn . Let (16.25), (16.26), (16.28), and (16.29) be satisfied. Then, H 2θ (Ω) if 0 ≤ θ < 14 ,

 θ 2 D(A ) = L2 (Ω), HD (Ω) θ = HD2θ (Ω) if 14 < θ ≤ 1, θ = 34 ,

560

16 Characterization of Domains of Fractional Powers

with norm equivalence C −1 uH 2θ ≤ Aθ uL2 ≤ CuH 2θ ,

u ∈ D(Aθ ),

˜ and the norms aij C1 , cL∞ . C > 0 being determined by Ω and by δ, c0 , δ,

6.2 Domains in Lp Spaces  We consider a realization Ap of the strongly elliptic operator − nij =1 Dj [aij (x) × Di ·]+c(x) in Lp (Ω), 1 < p < ∞, under the Dirichlet boundary conditions γ u = 0. We assume that Ω is a bounded domain in Rn with C2 boundary. The functions aij (x), 1 ≤ i, j ≤ n, are real valued and are assumed to satisfy conditions (16.25) and (16.29), and c(x) is assumed to satisfy (16.26). Under these conditions, we already know that, for 1 < p < ∞, the realization Ap is a sectorial operator of Lp (Ω), and its domain is given by D(Ap ) = u ∈ Hp2 (Ω); γ u = 0 on ∂Ω

!

with the estimate δ˜p uHp2 ≤ Ap uLp ,

u ∈ D(Ap ),

δ˜p > 0 being a positive constant. Indeed, recall Theorem 2.15. In view of Theorem 1.39, for p1 < s ≤ 2, we put ! s = u ∈ Hps (Ω); γ u = 0 on ∂Ω . Hp,D Let {Lp (Ω), Lp (Ω)} be the adjoint pair, where p1 + p1 = 1. Let Bp = A∗p be the adjoint operator of Ap with respect to {Lp (Ω), Lp (Ω)}; then, Bp is a sectorial operator of Lp (Ω) with domain ! D(Bp ) = v ∈ Hp2 (Ω); γ v = 0 on ∂Ω . The following estimate again holds: δp∗ vH 2 ≤ Bp vLp , p

v ∈ D(Bp ),

δp∗ > 0 being a positive constant. Concerning the domains D(Aθp ), 0 < θ < 1, of fractional powers, we can verify analogous results as Theorems 16.10 and 16.11. For the proof, however, see [Yag08a]. Theorem 16.14 Let Ω be a bounded domain in Rn with C2 boundary. Let (16.25), (16.26), and (16.29) be satisfied. Assume that Ap has bounded purely imaginary

Notes and Further Researches

561

powers in Lp (Ω), namely, Ap L(Lp ) ≤ Np eβp |y| , −∞ < y < ∞. Then, ⎧ 1  θ  ⎨Hp2θ (Ω) if 0 ≤ θ < 2p , D Ap ⊂ ⎩H 2θ (Ω) if 1 < θ ≤ 1, iy

p,D

2p

with continuous embedding   uHp2θ ≤ Cp Aθp uL , 2

  u ∈ D Aθp ,

Cp > 0 being determined by Ω and by δ˜p , Np , and βp . Theorem 16.15 Let Ω be a bounded domain in Rn with C2 boundary. Let (16.25), (16.26), and (16.29) be satisfied. Assume that Ap has a bounded H∞ functional calculus in Lp (Ω), namely, f (Ap )L(Lp ) ≤ Cω f ∞ , f ∈ H∞ (Σω ), for some angle ω such that ωAp < ω ≤ π , where ωAp denotes the angle of Ap . Then, ⎧ 1  θ  ⎨Hp2θ (Ω) if 0 ≤ θ < 2p , D Ap = ⎩H 2θ (Ω) if 1 < θ ≤ 1, θ = p+1 , p,D

2p

2p

with norm equivalence   Cp−1 uHp2θ ≤ Aθp uLp ≤ Cp uHp2θ ,

  u ∈ D Aθp ,

Cp > 0 being determined by Ω and by δ˜p , δp∗ , Cω , and the norms aij C1 , cL∞ .

