Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications (Monographs and Surveys in Pure and Applied Math)

CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics 139 SPECTRAL THEORY FOR RANDOM AND NONAUTON...

Author: Janusz Mierczynski | Wenxian Shen

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CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics

139

SPECTRAL THEORY FOR RANDOM AND NONAUTONOMOUS PARABOLIC EQUATIONS AND APPLICATIONS

CHAPMAN & HALL/CRC

Monographs and Surveys in Pure and Applied Mathematics Main Editors

H. Brezis, Université de Paris R.G. Douglas, Texas A&M University A. Jeffrey, University of Newcastle upon Tyne (Founding Editor)

Editorial Board

R. Aris, University of Minnesota G.I. Barenblatt, University of California at Berkeley H. Begehr, Freie Universität Berlin P. Bullen, University of British Columbia R.J. Elliott, University of Alberta R.P. Gilbert, University of Delaware R. Glowinski, University of Houston D. Jerison, Massachusetts Institute of Technology K. Kirchgässner, Universität Stuttgart B. Lawson, State University of New York B. Moodie, University of Alberta L.E. Payne, Cornell University D.B. Pearson, University of Hull G.F. Roach, University of Strathclyde I. Stakgold, University of Delaware W.A. Strauss, Brown University J. van der Hoek, University of Adelaide

CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics

139

SPECTRAL THEORY FOR RANDOM AND NONAUTONOMOUS PARABOLIC EQUATIONS AND APPLICATIONS

Janusz Mierczy´nski Wenxian Shen

Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487‑2742 © 2008 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid‑free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number‑13: 978‑1‑58488‑895‑6 (Hardcover) This book contains information obtained from authentic and highly regarded sources Reason‑ able efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The Authors and Publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978‑750‑8400. CCC is a not‑for‑profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data Mierczynski, Janusz. Spectral theory for random and nonautonomous parabolic equations and applications / Janusz Mierczynski and Wenxian Shen. p. cm. ‑‑ (Monographs and surveys in pure and applied mathematics) Includes bibliographical references and index. ISBN 978‑1‑58488‑895‑6 (alk. paper) 1. Differential equations, Parabolic. 2. Evolution equations. 3. Spectral theory (Mathematics) I. Shen, Wenxian, 1961‑ II. Title. III. Series. QA377.M514 2008 515’.3534‑‑dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

2007048063

To Janusz’s Mother and Edyta, and Ruijun, Bonny, Charles

Contents

Preface

xi

Symbol Description

xiii

1 Introduction 1.1 Outline of the Monograph . . . . . . . . . . . . . . . . . . . 1.2 General Notations and Concepts . . . . . . . . . . . . . . . . 1.3 Standing Assumptions . . . . . . . . . . . . . . . . . . . . . .

1 4 8 16

2 Fundamental Properties in the General Setting 2.1 Assumptions and Weak Solutions . . . . . . . . 2.2 Basic Properties of Weak Solutions . . . . . . . 2.3 The Adjoint Problem . . . . . . . . . . . . . . . 2.4 Perturbation of Coefficients . . . . . . . . . . . . 2.5 The Smooth Case . . . . . . . . . . . . . . . . . 2.6 Remarks on Equations in Nondivergence Form .

23 24 31 44 47 51 63

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

3 Spectral Theory in the General Setting 3.1 Principal Spectrum and Principal Lyapunov Exponents: Definitions and Properties . . . . . . . . . . . . . . . . . 3.2 Exponential Separation: Definitions and Basic Properties 3.3 Existence of Exponential Separation and Entire Positive Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Multiplicative Ergodic Theorems . . . . . . . . . . . . . . 3.5 The Smooth Case . . . . . . . . . . . . . . . . . . . . . . 3.6 Remarks on the General Nondivergence Case . . . . . . . 3.7 Appendix: The Case of One-Dimensional Spatial Domain

. . . . . .

. . . . . .

65 . . . . . . . . .

4 Spectral Theory in Nonautonomous and Random Cases 4.1 Principal Spectrum and Principal Lyapunov Exponents in Random and Nonautonomous Cases . . . . . . . . . . . . . 4.1.1 The Random Case . . . . . . . . . . . . . . . . . . . 4.1.2 The Nonautonomous Case . . . . . . . . . . . . . . . 4.2 Monotonicity with Respect to the Zero Order Terms . . . . 4.2.1 The Random Case . . . . . . . . . . . . . . . . . . . 4.2.2 The Nonautonomous Case . . . . . . . . . . . . . . . 4.3 Continuity with Respect to the Zero Order Coefficients . .

. . . . .

65 74 89 103 107 113 114 119

. . . . . . .

120 120 131 136 136 138 139

vii

viii

4.4

4.5

4.3.1 The Random Case . . . . . . . . . . . . . . . 4.3.2 The Nonautonomous Case . . . . . . . . . . . General Continuity with Respect to the Coefficients 4.4.1 The Random Case . . . . . . . . . . . . . . . 4.4.2 The Nonautonomous Case . . . . . . . . . . . Historical Remarks . . . . . . . . . . . . . . . . . . 4.5.1 The Time Independent and Periodic Case . . 4.5.2 The General Time Dependent Case . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

141 142 142 144 145 146 146 148

5 Influence of Spatial-Temporal Variations and the Shape of Domain 149 5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.1.1 Notions and Basic Assumptions . . . . . . . . . . . . . 150 5.1.2 Auxiliary Lemmas . . . . . . . . . . . . . . . . . . . . 153 5.2 Influence of Temporal Variation on Principal Lyapunov Exponents and Principal Spectrum . . . . . . . . . . . . . . 156 5.2.1 The Smooth Case . . . . . . . . . . . . . . . . . . . . 158 5.2.2 The Nonsmooth Case . . . . . . . . . . . . . . . . . . 170 5.3 Influence of Spatial Variation on Principal Lyapunov Exponents and Principal Spectrum . . . . . . . . . . . . . . 179 5.4 Faber–Krahn Inequalities . . . . . . . . . . . . . . . . . . . . 186 5.5 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . 189 6 Cooperative Systems of Parabolic Equations 6.1 Existence and Basic Properties of Mild Solutions in the General Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 The Nonsmooth Case . . . . . . . . . . . . . . . . . . 6.1.2 The Smooth Case . . . . . . . . . . . . . . . . . . . . 6.2 Principal Spectrum and Principal Lyapunov Exponents and Exponential Separation in the General Setting . . . . . . . . 6.2.1 Principal Spectrum and Principal Lyapunov Exponents 6.2.2 Exponential Separation: Basic Properties . . . . . . . 6.2.3 Existence of Exponential Separation and Entire Positive Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Principal Spectrum and Principal Lyapunov Exponents in Nonautonomous and Random Cases . . . . . . . . . . . . . . 6.3.1 The Random Case . . . . . . . . . . . . . . . . . . . . 6.3.2 The Nonautonomous Case . . . . . . . . . . . . . . . . 6.3.3 Influence of Time and Space Variations . . . . . . . . 6.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

191 192 193 208 215 215 220 223 232 233 237 240 245

7 Applications to Kolmogorov Systems of Parabolic Equations 247 7.1 Semilinear Equations of Kolmogorov Type: General Theory 249 7.1.1 Existence, Uniqueness, and Basic Properties of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 249

ix

7.2

7.3

7.4

7.5

7.1.2 Linearization at the Trivial Solution . . . . . . . . . 7.1.3 Global Attractor and Uniform Persistence . . . . . . Semilinear Equations of Kolmogorov Type: Examples . . . 7.2.1 The Random Case . . . . . . . . . . . . . . . . . . . 7.2.2 The Nonautonomous Case . . . . . . . . . . . . . . . Competitive Kolmogorov Systems of Semilinear Equations: General Theory . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Existence, Uniqueness, and Basic Properties of Solutions . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Linearization at Trivial and Semitrivial Solutions . . 7.3.3 Global Attractor and Uniform Persistence . . . . . . Competitive Kolmogorov Systems of Semilinear Equations: Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 The Random Case . . . . . . . . . . . . . . . . . . . 7.4.2 The Nonautonomous Case . . . . . . . . . . . . . . . Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

261 269 277 277 280

.

282

. . .

282 288 293

. . . .

297 298 301 302

References

305

Index

315

Preface

Spectral theory for linear parabolic equations plays a fundamental role in the study of nonlinear parabolic problems. It is well developed for smooth linear elliptic and periodic parabolic equations. It is also quite well understood for general linear elliptic and periodic parabolic equations. In recent years, much attention has been paid to the extension of spectral theory for linear elliptic and periodic parabolic equations to general time dependent and random linear parabolic equations. The goal of this monograph is to give a clear and essentially self-contained account of the spectral theory, in particular, principal spectral theory for general time dependent and random linear parabolic equations and systems of such equations. We establish a unified approach to the study of the principal spectral theory: we start to develop the abstract general theory, in the framework of weak solutions, and then specialize to the cases of random and nonautonomous equations. Among others, fundamental properties of the principal spectrum and principal Lyapunov exponents for nonautonomous and random linear parabolic equations are investigated and applications of the developed principal spectral theory to uniform persistence for competitive Kolmogorov systems of nonautonomous and random nonlinear parabolic equations are discussed. The monograph contains many new results, and puts already known results in a new perspective. The works by H. Amann ([3]–[6]), the works by D. Daners ([29]–[34]), the book by R. Dautray and J.-L. Lions ([36]), the book by D. Henry ([48]), the book by O. A. Ladyzhenskaya [O. A. Ladyˇzenskaja], V. A. Solonnikov and N. N. Ural0 tseva [N. N. Ural’ceva] ([70]), and the book by G. M. Lieberman ([73]) are the main sources for the fundamentals (mainly existence, uniqueness, continuous dependence of solutions and Harnack inequalities for positive solutions) for the development of the spectral theory and applications in this monograph. We have benefited a lot by reading the works by Z. Lian and K. Lu ([72]), J. H´ uska ([59]), J. H´ uska and P. Pol´aˇcik ([60]), and J. H´ uska, P. Pol´aˇcik and M. V. Safonov ([61]) on Lyapunov exponents for general random dynamical systems and on principal Floquet bundle and exponential separation for general time dependent parabolic equations. We are also indebted to many other people whose works provide the basics for the monograph. The second author has benefited greatly from the collaborations with V. Hutson and G. T.

xi

xii Vickers on spectral theory for parabolic as well as other types of evolution operators. This monograph was written under the partial support of NSF grants INT0341754 and DMS-0504166. The first author was also supported by the research funds for 2005–2008 (grant MENII 1 PO3A 021 29, Poland). During the preparation of this monograph, the first author visited Auburn University in the summers of 2004 and 2005 and in the spring of 2007. He thanks the faculty of the Department of Mathematics and Statistics for their hospitality. Both of the authors would like to thank Professors Tomasz Dlotko, Georg Hetzer, and Kening Lu for helpful discussions and references. We are very grateful to the people in Chapman & Hall/CRC, in particular, Sunil Nair, Marsha Pronin, Sarah Morris, Ari Silver, and Tom Skipp for their assistance and cooperation. The monograph is prepared in LATEX. Our thanks go to Shashi Kumar for invaluable technical assistance. Janusz Mierczy´ nski Institute of Mathematics and Computer Science Wroclaw University of Technology [email protected] Wenxian Shen Department of Mathematics and Statistics Auburn University [email protected]

Symbol Description

Bounded domain in RN Boundary of D Boundary operator The outer unit normal on ∂D (Ω, F, P) Probability space θ Metric flow on (Ω, F, P) B Family of Borel sets µ Invariant measure α0 Ellipticity constant Y Parameter space for a single linear equation or a general metric space Y Parameter space for a system of linear equations σ Translation flow on Y (or on Y) or general flow or semiflow on Y Z Parameter space for a single nonlinear equation Z Parameter space for a system of nonlinear equations ζ Translation flow on Z (or on Z)

D ∂D B ν

X X Π Φ

ρ− ρ+ Σ λmin λmax λ(µ) λprinc ϕprinc γ0 ψ

Phase space for scalar parabolic equations Phase space for systems of parabolic equations Topological or random linear skew-product semiflow Topological or random (nonlinear) skew-product semiflow Lower principal resolvent Upper principal resolvent Principal spectrum Minimum of the principal spectrum Maximum of the principal spectrum Principal Lyapunov exponent Principal eigenvalue Principal eigenfunction of an elliptic equation Separating exponent test function

xiii

Chapter 1 Introduction

Reaction–diffusion equations or systems in bounded domains have been used to model many evolution processes in science and engineering, for example, Lotka–Volterra competitive and predator–prey systems, color pattern formation in butterflies and sea shells, tumor growth, just to mention a few in biology. Regardless of the details of the model, one of the common requirements is to investigate the spectral problem for an associated linear evolution problem. This is often required as a tool for nonlinear problems, for example when considering stability or invasion (in the ecological context). Traditionally most evolution processes are considered in time independent and spatially homogeneous environments. However, in nature, many evolution processes are subject to various variations of the external environments, and the media of the processes are also heterogeneous. General time dependent and random parabolic equations and systems of such equations are therefore of great interest since they can take the above facts into account in modeling evolution processes. A vast amount of research has been carried out toward various dynamical aspects of nonautonomous and random parabolic equations (see, for a few examples, [7], [8], [17], [23], [24], [25], [26], [27], [37], [38], [54], [98], [110], [111]). As a basic tool for nonlinear problems, it is of great significance to investigate spectral theory for general nonautonomous linear parabolic equations of the form N

N

∂u X ∂ X ∂u = aij (t, x) + ai (t, x)u ∂t ∂xi j=1 ∂xj i=1 +

N X i=1

bi (t, x)

∂u + c0 (t, x)u, ∂xi

x ∈ D,

(1.0.1)

complemented with the boundary conditions B(t)u = 0,

on ∂D,

(1.0.2)

where D ⊂ RN is a bounded domain and B is a boundary operator of either

1

2

Spectral Theory for Parabolic Equations

the Dirichlet or Neumann or Robin type, that is,  u       X N X N  ∂u    + ai (t, x)u νi aij (t, x)  ∂xj B(t)u = i=1 j=1  N X N  X  ∂u    aij (t, x) + ai (t, x)u νi   ∂xj  i=1 j=1    + d0 (t, x)u.

(Dirichlet) (Neumann) (1.0.3)

(Robin)

(νν = (ν1 , ν2 , . . . , νN ) denotes the unit normal on the boundary ∂D pointing out of D, interpreted in a certain weak sense (in the regular sense if ∂D is sufficiently smooth)). It is also of great importance to investigate spectral theory for general random linear parabolic equations of the form N

N

∂u X ∂ X ∂u = aij (θt ω, x) + ai (θt ω, x)u ∂t ∂xi j=1 ∂xj i=1 +

N X

bi (θt ω, x)

i=1

∂u + c0 (θt ω, x)u, ∂xi

x ∈ D,

(1.0.4)

complemented with the boundary conditions B(θt ω)u = 0,

on ∂D,

(1.0.5)

where B is a boundary operator of either the Dirichlet or Neumann or Robin type, that is, B(θt ω) is of the same form as B(t) in (1.0.3) with aij (t, x) = aij (θt ω, x), ai (t, x) = ai (θt ω, x), and d0 (t, x) = d0 (θt ω, x), ω ∈ Ω and ((Ω, F, P), {θt }t∈R ) is an ergodic metric dynamical system. It is important as well to investigate spectral theory of nonautonomous and random linear parabolic equations in nondivergence form and coupled systems of nonautonomous and random linear parabolic equations. Spectral theory is well understood for smooth elliptic or time periodic parabolic equations. For example, it is well known that the eigenvalue λprinc to the eigenvalue problem  x ∈ D, ∆u(x) + h(x)u(x) = λu(x),  ∂u (x) = 0, ∂νν

x ∈ ∂D,

¯ → R are sufficiently smooth, where both the domain D and the function h : D having the largest real part (called the principal eigenvalue) is real, simple, and an eigenfunction ϕprinc corresponding to it (called principal eigenfunction) can be chosen so that ϕprinc (x) > 0 for x ∈ D. Hence all positive

1. Introduction

3

solutions of a time independent linear parabolic equation are attracted in the direction toward the one-dimensional space spanned by a principal eigenfunction of the associated elliptic eigenvalue problem (principal eigenspace) and the solutions lying in the complementary space of the principal eigenspace decay exponentially faster than positive solutions (which is referred to as an exponential separation property). The principal eigenvalue and principal eigenfunction theory is of special interest in applications since it provides necessary and/or sufficient conditions for exponential stability and/or instability in nonlinear problems. The concepts of principal eigenvalue and principal eigenfunction and their properties were extended in [50] to time-periodic parabolic equations (see also [35]). The extension to general time dependent and random parabolic equations is of great difficulty since many approaches which can be successfully applied to time-periodic problems fail to be useful for general time dependent and random problems. Nevertheless, quite a lot of linear theories for time almost-periodic, and general nonautonomous, or even random parabolic problems have been established in various publications, see for example [19], [20], [21], [22], [29], [30], [59], [60], [61], [62], [79], [81], [82], [84], [92], [94], and [97]. The established theories have also found great applications (see [51], [64], [83], [85], [93], [103]). In the past several years we studied nonautonomous/random linear parabolic equations. A general theory of principal spectrum for nonautonomous linear parabolic equations with certain smoothness (both the coefficients and the domain are sufficiently smooth) has been established, serving as a generalization of the well-known theory of principal eigenvalues and principal eigenfunctions for elliptic equations ([81], [82]). As a counterpart, a theory of principal Lyapunov exponents for random linear parabolic equations with certain smoothness has also been established ([81], [82]). For general nonautonomous linear parabolic equations, many fundamental results about existence, uniqueness, continuous dependence on coefficients of solutions are established in [29], [30], [36], [70], and various versions of Harnack inequalities are developed in [10], [40], [59], [60], [61], [69], [73], [88]. Recently a spectral theory for such equations has also been obtained in [59], [60], [61], mostly for the Dirichlet boundary condition case. There is surely a need to develop an adequate spectral theory for general nonautonomous and random parabolic equations with Neumann or Robin boundary conditions. As a basic tool for the study of nonlinear parabolic problems, it is also of great importance to collect existing as well as newly developed linear theories for general and smooth nonautonomous and random parabolic equations in a monograph. The objective of this monograph is to give a hopefully clear and essentially self-contained account of the spectral theory, in particular, principal spectral theory for general time dependent and random linear parabolic equations and systems. We follow the following unified approach for the investigation of the spectral theory: we start to develop the abstract general theory, in the framework of weak solutions (mild solutions in the case of systems of parabolic

4

Spectral Theory for Parabolic Equations

equations), and then specialize to the cases of random and nonautonomous equations. We treat all types of boundary conditions in the same manner. Our exposition focuses on equations in the divergence form, however we provide remarks on corresponding theories for equations in the nondivergence form. On the regularity of the coefficients, we assume the boundedness and measurability, that is, in (1.0.1)+(1.0.2), we assume that aij , ai , bi , and c0 are bounded and measurable on (−∞, ∞) × D and that d0 is nonnegative bounded and measurable on ∂D × (−∞, ∞). In (1.0.4)+(1.0.5), we assume ω ω ω that aω ij , ai , bi , and c are uniformly bounded in ω ∈ Ω and are measurable on (−∞, ∞) × D for each ω ∈ Ω and that dω 0 is nonnegative uniformly bounded in ω ∈ Ω and is measurable on ∂D × (−∞, ∞) for each ω ∈ Ω, where aω ij (t, x) = aij (θt ω, x), etc. As for the regularity of the domain D, no assumption is needed in the Dirichlet boundary condition case and it is assumed that D is Lipschitz in the Neumann or Robin boundary condition case. We prove various additional properties when the coefficients and the domain turn out to be smooth.

1.1

Outline of the Monograph

First of all, in Chapter 2 we establish fundamental theories in a general setting, i.e., for a general family of nonautonomous equations. To be more specific, let Y be a (norm-)bounded subset of L∞ (R × D, 2 RN +2N +1 ) × L∞ (R × ∂D, R) that is closed (hence, compact) in the weak-* topology of that space and is translation invariant (see Section 1.3). We then consider (1.0.1)+(1.0.2) for a whole family of coefficients a ∈ Y . The reason for considering (1.0.1)+(1.0.2) for a whole family of coefficients a ∈ Y is at least fourfold. First, even when we start with only one nonautonomous equation, in many proofs one has to use the procedure of passing to a limit of a sequence of time-translated equations, which can be most easily put in the context of linear skew-product semidynamical systems on a bundle whose base space consists of the closure of all the time translates of the coefficients of the original equation. Second, when considering random equations, their coefficients belong to some family. Third, sometimes we have to compare the properties of the principal spectrum for two equations or even investigate the continuity of the principal spectrum with respect to parameters. And fourth, to study the stability of an invariant set of a nonlinear equation, we need to consider the linearized equations along all the solutions in the invariant set. To emphasize the coefficients and the boundary terms in the problem (1.0.1)+(1.0.2), we will write (1.0.1)a +(1.0.2)a . We list in Chapter 2 some basic assumptions including the uniform ellipticity of the time dependent parabolic equations in the general setting introduced

1. Introduction

5

above and the Lipschitz continuity of the underlying domain of the equations in the Neumann or Robin boundary condition case. We introduce the concept of weak solutions of the equations in the general setting in the space L2 (D) and collect basic properties of weak solutions which will be needed in later chapters, including local regularity, Harnack inequalities, comparison properties, compactness, continuity with respect to initial data as well as the coefficients of the equations. The solutions of the equations in the general setting are shown to form a skew-product semiflow with fibre or phase space L2 (D). Additional properties are proved when both the domain and the coefficients are smooth. Several remarks are provided for the parabolic problems in nondivergence form. In Chapter 3, the concepts of principal spectrum and principal Lyapunov exponents and exponential separation of the skew-product semiflow induced from a family of equations in the general setting are introduced. Various basic properties of principal spectrum and principal Lyapunov exponents are presented. Existence of exponential separation and existence and uniqueness of entire positive solutions are shown under quite general assumptions, namely, that positive solutions satisfy appropriate Harnack inequalities. In addition, we present a multiplicative ergodic theorem for a family of equations in the general setting. We also collect several properties of parabolic equations on one-dimensional space domain in an appendix. Chapter 4 concerns principal spectrum and principal Lyapunov exponents of nonautonomous and random parabolic equations. First the concepts of principal spectrum and principal Lyapunov exponents of nonautonomous and random parabolic equations are introduced in terms of proper family of parabolic equations associated to the given nonautonomous and random equations, which extend the classical concept of principal eigenvalue of elliptic and periodic parabolic equations. Applying the theories developed in Chapters 2 and 3, fundamental properties are then proved, including continuity with respect to the perturbation of coefficients and monotonicity with respect to zero order terms. In Chapter 5, we investigate the effect of time (space) dependence and randomness of zero order terms on principal spectrum and principal Lyapunov exponents of nonautonomous and random parabolic equations. It is shown that neither time (space) dependence nor randomness will reduce principal spectrum and principal Lyapunov exponents and they are indeed increased except in degenerate cases. More precisely, we show that in the general case the principal spectrum (principal Lyapunov exponent) of a nonautonomous (random) parabolic equation is always greater than or equal to that of the corresponding time-averaged equation. We also show that in the smooth case, the principal spectrum (principal Lyapunov exponent) of a nonautonomous (random) parabolic equation is strictly greater than that of the time-averaged equation except in the case that the coefficient can be decomposed as the sum of a spatially dependent term and a time dependent term. Similar results are proved about the effect of space dependence of zero order terms on principal

6

Spectral Theory for Parabolic Equations

spectrum and principal Lyapunov exponents of nonautonomous and random parabolic equations. In the biological context these results mean that invasion by a new species is always easier in the time and space dependent case. In addition, we explore the effect of the shape of the domain on principal spectrum and principal Lyapunov exponents and extend the well-known Faber–Krahn inequalities for elliptic and time-periodic parabolic problems to general time dependent and random problems. Chapter 6 is to extend the linear theory for scalar nonautonomous and random parabolic equations to cooperative systems of such equations. More precisely, we consider the following cooperative systems of nonautonomous parabolic equations  N N X  ∂ X k ∂uk k  ∂uk =  a (t, x) + a (t, x)u k  ij i  ∂t ∂xi j=1 ∂xj   i=1  N K X ∂uk X k  + bki (t, x) + cl (t, x)ul ,   ∂xi   i=1 l=1    k B (t)uk = 0,

x ∈ D,

(1.1.1)

x ∈ ∂D,

where B k is a boundary operator of either the Dirichlet or Neumann or Robin type, that is, B k (t) = B(t) (B(t) is as in (1.0.3)) with aij (t, x) = akij (t, x), ai (t, x) = aki (t, x), and d0 (t, x) = dk0 (t, x), k = 1, 2, . . . , K, and the following cooperative systems of random parabolic equations  N N X  ∂uk ∂ X k ∂uk k   = a (θ ω, x) + a (θ ω, x)u t t k  i  ∂t ∂xi j=1 ij ∂xj   i=1  N K X ∂uk X k k  + b (θ ω, x) + cl (θt ω, x)ul , t  i  ∂xi   i=1 l=1    B k (θt ω)uk = 0,

x ∈ D,

(1.1.2)

x ∈ ∂D,

where k = 1, 2, . . . , K, ω ∈ Ω, ((Ω, F, P), {θt }t∈R ) is an ergodic metric dynamical system, and for each ω ∈ Ω, B k (θt ω) = B(t) with aij (t, x) = akij (θt ω, x), ai (t, x) = aki (θt ω, x), and d0 (t, x) = dk0 (θt ω, x) (B(t) is as in (1.0.3)). We extend the theories developed in Chapters 2–5 for nonautonomous and random parabolic equations to the above cooperative systems of nonautonomous and random parabolic equations. While doing so, a linear theory is first established for a general family of cooperative systems of parabolic equations, that is, (1.1.1) for all a = (akij , aki , bki , ckl , dk0 ) in a subset Y of L∞ (R × 2 D, RK(N +2N +K) ) × L∞ (R × ∂D, RK ). In the last chapter we consider the applications of the linear theory developed in Chapters 2 to 5 to the uniform persistence issue in systems of random and nonautonomous nonlinear parabolic equations of Kolmogorov type. We

1. Introduction

7

focus on the uniform persistence of the following two species competitive Kolmogorov systems of random partial differential equations:    ∂u1 = ∆u1 + f1 (θt ω, x, u1 , u2 )u1 , x ∈ D,   ∂t     ∂u2 = ∆u2 + f2 (θt ω, x, u1 , u2 )u2 , x ∈ D, (1.1.3) ∂t     Bu1 = 0, x ∈ ∂D,    Bu = 0, x ∈ ∂D, 2 where ω ∈ Ω, ((Ω, F, P), {θt }t∈R ) is an ergodic metric dynamical system, f = ¯ (f1 , f2 ) : Ω×D×[0, ∞)×[0, ∞) → R2 , and B is either the Dirichlet or Neumann boundary operator, i.e.,  (Dirichlet) Id (1.1.4) B := ∂ (Neumann) ∂νν and the uniform persistence of the following two species competitive Kolmogorov systems of nonautonomous partial differential equations :  ∂u1   = ∆u1 + f1 (t, x, u1 , u2 )u1 , x ∈ D,   ∂t     ∂u2 = ∆u2 + f2 (t, x, u1 , u2 )u2 , x ∈ D, (1.1.5) ∂t     Bu1 = 0, x ∈ ∂D,    Bu = 0, x ∈ ∂D, 2 ¯ × [0, ∞) × [0, ∞) → R2 , and B is as in (1.1.4). where f = (f1 , f2 ) : R × D To this end, by applying the linear theory in Chapters 1–5, we first establish uniform persistence theorems for the following random parabolic equation of Kolmogorov type:   ∂u = ∆u + f (θt ω, x, u)u, x ∈ D, ∂t (1.1.6)  Bu = 0, x ∈ ∂D, ¯ × [0, ∞) 7→ R and B is as in (1.1.4), and the following nonauwhere f : Ω × D tonomous parabolic equation of Kolmogorov type:   ∂u = ∆u + f (t, x, u)u, x ∈ D, ∂t (1.1.7)  Bu = 0, x ∈ ∂D, ¯ × [0, ∞) 7→ R and B is as in (1.1.4). Uniform persistence thewhere f : R × D orems are then established for (1.1.3) and (1.1.5). While doing all the above,

8

Spectral Theory for Parabolic Equations

global attracting dynamics and uniform persistence theories are first established for a general family of nonlinear parabolic equations of Kolmogorov type (i.e., (1.1.7) for all f in a set Z of certain admissible functions) and for a general family of competitive Kolmogorov systems of parabolic equations (i.e., (1.1.5) for all f in a set Z of certain admissible functions). We have chosen to provide the fundamentals (mainly existence, uniqueness, continuous dependence of solutions and Harnack inequalities for positive solutions) for the introduction of spectral theory rather than to actually carry out such analysis, and we supply appropriate references where specific results are quoted. For the exposition to be self-contained, we provide proofs for some (already known) spectral results in existing publications whenever we feel it would be more helpful for the reader.

1.2

General Notations and Concepts

R denotes the set of reals, Z denotes the set of integers, and N denotes the set of positive integers. For t ∈ R, btc stands for the greatest integer smaller than or equal to t. If E ⊂ Rm , it is always considered with the topology induced from the standard topology on Rm . For a measurable subset E ⊂ Rm , |E| denotes the m-dimensional Lebesgue measure of E. All Banach spaces are assumed to be real. For a Banach space B let k·kB denote its norm. Let B1 , B2 be Banach spaces. The symbol L(B1 , B2 ) denotes the space of all bounded linear operators from B1 into B2 endowed with the norm topology. The norm in L(B1 , B2 ) is denoted by k·kB1 ,B2 . The symbol Ls (B1 , B2 ) denotes the vector space of all bounded linear operators from B1 into B2 , but endowed with the strong operator topology. Instead of L(B, B) (Ls (B, B), resp.) we write L(B) (Ls (B), resp.). For B a Banach space and B ∗ its dual, we denote by h·, ·iB,B ∗ the duality pairing between them. For H a Hilbert space, h·, ·iH,H stands for the inner product in H. For E ⊂ Rm , B a Banach space and k = 0, 1, 2, . . . , we write C k (E, B) for the Banach space of k-times continuously differentiable B-valued functions defined on E whose derivatives up to order k are bounded on B. C k (E, B) is assumed to be endowed with the standard C k -norm. Instead of C 0 (E, B) we write C(E, B), and instead of C k (E, R) we write C k (E). ˚ For a compact E ⊂ Rm , we denote by C(E, B) the (closed) linear subspace of the Banach space C(E, B) consisting of functions taking value 0 on the boundary of E.

1. Introduction

9

For E ⊂ Rm , k = 0, 1, 2, . . . and α ∈ (0, 1), we denote by C k+α (E) the Banach space of functions in C k (E) whose derivatives up to order k are H¨older continuous with exponent α, uniformly in x ∈ E. C k+α (E) is assumed to be given the standard norm. Instead of C 0+α (E) we write C α (E). For E ⊂ Rm and k = 1, 2, . . . we denote by C k− (E) the Banach space of functions in C k−1 (E) whose derivatives of up to order k − 1 are Lipschitz continuous, uniformly in x ∈ E. C k− (E) is assumed to be given the standard norm. For E1 ⊂ Rm1 , E2 ⊂ Rm2 , k = 0, 1, 2, . . . and l = 0, 1, 2, . . . , we denote by C k,l (E1 × E2 ) the Banach space of real-valued functions u = u(x1 , x2 ) such that all the derivatives ∂ i+j u/∂xi1 ∂xj2 with 0 ≤ i ≤ k, 0 ≤ j ≤ l, are continuous and bounded on E1 × E2 . C k,l (E1 × E2 ) is assumed to be endowed with the standard norm. For E1 ⊂ Rm1 , E2 ⊂ Rm2 , k = 0, 1, 2, . . . , l = 0, 1, 2, . . . , α ∈ (0, 1) and β ∈ (0, 1), we denote by C k+α,l+β (E1 × E2 ) the Banach space of functions from C k,l (E1 ×E2 ) such that all the derivatives ∂ i+j u/∂xi1 ∂xj2 with 0 ≤ i ≤ k, 0 ≤ j ≤ l, are H¨ older continuous in x1 with exponent α and in x2 with exponent β, uniformly in (x1 , x2 ) ∈ E1 × E2 . C k+α,l+β (E1 × E2 ) is assumed to be endowed with the standard norm. For E1 ⊂ Rm1 , E2 ⊂ Rm2 , k = 1, 2, . . . and l = 1, 2, . . . , we denote by C k−,l− (E1 × E2 ) the Banach space of functions from C k−1,l−1 (E1 × E2 ) such that all the derivatives ∂ i+j u/∂xi1 ∂xj2 with 0 ≤ i ≤ k − 1, 0 ≤ j ≤ l − 1, are Lipschitz continuous in x1 and in x2 , uniformly in (x1 , x2 ) ∈ E1 × E2 . C k−,l− (E1 × E2 ) is assumed to be endowed with the standard norm. For E ⊂ Rm , let D(E) stand for the vector space of (real-valued) C ∞ functions with compact supports in E (test functions), and let D0 (E) stand for the corresponding vector space of distributions. We collect now some facts about measurable functions defined on a Lebesguemeasurable subset E of Rm and taking values in a Banach space B (for a reference, see [74] or [99]). We will assume B to be separable. To start with, recall that a function u : E → B is called simple if there are Lebesgue-measurable pairwise disjoint sets E1 , . . . , Em ⊂ E, E1 ∪ · · · ∪ Em = E, and elements u1 , . . . , um ∈ B such that u(x) = uj for any x ∈ Ej (1 ≤ j ≤ m). DEFINITION 1.2.1 A function u : E → B, where B is a separable Banach space, is called measurable if one of the following (mutually equivalent) conditions is satisfied: (i) There is a sequence (un )∞ n=1 of simple functions such that kun (x) − u(x)kB converges to 0 as n → ∞, for Lebesgue-a.e. x ∈ E, (ii) For any open subset A ⊂ B (or for any closed subset A ⊂ B), u−1 (A) is Lebesgue-measurable, (iii) For any v ∗ ∈ B ∗ the function [ E 3 x 7→ hu(x), v ∗ iB,B ∗ ∈ R ] is Lebesgue-measurable.

10

Spectral Theory for Parabolic Equations

A function satisfying (i) is usually referred to as strongly measurable, whereas a function satisfying (iii) is usually called weakly measurable. The equivalence of (i) and (iii) is a consequence of the Pettis theorem. The following two lemmas can be easily proved. LEMMA 1.2.1 A continuous function u : E → B is measurable. LEMMA 1.2.2 If u : E → B is measurable and f : B → B1 is continuous, where B and B1 are separable Banach spaces, then the composition f ◦ u : E → B1 is measurable. For 1 ≤ p < ∞, a measurable function u : E → B belongs to Lp (E, B) if the (measurable) function [ E 3 x 7→ (ku(x)kB )p ∈ R ] belongs to L1 (E, R). The norm in Lp (E, B) is defined as Z kukLp (E,B) :=

1/p (ku(x)kB )p dx .

E

A measurable function u : E → B belongs to L∞ (E, B) if the (measurable) function [ E 3 x 7→ ku(x)kB ∈ R ] belongs to L∞ (E, R). The norm in L∞ (E, B) is defined as kukL∞ (E,B) := ess sup { ku(x)kB : x ∈ E }. LEMMA 1.2.3 Let 1 ≤ p ≤ ∞. If u ∈ Lp (E, B), where the Banach space B is separable, then there is a sequence (un )∞ n=1 of simple functions such that ku1 kLp (E,B) ≤ ku2 kLp (E,B) ≤ · · · ≤ kukLp (E,B) , kun kLp (E,B) → kukLp (E,B) as n → ∞, and un (x) converges in B to u(x) as n → ∞, for Lebesgue-a.e. x ∈ E. PROOF

See [74, Lemma 21-2.5].

Instead of Lp (E, R) we write Lp (E). (But for the notation Lp (R × ∂D) see Section 1.3.) For 1 ≤ p ≤ ∞, E a Lebesgue-measurable subset of Rm and k = 1, 2, . . . we denote by Wpk (E) the Banach space of real-valued functions whose generalized derivatives up to order k belong to Lp (E). For 1 ≤ p ≤ ∞, E1 a Lebesgue-measurable subset of Rm1 , E2 a Lebesguemeasurable subset of Rm2 , and l = 0, 1, 2, . . . , we denote by Wpl,2l (E1 × E2 ) (Wpl,0 (E1 × E2 ), Wp0,l (E1 × E2 )) the Banach space of real-valued functions u = u(x1 , x2 ) such that all generalized derivatives ∂ i+j u/∂xi1 ∂xj2 (∂ i u/∂xi1 , ∂ j u/∂xj2 ) with 0 ≤ 2i + j ≤ 2l (0 ≤ i ≤ l, 0 ≤ j ≤ l) belong to Lp (E1 × E2 ).

1. Introduction

11

For 1 ≤ p ≤ ∞ and E a Lebesgue-measurable subset of Rm , Lp,loc (E) stands for the Fr´echet space consisting of real-valued functions such that for any compact subset E1 ⊂ E the restriction u|E1 belongs to Lp (E1 ). For a metric space S, by B(S) we denote the countably additive algebra of all Borel subsets of S, and by C(S) we denote the Banach space of all bounded continuous real functions on S with the supremum norm. A topological flow (or a topological dynamical system) on a metric space Y is a continuous mapping σ: R × Y → Y satisfying the following properties (where σt (·) stands for σ(t, ·)): (TF1) σ0 = IdY , (TF2) σs+t = σs ◦ σt for any s, t ∈ R. It follows from (TF1) and (TF2) that (TF3) (σt )−1 = σ−t for any t ∈ R. Sometimes for a topological flow σ we write σ = {σt }t∈R . Also, we can write (Y, σ) = (Y, {σt }t∈R ). If Y is compact, we call (Y, σ) a compact flow. A topological semiflow (or a topological semidynamical system) on a metric space Y is a mapping σ : [0, ∞) × Y → Y satisfying the following properties (where σt (·) stands for σ(t, ·)): (TSF0) σ restricted to (0, ∞) × Y is continuous; moreover, for each y ∈ Y the mapping [ [0, ∞) 3 t 7→ σt y ∈ Y ] is continuous, (TSF1) σ0 = IdY , (TSF2) σs+t = σs ◦ σt for any s, t ≥ 0. Sometimes for a topological semiflow σ we write σ = {σt }t≥0 . Also, we can write (Y, σ) = (Y, {σt }t≥0 ). From now on until revoking (Y, σ) denotes a topological semiflow. Let d(·, ·) stand for the metric on Y . A set A ⊂ Y is invariant if σt (A) = A for any t ≥ 0. Sometimes we say that A is invariant under σ (or σ-invariant). A closed invariant set A ⊂ Y is called an isolated invariant set if there is a neighborhood of U of A such that A is the largest invariant set contained in U. A set A ⊂ Y is forward invariant if σt (A) ⊂ A for any t ≥ 0. For y ∈ Y the forward orbit of y is defined as O+ (y) := { σt y : t ≥ 0 }. For A ⊂ Y the forward orbit of A is defined as O+ (A) := { σt (A) : t ≥ 0 }. It should be remarked that in the literature forward invariant sets, forward orbits, etc., are usually called positively invariant sets, positive orbits,

12

Spectral Theory for Parabolic Equations

etc. But in the present monograph “positive” is reserved for “belonging (or related) to the cone of nonnegative functions (or its interior).” We say that A ⊂ Y attracts B ⊂ Y if for each > 0 there is T = T () ≥ 0 such that σt (B) is contained in the -neighborhood of A, for any t ≥ T . The ω-limit set of y ∈ Y is defined as \ ω(y) := cl O+ (σt y). t≥0

There holds: z ∈ Y belongs to ω(y) if and only if there is a sequence (tn )∞ n=1 ⊂ R such that limn→∞ tn = ∞ and limn→∞ σ(tn , y) = z. ω(y) is closed and invariant. Moreover, ω(y) = ω(σt y) for any t ≥ 0. The following lemma follows from general theory of topological semiflows (see [46]). LEMMA 1.2.4 Assume that for some t ≥ 0 the forward orbit O+ (σt y) has compact closure. Then ω(y) is nonempty, compact and connected, and attracts y. The ω-limit set of A ⊂ Y is defined as \ ω(A) := cl O+ (σt (A)). t≥0

There holds: z ∈ Y belongs to ω(A) if and only if there are sequences ∞ (tn )∞ n=1 ⊂ R and (yn )n=1 ⊂ A such that tn → ∞ and σ(tn , yn ) → z as n → ∞. ω(A) is closed and invariant. Moreover, ω(A) = ω(σt (A)) for any t ≥ 0. Similar to Lemma 1.2.4, there holds (see [46]) LEMMA 1.2.5 Assume that for some t ≥ 0 the forward orbit O+ (σt (A)) has compact closure. Then ω(A) is nonempty and compact, and attracts A. Moreover, if A is connected then ω(A) is connected, too. A nonempty compact invariant set A ⊂ Y is an attractor if there is a neighborhood C of A such that A attracts C, or, equivalently, ω(C) = A. A nonempty compact invariant set Γ ⊂ Y is said to be the global attractor if it attracts each bounded B ⊂ Y . Observe that if (Y, σ) possesses a global attractor Γ then for each bounded B ⊂ Y there holds ∅ 6= ω(B) ⊂ Γ. Further, Γ is the maximal compact invariant set in the sense that if A ⊂ Y is compact invariant then necessarily A ⊂ Γ. Consequently, a global attractor is uniquely determined. Let A ⊂ Y be a forward invariant set. Then the restriction σ|[0,∞)×A satisfies all the properties (TSF0)–(TSF2) of a semiflow, and is referred to as the restriction of the semiflow σ to A (usually denoted simply by σ|A ).

1. Introduction

13

From now on, again until revoking, (Y, σ) will denote a compact flow. A set A ⊂ Y is invariant if σt (A) = A for each t ∈ R. For y ∈ Y the backward orbit of y is defined as O− (y) := { σt y : t ≤ 0 }. For A ⊂ Y the backward orbit of A is defined as O− (A) := { σt (A) : t ≤ 0 }. For y ∈ Y the orbit of y is defined as O(y) := { σt y : t ∈ R }. For A ⊂ Y the orbit of A is defined as O(A) := { σt (A) : t ∈ R }. The definitions of the ω-limit set of a point and of a set are the same as in the case of semiflows. We have that ω(y) = ω(σt y) for any t ∈ R, and ω(A) = ω(σt (A)) for any t ∈ R. The α-limit set of y ∈ Y is defined as \ α(y) := cl O− (σt y). t≤0

There holds: z ∈ Y belongs to α(y) if and only if there is a sequence (tn )∞ n=1 ⊂ R such that limn→∞ tn = −∞ and limn→∞ σ(tn , y) = z as n → ∞. α(y) is closed and invariant. Moreover, α(y) = α(σt y) for any t ∈ R. The α-limit set of A ⊂ Y is defined as \ α(A) := cl O− (σt (A)). t≤0

There holds: z ∈ Y belongs to α(A) if and only if there are sequences ∞ (tn )∞ n=1 ⊂ R and (yn )n=1 ⊂ A such that tn → −∞ and σ(tn , yn ) → z. α(A) is closed and invariant. Moreover, α(A) = α(σt (A)) for any t ∈ R. A set B ⊂ Y is a repeller if there is a neighborhood C of B such that α(C) = B. Consequently, B is compact and invariant. Let A be an attractor. We say that y ∈ Y belongs to the attraction basin of A if ω(y) ⊂ A. The complement in Y of the attraction basin of A is a repeller (called the repeller dual to A). Let B be a repeller. We say that y ∈ Y belongs to the repulsion basin of B if α(y) ⊂ B. The complement in Y of the repulsion basin of B is an attractor (called the attractor dual to B). A finite ordered family {A1 , . . . , Ak } of nonempty pairwise disjoint compact Sk invariant sets is called a Morse decomposition of Y if for each y ∈ Y \ i=1 Ai there are 1 ≤ i1 < i2 ≤ k such that α(y) ⊂ Ai1 and ω(y) ⊂ Ai2 . Let A ⊂ Y be an invariant set. Then the restriction σ|R×A satisfies the properties (TF1), (TF2) of a flow, and is referred to as the restriction of the flow σ to A (usually denoted simply by σ|A ). A compact flow (Y, σ) is called topologically transitive if there is y0 ∈ Y such that Y = cl O(y0 ). A compact flow (Y, σ) is called minimal if the only closed invariant nonempty set is Y itself. From now on until further notice, (Y, σ) has no specific meaning. Let (Ω, F) be a measurable space (that is, Ω is a set and F is a countably additive algebra of subsets of Ω). A (B(R)×F, F)-measurable mapping θ : R×

14

Spectral Theory for Parabolic Equations

Ω → Ω is a measurable flow (or a measurable dynamical system) on (Ω, F) if it satisfies the following properties (where θt (·) stands for θ(t, ·)). (MF1) θ0 = IdΩ , (MF2) θs+t = θs ◦ θt for any s, t ∈ R. As a section of a measurable mapping, θt : Ω → Ω is (F, F)-measurable, for each t ∈ R. Further, from (MF1) and (MF2) it follows that (θt )−1 is (F, F)-measurable, and (MF3) (θt )−1 = θ−t for any t ∈ R. Sometimes for a measurable flow θ we write θ = {θt }t∈R . Also, we can write ((Ω, F), θ) = ((Ω, F), {θt }t∈R ). For ((Ω, F), θ) a measurable flow, a set A ⊂ Ω is invariant if θt (A) = A for any t ∈ R. Sometimes we say that A is invariant under θ (or θ-invariant). We say that a triple (Ω, F, P) is a probability space if (Ω, F) is a measurable space and P is a probability measure on F. For a probability space (Ω, F, P) denote by L1 ((Ω, F, P)) the Banach space of all real-valued (F, B(R))-measurable functions that are integrable with respect to P, with the standard norm. Let (Ω1 , F1 ) and (Ω2 , F2 ) be measurable spaces, and let P1 be a probability measure on F1 . For a (F1 , F2 )-measurable mapping F : Ω1 → Ω2 , we define F P1 , the image of P1 with respect to F , by F P1 (A) := P1 (F −1 (A))

for any

A ∈ F2 .

F P1 so defined is a probability measure on F2 . For a measurable flow ((Ω, F), θ) we say that a probability measure P on F is θ-invariant if θt P = P for each t ∈ R. In such a case we will call ((Ω, F, P), θ) a metric dynamical system (or a metric flow ) . Sometimes we write ((Ω, F, P), {θt }t∈R ). For a metric dynamical system ((Ω, F, P), θ) we say that the invariant measure P is ergodic if for any θ-invariant set A ∈ F one has either P(A) = 0 or P(A) = 1. A metric dynamical system ((Ω, F, P), θ) is ergodic if the invariant measure P is ergodic. The following Birkhoff Ergodic Theorem will be often utilized throughout the monograph. LEMMA 1.2.6 (Birkhoff ’s Ergodic Theorem) Let ((Ω, F, P), θ) be an ergodic metric dynamical system and h ∈ L1 ((Ω, F, P)). ˜ ∈ F such that P(Ω) ˜ = 1 and Then there is a θ-invariant set Ω 1 t→∞ t

Z

t

Z h(θs ω) ds =

lim

0

h(·) dP(·) Ω

1. Introduction

15

˜ for any ω ∈ Ω. PROOF

See [7] or references therein.

For Y a compact metric space, by a measure on Y we mean a probability measure on B(Y ). For a compact flow (Y, σ) = (Y, {σt }t∈R ), a probability measure µ on Y is said to be an invariant measure of (Y, σ) if µ(σt (A)) = µ(A) for any t ∈ R and A ∈ B(Y ) (hence ((Y, B(Y ), µ), σ) is a metric dynamical system). An invariant measure µ of (Y, σ) is ergodic if for any σ-invariant set A ∈ B(Y ) one has either µ(A) = 0 or µ(A) = 1. A compact flow (Y, σ) is said to be uniquely ergodic if there is a unique (necessarily ergodic) invariant measure for σ. Regarding the existence of (ergodic) invariant measure of a compact flow (Y, σ), we have LEMMA 1.2.7 (Krylov–Bogolyubov Theorem) If σ is a topological flow on a compact metric space Y then there exist (ergodic) invariant measures of (Y, σ). PROOF

See [107, Theorem 6.10].

Consider a product bundle S × Y , where S and Y are metric spaces (notice that Y is the base and S is a model fiber). Let σ be a topological semiflow on Y . We say that Φ : [0, ∞) × S × Y → S × Y is a topological skew-product semiflow on S × Y covering σ if it can be written as Φ(t; u, a) = (φt (a, u), σt a), t ≥ 0, u ∈ S, a ∈ Y, and has the following properties (where Φt (·, ·) stands for Φ(t; ·, ·)): (TSP0) Φ restricted to (0, ∞) × S × Y is continuous; moreover, for each (z, a) ∈ S × Y the mapping [ [0, ∞) 3 t 7→ (φt (z, a), σt a) ∈ S × Y ] is continuous, (TSP1) Φ0 = IdS×Y , (TSP2) Φt+s = Φt ◦ Φs for any t, s ≥ 0. When B is a Banach space, a topological linear skew-product semiflow Π : [0, ∞) × B × Y → B × Y, Π(t; u, a) = (U (t, a)u, σt a)

16

Spectral Theory for Parabolic Equations

on the product Banach bundle B × Y covering a topological semiflow σ on Y is a topological skew-product semiflow with the property that for each t ≥ 0 and each a ∈ Y the mapping [ B 3 u 7→ U (t, a)u ∈ B ] belongs to L(B). We write that mapping as Ua (t, 0). When S is a subset of a Banach space, a topological C 1 skew-product semiflow Φ : [0, ∞) × S × Y → S × Y on the product bundle S × Y covering a topological semiflow σ on Y is a topological skew-product semiflow with the property that for each t ≥ 0 and each a ∈ Y the mapping [ S 3 u 7→ φt (u, a) ∈ S ] is of class C 1 , and, moreover, the derivatives in u depend continuously on (t, u, a) ∈ (0, ∞) × S × Y . Consider a measurable bundle S × Ω, where S is a metric space and (Ω, F) is a measurable space (notice that Ω is the base and S is a model fiber). Let θ be a metric flow on (Ω, F, P). We say that a mapping Φ : [0, ∞) × S × Ω → S × Ω is a continuous random skew-product semiflow on S × Ω covering θ if it can be written as Φ(t; u, ω) = (φt (ω, u), θt ω),

t ≥ 0, u ∈ S, ω ∈ Ω,

and has the following properties (where Φt (·, ·) stands for Φ(t; ·, ·)): (RSP0) Φ is (B([0, ∞)) × B(S) × F, B(S) × F)-measurable, (RSP1) Φ0 = IdX×Y , (RSP2) Φt+s = Φt ◦ Φs for any t, s ≥ 0, (RSP3) For any t ≥ 0 and ω ∈ Ω the mapping Φt (ω, ·) is continuous. When B is a Banach space, a random linear skew-product semiflow Π : [0, ∞) × B × Ω → B × Ω, Π(t; u, ω) = (U (t, ω)u, θt ω) on the measurable Banach bundle B × Ω covering a metric flow θ on (Ω, F, P) is a random skew-product semiflow with the property that for each t ≥ 0 and each ω ∈ Ω the mapping [ B 3 u 7→ U (t, ω)u ∈ B ] belongs to L(B). We write that mapping as Uω (t, 0).

1.3

Standing Assumptions

We assume that D ⊂ RN is a bounded domain (that is, an open connected set), with boundary ∂D.

1. Introduction

17

Subsets of R are always considered with the (one-dimensional) Lebesgue measure. In all the notations of the form “Lp (R × D, ·),” or “a.e. on D,” etc., the domain D is assumed to be endowed with the N -dimensional Lebesgue measure, whereas in all the notations of the form “Lp (R × ∂D, ·),” or “a.e. on ∂D,” etc., the boundary ∂D is assumed to be endowed with the (N − 1)-dimensional Hausdorff measure HN −1 . When the boundary ∂D is (at least) Lipschitz, then the (N − 1)-dimensional Hausdorff measure on ∂D equals the ordinary surface measure on ∂D. We denote the operator of differentiation in t (acting on functions of N + 1 ∂ , and we denote the operator of differenvariables u = u(t, x1 , . . . , xN )) by ∂t ∂ ∂ tiation in xi (1 = 1, . . . , N ) by ∂xi . Sometimes instead of ∂t we write ∂t , and ∂ instead of ∂xi we write ∂xi . Throughout Chapters 2 to 5 of the monograph we will write N N a = ((aij )N i,j=1 , (ai )i=1 , (bi )i=1 , c0 , d0 ),

where aij , ai , bi and c0 are the coefficients of the equation (1.0.1), and d0 is the coefficient in the boundary condition (in the Dirichlet or Neumann case d0 is set to be equal to zero). N N For any a = ((aij )N i,j=1 , (ai )i=1 , (bi )i=1 , c0 , d0 ) and any t ∈ R we define the time-translate a · t of a by N N a · t := ((aij · t)N i,j=1 , (ai · t)i=1 , (bi · t)i=1 , c0 · t, d0 · t),

where aij · t(τ, x) := aij (τ + t, x) for s, τ ∈ R, x ∈ D, etc. We fix a countable dense subset {g1 , g2 , . . . } of the unit ball in L1 (R × 2 D, RN +2N +1 ) × L1 (R × ∂D, R) such that for each k ∈ N there exists K = K(k) > 0 with the property that gk (t, ·) = 0 for a.e. t ∈ R \ [−K, K]. 2 For any a(1) , a(2) ∈ L∞ (R × D, RN +2N +1 ) × L∞ (R × ∂D, R) put d(a(1) , a(2) ) :=

∞ X 1 |hgk , (a(1) − a(2) )iL1 ,L∞ |. 2k

(1.3.1)

k=1

We make the following assumptions on Y , the family of admissible coefficients. 2

(A1-1) Y is a (norm-)bounded subset of L∞ (R×D, RN +2N +1 )×L∞ (R×∂D, R) that is closed (hence, compact) in the weak-* topology of that space. (A1-2) Y is translation invariant: If a ∈ Y then a · t ∈ Y , for each t ∈ R. N N (A1-3) d0 = 0 for all a = ((aij )N i,j=1 , (ai )i=1 , (bi )i=1 , c0 , d0 ) ∈ Y (in the Dirichlet N N or Neumann cases) or d0 ≥ 0 for all a = ((aij )N i,j=1 , (ai )i=1 , (bi )i=1 , c0 , d0 ) ∈ Y (in the Robin case).

18

Spectral Theory for Parabolic Equations

(Y, d) is a compact metric space. When speaking of convergence of sequences in Y , continuity of mappings from and/or to Y , etc., that space is assumed to be endowed with the weak-* topology, or, which is equivalent, with the topology generated by the metric d. The mapping [ Y × R 3 (a, t) 7→ a · t ∈ Y ] is continuous. For t ∈ R put σt : Y → Y to be σt a := a · t. The family σ = {σt }t∈R is a flow on the compact metrizable space Y . We will write simply k·k for the standard norm in L2 (D). Similarly, h·, ·i will stand for the standard inner product in the Hilbert space L2 (D). Let B1 , B2 be Banach spaces whose members are (equivalence classes of) real functions defined on D. We write B1 ,→ B2 if B1 continuously embeds into B2 , and B1 ,−,→ B2 if B1 ,→ B2 and the embedding is compact (completely continuous). Let X ,→ L1 (D). In X we define the nonnegative cone X + as X + := { u ∈ X : u(x) ≥ 0

for a.e.

x ∈ D }.

¯ then If X ,→ C(D) X + = { u ∈ X : u(x) ≥ 0

for all

¯ }. x∈D

The nonnegative cone X + is a closed and convex subset of X, having the following properties: • If u ∈ X + and r ≥ 0 then ru ∈ X + . • If u ∈ X + and −u ∈ X + then u = 0. For any u1 , u2 ∈ X we write u1 ≤ u2

if

u2 − u1 ∈ X + ,

u1 < u2

if

u2 − u1 ∈ X + \ {0}.

The reversed symbols ≥ and > are used in the usual way. ¯ the norm has In the Banach spaces X = Lp (D), 1 ≤ p ≤ ∞, or X = C(D), the following monotonicity property: For any u1 , u2 ∈ X, if 0 ≤ u1 ≤ u2 then ku1 kX ≤ ku2 kX . Sometimes it happens that the interior of the nonnegative cone X + is nonempty. We denote then that interior by X ++ . Also, for any u1 , u2 ∈ X we write u1 u2 if u2 − u1 ∈ X ++ .

1. Introduction

19

The reversed symbol is used in the usual way. ˚1 (D) ¯ the Banach If the boundary ∂D of D is of class C 1 , we denote by C 1 ¯ space consisting of u ∈ C (D) such that u(x) = 0 for all x ∈ ∂D. Recall that ν = (ν1 , ν2 , . . . , νN ) denotes the outer unit normal on the boundary ∂D pointing out of D. LEMMA 1.3.1 Assume additionally that D has a boundary ∂D of class C 2+α , for some 0 < α < 1. ¯ ++ of the nonnegative cone C 1 (D) ¯ + is nonempty, (1) The interior C 1 (D) and is characterized by ¯ ++ = { u ∈ C 1 (D) ¯ + : u(x) > 0 C 1 (D)

for all

¯ }. x∈D

(1.3.2)

˚1 (D) ¯ ++ of the nonnegative cone C ˚1 (D) ¯ + is nonempty, (2) The interior C and is characterized by ˚1 (D) ¯ ++ = { u ∈ C ˚1 (D) ¯ + : u(x) > 0 for all x ∈ D C and (∂u/∂νν )(x) < 0 for all x ∈ ∂D }. (1.3.3) PROOF We prove only (2), as the main idea of the proof of (1) is the same (but the details are much simpler). We apply a construction of a collar of the boundary ∂D of the manifold ¯ (see, e.g., [55]). We extend the C 1 vector field ν : ∂D → RN to a C 1 D vector field ν˜ defined on some compact (relative) neighborhood V of ∂D in ¯ Denote by ρ = ρ(t, x) the local flow of the vector field ν˜. The standard D. theorem on the C 1 dependence of solutions of systems of ordinary differential equations on initial values guarantees the existence of η > 0 such that the restriction of ρ to [−η, 0] × ∂D is a C 1 diffeomorphism into some compact ¯ (relative) neighborhood V1 of ∂D in D. Fix u belonging to the right-hand side of (1.3.3). For x ∈ V1 , x = ρ(s, x ˜), s ∈ [−η, 0], x ˜ ∈ ∂D, we write u ˜(s, x ˜) := u(x) (in other words, u ˜ is the representation of the restriction u|V1 in the (s, x ˜)-coordinates). The second condition in (1.3.3) translates into (∂ u ˜/∂s)(0, x ˜) < 0 for all x ˜ ∈ ∂D. By the compactness of ∂D, there exist δ > 0 and 1 > 0 such that for any v ∈ X with kv − ukC˚1 (D) v /∂s)(s, x ˜) < 0 for all s ∈ [−δ, 0] and all x ˜ ∈ ∂D. ¯ < 1 one has (∂˜ Denote V2 := ρ([−δ, 0] × ∂D). V2 is a compact (relative) neighborhood of ∂D ¯ We conclude that v(x) > 0 for any v ∈ X with kv − uk ˚1 ¯ < 1 and in D. C (D) ¯ \ V2 . Since u is positive on any x ∈ V2 \ ∂D. Denote by V20 the closure of D the compact set V20 , it is bounded away from zero on V20 . There is 2 > 0 such that for any v ∈ X, if kv − ukC˚1 (D) ¯ < 2 then v|V20 is positive and bounded

20

Spectral Theory for Parabolic Equations

away from zero. Consequently, if kv − ukC˚1 (D) ¯ < , where := min{1 , 2 }, ˚1 (D) ¯ + . This proves the “⊃” inclusion. then v ∈ C Let ϕprinc be some (nonnegative) principal eigenvalue of the elliptic equation ∆u = 0 on D with the Dirichlet boundary conditions. The standard regularity theory and maximum principles (see, e.g., [43]) guarantee that ϕprinc belongs ˚1 (D) ¯ ++ . Finally, let u ∈ C ˚1 (D) ¯ ++ . to the right-hand side of (1.3.3), hence to C 1 ¯ + ˚ There is > 0 such that u − ϕprinc ∈ C (D) , therefore u(x) ≥ ϕprinc > 0 ∂ϕprinc for all x ∈ D, which gives further that ∂u ν (x) ≤ ∂ν ν (x) < 0 for all x ∈ ∂D. ∂ν

Throughout Chapter 6 of the monograph, K ≥ 1 is a fixed integer. We write a = (akij , aki , bki , ckl , dk0 ) and σt a ≡ a · t := (akij · t, aki · t, bki · t, ckl · t, dk0 · t), where i, j = 1, 2, . . . , N , k, l = 1, 2, . . . , K, akij · t(τ, x) := akij (t + τ, x) for t, τ ∈ R, x ∈ D, etc. 2 We assume Y ⊂ L∞ (R × D, RK(N +2N +K) ) × L∞ (R × ∂D, RK ) satisfies 2

(A1-4) Y is a (norm-)bounded subset of L∞ (R × D, RK(N +2N +K) ) × L∞ (R × ∂D, RK ) that is closed (hence, compact) in the weak-* topology of that space. (A1-5) Y is translation invariant: If a ∈ Y then a · t ∈ Y, for each t ∈ R. (A1-6) dk0 = 0 for all a = (akij , aki , (bi )k , ckl , dk0 ) ∈ Y (in the Dirichlet or Neumann cases) or dk0 ≥ 0 for all a = (akij , aki , (bi )k , ckl , dk0 ) ∈ Y (in the Robin case). We assume that Y is endowed with the weak-* topology. Thus (Y, σ) is a compact flow, where σ = {σt }t∈R and σt a = a · t. For a given a = (akij , aki , bki , ckl , dk0 ) ∈ Y, let ak := (akij , aki , bki , 0, dk0 ) := k N k N k k k ((akij )N i,j=1 , (aj )i=1 , (bi )i=1 , 0, d0 ) and Ca := (cl )l,k=1,2,...,K . Let P (a) be k k k k defined by P (a) := a and Y := { P (a) : a ∈ Y } (k = 1, 2, . . . , K). Hence 2 Y k ⊂ L∞ (R × D, RN +2N +1 ) × L∞ (R × ∂D, R) satisfies (A1-1)–(A1-3) for k = 1, 2, . . . , K. For 1 ≤ p ≤ ∞ we denote Lp (D) := (Lp (D))K . If 1 ≤ p < ∞ we define the norm in Lp (D) by kukp :=

K Z X k=1

D

1/p |uk (x)|p dx .

1. Introduction

21

We define the norm in L∞ (D) by kuk∞ := max ess sup { |uk (x)| : x ∈ D }. 1≤k≤K

(In both cases, u = (u1 , . . . , uK ).) We write simply k·k for the standard norm in the Hilbert space L2 (D). For u, v ∈ RK , u = (u1 , . . . , uK ), v = (v1 , . . . , vK ), we write u≤v

if uk ≤ vk

u
for 1 ≤ k ≤ K,

if u ≤ v and u 6= v.

The reversed symbols ≥ and > are used in the usual way. Let X ,→ L1 (D). In X we define the nonnegative cone X+ as X+ := { u ∈ X : u(x) ≥ 0 for a.e.

x ∈ D }.

¯ RK ) then If X ,→ C(D, X+ = { u ∈ X : u(x) ≥ 0 for all

¯ }. x∈D

The nonnegative cone X+ is a closed and convex subset of X, having the following properties: • If u ∈ X+ and r ≥ 0 then ru ∈ X+ . • If u ∈ X+ and −u ∈ X+ then u = 0. For any u, v ∈ X we write u≤v

if

v − u ∈ X+ ,

u
if

v − u ∈ X+ \ {0}.

The reversed symbols ≥ and > are used in the usual way. Sometimes it happens that the interior of the nonnegative cone X+ is nonempty. We denote then that interior by X++ . Also, for any u, v ∈ X we write u v if v − u ∈ X++ . The reversed symbol is used in the usual way. Notice that, if X = (X)K then X+ = (X + )K . ˚1 (D, ¯ RK ) the Banach If the boundary ∂D of D is of class C 1 , we denote by C 1 ¯ K space consisting of u ∈ C (D, R ) such that u(x) = 0 for all x ∈ ∂D. We have the following corollary of Lemma 1.3.1. LEMMA 1.3.2 Assume additionally that the boundary ∂D of D is of class C 2+α , for some 0 < α < 1.

22

Spectral Theory for Parabolic Equations

¯ RK )++ of C 1 (D, ¯ RK )+ is nonempty, and is charac(1) The interior C 1 (D, terized by ¯ RK )++ = { u = (u1 , . . . , uK ) ∈ C 1 (D, ¯ RK )+ : uk (x) > 0 C 1 (D, ¯ and all 1 ≤ k ≤ K }. for all x ∈ D (1.3.4) ˚1 (D, ¯ RK )++ of C ˚1 (D, ¯ RK )+ is nonempty, and is charac(2) The interior C terized by ˚1 (D, ¯ RK )++ = { u = (u1 , . . . , uK ) ∈ C ˚1 (D, ¯ RK )+ : uk (x) > 0 C for all x ∈ D and all 1 ≤ k ≤ K, and (∂uk /∂νν )(x) < 0 for all x ∈ ∂D, 1 ≤ k ≤ K }. (1.3.5) Throughout Chapter 7 of the monograph, D is assumed to be C 3+α . ¯× We denote by Z the set of admissible functions for (1.1.7), Z = { g : R× D [0, ∞) → R : g satisfies certain conditions }, and by Z the set of admissible ¯ × [0, ∞) × [0, ∞) → R × R : g functions for (1.1.5), Z = { g = (g1 , g2 ) : R × D satisfies certain conditions }. X denotes some fractional power space of the Laplacian operator ∆ in Lp (D) with the corresponding boundary condition such that ¯ X ,−,→ C 1 (D). X := X × X.

Chapter 2 Fundamental Properties in the General Setting

Introduction In the present chapter we establish some fundamental properties for a general family of parabolic equations. 2 Let Y be a subset of L∞ (R × D, RN +2N +1 ) × L∞ (R × ∂D, R) satisfying (A1-1)-(A1-3) (see in Section 1.3). We may write a = (aij , ai , bi , c0 , d0 ) for N N a = ((aij )N i,j=1 , (ai )i=1 , (bi )i=1 , c0 , d0 ) ∈ Y if no confusion occurs. For a given a = (aij , ai , bi , c0 , d0 ), we may assume that aij (t, x), ai (t, x), bi (t, x), and c0 (t, x) are defined and bounded for all (t, x) ∈ R × D, and d0 (t, x) is defined and bounded for all (t, x) ∈ R × ∂D. For each a = (aij , ai , bi , c0 , d0 ) ∈ Y we consider N N ∂u X ∂ X ∂u = aij (t, x) + ai (t, x)u ∂t ∂xi j=1 ∂xj i=1 +

N X i=1

bi (t, x)

∂u + c0 (t, x)u, ∂xi

t > s, x ∈ D,

(2.0.1)

complemented with the boundary conditions Ba (t)u = 0,

t > s, x ∈ ∂D,

where s ∈ R is an initial time and Ba is a boundary Dirichlet, or Neumann, or Robin type, that is,   u      X N X N     aij (t, x)∂xj u + ai (t, x)u νi  Ba (t)u = i=1 j=1  N X N  X    aij (t, x)∂xj u + ai (t, x)u νi      i=1 j=1   + d0 (t, x)u

(2.0.2) operator of either the (Dirichlet) (Neumann) (2.0.3)

(Robin).

23

24

Spectral Theory for Parabolic Equations

To emphasize the coefficients and the boundary terms in the problem (2.0.1)+ (2.0.2), we will write (2.0.1)a +(2.0.2)a . We use standard notion of solutions of (2.0.1)+(2.0.2), i.e., weak solutions (see [10], [30], [70], [73]). First, in Section 2.1 we list basic assumptions, i.e., (A2-1) (the uniform ellipticity), (A2-2) (a very weak condition on the regularity of ∂D), and (A2-3) (a condition on perturbation of coefficients), and introduce the definition of weak solutions. Then, in Section 2.2 we collect basic properties of weak solutions of (2.0.1)+(2.0.2), including regularity, Harnack inequalities, monotonicity, joint continuity, compactness, etc., which are needed in the following chapters. It is shown that (2.0.1)+(2.0.2) generates a skew-product semiflow on the product bundle L2 (D) × Y . The adjoint problem of (2.0.1)+(2.0.2) is considered in Section 2.3. In Section 2.4 we discuss the satisfaction of the assumption (A2-3), and show that (A2-3) is fulfilled under some very general condition. We study in Section 2.5 the case that the coefficients and the domain of (2.0.1)+(2.0.2) are sufficiently smooth. This chapter is ended up with some remarks in Section 2.6 about the solutions of nonautonomous equations in nondivergence form.

2.1

Assumptions and Weak Solutions

In this section we list our basic assumptions (A2-1)–(A2-3) and introduce the concept of weak solutions. Consider (2.0.1)a +(2.0.2)a , a ∈ Y . First of all, we assume that the principal parts of all the elements of Y are uniformly elliptic: (A2-1) (Uniform ellipticity) There exists α0 > 0 such that for all a ∈ Y there holds N X

aij (t, x) ξi ξj ≥ α0

i,j=1

N X

ξi2

for a.e. (t, x) ∈ R × D and all ξ ∈ RN ,

i=1

(2.1.1) aij (t, x) = aji (t, x)

for a.e. (t, x) ∈ R × D,

i, j = 1, 2, . . . , N.

Let V be defined as follows  1 ˚ (D) W    2 V := W21 (D)    1 W2,2 (D, ∂D)

(Dirichlet) (Neumann) (Robin)

(2.1.2)

2. Fundamental Properties in the General Setting

25

˚ 1 (D) is the closure of D(D) in W 1 (D) and W 1 (D, ∂D) is the comwhere W 2 2 2,2 pletion of ¯ : v is C ∞ on D and kvkV < ∞ } V0 := { v ∈ W21 (D) ∩ C(D) with respect to the norm kvkV := (k∇vk22 + kvk22,∂D )1/2 . If no confusion occurs, we will write hu, u∗ i for the duality between V and ∗ V , where u ∈ V and u∗ ∈ V ∗ . LEMMA 2.1.1 If D is Lipschitz then 1 (1) W2,2 (D, ∂D) = W21 (D) up to norm equivalence;

˚ 1 (D) = { u ∈ W 1 (D) : u|∂D = 0 }. (2) W 2 2 PROOF (1) See [30, (2.6)]. (2) See [45, Theorem 1.5]. For s ≤ t let W = W (s, t; V, V ∗ ) := { v ∈ L2 ((s, t), V ) : v˙ ∈ L2 ((s, t), V ∗ ) }

(2.1.3)

equipped with the norm kvkW :=

Z s

t

kv(τ )k2V dτ +

Z

t

kv(τ ˙ )k2V ∗ dτ

12

,

s

where v˙ := dv/dτ is the time derivative in the sense of distributions taking values in V ∗ (see [36, Chapter XVIII] for definitions). LEMMA 2.1.2 Let s < t. Then W (s, t; V, V ∗ ) embeds continuously into C([s, t], L2 (D)). PROOF

See [36, Theorem 1, Chapter XVIII].

LEMMA 2.1.3 Let s < t and u ∈ W (s, t; V, V ∗ ). Then u ∈ L2 ((s, t) × D). PROOF

For given s < t, u ∈ W (s, t; V, V ∗ ), and n ∈ N, let

t0 = s < t1 = s +

t−s 2(t − s) < t2 = s + < · · · < tn = t, n n

26

Spectral Theory for Parabolic Equations ( u(s, x), τ =s un (τ, x) := u(ti , x), ti−1 < τ ≤ ti , i = 1, 2, . . . , n.

Then by Lemma 2.1.2, kun (τ, ·) − u(τ, ·)k → 0

as n → ∞

for all τ ∈ (s, t). Clearly, un (τ, x) is measurable in (τ, x) ∈ (s, t) × D. We claim that un (τ, x) → u(τ, x)

for a.e.

(τ, x) ∈ (s, t).

Assume this is false. Then there is an 0 > 0 such that |E0 | > 0, where E0 := { (τ, x) : |un (τ, x) − u(τ, x)| ≥ 0

for infinitely many n ∈ N }.

Let E0 (τ ) := { x ∈ D : (τ, x) ∈ E0 }. Then [ (s, t) 3 τ 7→ |E0 (τ )| ∈ R ] is measurable, and Z t |E0 | = |E0 (τ )| dτ s

(see [42]). Therefore there is τ0 ∈ (s, t) such that |E0 (τ0 )| > 0. This implies that kun (τ0 , ·) − u(τ0 , ·)k 6→ 0 as n → ∞. This is a contradiction. It then follows from the measurability of un (τ, x) that u(τ, x) is measurable in (τ, x) ∈ (s, t) × D. Therefore Z Z tZ 2 |u(τ, x)| dτ dx = |u(τ, x)|2 dx dτ < ∞ (s,t)×D

s

D

and u ∈ L2 ((s, t) × D). For a ∈ Y denote by Ba = Ba (t, ·, ·) the bilinear form on V associated with a, Z Ba (t, u, v) := aij (t, x)∂xj u + ai (t, x)u)∂xi v dx ZD − (bi (t, x)∂xi u + c0 (t, x)u)v dx, u, v ∈ V,

(2.1.4)

D

in the Dirichlet and Neumann boundary condition cases, and Z Ba (t, u, v) := (aij (t, x)∂xj u + ai (t, x)u)∂xi v dx D Z − (bi (t, x)∂xi u + c0 (t, x)u)v dx ZD + d0 (t, x)uv dHN −1 , u, v ∈ V, ∂D

(2.1.5)

2. Fundamental Properties in the General Setting

27

in the Robin boundary condition case, where HN −1 stands for the (N − 1)-dimensional Hausdorff measure (we used the summation convention in the above). LEMMA 2.1.4 Assume (A2-1). Then the following holds. (i) For any a ∈ Y and for any u, v ∈ V the function [ R 3 t 7→ Ba (t, u, v) ∈ R ] is (Lebesgue-)measurable. (ii) There exists M0 > 0 such that |Ba (t, u, v)| ≤ M0 kukV kvkV for any a ∈ Y , a.e. t ∈ R and any u, v ∈ V . PROOF

See [36, Section XVIII.4.4].

DEFINITION 2.1.1 (Weak solution) A function u ∈ L2 ((s, t), V ) is a weak solution of (2.0.1)+(2.0.2) on [s, t] × D (s < t) with initial condition u(s) = u0 if Z −

t

˙ ) dτ + hu(τ ), vi ψ(τ

s

Z

t

Ba (τ, u(τ ), v)ψ(τ ) dτ − hu0 , vi ψ(s) = 0 (2.1.6) s

for all v ∈ V and ψ ∈ D([s, t)), where D([s, t)) is the space of all smooth real functions having compact support in [s, t). PROPOSITION 2.1.1 If u is a weak solution of (2.0.1)+(2.0.2) with initial condition u(s) = u0 , then u ∈ W (s, t; V, V ∗ ). PROOF

See [30, Theorem 2.4].

Lemma 2.1.2 and Proposition 2.1.1 allow us to state the following. PROPOSITION 2.1.2 Assume (A2-1). If u is a weak solution of (2.0.1)+(2.0.2) on [s, t] × D with initial condition u(s) = u0 ∈ L2 (D) then u(s) = u0 . PROOF

See [36, Section XVIII.1.2].

PROPOSITION 2.1.3 (Equivalence) Assume (A2-1). Let u ∈ L2 ((s, t), V ), s < t, s ∈ R. u is a weak solution of (2.0.1)+(2.0.2) on [s, t] × D with u(s) = u0 if and only if u ∈ W (s, t; V, V ∗ )

28

Spectral Theory for Parabolic Equations

and for any v ∈ W (s, t; V, V ∗ ), Z −

t

Z

t

hu(τ ), v(τ ˙ )i dτ + s

Ba (τ, u(τ ), v(τ )) dτ s

+ hu(t), v(t)i − hu0 , v(s)i = 0

(2.1.7)

PROOF First note that u ∈ L2 ((s, t), V ) is a weak solution of (2.0.1)+ (2.0.2) on [s, t] × D with u(s) = u0 if and only if u ∈ W (s, t; V, V ∗ ) and for all v ∈ W (s, t; V, V ∗ ) satisfying v(t) = 0, Z

t

−

t

Z hu(τ ), v(τ ˙ )i dτ +

s

Ba (τ, u(τ ), v(τ )) dτ − hu0 , v(s)i = 0

(2.1.8)

s

(see [30, Remark 2.3]). Hence we only need to prove that if u ∈ W (s, t; V, V ∗ ) is a weak solution of (2.0.1)+(2.0.2) with u(s) = u0 , then for any v ∈ W (s, t; V, V ∗ ), (2.1.7) holds. By Lemma 2.1.2 and Proposition 2.1.1, a weak solution u of (2.0.1) +(2.0.2) on [s, t] × D belongs to C([s, t], L2 (D)). Next, assume that u ∈ W (s, t; V, V ∗ ) is a weak solution of (2.0.1)+(2.0.2) with u(s) = u0 . Let v ∈ W (s, t; V, V ∗ ) be such that v(t) ∈ V . Let v˜(τ ) := v(τ ) − v(t). Then v˜ ∈ W (s, t; V, V ∗ ) and v˜(t) = 0. Hence, Z −

t

hu(τ ), v˜˙ (τ )i dτ +

Z

Ba (τ, u(τ ), v˜(τ )) dτ − hu0 , v˜(s)i = 0.

s

s

Note that Z

t

hu(τ ), v˜˙ (τ )i dτ =

s

Z

t

t

hu(τ ), v(τ ˙ )i dτ, s

t

Z

t

Z

t

Ba (τ, u(τ ), v(τ )) dτ −

Ba (τ, u(τ ), v˜(τ )) dτ = s

Z

s

Ba (τ, u(τ ), v(t)) dτ s

and hu0 , v˜(s)i = hu0 , v(s)i − hu0 , v(t)i. Note also that

d hu(τ ), v(t)i + Ba (τ, u(τ ), v(t)) = 0 dτ

in the sense of distributions in D0 ((s, t)). Since [ [s, t] 3 τ 7→ u(τ ) ∈ L2 (D) ] is continuous (by Lemma 2.1.2), the function [ [s, t] 3 τ 7→ hu(τ ), v(t)i ] is continuous, and the function [ [s, t] 3 τ 7→ Ba (τ, u(τ ), v(t)) ∈ R ] belongs to L1 ((s, t)) (see Lemma 2.1.4). Consequently, we have Z

t

Ba (τ, u(τ ), v(t)) dτ = hu(s), v(t)i − hu(t), v(t)i. s

Therefore (2.1.7) holds for any v ∈ W (s, t; V, V ∗ ) with v(t) ∈ V .

2. Fundamental Properties in the General Setting

29

Now for any v ∈ W (s, t; V, V ∗ ), v(τ ) ∈ V for a.e. τ ∈ [s, t]. Hence there is a sequence (τn )∞ n=1 ⊂ [s, t] such that τn → t and v(τn ) ∈ V . By the above arguments, Z −

τn

Z hu(τ ), v(τ ˙ )i dτ +

s

τn

Ba (τ, u(τ ), v(τ )) dτ s

+ hu(τn ), v(τn )i − hu0 , v(s)i = 0

(2.1.9)

for n = 1, 2, . . . . Observe that, since the functions [ [s, t] 3 τ 7→ hu(τ ), v(τ ˙ )i ∈ R ] and [ [s, t] 3 τ 7→ Ba (τ, u(τ ), v(τ )) ∈ R ] are in L1 ((s, t)), there holds Rt Rt hu(τ ), v(τ ˙ )i dτ → 0, τn Ba (τ, u(τ ), v(τ )) dτ → 0, and hu(τn ), v(τn )i → τn hu(t), v(t)i as n → ∞. It then follows from (2.1.9) that (2.1.7) holds for any v ∈ W (s, t; V, V ∗ ). PROPOSITION 2.1.4 Assume (A2-1). Let u be a weak solution of (2.0.1)+(2.0.2) on [s, t] × D. Then Z t 2 2 ku(t)k − ku(s)k = −2 Ba (τ, u(τ ), u(τ )) dτ. (2.1.10) s

PROOF

Observe that ku(t)k2 − ku(s)k2 = 2

Z

t

hu(τ ), u(τ ˙ )i dτ. s

(see, e.g., [30, Lemma 3.3]), and apply Proposition 2.1.3 to v = u. DEFINITION 2.1.2 (Global weak solution) A function u ∈ L2,loc ((s, ∞), V ) is a global weak solution of (2.0.1)+(2.0.2) with initial condition u(s) = u0 , s ∈ R, if for each t > s its restriction u|[s,t] is a weak solution of (2.0.1)+(2.0.2) on [s, t]×D with initial condition u(s) = u0 . Global solutions of (2.0.1)+(2.0.2) exist under very general conditions. To be more specific, we make the following assumption. (A2-2) (Boundary regularity) For Dirichlet boundary conditions, D is a bounded domain. For Neumann or Robin boundary conditions, D is a bounded domain with Lipschitz boundary. Notice that under the assumption (A2-2), in the Robin boundary condition case the Hausdorff (N − 1)-dimensional measure on HN −1 on ∂D reduces to the ordinary surface measure.

30

Spectral Theory for Parabolic Equations

PROPOSITION 2.1.5 (Existence of global solution) Assume (A2-1) and (A2-2). Then for any s ∈ R and any u0 ∈ L2 (D) there exists a unique global weak solution of (2.0.1)+(2.0.2) with initial condition u(s) = u0 . PROOF

See [30, Theorem 2.4].

We write the unique global weak solution [ t 7→ u(t) ] of (2.0.1)+(2.0.2) with initial condition u(s) = u0 as Ua (t, s)u0 := u(t), t > s. We write Ua (s, s)u0 = u0 , for any a ∈ Y , s ∈ R and u0 ∈ L2 (D). Important properties are given by the following results. PROPOSITION 2.1.6 Assume (A2-1) and (A2-2). Then for any a ∈ Y and s ≤ t one has Ua (t, s) = Ua·s (t − s, 0).

(2.1.11)

PROOF For s = t there is nothing to prove. So assume s < t. Fix u0 ∈ L2 (D). Put u1 (τ ) := Ua (τ, s)u0 for τ ∈ [s, t], and u2 (˜ τ ) := Ua·s (˜ τ − s, 0)u0 for τ˜ ∈ [s, t]. For any v ∈ V and ψ ∈ D([s, t)) the function u1 satisfies the equation Z t Z t ˙ ) dτ + − hu1 (τ ), viψ(τ Ba (τ, u1 (τ ), v)ψ(τ ) dτ − hu0 , viψ(s) = 0. s

s

After the change of variables τ˜ = τ − s we obtain Z t−s Z t−s ˙ τ + s) d˜ − hu1 (˜ τ + s), viψ(˜ τ+ Ba·s (˜ τ , u1 (˜ τ + s), v)ψ(˜ τ + s) d˜ τ 0

0

− hu0 , viψ(s) = 0. By the uniqueness of weak solutions, u2 (τ ) = u1 (τ ) for τ ∈ [s, t]. PROPOSITION 2.1.7 Assume (A2-1) and (A2-2). Then for any a ∈ Y and s ≤ t1 ≤ t2 , one has Ua (t2 , t1 ) ◦ Ua (t1 , s) = Ua (t2 , s).

(2.1.12)

PROOF Assume s < t1 < t2 . Fix u0 ∈ L2 (D), and put u1 (t) := Ua (t, s)u0 for t ∈ [s, t1 ], u2 (t) := Ua (t, t1 )u1 (t1 ) for t ∈ [t1 , t2 ]. Let u(t) be defined by ( u1 (t) for t ∈ [s, t1 ] u(t) := u2 (t) for t ∈ [t1 , t2 ].

2. Fundamental Properties in the General Setting

31

It is clear that u ∈ L2 ((s, t2 ), V ) and u(t) satisfies (2.1.7) on [s, t2 ] for any v ∈ W (s, t2 ; V, V ∗ ). Note that for any ψ ∈ D([s, t2 )) and v ∈ V , ψv ∈ W (s, t2 ; V, V ∗ ) and dψ d dt (ψv) = dt v. By the fact that u(t) satisfies (2.1.7) on [s, t2 ], there holds Z −

t2

˙ ) dτ + hu(τ ), viψ(τ

Z

t2

Ba (τ, u(τ ), v)ψ(τ ) dτ − hu0 , viψ(s) = 0. s

s

It then follows from the definition of weak solution (Definition 2.1.1) that u(t) = Ua (t, s)u0 for t ∈ [s, t2 ] and then Ua (t2 , t1 ) ◦ Ua (t1 , s)u0 = Ua (t2 , s)u0 .

As a consequence of Propositions 2.1.6 and 2.1.7 we obtain the following cocycle equality. PROPOSITION 2.1.8 Assume (A2-1) and (A2-2). Then for any a ∈ Y and s, t ∈ [0, ∞) we have Ua·s (t, 0)Ua (s, 0) = Ua (s + t, 0).

(2.1.13)

As the set Y is assumed to have the property that a ∈ Y and t ∈ R implies a · t ∈ Y , the above results allow us to consider Ua (t, 0) for t > 0 instead of Ua (t, s) for t > s. A third assumption imposed on Y regards the continuous dependence of solutions on parameters: (A2-3) (Perturbation of coefficients) For any sequence (a(n) )∞ n=1 ⊂ Y and any real sequence (tn )∞ n=1 with tn > 0, if lim a(n) = a and lim tn = t > 0,

n→∞

n→∞

then for any u0 ∈ L2 (D), Ua(n) (tn , 0)u0 converges to Ua (t, 0)u0 in L2 (D). We remark that (A2-3) is satisfied under some very general condition (see Section 2.4).

2.2

Basic Properties of Weak Solutions

In this section, we present some basic properties about weak solutions. We will assume (A2-1) and (A2-2) throughout this section and assume (A2-3) at some places. First of all, the linear operator Ua (t, 0) can be extended to Lp (D). Indeed, we have:

32

Spectral Theory for Parabolic Equations

PROPOSITION 2.2.1 (Extension in Lp ) Under the assumptions (A2-1) and (A2-2), for any 1 ≤ p ≤ ∞, any a ∈ Y and any t > 0 there exists an operator Ua,p (t, 0) ∈ L(Lp (D)) such that Ua,p (t, 0)u0 = Ua (t, 0)u0 for all u0 ∈ L2 (D)∩Lp (D). Moreover, for 1 < p < ∞ and a ∈ Y the mapping [ [0, ∞) 3 t 7→ Ua,p (t, 0) ∈ Ls (Lp (D)) ] is continuous. PROOF

See [30, Theorem 5.1 and Corollary 5.3].

Ua,p (t, s), for s < t, is understood as Ua·s,p (t − s, 0). Further, Ua,p (s, s)u0 = u0 . In the following we may write Ua (t, s) instead of Ua,p (t, s). We may also write U (t, s) instead of Ua (t, s), if no confusion occurs. The next proposition gives us Lp –Lq estimates of Ua (t, 0). PROPOSITION 2.2.2 (Lp –Lq estimates) Assume (A2-1) and (A2-2). Then there are constants M > 0 and γ > 0 such that N

1

1

kUa (t, 0)kLp (D),Lq (D) ≤ M t− 2 ( p − q ) eγt

(2.2.1)

for a ∈ Y , 1 ≤ p ≤ q ≤ ∞, t > 0. PROOF

See [30, Corollary 7.2].

PROPOSITION 2.2.3 (Strong continuity in t) Assume (A2-1) and (A2-2). For any 1 ≤ p < ∞ and any a ∈ Y , the mapping [ (0, ∞) 3 t 7→ Ua,p (t, 0) ∈ Ls (Lp (D))] is continuous. PROOF The continuity of [ (0, ∞) 3 t 7→ Ua,p (t, 0) ∈ Ls (Lp (D))] for 1 0, let δ > 0 be such that t − δ > 0. Then by Proposition 2.2.2, for any u0 ∈ L1 (D), Ua,1 (δ, 0)u0 ∈ L2 (D). Hence Ua,1 (tn , 0)u0 = Ua·δ,1 (tn − δ, 0)Ua,1 (δ, 0)u0 = Ua·δ,2 (tn − δ, 0)Ua,1 (δ, 0)u0 → Ua·δ,2 (t − δ, 0)Ua,1 (δ, 0)u0 = Ua,1 (t, 0)u0 . PROPOSITION 2.2.4 (Local regularity) Assume (A2-1) and (A2-2). Let 1 ≤ p ≤ ∞. Then for any 0 < t1 < t2 there exists α ∈ (0, 1) such that for any a ∈ Y , any u0 ∈ Lp (D), and any compact subset D0 ⊂ D the function [ [t1 , t2 ]×D0 3 (t, x) 7→ (Ua (t, 0)u0 )(x) ] belongs to C α/2,α ([t1 , t2 ] × D0 ). Moreover, for fixed t1 , t2 , and D0 , the C α/2,α ([t1 , t2 ] × D0 )-norm of the above restriction is bounded above by a constant depending on ku0 kp only.

2. Fundamental Properties in the General Setting

33

PROOF It follows from Proposition 2.2.2 and from [70, Chapter III, Theorem 10.1]. PROPOSITION 2.2.5 (Compactness) Let (A2-1) through (A2-2) be satisfied and 1 ≤ p ≤ ∞, 1 ≤ q < ∞. Then for any given 0 < t1 ≤ t2 , if E is a bounded subset of Lp (D) then { Ua (τ, 0)u0 : a ∈ Y, τ ∈ [t1 , t2 ], u0 ∈ E } is relatively compact in Lq (D). (n) ∞ PROOF Let (τn )∞ )n=1 ⊂ Y , and (un )∞ n=1 ⊂ [t1 , t2 ], (a n=1 ⊂ E. From ∞ Proposition 2.2.4 it follows that there are subsequences (a(nk ) )∞ k=1 , (τnk )k=1 , ∞ and (unk )k=1 such that Ua(nk ) (τnk , 0)unk converges to some u∞ ∈ L∞ (D) uniformly on compact subsets of D. This together with Proposition 2.2.2 implies that Ua(nk ) (τnk , 0)unk converges to u∞ in Lq (D) for any 1 ≤ q < ∞.

PROPOSITION 2.2.6 (Joint continuity in t and u0 ) Assume (A2-1) and (A2-2). Let 1 ≤ p ≤ q < ∞ and a ∈ Y . For any ∞ real sequence (tn )∞ n=1 and any sequence (un )n=1 ⊂ Lp (D), if limn→∞ tn = t, where t > 0, and limn→∞ un = u0 in Lp (D), then Ua (tn , 0)un converges in Lq (D) to Ua (t, 0)u0 . PROOF

First observe that

kUa (tn , 0)un − Ua (t, 0)u0 kq ≤ kUa (tn , 0)un − Ua (tn , 0)u0 kq + kUa (tn , 0)u0 − Ua (t, 0)u0 kq . Next, by Proposition 2.2.2, 1 1 −N 2 ( p − q ) γtn

kUa (tn , 0)un − Ua (tn , 0)u0 kq ≤ M tn

e

kun − u0 kp .

Hence kUa (tn , 0)un −Ua (tn , 0)u0 kq → 0 as n → ∞. Now, by Proposition 2.2.3, kUa (tn , 0)u0 − Ua (t, 0)u0 kp → 0 as n → ∞. We deduce from Proposition 2.2.5 that also kUa (tn , 0)u0 − Ua (t, 0)u0 kq → 0. It then follows that kUa (tn , 0)un − Ua (t, 0)u0 kq → 0

as

n → ∞.

PROPOSITION 2.2.7 (Positivity) Assume (A2-1) and (A2-2). Let 1 ≤ p ≤ ∞. For any u0 ∈ Lp (D)+ there holds Ua (t, 0)u0 ∈ Lp (D)+ for all a ∈ Y and t ≥ 0. PROOF For the case 1 < p < ∞, the proposition follows from [30, Proposition 8.1]. The case p = ∞ then follows from the fact that L∞ (D)+ ⊂

34

Spectral Theory for Parabolic Equations

Lp (D)+ for any 1 0 and n = 1, 2, . . . . For any given t > 0, Ua (t, 0) ∈ L(L1 (D)), consequently kUa (t, 0)un − Ua (t, 0)u0 kL1 (D) → 0 as n → ∞ and there is subsequence (unk )∞ k=1 such that (Ua (t, 0)unk )(x) → (Ua (t, 0)u0 )(x) as k → ∞ for a.e. x ∈ D. We then also have Ua (t, 0)u0 ∈ L1 (D)+ for t > 0. In the following, B(x0 ; r) is defined by B(x0 ; r) := { x ∈ RN : kx − x0 k ≤ r }. PROPOSITION 2.2.8 (Interior Harnack inequality) Assume (A2-1) and (A2-2). Let 1 ≤ p ≤ ∞. (1) Given r > 0, there is Cr > 0 such that for any x0 ∈ D and t0 > 0 satisfying B(x0 ; 2r) ⊂ D and t0 − 2r2 > 0, and any a ∈ Y and u0 ∈ Lp (D) such that (Ua (t, 0)u0 )(x) is nonnegative in [t0 − 2r2 , t0 + 2r2 ] × B(x0 ; 2r), the following holds: sup{ (Ua (t, 0)u0 )(x) : t ∈ [t0 −(29/16)r2 , t0 −(7/4)r2 ], x ∈ B(x0 ; r/4) } ≤ Cr ·inf{ (Ua (t, 0)u0 )(x) : t ∈ [t0 +(31/16)r2 , t0 +2r2 ], x ∈ B(x0 ; r/4) }. (2) For any t0 > 0 there is δ0 = δ0 (t0 ), 0 < δ0 < 1, with the property that for any 0 < δ < δ0 there is Cδ > 0 such that (Ua (t, 0)u0 )(y) ≤ Cδ · (Ua (t + τ, 0)u0 )(x) for any a ∈ Y , t ≥ δ 2 , δ 2 ≤ τ ≤ t0 , u0 ∈ Lp (D)+ , and any x, y ∈ Dδ := { ξ ∈ D : d(ξ) > δ}, where d(ξ) denotes the distance of ξ ∈ D from the boundary ∂D of D. PROOF (1) First, (1) with p = 2 follows from [63, Theorem 1] (see also [68], [73]). Next for any p ≥ 2, Lp (D) ⊂ L2 (D), hence (1) also holds. Now, for 1 ≤ p < 2, Ua (t, 0)u0 = Ua·δ (t − δ, 0)Ua (δ, 0)u0 for any δ > 0. Note that u1 = Ua (δ, 0)u0 ∈ L2 (D). Let δ > 0 be so small that t0 − δ − 2r2 > 0. Then (Ua·δ (t, 0)u1 )(x) is nonnegative in [tδ − 2r2 , tδ + 2r2 ] × B(x0 ; 2r), where tδ = t0 − δ. This implies that sup { (Ua·δ (t, 0)u1 )(x) : t ∈ [tδ − (29/16)r2 , tδ − 47 r2 ], x ∈ B(x0 ; r/4) } ≤ Cr · inf { (Ua·δ (t, 0)u1 )(x) : [tδ + (31/16)r2 , tδ + 2r2 ], x ∈ B(x0 ; r/4) }, which is equivalent to the desired result. (2) By Proposition 2.2.7, Ua (t, 0)u0 ≥ 0 for all t > 0. (2) then follows from (1).

