Nonlinear Elliptic and Parabolic Problems: A Special Tribute to the Work of Herbert Amann

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Progress in Nonlinear Differential Equations and Their Applications Volume 66

Editor Haim Brezis Université Pierre et Marie Curie Paris and Rutgers University New Brunswick, N.J.

Editorial Board Antonio Ambrosetti, Scuola Internazionale Superiore di Studi Avanzati, Trieste A. Bahri, Rutgers University, New Brunswick Felix Browder, Rutgers University, New Brunswick Luis Cafarelli, Institute for Advanced Study, Princeton Lawrence C. Evans, University of California, Berkeley Mariano Giaquinta, University of Pisa David Kinderlehrer, Carnegie-Mellon University, Pittsburgh Sergiu Klainerman, Princeton University Robert Kohn, New York University P.L. Lions, University of Paris IX Jean Mahwin, Université Catholique de Louvain Louis Nirenberg, New York University Lambertus Peletier, University of Leiden Paul Rabinowitz, University of Wisconsin, Madison John Toland, University of Bath

Nonlinear Elliptic and Parabolic Problems A Special Tribute to the Work of Herbert Amann

Michel Chipot Joachim Escher Editors V nda Valent e Giorgi o Vergara Caffarelli Editors

Birkhäuser Basel Boston Berlin Ɣ

Ɣ

Editors: Michel Chipot Universität Zürich Angewandte Mathematik Winterthurerstr. 190 8057 Zürich Switzerland e-mail: [email protected]

Joachim Escher Institute of Applied Mathematics University of Hannover Welfengarten 1 30167 Hannover Germany e-mail: [email protected] Carlo Sbordone Dipartimento di Matematica e Applicazioni Università di Napoli “Federico II” Via Cintia 80126 Napoli, Italy [email protected] Itai Shafrir Department of Mathematics Technion – Israel Institute of Technology 32000 Haifa, Israel [email protected] Vanda Valente CNR-IAC Viale del Policlinico, 137 00161 Roma, Italy [email protected] Giorgio Vergara Caffarelli Dipartimento di metodi e modelli matematici per le Scienze Aplicate Università di Roma “La Sapienza” Via A. Scarpa 16 00161 Roma, Italy [email protected]

2000 Mathematics Subject Classification 35Jxx, 35Kxx, 37Gxx, 76xx A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at .

ISBN 3-7643-7266-4 Birkhäuser Verlag, Basel – Boston – Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained. © 2005 Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced of chlorine-free pulp. TCF ∞ Printed in Germany ISBN 10: 3-7643-7266-4 e-ISBN 3-7643-7385-7 ISBN 13: 978-3-7643-7266-8 987654321

www.birkhauser.ch

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

H. Abels Bounded Imaginary Powers and H∞ -Calculus of the Stokes Operator in Unbounded Domains . . . . . . . . . . . . . . . . . . . . . . . .

1

M. Borsuk and A. Zawadzka Exact Estimates of Solutions to the Robin Boundary Value Problem for Elliptic Non-divergent Second-order Equations in a Neighborhood of the Boundary Conical Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

D. Bothe, J. Pr¨ uss and G. Simonett Well-posedness of a Two-phase Flow with Soluble Surfactant . . . . . . . .

37

J. Brasche and M. Demuth Resolvent Diﬀerences for General Obstacles . . . . . . . . . . . . . . . . . . . . . . . . .

63

J.I. D´ıaz Special Finite Time Extinction in Nonlinear Evolution Systems: Dynamic Boundary Conditions and Coulomb Friction Type Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

M. Duelli and L. Weis Spectral Projections, Riesz Transforms and H ∞ -calculus for Bisectorial Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

R. Farwig, G.P. Galdi and H. Sohr Very Weak Solutions of Stationary and Instationary Navier-Stokes Equations with Nonhomogeneous Data . . . . . . . . . . . . . . .

113

M. Fila, J.J.L. Vel´ azquez and M. Winkler Grow-up on the Boundary for a Semilinear Parabolic Problem . . . . . .

137

H. Gajewski and I.V. Skrypnik Existence and Uniqueness Results for Reaction-diﬀusion Processes of Electrically Charged Species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

151

F. Colombo and D. Guidetti An Inverse Problem for a Phase-ﬁeld Model in Sobolev Spaces . . . . . .

189

vi

Contents

O. Anza Hafsa and M. Chipot Numerical Analysis of Microstructures: The Inﬂuence of Incompatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

211

J. Hern´ andez, F.J. Mancebo and J.M. Vega Nonlinear Singular Elliptic Problems: Recent Results and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

227

M. Hieber The Navier-Stokes Flow in the Exterior of Rotating Obstacles . . . . . . .

243

M. Kuˇcera, J. Eisner and L. Recke A Global Bifurcation Result for Variational Inequalities . . . . . . . . . . . . .

253

P.C. Kunstmann On Elliptic Non-divergence Operators with Measurable Coeﬃcients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

265

P. Lauren¸cot and D. Wrzosek A Chemotaxis Model with Threshold Density and Degenerate Diﬀusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

273

J. L´ opez-G´ omez and M. Molina-Meyer In the Blink of an Eye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

291

J. L´ opez-G´ omez and C. Mora-Corral Generalized Minimal Cardinal of the λ-slices of the Semi-bounded Components Arising in Global Bifurcation Theory . . . . . . . . . . . . . . . . . .