Notes and Further Researches The notion of H∞ functional calculus in Hilbert spaces was invented by McIntosh [McI86] with discovering equivalence with their various favorable properties presented by Yagi [Yag84]. Theorem 16.3 was obtained by [McI86] and [Yag84]. Theorem 16.4 is new. The H∞ functional calculus in reflexive Banach spaces was studied by Cowling–Doust–McIntosh–Yagi [CDMY96], where Theorem 16.5 was shown. The theory of H∞ functional calculus may be regarded as one of possible ways to develop the Fourier analysis for sectorial operators acting in Hilbert or Banach spaces. The theory is actually utilized for establishing all kinds of the optimal properties for them, including the maximal regularity of solutions for abstract evolution equations, see Dore–Venni [DV87], Giga–Giga–Sohr [GGS91], Kalton–Weis [KW01], Prüss [Pru93, Pru02], Arendt [Are04], Haase [Ha06], and so forth. When the underlying space X is a Hilbert space, we have some convenient sufficient conditions in order to know that a sectorial operator A has a bounded H∞ functional calculus. Every self-adjoint operator has a bounded H∞ functional calculus thanks to Remark 2.7. Kato [Kat61a, Kat62] proved that the maximal accretive operator A always has bounded purely imaginary powers, i.e., Theorem 2.30.

562

16 Characterization of Domains of Fractional Powers

More generally, if a sectorial operator A of a Hilbert space satisfies D(Aθ ) = (A∗θ ) for an exponent 0 < θ ≤ 1, then the H∞ functional calculus of A is bounded (see [Yag84]). On the other hand, the counterexample in Sect. 1.5 was constructed in McIntosh–Yagi [MY89]. Afterward, Baillon–Clément [BC91] constructed another counterexample in a different context. When the space X is a Banach space, we have so far no convenient sufficient conditions (cf. [BD05]). We have to investigate each sectorial operator under consideration in a direct way. There are already a great deal of literature [AHS94, AMIN97, DHP03, DDHPV, DV02, Duo89, Duo90, DMI96, DO99, DR96, DS97, DY02, HP98, MIN00, Sor99] giving positive results for many classes of elliptic differential operators equipped with suitable boundary operators acting in Lp spaces, where 1 < p < ∞. There are mainly two methods for establishing the boundedness of H∞ functional calculus. The first one is obtaining a keen representation of the resolvent and applying the theory of pseudo-differential operators. The second one is using integral kernels and estimating the kernels by employing the Calderón– Zygmund operator theory. The Stokes operator is handled by [DHP01]. On the other hand, Hieber [Hie96] presented a counterexample of a pseudo-differential operator in Lp which has unbounded purely imaginary powers and hence an unbounded H∞ functional calculus. For the characterization problem of the domains of fractional powers of strongly elliptic operators equipped with suitable boundary conditions, the H∞ functional calculus was first utilized by Nakaguchi–Yagi [NY02] and afterward in [Yag08a]. Before these papers, such a characterization problem was studied in the papers by Fujiwara [Fuj68, Fuj69] and by Seely [See71a, See72], and for Stokes operators, by Giga [Gig81, Gig85]. Firstly, the authors of those papers proved that iy the purely imaginary powers Ap (−∞ < y < ∞) are bounded operators to conclude that D(Aθp ) = [Lp (Ω), D(Ap )]θ for 0 < θ < 1. Secondly, they calculated precisely the interpolation spaces [Lp (Ω), D(Ap )]θ . The methods of proofs of Theorems 16.7, 16.9, and 16.11 then show that the boundedness of the H∞ functional calculus enables a direct calculation of D(Aθp ) without using the intermedium [Lp (Ω), D(Ap )]θ . Theorem 16.9 was presented in [NY02]. Theorems 16.10 and 16.11 are new. Theorems 16.12, 16.13, 16.14, and 16.15 are seen in [Yag08a]. For higher-order elliptic operators, characterization of domains of fractional powers by the direct method of using H∞ functional calculus remains to be done.

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Index

A Absorbing set, 254, 256–258, 281, 284, 303, 341, 400, 458 Adjoint form, 25, 27, 79, 80 Adjoint pair, 17–23, 27, 47, 79, 89, 101, 151, 545, 553, 554, 560 Adjoint semigroup, 90 Angle of analytic semigroup, 118, 132–134, 177, 450 Angle of sectorial operator, 56 Approximate operator, 318, 329 Approximate semigroup, 90 Attractor, 252, 255–258, 261–263, 270, 272– 275, 277–280, 285, 304, 315, 338, 339, 342, 343, 355, 356, 373, 389, 403, 434, 458, 460, 484, 525 A priori estimate, 186, 282, 287, 302, 425, 433, 450 B Bounded H∞ functional calculus, 527, 529, 530, 532, 541–543, 561, 562 Bounded purely imaginary powers, 105–107, 109, 112, 113, 175, 492, 553, 558, 561 C Compact perturbation of contraction operator, 270, 275 Complexified dynamical system, 387, 410, 438, 464, 520 Continuous semigroup, 91, 105, 253, 254, 267, 302, 410, 520 Contraction mapping, 12–14, 179, 181, 210, 265, 267