2. Fundamental Properties in the General Setting

35

PROPOSITION 2.2.9 (Monotonicity on initial data) Assume (A2-1) and (A2-2) and 1 ≤ p ≤ ∞. Let a ∈ Y , t > 0 and u1 , u2 ∈ Lp (D). (1) If u1 ≤ u2 then Ua (t, 0)u1 ≤ Ua (t, 0)u2 . (2) If u1 ≤ u2 , u1 6= u2 , then (Ua (t, 0)u1 )(x) < (Ua (t, 0)u2 )(x) for x ∈ D. PROOF (1) By Proposition 2.2.7, Ua (t, 0)(u2 − u1 ) ≥ 0 for all t > 0. It then follows that Ua (t, 0)u1 ≤ Ua (t, 0)u2 for all t > 0. (2) Let u0 := u2 − u1 and u ˜0 be given by ( u0 (x) if 0 ≤ u0 (x) ≤ 1 u ˜0 (x) := 1 if u0 (x) > 1. Then u ˜0 ∈ L2 (D) and u0 ≥ u ˜0 > 0. By Part (1), Ua (t, 0)u0 ≥ Ua (t, 0)˜ u0 ≥ 0 for all t ≥ 0. We claim that there is δ 0 > 0 such that for any δ ∈ (0, δ 0 ) there is xδ ∈ Dδ with the property that (Ua (δ 2 , 0)˜ u0 )(xδ ) > 0. Suppose not. 2 Then there is a sequence δn & 0 such that U ((δ ) u0 ≡ 0 on Dδn . ConseR a n , 0)˜ quently, for each compact subset K ⊂ D, K (Ua ((δnR)2 , 0)˜ u0 )(x) dx converge 2 to 0 as n → ∞. It follows from Proposition 2.2.1 that (U u0 )(x) dx a ((δn ) , 0)˜ K R converge in L2 (D) to K u ˜0 (x) dx, which gives u ˜0 = 0. This is a contradiction. p Fix now t0 > 0 and x0 ∈ D. Take δ ∈ (0, min {δ0 (t0 ), d(x0 , ∂D), t0 /2, δ 0 }), where δ0 (t0 ) is as in Proposition 2.2.8(2). We have thus δ 2 < t0 − δ 2 < t0 . Take y ∈ Dδ such that (Ua (δ 2 , 0)˜ u0 )(y) > 0. The interior Harnack inequality (Proposition 2.2.8(2)) implies (Ua (t0 , 0)˜ u0 )(x0 ) > 0, hence (Ua (t0 , 0)u0 )(x0 ) > 0 and then (Ua (t0 , 0)u1 )(x0 ) < (Ua (t0 , 0)u2 )(x0 ). In view of the above proposition, we say that a (global) weak solution u of (2.0.1)+(2.0.2) is a positive weak solution (on [s, ∞) × D) if u(t)(x) > 0 for all t > s and all x ∈ D. PROPOSITION 2.2.10 (Monotonicity on coefficients) Let (A2-1) and (A2-2) be satisfied and 1 ≤ p ≤ ∞. (1) Assume the Dirichlet boundary condition. Let a(k) , k = 1, 2, be such (1) (2) (1) (2) (1) (2) (1) (2) that aij = aij , ai = ai , bi = bi , but c0 ≤ c0 , where equalities and inequalities are to be understood a.e. on R × D. Then Ua(1) (t, 0)u0 ≤ Ua(2) (t, 0)u0 for any t > 0 and any u0 ∈ Lp (D)+ . (2) Assume the Neumann or Robin boundary condition. Let a(k) , k = 1, 2, (1) (2) (1) (2) (1) (2) (1) (2) (1) be such that aij = aij , ai = ai , bi = bi , but c0 ≤ c0 , d0 ≥

36

Spectral Theory for Parabolic Equations (2)

d0 , where equalities and inequalities are to be understood a.e. on R × D or a.e. on R × ∂D. Then Ua(1) (t, 0)u0 ≤ Ua(2) (t, 0)u0 for any t > 0 and any u0 ∈ Lp (D)+ . (1)

(2)

(1)

(3) Let a(k) , k = 1, 2, be such that aij = aij , ai (1)

(2)

(1)

(2)

(1)

= ai , bi

(2)

= bi ,

(2)

c0 = c0 , but d0 ≥ 0, d0 = 0, where equalities and inequalities are to be understood a.e. on R × D or a.e. on R × ∂D. Then UaR(1) (t, 0)u0 ≤ UaN(2) (t, 0)u0 for any t > 0 and any u0 ∈ Lp (D)+ , where UaR (t, 0)u0 and UaN (t, 0)u0 denote the solutions of (2.0.1)a +(2.0.2)a with Robin and Neumann boundary conditions, respectively. (1)

(2)

(1)

(4) Let a(k) , k = 1, 2, be such that aij = aij , ai (1)

(2)

(2)

(1)

= ai , bi

(2)

= bi ,

(2)

c0 = c0 , but d0 ≥ 0, where equalities and inequalities are to be understood a.e. on R × D or a.e. on R × ∂D. Then UaD(1) (t, 0)u0 ≤ UaR(2) (t, 0)u0 for any t > 0 and any u0 ∈ Lp (D)+ , where UaD (t, 0)u0 and UaR (t, 0)u0 denote the solutions of (2.0.1)a +(2.0.2)a with Dirichlet and Robin boundary conditions, respectively. PROOF First of all, note that we only need to prove the proposition in the case that u0 ∈ L2 (D)+ . In fact, if u0 ∈ Lp (D)+ with p > 2, we have u0 ∈ L2 (D)+ . If u0 ∈ Lp (D)+ with 1 ≤ p < 2, there are un ∈ L2 (D)+ such that kun − u0 kp → 0 as n → ∞. For any given a ∈ Y and t > 0, Ua (t, 0) ∈ L(Lp (D)). Hence Ua (t, 0)un → Ua (t, 0)u0 in Lp (D) as n → ∞ and then there is a subsequence nk such that (Ua (t, 0)unk )(x) → (Ua (t, 0)u0 )(x) for a.e. x ∈ D. Therefore we only need to prove the case that u0 ∈ L2 (D)+ . In the rest of the proof, we assume u0 ∈ L2 (D)+ . (1) follows from the arguments of [30, Proposition 8.1]. For completeness, we provide a proof here. Let u0 ∈ L2 (D)+ and v(t) := Ua(2) (t, 0)u0 − Ua(1) (t, 0)u0 . It follows from Proposition 2.1.3 that v(t) satisfies Z t Z t Ba(2) (τ, v(τ ), v˜(τ )) dτ − hv(τ ), v˜˙ (τ )i dτ + 0 0 Z t (2) (1) − h(c0 (τ, ·) − c0 (τ, ·))(Ua(1) (τ, 0)u0 ), v˜(τ )i dτ + hv(t), v˜(t)i = 0 0

˚ 1 (D). for any v˜ ∈ W (0, t; V, V ∗ ). Here and in the following, V = W 2

2. Fundamental Properties in the General Setting

37

Put δ0 := (α0 )−1

N N X X 1/2 1/2 (2) (2) (2) kai k2∞ + kbi k2∞ + k(c0 )− k∞ , i=1

i=1

(2)

(2)

where (c0 )− is the negative part of c0 and k·k∞ denotes the L∞ (R × D)norm. Let w(t) := e−δ0 t v(t) and w− (t) be the negative part of w(t). Note that D([0, t]; V ) is dense in W (0, t; V, V ∗ ) (see [36, Lemma 1 in Section XVIII.1.2]). Choose (wm )∞ m=1 ⊂ D([0, t], V ) such that wm → w in the W (0, t; V, V ∗ )-norm. By Lemma 2.1.3, wm , w ∈ L2 ((0, t) × D) and hence kwm − wkL2 ((0,t)×D) → 0 as m → ∞. Without loss of generality, we may then assume that wm (τ, x) → w(τ, x) as m → ∞ for a.e. (τ, x) ∈ (0, t) × D. For a given > 0 let f : R → R be defined by ( (ξ 2 + 2 )1/2 − if ξ < 0 f (ξ) := (2.2.2) 0 if ξ ≥ 0. 0

Observe that f ∈ C 1 (R) and f (ξ) is bounded in ξ ∈ R. For τ ∈ [0, t] we define f (wm )(τ )(x) := f (wm (τ )(x)),

f (w)(τ )(x) := f (w(τ )(x)),

f0 (wm )(τ )(x)

f0 (w)(τ )(x) := f0 (w(τ )(x))

:=

f0 (wm (τ )(x)),

for a.e. x ∈ D. Then the function [ [0, t] 3 τ 7→ f (wm )(τ ) ∈ V ] is in d f (wm )(τ ) ∈ L2 (D) ] is in L2 ((0, t), V ) and the function [ [0, t] 3 τ 7→ dτ L2 ((0, t), L2 (D)), and, moreover, there holds Z tD 0

E d w(τ ), f (wm )(τ ) dτ = dτ

Z tZ 0

w(τ )(x)f0 (wm )(τ )(x)

D

dwm (τ )(x) dx dτ. dt

Then we have f (wm ) ∈ W (0, t; V, V ∗ ), and Z tZ

w(τ )(x)f0 (wm )(τ )(x)

− 0

+

D

Z t

dwm (τ )(x) dx dτ dt

Ba(2) (τ, w(τ ), f (wm )(τ )) + δ0 hw(τ ), f (wm )(τ )i dτ

0

Z

t

(2)

(1)

h(c0 (τ, ·) − c0 (τ, ·))(Ua(1) (τ, 0)u0 ), f (wm )(τ ))i dτ

− 0

+ hw(t), f (wm )(t)i − hw(0), f (wm )(0)i = 0. Let g : R → R be given by ( g (ξ) :=

ξf0 (ξ)

=

ξ2 (ξ 2 +2 )1/2

if ξ < 0

0

if ξ ≥ 0.

38

Spectral Theory for Parabolic Equations

It is not difficult to see that g and g0 are continuous and |g (ξ)| ≤ |ξ| and |g0 (ξ)| is bounded in ξ ∈ R. This implies that g (wm )(·) := wm (·)f0 (wm )(·), g (w)(·) := w(·)f0 (w)(·) ∈ L2 ((0, t), V ). Moreover, by wm (τ, x) → w(τ, x) as m → ∞ for a.e. (τ, x) ∈ (0, t) × D, we have wm f0 (wm ) → wf0 (w) as m → ∞ dw m in L2 ((0, t), V ). This together with the fact that dw dt → dt as m → ∞ in ∗ L2 ((0, t), V ) implies Z tZ 0

→

Z tD

wm (τ )(x)f0 (wm )(τ )(x)

D

w(τ )f0 (w)(τ ),

0

dw E (τ ) dτ dt

dwm (τ )(x) dx dτ dt

as

m → ∞.

Observe that Z t Z dwm (τ )(x) dx dτ w(τ )(x)f0 (wm )(τ )(x) dt 0 D Z tZ dwm wm (τ )f0 (wm )(τ ) − (τ ) dx dτ dt 0 D Z t Z 1/2 Z t Z dw 1/2 m 2 ≤ |w(τ ) − wm (τ )|2 dx dτ (τ ) dx dτ dt 0 D 0 D →0 as m → ∞. We therefore have Z tZ 0

w(τ )(x)f0 (wm )(τ )(x)

D

dwm (τ )(x) dx dτ dt Z tD dw E → w(τ )f0 (w)(τ ), (τ ) dτ dt 0

as m → ∞ and then −

Z tD dw E (wf0 (w))(τ ), (τ ) dτ dt 0 Z t + Ba(2) (τ, w(τ ), f (w)(τ )) + δ0 hw(τ ), f (w)(τ )i dτ 0 Z t (2) (1) − h(c0 (τ, ·) − c0 (τ, ·))(Ua(1) (τ, 0)u0 ), f (w)(τ )i dτ 0

+ hw(t), f (w)(t)i − hw(0), f (w)(0)i = 0. Note that wf0 (w) → w− in L2 ((0, t), V ) as → 0. This together with

2. Fundamental Properties in the General Setting

39

f (w) → w− in L2 ((0, t), V ) as → 0 implies that Z tD dw E (τ ) dτ − w− (τ ), dt 0 Z t + Ba(2) (τ, w(τ ), w− (τ )) + δ0 hw(τ ), w− (τ )i dτ 0 Z t (2) (1) − hc0 (τ, ·) − c0 (τ, ·))(Ua(1) (τ, 0)u0 ), w− (τ )i dτ 0

+ hw(t), w− (t)i − hw(0), w− (0)i = 0. By [30, Lemma 3.3], there holds 1 − kw (t)k2 − kw− (0)k2 = − 2

Z tD dw E (τ ) dτ. w− (τ ), dt 0

We then have Z t kw− (t)k2 = kw− (0)k2 − 2 Ba(2) (τ, w− (τ ), w− (τ )) + δ0 kw− (τ )k2 dτ 0 Z t (2) (1) −2 hc0 (τ, ·) − c0 (τ, ·))(Ua(1) (τ, 0)u0 ), w− (τ )i dτ. 0 (2) (1) R t (2) (1) As c0 ≥ c0 , 0 hc0 (τ, ·) − c0 (τ, ·))(Ua(1) (τ, 0)u0 ), w− (τ )i dτ ≥ 0. By [30, Rt Lemma 3.1], 0 Ba (τ, w− (τ ), w− (τ )) + δ0 kw− (τ k2 dτ ≥ 0. We then have kw− (t)k ≤ kw− (0)k. But w− (0) = 0. Hence w− (t) = 0. It then follows that w(t) ≥ 0 and then v(t) ≥ 0, that is, Ua(2) (t, 0)u0 ≥ Ua(1) (t, 0)u0 . (2) and (3) can be proved by the arguments similar to those in (1). (2,n) (2) (4) First of all, let d0 (t, x) := d0 (t, x) + n for n = 1, 2, . . . . Then (2,n) ∞ (d0 )n=1 ⊂ L∞ (R × ∂D, R) and (2)

(2,1)

d0 (t, x) ≤ d0

(2,2)

(t, x) ≤ d0

(2,n)

(t, x) ≤ · · · ≤ d0

(t, x) ≤ · · ·

for a.e. (t, x) ∈ R × ∂D. Let (2)

(2)

(2)

(2)

(2,n)

a(2,n) := (aij , ai , bi , c0 , d0

)

for n = 1, 2, . . . . Then by (2) we have UaR(2,1) (t, 0)u0 ≥ UaR(2,2) (t, 0)u0 ≥ · · · ≥ UaR(2,n) (t, 0)u0 ≥ · · · ≥ 0,

(2.2.3)

hence kUaR(2,n) (t, 0)u0 k ≤ kUaR(2) (t, 0)u0 k for n = 1, 2, . . . .

(2.2.4)

40

Spectral Theory for Parabolic Equations Let un (·) := UaR(2,n) (·, 0)u0 . By [30, Lemma 3.1], t

Z

k∇un (τ )k2 dτ

α0 0

t

Z ≤2 0

Ba0(2,n) (τ, un (τ ), un (τ )) dτ + 2δ0

t

Z

kun (τ )k2 dτ

0

for n = 1, 2, . . . , where α0 is as in (2.1.1), Ba0(2,n) (τ, un (τ ), un (τ )) := Ba(2,n) (τ, un (τ ), un (τ )) Z (2,n) − d0 (τ, x)(un (τ )(x))2 dHN −1 , ∂D

and δ0 := (α0 )

−1

N N X X 1/2 (2) 2 1/2 (2) (2) kai k∞ kbi k2∞ + + k(c0 )− k∞ . i=1

(2,n)

Since d0

i=1

≥ 0, we have Ba0(2,n) (τ, un (τ ), un (τ )) ≤ Ba(2,n) (τ, un (τ ), un (τ )).

Hence Z t Z t Z t 2 α0 k∇un (τ )k dτ ≤ 2 Ba(2,n) (τ, un (τ ), un (τ )) dτ + 2δ0 kun (τ )k2 dτ. 0

0

0

Because un is a weak solution, we obtain with the help of (2.1.10) that Z

t 2

2

Z

2

k∇un (τ )k dτ ≤ −kun (t)k + ku0 k + 2δ0

α0 0

t

kun (τ )k2 dτ.

(2.2.5)

0

Rt By (2.2.4), 0 k∇un (τ )k2 dτ is bounded uniformly in n ∈ N. Hence { un |[0,t] : n ∈ N } is bounded in L2 ((0, t), W21 (D)). This makes sure that (un |[0,t] ) has a subsequence (denoted again by (un |[0,t] )) that converges weakly in L2 ((0, t), W21 (D)) to some u(·). ˚ 1 (D) for a.e. τ ∈ [0, t]. Note that, by PropoNext, we show that u(τ ) ∈ W 2 sition 2.1.4, Z t 1 Ba(2,n) (τ, un (τ ), un (τ )) dτ ≤ (kun (t)k2 + kun (0)k2 ) 2 0 and, by Lemma 2.1.4(ii), Z t Z t Ba0(2,n) (τ, un (τ ), un (τ )) dτ ≤ M0 kun (τ )k2W 1 (D) dτ. 2 0

0

2. Fundamental Properties in the General Setting

41

Hence Z tZ

(un (τ )(x))2 dHN −1 dτ

n 0

∂D

Z tZ

(2,n)

(τ, x)(un (τ )(x))2 dHN −1 dτ Z t Z t = Ba(2,n) (τ, un (τ ), un (τ )) dτ − Ba0(2,n) (τ, un (τ ), un (τ )) dτ 0 0 Z t 1 kun (τ )k2W 1 (D) dτ ≤ (kun (t)k2 + kun (0)k2 ) + M0 2 2 0 Rt for each n ∈ N. But 21 (kun (t)k2 + kun (0)k2 ) + M0 0 kun (τ )k2W 1 (D) dτ is 2 bounded uniformly in n ∈ N, by (2.2.4) and (2.2.5). We then must have Z tZ (un (τ )(x))2 dHN −1 dτ → 0 ≤

d0

0

∂D

0

∂D

as n → ∞, consequently Z tZ un (τ )(x) dHN −1 dτ → 0 0

∂D

as n → ∞. Observe that Z tZ hun , 1iL2 ((0,t),W21 (D)) =

un (τ )(x) dHN −1 dτ. 0

∂D

As un converge weakly in L2 ((0, t), W21 (D)) to u, we have Z tZ u(τ )(x) dHN −1 dτ = 0. 0

∂D

Further, since u(τ ) is nonnegative, its trace u(τ )|∂D is nonnegative for a.e. τ ∈ [0, t]. Therefore, u(τ )|∂D = 0 for a.e. τ ∈ [0, t]. By Lemma 2.1.1, we have ˚ 1 (D) for a.e. τ ∈ [0, t]. u(τ ) ∈ W 2 We now prove that u(t) = UaD(1) (t, 0)u0 . Note that for any n ∈ N and for any v ∈ W (0, t; V, V ∗ ), Z t Z t − hun (τ ), v(τ ˙ )i dτ + Ba0(2) (τ, un (τ ), v(τ )) dτ 0 0 Z tZ (2,n) + d0 (τ, x)(un (τ )v(τ ))(x) dHN −1 dτ + hun (t), v(t)i − hu0 , v(0)i 0 ∂D Z t Z t =− hun (τ ), v(τ ˙ )i dτ + Ba(1) (τ, un (τ ), v(τ )) dτ 0

0

+ hun (t), v(t)i − hu0 , v(0)i = 0,

42

Spectral Theory for Parabolic Equations

˚ 1 (D). Letting n → ∞ and using again the fact that un converge where V = W 2 weakly in L2 ((0, t), W21 (D)) to u, we have Z −

t

Z hu(τ ), v(τ ˙ )i dτ +

0

t

Ba(1) (τ, u(τ ), v(τ )) dτ + hu(t), v(t)i − hu0 , v(0)i = 0 0

for any v ∈ W (0, t; V, V ∗ ). This means that u(t) = UaD(1) (t, 0)u0 (see Proposition 2.1.3). Finally, by (2.2.3), we have UaD(1) (t, 0)u0 = u(t) ≤ UaR(2) (t, 0)u0 for t ≥ 0. (4) is thus proved. PROPOSITION 2.2.11 (Joint measurability) For any a ∈ Y , u0 ∈ L2 (D), and T > 0, u(·, ·) ∈ W20,1 ((0, T ) × D), where u(t, x) := (Ua (t, 0)u0 )(x). PROOF First of all, by Lemma 2.1.3, [ (0, ∞) × D 3 (t, x) 7→ u(t, x) ∈ R ] is measurable and for any T > 0, u ∈ L2 ((0, T ) × D). Next we prove that u ∈ W20,1 ((0, T ) × D). By the fact that [ (0, T ) 3 t 7→ u(t, ·) ] ∈ L2 ((0, T ), V ) and Lemma 1.2.3, there are simple functions φn ∈ L2 ((0, T ), V ) such that kφ1 kL2 ((0,T ),V ) ≤ kφ2 kL2 ((0,T ),V ) ≤ . . . , kφn kL2 ((0,T ),V ) → kukL2 ((0,T ),V ) , and φn (t) → u(t, ·) in V as n → ∞ for a.e. t ∈ (0, T ). Let φ˜n (t, x) := (φn (t))(x), t ∈ (0, T ), x ∈ D. It is clear that φ˜n ∈ W20,1 ((0, T ) × D) and kφ˜n kW 0,1 ((0,T )×D) = kφn kL2 ((0,T ),V ) ≤ kukL2 ((0,T ),V ) . 2 Hence, without loss of generality, we may assume that φ˜n weakly converges to φ˜ in W20,1 ((0, T ) × D). Therefore, for any ψ ∈ D((0, T ) × D), we have Z Z ˜ ˜ x)ψ(t, x) dt dx. φn (t, x)ψ(t, x) dt dx → φ(t, (0,T )×D

(0,T )×D

But Z

φ˜n (t, x)ψ(t, x) dt dx =

(0,T )×D

Z → 0

Z

T Z

0

T Z D

φ˜n (t, x)ψ(t, x) dx dt

D

Z u(t, x)ψ(t, x) dx dt =

u(t, x)ψ(t, x) dt dx.

(0,T )×D

˜ x) for a.e. (t, x) ∈ (0, T ) × D, hence u ∈ It then follows that u(t, x) = φ(t, W20,1 ((0, T ) × D).

2. Fundamental Properties in the General Setting

43

We proceed now to investigate consequences of (A2-3). As the mapping [ Y × R 3 (a, t) 7→ a · t ∈ Y ] is continuous, we have, in the light of Propositions 2.1.6 through 2.1.8, the following consequence of Proposition 2.2.6. PROPOSITION 2.2.12 Let (A2-1)–(A2-3) be satisfied. For any sequence (a(n) )∞ n=1 ⊂ Y , any real ∞ sequences (sn )∞ and (t ) , s < t , if n n=1 n n n=1 lim a(n) = a, lim sn = s, lim tn = t, where s < t,

n→∞

n→∞

n→∞

then for any u0 ∈ L2 (D), Ua(n) (tn , sn )u0 converges to Ua (t, s)u0 in L2 (D). The next result is much more important. PROPOSITION 2.2.13 (Joint continuity in t, u0 , and a) Assume (A2-1) through (A2-3). For any sequence (a(n) )∞ n=1 ⊂ Y , any real ∞ sequence (tn )∞ , and any sequence (u ) ⊂ L (D) (2 ≤ p < ∞), if n n=1 p n=1 limn→∞ a(n) = a, limn→∞ tn = t, where t > 0, and limn→∞ un = u0 in Lp (D), then Ua(n) (tn , 0)un converges in Lp (D) to Ua (t, 0)u0 . PROOF Since Lp (D) ,→ L2 (D) for 2 ≤ p < ∞, we assume limn→∞ un = u0 in L2 (D). Put K := sup {kUa(n) (tn , 0)k : n ∈ N }. By Proposition 2.2.2, K < ∞. There holds kUa(n) (tn , 0)un − Ua (t, 0)u0 k ≤ kUa(n) (tn , 0)un − Ua(n) (tn , 0)u0 k + kUa(n) (tn , 0)u0 − Ua (t, 0)u0 k. The first term on the right-hand side is bounded by Kkun − u0 k, hence it converges to 0, and the second one converges to 0 by virtue of (A2-3). We deduce from Proposition 2.2.5 that Ua(n) (tn , 0)un converges to Ua (t, 0)u0 in Lp (D), too. We put Π(t; u0 , a) = Πt (u0 , a) := (Ua (t, 0)u0 , a · t)

(2.2.6)

for t ≥ 0, a ∈ Y and u0 ∈ L2 (D). Proposition 2.2.13, Lemma 2.1.2 and (2.1.13) guarantee that the mapping Π = {Πt }t≥0 so defined is a topological linear skew-product semiflow on the product Banach bundle L2 (D) × Y covering the topological (semi)flow σ = {σt }t∈Z , σt a = a · t. We shall refer to Π defined by (2.2.6) as the (topological ) linear skew-product semiflow on L2 (D) × Y generated by (2.0.1)+(2.0.1).

44

Spectral Theory for Parabolic Equations

By Proposition 2.2.9(2), the solution operator Ua (t, 0) (a ∈ Y , t > 0) has the property that, for any u1 , u2 with u1 < u2 there holds Ua (t, 0)u1 < Ua (t, 0)u2 . By adjusting the terminology used for semiflows on ordered metric spaces (see, e.g., [57]) to skew-product semiflows with ordered fibers we can say that the (topological) linear skew-product semiflow Π is strictly monotone.

2.3

The Adjoint Problem

We consider in this section the adjoint problem of (2.0.1)+(2.0.2), i.e., the backward parabolic equation −

N N ∂u X ∂ X ∂u = aji (t, x) − bi (t, x)u ∂t ∂xi j=1 ∂xj i=1

−

N X

ai (t, x)

i=1

∂u + c0 (t, x)u, ∂xi

t < s, x ∈ D,

(2.3.1)

where s ∈ R is a final time, complemented with the boundary conditions: Ba∗ (t)u = 0,

t < s, x ∈ ∂D,

(2.3.2)

N N where Ba∗ (t)u = Ba∗ (t)u with a∗ := ((aji )N i,j=1 , −(bi )i=1 , −(ai )i=1 , c0 , a0 ) and Ba∗ (t) is as in (2.0.3) with a being replaced by a∗ . Denote Ua∗ (t, s) (t ≤ s) to be the (weak) solution operator of (2.3.1)+(2.3.2) (the weak solution of (2.3.1)+(2.3.2) is defined in a way similar to the weak solution of (2.0.1)+(2.0.2), see Definition 2.1.1). ˜a (t, s) (t ≥ s) be the (weak) solution operator of Let U N N ∂u X ∂ X ∂u = aji (−t, x) − bi (−t, x)u ∂t ∂xi j=1 ∂xj i=1

−

N X i=1

ai (−t, x)

∂u + c0 (−t, x)u, ∂xi

t > s, x ∈ D,

(2.3.3)

where s ∈ R is an initial time, complemented with the boundary conditions: B˜a∗ (t)u = 0,

t > s, x ∈ ∂D,

(2.3.4)

where B˜a∗ (t)u = Ba˜∗ (t)u with a ˜∗ (t, x) := a∗ (−t, x) and Ba˜∗ (t) is as in (2.0.3) ∗ with a being replaced by a ˜ . Then we have ˜a (−t, −s) (t ≤ s). Ua∗ (t, s) = U

2. Fundamental Properties in the General Setting

45

˜a = B ˜a (t, ·, ·) the bilinear form on V associated with For a ∈ Y denote by B a, ˜a (t, u, v) := Ba˜∗ (t, u, v) B

(2.3.5) ∗

where Ba˜∗ (t, u, v) is as in (2.1.4) with a being replaced by a ˜ in the Dirichlet and Neumann boundary condition cases, and Ba˜∗ (t, u, v) is as in (2.1.5) with a being replaced by a ˜∗ in the Robin boundary condition case. Similarly to Proposition 2.1.3, we have PROPOSITION 2.3.1 Assume (A2-1). Let u ∈ L2 ((s, t), V ). u is a weak solution of (2.3.3)+(2.3.4) on [s, t] × D (t > s) with u(s) = u0 if and only if u ∈ W (s, t; V, V ∗ ) and for any v ∈ W (s, t; V, V ∗ ), Z t Z t ˜a (τ, u(τ ), v(τ )) dτ B − hu(τ ), v(τ ˙ )i dτ + s

s

+ hu(t), v(t)i − hu0 , v(s)i = 0.

(2.3.6)

PROPOSITION 2.3.2 Assume (A2-1), (A2-2). If u and v are solutions of (2.0.1)+(2.0.2) and of (2.3.1)+(2.3.2) on [s, t] × D, respectively, then hu(τ ), v(τ )i is independent of τ for τ ∈ [s, t]. ˜a (−τ, −t)(v(t)) for PROOF First note that v(τ ) = Ua∗ (τ, t)(v(t)) = U s ≤ τ ≤ t. For any s ≤ τ ≤ t, by Proposition 2.1.3, Z τ Z τ hu(r), v(r)i ˙ dr = Ba (r, u(r), v(r)) dr s

s

+ hu(τ ), v(τ )i − hu(s), v(s)i.

(2.3.7)

By Proposition 2.3.1, we have Z τ hv(r), u(r)i ˙ dr s Z −s ˜ (r, −τ )v(τ ), u(−r)i = hU ˙ dr −τ

Z

−s

˜a (r, U ˜ (r, −τ )v(τ ), u(−r)) dr − hv(τ ), u(τ )i + hv(s), u(s)i B

=− −τ Z τ

=−

˜a (−r, U ˜ (−r, −τ )v(τ ), u(r)) dr − hv(τ ), u(τ )i + hv(s), u(s)i B

s

Z =− s

τ

˜a (−r, v(r), u(r)) dr − hu(τ ), v(τ )i + hu(s), v(s)i. B

(2.3.8)

46

Spectral Theory for Parabolic Equations

Note that Z

τ

Z

τ

˜a (−r, v(r), u(r)) dr = 0 B

Ba (r, u(r), v(r)) dr − s

s

for any s ≤ τ ≤ t. It then follows that Z τ Z τ hv(r), u(r)i ˙ dr + hu(r), v(r)i ˙ dr = 0 s

s

for s ≤ τ ≤ t. As both u and v are in W (s, t; V, V ∗ ), by [36, Section XVIII.1.2, Theorem 2] we have hu(τ ), v(τ )i = hu(s), v(s)i for any s ≤ τ ≤ t, so hu(τ ), v(τ )i is independent of τ . PROPOSITION 2.3.3 Assume (A2-1) and (A2-2). (Ua (t, s))∗ = Ua∗ (s, t)

for any a ∈ Y and any s < t.

PROOF First, recall that the dual (Ua (t, s))∗ (a ∈ Y , s < t) of the linear operator Ua (t, s) is defined by hu0 , (Ua (t, s))∗ v0 i = hUa (t, s)u0 , v0 i,

u0 , v0 ∈ L2 (D).

By Proposition 2.3.2, hu(t), v(t)i = hu(s), v(s)i where u and v are solutions of (2.0.1)+(2.0.2) and of (2.3.1)+(2.3.2) on [s, t]× D, respectively. Let u(·) := Ua (·, s)u0 and v(·) := Ua∗ (·, t)v0 . Then u(s) = u0 , v(t) = v0 , and hUa (t, s)u0 , v0 i = hu(t), v(t)i = hu(s), v(s)i = hu0 , Ua∗ (s, t)v0 i for any u0 , v0 ∈ L2 (D). Consequently, (Ua (t, s))∗ = Ua∗ (s, t)

for any a ∈ Y and any s < t.

Observe that if (2.0.1)+(2.0.2) satisfies (A2-3), then (2.3.1)+(2.3.2) also ∞ satisfies (A2-3). In fact, let (a(n) )∞ n=1 ⊂ Y and (tn )n=1 ⊂ R be such that (n) a → a and tn → t > 0 as n → ∞. Then for any v ∈ L2 (D), Ua(n) (tn , 0)v → Ua (t, 0)v in L2 (D) as n → ∞. Now for any u0 ∈ L2 (D) and any v ∈ L2 (D), hv, Ua∗(n) (0, −tn )u0 i = hu0 , Ua(n) (tn , 0)vi → hu0 , Ua (t, 0)vi = hv, Ua∗ (0, −t)u0 i

2. Fundamental Properties in the General Setting

47

as n → ∞. Therefore Ua∗(n) (0, −tn )u0 → Ua∗ (0, −t)u0 weakly in L2 (D). By ˜a (−t, −s) for any a ∈ Y and t < s, without Proposition 2.2.5 and Ua∗ (t, s) = U loss of generality, we may assume that Ua∗(n) (0, −tn ) → u∗ in L2 (D). We then have Ua∗ (0, −t)u0 = u∗ and Ua∗(n) (0, −tn )u0 → Ua∗ (0, −t)u0 in L2 (D). Hence having constructed a topological linear skew-product semiflow Π on the Banach bundle L2 (D) × Y , we have the dual topological linear skew-product semiflow Π∗ = {Π∗t }t≥0 , defined by the formula: Π∗ (t, v0 , a) = Π∗t (v0 , a) := ((Ua·(−t) (t, 0))∗ v0 , a·(−t)) = (Ua∗ (−t, 0)v0 , a·(−t)), where t ≥ 0, a ∈ Y , and v0 ∈ L2 (D).

2.4

Perturbation of Coefficients

In this section we discuss the satisfaction of the assumptions (A2-1)–(A2-3) presented in the previous sections. Consider (2.0.1)+(2.0.2). We first note that (A2-1) is a natural uniform ellipticity assumption and (A2-2) is a regularity condition of the domain D. We therefore assume throughout this section that (2.0.1)+(2.0.2) satisfies (A2-1) and (A2-2), and focus on the investigation on (A2-3) (perturbation property). In this section we make also another standing assumption: (A2-4) (Convergence almost everywhere) In the Dirichlet or Neumann case: (n) (n) For any sequence (a(n) ) converging in Y to a we have that aij → aij , ai → (n)

ai , bi → bi pointwise a.e. on R × D. In the Robin case: (n) (n) For any sequence (a(n) ) converging in Y to a we have that aij → aij , ai → (n)

ai , bi

(n)

→ bi pointwise a.e. on R×D, and d0

→ d0 pointwise a.e. on R×∂D.

THEOREM 2.4.1 Consider (2.0.1)+(2.0.2). Let V be as in (2.1.2). Let u0 ∈ L2 (D) and a(n) be as in (A2-4). Then for each t > 0 the following holds: (1) The restrictions Ua(n) (·, 0)u0 |[0,t] converge weakly in L2 ((0, T ), V ) to Ua (·, 0)u0 |[0,t] . (2) The functions [ [0, t] × D 3 (τ, x) 7→ (Ua(n) (τ, 0)u0 )(x) ] converge in the L2 ((0, t) × D)-norm to [ [0, t] × D 3 (τ, x) 7→ (Ua (τ, 0)u0 )(x) ]. (3) For any 0 < t0 < t, the restrictions Ua(n) (·, 0)u0 |[t0 ,t] converge in the C([t0 , t], L2 (D))-norm to Ua (·, 0)u0 |[t0 ,t] .

48

Spectral Theory for Parabolic Equations

PROOF We prove the theorem only for the Neumann or Robin boundary cases (V = W21 (D) in these cases), the proof of the theorem for the Dirichlet case being similar, but simpler (cf. [30, Lemma 8.4]). Put un (·) := Ua(n) (·, 0)u0 . First, by [30, Lemma 3.1], Z

t

k∇un k2 dτ ≤ 2

α0

Z

t

Ba0(n) (τ, un (τ ), un (τ )) dτ + 2δ0

0

0

Z

t

kun (τ )k2 dτ,

0

where α0 is as in (2.1.1), Ba0(n) (τ, un (τ ), un (τ ))

Z

(n)

:= Ba(n) (τ, un (τ ), un (τ ))−

d0 un (τ )un (τ ) dHN −1 , ∂Dn

and N N X X 1/2 1/2 (n) (n) (n) kai k2∞ kbi k2∞ δ0 := (α0 )−1 sup + sup + sup kc0 k∞ , n∈N i=1

n∈N i=1

n∈N

(n)

where k·k∞ denotes the L∞ (R × D)-norm. Since d0

≥ 0, we have

Ba0(n) (τ, un (τ ), un (τ )) ≤ Ba(n) (τ, un (τ ), un (τ )). Hence Z α0

t

k∇un k2 dτ ≤ 2

0

Z

t

Z Ba(n) (τ, un (τ ), un (τ )) dτ + 2δ0

0

t

kun (τ )k2 dτ.

0

Since un is a weak solution of (2.0.1)+(2.0.2), we obtain with the help of (2.1.10) that Z Z t

t

k∇un k2 dτ ≤ ku0 k2 − kun (t)k2 + 2δ0

α0 0

kun (τ )k2 dτ.

0

Rt

2

Note that by Proposition 2.2.2 0 kun (τ )k dτ is bounded uniformly in n ∈ N. Hence { un |[0,t] : n ∈ N } is bounded in L2 ((0, t), W21 (D)). This makes sure that (un |[0,t] ) has a subsequence (denoted again by (un |[0,t] )) that converges weakly in L2 ((0, t), W21 (D)) to some u(·). By Proposition 2.2.11, un (t, x) is integrable on [0, t]×D. It therefore follows that [ (0, t)×D 3 (τ, x) 7→ un (τ, x) ] converge weakly in L2 ((0, t) × D) to [ (0, t) × D 3 (τ, x) 7→ u(τ, x) ], where we write u(τ, x) = u(τ )(x). An application of Proposition 2.2.4 allows us, after possibly taking a subsequence, to conclude that there exists a continuous function v : (0, t) × D → R such that for any 0 < t0 < t and any compact D0 ⊂ D the functions [ [t0 , t] × D0 3 (τ, x) 7→ un (τ, x) ] converge uniformly to the restriction v|[t0 ,t]×D0 . Fix for the moment 0 < t0 < t. From the L2 –L∞ estimates in Proposition 2.2.2 it follows that there is M0 > 0 such that kun (τ )k∞ ≤ M0 for each τ ∈ [t0 , t] and each n ∈ N. As a consequence, kv(τ, ·)k∞ ≤ M0 for each

2. Fundamental Properties in the General Setting

49

τ ∈ [t0 , t] (here again k·k∞ means the L∞ (D)-norm). Take > 0. Let D0 ⊂ D be a compact set with |D \ D0 | < /8M02 . Consequently, Z

|un (τ, x) − v(τ, x)|2 dx < /2

D\D0

for any t0 ≤ τ ≤ t and any n ∈ N. Now we take n0 ∈ N so large that Z

|un (τ, x) − v(τ, x)|2 dx < /2

D0

for any t ≤ τ ≤ t and any n > n0 . Therefore it follows that un |[t0 ,t] (∈ C([t0 , t], L2 (D))) converge uniformly, as functions from [t0 , t] into L2 (D), to [ [t0 , t] 3 τ → v(τ, ·) ∈ L2 (D) ] (which belongs to C([t0 , t], L2 (D))). By the L2 –L2 estimates in Proposition 2.2.2, there exists M1 > 0 such that kun (τ )k ≤ M1 for any n ∈ N and any τ ∈ [0, t]. Let > 0, and take 0 < t0 < min{/(8M12 ), t}. One has t0

Z

Z

0

|un (τ, x) − v(τ, x)|2 dx dτ <

D

. 2

By the previous paragraph, there exists n1 ∈ N such that Z tZ t0

|un (τ, x) − v(τ, x)|2 dx dτ <

D

2

for all n > n1 . Therefore the functions [ (0, t) × D 3 (τ, x) 7→ un (τ )(x) ] converge in the L2 ((0, t)×D)-norm, hence weakly, to v. Consequently, u(τ ) = v(τ, ·) for each τ ∈ (0, t]. To conclude the proof it suffices to show that u(τ ) = Ua (τ, 0)u0 for any τ ∈ (0, t]. For any v ∈ W21 (D) and any ψ ∈ D([0, t)), Z −

t

˙ ) dτ + hun (τ ), viψ(τ

0

t

Z

Ba(n) (τ, un (τ ), v)ψ(τ ) dτ − hu0 , viψ(0) = 0. 0

Observe that Z

t

˙ ) dτ = hun (τ ), viψ(τ

0

Z t Z 0

˙ ) dτ. un (τ, x)v(x) dx ψ(τ

D

Hence Z 0

t

˙ ) dτ → hun (τ ), viψ(τ

Z t Z 0

D

Z t ˙ ) dτ. ˙ ) dτ = u(t, x)v(x) dx ψ(τ hu(τ ), viψ(τ 0

50

Spectral Theory for Parabolic Equations

Observe also that Z t Z (n) d0 (τ, x)un (τ, x)v(x) dHN −1 ψ(τ ) dτ 0 ∂D Z t Z − d0 (τ, x)u(τ, x)v(x) dHN −1 ψ(τ ) dτ 0 ∂D Z t Z (n) (d0 (τ, x) − d0 (τ, x))un (τ )(x)v(x) dHN −1 ψ(τ ) dτ = 0 ∂D Z t Z + d0 (τ, x)(un (τ, x) − u(τ )(x))v(x) dHN −1 ψ(τ ) dτ. ∂D

0

Note that Z t Z 0

d0 (τ, x)(un (τ, x) − u(τ, x))v(x) dHN −1 ψ(τ ) dτ → 0

∂D

by the fact that un → u weakly in L2 ((0, t), W21 (D)), hence the traces of un on ∂D converge weakly in L2 ((0, t), L2 (∂D)) to the trace of u on ∂D (see Lemma 2.1.1). Further, since { un |[0,t] : n ∈ N} is bounded in L2 ((0, t), W21 (D)), RtR { 0 ∂D (un (τ, x))2 dHN −1 dτ : n ∈ N } is a bounded sequence. We then have Z t Z 2 (n) (d0 (τ, x) − d0 (τ, x))un (τ, x)v(x) dHN −1 ψ(τ ) dτ ∂D 0 Z t Z (n) 2 2 2 ≤ |d0 (τ, x) − d0 (τ, x)| (v(x)) dHN −1 (ψ(τ )) dτ 0 ∂D Z t Z 2 · (un (τ, x)) dHN −1 dτ 0

∂D

→0 (n)

by the boundedness of d0 in the L∞ (R × ∂D)-norm and the convergence of (n) d0 to d0 a.e. on [0, t] × ∂D. Hence Z t Z

(n) d0 (τ, x)un (τ, x)v(x) dHN −1 ψ(τ ) dτ 0 ∂D Z t Z → d0 (τ, x)u(τ, x)v(x) dHN −1 ψ(τ ) dτ. 0

∂D

Similarly, we can prove that Z t Z t Z (n) 0 Ba(n) (τ, un (τ ), v)ψ(τ ) dτ + c0 (τ, x)un (τ, x)v(x) dx ψ(τ ) dτ 0 0 D Z t Z t Z → Ba0 (τ, u(τ ), v)ψ(τ ) dτ + c0 (τ, x)u(τ, x)v(x) dx ψ(τ ) dτ. 0

0

D

2. Fundamental Properties in the General Setting Now, Z tZ 0

(n) c0 (τ, x)un (τ, x)v(x)ψ(τ ) dx dτ

Z tZ −

D

c0 (τ, x)u(τ, x)v(x)ψ(τ ) dx dτ 0

Z tZ

51

D

(n)

c0 (τ, x)(un (τ, x) − u(τ, x))v(x)ψ(τ ) dx dτ

= 0

D

Z tZ

(n)

(c0 (τ, x) − c0 (τ, x))u(τ, x)v(x)ψ(τ ) dx dτ.

+ 0

D

We estimate Z t Z 2 (n) c0 (τ, x)(un (τ, x) − u(τ, x))v(x)ψ(τ ) dx dτ 0 D Z t Z ≤ |un (τ, x) − u(τ, x)|2 dx dτ 0 D Z t Z (n) |c0 (τ, x)|2 (v(x))2 (ψ(τ ))2 dx dτ 0

D

→ 0. (n)

By the weak-* convergence of c0 Z tZ

to c0 ,

(n)

(c0 (τ, x) − c0 (τ, x))u(τ, x)v(x)ψ(τ ) dx dτ → 0. 0

D

Consequently Z tZ Z tZ (n) c0 (τ, x)un (τ, x)v(x)ψ(τ ) dx dτ → c0 (τ, x)u(τ, x)v(x)ψ(τ ) dx dτ. 0

D

0

D

It then follows that Z t Z t ˙ ) dτ + − hu(τ ), viψ(τ Ba (τ, u(τ ), v)ψ(τ ) dτ − hu0 , viψ(0) = 0. 0

0

Therefore, u(·) is a weak solution of (2.0.1)+(2.0.2) on [0, t] × D with u(0) = u0 . In particular, the whole sequence (un ) (and not only a subsequence) converges to u as required.

2.5

The Smooth Case

In the present section we are considering the case when the domain and the coefficients are sufficiently regular for any solution to be a classical one.

52

Spectral Theory for Parabolic Equations We introduce the following assumption.

(A2-5) (Smoothness) ∂D is an (N − 1)-dimensional manifold of class C 3+α for some 0 < α < 1. Moreover, there is M > 0 such that for any a ∈ Y , the C 2+α,2+α (R × ¯ ¯ D)-norms of aij and ai (i, j = 1, 2, . . . , N ), the C 2+α,1+α (R × D)-norms of bi 2+α,2+α (i = 1, 2, · · · , N ) and c0 , and the C (R×∂D)-norms of d0 , are bounded by M . First of all, we have LEMMA 2.5.1 (n) Assume (A2-5). Then limn→∞ a(n) = a if and only if aij converge to aij , (n)

(n)

(n)

ai converge to ai , bi converge to bi , c0 converge to c0 , all uniformly ¯ and (in the Robin case) d(n) converge to d0 on compact subsets of R × D, 0 uniformly on compact subsets of R × ∂D. PROOF As the convergence in the open-compact topology implies convergence in the weak-* topology, the closure Y˜ of Y in the open-compact topology equals, as a set, Y . By the Ascoli–Arzel`a theorem, Y˜ is compact. Compact topologies on the same set are identical. In the remainder of the present section we assume that (A2-1) and (A2-5) are satisfied. (A2-5) implies (A2-2). Also, by Lemma 2.5.1 (A2-5) implies (A2-4), from which it follows that (A2-3) is satisfied. We will apply the theories developed in [3] to derive regularity properties, various estimates, and strong monotonicity of weak solutions of (2.0.1)+(2.0.2). Observe that (2.0.1)a +(2.0.2)a can be rewritten as N N X X ∂u ∂2u ˜bi (t, x) ∂u = aij (t, x) + ∂t ∂xi ∂xj ∂xi i,j=1 i=1

+ c˜0 (t, x)u,

t > s, x ∈ D,

(2.5.1)

complemented with the boundary conditions B(t)u = 0,

t > s, x ∈ ∂D,

(2.5.2)

where

B(t)u =

  u        N  X

aij (t, x)∂xj uνi + d˜0 (t, x)u  i,j=1    N  X    aij (t, x)∂xj uνi + d˜0 (t, x)u   i,j=1

(Dirichlet) (Neumann) (Robin),

2. Fundamental Properties in the General Setting

53

PN ∂a with ˜bi (t, x) := bi (t, x) + ai (t, x) + j=1 ∂xjij (t, x), c˜0 (t, x) := c(t, x) + PN ∂ai PN ˜ i=1 ∂xi (t, x), and d0 (t, x) := i=1 ai (t, x)νi in the Neumann case and PN d˜0 (t, x) := d0 (t, x) + i=1 ai (t, x)νi in the Robin case. Note that the boundary conditions in the Neumann and Robin cases are of the same form after rewriting. Note also that d˜0 (t, x) may change sign. We point out that the theory presented in [3] applies to such a general case. More precisely, to apply that theory we only need the smoothness of the coefficients and the domain and the uniform ellipticity of (2.5.1)+(2.5.2) (see [3] for detail). PROPOSITION 2.5.1 (Regularity up to boundary) Let 1 ≤ p ≤ ∞ and u0 ∈ Lp (D). Then for any α ∈ (0, 1/2), any a ∈ Y , and any 0 < t1 < t2 the restriction [ [t1 , t2 ] 3 t 7→ Ua (t, 0)u0 ] belongs to ¯ ∩ C([t1 , t2 ], C 2+α (D)). ¯ C 1 ([t1 , t2 ], C α (D)) Moreover, there is C = C(t1 , t2 , u0 ) > 0 such that k[ [t1 , t2 ] 3 t 7→ Ua (t, 0)u0 ]kC 1 ([t1 ,t2 ],C α (D)) ¯ ≤C and k[ [t1 , t2 ] 3 t 7→ Ua (t, 0)u0 ]kC([t1 ,t2 ],C 2+α (D)) ¯ ≤C for all a ∈ Y . PROOF For given 1 ≤ p ≤ ∞ and u0 ∈ Lp (D), for any t > 0 and 1 < q < ∞, one has Ua (t, 0)u0 ∈ Lq (D). The result then follows from [3, Corollary 15.3]. By Proposition 2.5.1, for any u0 ∈ Lp (D) (1 ≤ p ≤ ∞), Ua (t, 0)u0 turns out to be a classical solution of (2.5.1)+(2.5.2): for any t > 0 and x ∈ D the equation (2.5.1) is satisfied pointwise, and for any t > 0 and x ∈ ∂D the boundary condition (2.5.2) is satisfied pointwise. Next, we derive other regularity properties and various estimates. To do so, for a ∈ Y let A(a) be the operator given by N N X ∂ X ∂u A(a)u := aij (0, x) + ai (0, x)u ∂xi j=1 ∂xj i=1 +

N X

bi (0, x)

i=1

∂u + c0 (0, x)u, ∂xi

and B(a) := Ba (0), where Ba (·) is as in (2.0.3).

x ∈ D,

54

Spectral Theory for Parabolic Equations

For given 1 0, let Wps (D), Hps (D), and be the Sobolev–Slobodetski˘ı spaces, the Bessel potential spaces, and the Besov spaces, respectively (see [3], [105] for definitions). s Bp,q (D)

LEMMA 2.5.2 If 0 < s < ∞, then, up to equivalent norms, Wps (D)

PROOF

=

( s Hp (D) s (D) Bp,p

if

s∈N

if

s 6∈ N.

See [105].

For given 0 < β < 1 and 1 < p < ∞, let (·, ·)β,p and [·, ·]β be a real interpolation functor and a complex interpolation functor, respectively, (see [15], [105] for definitions), and let Vpβ := (Lp (D), Wp2 (D))β,p

(2.5.3)

V˜pβ := [Lp (D), Wp2 (D)]β .

(2.5.4)

and

LEMMA 2.5.3 Let 1 < p < ∞ and 0 < β < 1. Then the following holds. 2β (1) Vpβ = Bp,p (D);

(2) V˜pβ = Hp2β (D). PROOF

See [105, Theorem 2 in Section 4.3.1].

For given a ∈ Y , 0 < β < 1 and 1 < p < ∞, let Vpβ (a) := (Lp (D), Vp1 (a))β,p ,

(2.5.5)

V˜pβ (a) := [Lp (D), V˜p1 (a)]β ,

(2.5.6)

and where Vp1 (a) = V˜p1 (a) := { u ∈ Wp2 (D) : B(a)u = 0 }. LEMMA 2.5.4 Let 1 < p < ∞ and 0 < β < 1 with 2β − (1) Vpβ (a) is a closed subspace of Vpβ ;

1 p

6= 0, 1. Then the following holds.

2. Fundamental Properties in the General Setting

55

(2) V˜pβ (a) is a closed subspace of V˜pβ . PROOF (1) follows from [3, Lemma 14.4]. (2) follows from [4, Lemma 5.1]. Recall that, for given two Banach spaces X1 and X2 consisting of (equivalence classes of) real functions defined on D, X1 ,→ X2 means that X1 is continuously embedded into X2 , and X1 ,− ,→ X2 means that X1 is compactly embedded into X2 . We have LEMMA 2.5.5 N 2p

(1) For p > N/2 and

< β ≤ 1 there holds ¯ Vpβ ,− ,→ C(D)

(2.5.7)

¯ V˜pβ ,− ,→ C(D);

(2.5.8)

¯ Wp2 (D) ,−,→ C(D);

(2.5.9)

and in particular

(2) For p > N and

N 2p

+

1 2

< β ≤ 1 there holds ˜

¯ Vpβ ,− ,→ C 1+β (D)

(2.5.10)

˜ ¯ V˜pβ ,−,→ C 1+β (D),

(2.5.11)

and where 0 < β˜ < 2β − 1 −

N p;

in particular ˜

¯ Wp2 (D) ,−,→ C 1+β (D), where 0 < β˜ < 1 −

(2.5.12)

N p;

(3) For given 1 < p < ∞ and 0 < β1 < β2 < 1, Wp2 (D) ,−,→ Wp2β2 (D) ,− ,→ Wp2β1 (D) ,−,→ Lp (D),

(2.5.13)

Wp2 (D) ,−,→ Vpβ2 ,−,→ Vpβ1 ,−,→ Lp (D),

(2.5.14)

Wp2 (D) ,−,→ V˜pβ2 ,−,→ V˜pβ1 ,−,→ Lp (D);

(2.5.15)

and

56

Spectral Theory for Parabolic Equations

(4) For given 1 < p < ∞ and 0 < β1 < β2 < 1 with 2β1 − p1 , 2β2 − p1 6= 0, 1, Vp1 (a) ,−,→ Vpβ2 (a) ,− ,→ Vpβ1 (a) ,− ,→ Lp (D),

(2.5.16)

V˜p1 (a) ,−,→ V˜pβ2 (a) ,− ,→ V˜pβ1 (a) ,− ,→ Lp (D).

(2.5.17)

and

PROOF (1) follows from the fact that if mp > N then Wpj+m (D) ,−,→ ¯ (see [1, Theorem 6.2(7)]), together with Lemmas 2.5.2 and 2.5.3 (see C j (D) also [3, Theorem 11.5]). (2) follows from the fact that if mp > N > (m − 1)p and 0 < β˜ < m − ˜ ¯ ,→ C j+β (D) (see [1, Theorem 6.2(8)]), together with (N/p) then Wpj+m (D) ,− Lemmas 2.5.2 and 2.5.3 (see also [3, Theorem 11.5]). (3) (2.5.13) follows from [3, Theorem 11.5 and Corollary 15.2]. The continuity of the embeddings in (2.5.14) follows from [105, Theorem 4.6.1] and the compactness follows from (2.5.13) and Lemma 2.5.2. The continuity of the embeddings in (2.5.15) also follows from [105, Theorem 4.1.1]. By [105, Theorem 4.6.2], for any > 0 with 0 < β1 − < β1 + < β2 − < β2 + < 1, β1 − β2 − β1 + β2 + (D). (D) ,→ Bp,p (D) ,→ Hpβ1 (D) ,→ Bp,p Bp,p (D) ,→ Hpβ2 (D) ,→ Bp,p

This together with the compactness of the embeddings in (2.5.14) implies that the embeddings in (2.5.15) are also compact. (4) It follows from (3) and Lemma 2.5.4. By [3, Lemma 6.1 and Theorem 14.5] (see also [109]), we have PROPOSITION 2.5.2 (1) For any 1 0. (2) For any 1 0. (3) For any fixed T > 0 and 1 0 such that kUa (t, 0)kLp (D),Wp2 (D) ≤ for all a ∈ Y and 0 < t ≤ T . By [3, Theorems 7.1 and 14.5] we have PROPOSITION 2.5.3 Suppose that 2β − 1/p 6∈ N. Then

Cp t

2. Fundamental Properties in the General Setting

57

(1) for any a ∈ Y , t ≥ 0 and u0 ∈ Vpβ (a) there holds Ua (t, 0)u0 ∈ Vpβ (a · t); moreover, the mapping [ [0, ∞) 3 t 7→ Ua (t, 0)u0 ∈ Vpβ ] is continuous; (2) for any a ∈ Y , t ≥ 0 and u0 ∈ V˜pβ (a) there holds Ua (t, 0)u0 ∈ V˜pβ (a · t); moreover, the mapping [ [0, ∞) 3 t 7→ Ua (t, 0)u0 ∈ V˜pβ ] is continuous; (3) for any T > 0 there is Cp,β > 0 such that kUa (t, 0)u0 kVpβ ≤ Cp,β ku0 kVpβ for any a ∈ Y , 0 ≤ t ≤ T , and u0 ∈ Vpβ (a), and kUa (t, 0)u0 kV˜pβ ≤ Cp,β ku0 kV˜pβ for any a ∈ Y , 0 ≤ t ≤ T , and u0 ∈ V˜pβ (a). PROPOSITION 2.5.4 (Joint continuity in X) Assume that X is a Banach space such that, for some 1 0, and limn→∞ un = u0 in L2 (D), then Ua(n) (tn , 0)un converges in X to Ua (t, 0)u0 .

PROOF First of all, we have by Proposition 2.2.13 that Ua(n) (tn , 0)un converges in L2 (D) to Ua (t, 0)u0 . It follows from Propositions 2.2.2 and 2.5.2 and the assumption Wp2 (D) ,−,→ X that there is a subsequence (nk )∞ k=1 such that Ua(nk ) (tnk , 0)unk converges in X to some u∗ . We then must have u∗ = Ua (t, 0)u0 and Ua(nk ) (tnk , 0)unk converges in X to Ua (t, 0)u0 . This implies that Ua(n) (tn , 0)un converges in X to Ua (t, 0)u0 . Examples of Banach spaces X satisfying the assumptions of the above ¯ (when p > N/2), proposition include Vpβ , V˜pβ (p ≥ 2, 0 < β < 1), C(D) 1 ¯ C (D) (when p > N ), etc. (see Lemma 2.5.5). PROPOSITION 2.5.5 (Norm continuity in X) Assume that a Banach space X has the property that, for some 1 0, then Ua(n) (tn , 0) converges in L(X) to Ua (t, 0).

58

Spectral Theory for Parabolic Equations

PROOF Assume that Ua(n) (tn , 0) does not converge in L(X) to Ua (t, 0). Then we may assume that there are 0 > 0 and un ∈ X with kun kX = 1 such that kUa(n) (tn , 0)un − Ua (t, 0)un kX ≥ for n = 1, 2, . . . . By the assumption X ,− ,→ L2 (D), without loss of generality we may assume that there is u0 ∈ L2 (D) such that kun − u0 k → 0 as n → ∞. Then by Proposition 2.5.4 we have kUa(n) (tn , 0)un − Ua (t, 0)u0 kX → 0,

kUa (t, 0)un − Ua (t, 0)u0 kX → 0

as n → ∞. Hence kUa(n) (tn , 0)un − Ua (t, 0)u0 kX → 0, which is a contradiction. Therefore, Ua(n) (tn , 0) converges in L(X) to Ua (t, 0).

Examples of Banach spaces satisfying the assumption of the above propo¯ (when p > N ), etc. (see sition include Vpβ , V˜pβ (p ≥ 2, 0 < β < 1), C 1 (D) Lemma 2.5.5). We proceed now to investigate the strong monotonicity property of the solution operators Ua (t, 0). We will use the strong maximum principle and the Hopf boundary point principle for classical solutions. But before we do that we have to analyze whether the existing theory (as presented, e.g., in [44]) can be applied: notice that in the Robin case d˜0 may change sign. We show that the zero-order coefficient can be made nonnegative by an appropriate change of variables. Fix a ∈ Y and p > N . Take u0 to be a C ∞ real function whose support is a nonempty compact set contained in D (the existence of such a function follows by the C ∞ Urysohn lemma, see, e.g., [42, Lemma 8.18]). Let u∗ (·, ·) be the solution of  N X  ∂u ∂2u   = aij (t, x) , t > −1, x ∈ D,   ∂t ∂xi ∂xj i,j=1 N X     aij (t, x)∂xj uνi + u = 0, 

t > −1, x ∈ ∂D,

i,j=1

with the initial condition u(−1, ·) = u0 . The initial function u0 clearly belongs to Vp1 , and satisfies pointwise the boundary conditions at t = −1. Consequently, [4, Theorem 7.3(ii)] states that [ [−1, ∞) 3 t 7→ u∗ (t, ·) ∈ Vp1 ] is continuous, from which it follows via (2.5.12) that [ [−1, ∞) 3 t 7→ u∗ (t, ·) ∈ ¯ ] is continuous, too. In particular, u∗ is continuous on [−1, ∞) × D. ¯ C 1 (D) Further, u∗ is a classical solution, so the strong maximum principle and the Hopf boundary point principle for parabolic equations imply that u∗ (t, x) > 0 ¯ for t > −1 and x ∈ D.

2. Fundamental Properties in the General Setting ∗

59

∗

Now, let v(t, x) := eM u (t,x) u(t, x), where M ∗ is a positive constant (to be determined later). Then (2.5.1)+(2.5.2) with s = 0 becomes N N X X ∂v ∂2v ¯bi (t, x) ∂v = + aij (t, x) ∂t ∂x ∂x ∂xi i j i,j=1 i=1

+ c¯0 (t, x)v,

t > 0, x ∈ D,

(2.5.18)

complemented with the boundary conditions N X

aij (t, x)∂xj vνi + d¯0 (t, x)v = 0,

t > 0, x ∈ ∂D,

(2.5.19)

i,j=1

where ¯bi (t, x) := ˜bi (t, x) − M ∗

X N j=1

c¯0 (t, x) := c˜0 (t, x) − M ∗

∂u∗ ∂u∗ aij + aji , ∂xj ∂xj

N X

N ∗ X ∂u∗ ∂u∗ ˜bi (t, x) ∂u + (M ∗ )2 aij (t, x) , ∂xi ∂xi ∂xj i=1 i,j=1

∗ ∗ d¯0 (t, x) := d˜0 (t, x) + M ∗ eM u (t,x) u∗ (t, x).

We see that for any T > 0, there is M ∗ = M ∗ (T ) > 0 such that d¯0 (t, x) > 0 for ¯ Also, the coefficients ¯bi , c¯0 , and d¯0 are continuous, hence t ∈ [0, T ] and x ∈ D. ¯ So the existing theory for classical solutions of parabolic bounded on [0, T ]×D. equations can be applied to (2.5.18)+(2.5.19) and then to (2.5.1)+(2.5.2). In particular, we have the following result. PROPOSITION 2.5.6 (Strong monotonicity on initial data) Let 1 ≤ p ≤ ∞ and u1 , u2 ∈ Lp (D). If u1 < u2 , then (i) (Ua (t, 0)u1 )(x) < (Ua (t, 0)u2 )(x)

for a ∈ Y, t > 0, x ∈ D

and ∂ ∂ (Ua (t, 0)u1 )(x) > (Ua (t, 0)u2 )(x) ∂νν ∂νν

for a ∈ Y, t > 0, x ∈ ∂D

in the Dirichlet case, (ii) (Ua (t, 0)u1 )(x) < (Ua (t, 0)u2 )(x) in the Neumann or Robin case.

¯ for a ∈ Y, t > 0, x ∈ D

60

Spectral Theory for Parabolic Equations Recall that the nonnegative cone Vpβ (a)+ of Vpβ (a) is defined by Vpβ (a)+ := { u ∈ Vpβ (a) : u(x) ≥ 0

for a.e. x ∈ D }.

V˜pβ (a)+ := { u ∈ V˜pβ (a) : u(x) ≥ 0

for a.e. x ∈ D }.

Similarly

¯ then If p and β are such that Vpβ (a) ,→ C(D), Vpβ (a)+ = { u ∈ Vpβ (a) : u(x) ≥ 0

¯ }. for all x ∈ D

¯ then Similarly, if p and β are such that V˜pβ (a) ,→ C(D), V˜pβ (a)+ = { u ∈ V˜pβ (a) : u(x) ≥ 0 LEMMA 2.5.6 Assume that p > N and

N 2p

+

1 2

¯ }. for all x ∈ D

< β ≤ 1. Let a ∈ Y .

(1) In the case of the Dirichlet boundary conditions the interior Vpβ (a)++ of the nonnegative cone Vpβ (a)+ is nonempty, and is characterized by Vpβ (a)++ = { u ∈ Vpβ (a)+ : u(x) > 0 and

for all

(∂u/∂νν )(x) < 0

x∈D

for all

x ∈ ∂D }. (2.5.20)

(2) In the case of the Neumann or Robin boundary conditions the interior Vpβ (a)++ of the nonnegative cone Vpβ (a)+ is nonempty, and is characterized by Vpβ (a)++ = { u ∈ Vpβ (a)+ : u(x) > 0

for all

¯ }. x∈D

(2.5.21)

Analogous results hold for the complex interpolation spaces V˜pβ (a). PROOF We prove the lemma only for the real interpolation spaces Vpβ (a). Fix a ∈ Y . It follows from Lemmas 2.5.4 and 2.5.5(2) that ¯ Vpβ (a) ,→ C 1 (D). (1) In the Dirichlet case Vp1 (a) consists precisely of those elements of Wp2 (D) ¯ any u ∈ Vp1 (a) is a whose trace on ∂D is zero. Since Vp1 (a) ,→ C 1 (D), 1 C function vanishing on ∂D. By [3, Section 7], the image of the embed¯ we conclude that ding Vp1 (a) ,→ Vpβ (a) is dense. Because Vpβ (a) ,→ C 1 (D), β 1 ˚ (D). ¯ Vp (a) ,→ C

2. Fundamental Properties in the General Setting

61

˚1 (D). ¯ It follows from Lemma 1.3.1(2) Denote by I the embedding Vpβ (a) ,→ C −1 ˚1 ¯ ++ ˚1 (D) ¯ ++ is that the right-hand side of (2.5.20) equals I (C (D) ), where C 1 ˚ (D). ¯ This proves the “⊃” inclusion. Denote by ϕprinc an open subset of C some (nonnegative) principal eigenfunction of the elliptic equation N N X ∂u ∂ X + ai (0, x)u aij (0, x) 0= ∂xi j=1 ∂xj i=1 +

N X

bi (0, x)

i=1

∂u + c0 (0, x)u, ∂xi

x ∈ D,

with the Dirichlet boundary conditions. We have that ϕprinc ∈ Vp1 (a) ,→ Vpβ (a) and that it belongs to the right-hand side of (2.5.20), consequently to Vpβ (a)++ . Finally, let u ∈ Vpβ (a)++ . There is > 0 such that u − ϕprinc ∈ Vpβ (a)+ , therefore u(x) ≥ ϕprinc (x) > 0 for all x ∈ D, which gives further ∂ϕprinc that ∂u ν (x) ≤ ∂ν ν (x) < 0 for all x ∈ ∂D. ∂ν (2) In the Neumann or Robin cases, denote by I the embedding Vpβ (a) ,→ ¯ It follows from Lemma 1.3.1(1) that the right-hand side of (2.5.21) C 1 (D). ¯ ++ ), where C 1 (D) ¯ ++ is an open subset of C 1 (D). ¯ equals I −1 (C 1 (D) This proves the “⊃” inclusion. Denote by ϕprinc some (nonnegative) principal eigenfunction of the elliptic equation N N X ∂ X ∂u 0= aij (0, x) + ai (0, x)u ∂xi j=1 ∂xj i=1 +

N X i=1

bi (0, x)

∂u + c0 (0, x)u, ∂xi

with the boundary conditions  N X N X    aij (0, x)∂xj u + ai (0, x)u νi ,      i=1 j=1 N X N 0= X  a (0, x)∂ u + a (0, x)u νi  ij x i j    i=1 j=1    +d0 (0, x)u,

x ∈ D,

x ∈ ∂D

(Neumann)

x ∈ ∂D

(Robin).

We have that ϕprinc ∈ Vp1 (a) ,→ Vpβ (a) and that it belongs to the right-hand side of (2.5.21), consequently to Vpβ (a)++ . Finally, let u ∈ Vpβ (a)++ . There is > 0 such that u − ϕprinc ∈ Vpβ (a)+ , therefore u(x) ≥ ϕprinc (x) > 0 for ¯ all x ∈ D. In view of Lemma 2.5.6, the following result is a consequence of Propositions 2.5.6 and 2.5.3.

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Spectral Theory for Parabolic Equations

PROPOSITION 2.5.7 (1) For any 1 ≤ p ≤ ∞, any q > N and any

N 2q

+

1 2

< β ≤ 1 there holds

Ua (t, 0)(Lp (D)+ \ {0}) ⊂ Vqβ (a · t)++ and Ua (t, 0)(Lp (D)+ \ {0}) ⊂ V˜qβ (a · t)++ , for all a ∈ Y and t > 0. (2) There holds ˚1 (D) ¯ ++ Ua (t, 0)(Lp (D)+ \ {0}) ⊂ C in the Dirichlet case, or ¯ ++ Ua (t, 0)(Lp (D)+ \ {0}) ⊂ C 1 (D) in the Neumann or Robin cases, for all a ∈ Y and t > 0. Proposition 2.5.7(2) yields that ˚1 (D) ¯ + \ {0}) ⊂ C ˚1 (D) ¯ ++ Ua (t, 0)(C

(2.5.22)

for all a ∈ Y and all t > 0 (in the Dirichlet case), and ¯ + \ {0}) ⊂ C 1 (D) ¯ ++ Ua (t, 0)(C 1 (D)

(2.5.23)

for all a ∈ Y and all t > 0 (in the Neumann or Robin cases). The property described in (2.5.22) (resp. (2.5.23)) can be written as: For ˚1 (D) ¯ (resp. u1 , u2 ∈ C 1 (D)) ¯ and u1 < u2 each a ∈ Y and t > 0, if u1 , u2 ∈ C then u1 u2 . In the existing terminology (see [57]) the linear operator ˚1 (D) ¯ →C ˚1 (D) ¯ (resp. Ua (t, 0) : C 1 (D) ¯ → C 1 (D)) ¯ is, for a ∈ Y and Ua (t, 0) : C t > 0, strongly positive (or strongly monotone). PROPOSITION 2.5.8 For each a ∈ Y and each t > 0 the linear operator Ua (t, 0) is injective. PROOF

See [43, Chapter 6].

We finish the section with a remark on the adjoint problem. Observe that the adjoint equation (2.3.1)a with the corresponding boundary conditions (2.3.2)a can be rewritten as N X ∂u ∂2u = a∗ji (t, x) − ∂t ∂xi ∂xj i,j=1

+

N X j=1

b∗j (t, x)

∂u + c∗0 (t, x)u, ∂xj

t < s, x ∈ D,

(2.5.24)

2. Fundamental Properties in the General Setting

63

complemented with the boundary conditions B ∗ (t)u = 0,

t < s, x ∈ ∂D,

(2.5.25)

where

B ∗ (t)u =

 u       N   X

(Dirichlet) a∗ji (t, x)∂xi uνj + d∗0 (t, x)u

 j,i=1   N  X    a∗ji (t, x)∂xi uνj + d∗0 (t, x)u  

(Neumann) (Robin),

j,i=1

PN ∂a with a∗ji (t, x) := aij (t, x), b∗j (t, x) := −bj (t, x) − aj (t, x) + i=1 ∂xjii (t, x), PN ∂bi PN c∗0 (t, x) := c0 (t, x) − i=1 ∂x (t, x), d∗0 (t, x) := − j=1 bj (t, x)νj in the Neui P N mann case and d∗0 (t, x) := d0 (t, x) − j=1 bj (t, x)νj in the Robin case. All the results presented above in the present section carry over to the case of the adjoint problem.

2.6

Remarks on Equations in Nondivergence Form

In this section, we provide remarks on nonautonomous equations in nondivergence form. Consider N N X X ∂u ∂2u ∂u = aij (t, x) + bi (t, x) ∂t ∂x ∂x ∂x i j i i,j=1 i=1

+ c0 (t, x)u,

t > s, x ∈ D,

(2.6.1)

complemented with the boundary conditions B(t)u = 0,

t > s, x ∈ ∂D,

(2.6.2)

where D ⊂ RN is a bounded domain, s ∈ R is an initial time, and B is a boundary operator of either the Dirichlet or Neumann or Robin type, that is,   u (Dirichlet)       N  X ∂xi u¯ νi (t, x) (Neumann) B(t)u =  i=1  N  X    ∂xi u¯ νi (t, x) + d0 (t, x)u, (Robin)   i=1

64

Spectral Theory for Parabolic Equations

where (in the Neumann or Robin cases) (¯ ν1 , . . . , ν¯N ) is a (in general time dependent) vector field on ∂D pointing out of D. First of all, if both the domain D and the coefficients PN are sufficiently smooth and, in the Neumann or Robin cases, ν¯i (t, x) = j=1 aji (t, x)νj (x), 1 ≤ i ≤ N , (that is, the derivative is conormal), then (2.6.1)+(2.6.2) can be written in the divergence form and then the results in Section 2.5 apply. In general, a proper notion of solutions of (2.6.1)+(2.6.2) is strong solutions. Roughly speaking, a function u is a strong solution of (2.6.1)+(2.6.2) on ¯ is such that (2.6.1) holds (s, t) × D if u ∈ Wp1,2 ((s, t) × D) ∩ C([s, t] × D) (Lebesgue-) almost everywhere and (2.6.2) holds everywhere (see [59], [61], [73]). For the Dirichlet case, under the additional assumption that the coefficients aij are continuous on R × D, it is proved in [61, Proposition 5.4] that for any ¯ satisfying the boundary conditions, (2.6.1)+(2.6.2) has a unique u0 ∈ C(D) strong solution with initial condition u(s) = u0 (see also [73, Theorem 7.17] about the existence and uniqueness of solutions). We refer the reader to [61], [73], and references therein for various properties of strong solutions of (2.6.1)+(2.6.2), for example, maximum principle, a priori estimates, weak Harnack inequality, etc. For the Neumann or Robin boundary condition case, we do not have all the results as in the Dirichlet case (see [59]). But many important properties of strong solutions in Dirichlet case still hold (see [59], [73], etc.).

Chapter 3 Spectral Theory in the General Setting

In this chapter, we introduce the definitions of principal spectrum and principal Lyapunov exponents and exponential separation for a family of general parabolic equations and present their basic properties. We also present a multiplicative ergodic theorem for a family of general parabolic equations. This chapter is organized as follows. In Section 3.1 we introduce the definitions of principal spectrum and Lyapunov exponents of (2.0.1)+(2.0.2) and study their basic properties. We introduce the definition of exponential separation and investigate relevant basic properties in Section 3.2. The existence of exponential separation is explored in Section 3.3. In Section 3.4 we present a multiplicative ergodic theorem. Special properties for a family of general smooth parabolic equations are discussed in Section 3.5. Some remarks on parabolic equations in nondivergence form are given in Section 3.6. This chapter ends up with an appendix on parabolic equations on one-dimensional space domain.

3.1

Principal Spectrum and Principal Lyapunov Exponents: Definitions and Properties

In the present section we introduce the definitions of principal spectrum and Lyapunov exponents of (2.0.1)+(2.0.2), or of Π (see (2.2.6)), and study their basic properties. The standing assumption throughout the present section is that (2.0.1)+ (2.0.2) satisfies (A2-1)–(A2-3). Π shall denote the topological linear skewproduct semiflow generated on L2 (D) × Y by (2.0.1)+(2.0.2). Recall that, for a ∈ Y and t > 0, kUa (t, 0)k denotes the L(L2 (D))-norm of the linear operator Ua (t, 0). We introduce now a norm-like concept. For a ∈ Y and t > 0 let kUa (t, 0)k+ := sup{ kUa (t, 0)u0 k : u0 ∈ L2 (D)+ , ku0 k = 1 }.

65

66

Spectral Theory for Parabolic Equations

LEMMA 3.1.1 For any a ∈ Y and any t > 0 one has kUa (t, 0)k+ = kUa (t, 0)k. PROOF The inequality kUa (t, 0)k+ ≤ kUa (t, 0)k is obvious. To prove the other inequality, notice that any u0 ∈ L2 (D) can be represented as − + − u0 = u+ 0 − u0 , where u0 (x) = max {u0 (x), 0} for a.e. x ∈ D and u0 (x) = + − max {−u0 (x), 0} for a.e. x ∈ D. Notice that for |u0 | := u0 + u0 one has k |u0 | k = ku0 k. The inequalities − + − |Ua (t, 0)u0 | = |Ua (t, 0)u+ 0 − Ua (t, 0)u0 | ≤ |Ua (t, 0)u0 | + |Ua (t, 0)u0 | − = Ua (t, 0)u+ 0 + Ua (t, 0)u0 = Ua (t, 0)|u0 |

give us, after imposing the norms, the desired inequality. From now on until the end of Chapter 3 let Y0 be a nonempty compact connected invariant subset of Y . DEFINITION 3.1.1 (Principal resolvent) A real number λ belongs to the principal resolvent of Π over Y0 , denoted by ρ(Y0 ), if either of the following conditions holds: • There are > 0 and M ≥ 1 such that kUa (t, 0)k ≤ M e(λ−)t

for t > 0 and a ∈ Y0

(such λ are said to belong to the upper principal resolvent, denoted by ρ+ (Y0 )), • There are > 0 and M ∈ (0, 1] such that kUa (t, 0)k ≥ M e(λ+)t

for t > 0 and a ∈ Y0

(such λ are said to belong to the lower principal resolvent, denoted by ρ− (Y0 )). In view of Lemma 3.1.1, in the above inequalities the k·k-norms can be replaced with k·k+ -“norms,” with the same M and . DEFINITION 3.1.2 (Principal spectrum) The principal spectrum of the topological linear skew-product semiflow Π over Y0 , denoted by Σ(Y0 ), equals the complement in R of the principal resolvent of Π over Y0 . To study the basic properties of Σ(Y0 ), we first prove some auxiliary results.

3. Spectral Theory in the General Setting

67

LEMMA 3.1.2 (1) For any t0 > 0 there is K1 = K1 (t0 ) ≥ 1 such that kUa (t, 0)k ≤ K1 for all a ∈ Y0 and all t ∈ [0, t0 ]. (2) For any t0 > 0 there is K2 = K2 (t0 ) > 0 such that kUa (t, 0)k ≥ K2 for all a ∈ Y0 and all t ∈ [0, t0 ]. PROOF Part (1) is a consequence of the L2 –L2 estimates (Proposition 2.2.2). To prove (2), notice that by Proposition 2.2.9(2), kUa (t, 0)1k > 0 for all a ∈ Y0 and t > 0, where 1 is identified with the function constantly equal to one. Since Y0 × [t0 /2, t0 ] is compact, Proposition 2.2.12 implies that the set { kUa (t, 0)1k : a ∈ Y0 , t ∈ [t0 /2, t0 ] } is bounded away from zero. Hence there is M1 > 0 such that kUa (t, 0)k ≥ M1

for t ∈ [t0 /2, t0 ],

a ∈ Y0 .

Now, for any 0 < t < t0 /2, a ∈ Y0 , and u0 ∈ L2 (D) with ku0 k = 1, kUa·(−t0 /2) (t + t0 /2, 0)u0 k ≤ kUa·(−t0 /2) (t + t0 /2, t0 /2)k · kUa·(−t0 /2) (t0 /2, 0)u0 k ≤ K1 kUa·(−t0 /2) (t + t0 /2, t0 /2)k

(by Part (1))

= K1 kUa (t, 0)k. This implies that M1 ≤ kUa·(−t0 /2) (t + t0 /2, 0)k ≤ K1 kUa (t, 0)k for any a ∈ Y0 and 0 < t < t0 /2. Part (2) then follows with K2 = min{M1 , M1 /K1 } = M1 /K1 . LEMMA 3.1.3 A real number λ belongs to the lower principal resolvent if and only if for any ˜ > 0 such that δ0 > 0 there are > 0 and M ˜ e(λ+)t kUa (t, 0)k ≥ M

for t ≥ δ0 and a ∈ Y0 .

PROOF The “only if” part follows from Definition 3.1.1 in a straightforward way. The “if” part follows from Lemma 3.1.2(2). LEMMA 3.1.4 There exist δ1 > 0, M1 > 0, and a real λ such that kUa (t, 0)k ≥ M1 eλt for all a ∈ Y0 and all t ≥ δ1 .

68

Spectral Theory for Parabolic Equations 0

PROOF Pick δ 0 > 0 sufficiently small that Dδ := { x ∈ D : dist(x, ∂D) √> δ 0 } is a nonempty bounded domain. Further, put t0 := (δ 0 )2 and δ := δ 0 / 2. It follows from the interior Harnack inequality (Proposition 2.2.8(2)) that there is Cδ > 0 such that (Ua (δ 2 , 0)1)(y) ≤ Cδ · (Ua (t, 0)1)(x) 0

for any a ∈ Y0 , any t ∈ [2δ 2 , 3δ 2 ], and any x, y ∈ Dδ . Without loss of generality, we assume that Cδ > 1. By Proposition 2.2.9(2), 0

sup{ (Ua (δ 2 , 0)1)(x) : x ∈ Dδ } =: m(a) > 0. Then by the arguments of Proposition 2.2.5, inf{ m(a) : a ∈ Y0 } := m > 0. Consequently 0

inf{ (Ua (t, 0)1)(x) : x ∈ Dδ } ≥

m m(a) ≥ Cδ Cδ

for any a ∈ Y0 and any t ∈ [2δ 2 , 3δ 2 ]. Repeating the application of the interior Harnack inequality (Proposition 2.2.8(2)), we obtain that Cδk−1 (Ua (t, 0)1)(x) ≥ (Ua (δ 2 , 0)1)(y) 0

for any a ∈ Y , t ∈ [kδ 2 , (k +1)δ 2 ], k = 2, 3, . . . , and x, y ∈ Dδ . It then follows that 0 m inf{ (Ua (t, 0)1)(x) : x ∈ Dδ } ≥ (Cδ )k−1 for any a ∈ Y0 and any t ∈ [kδ 2 , (k + 1)δ 2 ], k = 2, 3, . . . . Thus the statement 0 holds with δ1 = 2δ 2 , M1 = mCδ |Dδ |1/2 , and λ = − ln(Cδ )/δ 2 . We start now to investigate the properties of the principal spectrum Σ(Y0 ). First of all, we have THEOREM 3.1.1 The principal spectrum of Π over Y0 is a compact nonempty interval [λmin , λmax ]. PROOF We prove first that the upper principal resolvent ρ+ (Y0 ) is nonempty. Indeed, by the L2 –L2 estimates (Proposition 2.2.2), there are M > 0 and γ > 0 such that kUa (t, 0)k ≤ M eγt for all a ∈ Y0 and t > 0, hence γ + 1 ∈ ρ+ (Y0 ). Further, ρ+ (Y0 ) is a right-unbounded open interval (λmax , ∞).

3. Spectral Theory in the General Setting

69

The lower principal resolvent ρ− (Y0 ) is nonempty, too, since it contains, by Lemma 3.1.4, the real number λ − 1. Consequently, as ρ− (Y0 ) ∪ ρ+ (Y0 ) = ρ(Y0 ) and ρ− (Y0 ) ∩ ρ+ (Y0 ) = ∅, one has Σ(Y0 ) = R \ ρ(Y0 ) = [λmin , λmax ]. The next theorem gives a characterization of the principal spectrum of Π. THEOREM 3.1.2 ∞ ∞ (1) For any sequence (a(n) )∞ n=1 ⊂ Y0 and any real sequences (tn )n=1 , (sn )n=1 such that tn − sn → ∞ as n → ∞ there holds

λmin ≤ lim inf n→∞

ln kUa(n) (tn , sn )k ln kUa(n) (tn , sn )k ≤ lim sup ≤ λmax . tn − sn tn − sn n→∞

∞ (2A) There exist a sequence (a(n,1) )∞ n=1 ⊂ Y0 and a sequence (tn,1 )n=1 ⊂ (0, ∞) such that tn,1 → ∞ as n → ∞, and

lim

n→∞

ln kUa(n,1) (tn,1 , 0)k = λmin . tn,1

∞ (2B) There exist a sequence (a(n,2) )∞ n=1 ⊂ Y0 and a sequence (tn,2 )n=1 ⊂ (0, ∞) such that tn,2 → ∞ as n → ∞, and

lim

n→∞

ln kUa(n,2) (tn,2 , 0)k = λmax . tn,2

PROOF Part (1) is a direct consequence of the definition of the principal spectrum. To prove (2A), notice that, since λmin ∈ / ρ− (Y0 ), it follows from Lemma 3.1.3 (with δ0 = 1) that for each n ∈ N there are a(n,1) ∈ Y0 and tn,1 > 0 such that kUa(n,1) (tn,1 , 0)k <

1 n

exp ((λmin + n1 )tn,1 ).

We claim that limn→∞ tn,1 = ∞. If not, there is a bounded subsequence (tnk ,1 )∞ k=1 , nk → ∞ as k → ∞. It follows that kUa(nk ,1) (tnk ,1 , 0)k → 0 as k → ∞, which contradicts Lemma 3.1.2(2). Thus we have lim sup n→∞

ln kUa(n,1) (tn,1 , 0)k ≤ λmin , tn,1

which together with Part (1) gives the desired result. To prove (2B), notice that, since λmax ∈ / ρ+ (Y0 ), it follows from Definition 3.1.1 that for each n ∈ N there are a(n,2) ∈ Y0 and tn,2 > 0 such that kUa(n,2) (tn,2 , 0)k > n exp ((λmax − n1 )tn,2 ).

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Spectral Theory for Parabolic Equations

We claim that limn→∞ tn,2 = ∞. If not, there is a bounded subsequence (tnk ,2 )∞ k=1 , nk → ∞ as k → ∞. It follows that kUa(nk ,2) (tnk ,2 , 0)k → ∞ as k → ∞, which contradicts Lemma 3.1.2(1). Thus we have

lim inf n→∞

ln kUa(n,2) (tn,2 , 0)k ≥ λmax , tn,2

which together with Part (1) gives the desired result. Recall that for any a ∈ Y we write a = (aij , ai , bi , c0 , d0 ).

THEOREM 3.1.3 Assume that for each a ∈ Y0 there holds: ai (t, x) = bi (t, x) = 0 for a.e. (t, x) ∈ R×D, and c0 (t, x) ≤ 0 for a.e. (t, x) ∈ R×D. Then Σ(Y0 ) ⊂ (−∞, 0]. PROOF Fix a ∈ Y0 and u0 ∈ L2 (D) with ku0 k = 1, and put u(t, x) := (Ua (t, 0)u0 )(x). It follows from Proposition 2.1.4 that

ku(t, ·)k2 − ku(0, ·)k2 = −2

Z

t

Ba (τ, u(τ, ·), u(τ, ·)) dτ 0

Z tZ ≤ −2 0

N X

aij (τ, x)∂xj u(τ, x)∂xj u(τ, x) dx dτ ≤ 0

D i,j=1

for any t > 0. Consequently, kUa (t, 0)u0 k ≤ ku0 k = 1 for all t > 0. Therefore (0, ∞) ⊂ ρ+ (Y0 ). In the case of the Dirichlet boundary conditions more can be said.

THEOREM 3.1.4 In the case of the Dirichlet boundary conditions, assume that for each a ∈ Y0 there holds: ai (t, x) = bi (t, x) = 0 for a.e. (t, x) ∈ R × D, and c0 (t, x) ≤ 0 for a.e. (t, x) ∈ R × D. Then λmax (Y0 ) < 0.

PROOF It follows by the Poincar´e inequality (see [39, Theorem 3 in ˚ 1 (D). Section 5.6]) that there is α1 > 0 such that kuk ≤ α1 k∇uk for any u ∈ W 2

3. Spectral Theory in the General Setting

71

Starting as in the proof of Theorem 3.1.3 we estimate Z t Ba (τ, u(τ, ·), u(τ, ·)) dτ ku(t, ·)k2 −ku(0, ·)k2 = −2 0

≤ −2

Z tZ X N 0

aij (τ, x)∂xj u(τ, x)∂xj u(τ, x) dx dτ

D i,j=1

Z Z t by (A2-1) −2α0 t ku(τ, ·)k2 dτ. ≤ −2α0 k∇u(τ, ·)k2 dτ ≤ (α1 )2 0 0 An application of the regular Gronwall inequality gives that kUa (t, 0)k ≤ e−λ0 t for all t ≥ 0, where λ0 := α0 /α12 > 0. Consequently, [−λ0 , ∞) ⊂ ρ+ (Y0 ) and λmax ≤ −λ0 . Assume that µ is an invariant ergodic Borel probability measure for the topological flow σ on Y0 (see Lemma 1.2.7 for the existence of invariant ergodic measures for (Y0 , σ)). We have THEOREM 3.1.5 There exist a Borel set Y1 ⊂ Y0 with µ(Y1 ) = 1 and a real number λ(µ) such that ln kUa (t, 0)k lim = λ(µ) t→∞ t for all a ∈ Y1 . PROOF We will prove the theorem by applying Kingman’s subadditive ergodic theorem. We define a sequence of functions fn : Y0 → R , n = 1, 2, 3, . . . , as fn (a) := ln kUa (n, 0)k,

n ∈ N, a ∈ Y0 .

For each n ∈ N the function fn is well-defined and bounded (by Lemma 3.1.2). Further, there holds fn+m (a) ≤ fn (a) + fm (σn a),

n, m ∈ N, a ∈ Y0 .

For each n ∈ N the function [ Y0 3 a 7→ kUa (n, 0)k ∈ (0, ∞) ] is easily seen to be lower semicontinuous. Therefore the functions fn are lower semicontinuous, hence (B(Y0 ), B(R))-measurable. Observe that by Lemma 3.1.4 there are M > 0 and λ ∈ R such that fn (a) > ln M + λn for any n ∈ N and any a ∈ Y0 . Consequently, n1Z o inf fn dµ : n ∈ N > −∞. n Y0

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We are now in a position to apply the subadditive ergodic theorem (see, e.g., [67, Theorem 5.3 in Chapter 1]) to conclude that there are a Borel set Y˜0 ⊂ Y0 ˜ : Y˜0 → R such that with µ(Y˜0 ) = 1 and a (B(Y0 ), B(R))-measurable function λ fn (a) n

˜ = λ(a) for any a ∈ Y˜0 , R R ˜ ∈ L1 ((Y0 , B(Y0 ), µ)), and ˜ dµ = inf 1 (ii) λ λ n Y0 fn dµ : n ∈ N , Y0 (i) limn→∞

˜ ˜ n a) for any a ∈ Y˜0 and any n ∈ N. (iii) λ(a) = λ(σ T Put Y˜1 := k∈Z σk (Y˜0 ). Y˜1 ⊂ Y˜0 is a Borel set with µ(Y˜1 ) = 1. We claim that for any s ∈ R and any a ∈ Y˜1 there holds lim

n→∞

ln kUa·s (n, 0)k ˜ = λ(a). n

By (i), (ii) and the construction of Y˜1 , the above equality holds for any s ∈ Z and any a ∈ Y˜1 . Let s ∈ R and let for the moment a be any member of Y0 . We estimate kUa·s (n, 0)k = kUa (n + s, s)k ≤ kUa (n + s, n + bsc)k · kUa (n + bsc, bsc + 1)k · kUa (bsc + 1, s)k for n = 2, 3, . . . . Since the first and the third term on the right-hand side are bounded above (by Lemma 3.1.2(1)), we have lim sup n→∞

ln kUa·(bsc+1) (n, 0)k ln kUa·s (n, 0)k ≤ lim sup . n n n→∞

Further, we estimate kUa (n + bsc + 1, bsc)k ≤ kUa (n + bsc + 1, n + s)k · kUa (n + s, s)k · kUa (s, bsc)k for n = 1, 2, . . . . Since the first and the third term on the right-hand side are bounded above (by Lemma 3.1.2(1)), we have lim inf n→∞

ln kUa·bsc (n, 0)k ln kUa·s (n, 0)k ≥ lim inf . n→∞ n n

Let now a ∈ Y˜1 . There holds ln kUa·bsc (n, 0)k ln kUa·s (n, 0)k ˜ λ(a) = lim ≤ lim inf n→∞ n→∞ n n ln kUa·(bsc+1) (n, 0)k ˜ ln kUa·s (n, 0)k ≤ lim sup ≤ lim = λ(a), n→∞ n n n→∞ which proves the claim.

3. Spectral Theory in the General Setting Next we show that lim

t→∞

73

ln kUa (t, 0)k ˜ = λ(a) t

for a ∈ Y˜1 . This follows from the estimates kUa (t, 0)k ≤ kUa (t, btc)k · kUa (btc, 0)k and kUa (btc + 1, 0)k ≤ kUa (btc + 1, t)k · kUa (t, 0)k and from Lemma 3.1.2(1). Finally we prove that there is a Borel set Y1 ⊂ Y0 with µ(Y1 ) = 1 such that ˜ λ(a) = const on Y1 . Let λ± (a) be defined as follows: λ+ (a) := lim sup n→∞

ln kUa (n, 0)k n

and λ− (a) := lim inf n→∞

ln kUa (n, 0)k . n

By the (B(Y0 ), B(R))-measurability of [ a 7→ fn (a) = kUa (n, 0)k ], both λ+ and λ− are (B(Y0 ), B(R))-measurable. From the subadditivity and Lemma 3.1.2(1) it follows that λ+ , consequently λ− , are bounded above. Lemma 3.1.4 implies that λ− , consequently λ+ , are bounded below. Therefore λ+ and λ− belong to L1 ((Y0 , B(Y0 ), µ)). It follows from the Birkhoff Ergodic Theorem (Lemma 1.2.6) that there is a Borel set Y2 ⊂ Y0 with µ(Y2 ) = 1 such that for each a ∈ Y2 , Z Z 1 t + lim λ (a · s) ds = λ+ dµ t→∞ t 0 Y0 and 1 t→∞ t

Z

t

lim

λ− (a · s) ds =

Z

0

λ− dµ.

Y0

Observe that for any a ∈ Y˜1 and s ∈ R, ˜ λ+ (a · s) = λ− (a · s) = λ(a). This implies that ˜ λ(a) =

Z Y0

λ+ dµ =

Z

λ− dµ

Y0

for any a ∈ Y1 := Y˜1 ∩ Y2 . The theorem is thus proved. DEFINITION 3.1.3 (Principal Lyapunov exponent) λ(µ) as defined above is called the principal Lyapunov exponent of Π for the ergodic invariant measure µ. THEOREM 3.1.6 For any ergodic µ supported on Y0 the principal Lyapunov exponent λ(µ) belongs to the principal spectrum [λmin , λmax ] of Π on Y0 .

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PROOF Suppose to the contrary that λ(µ) < λmin for some ergodic µ supported on Y0 . It follows by definition that there is > 0 such that lim inf t→∞ (1/t) ln kUa (t, 0)k ≥ λ(µ) + for all a ∈ Y0 , whereas Theorem 3.1.5 establishes the existence of a ˜ ∈ Y0 with limt→∞ (1/t) ln kUa˜ (t, 0)k = λ(µ). The case λ(µ) > λmax is excluded in a similar way.