329

S.A. Messaoudi Blow-up of Solutions of a Semilinear Heat Equation with a Visco-elastic Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

351

C.M. Murea and G. Hentschel Finite Element Methods for Investigating the Moving Boundary Problem in Biological Development . . . . . . . . . . . . . . . . . . . . . . .

357

J. Naumann Existence of Weak Solutions to the Equations of Stationary Motion of Heat-conducting Incompressible Viscous Fluids . . . . . . . . . . .

373

P. Pol´ aˇcik and P. Quittner Liouville Type Theorems and Complete Blow-up for Indeﬁnite Superlinear Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . .

391

D. Praˇza ´k On Reducing the 2d Navier-Stokes Equations to a System of Delayed ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

403

J. Rehberg Quasilinear Parabolic Equations in Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

413

Contents

vii

A. Rodr´ıguez-Bernal Parabolic Equations in Locally Uniform Spaces . . . . . . . . . . . . . . . . . . . . .

421

B. Scarpellini Bifurcation of Traveling Waves Related to the B´enard Equations with an Exterior Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

433

H.-J. Schmeißer and W. Sickel Vector-valued Sobolev Spaces and Gagliardo-Nirenberg Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

463

P. Souplet The Inﬂuence of Gradient Perturbations on Blow-up Asymptotics in Semilinear Parabolic Problems: A Survey . . . . . . . . . . . . . . . . . . . . . . . . . . .

473

Kenichiro Umezu Non-existence of Positive Solutions for Diﬀusive Logistic Equations with Nonlinear Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

497

A. Rodr´ıguez-Bernal and A. Vidal-L´ opez Extremal Equilibria and Asymptotic Behavior of Parabolic Nonlinear Reaction-diﬀusion Equations . . . . . . . . . . . . . . . . . . .

509

C. Walker A Remark on Continuous Coagulation-Fragmentation Equations with Unbounded Diﬀusion Coeﬃcients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

517

P. Weidemaier On Lp -Estimates of Optimal Type for the Parabolic Oblique Derivative Problem with VMO-Coeﬃcients – A Reﬁned Version . . . . .

529

Preface Celebrating the work of a renowned mathematician, it is our pleasure to present this volume containing the proceedings of the conference “Nonlinear Elliptic and Parabolic Problems: A Special Tribute to the Work of Herbert Amann”, held in Zurich, June 28–30, 2004. Herbert Amann had a signiﬁcant and decisive impact in developing Nonlinear Analysis and one goal of this conference was to reﬂect his broad scientiﬁc interest. It is our hope that this collection of papers gives the reader some idea of the subjects in which Herbert Amann had and still has a deep inﬂuence. Of particular importance are ﬂuid dynamics, reaction-diﬀusion systems, bifurcation theory, maximal regularity, evolution equations, and the theory of function spaces. The organizers thank the following institutions for provided support for the conference: • • • •

Swiss National Foundation Z¨ urcher Hochschulstiftung Z¨ urcher Universit¨atsverein Mathematisch-naturwissenschaftliche Fakult¨ at (MNF).

Finally, it is our pleasure to thank all contributors, referees, and Birkh¨ auser Verlag, particularly T. Hempﬂing for their help and cooperation in making possible this volume. The Editors

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 1–15 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Bounded Imaginary Powers and H∞ -Calculus of the Stokes Operator in Unbounded Domains Helmut Abels Abstract. In the present contribution we study the Stokes operator Aq = −Pq ∆ on Lqσ (Ω), 1 < q < ∞, where Ω is a suitable bounded or unbounded domain in Rn , n ≥ 2, with C 1,1 -boundary. We present some conditions on Ω and the related function spaces and basic equations which guarantee that c + Aq for suitable c ∈ R is of positive type and admits a bounded H∞ calculus. This implies the existence of bounded imaginary powers of c + Aq . Most domains studied in the theory of Navier-Stokes like, e.g., bounded, exterior, and aperture domains as well as asymptotically ﬂat layers satisfy the conditions. The proof is done by constructing an approximate resolvent based on the results of [3], which were obtained by applying the calculus of pseudodiﬀerential boundary value problems. Finally, the result is used to proof the existence of a bounded H∞ -calculus of the Stokes operator Aq on an aperture domain. Mathematics Subject Classiﬁcation (2000). 35 Q 30, 76 D 07, 47 A 60, 47 F 05. Keywords. Stokes equations, exterior domains, bounded imaginary powers, H∞ -calculus, aperture domain.

1. Introduction In this article we consider the Stokes operator Aq = −Pq ∆ on Lqσ (Ω) with domain D(Aq ) = {f ∈ Wq2 (Ω)n : f |∂Ω = 0} ∩ Lqσ (Ω) where Pq : Lq (Ω)n → Lqσ (Ω) denotes the Helmholtz projection, ∞ (Ω) Lqσ (Ω) := C0,σ

.q

∞ , C0,σ (Ω) := {u ∈ C0∞ (Ω)n : div u = 0}, and Ω ⊆ Rn , n ≥ 2,

is a domain speciﬁed in Assumption 1.1 below. Properties of the Stokes operator are important for the associated instationary Stokes and Navier-Stokes equations. Since the latter equations arise in mathematical ﬂuid mechanics, many diﬀerent kinds of bounded and unbounded domains are of interest and have been studied.