Contraction operator, 12, 252, 269, 270, 274, 275, 314, 486 Contraction semigroup, 88 Cross-diffusion, 414, 488 D Dirichlet boundary conditions, 60, 65, 81, 132, 165, 200, 346, 347, 371, 389, 393, 408, 409, 412, 525, 558–560 Dissipative condition, 36, 284, 303, 354, 367, 379, 386, 434, 461, 484, 514, 521 Drift diffusion, 345, 356 Dual space, 16–18, 47 Duality product, 17–23, 25, 47, 56, 67, 70, 89, 352, 453, 500, 503, 510, 518, 520, 542 Dunford integral, 11, 84, 92 E Equilibrium, 251, 254–256, 258, 259, 261, 263, 264, 267–270, 287, 289, 293, 304, 308, 312, 368–370, 373, 387– 389, 405, 410, 412, 413, 438, 441, 442, 463–465, 469, 521, 524 Error estimate, 326 Exponential attractor, 252, 262, 263, 270, 272, 273, 275, 277–280, 285, 304, 315, 338, 341, 342, 373, 434, 484 Exponential law, 85 Extrapolation space, 22, 23 Extrapolation theorem, 24 F Finite element method, 328 Fixed point theorem, 182, 210, 215, 269

A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-04631-5, © Springer-Verlag Berlin Heidelberg 2010

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580

Fractal dimension, 2, 3, 53, 252, 262, 270, 272, 277, 279, 280, 285, 304, 341, 373 Fractional power, 94, 95, 123, 149, 177, 182, 298, 450 Fréchet derivative, 13, 53, 263, 268, 289, 294, 308, 369, 387, 402, 411, 438, 465, 521 Fréchet differentiable, 13, 51, 53, 251, 263, 264, 268, 287–289, 291, 294, 305, 307, 308, 311, 369, 387, 402, 411, 438, 465, 521 Fréchet differential, 13, 51, 287 Functional calculus, 11, 175, 527, 529–533, 536, 541–543, 554, 555, 558, 561, 562 G Generalized semigroup property, 139, 144 Generation of analytic semigroup, 91 Generator of analytic semigroup, 119 Gierer-Meinhardt, 371 Global attractor, 252, 255–258, 261–263, 270, 274, 278–280, 341, 355, 356, 389, 403, 458, 460, 525 Global estimate of convergence, 337 Global existence, 185, 194, 199, 212, 225, 243, 247, 249, 282, 299, 345, 353, 355, 371, 373, 379, 385, 392, 398, 418, 423, 425, 433, 442, 446, 483, 488, 511, 525 Global solution, 185, 189, 194, 197, 199, 212, 225, 242, 243, 282, 299, 354, 366, 379, 385, 398, 406, 425, 426, 433, 455, 483, 512, 513, 520 H H∞ functional calculus, 175, 527, 529–533, 536, 541–543, 554, 555, 558, 561, 562 Hölder continuous function, 5 Hyperbolic equilibrium, 263, 268, 293, 312 I Integrable condition, 529, 532, 536, 543, 555 Interpolation property, 24, 43, 330, 331, 551 Interpolation space, 15, 237 Interpolation theorem, 43, 49, 50, 547, 554 Invariant set, 254–258, 261, 262, 274, 279, 285, 303, 367, 379, 386, 458