3.2

Exponential Separation: Definitions and Basic Properties

In the present section we introduce the definitions of exponential separation for (2.0.1)+(2.0.2), or for Π, as well as show basic properties of principal spectrum and principal Lyapunov exponents under the assumption that exponential separation holds. The standing assumption throughout the present section is that (2.0.1)+ (2.0.2) satisfies (A2-1)–(A2-3). Π shall denote the topological linear skewproduct semiflow generated on L2 (D) × Y by (2.0.1)+(2.0.2) (see (2.2.6)). Let Y0 be a closed connected invariant subset of Y . By a one-dimensional (trivial ) subbundle X (1) of L2 (D) × Y0 we understand a set { (rw(a), a) : r ∈ R, a ∈ Y0 }, where w : Y0 → L2 (D) is a continuous mapping with the property that kw(a)k = 1 for all a ∈ Y0 . For a ∈ Y0 we call the one-dimensional vector subspace span{w(a)} =: X (1) (a) the fiber of X (1) over a. A onecodimensional (trivial ) subbundle X (2) is usually defined as a family of onecodimensional subspaces continuously depending on a ∈ Y0 . For our purposes it suffices to define it as a set { (v, a) ∈ L2 (D) : hv, w∗ (a)i = 0, a ∈ Y0 }, where w∗ : Y0 → L2 (D)∗ is a continuous mapping satisfying kw∗ (a)k = 1 for all a ∈ Y0 . For a ∈ Y0 we call the one-codimensional vector subspace { v ∈ L2 (D) : hv, w∗ (a)i = 0 } =: X (2) (a) the fiber of X (2) over a. As we will consider only trivial subbundles, we drop the adjective “trivial” from now on. A one-dimensional subbundle X (1) and a one-codimensional subbundle X (2) are complementary if X (1) (a) ⊕ X (2) (a) = L2 (D) for each a ∈ Y0 , where ⊕ denotes direct sum in the Banach space sense (X (1) (a) and X (2) (a) need not be orthogonal). Notice that X (1) and X (2) are complementary if and only if hw(a), w∗ (a)i 6= 0 for all a ∈ Y0 . If X (1) and X (2) are complementary, we write X (1) ⊕ X (2) = L2 (D) × Y0 . A one-dimensional subbundle X (1) is said to be invariant (under Π) if for each a ∈ Y0 and each t > 0 there is a real ra (t) such that Ua (t, 0)w(a) = ra (t)w(a · t). This is equivalent to saying that Ua (t, 0)X (1) (a) ⊂ X (1) (a · t) for each a ∈ Y0 and each t > 0. A one-codimensional subbundle X (2) is said to be invariant (under Π) if for each a ∈ Y0 and each t < 0 there is a real ra∗ (t) such that Ua∗ (t, 0)w∗ (a) = ra∗ (t)w∗ (a · t). This is equivalent to saying that Ua (t, 0)X (2) (a) ⊂ X (2) (a · t)

3. Spectral Theory in the General Setting

75

for each a ∈ Y0 and each t > 0. For a given one-dimensional subbundle X (1) denote by (X (1) )∗ the onecodimensional subbundle given by (X (1) )∗ (a) := { v ∈ L2 (D) : hw(a), vi = 0 } = X (1) (a)⊥ , a ∈ Y0 . It is easy to see that X (1) is invariant under Π if and only if (X (1) )∗ is invariant under Π∗ . Similarly, for a one-codimensional subbundle X (2) denote by (X (2) )∗ the one-dimensional subbundle given by (X (2) )∗ (a) := span{w∗ (a)}. It is easy to see that X (2) is invariant under Π if and only if (X (2) )∗ is invariant under Π∗ . For more on subbundles, see [97]. DEFINITION 3.2.1 (Exponential separation) Let Y0 be a compact connected invariant subset of Y . We say that Π admits an exponential separation with separating exponent γ0 > 0 over Y0 if there are an invariant onedimensional subbundle X1 of L2 (D)×Y0 with fibers X1 (a) = span{w(a)}, and an invariant complementary one-codimensional subbundle X2 of L2 (D) × Y0 with fibers X2 (a) = { v ∈ L2 (D) : hv, w∗ (a)i = 0 } having the following properties: (i) w(a) ∈ L2 (D)+ for all a ∈ Y0 , (ii) X2 (a) ∩ L2 (D)+ = {0} for all a ∈ Y0 , (iii) There is M ≥ 1 such that for any a ∈ Y0 and any v ∈ X2 (a) with kvk = 1, kUa (t, 0)vk ≤ M e−γ0 t kUa (t, 0)w(a)k

(t > 0).

As a consequence of the invariance of X1 and of Proposition 2.2.9(2) we have the following stronger property. Ua (t, 0)X (1) (a) = X (1) (a · t),

a ∈ Y0 , t > 0.

(3.2.1)

Moreover, for each a ∈ Y0 and each t > 0 there is ra (t) > 0 such that Ua (t; 0)w(a) = ra (t)w(a·t). Since ra (t) = kUa (t, 0)w(a)k, by Proposition 2.2.13, the function [ Y0 × (0, ∞) 3 (a, t) 7→ ra (t) ∈ (0, ∞) ] is continuous. We claim that w∗ (a) can be taken to belong to L2 (D)+ for each a ∈ Y0 . If w∗ (a) ∈ −L2 (D)+ for each a ∈ Y0 then we replace w∗ (a) with −w∗ (a). Suppose to the contrary that there is a ∈ Y0 such that w∗ (a) ∈ / L2 (D)+ ∪ + −L2 (D) . This means that the Lebesgue measure of the set D1 := { x ∈ D : w∗ (a)(x) > 0 } is positive, as well as the Lebesgue measure of the set D2 := {R x ∈ D : w∗ (a)(x) < 0 } is positive. Let v be R a simple function taking value ( D1 w∗ (a)(x) dx)−1 on D1 , taking value −( D2 w∗ (a)(x) dx)−1 on D2 , and equal to zero elsewhere. Clearly v ∈ L2 (D)+ \ {0} and hv, w∗ (a)i = 0, that is, v ∈ X2 (a), which contradicts (ii). As the dual Π∗ of the topological linear skew-product semiflow Π is generated by the adjoint equation (2.3.1)+(2.3.2), we have that for each a ∈ Y0

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and each t < 0 there is ra∗ (t) > 0 such that Ua∗ (t, 0)w∗ (a) = ra∗ (t)w∗ (a · t). Further, the function [ Y0 × (0, ∞) 3 (a, t) 7→ ra∗ (t) ∈ (0, ∞) ] is continuous. LEMMA 3.2.1 Let Π admit an exponential separation over a compact connected invariant subset Y0 ⊂ Y . For any sequence (a(n) ) ⊂ Y0 and any positive real sequence (tn ), if limn→∞ a(n) = a and limn→∞ tn = t, where t ≥ 0, then Ua(n) (tn , 0)w(a(n) ) converge in L2 (D) to Ua (t, 0)w(a). PROOF

Observe that Ua(n) (tn , 0)w(a(n) ) =

Ua(n) ·(−1) (tn + 1, 0)w(a(n) · (−1)) , ra(n) ·(−1) (1)

which converges in L2 (D), by Proposition 2.2.12, to Ua·(−1) (t + 1, 0)w(a · (−1)) = Ua (t, 0)w(a). ra·(−1) (1)

Denote by Π|X1 the restriction of Π to the subbundle X1 . We extend Π|X1 to negative times in the following way: ( ((Ua·t (−t, 0)|X1 (a·t) )−1 w(a), a · t), t < 0 (Π|X1 )t (w(a), a) := (Ua (t, 0)w(a), a · t), t ≥ 0, or, in view of the invariance of X1 :  w(a · t)  ,a · t , kUa·t (−t, 0)w(a · t)k (Π|X1 )t (w(a), a) =  (kUa (t, 0)w(a)kw(a · t), a · t),

t<0 t ≥ 0.

One has (Π|X1 )0 = IdX1 and (Π|X1 )s ◦ (Π|X1 )t = (Π|X1 )s+t

for any s, t ∈ R.

Also, from Lemma 3.2.1 it follows that the mapping [ R × X1 3 (t, (v, a)) 7→ (Π|X1 )t (v, a) ∈ X1 ] is continuous. Such an object is called a topological linear skew-product flow on the bundle X1 covering the topological flow σ. For a theory of topological linear skew-product flows on (finite-dimensional) vector bundles see [65]. For a ∈ Y0 fixed, u ∈ L2,loc ((−∞, ∞), V ) is an entire positive weak solution of (2.0.1)a +(2.0.2)a if for any s < t, u|[s,t] is a weak solution of

3. Spectral Theory in the General Setting

77

(2.0.1)a +(2.0.2)a and for any t ∈ R, u(t) ∈ L2 (D)+ \ {0}. Note that the mapping defined as ( (Ua·t (−t, 0)|X1 (a·t) )−1 w(a), t < 0 va (t) := Ua (t, 0)w(a), t≥0 (that is, the projection onto the first axis of [ (−∞, ∞) 3 t 7→ (Π|X1 )t (w(a), a) ]) is an entire positive weak solution of the problem (2.0.1)a +(2.0.2)a . For a ∈ Y0 , denote by P1 (a) the projection of L2 (D) on X1 (a) along X1 (a), and by P2 (a) the projection of L2 (D) on X2 (a) along X1 (a), P2 (a) = IdL2 (D) −P1 (a). Notice that P1 (a)u =

hu, w∗ (a)i w(a), hw(a), w∗ (a)i

a ∈ Y0 , u ∈ L2 (D).

(3.2.2)

The mappings [ Y0 3 a 7→ P1 (a) ∈ L(L2 (D)) ] and [ Y0 3 a 7→ P2 (a) ∈ L(L2 (D)) ] are continuous. Indeed, notice that for any two a(1) , a(2) ∈ Y0 and any u0 ∈ L2 (D) the following estimate holds: kP1 (a(1) )u0 − P1 (a(2) )u0 k 1 1 |hu0 , w∗ (a(1) )i| kw(a(1) )k ≤ − (1) ∗ (1) (2) ∗ (2) hw(a ), w (a )i hw(a ), w (a )i +

|hu0 , w∗ (a(1) ) − w∗ (a(2) )i| kw(a(1) )k hw(a(2) ), w∗ (a(2) )i

+

|hu0 , w∗ (a(2) )i| kw(a(1) ) − w(a(2) )k, hw(a(2) ), w∗ (a(2) )i

which reduces the issue of the continuity of the former mapping to the continuity of the mappings w, w∗ : Y0 → L2 (D). The continuity of the latter mapping follows by the formula P2 (a) = IdL2 (D) −P1 (a). Recall that the dual topological linear skew-product semiflow {Π∗ (t)}t≥0 (denoted also by Π∗ ) is defined as Π∗t (v ∗ , a) := (Ua∗ (−t, 0)v ∗ , a · (−t)),

t ≥ 0, v ∗ ∈ L2 (D), a ∈ Y,

where Ua∗ (−t, 0) = (Ua·(−t) (t, 0))∗ ,

a ∈ Y, t ≥ 0.

THEOREM 3.2.1 A topological linear skew-product semiflow Π admits an exponential separation over Y0 , with a one-dimensional subbundle X1 and a one-codimensional subbundle X2 , if and only if its dual Π∗ admits an exponential separation over Y0 , with a one-dimensional subbundle X2∗ and a one-codimensional subbundle X1∗ . Separating exponents can be chosen to be equal.

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PROOF

Recall that, for any a ∈ Y0 ,

v1∗ ∈ X1∗ (a)

if and only if

hv1 , v1∗ i = 0 for each v1 ∈ X1 (a),

and X2∗ (a) = span{w∗ (a)}, where w∗ (a) ∈ L2 (D)+ for any a ∈ Y0 . Also, X1∗ (a) is, for any a ∈ Y0 , the subspace orthogonal to w(a) ∈ L2 (D)+ . The facts that X1∗ is a one-codimensional subbundle invariant under Π∗ and that X2∗ is a one-dimensional subbundle invariant under Π∗ , follow from the respective definitions and from the invariance of X1 and X2 under Π. We estimate first the norms of the restrictions of Ua∗ (−t, 0) to X2∗ . kUa∗ (−t, 0)w∗ (a)k = sup{ |hu, Ua∗ (−t, 0)w∗ (a)i| : kuk = 1 } = sup{ |hP1 (a)Ua·(−t) (t, 0)u, w∗ (a)i| : kuk = 1 } = sup{ |hUa·(−t) (t, 0)P1 (a · (−t))u, w∗ (a)i| : kuk = 1 } ≥ K1 kUa·(−t) (t, 0)w(a · (−t))k, where K1 := inf{ hw(˜ a), w∗ (˜ a)i : a ˜ ∈ Y0 } > 0. Next we estimate the norms of the restrictions of Ua∗ (−t, 0) to X1∗ . Let v1∗ ∈ X1∗ (a) with kv1∗ k = 1. kUa∗ (−t, 0)v1∗ k = sup{ |hu, Ua∗ (−t, 0)v1∗ i| : kuk = 1 } = sup{ |hP2 (a)Ua·(−t) (t, 0)u, v1∗ i| : kuk = 1 } = sup{ |hUa·(−t) (t, 0)P2 (a · (−t))u, v1∗ i| : kuk = 1 } ≤ K2 sup{ kUa·(−t) (t, 0)uk : u ∈ X2 (a · (−t)), kuk = 1 }, where K2 := sup{ kP2 (˜ a)k : a ˜ ∈ Y0 } < ∞. Consequently, K2 kUa·(−t) (t, 0)|X2 (a·(−t)) k kUa∗ (−t, 0)v1∗ k ≤ kUa∗ (−t, 0)w∗ (a)k K1 kUa·(−t) (t, 0)|X1 (a·(−t)) k for any a ∈ Y , t > 0 and v1∗ ∈ X1∗ (a) with v1∗ = 1. By Definition 3.2.1(iii), kUa·(−t) (t, 0)|X2 (a·(−t)) k ≤ M e−γ0 t kUa·(−t) (t, 0)|X1 (a·(−t)) k for any a ∈ Y and any t > 0. This together with the previous display gives a desired result. The reverse implication follows by the observation that (Π∗ )∗ = Π. The following easy result will be needed a couple of times, so we formulate it here.

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79

LEMMA 3.2.2 Assume that Π admits an exponential separation over a compact connected invariant subset Y0 ⊂ Y . Then for each nonzero u0 ∈ L2 (D)+ there exists K > 0 such that for each a ∈ Y0 the inequality kP2 (a)u0 k ≤ KkP1 (a)u0 k holds. PROOF Observe that for each a ∈ Y0 and each nonzero u0 ∈ L2 (D)+ one has P1 (a)u0 6= 0, since otherwise u0 would be in X2 (a), which contradicts the property in Definition 3.2.1(ii). Suppose to the contrary that for each positive integer n there is a(n) ∈ Y0 such that kP1 (a(n) )u0 k < n1 kP2 (a(n) )u0 k. We can choose a subsequence of (a(n) )∞ ˜ ∈ Y0 . But by n=1 converging to some a the continuous dependence of the projections P2 on the base point and the compactness of Y0 , the set { kP2 (a)u0 k : a ∈ Y0 } is bounded, consequently kP1 (˜ a)u0 k = 0, which is impossible. Sometimes we have an “exponential separation” only for the discrete time. For convenience, we introduce DEFINITION 3.2.2 (Exponential separation for discrete time) Let Y0 be a compact connected invariant subset of Y , and let T > 0. Π is said to admit an exponential separation with separating exponent γ00 > 0 for the discrete time T over Y0 if there are a one-dimensional subbundle X1 of L2 (D) × Y0 with fibers X1 (a) = span{w(a)}, and a one-codimensional subbundle X2 of L2 (D)×Y0 with fibers X2 (a) = { v ∈ L2 (D) : hv, w∗ (a)i = 0 } having the following properties: (a) Ua (T, 0)X1 (a) = X1 (a · T ) and Ua (T, 0)X2 (a) ⊂ X2 (a · T ) for all a ∈ Y0 , (b) w(a) ∈ L2 (D)+ for all a ∈ Y0 , (c) X2 (a) ∩ L2 (D)+ = {0} for all a ∈ Y0 , (d) there are M 0 ≥ 1 such that for any a ∈ Y0 and any v ∈ X2 (a) with kvk = 1, 0

kUa (nT, 0)vk ≤ M 0 e−γ0 n kUa (nT, 0)w(a)k

(n = 1, 2, 3, . . . ).

The next result shows that the exponential separation for some discrete time implies exponential separation. THEOREM 3.2.2 Assume that Π admits an exponential separation with separating exponent γ00 for some discrete time T > 0 over a compact connected invariant subset Y0 ⊂ Y . Then Π admits an exponential separation with separating exponent γ0 = γ00 over Y0 .

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PROOF We start by proving that the subbundles X1 and X2 having the above properties are invariant. Without loss of generality, assume that T = 1. Suppose by way of contradiction that X1 is not invariant, that is, there are a ∈ Y0 and τ > 0 such that Ua (τ, 0)w(a) ∈ / X1 (a · τ ). Define a continuous function f : R → R as  kP2 (a · t)Ua (t, 0)w(a)k    kP (a · t)U (t, 0)w(a)k 1 a f (t) :=  kP2 (a · t)Ua (t, btc)w(a · btc)k   kP1 (a · t)Ua (t, btc)w(a · btc)k

for t ≥ 0 for t < 0,

where P1 (·) is as in (3.2.2) and P2 (·) = IdL2 (D) − P1 (·). As, by Proposition 2.2.9(2) and (b), (Ua·s (t + s, s)w(a · s))(x) > 0 for all a ∈ Y , s ∈ R, t > 0 and a.e. x ∈ D, the function f is well defined. By part (a) f (k) = 0 for any integer k. Moreover, from the continuity and the positivity of the mapping [ Y × [0, 1] 3 (a, t) 7→ kP1 (Ua (t, 0)w(a))k ] and from the continuity of [ Y × [0, 1] 3 (a, t) 7→ kP2 (Ua (t, 0)w(a))k ] it follows that f is bounded from above. 0 Take a positive integer n0 so large that M 0 e−γ0 n0 < 1. From (d) it follows 0 that f (t + n) ≤ M 0 e−γ0 n f (t) for all t ∈ R and all n ∈ N. This implies that 0

f (t) ≤ (M 0 e−γ0 n0 )k f (t − n0 k) for all t ∈ R and k ∈ N. It then follows from the nonnegativity and boundedness of f that 0

0 ≤ f (t) ≤ lim sup (M 0 e−γ0 n0 )k f (t − n0 k) = 0 k→∞

for all t ∈ R. Hence f ≡ 0. This contradicts the assumption Ua (a · τ )w(a) ∈ / X1 (a · τ ). Therefore X1 is invariant. The proof of the invariance of X2 goes along the following lines. We prove first a discrete-time analog of Theorem 3.2.1, that is, that Π∗ admits an exponential separation for discrete time T > 0 over a compact invariant Y0 ⊂ Y , with one-dimensional subbundle X2∗ and one-codimensional subbundle X1∗ . The previous paragraph gives that X2∗ is invariant under Π∗ , which is equivalent to X2 being invariant under Π. The property (iii) in Definition 3.2.1 follows from (d) by applying the L2 (D)–L2 (D) estimate in Proposition 2.2.2. THEOREM 3.2.3 Assume that Π admits an exponential separation over Y0 with a one dimensional bundle X1 and a one-codimensional bundle X2 , and admits an ˜ 1 and a oneexponential separation over Y0 with a one-dimensional bundle X ˜ 2 . Then X1 = X ˜ 1 and X2 = X ˜2. codimensional bundle X

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81

PROOF For a ∈ Y0 denote by P1 (a) and P2 (a) the projections corresponding to the exponential separation with X1 and X2 , and by w(a) ˜ the unique ˜ 1 = span{w(a)}. element of L2 (D)+ such that kw(a)k ˜ = 1 and X ˜ Further, we define a function f : Y0 → [0, ∞) as f (a) := kP2 (a)w(a)k/kP ˜ ˜ 1 (a)w(a)k. Since kP1 (a)w(a)k ˜ > 0 (compare the proof of Lemma 3.2.2), f is well defined. The function f is clearly continuous, so it is bounded. The invari˜ 1 together with Definition 3.2.1 (for X1 and X2 ) ance of the subbundle X imply that f (a · t) ≤ M e−γ0 t f (a) for each a ∈ Y and t > 0, consequently f (a) ≤ M e−γ0 t f (a · (−t)) for each a ∈ Y and t > 0, which gives f ≡ 0. We ˜1. have obtained thus X1 = X ∗ By Theorem 3.2.1, Π admits an exponential separation over Y0 with a one-dimensional bundle X2∗ and a one-codimensional bundle X1∗ , as well as ˜∗ admits an exponential separation over Y0 with a one-dimensional bundle X 2 ∗ ˜ . The first part of the proof gives that and a one-codimensional bundle X 1 ˜ ∗ , which implies X2 = X ˜2. X2∗ = X 2 LEMMA 3.2.3 Assume that Π admits an exponential separation over a nonempty compact connected invariant subset Y0 of Y . (1) For each a ∈ Y0 , each u0 ∈ L2 (D) \ X2 (a), ku0 k = 1, and each δ0 > 0 there is M1 = M1 (a, u0 , δ0 ) ∈ (0, 1) such that kUa (t, 0)u0 k ≥ M1 kUa (t, 0)w(a)k for all t ≥ δ0 . (2) There is M2 ≥ 1 such that kUa (t, 0)u0 k ≤ M2 kUa (t, 0)w(a)k for all a ∈ Y0 , all t ≥ 0, and all u0 ∈ L2 (D) with ku0 k = 1. Consequently, for that same M2 ≥ 1, kUa (t, 0)w(a)k ≤ kUa (t, 0)k+ = kUa (t, 0)k ≤ M2 kUa (t, 0)w(a)k for all a ∈ Y0 and all t ≥ 0. PROOF (1) Let P1 (·) and P2 (·) denote the projections for the exponential separation. Fix a ∈ Y0 and u0 ∈ L2 (D) \ X2 (a) with ku0 k = 1. Put K :=

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kP2 (a)u0 k/kP1 (a)u0 k. We estimate kUa (t, 0)u0 k ≥ kP1 (a · t)Ua (t, 0)u0 k − kP2 (a · t)Ua (t, 0)u0 k kP2 (a · t)Ua (t, 0)u0 k kP1 (a · t)Ua (t, 0)u0 k = 1− kP1 (a · t)Ua (t, 0)u0 k kUa (t, 0)P2 (a)u0 k = 1− kUa (t, 0)P1 (a)u0 k kUa (t, 0)P1 (a)u0 k ≥ (1 − M Ke−γ0 t )kUa (t, 0)w(a)k kP1 (a)u0 k

(by Def. 3.2.1(iii))

−γ0 t

≥

1 − M Ke 1+K

kUa (t, 0)w(a)k

(since ku0 k ≤ (1 + K)kP1 (a)u0 k)

for any a ∈ Y0 and any t > 0. Take T > 0 to be such that (1−M Ke−γ0 t )/(1+ K) > 0 for all t ≥ T . If T ≤ δ0 we are done. Assume not. Notice that Ua (t, 0)u0 6= 0 for all t > 0. Indeed, P1 (a)u0 6= 0, consequently P1 (a · t)Ua (t, 0)u0 = Ua (t, 0)P1 (a)u0 6= 0. The equality Ua (t, 0)u0 = 0 for some t > 0 would imply P2 (a · t)Ua (t, 0)u0 = −P1 (a · t)Ua (t, 0)u0 6= 0. But then 0 6= P2 (a·t)Ua (t, 0)u0 ∈ X1 (a·t)∩X2 (a·t), which is impossible. Consequently kUa (t, 0)u0 k > 0 for all t > 0. Since Y0 × [δ0 , T ] is compact, Proposition 2.2.12 implies that the set { kUa (t, 0)u0 k : t ∈ [δ0 , T ] } is bounded away from zero. The set { kUa (t, 0)w(a)k : t ∈ [δ0 , T ] } is clearly bounded, hence the conclusion of Part (1) follows. (2) Put K1 := max{ kP1 (a)k : a ∈ Y0 }, K2 := max{ kP2 (a)k : a ∈ Y0 }. Fix a ∈ Y0 and u0 ∈ L2 (D) with ku0 k = 1. We estimate kUa (t, 0)u0 k ≤ kP1 (a · t)Ua (t, 0)u0 k + kP2 (a · t)Ua (t, 0)u0 k = kUa (t, 0)P1 (a)u0 k + kUa (t, 0)P2 (a)u0 k kUa (t, 0)P2 (a)u0 k = kP1 (a)u0 k + kUa (t, 0)w(a)k kUa (t, 0)w(a)k ≤ (kP1 (a)u0 k + M e−γ0 t kP2 (a)u0 k)kUa (t, 0)w(a)k (by Def. 3.2.1(iii)) ≤ (K1 + K2 M e−γ0 t )kUa (t, 0)w(a)k for any t > 0. LEMMA 3.2.4 Assume that Π admits an exponential separation over a nonempty compact connected invariant subset Y0 of Y . Then for each u0 ∈ L2 (D)+ , ku0 k = 1, and each δ0 > 0 there is M1 = M1 (u0 , δ0 ) ∈ (0, 1) such that kUa (t, 0)u0 k ≥ M1 kUa (t, 0)w(a)k for all a ∈ Y0 and all t ≥ δ0 . PROOF Let P1 (·) and P2 (·) denote the projections for the exponential separation. By Lemma 3.2.2 there is K = K(u0 ) > 0 such that kP2 (a)u0 k ≤

3. Spectral Theory in the General Setting

83

KkP1 (a)u0 k for all a ∈ Y0 . We estimate kUa (t, 0)u0 k ≥ kP1 (a · t)Ua (t, 0)u0 k − kP2 (a · t)Ua (t, 0)u0 k kP2 (a · t)Ua (t, 0)u0 k kP1 (a · t)Ua (t, 0)u0 k = 1− kP1 (a · t)Ua (t, 0)u0 k kUa (t, 0)P2 (a)u0 k = 1− kUa (t, 0)P1 (a)u0 k kUa (t, 0)P1 (a)u0 k ≥ (1 − M Ke−γ0 t )kUa (t, 0)w(a)k kP1 (a)u0 k

(by Def. 3.2.1(iii))

−γ0 t

≥

1 − M Ke 1+K

kUa (t, 0)w(a)k

(since ku0 k ≤ (1 + K)kP1 (a)u0 k)

for any a ∈ Y0 and any t > 0. Take T > 0 to be such that (1−M Ke−γ0 t )/(1+ K) > 0 for all t ≥ T . If T ≤ δ0 we are done. Assume not. Then notice that by Proposition 2.2.9(2), kUa (t, 0)u0 k > 0 for all a ∈ Y0 and t > 0. Since Y0 × [δ0 , T ] is compact, Proposition 2.2.12 implies that the set { kUa (t, 0)u0 k : a ∈ Y0 , t ∈ [δ0 , T ] } is bounded away from zero. The set { kUa (t, 0)w(a)k : a ∈ Y0 , t ∈ [δ0 , T ] } is clearly bounded, hence the conclusion follows. From now on until the end of Section 3.2 we assume that Y0 is a compact connected invariant subset of Y such that Π admits an exponential separation with separating exponent γ0 over Y0 . THEOREM 3.2.4 Let µ be an invariant ergodic Borel probability measure for σ|Y0 . Then there is a Borel set Y1 ⊂ Y0 , µ(Y1 ) = 1, with the following properties. (1) For any a ∈ Y1 and any u0 ∈ L2 (D) \ X2 (a) one has lim

t→∞

ln kUa (t, 0)u0 k = λ(µ). t

(3.2.3)

(2) For any a ∈ Y1 and any u0 ∈ X2 (a) \ {0} one has lim sup t→∞

PROOF such that

ln kUa (t, 0)u0 k ≤ λ(µ) − γ0 . t

(3.2.4)

By Theorem 3.1.5 there is a Borel set Y1 ⊂ Y0 with µ(Y1 ) = 1

ln kUa (t, 0)k = λ(µ) t for all a ∈ Y1 . Fix a ∈ Y0 and u0 ∈ L2 (D)\X2 (a) with ku0 k = 1. Lemma 3.2.3 yields the existence of M1 = M1 (a, u0 , 1) > 0 and M2 > 0 such that lim

t→∞

kUa (t, 0)u0 k ≥ M1 kUa (t, 0)w(a)k ≥

M1 M1 kUa (t, 0)k ≥ kUa (t, 0)u0 k M2 M2

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for all t ≥ 1. Part (1) follows immediately. Part (2) is a consequence of Part (1) and Definition 3.2.1(iii). COROLLARY 3.2.1 Let µ and Y1 be as in Theorem 3.2.4. Then for any a ∈ Y1 , u0 ∈ X2 (a) \ {0} if and only if ln kUa (t, 0)u0 k lim sup < λ(µ). t t→∞ The following result will be extensively used in Chapter 4. LEMMA 3.2.5 ∞ ∞ Let λ ∈ R, (a(n) )∞ n=1 ⊂ Y0 , and (sn )n=1 ⊂ R, (tn )n=1 ⊂ R with tn − sn → ∞. Then the following conditions are equivalent: ln kUa(n) (tn , sn )w(a(n) · sn )k = λ. n→∞ tn − sn

(1) lim

ln kUa(n) (tn , sn )u0 k = λ for any u0 ∈ L2 (D)+ \ {0}. n→∞ tn − sn

(2) lim

ln kUa(n) (tn , sn )k ln kUa(n) (tn , sn )k+ = lim = λ. n→∞ n→∞ tn − sn tn − sn

(3) lim

PROOF Fix u0 ∈ L2 (D)+ with ku0 k = 1. By Lemmas 3.2.4 and 3.2.3(2) there are M1 = M1 (u0 , 1) > 0 and M2 > 0 such that kUa(n) (tn , sn )u0 k ≥ M1 kUa(n) (tn , sn )w(a(n) · sn )k ≥

M1 kU (n) (tn , sn )u0 k M2 a

for any n ∈ N such that tn − sn ≥ 1. This implies the equivalence of (1) and (2). The equivalence of (1) and (3) is a consequence of Lemma 3.2.3(2). LEMMA 3.2.6 (1) λ ∈ R belongs to the upper principal resolvent of Π over Y0 if and only if there are > 0 and M ≥ 1 such that kUa (t, 0)w(a)k ≤ M e(λ−)t

for t > 0 and a ∈ Y0 ,

(2) λ ∈ R belongs to the lower principal resolvent of Π over Y0 if and only if there are > 0 and M ∈ (0, 1) such that kUa (t, 0)w(a)k ≥ M e(λ+)t

for t > 0 and a ∈ Y0 .

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85

PROOF Part (1) is a consequence of Lemma 3.2.3(2). Part (2) is a consequence of Lemma 3.2.3(2) together with Lemma 3.1.3. For the topological linear skew-product flow Π|X1 on the one-dimensional bundle X1 its dynamical spectrum (or the Sacker–Sell spectrum) is defined as the complement of the set of those λ ∈ R for which either of the conditions in Lemma 3.2.6 holds (see [65]). Therefore the principal spectrum of Π over Y0 equals the dynamical spectrum of Π|X1 , which allows us to make use of [65, Theorem 3.3] to prove the following important results. THEOREM 3.2.5 There exist ergodic invariant measures µmin and µmax for σ|Y0 such that λmin = λ(µmin ) and λmax = λ(µmax ). COROLLARY 3.2.2 If (Y0 , {σt }t∈R ) is uniquely ergodic then λmin = λmax . REMARK 3.2.1 When (2.0.1)a +(2.0.2)a is actually time independent or periodic, then the unique (by Corollary 3.2.2) element of the principal spectrum turns out to be the classical principal eigenvalue. THEOREM 3.2.6 Let µ be an invariant ergodic measure for σ|Y0 . Then there is a Borel set Y˜1 ⊂ Y0 , µ(Y˜1 ) = 1, with the property that lim

t→±∞

ln kUa (t, 0)w(a)k = λ(µ). t

(3.2.5)

for any a ∈ Y˜1 , where, for t < 0, Ua (t, 0)w(a) denotes the only element v ∈ X1 (a · t) such that Ua·t (−t, 0)v = w(a). PROOF It follows by an application of [65, Theorem 2.1] to the topological linear skew-product flow Π|X1 on the one-dimensional bundle X1 . The following result shows that in the case of (Y0 , σ) being topologically transitive, when analyzing the principal spectrum/principal Lyapunov exponent one can restrict oneself to considering solutions of (2.0.1)+(2.0.2) with only one parameter value. It will be extensively used in Chapter 4. THEOREM 3.2.7 If Y0 = cl { a(0) · t : t ∈ R } for some a(0) ∈ Y0 , where the closure is taken in the weak-* topology, then

86 (1)

Spectral Theory for Parabolic Equations 0 ∞ 0 0 (i) There are sequences (s0n )∞ n=1 , (tn )n=1 ⊂ R, tn −sn → ∞ as n → ∞, such that

ln kUa(0) (t0n , s0n )w(a(0) · s0n )k n→∞ t0n − s0n ln kUa(0) (t0n , s0n )u0 k ln kUa(0) (t0n , s0n )k = lim = lim 0 0 n→∞ n→∞ tn − sn t0n − s0n

λmin = lim

for each u0 ∈ L2 (D)+ \ {0}, 00 00 00 ∞ (ii) There are sequences (s00n )∞ n=1 , (tn )n=1 ⊂ R, tn −sn → ∞ as n → ∞, such that

ln kUa(0) (t00n , s00n )w(a(0) · s00n )k n→∞ t00n − s00n ln kUa(0) (t00n , s00n )k ln kUa(0) (t00n , s00n )u0 k = lim = lim 00 00 n→∞ n→∞ tn − sn t00n − s00n

λmax = lim

for each u0 ∈ L2 (D)+ \ {0}. (2) For any u0 ∈ L2 (D)+ \ {0} there holds ln kUa(0) (t, s)w(a(0) · s)k t−s→∞ t−s ln kUa(0) (t, s)u0 k ln kUa(0) (t, s)k = lim inf = lim inf t−s→∞ t−s→∞ t−s t−s ln kUa(0) (t, s)k ln kUa(0) (t, s)u0 k ≤ lim sup = lim sup t−s t−s t−s→∞ t−s→∞

λmin = lim inf

= lim sup t−s→∞

ln kUa(0) (t, s)w(a(0) · s)k = λmax . t−s

∞ (3) For each λ ∈ [λmin , λmax ] there are sequences (kn )∞ n=1 , (ln )n=1 ⊂ Z, ln − kn → ∞ as n → ∞, such that

ln kUa(0) (ln , kn )w(a(0) · kn )k n→∞ ln − kn ln kUa(0) (ln , kn )u0 k ln kUa(0) (ln , kn )k = lim = lim n→∞ n→∞ ln − k n ln − kn

λ = lim

for each u0 ∈ L2 (D)+ \ {0}. PROOF (1) We prove only (i), the other part being similar. By Theorem 3.2.5(2), there is an ergodic invariant measure µmin for the topological flow (Y0 , σ) such that λmin equals the principal Lyapunov exponent on Y0 for µmin .

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87

Theorem 3.1.5 and Lemma 3.2.5 provide the existence of a Borel set Y1 ⊂ Y0 with µmin (Y1 ) = 1 such that lim

t→∞

ln kUa˜ (t, 0)w(˜ a)k = λmin t

for any

a ˜ ∈ Y1 .

Fix a ˜ ∈ Y1 . For each n = 1, 2, 3, . . . , take τn ≥ n such that λmin −

ln kUa˜ (τn , 0)w(˜ a)k 1 1 < . < λmin + 2n τn 2n

As Y0 equals the closure of { a(0) · t : t ∈ R }, for any n = 1, 2, 3, . . . we can find s0n ∈ R with the property that a(0) · s0n is so close to a ˜ that | ln kUa˜ (τn , 0)w(˜ a)k − ln kUa(0) ·s0n (τn , 0)w(a · s0n )k | <

τn . 2n

0 ∞ Take t0n := s0n + τn . We have found two sequences (s0n )∞ n=1 , (tn )n=1 ⊂ R, 0 0 tn − sn → ∞ as n → ∞, such that

ln kUa(0) (t0n , s0n )w(a(0) · s0n )k . n→∞ t0n − s0n

λmin = lim

By Lemma 3.2.5 again, the statement follows. (2) It follows from (1) together with Theorem 3.1.2 and Lemma 3.2.5. (3) Let δ ∈ [0, 1] be such that λ = δλmin + (1 − δ)λmax . Let (s0n ), (t0n ), (s00n ), 00 (tn ) be sequences as in (1). For each n = 1, 2, 3, . . . consider the function [0, 1] 3 δ 7→

ln kUa(0) (δt0n + (1 − δ)t00n , δs0n + (1 − δ)s00n )w(a(0) · (δs0n + (1 − δ)s00n ))k . δ(t0n − s0n ) + (1 − δ)(t00n − s00n )

˜ The above function is continuous, so there exists δ˜ = δ(n) ∈ [0, 1] such that ˜ 0 + (1 − δ)t ˜ 0 + (1 − δ)s ˜ 00 , δs ˜ 0 + (1 − δ)s ˜ 00 )w(a(0) · (δs ˜ 00 ))k ln kUa(0) (δt n n n n n n 0 0 00 00 ˜ ˜ δ(tn − sn ) + (1 − δ)(tn − sn ) =δ

ln kUa(0) (t00n , s00n )w(a(0) · s00n )k ln kUa(0) (t0n , s0n )w(a(0) · s0n )k + (1 − δ) . t0n − s0n t00n − s00n

0 00 0 00 ˜ ˜ ˜ ˜ Taking sn := δ(n)s n + (1 − δ(n))sn and tn := δ(n)tn + (1 − δ(n))tn , we see ∞ ∞ that we have found sequences (sn )n=1 , (tn )n=1 ⊂ R, tn − sn → ∞ as n → ∞, such that ln kUa(0) (tn , sn )w(a(0) · sn )k = λ. lim n→∞ tn − sn

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Put kn := bsn c, ln := btn c. We have M1 kU (0) (tn , sn )w(a(0) · sn )k ≤ kUa(0) (ln , kn )w(a(0) · kn )k M2 a M2 kU (0) (tn , sn )w(a(0) · sn )k ≤ M1 a for n sufficiently large, where M1 := inf{ kUa˜ (s, 0)w(˜ a)k : s ∈ [0, 1], a ˜ ∈ Y0 } > 0, M2 := sup{ kUa˜ (s, 0)w(˜ a)k : s ∈ [0, 1], a ˜ ∈ Y0 } < ∞. After simple calculation we have that ln kUa(0) (ln , kn )w(a(0) · kn )k = λ. n→∞ ln − k n lim

An application of Lemma 3.2.5 gives the desired result. Recall that for a ∈ Y , Ba (t, u, v) is defined in (2.1.4) in the Dirichlet and Neumann boundary condition cases, and is defined in (2.1.5) in the Robin boundary condition case. LEMMA 3.2.7 Z t kUa (t, s)w(a)k = exp − Ba (τ, w(a·τ ), w(a·τ )) dτ

for any a ∈ Y0 , s < t.

s

PROOF Fix a ∈ Y0 and s < t. Let, for τ ∈ [s, t], η(τ ) := kUa (τ, s)w(a)k. By Proposition 2.1.4, 1 (η(τ ))2 − (η(s)))2 = − 2

Z

τ

Ba (r, w(a · r), w(a · r))(η(r))2 dr,

s

hence (η(·))2 is absolutely continuous. As it is bounded away from 0, ln η(·) is absolutely continuous on [s, t]. We deduce then that η(τ ˙ ) = −Ba (τ, w(a · τ ), w(a · τ ))η(τ ) for a.e. τ ∈ [s, t]. This implies the statement of the lemma. The following results are straightforward corollaries of Lemma 3.2.7 and Theorems 3.2.7 or 3.1.5, respectively. THEOREM 3.2.8 If Y0 = cl { a(0) · t : t ∈ R } for some a(0) ∈ Y0 , where the closure is taken in

3. Spectral Theory in the General Setting

89

the weak-* topology, then λmin

Z t −1 Ba(0) (τ, w(a(0) · τ ), w(a(0) · τ )) dτ = lim inf t−s→∞ t − s s Z t −1 Ba(0) (τ, w(a(0) · τ ), w(a(0) · τ )) dτ = λmax . ≤ lim sup t−s→∞ t − s s

THEOREM 3.2.9 Let µ be an ergodic invariant measure for σ|Y0 . Then there exists a Borel set Y1 ⊂ Y0 with µ(Y1 ) = 1 such that 1 λ(µ) = − lim t→∞ t

Z

t

Ba (τ, w(a · τ ), w(a · τ )) dτ 0

for any a ∈ Y1 .

3.3

Existence of Exponential Separation and Entire Positive Solutions

In this section we discuss the existence of exponential separation and the existence and uniqueness of positive solutions. We show that in general the existence of an entire positive solution and exponential separation follows from certain Harnack inequalities for nonnegative solutions of parabolic problems. In the present section we assume that assumptions (A2-1) through (A2-3) are satisfied. Y0 is a compact connected invariant subset of Y . For x ∈ D we denote by d(x) the distance of x from the boundary ∂D of D. We introduce now the assumptions (A3-1) and (A3-2). (A3-1) (Harnack type inequality for quotients): For each δ1 > 0 there is C1 = C1 (δ1 ) > 1 such that sup x∈D

(Ua (t, 0)u01 )(x) (Ua (t, 0)u01 )(x) ≤ C1 inf x∈D (Ua (t, 0)u02 )(x) (Ua (t, 0)u02 )(x)

for any a ∈ Y0 , t ≥ δ1 , any u01 , u02 ∈ L2 (D)+ , where u02 6= 0, and any x ∈ D. (A3-2) (Pointwise Harnack inequality) There is ς ≥ 0 such that for each δ2 > 0 there is C2 = C2 (δ2 ) > 0 with the property that (Ua (t, 0)u0 )(x) ≥ C2 (d(x))ς kUa (t, 0)u0 k∞

(3.3.1)

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for any a ∈ Y0 , t ≥ δ2 , u0 ∈ L2 (D)+ and x ∈ D. Both (2.0.1)+(2.0.2) and (2.3.1)+(2.3.2) satisfy (A3-1) and (A3-2) under quite general conditions. For example, it is proved in [61] that both (A31) and (A3-2) are satisfied in the Dirichlet boundary condition case for a Lipschitz domain (see [61, Theorem 2.1 and Lemma 3.9]). It is proved in [59] that (A3-2) is satisfied with ς = 0 in certain Neumann and Robin boundary condition cases (see [59, Theorem 2.5]). If (A3-2) is satisfied with ς = 0, then (A3-1) is also satisfied. More precisely, we have LEMMA 3.3.1 (A3-2) with ς = 0 implies (A3-1). PROOF Let u01 and u02 be as in (A3-1). Put ui (t, x) := (Ua (t, 0)u0i )(x), i = 1, 2. By (A3-2) (with ς = 0) we have sup x∈D

supx∈D u1 (t, x) 1 inf x∈D u1 (t, x) 1 u1 (t, x) u1 (t, x) ≤ ≤ 2 ≤ 2 inf u2 (t, x) inf x∈D u2 (t, x) C2 supx∈D u2 (t, x) C2 x∈D u2 (t, x)

for t ≥ δ2 . Hence (A3-1) holds with δ1 = δ2 and C1 = 1/C22 . We say that (A3-1) and/or (A3-2) are satisfied by (2.3.1)+(2.3.2) if they are satisfied with Ua (t, 0) being replaced by Ua∗ (−t, 0). We have the following theorems about the existence and uniqueness of an entire positive weak solution and the existence of exponential separation. THEOREM 3.3.1 Consider (2.0.1)+(2.0.2) and assume (A3-1)–(A3-2). Then there exists a continuous function w : Y0 → L2 (D)+ , kw(a)k = 1 for each a ∈ Y0 , having the property that for each a ∈ Y0 the function [ t 7→ va (t) = ra (t)w(a · t) ], where ( kUa·t (−t, 0)w(a · t)k−1 t < 0, ra (t) := kUa (t, 0)w(a)k t ≥ 0, is an entire positive weak solution of (2.0.1)a +(2.0.2)a . Moreover, for any entire positive weak solution v of (2.0.1)a +(2.0.2)a one has v(t) = kv(0)kva (t), t ∈ R. THEOREM 3.3.2 Consider (2.3.1)+(2.3.2) and assume that (A3-1)–(A3-2) are satisfied by (2.3.1)+(2.3.2). Then there exists a continuous function w∗ : Y0 → L2 (D)+ , kw∗ (a)k = 1 for each a ∈ Y0 , having the property that for each a ∈ Y0 the

3. Spectral Theory in the General Setting function [ t 7→ va∗ (t) = ra∗ (t)w∗ (a · t) ], where ( kUa∗ (t, 0)w∗ (a)k ∗ ra (t) := ∗ kUa·t (−t, 0)w∗ (a · t)k−1

91

t ≤ 0, t > 0,

is an entire positive weak solution of (2.3.1)a +(2.3.2)a . Moreover, for any entire positive weak solution v ∗ of (2.3.1)a +(2.3.2)a one has v ∗ (t) = kv ∗ (0)kva∗ (t), t ∈ R. THEOREM 3.3.3 Assume that (A3-1) and (A3-2) are satisfied by both (2.0.1) +(2.0.2) and (2.3.1) +(2.3.2). Let w and w∗ be as in Theorems 3.3.1 and 3.3.2, respectively. Then Π admits an exponential separation over Y0 with an invariant one-dimensional subbundle given by X1 (a) = span{w(a)} and an invariant one-codimensional subbundle given by X2 (a) = {v ∈ L2 (D) : hv, w∗ (a)i = 0}. Moreover, for any a ∈ Y0 the fiber X2 (a) is characterized as the set of those u0 ∈ L2 (D) such that the global weak solution [ [0, ∞) 3 t 7→ Ua (t, 0)u0 ] is neither eventually positive nor eventually negative (plus the trivial solution). We remark that Theorems 3.3.1–3.3.3 have been proved for the Dirichlet boundary conditions case in [60] and [61] (see Section 3.6). We shall provide unified proofs of Theorems 3.3.1–3.3.3 (i.e., proofs which apply to all three, Dirichlet, Neumann, Robin, boundary conditions cases). In order to do so, we first introduce three positive constants M1 , M2 , M3 by the following:  kUa (t, 0)u0 k ≤ M1 ku0 k    kUa (t, 0)u0 k∞ ≤ M1 ku0 k∞    kUa (t + 1/2, 0)u0 k∞ ≤ M1 ku0 k

if u0 ∈ L2 (D) if u0 ∈ L∞ (D)

(3.3.2)

if u0 ∈ L2 (D)

for any a ∈ Y0 and 0 ≤ t ≤ 1, kuk ≤ M2 kuk∞

(3.3.3)

M3 := k(d(·))ς k,

(3.3.4)

for any u ∈ L∞ (D), and where ς is a nonnegative constant as in (A3-2). The existence of M1 is guaranteed by Proposition 2.2.2. Observe that for any u0 ∈ L2 (D), by Proposition 2.2.2, Ua (t, 0)u0 ∈ L∞ (D) for t > 0. Notice that if (A3-2) holds, then for any positive weak solution u on [0, ∞) × D ku(t)k ≥ M4 ku(t)k∞ (3.3.5) for t ≥ δ2 , where M4 = C2 M3 .

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We now show several lemmas, among which, some lemmas follow from the arguments in [60] and [61]. For convenience, we provide proofs here. They will be formulated for the problem (2.0.1)+(2.0.2) only. Their analogs for the adjoint problem (2.3.1)+(2.3.2) are straightforward. First of all, for convenience, we restate the interior Harnack inequality (see Proposition 2.2.8) LEMMA 3.3.2 For any t1 > 0 there is 0 < δ3 < 1 such that for any 0 < δ < δ3 there is C3 > 0 with the property that (Ua (t, 0)u0 )(y) ≤ C3 (Ua (t + τ, 0)u0 )(x) for any a ∈ Y0 , t ≥ δ 2 , δ 2 ≤ τ ≤ t1 , u0 ∈ L2 (D)+ , and any x, y ∈ Dδ := { ξ ∈ D : d(ξ) > δ }. LEMMA 3.3.3 Assume (A3-2). Then there are 0 < δ4 ≤ 1 and C4 > 0 such that kUa (t + τ, 0)u0 k∞ ≥ C4 kUa (t, 0)u0 k∞

(3.3.6)

for any a ∈ Y0 , t ≥ δ4 , τ ∈ [0, 2], and u0 ∈ L2 (D)+ . PROOF In Lemma 3.3.2 take t1 = 2, fix 0 < δ < δ3 and fix an x0 ∈ Dδ . We have the existence of C3 > 0 such that (Ua (t + τ, 0)u0 )(x) ≥

1 (Ua (t, 0)u0 )(x0 ) C3

for any a ∈ Y0 , t ≥ δ 2 , δ 2 ≤ τ ≤ 2, u0 ∈ L2 (D)+ , and x ∈ Dδ . Further, fix 0 < δ2 ≤ δ 2 . By (A3-2) there is C2 > 0 such that (Ua (t, 0)u0 )(x0 ) ≥ C2 (d(x0 ))ς kUa (t, 0)u0 k∞ for any a ∈ Y0 , any u0 ∈ L2 (D), and any t ≥ δ2 . Consequently, (Ua (t + τ, 0)u0 )(x) ≥

C2 (d(x0 ))ς kUa (t, 0)u0 k∞ C3

for any a ∈ Y0 , t ≥ δ 2 , δ 2 ≤ τ ≤ 2, u0 ∈ L2 (D)+ , and x ∈ Dδ , hence kUa (t + τ, 0)u0 k∞ ≥

C2 (d(x0 ))ς kUa (t, 0)u0 k∞ C3

(3.3.7)

for any a ∈ Y0 , t ≥ δ 2 , δ 2 ≤ τ ≤ 2, and u0 ∈ L2 (D)+ . On the other hand, by (3.3.2) there holds kUa (t + δ 2 , 0)u0 k∞ ≤ M1 kUa (t + τ, 0)u0 k∞

(3.3.8)

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for any a ∈ Y0 , t > 0, 0 ≤ τ ≤ δ 2 , and u0 ∈ L2 (D)+ . It then follows that (3.3.6) holds with δ4 = δ 2 and C4 = C2 (d(x0 ))ς /C3 (notice that M1 ≥ 1). LEMMA 3.3.4 Assume (A3-2). Then there are 0 < δ4 ≤ 1 and C˜4 > 0 such that kUa (t + 1, 0)u0 k ≥ C˜4 kUa (t, 0)u0 k for any a ∈ Y0 , t ≥ δ4 , and u0 ∈ L2 (D)+ . PROOF

By (3.3.6) there is δ4 > 0 such that kUa (t + 2, 0)u0 k∞ ≥ C4 kUa (t, 0)u0 k∞

for t ≥ δ4 . (3.3.2) gives kUa (t + 2, 0)u0 k∞ ≤ M1 kUa (t + 1, 0)u0 k for t ≥ 0. By (3.3.3), kUa (t, 0)u0 k ≤ M2 kUa (t, 0)u0 k∞ for t ≥ δ4 . It then follows that kUa (t + 1, 0)u0 k ≥ C˜4 kUa (t, 0)u0 k for t ≥ δ4 , where C˜4 =

C4 M1 M2 .

LEMMA 3.3.5 Assume (A3-1). Then for each δ1 > 0 there is C1 > 1 such that for any a ∈ Y0 , if u1 , u2 are global weak solutions of (2.0.1)a +(2.0.2)a on [0, ∞) × D, with u2 being positive (u1 (0), u2 (0) ∈ L2 (D)), then the functions %min , %max : (0, ∞) → R defined as %min (t) := inf

x∈D

u1 (t)(x) , u2 (t)(x)

%max (t) := sup x∈D

u1 (t)(x) u2 (t)(x)

enjoy the following properties: (1) %min (·) is nondecreasing and %max (·) is nonincreasing on (0, ∞). (2) %max (t) − %min (t) ≤ 1 − C11 (%max (τ ) − %min (τ )) for 0 < τ < τ + δ1 ≤ t. PROOF (1) Fix a t0 > 0 and put w(t) := u1 (t) − %min (t0 )u2 (t). Then w(t0 ) ≥ 0, hence by Proposition 2.2.7, w(t) ≥ 0 for t > t0 . It then follows

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that %min (t) ≥ %min (t0 ) for t > t0 , consequently, %min (·) is nondecreasing. In a similar way we prove that %max (·) is nonincreasing. (2) Put %(t) := %max (t) − %min (t), t > 0. Fix a τ > 0. Put u ˆ1 (t) := u1 (t) − %min (τ )u2 (t), and define %ˆmin (t) := inf

x∈D

u ˆ1 (t)(x) , u2 (t)(x)

%ˆmax (t) := sup x∈D

u ˆ1 (t)(x) , u2 (t)(x)

%ˆ(t) := %ˆmax (t)−ˆ %min (t),

for t ≥ τ . We have u ˆ1 (τ ) ∈ L2 (D)+ . Assume first that u ˆ1 (τ ) 6= 0. Then u ˆ1 and u2 are positive weak solutions on [τ, ∞) × D. Notice that %ˆmin (t) = %min (t) − %min (τ ) and %ˆmax (t) = %max (t) − %min (τ ) for t ≥ τ , consequently %ˆ(t) = %(t) for any t ≥ τ . With the help of (A3-1) we obtain 1 1 1 %(t) = %ˆ(t) ≤ 1− %ˆmax (t) = 1− (%max (t)−%min (τ )) ≤ 1− %(τ ) C1 C1 C1 for any t ≥ τ + δ1 . The case u ˆ1 (τ ) = 0 means that the solutions u1 and u2 are proportional on [τ, ∞), which implies that %(t) = 0 for all t ∈ [τ, ∞). LEMMA 3.3.6 Assume (A3-2). There are C˜1 , C˜2 > 0 such that if u1 is a weak solution on [0, ∞) × D with u1 (0) ∈ L2 (D) and u2 is a positive weak solution on [−1, ∞) × D with u2 (−1) ∈ L2 (D)+ then (1)

ku1 (t)k∞ ku1 (1/2)k for t ∈ [1, 2], ≤ C˜1 ku2 (t)k∞ ku2 (1/2)k

(2)

ku1 (t)k ku1 (0)k ≤ C˜2 for t ∈ [0, 1]. ku2 (t)k ku2 (0)k

PROOF (1) First of all, by (3.3.2) ku1 (t)k∞ ≤ M1 ku1 (1/2)k for t ∈ [1, 2]. By (3.3.6) and (3.3.3), ku2 (t)k∞ ≥ C4 ku2 (1/2)k∞ ≥

C4 ku2 (1/2)k M2

2 for t ∈ [1, 2]. (1) therefore follows with C˜1 = MC1 M . 4 (2) By (3.3.2), ku1 (t)k ≤ M1 ku1 (0)k for t ∈ [0, 1]. By (3.3.5) with δ2 = 1, (3.3.6) and (3.3.3),

ku2 (t)k ≥ M4 ku2 (t)k∞ ≥ C4 M4 ku2 (0)k∞ ≥ for t ∈ [0, 1]. (2) then follows with C˜2 =

C4 M4 ku2 (0)k M2

M1 M2 C3 M4 .

LEMMA 3.3.7 Assume (A3-1). There is C˜0 > 0 such that if u1 is a weak solution on [0, ∞)×D (u1 (0) ∈ L2 (D)) which is neither eventually positive nor eventually

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95

negative, u2 is a positive weak solution on [−1, ∞) × D (u2 (−1) ∈ L2 (D)+ ), and ku1 (k)k∞ ≤ η0 ku1 (k + 1)k∞ for some η0 > 0 and some k ≥ 1, then sup x∈D

ku1 (k + 1)k∞ |u1 (k + 1)(x)| ≤ C˜0 η0 . u2 (k + 1)(x) ku2 (k + 1)k∞

PROOF It can be proved by arguments similar to those in [61, Lemma 6.1] (for parabolic equations with Dirichlet boundary conditions). For completeness, we provide a proof here. Put u(t) := Ua (t, k)(u1 )+ (k), v(t) := Ua (t, k)(u1 )− (k), t ≥ k. Then by (A3-1) with δ1 = 1 sup x∈D

v(k + 1)(x) v(k + 1)(x) ≤ C1 inf . x∈D u(k + 1)(x) u(k + 1)(x)

Without loss of generality we may assume that inf

x∈D

v(k + 1)(x) ≤1 u(k + 1)(x)

(otherwise, we just exchange the roles of u and v). By (3.3.2) and the assumption of the lemma, ku(k + 1)k∞ ≤ M1 ku(k)k∞ = M1 k(u1 )+ (k)k∞ ≤ M1 ku1 (k)k∞ ≤ M1 η0 ku1 (k + 1)k∞ . Note that inf

x∈D

u(k + 1)(x) ku(k + 1)k∞ u(k + 1)(x) ≤ . ≤ sup u2 (k + 1)(x) ku2 (k + 1)k∞ u x∈D 2 (k + 1)(x)

Consequently sup x∈D

u(k + 1)(x) |u (k + 1)(x)| |u1 (k + 1)(x)| 1 = sup u2 (k + 1)(x) x∈D u2 (k + 1)(x) u(k + 1)(x) u(k + 1)(x) |u1 (k + 1)(x)| ≤ sup · sup x∈D u2 (k + 1)(x) x∈D u(k + 1)(x) u(k + 1)(x) u(k + 1)(x) + v(k + 1)(x) ≤ C1 inf · sup x∈D u2 (k + 1)(x) x∈D u(k + 1)(x) ku(k + 1)k∞ ≤ C1 (1 + C1 ) ku2 (k + 1)k∞ ku1 (k + 1)k∞ ≤ M1 C1 (1 + C1 )η0 . ku2 (k + 1)k∞

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The lemma then follows with C˜0 = M1 C1 (1 + C1 ). LEMMA 3.3.8 ˜ > 0 and γ0 > 0 such that if u1 is a weak Assume (A3-1)–(A3-2). There are M solution on [0, ∞) × D (u1 (0) ∈ L2 (D)) which is neither eventually positive nor eventually negative and u2 is a positive weak solution on [−1, ∞) × D (u2 (−1) ∈ L2 (D)+ ) then ku1 (t)k∞ ˜ e−γ0 t ku1 (1)k∞ ≤M ku2 (t)k∞ ku2 (1)k∞

for

t > 1.

PROOF It can be proved by arguments similar to those in [61, Theorem 2.2] (for parabolic equations with Dirichlet boundary conditions). For completeness, we provide a proof here. First of all, by (3.3.2), ku1 (t)k∞ ≤ M1 ku1 ([t])k∞ for any t ≥ 1, which together with (3.3.6) implies that ku1 (t)k∞ M1 ku1 (btc)k∞ ≤ , ku2 (t)k∞ C4 ku2 (btc)k∞

t ≥ 1.

(3.3.9)

Put %0 := 1 − C11 , where C1 is as in (A3-1) with δ1 = 1. Clearly we have either 1 (k+1)k∞ (i) For all k ∈ N, kuku < C4 %0 , 1 (k)k∞ or 1 (k+1)k∞ ≥ C4 %0 . (ii) For some k ∈ N, kuku 1 (k)k∞ Assume that (i) holds. Note that by (3.3.6),

ku2 (k + 1)k∞ ≥ C4 ku2 (k)k∞ for k ∈ N. Hence

ku1 (k + 1)k∞ ku1 (k)k∞ ≤ %0 ku2 (k + 1)k∞ ku2 (k)k∞

for k ∈ N. Therefore ku1 (t)k∞ M1 ku1 (btc)k∞ M1 btc−1 ku1 (1)k∞ ≤ ≤ % ku2 (t)k∞ C4 ku2 (btc)k∞ C4 0 ku2 (1)k∞ M1 −2 −(− ln %0 )t ku1 (1)k∞ %0 e ≤ C4 ku2 (1)k∞ for any t ≥ 1. Assume that (ii) holds. Let k0 := inf{ k ∈ N : ku1 (k + 1)k∞ ≥ C4 %0 ku1 (k)k∞ }.

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97

Arguments as in (i) give ku1 (t)k∞ M1 −2 −(− ln %0 )t ku1 (1)k∞ ≤ %0 e ku2 (t)k∞ C4 ku2 (1)k∞ for 1 ≤ t ≤ k0 + 1. By (3.3.9) and Lemma 3.3.5 with δ1 = 1, for t ≥ k0 + 1 we have ku1 (t)k∞ M1 ku1 (btc)k∞ M1 |u1 (btc)(x)| ≤ ≤ sup ku2 (t)k∞ C4 ku2 (btc)k∞ C4 x∈D u2 (btc)(x) u1 (btc)(x) u1 (btc)(x) M1 sup − inf ≤ C4 x∈D u2 (btc)(x) x∈D u2 (btc)(x) M1 btc−k0 −1 u1 (k0 + 1)(x) u1 (k0 + 1)(x) ≤ − inf %0 sup x∈D u2 (k0 + 1)(x) C4 x∈D u2 (k0 + 1)(x) M1 btc−k0 −1 |u1 (k0 + 1)(x)| . ≤2 % sup C4 0 x∈D u2 (k0 + 1)(x) This together with Lemma 3.3.7 implies that ku1 (t)k∞ btc−k0 ku1 (k0 + 1)k∞ ≤ 2M1 C˜0 %0 . ku2 (t)k∞ ku2 (k0 + 1)k∞ But we have already proved that M1 k0 −1 ku1 (1)k∞ ku1 (k0 + 1)k∞ ≤ %0 , ku2 (k0 + 1)k∞ C4 ku2 (1)k∞ which gives ku1 (t)k∞ 2M12 C˜0 btc−1 ku1 (1)k∞ ≤ %0 ku2 (t)k∞ C4 ku2 (1)k∞ 2M12 C˜0 −2 −(− ln %0 )t ku1 (1)k∞ ≤ %0 e C4 ku2 (1)k∞ ˜ = for t ≥ k0 + 1. The lemma then follows with M

M1 %20 C4

max{1, 2M1 C˜0 } and

γ0 = − ln %0 . PROOF (Proof of Theorem 3.3.1) We prove first the existence of an entire positive solution. Fix a ∈ Y0 and u0 ∈ L2 (D)+ with ku0 k = 1. Define + a sequence (un )∞ n=1 ⊂ L2 (D) \ {0} by un :=

Ua (0, −n)u0 Ua (−1, −n)u0 = Ua (0, −1) . kUa (−1, −n)u0 k kUa (−1, −n)u0 k

By Proposition 2.2.2, the set { kun k : n = 1, 2, . . . } is bounded above. Moreover, from the local regularity (Proposition 2.2.4) we deduce that there is a

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sequence (nk )∞ ˜0 in L2 (D). k=1 such that limk→∞ nk = ∞ and limk→∞ unk = u By Lemma 3.3.4 the set { kun k : n = 1, 2, . . . } is bounded below by C˜4 > 0, (0) consequently u ˜0 ∈ L2 (D)+ \ {0}. We claim that there is an entire positive weak solution u ˆ of (2.0.1)a +(2.0.2)a (0) (−l) + such that u ˆ(0) = u ˜0 . We start by finding a sequence (˜ u0 )∞ l=1 ⊂ L2 (D) \ (−l) (−l+1) ∞ {0} and a sequence (rl )l=1 ⊂ (0, ∞) such that Ua (−l + 1, −l)˜ u0 = rl u ˜0 for l = 1, 2, . . . . Such sequences are constructed by induction on l. We show (−1) only the first step (that is, finding u ˜0 and r1 ), the remaining being similar. Put Ua (−2, −nk )u0 Ua (−1, −nk )u0 = Ua (−1, −2) . vk := kUa (−2, −nk )u0 k kUa (−2, −nk )u0 k By the same reason as above, there are a subsequence (km )∞ ˜0 ∈ m=1 and v L2 (D)+ with k˜ v0 k ≥ C˜4 such that limm→∞ km = ∞ and limm→∞ vkm = v˜ in L2 (D). Note that unk = r(k) · Ua (0, −1)vk for each k ∈ N, where r(k) =

kUa (−2, −nk )u0 k . kUa (−1, −nk )u0 k

From (3.3.2) it follows that the set { r(k) ; k = 1, 2, . . . } is bounded below by (M1 )−1 > 0, and from Lemma 3.3.4 it follows that the set { r(k) : k = 1, 2, . . . } is bounded above by (C˜4 )−1 . Therefore we may assume without loss of generality that limm→∞ r(km ) =: r > 0. Then we have (0)

u ˜0 = r · Ua (0, −1)˜ v0 (−1)

It suffices to take u ˜0 := v˜0 and r1 := r. By a diagonal process we obtain (−l) sequences (˜ u0 ) and a sequence (rl ) sought for. The function u ˆ(·) defined as  1 (btc)  Ua (t, btc)˜ u0 t < 0, r r . . . r 1 u ˆ(·) := btc btc−1  (0) Ua (t, 0)˜ u0 t≥0 is an entire positive weak solution of (2.0.1)a +(2.0.2)a . To prove uniqueness, fix a ∈ Y0 and suppose that u ˆ1 and u ˆ2 are two entire positive weak solutions of (2.0.1)a +(2.0.2)a . Without loss of generality, assume that kˆ u1 (0)k = kˆ u2 (0)k = 1. Let u ˆ(t, x) := u ˆ1 (t)(x) − u ˆ2 (t)(x). We first claim that for any t < 0, there is x(t) ∈ D such that u ˆ(t, x(t)) = 0. For otherwise, if there is t < 0 such that u ˆ(t, x) > 0 for all x ∈ D, then u ˆ(0, x) > 0 or u ˆ1 (0)(x) > u ˆ2 (0)(x) for all x ∈ D. This implies kˆ u1 (0)k > kˆ u2 (0)k. This is a contradiction. Similarly, if u ˆ(t, x) < 0 for some t < 0 and all x ∈ D, then u ˆ1 (0)(x) < u ˆ2 (0)(x) for all x ∈ D. This implies that kˆ u1 (0)k < kˆ u2 (0)k,

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99

a contradiction again. Therefore for any t < 0, there is x(t) ∈ D such that u ˆ(t, x(t)) = 0. This implies that %min (t) := inf

x∈D

u ˆ1 (t)(x) ≤1 u ˆ2 (t)(x)

for all t < 0. Then by (A3-1), %max (t) := sup x∈D

u ˆ1 (t)(x) ≤ C1 u ˆ2 (t)(x)

for all t < 0. Hence %max (t) − %min (t) is bounded for t < 0. Define %(t) := %max (t) − %min (t), (t > 0). By Lemma 3.3.5(2) with δ1 = 1 1 %(t) ≤ 1 − %(s) C1 for t ≥ s + 1, t, s ∈ R. This implies that %max (t) = %min (t) for t ∈ R, hence u ˆ1 (t)(x) = u ˆ2 (t)(x) for all t ∈ R and all x ∈ D. Now, for each a ∈ Y0 denote by w(a) the value at time 0 of the unique positive entire weak solution of (2.0.1)a +(2.0.2)a , normalized so that kw(a)k = 1. We want to show that w : Y0 → L2 (D) is continuous. Fix a sequence (a(n) ) ⊂ Y0 converging, as n → ∞, to a ˜ ∈ Y0 . By the uniqueness of entire positive solutions, Uan (0, −1)w(a(n) · (−1)) w(a(n) ) = kUa(n) (0, −1)w(a(n) · (−1))k and Ua(n) (0, −1)w(a(n) · (−1)) =

Ua(n) (0, −2)w(a(n) · (−2)) . kUa(n) (−1, −2)w(a(n) · (−2))k

From the local regularity (Proposition 2.2.4) we obtain that there is a sequence (nk ) (nk )∞ ·(−1)) = k=1 such that limk→∞ nk = ∞ and limk→∞ Ua(nk ) (0, −1)w(a u ˜0 in L2 (D). Lemma 3.3.3 implies kUa(n) (0, −1)w(a(n) · (−1))k ≥ C˜4 for all n ∈ N. Consequently, by extracting again a subsequence, if necessary, we can assume that w(a(nk ) ) → w0 in L2 (D), where kw0 k = 1. By a diagonal process as in the proof of the existence we obtain that there are a subsequence (nkm )∞ ˇ m=1 with limit ∞, a positive entire weak solution u of (2.0.1)a +(2.0.2)a and a sequence (rl )∞ such that l=0 u ˇ(−l) = rl lim w(a(nkm ) · (−l)) m→∞

for each l = 0, 1, 2, . . . . In particular, u ˇ(0) = w0 . By uniqueness, w0 = w(˜ a).

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PROOF (Proof of Theorem 3.3.2) similar to those in Theorem 3.3.1.

It can be proved by arguments

PROOF (Proof of Theorem 3.3.3) First, we prove that Π admits an exponential separation over Y0 with an invariant one-dimensional subbundle given by X1 (a) = span{w(a)} and an invariant one-codimensional subbundle given by X2 (a) = { v ∈ L2 (D) : hv, w∗ (a)i = 0 }. First of all, hw(a), w∗ (a)i > 0 for any a ∈ Y0 . Therefore L2 (D) = X1 (a) ⊕ X2 (a) for any a ∈ Y0 . Clearly, X1 and X2 are invariant. Now, for any u0 ∈ X2 (a), by Lemma 3.3.6(2), ku0 k kUa (t, 0)u0 k ≤ C˜2 kUa (t, 0)w(a)k kw(a)k for 0 < t ≤ 1. By Lemma 3.3.8, kUa (t, 0)u0 k∞ ˜ e−γ0 t kUa (1, 0)u0 k∞ ≤M kUa (t, 0)w(a)k∞ kUa (1, 0)w(a)k∞ for t ≥ 1. Applying Lemma 3.3.6 again we get kUa (1, 0)u0 k∞ ku0 k kUa (1/2, 0)u0 k ≤ C˜1 C˜2 . ≤ C˜1 kUa (1, 0)w(a)k∞ kUa (1/2, 0)w(a)k kw(a)k Note that by (3.3.3), kUa (t, 0)u0 k ≤ M2 kUa (t, 0)u0 k∞ and by (3.3.5), M4 kUa (t, 0)w(a)k∞ ≤ kUa (t, 0)w(a)k for t ≥ 1. Therefore we have ˜ kUa (t, 0)u0 k M2 C˜1 C˜2 M ku0 k ≤ e−γ0 t kUa (t, 0)w(a)k M4 kw(a)k for t > 1. It then follows that ku0 k kUa (t, 0)u0 k ≤ M e−γ0 t kUa (t, 0)w(a)k kw(a)k ˜ ˜ ˜

2M for t > 0, where M = max{C˜2 eγ0 , M2 CM1 C }. Therefore Π admits an ex4 ponential separation over Y0 with an invariant one-dimensional subbundle X1 (a) = span{w(a)} and an invariant one-codimensional subbundle X2 (a) = { v ∈ L2 (D) : hv, w∗ (a)i = 0 }.

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101

Next, we prove that X2 (a) is characterized as the set of those u0 ∈ L2 (D) such that the global weak solution [ [0, ∞) 3 t 7→ Ua (t, 0)u0 ] is neither eventually positive nor eventually negative (plus the trivial solution). First, for a given a ∈ Y0 , by the invariance of X2 , if a nonzero u0 ∈ X2 (a) then Ua (t, 0)u0 6∈ L2 (D)+ ∪ (−L2 (D)+ ) for any t ≥ 0. Conversely, suppose that a ∈ Y0 and u0 ∈ L2 (D) are such that Ua (t, 0)u0 6∈ L2 (D)+ ∪ (−L2 (D)+ ) for any t ≥ 0. If u0 6∈ X2 (a) then there is a nonzero c ∈ R such that u0 − cw(a) ∈ X2 (a). It then follows that ku0 − cw(a)k kUa (t, 0)(u0 − cw(a))k ≤ M e−γ0 t , kUa (t, 0)w(a)k kw(a)k which gives kUa (t, 0)u0 k ≥ kUa (t, 0)(cw(a))k − kUa (t, 0)(u0 − cw(a))k −γ0 t ku0 − cw(a)k ≥ |c| − e kUa (t, 0)w(a)k, kw(a)k for t > 0. This implies that lim inf t→∞

1 1 ln kUa (t, 0)u0 k ≥ lim inf ln kUa (t, 0)w(a)k. t→∞ t t

On the other hand, by Lemma 3.3.8, we have 1 1 ln kUa (t, 0)u0 k∞ < lim inf ln kUa (t, 0)w(a)k∞ . t→∞ t t

lim inf t→∞

Note that, by the inequality kUa (t, 0)w(a)k ≤ |D|1/2 kUa (t, 0)w(a)k∞ and Eq. (3.3.5) with δ2 = 1, there holds lim inf t→∞

1 1 ln kUa (t, 0)w(a)k = lim inf ln kUa (t, 0)w(a)k∞ . t→∞ t t

Further, by the inequality kUa (t, 0)u0 k ≤ |D|1/2 kUa (t, 0)u0 k∞ we have lim inf t→∞

1 1 ln kUa (t, 0)u0 k ≤ lim inf ln kUa (t, 0)u0 k∞ . t→∞ t t

Consequently, 1 1 ln kUa (t, 0)u0 k ≤ lim inf ln kUa (t, 0)u0 k∞ t→∞ t t 1 1 < lim inf ln kUa (t, 0)w(a)k∞ = lim inf ln kUa (t, 0)w(a)k t→∞ t t→∞ t 1 ≤ lim inf ln kUa (t, 0)u0 k, t→∞ t lim inf t→∞

which is a contradiction. Therefore u0 ∈ X2 (a).

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THEOREM 3.3.4 Assume that (A3-1) and (A3-2) are satisfied by both (2.0.1)+(2.0.2) and (2.3.1)+(2.3.2). Then for each δ0 > 0 there exists K = K(δ0 ) > 0 such that kP2 (a · t)Ua (t, 0)u0 k ≤K kP1 (a · t)Ua (t, 0)u0 k for all t ≥ δ0 , all a ∈ Y0 , and all u0 ∈ L2 (D)+ \ {0}. PROOF

Eq. (3.2.2) gives P1 (a)u =

hu, w∗ (a)i w(a) hw(a), w∗ (a)i

for any a ∈ Y0 and any u ∈ L2 (D). By the pointwise Harnack inequality (A3-2), there are C2 > 0 and ς ≥ 0, ς 0 ≥ 0 such that (Ua (t, 0)u0 )(x) ≥ C2 (d(x))ς kUa (t, 0)u0 k∞ for any t ≥ δ0 /2, a ∈ Y0 , x ∈ D and nonzero u0 ∈ X + , and 0

w∗ (a)(x) ≥ C2 (d(x))ς kw∗ (a)k∞ for any a ∈ Y0 and x ∈ D. Further, (3.3.3) yields kw(a)k∞ ≥ (M2 )−1 and kw∗ (a)k∞ ≥ (M2 )−1 for all a ∈ Y0 . It follows that Z ˜ 1 kUa (t, 0)u0 k∞ hUa (t, 0)u0 , w∗ (a · t)i = w∗ (a · t)(x) (Ua (t, 0)u0 )(x) dx ≥ M D

for any a ∈ Y0 , t ≥ δ0 /2 and any nonzero u0 ∈ L2 (D)+ , where Z 0 (C2 )2 ˜ (d(x))ς+ς dx > 0. M1 := M2 D We have thus obtained that ˜ 2 kUa (t, 0)u0 k∞ w(a)(x) (P1 (a · t)Ua (t, 0)u0 )(x) ≥ M for all u0 ∈ L2 (D)+ \ {0}, a ∈ Y0 , t ≥ δ0 /2 and a.e. x ∈ D, where ˜ 2 := M

˜1 M > 0. sup{ hw(a), w∗ (a)i : a ∈ Y0 }

Consequently, ˜ 2 kUa (δ0 /2, 0)u0 k∞ kP1 (a · (δ0 /2))Ua (δ0 /2, 0)u0 k ≥ M ˜2 M ≥ kUa (δ0 /2, 0)u0 k M2

by (3.3.3)

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103

˜ 3 := sup{ kP2 (a)k : a ∈ Y0 } for all u0 ∈ L2 (D)+ \ {0} and a ∈ Y0 . Let M (< ∞). There holds ˜3 M2 M kP2 (a · (δ0 /2))Ua (δ0 /2, 0)u0 k ≤ ˜ kP1 (a · (δ0 /2))Ua (δ0 /2, 0)u0 k M2 for all nonzero u0 ∈ L2 (D)+ and all a ∈ Y0 . An exponential separation (Definition 3.2.1(iii)) gives ˜3 M M2 M kP2 (a · t)Ua (t, 0)u0 k ≤ e−γ0 δ0 /2 ˜ kP1 (a · t)Ua (t, 0)u0 k M2 for all nonzero u0 ∈ L2 (D)+ , all t ≥ δ0 and all a ∈ Y0 .

3.4

Multiplicative Ergodic Theorems

In this section we will give applications of multiplicative ergodic theorems (Oseledets-type theorems) to the linear topological skew-product semiflow Π = {Πt }t≥0 generated on L2 (D) × Y by (2.0.1)+(2.0.2), Πt (u0 , a) = (Ua (t, 0)u0 , a · t) for u0 ∈ L2 (D) and a ∈ Y (see (2.2.6)). Throughout the present section we assume that (2.0.1)+(2.0.2) satisfies (A2-1)–(A2-3). Assume that Y0 is a compact connected invariant subset of Y , and µ is an ergodic invariant measure on Y0 . Moreover, we assume that Π admits an exponential separation over Y0 with an invariant one-dimensional subbundle X1 and an invariant one-codimensional subbundle X2 . We also assume that for each a ∈ Y and t > 0 the linear operator Ua (t, 0) is injective. For t < 0, a ∈ Y0 and u0 ∈ L2 (D) the symbol Ua (t, 0)u0 stands for v0 ∈ L2 (D) such that Ua·t (−t, 0)v0 = u0 . By injectivity, such a v0 , if it exists, is unique. Let λ(µ) be the principal Lyapunov exponent of Π for the ergodic invariant measure µ. For a ∈ Y0 let w(a) ∈ L2 (D)+ be as in Definition 3.2.1. THEOREM 3.4.1 There exists a Borel set Y˜0 ⊂ Y0 , µ(Y˜0 ) = 1, with the property that one of the following (mutually exclusive) cases holds: (1) There are: (a) k (≥ 1) real numbers λ1 (µ) > · · · > λk (µ), and

104

Spectral Theory for Parabolic Equations (b) k measurable families {E1 (µ; a)}a∈Y˜0 , . . . , {Ek (µ; a)}a∈Y˜0 , of vector subspaces of constant finite dimensions, and a measurable family {F∞ (µ; a)}a∈Y˜0 of infinite dimensional vector subspaces such that • Ua (t, 0)Ei (µ; a) = Ei (µ; a·t) (i = 1, 2, . . . ) and Ua (t, 0)F∞ (µ; a) ⊂ F∞ (µ; a · t), for any a ∈ Y˜0 and t ≥ 0, • E1 (µ; a) ⊕ · · · ⊕ Ek (µ; a) ⊕ F∞ (µ; a) = L2 (D) for any a ∈ Y˜0 , • limt→∞ (1/t) ln kUa (t, 0)u0 k = λi (µ) for any a ∈ Y˜0 and any nonzero u0 ∈ Ei (µ; a) (i = 1, . . . , k), and • limt→∞ (1/t) ln kUa (t, 0)u0 k = −∞ for any a ∈ Y˜0 and any nonzero u0 ∈ F∞ (µ; a). Further, for each i = 1, . . . , k and each a ∈ Y˜0 , Ei (µ; a) \ {0} is characterized as the set of those nonzero u0 ∈ L2 (D) for which Ua (t, 0)u0 exists for all t ∈ R and lim

t→±∞

ln kUa (t, 0)u0 k = λi (µ) t

holds. (2) There are: (a) a sequence of real numbers λ1 (µ) > · · · > λi (µ) > λi+1 (µ) > . . . having limit −∞, and (b) countably many measurable families {E1 (µ; a)}a∈Y˜0 , {E2 (µ; a)}a∈Y˜0 , . . . , of vector subspaces of constant finite dimensions, and countably many measurable families {F1 (µ; a)}a∈Y˜0 , {F2 (µ; a)}a∈Y˜0 , . . . , of vector subspaces of constant finite codimensions such that there holds: • Ua (t, 0)Ei (µ; a) = Ei (µ; a · t) and Ua (t, 0)Fi (µ; a) ⊂ Fi (µ; a · t) (i = 1, 2, . . . ), for any a ∈ Y˜0 and t ≥ 0, • E1 (µ; a) ⊕ . . . Ei (µ; a) ⊕ Fi (µ; a) = L2 (D) and Fi (µ; a) = Ei+1 (µ; a) ⊕ Fi+1 (µ; a) for any a ∈ Y˜0 (i = 1, 2, . . . ), • limt→∞ (1/t) ln kUa (t, 0)u0 k = λi (µ) for any a ∈ Y˜0 and any nonzero u0 ∈ Ei (µ; a) (i = 1, 2, . . . ), and • limt→∞ (1/t) ln kUa (t, 0)|Fi (µ;a) k = λi+1 (µ) for any a ∈ Y˜0 (i = 1, 2, . . . ). Further, for each i ∈ N and each a ∈ Y˜0 , Ei (µ; a) \ {0} is characterized as the set of those nonzero u0 ∈ L2 (D) for which Ua (t, 0)u0 exists for all t ∈ R and ln kUa (t, 0)u0 k lim = λi (µ) t→±∞ t holds.

3. Spectral Theory in the General Setting

105

In both cases λ1 (µ) = λ(µ). For the meaning of measurability, see [72]; compare [75]. The decompositions of L2 (D) as in Theorem 3.4.1 are called the Oseledets splittings for µ. The real numbers λ1 (µ) > λ2 (µ) > . . . are referred to as the Lyapunov exponents for µ. PROOF (Proof of Theorem 3.4.1) The theorem follows from the Multiplicative Ergodic Theorem for Continuous Time Linear Random Dynamical Systems in [72, Theorem 3.3]. We check the applicability of that theorem, namely, the measurability of [ Y0 3 a 7→ Ua (1, 0)u0 ∈ L2 (D) ] for u0 ∈ L2 (D), the injectivity of Ua (1, 0), and the integrability of f1 (·) and f2 (·), where f1 (a) := sup ln+ kUa (s, 0)k,

a ∈ Y0 ,

0≤s≤1

and f2 (a) := sup ln+ kUa·s (1 − s, 0)k,

a ∈ Y0 ,

0≤s≤1

where ln+ (·) := max {ln(·), 0}. First, for each u0 ∈ L2 (D) the mapping [ Y0 3 a 7→ Ua (1, 0)u0 ∈ L2 (D) ] is continuous (by Proposition 2.2.12), hence (B(Y0 ), B(L2 (D)))-measurable. The injectivity of Ua (1, 0) is just an assumption. We proceed now to show that the mappings f1 (·) and f2 (·) belong to L1 ((Y0 , B(Y0 ), µ)). We show that both f1 and f2 are lower semicontinuous, consequently (B(Y0 ), B(R))-measurable. Assume a(n) → a in Y0 . For any > 0 there are s ∈ [0, 1] and u ∈ L2 (D) with ku k = 1 such that ln+ kUa (s , 0)u k + ≥ f1 (a). Since kUa(n) (s , 0)u k → kUa (s , 0)u k as n → ∞ (this is obvious if s = 0, and follows from Proposition 2.2.13 otherwise), there is n1 > 0 such that for n ≥ n1 , ln+ kUa(n) (s , 0)u k + 2 ≥ f1 (a). Therefore f1 (a(n) ) + 2 ≥ f1 (a) for n ≥ n1 , which implies that f1 is lower semicontinuous. Note that f2 (a) = ln+ kUa (1, s)k.

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By similar arguments we prove that f2 is lower semicontinuous, too. By the L2 –L2 estimates (see Proposition 2.2.2), we have that f1 and f2 are bounded. Therefore they belong to L1 ((Y0 , B(Y0 ), µ)). The final part of the proof is to exclude the case that lim

t→∞

ln kUa (t, 0)k = −∞ t

for µ-a.e. a ∈ Y0 , which is done by applying Lemma 3.1.4. In the light of the properties of Lyapunov exponents contained in Theorem 3.4.1 the following corollary is a consequence of Theorem 3.2.4. COROLLARY 3.4.1 There exists a Borel set Yˆ1 ⊂ Y0 , µ(Yˆ1 ) = 1, such that either • X1 (a) = E1 (µ; a) and X2 (a) = F∞ (µ; a) for all a ∈ Yˆ1 (if (1) in Theorem 3.4.1 holds with k = 1), or • X1 (a) = E1 (µ; a) and X2 (a) = E2 (µ; a) ⊕ · · · ⊕ Ek (µ; a) ⊕ F∞ (µ; a) for all a ∈ Yˆ1 (if (1) in Theorem 3.4.1 holds with k > 1), or else • X1 (a) = E1 (µ; a) and X2 (a) = F1 (µ; a) for all a ∈ Yˆ1 (if (2) in Theorem 3.4.1 holds). PROOF Put Yˆ1 := Y1 ∩ Y˜1 ∩ Y˜0 , where Y1 is as in Theorem 3.2.4, Y˜1 is as in Theorem 3.2.6 and Y˜0 is as in Theorem 3.4.1. The characterization of E1 (µ; a) given in Theorem 3.4.1 together with Theorem 3.2.6 implies that E1 (µ; a) = X1 (a) for all a ∈ Yˆ1 . For each a ∈ Yˆ1 define   F∞ (µ; a) X20 (a) := E2 (µ; a) ⊕ · · · ⊕ Ek (µ; a) ⊕ F∞ (µ; a)   F1 (µ; a), depending on which property in Theorem 3.4.1 holds. In each case X20 (a) is a subspace of L2 (D) of codimension one. Fix a ∈ Yˆ0 . Take any nonzero u0 ∈ X20 (a). If in Theorem 3.4.1 (2) holds then lim supt→∞ (1/t) ln kUa (t, 0)u0 k ≤ λ2 (µ) < λ(µ), hence u0 ∈ X2 (a) by Corollary 3.2.1. If in Theorem 3.4.1 (1) holds with k = 1 then we have limt→∞ (1/t) ln kUa (t, 0)u0 k = −∞, hence u0 ∈ X2 (a) by Corollary 3.2.1. Assume now that (1) in Theorem 3.4.1 holds with k > 1. Write u0 in the E2 (µ; a)⊕· · ·⊕Ek (µ; a)⊕F∞ (µ; a) decomposition as uj(1) +· · ·+uj(m) , where 1 < j(1) < · · · < j(m) ≤ ∞ and all uj(1) , . . . , uj(m) are nonzero. Take some ∗ λ∗ ∈ (λ2 (µ), λ1 (µ)). There exists T ≥ 0 such that kUa (t, 0)uj(l) k ≤ eλ t for all

3. Spectral Theory in the General Setting

107 ∗

t ≥ T and all l = 1, . . . , m. Consequently, kUa (t, 0)u0 k ≤ meλ t for all t ≥ T , hence lim supt→∞ (1/t) ln kUa (t, 0)u0 k ≤ λ∗ < λ(µ). Therefore u0 ∈ X2 (a) by Corollary 3.2.1. We have proved that X20 (a) is a one-codimensional subspace contained in the one-codimensional subspace X2 (a). As a consequence, X20 (a) = X2 (a).

3.5

The Smooth Case

Consider (2.0.1)+(2.0.2) and assume (A2-5). Then (A2-1)–(A2-3) as well as (A3-1) and (A3-2) are satisfied for both (2.0.1)+(2.0.2) and (2.3.1)+ (2.3.2) (see [59] and [61]). Let Y0 be a compact connected invariant subset of Y . By Theorem 3.3.3, Π admits an exponential separation over Y0 with invariant complementary subbundles X1 and X2 , where X1 (a) = span {w(a)} and X2 (a) = {v ∈ L2 (D) : hv, w∗ (a)i = 0}, with kw(a)k = kw∗ (a)k = 1 for all a ∈ Y0 . From now on, until the end of the section, X stands for a Banach space such that Wp2 (D) ,−,→ X ,→ L2 (D) (3.5.1) (recall that ,− ,→ denotes a compact embedding). Examples of spaces X are interpolation spaces Vpβ and V˜pβ (when p ≥ 2 ¯ (when p > N/2), and C 1 (D) ¯ (when p > N ), see and 0 < β < 1), C(D) Lemma 2.5.5. For a space X satisfying (3.5.1) we have the following stronger exponential separation theorem. THEOREM 3.5.1 Assume that X is a Banach space satisfying (3.5.1). Then for any δ0 > 0 ˇ =M ˇ (δ0 , X) > 0 such that there is M kUa (t, 0)u0 kX ˇ e−γ0 t ku0 k ≤M kUa (t, 0)w(a)kX kw(a)k for any a ∈ Y0 , u0 ∈ X2 (a), and t ≥ δ0 , where γ0 > 0 is as in Theorem 3.3.3. THEOREM 3.5.2 Assume that X = Vpβ or X = V˜pβ with p ≥ 2 and 0 < β < 1, and denote, for each a ∈ Y , X(a) := Vpβ (a) or X(a) := V˜pβ (a), respectively. Then there is c=M c(X) > 0 such that M kUa (t, 0)u0 kX ce−γ0 t ku0 kX ≤M kUa (t, 0)w(a)kX kw(a)kX

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for any a ∈ Y0 , u0 ∈ X2 (a) ∩ X(a), and t ≥ 0, where γ0 > 0 is as in Theorem 3.3.3. REMARK 3.5.1 We remark that the exponential separation in the smooth case does not follow from the abstract result in [94] directly due to the lack of the proper continuity of Ua (t, 0) at t = 0, which results from the time dependence of the boundary conditions. But under some additional assumption on the continuous dependence of the evolution operator Ua (t, 0)u0 on a ∈ Y0 , t ≥ 0, and u0 ∈ L2 (D), the existence of exponential separation can be proved by the abstract results in [94] together with Theorem 3.2.2 (this approach is utilized in Chapter 6 and in the paper [84]). To prove Theorems 3.5.1 and 3.5.2, we first show the following two lemmas. LEMMA 3.5.1 The functions w, w∗ : Y0 → X are continuous. PROOF Recall that ra·(−1) (1) = kUa·(−1) (1, 0)w(a · (−1))k. The function [ Y0 3 a 7→ ra·(−1) (1) ] is bounded above and bounded away from 0. Now w(a) = Ua·(−1) (1, 0)w(a · (−1))/ra·(−1) (1). It then follows from Proposition 2.5.2 that the set { w(a) : a ∈ Y0 } is bounded in Wp2 (D), consequently is relatively compact in X. Assume that a(n) → a. Then w(a(n) ) → w(a) in L2 (D). By the above ∗ (nk ) arguments, there are a subsequence (nk )∞ )→ k=1 and u ∈ X such that w(a ∗ ∗ (n) u in X. Therefore we must have u = w(a) and w(a ) → w(a) in X. In a similar way we prove that w∗ : Y0 → X is continuous. f1 = Observe that by the above lemma and the compactness of Y0 there is M f M1 (X) ≥ 1 such that 1 f1 kw(a)k kw(a)k ≤ kw(a)kX ≤ M f M1

for any

x ∈ Y0 .

(3.5.2)

LEMMA 3.5.2 For each δ0 > 0 there is C ∗ = C ∗ (δ0 ) > 0 such that kUa (δ0 , 0)kL2 (D),Wp2 (D) ≤ C ∗ (δ0 )

for any a ∈ Y.

PROOF It is a consequence of the L2 –Lp estimates (Proposition 2.2.2) and Proposition 2.5.2.

3. Spectral Theory in the General Setting PROOF (Proof of Theorem 3.5.1)

109

By Lemma 3.5.2

kUa (t, 0)u0 kWp2 (D) ≤ C ∗ (δ0 )kUa (t − δ0 , 0)u0 k for t ≥ δ0 . Further, we have ˇ 1 kUa (t, 0)u0 kW 2 (D) kUa (t, 0)u0 kX ≤ M p ˇ 1 denotes the norm of the embedding Wp2 (D) ,− for t ≥ δ0 , where M ,→ X. Since kUa (t, 0)w(a)kX ≥

ˇ2 M 1 kUa (t, 0)w(a)k ≥ kUa (t − δ0 , 0)w(a)k f f M1 M1

ˇ 2 := inf{ kUa (δ0 , 0)w(a)k : a ∈ Y0 } > 0, we have for all t ∈ R, where M ˇ 1M f1 kUa (t − δ0 , 0)u0 k kUa (t, 0)u0 kX C ∗ (δ0 )M ≤ ˇ kUa (t, 0)w(a)kX kUa (t − δ0 , 0)w(a)k M2 for t ≥ δ0 . Theorem 3.3.3 provides the existence of M > 0 and γ0 > 0 such that kUa (t − δ0 , 0)u0 k ku0 k ≤ M e−γ0 (t−δ0 ) kUa (t − δ0 , 0)w(a)k kw(a)k for t ≥ δ0 . Consequently, kUa (t, 0)u0 kX ˇ e−γ0 t ku0 k ≤M kUa (t, 0)w(a)kX kw(a)k ˇ = M C ∗ (δ0 )M ˇ 1M ˜ 1 eγ0 /M ˇ 2. for t ≥ δ0 , where M PROOF (Proof of Theorem 3.5.2) By Proposition 2.5.3, there is C > 0 such that kUa (t, 0)u0 kX ≤ Cku0 kX for any a ∈ Y , 0 ≤ t ≤ 2 and u0 ∈ X(a). Consequently kUa (t, 0)u0 kX ≤ Ce2γ0 e−γ0 t ku0 kX for a ∈ Y , 0 ≤ t ≤ 2 and u0 ∈ X(a). On the other hand, kUa (t, 0)w(a)kX ≥

ˇ3 M kw(a)kX f2 M 1

ˇ 3 := inf{ kUa (t, 0)w(a)k : a ∈ Y0 , t ∈ [0, 2] } > 0. for 0 ≤ t ≤ 2, where M Hence f2 e2γ0 CM kUa (t, 0)u0 kX ku0 kX 1 ≤ e−γ0 t ˇ kUa (t, 0)w(a)kX kw(a)kX M3

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Spectral Theory for Parabolic Equations

for 0 ≤ t ≤ 2, provided that u0 ∈ X(a). For t > 2 we have, with the help of Lemma 3.5.2, ˇ 4M f1 C ∗ (1) kUa (t − 1, 0)u0 k kU (t, 0)u0 kWp2 (D) M kUa (t, 0)u0 kX ˇ4 a ≤M ≤ , ˇ3 kUa (t, 0)w(a)kX kUa (t, 0)w(a)kX kUa (t − 1, 0)w(a)k M ˇ 4 > 0 stands for the norm of the embedding Wp2 (D) ,−,→ X. By where M Theorem 3.3.3, kUa (t − 1, 0)u0 k kUa (1, 0)u0 k ≤ M e−γ0 (t−2) . kUa (t − 1, 0)w(a)k kUa (1, 0)w(a)k Further, ˇ 5M f1 ku0 kX M kUa (1, 0)u0 k ≤ , ˇ 3 kw(a)kX kUa (1, 0)w(a)k M ˇ 5 := sup{ kUa (1, 0)kX,L (D) : a ∈ Y0 } < ∞ (by X ,→ L2 (D) and the where M 2 L2 –L2 estimates in Proposition 2.2.2). As a consequence, kUa (t, 0)u0 kX ce−γ0 t ku0 kX ≤M kUa (t, 0)w(a)kX kw(a)kX c= for t ≥ 0 and u0 ∈ X(a), where M

f2 e2γ0 M 1 ˇ3 M

max {C, C

∗

ˇ5 ˇ 4M (1)M }. ˇ3 M

For each a ∈ Y0 , let κ(a) be defined by κ(a) := −Ba (0, w(a), w(a)).