2

H. Abels

The purpose of the present contribution is to present some conditions on Ω and the related function spaces which guarantee that c + Aq for suitable c ∈ R is of positive type and admits a bounded H∞ -calculus w.r.t. δ ∈ (0, π). Here c + Aq is of positive type w.r.t. δ if and only if Σδ ∪ {0} ⊆ ρ(−c − Aq ) and (λ + c + Aq )−1 L(Lqσ (Ω)) ≤

Cq,δ , |λ|

λ ∈ Σδ ,

(1.1)

where Σδ := {z ∈ C \ {0} : | arg z| < δ}. – Note that, if δ > π2 , (1.1) implies that −c − Aq generates a bounded, strongly continuous, analytic semi-group. – Moreover, A := c + Aq is said to admit a bounded H∞ -calculus w.r.t. δ if and only if 1 h(A) := h(−λ)(λ + A)−1 dλ (1.2) 2πi Γ is a bounded operator satisfying h(A)L(Lqσ (Ω)) ≤ Cq,δ h∞

for all h ∈ H∞ (δ),

(1.3)

where H∞ (δ) denotes the Banach algebra of all bounded holomorphic functions h : Σπ−δ → C, cf. McIntosh [25], and Γ is the negatively orientated boundary of Σδ . We note that in order to prove (1.3) for all h ∈ H∞ (δ) it is suﬃcient to show the estimate for h ∈ H(δ), which consists of all h ∈ H∞ (δ) such that |h(z)| ≤ C

|z|s 1 + |z|2s

for all z ∈ Σπ−δ

for some s > 0, cf. [8, Lemma 2.1]. For h ∈ H(δ) the integral (1.2) is well deﬁned as a Bochner integral. The property of admitting a bounded H∞ -calculus is a generalization of possessing bounded imaginary powers since hy (z) = z iy ∈ H∞ (δ) for all δ ∈ (0, π), which has many important consequences. In particular, (1.3) for δ > π2 yields the maximal regularity of −c − Aq by the result of Dore and Venni [13]. The resolvent estimate (1.1) with arbitrary δ ∈ (0, π), 1 < q < ∞ and c = 0, has been proved for various kinds of domains Ω ⊂ Rn , n ≥ 2, cf. Giga [19] for bounded domains, Borchers and Sohr [9] and Borchers and Varnhorn [9] for exterior domains, Farwig and Sohr [16] for aperture domains, Abels and Wiegner [6] for an inﬁnite layer Ω = Rn−1 × (−1, 1), and Abels [2] for asymptotically ﬂat layers. Moreover, see Farwig and Sohr [15] for a general treatment and Desch, Hieber, and Pr¨ uss [12] for the case of a half-space Rn+ and Rn , where also the borderline cases q = 1, ∞ have been studied. The fact that Aq possesses bounded imaginary powers and admits a bounded H∞ -calculus was proved by Giga [20], Giga and Sohr [21], and Noll and Saal [27] for bounded domains and for exterior domains in Rn , n ≥ 3, by Giga and Sohr [22] for the half-space Rn+ , see also [12], and by Abels [5, 4, 3] for two-dimensional exterior domains, an inﬁnite layer, and asymptotically ﬂat layers.