Index

K Keller-Segel, 417, 445 L Lebesgue space, 43 Local estimate of convergence, 343 Local existence, 225, 299, 324, 355, 477, 489 Local solution, 178, 183, 185, 186, 188–194, 196, 197, 203, 210–213, 215, 219, 221, 225, 226, 228, 230, 233, 235, 238–240, 242, 243, 246, 247, 281, 282, 299, 322, 324, 325, 349, 350, 352, 361–363, 365, 366, 375–377, 379, 383, 384, 393–395, 421–423, 428, 448–450, 455, 477, 479, 492– 495, 498, 505, 511, 512, 516, 517 Local stable manifold, 260, 267 Local unstable manifold, 260, 267, 293, 312, 368–370, 388, 389, 410, 413, 441, 442, 469, 524 Lyapunov function, 392, 400–402, 405, 408, 414, 426, 442 M Maximal accretive operator, 106–109, 561 Maximal regularity, 117, 126, 129, 153, 159, 174, 175, 305, 310, 561 Moment inequality, 97–99, 101, 123, 139, 171, 182, 222, 285, 300, 304, 321, 328, 336, 431 Multivalued linear operator, 9, 10, 76, 79 N Negative trajectory, 258, 259, 261 Neumann boundary conditions, 62, 71, 192, 195, 236, 237, 244, 294, 295, 346, 347, 360, 369, 373, 374, 382, 388, 415, 418, 419, 439, 446, 450, 465, 472, 521 Neumann-type boundary conditions, 61, 63, 78, 81, 91, 133, 151, 166, 167, 191, 234, 329, 545, 546, 548, 551–553 Nicoli-Prigogine, 470 O ω-limit set, 256, 403–405, 414, 415 P Part of linear operator, 8 Perturbation of analytic semigroup, 123

Index

Phase space 251, 254, 257, 258, 263, 274, 277, 278, 280, 283, 285, 302, 303, 366, 367, 385, 400, 433, 457, 461, 462, 483, 513, 520 Poincaré inequality, 62 Positive trajectory, 258 Product space, 17, 48, 67, 192, 195, 236, 249, 267, 348, 359, 360, 382, 419, 420, 427, 471, 472, 489, 490, 541 Pseudo-distance, 1, 254, 285, 304, 460 Pseudo-resolvent, 9, 10, 76, 79, 96, 120 Purely imaginary power, 550 Q Quantum drift diffusion, 356 R Realization, 62, 65, 71, 78, 79, 81, 91, 131– 133, 164–166, 191, 233, 237, 244, 295, 329, 347, 393, 408, 439, 450, 451, 465, 491, 516, 521, 545, 552, 559, 560 Ritz operator, 318, 323, 334 S Self-diffusion, 243, 249, 488 Semigroup property, 88, 118, 139, 144, 180, 251, 253, 262 Sesquilinear form, 17, 19, 23, 25–27, 56–60, 67, 69, 71, 79, 90, 109, 110, 112, 113, 116, 328, 329, 346, 489, 492, 515, 527, 534, 545, 547, 558, 559 Shift property, 62, 63, 66, 81, 115, 133, 347, 355, 356, 434, 447, 484, 500, 510, 518, 559 Shockley, 345, 355 Simon and Łojasiewicz method, 415, 442 Sobolev space, 43, 233, 419, 446, 472 Spectral mapping theorem, 11, 292 Spectrum separation, 11, 251, 293, 312, 369, 388, 438, 465, 521

581

Splitting boundary conditions, 62, 66 Square function estimate, 528, 534 Square root problem, 112, 113, 116 Squeezing property, 252, 277, 278, 286, 287, 314, 367, 380, 386, 434, 442, 484, 486 Stable manifold, 251, 259, 260, 267 Strictly invariant set, 254, 256, 257 Strong topology, 8, 34, 35, 52, 53, 170 Strongly elliptic operator, 79, 81, 131, 328, 545, 552, 560 Structural assumptions, 115, 117, 134, 142, 146, 167, 205, 219, 222, 248, 298, 299, 317–319, 324, 328, 343, 420, 475, 527 Symmetric form, 112 T Trajectory, 251, 254, 256, 258, 259, 261, 270, 368, 387, 401–403, 405, 408, 410, 413, 438, 464 Triplet of spaces, 22, 24, 25, 56, 66, 90, 109, 110, 112, 149, 163, 189, 232, 329, 330, 346 Truncation method, 193, 246, 351, 361, 376, 383, 422, 428, 449, 450, 477 U Uniform boundedness theorem, 7, 106, 118 Uniform topology, 8, 145, 146 Universal space, 251, 254, 283, 302, 337, 354, 366, 367, 379, 385, 386, 400, 433, 434, 457, 458, 517, 520 Unstable manifold, 251, 259–261, 267, 293, 312, 368–370, 388, 389, 410, 413, 441, 442, 469, 524, 526 W Weak* compact, 214, 284 Weak singularity, 29–31, 34, 155 Weighted Hölder continuous function, 5