(3.5.3)

By (A2-5) and Lemma 3.5.1(2), the function κ : Y0 → R is well defined and continuous. Furthermore, we have the following two lemmas. LEMMA 3.5.3 For a ∈ Y and t ∈ R, put η(t; a) := kUa (t, 0)w(a)k. Then η(t; ˙ a) = κ(a · t)η(t; a) for any a ∈ Y and t ∈ R, where dot denotes the derivative in t. PROOF

Let u(t) := Ua (t, 0)w(a) (recall that for t < 0, Ua (t, 0)w(a) = Then by Proposition 2.1.4, we have Z t 2 2 (η(t; a)) − (η(s; a)) = −2 Ba (τ, u(τ ), u(τ ) dτ s Z t = −2 Ba·τ (0, η(τ ; a)w(a · τ ), η(τ ; a)w(a · τ )) dτ s Z t =2 κ(a · τ )(η(τ ; a))2 dτ

w(a·t) kUa·t (−t,0)w(a·t)k ).

s

3. Spectral Theory in the General Setting

111

for any a ∈ Y and s ≤ t. It then follows that η(t; ˙ a) = κ(a · t)η(t; a) for any a ∈ Y and t ∈ R. LEMMA 3.5.4 ¯ put v(t, x; a) := w(a · t)(x). Then v(·, ·) is an For a ∈ Y , t ∈ R and x ∈ D, entire classical solution of the parabolic equation  N N  ∂v X ∂ X ∂v   = a (t, x) + a (t, x)v ij i   ∂xi j=1 ∂xj   ∂t i=1  N X ∂v  bi (t, x) + + c0 (t, x)v − κ(a · t)v, t ∈ R, x ∈ D   ∂xi   i=1    Ba (t)v = 0, t ∈ R, x ∈ ∂D. PROOF Observe that Ua (t, 0)w(a) = kUa (t, 0)w(a)kw(a · t) = η(t; a)w(a · t). The lemma then follows from Lemma 3.5.3. Similarly to Theorems 3.1.5 and 3.2.4 we have THEOREM 3.5.3 Let µ be an ergodic invariant measure for σ|Y0 . There exists a Borel set Y˜ ⊂ Y0 with µ(Y˜ ) = 1 such that λ(µ) = lim

t→∞

ln kUa (t, 0)w(a)k ln kUa (t, 0)w(a)kX = lim t→∞ t t Z Z 1 t = lim κ(a · τ ) dτ = κ dµ t→∞ t 0 Y0

for all a ∈ Y˜ . COROLLARY 3.5.1 Assume that µ is an ergodic invariant measure for σ|Y0 . Then for µ-a.e. a ∈ Y0 and each nonzero u0 ∈ L2 (D)+ one has λ(µ) = lim

t→∞

ln kUa (t, 0)u0 k ln kUa (t, 0)u0 kX = lim . t→∞ t t

PROOF (Proof of Theorem 3.5.3) with µ(Y1 ) = 1 such that

The existence of a Borel set Y1 ⊂ Y0

ln kUa (t, 0)w(a)k 1 = lim t→∞ t→∞ t t

Z

t

κ(a · τ ) dτ

λ(µ) = lim

0

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Spectral Theory for Parabolic Equations

for all a ∈ Y1 follows by Theorem 3.1.5 and Proposition 3.2.9. The fact that lim

t→∞

ln kUa (t, 0)w(a)k ln kUa (t, 0)w(a)kX = lim t→∞ t t

is a consequence of (3.5.2). The use of Birkhoff’s Ergodic Theorem (Lemma 1.2.6) establishes the existence of a Borel set Y2 ⊂ Y0 with µ(Y2 ) = 1 such that 1 lim t→∞ t

Z

t

Z κ(a · τ ) dτ =

0

κ dµ Y0

for all a ∈ Y2 . It suffices now to put Y˜ := Y1 ∩ Y2 . PROOF (Proof of Corollary 3.5.1) The proof goes along the lines of the proof of Theorem 3.2.4, with the L2 (D)-norm replaced by the X-norm, and Definition 3.2.1 replaced by Theorem 3.5.1. The following result is an analog of Theorem 3.3.4. THEOREM 3.5.4 e = K(δ e 0 , X) > 0 such that For any δ0 > 0 there exists K kP2 (a · t)Ua (t, 0)u0 kX e ≤K kP1 (a · t)Ua (t, 0)u0 kX for all t ≥ δ0 , all a ∈ Y0 , and all nonzero u0 ∈ L2 (D)+ . PROOF

By Theorem 3.3.4, there exists K > 0 such that kP2 (a · kP1 (a ·

δ0 δ0 2 )Ua ( 2 , 0)u0 k δ0 δ0 2 )Ua ( 2 , 0)u0 k

≤K

for any a ∈ Y0 and any nonzero u0 ∈ L2 (D)+ . Theorem 3.5.1 implies the c > 0 such that existence of M δ kP2 (a · t)Ua (t, 0)u0 kX ce−γ0 (t− 20 ) kP2 (a · ≤M kP1 (a · t)Ua (t, 0)u0 kX kP1 (a ·

δ0 δ0 2 )Ua ( 2 , 0)u0 k δ0 δ0 2 )Ua ( 2 , 0)u0 k

for any a ∈ Y0 , any nonzero u0 ∈ L2 (D)+ , and any t ≥ δ0 . It suffices to put e := K M ceγ0 δ0 /2 . K

3. Spectral Theory in the General Setting

3.6

113

Remarks on the General Nondivergence Case

Consider N N X X ∂u ∂2u ∂u = + + c0 (t, x)u, aij (t, x) bi (t, x) ∂t ∂x ∂x ∂x i j i i,j=1 i=1

t > s, x ∈ D, (3.6.1)

complemented with the boundary conditions B(t)u = 0,

t > s, x ∈ ∂D,

(3.6.2)

where D ⊂ RN is a bounded domain, s ∈ R is an initial time, and B is a boundary operator of either the Dirichlet or Neumann or Robin type, that is,   u (Dirichlet)      X  N ∂xi u¯ νi (t, x) (Neumann) B(t)u =  i=1   N  X    ∂xi u¯ νi (t, x) + d0 (t, x)u, (Robin)   i=1

where (in the Neumann or Robin cases) (¯ ν1 , . . . , ν¯N ) is (in general time dependent) a vector field on ∂D pointing out of D. First of all, if both the domain D and the coefficients PN are sufficiently smooth and, in the Neumann or Robin cases, ν¯i (t, x) = j=1 aji (t, x)νj (x), 1 ≤ i ≤ N , (that is, the derivative is conormal), then (3.6.1)+(3.6.2) can be written in the divergence form and then the results in the previous sections hold for (3.6.1)+(3.6.2). Historically, when the domain D and the coefficients are sufficiently smooth and the boundary conditions are independent of time, the existence of exponential separation has been proved in [94] (see also [76], [102]; compare [95] for a finite-dimensional counterpart). The existence and uniqueness of entire positive solutions has been proved in [77], [78], [92]. Recently, in [84] the authors proved the exponential separation as well as the existence and uniqueness of entire positive solutions in a general nondivergence case and with time dependent boundary conditions but assuming that the domain and the coefficients are smooth enough. In [60] H´ uska and Pol´ aˇcik proved the existence of exponential separation and existence and uniqueness of entire positive solutions in a general divergence case with the Dirichlet boundary condition, with weak assumptions on the regularity of the coefficients. As regards a general nondivergence case, recently H´ uska, Pol´aˇcik, and Safonov in [61] proved the existence of exponential separation and existence and

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Spectral Theory for Parabolic Equations

uniqueness of entire positive solutions of (3.6.1)+(3.6.2) when the boundary condition is Dirichlet. In [59], H´ uska studied (3.6.1) with a general oblique boundary conditions and showed the exponential separation between a positive solution and sign-changing solutions and the uniqueness of entire positive solutions, both under the condition that an entire positive solution exists. However, the question of existence of entire positive solutions was not addressed in [59].

3.7

Appendix: The Case of One-Dimensional Spatial Domain

In the present appendix we consider the case when the domain D is onedimensional. Let Y0 = Y be defined as Y := { a = (1, 0, c, 0) : kckL∞ (R×[0,π]) ≤ R } for some R > 0. The set Y is (norm-)bounded in L∞ (R × [0, π], R4 ). Y is considered with the weak-* topology. It is a compact connected metrizable space. For a ∈ Y and t ∈ R there holds a · t ∈ Y . Consider the family of partial differential equations  2   ∂u = ∂ u + c(t, x)u, t > 0, x ∈ (0, π), ∂t ∂x2 (3.7.1)  u(t, 0) = u(t, π) = 0, t > 0, parameterized by a = (1, 0, c, 0) ∈ Y . For a ∈ Y and u0 ∈ L2 ((0, π)) denote by [ [0, ∞) 3 t 7→ Ua (t, 0)u0 ∈ L2 (0, π)) ] the unique (weak) solution of (3.7.1) satisfying the initial condition u(0, ·) = u0 . For a fixed a ∈ Y we will write (3.7.1)a . 2 Denote by A2 the realization in L2 ((0, π)) of the operator [ u 7→ ∂∂xu2 ] with the Dirichlet boundary conditions. Denote by {eA2 t }t≥0 the analytic semigroup generated on L2 ((0, π)) by A2 . A continuous function u : [0, ∞) → L2 ((0, π)) is a mild solution of (3.7.1)a satisfying the initial condition u(0, ·) = u0 if for any t > 0 there holds u(t) = eA2 t u0 +

Z

t

eA2 (t−τ ) (C(τ )u(τ )) dτ,

0

where, for τ ∈ (0, τ ], (C(τ )u(τ ))(x) := c(t, x)u(t)(x) for a.e. x ∈ (0, π). We claim that for each a ∈ Y and u0 ∈ L2 ((0, π)) the weak and mild solutions are

3. Spectral Theory in the General Setting

115

the same. Indeed, if c is sufficiently smooth then both are classical solutions, and the claim follows from the uniqueness of classical solutions. For a general c we use approximation along the lines of the proof of [33, Proposition 4.2]. Consequently, we can use results in [22], where solutions of (3.7.1) were defined as mild solutions. ˚1 ([0, π]) denotes the (closed) vector subspace consisting of Recall that C 1 those φ ∈ C ([0, π]) for which φ(0) = φ(π) = 0. The following result was proved as [22, Theorem 3.3]. PROPOSITION 3.7.1 (1) For each T > 0 the mapping [ Y × L2 ((0, π)) 3 (a, u0 ) 7→ Ua (·, 0)u0 |[0,T ] ∈ C([0, T ], L2 ((0, π))) ] is continuous. (2) For any a ∈ Y , t > 0, and u0 ∈ L2 ((0, π)) there holds Ua (t, 0)u0 ∈ ˚1 ([0, π]). Moreover, for any 0 < t1 ≤ t2 the mapping C ˚1 ([0, π])) ] [ Y × L2 ((0, π)) 3 (a, u0 ) 7→ Ua (·, 0)u0 |[t1 ,t2 ] ∈ C([t1 , t2 ], C is continuous, and there is M = M (t1 , t2 ) > 0 such that kUa (t, 0)kL2 ((0,π)),C˚1 ([0,π]) ≤ M for each t ∈ [t1 , t2 ]. For a given nonzero φ ∈ C([0, π]) such that φ(0) = φ(π) = 0 define the lap or Matano number of φ to be z(φ) = sup{ l ≥ 1 : there exist 0 < x1 < x2 < · · · < xl < π such that φ(xk )φ(xk+1 ) < 0

for

1 ≤ k ≤ l − 1}

with z(φ) = 1 if either φ(x) ≥ 0 or φ(x) ≤ 0 for all x ∈ [0, π]. We have the following result (see [22, p. 257] and the references contained therein). LEMMA 3.7.1 (i) z(Ua (t, 0)u0 ) < ∞ for any a ∈ Y , any t > 0, and any nonzero u0 ∈ L2 ((0, π)). (ii) For any a ∈ Y and any nonzero u0 ∈ L2 ((0, π)) the function [ (0, ∞) 3 t 7→ z(Ua (t, 0)u0 ) ∈ N ] is nonincreasing.

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Spectral Theory for Parabolic Equations

As a byproduct of Lemma 3.7.1(i) we have that for each a ∈ Y and each t > 0 the operator Ua (t, 0) is injective. Define an exponentially bounded solution of (3.7.1) to be an entire solution u : R → L2 ((0, π)) such that there are constants K1 , K2 with ku(t)k ≤ K1 eK2 |t|

for t ∈ R.

For a ∈ Y and i = 1, 2, . . . let Xi0 (a) ⊂ L2 ((0, π)) be defined by Xi0 (a) := { φ ∈ L2 ((0, π)) : φ = u(0) for some exponentially bounded solution u : R → L2 ((0, π)) of (3.7.1)a satisfying z(u(t)) = i for all t ∈ R } ∪ {0}. By [22, Theorem 5.1 and Proposition 5.2], Xi0 (a) is a one-dimensional vector subspace, for each i = 1, 2, . . . and each a ∈ Y . This allows us to define, for a ∈ Y and i = 1, 2, . . . , ! ∞ M 00 0 Xi (a) := cl Xj (a) , j=i

where the closure is considered in the L2 ((0, π))-norm. PROPOSITION 3.7.2 For each i = 1, 2, . . . the following holds. 00 00 (i) Ua (t, 0)Xi0 (a) = Xi0 (a · t) and Ua (t, 0)Xi+1 (a) ⊂ Xi+1 (a · t) for each a ∈ Y and each t ≥ 0.

(ii) There exists a continuous function wi : Y → L2 ((0, π)) such that for i (a) (0) > 0, and Xi0 (a) = span{wi (a)}; each a ∈ Y , kwi (a)k = 1, ∂w∂x moreover, such a function is unique. 00 (iii) z(φ) ≥ i + 1 for any a ∈ Y and any nonzero φ ∈ Xi+1 (a). 00 (iv) L2 (D) = X10 (a) ⊕ X20 (a) ⊕ · · · ⊕ Xi0 (a) ⊕ Xi+1 (a) for any a ∈ Y .

(v) There are constants Ki > 0 and γi > 0 such that kUa (t, 0)u0 k ≤ Ki e−γi t kUa (t, 0)wi (a)k 00 for any a ∈ Y , any t ≥ 0, and any u0 ∈ Xi+1 (a) with ku0 k = 1.

S From (i) and (ii) it follows that Xi0 := a∈Y ({a} × Xi0 (a)) is a trivial one-dimensional subbundle of L2 (D) × Y , invariant under Π. PROOF orem 7.1]

See [22, Proposition 4.6, Theorem 5.1, Proposition 6.2 and The-

3. Spectral Theory in the General Setting

117

Notice that from the above proposition it follows that Π admits an exponential separation. Indeed, we take X1 (a) = X10 (a) and X2 (a) = X200 (a) (in fact, X200 (a) being equal to { v ∈ L2 ((0, π)) : hv, w∗ (a)i = 0 } for some w∗ (a) ∈ L2 ((0, π))+ is not explicitly mentioned in Proposition 3.7.2, but it follows from the method of proving relevant theorems in [22].) THEOREM 3.7.1 Assume that µ is an ergodic invariant measure for σ|Y0 . Then condition (2 ) in Theorem 3.4.1 holds. More precisely, there exist: • a Borel set Y˜0 ⊂ Y0 , µ(Y˜0 ) = 1, and • a sequence of real numbers λ1 (µ) > · · · > λi (µ) > λi+1 (µ) > . . . having limit −∞, such that lim

t→±∞

ln kUa (t, 0)wi (a)k = λi (µ) t

for each a ∈ Y˜0 and each i = 1, 2, . . . . Further, for each a ∈ Y˜0 and each i = 1, 2, . . . there holds Ei (µ; a) = Xi0 (a)

and

00 Fi (µ; a) = Xi+1 (a).

PROOF For each i = 1, 2, . . . an application of [65, Theorem 2.1] to the topological linear skew-product flow Π|Xi0 on the one-dimensional bundle Xi0 gives the existence of Borel set Yˆ0,i ⊂ Y0 , µ(Yˆ0,i ) = 1, and a real number λ0i such that ln kUa (t, 0)wi (a)k = λ0i lim t→±∞ t T∞ for each a ∈ Yˆ0,i . Put Yˆ0 := i=1 Yˆ0,i . Obviously Yˆ0 is a Borel set, with µ(Yˆ0 ) = 1. It is a consequence of Corollary 3.4.1 and the remarks below Proposition 3.7.2 that λ01 = λ1 (µ), E1 (µ; a) = X10 (a) and F1 (µ; a) = X200 (a), for µ-a.e. a ∈ Yˆ0 . Assume that we already have that, for some i = 1, 2, . . . , there holds 00 λ0i = λi (µ), Ei (µ; a) = Xi0 (a) and Fi (µ; a) = Xi+1 (a), for µ-a.e. a ∈ Yˆ0 . By Theorem 3.4.1, λi+1 (µ) = lim (1/t) ln kUa (t, 0)|Fi (µ;a) k t→∞

for µ-a.e. a ∈ Yˆ0 . From Proposition 3.7.2 we deduce that 00 (µ;a) k λ0i+1 = lim (1/t) ln kUa (t, 0)|Xi+1

t→∞

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Spectral Theory for Parabolic Equations

for all a ∈ Yˆ0 . So our induction assumption gives that λ0i+1 = λi+1 (µ). 0 Further, from Theorem 3.4.1 it follows that Ei+1 (µ; a) = Xi+1 (a) for µˆ a.e. a ∈ Y0 . Suppose to the contrary that for some a ∈ Yˆ0 there is v ∈ Fi+1 (µ; a) \ 00 0 Xi+2 (a). Decompose v = vi+1 + vi+2 , where vi+1 ∈ Xi+1 (a) \ {0} and vi+2 ∈ 00 Xi+2 (a). It follows from Proposition 3.7.2 that limt→∞ (1/t) ln kUa (t, 0)vk = limt→∞ (1/t) ln kUa (t, 0)vi+1 k = λi+1 , consequently lim (1/t) ln kUa (t, 0)|Fi+1 (µ;a) k ≥ λi+1 > λi+2 ,

t→∞

00 which is impossible. Therefore, Fi+1 (µ; a) ⊂ Xi+2 (a). But both are subspaces 00 of Xi+1 (a), of relative codimension one, so they must be equal.

Chapter 4 Spectral Theory in Nonautonomous and Random Cases

In this chapter, we consider principal spectrum and principal Lyapunov exponents of nonautonomous and random parabolic equations. First in Section 4.1, we introduce basic assumptions for a given random (nonautonomous) parabolic equation and associate a proper family of parabolic equations with the given equation. Then based on the notions introduced in Chapters 2 and 3 in the general setting, we introduce the concepts of principal spectrum and principal Lyapunov exponents for a given random (nonautonomous) parabolic equation in terms of the associated family of parabolic equations, which naturally extends the classical concept of principal eigenvalue for the elliptic and periodic parabolic problems. Also applying the general theories developed in Chapters 2 and 3, we present basic properties of principal spectrum and principal Lyapunov exponents of random (nonautonomous) parabolic equations. In addition, we provide some examples which satisfy the basic assumptions in this section. In Section 4.2, we investigate the monotonicity of principal spectrum and principal Lyapunov exponents of random (nonautonomous) parabolic equations with respect to zero order terms. We also study the relation among the principal spectrum and principal Lyapunov exponents for random (nonautonomous) parabolic equations with different types of boundary conditions. Sections 4.3 and 4.4 concern the continuous dependence of principal spectrum and principal Lyapunov exponents of random (nonautonomous) parabolic equations with respect to the coefficients. Because of the speciality of the zero order coefficients, the continuous dependence with respect to these coefficients are considered in Section 4.3 first. In Section 4.4, the general continuous dependence is then discussed. Throughout Section 4.1 to Section 4.4, many results and arguments for random and nonautonomous equations are similar. However, considering that different readers may be interested in different types of equations, for convenience, in each section from Section 4.1 to Section 4.4, we treat these two types of equations in different subsections and provide proofs for most similar results. This chapter ends up with some historical remarks in Section 4.5.

119

120

4.1

Spectral Theory for Parabolic Equations

Principal Spectrum and Principal Lyapunov Exponents in Random and Nonautonomous Cases

This section is to introduce basic assumptions, concepts, and properties. We first introduce basic assumptions for a given random (nonautonomous) parabolic equation and associate a proper family of parabolic equations with the given equation. Next, based on the notions introduced in Chapters 2 and 3 in the general setting, we introduce the concepts of principal spectrum and principal Lyapunov exponents for a given random (nonautonomous) parabolic equation in terms of the associated family of parabolic equations, which naturally extends the classical concept of principal eigenvalue for the elliptic and periodic parabolic problems. We provide some examples which satisfy the basic assumptions. Then applying the general theories developed in Chapters 2 and 3, we present basic properties of principal spectrum and principal Lyapunov exponents of random (nonautonomous) parabolic equations. For the reader’s convenience, random and nonautonomous cases are treated separately.

4.1.1

The Random Case

Assume that ((Ω, F, P), {θt }t∈R ) is an ergodic metric dynamical system. Consider the following random linear parabolic equation: N N ∂u ∂u X ∂ X = aij (θt ω, x) + ai (θt ω, x)u ∂t ∂xi j=1 ∂xj i=1 +

N X i=1

bi (θt ω, x)

∂u + c0 (θt ω, x)u, ∂xi

x ∈ D,

(4.1.1)

endowed with the boundary condition Bω (t)u = 0,

x ∈ ∂D,

(4.1.2)

where Bω (t) = Baω (t), Baω (t) is as in (2.0.3) with a being replaced by aω , aω (t, x) = (aij (θt ω, x), ai (θt ω, x), bi (θt ω, x), c0 (θt ω, x), d0 (θt ω, x)), and d0 (ω, x) ≥ 0 for all ω ∈ Ω and a.e. x ∈ ∂D. To emphasize the coefficients in (4.1.1)+(4.1.2), we will write (4.1.1)a +(4.1.2)a . Our first assumption in the present subsection concerns measurability of the coefficients of the equation and of the boundary conditions (recall that for a metric space S the symbol B(S) stands for the countably additive algebra of Borel sets): (A4-R1) (Measurability) The functions aij (= aji ) (i, j = 1, . . . , N ), ai (i = 1, . . . , N ), bi (i = 1, . . . , N ), and c0 are (F × B(D), B(R))-measurable, and the function d0 is (F × B(∂D), B(R))-measurable.

4. Spectral Theory in Nonautonomous and Random Cases

121

Among others, (4.1.1)+(4.1.2) arise from linearization of random nonlinear parabolic equations at a certain entire solution (i.e., a solution which exists for all t ∈ R) as well as from linearization of autonomous nonlinear equations at some invariant set of solutions. ω ω For each ω ∈ Ω, let aω ij (t, x) := aij (θt ω, x), ai (t, x) := ai (θt ω, x), bi (t, x) := ω ω bi (θt ω, x), c0 (t, x) := c0 (θt ω, x), d0 (t, x) := d0 (θt ω, x). The functions [ Ω × R × D 3 (ω, t, x) 7→ aω ij (t, x) ∈ R ] are (F × B(R) × B(D), B(R))-measurable (as composites of Borel measurω ω able functions). Similarly, the functions aω i (t, x), bi (t, x), and c0 are (F × B(R) × B(D), B(R))-measurable and the function dω (t, x) is (F × B(R) × 0 B(∂D), B(R))-measurable. ω ω As sections of Borel measurable functions, the functions aω ij , ai , bi , and ω c0 are (B(R) × B(D), B(R))-measurable, for any fixed ω ∈ Ω. Similarly, the function dω 0 is (B(R) × B(∂D), B(R))-measurable, for any fixed ω ∈ Ω. N ω N ω N ω ω We write aω := ((aω ij )i,j=1 , (ai )i=1 , (bi )i=1 , c0 , d0 ). Sometimes we write the random problem (4.1.1)+(4.1.2) as (4.1.1)a +(4.1.2)a . Our second assumption regards uniform boundedness of the coefficients of the equations (and of the boundary conditions): (A4-R2) (Boundedness) 2 For each ω ∈ Ω, aω belongs to L∞ (R × D, RN +2N +1 ) × L∞ (R × ∂D, R). 2 Moreover, the set { aω : ω ∈ Ω } is bounded in the L∞ (R × D, RN +2N +1 ) × L∞ (R × ∂D, R)-norm by M ≥ 0. Define the mapping Ea : Ω → L∞ (R × D, RN

2

+2N +1

) × L∞ (R × ∂D, R) as

Ea (ω) := aω . Put Y˜ (a) := cl { Ea (ω) : ω ∈ Ω }

(4.1.3)

with the weak-* topology, where the closure is taken in the weak-* topology. The set Y˜ (a) is a compact metrizable space and (Y˜ (a), {σt }t∈R ) is a compact flow, where σt a ˜(·, ·) = a ˜(· + t, ·). LEMMA 4.1.1 The mapping Ea is (F, B(Y˜ (a)))-measurable. PROOF Recall that {g1 , g2 , . . . } is a countable dense subset of the unit 2 ball in L1 (R × D, RN +2N +1 ) × L1 (R × ∂D, R) (see (1.3.1)). It is clear that for each a ˜ ∈ Y˜ (a) and k ∈ N the function [ Ω 3 ω 7→ hgk , (˜ a − aω )iL1 ,L∞ ∈ R ]

122

Spectral Theory for Parabolic Equations

is (F, B(R))-measurable. This implies that Ea is (F, B(Y˜ (a)))-measurable. An important property of the mapping Ea is the following σt ◦ Ea = Ea ◦ θt

for each t ∈ R.

(4.1.4)

It follows that Ea (Ω) is {σt }-invariant. Consequently, Y˜ (a) is {σt }-invariant, too. The mapping Ea is a homomorphism of the measurable flow ((Ω, F), {θt }t∈R ) ˜ the iminto the measurable flow ((Y˜ (a), B(Y˜ (a))), {σt }t∈R ). Denote by P ˜ ˜ age of the measure P under Ea : for any Borel set A ∈ B(Y (a)), P(A) := −1 ˜ ˜ P(Ea (A)). P is a {σt }-invariant ergodic Borel measure on Y (a). So, Ea is a homomorphism of the metric flow ((Ω, F, P), {θt }t∈R ) into the metric flow ˜ {σt }t∈R ). ((Y˜ (a), B(Y˜ (a)), P), We will consider (Y˜ (a), {σt }t∈R ) a topological flow, with an ergodic invariant ˜ Put measure P. ˜ Y˜0 (a) := supp P (4.1.5) ˜ ˜ ˜ (˜ a ∈ Y0 (a) if and only if for any neighborhood U of a ˜ in Y (a) one has P(U ) > 0). Y˜0 (a) is a closed (hence compact) and {σt }-invariant subset of Y˜ (a), with ˜ Y˜0 (a)) = 1. Also, Y˜0 (a) is connected, since otherwise there would exist two P( open sets U1 , U2 ⊂ Y˜ (a) such that Y˜0 (a) ∩ U1 and Y˜0 (a) ∩ U2 are nonempty, compact and disjoint, and their union equals Y˜0 (a). The sets Y˜0 (a) ∩ U1 and Y˜0 (a) ∩ U2 are invariant, and, by the definition of support, each of them has ˜ ˜ P-measure positive, which contradicts the ergodicity of P. LEMMA 4.1.2 There exists Ω0 ⊂ Ω with P(Ω0 ) = 1 such that Y˜0 (a) = cl { E(θt ω) : t ∈ R } for any ω ∈ Ω0 , where the closure is taken in the weak-* topology on Y . PROOF By [89, Theorem 9.27], there exists a Borel set Y 0 ⊂ Y˜0 (a) with ˜ P(Y 0 ) = 1 with the property that for each a ˜ ∈ Y 0 there holds Z Z 1 t ˜ h(σs a ˜) ds = h(·) dP(·), lim t→∞ t 0 Y˜0 (a) for any h ∈ C(Y˜0 (a)). We claim that cl { σt a ˜ : t ≥ 0 } = Y˜0 (a) for any a ˜ ∈ Y 0 . Suppose not. Then there are a ˜ ∈ Y 0 and a ¯ ∈ Y˜0 (a) such that a ¯ 6∈ cl { σt a ˜ : t ≥ 0 } =: Y 00 . By the Urysohn lemma, there is a nonnegative h ∈ ˜ C(Y0 (a)) such that h(ˆ a) = 0 for any a ˆ ∈ Y 00 and h(¯ a) > 0. From the former R 1 t property it follows that limt→∞ t 0 h(σs a ˜) ds = 0. By continuity, there is a relative neighborhood V of a ¯ in Y˜0 (a) such a) > 0 for ˇ ∈ V . Since R that h(ˇ R any a ˜ ≥ ˜ > 0, a a ¯ belongs to the support of µ, we have Y˜0 (a) h(·) dP(·) h(·) dP(·) V contradiction. It suffices to put Ω0 := Ea−1 (Y 0 ).

4. Spectral Theory in Nonautonomous and Random Cases

123

If Assumptions (A2-1)–(A2-3) are satisfied for Y replaced with Y˜ (a), we will denote by Π(a) = {Π(a)t }t≥0 the topological linear skew-product semiflow generated by (4.1.1)+(4.1.2) on the product Banach bundle L2 (D) × Y˜ (a): Π(a)(t; u0 , a ˜) = Π(a)t (u0 , a ˜) := (Ua˜ (t, 0)u0 , σt a ˜) for t ≥ 0, a ˜ ∈ Y˜ (a), and u0 ∈ L2 (D), where Ua˜ (t, 0)u0 stands for the weak solution of (2.0.1)a˜ +(2.0.2)a˜ with the initial condition u(0, x) = u0 (x), x ∈ D. (Here, a ˜ = (˜ aij , a ˜i , ˜bi , c˜0 , d˜0 ).) Moreover, define ˜ u0 , ω) := (UE (ω) (t, 0)u0 , θt ω), Π(t; a

t ≥ 0, ω ∈ Ω, u0 ∈ L2 (D).

We have LEMMA 4.1.3 ˜ is a random linear skew-product semiIf (A2-1)–(A2-3) are satisfied, then Π flow on the measurable Banach bundle L2 (D) × Ω, covering the metric flow ((Ω, F, P), {θt }t∈R ). ˜ that for PROOF It follows from (4.1.4) and the definitions of Π and Π each t ≥ 0 the diagram ˜ Π

t → L2 (D) × Ω L2 (D) × Ω −−−−    (Id (IdL2 (D) ,E)y y L2 (D) ,E)

Π

t L2 (D) × Y −−−− → L2 (D) × Y

commutes. Consequently, the properties (RSP1) and (RSP2) of the random skew-product semiflow are satisfied. Obviously, for any t ≥ 0 and ω ∈ Ω the mapping UEa (ω) (t, 0) belongs to L(L2 (D)). It remains to prove that the mapping [ (t, u0 , ω) 7→ UEa (ω) (t, 0)u0 ] is (B([0, ∞)) × B(L2 (D)) × F, B(L2 (D)))-measurable. Indeed, for n ∈ N denote ˜ [n] (t; u0 , ω) := (U ˜ [n] (t; u0 , ω), θt ω) Π

for

t ≥ 0, u0 ∈ L2 (D), ω ∈ Ω,

where ( ˜ [n]

U

(t; u0 , ω) :=

UEa (ω) (1/n, 0)u0 UEa (ω) (t, 0)u0

for t ∈ [0, 1/n], u0 ∈ L2 (D), ω ∈ Ω, for t ∈ [1/n, ∞), u0 ∈ L2 (D), ω ∈ Ω,

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Spectral Theory for Parabolic Equations

and Π[n] (t; u0 , a ˜) := (U [n] (t; u0 , a ˜), σt a ˜)

for t ≥ 0, u0 ∈ L2 (D), a ˜ ∈ Y˜ (a),

where U

[n]

( Ua˜ (1/n, 0)u0 (t; u0 , a ˜) := Ua˜ (t, 0)u0

for t ∈ [0, 1/n], u0 ∈ L2 (D), a ˜ ∈ Y˜ (a), for t ∈ [1/n, ∞), u0 ∈ L2 (D), a ˜ ∈ Y˜ (a).

One has ˜ [n] = U [n] ◦ (IdR , IdL (D) , E). U + 2 Since Assumption (A2-3) is satisfied, the mapping U [n] is continuous. Further, (Id[0,∞) , IdL2 (D) , E) is (B([0, ∞)) × B(L2 (D)) × F, B([0, ∞)) × B(L2 (D)) × ˜ [n] is (B([0, ∞)) × B(L2 (D)) × F, B(Y˜ (a)))-measurable. Consequently, Π [n] ˜ ˜ the latter is B(L2 (D)) ×F)-measurable. As Π converge pointwise to Π, (B([0, ∞)) × B(L2 (D)) × F, B(L2 (D)) × F)-measurable, too. A next assumption regards the satisfaction of (A2-1)–(A2-3) by (4.1.1)a + (4.1.2)a or Y˜ (a): (A4-R) (Satisfaction of (A2-1)–(A2-3) and (A4-R1), (A4-R2)) The assumptions (A4-R1)–(A4-R2) are fulfilled and assumptions (A2-1)–(A2-3) are satisfied for Y replaced with Y˜ (a) defined by (4.1.3). ˜ or Π satisfies Sometimes we say simply that a or (4.1.1)a +(4.1.2)a or Π property (A4-R). DEFINITION 4.1.1 The principal spectrum of the random problem (4.1.1)a +(4.1.2)a satisfying property (A4-R) equals the principal spectrum of the topological linear skew-product semiflow Π(a) over Y˜0 (a). We will denote the principal spectrum by Σ(a) = [λmin (a), λmax (a)]. DEFINITION 4.1.2 The principal Lyapunov exponent of the random problem (4.1.1)a +(4.1.2)a satisfying property (A4-R) equals the principal Lyapunov exponent of the topological linear skew-product semiflow Π(a) over Y˜0 (a) ˜ We will denote the principal Lyapunov for the ergodic invariant measure P. exponent by λ(a). LEMMA 4.1.4 There exists Ω1 ⊂ Ω0 with P(Ω1 ) = 1, where Ω0 is as in Lemma 4.1.2, such that ln kUE(ω) (t, 0)k lim = λ(a) for any ω ∈ Ω1 . t→∞ t

4. Spectral Theory in Nonautonomous and Random Cases PROOF such that

125

˜ 1) = 1 By Theorem 3.1.5, there is a Borel set Y1 ⊂ Y˜0 with P(Y lim

t→∞

ln kUa˜ (t, 0)k = λ(a) t

for any a ˜ ∈ Y1 . It suffices to take Ω1 := E −1 (Y1 ) ∩ Ω0 . The following assumption is about exponential separation: (A4-R-ES) (Exponential separation) (4.1.1)a +(4.1.2)a has property (A4R) and, moreover, the topological linear skew-product semiflow Π(a) generated by (4.1.1)a +(4.1.2)a on L2 (D) × Y˜ (a) admits an exponential separation over Y˜0 (a) defined by (4.1.5). Sometimes we say simply that a satisfies property (A4-R-ES). We now give two examples of a satisfying the property (A4-R-ES). EXAMPLE 4.1.1 (Only zero-order terms depend on t) Consider the following random linear parabolic equation: N N ∂u X ∂ X ∂u = aij (x) + ai (x)u ∂t ∂xi j=1 ∂xj i=1 +

N X

bi (x)

i=1

∂u + c0 (θt ω, x)u, ∂xi

x ∈ D,

(4.1.6)

endowed with the boundary condition Bω (t)u = 0,

x ∈ ∂D,

(4.1.7)

where Bω (t) = Baω (t) and aω (t, x) = (aij (x), ai (x), bi (x), c0 (θt ω, x), d0 (x)). We make the following assumptions: In the Dirichlet case: • D ⊂ RN is a bounded domain with Lipschitz boundary. • aij (= aji ), ai , bi : D → R are (B(D), B(R))-measurable and aij (= aji ), ai , bi ∈ L∞ (D). Moreover, (A2-1) is satisfied. • c0 : Ω × D → R is (F × B(D), B(R))-measurable. Moreover, for any ω ∈ Ω, the function cω 0 (·) belongs to L∞ (R × D), with the L∞ (R × D)-norm bounded uniformly in ω ∈ Ω. In the Neumann or Robin cases: • D ⊂ RN is a bounded domain, where its boundary is an (N − 1)-dimensional manifold of class C 2 .

126

Spectral Theory for Parabolic Equations ¯ d0 ∈ C 1 (∂D). Moreover, (A2-1) is satisfied, • aij (= aji ), ai , bi ∈ C 1 (D), and d0 (x) ≥ 0 for all x ∈ ∂D. • c0 : Ω × D → R is (F × B(D), B(R))-measurable. Moreover, for any ω ∈ Ω, the function cω 0 (·) belongs to L∞ (R × D), with the L∞ (R × D)-norm bounded uniformly in ω ∈ Ω.

We claim that a random problem (4.1.6)+(4.1.7) satisfying the above requirements has the property (A4-R-ES). Indeed, assumptions (A4-R1)–(A4R2) and (A2-1)–(A2-2) are formulated explicitly. As aij , ai , bi , and d0 are independent of time, the condition (A2-4) is satisfied, so, by Theorem 2.4.1, the assumption (A2-3) holds. In the Dirichlet case, the inequalities (A3-1) and (A3-2) hold for any a ˜∈ Y˜0 (a), for both (2.0.1)a˜ +(2.0.2)a˜ and its adjoint, by [61, Theorem 2.1 and Lemma 3.9], so the topological linear skew-product flow Π(a) admits an exponential separation over Y˜0 (a). In the Neumann and Robin cases, we first show that any weak solution of (2.0.1)a˜ +(2.0.2)a˜ , as well as of its adjoint equation, is in fact a strong solution on any interval away from the initial time. In order to do so, fix a ˜ ∈ Y˜0 . We approximate a ˜ = (aij , ai , bi , c˜0 , d0 ) by 2 (n) ∞ a sequence (a )n=1 ⊂ L∞ (R × D, RN +2N +1 ) × L∞ (R × ∂D, R) such that (n) (n) (n) (n) (n) (where we write a(n) = (aij , ai , bi , c0 , d0 )): (n)

(n)

(n)

(n)

(i) aij (= aji ), ai , bi (n) d0

¯ (i, j = 1, 2, . . . , N ), c(n) ∈ C 2 (R × D), ¯ ∈ C 2 (D) 0 (n)

∈ C 2 (∂D); moreover, d0 (x) ≥ 0 for all n ∈ N and all x ∈ ∂D,

(ii) sup{ ka(n) k∞ : n ∈ N } < ∞, where k·k∞ denotes the L∞ (R × D, 2 RN +2N +1 ) ×L∞ (R × ∂D, R)-norm, (n)

(n)

(n)

(iii) aij (x), ai (x), bi (x) (i, j = 1, 2, . . . , N ) converge respectively to aij (x), ai (x), bi (x), for a.e. x ∈ D, (n) ¯ sense; moreover, the sections (iv) c0 converge to c˜0 in the L2,loc (R × D) (n) c0 (t, ·) converge in the L2 (D)-norm to the section c˜0 (t, ·) for a.e. t > 0, (n)

(v) d0 (x) converge to d0 (x) for a.e. x ∈ ∂D. Let Ua(n) (t, s) stand for the solution operator of (2.0.1)+(2.0.2) with a replaced by a(n) . Further, let Ua0 (t, s) denote the solution operator of (2.0.1)+ (2.0.2) with a replaced by (aij , ai , bi , 0, d0 ), and let Ua0(n) (t, s) denote the so(n)

(n)

(n)

(n)

lution operator of (2.0.1)+(2.0.2) with a replaced by (aij , ai , bi , 0, d0 ). ˚ ¯ For a given u0 ∈ L2 (D), let (un )∞ n=1 ⊂ C(D) be such that un → u0 in L2 (D). Ua(n) (·, 0)un is, for any n ∈ N, a classical solution, consequently it satisfies the following integral equation Z t (n) Ua(n) (t, 0)un = Ua0(n) (t, 0)un + Ua0(n) (t, τ )(c0 (τ, ·)Ua(n) (τ, 0)un ) dτ 0

4. Spectral Theory in Nonautonomous and Random Cases

127

for all t > 0 (here and in the sequel, we identify the operator of multiplying a (n) function from L2 (D) by the section c0 (τ, ·) with that section). Applying the ideas used in the proofs of Theorem 2.4.1 and Proposition 2.2.13 we see that kUa(n) (t, 0)un − Ua˜ (t, 0)u0 k → 0 and kUa0(n) (t, 0)un − Ua0 (t, 0)u0 k → 0 for any t > 0. Put un (t) := Ua(n) (t, 0)un and u(t) := Ua˜ (t, 0)u0 , for any t ≥ 0. Fix t > 0. The sequence (a(n) ) is so chosen that we get, via the L2 –L2 estimates (Proposition 2.2.2) the following: (n)

kUa0(n) (t, τ )(c0 (τ, ·)un (τ )) − Ua0 (t, τ )(˜ c0 (τ, ·)u(τ ))k (n)

≤ kUa0(n) (t, τ )((c0 (τ, ·) − c˜0 (τ, ·))un (τ ))k + kUa0(n) (t, τ )(˜ c0 (τ, ·)un (τ )) − Ua0 (t, τ )(˜ c0 (τ, ·)u(τ ))k (n)

≤ M eγ(t−τ ) k(c0 (τ, ·) − c˜0 (τ, ·))(un (τ ))k + kUa0(n) (t, τ )(˜ c0 (τ, ·)un (τ )) − Ua0 (t, τ )(˜ c0 (τ, ·)u(τ ))k (n)

≤ M eγ(t−τ ) kc0 (τ, ·) − c˜0 (τ, ·)k kun (τ )k + kUa0(n) (t, τ )(˜ c0 (τ, ·)un (τ )) − Ua0 (t, τ )(˜ c0 (τ, ·)u(τ ))k →0 as n → ∞ for a.e. τ ∈ (0, t). By the L2 –L2 estimates (Proposition 2.2.2), the set (n) { kUa0(n) (t, τ )(c0 (τ, 0)un (τ ))k : τ ∈ [0, t], n ∈ N } is bounded. It then follows that Ua˜ (t, 0)u0 = Ua0 (t, 0)u0 +

Z

t

Ua0 (t, τ )(˜ c0 (τ, ·)Ua˜ (τ, 0)u0 ) dτ

0

for t > 0 (in other words, [ t 7→ Ua˜ (t, 0)u0 ] is a mild solution). Therefore by the arguments in [92, Section 2] and the Sobolev embedding theorems we have that [ [0, T ] 3 t 7→ Ua˜ (t, 0)u0 ] ∈ Wp1,2 ((0, T ) × D) for any T > 0 and p > 1, and Ua˜ (t, 0)u0 is a strong solution on (t0 , T ) for any 0 < t0 < T . Now we use [59, Theorem 2.5] to conclude that the inequality (A3-2) holds with ς = 0, which by Lemma 3.3.1 implies the assumption (A3-1). Similarly, (A3-1) and (A3-2) hold for the adjoint problem of (2.0.1)a˜ +(2.0.2)a˜ . Consequently, the topological linear skew-product flow Π(a) admits an exponential separation over Y˜0 (a).

128

Spectral Theory for Parabolic Equations

EXAMPLE 4.1.2 (The classical case) Consider the linear random parabolic equation (4.1.1) endowed with the boundary conditions (4.1.2), where we make the following assumptions: • D ⊂ RN is a bounded domain, where its boundary is an (N − 1)-dimensional manifold of class C 3+α for some α > 0. • The functions aij (= aji ), ai , bi , and c0 are (F × B(D), B(R))-measurable, and the function d0 is (F × B(∂D), B(R))-measurable. • There is α ∈ (0, 1) such that for any ω ∈ Ω, the functions aω ij and 2+α,2+α ¯ the functions bω aω (i, j = 1, 2, . . . , N ) belong to C (R × D), i i 2+α,1+α ¯ and the function dω (i = 1, 2, . . . , N ) and cω (R × D) 0 belong to C 0 belongs to C 2+α,2+α (R × ∂D). Moreover, there is M > 0 such that for ω ¯ any ω ∈ Ω, the C 2+α,2+α (R× D)-norms of aω ij and ai (i, j = 1, 2, . . . , N ), 2+α,1+α ω ¯ the C (R × D)-norms of bi (i = 1, 2, . . . , N ) and cω 0 , and the 2+α,2+α C (R × ∂D)-norms of dω , are bounded by M . 0 PN PN • There exists α0 > 0 such that i,j=1 aij (ω, x) ξi ξj ≥ α0 i=1 ξi2 for all ¯ and ξ ∈ RN , and d0 (ω, x) ≥ 0 for all ω ∈ Ω and x ∈ ∂D. ω ∈ Ω, x ∈ D, The problem (4.1.1)+(4.1.2) satisfies (A4-R-ES). Indeed, (A2-1)–(A2-2) and (A4-R1)–(A4-R2) are explicitly stated. By a standard reasoning making use of the Ascoli–Arzel`a theorem we see that all the estimates on the ¯ C 2+α,2+α (R × D)-norms, etc., carry over to the elements of Y˜ (a). Consequently, (A2-5) is satisfied (see Section 2.5), so (A2-3) is fulfilled. As in Example 4.1.1, in the Dirichlet case, the inequalities (A3-1) and (A32) hold for any a ˜ ∈ Y˜0 (a), for both (2.0.1)a˜ +(2.0.2)a˜ and its adjoint, by [61, Theorem 2.1 and Lemma 3.9], so the topological linear skew-product flow Π(a) admits an exponential separation over Y˜0 (a). In the Neumann and Robin cases, by Proposition 2.5.1, Ua˜ (t, 0)u0 is a classical solution on [t0 , T ] for any 0 < t0 < T . Then we use [59, Theorem 2.5] again to conclude that the inequality (A3-2) holds with ς = 0, which by Lemma 3.3.1 implies the assumption (A3-1). Similarly, (A3-1) and (A3-2) hold for the adjoint problem of (2.0.1)a˜ +(2.0.2)a˜ . Consequently, the topological linear skew-product flow Π(a) admits an exponential separation over Y˜0 (a). From now until the end of the present subsection we assume that a is such that property (A4-R-ES) holds. As a is fixed, we will suppress its symbol: We write E, Y˜ , Y˜0 , Π for Ea , ˜ Y (a), Y˜0 (a), Π(a), respectively. Also, instead of UE(ω) (t, s) we will write Uω (t, s). The following results are simple consequences of the results in Chapter 3.

4. Spectral Theory in Nonautonomous and Random Cases

129

PROPOSITION 4.1.1 Let Ω1 be as in Lemma 4.1.4. Then for any ω ∈ Ω1 and any u0 ∈ L2 (D)+ \{0} one has ln kUω (t, 0)u0 k = λ(a). (4.1.8) lim t→∞ t PROOF

See Lemmas 4.1.4 and 3.2.5.

PROPOSITION 4.1.2 For any sequence (ω (n) )∞ n=1 ⊂ Ω0 , where Ω0 is as in Lemma 4.1.2, any u0 ∈ ∞ L2 (D)+ \ {0}, and any real sequences (sn )∞ n=1 , (tn )n=1 such that tn − sn → ∞ one has λmin (a) ln kUω(n) (tn , sn )w(E(ω (n) ) · sn )k ln kUω(n) (tn , sn )u0 k = lim inf n→∞ n→∞ tn − sn tn − sn ln kUω(n) (tn , sn )k ln kUω(n) (tn , sn )k = lim inf ≤ lim sup n→∞ tn − sn tn − sn n→∞ ln kUω(n) (tn , sn )u0 k ln kUω(n) (tn , sn )w(E(ω (n) ) · sn )k = lim sup = lim sup tn − sn tn − sn n→∞ n→∞ ≤ λmax (a). ≤ lim inf

PROOF

See Theorem 3.1.2(1) and Lemma 3.2.5.

PROPOSITION 4.1.3 For each ω ∈ Ω0 , where Ω0 is as in Lemma 4.1.2, 0 ∞ 0 0 (i) there are sequences (s0n )∞ n=1 , (tn )n=1 ⊂ R, tn − sn → ∞ as n → ∞, such that

ln kUω (t0n , s0n )w(E(ω) · s0n )k n→∞ t0n − s0n ln kUω (t0n , s0n )u0 k ln kUω (t0n , s0n )k = lim = lim n→∞ n→∞ t0n − s0n t0n − s0n

λmin (a) = lim

for each u0 ∈ L2 (D)+ \ {0}, 00 ∞ 00 00 (ii) there are sequences (s00n )∞ n=1 , (tn )n=1 ⊂ R, tn − sn → ∞ as n → ∞, such that

ln kUω (t00n , s00n )w(E(ω) · s00n )k n→∞ t00n − s00n ln kUω (t00n , s00n )u0 k ln kUω (t00n , s00n )k = lim = lim n→∞ n→∞ t00n − s00n t00n − s00n

λmax (a) = lim

130

Spectral Theory for Parabolic Equations for each u0 ∈ L2 (D)+ \ {0}.

PROOF Note that for each ω ∈ Ω0 , Y˜0 = cl{ E(ω) · t : t ∈ R }. The proposition then follows from Theorem 3.2.7(1). A consequence of Propositions 4.1.2 and 4.1.3 is the following. PROPOSITION 4.1.4 For any ω ∈ Ω0 , where Ω0 is as in Lemma 4.1.2, and any u0 ∈ L2 (D)+ \ {0} there holds ln kUω (t, s)w(E(ω) · s)k t−s ln kUω (t, s)u0 k ln kUω (t, s)k = lim inf = lim inf t−s→∞ t−s→∞ t−s t−s lnkUω (t, s)k ln kUω (t, s)u0 k ≤ lim sup = lim sup t−s t−s t−s→∞ t−s→∞ ln kUω (t, s)w(E(ω) · s)k = lim sup = λmax (a). t−s t−s→∞

λmin (a) = lim inf

t−s→∞

PROPOSITION 4.1.5 For each ω ∈ Ω0 , where Ω0 is as in Lemma 4.1.2, and each λ ∈ [λmin (a), ∞ λmax (a)] there are sequences (kn )∞ n=1 , (ln )n=1 ⊂ Z, ln − kn → ∞ as n → ∞, such that λ = lim

n→∞

ln kUω (ln , kn )w(E(ω) · kn )k ln − kn ln kUω (ln , kn )u0 k ln kUω (ln , kn )k = lim = lim n→∞ n→∞ ln − kn ln − k n

for each u0 ∈ L2 (D)+ \ {0}. PROOF Note that for ω ∈ Ω0 , Y˜0 = cl { E(ω) · t : t ∈ R }. The proposition then follows from Theorem 3.2.7(3). In the light of Proposition 4.1.5, Proposition 4.1.4 has the following strengthening. PROPOSITION 4.1.6 For any ω ∈ Ω0 , where Ω0 is as in Lemma 4.1.2, and any u0 ∈ L2 (D)+ \ {0}

4. Spectral Theory in Nonautonomous and Random Cases

131

there holds λmin (a) = lim inf

t−s→∞

ln kUω (t, s)w(E(ω) · s)k ln kUω (l, k)w(E(ω) · k)k = lim inf l−k→∞ t−s l−k k,l∈Z

ln kUω (t, s)u0 k ln kUω (l, k)u0 k = lim inf = lim inf t−s→∞ l−k→∞ t−s l−k k,l∈Z

ln kUω (l, k)u0 k ln kUω (t, s)u0 k ≤ lim sup = lim sup l−k t−s t−s→∞ l−k→∞ k,l∈Z

ln kUω (t, s)w(E(ω) · s)k ln kUω (l, k)w(E(ω) · k)k = lim sup = lim sup l−k t−s t−s→∞ l−k→∞ k,l∈Z

= λmax (a).

4.1.2

The Nonautonomous Case

Consider the following nonautonomous linear parabolic equation: N N ∂u X ∂ X ∂u = aij (t, x) + ai (t, x)u ∂t ∂xi j=1 ∂xj i=1 +

N X i=1

bi (t, x)

∂u + c0 (t, x)u, ∂xi

x ∈ D,

(4.1.9)

endowed with the boundary condition Ba (t)u = 0,

x ∈ ∂D,

(4.1.10)

where Ba (t) is a boundary operator of either the Dirichlet or Neumann or Robin type as in (2.0.3), a = (aij , ai , bi , c0 , d0 ) and d0 (t, x) ≥ 0 for a.e. (t, x) ∈ R × ∂D. Sometimes we write the nonautonomous problem (4.1.9)+(4.1.10) as (4.1.9)a +(4.1.10)a . Our first assumption regards boundedness of the coefficients of the equations (and of the boundary conditions): (A4-N1) (Boundedness) a belongs to L∞ (R × D, RN ∂D, R).

2

+2N +1

) × L∞ (R ×

Among others, (4.1.9)+(4.1.10) arise from linearization of nonautonomous nonlinear parabolic equations at certain entire solution as well as from linearization of autonomous nonlinear parabolic equations at some entire time dependent solution. In the rest of the present subsection it is assumed that a satisfies (A4-N1).

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Put Y˜ (a) := cl { a · t : t ∈ R }

(4.1.11)

with the weak-* topology, where the closure is taken in the weak-* topology. The set Y˜ (a) is a compact connected metrizable space. If Assumptions (A2-1)–(A2-3) are satisfied for Y replaced with Y˜ (a), we will denote by Π(a) = {Π(a)t }t≥0 the topological linear skew-product semiflow generated by (4.1.9)+(4.1.10) on the product Banach bundle L2 (D) × Y˜ (a): Π(a)(t; u0 , a ˜) = Π(a)t (u0 , a ˜) := (Ua˜ (t, 0)u0 , σt a ˜) for t ≥ 0, a ˜ ∈ Y˜ (a), u0 ∈ L2 (D), where Ua˜ (t, 0)u0 stands for the weak solution of (2.0.1)a˜ +(2.0.2)a˜ with initial condition u(0, x) = u0 (x). The next assumption is about the satisfaction of (A2-1)–(A2-3). (A4-N) (Satisfaction of (A2-1)–(A2-3) and (A4-N1)) The assumption (A4-N1) is fulfilled, and assumptions (A2-1)–(A2-3) are satisfied for Y replaced with Y˜ (a) defined by (4.1.11). Sometimes we say simply that a or (4.1.9)a +(4.1.10)a or Π(a) satisfies property (A4-N). DEFINITION 4.1.3 The principal spectrum of the nonautonomous problem (4.1.9)a +(4.1.10)a satisfying property (A4-N) equals the principal spectrum of the topological linear skew-product semiflow Π(a) over Y˜ (a). We will denote the principal spectrum by Σ(a) = [λmin (a), λmax (a)]. The following assumption is about the exponential separation. (A4-N-ES) (Exponential separation) (4.1.9)a +(4.1.10)a has property (A4N) and, moreover, the topological linear skew-product semiflow Π(a) generated by (4.1.9)a +(4.1.10)a on L2 (D) × Y˜ (a) admits an exponential separation over Y˜ (a). Sometimes we say simply that a satisfies property (A4-N-ES). We now give two examples of a satisfying the property (A4-N-ES). EXAMPLE 4.1.3 (Only zero-order terms depend on t) Consider the following nonautonomous linear parabolic equation: N N ∂u ∂u X ∂ X = aij (x) + ai (x)u ∂t ∂xi j=1 ∂xj i=1 +

N X i=1

bi (x)

∂u + c0 (t, x)u, ∂xi

x ∈ D,

(4.1.12)

4. Spectral Theory in Nonautonomous and Random Cases

133

endowed with the boundary condition Ba (t)u = 0,

x ∈ ∂D,

(4.1.13)

where Ba is as in (2.0.3) with a(t, x) = (aij (x), ai (x), bi (x), c0 (t, x), d0 (x)). We make the following assumptions: In the Dirichlet case: • D ⊂ RN is a bounded domain with Lipschitz boundary, • aij (= aji ), ai , bi ∈ L∞ (D). Moreover, (A2-1) is satisfied, • c0 ∈ L∞ (R × D). In the Neumann or Robin cases: • D ⊂ RN is a bounded domain, where its boundary is an (N − 1)-dimensional manifold of class C 2 , ¯ d0 ∈ C 1 (∂D). Moreover, (A2-1) is satisfied, • aij (= aji ), ai , bi ∈ C 1 (D), and d0 (x) ≥ 0 for all x ∈ ∂D, • c0 ∈ L∞ (R × D). By arguments similar to those in Example 4.1.1, the nonautonomous problem (4.1.12)+(4.1.13) satisfying the above requirements has the property (A4N-ES). EXAMPLE 4.1.4 (The classical case) Consider the linear nonautonomous parabolic equation (4.1.9) endowed with the boundary conditions (4.1.10), where we make the following assumptions: • D ⊂ RN is a bounded domain, where its boundary is an (N − 1)-dimensional manifold of class C 3+α for some α > 0. • There is α > 0 such that the functions aij (= aji ) and ai (i, j = ¯ the functions bi (i = 2, · · · , N ) 1, 2, . . . , N ) belong to C 2+α,2+α (R × D), 2+α,1+α ¯ and c0 belong to C (R × D), and the function d0 belongs to C 2+α,2+α (R × ∂D). PN PN • There exists α0 > 0 such that i,j=1 aij (t, x) ξi ξj ≥ α0 i=1 ξi2 for all ¯ and ξ ∈ RN , and d0 (t, x) ≥ 0 for all t ∈ R and x ∈ ∂D. t ∈ R, x ∈ D, By similar arguments to those in Example 4.1.2, the nonautonomous problem (4.1.12)+(4.1.13) satisfying the above requirements has the property (A4N-ES). From now until the end of the present subsection we assume that a is such that property (A4-N-ES) holds.

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As a is fixed, we will suppress its symbol: We write Y˜ , Π for Y˜ (a), Π(a), respectively. Also, instead of Ua (t, s) we will write U (t, s). The following results are simple consequences of the results in Chapter 3. PROPOSITION 4.1.7 ∞ For any u0 ∈ L2 (D)+ \ {0} and any real sequences (sn )∞ n=1 , (tn )n=1 such that tn − sn → ∞ one has ln kU (tn , sn )u0 k ln kU (tn , sn )w(a · sn )k = lim inf n→∞ n→∞ tn − sn tn − sn ln kU (tn , sn )k ln kU (tn , sn )k ≤ lim sup = lim inf n→∞ tn − sn tn − sn n→∞ lnkU (tn , sn )u0 k ln kU (tn , sn )w(a · sn )k = lim sup = lim sup ≤ λmax (a). tn − sn tn − sn n→∞ n→∞

λmin (a) ≤ lim inf

PROOF

See Theorem 3.1.2(1) and Lemma 3.2.5.

PROPOSITION 4.1.8 0 0 0 ∞ (i) There are sequences (s0n )∞ n=1 , (tn )n=1 ⊂ R, tn − sn → ∞ as n → ∞, such that

ln kU (t0n , s0n )w(a · s0n )k n→∞ t0n − s0n ln kU (t0n , s0n )u0 k ln kU (t0n , s0n )k = lim = lim 0 0 n→∞ n→∞ tn − sn t0n − s0n

λmin (a) = lim

for each u0 ∈ L2 (D)+ \ {0}. 00 ∞ 00 00 (ii) There are sequences (s00n )∞ n=1 , (tn )n=1 ⊂ R, tn − sn → ∞ as n → ∞, such that

ln kU (t00n , s00n )w(a · s00n )k n→∞ t00n − s00n ln kU (t00n , s00n )u0 k ln kU (t00n , s00n )k = lim = lim 00 00 n→∞ n→∞ tn − sn t00n − s00n

λmax (a) = lim

for each u0 ∈ L2 (D)+ \ {0}. PROOF Note that Y˜ = cl { a · t : t ∈ R }. The proposition then follows from Theorem 3.2.7(1). A consequence of Propositions 4.1.7 and 4.1.8 is the following.

4. Spectral Theory in Nonautonomous and Random Cases

135

PROPOSITION 4.1.9 For any u0 ∈ L2 (D)+ \ {0} there holds ln kU (t, s)w(a · s)k ln kU (t, s)u0 k = lim inf t−s→∞ t−s→∞ t−s t−s ln kU (t, s)k ln kU (t, s)k ln kU (t, s)u0 k = lim inf ≤ lim sup = lim sup t−s→∞ t−s t − s t−s t−s→∞ t−s→∞ ln kU (t, s)w(a · s)k = lim sup = λmax (a). t−s t−s→∞

λmin (a) = lim inf

PROPOSITION 4.1.10 ∞ For each λ ∈ [λmin (a), λmax (a)] there are sequences (kn )∞ n=1 , (ln )n=1 ⊂ Z, ln − kn → ∞ as n → ∞, such that λ = lim

n→∞

ln kU (ln , kn )w(a · kn )k ln − k n = lim

n→∞

ln kU (ln , kn )u0 k ln kU (ln , kn )k = lim n→∞ ln − k n ln − kn

for each u0 ∈ L2 (D)+ \ {0}. PROOF Note that Y˜ = cl { a · t : t ∈ R }. The proposition then follows from Theorem 3.2.7(3). In the light of Proposition 4.1.10, Proposition 4.1.9 has the following strengthening. PROPOSITION 4.1.11 For any u0 ∈ L2 (D)+ \ {0} there holds λmin (a) = lim inf

t−s→∞

ln kU (t, s)w(a · s)k ln kU (l, k)w(a · k)k = lim inf l−k→∞ t−s l−k k,l∈Z

= lim inf

t−s→∞

ln kU (l, k)u0 k ln kU (t, s)u0 k = lim inf l−k→∞ t−s l−k k,l∈Z

ln kU (l, k)u0 k ln kU (t, s)u0 k ≤ lim sup = lim sup l−k t−s t−s→∞ l−k→∞ k,l∈Z

ln kU (l, k)w(a · k)k ln kU (t, s)w(a · s)k = lim sup = lim sup = λmax (a). l − k t−s t−s→∞ l−k→∞ k,l∈Z

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Spectral Theory for Parabolic Equations

4.2

Monotonicity with Respect to the Zero Order Terms

In this section, we investigate the monotonicity of principal spectrum and principal Lyapunov exponents of random (nonautonomous) parabolic equations with respect to zero order terms. We also study the relation among the principal spectrum and principal Lyapunov exponents for random (nonautonomous) parabolic equations with different types of boundary conditions. Similarly, we treat random equations and nonautonomous equations separately.

4.2.1

The Random Case

Assume that ((Ω, F, P), {θt }t∈R ) is an ergodic metric dynamical system. Let a(1) , a(2) satisfy property (A4-R). ˜ ⊂Ω Throughout the present subsection we assume moreover that there is Ω ˜ = 1 such that for each ω ∈ Ω: ˜ with P(Ω) (1),ω

(2),ω

(1),ω

• aij (·, ·) = aij (·, ·), ai for a.e. (t, x) ∈ R × D, and

(2),ω

(·, ·) = ai

(1),ω

(·, ·), bi

(2),ω

(·, ·) = bi

(·, ·),

• one of the following conditions, (M-Ra), (M-Rb), (M-Rc), (M-Rd), or (M-Re), holds: (M-Ra) both a(1) and a(2) are endowed with the Dirichlet boundary conditions, and (1),ω

∗ c0

(2),ω

(·, ·) ≤ c0

(·, ·)

for a.e. (t, x) ∈ R × D,

(M-Rb) both a(1) and a(2) are endowed with the Robin boundary conditions, and (1),ω

∗ c0 ∗

(2),ω

(·, ·) ≤ c0

(1),ω d0 (·, ·)

≥

(·, ·)

(2),ω d0 (·, ·)

for a.e. (t, x) ∈ R × D, for a.e. (t, x) ∈ R × ∂D,

(M-Rc) both a(1) and a(2) are endowed with the Neumann boundary conditions, and (1),ω

∗ c0

(2),ω

(·, ·) ≤ c0

(·, ·)

for a.e. (t, x) ∈ R × D,

(M-Rd) a(1) is endowed with the Dirichlet boundary conditions and a(2) is endowed with Robin boundary conditions, and (1),ω

∗ c0

(2),ω

(·, ·) = c0

(·, ·)

for a.e. (t, x) ∈ R × D,

(M-Re) a(1) is endowed with the Robin boundary conditions and a(2) is endowed with the Neumann conditions, and (1),ω

∗ c0

(2),ω

(·, ·) = c0

(·, ·)

for a.e. (t, x) ∈ R × D.

4. Spectral Theory in Nonautonomous and Random Cases

137

(k) For a ˜ ∈ Y˜ (a(k) ), s < t, and u0 ∈ L2 (D), denote by Ua˜ (t, s)u0 , k = 1, 2, the weak solution of (2.0.1)a˜ +(2.0.2)a˜ with initial condition u(0, x) = u0 (x). (k) (k) For ω ∈ Ω, instead of UE (k) (ω) (t, s)u0 we write Uω (t, s)u0 . a

THEOREM 4.2.1 λ(a(1) ) ≤ λ(a(2) ). PROOF

(k)

(k)

Let Ω1 ⊂ Ω, k = 1, 2, be sets such that P(Ω1 ) = 1 and (k)

ln kUω (t, 0)u0 k t→∞ t

λ(a(k) ) = lim (k)

for any ω ∈ Ω1 and any u0 ∈ L2 (D)+ \ {0} (see Lemmas 4.1.4 and 3.1.1). (1) (2) ˜ As a consequence of Proposition 2.2.10, 0 ≤ Fix ω ∈ Ω1 ∩ Ω1 ∩ Ω. (2) (1) (Uω (t, 0)u0 )(x) ≤ (Uω (t, 0)u0 )(x) for each nonzero u0 ∈ L2 (D)+ , each t > 0, and each x ∈ D. The monotonicity of the L2 (D)-norm gives the desired result. THEOREM 4.2.2 λmin (a(1) ) ≤ λmin (a(2) ) and λmax (a(1) ) ≤ λmax (a(2) ). PROOF We prove only the first inequality, the proof of the other being (k) (k) similar. Let Ω0 ⊂ Ω, k = 1, 2, be sets such that P(Ω0 ) = 1 and Y˜0 (a(k) ) = (k) (1) (2) ˜ cl{ Ea(1) (θt ω) : t ∈ R } for ω ∈ Ω0 (see Lemma 4.1.2). Fix ω ∈ Ω0 ∩Ω0 ∩ Ω. ˜ (2) ), (tn )∞ ⊂ By Theorem 3.1.2(2A), there are sequences (˜ a(n) )∞ n=1 ⊂ Y0 (a n=1 R, (sn )∞ ⊂ R, with t − s → ∞ as n → ∞, such that n n n=1 (2)

ln kUa˜(n) (tn , sn )k+ = λmin (a(2) ). n→∞ tn − sn lim

For each a ˜(n) there is a real sequence (τl )∞ l=1 (depending on n) such that Ea(2) (ω)·τl converge in Y˜ (a(2) ) to a ˜(n) . From (τl ) we can extract a subsequence (denoted again by (τl )) such that Ea(1) (ω)·τl converge in Y˜ (a(1) ) to some a ˆ(n) . + Proposition 2.2.10 implies that for each u0 ∈ L2 (D) there holds (1)

kUE

a(1)

(ω)·τl (tn , sn )u0 k

(2)

≤ kUE

(ω)·τl a(2)

(tn , sn )u0 k.

(1)

(2)

From Proposition 2.2.13 we deduce that kUaˆ(n) (tn , sn )u0 k ≤ kUa˜(n) (tn , sn )u0 k (1)

(2)

for each u0 ∈ L2 (D)+ , which implies kUaˆ(n) (tn , sn )k+ ≤ kUa˜(n) (tn , sn )k+ . By

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Spectral Theory for Parabolic Equations

Theorem 3.1.2 and Lemma 3.1.1, (1)

(1)

λmin (a

ln kUaˆ(n) (tn , sn )k+ ) ≤ lim inf n→∞ tn − sn (2)

ln kUa˜(n) (tn , sn )k+ = λmin (a(2) ). n→∞ tn − sn

≤ lim

4.2.2

The Nonautonomous Case

Let a(1) , a(2) satisfy property (A4-N). Throughout the present subsection we assume moreover that: (1)

(2)

(1)

(2)

(1)

(2)

• aij (·, ·) = aij (·, ·), ai (·, ·) = ai (·, ·), bi (·, ·) = bi (·, ·), for a.e. (t, x) ∈ R × D, and • one of the following conditions, (M-Na), (M-Nb), (M-Nc), (M-Nd), or (M-Ne) holds: (M-Na) both a(1) and a(2) are endowed with the Dirichlet boundary conditions, and (1)

(2)

∗ c0 (·, ·) ≤ c0 (·, ·)

for a.e. (t, x) ∈ R × D,

(M-Nb) both a(1) and a(2) are endowed with the Robin boundary conditions, and (1)

(2)

∗ c0 (·, ·) ≤ c0 (·, ·) ∗

(1) d0 (·, ·)

≥

(2) d0 (·, ·)

for a.e. (t, x) ∈ R × D, for a.e. (t, x) ∈ R × ∂D,

(M-Nc) both a(1) and a(2) are endowed with the Neumann boundary conditions, and (1)

(2)

∗ c0 (·, ·) ≤ c0 (·, ·)

for a.e. (t, x) ∈ R × D,

(M-Nd) a(1) is endowed with the Dirichlet boundary conditions and a(2) is endowed with the Robin boundary conditions, and (1)

(2)

∗ c0 (·, ·) = c0 (·, ·)

for a.e. (t, x) ∈ R × D,

(M-Ne) a(1) is endowed with the Robin boundary conditions and a(2) is endowed with the Neumann boundary conditions, and (1)

(2)

∗ c0 (·, ·) = c0 (·, ·)

for a.e. (t, x) ∈ R × D.

4. Spectral Theory in Nonautonomous and Random Cases

139

(k) For a ˜ ∈ Y˜ (a(k) ), s < t, and u0 ∈ L2 (D), denote by Ua˜ (t, s)u0 , k = 1, 2, the weak solution of (2.0.1)a˜ +(2.0.2)a˜ with initial condition u(0, x) = u0 (x).

THEOREM 4.2.3 λmin (a(1) ) ≤ λmin (a(2) ) and λmax (a(1) ) ≤ λmax (a(2) ). PROOF We prove only the second inequality, the proof of the other being similar. ˜ (1) ), (tn )∞ ⊂ By Theorem 3.1.2(2A), there are sequences (˜ a(n) )∞ n=1 ⊂ Y (a n=1 ∞ R, (sn )n=1 ⊂ R, with tn − sn → ∞ as n → ∞, such that (1)

ln kUa˜(n) (tn , sn )k+ = λmax (a(1) ). n→∞ tn − sn lim

For each a ˜(n) there is a real sequence (τl )∞ l=1 (depending on n) such that (1) a · τl converge in Y˜ (a(1) ) to a ˜(n) . From (τl ) we can extract a subsequence (denoted again by (τl )) such that a(2) · τl converge in Y˜ (a(2) ) to some a ˆ(n) . + Proposition 2.2.10 implies that for each u0 ∈ L2 (D) there holds (1)

(2)

kUa(1) ·τl (tn , sn )u0 k ≤ kUa(2) ·τl (tn , sn )u0 k. From Proposition 2.2.13 we deduce that (1)

(2)

kUa˜(n) (tn , sn )u0 k ≤ kUaˆ(n) (tn , sn )u0 k (1)

(2)

for each u0 ∈ L2 (D)+ , which implies kUa˜(n) (tn , sn )k+ ≤ kUaˆ(n) (tn , sn )k+ . By Theorem 3.1.2 and Lemma 3.1.1, (1)

ln kUa˜(n) (tn , sn )k+ n→∞ tn − sn

λmax (a(1) ) = lim

(2)

ln kUaˆ(n) (tn , sn )k+ ≤ lim sup ≤ λmax (a(2) ). tn − sn n→∞

4.3

Continuity with Respect to the Zero Order Coefficients

In the present section we investigate the continuous dependence of the principal spectrum/principal Lyapunov exponent on the zero order terms.

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Spectral Theory for Parabolic Equations

2 For any a ˜ = (˜ aij , a ˜i , ˜bi , c˜0 , d˜0 ) ∈ L∞ (R × D, RN +2N +1 ) × L∞ (R × ∂D, R) and any R ∈ R, put

a ˜ + R := (˜ aij , a ˜i , ˜bi , c˜0 + R, d˜0 ). It is straightforward that if a ˜(n) converges to a ˜ in the weak-* topology, then (n) a ˜ + R converges to a ˜ + R in the weak-* topology. For R ∈ R we write sometimes TR a ˜ instead of a ˜ + R. We collect now some facts that will be useful in the sequel. LEMMA 4.3.1 Assume that Y is such that (A2-1)–(A2-3) are satisfied for Y . Let R ∈ R. Then (i) The assumptions (A2-1)–(A2-3) are satisfied for TR Y . (ii) For any a ˜ ∈ Y and any s ≤ t, Ua˜+R (t, s) = eR(t−s) Ua˜ (t, s).

(4.3.1)

PROOF The fulfillment of (A2-1)–(A2-2) for TR Y is obvious. The bilinear forms Ba˜ (·, ·, ·) and Ba˜+R (·, ·, ·) are related, for a.e. t ∈ R, in the following way: Ba˜+R (t, u, v) = Ba˜ (t, u, v) − Rhu, vi, u, v ∈ V (see Section 2.1). Fix s ∈ R and u0 ∈ L2 (D), and denote u(t) := Ua˜ (t, s)u0 , t ≥ s. One has Z t Z t ˙ − hu(τ ), vi ψ(τ ) dτ + Ba˜ (τ, u(τ ), v)ψ(τ ) dτ − hu0 , vi ψ(s) = 0 (4.3.2) s

s

for all v ∈ V and ψ ∈ D([s, t)) (see Definition 2.1.6). We have to show that Z −

t

˙ ) dτ + heR(τ −s) u(τ ), vi ψ(τ

s

Z

t

Ba˜+R (τ, eR(τ −s) u(τ ), v)ψ(τ ) dτ

s

− hu0 , vi ψ(s) = 0, that is, Z t Z t ˙ ) dτ + − heR(τ −s) u(τ ), vi ψ(τ Ba˜ (τ, eR(τ −s) u(τ ), v)ψ(τ ) dτ s s Z t R(τ −s) − hRe u(τ ), vi ψ(τ ) dτ − hu0 , vi ψ(s) = 0 s

for all v ∈ V and ψ ∈ D([s, t)). This follows from (4.3.2) by replacing the test function ψ with eR(t−s) ψ(t). To prove that (A2-3) is satisfied for TR Y , notice that a ˜(n) → a ˜ in Y if and (n) only if a ˜ +R→a ˜ + R in TR Y , and apply (4.3.1).

4. Spectral Theory in Nonautonomous and Random Cases

4.3.1

141

The Random Case

Assume that ((Ω, F, P), {θt }t∈R ) is an ergodic metric dynamical system. Let a(1) , a(2) satisfy property (A4-R). ˜ ⊂ Throughout the present subsection we assume moreover that there is Ω ˜ = 1 such that for each ω ∈ Ω ˜ the following holds: Ω with P(Ω) (1),ω

(2),ω

• aij (·, ·) = aij a.e. on R × D, (2),ω

• d0

(1),ω

(·, ·) = d0

(1),ω

(·, ·), ai

(·, ·)

(2),ω

(·, ·) = ai

(1),ω

(·, ·), bi

(2),ω

(·, ·) = bi

(·, ·)

a.e. on R × ∂D.

Denote (2),ω

r := ess sup { |c0

(1),ω

(t, x) − c0

˜ t ∈ R, x ∈ D }. (t, x)| : ω ∈ Ω,

THEOREM 4.3.1 |λ(a(1) ) − λ(a(2) )| ≤ r. THEOREM 4.3.2 |λmin (a(1) ) − λmin (a(2) )| ≤ r and |λmax (a(1) ) − λmax (a(2) )| ≤ r. PROOF (Proofs of Theorems 4.3.1 and 4.3.2) The assumptions (A4-R1) and (A4-R2) are satisfied for a(1) − r, a(1) + r. As the mappings T±r are continuous in the weak-* topology and T±r ◦ σt = σt ◦ T±r for all t ∈ R, we have that Y˜ (a(1) ± r) = T±r Y˜ (a(1) ). Further, it follows that T±r sends, in a one-to-one way, invariant measures on Y˜ (a(1) ) onto invariant measures on T±r Y˜ (a(1) ), consequently Y˜0 (a(1) ± r) = T±r Y˜0 (a(1) ). By Lemma 4.3.1, a(1) ± r satisfy (A4-R), and λ(a(1) ± r) = λ(a(1) ) ± r, λmin (a(1) ± r) = λmin (a(1) ) ± r, λmin (a(1) ± r) = λmin (a(1) ) ± r. It follows from Theorems 4.2.1 and 4.2.2 that λ(a(1) − r) ≤ λ(a(2) ) ≤ λ(a(1) + r) as well as λmin (a(1) − r) ≤ λmin (a(2) ) ≤ λmin (a(1) + r) and λmax (a(1) − r) ≤ λmax (a(2) ) ≤ λmax (a(1) + r), which gives the desired result.

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Spectral Theory for Parabolic Equations

4.3.2

The Nonautonomous Case (1)

Let a , a(2) satisfy property (A4-N). Throughout the present subsection we assume moreover that the following holds: (1)

(2)

(1)

(2)

(1)

(2)

• aij (·, ·) = aij (·, ·), ai (·, ·) = ai (·, ·), bi (·, ·) = bi (·, ·), R × D, (2)

(1)

• d0 (·, ·) = d0 (·, ·)

a.e. on

a.e. on R × ∂D.

Denote (2)

(1)

r := ess sup { |c0 (t, x) − c0 (t, x)| : t ∈ R, x ∈ D }. THEOREM 4.3.3 |λmin (a(1) ) − λmin (a(2) )| ≤ r and |λmax (a(1) ) − λmax (a(2) )| ≤ r. PROOF The assumption (A4-N1) is satisfied for a(1) − r, a(1) + r. As the mappings T±r are continuous in the weak-* topology and T±r ◦ σt = σt ◦ T±r for all t ∈ R, we have that Y˜ (a(1) ± r) = T±r Y˜ (a(1) ). By Lemma 4.3.1, a(1) ± r satisfy (A4-N), and λmin (a(1) ± r) = λmin (a(1) ) ± r, λmin (a(1) ± r) = λmin (a(1) ) ± r. It follows from Theorem 4.2.3 that λmin (a(1) − r) ≤ λmin (a(2) ) ≤ λmin (a(1) + r) and λmax (a(1) − r) ≤ λmax (a(2) ) ≤ λmax (a(1) + r), which gives the desired result.

4.4

General Continuity with Respect to the Coefficients

In the present section, L∞ denotes the Banach space L∞ (R×D, RN ×L∞ (R × ∂D, R), and k·k∞ stands for the norm in L∞ . We start with the following simple result.

2

+2N +1

)

LEMMA 4.4.1 For any a(1) , a(2) ∈ L∞ one has d(a(1) · t, a(2) · t) ≤ ka(1) − a(2) k∞ for each t ∈ R, where d stands for the metric given by (1.3.1).

4. Spectral Theory in Nonautonomous and Random Cases

143

PROOF (1)

d(a

(2)

· t, a

∞ X 1 |hgk , a(1) · t − a(2) · tiL1 ,L∞ | ≤ ka(1) − a(2) k∞ . · t) = 2k k=1

In the rest of the present subsection let Y be such that (A2-1)–(A2-3) are satisfied and that the linear skew-product flow Π admits an exponential separation over Y . Recall that, for a ˜ ∈ Y and u0 ∈ L2 (D), [ [0, ∞) 3 t 7→ Ua˜ (t, 0)u0 ∈ L2 (D) ] denotes the weak solution of (2.0.1)a˜ +(2.0.2)a˜ with the initial condition u(0, x) = u0 (x) (x ∈ D). For s < t, Ua˜ (t, s) stands for Ua˜·s (t − s, 0). LEMMA 4.4.2 For each > 0 there is δ > 0 with the following property. Let a ˆ, a ˇ ∈ Y be such that d(ˆ a · t, a ˇ · t) < δ for all t ∈ R. Then, for any integer sequences (kn )∞ n=1 , (ln )∞ , such that l − k → ∞ as n → ∞ and n n n=1 lim

n→∞

ln kUaˆ (ln , kn )w(ˆ a · kn )k = λ, ln − k n

one has ln kUaˇ (ln , kn )w(ˇ a · kn )k ln − k n ln kUaˇ (ln , kn )w(ˇ a · kn )k ≤ lim sup ≤ λ + . ln − k n n→∞

λ − ≤ lim inf n→∞

PROOF As the mapping [ Y 3 a ˜ 7→ kUa˜ (1, 0)w(˜ a)k ∈ (0, ∞) ] is continuous on a compact set, we have M1 := inf { kUa˜ (1, 0)w(˜ a)k : a ˜ ∈ Y } > 0. Moreover, that mapping is uniformly continuous. Fix > 0, and take δ > 0 such that for any a ˜(1) , a ˜(2) ∈ Y , if d(˜ a(1) , a ˜(2) ) < δ then | kUa˜(1) (1, 0)w(˜ a(1) )k− (2) − kUa˜(2) (1, 0)w(˜ a )k | < M1 (1 − e ). We have e− kUaˇ (k + 1, k)w(ˇ a · k)k ≤ kUaˇ (k + 1, k)w(ˇ a · k)k − M1 (1 − e− ) < kUaˆ (k + 1, k)w(ˆ a · k)k < kUaˇ (k + 1, k)w(ˇ a · k)k + M1 (1 − e− ) ≤ (2 − e− )kUaˇ (k + 1, k)w(ˇ a · k)k < e kUaˇ (k + 1, k)w(ˇ a · k)k for all k ∈ Z. Since Uaˇ (l, k)w(ˇ a ·k) = Uaˇ (l, l−1)U (l−1, l−2) . . . U (k+1, k)w(ˇ a· k)) and Uaˆ (l, k)w(ˆ a · k) = Uaˆ (l, l − 1)U (l − 1, l − 2) . . . U (k + 1, k)w(ˆ a · k)), for any k < l, k, l ∈ Z, there holds e−(l−k) kUaˇ (l, k)w(ˇ a · k)k < kUaˆ (l, k)w(ˆ a · k)k < e(l−k) kUaˇ (l, k)w(ˇ a · k)k for any k, l ∈ Z, k < l. The lemma follows easily.

144

4.4.1

Spectral Theory for Parabolic Equations

The Random Case

Let ((Ω, F, P), {θt }t∈R ) be an ergodic metric dynamical system. We say that the random problem (4.1.1)a +(4.1.2)a (or, simply, a) is Y -admissible if it satisfies property (A4-R-ES) and Y˜0 (a) ⊂ Y . In the present subsection we will investigate the continuous dependence of the principal Lyapunov exponent and the principal spectrum on the coefficients in the norm topology in the random case. For the rest of the subsection we fix a Y -admissible a(0) . (0) For ω ∈ Ω, we write Uω (t, s) instead of UEa(0) (ω) (t, s). THEOREM 4.4.1 For each > 0 there is δ > 0 such that for any Y -admissible a, if kEa (ω) − Ea(0) (ω)k∞ < δ for P-a.e. ω ∈ Ω then |λ(a) − λ(a(0) )| < . THEOREM 4.4.2 For each > 0 there is δ > 0 such that for any Y -admissible a, if kEa (ω) − Ea(0) (ω)k∞ < δ for P-a.e. ω ∈ Ω then |λmin (a) − λmin (a(0) )| <

and

|λmax (a) − λmax (a(0) )| < .

PROOF (Proof of Theorems 4.4.1 and 4.4.2) Fix > 0, and fix a Y -admissible a such that kEa (ω) − Ea(0) (ω)k∞ < δ for P-a.e. ω ∈ Ω, where δ > 0 is as in Lemma 4.4.2. For ω ∈ Ω, we write Uω (t, s) instead of UEa (ω) (t, s). (0)

(0)

We start with the proof of Theorem 4.4.2. Let Ω0 ⊂ Ω, P(Ω0 ) = 1, (0) be such that for any ω ∈ Ω0 and any λ ∈ [λmin (a(0) ), λmax (a(0) )] there are ∞ sequences (kn )∞ n=1 , (ln )n=1 ⊂ Z, ln − kn → ∞ as n → ∞, such that (0)

ln kUω (ln , kn )w(Ea(0) (ω) · kn )k n→∞ ln − kn

λ = lim

(see Proposition 4.1.5). Similarly, let Ω0 ⊂ Ω, P(Ω0 ) = 1, be such that ln kUω (t, s)w(Ea (ω) · s)k t−s ln kUω (t, s)w(Ea (ω) · s)k = λmax (a) ≤ lim sup t−s t−s→∞

λmin (a) = lim inf

t−s→∞

for any ω ∈ Ω0 (see Proposition 4.1.4). (0) Fix ω ∈ Ω0 ∩ Ω0 such that kEa (ω) − Ea(0) (ω)k∞ < δ. It is a consequence of Lemma 4.4.1 that d(Ea(0) (ω) · t, Ea (ω) · t) < δ for all t ∈ R. It follows ˜ ∈ from Lemma 4.4.2 that for each λ ∈ [λmin (a(0) ), λmax (a(0) )] there is λ ˜ − λ| < . [λmin (a), λmax (a)] with |λ

4. Spectral Theory in Nonautonomous and Random Cases

145

˜ ∈ [λmin (a), By interchanging the roles of a(0) and a we obtain that for each λ (0) (0) ˜ λmax (a)] there is λ ∈ [λmin (a ), λmax (a )] with |λ − λ| < . Hence the Hausdorff distance between [λmin (a), λmax (a)] and [λmin (a(0) ), λmax (a(0) )] is less than , which is equivalent to the statement of Theorem 4.4.2. (0) We proceed now to the proof of Theorem 4.4.1. Let Ω1 be such that (0) (0) P(Ω1 ) = 1 and for any ω ∈ Ω1 there holds (0)

ln kUω (t, 0)w(Ea(0) (ω))k = λ(a(0) ). t→∞ t lim

Similarly, let Ω1 be such that P(Ω1 ) = 1 and for any ω ∈ Ω1 there holds ln kUω (t, 0)w(Ea (ω))k = λ(a) t→∞ t lim

(see Lemma 4.1.4). (0) Fix ω ∈ Ω1 ∩ Ω1 such that kEa (ω) − Ea(0) (ω)k∞ < δ. It is a consequence of Lemma 4.4.1 that d(Ea(0) (ω) · t, Ea (ω) · t) < δ for all t ∈ R. The statement of Theorem 4.4.1 follows now from Lemma 4.4.2.

4.4.2

The Nonautonomous Case

We say that the nonautonomous problem (4.1.9)a +(4.1.10)a (or, simply, a) is Y -admissible if it satisfies property (A4-N-ES) and Y˜ (a) ⊂ Y . In the present subsection we will investigate the continuous dependence of principal spectrum on the coefficients in the norm topology in the nonautonomous case. For the rest of the subsection we fix a Y -admissible a(0) . We write U (0) (t, s) instead of Ua(0) (t, s). THEOREM 4.4.3 For each > 0 there is δ > 0 such that for any Y -admissible a, if ka − a(0) k∞ < δ then |λmin (a) − λmin (a(0) )| <

and

|λmax (a) − λmax (a(0) )| < .

PROOF Fix > 0, and fix a Y -admissible a such that ka − a(0) k∞ < δ, where δ > 0 is as in Lemma 4.4.2. We write U (t, s) instead of Ua (t, s). ∞ For any λ ∈ [λmin (a(0) ), λmax (a(0) )] there are sequences (kn )∞ n=1 , (ln )n=1 ⊂ Z, ln − kn → ∞ as n → ∞, such that ln kU (0) (ln , kn )w(a(0) · kn )k n→∞ ln − kn

λ = lim

146

Spectral Theory for Parabolic Equations

(see Proposition 4.1.10). Further, we have ln kU (t, s)w(a · s)k t−s ln kU (t, s)w(a · s)k ≤ lim sup = λmax (a) t−s t−s→∞

λmin (a) = lim inf

t−s→∞

(see Proposition 4.1.9). It is a consequence of Lemma 4.4.1 that d(a(0) · t, a · t) < δ for all t ∈ R. It follows from Lemma 4.4.2 that for each λ ∈ [λmin (a(0) ), λmax (a(0) )] there is ˜ ∈ [λmin (a), λmax (a)] with |λ ˜ − λ| < . λ ˜ ∈ [λmin (a), By interchanging the roles of a(0) and a we obtain that for each λ ˜ − λ| < . Hence the λmax (a)] there is λ ∈ [λmin (a(0) ), λmax (a(0) )] with |λ Hausdorff distance between [λmin (a), λmax (a)] and [λmin (a(0) ), λmax (a(0) )] is less than , which is equivalent to the statement of Theorem 4.4.3.

4.5

Historical Remarks

In this section, we present some historical works on the principal spectrum of time independent, periodic, as well as general time dependent parabolic problems. Consider the following nonautonomous linear parabolic equation: N N ∂u X ∂ X ∂u = aij (t, x) + ai (t, x)u ∂t ∂xi j=1 ∂xj i=1 +

N X i=1

bi (t, x)

∂u + c0 (t, x)u, ∂xi

x ∈ D,

(4.5.1)

endowed with the boundary condition Ba (t)u = 0,

x ∈ ∂D,

(4.5.2)

where Ba (t) is a boundary operator of either the Dirichlet or Neumann or Robin type as in (2.0.3) with a(t, x) = (aij (t, x), ai (t, x), bi (t, x), c0 (t, x), d0 (t, x)), d0 (t, x) ≥ 0 for a.e. (t, x) ∈ R × ∂D. We first outline the principal eigenvalue theory for (4.5.1)+(4.5.2) in the time independent and periodic cases, and then give a review of the principal spectrum theory in the general time dependent case.

4.5.1

The Time Independent and Periodic Case

In this subsection, we assume that all the coefficients in (4.5.1)+(4.5.2) are time independent or periodic with period T . Recall that D is a domain, hence

4. Spectral Theory in Nonautonomous and Random Cases

147

is connected. The eigenvalue problem associated to (4.5.1)+(4.5.2) with time T -periodic coefficients reads as follows:  N N  ∂u ∂u X ∂ X  − + + ai (t, x)u aij (t, x)   ∂t ∂xi j=1 ∂xj   i=1    N  X  ∂u + c0 (t, x)u = λu, x ∈ D, + bi (t, x) ∂xi  i=1     Ba (t)u = 0, x ∈ ∂D,       u(0, ·) = u(T, ·).

(4.5.3)

When the domain D and the coefficients of (4.5.1)+(4.5.2) are sufficiently smooth, based on the Kre˘ın–Rutman theorem, it can be proved that there is a unique λprinc ∈ R such that (4.5.3) with λ = λprinc has a positive solution u (λprinc is called the principal eigenvalue of (4.5.1)+(4.5.2) and u is a principal eigenfunction) (see [50, Proposition 14.4]). It is not difficult to prove that [λmin (a), λmax (a)] = {λprinc }. Observe that for any other eigenvalue λ of (4.5.3), Re λ < λprinc . When D is a general bounded domain and a = (aij , ai , bi , c0 , d0 ) is a time in2 dependent or periodic function in L∞ (R×D, RN +2N +1 )×L∞ (R×∂D, R), the associated eigenvalue problem, in particular, the associated principal eigenvalue problem of (4.5.1)+(4.5.2) has also been extensively studied (see [11], [14], [29], [31], [32], etc.). Under quite general conditions, it is shown that the principal eigenvalue λprinc of (4.5.1)+(4.5.2) exists (λprinc is called a principal eigenvalue of (4.5.3) if (4.5.3) with λ = λprinc has a nontrivial nonnegative solution (in weak sense), such an eigenvalue if exists is unique). Write λprinc as λprinc (a, D). The continuous dependence of λprinc (a, D) with respect to a and D has also been widely studied (see [11], [14], [29], [31], [32] and references therein). For use in Chapter 5 we state some results from [29]. (n)

(n)

(n)

(n)

Assume that a = (aij , ai , bi , c0 , 0) and a(n) = (aij , ai , bi , c0 , 0) satisfy 2

(A4-N). Moreover, we assume that the L∞ (R × D, RN +2N +1 ) × L∞ (R × ∂D, R)-norms of a(n) are bounded in n ∈ N. Let Dn ⊂ RN be bounded domains. We say that Dn converges to D, denoted by Dn → D, if ¯ = 0, where |·| denotes the N -dimensional Lebesgue • limn→∞ |Dn \ D| measure. • There exists a compact set K ⊂ D of capacity zero such that for each compact set K 0 ⊂ D \ K there exists n0 ∈ N such that K 0 ⊂ Dn for n ≥ n0 .

148

Spectral Theory for Parabolic Equations

Recall that the capacity of a compact subset K ⊂ RN with respect to a set B0 , denoted by capB0 (K), is defined by capB0 (K) := inf{ kφk2W 1 (B0 ) : φ ∈ D(B0 ) 2

and φ ≥ 1

on K }

(see [28, (2.15)]). If for one open set B0 , capB0 (K) = 0, then for any open set B, capB (K) = 0. In this case, K is said to have zero capacity. Observe that if D is a Lipschitz domain, Dn ⊂ D, and for each compact set K ⊂ D there exists n0 ∈ N such that K ⊂ Dn for n ≥ n0 , then Dn → D. LEMMA 4.5.1 Consider a Dirichlet boundary condition problem. Assume that D is a Lips(n) (n) (n) chitz domain, Dn ⊂ D converge to D, and aij , ai , bi (i, j = 1, 2, . . . , N ), (n)

c0 converge respectively to aij , ai , bi , c0 in L2,loc ((0, T ) × RN ). Then λprinc (a(n) , Dn ) → λprinc (a, D). In the above, the coordinates of a(n) are understood to be equal to 0 outside Dn . PROOF

See [29, Theorem 2.10].

LEMMA 4.5.2 Let D be a Lipschitz domain in RN . Then there is a sequence (Dn )∞ n=1 of C ∞ domains satisfying Dn ⊂ D and Dn → D as n → ∞. PROOF

4.5.2

See the proof of [60, Lemma 4.1].

The General Time Dependent Case

In this subsection, we assume that a in (4.5.1)+(4.5.2) is a general time dependent function. When D and a are sufficiently smooth, based on the abstract work [94] the principal spectrum and exponential separation theory for (4.5.1)+(4.5.2) has been well established (see [62], [79], [81], [82], [84], [92], etc.), which extends the principal eigenvalue and principal eigenfunction theory for elliptic and time periodic parabolic problems to general nonautonomous problems. Recently, principal spectrum and exponential separation for general time dependent parabolic problems on general domain have been investigated in several papers (see [59], [60], [61], etc.), mostly for the Dirichlet boundary condition case. It should be remarked that in some of the papers mentioned above the equations in the nondivergence form are allowed, also the derivatives in the Neumann or Robin boundary conditions need not be conormal.

Chapter 5 Influence of Spatial-Temporal Variations and the Shape of Domain

Consider the following random linear parabolic equation:  N N  ∂u X ∂ X ∂u    = aij (θt ω, x) + ai (θt ω, x)u   ∂xi j=1 ∂xj   ∂t i=1 N X ∂u  + c0 (θt ω, x)u, bi (θt ω, x) +    ∂xi  i=1   B (t)u = 0, ω

x ∈ D,

(5.0.1)

x ∈ ∂D,

where Bω (t) = Baω (t), Baω is the boundary operator in (2.0.3) with a being replaced by aω (t, x) = (aij (θt ω, x), ai (θt ω, x), bi (θt ω, x), c0 (θt , x), d0 (θt ω, x)), d0 (ω, x) ≥ 0 for all ω ∈ Ω and a.e. x ∈ ∂D, ((Ω, F, P), {θt }t∈R ) is an ergodic metric dynamical system, and the functions aij (i, j = 1, . . . , N ), ai (i = 1, . . . , N ), bi (i = 1, . . . , N ) and c0 are (F × B(D), B(R))-measurable, and the function d0 is (F × B(∂D), B(R))-measurable; and consider the following nonautonomous linear parabolic equation:  N N  ∂u X ∂ X ∂u    = aij (t, x) + ai (t, x)u   ∂xi j=1 ∂xj   ∂t i=1 N X (5.0.2) ∂u  + bi (t, x) + c0 (t, x)u, x ∈ D,    ∂xi  i=1   B (t)u = 0, x ∈ ∂D, a

where Ba is the boundary operator in (2.0.3) with a(t, x) = (aij (t, x), ai (t, x), bi (t, x), c0 (t, x), d0 (t, x)), d0 (t, x) ≥ 0 for a.e. (t, x) ∈ R × ∂D. In the present chapter, we investigate the influence of spatial and temporal variations of the zero order terms of (5.0.1) and (5.0.2) on their principal spectrum and principal Lyapunov exponents. We show that spatial and temporal variations cannot reduce the principal spectrum and principal Lyapunov exponents. Indeed, if the coefficients and the domain are sufficiently regular, spatial and temporal variations increase the principal spectrum and principal Lyapunov exponents except in the degenerate cases. In the biological context these results mean that invasion by a new species (see [16], p. 220) is always easier in the space and time dependent case.

149

150

Spectral Theory for Parabolic Equations

We also investigate the influence of the shape of the domain of (5.0.1) and (5.0.2) on their principal spectrum and principal Lyapunov exponents and extend the so called Faber–Krahn inequalities for elliptic and periodic parabolic problems to general time dependent and random ones. This chapter is organized as follows. In Section 5.1, we introduce notions and basic assumptions and present some lemmas for the use in later sections. We study the influence of temporal variations of the zero order terms of (5.0.1) and (5.0.2) on their principal spectrum and principal Lyapunov exponents in Section 5.2. Section 5.3 is devoted to the investigation of the influence of spatial variations of the zero order terms of (5.0.1) and (5.0.2) on their principal spectrum and principal Lyapunov exponents. In Section 5.4 the influence of the shape of the domain of (5.0.1) and (5.0.2) on their principal spectrum and principal Lyapunov exponents is explored.

5.1

Preliminaries

In this section, we introduce notions and basic assumptions and establish lemmas which will be used in later sections.