Stokes Operator in Unbounded Domains

3

In the following we will present an approach, proving (1.1) for large λ and (1.3) for suitable c simultaneously for a class of domains, which includes all previously mentioned cases provided that some auxiliary results are known. More precisely, we make the following assumption. Assumption 1.1. Let 1 < q < ∞ and δ ∈ (0, π) be ﬁxed. Moreover, let Ω ⊆ Rn , n ≥ 2, be a domain satisfying the following conditions: (A1) There is a ﬁnite covering of Ω with relatively open sets Uj , j = 1, . . . , m, such that Uj coincides (after rotation) with a relatively open set of Rnγj , where Rnγj := {(x , xn ) ∈ Rn : xn > γj (x )}, γj ∈ C 1,1 (Rn−1 ). Moreover, suppose that there are cut-oﬀ functions ϕj , ψj ∈ Cb∞ (Ω), j = 1, . . . , m, such that ϕj , j = 1, . . . , m, is a partition of unity, ψj ≡ 1 on supp ϕj , and supp ψj ⊂ Uj , j = 1, . . . , m. (A2) The Helmholtz decomposition is valid for Lr (Ω)n with r = q, q , i.e., for every f ∈ Lr (Ω)n there is a unique decomposition f = f0 + ∇p with f0 ∈ Lrσ (Ω) ˙ r1 (Ω). Moreover, and p ∈ W Lqσ (Ω) = {f ∈ Lq (Ω)n : div f = 0, γν f = 0}. (1.4) 1 ˙ (A3) For every p ∈ Wr (Ω), r = q, q , there is a decomposition p = p1 +p2 such that p1 ∈ Wr1 (Ω), p2 ∈ Lrloc (Ω) with ∇p2 ∈ Wr1 (Ω) and (p1 , ∇p2 )1,r ≤ C∇pr . We refer to Section 2 below for the deﬁnitions of the function spaces and the normal trace γν . Remark 1.2. First of all, we note that (A1) can be generalized to the case of a locally ﬁnite covering Uj , j ∈ N, if uniform bounds on γj , ϕj , ψj in C 1,1 norm are assumed and if for every x ∈ Ω the number of sets Uj containing x is bounded by a constant independent of x. Moreover, it is easy to see that (A1) is fulﬁlled for all kinds of domains with C 1,1 -boundary mentioned above. We refer to [28, 15, 16, 26, 14, 2] for the validity of the Helmholtz decomposition for these types of domains. The characterization (1.4) holds as well in these cases except for n for which the characterization is diﬀerent, cf. [16, an aperture domain if q > n−1 Lemma 3.1]. The identity (1.4) is used in Lemma 3.1 below. Moreover, (A3) is a technical condition needed in the proof of Lemma 4.2 below. It is satisﬁed if the ˙ 1 (Ω) there is an extension following extension property is valid: For every p ∈ W q 1 n ˙ pq ≤ C∇pq . This is the case for every p˜ ∈ Wq (R ) such that p˜|Ω = p and ∇˜ (ε, ∞)-domain, cf. [10], in particular, for exterior domains. This extension property does not hold for layer-like domains, cf. [2, Section 2.4]. Nevertheless (A3) is also valid in layer-like domains, cf. [2, Lemma 2.4]. The main result is the following: Theorem 1.3. Let Ω ⊂ Rn , n ≥ 2, δ ∈ (0, π), and 1 < q < ∞ satisfy Assumption 1.1. Then there is an R > 0 such that (λ + Aq )−1 exists and Cq,δ (λ + Aq )−1 q ≤ 1 + |λ|

4

H. Abels

for all λ ∈ Σδ with |λ| ≥ R. Moreover, −1 h(−λ)(λ + Aq ) dλ ΓR

L(Lqσ (Ω))

≤ Cq,δ h∞

(1.5)

for every h ∈ H(δ), where ΓR = Γ \ BR (0). In particular, for every c ∈ R and 0 < δ ≤ δ such that c + Σδ ⊂ ρ(−Aq ) the shifted Stokes operator c + Aq admits a bounded H∞ -calculus with respect to δ . Remark: The statement of Theorem 1.3. also holds for the reduced Stokes operator introduced in Section 3 below, which is shown in the proof. We note that the characterization (1.4) is not needed in this case. The proof of Theorem 1.3 is based on the construction of an approximative resolvent Rλ that coincides with (λ + Aq )−1 modulo an operator which decays of order O(|λ|−1−ε ) as |λ| → ∞ for some ε > 0. The construction is based on a localization procedure and a suitable result for the reduced Stokes operator on a curved half-space Rnγ , γ ∈ C 1,1 (Rn−1 ), cf. Theorem 4.1 below. The latter result was basically obtained in [3] and is achieved using the calculus of pseudodiﬀerential boundary value problems developed by Grubb [23] in a non-smooth version, cf. [3, Section 4] and [1]. In particular, if it is known that σ(−Aq ) ⊆ (−∞, 0], then Theorem 1.3 implies that c + Aq admits a bounded H∞ -calculus for every c > 0. In order to prove that this is also true for c = 0 it remains to analyze the resolvent near 0. This is done for the case of an aperture domain in Rn , n ≥ 2, in Section 5 below, where the following result is proved: Theorem 1.4. Let 1 < q < ∞, δ ∈ (0, π), and let Ω ⊂ Rn , n ≥ 2, be an aperture domain with C 2,µ -boundary, µ > 0, as deﬁned in Section 5 below. Then the Stokes operator Aq admits a bounded H∞ -calculus with respect to δ. In particular, this yields that the Stokes operator has maximal regularity on Lqσ (Ω), which was also obtained by Fr¨ ohlich [18] in the context of weighted Lq -spaces.

2. Preliminaries Let Ω ⊆ Rn , n ≥ 2, be a domain. Then C0∞ (Ω) denotes the set of all smooth f : Ω → C with compact support, and ∞ (Ω) := {u = v|Ω : v ∈ C0∞ (Rn )}. C(0)

Moreover, Cb∞ (Ω) denotes the set of all smooth and bounded f : Ω → C with bounded derivatives and C 1,1 (Rn ) is the set of all bounded functions f : Rn → C with bounded and globally Lipschitz continuous ﬁrst-order derivatives. The usual Lebesgue space will be denoted by Lq (Ω) and Lq (∂Ω), 1 ≤ q ≤ ∞, normed by .q ≡ .Lq (Ω) and .q,∂Ω ≡ .Lq (∂Ω) , respectively. Furthermore, Lqloc (Ω) consists of all f : Ω → C such that f ∈ Lq (B ∩ Ω) for all balls B with

Stokes Operator in Unbounded Domains

5

B ∩ Ω = ∅. The scalar product on L2 (M ) is denoted by (., .)M for M = Ω or M = ∂Ω. Moreover, the usual Sobolev-Slobodeckij spaces based on Lq , 1 < q < ∞, are denoted by Wqs (Ω) and Wqs (∂Ω), s ≥ 0, with norms .s,q and .s,q,∂Ω , resp., s cf., e.g., [7]. As usual Wq,0 (Ω), s ≥ 0 with s − 1q ∈ N, is deﬁned as the closure of ∞ s C0 (Ω) in Wq (Ω), and −s Wq,0 (Ω) := (Wqs (Ω)) ,

Wq−s (Ω) := (Wqs ,0 (Ω)) ,

Wq−t (∂Ω) := (Wqt (∂Ω))

for s, t > 0 with s − q1 ∈ N where 1q + q1 = 1. Finally, the homogeneous Sobolev space of order 1 is deﬁned as ˙ q1 (Ω) := p ∈ Lq (Ω) : ∇p ∈ Lq (Ω) W loc

normed by ∇ · q . In the following let Ω ⊆ Rn , n ≥ 2, be a domain satisfying (A1). Using the cut-oﬀ functions ϕj , ψj , j = 1, . . . , m, many properties of the Sobolev-Slobodeckij spaces on Ω and ∂Ω can be reduced to the case of a curved half-space Rnγ , γ ∈ C 1,1 (Rn−1 ). Then the diﬀeomorphism F : Rn+ → Rnγ : x → (x , xn + γ(x ))

(2.1)

on Rnγ ∗

can be used to reduce the statement to the case of a half-space Rn+ = n {x ∈ R : xn > 0}. More precisely, if (F f )(x) := f (F (x)) and (F ∗,−1 f )(x) := f (F −1 (x)) denote the pull-back and push-forward, resp., of a scalar function f , then F ∗ : Wqs (Rn+ ) → Wqs (Rnγ ),

s s F ∗ : Wq,0 (Rn+ ) → Wq,0 (Rnγ ),

F0∗ : Wqs (Rn−1 ) → Wqs (∂Rnγ ) are linear isomorphisms for |s| ≤ 2, where F0 denotes the restriction of F on ∂Rn+ and ∂Rn+ is identiﬁed with Rn−1 . In particular, we note that, if 2 ≥ s > j + 1q , j = 0, 1, with s − 1q ∈ N, there is a bounded and surjective linear operator s− q1

γj : Wqs (Ω) → Wq

(∂Ω)

(2.2)

∞ such that γj u = ∂νj u|∂Ω for all f ∈ C(0) (Ω), cf., e.g., [7], where ν denotes the s exterior normal. Moreover, if 0 < s < 1q , by [29, Theorem 2.9.3] Wq,0 (Ω) = Wqs (Ω), where Ω is as in Assumption 1.1. Thus 1 1 (Wqs (Ω)) = Wq−s for all s ∈ − , . (2.3) (Ω) q q

Furthermore, recall that for f ∈ Lq (Ω) with div f = 0 it is possible to deﬁne a −1 ˙ 1 (Ω) by weak trace of the normal component γν f ∈ Wq,νq (∂Ω) := γ0 W q γν f, γ0 v∂Ω := (f, ∇v)Ω

˙ q1 (Ω), for all v ∈ W

(2.4)

˙ 1 (Ω) := {a ∈ Lq (∂Ω) : a = A|∂Ω , A ∈ W ˙ 1 (Ω)} is equipped with the where γ0 W q q loc ∞ quotient norm. Of course, if f ∈ C(0) (Ω) with div f = 0, the deﬁnition of γν f by

6

H. Abels

(2.4) coincides with the usual trace ν · f |∂Ω , i.e., γν f, γ0 v∂Ω = (ν · f, v)∂Ω for all ˙ 1 (Ω). v∈W q

˙ q1 (Ω), is the Moreover, we note that, if f = f0 + ∇p, f0 ∈ Lqσ (Ω), p ∈ W q n Helmholtz decomposition of f ∈ L (Ω) and (1.4) is valid, then p is uniquely determined as solution of the weak Neumann problem ∆p = div f ∂ν p|∂Ω = ν · f |∂Ω

in Ω,

(2.5)

on ∂Ω,

(2.6)

where (2.5) is understood in the sense of distributions and (2.6) is understood ˙ q1 (Ω) solves as γν (f − ∇p) = 0, cf. [28]. Because of the deﬁnition of γν , p ∈ W (2.5)–(2.6) if and only if ˙ 1 (Ω). for all v ∈ W q

(∇p, ∇v)Ω = (f, ∇v)Ω

(2.7)

Moreover, the existence of the (unique) Helmholtz decomposition is equivalent to ˙ q1 (Ω) of (2.7) for every f ∈ Lq (Ω)n . We note the existence of a unique solution p ∈ W ˙ 1 (Ω)) can be represented as F, vΩ = (f, ∇v)Ω ˙ −1 (Ω) := (W that every F ∈ W q,0 q q n for some f ∈ L (Ω) with f q ≤ CF W˙ −1 , which is a consequence of the Hahnq,0

Banach theorem. Finally, let KN :

−1 Wq,νq (∂Ω)

(∇KN a, ∇v)Ω = a, γ0 v∂Ω

˙ 1 (Ω) be deﬁned by →W q ˙ q1 (Ω). for all v ∈ W

(2.8)

−1

By deﬁnition of Wq,νq (∂Ω), a = γν A for some A ∈ Lqσ (Ω). Hence (2.8) is equivalent to ∆KN a = 0 and ∂ν KN a|∂Ω = a.