5.1.1

Notions and Basic Assumptions

N ω N ω N ω ω First, consider (5.0.1). Write aω := ((aω ij )i,j=1 , (ai )i=1 , (bi )i=1 , c0 , d0 ), ω ω where aij (t, x) = aij (θt ω, x), etc. We assume that for each ω ∈ Ω, a belongs 2 to L∞ (R × D, RN +2N +1 ) × L∞ (R × ∂D, R). Moreover, the set { aω : ω ∈ Ω } 2 is bounded in the L∞ (R × D, RN +2N +1 ) × L∞ (R × ∂D, R)-norm by M ≥ 0. 2 Define the mapping Ea : Ω → L∞ (R × D, RN +2N +1 ) × L∞ (R × ∂D, R) as

Ea (ω) := aω . Put Y˜ (a) := cl { Ea (ω) : ω ∈ Ω }

(5.1.1)

with the weak-* topology, where the closure is taken in the weak-* topology. The set Y˜ (a) is a compact metrizable space. Note that Ea is (F, B(Y˜ (a)))-measurable (see Lemma 4.1.1). ˜ the image of the measure P under Ea : for any Borel set A ∈ Denote by P ˜ ˜ is a {σt }-invariant ergodic Borel measure on B(Y˜ (a)), P(A) := P(Ea−1 (A)). P Y˜ (a). Put ˜ Y˜0 (a) := supp P. (5.1.2) Then Y˜0 (a) is a closed (hence compact) and {σt }-invariant subset of Y˜ (a), ˜ Y˜0 (a)) = 1. Also, Y˜0 (a) is connected (see Subsection 4.1.1 for detail). with P(

5. Influence of Spatial-Temporal Variations and the Shape of Domain

151

If Assumptions (A2-1)–(A2-3) are satisfied for Y replaced with Y˜ (a), we will denote by Π(a) = {Π(a)t }t≥0 the topological linear skew-product semiflow generated by (5.0.1) on the product Banach bundle L2 (D) × Y˜ (a): Π(a)(t; u0 , a ˜) = Π(a)t (u0 , a ˜) := (Ua˜ (t, 0)u0 , σt a ˜) for t ≥ 0, a ˜ ∈ Y˜ (a), u0 ∈ L2 (D). Throughout this chapter, when speaking about the random equation (5.1.1) we assume that (A5-R1) The assumptions (A2-1)–(A2-3) are satisfied for Y replaced with Y˜ (a) defined by (5.1.1), and the topological linear skew-product semiflow Π(a) generated by (5.0.1) on L2 (D) × Y˜ (a) satisfies (A3-1) and (A3-2) and admits an exponential separation over Y˜0 (a) defined by (5.1.2). In the case of the Neumann or Robin boundary conditions, the exponent ς in (A3-2) equals zero. It should be remarked that from (A5-R1) the assumption (A4-R-ES) follows. We denote by λ(a) the principal Lyapunov exponent of (5.0.1) (see Section 4.1 for definitions). Sometimes we may denote λ(a) by λ(a, D) to indicate the dependence of λ(a) on the domain D. We denote by w : Y˜0 (a) → L2 (D)+ with kw(˜ a)k = 1 for any a ˜ ∈ Y˜0 (a) the unique function such that for any ˜ a ˜ ∈ Y0 (a) the fiber X1 (˜ a) of the one-dimensional bundle X1 in the exponential separation equals span{w(˜ a)}. We observe that, as (A3-1) and (A3-2) hold, it follows by Theorem 3.3.1 that if, for some ω ∈ Ω with Ea (ω) ∈ Y˜0 (a), a function u = u(t, x) is an entire positive solution of (5.0.1) then w(Ea (ω) · t) = u(t, ·)/ku(t, ·)k for each t ∈ R. At some places we also assume the following: (A5-R2) In the case of Dirichlet boundary conditions, D is a Lipschitz domain. In the case of Neumann and Robin boundary conditions, for each a ˜ ∈ Y˜0 (a), w(·, ·; a ˜) ∈ W 1,0 ((S, T ) × D) and w(·, ·; a ˜)|∂D ∈ L2 ((S, T ) × ∂D) for any S < T , where w(t, x; a ˜) := w(˜ a · t)(x). It will be pointed out explicitly where (A5-R2) is assumed. Recall that by Proposition 2.2.11, for any a ˜ ∈ Y˜0 (a), w(·, ·; a ˜) ∈ W 0,1 ((S, T ) × D). Consider (5.0.2). We write a = (aij , ai , bi , c0 , d0 ) and assume that a belongs 2 to L∞ (R × D, RN +2N +1 ) × L∞ (R × ∂D, R). Put Y˜ (a) := cl { a · t : t ∈ R } (5.1.3) with the weak-* topology, where the closure is taken in the weak-* topology. The set Y˜ (a) is a compact connected metrizable space. If Assumptions (A2-1)–(A2-3) are satisfied for Y replaced with Y˜ (a), we will denote by Π(a) = {Π(a)t }t≥0 the topological linear skew-product semiflow generated by (5.0.2) on the product Banach bundle L2 (D) × Y˜ (a): Π(a)(t; u0 , a ˜) = Π(a)t (u0 , a ˜) := (Ua˜ (t, 0)u0 , σt a ˜)

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for t ≥ 0, a ˜ ∈ Y˜ (a), u0 ∈ L2 (D). Throughout this chapter, when speaking about the nonautonomous equation (5.1.3) we assume that (A5-N1) The assumptions (A2-1)–(A2-3) are satisfied for Y replaced with Y˜ (a) defined by (5.1.3) and the topological linear skew-product semiflow Π(a) generated by (5.0.2) on L2 (D) × Y˜ (a) satisfies (A3-1) and (A3-2) and admits an exponential separation over Y˜ (a). In the case of the Neumann or Robin boundary conditions, the exponent ς in (A3-2) equals zero. It should be remarked that from (A5-N1) the assumption (A4-N-ES) follows. We denote by [λmin (a), λmax (a)] the principal spectrum interval of (5.0.2) (see Section 4.1 for definitions). Sometimes we may denote λmin (a) and λmax (a) by λmin (a, D) and λmax (a, D) to indicate the dependence of λmin (a) and λmax (a) on the domain D. We denote by w : Y˜ (a) → L2 (D)+ with kw(˜ a)k = 1 for any a ˜ ∈ Y˜ (a) the unique function such that for any a ˜ ∈ Y˜ (a) the fiber X1 (a) of the one-dimensional bundle X1 in the exponential separation of (5.0.2) equals span{w(˜ a)}. We observe that, as (A3-1) and (A3-2) hold, it follows by Theorem 3.3.1 that if a function u = u(t, x) is an entire positive solution of (5.0.2) then w(a · t) = u(t, ·)/ku(t, ·)k for each t ∈ R. At some places we also assume the following: (A5-N2) In the case of Dirichlet boundary conditions, D is a Lipschitz domain. In the case of Neumann and Robin boundary conditions, for each a ˜ ∈ Y˜ (a), w(·, ·; a ˜) ∈ W 1,0 ((S, T ) × D) and w(·, ·; a ˜)|∂D ∈ L2 ((S, T ) × ∂D) for any S < T , where w(t, x; a ˜) := w(˜ a · t)(x). Again, it will be pointed out explicitly where (A5-N2) is assumed. Recall also that by Proposition 2.2.11, for any a ˜ ∈ Y˜ (a), w(·, ·; a ˜) ∈ W 0,1 ((S, T )×D). DEFINITION 5.1.1 (1) Let a be as in (5.0.2). We say that a is uniquely ergodic if the compact flow (Y˜ (a), σ) is uniquely ergodic. (2) Let a be as in (5.0.2). We say that a is minimal or recurrent if the compact flow (Y˜ (a), σ) is minimal. (3) Let g ∈ L∞ (R, R) and H(g) := cl { g(t + ·) : t ∈ R } with the weak-* topology, where the closure is taken under the weak-* topology. We say g is minimal or recurrent if the compact flow (H(g), σ ˜ ) is minimal, where σ ˜t g˜(·) := g˜(t + ·) for any g˜ ∈ H(g) and t ∈ R. DEFINITION 5.1.2 Let g ∈ L∞ (R, R) and a be as in (5.0.2). We say that g is recurrent with at least the same recurrence as a if both g and a

5. Influence of Spatial-Temporal Variations and the Shape of Domain

153

are recurrent and for any (tn )∞ n=1 ⊂ R, if limn→∞ a · tn exists, then so does limn→∞ g · tn (in the weak-* topology of L∞ (R, R)).

5.1.2

Auxiliary Lemmas

LEMMA 5.1.1 (1) Let hi : [0, T ] × D → R (i = 1, 2, . . . , N ) be square-integrable in t ∈ [0, T ] and aij = aji : D → R (i, j = 1, 2, . . . , N ) satisfy N X

N X

aij (x)ξi ξj ≥ α0

i,j=1

ξi2

i=1

for some α0 > 0 and a.e. x ∈ D, ξ = (ξ1 , ξ2 , . . . , ξN )> ∈ RN . Then for a.e. x ∈ D, N X

Z

1 aij (x) T i,j=1

0

T

Z

1 hi (t, x) dt T

N X

1 ≤ aij (x) T i,j=1

Z

T

hj (t, x) dt 0

T

hi (t, x)hj (t, x) dt. 0

Moreover, the equality holds at some x0 ∈ D if and only if hi (t, x0 ) = ˜ i (x0 ) for some h ˜ i (x0 ) (i = 1, 2, . . . , N ) and a.e. t ∈ [0, T ]. h (2) Let hi : Ω × D → R (i = 1, 2, · · · , N ) be square-integrable in ω ∈ Ω and aij = aji : D → R (i, j = 1, 2, . . . , N ) satisfy N X

N X

aij (x)ξi ξj ≥ α0

i,j=1

ξi2

i=1

for some α0 > 0 and a.e. x ∈ D, ξ = (ξ1 , ξ2 , · · · , ξN )> ∈ RN . Then for any a.e. x ∈ D, N X

Z aij (x)

Z hi (ω, x) dP(ω)

Ω

i,j=1

≤

N X i,j=1

hj (ω, x) dP(ω) Ω

Z aij (x)

hi (ω, x)hj (ω, x) dP(ω). Ω

Moreover, the equality holds at some x0 ∈ D if and only if hi (ω, x0 ) = ˜ i (x0 ) for some h ˜ i (x0 ) (i = 1, 2, . . . , N ) and P-a.e. ω ∈ Ω. h

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PROOF (1) is proved in [62, Lemma 2.2] and (2) is proved in [81, Lemma 3.5]. For completeness, we provide a proof of (2) in the following. (1) can be proved by similar arguments. (2) First, note that for any fixed x ∈ D, there is an orthogonal matrix L such that A = (aij (x))N ×N = L> diag (di ) L, where di > 0. Let (y1 (ω, x), y2 (ω, x), . . . , yN (ω, x))> := L(h1 (ω, x), h2 (ω, x), . . . , hN (ω, x))> . Then we have Z Z N Z X hi (ω, x) dP(ω) hj (ω, x) dP(ω) − hi (ω, x)hj (ω, x) dP(ω) aij (x) i,j=1

Ω

=

Ω N X

di

Z

Ω

2 yi (ω, x) dP(ω) −

Z

Ω

i=1

yi2 (ω, x) dP(ω) .

Ω

By the Schwarz inequality, 2 Z Z yi (ω, x) dP(ω) ≤ yi2 (ω, x) dP(ω) Ω

Ω

and the equality holds for some x0 ∈ D if and only if yi (ω, x0 ) = y˜i (x0 ) for some y˜i (x0 ) (i = 1, 2, . . . , N ) and P-a.e. ω ∈ Ω. Hence N X

di

Z Ω

i=1

and then

2 yi (ω, x) dP(ω) −

N X

Z Ω

Z aij (x)

≤

Z hi (ω, x) dP(ω)

Ω

i,j=1 N X i,j=1

yi2 (ω, x) dP(ω) ≤ 0

hj (ω, x) dP(ω) Ω

Z aij (x)

hi (ω, x)hj (ω, x) dP(ω) Ω

˜ i (x0 ) for and the equality holds at some x0 ∈ D if and only if hi (ω, x0 ) = h ˜ some hi (x0 ) (i = 1, 2, . . . , N ) and P-a.e. ω ∈ Ω. LEMMA 5.1.2 Let (Ω, F, P) be a probability space, and let E ⊂ RN . Assume that h : Ω×E → ¯ → R) has the following properties: R (resp. h : Ω × E (i) h(·, x) belongs to L1 ((Ω, F, P)), for each x ∈ E, ¯ and each > 0 there is δ > 0 such that if (ii) For each x ∈ E (resp. x ∈ E) ¯ ω ∈ Ω and kx − yk < δ then |h(ω, x) − h(ω, y)| < , y ∈ E (resp. y ∈ E), where k·k stands for the norm in RN .

5. Influence of Spatial-Temporal Variations and the Shape of Domain ˆ ¯ h(x) Denote, for each x ∈ E (resp. x ∈ E), :=

R Ω

155

h(ω, x) dP(ω). Then

¯ and any > 0 there is δ > 0 (the same as (a) for any x ∈ E (resp. x ∈ E) ¯ ω ∈ Ω and kx − yk < δ then in (ii)) such that if y ∈ E (resp. y ∈ E), ˆ ˆ |h(x) − h(y)| < , (b) there is a measurable Ω0 ⊂ Ω with P(Ω0 ) = 1 such that Z 1 T ˆ h(θt ω, x) dt = h(x) lim T →∞ T 0 ¯ Moreover the convergence is for all ω ∈ Ω0 and all x ∈ E (resp. x ∈ E). ¯ uniform in x ∈ E0 , for any compact E0 ⊂ E (resp. uniform in x ∈ E). PROOF This is, in fact, [84, Lemma 2.3]. For completeness we give a proof here. Part (a) follows easily by the fact that the continuity is uniform in ω ∈ Ω. To prove (b), take a countable dense set {xl }∞ l=1 in E. By Birkhoff’s Ergodic Theorem (Lemma 1.2.6), for each l ∈ N there is Ωl ⊂ Ω with P(Ωl ) = 1 such that Z 1 T ˆ l) lim h(θt ω, xl ) dt = h(x T →∞ T 0 T∞ for each ω ∈ Ωl . Take Ω0 := l=1 Ωl . ¯ For > 0 take δ > 0 such that if kx − yk < δ Fix x ∈ E (resp. x ∈ E). ˆ ˆ then |h(ω, x) − h(ω, y)| < /3 and |h(x) − h(y)| < /3. Let xl be such that kx − xl k < δ, and let T0 > 0 be such that Z 1 T ˆ h(θt ω, xl ) dt − h(xl ) < 3 T 0 for all T > T0 . Then Z 1 T ˆ < h(θt ω, x) dt − h(x) T 0 for all T > T0 . (b) then follows. For a given bounded Lipschitz domain D ⊂ RN , the so-called Schwarz symmetrized domain Dsym ⊂ RN of D is the open ball in RN with center 0 and the same volume as D. For a Lebesgue measurable function u : D → R+ , usym defined by usym (x) := sup { c ∈ R : x ∈ (Dc )sym }

for x ∈ Dsym

is called the Schwarz symmetrization of u, where Dc := { x ∈ D : u(x) ≥ c } (see [66]).

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LEMMA 5.1.3 (Symmetrization) (1) For every u ∈ L2 (D)+ ,

R D

(u(x))2 dx =

R Dsym

(usym (x))2 dx.

R R ˚ 1 (D)+ , (2) For every u ∈ W |∇u(x)|2 dx ≥ Dsym |∇usym (x)|2 dx. More2 D over, if u is analytic in D and ∂D is analytic, then the equality holds if and only if D = Dsym up to translation and u = usym up to phase shift. PROOF (1) See [66, Properties (C) and (P1)]. (2) See [66, Properties (G1) and (G2g)].

5.2

Influence of Temporal Variation on Principal Lyapunov Exponents and Principal Spectrum

In this section, we study the influence of temporal variations of the zeroth order terms in (5.0.1) and (5.0.2) on their principal Lyapunov exponents and principal spectrum. We assume aij (ω, x) = aij (x), ai (ω, x) = ai (x), bi (ω, x) = bi (x) in (5.0.1) and aij (t, x) = aij (x), ai (t, x) = ai (x), bi (t, x) = bi (x) in (5.0.2). That is, in this section, we consider the random equation of the form  N N  ∂u X ∂ X ∂u   = a (x) + a (x)u  ij i  ∂t ∂xi j=1 ∂xj   i=1    N X ∂u + bi (x) + c0 (θt ω, x)u,  ∂x  i  i=1        Bω (t)u = 0,

x ∈ D,

(5.2.1)

x ∈ ∂D,

where Bω (t)u is as in (5.0.1) with aij (ω, x) = aij (x), ai (ω, x) = ai (x), bi (ω, x) = bi (x), and ((Ω, F, P), θt ) is an ergodic metric dynamical system, and consider the nonautonomous equation of the form  N N  ∂u  ∂u X ∂ X  = a (x) + a (x)u  ij i  ∂xi j=1 ∂xj  ∂t  i=1    N X ∂u + c0 (t, x)u, + bi (x)  ∂xi   i=1        Ba (t)u = 0,

x ∈ D,

x ∈ ∂D,

(5.2.2)

5. Influence of Spatial-Temporal Variations and the Shape of Domain

157

where Ba (t)u is as in (5.0.2) with aij (t, x) = aij (x), ai (t, x) = ai (x), bi (t, x) = bi (x). Consider (5.2.1). Write a = (aij (·), ai (·), bi (·), c0 (·, ·), d0 (·, ·)). We assume that a satisfies (A5-R1) and ˜ ⊂ Ω with P(Ω) ˜ = 1 such that for each ω ∈ Ω ˜ the (A5-R3) There exists Ω R R 1 T 1 T limits limT →∞ T 0 c0 (θt ω, x) dt and limT →∞ T 0 d0 (θt ω, x) dt exist for a.e. x ∈ D and x ∈ ∂D, respectively; moreover, Z Z 1 T lim c0 (θt ω, x) dt = c0 (·, x) dP(·) for a.e. x ∈ D, T →∞ T 0 Ω Z Z 1 T lim d0 (θt ω, x) dt = d0 (·, x) dP(·) for a.e. x ∈ ∂D. T →∞ T 0 Ω 2 We call a ˆ = (aij (·), ai (·), bi (·), cˆ0 (·), dˆ0 (·)) ∈ L∞ (D, RN +2N +1 ) × L∞ (∂D, R) the time averaged function of a if Z cˆ0 (x) = c0 (·, x) dP(·) for a.e. x ∈ D,

Ω

dˆ0 (x) =

Z d0 (·, x) dP(·)

for a.e. x ∈ ∂D.

Ω

The time independent equation  N N  ∂u X ∂ X ∂u   = a (x) + a (x)u  ij i   ∂t ∂xi j=1 ∂xj  i=1  N X ∂u  bi (x) + + cˆ0 (x)u,   ∂x  i  i=1    Baˆ u = 0,

(5.2.3)

x ∈ D, x ∈ ∂D,

where Baˆ ≡ Baˆ (t) is as in (2.0.3) with a being replaced by a ˆ = (aij (·), ai (·), ˆ bi (·), cˆ0 (·), d0 (·)) is called the time averaged equation of (5.2.1) if a ˆ is the time averaged function of a. Note that under assumption (A5-R3), the averaged equation of (5.2.1) exists. Consider (5.2.2). Let a = (aij (·), ai (·), bi (·), c0 (·, ·), d0 (·, ·)). We assume that a satisfies (A5-N1) and (A5-N3) The weak-* convergence of Z Tn 1 c0 (t, x) dt and lim n→∞ Tn − Sn S n

1 n→∞ Tn − Sn

Z

Tn

lim

d0 (t, x) dt Sn

in L∞ (D, R) and L∞ (∂D, R) imply pointwise convergence for a.e. x ∈ D and x ∈ ∂D, respectively, for any Tn − Sn → ∞.

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2 We call a ˆ = (aij (·), ai (·), bi (·), cˆ0 (·), dˆ0 (·)) ∈ L∞ (D, RN +2N +1 ) × L∞ (∂D, R) a time averaged function of a if

1 n→∞ Tn − Sn

Z

1 n→∞ Tn − Sn

Z

Tn

cˆ0 (x) = lim dˆ0 (x) = lim

c0 (t, x) dt

for a.e. x ∈ D,

d0 (t, x) dt

for a.e. x ∈ ∂D

Sn Tn

Sn

for some Tn − Sn → ∞. Equation (5.2.3) is called a time averaged equation of (5.2.2) if a ˆ = (aij (·), ai (·), bi (·), cˆ0 (·), dˆ0 (·)) is a time averaged function of a. Note that under assumption (A5-N3), the averaged equations of (5.2.2) exist. Observe that the eigenvalue problem associated to (5.2.3) reads as follows: N N X ∂ X  ∂u   aij (x) + ai (x)u    ∂xi j=1 ∂xj   i=1 N X (5.2.4) ∂u  + b (x) + c ˆ (x)u = λu, x ∈ D,  i 0  ∂xi   i=1    Baˆ u = 0, x ∈ ∂D. It is well known that (5.2.4) has a unique eigenvalue, denoted by λprinc (ˆ a), which satisfies that it is real, simple, has an eigenfunction ϕprinc (ˆ a) ∈ L2 (D)+ associated to it, and for any other eigenvalue λ of (5.2.4), Re λ < λprinc (ˆ a) (see [29], [31]). We call λprinc (ˆ a) the principal eigenvalue of (5.2.3) and ϕprinc (ˆ a) a principal eigenfunction (in the literature, sometimes, −λprinc (a) is called the principal eigenvalue of (5.2.3)). Our objective in this section is to compare the principal Lyapunov exponents and principal spectrum of (5.2.1) and (5.2.2) with the principal eigenvalue of their averaged equations. We will consider the smooth case (both the domain and the coefficients are sufficiently smooth) and the nonsmooth case separately.

5.2.1

The Smooth Case

In this subsection we assume that (5.2.1) satisfies (A5-R1) and (A2-5) (i.e., Y˜ (a) satisfies (A2-5), which implies that both (A5-R2) and (A5-R3) are satisfied), and that (5.2.2) satisfies (A5-N1)and (A2-5) (i.e., Y˜ (a) satisfies (A2-5), which also implies that both (A5-N2) and (A5-N3) are satisfied). We show that spatial and temporal variations cannot reduce the principal spectrum and principal Lyapunov exponents, and indeed, spatial and temporal variations increase the principal spectrum and principal Lyapunov exponents except in the degenerate cases. Consider (5.2.1). Let λ be the principal Lyapunov exponent. An application of Lemma 5.1.2 to appropriate derivatives of c0 and d0 , together with (A2-5),

5. Influence of Spatial-Temporal Variations and the Shape of Domain

159

ˆ := λprinc (ˆ ¯ and dˆ0 ∈ C 2+α (∂D). Denote λ gives that cˆ0 ∈ C 1+α (D) a), where λprinc (ˆ a) is the principal eigenvalue of (5.2.3). Let κ(˜ a) = −Ba˜ (0, w(˜ a), w(˜ a)) (5.2.5) ˜ ˜ for a ˜ = (aij , ai , bi , c˜0 , d0 ) ∈ Y0 (a), where Ba (·, u, v) is as in (2.1.4) in the Dirichlet and Neumann boundary condition cases, and is as in (2.1.5) in the Robin boundary condition case. Note that κ(˜ a) is well defined under the smoothness assumption (A2-5). Moreover, by the fact that w(˜ a) = Ua˜·(−1) (1, 0)w(˜ a · (−1))/kUa˜·(−1) (1, 0)w(˜ a · (−1))k and Proposition 2.5.4, the function [ Y˜0 (a) 3 a ˜ 7→ κ(˜ a) ∈ (0, ∞) ] is continuous. We have THEOREM 5.2.1 Consider (5.2.1). ˆ (1) λ ≥ λ. ˆ if and only if c0 (θt ω, x) = c01 (x) + c02 (θt ω) for P-a.e. ω ∈ Ω and (2) λ = λ d0 (θt ω, x) = d0 (x) for P-a.e. ω ∈ Ω. Consider (5.2.2). Let a = (aij (·), ai (·), bi (·), c0 (·, ·), d0 (·, ·)), with d0 ≡ 0 in the Dirichlet and Neumann boundary condition cases. Let Yˆ (a) := { a ˆ :∃Sn < Tn

Tn − Sn → ∞ such that Z Tn 1 c0 (t, x) dt for x ∈ D, cˆ0 (x) = lim n→∞ Tn − Sn S n Z Tn 1 d0 (t, x) dt for x ∈ ∂D }. dˆ0 (x) = lim n→∞ Tn − Sn S n with

It follows from (A2-5), via the Ascoli–Arzel`a theorem, that Yˆ (a) is nonempty; ¯ further, the C 1+α (D)-norms of cˆ0 and the C 2+α (∂D)-norms of dˆ0 are bounded ˆ uniformly in Y (a). Denote by Σ(a) = [λmin (a), λmax (a)] the principal spectrum interval of (5.2.2). We have THEOREM 5.2.2 Consider (5.2.2). (1) There is a ˆ ∈ Yˆ (a) such that λmin (a) ≥ λprinc (ˆ a). (2) λmax (a) ≥ λprinc (ˆ a) for any a ˆ ∈ Yˆ (a). (3) If a is uniquely ergodic and minimal, then λmin (a) = λmax (a) and λmin (a) = λprinc (ˆ a) for some a ˆ ∈ Yˆ (a) (Yˆ (a) is necessarily a singleton in this case) if and only if c0 (t, x) = c01 (x) + c02 (t) and d0 (t, x) = d0 (x).

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The above theorems are proved in [84] for general smooth case (see also [81] for the case that the boundary condition is Dirichlet type or Neumann type or Robin type with d0 being independent of t). For the completeness, we will provide proofs of the theorems. To do so, we first prove the following lemma. LEMMA 5.2.1 Let Y0 = Y˜ (a) in the nonautonomous case and Y0 = Y˜0 (a) in the random case. For any a ˜ ∈ Y0 and S < T , let v˜(t, x) = v˜(t, x; a ˜) := w(˜ a · t)(x) and w(x; ˆ S, T ) = w(x; ˆ S, T, a ˜) := exp

1 T −S

Z

T

ln w(˜ a · t)(x) dt .

S

Then w(x; ˆ S, T ) satisfies N N N X X ∂ X ∂w ˆ ∂w ˆ aij (x) + ai (x)w ˆ + bi (x) ∂x ∂x ∂x i j i i=1 j=1 i=1 Z T 1 v 1 ∂˜ (t, x) dt w ˆ ≤ T − S S v˜ ∂t Z T 1 Z T 1 + κ(˜ a · t) dt − c˜0 (t, x) dt w ˆ T −S S T −S S

(5.2.6)

for x ∈ D and Bˆa˜ (S, T )w ˆ=0 for x ∈ ∂D, where

Bˆa˜ (S, T )w ˆ :=

 w ˆ       N X N  X    a (x)∂ w ˆ + a (x) w ˆ νi  ij x i j  

(Dirichlet) (Neumann)

i=1 j=1 N X N X

   a (x)∂ w ˆ + a (x) w ˆ νi  ij x i j    i=1 j=1    1 Z T    + d˜0 (t, x) dt w ˆ T −S S

(5.2.7)

(Robin).

PROOF First, fix a ˜ ∈ Y0 . For given S < T , let η(t; S) := kUa˜·S (t − S, 0)w(˜ a · S)k. Let v¯(t, x; S) = v˜(t + S, x). Then η(t; S) satisfies ηt (t; S) = κ(˜ a · (t + S))η(t; S)

(5.2.8)

5. Influence of Spatial-Temporal Variations and the Shape of Domain

161

where κ(·) is defined in (5.2.5) (see Lemma 3.5.3) and v¯(t, x; S) satisfies  N N  ∂¯ v ∂¯ v X ∂ X   = + a (x)¯ v a (x) i ij   ∂t ∂xi j=1 ∂xj   i=1  N X ∂¯ v  + + c˜0 (t + S, x)¯ v − κ(˜ a · (t + S))¯ v , x ∈ D, bi (x)   ∂x i   i=1    Ba˜ (t + S)¯ v = 0, x ∈ ∂D, (see Lemma 3.5.4). By Proposition 2.5.1, we can differentiate w ˆ twice and have Z T ∂w(˜ a · t)(x) 1 1 ∂w ˆ (x; S, T ) = w(x; ˆ S, T ) dt, ∂xi T − S S w(˜ a · t)(x) ∂xi

(5.2.9)

(5.2.10)

! Z T ∂2w ˆ 1 ∂w(˜ a · t) 1 (x; S, T ) = w(x; ˆ S, T ) (x) dt ∂xi ∂xj (T − S)2 S w(˜ a · t)(x) ∂xi ! Z T ∂w(˜ a · t) 1 (x) dt · w(˜ a · t)(x) ∂xj S Z T 1 ∂ 2 w(˜ a · t) 1 + w(x; ˆ S, T ) (x) T − S S w(˜ a · t)(x) ∂xi ∂xj ∂w(˜ a · t) 1 ∂w(˜ a · t) − 2 (x) (x) dt (5.2.11) w (˜ a · t)(x) ∂xi ∂xj for x ∈ D. Hence w ˆ = w(x; ˆ S, T ) satisfies N N N X X ∂ X ∂w ˆ ∂w ˆ aij (x) + ai (x)w ˆ + bi (x) ∂x ∂x ∂x i j=1 j i i=1 i=1 Z T −S Z T Z T 1 1 ∂¯ v = (t, x; S) dt + κ(˜ a · t) dt − c˜0 (t, x) dt w ˆ T −S 0 v¯ ∂t S s N hZ T 1 X 1 ∂w(˜ a · t) +w ˆ aij (x) dt (T − S)2 S w(˜ a · t) ∂xi i,j=1 Z T ∂w(˜ a · t) i 1 dt · w(˜ a · t) ∂xj S Z T N X 1 1 ∂w(˜ a · t) ∂w(˜ a · t) −w ˆ aij (x) dt (5.2.12) 2 T − S S w (˜ a · t) ∂xi ∂xj i,j=1

for x ∈ D and Bˆa˜ (S, T )w ˆ=0

162

Spectral Theory for Parabolic Equations

on ∂D, where Bˆa˜ (S, T ) is as defined in (5.2.7). By Lemma 5.1.1 we have N N N X X ∂ X ∂w ˆ ∂w ˆ + ai (x)w ˆ + aij (x) bi (x) ∂x ∂x ∂x i j=1 j i i=1 i=1 Z T −S Z T Z T 1 1 ∂¯ v ≤ (t, x; S) dt + κ(˜ a · t) dt − c˜0 (t, x) dt w ˆ T −S 0 v¯ ∂t S s

for x ∈ D. The lemma thus follows. PROOF (Proof of Theorem 5.2.1) In the following proof, we write Uω (t, 0) for UEa (ω) (t, 0), write w(ω) for w(Ea (ω)), and write κ(ω) for κ(Ea (ω)) for ω ∈ Ω with Ea (ω) ∈ Y˜0 (a). (1) First of all, since c0 (ω, x) and d0 (ω, x) are continuous in x uniformly with respect to ω, an application of Lemma 5.1.2 to c0 and d0 gives the existence of Ω1 ⊂ Ω with P(Ω1 ) = 1 such that for each ω ∈ Ω1 , Z Z 1 T 1 T c0 (θt ω, x) dt and dˆ0 (x) = lim d0 (θt ω, x) dt cˆ0 (x) = lim T →∞ T 0 T →∞ T 0 ¯ and x ∈ ∂D, respectively. Let η(t; ω) := kUω (t, 0)w(ω)k. uniformly for x ∈ D Then Uω (t, 0)w(ω) = η(t; ω)w(ω)(·) and by (5.2.8) η(t; ω) satisfies ηt (t; ω) = κ(θt ω)η(t; ω).

(5.2.13)

It follows from Theorem 3.5.3 that there is Ω2 ⊂ Ω with P(Ω2 ) = 1 such that Z Z 1 T κ(θt ω) dt = κ(·) dP(·) for ω ∈ Ω2 . (5.2.14) λ = lim T →∞ T 0 Ω Let a ˆ := (aij , ai , bi , cˆ0 , dˆ0 ). We claim that λ(a) ≥ λ(ˆ a). In fact, take RT (n) (n) ω ∈ Ω1 ∩ Ω2 and Tn → ∞. Let cˆ0 (x) := T1n 0 n c0 (θt ω, x) dt, dˆ0 (x) := R Tn (n) (n) 1 ˆ(n) := (aij , ai , bi , cˆ0 , dˆ0 ). Then a ˆ(n) → a ˆ in the Tn 0 d0 (θt ωt, x) dt, and a R Tn 1 open-compact topology as n → ∞ and Tn 0 κ(θt ω) dt → λ(a) and n → ∞. Recall that v˜(t, x) = w(θt ω)(x). It follows from Proposition 2.5.1 that for each x ∈ D the set { w(θt ω)(x) : t ∈ R } is bounded away from zero. Consequently, Z 1 T 1 ∂˜ v 1 lim (t, x) dt = lim (ln v˜(T, x) − ln v˜(0, x)) = 0 (5.2.15) T →∞ T T →∞ T 0 v ˜ ∂t for any x ∈ D. Again by an application of Proposition 2.5.1 we may assume without loss of generality that there is w∗ (x) (w∗ (x) = 0 for x ∈ ∂D in the Dirichlet boundary condition case) such that lim w(x; ˆ Tn ) = w∗ (x)

n→∞

(5.2.16)

5. Influence of Spatial-Temporal Variations and the Shape of Domain

163

∗

∂w ∂w ˆ (x; Tn ) = (x) n→∞ ∂xi ∂xi

(5.2.17)

∂2w ˆ ∂ 2 w∗ (x; Tn ) = (x) n→∞ ∂xi ∂xj ∂xi ∂xj

(5.2.18)

lim

lim

for i, j = 1, 2, . . . , N and x ∈ D, where w(x; ˆ Tn ) = w(x; ˆ 0, Tn , aω ). Moreover, ¯ and the limits in (5.2.17), (5.2.18) the limit in (5.2.16) is uniform for x in D, are uniform for x in any compact subset D0 of D. In the Neumann and Robin ¯ cases, the limit in (5.2.17) is also uniform for x ∈ D. Then by Lemma 5.2.1 and (5.2.15)–(5.2.18) we have N N N X X ∂ X  ∂w∗ ∂w∗  ∗  a (x) + a (x)w + bi (x)  ij i   ∂xj ∂xi  i=1 ∂xi j=1 i=1 ∗ (5.2.19) + (ˆ c0 (x) − λ)w ≤ 0, x ∈ D,        Baˆ w∗ = 0, x ∈ ∂D, where Baˆ is as in (5.2.3). This implies that w(t, x) := w∗ (x) is a supersolution of  N N N X  ∂u ∂u  ∂u X ∂ X   = a (x) + a (x)u + bi (x) ij i   ∂t ∂x ∂x ∂x i j i  i=1 j=1 i=1 + (ˆ c0 (x) − λ)u,       B u = 0, a ˆ

x ∈ D,

x ∈ ∂D. (5.2.20) Let u(t, x; w∗ ) be the solution of (5.2.20) satisfying the initial condition u(0, ·; w∗ ) = w∗ . Then u(t, x; w∗ ) ≤ w∗ (x)

¯ for t > 0, x ∈ D.

(5.2.21)

ˆ − λ is the principal eigenvalue of (5.2.20). Then by (5.2.21), Note that λ ˆ − λ ≤ 0, hence λ ˆ ≤ λ. we must have λ (2) First, suppose that c0 (θt ω, x) = c01 (x) + c02 (θt ω) for any t ∈ R, ∗ ∗ x ∈ D and R ω ∈ Ω , where P(Ω ) = 1. Without loss of generality we can c01 (x) with c01 (x) + Rassume Ω c02 (·) dP(·) = 0 (for otherwise we replace R c (·) dP(·) and replace c (ω) with c (ω) − c (·) dP(·)). Suppose also 02 02 02 02 Ω Ω ¯ and dˆ0 (x) = that d0 (θt ω, x) = d0 (x). One has cˆ0 (x) = c01 (x) for x ∈ D, d0 (x) for x ∈ ∂D. Let u = ϕprinc (ˆ a)(x) be the positive principal eigenfunction of (5.2.3) normalized so that kϕprinc (ˆ a)k = 1. Define v(t, x; ω) := Rt ˆ ¯ and ω ∈ Ω∗ . It is easy ϕprinc (ˆ a)(x) exp λt + 0 c02 (θs ω) ds for t ∈ R, x ∈ D ∗ to see that for each ω ∈ Ω there holds v(t, x; ω) = (Uω (t, 0)u)(x),

¯ t ≥ 0, x ∈ D.

164

Spectral Theory for Parabolic Equations

By Proposition 4.1.1, λ = limt→∞ (1/t) ln kv(t, ·; ω)k for P-a.e. ω ∈ Ω. An application of Birkhoff’s Ergodic Theorem (Lemma 1.2.6) to c02 (bounded, ˆ for P-a.e. hence ∈ L1 ((Ω, F, P))) yields that limt→∞ (1/t) ln kv(t, ·; ω)k = λ ω ∈ Ω. ˆ Let Ω1 and Ω2 be as (1). Let v¯(s, x; ω) := Conversely, suppose that λ = λ. w(θs ω)(x) and Z t 1 ¯ ln w(θs ω)(x) ds for x ∈ D φ(x; ω) := lim sup exp t 0 t→∞ in the case of Neumann or Robin boundary condition, Z t  lim sup exp 1 ln w(θs ω)(x) ds φ(x; ω) := t 0 t→∞  0

for x ∈ D for x ∈ ∂D

in the case of Dirichlet boundary condition. Since w(ω)(x) are continuous in x uniformly in ω, an application of Lemma 5.1.2 provides the existence of Ω3 ⊂ Ω with P(Ω3 ) = 1 such that 1 Z t φ(x; ω) = lim exp ln w(θs ω)(x) ds t→∞ t 0 Z = exp ln w(·)(x) dP(·)

(5.2.22)

Ω

for any ω ∈ Ω3 and x ∈ D. Clearly, φ(x; ω) > 0 for x ∈ D and is independent of ω ∈ Ω3 . ∂ 2 w(ω)(x) ∂w(ω)(x) (i = 1, 2, . . . , N ) ( , i, j = 1, 2, . . . , N ) ∂xi ∂xi ∂xj ¯ (x ∈ D) uniformly in ω ∈ Ω and are are locally H¨ older continuous in x ∈ D integrable in ω ∈ Ω. Hence, again by Lemma 5.1.2, there is Ω4 ⊂ Ω with P(Ω4 ) = 1 such that Observe that

Z ∂φ 1 t 1 ∂w(θs ω)(x) (x; ω) = φ(x; ω) lim ds t→∞ t 0 ∂xi w(θs ω)(x) ∂xi Z 1 ∂w(·)(x) = φ(x; ω) dP(·), (5.2.23) ∂xi Ω w(·)(x)

5. Influence of Spatial-Temporal Variations and the Shape of Domain 165 Z t 1 1 ∂w(θs ω) ∂2φ (x; ω) = φ(x; ω) lim 2 (x) ds · t→∞ t ∂xi ∂xj w(θs ω)(x) ∂xi 0 Z t 1 ∂w(θs ω) (x) ds w(θs ω)(x) ∂xj 0 Z 1 1 t ∂ 2 w(θs ω) + φ(x; ω) lim (x) t→∞ t 0 w(θs ω)(x) ∂xi ∂xj 1 ∂w(θs ω) ∂w(θs ω) − 2 (x) (x) ds w (θs ω)(x) ∂xi ∂xj hZ 1 ∂w(·) = φ(x; ω) (x) dP(·) w(·)(x) ∂xi Z Ω i 1 ∂w(·) · (x) dP(·) Ω w(·)(x) ∂xj Z 1 ∂ 2 w(·) (x) + φ(x; ω) Ω w(·)(x) ∂xi ∂xj 1 ∂w(·) ∂w(·) − 2 (x) (x) dP(·) (5.2.24) w (·)(x) ∂xi ∂xj for ω ∈ Ω4 , x ∈ D, and Baˆ φ = 0

for x ∈ ∂D,

ω ∈ Ω4 ,

where Baˆ is as in (5.2.3). Let Ω0 := Ω1 ∩ Ω2 ∩ Ω3 ∩ Ω4 . By Lemma 3.5.4, v¯(t, x; ω) := w(θt ω)(x) satisfies  N N N X  ∂¯ v X ∂ X ∂¯ v ∂¯ v   = aij (x) bi (x) + ai (x)¯ v +   ∂x ∂x ∂x  ∂t i j=1 j i i=1 i=1 (5.2.25) + c0 (θt ω, x)¯ v − κ(θt ω)¯ v, x∈D       Bω (t)¯ v = 0, x ∈ ∂D for all ω ∈ Ω0 . (5.2.22)–(5.2.25) yield that N N N X X ∂φ ∂φ ∂ X aij (x) + ai (x)φ + bi (x) ∂x ∂x ∂x i j=1 j i i=1 i=1

= (λ − cˆ0 (x))φ N X

Z Z 1 ∂w(·) 1 ∂w(·) dP(·) dP(·) +φ aij (x) Ω w(·) ∂xi Ω w(·) ∂xj i,j=1 N X

Z 1 ∂w(·) ∂w(·) −φ aij (x) dP(·) 2 ∂xj Ω w (·) ∂xi i,j=1

(5.2.26)

166

Spectral Theory for Parabolic Equations

for ω ∈ Ω0 and x ∈ D, and Baˆ φ = 0

for ω ∈ Ω0 , x ∈ ∂D.

Consider  N N  ∂u ∂u X ∂ X   = a (x) + a (x)u  ij i   ∂t ∂xi j=1 ∂xj  i=1  N X ∂u  + bi (x) + (ˆ c0 (x) − λ)u, x ∈ D,   ∂xi   i=1    Baˆ u = 0, x ∈ ∂D.

(5.2.27)

ˆ we have that 0 is the principal eigenvalue of (5.2.27). Let ϕˆprinc be By λ = λ, a positive principal eigenfunction of (5.2.27), and let u(t, x; φ) be the solution of (5.2.27) with initial condition u(0, x; φ) = φ(x). By Lemma 5.1.1(2), N X i,j=1

aij (x)

Z Z 1 ∂w(·) 1 ∂w(·) dP(·) dP(·) Ω w(·) ∂xi Ω w(·) ∂xj −

N X

aij (x)

i,j=1

Z 1 ∂w(·) ∂w(·) dP(·) ≤ 0 2 ∂xj Ω w (·) ∂xi

for all x ∈ D. This together with (5.2.26) implies that φ(x) is a supersolution of (5.2.27) and hence u(t, x; φ) ≤ φ(x)

for

x ∈ D,

t ≥ 0.

(5.2.28)

Let w∗ (x) be a positive principal eigenfunction of the adjoint problem of (5.2.27). We then have that hφ, w∗ i > 0 and hϕˆprinc , w∗ i > 0. By taking α := hφ, w∗ i/hϕˆprinc , w∗ i (> 0) we see that φ = αϕˆprinc + vˆ, where vˆ ∈ L2 (D) is such that hˆ v , w∗ i = 0. Note that u(t, x; φ) = αϕˆprinc (x) + u(t, x; vˆ), where u(t, x; vˆ) is the solution of (5.2.27) with u(0, x; vˆ) = vˆ(x). Due to the exponential separation, ku(t, ·; vˆ)k → 0 as t → ∞. It then follows from (5.2.28) that αϕˆprinc (x) ≤ φ(x) for x ∈ D and then vˆ(x) ≥ 0 for x ∈ D. This implies that vˆ(x) = 0 for x ∈ D, hence αϕˆprinc (x) = φ(x) for x ∈ D. Therefore we must have Z Z N X 1 ∂w(ω) 1 ∂w(ω) dP(ω) dP(ω) aij (x) Ω w(ω) ∂xi Ω w(ω) ∂xj i,j=1 =

N X i,j=1

aij (x)

Z Ω

1 ∂w(ω) ∂w(ω) dP(ω) w2 (ω) ∂xi ∂xj

5. Influence of Spatial-Temporal Variations and the Shape of Domain

167

1 ∂w(ω)(x) in w(ω)(x) ∂xi x ∈ D, there are Ω5 ⊂ Ω0 with P(Ω5 ) = 1 and Fi = Fi (x) such that

for all x ∈ D. Then by Lemma 5.1.1 and continuity of

∂w(ω)(x) 1 = Fi (x) w(ω)(x) ∂xi for i = 1, 2, . . . , N , x ∈ D and ω ∈ Ω5 . Hence ∇ln w(ω)(x) = (F1 (x), F2 (x), . . . , FN (x))> ¯ →R for x ∈ D and ω ∈ Ω5 . This implies that there are a continuous F : D and a measurable G : Ω5 → R such that F (x) > 0, G(ω) > 0 and w(ω)(x) = F (x)G(ω) for T any x ∈ D and any ω ∈ Ω5 . Let Ω6 := r∈Q θr Ω5 , where Q is the set of rational numbers. Clearly, P(Ω6 ) = 1 and w(θt ω)(x) = F (x)G(θt ω) for t ∈ Q, x ∈ D, and ω ∈ Ω6 . The continuity of w(θt ω)(x) in t ∈ R then implies that w(θt ω)(x) = F (x)G(θt ω) for any t ∈ R, x ∈ D, and ω ∈ Ω6 . Therefore, by the first equation in (5.2.25), dG(θt ω) = F (x) dt

X N

∂ ∂F aij (x) (x) + ai (x)F (x) ∂xi ∂xj i,j=1

N X

∂F + bi (x) (x) + c0 (θt ω, x)F (x) − κ(θt ω)F (x) G(θt ω) ∂xi i=1 for t ∈ R, x ∈ D and ω ∈ Ω6 . ¯ → R After simple calculation we obtain that there are functions c01 : D and c02 : Ω6 → R such that c0 (ω, x) = c01 (x) + c02 (θt ω) for all t ∈ R, ω ∈ Ω6 ¯ Both functions c01 and c02 are bounded, consequently c02 is Pand x ∈ D. integrable. Applying a similar reasoning to the boundary condition Bω (t)F = 0, we see that d0 (θt ω, x) = d0 (x) for x ∈ ∂D, t ∈ R and P-a.e. ω ∈ Ω. In order not to interrupt the presentation, before giving the proof of Theorem 5.2.2 we formulate and prove the following result. LEMMA 5.2.2 Consider (5.2.2). Assume that a is uniquely ergodic. Then the limits Z T 1 lim c0 (t, x) dt =: cˆ0 (x) T −S→∞ T − S S and

Z T 1 d0 (t, x) dt =: dˆ0 (x) T −S→∞ T − S S ¯ and each x ∈ ∂D, respectively. In particular, it follows exist for each x ∈ D that Yˆ (a) = {(aij , ai , bi , cˆ0 , dˆ0 )}. lim

168

Spectral Theory for Parabolic Equations

PROOF Let P be the unique ergodic measure on (Y˜ (a), σ). Write c0 (˜ a, x) for c˜0 (0, x) and write d0 (˜ a, x) for d˜0 (0, x), where a ˜ = (aij , ai , bi , c˜0 , d˜0 ). Put R R ¯ and dˆ0 (x) := ˜ d0 (˜ cˆ0 (x) := Y˜ (a) c0 (˜ a, x) dP(˜ a) (x ∈ D) a, x) dP(˜ a) (x ∈ Y (a) ¯ ˜ ˜ D). As (Y (a), σ) is uniquely ergodic, for any g ∈ C(Y (a)) and any > 0 there is T0 = T0 (g, ) > 0 such that Z t Z 1 < g(˜ a · s) ds − g(·) dP(·) t 0 Y˜ (a) for each t > T0 and each a ˜ ∈ Y˜ (a) (see [90]). In particular, for any g ∈ C(Y˜ (a)) there holds 1 lim T −S→∞ T − S

Z

T

Z g(a · t) dt =

g(˜ a) dP(˜ a). Y˜ (a)

S

¯ a function g ∈ C(Y˜ (a)) given by g(˜ By taking, for a fixed x ∈ D, a) := c0 (˜ a, x) we obtain Z T 1 lim c0 (t, x) dt = cˆ0 (x). T −S→∞ T − S S Similarly, by taking, for a fixed x ∈ ∂D, a function g ∈ C(Y˜ (a)) given by g(˜ a) := d0 (˜ a, x) we obtain 1 T −S→∞ T − S

Z

T

lim

d˜0 (t, x) dt = dˆ0 (x).

S

PROOF (Proof of Theorem 5.2.2)

(1) First, for given S < T , let

η(t; S) := kUa (t, S)w(a · S)k and w(x; ˆ S, T ) := exp

1 T −S

Z

T

ln w(a · t)(x) dt .

S

n ;Sn ) There are Sn < Tn with Tn − Sn → ∞ such that ln Tη(T → λmin (a). It n −Sn follows from (A2-5) with the help of the Ascoli–Arzel`a theorem that withR Tn 1 out loss of generality we may assume that limn→∞ Tn −S c (t, x) dt and Sn 0 n R Tn 1 ¯ and limn→∞ Tn −Sn Sn d0 (t, x) dt exist, and the limits are uniform in x ∈ D in x ∈ ∂D, respectively. Denote Z Tn 1 cˆ0 (x) := lim c0 (t, x) dt n→∞ Tn − Sn S n

5. Influence of Spatial-Temporal Variations and the Shape of Domain

169

and 1 n→∞ Tn − Sn

dˆ0 (x) := lim

Z

Tn

d0 (t, x) dt. Sn

Let a ˆ := (aij , ai , bi , cˆ0 , dˆ0 ). We claim that λmin (a) ≥ λprinc (ˆ a). Denote v¯(t, x) := w(a · t)(x). It follows from Proposition 2.5.1 that for each x ∈ D the set { w(a · t)(x) : t ∈ R } is bounded away from zero. Consequently, 1 T −S→∞ T − S

Z

T

lim

S

1 ∂¯ v 1 (t, x) dt = lim (ln v¯(T, x) − ln v¯(S, x)) T −S→∞ T − S v¯ ∂t =0 (5.2.29)

for any x ∈ D. Again by an application of Proposition 2.5.1 we may assume without loss of generality that there is w∗ (x) (w∗ (x) = 0 for x ∈ ∂D in the Dirichlet boundary condition case) such that lim w(x; ˆ Sn , Tn ) = w∗ (x)

(5.2.30)

∂w ˆ ∂w∗ (x; Sn , Tn ) = (x) n→∞ ∂xi ∂xi

(5.2.31)

∂2w ˆ ∂ 2 w∗ (x; Sn , Tn ) = (x) n→∞ ∂xi ∂xj ∂xi ∂xj

(5.2.32)

n→∞

lim

lim

for i, j = 1, 2, . . . , N and x ∈ D. Moreover, the limit in (5.2.30) is uniform for ¯ and the limits in (5.2.31), (5.2.32) are uniform for x in any compact x in D, subset D0 of D. In the Neumann and Robin cases, the limit in (5.2.31) is also ¯ uniform for x in D. Then by Lemma 5.2.1 and (5.2.29)–(5.2.32) we have N N N X X ∂ X  ∂w∗ ∂w∗  ∗  a (x) + a (x)w + b (x)  ij i i   ∂xi ∂xj ∂xi  i=1

j=1

+ (ˆ c0 (x) − λmin (a))w∗ ≤ 0,

      

Baˆ w∗ = 0,

i=1

x ∈ D,

(5.2.33)

x ∈ ∂D.

It then follows from arguments similar to those in the proof of Theorem 5.2.1(1) that λmin (a) ≥ λprinc (ˆ a). (2) For any a ˆ = (aij , ai , bi , cˆ0 , dˆ0 ) ∈ Yˆ (a) there are Sn < Tn with Tn − Sn → ∞ such that Z Tn Z Tn 1 1 c0 (t, x) dt → cˆ0 (x) and d0 (t, x) dt → dˆ0 (x) Tn − Sn Sn Tn − Sn Sn

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Spectral Theory for Parabolic Equations

¯ and in x ∈ ∂D, respectively. Without loss of generality, uniformly in x ∈ D assume that there is λ0 such that 1 lim n→∞ Tn − Sn

Z

Tn

κ(a · t) dt = λ0 . Sn

By arguments similar to those in the proof of (1), λ0 ≥ λprinc (ˆ a). It follows from Theorem 3.1.2 and Lemmas 3.2.5 and 3.2.7 that λmax (a) ≥ λ0 . Then we have λmax (a) ≥ λprinc (ˆ a). (3) Let P be the unique ergodic measure on Y˜ (a). By Lemma 5.2.2, Yˆ (a) = {ˆ a}, where a ˆ = (aij , ai , bi , cˆ0 , dˆ0 ). Assume that the equality λmin (a) = λprinc (ˆ a) holds. By Theorem 5.2.1, ¯ → R, a there are Y0 (a) ⊂ Y˜ (a) with P(Y0R(a)) = 1, a continuous c01 : D P-integrable c¯02 : Y0 (a) → R with Y0 (a) c¯02 (˜ a) dP(˜ a) = 0, and a continuous d0 : ∂D → R such that for any a ˜ = (aij , ai , bi , c˜0 , d˜0 ) ∈ Y0 (a) there holds c˜0 (t, x) = c01 (x) + c¯02 (˜ a · t) for any t ∈ R and x ∈ D, and d˜0 (t, x) = d0 (x) for any t ∈ R and x ∈ ∂D. Fix some a ˜ ∈ Y0 (a). Since the compact flow (Y˜ (a), σ) is minimal, the orbit { a ˜ · t : t ∈ R } is dense in Y˜ (a), consequently there is a real sequence ∞ (sn )n=1 such that a ˜ · sn converges in the topology of Y˜ (a) to a, as n → ∞. ¯ In particular, c˜0 (t + sn , x) converges to c0 (t, x), for any t ∈ R and any x ∈ D. As a consequence, c¯02 (˜ a · (t + sn )) converges, for each t ∈ R, to some c02 (t). ¯ The fact that Therefore c0 (t, x) = c01 (x) + c02 (t) for all t ∈ R and all x ∈ D. d0 (t, x) = d0 (x) for all t ∈ R and all x ∈ ∂D follows in much the same way. Let c0 (t, x) = c01 (x)+c02 (t) and d0 (t, x) = d0 (x). Lemma 5.2.2 implies that Rt limt→∞ 1t 0 c02 (s) ds exists. Without loss of generality we can assume that ¯ and dˆ0 (x) = d0 (x) for this limit equals 0. One has cˆ0 (x) = c01 (x) for x ∈ D, x ∈ ∂D. Let u = u(x) be the positive principal eigenfunction of (5.2.3) nor Rt malized so that kuk = 1. Define v(t, x) := u(x) exp λprinc (ˆ a)t + 0 c02 (s) ds ¯ A straightforward computation shows that for t ∈ R and x ∈ D. lim

t→∞

1 ln kv(t, ·)k = λprinc (ˆ a). t

It is easy to see that there holds v(t, x) = (Ua (t, 0)u)(x),

¯ t ≥ 0, x ∈ D.

Then by Proposition 4.1.7, λmin (a) = λmax (a) = limt→∞

5.2.2

1 t

ln kv(t, ·)k.

The Nonsmooth Case

In this subsection, we study the extension of Theorems 5.2.1 and 5.2.2 about influence of temporal variations on principal spectrum and principal Lyapunov exponents of (5.2.1) and (5.2.2) to the nonsmooth case.

5. Influence of Spatial-Temporal Variations and the Shape of Domain

171

First, consider (5.2.1). Under the assumptions (A5-R1) and (A5-R3) the functions Z Z ˆ cˆ0 (x) = c0 (ω, x) dP(ω) and d0 (x) = d0 (ω, x) dP(ω) Ω

Ω

are defined for a.e. x ∈ D and a.e. x ∈ ∂D, respectively. Recall that a ˆ = (aij , ai , bi , cˆ0 , dˆ0 ) and that λprinc (ˆ a) stands for the principal eigenvalue of the time averaged equation (5.2.3). As in the smooth case we write λ for λ(a), ˆ for λprinc (ˆ and λ a). We have THEOREM 5.2.3 ˆ Consider (5.2.1) and assume (A5-R1)–(A5-R3). There holds λ ≥ λ. Next, consider (5.2.2). Note that for any sn < tn with tn − sn → ∞, R tn 1 k c (t, x) dt and there are subsequences snk and tnk such that tn −s snk 0 nk k R t nk 1 tnk −snk snk d0 (t, x) dt converge in the weak-* topology. Under (A5-N3), the R tn R tn 1 1 k k weak-* convergence of tn −s c0 (t, x) dt and tn −s d (t, x) dt ims snk 0 n n n k k k k k plies the pointwise convergence almost everywhere. Let Yˆ (a) := { a ˆ = (aij (·), ai (·), bi (·), cˆ0 (·), dˆ0 (·)) : ∃sn < tn with tn − sn → ∞ such that Z tn 1 cˆ0 (x) = lim c0 (t, x) dt for a.e. x ∈ D n→∞ tn − sn s n Z tn 1 d0 (t, x) dt for a.e. x ∈ ∂D }. dˆ0 (x) = lim n→∞ tn − sn s n We have THEOREM 5.2.4 Consider (5.2.2) and assume (A5-N1)–(A5-N3). (1) There is a ˆ ∈ Yˆ (a) such that λmin (a) ≥ λprinc (ˆ a). (2) λmax (a) ≥ λprinc (ˆ a) for any a ˆ ∈ Yˆ (a). To prove the theorems, we first show three lemmas. In the following, let Y0 = Y˜0 (a) in the random case, where Y˜0 (a) is defined in (5.1.2), and Y0 = Y˜ (a) in the nonautonomous case, where Y˜ (a) is defined in (5.1.3).

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LEMMA 5.2.3 (1) In the Dirichlet boundary condition case, there is M > 0 such that for any a ˜ ∈ Y0 , w(˜ a)(x) ≤ M for a.e. x ∈ D. (2) In the Neumann or Robin boundary condition case, there are M, m > 0 such that for any a ˜ ∈ Y0 , m ≤ w(˜ a)(x) ≤ M for a.e. x ∈ D. PROOF First by the L2 –L∞ estimates (Proposition 2.2.2), there is C1 > 0 such that for any a ˜ ∈ Y0 , kUa˜ (1, 0)w(˜ a)k∞ ≤ C1 . Further, by the compactness of Y0 and the continuity of w, there is C2 > 0 such that for any a ˜ ∈ Y0 , kUa˜ (1, 0)w(˜ a)k ≥ C2 . Note that w(˜ a) = Ua˜·(−1) (1, 0)w(˜ a · (−1))/kUa˜·(−1) (1, 0)w(˜ a · (−1))k. Hence for any a ˜ ∈ Y0 , w(˜ a)(x) ≤ M for a.e. x ∈ D, where M = C1 /C2 . It remains to prove the first inequality in (2). Since kUa˜ (1, 0)w(˜ a)k ≥ C2 for all a ˜ ∈ Y0 , there is C3 > 0 such that kUa˜ (1, 0)w(˜ a)k∞ ≥ C3 for any a ˜ ∈ Y0 . By (A3-2) with ς = 0, there is C4 > 0 such that for any a ˜ ∈ Y0 , (Ua˜ (1, 0)w(˜ a))(x) ≥ C4 for a.e. x ∈ D. By the L2 –L2 estimates (Proposition 2.2.2), there is C5 > 0 such that for any a ˜ ∈ Y0 , kUa˜ (1, 0)w(˜ a)k ≤ C5 . Hence for any a ˜ ∈ Y0 , w(˜ a)(x) ≥ m for a.e. x ∈ D, where m = C4 /C5 . For any a ˜ ∈ Y0 and any T > S, let 1 w(x; ˆ S, T, a ˜) := exp T −S

Z

T

ln w(˜ a · τ )(x) dτ

S

for x ∈ D. We claim that the function is well defined. By Proposition 2.2.9, w(˜ a)(x) > 0 for all x ∈ D. Further, Proposition 2.2.4 implies that we can integrate in the definition. It is a simple consequence of Lemma 5.2.3 that 0 < w(x; ˆ S, T, a ˜) ≤ M

(5.2.34)

for a.e. x ∈ D, in the Dirichlet case, and m ≤ w(x; ˆ S, T, a ˜) ≤ M

(5.2.35)

for a.e. x ∈ D, in the Neumann and Robin cases. In particular, w(·; ˆ S, T, a ˜) ∈ L∞ (D)+ for any S < T and any a ˜ ∈ Y0 . Moreover, as a consequence of Lemma 5.2.3(2) we have LEMMA 5.2.4 Assume the Neumann or Robin boundary conditions and (A5-R1)–(A5-R3) or (A5-N1)–(A5-N3). For given a ˜ ∈ Y0 and S < T we have that the derivatives

5. Influence of Spatial-Temporal Variations and the Shape of Domain ∂w ˆ ˜) ∂xi (x; T, S, a

173

=: ∂xi w(x; ˆ T, S, a ˜) (i = 1, . . . , N ) are well defined and satisfy

∂w ˆ w(x; ˆ T, S, a ˜) (x; T, S, a ˜) = ∂xi T −S

Z

T

S

1 ∂w(˜ a · τ) (x) dτ w(˜ a · τ )(x) ∂xi

for any x ∈ D. Further, ∂xi w(·; ˆ S, T, a ˜) ∈ L2 (D). LEMMA 5.2.5 Assume the Neumann or Robin boundary conditions and (A5-R1)–(A5-R3) or (A5-N1)–(A5-N3). For given a ˜ ∈ Y0 , S < T , and v(·) ∈ V ∩ L∞ (D) (see w(·) ˆ 1 ∗ , v(·) (2.1.2) for the definition of V ), there holds u(·,·) u(·,·) ∈ W (S, T ; V, V ) ∗ (see (2.1.3) for the definition of W (S, T ; V, V )), where w(x) ˆ = w(x; ˆ T, S, a ˜) and u(t, x) = (Ua˜ (t, S)w(˜ a · S))(x). PROOF We only prove that Observe that ∂ v(x)w(x) ˆ = ∂xi u(t, x)

v(·)w(·) ˆ u(·,·)

∂v(x) ˆ x) ∂xi w(x)u(t,

∈ W (S, T ; V, V ∗ ).

w(x) ˆ ∂u(t,x) + v(x) ∂∂x u(t, x) − v(x)w(x) ˆ ∂xi i

u2 (t, x)

and

∂u(t,x)

v(x)w(x) ˆ ∂ v(x)w(x) ˆ ∂t =− ∂t u(t, x) u2 (t, x)

.

It then follows from (A5-R2) or (A5-N2), Lemmas 5.2.3 and 5.2.4, andthe w(x) ˆ ˆ ∂ v(x)w(x) boundedness of v(·), that v(x) ∈ L ((S, T ), V ) and ∈ 2 u(t,x) ∂t u(t,x) L((S, T ), V ∗ ). Therefore

v(x)w(x) ˆ u(t,x)

∈ W (S, T ; V, V ∗ ).

In the following, we first prove Theorem 5.2.4. We consider the Dirichlet boundary condition and the Neumann and Robin boundary conditions separately. PROOF (Proof of Theorem 5.2.4 in the Dirichlet boundary condition case) (1) By Lemma 3.2.3, there is M2 ≥ 1 such that kUa˜ (T, S)uk ≤ M2 kUa˜ (T, S)w(˜ a · S)k for all a ˜ ∈ Y˜ (a), all S < T , and all u ∈ L2 (D) with kuk = 1. Therefore, for any > 0, there is K > 0 such that for any S < T with T − S > K there holds ln kUa˜ (T, S)w(˜ a · S)k ln kUa˜ (T, S)uk ≤ + (5.2.36) T −S T −S for all a ˜ ∈ Y˜ (a) and u ∈ L2 (D) with kuk = 1.

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Spectral Theory for Parabolic Equations

Next, it follows from Proposition 4.1.8 that for a given > 0 there are S < T with T − S > K such that ln kUa (T , S )w(a · S )k − . T − S

λmin (a) ≥

(5.2.37)

2

Let a ∈ L∞ (R×D, RN +2N +1 )×L∞ (R×∂D, R), a = a (t, x), be the function periodic in t with period T − S such that a (t, x) = a(t, x) for S ≤ t < T . We then have Ua (T , S ) = Ua (T , S ), where the symbol Ua (·, ·) has the obvious meaning. Let λ := λprinc (a ) and let u be the nonnegative principal eigenfunction associated to λ , normalized so that ku k = 1. Then Ua (T , S )u = eλ (T −S ) u . It then follows from (5.2.36) and (5.2.37) that λ =

ln kUa (T , S )u k ≤ λmin (a) + 2. T − S

(5.2.38)

By [39, Appendix C, Theorem 6], there is a sequence of C ∞ functions which are periodic in t with period T − S such that

(n) a (t, x)

a(n) → a

as

n → ∞ in L2 ([0, T − S ] × D0 )

for any compact subset D0 of D. Let Dn ⊂ D be a sequence of C ∞ subdomains of D such that Dn → D as n → ∞ (see Lemma 4.5.2). Let (n) (n) (n) λ := λprinc (a , Dn ). By Lemma 4.5.1, λ → λ as n → ∞. Therefore, for a given > 0, there is n1 = n1 () > 0 such that λ(n) ≤ λmin (a) + 3 for n ≥ n1 . Define a ˆ(n) (x)

1 := T − S

Z

T

a(n) (t, x) dt,

(5.2.39)

¯ x ∈ D.

S

(n) ˆ (n) Let λ := λprinc (ˆ a , Dn ). By Theorem 5.2.2,

ˆ (n) ≤ λ(n) . λ

(5.2.40)

Note that for any compact subset D0 of D, Z Z Z T

2 1

2 (n) kˆ a(n) (x) − a ˆ (x)k dx = (a (t, x) − a (t, x)) dt

dx (T − S )2 D0 S D0 Z Z T 1 2 ≤ ka(n) (t, x) − a (t, x)k dt dx T − S D0 S →0

as

n→∞

5. Influence of Spatial-Temporal Variations and the Shape of Domain

175

2

(in the above display, k·k stands for the standard norm in RN +2N +1 ). Then ˆ (n) → λ ˆ as n → ∞. Hence there is n2 ≥ n1 such by Lemma 4.5.1 again, λ that ˆ ≤ λ ˆ (n) + for n ≥ n2 . λ (5.2.41) It then follows that ˆ ≤ λmin (a) + 4. λ

(5.2.42)

Finally, take a sequence k → 0 such that cˆ0 (x) := lim

k→∞

Tk

1 − Sk

Z

Tk

c0 (t, x) dt Sk

ˆ → λprinc (ˆ exists for a.e. x ∈ D. By Lemma 4.5.1, we have λ a), where a ˆ= k (aij , ai , bi , cˆ0 , 0) ∈ Yˆ (a). This, together with (5.2.42), implies that λmin (a) ≥ λprinc (ˆ a), which proves (1). (2) Take any Sn < Tn with Tn − Sn → ∞ such that 1 n→∞ Tn − Sn

Z

Tn

c0 (t, x) dt

cˆ0 (x) := lim

Sn

exists for a.e. x ∈ D. Put a ˆ = (aij , ai , bi , cˆ0 , 0). We claim that λmax (a) ≥ λprinc (ˆ a). In fact, without loss of generality, we may assume that λ = 1 limn→∞ Tn −S ln kUa (Tn , Sn )w(a · Sn )k exists. It then follows from argun ments as above that λmax (a) ≥ λ ≥ λprinc (ˆ a). PROOF (Proof of Theorem 5.2.4 in the Neumann and Robin boundary conditions cases) (1) First, w ˆ denotes w(·; ˆ S, T, a) unless specified explicitly. Also, we use the summation convention. Let η(t; S) := kUa (t, S)w(a · S)k. For any fixed T > S and any v ∈ V ∩ L∞ (D), we have −

1 T −S

Z

T

hw(a · t), ∂t (v w/w(a ˆ · t))i dt S

1 =− T −S

Z

T

hw(a · t)η(t; S), ∂t ( S

1 − T −S

Z

T

S

vw ˆ )i dt η(t; S)w(a · t)

∂t η(t; S) hv, wi ˆ dt. η(t; S)

wi ˆ ∗ By Lemma 5.2.5, hv, ηw ∈ W (S, T ; V, V ). Hence by Proposition 2.1.3 and Eq. (2.1.4) or Eq. (2.1.5) we have (recall that the boundary ∂D of the domain D is assumed to be Lipschitz, so the Hausdorff measure on ∂D reduces to the

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Spectral Theory for Parabolic Equations

ordinary surface measure) 1 − T −S

Z

TD

η(t; S)w(a · t), ∂t

S

Z

1 =− T −S

E vw ˆ dt η(t; S)w(a · t)

T

vw ˆ dt η(t; S)w(a · t) S Z TZ 1 vw ˆ aij ∂xj (η(t; S)w(a · t))∂xi =− dx dt T −S S D η(t; S)w(a · t) Z TZ vw ˆ 1 dx dt ai η(t; S)w(a · t) ∂xi − T −S S D η(t; S)w(a · t) Z TZ 1 vw ˆ + bi ∂xi (η(t; S)w(a · t)) dx dt T −S S D η(t; S)w(a · t) Z TZ Z TZ 1 1 + c0 v w ˆ dx dt − d0 v w ˆ dx dt T −S S D T − S S ∂D Z Z T ∂ w ∂ w ∂x w ∂x w 1 xj xi vw ˆ − j ∂xi v w ˆ − j v ∂xi w ˆ dt dx = aij T −S D S w w w w Z Z T 1 ∂xi w vw ˆ − ∂xi v w ˆ − v ∂xi w ˆ dt dx + ai T −S D S w Z Z T Z T 1 ∂x w + bi i v w ˆ dt + c0 v w ˆ dt dx T −S D S w S Z Z T 1 d0 v w ˆ dt dx. − T − S ∂D S Ba t, η(t; S)w(a · t),

By Lemma 5.1.1, for v ∈ V ∩ L∞ (D)+

1 T −S

Z Z

T

∂xj w ∂xi w vw ˆ dt dx w w Z 1 Z T ∂ w 1 Z T ∂ w xj xi ≥ aij v w ˆ dt dt dx. T − S w T − S w D S S aij

D

S

Observe that ∂xi w ˆ 1 = w ˆ T −S

Z

T

S

∂xi w dt. w

It then follows from v ∈ V ∩ L∞ (D)+ , Proposition 2.2.11 and (A5-N2) (which

5. Influence of Spatial-Temporal Variations and the Shape of Domain

177

allows us to change the order of integration) that there holds Z TD vw ˆ E 1 w(a · t), ∂t dt − T −S S w(a · t) Z Z T 1 ∂t w(a · t) = dt v w ˆ dx T − S D S w(a · t) Z Z Z ∂xj w ˆ ∂xi w ∂xj w ˆ ∂x w ˆ ˆ ≥ aij vw ˆ dx − aij (∂xi v)w ˆ− aij j v(∂xi w) ˆ dx w ˆ w ˆ w ˆ w ˆ D D D Z Z ∂x w ˆ ˆ dx + ai ∂xi w ˆ v − ∂xi v w ˆ − ∂xi w ˆ v dx + bi i v w w ˆ D D Z Z Z Z T 1 T ∂t η(t; S) + c0 − dt v w ˆ dx − d0 v w ˆ dt dx T −S D S η(t; S) ∂D S Z = −aij ∂xj w ˆ ∂xi v − ai w ˆ ∂xi v + bi ∂xi w ˆ v dx D

Z Z ∂t η(t; S) 1 T c0 − dt wv ˆ dx + T −S D S η(t; S) Z Z T − d0 v w ˆ dt dx . ∂D

(5.2.43)

S

Note that there are Sn < Tn with Tn − Sn → ∞ such that Z Tn ∂t η(t; Sn ) 1 dt → λmin (a) as n → ∞. Tn − Sn Sn η(t; Sn ) Without loss of generality, we may assume that Z Tn 1 c0 (t, x) dt → cˆ0 (x) for a.e. Tn − Sn Sn and 1 Tn − Sn

Z

Tn

d0 (t, x) dt → dˆ0 (x)

for a.e.

(5.2.44)

x∈D

x ∈ ∂D

Sn

as n → ∞. Let a ˆ := (aij , ai , bi , cˆ0 , dˆ0 ). We prove that λmin (a) ≥ λ(ˆ a). In fact, for any > 0 there is n0 = n0 () ∈ N such that for any v ∈ V ∩ L∞ (D)+ with kvk ≤ 1 and any n ≥ n0 we have Z Tn 1 ∂t η(t; Sn ) dt ≤ λmin (a) + , Tn − Sn Sn η(t; Sn ) Z Tn 1 c0 (t, x) − cˆ0 (x) dt w(x; ˆ Sn , Tn )v dx ≥ −, D Tn − Sn Sn Z Z Tn 1 d0 (t, x) − dˆ0 (x) dt w(x; ˆ Sn , Tn )v dx ≤ , ∂D Tn − Sn Sn

Z

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Spectral Theory for Parabolic Equations

and Z D

1 Tn − Sn

Z

Tn

Sn

∂t w(a · t) dt v(x)w(x; ˆ Sn , Tn ) dx ≤ w(a · t)

(for the last display, see Lemma 5.2.3). It then follows from (5.2.43) that Z D

−aij ∂xj w ˆ ∂xi v − ai w ˆ ∂xi v + bi ∂xi w ˆ v + (ˆ c0 − λprinc (ˆ a))wv ˆ dx Z Z ˆ − d0 wv ˆ dx ≤ (λmin (a) − λprinc (ˆ a) + ) wv ˆ dx + 3 ∂D

D

for any v ∈ V ∩ L∞ (D)+ with kvk ≤ 1 and any n ≥ n0 , where w(x) ˆ = w(x; ˆ Sn , Tn , a). We specialize v(·) to be the principal eigenfunction of the adjoint equation associated to (5.2.3) with cˆ0 being replaced by cˆ0 − λprinc (ˆ a). Then Z 0= −aij ∂xj w ˆ ∂xi v − ai w ˆ ∂xi v + bi ∂xi w ˆ v + (ˆ c0 − λprinc (ˆ a))wv ˆ dx D Z − dˆ0 wv ˆ dx. ∂D

Further, with the help of (5.2.35) and taking into account that v satisfy a similar estimate, we see that there is m1 > 0 such that Z wv ˆ dx ≥ m1 D

for all n ≥ n0 , where w(x) ˆ = w(x; ˆ Sn , Tn , a). Suppose to the contrary that λmin (a) < λprinc (ˆ a). Let > 0 be so small that (λmin (a) − λprinc (ˆ a) + )m1 + 3 < 0. Then Z 0= D

−aij ∂xj w ˆ ∂xi v − ai w ˆ ∂xi v + bi ∂xi w ˆ v + (ˆ c0 − λprinc (ˆ a))wv ˆ dx Z Z ˆ − d0 wv ˆ dx ≤ (λmin (a) − λprinc (ˆ a) + ) wv ˆ dx + 3 < 0 ∂D

D

for n sufficiently large, where w(x) ˆ = w(x; ˆ Sn , Tn , a). This is a contradiction. Therefore we must have λmin (a) ≥ λprinc (ˆ a). (2) For any a ˆ = (aij , bi , bi , cˆ0 , dˆ0 ) ∈ Yˆ (a), there is Sn < Tn with Tn − Sn → ∞ such that Z Tn 1 c0 (t, x) dx → cˆ0 (x) for a.e. x ∈ D Tn − Sn Sn

5. Influence of Spatial-Temporal Variations and the Shape of Domain

179

and 1 Tn − Sn

Z

Tn

d0 (t, x) dx → dˆ0 (x)

for a.e. x ∈ ∂D.

Sn

Observe that for any > 0 there is n0 = n0 () > 0 such that 1 Tn − Sn

Z

Tn

∂t η(t; Sn ) dt ≤ λmax (a) + η(t; Sn )

Sn

for n > n0 . Then by arguments similar to those in (1), we have λmax (a) ≥ λprinc (ˆ a). PROOF (Proof of Theorem 5.2.3) λ = lim

T →∞

Let ω ∈ Ω be such that

ln kUEa (ω) (T, 0)w(Ea (ω))k T

and 1 T →∞ T

T

Z

c0 (θt ω, x) dt

cˆ0 (x) = lim

1 dˆ0 (x) = lim T →∞ T

for a.e. x ∈ D

0

Z

T

d0 (θt ω, x) dt

for a.e. x ∈ ∂D.

0

Then by arguments similar to those in the proof of Theorem 5.2.4, we have ˆ λ ≥ λprinc (ˆ a) (= λ).

5.3

Influence of Spatial Variation on Principal Lyapunov Exponents and Principal Spectrum

In this section, we study the influence of spatial variation of the zeroth order terms in (5.0.1) and (5.0.2) on their principal spectrum and principal Lyapunov exponents when the boundary conditions are Neumann. We assume aij (ω, x) = aij (ω), ai (ω, x) ≡ 0, bi (ω, x) ≡ 0 in (5.0.1) and aij (t, x) = aij (t), ai (t, x) ≡ 0, bi (t, x) ≡ 0 in (5.0.2). That is, we consider the nonautonomous equation of form  N N  ∂u X ∂ X ∂u   = aij (t) + c0 (t, x)u, x ∈ D,   ∂t ∂xi j=1 ∂xj i=1 N X N X     a (t)∂ u νi = 0, x ∈ ∂D, ij x j  i=1

j=1

(5.3.1)

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Spectral Theory for Parabolic Equations

and the random equation of form  N N  ∂u ∂u X ∂ X   + c0 (θt ω, x)u, x ∈ D, = aij (θt ω)   ∂t ∂xi j=1 ∂xj i=1 N X N X     aij (θt ω)∂xj u νi = 0, x ∈ ∂D,  i=1

(5.3.2)

j=1

where ((Ω, F, P), θ)R is an ergodic metric dynamical system. R 1 1 c (t, x) dx in the case of (5.3.1) and c¯0 (ω) := |D| c (ω, Let c¯0 (t) := |D| D 0 D 0 x) dx in the case of (5.3.2). Let a ¯ = (aij , 0, 0, c¯0 , 0). We call a ¯ the space average of a, and call the equations  N N  ∂u X ∂ X ∂u   = aij (t) + c¯0 (t)u, x ∈ D,   ∂t ∂xi j=1 ∂xj i=1 N X N X     a (t)∂ u νi = 0, x ∈ ∂D, ij x j 

(5.3.3)

 N N  ∂u X ∂ X ∂u   = aij (θt ω) + c¯0 (θt ω)u, x ∈ D,   ∂t ∂xi j=1 ∂xj i=1 N X N X     aij (θt ω)∂xj u νi = 0, x ∈ ∂D, 

(5.3.4)

i=1

j=1

and

i=1

j=1

the space averaged equations of (5.3.1) and (5.3.2), respectively. In the sequel we will speak of exponential separation, principal eigenvalues, etc., for space averaged equations. We assume (A5-N4) (5.3.1) satisfies (A5-N1), (A5-N2), and (5.3.3) satisfies (A5-N1). (A5-R4) (5.3.2) satisfies (A5-R1), (A5-R2), and (5.3.4) satisfies (A5-R1). Consider the space averaged equation (5.3.3). It is straightforward to see Rt ¯ is an entire that the function v(t, x) := exp 0 c¯0 (τ ) dτ (t ∈ R, x ∈ D) −1/2 positive solution of (5.3.3). Consequently, w(¯ a · t) = |D| 1 for each t ∈ R, where 1 means the function constantly equal to one (see a remark below Assumption (A5-N1)). This implies that (5.3.3) automatically satisfies (A5N2). Consider the space averaged equation (5.3.4). It is also straightforward to Rt see that, for P-a.e. ω ∈ Ω, the function v(t, x) := exp 0 c¯0 (θτ ω) dτ (t ∈ R, ¯ is an entire positive solution of (5.3.4). Consequently, w(Ea¯ (ω) · t) = x ∈ D) |D|−1/2 1 for each t ∈ R (see a remark below Assumption (A5-R1)). This also implies that (5.3.4) automatically satisfies (A5-R2).

5. Influence of Spatial-Temporal Variations and the Shape of Domain

181

Denote by [λmin (¯ a), λmax (¯ a)] and by λ(¯ a) the principal spectrum interval and principal Lyapunov exponent of (5.3.3) and (5.3.4), respectively. Then we have THEOREM 5.3.1 Consider (5.3.1) and assume (A5-N4). (1) [λmin (¯ a), λmax (¯ a)] = { λ : ∃Sn < Tn with Tn − Sn → ∞ such that R Tn 1 c¯ (t) dt }. λ = limn→∞ Tn −S Sn 0 n (2) λmin (a) ≥ λmin (¯ a) and λmax (a) ≥ λmax (¯ a). (3) Assume the smoothness assumption (A2-5). Then the equalities in (2) hold if and only if c0 (t, x) = c¯0 (t) for all x ∈ D and ∈ R. THEOREM 5.3.2 Consider (5.3.2) and assume (A5-R4). R (1) λ(¯ a) = Ω c¯0 (ω) dP(ω). (2) λ(a) ≥ λ(¯ a). (3) Assume the smoothness assumption (A2-5). Then the equality in (2) holds if and only if c0 (ω, x) = c¯0 (ω) for all x ∈ D and P-a.e. ω ∈ Ω. COROLLARY 5.3.1 Consider (5.3.1) and assume that aij (t) = aij and (A5-N1)–(A5-N4). (1) λmax (a) ≥ λprinc (ˆ a) ≥ λmin (¯ a) for all a ˆ ∈ Yˆ (a). (2) λmin (a) ≥ λprinc (ˆ a) for some a ˆ ∈ Yˆ (a). (3) λprinc (ˆ a) ≥ λmax (¯ a) for some a ˆ ∈ Yˆ (a). (4) Assume the smoothness assumption (A2-5). If a is minimal and uniquely ergodic, then 1 T →∞ T

Z

λ(a) ≥ λprinc (ˆ a) ≥ λ(¯ a) = lim

0

T

1 |D|

Z

c0 (t, x) dx dt.

D

COROLLARY 5.3.2 Consider (5.3.2) and assume that aij (ω) = aij and (A5-R1)–(A5-R4) hold. Then Z Z 1 λ(a) ≥ λprinc (ˆ a) ≥ λ(¯ a) = c0 (ω, x) dx dP(ω). Ω |D| D

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Spectral Theory for Parabolic Equations

We prove the above theorems and corollaries in the following order. First we prove Theorem 5.3.1(1), (2). Next we prove Theorem 5.3.2. Then we prove Theorem 5.3.1(3). Finally we prove Corollaries 5.3.1 and 5.3.2. PROOF (Proof of Theorem 5.3.1(1))

Since

Ua¯ (T, S)w(¯ a · S) = |D|−1/2 e

RT S

c¯0 (t) dt

for any S < T , there holds ln kUa¯ (T, S)w(¯ a · S)k = T −S

RT

c¯0 (t) dt . T −S

S

(1) then follows. PROOF (Proof of Theorem 5.3.1(2)) Note that u(t, ·; w(a·S), a·S) = 1 Ua (t, S)w(a · S). By Lemma 5.2.5, u(t,·;w(a·S),a·S) ∈ W (S, T ; V, V ∗ ). Hence by Proposition 2.1.3 and Eq. (2.1.4) we have −

1 T −S

T

1 dx dt u(t, x; w(a · S), a · S) D S Z T Z X N ∂u ∂u 2 1 1 aij (t) = /u dx dt T − S S |D| D i,j=1 ∂xi ∂xj Z T 1 + c¯0 (t) dt. (5.3.5) T −S S

Z

1 |D|

Z

u(t, x; w(a · S), a · S) ∂t

This implies that 1 T −S

Z

T

Z TZ 1 1 ∂t u(t, x; w(a · S), a · S) dx dt |D| T − S S D u(t, x; w(a · S), a · S) Z Z T 1 ∂t u(t, x; w(a · S), a · S) 1 dt dx = |D| T − S D S u(t, x; w(a · S), a · S) Z 1 1 = ln u(T, x; w(a · S), a · S) − ln w(a · S)(x) dx |D| T − S D ln ku(T, ·; w(a · S), a · S)k = T −Z S 1 1 ln w(a · T )(x) − ln w(a · S)(x) dx. + |D| T − S D

c¯0 (t) dt ≤ S

Note that λmin (a) = lim inf

T −S→∞

ln ku(T, ·; w(a · S), a · S)k T −S

5. Influence of Spatial-Temporal Variations and the Shape of Domain and

183

ln ku(T, ·; w(a · S), a · S)k . T −S T −S→∞

λmax (a) = lim sup Therefore,

λmin (a) ≥ λmin (¯ a),

λmax (a) ≥ λmax (¯ a).

PROOF (Proof of Theorem 5.3.2) (1) It follows from Proposition 4.1.1 and Lemma 1.2.6 that Z Z 1 T λ(¯ a) = lim c¯0 (θt ω) dt = c¯0 (·) dP(·) T →∞ T 0 Ω for P-a.e. ω ∈ Ω. (2) By Proposition 4.1.1 again, for P-a.e. ω ∈ Ω, λ(a) = lim

T →∞

1 ln kUEa (ω) (T, 0)w(Ea (ω))k T

and 1 λ(¯ a) = lim T →∞ T

Z

T

c¯0 (θt ω) dt. 0

It then follows from Theorem 5.3.1(2) that λ(a) ≥ λ(¯ a). (3) The “if” part is straightforward. Theorem 3.5.3 together with Part (1) imply that N X

Z Z λ(a) = D

aij (ω)

Ω i,j=1

∂w(ω) ∂w(ω) dP(ω) dx + λ(¯ a). ∂xi ∂xj

It then follows that λ(a) = λ(¯ a) if and only if Z Z D

N X

Ω i,j=1

aij (ω)

∂w(ω) ∂w(ω) dP(ω) dx = 0. ∂xi ∂xj

Then by the ellipticity we must have Z

N X

Ω i,j=1

aij (ω)

∂w(ω) ∂w(ω) dP(ω) = 0 ∂xi ∂xj

for all x ∈ D.

¯ we have that for P-a.e. ω ∈ Ω there holds Since w(ω) ∈ C 1 (D) ∂w(ω) ≡0 ∂xi

¯ on D,

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Spectral Theory for Parabolic Equations

which implies that w(ω) = const for P-a.e. ω ∈ Ω. Lemma 3.5.4 gives that c0 (θt ω) = κ(Ea (ω) · t) ¯ Consequently for P-a.e. ω ∈ Ω, t ∈ R and x ∈ D. c0 (ω, x) = c¯0 (ω) for all x ∈ D and P-a.e. ω ∈ Ω. PROOF (Proof of Theorem 5.3.1(3)) Let P be the unique ergodic measure on Y˜ (a). Then by Theorem 5.3.2(3), for P-a.e. a ˜ = (˜ aij , 0, 0, c˜0 , 0) ∈ ˜ ˜ Y (a), c˜0 is independent of x. By the minimality of (Y (a), σ), there is (tn )∞ n=1 ⊂ R such that c˜0 · tn → c0 in the open-compact topology. This implies that c0 is independent of x, and hence c0 (t, x) = c¯0 (t). PROOF (Proof of Corollary 5.3.1) (1) First, by Theorem 5.2.4(2), λmax (a) ≥ λprinc (ˆ a)

for all a ˆ ∈ Yˆ (a).

Now, for any a ˆ ∈ Yˆ (a), there are Sn , Tn ∈ R with Sn < Tn such that Tn −Sn → ∞ and Z Tn 1 cˆ0 (x) = lim c0 (t, x) dt for a.e. x ∈ D. n→∞ Tn − Sn S n Without loss of generality we can assume that Z Tn 1 cˇ0 := lim c¯0 (t) dt n→∞ Tn − Sn S n exists. Note that Z Z Z Tn 1 1 1 cˆ0 (x) dx = lim c0 (t, x) dt dx |D| D |D| D n→∞ Tn − Sn Sn Z Tn 1 c¯0 (t) dt = lim n→∞ Tn − Sn S n = cˇ0 . This together with Theorem 5.3.1(2) implies that λprinc (ˆ a) ≥ λprinc (ˇ a) ≥ λmin (¯ a) where a ˇ = (aij , 0, 0, cˇ0 , 0).

5. Influence of Spatial-Temporal Variations and the Shape of Domain

185

(2) This is Theorem 5.2.4(2). (3) There are Sn , Tn ∈ R such that Tn − Sn → ∞ and Z Tn 1 c¯0 (t) dt. λmax (¯ a) = lim n→∞ Tn − Sn S n Without loss of generality we can assume that Z Tn 1 cˆ0 (x) = lim c0 (t, x) dt n→∞ Tn − Sn S n exists for a.e. x ∈ D. Then by arguments as in (1) we have λprinc (ˆ a) ≥ λmax (¯ a) where a ˆ = (aij , 0, 0, cˆ0 , 0). (4) When a is minimal and uniquely ergodic then, by Lemma 5.2.2, λmax (a) = λmin (a) := λ(a), Yˆ (a) = singleton and Z

1 T →∞ T

λmax (¯ a) = λmin (¯ a) := λ(¯ a) = lim

0

T

Z

1 |D|

c0 (t, x) dx dt. D

Then by (1)–(3), 1 λ(a) ≥ λprinc (ˆ a) ≥ λ(¯ a) = lim T →∞ T

Z 0

T

1 |D|

Z c0 (t, x) dx dt. D

PROOF (Proof of Corollary 5.3.2) By Proposition 4.1.1 and Lemma 1.2.6, there is Ω0 ⊂ Ω with P(Ω0 ) = 1 such that ln ku(T, ·; w(ω), w)k , T →∞ T Z 1 T cˆ0 (x) = lim c0 (θt ω, x) dt, T →∞ T 0

λ(a) = lim

and

Z Z 1 T c¯0 (θt ω) dt = c¯0 (·) dP(·) T →∞ T 0 Ω for ω ∈ Ω0 . It then follows from Corollary 5.3.1 that Z Z 1 λ(a) ≥ λprinc (ˆ a) ≥ λ(¯ a) = c0 (ω, x) dx dP(ω). Ω |D| D λ(¯ a) = lim

186

5.4

Spectral Theory for Parabolic Equations

Faber–Krahn Inequalities

In this section we consider the influence of the shape of domain on the principal spectrum and principal Lyapunov exponent for the following nonautonomous equation with Dirichlet boundary condition  N N  X  ∂ X ∂u  ∂u = , x ∈ D, aij (t, x) ∂t ∂xi j=1 ∂xj (5.4.1) i=1   u = 0, x ∈ ∂D, and for the following random equation with Dirichlet boundary condition  N N   ∂u X ∂ X ∂u  = aij (θt ω, x) , x ∈ D, ∂t ∂xi j=1 ∂xj (5.4.2) i=1   u = 0, x ∈ ∂D. The standing assumption in the present section is that (A5-N1), (A5-N2) hold when we consider (5.4.1), and that (A5-R1), (A5-R2) hold when we consider (5.4.2). Denote by Σ(a) = [λmin (a), λmax (a)] the principal spectrum interval of (5.4.1), and by λ(a) the principal Lyapunov exponent of (5.4.2), respectively. Let λsym be the principal eigenvalue of ( ∆u = λu, x ∈ Dsym (5.4.3) u = 0, x ∈ ∂Dsym , where Dsym is the ball in RN with center 0 which has the same volume as D. We extend the so called Faber–Krahn inequalities for elliptic and periodic parabolic problems to general time dependent and random ones. THEOREM 5.4.1 Consider (5.4.1). Let δ : R → R be a bounded continuous function with δ(t) > 0 for all t ∈ R such that N X i,j=1

aij (t, x)ξi ξj ≥ δ(t)

N X

ξi2

i=1

for a.e. (t, x) ∈ R × D and any (ξ1 , ξ2 , . . . , ξN )> ∈ RN . Then Z t 1 (1) λmax (a) ≤ δ1 λsym , where δ1 := lim supt−s→∞ δ(τ ) dτ . t−s s

5. Influence of Spatial-Temporal Variations and the Shape of Domain

187

(2) Assume moreover that a = (aij ) is uniquely ergodic and recurrent, δ is ¯ for recurrent with at least the same recurrence as a, aij ∈ C α (R × D) some 0 < α < 1, aij (t, x) is analytic in x, and that the boundary ∂D is analytic. Then λmin (a) = λmax (a) and λmax (a) = δ1 λsym if and only if D = Dsym up to translation, wsym (a · t) = w(a · t) = ϕsym up to phase shift and N X ∂w(a · t) ∂ aij (t, ·) = δ(t) ∆w(a · t) ∂xi ∂xj i,j=1 for t ∈ R, where ϕsym is the positive principal eigenfunction of (5.4.3) with kϕsym k = 1 and wsym is the Schwarz symmetrization of w. THEOREM 5.4.2 Consider (5.4.2). Let δ : Ω → R be a P-integrable function with δ(ω) > 0 for P-a.e. ω ∈ Ω such that N X

aij (ω, x)ξi ξj ≥ δ(ω)

N X

ξi2

i=1

i,j=1

for P-a.e. ω ∈ Ω, a.e. x ∈ D and any (ξ1 , ξ2 , . . . , ξN )> ∈ RN . Then R (1) λ ≤ δ2 λsym , where δ2 := Ω δ(ω) dP(ω). α ¯ (2) Assume moreover that aω ij (·, ·) ∈ C (R× D) for some 0 < α < 1, with the α ¯ C (R × D)-norms bounded uniformly in ω ∈ Ω (aω ij (t, x) = aij (θt ω, x)), aij (ω, x) is analytic in x for any ω ∈ Ω, and the boundary ∂D is analytic. Then λ = δ2 λsym if and only if D = Dsym up to translation and there is Ω0 ⊂ Ω with P(Ω0 ) = 1 such that wsym (ω) = w(ω) = ϕsym up to phase shift and N X ∂ ∂w(ω) aij (ω, ·) = δ(ω) ∆w(ω) ∂xi ∂xj i,j=1

for all ω ∈ Ω0 , where ϕsym is the positive principal eigenfunction of (5.4.3) with kϕsym k = 1 and wsym is the Schwarz symmetrization of w. The above theorems have been proved in [82] for the smooth case. We shall then only prove Theorem 5.4.1(1) and Theorem 5.4.2(1) for the general case. PROOF (Proof of Theorem 5.4.1) with Tn − Sn → ∞ such that λmax (a) = lim

n→∞

(1) First of all, there are Sn < Tn

1 ln kUa (Tn , Sn )w(a · Sn )k. Tn − Sn

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Spectral Theory for Parabolic Equations

Let ηn (t; a) := kUa (t, Sn )w(a · Sn )k and v(t, x; a) := w(a · t)(x). Then Ua (t, Sn )w(a · Sn ) = ηn (t; a)v(t, ·; a). By Lemma 3.2.7, we have Z t ηn (t; a) = exp − Ba (τ, w(a · τ ), w(a · τ )) dτ for any Sn < t. Sn

For each > 0 let B (·) : R → R be a C ∞ function such that B (t) → Ba (t, w(a · t), w(a · t))

for a.e.

t∈R

as → 0

and B (t) → B(t, w(a · t), w(a · t))

in L1,loc (R)

Put Z ηn, (t) := exp −

t

as

→ 0.

B (τ ) dτ .

Sn

Then it is not difficult to prove that ηn, (t) → ηn (t; a)

uniformly on compact subsets of

R

as

→0

and 0 ηn, (t) → ηn0 (t; a)

for a.e.

t∈R

as

→ 0.

Consequently, Z Tn D Z Tn D v E v E ∂ ∂ dt → dt as vηn (t; a), vηn (t; a), ∂t ηn, (t) ∂t ηn (t; a) Sn Sn

→ 0.

Note that for any fixed > 0, v/ηn, ∈ W (Sn , Tn ; V, V ∗ ). By Proposition 2.1.3, we have Z Tn D Z Tn ∂ v E − vηn (t; a), dt = − Ba (t, vηn , v/ηn, ) dt ∂t ηn, (t) Sn Sn + ηn (Sn )/ηn, (Sn ) − ηn (Tn )/ηn, (Tn ). Letting → 0, we obtain Z Tn D Z Tn v E ∂ dt = − Ba (t, vηn , v/ηn ) dt. − vηn (t; a), ∂t ηn (t; a) Sn Sn Therefore Z Tn D Z Tn D Z Tn v E ∂ηn 1 ∂ ∂v E − v, dt = − vηn (t; a), dt − dt ∂t ∂t ηn (t; a) ∂t ηn Sn Sn Sn Z Tn =− Ba (t, vηn , v/ηn ) dt − ln kUa (Tn , Sn )w(a · Sn )k Sn Tn

Z =−

Sn Z Tn

Z X

aij ∂xj v ∂xi v dx dt − ln kUa (Tn , Sn )w(a · Sn )k

D

Z

≤−

δ(t) Sn

D

X

∂xi v ∂xi v dx dt − ln kUa (Tn , Sn )w(a · Sn )k.

5. Influence of Spatial-Temporal Variations and the Shape of Domain

189

By Lemma 5.1.3, k∇vsym (t, x; a)k2 ≤ k∇v(t, x; a)k2 for a.e. t ∈ R, where vsym (t, x; a) is the Schwarz symmetrization of v(t, x; a). Then by the variational characterization of the principal eigenvalue of the Laplace operator with Dirichlet boundary conditions we have −k∇vsym (t, ·; a)k2 ≤ λsym kvsym (t, ·; a)k2 = λsym for a.e. t ∈ R. It then follows that Z Tn D ∂v E 1 v, dt − Tn − Sn Sn ∂t Z Tn λsym ln kUa (Tn , Sn )w(a · Sn )k ≤ δ(t) dt − . Tn − Sn Sn Tn − Sn This implies that 1 T −S→∞ T − S

Z

T

λmax (a) ≤ λsym lim sup

δ(t) dt = δ1 λsym . S

(2) See Theorem B in [82]. PROOF (Proof of Theorem 5.4.2) Lemma 1.2.6, there is ω0 ∈ Ω such that λ(a) = lim

T →∞

(1) By Proposition 4.1.1 and

1 ln kUEa (ω0 ) (T, 0)w(Ea (ω0 ))k T

and Z δ2 =

1 T →∞ T

Z

Ω

T

δ(θt ω0 ) dt.

δ(ω) dP(ω) = lim

0

It then follows from the arguments in Theorem 5.4.1(1) with a = aω that λ ≤ δ2 λsym . (2) See Theorem A in [82].

5.5

Historical Remarks

Spectral theory for linear parabolic equations is a basic tool for the study of nonlinear parabolic equations. The influence of spatial and temporal variations and the shape of the domain of linear parabolic equations on their

190

Spectral Theory for Parabolic Equations

principal spectrum and principal Lyapunov exponents is of great interest for a lot of applied problems and has been investigated in many papers for the smooth case (both the domain and the coefficients are sufficiently smooth). For example, in [62], the authors studied the influence of temporal variations of periodic and almost periodic smooth parabolic equations on the principal eigenvalue and principal spectrum point and proved some results similar to those in Theorem 5.2.2. The results of [62] for periodic and almost periodic parabolic equations were extended to general nonautonomous and random smooth parabolic equations in [81] and [84]. In [13], the authors studied the influence of spatial variations of some time independent smooth parabolic equations on their principal eigenvalues. In [49], the Faber–Krahn inequality for elliptic equations was extended to periodic parabolic equations. The Faber–Krahn inequality for elliptic and periodic parabolic equations was further extended to general time dependent and random parabolic equations in [82]. The results in this chapter extend all the above mentioned works to general time dependent and random (nonsmooth) parabolic equations.

Chapter 6 Cooperative Systems of Parabolic Equations

The purpose of this chapter is to extend the theories developed in the previous chapters for scalar parabolic equations to cooperative systems of nonautonomous and random parabolic equations. To do so, we first consider cooperative systems of parabolic equations in the general setting. We then study the principal spectrum and principal Lyapunov exponent for cooperative systems of nonautonomous and random parabolic equations. To be more precise, let D ⊂ RN be a bounded domain and Y be a bounded 2 subset of L∞ (R × D, RK(N +2N +K) ) × L∞ (R × ∂D, RK ) satisfying (A1-4) and (A1-5). Recall that for a ∈ Y, we write it as a = (akij , aki , bki , ckl , dk0 ), where i, j = 1, 2, . . . , N and k, l = 1, 2, . . . , K. We first consider the following cooperative systems of parabolic equations on D, N

N

∂uk X ∂ X k ∂uk = aij (t, x) + aki (t, x)uk ∂t ∂xi j=1 ∂xj i=1 +

N X i=1

K

bki (t, x)

∂uk X k + cl (t, x)ul , ∂xi

t > s, x ∈ D,

(6.0.1)

l=1

complemented with the boundary conditions Bak (t)uk = 0

t > s, x ∈ ∂D

(6.0.2)

for all a = (akij , aki , bki , ckl , dk0 ) ∈ Y, where Bak is as in (2.0.3) with a being k N k N k replaced by ak = ((akij )N i,j=1 , (ai )i=1 , (bi )i=1 , 0, d0 ), k = 1, 2, . . . , K. We also k assume that Y satisfies (A1-6), i.e., d0 = 0 for all a = (akij , aki , bki , ckl , dk0 ) ∈ Y in the Dirichlet or Neumann cases and dk0 ≥ 0 for all a = (akij , aki , bki , ckl , dk0 ) ∈ Y in the Robin case. For convenience, we use the notion of mild solutions of (6.0.1)+(6.0.2) in this chapter. We introduce the concept of a mild solution of (6.0.1)+(6.0.2) and investigate the basic properties of the mild solutions in Section 6.1, which extends the theories developed in Chapter 2 for weak solutions of scalar parabolic equations in the general setting to mild solutions of cooperative systems of parabolic equations in the general setting. We introduce the concept of principal spectrum and principal Lyapunov exponent and exponential

191

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Spectral Theory for Parabolic Equations

separation of (6.0.1)+(6.0.2), investigate their basic properties and show the existence of exponential separation and existence and uniqueness of entire positive solutions in Section 6.2, which extends the theories established in Chapter 3 for scalar parabolic equation in general setting to cooperative systems of parabolic equations in general setting. We then consider the following cooperative systems of nonautonomous parabolic equations  N N X  ∂ X k ∂uk k  ∂uk =  a (t, x) + a (t, x)u k  ij i  ∂t ∂xi j=1 ∂xj   i=1  K N X ∂uk X k  cl (t, x)ul , t > 0, x ∈ D bki (t, x) + +   ∂xi   i=1 l=1    t > 0, x ∈ ∂D, Bak (t)uk = 0,

(6.0.3)

where Bak is of the same form as in (6.0.2), k = 1, 2, . . . , K and a = (akij , aki , bki , ckl , dk0 ) is a given element of Y, and consider the following cooperative systems of random parabolic equations  N N X  ∂uk ∂ X k ∂uk k   = a (θ ω, x) + a (θ ω, x)u t t k  i  ∂t ∂xi j=1 ij ∂xj   i=1  N K X ∂uk X k k  + b (θ ω, x) + cl (θt ω, x)ul , t  i  ∂xi   i=1 l=1    Bak,ω (t)uk = 0,

t > 0, x ∈ D

t > 0, x ∈ ∂D, (6.0.4) where k = 1, 2, . . . , K, ω ∈ Ω, ((Ω, F, P), {θt }t∈R ) is an ergodic metric dynamical system, and for each ω ∈ Ω, aω (t, x) = (akij (θt ω, x), aki (θt ω, x), bki (θt ω), ckl (θt ω, x), dk0 (θt ω, x)) ∈ Y and Bak,ω (t) is of the same form as in (6.0.2) with ak being replaced by ak,ω (t, x) = (akij (θt ω, x), aki (θt ω, x), bki (θt ω, x), 0, dk0 (θt ω, x)). We extend the theories developed in Chapters 4 and 5 for nonautonomous and random parabolic equations to cooperative systems of nonautonomous and random parabolic equations in Section 6.3. This chapter is ended with some remarks in Section 6.4.

6.1

Existence and Basic Properties of Mild Solutions in the General Setting

In this section, we extend the theories developed in Chapter 2 for scalar parabolic equations in the general setting to cooperative systems of parabolic

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193

equations in the general setting. We first consider the nonsmooth case (both the coefficients and the domain are not smooth) and then consider the smooth case (both the coefficients and the domain are sufficiently smooth).

6.1.1

The Nonsmooth Case

Consider (6.0.1)+(6.0.2). Recall that for a given a = (akij , aki , bki , ckl , dk0 ) ∈ Y, ak = (akij , aki , bki , 0, dk0 ) denotes an element in Y , which is a subset of 2 L∞ (R × D, RN +2N +1 ) × L∞ (R × ∂D, R) satisfying (A1-1)–(A1-3) (see Section 1.3) (hence for ak , 1 ≤ k ≤ K is fixed and i, j = 1, 2, . . . , N ), and Ca = (ckl )l,k=1,2,...,K . For a fixed 1 ≤ k ≤ K, P k : Y → Y is defined by P k (a) := ak and Y k := { P k (a) : a ∈ Y }. Then for each k (1 ≤ k ≤ K), there corresponds the following family of scalar parabolic equations  N N X  ∂uk ∂ X k ∂uk k   = a (t, x) + a (t, x)u k  i  ∂t ∂xi j=1 ij ∂xj   i=1  N X (6.1.1) ∂uk  + bki (t, x) , t > 0, x ∈ D,   ∂xi   i=1    Bak (t)uk = 0, t > 0, x ∈ ∂D, where ak = (akij , aki , bki , 0, dk0 ) ∈ Y k . We start by introducing the following standing assumptions. (A6-1) (akij ) satisfies (A2-1) and D satisfies (A2-2). (n) (A6-2) For any sequence (a(n) )∞ = n=1 ⊂ Y convergent to a ∈ Y, where a k,(n) k,(n) k,(n) k,(n) k,(n) k,(n) k k k k k (aij , ai , bi , cl , d0 ) and a = (aij , ai , bi , cl , d0 ), one has aij (t, k,(n)

x), ai

k,(n)

(t, x), bi

k,(n)

(t, x), and cl

(t, x) converge to akij (t, x), aki (t, x), bki (t, k,(n)

x), and ckl (t, x), respectively, for a.e. (t, x) ∈ R × D, and d0 (t, x) converges to dk0 (t, x) for a.e. (t, x) ∈ R × ∂D (i, j = 1, 2, . . . , N , k, l = 1, 2, . . . , K). (A6-3) (Cooperativity) For any a = (akij , aki , bki , ckl , dk0 ) ∈ Y, ckl (t, x) ≥ 0 for (t, x) ∈ R × D and k, l = 1, 2, . . . , K with k 6= l. (A6-4) (Irreducibility) For any a = (akij , aki , bki , ckl , dk0 ) ∈ Y, any t0 ∈ R and any partition I = {i1 , i2 , . . . , ik } and J = {j1 , j2 , . . . , jl } of {1, 2, . . . , K} (i.e., I 6= ∅, J 6= ∅, I ∩ J = ∅, I ∪ J = {1, 2, . . . , K}) there are i ∈ I and j ∈ J such that cji (t, ·) ≥ 0 and cji (t, ·) 6≡ 0 for t in any sufficiently small neighborhood of t0 . Throughout this section, we assume (A6-1). At some places, we will also assume (A6-2) and/or (A6-3) and/or (A6-4), which will be pointed out explicitly. For a given u = (u1 , u2 , . . . , uK ) ∈ RK , u± := (u1± , u2± , . . . , uK± ), where ul+ := ul (ul− := 0) if ul > 0 and ul+ := 0 (ul− := −ul ) if ul ≤ 0, l = 1, 2, . . . , K. Hence u = u+ − u− . We write |u| for (|u1 |, |u2 |, . . . , |uK |).

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For 1 ≤ p < ∞ we write kukp for (|u1 |p + · · · + |uK |p )1/p , and we write kuk∞ for sup { |ul | : 1 ≤ l ≤ K }. For a given u : D → RK , u ∈ Lp (D, RK ) if and only if ul ∈ Lp (D) for all l = 1, 2, . . . , K. Recall that if 1 ≤ p < ∞ we denote the norm in Lp (D) by kukp :=

K Z X k=1

1/p |uk (x)|p dx .

D

We denote the norm in L∞ (D) by kuk∞ := max ess sup { |uk (x)| : x ∈ D }. 1≤k≤K

First of all, by Propositions 2.1.5, 2.2.1, and 2.2.2, we have LEMMA 6.1.1 For any k (1 ≤ k ≤ K), any u0k ∈ L2 (D), and any s ∈ R, (6.1.1) with initial condition uk (s) = u0k has a unique global weak solution [ [s, ∞) 3 t 7→ Uak (t, s)u0k ∈ L2 (D) ]. Moreover, for any s < t the linear operator Uak (t, s) can be extended to an operator in L(Lp (D), Lp (D)) (1 ≤ p ≤ ∞) and there are M > 0 and γ > 0 such that N

1

1

kUak (t, 0)kp,q ≤ M t− 2 ( p − q ) eγt

(6.1.2)

for ak ∈ Y k , 1 ≤ p ≤ q ≤ ∞, t > 0, and k = 1, 2, . . . , K. When 1 < p < ∞, the mapping [ [0, ∞) 3 t 7→ Uak (t, 0) ∈ Ls (Lp (D)) ] is continuous. By Proposition 2.2.4, we have LEMMA 6.1.2 For any t1 < t2 , there exists α ∈ (0, 1) such that for any a ∈ Y, any 1 ≤ k ≤ K, any u0k ∈ Lp (D), and any compact subset D0 ⊂ D the function [ [t1 , t2 ] × D0 3 (t, x) 7→ (Uak (t, 0)u0k )(x) ] belongs to C α/2,α ([t1 , t2 ] × D0 ). Moreover, for fixed t1 , t2 and D0 , the C α/2,α ([t1 , t2 ] × D0 )-norm of the above restriction is bounded above by a constant depending on ku0k kp only. LEMMA 6.1.3 Denote M1 := sup{ kCa kL∞ (R×D,RK 2 ) : a ∈ Y }. (1) For all a ∈ Y and a.e. t ∈ R there hold 2

Ca (t)(·) := (ckl (t, ·))k,l=1,2,...,K ∈ L∞ (D, RK ) and kCa (t)(·)kL∞ (D,RK 2 ) ≤ M1 .

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195

(2) If 1 ≤ p ≤ ∞ then Ca (t, u)(·) := Ca (t)(·)u ∈ Lp (D, RK ) for any a ∈ Y, a.e. t ∈ R, and any u ∈ Lp (D, RK ). Moreover kCa (t, u)(·)kp ≤ KM1 kukp . (3) For any −∞ ≤ t1 < t2 ≤ ∞, any 1 ≤ p < ∞, and any 1 ≤ q ≤ ∞, if u(·) ∈ Lq ((t1 , t2 ), Lp (D, RK )) then [t 7→ Ca (t, u) ] ∈ Lq ((t1 , t2 ), Lp (D, RK )). Further, the linear operator [ u 7→ Ca (·, u)) ] belongs to L(Lq ((t1 , t2 ), Lp (D, RK ))) and has norm ≤ KM1 . Recall that necessary facts on measurability, etc., of functions taking values in a separable Banach space are collected in Chapter 1. PROOF (Proof of Lemma 6.1.3) (1) For a.e. t ∈ R the function 2 Ca (t)(·), as the section of a function in L∞ (R × D, RK ), is Lebesgue measurable. By the fact that ckl (·, ·) ∈ L∞ (R × D), for any > 0 there is a measurable set E ⊂ R × D with |E| = 0 such that |ckl (t, x)| ≤ kckl k∞ + for (t, x) ∈ (R × D) \ E. LetR E(t) := { x ∈ D : (t, x) ∈ E }. Then by [42, Theorem 2.36], |E| = R |E(t)| dt = 0. Therefore |E(t)| = 0 for 2 a.e. t ∈ R. This implies that for a.e. t ∈ R, Ca (t)(·) ∈ L∞ (D, RK ) and kCa (t)kL∞ (D,RK 2 ) ≤ kCa (·, ·)kL∞ (R×D,RK 2 ) . (2) The first statement is a consequence of (1) and the fact that for any v ∈ 2 L∞ (D, RK ) and any u ∈ Lp (D, RK ) their product vu belongs to Lp (D, RK ). To prove the second statement for 1 < p < ∞, let p1 + p1∗ = 1. Then we have K K Z X p p1 X ckl (t, x)ul (x) dx kCa (t, u)(·)kp = k=1

≤

D l=1

K Z X k=1

≤ KM1

K K X X p1 ∗ p |ckl (t, x)|p p∗ · |ul (x)|p dx

D l=1 K Z X l=1

l=1

p1 |ul (x)|p dx

D

= KM1 kukp . The second statement for p = 1 and p = ∞ can be proved in a similar way. (3) Assume that u(·) ∈ Lq ((t1 , t2 ), Lp (D, RK )). There exists a sequence K (un )∞ n=1 of simple functions such that un (t) converge in Lp (D, R ) to u(t) as n → ∞, for a.e. t ∈ (t1 , t2 ). For n = 1, 2, 3, . . . , let un = χE1 u1 + · · · + χEmn umn be the representation of the simple function un . We prove first that the function [ t 7→ Ca (t, un )(·) ∈ Lp (D, RK ) ] is measurable. By the Pettis theorem (see Definition 1.2.1(iii)), it suffices to prove that the function in question is weakly measurable. Further,

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Spectral Theory for Parabolic Equations

it reduces to showing that its restrictions to Ei , 1 ≤ i ≤ mn , are weakly measurable. Let v∗ ∈ Lp∗ (D, RK ), where p1 + p1∗ = 1. The function [ Ei 3 R t 7→ D Ca (t)(x)ui (x)v∗ (x) dx ] is measurable. It is a consequence of Part (2) that for a.e. t ∈ (t1 , t2 ), Ca (t)un (t) converge, as n → ∞, to Ca (t)u(t), in Lp (D, RK ). Therefore the function [ t 7→ Ca (t)u(t) ∈ Lp (D, RK ) ], as the a.e. pointwise limit of a sequence of measurable functions, is measurable. The fact that the mapping [ t 7→ kCa (t)u(t)kLp (D,RK ) ] belongs to Lq ((t1 , t2 ), R) follows again from (2). Let U0a (t, s) be defined by U0a (t, s) := (Ua1 (t, s), Ua2 (t, s), . . . , UaK (t, s)).

(6.1.3)

Observe that U0a (t, s) = U0a·s (t − s, 0)

for any t ≥ s

(6.1.4)

and U0a (t + s, s) ◦ U0a (s, 0) = U0a (t + s, 0)

for all s, t ≥ 0

(6.1.5)

(see Propositions 2.1.6 and 2.1.7). Recall that Lp (D) = (Lp (D))K = Lp (D, RK ). We denote the norm in L2 (D) by k·k, and the norm in Lp (D) by k·kp . The symbol k·kp,q stands for the norm in L(Lp (D), Lq (D)). LEMMA 6.1.4 For any 1 ≤ p ≤ q ≤ ∞, a ∈ Y and s < t there holds N

1

1

kU0a (t, s)kp,q ≤ M (t − s)− 2 ( p − q ) eγ(t−s)

(6.1.6)

where M and γ are as in Lemma 6.1.1. PROOF First, let 1 ≤ p ≤ q < ∞. Then for any a ∈ Y, s < t, and u0 ∈ Lp (D), by Lemma 6.1.1, U0a (t, s)u0 ∈ Lq (D) and kU0a (t, s)u0 kqq

=

K Z X k=1

≤

K X

|(Uak (t, s)u0k )(x)|q dx

D N

1

1

M (t − s)− 2 ( p − q ) eγ(t−s)

q Z D

k=1

q/p |u0k (x)|p dx

N

1

1

= M (t − s)− 2 ( p − q ) eγ(t−s)

K Z q X k=1

|u0k (x)|p dx

q/p

D

K Z q X q/p N 1 1 ≤ M (t − s)− 2 ( p − q ) eγ(t−s) |u0k (x)|p dx , k=1

D

6. Cooperative Systems of Parabolic Equations

197

which implies N

1

1

kU0a (t, s)u0 kq ≤ M (t − s)− 2 ( p − q ) eγ(t−s) ku0 kp , hence (6.1.6) holds. Now, let 1 ≤ p ≤ q = ∞. For any a ∈ Y, s < t, and u0 ∈ Lp (D), by Lemma 6.1.1 again, we have kU0a (t, s)u0 k∞ = max kUak (t, s)u0k k∞ 1≤k≤K

N

1

1

≤ max M (t − s)− 2 ( p − q ) eγ(t−s) ku0k kp 1≤k≤K

N

1

1

≤ M (t − s)− 2 ( p − q ) eγ(t−s) ku0 kp . Hence (6.1.6) also holds. DEFINITION 6.1.1 (Mild solution) For given 1 < p < ∞ and u0 ∈ Lp (D), a continuous function u : [s, T ) → Lp (D) is called a mild (Lp (D)-) solution of (6.0.1)+(6.0.2) on [s, T ) (s < T ≤ ∞) with u(s) = u0 if Z t 0 0 u(t) = Ua (t, s)u + U0a (t, τ )(Ca (τ )u(τ )) dτ for any s < t < T. (6.1.7) s

THEOREM 6.1.1 (Existence of mild solution) For any given 1 < p < ∞, a ∈ Y, u0 ∈ Lp (D) and s ∈ R, there is a unique mild Lp (D)-solution u(·) of (6.0.1)+(6.0.2) on [s, ∞) with u(s) = u0 . PROOF As in Lemma 6.1.3, let M1 = sup{ kCa kL∞ (R×D,RK 2 ) : a ∈ Y }. From now until the end of the proof, we consider 1 0 and r > 0, let Mr (h, s) := { u ∈ C([s, s + h], Lp (D)) : ku(τ )kp ≤ r for all τ ∈ [s, s + h] }

(6.1.8)

equipped with the topology given by the supremum norm. Mr (h, s) is a complete metric space. For a given u0 ∈ Lp (D) we define an operator Gu0 , acting from Mr (h, s) into the set of functions from [s, s + h] into Lp (D), by the formula Z t Gu0 (u)(t) := U0 (t, s)u0 + U0 (t, τ )(C(τ )u(τ )) dτ, t ∈ [s, s + h]. (6.1.9) s

The integral on the right-hand side of (6.1.9) is well defined. Indeed, for any u ∈ Mr (h, s) and any s < t < s + h there holds u|(s,t) ∈ L∞ ((s, t), Lp (D)).

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By Lemma 6.1.3, the function [ (s, t) 3 τ 7→ C(τ )u(τ ) ] belongs to L∞ ((s, t), Lp (D)). Proposition 2.2.6 states that the mapping [ (τ, v) 3 (s, t) × Lp (D) 7→ U0 (t, τ )v ∈ Lp (D) ] is continuous, which implies, via Lemma 1.2.2, that the mapping [ (s, t) 3 τ 7→ U0 (t, τ )(C(τ )u(τ )) ∈ Lp (D) ] is measurable. Proposition 2.2.2 (Lp –Lp estimates) yields that this function is in L∞ ((s, t), Lp (D)). We prove now that for any u ∈ Mr (h, s) the function [ [s, s + h] 3 t 7→ Gu0 (u)(t) ∈ Lp (D) ] is continuous. Indeed, the mapping [ [s, s + h] 3 t 7→ U0 (t, s)u0 ∈ Lp (D) ] is continuous by Lemma 6.1.1. Denote v(τ ) := C(τ )u(τ ), e 0 (t, τ ) ∈ τ ∈ [s, s + h]. For t ∈ [s, s + h] we define a function [ [s, s + h] 3 τ 7→ U L(Lp (D)) ] by the formula ( e 0 (t, τ ) := U

U0 (t, τ ) 0

for τ ∈ [s, t] for τ ∈ (t, s + h].

Let s ≤ t1 < t2 ≤ s + h. There holds Z

t2

U0 (t2 , τ )v(τ ) dτ −

s

t1

Z

U0 (t1 , τ )v(τ ) dτ

s

Z

s+h

=

e 0 (t2 , τ ) − U e 0 (t1 , τ ) v(τ ) dτ. U

s

We consider separately two cases: first, t1 is fixed and t2 is let to approach t1 from the right, second, t2 is fixed and t1 is let to approach t2 from the left. In both cases, we deduce from Lemma 6.1.1 that the integrand converges for a.e. t ∈ [s, s + h] to zero in the Lp (D)-norm, which gives with the help of the Dominated Convergence Theorem Rthat the integral converges to zero. t Further, it follows from Lemma 6.1.1 that s U0 (t, τ )v(τ ) dτ converges to zero in Lp (D), as t & s. We claim that for any % > 0, there are h > 0 and r > 0 such that for any u0 ∈ Lp (D) with ku0 kp < % there holds Gu0 : Mr (h, s) → Mr (h, s). In fact, by Lemma 6.1.4, kU0 (t, s)kp ≤ M eγ(t−s) for all t ≥ s. An application of Lemma 6.1.3(3) with q = ∞ gives that if u(·) ∈ Mr (h, s) then kC(τ )u(τ )kp ≤ rKM1 for a.e. τ ∈ (s, s + h). Consequently, we have kGu0 (u)(t)kp ≤ %M e

γ(t−s)

Z + rM M1

t

eγ(t−τ ) dτ

s γ(t−s)

= %M eγ(t−s) + rM M1 (e

− 1)/γ

(6.1.10)

for all t ∈ [s, s + h]. Therefore there are h > 0 (h is such that M M1 (eγh − γh 1)/γ < 1/2) and r > 0 (r = M e2 % ) such that kGu0 (u)(t)kp ≤ r

for all t ∈ [s, s + h].

(6.1.11)

We claim that Gu0 : Mr (h, s) → Mr (h, s) is a contraction. In fact, for any

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199

given u, v ∈ Mr (h, s) we have

Z t

U0 (t, τ )(C(τ )(u(τ ) − v(τ ))) dτ kGu0 (u)(t) − Gu0 (v)(t)kp = p s Z t M eγ(t−τ ) KM1 ku(τ ) − v(τ )kp dτ ≤ s Z t eγ(t−τ ) dτ ≤ M KM1 ku − vkC([s,s+h],Lp (D)) s

M KM1 (eγ(t−s) − 1) ≤ ku − vkC([s,s+h],Lp (D)) γ (6.1.12) for all t ∈ [s, s + h]. By reducing h, if necessary, we can obtain that Gu0 is a contraction. Let u ∈ Mr (h, s) be the unique fixed point of Gu0 , that is, Z t 0 0 u(t) = U (t, s)u + U0 (t, τ )(C(τ )u(τ )) dτ, t ∈ [s, s + h]. s

It follows from Lemma 6.1.1 that the U0 (t, s)u0 converges to u0 in Lp (D), as t & s. Again, by the estimate (6.1.2) and Lemma 6.1.3(2), the second term on the right-hand side converges to zero in Lp (D), as t & s. Therefore, u(·) is a (necessarily unique) mild Lp (D)-solution (6.0.1)+(6.0.2) on [s, s + h]. Moreover, by the arguments of [34, Theorem 3.8], this solution can be extended to [s, ∞). The theorem is thus proved. Theorem 6.1.1 allows us, for any a ∈ Y, any 1 < p < ∞, any s ∈ R and any u0 ∈ Lp (D), to denote by [ [s, ∞) 3 t 7→ Ua,p (t, s)u0 ∈ Lp (D) ] the unique mild Lp (D)-solution of (6.0.1)+(6.0.2) with u(s) = u0 . LEMMA 6.1.5 For any a ∈ Y and any 1 < p < ∞ we have (1) Ua,p (t, s) = Ua·s,p (t − s, 0) for all t ≥ s. (2) Ua,p (t + s, s) ◦ Ua,p (s, 0) = Ua,p (t + s, 0) for all s, t ≥ 0. PROOF Fix 1 < p < ∞ and a ∈ Y. We will write Ua instead of Ua,p . (1) For s ∈ R and u0 ∈ Lp (D) we denote u1 (t) := Ua·s (t − s, 0)u0 , t ≥ s. By (6.1.7) the function u1 (·) satisfies Z t−s 0 0 u1 (t) = Ua·s (t − s, 0)u + U0a·s (t − s, τ )(Ca·s (τ )u1 (τ + s)) dτ 0

for all t ≥ s. It follows from (6.1.4) and (6.1.5) that U0a·s (t−s, τ ) = U0a (t, τ +s) for any τ ∈ [0, t − s]. Further, Ca·s (τ )u1 (τ + s) = Ca (τ + s)u1 (τ + s) for a.e.

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Spectral Theory for Parabolic Equations

τ ∈ [0, t − s]. We have thus u1 (t) =

U0a (t, s)u0

t

Z

U0a (t, r)(Ca (r)u1 (r)) dr

+

for all t > s

s

(r = τ + s). By uniqueness, u1 (t) = Ua (t, s)u0 for all t ≥ s. (2) Fix u0 ∈ Lp (D), and put u(t) := Ua (t, 0)u0 , t ≥ 0. By (6.1.4) and (6.1.5), we have, for s ≥ 0, t ≥ 0, Z t+s Ua (t + s, 0)u0 = U0a (t + s, 0)u0 + U0a (t + s, τ )(Ca (τ )u(τ )) dτ 0 Z s 0 0 = Ua (t + s, s) Ua (s, 0)u0 + U0a (s, τ )(Ca (τ )u(τ )) dτ 0 Z t+s 0 + Ua (t + s, τ )(Ca (τ )u(τ )) dτ s Z t+s 0 = Ua (t + s, s)u(s) + U0a (t + s, τ )(Ca (τ )u(τ )) dτ s

= Ua (t + s, s)(Ua (s, 0)u0 ).

Since the set Y is translation invariant, in view of Lemma 6.1.5(1) we will formulate (when possible) results concerning properties of the solution operator for s = 0 only. THEOREM 6.1.2 (Lp –Lq estimates) (1) There are M > 0 and γ¯ such that kUa,p (t, 0)kp,p ≤ M eγ¯ t

(6.1.13)

for all 1 0. (2) For any 1 0. Moreover, for any 1 0 such that M f kUa,p (t, 0)kp,q ≤ M for all q ∈ (p, ∞], all a ∈ Y, and all t ∈ [T0 , T ]. PROOF

(1) We write Ua (t, 0) for Ua,p (t, 0). Also, we denote M1 := sup { kCa kL∞ (R×D,RK 2 ) : a ∈ Y }.

(6.1.14)

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201

Applying Lemma 6.1.4 with q = p and Lemma 6.1.3 to (6.1.7) we have that Z t eγ(t−τ ) kUa (τ, 0)u0 kp dτ kUa (t, 0)u0 kp ≤ M eγt ku0 kp + KM1 M 0

for all a ∈ Y and all t > 0. It then follows from the regular Gronwall inequality that there are M > 0 and γ¯ such that kUa (t, 0)u0 kp ≤ M eγ¯ t ku0 kp

for t > 0 and a ∈ Y.

Hence (6.1.13) holds. (2) We first assume that q < ∞. By Lemmas 6.1.4, 6.1.3(2) and Part (1), we have that N

1

1

kU0a (t, 0)u0 kq ≤ M t− 2 ( p − q ) eγt ku0 kp

for t > 0

(6.1.15)

and kU0a (t, τ )(Ca (τ )Ua (τ, 0)u0 )kq N

1

1

≤ M KM1 M (t − τ )− 2 ( p − q ) eγ(t−τ ) eγ¯ τ ku0 kp

(6.1.16) for 0 < τ < t.

Fix for the moment t > 0. By Lemma 6.1.3(3), the function [ (0, t) 3 τ 7→ Ca (τ )Ua (τ, 0)u0 ∈ Lp (D) ] is measurable. Proposition 2.2.6 states that the mapping [ (τ, v) 3 (0, t) × Lp (D) 7→ U0 (t, τ )v ∈ Lq (D) ] is continuous, consequently, by Lemma 1.2.2, the mapping [ (0, t) 3 τ 7→ U0a (t, τ )(Ca (τ )Ua (τ, 0)u0 ) ∈ Lq (D) ] is measurable. (6.1.16) implies that the above mapping is in L1 ((0, t), Lq (D)), provided that 1 0 and for such p, q. Fix T > 0. An application of (6.1.15) and (6.1.16) to (6.1.7) gives N

1

1

kUa (t, 0)u0 kq ≤ M t− 2 ( p − q ) eγt ku0 kp Z t N 1 1 + M KM1 M (t − τ )− 2 ( p − q ) eγ(t−τ ) eγ¯ τ ku0 kp dτ 0 N

1

1

f0 t− 2 ( p − q ) ku0 kp ≤M for all 0 < t ≤ T , provided that 1 1 there is J ∈ N (J is independent of q ∈ (p, ∞)) such that for any q ∈ (p, ∞) there are q0 = p < q1 < q2 < qJ−1 < qJ = q with the property that N 1 1 1 − < for j = 1, 2, . . . , J. 2 qj−1 qj 2

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Spectral Theory for Parabolic Equations

By (6.1.17), there holds N

1

1

f0 t− 2 ( qj−1 − qj ) , kUa (t, 0)kqj−1 ,qj ≤ M

0 < t ≤ T,

(6.1.18)

where j = 1, 2, . . . , J. Fix T0 ∈ (0, T ]. Let 0 < δ1 < δ2 < · · · < δJ < T0 and u0 ∈ Lp (D). Then kUa (t, 0)u0 kq = kUa (t, δJ )Ua (δJ , 0)u0 kq 1

N

1

f0 (t − δJ )− 2 ( qJ−1 − q ) kUa (δJ , 0)u0 kq ≤M J−1 N

1

1

f0 )2 (t − δJ )− 2 ( qJ−1 − q ) ≤ (M −N 2 (q

· (δJ − δJ−1 )

1 J−2

−q

1 J−1

)

kUa (δJ−1 , 0)u0 kqJ−2

≤ ... fku0 kp ≤M for T0 < t ≤ T , where N

1

1

f := (M f0 )J (T0 − δJ )− 2 ( qJ−1 − q ) M · (δJ − δJ−1 )

1 J−2

−N 2 (q

−q

1 J−1

)

N

1

1

1 1 −N 2 (p−q )

· · · (δ2 − δ1 )− 2 ( q1 − q2 ) δ1

1

.

f > 0 depending on p, T0 , and T only, but (6.1.14) then follows, with M independent of q ∈ (p, ∞). It remains now to prove (2) for q = ∞. Suppose to the contrary that there are a ∈ Y, u0 ∈ Lp (D) (1 0 such that Ua (t, 0)u0 is not in L∞ (D). This means that for any n ∈ N there is En ⊂ D with |En | > 0 and kn (1 ≤ kn ≤ K) such that |(Ua (t, 0)u0 )kn (x)| ≥ n for all x ∈ En . Consequently, lim inf kUa (t, 0)u0 kq ≥ n lim |En |1/q = n q→∞

q→∞

for all n ∈ N, which contradicts the already proven fact that { kUa (t, 0)u0 kq : q ∈ (p, ∞) } is bounded. The satisfaction of (6.1.14) for q = ∞ follows from the observation that kUa (t, 0)u0 k∞ = limq→∞ kUa (t, 0)u0 kq and that (6.1.14) f independent of q. holds for all q ∈ (p, ∞) with M In view of the above theorem we can (and will) legitimately speak of a mild solution of (6.0.1)+(6.0.2). Accordingly, we write simply Ua (t, 0) for the solution operator of (6.0.1)+(6.0.2). THEOREM 6.1.3 (Compactness) For any given 0 < t1 ≤ t2 , if E is a bounded subset of Lp (D) (1 < p < ∞) then { Ua (τ, 0)u0 : a ∈ Y, τ ∈ [t1 , t2 ], u0 ∈ E } is relatively compact in Lp (D).

6. Cooperative Systems of Parabolic Equations

203

(n) ∞ Let (τn )∞ )n=1 ⊂ Y, and (un )∞ n=1 ⊂ [t1 , t2 ], (a n=1 ⊂ E. Note

PROOF that

Ua(n) (τn , 0)un = U0a(n) (τn , 0)un +

Z

τn

U0a(n) (τn , s)(Ca(n) (s)Ua(n) (s, 0)un ) ds.

0

Since { U0a(n) (τn , 0)un : n ∈ N } is relatively compact in Lp (D), we only o nR τn 0 n U (τ , s)(C (s)U (s, 0)u ) ds : n ∈ N is need to prove that (n) (n) n (n) a a 0 a relatively compact in Lp (D). Put vn (t) := Ca(n) (t)Ua(n) (t, 0)un . We claim that for any fixed m ∈ N the set o n Z τn − m1 e U0a(n) (τn , s)vn (s) ds : n ∈ N Em := =

nZ

0 1 τn − m

0

U0a(n) ·s (τn − s, 0)vn (s) ds : n ∈ N

o

is relatively compact in Lp (D). By Theorem 6.1.2 and Lemma 6.1.3(3), there is M0 > 0 such that the L∞ ((0, t2 ), Lp (D))-norms of vn are ≤ M0 for any b m the closure in Lp (D) of the set n ∈ N. Denote by E 1 ˜ ∈ Y, s ∈ [ m u:a { U0a˜ (s, 0)˜ , t2 ], k˜ ukp ≤ M0 }.

b m is compact. It is straightforward that for any simple function The set E Rt − 1 w whose L∞ ((0, t2 ), Lp (D))-norm is ≤ M0 , 0 n m U0a(n) (τn , s)w(s) ds ∈ t2 · b m for any n ∈ N, where co stands for the closed (in Lp (D)) convex hull co E of a set. As, by Lemma 1.2.3, any vn can be approximated by such simple e m ⊂ t2 · co E b m. functions, we see that E b m is compact, for any m ∈ N. Consequently, by a diagonal Note that t2 ·co E process we can assume without loss of generality that, for each m = 1, 2, . . . , Rτ − 1 the integrals 0 n m Ua(n) (τn , s)vn (s) ds converge, as n → ∞, in Lp (D). For any > 0 there is m0 ∈ N such that

Z τn

U0a(n) (τn , s)vn (s) ds <

p

τn − m1

0

for all n ∈ N. Take n0 ∈ N so large that

Z

τn1 − m1

0

0

U0a(n1 ) (τn1 , s)vn1 (s) ds

0

− 0

for any n1 , n2 ≥ n0 . Therefore Z

Z τn1

U0a(n1 ) (τn1 , s)vn1 (s) ds −

0

τn2 − m1

Z

0

τn 2

U0a(n2 ) (τn2 , s)vn2 (s) ds < p

U0a(n2 ) (τn2 , s)vn2 (s) ds < 3 p

204

Spectral Theory for Parabolic Equations

Rτ for any n1 , n2 ≥ n0 . Hence ( 0 n U0a(n) (τn , s)(Ca(n) (s)Ua(n) (s, 0)un ) ds)∞ n=1 is a Cauchy sequence in Lp (D). The theorem then follows. THEOREM 6.1.4 (Joint continuity) Assume (A6-1) and (A6-2). For any sequence {a(n) }∞ n=1 ⊂ Y, any sequence n ∞ (n) (tn )∞ → a, n=1 , and any sequence (u )n=1 ⊂ Lp (D) (2 ≤ p < ∞), if a n 0 tn → t, where t > 0, and u → u in Lp (D) as n → ∞, then Ua(n) (tn , 0)un converges in Lp (D) to Ua (t, 0)u0 . PROOF Take T > 0 such that tn , t ≤ T (n = 1, 2, . . . ). For a given u(·) ∈ C((0, T ], Lp (D)) define Z t 0 n Gn (u)(t) := Ua(n) (t, 0)u + U0a(n) (t, τ )(Ca(n) (τ )u(τ )) dτ, t ∈ (0, T ], 0

and G(u)(t) :=

U0a (t, 0)u

Z +

t

U0a (t, τ )(Ca (τ )u(τ )) dτ,

t ∈ (0, T ].

0

By [34, Theorem 4.4], if sup { kGn (u)(t) − G(u)(t)kp : t ∈ [τ, T ] } → 0 as n → ∞ for any 0 < τ ≤ T and any u(·) ∈ C((0, T ], Lp (D)), then Ua(n) (t, 0)un → Ua (t, 0)u0 as n → ∞, uniformly for t in compact intervals of (0, T ]. It therefore suffices to verify that Gn (u)(t) → G(u)(t) in Lp (D), uniformly for t in compact subsets of (0, T ]. By Proposition 2.2.13, we have that U0a(n) (t, 0)un → U0a (t, 0)u0 as n → ∞ in Lp (D), uniformly for t in compact subsets of (0, T ]. Note that for any u ∈ C((0, T ], Lp (D)), by (A6-2) we have ((Ca(n) (τ ) − Ca (τ ))u(τ ))(x) → 0 Hence Z

for a.e. (τ, x) ∈ (0, T ) × D.

T

k(Ca(n) (τ ) − Ca (τ ))u(τ )kp dτ → 0 0

and then (by the Lp –Lp estimates in Lemma 6.1.4)

Z t

(U0a(n) (t, τ )((Ca(n) (τ ) − Ca (τ ))u(τ )) dτ → 0,

(6.1.19)

p

0

uniformly for t in compact subsets of (0, T ]. For any 0 < τ < t we have k(U0a(n) (t, τ ) − U0a (t, τ ))(Ca (τ )u(τ ))kp → 0 as n → ∞. This, together Lp –Lp estimates (Lemma 6.1.4), implies that Z t k(U0a(n) (t, τ ) − U0a (t, τ ))(Ca (τ )u(τ ))kp dτ → 0 (6.1.20) 0

6. Cooperative Systems of Parabolic Equations

205

Z t

(U0a(n) (t, τ ) − U0a (t, τ ))(Ca (τ )u(τ )) dτ → 0

(6.1.21)

and then

p

0

as n → ∞, uniformly for t in compact subsets of (0, T ]. By (6.1.19) and (6.1.21), Z t Z t 0 Ua(n) (t, τ )(Ca(n) (τ )u(τ )) dτ − U0a (t, τ )(Ca (τ )u(τ )) dτ 0 0 Z t = U0a(n) (t, τ )((Ca(n) (τ ) − Ca (τ ))u(τ )) dτ 0 Z t + (U0a(n) (t, τ ) − U0a (t, τ ))(Ca (τ )u(τ )) dτ 0

→0 in Lp (D) as n → ∞, uniformly for t in compact subsets of (0, T ]. It then follows from [34, Theorem 4.4] that kUa(n) (tn , 0)un − Ua (t, 0)u0 kp ≤ kUa(n) (tn , 0)un − Ua (tn , 0)u0 kp + kUa (tn , 0)u0 − Ua (t, 0)u0 kp →0 as n → ∞. For given a = (akij , aki , bki , ckl , dk0 ) ∈ Y and u0 ∈ Lp (D) (1 < p < ∞), we 0 denote by UD a (t, s)u the mild solution of (6.0.1) with the Dirichlet boundary condition: uk (t, x) = 0 for x ∈ ∂D and 1 ≤ k ≤ K, denote by UR (t, s)u0 the PN a PN k mild solution of (6.0.1) with the Robin boundary condition: i=1 ( j=1 aij (t, x)∂xj uk + aki (t, x)uk )νi + dk0 (t, x)uk = 0 for x ∈ ∂D and 1 ≤ k ≤ K, and 0 denote by UN the mild solution of (6.0.1) with the Neumann boundary a (t, s)u P PN k N k ( condition: i=1 j=1 aij (t, x)∂xj uk + ai (t, x)uk )νi = 0 for x ∈ ∂D and 1 ≤ k ≤ K. THEOREM 6.1.5 (Monotonicity of mild solution) Let 1 0 then Ua (t, 0)u0 > 0 for all a ∈ Y and t > 0, and Ua (t, 0)u0 k1 (x) > 0 for all a ∈ Y, t > 0, and x ∈ D. It follows that if u0 ∈ Lp (D)+ then Ua (t, 0)u0 ∈ Lp (D)+ for all a ∈ Y and all t > 0. (2) Assume (A6-1), (A6-3), and (A6-4). If u0 ∈ Lp (D)+ \ {0} then (Ua (t, 0)u0 )k (x) > 0 for all a ∈ Y, 1 ≤ k ≤ K, x ∈ D, and t > 0.

206

Spectral Theory for Parabolic Equations

(3) Assume (A6-1) and (A6-3). Assume also that a(1) , a(2) ∈ Y satisfy k,(1) k,(2) k,(1) k,(2) k,(1) k,(2) aij = aij , ai = ai , bi = bi (i, j = 1, 2, . . . , N , k = k,(2)

k,(1)

k,(2)

k,(1)

1, 2, . . . , K), and cl ≥ cl and d0 ≤ d0 (l, k = 1, 2, . . . , K), where the equalities and inequalities are considered a.e. on R × D, or on R × ∂D. Then Ua(2) (t, 0)u0 ≥ Ua(1) (t, 0)u0 for any u0 ∈ Lp (D)+ and t ≥ 0. (4) Assume (A6-1) and (A6-3). Then 0 R 0 N 0 UD a (t, 0)u ≤ Ua (t, 0)u ≤ Ua (t, 0)u

for any a ∈ Y, u0 ∈ Lp (D)+ , and t ≥ 0. PROOF (1) Recall that in the proof of Theorem 6.1.1, Ua (t, 0)u0 |[0,h] is a fixed point of Gu0 in the space Mr (h, s) = { u ∈ C([0, h], Lp (D)) : ku(τ )kp ≤ r for all τ ∈ [0, h] }, for some h > 0 and r > 0, where Gu0 is defined by Z t Gu0 (u)(t) = U0a (t, 0)u0 + U0a (t, τ )(Ca (τ )u(τ )) dτ,

t ∈ [0, h].

0

Assume first that ckk (t, x) ≥ 0 for all 1 ≤ k ≤ K and a.e. (t, x) ∈ R × D. Then, for any u(τ ) ≥ 0, Ca (τ )u(τ ) ≥ 0. By Proposition 2.2.9, Uak (t, 0)u0k ≥ 0 for all 1 ≤ k ≤ K and (Uak1 (t, 0)u0k1 )(x) > 0 for all x ∈ D, which implies U0a (t, 0)u0 > 0, for all t > 0. Then by Proposition 2.2.9(1), for u(τ ) ≥ 0 for all τ ∈ [0, t], U0a (t, τ )(Ca (τ )u(τ )) ≥ 0 for all τ ∈ (0, h]. This together with the arguments of Theorem 6.1.1 implies that the fixed point of Gu0 is nonnegative. Therefore Ua (t, 0)u0 k (x) > 0 (x ∈ D) and Ua (t, 0)u0 > 0 1 for t ∈ (0, h], consequently Ua (t, 0)u0 k (x) > 0 (x ∈ D) and Ua (t, 0)u0 > 0 1 for all t > 0. We proceed now to the general case. For a = (akij , aki , bki , ckl , dk0 ) ∈ Y and r0 ∈ R denote a + r0 := (akij , aki , bki , c˜kl , dk0 ), where c˜kl = ckl for k 6= l and c˜kk := ckk + r0 . Further, for 1 ≤ k ≤ K put (a + r0 )k := (akij , aki , bki , r0 , dk0 ), and denote U0a+r0 := (U(a+r0 )1 , . . . , U(a+r0 )K ). By Lemma 4.3.1(ii), U0a+r0 (t, 0) = er0 t U0a (t, 0). Notice that Ua+r0 (t, 0)u0 satisfies Z t 0 0 0 Ua+r0 (t, 0)u = Ua+r0 (t, 0)u + U0a+r0 (t, τ )(Ca (τ )Ua+r0 (τ, 0)u0 ) dτ. 0

r0 t

Therefore Ua+r0 (t, 0) = e Ua (t, 0) for all t ≥ 0. It suffices now to take r0 > 0 so large that ckk + r0 ≥ 0 a.e. on R × D, and apply the reasoning from the above paragraph.

6. Cooperative Systems of Parabolic Equations

207

(2) Without loss of generality we may assume that ckk (t, x) ≥ 0 for all 1 ≤ k ≤ K and a.e. (t, x) ∈ R × D. For otherwise, we may replace a with a + r0 for some sufficiently large number r0 , as in the above arguments. Let u0 ∈ Lp (D) \ {0}, and let 1 ≤ k1 ≤ K be such that u0k1 > 0. By Part (1), (Ua (t, 0)u0 )k1 (x) > 0 for all t > 0 and x ∈ D. Now, for any t2 > 0, by (A6-4), there is 1 ≤ k2 ≤ K with k1 6= k2 such that ckk21 (t, ·) ≥ 0 and ckk21 (t, ·) 6≡ 0 for t in any sufficiently small neighborhood of t2 . Then, again by Proposition 2.2.9(2), for given t > t2 , U0a (t, τ )(Ca (τ )u(τ )) k2 (x) > 0 for τ (< t) in a sufficiently small neighborhood of t2 and each x ∈ D. This implies that Ua (t, 0)u0 k2 (x) > 0 for t > t2 , x ∈ D. For any t3 > t2 , by (A6-4) again, there is 1 ≤ k3 ≤ K with k3 6= k1 , k2 such that ckk31 (t, ·) ≥ 0 and ckk31 (t, ·) 6≡ 0 for t in any sufficiently small neighborhood of t3 or ckk32 (t, ·) ≥ 0 and ckk32 (t, ·) 6≡ 0 for t in a sufficiently small neighborhood of t3 . This implies that (Ua (t, 0)u0 )i3 (x) > 0 for t > t3 and x ∈ D. By induction, there are tK > tK−1 > · · · > t2 > 0 = t1 and kK , kK−1 , . . . , k1 with km 6= kn for m 6= n such that (Ua (t, 0)u0 )km (x) > 0 for t > tm and x ∈ D. Then by the arbitrariness of t2 , t3 , . . . , tK , we have (Ua (t, 0)u0 )k (x) > 0 for any t > 0, x ∈ D and 1 ≤ k ≤ K. (3) Again, without loss of generality, we assume ckk (t, x) ≥ 0 for all 1 ≤ k ≤ K and a.e. (t, x) ∈ R × D. Let Z t 0 0 G(m) (u) = U (t, 0)u + U0a(m) (t, τ )(Ca(m) (τ )u(τ )) dτ u0 a(m) 0

for m = 1, 2, u ∈ Mr (h, s), where Mr (h, s) is as in (6.1.8). By Proposition 2.2.10, U0a(2) (t, 0)u0 ≥ U0a(1) (t, 0)u0 for any u0 ≥ 0 and t ≥ 0. Hence for any u(·) ∈ Mr (h, s) with u(t) ≥ 0 for all τ ∈ [0, h], (2) (1) Gu0 (u) ≥ Gu0 (u). This together with the arguments in (1) implies that Ua(2) (t, 0)u0 ≥ Ua(1) (t, 0)u0 for any u0 ≥ 0 and t ≥ 0. (4) It can be proved by arguments similar to those in (3). By the above theorems, under the assumptions (A6-1), (A6-2), (6.0.1)+ (6.0.2) generate a topological linear skew-product semiflow Π = {Πt }t≥0 Πt : L2 (D) × Y → L2 (D) × Y,

(6.1.22)

208

Spectral Theory for Parabolic Equations

where Πt (u0 , a) := (Ua (t, 0)u0 , σt a).

(6.1.23)

Under additional assumption (A6-3) it follows from Theorem 6.1.5(1) that the solution operator Ua (t, 0) (a ∈ Y, t > 0) has the property that, for any u1 , u2 ∈ L2 (D) with u1 < u2 there holds Ua (t, 0)u1 < Ua (t, 0)u2 . Adjusting the terminology used for semiflows on ordered metric spaces (see, e.g., [57]) to skew-product semiflows with ordered fibers we can say that then the (topological) linear skew-product semiflow Π is strictly monotone. In view of Theorem 6.1.5(2), we say that a (global) mild solution u(·) = (u1 (·), . . . , uK (·)) of (6.0.1)+(6.0.2) satisfying (A6-1), (A6-3), and (A6-4) is a positive mild solution on [s, ∞) × D if uk (t)(x) > 0 for all t > s, all x ∈ D and all k = 1, . . . , K.

6.1.2

The Smooth Case

In this subsection we show that if both a(t, x) and ∂D are sufficiently smooth, then Ua (t, 0)u0 is also smooth. To be more precise, we make the following standing assumption. (A6-5) ∂D is an (N − 1)-dimensional manifold of class C 3+α . There is ¯ M > 0 such that for any a ∈ Y, the C 2+α,2+α (R × D)-norms of akij , aki 2+α,1+α ¯ (i, j = 1, 2, . . . , N , k = 1, 2, . . . , K), the C (R × D)-norm of bki and ckl 2+α,2+α (i, j = 1, 2, . . . , N , l, k = 1, 2, . . . , K), and the C (R × ∂D)-norms of dk0 (k = 1, 2, . . . , K) are bounded by M . First of all, similar to Lemma 2.5.1, we have LEMMA 6.1.6 k,(n) k,(n) k,(n) k,(n) k,(n) Assume (A6-5). Let a(n) = (aij , ai , bi , cl , d0 ) ∈ Y and k,(n)

a = (akij , aki , bki , ckl , dk0 ) ∈ Y. Then limn→∞ a(n) = a if and only if aij k,(n)

converge to akij , ai

k,(n)

converge to aki , bi

k,(n)

converge to bki , cl converge to k,(n) k ¯ cl , all uniformly on compact subsets of R × D, and (in the Robin case) d0 converge to dk0 uniformly on compact subsets of R × ∂D. Throughout this section, we assume (A6-1) and (A6-5). Note that (A6-5) implies (A6-2). THEOREM 6.1.6 (Regularity up to boundary) Let 1 < p < ∞ and u0 ∈ L2 (D). Then for any α ∈ (0, 1/2), any a ∈ Y, and any 0 < t1 < t2 the restriction [ [t1 , t2 ] 3 t 7→ Ua (t, 0)u0 ] belongs ¯ K ) ∩ C([t1 , t2 ], (C 2+α (D)) ¯ K ). Moreover, there is C = to C 1 ([t1 , t2 ], (C α (D)) C(t1 , t2 , u0 ) > 0 such that k[ [t1 , t2 ] 3 t 7→ Ua (t, 0)u0 ]kC 1 ([t1 ,t2 ],(C α (D)) ¯ K) ≤ C

6. Cooperative Systems of Parabolic Equations

209

and k[ [t1 , t2 ] 3 t 7→ Ua (t, 0)u0 ]kC([t1 ,t2 ],(C 2+α (D)) ¯ K) ≤ C for all a ∈ Y. PROOF For given 1 0 and 1 < q < ∞, one has Ua (t, 0)u0 ∈ Lq (D) (Theorem 6.1.2). The result then follows from [3, Corollary 15.3]. Given 1 < p < ∞, let Vp1 (ak ) be as in Section 2.5, i.e., Vp1 (ak ) := { uk ∈ Wp2 (D) : Bak (0)uk = 0 }. Let Vp1 (a) := Vp1 (a1 ) × Vp1 (a2 ) × · · · × Vp1 (aK ). For given 0 < β < 1 and 1 < p < ∞, let  (Lp (D), Wp2 (D))β,p β Vp :=  [Lp (D), Wp2 (D)]β

if 2β 6∈ N if 2β ∈ N,

Vpβ := (Vpβ )K , and Vpβ (ak ) :=

 (Lp (D), Vp1 (ak ))β,p

if 2β 6∈ N

[Lp (D), Vp1 (ak )]β

if 2β ∈ N,



Vpβ (a) := Vpβ (a1 ) × Vpβ (a2 ) × · · · × Vpβ (aK ), where (·, ·)β,p is a real interpolation functor and [·, ·]β is a complex interpolation functor (see [15], [105] for more detail). By Lemmas 2.5.2 and 2.5.3, we have LEMMA 6.1.7 (1) Vpβ = (Wp2β (D))K . (2) If 2β −

1 p

6= 0, 1 then Vpβ (a) is a closed subspace of Vpβ .

Also, we have the following compact embeddings: ¯ Wpj+m (D) ,−,→ C j+λ (D)

(6.1.24)

if mp > N > (m − 1)p and 0 < λ < m − (N/p), and (Wp2 (D))K ,−,→ Vpβ ,−,→ Lp (D),

(6.1.25)

210

Spectral Theory for Parabolic Equations Vp1 (a) ,−,→ Vpβ (a) ,−,→ Lp (D)

(6.1.26)

for any 0 < β < 1 and a ∈ Y. By [3, Lemma 6.1 and Theorem 14.5] (see also [109]), we have THEOREM 6.1.7 For any 1 0. Moreover, for any fixed T > 0 and 1 0 such that kUa (t, 0)kLp (D),(Wp2 (D))K ≤

Cp t

for all a ∈ Y and 0 < t ≤ T . COROLLARY 6.1.1 For any 1 0. Moreover, for any fixed 0 < δ < T and 1 0 such that kUa (t, 0)kL2 (D),(Wp2 (D))K ≤ Cδ,p for all a ∈ Y and δ ≤ t ≤ T . PROOF

It follows from L2 –Lp estimates and Theorem 6.1.7.

By [3, Theorems 7.1 and 14.5] we have THEOREM 6.1.8 Suppose that 2β −1/p 6∈ N. Then for any t ≥ 0 and u0 ∈ Vpβ (a), Ua (t, 0)u0 ∈ Vpβ (a · t). Moreover, for any T > 0 there is Cp,β = Cp,β (T ) > 0 such that kUa (t, 0)u0 kVpβ ≤ Cp,β ku0 kVpβ for any a ∈ Y, 0 ≤ t ≤ T , and u0 ∈ Vpβ (a). THEOREM 6.1.9 For any given 0 < t1 ≤ t2 , if E is a bounded subset of L2 (D) then { Ua (τ, 0)u0 : ¯ RK ). a ∈ Y, τ ∈ [t1 , t2 ], u0 ∈ E } has compact closure in C 1 (D, PROOF

It is a consequence of Corollary 6.1.1 and Eq. (6.1.24) for p > N .

THEOREM 6.1.10 (Joint continuity in Vpβ ) Let 0 ≤ β < 1 and 1 < p < ∞ with 2β − 1/p 6∈ N. For any sequence ∞ n ∞ (a(n) )∞ n=1 ⊂ Y, any real sequence (tn )n=1 , and any sequence (u )n=1 ⊂

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L2 (D), if limn→∞ a(n) = a, limn→∞ tn = t, where t > 0, and limn→∞ un = u0 in L2 (D), then Ua(n) (tn , 0)un converges in Vpβ to Ua (t, 0)u0 . PROOF First of all, by Theorem 6.1.4, we have that Ua(n) (tn , 0)un converges in L2 (D) to Ua (t, 0)u0 . Now Corollary 6.1.1 and (6.1.25) imply that there is a subsequence (nk )∞ k=1 such that Ua(nk ) (tnk , 0)unk converges in Vpβ to some u∗ . We then must have u∗ = Ua (t, 0)u0 and Ua(nk ) (tnk , 0)unk converges in Vpβ to Ua (t, 0)u0 . This implies that Ua(n) (tn , 0)un converges in Vpβ to Ua (t, 0)u0 . ¯ RK )) THEOREM 6.1.11 (Joint continuity in C 1 (D, (n) ∞ For any sequence (a )n=1 ⊂ Y, any real sequence (tn )∞ n=1 , and any se(n) quence (un )∞ = a, limn→∞ tn = t, where t > 0, n=1 ⊂ L2 (D), if limn→∞ a ¯ RK ) and limn→∞ un = u0 in L2 (D), then Ua(n) (tn , 0)un converges in C 1 (D, 0 to Ua (t, 0)u . PROOF

It follows by (6.1.24) and similar arguments as in Theorem 6.1.10.

¯ RK )) THEOREM 6.1.12 (Norm continuity in C 1 (D, (n) ∞ (n) For any sequence (a )n=1 ⊂ Y and any real sequence (tn )∞ →a n=1 , if a and tn → t as n → ∞, where t > 0, then kUa(n) (tn , 0) − Ua (t, 0)kC 1 (D,R ¯ K) converges to 0 as n → ∞. PROOF Assume that kUa(n) (tn , 0) − Ua (t, 0)kC 1 (D,R ¯ K ) does not con¯ RK ) with verge to 0 as n → ∞. Then there are 0 > 0 and unk ∈ C 1 (D, nk ku kC 1 (D,R ¯ K ) = 1 such that kUa(nk ) (tnk , 0)unk − Ua (tnk , 0)unk kC 1 (D,R ¯ K ) ≥ 0

(6.1.27)

¯ RK ) ,−,→ L2 (D), we may assume without loss of for any nk . Since C 1 (D, generality that there is u0 ∈ L2 (D) such that kunk − u0 k → 0 as k → ∞. Then by Theorem 6.1.11, we have kUa(nk ) (tnk , 0)unk − Ua (tnk , 0)unk kC 1 (D,R ¯ K) → 0

(6.1.28)

as k → ∞. This contradicts (6.1.27). The theorem is thus proved. We now investigate the strong monotonicity property of the solution operator Ua (t, 0). By Theorem 6.1.5 and arguments similar to those in Proposition 2.5.6, we have

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THEOREM 6.1.13 (Strong monotonicity on initial data) Assume (A6-1), (A6-3)–(A6-5). Let 1 0, x ∈ D

and ∂ ∂ Ua (t, 0)u1 k (x) > Ua (t, 0)u2 k (x) ∂νν ∂νν

for a ∈ Y, t > 0, x ∈ ∂D

in the Dirichlet case, (ii) Ua (t, 0)u1

k

(x) < Ua (t, 0)u2

k

(x)

¯ for a ∈ Y, t > 0, x ∈ D

in the Neumann or Robin case. In view of Lemma 1.3.2 the following result is a consequence of Theorem 6.1.13. PROPOSITION 6.1.1 Assume (A6-1) and (A6-3)–(A6-5). Let 1 0. ˚1 (D, ¯ RK ) (resp. u1 , u2 ∈ C 1 (D, ¯ RK )), if The property that, for u1 , u2 ∈ C 2 1 2 u 0), can be expressed ˚1 (D, ¯ RK ) → as: For each a ∈ Y and t > 0 the linear operator Ua (t, 0) : C 1 ¯ K 1 ¯ K 1 ¯ K ˚ C (D, R ) (resp. Ua (t, 0) : C (D, R ) → C (D, R )) is strongly positive (or strongly monotone), see, e.g., [57]. In the rest of this section, we consider the adjoint problem of (6.0.1)+(6.0.2), 1

N N ∂uk ∂uk X ∂ X k k = a (t, x) − bi (t, x)uk − ∂t ∂xi j=1 ji ∂xj i=1 −

N X i=1

K

aki (t, x)

∂uk X l + ck (t, x)ul , ∂xi l=1

t < s, x ∈ D,

(6.1.29)

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complemented with the boundary conditions Ba∗k (t)uk = 0

on ∂D

for t < s,

(6.1.30)

k N k N k where Ba∗k = B(a∗ )k and (a∗ )k = ((akji )N j,i=1 , −(bi )i=1 , −(ai )i=1 , 0, d0 ), k = 1, 2, . . . , K. We study mild solutions as well as weak solutions of (6.0.1)+(6.0.2) and (6.1.29)+(6.1.30), and their relations. First, similar to Definition 6.1.1 for mild solutions of (6.0.1)+(6.0.2), we can define mild solutions of (6.1.29)+(6.1.30). Then by similar arguments as in Theorem 6.1.1, we can prove that for any given 1 < p < ∞, u0 ∈ Lp (D), and s ∈ R, there is a unique mild solution u(·) of (6.1.29)+(6.1.30) on (−∞, s] with u(s) = u0 . Denote U∗a,p (t, s) (t < s) to be the mild solution operator of (6.1.29)+(6.1.30) in Lp (D). We write U∗a (t, s) for U∗a,2 (t, s) (t < s). Let V := (V )K , (6.1.31)

where V is as in (2.1.2). For given a ∈ Y and u, v ∈ V, define the bilinear form Ba = Ba (t, u, v) by, Z (akij (t, x)∂xj uk + aki (t, x)uk )∂xi vk dx Ba (t, u, v) := D Z − (bki (t, x)∂xi uk + ckl (t, x)ul )vk dx (6.1.32) D

in the Dirichlet and Neumann boundary condition cases, and Z Ba (t, u, v) := (akij (t, x)∂xj uk + aki (t, x)uk )∂xi vk dx Z D − (bki (t, x)∂xi uk + ckl (t, x)ul )vk dx ZD + dk0 (t, x)uk vk dx

(6.1.33)

∂D

in the Robin boundary condition case. Also, define Ba∗ (t, u, v) by Z Ba∗ (t, u, v) := (akji (t, x)∂xj uk ∂xi vk dx Z D − bi (t, x)uk )∂xi vk + (aki (t, x)∂xi uk − clk (t, x)ul )vk dx

(6.1.34)

D

in the Dirichlet and Neumann boundary condition cases, and Z ∗ Ba (t, u, v) := (akji (t, x)∂xj uk ∂xi vk dx Z D − bki (t, x)uk )∂xi vk + (aki (t, x)∂xi uk − clk (t, x)ul )vk dx D Z + dk0 (t, x)uk vk dx (6.1.35) ∂D

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in the Robin boundary condition case. (We used the summation convention in the above). DEFINITION 6.1.2 (Weak solution) (1) A function u(·) ∈ L2 ((s, t), V) is a weak solution of (6.0.1)+(6.0.2) on [s, t] × D with initial condition u(s) = u0 (u0 ∈ L2 (D)) if t

Z

˙ ) dτ + hu(τ ), viψ(τ

−

t

Z

s

Ba (τ, u(τ ), v)ψ(τ ) dτ − hu0 , viψ(s) = 0

s

for all v ∈ V and ψ(·) ∈ D([s, t)). (2) A function u(·) ∈ L2 ((s, t), V) is a weak solution of (6.1.29)+(6.1.30) on [s, t] × D with final condition u(t) = u0 (u0 ∈ L2 (D)) if Z s

t

˙ ) dτ + hu(τ ), viψ(τ

Z

t

Ba∗ (τ, u(τ ), v)ψ(τ ) dτ − hu0 , viψ(t) = 0

s

for all v ∈ V and ψ(·) ∈ D((s, t]). THEOREM 6.1.14 For any given a ∈ Y and u0 ∈ L2 (D), u(t) := Ua (t, 0)u0 is a weak solution of (6.0.1)+(6.0.2) on (0, ∞) with initial condition u(0) = u0 and u∗ (t) := U∗a (t, 0)u0 is a weak solution of (6.1.29)+(6.1.30) on (−∞, 0) with final condition u∗ (0) = u0 . PROOF By Theorem 6.1.6, u(t) = Ua (t, 0)u0 is a classical solution of (6.0.1) +(6.0.2) on (0, ∞) with u(0) = u0 and u∗ (t) = U∗a (t, 0)u0 is a classical solution of (6.1.29)+(6.1.30) on (−∞, 0) with u∗ (0) = u0 . The theorem then follows. Thanks to Theorem 6.1.14, throughout the following sections, when both the coefficients and the domain of (6.0.1)+(6.0.2) are sufficiently smooth, a solution u(·) of (6.0.1)+(6.0.2) ((6.1.29)+(6.1.30)) means either a mild or a weak solution. By arguments similar to those in the proof of Proposition 2.3.2, we have THEOREM 6.1.15 If u and v are solutions of (6.0.1)+(6.0.2) and (6.1.29)+ (6.1.30) on [s, t]×D, respectively, then hu(τ ), v(τ )i is independent of τ for τ ∈ [s, t]. Also, by arguments similar to those in Proposition 2.3.3, we have

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THEOREM 6.1.16 (Ua (t, s))∗ = U∗a (s, t)

6.2

for any a ∈ Y and any s < t.

Principal Spectrum and Principal Lyapunov Exponents and Exponential Separation in the General Setting

In this section, we introduce the concepts of principal spectrum, principal Lyapunov, and exponential separation of (6.0.1)+(6.0.2) and investigate their basic properties and show the existence of exponential separation and existence and uniqueness of entire positive solutions, which extends the theories established in Chapter 3 for scalar parabolic equation in the general setting to cooperative systems of parabolic equations in the general setting.

6.2.1

Principal Spectrum and Principal Lyapunov Exponents

Throughout this subsection, we assume (A6-1)–(A6-3). Let Ua (t, s) : L2 (D) → L2 (D) be the mild solution operator of (6.0.1)+(6.0.2) and let Uak (t, s) : L2 (D) → L2 (D), 1 ≤ k ≤ K, be the weak solution operator of (6.1.1). Let Y0 ⊂ Y be a compact connected translation invariant subset of Y. For given u0 ∈ L2 (D) and a ∈ Y0 , let Πkt (u0k , ak ) = (Uak (t, 0)u0k , ak · t).

(6.2.1)

Πt (u0 , a) = (Ua (t, 0)u0 , a · t).

(6.2.2)

Recall that ckk (t, x)

(akij , aki , bki , ckl , dk0 )

Let r0 ≥ 0 be such that ≥ −r0 for any a = ∈ Y0 , a.e. t ∈ R, a.e. x ∈ D, and k = 1, 2, . . . , K. By (A6-3) and arguments as in the proof of Theorem 6.1.5, for u0 ≥ 0, (Ua (t, 0)u0 )k ≥ e−r0 t Uak (t, 0)u0k

for t > 0, 1 ≤ k ≤ K.

For a ∈ Y0 and t > 0, we define kUa (t, 0)k+ := sup { kUa (t, 0)u0 k : u0 ∈ L2 (D)+ , ku0 k = 1 }. LEMMA 6.2.1 For any a ∈ Y and t > 0 one has kUa (t, 0)k+ = kUa (t, 0)k. PROOF

It can be proved by arguments as in Lemma 3.1.1.

(6.2.3)

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DEFINITION 6.2.1 (Principal resolvent) A real number λ belongs to the principal resolvent of Π over Y0 , denoted by ρ(Y0 ), if either of the following conditions holds: • There are > 0 and M ≥ 1 such that kUa (t, 0)k ≤ M e(λ−)t

for t > 0 and a ∈ Y0

(such λ is said to belong to the upper principal resolvent, denoted by ρ+ (Y0 )), • There are > 0 and M > 0 such that kUa (t, 0)k ≥ M e(λ+)t

for t > 0 and a ∈ Y0

(such λ is said to belong to the lower principal resolvent, denoted by ρ− (Y0 )). In view of Lemma 6.2.1, in the above inequalities the k·k-norms can be replaced with k·k+ -“norms,” with the same M and . DEFINITION 6.2.2 (Principal spectrum) The principal spectrum of the topological linear skew-product semiflow Π over Y0 , denoted by Σ(Y0 ), equals the complement in R of the principal resolvent of Π over Y0 . LEMMA 6.2.2 (i) For any t2 > 0 there is K1 = K1 (t2 ) > 0 such that kUa (t, 0)k ≤ K1 for all a ∈ Y0 and all t ∈ [0, t2 ]. (ii) For any t2 > 0 there is K2 = K(t2 ) > 0 such that kUa (t, 0)k ≥ K2 for all a ∈ Y0 and all t ∈ [0, t2 ]. PROOF Part (i) is a consequence of the L2 –L2 estimates (Theorem 6.1.2), To prove (ii), take φ := (φ1 , φ2 , . . . , φK ) with φk ≡ 1 (k = 1, 2, . . . , K), and φ)k ≥ e−r0 t Uak (t, 0)φk . (ii) then follows from Lemma 3.1.2(ii). notice (Ua (t, 0)φ

LEMMA 6.2.3 There exist δ0 > 0, M1 > 0, and a real λ such that kUa (t, 0)k ≥ M1 eλt for all a ∈ Y0 and all t ≥ δ0 . PROOF Note that for any u0 = (u01 , . . . , u0K ) ≥ 0, (Ua (t, 0)u0 )k ≥ e Uak (t, 0)u0k . The lemma then follows from Lemma 3.1.4. −r0 t

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THEOREM 6.2.1 The principal spectrum of Π over Y0 is a compact nonempty interval [λmin , λmax ]. PROOF We prove first that the upper principal resolvent ρ+ (Y0 ) is nonempty. Indeed, by the L2 –L2 estimates (see Theorem 6.1.2), there are ¯ > 0 and γ¯ > 0 such that kUa (t, 0)k ≤ M ¯ eγ¯ t for all a ∈ Y0 and t > 0, M hence γ¯ + 1 ∈ ρ+ (Y0 ). Further, ρ+ (Y0 ) is a right-unbounded open interval (λmax , ∞). The lower principal resolvent ρ− (Y0 ) is nonempty, too, since it contains, by Lemma 6.2.3, the real number λ − 1. Consequently, as ρ− (Y0 ) ∪ ρ+ (Y0 ) = ρ(Y0 ) and ρ− (Y0 ) ∩ ρ+ (Y0 ) = ∅, one has Σ(Y0 ) = R \ ρ(Y0 ) = [λmin , λmax ]. Similarly to Theorem 3.1.2 we have THEOREM 6.2.2 ∞ ∞ (1) For any sequence (a(n) )∞ n=1 ⊂ Y0 and any real sequences (tn )n=1 , (sn )n=1 such that tn − sn → ∞ as n → ∞ there holds

λmin ≤ lim inf n→∞

ln kUa(n) (tn , sn )k ln kUa(n) (tn , sn )k ≤ lim sup ≤ λmax . tn − sn tn − sn n→∞

∞ (2A) There exist a sequence (a(n,1) )∞ n=1 ⊂ Y0 and a sequence (tn,1 )n=1 ⊂ (0, ∞) such that tn,1 → ∞ as n → ∞, and

ln kUa(n,1) (tn,1 , 0)k = λmin . n→∞ tn,1 lim

∞ (2B) There exist a sequence (a(n,2) )∞ n=1 ⊂ Y0 and a sequence (tn,2 )n=1 ⊂ (0, ∞) such that tn,2 → ∞ as n → ∞, and

lim

n→∞

ln kUa(n,2) (tn,2 , 0)k = λmax . tn,2

Assume that µ is an invariant ergodic Borel probability measure for the topological flow σ on Y0 . Similarly to Theorem 3.1.5 we have THEOREM 6.2.3 There exist a Borel set Y1 ⊂ Y0 with µ(Y1 ) = 1 and a real number λ(µ) such that ln kUa (t, 0)k = λ(µ) for all a ∈ Y1 . lim t→∞ t

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DEFINITION 6.2.3 (Principal Lyapunov exponent) λ(µ) as defined above is called the principal Lyapunov exponent of Π for the ergodic invariant measure µ. Similarly to Theorem 3.1.6 we have THEOREM 6.2.4 For any ergodic invariant measure µ for σ|Y0 the principal Lyapunov exponent λ(µ) belongs to the principal spectrum [λmin , λmax ] of Π on Y0 . In the rest of this subsection, we assume (A6-5). We define ( ˚1 (D, ¯ RK ) C (Dirichlet) X := 1 ¯ C (D, RK ) (Neumann or Robin). By Corollary 6.1.1, there is C1,p > 0 such that kUa (1, 0)kL2 (D),(Wp2 (D))K ≤ C1,p

for alla ∈ Y.

This implies, via (6.1.24), that for p sufficiently large there is C˜p > 0 such that kUa (1, 0)kL2 (D),X ≤ C˜p for all a ∈ Y. LEMMA 6.2.4 There are M1 , M2 > 0 such that kUa (t, 0)kX ≤ M1 kUa (t − 1, 0)k for all a ∈ Y and all t > 1, and kUa (t, 0)k ≤ M2 kUa·1 (t − 1, 0)kX for all a ∈ Y and all t > 1. ˆ p the norm of the embedding PROOF For 1 ≤ p ≤ ∞, denote by M X ,→ Lp (D). ¯ RK ). Denote Fix some p ∈ (N, ∞). Then, by (6.1.24), (Wp2 (D))K ,→ C 1 (D, ˇ the norm of this embedding. We estimate by M ˇ kUa (t, 0)u0 k(W 2 (D))K kUa (t, 0)u0 kX ≤ M p ˇ kUa (t, t − 1)Ua (t − 1, 0)u0 k(W 2 (D))K =M p ˇ C1,p kUa (t − 1, 0)u0 k ≤M ˇ C1,p kUa (t − 1, 0)k · ku0 k ≤M ˇ C1,p M ˆ 2 kUa (t − 1, 0)k · ku0 kX ≤M

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for all a ∈ Y, u0 ∈ X and all t > 1, which gives the first inequality. Further, we estimate ˆ 2 kUa (t, 0)u0 kX ≤ M ˆ 2 kUa (t, 1)kX · kUa (1, 0)u0 kX kUa (t, 0)u0 k ≤ M ˆ 2 C˜p kUa (t, 1)kX · ku0 k = M ˆ 2 C˜p kUa·1 (t − 1, 0)kX · ku0 k ≤M for all a ∈ Y, u0 ∈ L2 (D) and all t > 1, which gives the second inequality. THEOREM 6.2.5 (1) λ belongs to the upper principal resolvent of Π over Y0 if and only if ˜ ≥ 1 such that there are > 0 and M ˜ e(λ−)t kUa (t, 0)kX ≤ M

for t > 1 and a ∈ Y0 .

(2) λ belongs to the lower principal resolvent of Π over Y0 if and only if ˜ > 0 such that there are > 0 and M ˜ e(λ+)t kUa (t, 0)kX ≥ M

for t > 0 and a ˜ ∈ Y0

PROOF It is an application of Lemma 6.2.4 and the Lp –Lq estimates to Definition 6.2.1. The next two results follow by applying Lemma 6.2.4 to Theorems 6.2.2 and 6.2.3, respectively. THEOREM 6.2.6 ∞ ∞ (1) For any sequence (a(n) )∞ n=1 ⊂ Y0 and any real sequences (tn )n=1 , (sn )n=1 such that tn − sn → ∞ as n → ∞ there holds

λmin ≤ lim inf n→∞

ln kUa(n) (tn , sn )kX ln kUa(n) (tn , sn )kX ≤ lim sup ≤ λmax . tn − sn tn − sn n→∞

∞ (2A) There exist a sequence (a(n,1) )∞ n=1 ⊂ Y0 and a sequence (tn,1 )n=1 ⊂ (0, ∞) such that tn,1 → ∞ as n → ∞, and

ln kUa(n,1) (tn,1 , 0)kX = λmin . n→∞ tn,1 lim

∞ (2B) There exist a sequence (a(n,2) )∞ n=1 ⊂ Y0 and a sequence (tn,2 )n=1 ⊂ (0, ∞) such that tn,2 → ∞ as n → ∞, and

lim

n→∞

ln kUa(n,2) (tn,2 , 0)kX = λmax . tn,2

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THEOREM 6.2.7 Let µ be an ergodic invariant measure for σ|Y0 . There exist a Borel set Y1 ⊂ Y0 with µ(Y1 ) = 1 such that lim

t→∞

6.2.2

ln kUa (t, 0)kX = λ(µ) t

for all

a ∈ Y1 .

Exponential Separation: Basic Properties

Throughout this subsection, we assume (A6-1)–(A6-4). Let Y0 ⊂ Y be a compact connected translation invariant subset of Y. The concepts of onedimensional subbundle of L2 (D) × Y0 and one-codimensional subbundle of L2 (D) × Y0 are defined in a way similar to that in Section 3.2. DEFINITION 6.2.4 (Exponential separation) We say that Π admits an exponential separation with separating exponent γ0 > 0 over Y0 if there are an invariant one-dimensional subbundle X1 of L2 (D) × Y0 with fibers X1 (a) = span{w(a)}, and an invariant complementary one-codimensional subbundle X2 of L2 (D) × Y0 with fibers X2 (a) = { v ∈ L2 (D) : hv, w∗ (a)i = 0 }, where w, w∗ : Y0 → L2 (D) are continuous with kw(a)k = kw∗ (a)k = 1 for all a ∈ Y0 , having the following properties: (i) w(a) ∈ L2 (D)+ for all a ∈ Y0 , (ii) X2 (a) ∩ L2 (D)+ = {0} for all a ∈ Y0 , (iii) there is M ≥ 1 such that for any a ∈ Y0 and any v ∈ X2 (a) with kvk = 1, kUa (t, 0)vk ≤ M e−γ0 t kUa (t, 0)w(a)k

(t > 0).

Similarly, we can define exponential separation for discrete time as in Definition 3.2.2. DEFINITION 6.2.5 (Exponential separation for discrete time) Let Y0 be a compact connected invariant subset of Y, and let T > 0. Π is said to admit an exponential separation with separating exponent γ00 for the discrete time T over Y0 if there are a one-dimensional subbundle X1 of L2 (D) × Y0 with fibers X1 (a) = span{w(a)}, and a one-codimensional subbundle X2 of L2 (D) × Y0 with fibers X2 (a) = { v ∈ L2 (D) : hv, w∗ (a)i = 0 }, where w, w∗ : Y0 → L2 (D) are continuous with kw(a)k = kw∗ (a)k = 1 for all a ∈ Y0 , having the following properties: (a) Ua (T, 0)X1 (a) = X1 (a · T ) and Ua (T, 0)X2 (a) ⊂ X2 (a · T ) for all a ∈ Y0 .

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(b) w(a) ∈ L2 (D)+ for all a ∈ Y0 , (c) X2 (a) ∩ L2 (D)+ = {0} for all a ∈ Y0 , (d) there is M 0 ≥ 1 such that for any a ∈ Y0 and any v ∈ X2 (a) with kvk = 1, 0

kUa (nT, 0)vk ≤ M 0 e−γ0 n kUa (nT, 0)w(a)k

(n = 1, 2, 3, . . . ).

The following lemma directly follows from Theorem 6.1.5(2). LEMMA 6.2.5 The function w = (w1 , . . . , wK ) in Definitions 6.2.4 and 6.2.5 satisfies: wk (a) ∈ L2 (D)+ \ {0} for any a ∈ Y and any 1 ≤ k ≤ K. By arguments as in Theorem 3.2.2, we have THEOREM 6.2.8 Assume (A6-1)–(A6-4) as well as (A6-5). If Π admits an exponential separation with separating exponent γ00 > 0 for some discrete time T > 0 over a compact invariant Y0 ⊂ Y, then Π admits an exponential separation over Y0 , with separating exponent γ0 = γ00 . We remark that, to prove Theorem 6.2.8 by using the arguments as in Theorem 3.2.2, we need that the adjoint operator (Ua (t, s))∗ of Ua (t, s), s < t, is the same as the mild solution operator U∗a (s, t) of (6.1.29)+(6.1.30). Under (A6-5), it is proved in Theorem 6.1.16 that (Ua (t, s))∗ = U∗a (s, t). We conjecture that Theorem 6.2.8 holds under (A6-1)–(A6-4) only. In the rest of this subsection, we assume that Π admits an exponential separation over some Y0 . Similarly to Lemma 3.2.5 we have LEMMA 6.2.6 ∞ ∞ Let λ ∈ R, (a(n) )∞ n=1 ⊂ Y0 , and (sn )n=1 ⊂ R, (tn )n=1 ⊂ R with tn −sn → ∞. Then the following conditions are equivalent: ln kUa(n) (tn , sn )w(a(n) · sn )k = λ. n→∞ tn − sn

(i) lim

ln kUa(n) (tn , sn )u0 k = λ for any u0 ∈ L2 (D)+ \ {0}. n→∞ tn − sn

(ii) lim

ln kUa(n) (tn , sn )k ln kUa(n) (tn , sn )k+ = lim = λ. n→∞ n→∞ tn − sn tn − sn

(iii) lim

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For the topological linear skew-product flow Π|X1 on the one-dimensional bundle X1 its dynamical spectrum (or the Sacker–Sell spectrum) is defined as the complement of the set of those λ ∈ R for which either of the following conditions holds: • There are > 0 and M ≥ 1 such that kUa (t, 0)w(a)k ≤ M e(λ−)t

for t > 0 and a ∈ Y0 ,

• There are > 0 and M > 0 such that kUa (t, 0)w(a)k ≥ M e(λ+)t

for t > 0 and a ∈ Y0 .

THEOREM 6.2.9 The Sacker–Sell spectrum of Π|X1 equals Σ(Y0 ). THEOREM 6.2.10 For an ergodic invariant measure µ for σ|Y0 there is Y1 ⊂ Y0 with µ(Y1 ) = 1 such that the following holds. (a) For any a ∈ Y1 and any u0 ∈ L2 (D) \ X2 (a) (in particular, for any u0 ∈ L2 (D)+ \ {0}) one has ln kUa (t, 0)u0 k = λ(µ) ∈ Σ(Y0 ). t→∞ t lim

(6.2.4)

(b) For any a ∈ Y1 and any u0 ∈ X2 (a) \ {0} one has lim sup t→∞

ln kUa (t, 0)u0 k ≤ λ(µ) − γ0 . t

(6.2.5)

THEOREM 6.2.11 There exist ergodic invariant measures µmin and µmax for σ|Y0 such that λmin = λ(µmin ) and λmax = λ(µmax ). Similarly to Theorem 3.2.7, we have THEOREM 6.2.12 If Y0 = cl { a(0) · t : t ∈ R } for some a(0) ∈ Y0 , where the closure is taken in the weak-* topology, then (1)

0 0 0 ∞ (i) There are sequences (s0n )∞ n=1 , (tn )n=1 ⊂ R, tn −sn → ∞ as n → ∞, such that

ln kUa(0) (t0n , s0n )w(a(0) · s0n )k n→∞ t0n − s0n ln kUa(0) (t0n , s0n )u0 k ln kUa(0) (t0n , s0n )k = lim = lim n→∞ n→∞ t0n − s0n t0n − s0n

λmin = lim

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223

for each u0 ∈ L2 (D)+ \ {0}. 00 ∞ 00 00 (ii) There are sequences (s00n )∞ n=1 , (tn )n=1 ⊂ R, tn −sn → ∞ as n → ∞, such that

ln kUa(0) (t00n , s00n )w(a(0) · s00n )k n→∞ t00n − s00n ln kUa(0) (t00n , s00n )u0 k ln kUa(0) (t00n , s00n )k = lim = lim n→∞ n→∞ t00n − s00n t00n − s00n

λmax = lim

for each u0 ∈ L2 (D)+ \ {0}. (2) For any u0 ∈ L2 (D)+ \ {0} there holds ln kUa(0) (t, s)w(a(0) · s)k t−s→∞ t−s ln kUa(0) (t, s)u0 k ln kUa(0) (t, s)k = lim inf = lim inf t−s→∞ t−s→∞ t−s t−s ln kUa(0) (t, s)k ln kUa(0) (t, s)u0 k ≤ lim sup = lim sup t−s t−s t−s→∞ t−s→∞

λmin = lim inf

= lim sup t−s→∞

ln kUa(0) (t, s)w(a(0) · s)k = λmax . t−s

∞ (3) For each λ ∈ [λmin , λmax ] there are sequences (kn )∞ n=1 , (ln )n=1 ⊂ Z, ln − kn → ∞ as n → ∞, such that

ln kUa(0) (ln , kn )w(a(0) · kn )k n→∞ ln − k n ln kUa(0) (ln , kn )k ln kUa(0) (ln , kn )u0 k = lim = lim n→∞ n→∞ ln − k n ln − kn

λ = lim

for each u0 ∈ L2 (D)+ \ {0}.

6.2.3

Existence of Exponential Separation and Entire Positive Solutions

In this subsection, we study the existence of exponential separation and the existence and uniqueness of entire positive solutions. We restrict ourselves to the smooth case. Let (A6-6) denote the following standing assumption: (A6-6) For any T > 0 the mapping [ Y 3 a 7→ [ [0, T ] 3 t 7→ Ua (t, 0) ] ∈ B([0, T ], L(L2 (D), L2 (D))) ] is continuous, where L(L2 (D), L2 (D)) represents the space of all bounded linear operators from L2 (D) into itself, endowed with the norm topology and

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B(·, ·) stands for the Banach space of bounded functions, endowed with the supremum norm. It should be pointed out that in [6] and [91] conditions, for some special cases (for example, the Dirichlet boundary condition case and the case with infinitely differentiable coefficients), are given that guarantee the continuous dependence of [ [0, T ] 3 t 7→ Ua (t, 0) ] ∈ B([0, T ], L(L2 (D), L2 (D))) on the coefficients. Let Y0 be a compact connected invariant subset of Y. We first study the existence of exponential separation. THEOREM 6.2.13 (Existence of exponential separation) Assume (A6-1)–(A6-6). Then Π admits an exponential separation over Y0 . To prove the above theorem, we first show the following lemma. LEMMA 6.2.7 Assume (A6-1)–(A6-6). Then the map [ Y0 × (0, ∞) 3 (a, t) 7→ Ua (t, 0) ∈ L(L2 (D), X) ] is continuous. PROOF Assume that a(n) converges to a in Y0 and that tn converges to t > 0. Suppose to the contrary that kUa(n) (tn , 0) − Ua (t, 0)kL2 (D),X 6→ 0 as n → ∞. Then there are 0 > 0 and a sequence (un )∞ n=1 ⊂ L2 (D) with kun k = 1 such that kUa(n) (tn , 0)un − Ua (t, 0)un kX ≥ 0 for all n. By Corollary 6.1.1 and Eq. (6.1.24), there are u∗ , u∗∗ ∈ X such that (after possibly extracting a subsequence) Ua(n) (tn , 0)un → u∗

and

Ua (t, 0)un → u∗∗

in X, as n → ∞. Without loss of generality, we may also assume that there ˜ ∗ ∈ X such that Ua (t/2, 0)un → u ˜ ∗ in X as n → ∞. is u By the property (A6-6) we have kUa(n) (tn , 0)un − Ua (tn , 0)un k → 0 and by Theorem 6.1.4 we have kUa (tn , 0)un − Ua (t, 0)un k = kUa (tn , t/2)Ua (t/2, 0)un − Ua (t, t/2)Ua (t/2, 0)un k →0

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225

as n → ∞. Therefore kUa(n) (tn , 0)un − Ua (t, 0)un k ≤ kUa(n) (tn , 0)un − Ua (tn , 0)un k + kUa (tn , 0)un − Ua (t, 0)un k →0 as n → ∞. Then we must have u∗ = u∗∗ , hence kUa(n) (tn , 0)un − Ua (t, 0)un kX → 0 as n → ∞, a contradiction. PROOF (Proof of Theorem 6.2.13) We show that Π admits an exponential separation over Y0 for discrete time T = 1, and apply Theorem 6.2.8. ˜ : X × Y0 → X × Y0 by the formula We define a mapping Π ˜ 0 , a) := (Ua (1, 0)u0 , a · 1), Π(u

a ∈ Y0 , u0 ∈ X.

We have that, for each a ∈ Y0 , Ua (1, 0) is a compact (completely continuous) operator in L(X) (Theorem 6.1.9), having the property that Ua (1, 0)(X+ \ {0}) ⊂ X++ (Proposition 6.1.1). Moreover, the mapping [ Y0 3 a 7→ Ua (1, 0) ∈ L(X) ] is continuous (by Lemma 6.2.7). These allow us to use the results ˜ : Y0 → X, contained in [94] to conclude that there are continuous functions w ˜ ∗ : Y0 → X∗ , kw(a)k ˜ ˜ ∗ (a)kX∗ = 1 for all a ∈ Y0 , such that w X = kw ˜ (i) w(a) ∈ X++ , for each a ∈ Y0 . ˜ ∗ (a))X,X∗ > 0 for each a ∈ Y0 and each nonzero v ∈ X+ . (ii) (v, w ˜ (iii) For each a ∈ Y0 there is d1 = d1 (a) > 0 such that Ua (1, 0)w(a) = ˜ · 1). d1 w(a ˜ ∗ (a·1) = (iv) For each a ∈ Y0 there is d∗2 = d∗2 (a) > 0 such that (Ua (1, 0))∗ w ∗˜∗ ∗ ∗ ∗ d2 w (a), where (Ua (1, 0)) : X → X stands for the linear operator dual to Ua (1, 0). (v) There are constants C˜ > 0 and 0 < γ < 1 such that ˜ n kUa (n, 0)w(a)k ˜ kUa (n, 0)u0 kX ≤ Cγ X

(6.2.6)

˜ ∗ (a))X,X∗ for any a ∈ Y0 , any u0 ∈ X with ku0 kX = 1 satisfying (u0 , w = 0, and any n ∈ N. ˜ ˜ Define w : Y0 → L2 (D) by w(a) := w(a)/k w(a)k. Since X ,→ L2 (D), the function w is well defined and continuous. Further, we deduce from Lemma 6.2.7 that the mapping [ Y0 3 a 7→ (Ua (1, 0))∗ ∈ L(X∗ , L2 (D)) ] is

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continuous. This together with (iv) gives that the function w∗ : Y0 → L2 (D), ˜ ∗ (a)/kw ˜ ∗ (a)k, is well defined and continuous. w∗ (a) := w For a ∈ Y0 put X1 (a) := span{w(a)}, X2 (a) := { v ∈ L2 (D) : hv, w∗ (a)i = 0 }. We see that the properties (a), (b), and (c) in Definition 6.2.5 are satisfied. Let M1 denote the norm of the embedding X ,→ L2 (D). Also, by the ˜ on the compact space Y0 there is M2 > 0 such that kw(a)k ˜ continuity of w X ≤ M2 kw(a)k for all a ∈ Y0 . Further, put D1 := sup { kUa (1, 0)kL2 (D),X : a ∈ Y0 } (< ∞), ˜ D2 := inf { kUa (1, 0)w(a)k X : a ∈ Y0 } (> 0). Take a ∈ Y0 and u0 ∈ X2 with ku0 k = 1. It follows from (iv) that Ua (1, 0)u0 ∈ X ∩ X2 (a · 1). For n = 2, 3, 4, . . . , we estimate kUa (n, 0)u0 k ≤ M1 kUa (n, 1)(Ua (1, 0)u0 )kX ˜ n−1 kUa (n, 1)w(a ˜ · 1)kX kUa (1, 0)u0 kX ≤ M1 Cγ

(by (v))

˜ M1 C˜ n kUa (n, 0)w(a)k X γ kUa (1, 0)u0 kX ˜ γ kUa (1, 0)w(a)k X M1 M2 D1 C˜ n ≤ γ kUa (n, 0)w(a)k. D2 γ =

M2 D1 kUa (1, 0)w(a)k for all a ∈ Y0 and all u0 ∈ Clearly, kUa (1, 0)u0 k ≤ M1 D 2 0 L2 (D) with ku k = 1. Therefore (d) in Definition 6.2.5 is satisfied.

We note that exponential separation and entire positive solutions defined in the following are strongly related. DEFINITION 6.2.6 (Entire positive solution) A function [ R 3 t 7→ u(t) ∈ L2 (D) ] is called an entire positive solution of (6.0.1)+(6.0.2) if Z t u(t) = U0a (t, s)u(s) + U0a (t, τ )(Ca (τ )u(τ )) dτ s

for any s < t, and uk (t)(x) > 0 for all t ∈ R, x ∈ D, and k = 1, 2, . . . , K. The following lemma follows from Theorems 3.3.1 and 3.3.3. LEMMA 6.2.8 (1) For each k (1 ≤ k ≤ K) and ak ∈ Y0k , (6.1.1) has a unique (up to multiplication by positive scalars) entire positive solution. (2) For each k (1 ≤ k ≤ K), Πk admits an exponential separation over Y0k (with the one-dimensional invariant subbundle X1k and the onecodimensional invariant subbundle X2k ).