3. The reduced Stokes operator In the following let Ω ⊆ Rn , n ≥ 2, δ ∈ (0, π), and let 1 < q < ∞ be as in Assumption 1.1. In order to apply the results from [3] for the construction of an approximate resolvent for large λ, we need the reduced Stokes operator A0,q deﬁned as A0,q u :=

(−∆ + ∇KN T )u,

T u := γν (∆ − ∇ div)u,

1 for u ∈ D(A0,q ) := Wq2 (Ω)n ∩ Wq,0 (Ω)n , where KN is deﬁned by (2.8). Since div(∆ − ∇ div)u = 0, we conclude that −1

T : D(A0,q ) → Wq,νq (∂Ω). Hence ∇KN T : D(A0,q ) → Lq (Ω)n is well deﬁned. – Note that A0,q is a densely deﬁned unbounded operator on Lq (Ω)n in contrast to the Stokes operator, which acts on the subspace Lqσ (Ω). We refer to [2, Section 3] for explanations of the relation between the Stokes and the reduced Stokes operator.

Stokes Operator in Unbounded Domains

7

Introducing local coordinates T u = divτ γ1 uτ for every u ∈ C ∞ (Ω) with u|∂Ω = 0 where divτ denotes the tangential divergence, cf. [24, Lemma A.1]. Hence by (2.2) s− 1q

1 T : Wq2+s (Ω)n ∩ Wq,0 (Ω)n → Wq

1 (∂Ω) for all 0 ≥ s > −1 + . q

(3.1)

We will use the following additional assumption: (A4) There is an R > 0 such that for every λ ∈ Σδ with |λ| ≥ R there is no non-trivial solution g ∈ Wq1 (Ω) of λ(g, v)Ω + (∇g, ∇v)Ω = 0

for all v ∈ Wq1 (Ω).

(3.2)

This assumption is needed in the proof of Lemma 3.1 below. In the following we will show that (A4) is a consequence of Assumption 1.1, cf. proof of Theorem 1.3 below. The construction of the approximate resolvent is based on the following lemma. Lemma 3.1. Let Ω ⊆ Rn , n ≥ 2, 1 < q < ∞, and δ ∈ (0, π) be as in Assumption 1.1. Moreover, assume that (A4) holds and that (λ + A0,q )−1 exists for some λ ∈ Σδ with |λ| ≥ R. Then (λ + Aq )−1 exists and A0,q |Lqσ (Ω) = Aq ,

(λ + A0,q )−1 |Lqσ (Ω) = (λ + Aq )−1 .

(3.3)

Proof. The ﬁrst statement can be seen as follows: If u ∈ D(A0,q ) ∩ Lqσ (Ω), then div(−∆u + ∇KN T u) = 0 in the sense of distributions and γν (−∆u + ∇KN T u) = −γν ∆u + ∂ν KN T u|∂Ω = 0 in the sense of (2.4). Hence −∆u = (−∆ + ∇KN T )u − ∇KN T u is the Helmholtz decomposition of −∆u by (A2), i.e., (−∆ + ∇KN T )u = Pq (−∆)u = Aq u. In order to prove the second relation let u = (λ + A0,q )−1 f with f ∈ Lqσ (Ω). Then multiplying (λ + A0,q )u = f by ∇v, v ∈ Wq1 (Ω), and using (2.8) and (2.4) we obtain that g = div u solves (3.2), which implies div u = 0 by (A4). Therefore, u ∈ Lqσ (Ω) by (1.4) and (λ + Aq )u = (λ + A0,q )u = f . Since by the ﬁrst statement λ + Aq = (λ + Aq,0 )|Lqσ (Ω) is injective, we ﬁnally conclude that (λ + Aq )−1 f = u = (λ + Aq,0 )−1 f for every f ∈ Lqσ (Ω). Lemma 3.2. Let Ω ⊂ Rn , n ≥ 2, and 1 < q < ∞ be as in Assumption 1.1. If λ + Aq is surjective for λ ∈ (−∞, 0], then there is no non-trivial solution of (3.2).

Proof. Let f ∈ Lq (Ω) be arbitrary and let u ∈ D(A0,q ) such that (λ+A0,q )u = f . Then multiplying f with ∇g we observe that div u ∈ Wq1 (Ω) solves −λ(div u, g) − (∇ div u, ∇g) = (f, ∇g),

for all g ∈ Wq1 (Ω).

Hence, if g ∈ Wq1 (Ω) solves (3.2), then (f, ∇g) = 0 for all f ∈ Lq (Ω) and therefore ∇g = 0. Because of (3.2) and λ = 0, we conclude g = 0.