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227

For 1 ≤ k ≤ K, let wk , wk∗ : ak → L2 (D) be continuous functions such that kwk (ak )k = kwk∗ (ak )k = 1, X1k (ak ) = span {wk (ak )}, X2k (ak ) = { v ∈ L2 (D) : hv, wk∗ (ak )i = 0 } (ak ∈ Y0k ). Recall that for any ak ∈ Y0k the value at time t = 0 of the unique normalized entire positive solution of (6.1.1) equals wk (ak ). THEOREM 6.2.14 (Existence of entire positive solution) Let (A6-1) through (A6-4) be satisfied. Then for each a ∈ Y0 , (6.0.1)+(6.0.2) has an entire positive solution. PROOF We apply the same idea as in the proof of Theorem 3.3.1. Fix a ∈ Y0 and u0 ∈ L2 (D)+ with ku0 k = 1. Define a sequence (un )∞ n=1 by Ua (−1, −n)u0 Ua (0, −n)u0 = U (0, −1) . un := a kUa (−1, −n)u0 k kUa (−1, −n)u0 k By Theorem 6.1.2, the set { kun k : n = 1, 2, . . . } is bounded above. Moreover, from Theorem 6.1.3 we deduce that there is a sequence (nk )∞ k=1 such that ˜ 0 in L2 (D). By (6.2.3) and Lemma limk→∞ nk = ∞ and limk→∞ unk = u 3.3.4, the set { kun k : n = 1, 2, . . . } is bounded away from zero, consequently ˜ 0 ∈ L2 (D)+ \ {0}. u By arguments as in Theorem 3.3.1, we have that there is an entire positive ˆ of (6.0.1)+(6.0.2) such that u ˆ (0) = u ˜ 0. weak solution u THEOREM 6.2.15 (Uniqueness of entire positive solution) Let (A6-1) through (A6-5) hold. Then for each a ∈ Y0 , an entire positive solution of (6.0.1) +(6.0.2) is unique in the following sense: If u1 and u2 are ˜ 1 (t) for all entire positive solutions then there is β˜ > 0 such that u2 (t) = βu t ∈ R. Theorem 6.2.15 can be proved by arguments similar to those in [92] for scalar parabolic equations. For convenience, we provide a proof here. To do so, we first show some lemmas. We assume until the end of the subsection that (A6-1)–(A6-5) are satisfied. Choose a φ ∗ ∈ X++ . For any u ∈ L2 (D), define kukφ ∗ := h|u|, φ ∗ i. LEMMA 6.2.9 Given τ > δ > 0, there exists a constant C = C(τ, δ) > 0 such that for any a ∈ Y and any u ∈ L2 (D) there holds kUa (t, s)ukX ≤ Ckukφ ∗

for

t, s ∈ R, τ ≥ t − s ≥ δ,

kU∗a (t, s)ukX ≤ Ckukφ ∗

f or

t, s ∈ R, τ ≥ s − t ≥ δ.

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PROOF First, by Theorem 6.1.11, for given τ > δ > 0 there is C˜ = ˜ C(τ, δ) > 0 such that ˜ a (t − δ/2, s)uk kUa (t, s)ukX ≤ CkU for all a ∈ Y0 and τ ≥ t − s ≥ δ. Note that kUa (t − δ/2, s)uk = sup { hUa (t − δ/2, s)u, vi : v ∈ L2 (D), kvk = 1 } = sup { hu, U∗a (s, t − δ/2)vi : v ∈ L2 (D), kvk = 1 } ≤ sup { h|u|, |U∗a (s, t − δ/2)v|i : v ∈ L2 (D), kvk = 1 }. ˆ δ) > 0 such that We claim that there is Cˆ = C(τ, ˆ φ∗ |U∗a (s, t − δ/2)v| ≤ Cφ for all a ∈ Y0 , τ ≥ t − s ≥ δ and v ∈ L2 (D) with kvk = 1. Since |U∗a (s, t − δ/2)v| ≤ U∗a (s, t − δ/2)|v| and kvk = k |v| k, it suffices to show the inequality for v ∈ L2 (D)+ . And this is a consequence of Theorems 6.1.11 and 6.1.13. It then follows that kUa (t, s)ukX ≤ Ckukφ ∗ , ˆ with C = C˜ C. The inequality for the adjoint equation is proved in an analogous way. LEMMA 6.2.10 (1) There are constants M , %0 such that the following holds: If v is a positive solution of (6.1.29)+(6.1.30) on the interval (−∞, τ ), then kv(t)kφ ∗ ≤ M e%0 |t−s| kv(s)kφ ∗

for t, s < τ.

(2) For each positive solution v of (6.1.29)+(6.1.30) on (−∞, τ ), there is a constant η0 > 0 such that v(t) ≥ η0 φ ∗ kv(t)kφ ∗

for t ≤ τ − 1.

Analogous results hold for positive solutions of (6.0.1)+(6.0.2) on intervals (τ, ∞), τ ∈ R. PROOF (1) For a given a ∈ Y0 , let φk (a) := wk (P k (a)). It fol¯ is continuous. Put φ (a) := lows from Lemma 6.2.8 that φk : Y0 → C 1 (D)

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229

(φ1 (a), . . . , φK (a)), a ∈ Y0 . Proposition 2.5.7(2) implies the existence of 0 < q1 < q2 such that q1φ (a) ≤ φ ∗ ≤ q2φ (a) for any a ∈ Y0 . Further, let κk (a) := Bak (0, φk (a), φk (a)), where Bak (·, ·, ·) is as in (2.1.4) with a being replaced by ak in the Dirichlet and Neumann boundary condition cases, and is as in (2.1.5) with a being replaced by ak in the Robin boundary condition case. The functions κk : Y0 → R are continuous. Moreover, by Lemma 3.5.4, uk (t) := φk (a · t) satisfies  N N X  ∂uk ∂ X k ∂uk  k   = a (t, x) + a (t, x)u k i   ∂xi j=1 ij ∂xj   ∂t i=1 N X ∂uk  bki (t, x) − κk (a · t)uk , +    ∂xi  i=1   B k (t)u = 0, k a

t > 0, x ∈ D

(6.2.7)

t > 0, x ∈ ∂D.

Choose %0 > 0 such that |κk (a)| + KkCa (·, ·)kL∞ (R×D) ≤ %0

for all

a ∈ Y0 .

Observe that, by Proposition 2.1.3, K Z X d vk (t)κk (a · t)φk (a · t) dx + hC∗a (t)v(t), φ (a · t)i hv(t), φ (a · t)i = − dt D k=1

≤ %0 hv(t), φ (a · t)i where C∗a (t, x) = (clk (t, x)) (recall that Ca (t, x) = (ckl (t, x)). It then follows that hv(t), φ (a · t)i ≤ hv(s), φ (a · s)ie%0 |t−s| . Therefore there is M > 0 such that hv(t), φ ∗ i ≤ M hv(s), φ ∗ ie%0 |t−s| . (1) then follows. (2) Suppose that the statement is not true. Then there are tn ≤ τ − 1 (n = 1, 2, . . . ) such that ηn := sup { η0 ≥ 0 :

v(tn ) ≥ η0 φ ∗ } → 0 kv(tn )kφ ∗

as n → ∞. By v(t) ∈ X++ , we must have tn → −∞. Let vn (t) :=

v(tn + t) kv(tn )kφ ∗

for t < τ − tn .

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Then vn (t) is solution of (6.1.29)+(6.1.30) with a replaced by a · tn and v(tn ) vn (0) = kv(t (hence kv(n) (0)kφ ∗ = 1). n )kφ ∗ Without loss of generality, we may assume that a · tn → a∗ ∈ Y0 . By (1), Lemma 6.2.9, and the embedding X ,→ L2 (D), kvn (2)k is bounded in n. Then by Theorem 6.1.3, we may assume that vn (3/2) → v∗ in L2 (D). By Theorem 6.1.11, we have that vn (t) → v∗ (t) in X for t ∈ [−1, 1], where v∗ (t) is solution of (6.1.29)+(6.1.30) with a replaced by a∗ and v∗ (3/2) = v∗ . We therefore have kv∗ (0)kφ ∗ = 1 and v∗ (0) ≥ η0∗φ ∗ for some η0∗ > 0. This implies that η∗ v(tn ) ≥ 0 φ ∗ for n sufficiently large, kv(tn )kφ ∗ 2 which contradicts ηn → 0 as n → ∞. (2) is thus proved. A solution u : J → L2 (D) of (6.0.1)+(6.0.2) is nontrivial if, for each t ∈ J, there holds u(t) 6= 0. LEMMA 6.2.11 (1) If u is a nontrivial solution of (6.0.1)+(6.0.2) on J and v is a positive solution of (6.1.29)+(6.1.30) on the same interval J such that hu(t), v(t)i = 0 for some (hence every) t ∈ J, then ξ(t) := h|u(t)|, v(t)i is a decreasing function on J. (2) If u is a nontrivial solution of (6.0.1)+(6.0.2) on (−∞, t0 ], v is a positive solution of (6.1.29)+(6.1.30) on (−∞, t0 ], and hu(t0 ), v(t0 )i = 0, then there is some 0 < %0 < 1 such that ξ(t + 1) ≤ %0 ξ(t) PROOF

for t ≤ t0 − 1.

(1) Note that u(t) = u+ (t) − u− (t) and u(t) = Ua (t, s)u+ (s) − Ua (t, s)u− (s)

for t > s. We then have u+ (t) ≤ Ua (t, s)u+ (s),

u− (t) ≤ Ua (t, s)u− (s)

for t > s. It follows that |u(t)| ≤ Ua (t, s)|u(s)|. Since v(t) > 0 for t ∈ J and u is a nontrivial solution, hu(t), v(t)i = 0 implies that u(t) changes sign. We therefore must have |u(t)| < Ua (t, s)|u(s)|. Consequently, ξ(t) < hUa (t, s)|u(s)|, v(t)i = h|u(s)|, v(s)i = ξ(s). Therefore ξ is a decreasing function on J.

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231

(2) Suppose that the statement is false. Then there are tn → −∞ such that 1≥

ξ(tn + t) ξ(tn + 1) ≥ →1 ξ(tn ) ξ(tn )

as n → ∞. Let

(t ∈ [0, 1])

un (t) :=

u(tn + t) ku(tn )kφ ∗

(t < t0 − tn )

vn (t) :=

v(tn + t) kv(tn )kφ ∗

(t < t0 − tn ).

and

(6.2.8)

Then un is a solution of (6.0.1)+(6.0.2) on (−∞, t0 − tn ) with a replaced u(tn ) by a · tn and un (0) = ku(t , and vn is a solution of (6.1.29)+(6.1.30) on n )kφ ∗ v(tn ) (−∞, t0 − tn ) with a replaced by a · tn and vn (0) = kv(t . Moreover, n )kφ ∗ n n n n ku (0)kφ ∗ = kv (0)kφ ∗ = 1, hu (t), v (t)i = 0 for t < t0 − tn , and vn (t) ≥ 0 for t < t0 − tn . Without loss of generality we may assume that a · tn → a∗ ∈ Y. By Lemma 6.2.10(1), kvn (2)kφ ∗ is bounded in n, and then by Lemma 6.2.9, kvn (3/2)k is bounded in n. Hence, by Theorems 6.1.3 and 6.1.11, we may assume that vn (t) → v∗ (t) in X for t ∈ [0, 1], where v∗ is a solution of (6.1.29)+(6.1.30) on [0, 1] with a replaced by a∗ . By Lemma 6.2.9, kun (δ)k is bounded in n, for any 0 < δ < 1. Then, by Theorems 6.1.3 and 6.1.11 again, we may assume that un (t) → u∗ (t) in X for 0 < t ≤ 1, where u∗ is a solution of (6.0.1)+(6.0.2) on (0, 1] with a replaced by a∗ , and hu∗ (t), v∗ (t)i = 0 for 0 < t ≤ 1. We claim that u∗ and v∗ are nontrivial solutions. In fact, by (6.2.8), for any s ∈ (0, 1) and sufficiently large n, ξ(tn + s) ≥ ξ(tn )/2, and hence

h|u∗ (s)|, v∗ (s)i = lim h|u(tn + s)|/ku(tn )kφ ∗ , v(tn + s)/kv(tn )kφ ∗ i n→∞

1 ≥ lim sup h|u(tn )|/ku(tn )kφ ∗ , v(tn )/kv(tn )kφ ∗ i. n→∞ 2 By Lemma 6.2.10(2), there holds h|u(tn )|, v(tn )/kv(tn )kφ ∗ i ≥ η0 h|u(tn )|, φ ∗ i = η0 ku(tn )kφ ∗ for some η0 > 0. It then follows that h|u∗ (s)|, v∗ (s)i ≥

η0 , 2

hence u∗ and v∗ are nontrivial solutions. Now, for any s, τ ∈ (0, 1) with s < τ , by Part (1) h|u∗ (s)|, v∗ (s)i > 1. h|u∗ (τ )|, v∗ (τ )i

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By (6.2.8), we have h|u∗ (s)|, v∗ (s)i ξ(tn + s) = lim = 1, h|u∗ (τ )|, v∗ (τ )i n→∞ ξ(tn + τ ) which is a contradiction. This proves (2). PROOF (Proof of Theorem 6.2.15) Suppose that u1 and u2 are two entire positive solutions of (6.0.1)+(6.0.2). Choose a nonzero v0 ∈ X+ , and let v(t) := U∗a (t, t0 )v0 . Clearly, there is a constant q such that hu1 (t0 ) − qu2 (t0 ), v0 i = 0. Let u(t) := u1 (t) − qu2 (t). We then have u(t) = 0 for all t ∈ R or u(t) = 6 0 for all t ≤ t1 and some t1 ≤ t0 . Then by Lemma 6.2.11(2), there are C ∗ > 0 and γ ∗ > 0 such that h|u(t)|, v(t)i ≤ C ∗ e−γ

∗

(t−s)

h|u(s)|, v(s)i t1 > t > s.

On the other hand, we have h|u(t)|, v(t)i = h|u1 (t) − qu2 (t)|, v(t)i ≤ hu1 (t), v(t)i + |q|hu2 (t), v(t)i = const

for t ≤ t1 .

It then follows that we must have u(t) = 0 and then u1 (t) = qu2 (t), for all t ∈ R. Therefore an entire positive solution of (6.0.1)+(6.0.2) is unique up to a constant positive multiple.

6.3

Principal Spectrum and Principal Lyapunov Exponents in Nonautonomous and Random Cases

In this section, we study the principal spectrum and principal Lyapunov exponents of (6.0.3) and (6.0.4) and extend the theories developed in Chapters 4 and 5 for scalar parabolic equations to cooperative systems of parabolic equations.

6. Cooperative Systems of Parabolic Equations

6.3.1

233

The Random Case

In this subsection, we consider (6.0.4), i.e.,  N N X  ∂ X k ∂uk  ∂uk k   = + a a (θ ω, x) (θ ω, x)u t t k i   ∂t ∂xi j=1 ij ∂xj   i=1        B

+

N X

K

bki (θt ω, x)

i=1

∂uk X k + cl (θt ω, x)ul , ∂xi

t > 0, x ∈ D

l=1

t > 0, x ∈ ∂D, (6.3.1) where k = 1, 2, . . . , K, ω ∈ Ω, ((Ω, F, P), {θt }t∈R ) is an ergodic metric dynamical system, and for each ω ∈ Ω, aω (t, x) := (akij (θt ω, x), aki (θt ω, x), bki (θt ω, x), ckl (θt ω, x), dk0 (θt ω, x)) ∈ Y and Bak,ω (t) is of the same form as in (6.0.2) with ak being replaced by ak,ω (t, x) = (akij (θt ω, x), aki (θt ω, x), bki (θt ω, x), 0, dk0 (θt ω, x)). For a given ω ∈ Ω, let ak,ω (t)uk

= 0,

Ea (ω)(t, x) := aω (t, x) = a(θt ω, x).

(6.3.2)

Throughout this subsection, we assume ˜ (A6-7) { Ea (ω) : ω ∈ Ω } ⊂ Y and Ea : Ω → Y(a) := cl { Ea (ω) : ω ∈ Ω } are measurable, where the closure is taken in the weak-* topology. Moreover, ˜ (A6-1)–(A6-4) are satisfied with Y replaced by Y(a). We say a is Y-admissible if a satisfies (A6-7). ˜ be the image of the measure P under Ea : ∀A ∈ B(Y(a)), ˜ ˜ Let P P(A) := ˜ {σt }t∈R ) is a topological dynamical system with an P(Ea−1 (A)). Then (Y(a), ˜ Put ergodic invariant measure P. ˜ ˜ 0 (a) := supp P. Y ˜ 0 (a) is a closed (hence compact) and {σt }–invariant subset of Y(a), ˜ Then Y ˜ ˜ ˜ with P(Y0 (a)) = 1. Moreover, Y0 (a) is connected. (See Chapter 4 for the reasonings.) Similarly to Lemma 4.1.2, we have LEMMA 6.3.1 There exists Ω0 ⊂ Ω with P(Ω0 ) = 1 such that ˜ 0 (a) = cl { Ea (θt ω) : t ∈ R } Y for any ω ∈ Ω0 , where the closure is taken in the weak-* topology. Denote by Π(a) = {Πt (a)}t≥0 the topological skew-product semiflow generated by (6.3.1), ˜) Πt (a)(u0 , ω) := (Ua˜ (t, 0)u0 , σt a

˜ 0 (a), u0 ∈ L2 (D), t > 0. ˜∈Y for a

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Spectral Theory for Parabolic Equations

Instead of UEa (ω) (t, s) we will write Uω (t, s). DEFINITION 6.3.1 (Principal spectrum and principal Lyapunov exponent) (1) The principal spectrum of (6.3.1), denoted by Σ(a) = [λmin (a), λmax (a)], ˜ 0 (a). is defined to be the principal spectrum of Π over Y (2) The principal Lyapunov exponent of (6.3.1), denoted by λ(a), is defined ˜ 0 (a) for the ergodic to be the principal Lyapunov exponent of Π over Y ˜ invariant measure P. THEOREM 6.3.1 Let Ω0 be as in Lemma 6.3.1. (1) There is Ω1 ⊂ Ω0 with P(Ω1 ) = 1 such that for any ω ∈ Ω1 one has ln kUω (t, 0)k = λ(a). t→∞ t lim

(6.3.3)

∞ ∞ (2) For any sequence (ω (n) )∞ n=1 ⊂ Ω0 and any real sequences (sn )n=1 , (tn )n=1 such that tn − sn → ∞ one has

λmin (a) ≤ lim inf n→∞

ln kUω(n) (tn , sn )k tn − sn ≤ lim sup n→∞

ln kUω(n) (tn , sn )k ≤ λmax (a). tn − sn

∞ (3A) There exist a sequence (ω (n) )∞ n=1 ⊂ Ω0 and a sequence (tn,1 )n=1 ⊂ (0, ∞) such that tn,1 → ∞ as n → ∞, and

lim

n→∞

ln kUωn (tn,1 , 0)k = λmin (a). tn,1

∞ (3B) There exist a sequence (ω (n) )∞ n=1 ⊂ Ω0 and a sequence (tn,2 )n=1 ⊂ (0, ∞) such that tn,2 → ∞ as n → ∞, and

lim

n→∞

PROOF that

ln kUω(n) (tn,2 , 0)k = λmax (a). tn,2

˜ Y ˜1 ⊂ Y ˜ 0 with P( ˜ 1 ) = 1 such (1) By Theorem 6.2.3, there is Y lim

t→∞

ln kUω (t, 0)k = λ(a) t

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235

˜ 1 . (1) then follows with Ω1 = Ω0 ∩ E −1 (Y ˜ 1 ). for any ω ∈ Ω with Ea (ω) ∈ Y a (2) follows from Theorem 6.2.2(1). (3) follows from Theorem 6.2.2(2A) and (2B). ˜ ⊂ Ω with P(Ω) ˜ = Let a(1) , a(2) be Y-admissible. Assume there is Ω k,ω,(m) ω,(m) (m) ˜ a 1 such that for each ω ∈ Ω, (t, x) = a (θt ω, x) = (aij (t, x), k,ω,(m)

ai

k,ω,(m)

(t, x), bi

k,ω,(1)

• aij

k,ω,(2)

(t, x) = aij

k,ω,(2) bi (t, x), k,ω,(1)

• cl

k,ω,(m)

(t, x), d0 k,ω,(1)

(t, x), ai

(t, x)) (m = 1, 2) satisfies k,ω,(2)

(t, x) = ai

k,ω,(1)

(t, x), bi

(t, x) =

for a.e. (t, x) ∈ R × D, k,ω,(2)

(t, x) ≤ cl

k,ω,(1)

• d0

k,ω,(m)

(t, x), cl

(t, x)

k,ω,(2)

(t, x) ≥ d0

(t, x)

for a.e. (t, x) ∈ R × D, for a.e. (t, x) ∈ R × ∂D.

THEOREM 6.3.2 (Monotonicity with respect to zero order terms)

(1) λ(a(1) ) ≤ λ(a(2) ). (2) λmin (a(1) ) ≤ λmin (a(2) ) and λmax (a(1) ) ≤ λmax (a(2) ). PROOF

It follows from Theorem 6.1.5 and Theorem 6.3.1.

For a given Y-admissible a, denote by λD (a), λR (a), and λN (a) the Lyapunov exponents of (6.3.1) with Dirichlet, Robin, and Neumann boundary D R R conditions, respectively. Denote by [λD min (a), λmax (a)], [λmin (a), λmax (a)], and N N [λmin (a), λmax (a)] the principal spectrum intervals of (6.3.1) with Dirichlet, Robin, and Neumann boundary conditions, respectively. THEOREM 6.3.3 (Monotonicity with respect to boundary conditions) (1) λD (a) ≤ λR (a) ≤ λN (a). R N D R N (2) λD min (a) ≤ λmin (a) ≤ λmin (a) and λmax (a) ≤ λmax (a) ≤ λmax (a).

PROOF

It also follows from Theorem 6.1.5 and Theorem 6.3.1.

In the rest of this subsection, we assume that (A6-1)–(A6-6) are satisfied. ˜ 0 (a). Then Π admits an exponential separation on Y

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Spectral Theory for Parabolic Equations

THEOREM 6.3.4 Let Ω0 be as in Lemma 6.3.1. (1) There is Ω1 ⊂ Ω0 with P(Ω1 ) = 1 such that for any ω ∈ Ω1 and any u0 ∈ L+ 2 (D) \ {0} one has ln kUω (t, 0)u0 k = λ(a). t→∞ t lim

(6.3.4)

∞ ∞ (2) For any sequence (ω (n) )∞ n=1 ⊂ Ω0 and any real sequences (sn )n=1 , (tn )n=1 such that tn − sn → ∞ one has

ln kUω(n) (tn , sn )w(E(ω (n) ) · sn )k n→∞ tn − sn ln kUω(n) (tn , sn )u0 k = lim inf n→∞ tn − sn ln kUω(n) (tn , sn )w(E(ω (n) ) · sn )k ≤ lim sup tn − sn n→∞ ln kUω(n) (tn , sn )u0 k = lim sup ≤ λmax (a) tn − sn n→∞

λmin (a) ≤ lim inf

for each u0 ∈ L2 (D)+ \ {0}. 0 ∞ 0 0 (3) For each ω ∈ Ω0 , there are sequences (s0n )∞ n=1 , (tn )n=1 ⊂ R, tn −sn → ∞ as n → ∞, such that

ln kUω (t0n , s0n )u0 k ln kUω (t0n , s0n )w(E(ω) · s0n )k = lim n→∞ n→∞ t0n − s0n t0n − s0n

λmin (a) = lim

for each u0 ∈ L2 (D)+ \ {0}. 00 ∞ 00 00 (4) For each ω ∈ Ω0 , there are sequences (s00n )∞ n=1 , (tn )n=1 ⊂ R, tn −sn → ∞ as n → ∞, such that

ln kUω (t00n , s00n )w(E(ω) · s00n )k ln kUω (t00n , s00n )u0 k = lim 00 00 n→∞ n→∞ tn − sn t00n − s00n

λmax (a) = lim

for each u0 ∈ L2 (D)+ \ {0}. PROOF (1) It follows from Lemma 6.2.6 and Theorem 6.3.1(1). (2) It follows from Lemma 6.2.6 and Theorem 6.3.1(2). (3) It follows from Lemma 6.2.6 and Theorem 6.2.12. In the theorem below a Y-admissible a(0) is fixed. k·k∞ stands for the norm 2 in L∞ (R × D, RK(N +2N +K) ) × L∞ (R × ∂D, RK ).

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237

THEOREM 6.3.5 (Continuous dependence on coefficients) (1) For each > 0 there is δ > 0 such that for any Y-admissible a, if kEa (ω) − Ea(0) (ω)k∞ < δ for P-a.e. ω ∈ Ω then |λ(a) − λ(a(0) )| < . (2) For each > 0 there is δ > 0 such that for any Y-admissible a, if kEa (ω) − Ea(0) (ω)k∞ < δ for P-a.e. ω ∈ Ω then |λmin (a) − λmin (a(0) )| < PROOF 4.4.2.

and

|λmax (a) − λmax (a(0) )| < .

It follows along the lines of the proofs of Theorems 4.4.1 and

The above theorems extend the theories developed in Chapter 4 for scalar random parabolic equations to cooperative systems of random parabolic equations.

6.3.2

The Nonautonomous Case

In this subsection, we consider (6.0.3), i.e.,  N N X  ∂ X k ∂uk ∂uk  k   = a (t, x) + a (t, x)u k i   ∂xi j=1 ij ∂xj   ∂t i=1 N K X ∂uk X k k  + b (t, x) + cl (t, x)ul ,  i   ∂xi  i=1 l=1   B k (t)u = 0 k a

t > 0, x ∈ D,

(6.3.5)

t > 0, x ∈ ∂D,

where Bak (t) is of the same form as in (6.0.2), k = 1, 2, . . . , K and a = (akij , aki , bki , ckl , dk0 ) is a given element in Y. Throughout this subsection, we assume ˜ (A6-8) Y(a) := cl { a · t : t ∈ R }, where the closure is taken in the weak-* ˜ topology, satisfies (A6-1)–(A6-4) with Y replaced by Y(a). We say a is Y-admissible if it satisfies (A6-8). Let Π = {Πt (a)}t≥0 be the topological linear skew-product semiflow generated by (6.3.5), ˜) := (Ua˜ (t, 0)u0 , σt a ˜) Πt (a)(u0 , a ˜ ˜ ∈ Y(a). where u0 ∈ L2 (D) and a DEFINITION 6.3.2 (Principal spectrum) The principal spectrum of (6.3.5), denoted by Σ(a) = [λmin (a), λmax (a)], is defined to be the principal ˜ spectrum of Π over Y(a).

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Spectral Theory for Parabolic Equations

THEOREM 6.3.6 ∞ ˜ (1) For any sequence (˜ a(n) )∞ n=1 ⊂ Y(a), and any real sequences (sn )n=1 , ∞ (tn )n=1 such that tn − sn → ∞ one has

λmin (a) ≤ lim inf n→∞

ln kUa˜(n) (tn , sn )k tn − sn ≤ lim sup n→∞

ln kUa˜(n) (tn , sn )k ≤ λmax (a). tn − sn

∞ ˜ (2A) There exist a sequence (˜ a(n) )∞ n=1 ⊂ Y(a) and a sequence (tn,1 )n=1 ⊂ (0, ∞) such that tn,1 → ∞ as n → ∞, and

lim

n→∞

ln kUa˜n (tn,1 , 0)k = λmin (a). tn,1

∞ ˜ (2B) There exist a sequence (˜ a(n) )∞ n=1 ⊂ Y(a) and a sequence (tn,2 )n=1 ⊂ (0, ∞) such that tn,2 → ∞ as n → ∞, and

ln kUa˜n (tn,2 , 0)k = λmax (a). n→∞ tn,2 lim

PROOF (1) follows from Theorem 6.2.2 (1). (2) follows from Theorem 6.2.2 (2A) and (2B). Let Y-admissible a(1) , a(2) satisfy the following: k,(1)

k,(2)

k,(1)

k,(2)

k,(1)

• aij (t, x) = aij (t, x), ai for a.e. (t, x) ∈ R × D, • cl

(t, x) ≤ cl

k,(1)

• d0

k,(1)

(t, x), bi

k,(2)

(t, x) = bi

(t, x),

(t, x) for a.e. (t, x) ∈ R × D,

k,(2)

(t, x) ≥ d0

k,(2)

(t, x) = ai

(t, x) for a.e. (t, x) ∈ R × ∂D.

THEOREM 6.3.7 (Monotonicity with respect to zero order terms) λmin (a(1) ) ≤ λmin (a(2) ) and λmax (a(1) ) ≤ λmax (a(2) ). PROOF

It follows from Theorem 6.1.5 and Theorem 6.3.6.

D R R For a given Y-admissible a, denote by [λD min (a), λmax (a)], [λmin (a), λmax (a)], N N and [λmin (a), λmax (a)] the principal spectrum intervals of (6.3.5) with Dirichlet, Robin, and Neumann boundary conditions, respectively.

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239

THEOREM 6.3.8 (Monotonicity with respect to boundary conditions) R N D R N λD min (a) ≤ λmin (a) ≤ λmin (a) and λmax (a) ≤ λmax (a) ≤ λmax (a). PROOF

It follows from Theorem 6.1.5 and Theorem 6.3.6.

In the rest of this subsection, we assume that (A6-1)–(A6-6) are satisfied. ˜ Hence Π admits an exponential separation over Y(a). THEOREM 6.3.9 ∞ (1) For any u0 ∈ L2 (D)+ \ {0} and any real sequences (sn )∞ n=1 , (tn )n=1 such that tn − sn → ∞ one has

ln kUa (tn , sn )w(a · sn )k tn − sn ln kUa (tn , sn )u0 k = lim inf n→∞ tn − sn ln kUa (tn , sn )w(a · sn )k ≤ lim sup tn − sn n→∞ ln kUa (tn , sn )u0 k = lim sup ≤ λmax (a). tn − sn n→∞

λmin (a) ≤ lim inf n→∞

0 0 0 ∞ (2) There are sequences (s0n )∞ n=1 , (tn )n=1 ⊂ R, tn − sn → ∞ as n → ∞, such that

ln kUa (t0n , s0n )w(a · s0n )k ln kUa (t0n , s0n )u0 k = lim 0 0 n→∞ n→∞ tn − sn t0n − s0n

λmin (a) = lim

for each u0 ∈ L2 (D)+ \ {0}. 00 ∞ 00 00 (3) There are sequences (s00n )∞ n=1 , (tn )n=1 ⊂ R, tn − sn → ∞ as n → ∞, such that

ln kUa (t00n , s00n )w(a · s00n )k ln kUa (t00n , s00n )u0 k = lim n→∞ n→∞ t00n − s00n t00n − s00n

λmax (a) = lim

for each u0 ∈ L2 (D)+ \ {0}. PROOF (1) It follows from Lemma 6.2.6 and Theorem 6.3.6. (2) It follows from Lemma 6.2.6 and Theorem 6.2.12. In the theorem below a Y-admissible a(0) is fixed. k·k∞ stands for the norm 2 in L∞ (R × D, RK(N +2N +K) ) × L∞ (R × ∂D, RK ).

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Spectral Theory for Parabolic Equations

THEOREM 6.3.10 (Continuous dependence on coefficients) For each > 0 there is δ > 0 such that for any Y-admissible a, if ka − a(0) k∞ < δ then |λmin (a) − λmin (a(0) )| < PROOF 4.4.3.

and

|λmax (a) − λmax (a(0) )| < .

It follows by arguments similar to those in the proof of Theorem

The above theorems extend the theories developed in Chapter 4 for scalar nonautonomous parabolic equations to cooperative systems of nonautonomous parabolic equations.

6.3.3

Influence of Time and Space Variations

In this subsection we study the influence of time and space variations of the zero-order terms on principal spectrum and principal Lyapunov exponent. We assume that akij , aki , bki , ckl (l 6= k), and dk0 are independent of t, i.e., we consider  N N X  ∂uk ∂ X k ∂uk  k  = a (x) + a (x)u  k i   ∂t ∂xi j=1 ij ∂xj  i=1  N X (6.3.6) ∂uk X k  bki (x) + cl (x)ul + ckk (t, x)uk , t > 0, x ∈ D, +   ∂xi   i=1 l6=k    t > 0, x ∈ ∂D, Bak uk = 0, and  N N X  ∂uk ∂uk ∂ X k  k  a (x) = + a (x)u  k i   ∂t ∂xi j=1 ij ∂xj  i=1  N X ∂uk X k k  + b (x) + cl (x)ul + ckk (θt ω, x)uk ,  i  ∂x  i  i=1 l6=k    Bak uk = 0,

t > 0, x ∈ D,

t > 0, x ∈ ∂D, (6.3.7) where Bak ≡ Bak (t) and Bak (t) is as in (2.0.3) with a = ak = (akij (·), aki (·), bki (·), 0, dk0 (·)), k = 1, 2, . . . , K. Throughout this subsection, we make the following assumption. ˜ (A6-9) Y(a) induced by (6.3.6) (or by (6.3.7)) satisfies (A6-1)–(A6-6). ˆ(x) = (ˆ In the case of (6.3.6), a function c ckk (x))K k=1 is called a time averaged function of c(t, x) = (ckk (t, x))K if k=1 Z tn 1 k cˆk (x) = lim ckk (t, x) dt tn −sn →∞ tn − sn s n

6. Cooperative Systems of Parabolic Equations

241

¯ for some sequence tn − sn → ∞, uniformly for x ∈ D. k K ˆ(x) = (ˆ In the case of (6.3.7), c ck (x))k=1 is called the time averaged function of c(θt ω, x) = (ckk (θt ω, x))K k=1 if Z cˆkk (x) = ckk (ω, x) dP(ω) Ω

¯ for x ∈ D. We call the following cooperative system of parabolic equations,  N N X  ∂uk ∂ X k ∂uk  k  = a (x) + ai (x)uk    ∂t ∂xi j=1 ij ∂xj  i=1  N X ∂uk X k k  + cl (x)ul + cˆkk (x)uk , t > 0, x ∈ D, b (x) +  i  ∂x  i  i=1 l6=k    t > 0, x ∈ ∂D Bak uk = 0,

(6.3.8)

a time averaged equation of (6.3.6) (the time averaged equation of (6.3.7)) ˆ = (ˆ if c ckk (x)) is an averaged function of c(t, x) = (ckk (t, x)) (the averaged function of c(ω, x) = (ckk (ω, x))). Let [λmin (a), λmax (a)] be the principal spectrum of (6.3.6), and let λ(a) be ˆ) be the the principal Lyapunov exponent of (6.3.7). Further, let λprinc (a, c principal eigenvalue of (6.3.8). Then we have THEOREM 6.3.11 (Influence of temporal variation in the nonautonomous case) Consider (6.3.6). ˆ(x) of c(t, x) such that λmin (a) ≥ (1) There is a time averaged function c ˆ). λprinc (a, c ˆ) for any time averaged function c ˆ(x) of c(t, x). (2) λmax (a) ≥ λprinc (a, c THEOREM 6.3.12 (Influence of time variations in the random case) ˆ). Consider (6.3.7). Then λ(a) ≥ λprinc (a, c To prove the above theorems, we first show a few lemmas. Let Au be defined by (Au)k :=

N N X ∂ X k ∂uk aij (x) + aki (x)uk ∂xi j=1 ∂xj i=1

+

N X i=1

bki (x)

∂uk X k + cl (x)ul ∂xi l6=k

(6.3.9)

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together with boundary conditions Bak uk = 0, 1 ≤ k ≤ K. We denote ( ˚ D, ¯ RK ) C( (Dirichlet) e X := K ¯ C(D, R ) (Neumann or Robin). LEMMA 6.3.2 A together with boundary conditions Bak uk = 0, 1 ≤ k ≤ K, generates an e Moreover, eAt u0 ≥ 0 for any u0 ≥ 0, analytic semigroup {eAt }t≥0 on X. 0 u ∈ D(A), and any t > 0. PROOF For the first statement, see [87]. The second statement follows along the lines of Theorem 6.1.5(1). Denote by D(A) the domain of A. LEMMA 6.3.3 Assume u0 ∈ D(A) and u0 ≥ 0. If u0k (x∗ ) = 0 for some x∗ ∈ D and 1 ≤ k ≤ K, then Au0 k (x∗ ) ≥ 0. PROOF

Since u0 ∈ D(A), we have (eAt u0 )(x) − u0 (x) = (Au0 )(x) t→0+ t lim

¯ But eAt u0 ≥ 0. It then follows that (Au0 )k (x∗ ) ≥ 0. for any x ∈ D. For given S < T , let η(t; S) := kUa (t, S)w(a·S)k, v(t, x; S) := (Ua (t, S)w(a· ˆ S))(x)/η(t; S), and w(x; S, T ) := (w ˆ1 (x; S, T ), w ˆ2 (x; S, T ), . . . , w ˆK (x; S, T )), where ! Z T 1 w ˆl (x; S, T ) := exp ln vl (t, x; S) dt (6.3.10) T −S S for x ∈ D and w ˆl (x; S, T ) = 0 for x ∈ ∂D in the Dirichlet case, and ! Z T 1 w ˆl (x; S, T ) := exp ln vl (t, x; S) dt (6.3.11) T −S S ¯ in the Neumann and Robin cases (l = 1, 2, . . . , K). for x ∈ D For S < T , v(T, ·; S) = w(a · T ) ∈ D(A). Also, it follows from Theoˆ S, T ) ∈ D(A), for S < T . rem 6.1.6 that w(·; We have

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243

LEMMA 6.3.4 Z

1 T −S

T

S

ˆ S, T ))(x) Al (v(t, ·; S))(x) Al (w(·; dt ≥ vl (t, x; S) w ˆl (x; S, T )

for all

x ∈ D, (6.3.12)

where Al u = (Au)l , l = 1, 2, . . . , K. PROOF It follows from the arguments of [100, Proposition 2.2]. For convenience, we provide a proof here. First of all, recall the Jensen inequality ! Z T Z T 1 1 f (t) dt ≥ exp ln (f (t)) dt (6.3.13) T −S S T −S S for any positive continuous function defined on [S, T ], with the equality if and only if f is a constant function. This implies that 1 T −S

Z

T

S

ˆ v(t, x; S) w(x; S, T ) dt ≥ vl (t, x∗ ; S) w ˆl (x∗ ; S, T )

for any x, x∗ ∈ D and 1 ≤ l ≤ K, where the inequality ≥ is to be understood coordinatewise. Let Z T ˆ v(t, x; S) w(x; S, T ) 1 l ∗ dt − . v (x, x ) := T − S S vl (t, x∗ ; S) w ˆl (x∗ ; S, T ) Then vl (·, x∗ ) ≥ 0 and vll (x∗ , x∗ ) = 0. Observe that vl (·, x∗ ) ∈ D(A). Then by Lemma 6.3.3, (6.3.12) holds at x∗ . Since x∗ ∈ D is arbitrary, we have that (6.3.12) holds for any x ∈ D. PROOF (Proof of Theorem 6.3.11) ˆ (1) Let η(t; S), v(t, x; S), and w(x; S, T ) be as above. Then we have ∂vk ∂η η + vk = η(Av)k + ckk (t, x)ηvk , ∂t ∂t

x ∈ D,

(6.3.14)

and Bak vk = 0,

x ∈ ∂D.

(6.3.15)

By (6.3.14), we have 1 T −S

Z

T

∂vk 1 1 dt + ln η(T ; S) ∂t v T − S k S Z T Z T 1 (Av)k 1 = dt + ck (t, x) dt, T −S S vk T −S S k

x ∈ D.

(6.3.16)

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By Lemma 6.3.4, Z T ∂vk 1 1 1 dt + ln η(T ; S) T − S S ∂t vk T −S Z T ˆ S, T )))k 1 A(w(·; ck (t, x) dt, + ≥ ˆ S, T )k w(·; T −S S k

x ∈ D.

(6.3.17)

Let Tn − Sn → ∞ be such that lim

n→∞

1 ln η(Tn ; Sn ) = λmin (a). Tn − Sn

Without loss of generality, assume that 1 n→∞ Tn − Sn

Z

Tn

cˆk (x) = lim

ckk (t, x) dt

Sn

exists for all x ∈ D and k = 1, 2, . . . , K. We may also assume that there is w∗ = w∗ (x) such that ˆ w(x; Sn , Tn ) → w∗ (x) ¯ uniformly for x ∈ D,

ˆ ∂w ∂w∗ (x; Sn , Tn ) → (x) ∂xi ∂xi

uniformly for x in compact subsets D0 ⊂ D (this limit is also uniform for x ¯ in the Neumann and Robin cases), and in D ˆ ∂ 2 w∗ ∂2w (x; Sn , Tn ) → (x) ∂xi ∂xj ∂xi ∂xj uniformly for x in compact subsets D0 ⊂ D (this is possible by Theorem 6.1.6). Proceeding as in the proof of Theorem 5.2.1 we see that Z Tn 1 ∂vk 1 lim (t, x; Sn ) dt = 0 n→∞ Tn − Sn S ∂t vk (t, x; Sn ) n for all x ∈ D. ˆ). In fact, by the above arguments, We claim that λmin (a) ≥ λprinc (a, c λmin (a) ≥ (Ak w∗ )(x)/wk∗ (x) + cˆk (x),

x ∈ D,

(6.3.18)

and Bak wk∗ (x) = 0

for x ∈ ∂D,

(6.3.19)

1 ≤ k ≤ K. This implies that w(t, x) = w∗ (x) is a supersolution of ( wt = Aw + (ˆ c − λmin (a))w, t > 0, x ∈ D, (6.3.20) Bak wk = 0, t > 0, x ∈ ∂D, 1 ≤ k ≤ K.

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245

By the fact that w∗ ≥ 0, we have that the principal eigenvalue of (6.3.20) ˆ) − λmin (a)) is less than or equal to zero. Hence (i.e., λprinc (a, c ˆ). λmin (a) ≥ λprinc (a, c ˆ(x) of c(t, x), there is Tn − Sn → ∞ such (2) For any averaged function c that Z Tn 1 ckk (t, x) dt cˆk (x) = lim n→∞ Tn − Sn S n ¯ Let η(t; S), v(t, x; S), and w(x; ˆ for k = 1, 2, . . . , K and x ∈ D. S, T ) be as 1 in (1). Without loss of generality we may assume that Tn −S ln η(Tn ; Sn ) n converges as n → ∞ exists. Note that λmax (a) ≥ lim

n→∞

1 ln η(Tn ; Sn ). Tn − Sn

It then follows from arguments as in (1) that ˆ) λmax (a) ≥ λprinc (a, c ˆ(x) of c(t, x). for any averaged function c PROOF (Proof of Theorem 6.3.12)

Let

η(t; ω) := kUEa (ω) (t, 0)w(Ea (ω))k. By Theorem 6.3.4(1), for P-a.e. ω ∈ Ω there holds 1 ln η(T ; ω) T →∞ T

λ(a) = lim and 1 T →∞ T

Z

ˆ(x) = lim c

T

c(θt ω, x) dt. 0

It then follows from arguments as in Theorem 6.3.11(1) that ˆ). λ(a) ≥ λprinc (a, c The theorem is thus proved.

6.4

Remarks

In this chapter, principal spectrum, principal Lyapunov, and exponential separation for cooperative systems of nonautonomous and random parabolic

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equations are investigated. The notion of mild solution is adopted for convenience. Many results on principal spectrum for single nonautonomous and random parabolic equations are extended to cooperative systems of nonautonomous and random parabolic equations. In the smooth case (both the coefficients and the domain of the systems are sufficiently smooth), it is proved that mild solutions are also weak solutions (in fact, they are classical solutions). Almost all of the principal spectrum theories for single parabolic equations established in Chapters 3, 4, and 5 are extended to cooperative systems of parabolic equations. In the nonsmooth case, it can also be proved that mild solutions are weak solutions in the Dirichlet boundary condition case and in the Neumann and Robin boundary conditions cases with sufficiently smooth domain (see the arguments in [33, Proposition 4.2]). We do not go into detail about this issue in the monograph. The existence of exponential separation and existence and uniqueness of entire positive solution for general single parabolic equations are proved in Chapter 3 provided that their positive solutions satisfy certain Harnack inequalities (see (A3-1) and (A3-2)). It is expected that these properties for general single parabolic equations can be extended to general cooperative systems of parabolic equations under proper conditions. We do not go into detail either about this issue in the monograph.

Chapter 7 Applications to Kolmogorov Systems of Parabolic Equations

Spectral theory for linear parabolic problems is a basic tool for the study of nonlinear parabolic problems. In this chapter, we discuss some applications of the principal spectral theory developed in previous chapters to uniform persistence of systems of random and nonautonomous nonlinear equations of Kolmogorov type. We first consider applications to random and nonautonomous nonlinear equations of Kolmogorov type, and then consider applications to systems of such equations. To be more precise, let D ⊂ RN be a sufficiently smooth domain and B be either the Dirichlet or Neumann boundary operator, i.e.,  (Dirichlet) Id B := (7.0.1) ∂ (Neumann). ∂νν Let ((Ω, F, P), (θt )t∈R ) be an ergodic metric dynamical system. We first study the following random and nonautonomous equations of Kolmogorov type,   ∂u = ∆u + f (θ ω, x, u)u, x ∈ D, t ∂t (7.0.2)  Bu = 0, x ∈ ∂D, ¯ × [0, ∞) 7→ R, where f : Ω × D   ∂u = ∆u + f (t, x, u)u, ∂t  Bu = 0,

x ∈ D,

(7.0.3)

x ∈ ∂D,

¯ × [0, ∞) 7→ R. In particular, we utilize the theories developed where f : R × D in the previous chapters to study the uniform persistence of (7.0.2) and (7.0.3). Among other problems, (7.0.2) and (7.0.3) are used to model population growth problem. Due to the biological reason, we are only interested in the nonnegative solutions of (7.0.2) and (7.0.3). Note that in the nonautonomous case, we are interested in the solutions with initial conditions at any t0 ∈ R. Both random and nonautonomous cases take certain temporal variations of the underline systems into account and are of great interest in practice.

247

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Spectral Theory for Parabolic Equations

As (7.0.2) and (7.0.3) can be embedded into proper families of nonlinear equations, to study their uniform persistence we start in Section 7.1 by investigating a general family of nonlinear equations of Kolmogorov type:   ∂u = ∆u + g(t, x, u)u, x ∈ D, ∂t (7.0.4)  Bu = 0, x ∈ ∂D, were g belongs to a set Z of functions satisfying certain conditions (see (A71)–(A7-3) in Section 7.1) and is considered as a parameter. We collect the existence, uniqueness, and basic properties of solutions of (7.0.4) in Subsection 7.1.1. Based on the spectral theory developed in previous chapters, the linear theory of the linearization of (7.0.4) at the trivial solution (i.e., u ≡ 0) is presented in Subsection 7.1.2. The global attractor and uniform persistence of (7.0.4) is explored in Subsection 7.1.3 in terms of the linear theory established in 7.1.2. We then in Section 7.2 introduce the definitions of uniform persistence for (7.0.2) and (7.0.3), and establish a uniform persistence theorem for each case based on the uniform persistence for general families of nonlinear parabolic equations of Kolmogorov type. As for the scalar equations case, to study uniform persistence for competitive Kolmogorov systems of random and nonautonomous parabolic equations, we start in Section 7.3 by considering a family of competitive Kolmogorov systems of parabolic equations:  ∂u 1  = ∆u1 + g1 (t, x, u1 , u2 )u1 , x ∈ D,    ∂t     ∂u2 = ∆u2 + g2 (t, x, u1 , u2 )u2 , x ∈ D, (7.0.5) ∂t     Bu = 0, x ∈ ∂D,   1   Bu2 = 0, x ∈ ∂D, where g = (g1 , g2 ) belongs to a set Z of functions satisfying certain conditions (see (A7-5)–(A7-8) in Section 7.3) and is considered as a parameter. We collect the existence, uniqueness, and basic properties of solutions (7.0.5) in Subsection 7.3.1. The linear theory of the linearization of (7.0.5) at trivial and semitrivial solutions (i.e., solutions (u1 (t), u2 (t)) satisfying u1 (t) ≡ 0 or u2 (t) ≡ 0) is investigated in Subsection 7.3.2 based on the general spectral theory developed in previous chapters. Global attractor and uniform persistence for (7.0.5) is studied in Subsection 7.3.3 in terms of the linear theory established in 7.3.2. We then consider in Section 7.4 competitive Kolmogorov systems of random and nonautonomous parabolic equations. We introduce the definition of uniform persistence and establish a uniform persistence theorem for either case.

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249

This chapter ends up with some remarks on the existing works about uniform persistence and global dynamics in Section 7.5. Throughout this chapter, we assume the following smoothness of the domain D. (A7-D) (Boundary smoothness) ∂D is an (N − 1)-dimensional manifold of class C 3+α , for some α > 0.

7.1

Semilinear Equations of Kolmogorov Type: General Theory

In this section we consider families of semilinear second order parabolic equations of Kolmogorov type   ∂u = ∆u + g(t, x, u)u, ∂t  Bu = 0,

x ∈ D,

(7.1.1)

x ∈ ∂D,

where B is a boundary operator of either the Dirichlet or Neumann type as in (7.0.1). Here g is considered a parameter. Sometimes we write (7.1.1) as (7.1.1)g . First, we present the existence, uniqueness, and basic properties of solutions of (7.1.1) in Subsection 7.1.1. We study the linearized problem at trivial solution of (7.1.1) in Subsection 7.1.2. In Subsection 7.1.3, we establish global attractor and uniform persistence theory of (7.1.1).

7.1.1

Existence, Uniqueness, and Basic Properties of Solutions

In this subsection, we present the existence, uniqueness, and basic properties of solutions of (7.1.1) in appropriate fractional power spaces of the operator ∆ (with corresponding boundary conditions) with admissible g(·, ·, ·)s. Most properties presented in this subsection can be found in literature. For convenience, we either provide proofs or references. ¯ × [0, ∞) → R and t ∈ R denote First, for a continuous function g : R × D ¯ by g · t the time-translate of g, g · t(s, x, u) := g(s + t, x, u) for s ∈ R, x ∈ D, and u ∈ [0, ∞). ¯ Let g (n) (n ∈ N) and g be continuous real functions defined on R×D×[0, ∞). (n) Recall that a sequence g converges to g in the open-compact topology if ¯ × [0, M ] and only if for any M > 0 the restrictions of g (n) to [−M, M ] × D ¯ converge uniformly to the restriction of g to [−M, M ] × D × [0, M ]. We state the following well-known result.

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LEMMA 7.1.1 If g (n) converge to g in the open-compact topology and tn converge to t then g (n) · tn converge to g · t in the open-compact topology. We shall denote by Z the set of admissible parameters of the equation (7.1.1). ¯ × [0, ∞) → A generic element of Z is a (at least) continuous function g : R × D R. Z is always considered with the open-compact topology. The standing assumptions on Z are the following: (A7-1) (1) Z is compact in the open-compact topology. (2) Z is translation invariant: If g ∈ Z then g · t ∈ Z, for each t ∈ R. For t ∈ R and g ∈ Z put ζt g := g · t. It follows from Lemma 7.1.1 that (Z, ζ) = (Z, {ζt }t∈R ) is a compact flow. The assumption below concerns the regularity of the functions g. (A7-2) (Regularity) For any g ∈ Z and any M > 0 the restrictions to R × ¯ × [0, M ] of g and its derivatives ∂t g, ∂x g, and ∂u g belong to C 1−,1−,1− (R × D ¯ × [0, M ]). Moreover, for M > 0 fixed the C 1−,1−,1− (R × D ¯ × [0, M ])-norms D of the restrictions of those functions are bounded uniformly in Z. For each g ∈ Z we denote: G(t, x, u) := g(t, x, u)u,

¯ u ∈ [0, ∞). t ∈ R, x ∈ D,

LEMMA 7.1.2 Assume (A7-1)–(A7-2). For any sequence (g (n) ) converging in Z to g all the derivatives of the functions G(n) up to order 1 converge to the respective ¯ × [0, ∞). derivatives of G, uniformly on compact subsets of R × D Denote by N the Nemytski˘ı (substitution) operator: N(t, u, g)(x) := G(t, x, u(x)),

¯ x ∈ D,

¯ → R, and g ∈ Z. where t ∈ R, u : D ¯ + × Z. It is straightWe consider N to be a mapping defined on R × C(D) + ¯ ¯ forward to see that N takes R × C(D) × Z into C(D). We proceed to the issue of the differentiability of the Nemytski˘ı operator N with respect to t and u. Denote by ∂1 the differentiation with respect to t, and denote by ∂2 the differentiation with respect to u. It is easy to see that LEMMA 7.1.3 Assume (A7-1)–(A7-2).

7. Applications to Kolmogorov Systems of Parabolic Equations

251

¯ ¯ +× • The derivative ∂1 N (∈ L(R, C(D))) is defined everywhere on R×C(D) Z, and is given by the formula (∂1 N(t, u, g)1)(x) =

∂G (t, x, u(x)) ∂t

¯ x ∈ D,

where 1 is the vector tangent at (t, u, g) to the R-axis. ¯ C(D))) ¯ • The derivative ∂2 N (∈ L(C(D), is defined everywhere on R × + ¯ C(D) × Z, and is given by the formula (∂2 N(t, u, g)v)(x) =

∂G (t, x, u(x)) · v(x) ∂u

¯ x ∈ D,

¯ is a vector tangent at (t, u, g) to the C(D)-axis. ¯ where v ∈ C(D) Regarding the differentiability with respect to u, notice that formally we ¯ + × Z in R × C(D) ¯ × Z. need to extend N to some open subset of R × C(D) ∂g Indeed, we can do that by putting g(t, x, u) := g(t, x, 0) + ∂u (t, x, 0)u for ¯ and u < 0. t ∈ R, x ∈ D In view of Lemma 7.1.3 the derivatives ∂1 N and ∂2 N can (and will) be identified with the functions ∂G/∂t and ∂G/∂u, respectively. ˜ stand for any of the mappings N, ∂1 N or In the following two lemmas N ∂2 N. LEMMA 7.1.4 Let (A7-1)–(A7-2) be satisfied. Then ˜ + s, u, g) = N(t, ˜ u, g · s) N(t

(7.1.2)

¯ + , and g ∈ Z. for any t, s ∈ R, u ∈ C(D) PROOF It follows immediately from the definition of the time-translate and from Lemma 7.1.3. LEMMA 7.1.5 Assume (A7-1) and (A7-2). ˜ u, g) ∈ C(D) ¯ + × Z 3 (t, u, g) 7→ N(t, ¯ ] is continuous. (i) [ R × C(D) ˜ satisfies the Lipschitz ¯ + , the mapping N (ii) For any bounded B ⊂ C(D) condition with respect to (t, u), uniformly in (t, u, g) ∈ R × B × Z. ˜ ¯ + , the image N(R (iii) For any bounded B ⊂ C(D) × B × Z) is bounded in ¯ C(D).

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PROOF The proofs of (i) and (ii) are standard. ˜ × B × Z) = N({0} ˜ To prove (iii) observe that, by Eq. (7.1.2), N(R × B × Z), and apply Part (ii). We collect now some basic properties of fractional power spaces. For proofs see [48]. For 1 < p < ∞, let Ap stand for the realization of the operator ∆ (with the corresponding boundary conditions) in Lp (D). The operator −Ap is sectorial. Denote by {eAp t }t≥0 the analytic semigroup generated on Lp (D) by Ap . For 1 < p < ∞ and β ≥ 0 denote by Fpβ the fractional power space of the sectorial operator −Ap . We have Fp0 = Lp (D), and Fp1 equals the domain of −Ap . Also Fp1 ⊂ Wp2 (D). LEMMA 7.1.6 The following embeddings hold: ,→ Fpβ1 for any 1 < p < ∞ and 0 ≤ β1 < β2 . (1) Fpβ2 ,− ˜ (2) Fpβ ,→ C β (D), for any 1 < p < ∞, β ≥ 0, and 0 ≤ β˜ < min{1, 2β − Np }. ˜ (3) Fpβ ,→ C 1+β (D), for any 1 0 there is M = M (p, β, T ) > 0 with the property that keAp t kLp (D),Fpβ ≤ M t−β

and

keAp t kFpβ ≤ M

for all 0 < t ≤ T . PROOF

See [48, 1.4 and 1.5].

Let ϕprinc be the unique (nonnegative) principal eigenfunction of the elliptic boundary value problem ( ∆u = 0 on D Bu = 0 on ∂D

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253

normalized so that kϕprinc kC(D) ¯ = 1. It follows from the regularity theory for ¯ ∩ Fp1 , for any 1 0 for each x ∈ D and (∂ϕprinc /∂νν )(x) < 0 for each x ∈ ∂D. In the Neumann case ϕprinc ≡ 1. N + 12 < Until the end of the present section we fix 1 N and 2p β < 1, and put X := Fpβ . (7.1.3) There holds ¯ X ,− ,→ C 1 (D). Indeed, p and β are so chosen that by Lemma 7.1.6 we have ˜

¯ Fpβ ,− ,→ Fpβ1 ,→ C 1+β (D) ,→ C 1 (D), N < β1 < β and 0 < β˜ < 2β1 − Np − 1. where 21 + 2p Recall that by X + we denote the nonnegative cone in X, X + = { u ∈ X : ¯ }. u(x) ≥ 0 for all x ∈ D We proceed now to the investigation of the interior X ++ of the nonnegative cone X + .

LEMMA 7.1.8 (1) In the case of the Dirichlet boundary conditions X ++ is nonempty, and is characterized by X ++ = { u ∈ X + : u(x) > 0 for all x ∈ D and (∂u/∂νν )(x) < 0 for all x ∈ ∂D }.

(7.1.4)

(2) In the case of the Neumann boundary conditions X ++ is nonempty, and is characterized by X ++ = { u ∈ X + : u(x) > 0

for all

¯ }. x∈D

(7.1.5)

PROOF We prove the lemma only for the Dirichlet case, the proof for the Neumann case being similar, but simpler. Fp1 consists precisely of those ¯ any elements of Wp2 (D) whose trace on ∂D is zero. Since Fp1 ,→ C 1 (D), u ∈ Fp1 is a C 1 function vanishing on ∂D. By [48, Theorem 1.4.8], the image ¯ we conclude that of the embedding Fp1 ,→ X is dense. Because X ,→ C 1 (D), 1 ˚ (D). ¯ X ,→ C ˚1 (D). ¯ It follows from Lemma 1.3.1(2) Denote by I the embedding X ,→ C ˚1 (D) ¯ ++ ), where C ˚1 (D) ¯ ++ is an that the right-hand side of (7.1.4) equals I −1 (C 1 ˚ (D). ¯ This proves the “⊃” inclusion. We have that ϕprinc ∈ open subset of C

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Fp1 ,→ X and that it belongs to the right-hand side of (7.1.4), consequently to X ++ . Finally, let u ∈ X ++ . There is > 0 such that u − ϕprinc ∈ X + , therefore u(x) ≥ ϕprinc (x) > 0 for all x ∈ D, which gives further that ∂ϕprinc ∂u ν (x) ≤ ∂ν ν (x) < 0 for all x ∈ ∂D. ∂ν Recall that for any u1 , u2 ∈ X we write u1 u2

if and only if

u2 − u1 ∈ X ++ .

The symbol is used in an analogous way. We write ∂X + for X + \ X ++ . DEFINITION 7.1.1 (Solution) For t0 ∈ R, u0 ∈ X + , and g ∈ Z by a solution of (7.1.1)g satisfying the initial condition u(t0 , ·) = u0 we mean a continuous function u : J → X, where J is a nondegenerate interval with inf J = t0 ∈ J, satisfying the following: • u(t0 ) = u0 , • u(t) ∈ Fp1 for each t ∈ J \ {t0 }, • u is differentiable, as a function into Lp (D), on J \ {t0 }, • there holds u(t) ˙ = Ap u(t) + N(t, u(t), g)

for each

t ∈ J \ {t0 }.

(See [86].) A solution u of (7.1.1)g satisfying u(t0 , ·) = u0 is nonextendible if there is no solution u∗ of (7.1.1)g defined on an interval J ∗ with sup J ∗ > sup J such that u∗ |J ≡ u. PROPOSITION 7.1.1 (Existence and uniqueness of solution) Let (A7-1)–(A7-2) be satisfied. Then for each g ∈ Z, each t0 ∈ R, and each u0 ∈ X + there exists a unique nonextendible solution of (7.1.1)g satisfying the initial condition u(t0 , ·) = u0 , defined on an interval of the form [t0 , τmax ), where τmax = τmax (t0 , u0 , g) > t0 . PROOF

See [48, Theorem 3.3.3].

We will denote the solution of (7.1.1)g satisfying the initial condition u(t0 , ·) = u0 by u(·; t0 , u0 , g). PROPOSITION 7.1.2 (Positivity) Given t0 ∈ R, u0 ∈ X + , and g ∈ Z, there holds u(t; t0 , u0 , g) ∈ X + for any t ∈ (t0 , τmax (t0 , u0 , g)).

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255

PROOF Note that for given t0 ∈ R and g ∈ Z, u(t; t0 , 0, g) = 0 for all t ≥ 0. The proposition then follows from the comparison principle for parabolic equations. In a couple of places we will make use of the fact that, in [48], a solution u(·; t0 , u0 , g) is defined initially as a mild solution. We formulate that as the following. PROPOSITION 7.1.3 (Variation of constant formula) Assume (A7-1)–(A7-2). Then for any t0 ∈ R, u0 ∈ X + , and g ∈ Z the unique nonextendible solution u(·) := u(·; t0 , u0 , g) satisfies Z t u(t) = eAp (t−t0 ) u0 + eAp (t−s) N(s, u(s), g) ds (7.1.6) t0

for t0 < t < τmax (t0 , u0 , g). PROPOSITION 7.1.4 (Regularity) Let (A7-1) and (A7-2) be satisfied. Then for each u0 ∈ X + and each g ∈ Z, u(t, x) = u(t; t0 , u0 , g)(x) is a classical solution, that is, •

∂u ∂t (t, ·)

¯ for each t ∈ (t0 , τmax (t0 , u0 , g)), ∈ C(D)

¯ for each t ∈ (t0 , τmax (t0 , u0 , g)), • u(t, ·) ∈ C 2 (D) • for any t ∈ (t0 , τmax (t0 , u0 , g)) and x ∈ D the equation (7.1.1)g is satisfied pointwise, • for any t ∈ (t0 , τmax (t0 , u0 , g)) and x ∈ ∂D the boundary condition in (7.1.1) is satisfied pointwise. PROOF

See [48, Sections 3.5 and 3.6].

We introduce now an assumption which guarantees, among others, that any solution is global , that is, defined on [0, ∞): (A7-3) There is P > 0 such that g(t, x, u) < 0 for any g ∈ Z, any t ∈ R, any ¯ and any u ≥ P . x ∈ D, PROPOSITION 7.1.5 (Global existence) Assume (A7-1) through (A7-3). Then for any t0 ∈ R, any u0 ∈ X + , and any g ∈ Z the unique nonextendible solution u(·; t0 , u0 , g) is defined on [t0 , ∞). ¯ PROOF By Lemma 7.1.5(iii), for any K > 0 and any bounded B ⊂ C(D) ¯ consequently is bounded the image N([−K, K] × B × Z) is bounded in C(D),

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in Lp (D). In view of [48, Theorem 3.3.4] it suffices to show that for any t0 ∈ R, u0 ∈ X + and g ∈ Z the set { ku(t; t0 , u0 , g)kX : t ∈ [t0 , τmax (t0 , u0 , g)) } is bounded. For u0 ∈ X + , take m := max {P, ku0 kC(D) ¯ }. The constant function m is a supersolution of (7.1.1)g with u0 ≤ m, hence (0 ≤) u(t; t0 , u0 , g)(x) ≤ m for all ¯ consequently ku(t; t0 , u0 , g)kC(D) t ∈ [t0 , τmax (t0 , u0 , g)) and all x ∈ D, ¯ ≤ m for all t ∈ [t0 , τmax (t0 , u0 , g)). By Lemma 7.1.7 there is M1 > 0 such that keAp t kX ≤ M1

and keAp t kLp (D),X ≤ M1 t−β

for all t ∈ (0, τmax (t0 , u0 , g) − t0 ). Further, M2 := sup{ kN(t, u, g)kC(D) ¯ : t ∈ [t0 , τmax (t0 , u0 , g)), ku0 kC(D) ¯ ≤ m, g ∈ Z } < ∞. Finally, put M3 to be the ¯ ,→ Lp (D). It follows from (7.1.6) that norm of the embedding C(D) Z

t

ku(t; t0 , u0 , g)kX ≤ M1 ku0 kX + M1 M2 M3

(t − s)−β ds,

t0

which is bounded for t ∈ (t0 , τmax (t0 , u0 , g)). LEMMA 7.1.9 Assume (A7-1)–(A7-3). Then for any t ≥ 0, t0 ∈ R, u0 ∈ X + , and g ∈ Z the following holds: u(t + t0 ; t0 , u0 , g) = u(t; 0, u0 , g · t0 ). PROOF

(7.1.7)

Fix t0 ∈ R, u0 ∈ X + , and g ∈ Z. By Proposition 7.1.3 we have

u(t + t0 ; t0 , u0 , g) = e

Ap t

Z

t+t0

u0 +

eAp (t+t0 −s) N(s, u(s; t0 , u0 , g), g) ds

t0

for all t > 0, which can be written as u(t + t0 ; t0 , u0 , g) = eAp t u0 +

Z

t

eAp (t−s) N(s + t0 , u(s + t0 ; t0 , u0 , g), g) ds

0

for all t > 0. Put u1 (t) := u(t + t0 ; t0 , u0 , g) for t > 0. We have thus u1 (t) = eAp t u0 +

Z

t

eAp (t−s) N(s + t0 , u1 (s), g) ds

0

= eAp t u0 +

Z

t

eAp (t−s) N(s, u1 (s), g · t0 ) ds

(by Eq. (7.1.2))

0

for all t > 0. Consequently, u1 (t) = u(t; 0, u0 , g · t0 ) for all t > 0.

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As g · t belongs to Z for any g ∈ Z and any t ∈ R, the above lemma allows us to restrict ourselves to considering the initial moment t0 to be equal to 0. LEMMA 7.1.10 Assume (A7-1)–(A7-3). Then for any 0 ≤ t0 ≤ t, u0 ∈ X + , and g ∈ Z the following holds: u(t; t0 , u(t0 ; 0, u0 , g), g) = u(t; 0, u0 , g). PROOF

(7.1.8)

Fix u0 ∈ X + and g ∈ Z, and put u(t) := u(t; 0, u0 , g). We have Z t Ap t u(t) = e u0 + eAp (t−s) N(s, u(s), g) ds t > 0, 0

which can be transformed, for t > t0 , into Z t0 u(t) = eAp (t−t0 ) eAp t u0 + eAp (t0 −s) N(s, u(s), g) ds 0 Z t Z t0 + eAp (t−s) N(s, u(s), g) ds − eAp (t−s) N(s, u(s), g) ds 0 0 Z t = eAp (t−t0 ) u(t0 ) + eAp (t−s) N(s, u(s), g) ds. t0

We write u(t; u0 , g) instead of u(t; 0, u0 , g). A consequence of Lemmas 7.1.9 and 7.1.10 is the following cocycle property: For all t, s ≥ 0, u0 ∈ X + , and g ∈ Z there holds u(t + s; u0 , g) = u(t; u(s; u0 , g), ζs g).

(7.1.9)

PROPOSITION 7.1.6 (Continuous dependence) Let (A7-1)–(A7-3) be satisfied. Then the mapping [ [0, ∞) × X + × Z 3 (t, u0 , g) 7→ u(t; u0 , g) ∈ X + ] is continuous. PROOF

See [48, Theorem 3.4.1].

PROPOSITION 7.1.7 (Monotonicity) Assume (A7-1)–(A7-3). Let g ∈ Z. (1) If u1 , u2 ∈ X + , u1 ≤ u2 , then u(t; u1 , g) ≤ u(t; u2 , g) for each t ≥ 0.

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(2) If u1 , u2 ∈ X + , u1 < u2 , then u(t; u1 , g) u(t; u2 , g) for each t > 0. PROOF Fix u1 , u2 ∈ X + and g ∈ Z, and denote v(t, x) := u(t; u2 , g)(x) − ¯ The function v = v(t, x) is the classical solution u(t; u1 , g)(x), t ≥ 0, x ∈ D. of the nonautonomous linear parabolic partial differential equation ( ∂v ˜ t > 0, x ∈ D ∂t = ∆v + G(t, x)v, Bv = 0, t > 0, x ∈ ∂D ¯ where with the initial condition v(0, x) = u2 (x) − u1 (x) for x ∈ D, Z 1 ∂G ˜ (t, x, u1 (t; x, g) + s(u2 (t; x, g) − u1 (t; x, g))) ds G(t, x) := 0 ∂u ¯ for t ≥ 0 and x ∈ D. ˜ is continuous on [0, ∞) × D, ¯ so the standard The zero-order coefficient G theory of maximum principles applies. In the existing terminology we can express Proposition 7.1.7(2) in the following way: For any g ∈ Z and any t > 0 the mapping [ X + 3 u0 7→ u(t; u0 , g) ∈ X + ] is strongly monotone (see, e.g., [56], or [57]). PROPOSITION 7.1.8 (Compactness) Assume (A7-1) through (A7-3). Then for any δ0 > 0 and any B ⊂ X + ¯ bounded in the C(D)-norm, the set { u(t; u0 , g) : t ≥ δ0 , u0 ∈ B, g ∈ Z } has compact closure in the X-norm. PROOF Take m := max{P, sup { ku0 kC(D) ¯ : u0 ∈ B }}. For any u0 ∈ B and g ∈ Z the constant function m is a supersolution of (7.1.1)g , hence ¯ 0 ≤ u(t; u0 , g)(x) ≤ m for all t ∈ [0, ∞) and all x ∈ D. Pick λ ≥ 0 larger than the supremum of the real parts of eigenvalues of the operator Ap . Then there is > 0 such that for any 0 < β1 < 1 there is M = M (β1 ) > 0 with the property that ke(Ap −λ)t kLp (D),F β1 ≤ M t−β1 e−t p

for all t > 0 (see [48, Section 1.5]; in fact, in the Dirichlet case we can take λ = 0, whereas in the Neumann case any λ > 0 will do). For t ∈ R, u ∈ ¯ + , and g ∈ Z put Nλ (t, u, g) := N(t, u, g) + λu. There holds C(D) Z t (Ap −λ)t u(t; u0 , g) = e u0 + e(Ap −λ)(t−s) Nλ (s, u(s; u0 , g), g) ds, t > 0. 0

It follows from Lemma 7.1.5(iii) that M1 := sup{ kNλ (t, u, g)kC(D) ¯ : t ∈ R, 0 ≤ u ≤ m, g ∈ Z } < ∞. Further, denote by M2 the norm of the

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¯ ,→ Lp (D). Fix β1 ∈ (β, 1). We see that embedding C(D) Z t ku(t; u0 , g)kF β1 ≤ mM M2 t−β1 e−t + M M1 M2 (t − s)−β1 e−(t−s) ds p

0

for all t > 0, u0 ∈ B, and g ∈ Z. Now it suffices to notice that the first term on the right-hand side of the above inequality is bounded (by mM M2 δ0 −β1 ) for all t ≥ δ0 , whereas the second term is bounded for all t > 0. Finally, we apply the compact embedding Fpβ1 ,− ,→ X (see Lemma 7.1.6(1)). PROPOSITION 7.1.9 (Backward uniqueness) Assume (A7-1)–(A7-3). For any u1 , u2 ∈ X + and any g ∈ Z, if u1 6= u2 then u(t; u1 , g) 6= u(t; u2 , g) for all t ≥ 0. PROOF

See [43, Chapter 6] or [44, Part II, Chapter 18].

We proceed now to the question of the differentiability of the solution operator. We will be interested in the differentiability of the first order with respect to u0 as well as the continuous dependence of the respective derivatives. PROPOSITION 7.1.10 (Differentiability) Assume (A7-1)–(A7-3). Then the following holds: (1) The derivative ∂2 u of the mapping [ [0, ∞) × X + × Z 3 (t, u0 , g) 7→ u(t; u0 , g) ∈ X + ] with respect to the u0 -variable exists and is continuous on the set (0, ∞)× Z × X +. (2) For u0 ∈ X + , g ∈ Z, and v0 ∈ X the mapping [ (0, ∞) 3 t 7→ ∂2 u(t; u0 , g)v0 ∈ X ] is the unique solution of the integral equation (where v(·) = ∂2 u(·; u0 , g)v0 ) Z t Ap t v(t) = e v0 + eAp (t−s) (∂2 N(s, u(s; u0 , g), g)v(s)) ds (7.1.10) 0

for t > 0. Moreover, it is the (classical ) solution of the nonautonomous linear parabolic equation:   ∂v = ∆v + ∂G (t, x, u(t; u0 , g))v, t > 0, x ∈ D ∂t ∂u (7.1.11)  Bv = 0, t > 0, x ∈ ∂D, with the initial condition v(0, x) = v0 (x) for x ∈ D.

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PROOF The fact that, for a fixed g ∈ Z, the statement is fulfilled, follows from [48, Theorem 3.4.4]. In particular, Part (2) holds. The continuous dependence of ∂2 u on g is a consequence of the fact that the solution u(·; u0 , g) is obtained as the fixed point of the operator S(u0 , g) : C([0, T ], X) → C([0, T ], X)) (T > 0) defined by (S(u0 , g)u)(t) := eAp (t) u0 +

Z

t

eAp (t−s) N(s, u(s), g) ds

0

(see Proposition 7.1.3). In a manner analogous to the proof of Eq. (7.1.9) one proves the following cocycle property: For all t, s ≥ 0, u0 ∈ X + and g ∈ Z there holds ∂2 u(t + s; u0 , g) = ∂2 u(t; u(s; u0 , g), ζs g) ◦ ∂2 u(s; u0 , g).

(7.1.12)

PROPOSITION 7.1.11 (Positivity of the derivative) Assume (A7-1)–(A7-3). Let u0 ∈ X + and g ∈ Z. (1) If v0 ∈ X + then ∂2 u(t; u0 , g)v0 ∈ X + for each t ≥ 0. (2) If v0 ∈ X + \ {0} then ∂2 u(t; u0 , g)v0 ∈ X ++ for each t > 0. PROOF In view of Proposition 7.1.10(2) this is an application of the standard theory of maximum principles for classical solutions. PROPOSITION 7.1.12 (Backward uniqueness of the derivative) Assume (A7-1)–(A7-3). For any u0 ∈ X + , any v1 , v2 ∈ X, and any g ∈ Z, if v1 6= v2 then ∂2 u(t; u0 , g)v1 6= ∂2 u(t; u0 , g)v2 for all t ≥ 0. PROOF

See [43, Chapter 6] or [44, Part II, Chapter 18].

It should be remarked that Lemmas 7.1.9 and 7.1.10 as well as Propositions 7.1.6 through 7.1.12 would hold in fact without Assumption (A7-3), a difference being that instead of the formulation “for all t ∈ [0, ∞)” one would have “for all t ∈ [0, τmax (0, u0 , g)),” etc. We put Φ(t; u0 , g) = Φt (u0 , g) := (u(t; u0 , g), ζt g), (7.1.13) where t ≥ 0, u0 ∈ X + , and g ∈ Z. Proposition 7.1.6 and Eq. (7.1.9) guarantee that Φ = {Φt }t≥0 is a topological skew-product semiflow on the product bundle X + × Z covering the topological flow (Z, ζ). (Notice that we have the joint continuity at t = 0.) It should be remarked that the property mentioned in Proposition 7.1.7(2) can be written as: The (topological) skew-product

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semiflow Φ is strongly monotone (this is an adjustment of the terminology used for semiflows on ordered metric spaces, see, e.g., [57], to skew-product semiflows with ordered fibers). Further we put ∂Φ(t; v0 , (u0 , g)) = ∂Φt (v0 , (u0 , g)) := (∂2 u(t; u0 , g)v0 , (u(t; u0 , g), ζt g)),

(7.1.14)

where t ≥ 0, v0 ∈ X, u0 ∈ X + , and g ∈ Z. Proposition 7.1.10 and Eq. (7.1.12) guarantee that ∂Φ = {∂Φt }t≥0 is a topological linear skew-product semiflow on the product Banach bundle X ×(X + ×Z) covering the topological semiflow Φ on X + × Z. Again, the property mentioned in Proposition 7.1.11(2) can be written as: The (topological) linear skew-product semiflow ∂Φ is strongly monotone (or strongly positive).

7.1.2

Linearization at the Trivial Solution

In the present subsection we investigate the linearization of the skew-product semiflow Φ at the trivial solution. Most results follow from the general theories developed in Chapters 2, 3, and 4. For convenience, we provide proofs for some results. We assume throughout this subsection that Assumptions (A7-1)–(A7-3) are satisfied. The compact set {0} × Z is invariant under Φ. Consider the restriction of the topological linear skew-product semiflow ∂Φ to X × ({0} × Z): ∂Φt (v0 , (0, g)) = (∂2 u(t; 0, g)v0 , (0, g · t)),

t ≥ 0, v0 ∈ X, g ∈ Z.

By Proposition 7.1.10, for any v0 ∈ X and any g ∈ Z the function [ (0, ∞) 3 t 7→ ∂2 u(t; 0, g)v0 ∈ X ] is given by the classical solution of the nonautonomous linear parabolic equation ( ∂v t > 0, x ∈ D ∂t = ∆v + g(t, x, 0)v, (7.1.15) Bv = 0, t > 0, x ∈ ∂D with the initial condition v(0, x) = v0 (x) for x ∈ D. 2 Define a mapping p˜0 : Z → L∞ (R × D, RN +2N +1 ) × L∞ (R × ∂D, R) by N N p˜0 (g) := ((δij )N i,j=1 , (0)i=1 , (0)i=1 , g0 , 0),

and p0 : Z → L∞ (R × D, R) by p0 (g) := g0 , ¯ where δij is the Kronecker symbol and g0 (t, x) := g(t, x, 0), t ∈ R, x ∈ D.

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Let Y˜ and Y stand for the images of Z under p˜0 and p0 , respectively. We identify p˜0 (g) with p0 (g) = g0 and identify Y˜ with Y . For g0 ∈ Y and t ∈ R we denote g0 · t(s, x) := g0 (t + s, x), s ∈ R, x ∈ ∂D. We write σt g0 for g0 · t. Y will be always considered with the open-compact topology. LEMMA 7.1.11 (1) The mapping p0 : Z → Y is continuous. (2) Y is compact. (3) g0 · t ∈ Y for any g0 ∈ Y and any t ∈ R. (4) p0 ◦ ζt = σt ◦ p0 for any t ∈ R. (5) The mapping [ R × Y 3 (t, g0 ) 7→ σt g0 ∈ Y ] is continuous. PROOF Part (1) follows by the definition of the open-compact topology. Part (2) is a consequence of Part (1) and the compactness of Z. Parts (3) and (4) are obvious. Part (5) is well known (compare Lemma 7.1.1). By Lemma 7.1.11(5) (Y, σ) is a compact flow. For g ∈ Z (or for g0 ∈ Y ) we write (7.1.15) as (7.1.15)g0 . From Assumptions (A7-1) and (A7-2) it follows that (A2-1)–(A2-4) and (A3-1), (A3-2) are satisfied by both (7.1.15) and its adjoint problem. Consequently, we can apply to (7.1.15) the theories presented in Sections 2.1–2.3 and Sections 3.1–3.3. For t ≥ 0, v0 ∈ L2 (D), and g ∈ Z (or for g0 ∈ Y ) we denote by Ug0 (t, 0)v0 the value at time t of the (weak or, what is equivalent, classical) solution of (7.1.15)g0 satisfying the initial condition v(0, ·) = v0 . Denote by Π = {Πt }t≥0 the topological linear skew-product semiflow defined on L2 (D) × Y by (7.1.15): Π(t; v0 , g0 ) = Πt (v0 , g0 ) := (Ug0 (t, 0)v0 , σt g0 ),

t ≥ 0, v0 ∈ L2 (D), g0 ∈ Y.

From now on, we assume that Z0 is a nonempty compact connected invariant subset of Z. By Lemma 7.1.11, Y0 := p0 (Z0 ) is a nonempty compact connected invariant subset of Y . First of all, we have the following result stating that Π admits an exponential separation over Y0 . THEOREM 7.1.1 There exist

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• an invariant (under Π) one-dimensional subbundle X1 of L2 (D) ×Y0 with fibers X1 (g0 ) = span{w(g0 )}, where w : Y0 → L2 (D) is continuous, with kw(g0 )k = 1 for all g0 ∈ Y0 , and • an invariant (under Π) complementary one-codimensional subbundle X2 of L2 (D)×Y0 with fibers X2 (g0 ) = { v ∈ L2 (D) : hv, w∗ (g0 )i = 0 }, where w∗ : Y0 → L2 (D) is continuous, with kw∗ (g0 )k = 1 for all g0 ∈ Y0 , having the following properties: (i) w(g0 ) ∈ L2 (D)+ for all g0 ∈ Y0 , (ii) X2 (g0 ) ∩ L2 (D)+ = {0} for all g0 ∈ Y0 , (iii) there are M ≥ 1 and γ0 > 0 such that for any g0 ∈ Y0 and any u0 ∈ X2 (g0 ) with ku0 k = 1, kUg0 (t, 0)u0 k ≤ M e−γ0 t kUg0 (t, 0)w(g0 )k PROOF

for

t > 0.

See Theorem 3.3.3.

Observe that by the uniqueness of solutions we have the following equality: Π(t; v0 , g0 ) = ∂Φ(t; v0 , (0, g)),

t ≥ 0, v0 ∈ X, g ∈ Z.

Recall that ∂Φ is continuous as a function from (0, ∞) × X × Z into X × Z. Next we show that we have a stronger exponential separation property. To do so, we first show some lemmas. LEMMA 7.1.12 For any T > 0 there is C = C(T ) > 0 such that kUg0 (t, 0)v0 kX ≤ Ckv0 kX for all g0 ∈ Y , 0 ≤ t ≤ T , and v0 ∈ X. PROOF By Proposition 7.1.10(2), for g ∈ Z and v0 ∈ X the function [ [0, ∞) 3 t 7→ Ug0 (t, 0)v0 ∈ X ] satisfies Ug0 (t, 0)v0 = eAp t v0 +

Z

t

eAp (t−s) (∂2 N(s, 0, g)(Ug0 (s, 0)v0 )) ds

(7.1.16)

0

for all t > 0. Fix T > 0. By Lemma 7.1.7 there is M > 0 such that keAp t kLp (D),X ≤ M t−β and keAp t kX ≤ M , for all t ∈ (0, T ]. It follows from Lemma 7.1.5(2) that M1 := sup{ k∂2 N(t, 0, g)kL(C(D)) : t ∈ [0, T ], g ∈ Z } is < ∞. Finally, ¯

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¯ and M3 to be the norm put M2 to be the norm of the embedding X ,→ C(D) ¯ of the embedding C(D) ,→ Lp (D). We estimate Z kUg0 (t, 0)v0 kX ≤ M kv0 kX + M M1 M2 M3

t

(t − s)−β kUg0 (s, 0)v0 kX ds

0

for all t ∈ (0, T ]. An application of the singular Gronwall lemma (see [48, 1.2.1]) gives the existence of C > 0 such that the desired inequality is satisfied.

LEMMA 7.1.13 For any v0 ∈ L2 (D), g0 ∈ Y , and t > 0, Ug0 (t, 0)v0 ∈ X. Moreover, for any 0 < T1 < T2 , there is C(T1 , T2 ) > 0 such that kUg0 (t, 0)v0 kX ≤ C(T1 , T2 )kv0 k for all g0 ∈ Y , T1 ≤ t ≤ T2 , and v0 ∈ L2 (D). N PROOF Let p and β with 1 0 such that kUg0 (T1 /2, 0)v0 kp ≤ C1 kv0 k for any g0 ∈ Y, v0 ∈ L2 (D).

For a given g0 ∈ Y , let v1 := Ug0 (T1 /2, 0)v0 and g1 := g0 · (T1 /2). Then Ug0 (t, 0)v0 = Ug1 (t − T1 /2, 0)v1

for

t ≥ T1 /2.

Note that Ug1 (t, 0)v1 ∈ Fp1 ,→ X for t > 0 (see [48, Theorem 3.3.3]). By the arbitrariness of T1 > 0, we have Ug0 (t, 0)v0 ∈ X for t > 0. By the density of X in Lp (D) and (7.1.16), we have Ug1 (t, 0)v1 = eAp t v1 +

Z

t

eAp (t−s) (∂2 N(s, 0, g1 )(Ug1 (s, 0)v1 )) ds

(7.1.17)

0

for all t > 0. Then by arguments similar to those in the proof of Lemma 7.1.12, we have Z t −β kUg1 (t, 0)v1 kX ≤ M t kv1 kLp (D) +M M1 M2 M3 (t−s)−β kUg1 (s, 0)v1 kX ds. 0

This together with the singular Gronwall lemma (see [48, 1.2.1]) implies that there is C2 > 0 such that kUg1 (t, 0)v1 kX ≤ C2 kv1 kp for t ∈ [T1 /2, T2 − T1 /2], v1 ∈ Lp (D), and g1 ∈ Y . It then follows that kUg0 (t, 0)v0 kX ≤ C(T1 , T2 )kv0 k

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for any t ∈ [T1 , T2 ], v0 ∈ L2 (D), and g ∈ Y , where C(T1 , T2 ) = C1 C2 . LEMMA 7.1.14 For any v0 ∈ L2 (D)+ \ {0}, g0 ∈ Y and t > 0, Ug0 (t, 0)v0 ∈ X ++ . PROOF By Lemma 7.1.13, Ug0 (t, 0)v0 ∈ X for t > 0. Then by Proposition 2.2.7, we have Ug0 (t, 0)v0 ∈ X + for t > 0. It then follows from Proposition 7.1.11 that Ug0 (t, 0)v0 ∈ X ++ . LEMMA 7.1.15 For any bounded set B ⊂ L2 (D) and 0 < T1 < T2 , the set { Ug0 (t, 0)v0 : t ∈ [T1 , T2 ], g0 ∈ Y, v0 ∈ B } is relatively compact in X. PROOF First, by Lemma 7.1.13, B1 := { Ug0 (T1 /2, 0)v0 : g0 ∈ Y, v0 ∈ B } is bounded in X. Then proceeding along the lines of the proof of Proposition 7.1.8 we obtain that { Ug0 ·T1 /2 (t, 0)v1 : g0 ∈ Y, t ∈ [T1 /2, T2 −T1 /2], v1 ∈ B1 } is relatively compact in X. Now we have THEOREM 7.1.2 (i) For each g0 ∈ Y0 , w(g0 ) ∈ X ++ . (ii) The function [ Y0 3 g0 7→ w(g0 ) ∈ X ] is continuous. (iii) There is M1 ≥ 1 such that 1 kw(g0 )k ≤ kw(g0 )kX ≤ M1 kw(g0 )k M1

for all

g0 ∈ Y0 .

PROOF Define a function r : Y0 → R as r(g0 ) := 1/kUg0 ·(−1) (1, 0)w(g0 · (−1)k. The function r is positive and continuous, hence bounded above and bounded away from zero. For each g0 ∈ Y0 there holds w(g0 ) = r(g0 )Ug0 ·(−1) (1, 0)w(g0 · (−1)). By Lemma 7.1.13, Ug0 ·(−1) (1, 0)w(g0 · (−1)), hence w(g0 ), belongs to X. Also, w(g0 · (−1)) ∈ L2 (D)+ \ {0}, hence it follows from By Lemma 7.1.14, that w(g0 ) ∈ X ++ . This concludes the proof of Part (i). By Lemma 7.1.15, { w(g0 ) : g0 ∈ Y0 } has compact closure in X. Assume (n) (n) that g0 → g0 . Then w(g0 ) → w(g0 ) in L2 (D). By the above arguments, (nk ) ∗ there are a subsequence (nk )∞ ) → u∗ in X. k=1 and u ∈ X such that w(g0

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Therefore we must have u∗ = w(g0 ) and w(g0 ) → w(g0 ) in X. This proves Part (ii) Part (iii) follows by Part (ii) and the compactness of Y0 . THEOREM 7.1.3 c ≥ 1 and γ0 > 0 (γ0 is the same as in Theorem 7.1.1) such that There are M kUg0 (t, 0)u0 kX ce−γ0 t ku0 kX ≤M kUg0 (t, 0)w(g0 )kX kw(g0 )kX for each t > 0, g0 ∈ Y0 , and u0 ∈ X2 (g0 ) ∩ X. PROOF It can be proved by arguments similar to those in the proof of in Theorem 3.5.2. For the reader’s convenience we give a proof here. By Lemma 7.1.12 there is C > 0 such that kUg0 (t, 0)u0 kX ≤ Cku0 kX for any g0 ∈ Y0 , 0 ≤ t ≤ 2, and u0 ∈ X. Consequently kUg0 (t, 0)u0 kX ≤ Ce2γ0 e−γ0 t ku0 kX for g0 ∈ Y0 , 0 ≤ t ≤ 2, and u0 ∈ X. On the other hand, kUg0 (t, 0)w(g0 )kX ≥

M2 kw(g0 )kX M12

for 0 ≤ t ≤ 2, where M2 := inf { kUg0 (t, 0)w(g0 )k : g0 ∈ Y0 , t ∈ [0, 2] } > 0. Hence CM12 e2γ0 −γ0 t ku0 kX kUg0 (t, 0)u0 kX ≤ e kUg0 (t, 0)w(g0 )kX M2 kw(g0 )kX for 0 ≤ t ≤ 2, provided that u0 ∈ X. Assume now t > 2. As a consequence of Lemma 7.1.13 there is C1 > 0 such that kUg0 (t, 0)u0 kX ≤ C1 kUg0 (t − 1, 0)u0 k for each g0 ∈ Y0 , t > 2, and u0 ∈ X. Further, an application of Theorem 7.1.2(iii) gives C1 M1 kUg0 (t − 1, 0)u0 k kUg0 (t, 0)u0 kX ≤ kUg0 (t, 0)w(g0 )kX M2 kUg0 (t − 1, 0)w(g0 )k for t > 2. By Theorem 7.1.1, kUg0 (t − 1, 0)u0 k kUg0 (1, 0)u0 k ≤ M e−γ0 (t−2) . kUg0 (t − 1, 0)w(g0 )k kUg0 (1, 0)w(g0 )k Further, kUg0 (1, 0)u0 k M1 M3 ku0 kX ≤ , kUg0 (1, 0)w(g0 )k M2 kw(g0 )kX where M3 := sup { kUg0 (1, 0)kX,L2 (D) : g0 ∈ Y0 } < ∞ (by X ,→ L2 (D) and the L2 –L2 estimates in Proposition 2.2.2).

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As a consequence, kUg0 (t, 0)u0 kX ce−γ0 t ku0 kX ≤M kUg0 (t, 0)w(g0 )kX kw(g0 )kX c= for t ≥ 0 and u0 ∈ X2 (g0 ) ∩ X, where M

e2γ0 M12 M2

1 M3 max {C, CM }. 2

For g0 ∈ Y0 we put w(g ˜ 0 ) := w(g0 )/kw(g0 )kX . Recall that by Definitions 3.1.1, 3.1.2, and Lemma 3.2.6, the principal spectrum of Π over Y0 equals the complement of the set of those λ ∈ R for which either of the following conditions holds: • There are > 0 and M ≥ 1 such that kUg0 (t, 0)w(g0 )k ≤ M e(λ−)t

for t > 0 and g0 ∈ Y0

(such λ ∈ R are members of the upper principal resolvent of Π over Y0 , denoted by ρ+ (Y0 )). • There are > 0 and M ∈ (0, 1] such that kUg0 (t, 0)w(g0 )k ≥ M e(λ+)t

for t > 0 and g0 ∈ Y0

(such λ ∈ R are members of the lower principal resolvent of Π over Y0 , denoted by ρ− (Y0 )). In view of Theorem 7.1.2 we have the following result. THEOREM 7.1.4 (1) λ ∈ R belongs to the upper principal resolvent of Π over Y0 if and only if there are > 0 and M ≥ 1 such that kUg0 (t, 0)w(g ˜ 0 )kX ≤ M e(λ−)t

for t > 0 and g0 ∈ Y0 .

(2) λ ∈ R belongs to the lower principal resolvent of Π over Y0 if and only if there are > 0 and M ∈ (0, 1] such that kUg0 (t, 0)w(g ˜ 0 )kX ≥ M e(λ+)t

for t > 0 and g0 ∈ Y0 .

As a consequence of Proposition 7.1.10 we have the following, which characterizes the closeness between solutions of nonlinear equations and solutions of the linearized equation at the trivial solution. THEOREM 7.1.5 For each t > 0 there holds ku(t; %u0 , g) − Ug0 (t, 0)(%u0 )kX →0 %

as % → 0+

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uniformly in g ∈ Z and u0 ∈ X + with ku0 kX = 1. PROOF Let t > 0 be fixed. The statement of the proposition is equivalent to saying that for each > 0 there is δ > 0 such that ku(t; u0 , g) − Ug0 (t, 0)u0 kX < ku0 kX for any g ∈ Z and any u0 ∈ X + with ku0 kX < δ. Let > 0 be fixed. Proposition 7.1.10 implies that for each g ∈ Z there is δ1 = δ1 (g) > 0 such that for any h ∈ Z and any u0 ∈ X + , if d(g, h) < δ1 and ku0 k < δ1 then k∂2 u(t; u0 , h) − ∂2 u(t; 0, g)kL(X) < , 2 where d(·, ·) denotes the distance in Z. Since Z is compact, there are finitely many g (1) , . . . , g (n) ∈ Z such that the open balls (in Z) with center g (k) and radius δ1 (g (k) ) cover Z. Set δ := min{δ1 (g (1) ), . . . , δ1 (g (n) )}. For g ∈ Z let g (k) be such that d(g, g (k) ) ≤ δ1 (g (k) ). For u0 ∈ X + with ku0 kX < δ we estimate k∂2 u(t; u0 , g) − ∂2 u(t; 0, g)kL(X) ≤ k∂2 u(t; u0 , g) − ∂2 u(t; 0, g (k) )kL(X) + k∂2 u(t; 0, g (k) ) − ∂2 u(t; 0, g)kL(X) < . X + is convex, hence there holds Z u(t; u0 , g) =

1

∂2 u(t; su0 , g)u0 ds 0

for any u0 ∈ X + and any g ∈ Z. Therefore ku(t; u0 , g) − Ug0 (t, 0)u0 kX

Z 1

= (∂2 u(t; su0 , g) − ∂2 u(t; 0, g))u0 ds

X

0

< ku0 kX

for any g ∈ Z and any u0 ∈ X + with ku0 kX < δ. The following result will be used several times, so we formulate it as a separate result. THEOREM 7.1.6 Let E ⊂ X ++ × Y0 be compact. Then there are 0 < c1 ≤ c2 < ∞ such that c1 ϕprinc u0 c2 ϕprinc

for all

(u0 , g0 ) ∈ E.

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269 (n)

PROOF Suppose to the contrary that for each n ∈ N there is (un , g0 ) ∈ E such that n1 ϕprinc 6 un . Without loss of generality we can assume that (n) (un , g0 ) converge to some (u0 , g0 ) ∈ E. But then 0 6 u0 , which contradicts the assumption. (n) Now, suppose to the contrary that for each n ∈ N there is (un , g0 ) ∈ E such that n1 un 6 ϕprinc . Without loss of generality we can assume that (n) (un , g0 ) converge to some (u0 , g0 ) ∈ E. The set E is compact, hence the set { un : n ∈ N } is bounded in the X-norm, consequently u0 = limn→∞ n1 un = 0. But then 0 6 ϕprinc , which is impossible.

7.1.3

Global Attractor and Uniform Persistence

In this subsection, we study global attractor and uniform persistence for the skew-product semiflow Φ. Throughout this subsection, we assume (A71)–(A7-3). Some results presented in this subsection can be proved by applying the general theories for dissipative systems in [46] and general theories for persistence in [47]. We choose to provide elementary proofs for all the results here. We will apply the theories in [46] and [47] in Subsection 7.3.3 to competitive Kolmogorov systems of parabolic equations. First we study the global attractor. We denote ¯ }. [0, P ]X := { u ∈ X : 0 ≤ u(x) ≤ P for all x ∈ D The set [0, P ]X is convex and closed (in X). THEOREM 7.1.7 (Absorbing set) ¯ Assume (A7-1)–(A7-3). Let B ⊂ X + be bounded in the C(D)-norm. Then there is T = T (B) ≥ 0 such that u(t; u0 , g)(x) ≤ P for all t ≥ T , u0 ∈ B, ¯ Moreover, if B ⊂ [0, P ]X then T (B) can be taken to be g ∈ Z, and x ∈ D. zero. PROOF

Define a function g˜ : [0, ∞) → R by ¯ }. g˜(w) := sup { g(t, x, w) : g ∈ Z, t ∈ R, x ∈ D

By Assumption (A7-2) the function g˜ is well defined. We claim that it is locally Lipschitz continuous. To do that, fix W > 0. Again by (A7-2) there is L > 0 such that |g(t, x, w1 ) − g(t, x, w2 )| ≤ L|w1 − w2 | ¯ and w1 , w2 ∈ [0, W ]. There holds for any g ∈ Z, t ∈ R, x ∈ D, −L(w2 − w1 ) ≤ g˜(w1 ) − g˜(w2 ) ≤ L(w2 − w1 )

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¯ we for any two 0 ≤ w1 < w2 ≤ W . Indeed, for any g ∈ Z, t ∈ R, and x ∈ D have g(t, x, w2 ) ≤ g(t, x, w1 ) + L(w2 − w1 ) ≤ g˜(w1 ) + L(w2 − w1 ), from which it follows that g˜(w2 ) ≤ g˜(w1 ) + L(w2 − w1 ). The inequality g˜(w1 ) ≤ g˜(w2 ) + L(w2 − w1 ) is proved in a similar way. Further, g˜(w) < 0 for all w ≥ P . Fix B ⊂ X + as in the hypothesis, and set w0 := sup { ku0 kC(D) ¯ : u0 ∈ B }. For any u0 ∈ B and g ∈ Z, the unique solution w(·) of the ODE w˙ = g˜(w)w satisfying the initial condition w(0) = w0 is a supersolution of (7.1.1)g . It suffices to observe that there is T ≥ 0 such that 0 ≤ w(t) ≤ P for all t ≥ T , and that if w0 ∈ [0, P ] then T = 0. THEOREM 7.1.8 (Global attractor in X + ) Let (A7-1) through (A7-3) be satisfied. Then the topological skew-product semiflow Φ possesses a global attractor Γ contained in [0, P ]X ×Z. In addition, ¯ for any B ⊂ X + bounded in the C(D)-norm one has (1) ∅ = 6 ω(B × Z) (⊂ Γ), (2) Γ attracts B × Z. PROOF We first define the set Γ as ω([0, P ]X × Z). Consequently, Γ is closed and invariant. It follows from Proposition 7.1.8 that O+ (Φ1 ([0, P ]X × Z)) has compact closure. Consequently, Γ is, by Lemma 1.2.5, nonempty and compact. Theorem 7.1.7 implies that cl O+ ([0, P ]X × Z) ⊂ cl([0, P ]X ×Z) = [0, P ]X × Z, consequently Γ ⊂ [0, P ]X × Z. ¯ Let B ⊂ X + be bounded in the C(D)-norm. Proposition 7.1.8 implies that O+ (Φ1 (B × Z)) has compact closure, therefore ω(B × Z) is, by Lemma 1.2.5, nonempty and compact. Theorem 7.1.7 guarantees the existence of T = T (B) ≥ 0 such that O+ (ΦT (B × Z)) ⊂ [0, P ]X × Z. Consequently, ω(B × Z) = ω(ΦT (B × Z)) ⊂ ω([0, P ]X × Z) = Γ. Since, by Lemma 1.2.5, ω(B × Z) attracts B × Z, Γ attracts B × Z, too. In particular, the (nonempty compact invariant) set Γ attracts any B × Z, where B ⊂ X + is bounded in the X-norm, consequently is the global attractor for the semiflow Φ. By the invariance of Γ and Proposition 7.1.9, for any (u0 , g) ∈ Γ and any t < 0 there is a unique u ˜(t; u0 , g) ∈ X + such that (˜ u(t; u0 , g), ζt g) ∈ Γ and Φ−t (˜ u(t; u0 , g), ζt g) = (u0 , g). Define a mapping Φ|Γ : R × Γ → Γ by ( (˜ u(t; u0 , g), ζt g) for t < 0, Φ|Γ (t, (u0 , g)) := (u(t; u0 , g), ζt g) for t ≥ 0.

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PROPOSITION 7.1.13 Assume (A7-1) through (A7-3). Then Φ|Γ is a topological flow. PROOF The satisfaction of the algebraic properties (TF2) and (TF3) follows from the definition and the fact that Φ is a semiflow. For each t ≥ 0 the mapping (Φ|Γ )t is a continuous bijection of a compact metric space onto itself, hence it is a homeomorphism. Further, for (u0 , g) ∈ Γ fixed the mapping [ R 3 t 7→ u ˜(t; u0 , g) ∈ X + ] is continuous (notice that for any t ∈ R, u ˜(t; u0 , g) = u(t; t0 , u ˜(t0 ; u0 , g), g) for any t0 < t, so the mapping is continuous at t). We have thus the separate continuity of Φ|Γ . The (joint) continuity follows from standard results on joint continuity of actions of a Baire topological group (R in our case) on a compact (even locally compact) metric space, see e.g. [18]. Next we study the uniform persistence. Let Z0 be a nonempty connected compact invariant subset of Z and Y0 := p0 (Z0 ). DEFINITION 7.1.2 (Uniform persistence) The skew-product semiflow Φ on X + × Z is said to be uniformly persistent over Z0 if there exists η0 > 0 such that for any u0 ∈ X + \ {0} there is τ = τ (u0 ) > 0 with the property that u(t; u0 , g) ≥ η0 ϕprinc

for all

g ∈ Z0 , t ≥ τ.

In the following, we formulate an assumption on the principal spectrum of the linearization at the trivial solution. (A7-4) The principal spectrum [λmin , λmax ] for Π over Y0 is contained in (0, ∞). In other words, (A7-4) holds if and only if 0 ∈ ρ− (Y0 ). LEMMA 7.1.16 Assume (A7-1)–(A7-4). Then there exists T > 0 such that Ug0 (T, 0)ϕprinc 2ϕprinc PROOF such that

for all

g ∈ Z0 .

By (A7-4) and Theorem 7.1.4(2) there are > 0 and 0 < M ≤ 1

kUg0 (t, 0)w(g ˜ 0 )kX ≥ M et

for all

t > 0, g ∈ Z0 .

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Theorem 7.1.6 applied to the compact set E = { (w(g ˜ 0 ), g0 ) : g0 ∈ Y0 } gives the existence of 0 < c1 ≤ c2 < ∞ such that 1 1 w(g ˜ 0 ) ϕprinc w(g ˜ 0) c2 c1

for all

g ∈ Z0 .

Take some T > 0 such that M eT > 2c2 /c1 . We have kUg0 (T, 0)w(g ˜ 0 )kX > ˜ 0 ) belongs to span{w(g ˜ 0 · T )}, there holds 2 cc12 . Since Ug0 (T, 0)w(g Ug0 (T, 0)w(g ˜ 0) 2

c2 w(g ˜ 0 · T ). c1

Consequently, an application of Lemma 7.1.14 yields Ug0 (T, 0)ϕprinc

1 2 Ug (T, 0)w(g ˜ 0 ) w(g ˜ 0 ·T ) 2ϕprinc c2 0 c1

for all

g ∈ Z0 .

THEOREM 7.1.9 (Uniform persistence) Assume (A7-1)–(A7-4). Then there exists r0 > 0 with the following properties. (1) u(T ; rϕprinc , g) 2rϕprinc for all r ∈ (0, r0 ] and all g ∈ Z0 , where T > 0 is as in Lemma 7.1.16. (2) For each compact E ⊂ (X + \ {0}) × Z0 there is T0 = T0 (E) > 0 such that u(t; u0 , g) 2r0 ϕprinc for all t ≥ T0 and all (u0 , g) ∈ E (hence {Φt }t≥0 is uniformly persistent over Z0 ). PROOF (1) Lemma 7.1.15 implies that the set { Ug0 (T, 0)ϕprinc − 2ϕprinc : g0 ∈ Y0 } (contained, by Lemma 7.1.16, in an open set X ++ ) is compact as a subset of X. Therefore 0 := inf{ k(Ug0 (T, 0)ϕprinc − 2ϕprinc ) − vkX : g0 ∈ Y0 , v ∈ ∂X + } is positive. By linearity inf{ k(Ug0 (T, 0)(rϕprinc ) − 2rϕprinc ) − vkX : g0 ∈ Y0 , v ∈ ∂X + } = r0 (7.1.18) for any r > 0. It follows from Theorem 7.1.5 that there is r0 > 0 such that ku(T ; rϕprinc , g) − Ug0 (T, 0)(rϕprinc )kX ≤

r0 2

for all

g ∈ Z0 , r ∈ (0, r0 ].

We estimate k(u(T ; rϕprinc , g) − 2rϕprinc ) − (Ug0 (T, 0)(rϕprinc ) − 2rϕprinc )kX = ku(T ; rϕprinc , g) − Ug0 (T, 0)(rϕprinc )kX ≤

r0 2

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273

for any g ∈ Z0 and 0 < r ≤ r0 . Eq. (7.1.18) gives u(T ; rϕprinc , g) − 2rϕprinc ∈ X ++ , that is, u(T ; rϕprinc , g) 2rϕprinc . This proves Part (1). Let a compact E ⊂ (X + \ {0}) × Z0 . Denote E1 := Φ([1, T + 1] × E). The set E1 is compact and contained in X ++ × Z0 (by Proposition 7.1.7(2)). An application of Theorem 7.1.6 to the compact set (IdX , p0 )E1 (⊂ X ++ × Y0 ) gives the existence of r˜ > 0 such that u(t; u0 , g) r˜ϕprinc for any t ∈ [1, T +1] and any (u0 , g) ∈ E. If r˜ ≥ r0 then we put T0 = 1. If not then, for instance, T0 =

j ln r − ln r˜ k 0 +2 T +1 ln 2

will do. Recall that it is proved in Theorem 7.1.8 that under (A7-1)-(A7-3), Φ possesses a global attractor Γ contained in [0, P ]X × Z. In the following we go back to study more properties of the global attractor Γ under (A7-1)-(A7-4). For each r > 0 we define rW + := (X + + rϕprinc ) × Z0

and rW ++ := (X ++ + rϕprinc ) × Z0 .

The following result is straightforward. LEMMA 7.1.17 (i) For any r > 0 the set rW + is closed (in X + × Z0 ). (ii) For any r > 0, rW ++ equals the relative interior of rW + in X + × Z0 . (iii) For any 0 < r1 < r2 there holds r2 W + ⊂ r1 W ++ . THEOREM 7.1.10 Assume (A7-1)–(A7-4). Then for the flow Φ|Γ∩(X + ×Z0 ) the compact invariant set {0} × Z0 is a repeller, with its dual attractor Γ++ equal to ω(r0 W ++ ∩ Γ) and contained in 2r0 W + , where r0 > 0 is as in Theorem 7.1.9. PROOF We start by showing that the set r0 W ++ ∩ Γ is nonempty. Pick any (u0 , g) ∈ r0 W ++ . Theorem 7.1.8 yields ∅ 6= ω((u0 , g)) ⊂ Γ. By Theorem 7.1.9(2), (u(t; u0 , g), ζt g) belong to 2r0 W ++ for t sufficiently large, consequently ω((u0 , g)) ⊂ cl 2r0 W ++ ⊂ 2r0 W + ⊂ r0 W ++ . The set r0 W + ∩ Γ is compact and contained in (X + \ {0}) × Z0 . Theorem 7.1.9(2) gives the existence of T0 > 0 such that Φt (r0 W ++ ∩ Γ) ⊂ 2r0 W ++ ∩ Γ for all t ≥ T0 . Thus Γ++ := ω(r0 W ++ ∩ Γ) = ω(ΦT0 (r0 W ++ ∩ Γ)) ⊂ 2r0 W + ∩ Γ.

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It follows from Theorem 7.1.2(ii) that r0 W ++ ∩Γ is relatively open in Γ∩(X + × Z0 ), so it is a neighborhood of Γ++ in the relative topology of Γ ∩ (X + × Z0 ). We have thus proved that Γ++ is an attractor for the flow Φ|Γ∩(X + ×Z0 ) . We prove now that the attraction basin of Γ++ equals Γ∩((X + \{0})×Z0 ), that is, the complement of {0} × Z0 in Γ ∩ (X + × Z0 ). Take any nonzero u0 ∈ X + and any g ∈ Z0 such that (u0 , g) ∈ Γ. Theorem 7.1.9(2) yields the existence of T0 > 0 such that Φ(t, u0 , g) ∈ 2r0 W ++ ⊂ r0 W + for t ≥ T0 . Consequently, ω((u0 , g)) = ω(Φ(T0 ; u0 , g)) ⊂ ω(r0 W + ∩ Γ) = Γ++ . The above theorem deals only with the restriction of Π to Γ ∩ (X + × Z0 ). That set, as a compact subset of a bundle with fibers being modeled on a subset of an infinite-dimensional Banach space with nonempty interior, is rather small. What is more, usually we do not have any useable characterization of members of Γ. These are reasons why we are interested in finding larger sets attracted by Γ++ . The next result gives two families of such sets. THEOREM 7.1.11 Let (A7-1)–(A7-4) be fulfilled. Assume that B ⊂ X + \ {0} satisfies one of the following conditions: (a) B is compact. ¯ (b) B is bounded in the C(D)-norm and there is r˜ > 0 such that u0 ≥ r˜ϕprinc for each u0 ∈ B. Then Γ++ attracts B × Z0 . PROOF From Proposition 7.1.8 it follows that O+ (Φ1 (B × Z0 )) has compact closure (in X + × Z0 ), consequently, by Lemma 1.2.5, ω(B × Z0 ) is compact nonempty and attracts B × Z0 . The remainder of the proof is devoted to showing that ω(B × Z0 ) ⊂ Γ++ . Put E1 := cl(Φ1 (B × Z0 )), where the closure is taken in the (X × Z0 )-topology. The set E1 is compact (in case (a) as a consequence of the continuity of Φ1 , in case (b) by Proposition 7.1.8). We claim that E1 ⊂ X ++ × Z0 . In case (a) this is a direct consequence of Proposition 7.1.7(2) and the fact that now E1 = Φ1 (B × Z0 ). In case (b), by Proposition 7.1.7(2) u(1; r˜ϕprinc , g) 0 for each g ∈ Z0 . An application of Theorem 7.1.6 to the compact set { u(1; r˜ϕprinc , g) : g ∈ Z0 } × Y0 gives the existence of r˜1 > 0 such that u(1; r˜ϕprinc , g) r˜1 ϕprinc for all g ∈ Z0 . By Proposition 7.1.7(1), u(1; u0 , g) ≥ u(1; r˜ϕprinc , g) r˜1 ϕprinc for all u0 ∈ B and all g ∈ Z0 , that is, Φ1 (B × Z0 ) ⊂ r˜1 W ++ , which yields E1 ⊂ r˜1 W + ⊂ X ++ × Z0 . Theorem 7.1.9(2) guarantees the existence of T0 = T0 (E1 ) > 0 such that Φt (E1 ) ⊂ 2r0 W ++ for all t ≥ T0 . We have ω(B × Z0 ) ⊂ ω(E1 ) = ω(ΦT0 (E1 )) ⊂ 2r0 W + ⊂ r0 W ++ .

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Since ω(B × Z0 ) ⊂ Γ, it follows from Theorem 7.1.10 that ω(B × Z0 ) = ω(ω(B × Z0 )) = ω(ω(B × Z0 ) ∩ Γ) ⊂ ω(r0 W ++ ∩ Γ) = Γ++ .

THEOREM 7.1.12 (Structure of Γ++ ) Assume (A7-1)–(A7-4). Assume also that ∂u g(t, x, u) < 0 for any g ∈ Z, ¯ and u ≥ 0. Then there is a continuous ξ : Z0 → X ++ such that t ∈ R, x ∈ D, Γ++ = { (ξ(g), g) : g ∈ Z0 }. Moreover, for any u0 ∈ X + \ {0} and g ∈ Z0 , u(t; u0 , g) − ξ(g · t) → 0 in X as t → ∞. PROOF By Theorem 7.1.6, there are 0 < c1 < c2 such that c1 ϕprinc u0 c2 ϕprinc for each (u0 , g) ∈ Γ++ . Put u∗ := 2c2 ϕprinc . The set { u0 ∈ X : 0 ≤ u0 c2 ϕprinc } × Z0 is an open neighborhood of Γ++ in the relative topology of X + × Z0 , consequently we deduce from Theorem 7.1.11 that there is T > 0 such that u(T ; u∗ , g) ≤ u∗ for all t ≥ T and all g ∈ Z0 . For n ∈ N and g ∈ Z0 define ξ (n) (g) := u(nT ; u∗ , g · (−nT )). For a fixed g ∈ Z0 the sequence (ξ (n) (g))∞ n=1 is monotone (decreasing), and bounded in L2 (D), consequently it has a limit in L2 (D) (denoted by ξ + (g)). This sequence is, by Proposition 7.1.8, relatively compact in the X-norm. As a consequence, we have that kξ (n) (g) − ξ + (g)kX → 0 as n → ∞, for each g ∈ Z0 . Clearly (ξ + (g), g) ∈ ω({ (u∗ , g · (−nT0 )) : n ∈ N }) ⊂ Γ++ . Fix for the moment (u0 , g) ∈ Γ++ . We have (u(−nT ; u0 , g), g · (−nT )) ∈ ++ Γ for all n ∈ N, hence u(−nT ; u0 , g) ≤ u∗ for all n ∈ N. Consequently u0 = u(nT ; u(−nT ; u0 , g), g·(−nT )) ≤ u(nT ; u∗ , g·(−nT ))

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

therefore u0 ≤ ξ + (g). In a similar way we prove, for each g ∈ Z0 , the existence of ξ − (g) such that − (ξ (g), g) ∈ Γ++ and ξ − (g) ≤ u0 for any (u0 , g) ∈ Γ++ . We claim that u(t; ξ + (g), g) = ξ + (g · t) and u(t; ξ − (g), g) = ξ − (g · t) for any g ∈ Z0 and t ∈ R. Suppose first that for some g ∈ Z0 and t < 0 we have u(t; ξ + (g), g) < ξ + (g · t). Then ξ + (g) = u(−t; u(t; ξ + (g), g), g · t) u(−t; ξ + (g · t), g · t), which contradicts the characterization of ξ + (g). It remains now to observe that, by construction, u(nT ; ξ + (g), g) = ξ + (g · nT ) and u(nT ; ξ − (g), g) = ξ − (g · nT ) for all n ∈ N and apply the previous reasoning. Next we show that ξ + (g) = ξ − (g) for any g ∈ Z0 . To do so we introduce the so called part metric ρ(·, ·) on X ++ defined by ρ(u, v) := inf { ln α : α > 1,

1 v ≤ u ≤ αv }, α

u, v ∈ X ++ .

(7.1.19)

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By arguments as in [83, Lemma 3.2], ρ(u(t; u0 , g), u(t; v0 , g)) < ρ(u(s; u0 , g), u(s; v0 , g))

(7.1.20)

for any u0 , v0 ∈ X ++ with u0 6= v0 , 0 < s < t, and g ∈ Z0 . This implies that if g˜ ∈ Z0 is such that ξ + (˜ g ) 6= ξ − (˜ g ), then ρ(ξ + (˜ g · t), ξ − (˜ g · t) < ρ(ξ + (˜ g · s), ξ − (˜ g · s)) for any s < t. Put ρ−∞ := lims→−∞ ρ(ξ + (˜ g ·s), ξ − (˜ g ·s)) (6= 0). Let sn → −∞. Without loss of generality we may assume that g˜ · sn → g∗ , ξ + (˜ g · sn ) → − + − g · sn ) → u− u+ ∗ (in X), as n → ∞. Then u∗ 6= u∗ and ∗ (in X) and ξ (˜ − + ρ(u(t; u∗ , g∗ ), u(t; u∗ , g∗ )) = ρ−∞ for all t ≥ 0, which contradicts (7.1.20). Therefore, ξ + (g) = ξ − (g) =: ξ(g) for any g ∈ Z0 . Since the compact set Γ++ equals the graph of the function ξ (with compact domain Z0 ), the continuity of φ follows. The last property in the statement of the theorem follows from the fact that Γ++ attracts any (u0 , g) ∈ (X + \ {0}) × Z0 . Finally, we provide some sufficient conditions which guarantee (A7-4) holds. ¯ is called a time averaged function of g0 ∈ Y0 if there A function gˆ0 ∈ C(D) are sn < tn with tn − sn → ∞ as n → ∞ such that Z tn 1 gˆ0 (x) = lim g0 (t, x) dt n→∞ tn − sn s n ¯ Let Yˆ0 be defined as follows: uniformly for x ∈ D. Yˆ0 := { gˆ0 : gˆ0 is an averaged function of some g0 ∈ Y0 } For a given gˆ0 ∈ Yˆ0 , let λ(ˆ g0 ) be the principal eigenvalue of ( ∂u ˆ0 (x)u, x∈D ∂t = ∆u + g Bu = 0, x ∈ ∂D

(7.1.21)

where Bu is as in (7.1.1). THEOREM 7.1.13 If λ(ˆ g0 ) > 0 for any gˆ0 ∈ Yˆ0 , then (A7-4) holds, and hence Theorems 7.1.9, 7.1.10, and 7.1.11 hold. PROOF

First of all, by Theorem 3.2.5, there are g0− such that 1 ln kUg− (t, 0)w(g0− )k. 0 t→∞ t

λmin = lim

Then by Theorem 5.2.2, there is gˆ0− ∈ Yˆ0 such that λmin ≥ λ(ˆ g0− ) > 0. This implies that (A7-4) holds. The theorem thus follows.

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7.2

277

Semilinear Equations of Kolmogorov Type: Examples

In this section, we discuss applications of the general theory established in the previous section to some random and nonautonomous semilinear equations of Kolmogorov type.

7.2.1

The Random Case

Assume that ((Ω, F, P), {θt }t∈R ) is an ergodic metric dynamical system. Consider the following random parabolic equation of Kolmogorov type:   ∂u = ∆u + f (θt ω, x, u)u, t > 0, x ∈ D, ∂t (7.2.1)  Bu = 0, t > 0, x ∈ ∂D, ¯ × [0, ∞) 7→ R. where f : Ω × D The first assumption in the present subsection concerns the measurability of the function f (recall that for a metric space S the symbol B(S) stands for the countably additive algebra of Borel sets): (A7-R1) (Measurability) The function f is (F×B(D)×B([0, ∞)), B(R))measurable. For each ω ∈ Ω, let f ω (t, x, u) := f (θt ω, x, u). The function [ Ω × R × D × [0, ∞) 3 (ω, t, x, u) 7→ f ω (t, x, u) ∈ R ] is (F × B(R) × B(D) × B([0, ∞)), B(R))-measurable (as a composite of Borel measurable functions). As a section of a Borel measurable function, the function f ω , is (B(R) × B(D) × B([0, ∞)), B(R))-measurable, for any fixed ω ∈ Ω. The next assumption regards regularity of the function f : (A7-R2) (Regularity) For each ω ∈ Ω and any M > 0 the restrictions ¯ × [0, M ] of f ω and all the derivatives of the functions f ω up to to R × D ¯ × [0, M ]). Moreover, for M > 0 fixed the order 1 belong to C 1−,1−,1− (R × D 1−,1−,1− ¯ C (R×D×[0, M ])-norms of the restrictions of f ω and those derivatives are bounded uniformly in ω ∈ Ω. The last assumption in the present subsection concerns the negativity of the function f for large values of u. (A7-R3) There are P > 0 and a function m : [P, ∞) → (0, ∞) such that ¯ and any u ≥ P . f (ω, x, u) ≤ −m(u) for any ω ∈ Ω, any x ∈ D,

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From now on, until the end of the present subsection, assume that (A7-R1) through (A7-R3) are satisfied. Define the mapping E from Ω into the set of continuous real functions ¯ × [0, ∞) as defined on R × D E(ω) := f ω . Put Z := cl { E(ω) : ω ∈ Ω }

(7.2.2)

with the open-compact topology, where the closure is taken in the opencompact topology. It is a consequence of (A7-R2) via the Ascoli–Arzel`a theorem that the set Z is a compact metrizable space. The following result follows immediately from the measurability properties of f (Assumption (A7-R1)). LEMMA 7.2.1 The mapping E is (F, B(Z))-measurable. An important property of the mapping E is the following ζt ◦ E = E ◦ θt

for each t ∈ R.

(7.2.3)

It follows that if g ∈ E(Ω) then g · t ≡ ζt g ∈ E(Ω) for all t ∈ R. Further, we deduce from Lemma 7.1.1 that if g ∈ Z then g · t ∈ Z for all t ∈ R. Hence (Z, {ζt }t∈R ) is a compact flow. The mapping E is a homomorphism of the measurable flow ((Ω, F), {θt }t∈R ) ˜ the image of into the measurable flow ((Z, B(Z)), {ζt }t∈R ). Denote by P ˜ ˜ is a the measure P under E: for any A ∈ B(Z), P(A) := P(E −1 (A)). P {ζt }-invariant ergodic Borel measure on Z. So, E is a homomorphism of the ˜ {ζt }t∈R ). metric flow ((Ω, F, P), {θt }t∈R ) into the metric flow ((Z, B(Z), P), We will consider a family of Eqs. (7.1.1) parameterized by g ∈ Z. We claim that Assumptions (A7-1) through (A7-3) are fulfilled. Indeed, (A7-1) is clearly satisfied. It follows from (A7-R2) through the Ascoli–Arzel`a theorem that (A7-2) is satisfied. The satisfaction of (A7-3) follows from (A7-R3). We denote by Φ = {Φt }t≥0 the topological skew-product semiflow generated by (7.2.1) on the product Banach bundle X + × Z: Φ(t; u0 , g) = Φt (u0 , g) := (u(t; u0 , g), ζt g),

t ≥ 0, g ∈ Z, u0 ∈ X + , (7.2.4)

where u(t; u0 , g) stands for the solution of (7.1.1) with initial condition u(0; u0 , g)(x) = u0 (x). Moreover, define ˜ u0 , ω) := (u(t; u0 , E(ω)), θt ω), Φ(t; We have

t ≥ 0, ω ∈ Ω, u0 ∈ X + .

(7.2.5)

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LEMMA 7.2.2 ˜ is a continuous random skew-product semiflow on the measurable bundle Φ X + × Ω, covering the metric flow ((Ω, F, P), {θt }t∈R ). ˜ that for PROOF It follows from (7.2.3) and the definitions of Φ and Φ each t ≥ 0 the diagram ˜ Φ

t → X+ × Ω X + × Ω −−−−    (Id ,E) (IdX + ,E)y y X+

Φ

t X + × Z −−−− → X + × Z,

commutes. Consequently, the properties (RSP1) and (RSP2) of the random skew-product semiflow are satisfied. ˜ its second coordinate As regards the measurability of the mapping Φ, [ [0, ∞) × Ω 3 (t, ω) 7→ θt ω ∈ Ω ] is (B([0, ∞)) × F, F)-measurable. The mapping [ [0, ∞) × X + × Ω 3 (t, u0 , ω) 7→ u(t; u0 , E(ω)) ∈ X + ] is the composition of the mapping (Id[0,∞) , IdX + , E) (which is (B([0, ∞)) × B(X + ) × F, B([0, ∞)) × B(X + ) × B(Z))-measurable) and the continuous mapping [ [0, ∞) × X + × Z 3 (t, u0 , g) 7→ u(t; u0 , g) ∈ X + ]. For t ≥ 0, u0 ∈ X + , and ω ∈ Ω we will write u(t; u0 , ω) instead of u(t; u0 , E(ω)). Similarly, for t0 ∈ R, t ≥ t0 , u0 ∈ X + , and ω ∈ Ω we will write u(t; t0 , u0 , ω) instead of u(t − t0 ; u0 , E(θt0 ω)). DEFINITION 7.2.1 (Uniform persistence) (7.2.1) is said to be uniformly persistent if there is η0 > 0 such that for any u0 ∈ X + \ {0} there is τ = τ (u0 ) > 0 with the property that u(t; t0 , u0 , ω) ≥ η0 ϕprinc

for P-a.e. ω ∈ Ω, all t0 ∈ R, and all t ≥ t0 + τ.

It should be noted that the adjective “uniform” in the above definition means that η0 > 0 is independent of u0 ∈ X + \ {0} (sometimes such a property is referred to as permanence). However, in Definition 7.2.1 we have also uniformity with respect to the initial time t0 ∈ R. It follows that our uniform persistence is both in the pullback as well as in the forward sense (compare, e.g., [71]). Put ˜ Z0 := supp P (7.2.6) ˜ ) > 0). Z0 (g ∈ Z0 if and only if for any neighborhood U of g in Z one has P(U ˜ 0 ) = 1. is a closed (hence compact) and {ζt }-invariant subset of Z, with P(Z

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Also, Z0 is connected, since otherwise there would exist two open sets U1 , U2 ⊂ Z such that Z0 ∩ U1 and Z0 ∩ U2 are nonempty, compact and disjoint, and their union equals Z0 . The sets Z0 ∩ U1 and Z0 ∩ U2 are invariant, and, by the ˜ definition of support, each of them has P-measure positive, which contradicts ˜ the ergodicity of P. LEMMA 7.2.3 There exists Ω0 ⊂ Ω with P(Ω0 ) = 1 such that Z0 = cl {E(θt ω) : t ∈ R } for any ω ∈ Ω0 , where the closure is taken in the open-compact topology on Z. PROOF

A proof is a copy of the proof of Lemma 4.1.2.

Note that {0} × Z0 is invariant under Φ. Consider the linearization of Φ at {0} × Z0 . Let Y0 := p0 (Z0 ). Let Π(t; v0 , g0 ) := (Ug0 (t, 0)v0 , σt g0 ),

t ≥ 0,

v0 ∈ X,

g0 ∈ Z0 ,

where Ug0 (t, 0)v0 = ∂2 u(t; 0, g)v0 . Denote the principal spectrum of Π over Y0 by [λmin (f ), λmax (f )]. Let Ω0 be as in Lemma 7.2.3. THEOREM 7.2.1 (Uniform persistence) Assume that λmin (f ) > 0. Then there is η0 > 0 such that for each nonzero u0 ∈ X + there exists τ = τ (u0 ) > 0 with the property that u(t; t0 , u0 , ω)(x) ≥ ¯ η0 ϕprinc (x) for all t0 ∈ R, t ≥ t0 + τ , ω ∈ Ω0 , and x ∈ D. PROOF An application of Theorem 7.1.9 to the compact set {u0 } × Z0 ⊂ (X + \{0})×Z0 gives the existence of T > 0 such that u(t; t0 , u0 , ω) ≥ η0 ϕprinc for any t0 ∈ R, any t ≥ t0 + T and any ω ∈ Ω0 .

7.2.2

The Nonautonomous Case

Consider the following nonautonomous parabolic equation of Kolmogorov type:   ∂u = ∆u + f (t, x, u)u, t > 0, x ∈ D, ∂t (7.2.7)  Bu = 0, t > 0, x ∈ ∂D, ¯ × [0, ∞) → R. where f : R × D We assume ¯ (A7-N1) (Regularity) For any M > 0 the restrictions to R× D×[0, M ] of f and all the derivatives of the function f up to order 1 belong to C 1−,1−,1− (R× ¯ × [0, M ]). D

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(A7-N2) There are P > 0 and a function m : [P, ∞) → (0, ∞) such that ¯ and any u ≥ P . f (t, x, u) ≤ −m(u) for any t ∈ R, any x ∈ D From now on, until the end of the present subsection, assume that (A7-N1) and (A7-N2) are satisfied. Put Z := cl { f · t : t ∈ R } (7.2.8) with the open-compact topology, where the closure is taken in the open-compact topology. It is a consequence of (A7-N1) via the Ascoli–Arzel`a theorem that the set Z is a compact metrizable space. We deduce from Lemma 7.1.1 that if g ∈ Z then g · t ∈ Z for all t ∈ R. Hence (Z, {ζt }t∈R ) is a compact flow. We will consider a family of Eqs. (7.1.1) parameterized by g ∈ Z. We claim that Assumptions (A7-1) through (A7-3) are fulfilled. Indeed, (A7-1) is clearly satisfied. It follows from (A7-N1) through the Ascoli–Arzel`a theorem that (A7-2) is satisfied. The satisfaction of (A7-3) follows from (A7-N2). We denote by Φ = {Φt }t≥0 the topological skew-product semiflow generated by (7.2.7) on the product Banach bundle X + × Z: Φ(t; u0 , g) = Φt (u0 , g) := (u(t; u0 , g), ζt g),

t ≥ 0, g ∈ Z, u0 ∈ X + , (7.2.9)

where u(t; u0 , g) stands for the solution of (7.1.1) with initial condition u(0; u0 , g)(x) = u0 (x). DEFINITION 7.2.2 (Uniform persistence) (7.2.7) is said to be uniformly persistent if there exists η0 > 0 such that for any u0 ∈ X + \ {0} there is τ = τ (u0 ) > 0 with the property that u(t; t0 , u0 , f ) ≥ η0 ϕprinc

for all

t0 ∈ R

and all

t ≥ t0 + τ (u0 ).

Note that Z is connected and {0} × Z is invariant under Φ. Consider the linearization of Φ at {0} × Z. Let Y := p0 (Z). Let Π(t; v0 , g0 ) := (Ug0 (t, 0)v0 , σt g0 ),

t ≥ 0,

v0 ∈ X,

g0 ∈ Y,

where Ug0 (t, 0)v0 = ∂2 u(t; 0, g)v0 . Denote the principal spectrum of Π over Y by [λmin (f ), λmax (f )]. THEOREM 7.2.2 (Uniform persistence) Assume that λmin (f ) > 0. Then there is η0 > 0 such that for each nonzero u0 ∈ X + there exists τ = τ (u0 ) > 0 with the property that u(t; t0 , u0 , f )(x) ≥ ¯ η0 ϕprinc (x) for all t0 ∈ R, t ≥ t0 + τ , and x ∈ D. PROOF

We apply Theorem 7.1.9(2).

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COROLLARY 7.2.1 If for any time averaged function fˆ0 of f0 (t, x) := f (t, x, 0) there holds λ(fˆ0 ) > 0, then there exists η0 > 0 such that for each nonzero u0 ∈ X + there is τ = τ (u0 ) > 0 with the property that u(t; t0 , u0 , f )(x) ≥ η0 ϕprinc (x) for all ¯ t0 ∈ R, t ≥ t0 + τ and x ∈ D. PROOF

7.3

It follows from Theorems 5.2.2 and 7.2.2.

Competitive Kolmogorov Systems of Semilinear Equations: General Theory

In the present section we consider families of competitive Kolmogorov systems of semilinear second order parabolic equations:    ∂u1 = ∆u1 + g1 (t, x, u1 , u2 )u1 , x ∈ D,   ∂t     ∂u2 = ∆u2 + g2 (t, x, u1 , u2 )u2 , x ∈ D, (7.3.1) ∂t     Bu1 = 0, x ∈ ∂D,    Bu = 0, x ∈ ∂D, 2 where B is a boundary operator of either the Dirichlet or Neumann type as in (7.0.1), and g = (g1 , g2 ) belongs to a certain set of functions. Sometimes we write (7.3.1) as (7.3.1)g . We consider the existence, uniqueness, and basic properties of solutions in Subsection 7.3.1. In Subsection 7.3.2 we study the linearizations of (7.3.1) at trivial and semitrivial solutions. Global attractor and uniform persistence of (7.3.1) are investigated in Subsection 7.3.3.

7.3.1

Existence, Uniqueness, and Basic Properties of Solutions

In this subsection, we present the existence, uniqueness, and basic properties of solutions of (7.3.1) in appropriate fractional power spaces of the operator ∆ × ∆ (with corresponding boundary conditions) with admissible g(·, ·, ·) = (g1 (·, ·, ·), g2 (·, ·, ·)). Those properties which are similar to the scalar equations case will be stated without proofs. ¯ × [0, ∞) → R2 As in Subsection 7.1.1, for a continuous function g : R × D and t ∈ R denote by g · t the time-translate of g, g · t(s, x, u) := g(s + t, x, u) ¯ and u ∈ [0, ∞) × [0, ∞). for s ∈ R, x ∈ D, ¯ × [0, ∞) × Let g(n) (n ∈ N) and g be continuous functions from R × D [0, ∞) to R2 . Recall that a sequence g(n) converges to g in the open-compact

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topology if and only if for any M > 0 the restrictions of g(n) to [−M, M ] × ¯ × [0, M ] × [0, M ] converge uniformly to the restriction of g to [−M, M ] × D ¯ × [0, M ] × [0, M ]. D Denote by Z the set of admissible functions g = (g1 , g2 ) for (7.3.1). We assume that Z satisfies (A7-5) (1) Z is compact in the open-compact topology. (2) Z is translation invariant: If g ∈ Z then g · t ∈ Z, for each t ∈ R. By (A7-5), (Z, {ζt }t∈R ) is a compact flow, where ζt g = g · t for g ∈ Z and t ∈ R. (A7-6) (Regularity) For any g = (g1 , g2 ) ∈ Z and any M > 0 the restric¯ × [0, M ] × [0, M ] of g1 , g2 , and all the derivatives of g1 and tions to R × D ¯ × [0, M ] × [0, M ]). Moreover, g2 up to order 1 belong to C 1−,1−,1−,1− (R × D 1−,1−,1−,1− ¯ for M > 0 fixed the C (R × D × [0, M ])-norms of the restrictions of g1 , g2 , and those derivatives are bounded uniformly in Z. For each g = (g1 , g2 ) ∈ Z we denote: G(t, x, u) = (G1 (t, x, u1 , u2 ), G2 (t, x, u1 , u2 )) := (g1 (t, x, u1 , u2 )u1 , g2 (t, x, u1 , u2 )u2 ) ¯ u = (u1 , u2 ) ∈ [0, ∞) × [0, ∞). for t ∈ R, x ∈ D, As in Subsection 7.1.1, we denote by N the Nemytski˘ı (substitution) operator: N(t, u, g)(x) = (N1 (t, u, g)(x), N2 (t, u, g)(x)) := G(t, x, u(x)),

¯ x ∈ D,

¯ → R × R, and g ∈ Z. where t ∈ R, u : D ¯ + × C(D) ¯ + ) × Z. We consider N to be a mapping defined on R × (C(D) + + ¯ ¯ It is straightforward to see that N takes R × (C(D) × C(D) ) × Z into ¯ × C(D). ¯ C(D) Let Ap stand the realization of the operator ∆ (with corresponding boundary conditions) in Lp (D). For 1 N and 21 + 2p β < 1, and put X := Fpβ . (7.3.2) There holds ¯ X ,− ,→ C 1 (D). Let X = X × X, +

+

(7.3.3) +

X =X ×X , and X++ = X ++ × X ++ .

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Note that X++ is the interior of X+ . Recall that for u = (u1 , u2 ), v = (v1 , , v2 ) ∈ X, u≤v u
if

if

(v1 − u1 , v2 − u2 ) ∈ X+ ,

(v1 − u1 , v2 − u2 ) ∈ X+ \ {(0, 0)},

and uv

if

(u1 − v1 , u2 − v2 ) ∈ X++ .

Let ≤2 be the order in X defined as follows: for u = (u1 , u2 ), v = (v1 , , v2 ) ∈ X, u ≤2 v if (v1 − u1 , u2 − v2 ) ∈ X+ , u <2 v

if

(v1 − u1 , u2 − v2 ) ∈ X+ \ {(0, 0)},

and u 2 v

if

(v1 − u1 , u2 − v2 ) ∈ X++ .

DEFINITION 7.3.1 For t0 ∈ R, u0 = (u10 , u20 ) ∈ X+ , and g = (g1 , g2 ) ∈ Z by a solution of (7.3.1)g satisfying the initial condition u(t0 ) = (u1 (t0 , ·), u2 (t0 , ·)) = u0 we mean a continuous function u = (u1 , u2 ) : J → X, where J is a nondegenerate interval with inf J = t0 ∈ J, satisfying the following: • u(t0 ) = u0 , • u(t) ∈ Fp1 × Fp1 for each t ∈ J \ {t0 }, • u(·) is differentiable, as a function into Lp (D) × Lp (D), on J \ {t0 }, • there holds (

u˙ 1 (t) = Ap u1 (t) + N1 (t, u1 (t), u2 (t), g) u˙ 2 (t) = Ap u2 (t) + N2 (t, u1 (t), u2 (t), g)

for each t ∈ J \ {t0 }. A solution u = (u1 , u2 ) of (7.3.1)g satisfying u(t0 ) = u0 is nonextendible if there is no solution u∗ of (7.3.1)g defined on an interval J ∗ with sup J ∗ > sup J such that u∗ |J ≡ u. Similarly to Propositions 7.1.1 and 7.1.3, we have PROPOSITION 7.3.1 (Existence and uniqueness of solution) Let (A7-5)–(A7-6) be satisfied. Then for each g ∈ Z, each t0 ∈ R, and each u0 = (u10 , u20 ) ∈ X+ there exists a unique nonextendible solution of (7.3.1)g satisfying the initial condition u(t0 , ·) = u0 , defined on an interval of the form

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[t0 , τmax ), where τmax = τmax (t0 , u0 , g) > t0 . Moreover, for any 1 < p < ∞ the unique nonextendible solution u(·) := u(·; t0 , u0 , g) satisfies Z t u(t) = e(Ap ×Ap )(t−t0 ) u0 + e(Ap ×Ap )(t−s) N(s, u(s), g) ds, t0

for t ∈ (t0 , τmax (t0 , u0 , g)). Now we formulate several analogs of results from Subsection 7.1.1. We give (indications of) proofs only when they differ from the proofs of the corresponding results for scalar equations. Similarly to Proposition 7.1.2, we have PROPOSITION 7.3.2 (Positivity) Assume (A7-5)–(A7-6). Given t0 ∈ R, u0 ∈ X+ , and g ∈ Z, there holds u(t; t0 , u0 , g) ∈ X+ for any t ∈ [t0 , τmax (t0 , u0 , g)). By arguments as in Lemmas 7.1.9 and 7.1.10, we can prove u(t + t0 ; t0 , u0 , g) = u(t; 0, u0 , g · t0 )

(7.3.4)

for t ∈ (0, τmax (t0 , u0 , g) − t0 ) and u(t; t0 , u(t0 ; 0, u0 , g), g) = u(t; 0, u0 , g)

(7.3.5)

for t0 , t ∈ (0, τmax (0, u0 , g)) with t0 ≤ t. In the following we write u(t; u0 , g) for u(t; 0, u0 , g). By (7.3.4) and (7.3.5), for any u0 ∈ X+ and g ∈ Z there holds u(t + s; u0 , g) = u(t; u(s; u0 , g), ζs g) for s ∈ [0, τmax (0, u0 , g)) and t ∈ [0, τmax (s, u(s; u0 , g), ζs g)). τmax (u0 , g) for τmax (0, u0 , g). Similarly to Proposition 7.1.4, we have

(7.3.6) We write

PROPOSITION 7.3.3 (Regularity) Let (A7-5) and (A7-6) be satisfied. Then for each u0 ∈ X+ and each g ∈ Z, u(t, x) = u(t; u0 , g)(x) is a classical solution, that is, •

∂u ∂t (t, ·)

¯ × C(D) ¯ for each t ∈ (0, τmax (u0 , g)), ∈ C(D)

¯ × C 2 (D) ¯ for each t ∈ (0, τmax (u0 , g)), • u(t, ·) ∈ C 2 (D) • for any t ∈ (0, τmax (u0 , g)) and x ∈ D the equation in (7.3.1) is satisfied pointwise, • for any t ∈ (0, τmax (u0 , g)) and x ∈ ∂D the boundary condition in (7.3.1) is satisfied pointwise.

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Similarly to Proposition 7.1.6 we have PROPOSITION 7.3.4 (Continuous dependence) Let (A7-5)–(A7-6) be satisfied. Then the mapping [ [0, τmax (u0 , g)) × X+ × Z 3 (t, u0 , g) 7→ u(t; u0 , g) ∈ X+ ] is continuous. Similarly to Proposition 7.1.10, we have PROPOSITION 7.3.5 (Differentiability) Assume (A7-5)–(A7-6). Then the derivative ∂2 u of the mapping [ [0, τmax (u0 , g)) × X+ × Z 3 (t, u0 , g) 7→ u(t; u0 , g) ∈ X+ ] with respect to the u0 -variable exists and is continuous on the set (0, τmax (u0 , g))× X+ ×Z. For u0 = (u01 , u02 ) ∈ X+ and v0 = (v01 , v02 ) ∈ X the function [ t 7→ v(t) ], where v(t) = (v1 (t), v2 (t)) := ∂2 u(t; u0 , g)v0 , is a classical solution of the system of parabolic equations  ∂v ∂G1 1  ∂t = ∆v1 + ∂u1 (t, x, u(t; u0 , g)(x))v1     1  + ∂G t > 0, x ∈ D,  ∂u2 (t, x, u(t; u0 , g)(x))v2 ,     ∂v2 = ∆v + ∂G2 (t, x, u(t; u , g)(x))v 2 0 1 ∂t ∂u1 (7.3.7) ∂G2   (t, x, u(t; u , g)(x))v , t > 0, x ∈ D, + 0 2  ∂u 2      Bv = 0, t > 0, x ∈ ∂D, 1    Bv2 = 0, t > 0, x ∈ ∂D, with initial conditions v1 (0) = v01 , v2 (0) = v02 . The following is a competition assumption. (A7-7) (Strong competitiveness) (∂g1 /∂u2 )(t, x, u1 , u2 ) < 0 and (∂g2 /∂u1 )(t, x, u1 , u2 ) < 0 for all g ∈ Z, t ∈ R, ¯ and (u1 , u2 ) ∈ [0, ∞) × [0, ∞). x ∈ D,

PROPOSITION 7.3.6 (Order preserving) Assume (A7-5)–(A7-7). Let g ∈ Z. (1) If (u1 , u2 ), (v1 , v2 ) ∈ X+ , (u1 , u2 ) ≤2 (v1 , v2 ), then u(t; (u1 , u2 ), g) ≤2 u(t; (v1 , v2 ), g) for each t ∈ [0, τmax ((u1 , u2 ), g)) ∩ [0, τmax ((v1 , v2 ), g)).

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(2) If (u1 , u2 ), (v1 , v2 ) ∈ X + , (u1 , u2 ) <2 (v1 , v2 ) and v1 > 0 or u2 > 0, then u(t; (u1 , u2 ), g) 2 u(t; (v1 , v2 ), g) for each t ∈ (0, τmax ((u1 , u2 ), g)) ∩ (0, τmax ((v1 , v2 ), g)). PROOF It follows from Proposition 7.3.2 and comparison principle for parabolic equations. Here is the assumption which guarantees the global existence of solutions. (A7-8) There is P > 0 such that g1 (t, x, u1 , u2 ) < 0 for any g ∈ Z, any t ∈ R, ¯ and any u1 ≥ P , any u2 ∈ [0, ∞); and g2 (t, x, u1 , u2 ) < 0 for any any x ∈ D, ¯ and any u1 ∈ [0, ∞), u2 ≥ P . g ∈ Z, any t ∈ R, any x ∈ D, The following proposition follows from Proposition 7.1.5. PROPOSITION 7.3.7 (Semitrivial solutions) Assume (A7-5)–(A7-8). (1) For any u0 ∈ X+ × {0}, u(t; u0 , g) exists and u(t; u0 , g) ∈ X+ × {0} for all t ≥ 0 and g ∈ Z (such u(t; u0 , g) is called a semitrivial solution). (2) For any u0 ∈ {0} × X + , u(t; u0 , g) exists and u(t; u0 , g) ∈ {0} × X + for all t ≥ 0 and g ∈ Z (such u(t; u0 , g) is also called a semitrivial solution). PROPOSITION 7.3.8 (Global existence) Assume (A7-5) through (A7-8). Then for any u0 ∈ X+ and any g ∈ Z the unique nonextendible solution u(·; u0 , g) is defined on [0, ∞). PROOF we have

For any u0 = (u01 , u02 ) ∈ X+ and g ∈ Z, by Proposition 7.3.6,

u(t; (0, u02 ), g) ≤2 u(t; (u01 , u02 ), g) ≤2 u(t; (u01 , 0), g) for all t ∈ [0, τmax (u0 , g)). An application of Theorem 7.1.7 to both the first coordinate of u(·; (u01 , 0), g) and the second coordinate of u(·; (0, u02 ), g) gives together with the above inequality that the set { ku(t; u0 , g)kC(D)×C( ¯ ¯ : t ∈ D) [0, τmax (u0 , g)) } is bounded. The remainder of the proof goes along the lines of the proof of Proposition 7.1.5. Similarly to Proposition 7.1.8, we have

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PROPOSITION 7.3.9 (Compactness) Assume (A7-5) through (A7-8). Then for any δ0 > 0 and any B ⊂ X+ ¯ × C(D)-norm, ¯ bounded in the C(D) the set { u(t; u0 , g) : t ≥ δ0 , u0 ∈ B, g ∈ Z } has compact closure in the X-norm. In the rest of this subsection, we assume (A7-5)-(A7-8). Let Φ(t; u0 , g) = Φt (u0 , g) := (u(t; u0 , g), ζt g)

(7.3.8)

for t ≥ 0, u0 ∈ X+ , and g ∈ Z. Then Φ = {Φt }t≥0 is a topological skewproduct semiflow on the product bundle X+ × Z covering the topological flow (Z, ζ). In the sequel an important role will be played by the set e + := (X + × {0}) ∪ ({0} × X + ). ∂X e + is a proper subset of X+ \ X++ (= the boundary of X+ in Notice that ∂X X). Indeed, let a nonzero u0 ∈ X + \ X ++ . Then (u0 , u0 ) ∈ X+ \ X++ but e +. (u0 , u0 ) 6∈ ∂X e + × Z and (X+ \ ∂X e +) × Z By Propositions 7.3.7 and 7.3.6(2), the sets ∂X are forward invariant. The property in Proposition 7.3.6(2) can be written as: The restriction Φ|(X+ \∂X e + )×Z is strongly monotone with respect to the order relation ≤2 (cp. [57]). Further, let ∂Φ(t; v0 , (u0 , g)) = ∂Φt (v0 , (u0 , g)) := (∂2 u(t; u0 , g)v0 , (u(t; u0 , g), ζt g)),

(7.3.9)

where t ≥ 0, v0 ∈ X, u0 ∈ X+ , and g ∈ Z. Proposition 7.3.5 guarantees that ∂Φ = {∂Φt }t≥0 is a topological linear skew-product semiflow on the product Banach bundle X × (X+ × Z) covering the topological semiflow Φ on X+ × Z.

7.3.2

Linearization at Trivial and Semitrivial Solutions

In this subsection, we consider the linearization of (7.3.1) at trivial and semitrivial solutions. Throughout this subsection we assume (A7-5)–(A7-8). Most results in this subsection follow from the general theories developed in Chapters 2, 3, and 4. We start by considering the linearization at trivial solution. Note that {0} × Z is invariant under Φ. We introduce p0 : Z → L∞ (R × D, R) by p0 (g) := g0 , where g0 (t, x) := g(t, x, 0, 0) for g ∈ Z. We further introduce p01 : Z → L∞ (R × D, R) by p01 (g) := g01 ,

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where g01 (t, x) := g1 (t, x, 0, 0) for g = (g1 , g2 ) ∈ Z, and p02 : Z → L∞ (R × D, R) by p02 (g) := g02 , where g02 (t, x) := g2 (t, x, 0, 0) for g = (g1 , g2 ) ∈ Z. Denote by Y0 , Y 01 , and Y 02 the images of p0 , p01 , and p02 , respectively. Consider the restriction of the topological linear skew-product semiflow ∂Φ to X × ({0} × Z): ∂Φt (v0 , (0, g)) = (∂2 u(t; 0, g)v0 , (0, g · t)),

t ≥ 0, v0 ∈ X, g ∈ Z.

It follows from Proposition 7.1.10 that for any v0 = (v01 , v02 ) ∈ X and any g = (g1 , g2 ) ∈ Z the function [ (0, ∞) 3 t 7→ ∂2 u(t; 0, g)v0 ∈ X ] is given by the classical solution v(·) = (v1 (·), v2 (·)) of ( ∂vi t > 0, x ∈ D, ∂t = ∆vi + g0i (t, x)vi , (7.3.10) Bvi = 0, t > 0, x ∈ ∂D, with initial condition vi (0) = v01 , i = 1, 2. We sometimes write (7.3.10) as (7.3.10)i , i = 1, 2. Let Ug0i0i (t, 0) : L2 (D) → L2 (D), i = 1, 2, be the weak solution operator of (7.3.10)i . We have ∂2 u(t; 0, g)v0 = (Ug01 (t, 0)v01 , Ug02 (t, 0)v02 ) 01 02

(7.3.11)

for any v0 = (v01 , v02 ) ∈ X and t ≥ 0. (t, 0), respec(t, 0) and Ug02 We may write Ug010 (t, 0) and Ug020 (t, 0) for Ug01 02 01 tively, if no confusion occurs, where g0 = (g01 , g02 ) = p0 (g). Let Z0 be a nonempty connected compact translation invariant subset of 0 0 Z. Denote Y01 := p01 (Z0 ) and Y02 := p02 (Z0 ). Let Π0i (t; v0i , g0i ) := (Ug0i0i (t, 0)v0i , g0i · t) for t ≥ 0, v0i ∈ L2 (D), and g0i ∈ Y0i0 , i = 1, 2. Similarly to Theorem 7.1.1, following from Theorem 3.3.3 we have THEOREM 7.3.1 Let i = 1 or 2. There exist 0 • an invariant (under Π0i ) one-dimensional subbundle Xi,1 of L2 (D) × 0 0 0 0 Y0i with fibers Xi,1 (g0i ) = span{wi (g0i )}, where wi : Y0i0 → L2 (D) is continuous, with kwi0 (g0i )k = 1 for all g0i ∈ Y0i0 , and

• an invariant (under Π0i ) complementary one-codimensional subbundle 0 0 Xi,2 of L2 (D) × Y0i0 with fibers Xi,2 (g0i ) = { v ∈ L2 (D) : hv, wi0,∗ (g0i )i = 0 }, where wi0,∗ : Y0i0 → L2 (D) is continuous, and for all g0i ∈ Y0i0 , kwi0,∗ (g0i )k = 1,

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having the following properties: (i) wi0 (g0i ) ∈ L2 (D)+ for all g0i ∈ Y0i0 , 0 (ii) Xi,2 (g01 ) ∩ L2 (D)+ = {0} for all g0i ∈ Y0i0 ,

(iii) there are Mi0 ≥ 1 and γi0 > 0 such that for any g0i ∈ Y0i0 and any 0 v0i ∈ Xi,2 (g0i ) with kv0i k = 1, 0

kUg0i0i (t, 0)v0i k ≤ Mi0 e−γi t kUg0i0i (t, 0)wi0 (g0i )k

for

t > 0.

For i = 1, 2, denote by [λ0i,min , λ0i,max ] the principal spectrum interval of (7.3.10)i over Y0i0 . [λ0i,min , λ0i,max ] are referred to as the principal spectrum intervals of Π on Y00 := p0 (Z0 ), or the principal spectrum intervals associated to {0} × Z0 . THEOREM 7.3.2 Assume (A7-5)–(A7-8). For any > 0, there is δ > 0 such that for any ¯ → R2 satisfying |u∗ (t, x)|, |u∗ (t, x)| ≤ δ for all continuous (u∗1 , u∗2 ) : R × D 1 2 ¯ there holds sufficiently large t and all x ∈ D lim inf t→∞

1 ln k˜ u1 (t; t0 , u01 , g1∗ )k ≥ λ01,min − t

and

1 ln k˜ u2 (t; t0 , u02 , g2∗ )k ≥ λ02,min − t for any t0 ∈ R and any u01 , u02 ∈ X + \ {0}, where u ˜i (·; t0 , u0i , gi∗ ), i = 1, 2, denotes the solution of (7.3.10)i with g0i (t, x) replaced by gi∗ (t, x) = gi (t, x, u∗1 (t, x), u∗2 (t, x)) and the initial condition u ˜i (t0 ; t0 , u0i , gi∗ ) = u0i . lim inf t→∞

PROOF Fix g = (g1 , g2 ) ∈ Z0 . Since, by (A7-6), the derivatives ∂gi /∂ui , ¯ × [0, M ] × [0, M ], i, j = 1, 2, are bounded uniformly on sets of the form R × D ¯ and for any > 0 there is δ > 0 such that if |u∗1 (t, x)|, |u∗2 (t, x)| ≤ δ for x ∈ D t sufficiently large, then gi∗ (t, x) ≥ gi (t, x, 0, 0) − for i = 1, 2 and t sufficiently large. Without loss of generality we may assume ¯ Then by the that gi∗ (t, x) ≥ gi (t, x, 0, 0) − for all t ≥ t0 and x ∈ D. comparison principle for parabolic equations, we have u ˜1 (t; t0 , u01 , g1∗ ) ≥ e−(t−t0 ) Ug010 (t, t0 )u01 and u ˜2 (t; t0 , u02 , g2∗ ) ≥ e−(t−t0 ) Ug020 (t, t0 )u02

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for t > t0 . By Theorem 3.1.2 and Lemma 3.2.5, lim inf

1 ln kUg010 (t, t0 )u01 k ≥ λ01,min t

lim inf

1 ln kUg020 (t, t0 )u02 k ≥ λ02,min . t

t→∞

and t→∞

The theorem thus follows. The following assumption is about the repelling property of the trivial solution (u1 (t), u2 (t)) ≡ (0, 0). (A7-9) λ0i,min > 0, i = 1, 2. By Theorem 7.1.9, we have THEOREM 7.3.3 Assume (A7-5)–(A7-9). There are nonempty compact invariant sets Γ1 ⊂ (X ++ × {0}) × Z0 and Γ2 ⊂ ({0} × X ++ ) × Z0 such that Γ1 attracts any point in ((X + \{0})×{0})×Z0 and Γ2 attracts any point in ({0}×(X + \{0}))×Z0 . In the rest of this subsection we assume that (A7-5)–(A7-9) hold. We also make the following assumption. (A7-10) (∂g1 /∂u1 )(t, x, u1 , u2 ) < 0 and (∂g2 /∂u2 )(t, x, u1 , u2 ) < 0 for all ¯ and (u1 , u2 ) ∈ [0, ∞) × [0, ∞). g ∈ Z0 , t ∈ R, x ∈ D, In the literature, systems satisfying both (A7-7) and (A7-10) are called totally competitive. We will investigate now the linearization of Φ at semitrivial solutions in Γ1 . By Theorem 7.1.12, there exists a continuous function ξ 1 : Z0 → X ++ × {0} such that Γ1 = { (ξξ 1 (g), g) : g ∈ Z0 }. For g ∈ Z0 denote g12 (t, x) := g2 (t, x, u1 (t; ξ 1 (g), g)(x), 0)

¯ t ∈ R, x ∈ D.

Further, define a mapping p12 : Γ1 → L∞ (R × D, R) as p12 (ξξ 1 (g), g) := g12 (t, x). Put Y1 := p12 (Γ1 ). Observe that at any (ξξ 1 (g), g) ∈ Γ1 the second coordinate of the linearized equation (7.3.7) takes the form ( ∂v2 ∂t = ∆v2 + g12 (t, x)v2 , t > 0, x ∈ D, (7.3.12) Bv2 = 0, t > 0, x ∈ ∂D. For g12 ∈ Y1 denote by Ug12 (t, 0)v02 the weak solution operator of (7.3.12). 12 Let Π12 (t; v02 , g12 ) := (Ug12 (t, 0)v02 , g12 · t) 12

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for t ≥ 0, v02 ∈ L2 (D), and g12 ∈ Y1 . By the theory developed in Chapter 3, (7.3.12) or Π12 admits an exponential separation over Y1 . Namely, we have THEOREM 7.3.4 There exist • an invariant (under Π12 ) one-dimensional subbundle X11 of L2 (D) × Y1 with fibers X11 (g12 ) = span {w1 (g12 )}, where w1 : Y1 → L2 (D) is continuous, with kw1 (g12 )k = 1 for all g12 ∈ Y1 , and • an invariant (under Π12 ) complementary one-codimensional subbundle X21 of L2 (D) × Y1 with fibers X21 (g12 ) = { v ∈ L2 (D) : hv, w1,∗ (g12 )i = 0 }, where w1,∗ : Y1 → L2 (D) is continuous, with kw1,∗ (g12 )k = 1 for all g12 ∈ Y1 , having the following properties: (i) w1 (g12 ) ∈ L2 (D)+ for all g12 ∈ Y1 , (ii) X21 (g12 ) ∩ L2 (D)+ = {0} for all g12 ∈ Y1 , (iii) there are M 1 ≥ 1 and γ 1 > 0 such that for any g12 ∈ Y1 and any v0 ∈ X21 (g12 ) with kv0 k = 1, 1

kUg12 (t, 0)v0 k ≤ M 1 e−γ t kUg12 (t, 0)w1 (g12 )k 12 12

for

t > 0.

We denote by [λ1min , λ1max ] the principal spectrum interval of (7.3.12) over Y1 . [λ1min , λ1max ] is also referred to as the principal spectrum interval associated to Γ1 . THEOREM 7.3.5 ¯ → R satisfies For any > 0, if a continuous g2∗ : R × D |g2∗ (t, x) − g12 (t, x)| < ¯ and t sufficiently large, then there holds for some g ∈ Z0 , all x ∈ D, lim inf t→∞

1 ln k˜ u2 (t; t0 , u02 , g2∗ )k > λ1min − , t

where u02 ∈ X + \ {0} and u ˜2 (t; t0 , u02 , g2∗ ) is the solution of (7.3.12) with g12 ∗ replaced by g2 and u ˜2 (t0 ; t0 , u02 , g2∗ ) = u02 . ¯ PROOF If g2∗ satisfies |g2∗ (t, x) − g12 (t, x)| < for some g ∈ Z0 , all x ∈ D, and sufficiently large t, then g2∗ (t, x) ≥ g12 (t, x) −

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¯ and large t. Without loss of generality, we may assume that for x ∈ D ∗ ¯ and all t ≥ t0 . Then we have g2 (t, x) ≥ g12 (t, x) − for all x ∈ D u ˜2 (t; t0 , u02 , g2∗ ) ≥ e−(t−t0 ) Ug12 (t, t0 )u02 12 for any u01 ∈ X + and t ≥ t0 . By Theorem 3.1.2 and Lemma 3.2.5, lim inf t→∞

1 ln kUg12 (t, t0 )u02 k ≥ λ1min . 12 t

The theorem thus follows. Results analogous to those presented above hold for semi-trivial solutions in Γ2 . We will not write them down. For reference, we denote by [λ2min , λ2max ] the principal spectrum interval associated to Γ2 . The next assumption will be used in the investigation of uniform persistence. (A7-11) λimin > 0, i = 1, 2.

7.3.3

Global Attractor and Uniform Persistence

In this subsection, we study the global attractor and uniform persistence for (7.3.1). First we study global attractor. We denote [0, P ]X × [0, P ]X := { u = (u1 , u2 ) ∈ X : ¯ i = 1, 2 }. 0 ≤ ui (x) ≤ P for all x ∈ D, The set [0, P ]X × [0, P ]X is convex and closed (in X). THEOREM 7.3.6 (Absorbing set) ¯ Assume (A7-5)–(A7-8). Suppose that B ⊂ X+ is bounded in the C(D)× ¯ C(D)-norm. Then there is T = T (B) ≥ 0 such that ui (t; u0 , g)(x) ≤ P for all ¯ i = 1, 2. Moreover, if B ⊂ [0, P ]X ×[0, P ]X t ≥ T , u0 ∈ B, g ∈ Z, and x ∈ D, then T (B) can be taken to be zero. PROOF By Theorem 7.1.7, there is T (B) > 0 such that for any u0 = (u01 , u02 ) ∈ B and g ∈ Z, Φt ((u01 , 0), g) ∈ [0, P ]X × {0} and Φt ((0, u02 ), g) ∈ {0} × [0, P ]X for any t ≥ T (B) and g ∈ Z. Now by Proposition 7.3.6, Φt ((0, u02 ), g) ≤2 Φt ((u01 , u02 ), g) ≤2 Φt ((u01 , 0), g) for all t > 0. It then follows that Φt (u0 , g) ∈ [0, P ]X × [0, P ]X for any t ≥ T (B), u0 ∈ B and g ∈ Z.

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THEOREM 7.3.7 (Global attractor) Assume (A7-5)–(A7-8). Then the topological skew-product semiflow Φ possesses a global attractor Γ contained in ([0, P ]X × [0, P ]X ) × Z. In addition, ¯ × C(D)-norm ¯ for any B ⊂ X+ bounded in the C(D) one has (1) ∅ = 6 ω(B × Z) (⊂ Γ), (2) Γ attracts B × Z. PROOF A proof can be obtained by rewriting the proof of Theorem 7.1.10 word for word, only with Proposition 7.1.8 replaced by Proposition 7.3.9 and Theorem 7.1.7 replaced by Theorem 7.3.6. e + = (X + × {0}) ∪ ({0} × X + ). Put Recall that ∂X e + × Z). Γ∂e := Γ ∩ (∂X It follows immediately from Theorem 7.3.7 that Γ∂e is the global attractor for the restriction of the skew-product semiflow Φ to the forward invariant set e + × Z. By Proposition 7.1.13, Φ|Γ is a topological flow. ∂X ˜ ∂ Let Z0 be a nonempty connected compact invariant subset of Z. If (A7-9) is fulfilled then Γ1 , Γ2 ⊂ Γ ∩ (X+ × Z0 ). We proceed now to study uniform persistence. DEFINITION 7.3.2 (Uniform persistence) (7.3.1) is said to be uniformly persistent over Z0 if there is η0 > 0 such that for any u0 ∈ X+ \ ((X + × {0}) ∪ ({0} × X + )) there is τ (u0 ) > 0 with the property that ui (t; u0 , g) ≥ η0 ϕprinc

for

i = 1, 2,

all

t ≥ τ (u0 ), g ∈ Z0 .

THEOREM 7.3.8 (Uniform persistence) Let (A7-5) through (A7-11) be satisfied. Then (7.3.1) is uniformly persistent over Z0 . To prove the theorem, we first prove some lemmas. Define Γ0 := { (0, g) : g ∈ Z0 }. Note that Γ0 , Γ1 and Γ2 are compact invariant subsets of Γ∂e ∩ (X+ × Z0 ). LEMMA 7.3.1 Assume that the conditions in Theorem 7.3.8 hold. Then there is δ0 > 0 such that if for some u0 ∈ X+ and g ∈ Z0 there holds ku(t; u0 , g)kX < δ0 for all t ≥ 0, then u0 = 0. In particular, Γ0 is an isolated invariant set for Φ|X+ ×Z0 .

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PROOF Take 0 > 0 be such that λ0i,min − 0 > 0 for i = 1, 2. Let δ0 = 0 . Suppose to the contrary that there are u0 = (u01 , u02 ) ∈ X+ \ {0} with ku0 kX < δ0 and g ∈ Z0 such that ku(t; u0 , g)kX < δ0 for all t ≥ 0. Without loss of generality, assume u02 6= 0. Then by Theorem 7.3.2, we have lim inf t→∞

1 ln ku2 (t; u0 , g)k > 0. t

This contradicts the fact that the set { ku2 (t; u0 , g)kX : t ≥ 0 } is bounded. LEMMA 7.3.2 Suppose that the conditions in Theorem 7.3.8 hold. Then, for each i = 1, 2, (i) there is δi > 0 such that the situation is impossible that d(Π(t, u0 , g), Γi ) < e +, δi for all t ≥ 0 but u0 6∈ ∂X (ii) Γi is an isolated invariant set for Φ|X+ ×Z0 . PROOF We prove the lemma for Γ1 . Similarly, we can prove that the corresponding results hold for Γ2 . (i) Take 1 > 0 such that λ1min − 1 > 0. Let δ1 > 0 be such that |g2 (t, x, u1 , u2 ) − g2 (t, x, u1 , 0)| <

1 2

for all g ∈ Z0 , t ∈ R, u1 ∈ [0, P ], and u2 ∈ [0, δ1 ] (the existence of such a δ1 follows by (A7-6)). Suppose to the contrary that there are u0 = (u01 , u02 ) ∈ X+ with u02 6= 0 and g ∈ Z0 such that d((u(t; u0 , g), ζt g), Γ1 ) < δ1

for all

t ≥ 0.

Let u∗0 := (u01 , 0). Then u0 ≤2 u∗0 . By Proposition 7.3.6, we have u(t; u0 , g) ≤2 u(t; u∗0 , g)

for all

t > 0,

u1 (t; u0 , g) ≤ u1 (t; u∗0 , g)

for all

t > 0.

hence There holds g2 (t, x,u1 (t; u0 , g)(x), u2 (t; u0 , g)(x)) ≥ g2 (t, x, u1 (t; u∗0 , g)(x), u2 (t; u0 , g)(x))

(7.3.13)

¯ It follows from Theorem 7.3.6 that u1 (t; u∗ , g)(x) ≤ for all t > 0 and all x ∈ D. 0 ¯ Since 0 < u2 (t; u0 , g)(x) < δ1 for P for sufficiently large t > 0 and all x ∈ D. all t > 0, we have 1 2 (7.3.14)

g2 (t, x, u1 (t; u∗0 , g)(x), u2 (t; u0 , g)(x)) > g2 (t, x, u1 (t; u∗0 , g)(x), 0) −

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¯ Further, by Theorem 7.1.12, for sufficiently large t > 0 and all x ∈ D. ∗ ku1 (t; u0 , g) − ξ(ζt g1 )kX → 0 as t → ∞. Therefore g2 (t, x, u1 (t; u∗0 , g)(x), 0) = g2 (t, x, u(t; u∗0 , g)(x)) > g2 (t, x, u(t; ξ(ζt g1 ), 0)(x)) −

1 2

(7.3.15)

¯ Combining Eqs. (7.3.13)–(7.3.15) for sufficiently large t > 0 and all x ∈ D. we obtain that g2 (t, x, u(t; u0 , g)(x)) > g12 (t, x) − 1 (7.3.16) ¯ for sufficiently large t > 0 and all x ∈ D. Then by Theorem 7.3.5, lim inf t→∞

1 ln ku2 (t; u0 , g)k > 0, t

which is a contradiction. This proves (i). (ii) To prove (ii), observe that, by (i), it suffices to show that Γ1 is an isolated invariant set in (X + × {0}) × Z0 . This is so, since Γ1 attracts any point in (X ++ × {0}) × Z0 . LEMMA 7.3.3 Assume (A7-5)–(A7-9). Then {Γ0 , Γ1 , Γ2 } is a Morse decomposition of Γ∂e ∩ (X+ × Z0 ). PROOF Take (u0 , g) ∈ Γ∂e ∩ (X+ × Z0 ) \ (Γ0 ∪ Γ1 ∪ Γ2 ). Then either (u0 , g) ∈ Γ ∩ (((X + \ {0}) × {0}) × Z0 ), in which case ω((u0 , g)) ⊂ Γ1 , or (u0 , g) ∈ Γ ∩ (({0} × (X + \ {0})) × Z0 ), in which case ω((u0 , g)) ⊂ Γ2 . In both cases α((u0 , g)) ⊂ Γ0 . From now on, let d(·, ·) stand for the distance between a point in and a subset of X+ × Z. LEMMA 7.3.4 Suppose the conditions in Theorem 7.3.8 hold. If there is some η1 > 0 such that lim sup d(Φ(t, u0 , g), Γi ) ≥ η1 (7.3.17) t→∞

e + , g ∈ Z0 , and i = 0, 1, 2, then there is η2 > 0 such that for all u0 ∈ X+ \ ∂X e + ) ≥ η2 lim inf d(Φ(t, u0 , g), ∂X t→∞

(7.3.18)

e + and g ∈ Z0 . for all u0 ∈ X+ \ ∂X PROOF It follows from Lemmas 7.3.1, 7.3.2, 7.3.3, and [58, Theorem 4.3] (see also [47, Theorem 4.1]).

7. Applications to Kolmogorov Systems of Parabolic Equations PROOF (Proof of Theorem 7.3.8) (7.3.17). Next, by Lemma 7.3.4, we have

297

First, Lemma 7.3.2(i) implies

e + ) ≥ η2 lim inf d(Φ(t, u0 , g), ∂X t→∞

e + and g ∈ Z0 . This together with Theorem 7.3.6 yields for all u0 ∈ X+ \ ∂X e + and each g ∈ Z0 there is τ1 = τ1 (u0 , g) > 0 that for each u0 ∈ X+ \ ∂X e + ) ≥ η2 for all such that u(t; u0 , g) ∈ [0, P ]X × [0, P ]X and d(Φ(t, u0 , g), ∂X 2 t ≥ τ1 . Applying ideas in the proof of [47, Theorem 3.2] to the restriction Φ|(X+ \∂X e + )×Z0 we obtain the existence of a nonempty compact invariant set ++ e + ) × Z0 attracting any compact B × Z0 with B ⊂ X+ \ ∂X e +. Γ ⊂ (X+ \ ∂X ++ ++ We claim that Γ ⊂X × Z0 . By invariance, for any (u0 , g) ∈ Γ++ , ++ there is (u−1 , g · (−1)) ∈ Γ such that u(t; u−1 , g · (−1)) = u0 . We write e + ) × Z0 , u0 = (u01 , u02 ) and u−1 = (u−1,1 , u−1,2 ). Since Γ++ ⊂ (X+ \ ∂X + u−1,1 , u−1,2 ∈ X \ {0}. Consequently, there holds (0, u−1,2 ) <2 u−1 <2 (u−1,1 , 0). It then follows from Proposition 7.3.6(2) that u(1; (0, u−1,2 ), g · (−1)) 2 u0 2 u(1; (u−1,1 , 0), g · (−1)). As the first and the third term are semitrivial solutions, Proposition 7.1.7(2) gives that u01 ∈ X ++ and u02 ∈ X ++ , that is, u0 ∈ X++ . We define p : X+ × Z0 → [0, ∞) by p(u, g) := sup { δ ≥ 0 : ui ≥ δϕprinc , i = 1, 2 }. Clearly, p(u, g) > 0 if and only if u ∈ X++ . Further, it follows from the openness of X++ that p is lower semicontinuous. Since p(u, g) > 0 for any (u, g) ∈ Γ++ , the compactness of Γ++ and the lower semicontinuity of p imply the existence of an open (in the relative topology of X++ × Z0 ) neighborhood V of Γ++ and of a positive number η0 such that p(u, g) ≥ η0

for all (u, g) ∈ V.

e + there is We conclude the proof by noting that for a given u0 ∈ X+ \ ∂X τ > 0 such that Πt ({u0 } × Z0 ) ∈ V for all t ≥ τ .

7.4

Competitive Kolmogorov Systems of Semilinear Equations: Examples

In this section, we discuss applications of the general theory established in Section 7.3 to some competitive Kolmogorov systems of random and nonautonomous semilinear equations.

298

7.4.1

Spectral Theory for Parabolic Equations

The Random Case

Assume that ((Ω, F, P), {θt }t∈R ) is an ergodic metric dynamical system. Consider the following competitive Kolmogorov systems of random partial differential equations:    ∂u1 = ∆u1 + f1 (θt ω, x, u1 , u2 )u1 , x ∈ D,   ∂t     ∂u2 = ∆u2 + f2 (θt ω, x, u1 , u2 )u2 , x ∈ D, (7.4.1) ∂t     Bu1 = 0, x ∈ ∂D,    Bu = 0, x ∈ ∂D, 2 ¯ × [0, ∞) × [0, ∞) → R2 . where f = (f1 , f2 ) : Ω × D We assume (A7-R4) (Measurability) The function f is (F×B(D)×B([0, ∞)×[0, ∞)), B(R2 ))-measurable. For each ω ∈ Ω, let f ω (t, x, u1 , u2 ) = (f1ω (t, x, u1 , u2 ), f2ω (t, x, u1 , u2 )) := f (θt ω, x, u1 , u2 ). (A7-R5) (Regularity) For each ω ∈ Ω and any M > 0 the restrictions to ¯ × [0, M ] × [0, M ] of f ω , f ω , and all the derivatives of the functions f ω , R×D 1 2 1 ω ¯ × [0, M ] × [0, M ]). Moreover, f2 , up to order 1 belong to C 1−,1−,1−,1− (R × D ¯ × [0, M ] × [0, M ])-norms of the for M > 0 fixed the C 1−,1−,1−,1− (R × D ω ω restrictions of f1 , f2 , and those derivatives are bounded uniformly in ω ∈ Ω. (A7-R6) There are P > 0 and a function m : [P, ∞) → (0, ∞) such that ¯ and any u1 ≥ P , f1 (ω, x, u1 , u2 ) ≤ −m(u1 ) for any ω ∈ Ω, any x ∈ D, ¯ and any u2 ≥ 0, and f2 (ω, x, u1 , u2 ) ≤ −m(u2 ) for any ω ∈ Ω, any x ∈ D, u1 ≥ 0, u2 ≥ P . (A7-R7) (Total competitiveness) There is a function m ˜ : [0, ∞) → (0, ∞) such that ∂u1 f1 (ω, x, u1 , u2 ) ≤ −m(u ˜ 1 ), ∂u2 f1 (ω, x, u1 , u2 ) ≤ −m(u ˜ 2 ) and ∂u1 f2 (ω, x, u1 , u2 ) ≤ −m(u ˜ 1 ), ∂u2 f2 (ω, x, u1 , u2 ) ≤ −m(u ˜ 2 ) for any ω ∈ Ω, ¯ and any (u1 , u2 ) ∈ [0, ∞) × [0, ∞). any x ∈ D, From now on, until the end of the present subsection, assume that (A7R4)–(A7-R7) are satisfied. Define the mapping E from Ω into the set of continuous real functions ¯ × [0, ∞) × [0, ∞) as defined on R × D E(ω) := f ω . Put Z := cl { E(ω) : ω ∈ Ω }

(7.4.2)

7. Applications to Kolmogorov Systems of Parabolic Equations

299

with the open-compact topology, where the closure is taken in the opencompact topology. It is a consequence of (A7-R5) via the Ascoli–Arzel`a theorem that the set Z is a compact metrizable space. By arguments similar to those in Subsection 7.2.1, (Z, {ζt }t∈R ) is a compact flow, where ζt g(τ, x, u1 , u2 ) = g · t(τ, x, u1 , u2 ) = g(τ + t, x, u1 , u2 ). The mapping E is a homomorphism of the measurable flow ((Ω, F), {θt }t∈R ) ˜ the image of into the measurable flow ((Z, B(Z)), {ζt }t∈R ). Denote by P ˜ ˜ is a the measure P under E: for any A ∈ B(Z), P(A) := P(E −1 (A)). P {ζt }-invariant ergodic Borel measure on Z. So, E is a homomorphism of the ˜ {ζt }t∈R ). metric flow ((Ω, F, P), {θt }t∈R ) into the metric flow ((Z, B(Z), P), We will consider a family of Eqs. (7.3.1) parameterized by g ∈ Z. By (A7-R4)–(A7-R7), (A7-5) through (A7-8) as well as (A7-10) are fulfilled. We denote by Φ = {Φt }t≥0 the topological skew-product semiflow generated by (7.4.1) on the product Banach bundle X+ × Z: Φ(t; u0 , g) = Φt (u0 , g) := (u(t; u0 , g), ζt g)

(7.4.3)

for t ≥ 0, g ∈ Z, u0 ∈ X+ , where u(t; u0 , g) stands for the solution of (7.3.1)g ¯ with the initial condition u(0; u0 , g)(x) = u0 (x) for x ∈ D. Moreover, define ˜ u0 , ω) := (u(t; u0 , E(ω)), θt ω), Φ(t;

t ≥ 0, ω ∈ Ω, u0 ∈ X + .

(7.4.4)

We have LEMMA 7.4.1 ˜ is a continuous random skew-product semiflow on the measurable bundle Φ X+ × Ω, covering the metric flow ((Ω, F, P), {θt }t∈R ). For t ≥ 0, u0 ∈ X+ , and ω ∈ Ω we will write u(t; u0 , ω) instead of u(t; u0 , E(ω)). Similarly, for t0 ∈ R, t ≥ t0 , u0 ∈ X+ , and ω ∈ Ω we will write u(t; t0 , u0 , ω) instead of u(t − t0 ; u0 , E(θt0 ω)). DEFINITION 7.4.1 (Uniform persistence) (7.4.1) is said to be uniformly persistent if there is η0 > 0 such that for any u0 ∈ (X + \ {0}) × (X + \ {0}) there is τ (u0 ) > 0 with the property that ui (t; t0 , u0 , ω) ≥ η0 ϕprinc for i = 1, 2, P-a.e. ω ∈ Ω, any t0 ∈ R, and any t ≥ t0 + τ (u0 ). Put ˜ Z0 := supp P

(7.4.5)

˜ ) > 0). Z0 (g ∈ Z0 if and only if for any neighborhood V of g in Z one has P(V ˜ 0 ) = 1. is a closed (hence compact) and {ζt }-invariant subset of Z, with P(Z Also, Z0 is connected.

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Spectral Theory for Parabolic Equations

Similarly to Lemma 7.2.3 we have

LEMMA 7.4.2 There exists Ω0 ⊂ Ω with P(Ω0 ) = 1 such that Z0 = cl { E(θt ω) : t ∈ R } for any ω ∈ Ω0 , where the closure is taken in the open-compact topology on Z. Observe that the set {0} × Z0 is invariant under the semiflow Φ. Consider the linearization of Φ at {0} × Z0 ,

∂Φt (v0 , (0, g)) = (∂2 u(t; 0, g)v0 , (0, g · t)),

t ≥ 0, v0 ∈ X, g ∈ Z0 , (7.4.6)

where ∂2 u(t; 0, g)v0 = (Ug01 (t, 0)v01 , Ug02 (t, 0)v02 ), and Ug0i (t, 0), i = 1, 2, is the solution operator of (7.3.10)i . Let [λ0i,min , λ0i,max ], i = 1, 2, be the principal spectrum intervals of (7.3.10)i over p0i (Z0 ).

THEOREM 7.4.1 Suppose that λ0i,min > 0 for i = 1, 2. Then there are nonempty compact invariant sets Γ1 ⊂ (X ++ × {0}) × Z0 and Γ2 ⊂ ({0} × X ++ ) × Z0 such that Γ1 attracts any point in ((X + \ {0}) × {0}) × Z0 and Γ2 attracts any point in ({0} × (X + \ {0})) × Z0 .

PROOF

It follows from Theorem 7.3.3.

Let [λ1min , λ1max ] 2 [λmin , λ2max ] be the

be the principal spectrum interval associated to Γ1 and principal spectrum interval associated to Γ2 .

THEOREM 7.4.2 (Uniform persistence) Suppose that λ0i,min > 0 and λimin > 0 for i = 1, 2. Then (7.4.1) is uniformly persistent.

PROOF

It follows from Theorem 7.3.8.

7. Applications to Kolmogorov Systems of Parabolic Equations

7.4.2

301

The Nonautonomous Case

Consider the following competitive Kolmogorov systems of nonautonomous partial differential equations:  ∂u1   = ∆u1 + f1 (t, x, u1 , u2 )u1 , x ∈ D,   ∂t     ∂u2 = ∆u2 + f2 (t, x, u1 , u2 )u2 , x ∈ D, (7.4.7) ∂t     Bu1 = 0, x ∈ ∂D,    Bu = 0, x ∈ ∂D, 2 ¯ × [0, ∞) × [0, ∞) → R2 . where f = (f1 , f2 ) : R × D We assume ¯ × [0, M ] × (A7-N3) (Regularity) For any M > 0 the restrictions to R × D [0, M ] of f1 , f2 , and all the derivatives of the functions f1 , f2 up to order 1 ¯ × [0, M ] × [0, M ]). belong to C 1−,1−,1−,1− (R × D (A7-N4) There are P > 0 and a function m : [P, ∞) → (0, ∞) such that ¯ and any u1 ≥ P , u2 ≥ 0 f1 (t, x, u1 , u2 ) ≤ −m(u1 ) for any t ∈ R, any x ∈ D, ¯ and any u1 ≥ 0, and f2 (t, x, u1 , u2 ) ≤ −m(u2 ) for any t ∈ R, any x ∈ D, u2 ≥ P . (A7-N5) (Total competitiveness) There is a function m ˜ : [0, ∞) → (0, ∞) such that ∂u1 f1 (t, x, u1 , u2 ) ≤ −m(u ˜ 1 ), ∂u2 f1 (t, x, u1 , u2 ) ≤ −m(u ˜ 2 ), and ∂u1 f2 (t, x, u1 , u2 ) ≤ −m(u ˜ 1 ), ∂u2 f2 (t, x, u1 , u2 ) ≤ −m(u ˜ 2 ) for all t ∈ R, x ∈ ¯ and u1 , u2 ∈ [0, ∞). D, Throughout this subsection, we assume (A7-N3)–(A7-N5). Put Z := cl { f · t : t ∈ R }

(7.4.8)

with the open-compact topology, where the closure is taken in the open-compact topology. It is a consequence of (A7-N3) via the Ascoli–Arzel`a theorem that the set Z is a compact metrizable space. We deduce from Lemma 7.1.1 that if g ∈ Z then g · t ∈ Z for all t ∈ R. Hence (Z, {ζt }t∈R ) is a compact flow, where ζt g = g · t. We will consider a family of Eqs. (7.3.1) parameterized by g ∈ Z. By (A7-N3)–(A7-N5), (A7-5) through (A7-8) as well as (A7-10) hold. We denote by Φ = {Φt }t≥0 the topological skew-product semiflow generated by (7.4.7) on the product Banach bundle X+ × Z: Φ(t; u0 , g) = Φt (u0 , g) := (u(t; u0 , g), ζt g)

(7.4.9)

for t ≥ 0, g ∈ Z, u0 ∈ X+ , where u(t; u0 , g) represents the solution of (7.3.1)g with initial condition u(0; u0 , g)(x) = u0 (x) for x ∈ D. For t0 ∈ R, t ≥ t0 , and u0 ∈ X+ we write u(t; t0 , u0 , f ) instead of u(t − t0 ; u0 , f · t0 ).

302

Spectral Theory for Parabolic Equations

DEFINITION 7.4.2 (Uniform persistence) (7.4.7) is said to be uniformly persistent if there is η0 > 0 such that for any u0 ∈ (X + \ {0}) × (X + \ {0}) there is τ (u0 ) > 0 with the property that ui (t; t0 , u0 , f ) ≥ η0 ϕprinc for i = 1, 2, all t0 ∈ R, and t ≥ t0 + τ (u0 ). Note that Z is connected and Γ0 = {0} × Z is invariant under Φ. Let [λ0i,min , λ0i,max ], i = 1, 2, be the principal spectrum intervals of (7.3.10)i over p0i (Z). THEOREM 7.4.3 Suppose that λ0i,min > 0 for i = 1, 2. Then there are nonempty compact invariant sets Γ1 ⊂ (X ++ × {0}) × Z and Γ2 ⊂ ({0} × X ++ ) × Z such that Γ1 attracts any point in ((X + \ {0}) × {0}) × Z and Γ2 attracts any point in ({0} × (X + \ {0})) × Z. PROOF

It follows from Theorem 7.3.3.

Let [λ1min , λ1max ] 2 [λmin , λ2max ] be the

be the principal spectrum interval associated to Γ1 and principal spectrum interval associated to Γ2 .

THEOREM 7.4.4 (Uniform persistence) Suppose that λ0i,min > 0 and λimin > 0 for i = 1, 2. Then (7.4.7) is uniformly persistent. PROOF

7.5

It follows from Theorem 7.3.8.

Remarks

As mentioned in the introduction of this monograph, principal spectral theory for linear parabolic equations under various special conditions has been studied in a lot of papers (see [29], [30], [31], [32], [28], [35], [49], [50], [59], [60], [61], [62], [79], [81], [82], [83], [84], [92], [93], [94], etc.) and has found many applications (see [51], [64], [83], [85], [93], etc.). In the previous chapters of this monograph, we developed the principal spectral theory for general random and nonautonomous parabolic equations. This theory will certainly also find great applications to lots of nonlinear problems. In the present chapter, we considered its application to the uniform persistence problem in random and

7. Applications to Kolmogorov Systems of Parabolic Equations

303

nonautonomous semilinear parabolic equations of Kolmogorov type and two dimensional competitive systems of such equations. It should be pointed out that uniform persistence as well as many other dynamical aspects in semilinear parabolic equations of Kolmogorov type and competitive Kolmogorov systems of parabolic equations have been widely studied. See for example, [9], [23], [25], [41], [52], [54], [101], [106], [111], etc., for scalar parabolic equations of Kolmogorov type, and [47], [53], [58], [80], [104], [108], [110], etc., for two species competitive Kolmogorov systems of parabolic equations. However, most equations in the literature are not as general as those in the present chapter, except [85], which in fact utilized principal spectral theory to consider the uniform persistence in quite general n-dimensional nonautonomous and random parabolic Kolmogorov systems.

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Index

Conormal derivative, 113 Continuous dependence, 139, 144, 145, 147, 237, 240, 257, 286 Continuous random skew-product semiflow, 16, 279, 299 Convex hull, 203 Cooperative system, 6, 191, 192 Cooperativity, 193

Lp –Lq estimates, 32 α-limit set, 13 Lp –Lq estimates, 200 ω-limit set, 12 Absorbing set, 269, 293 Adjoint problem, 44, 62, 166 Analytic semigroup, 114, 252 Ascoli–Arzel` a theorem, 52, 128, 159, 278, 281, 299, 301 Attraction basin, 13 Attractor, 12 Attractor dual to a repeller, 13

Differentiability, 259, 286 Distribution(s), 25 Dominated Convergence Theorem, 198 Dual attractor, 273 Dual topological linear skew-product semiflow, 47, 77 Duality pairing, 8 Dynamical spectrum, 85, 222

Backward parabolic equation, 44 Backward uniqueness, 259, 260 Backward orbit, 13 Besov space, 54 Bessel potential space, 54 Bijection, 271 Bilinear form, 26, 45, 140, 213 Birkhoff Ergodic Theorem, 14, 73, 155, 164 Boundary operator, 1, 2, 7, 23, 131, 247, 249, 282 Boundary regularity, 29 Classical solution, 53, 126, 214, 255, 285 Cocycle equality, 31 Cocycle property, 257 Compact flow, 11, 121 Compactness, 33, 202, 258, 288 Comparison principle, 255 Competitive Kolmogorov system, 7, 282, 298, 301 Complex interpolation functor, 54, 209 Conormal, 64

315

Elliptic equation(s), 2 Entire positive solution, 151, 152, 180, 226, 227 Entire positive weak solution, 76, 90, 91 Ergodic invariant measure, 15 Ergodic metric dynamical system, 14 Exponential separation, 75, 77, 79, 91, 125, 126, 132, 220, 221, 224, 262 Exponentially bounded solution, 116 Extension in Lp , 32 Faber–Krahn inequalities, 150, 186 Fiber, 74 Fixed point, 199 Forward invariant, 11 Forward orbit, 11 Fractional power space, 252, 282, 283 Generalized derivative(s), 10

316 Global attractor, 12, 270, 294 Global existence, 255, 287 Global weak solution, 29, 30, 194 Gronwall inequality, 71, 201 Harnack inequality, 34, 35, 68, 89 Hausdorff measure, 17 Hopf boundary point principle, 58, 253 Influence of spatial variation, 179 Influence of temporal variation, 156, 170, 241 Influence of the shape of domain, 186 Injective, 62, 103 Interior of the nonnegative cone, 18, 21 Invariant, 11, 14 Invariant measure, 15 Irreducibility, 193 Isolated invariant set, 11, 294, 295 Jensen inequality, 243 Joint continuity, 33, 43, 57, 204, 210, 211 Joint measurability, 42 Kingman’s subadditive ergodic theorem, 71 Kre˘ın–Rutman theorem, 147 Krylov–Bogolyubov Theorem, 15 Lap number, 115 Local regularity, 32 Matano number, 115 Maximum principle, 258 Measurable bundle, 16 Measurable dynamical system, 14 Measurable flow, 14, 122, 278 Metric dynamical system, 14 Metric flow, 14, 122, 278 Mild solution, 114, 127, 191, 197, 202, 213, 255 Minimal, 13, 152, 159, 181

Index Monotonicity, 18, 35, 136, 137, 205, 235, 238, 239, 257 Morse decomposition, 13, 296 Multiplicative Ergodic Theorem, 103, 105 Nemytski˘ı (substitution) operator, 250, 283 Nonautonomous linear parabolic equation, 1, 131, 132, 149, 261 Nonautonomous parabolic equation of Kolmogorov type, 7, 280 Nondivergence, 113, 148 Nondivergence form, 63 Nonextendible solution, 254, 255, 284 Nonnegative cone, 18, 21, 253 Nontrivial solution, 230 Norm continuity, 57, 211 One-codimensional subbundle, 74, 220 One-dimensional spatial domain, 114 One-dimensional subbundle, 74, 220 Orbit, 13 Order preserving, 286 Oseledets splittings, 105 Part metric, 275 Perturbation, 31, 47 Pettis theorem, 10, 195 Poincar´e inequality, 70 Positive mild solution, 208 Positive weak solution, 35 Positivity, 33, 254, 260, 285 Principal eigenfunction, 2, 61, 158, 163, 166, 252 Principal eigenvalue, 2, 85, 147, 158, 163, 166, 186, 245 Principal Lyapunov exponent, 73, 124, 218, 234 Principal resolvent, 66, 84, 216, 267 Principal spectrum, 66, 68, 124, 132, 216, 234, 237 Principal spectrum interval, 290, 292, 293, 300, 302 Probability space, 14

Index Product bundle, 15 Random linear parabolic equation, 2, 120, 125, 149 Random linear skew-product semiflow, 16 Random parabolic equation of Kolmogorov type, 7, 277 Real interpolation functor, 54, 209 Recurrent, 152 Regularity, 255, 285 Regularity up to boundary, 53, 208 Repeller, 13, 273 Repeller dual to an attractor, 13 Repulsion basin, 13 Sacker–Sell spectrum, 85, 222 Same recurrence, 152 Schwarz symmetrization, 155 Schwarz symmetrized domain, 155 Sectorial operator, 252 Semitrivial solution, 287, 288, 291 Simple function, 9, 195, 203 Singular Gronwall lemma, 264 Smoothness, 52 Sobolev embedding theorem, 127 Sobolev–Slobodetski˘ı space, 54 Space average, 180 Space averaged equation, 180 Strictly monotone, 44, 208 Strong competitiveness, 286 Strong continuity, 32 Strong maximum principle, 58, 253 Strong monotonicity, 59, 212 Strong solution, 64, 126, 127 Strongly measurable, 10 Strongly monotone, 62, 212, 258, 261, 288 Strongly positive, 62, 212, 261 Subadditive ergodic theorem, 72 Time averaged equation, 157, 158, 241 Time averaged function, 157, 158, 240, 241, 276, 282

317 Time periodic parabolic equation(s), 2, 3 Topological C 1 skew-product semiflow, 16 Topological dynamical system, 11 Topological flow, 11, 122, 271 Topological linear skew-product semiflow, 15, 43, 123, 132, 207, 237, 261, 262, 288 Topological semidynamical system, 11 Topological semiflow, 11 Topological skew-product semiflow, 15, 233, 260, 278, 281, 299, 301 Topologically transitive, 13 Total competitiveness, 298, 301 Totally competitive, 291 Trivial solution, 261, 267, 288 Uniform ellipticity, 24 Uniform persistence, 271, 272, 279– 281, 294, 299, 300, 302 Uniquely ergodic, 15, 85, 152, 159, 167, 181 Urysohn lemma, 58, 122 Variation of constant formula, 255 Weak solution, 27, 44, 214 Weak-* topology, 18 Weakly measurable, 10

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