8

H. Abels

4. Construction of the approximative resolvent The proof of Theorem 1.3 is based on the following result. Theorem 4.1. Let Rnγ , n ≥ 2, γ ∈ C 1,1 (Rn−1 ), be a curved half-space, 1 < q < ∞, and let δ ∈ (0, π). Then there is a bounded operator Rγ,λ : Lq (Rnγ )n → Wq2 (Rnγ )n such that γ,N T )Rγ,λ f = f + Sγ,λ f (λ − ∆ + ∇K

in Rnγ ,

Rγ,λ f = 0

on

(4.1)

∂Rnγ

(4.2)

for every f ∈ Lq (Rnγ )n and λ ∈ C\(−∞, 0], where Sγ,λ L(Lq (Rnγ )) ≤ Cq,δ (1+|λ|)−ε 1

γ,N : Wqs− q (∂Rnγ ) → Wqs+1 (Rnγ ), uniformly in λ ∈ Σδ for some ε > 0. Here K s ∈ (− q1 , 1 + 1q ), is a bounded operator satisfying γ,N a = Rγ a ∆K γ,N a|∂Ω = a + ∂ν K

Sγ a

− 1 −ε

in Rnγ , on

− 1q −ε

−1 where Rγ : Wq q (∂Rnγ ) → Wq,0 (Rnγ ) and Sγ : Wq bounded operators. Moreover, for every R > 0

(4.3)

∂Rnγ ,

(4.4) − 1q

(∂Rnγ ) → Wq

(1 + |λ|)Rγ,λ L(Lq (Rnγ )) + ∇2 Rγ,λ L(Lq (Rnγ )) ≤ Cq,δ , h(−λ)R dλ ≤ Cq,δ h∞ , γ,λ

(∂Rnγ ) are

λ ∈ Σδ ,

(4.5)

h ∈ H(δ).

(4.6)

L(Lq (Rn γ ))

ΓR

Proof. The theorem is proved in [5, Theorem 4.1] but only for s = 0 in the mapping γ,N , which remains to be extended. properties of K γ,N ≡ K 1 is deﬁned in [3, Section 5.5] as The operator K γ,N K

=

F ∗,−1 k 1 (Dx , x )F0∗ ,

where k 1 (Dx , x ) is a Poisson operator of order −1 in R-form with C 0,1 -coeﬃcients in the sense of [3, Section 4] and F ∗,−1 , F0∗ are as in Section 2. By duality (∂xα k 1 (Dx , x )a, f )Ω = (−1)|α| (a, t(x , Dx )∂xα f )∂Ω

(4.7)

∞ (Rn+ ), and α ∈ Nn0 , where for a ∈ C0∞ (Rn−1 ), f ∈ C(0)

∞ t(x , Dx )f := Fξ−1 k˜1 (x , ξ , yn )f˜(ξ , yn )dyn , →x 0

f˜(ξ , xn ) = Fx →ξ [f (., xn )], and k˜1 (x , ξ , yn ) denotes the symbol-kernel of k 1 (Dx , x ), cf. [3, Section 4]. Using (4.7) for suitable α and [1, Theorem 4.8] for t(x , Dx ), the mapping properties of k 1 (Dx , x ) are obtained by duality and interpolation.

Stokes Operator in Unbounded Domains

9

Now we deﬁne the approximate resolvent Rλ on a domain Ω satisfying (A1) as Rλ f =

m

f ∈ Lq (Ω)n ,

ψj Rγj ,λ (ϕj f ),

j=1

where Rγj ,λ , j = 1, . . . , m, is the approximate resolvent on Rnγj due to Theorem 4.1. Moreover, we deﬁne the approximate Poisson operator N a = K

m

− 1q

γj ,N (ϕj a), ψj K

a ∈ Wq

(∂Ω),

j=1

γj ,N is the operator due to Theorem 4.1 for Rnγ . Now we have where K j Lemma 4.2. Let 1 < q < ∞ and Ω ⊆ Rn be as in Assumption 1.1 and let r = q or r = q . Moreover, let KN be the Poisson operator of the Neumann problem as deﬁned in (2.8). Then there is some ε > 0 such that N )T ur ≤ Cr u2−ε,r ∇(KN − K

1 for all u ∈ Wr2−ε (Ω)n ∩ Wr,0 (Ω)n .

Proof. For simplicity let r = q. Let f ∈ Lq (Ω)n be arbitrary and let f = f0 + ˙ 1 (Ω) be its Helmholtz decomposition. By (A3) we have ∇p, f0 ∈ Lqσ (Ω), p ∈ W q

p = p1 + p2 , where p1 ∈ Wq1 (Ω) and p2 ∈ Lqloc (Ω) with ∇p2 ∈ Wq1 (Ω) and (p1 , ∇p2 )1,q ≤ Cq ∇p1,q . Then by (4.3)–(4.4) N )T u, f )Ω = (∇(KN − K N )T u, ∇p)Ω (∇(KN − K N )T u, p1 ∂Ω − (∆K N )T u, ∇p2 )Ω + (I − ∂ν K N T u, p1 )Ω . = (∇(KN − K N )T u, p1 ∂Ω can be estimated by u2−ε,q f q for some ε > 0 The term (I − ∂ν K in a straightforward manner using Theorem 4.1 and (3.1). Moreover, N T u, p1 )Ω | ≤ |(∆K

m

|(ψj Rγ j (ϕj T u), p1 )Ω |

j=1

+

m

γj ,N (ϕj T u) + (∆ψj )K γj ,N (ϕj T u), p1 )Ω |, |(2(∇ψj ) · ∇K

j=1

where |(ψj Rγ j (ϕj T u), p1 )Ω | ≤ CRγ j ϕj T uW −1 (Rn ) p1 1,q ≤ Cu2−ε,q f q q,0

γj

for some ε > 0. Since p1 ∈ Wq1 (Ω) → Wqε (Ω) with 0 < ε < γj ,N : property K

−ε− 1q Wq (∂Rnγj )

1 q ,

by the mapping

→ Wq1−ε (Rnγ ), and by (2.3)

γj ,N (ϕj T u) + (∆ψj )K γj ,N (ϕj T u), p1)Ω | ≤ Cu2−ε,q f q |(2(∇ψj ) · ∇K

10

H. Abels

N T u, ∇p2 )Ω is estimated in the same way using for 0 < ε < q1 . The term (∇K 1 ∇p2 ∈ Wq (Ω). Finally, by the deﬁnitions of KN and T |(∇KN T u, ∇p2 )Ω | = |((∆ − ∇ div)u, ∇p2 )Ω | ≤

|(∇u, ∇2 p2 )Ω | + |(∂ν u, ∇p2 )∂Ω | + |(div u, ∆p2 )Ω | + |(div u, ∂ν p2 )∂Ω |

≤

Cu2−ε,q ∇p2 1,q ≤ Cu2−ε,q f q

for some ε > 0. – The proof for r = q is done in the same way.

Proof of Theorem 1.3. The proof is the same as in [5, Theorem 4.4] with minor modiﬁcations. We include it for the convenience of the reader. First of all, by (4.5) and interpolation Rj,λ (ϕj f )s,q ≤ Cq,δ,R (1 + |λ|)−1+ 2 f q , s

λ ∈ Σδ ,

(4.8)

for all s ∈ [0, 2] and f ∈ Lq (Ω)n , j = 1, . . . , m. Moreover, by (4.1) (λ − ∆ + ∇KN T )Rλ f m m ψj Sγj ,λ (ϕj f ) − (2(∇ψj ) · ∇Rj,λ (ϕj f ) + (∆ψj )Rj,λ (ϕj f )) = f+ j=1

j=1

N T )Rλ f. +(∇KN T − ∇K Hence (4.8), Theorem 4.1, and Lemma 4.2 imply (λ − ∆ + ∇KN T )Rλ = I + Sλ , where Sλ L(Lq (Ω)) ≤ Cq,δ (1+|λ|)−ε uniformly in λ ∈ Σδ for some ε > 0. Therefore (λ + A0,q )−1 exists for all λ ∈ Σδ with |λ| ≥ R for some R > 0 and (λ + A0,q )−1 = Rλ + Sλ , where Sλ L(Lq (Ω)) ≤ Cq,δ (1+|λ|)−1−ε uniformly in λ ∈ Σδ , |λ| ≥ R. Furthermore, for Bλ = Rλ , Sλ we have h(−λ)B dλ ≤ Cq,δ h∞ , h ∈ H(δ), λ ΓR

L(Lq (Ω))

because of (4.6) and Sλ ≤ Cq,δ (1 + |λ|)−1−ε . The same arguments apply to A0,q instead of A0,q . Hence we can assume that also (λ + A0,q )−1 exists for all λ ∈ Σδ with |λ| ≥ R. In particular, this implies that (A4) is valid because of Lemma 3.2. Now (λ + A0,q )−1 |Lqσ (Ω) = (λ + Aq )−1 by Lemma 3.1, which proves (1.5). The rest of the theorem follows from (1.5) and the fact that f (λ) = (λ + Aq )−1 is uniformly bounded on compact subsets of ρ(−Aq ).

Stokes Operator in Unbounded Domains

11

5. Bounded H∞ -calculus for an aperture domain Roughly speaking an aperture domain is a domain separated by a wall with a hole (aperture) inside. More precisely, an aperture domain Ω ⊂ Rn , n ≥ 2, is a domain such that Ω ∪ B = Rn+ ∪ Rn− ∪ B for some ball B, where Rn− := {x ∈ Rn : xn < −d} for some d > 0. Concerning the Stokes equations the aperture domain is of particular interest since under certain circumstances depending on the dimension n and the integral exponent q an additional condition has to be posed to get unique solutions, cf. [16]. This may be done by prescribing the ﬂux φ(u) of the velocity ﬁeld u through the hole, i.e., ν · udσ, φ(u) := M

where M is an (n − 1)-dimensional compact manifold dividing Ω into an upper and lower part Ω+ , Ω− , resp., such that Ω± ∪ B = Rn± ∪ B. In the following let ϕ± , ψ± ∈ Cb∞ (Rn± ) be cut-oﬀ functions such that ϕ± (x) = 1 for x ∈ Rn± with |x| ≥ R for some suitable R > 0, ψ± ≡ 1 on supp ϕ± , and supp ψ± ⊂ Rn± \ B. In order to apply Theorem 1.3, it remains to verify Assumption 1.1. Proposition 5.1. Let Ω ⊂ Rn , n ≥ 2, be an aperture domain with C 1,1 -boundary. n Then Assumption 1.1 is valid for Ω and every 1 < q ≤ n−1 and δ ∈ (0, π). Proof. Obviously, (A1) is true. The ﬁrst part of (A2) is true for every 1 < q < ∞ n but (1.4) only holds if 1 < q ≤ n−1 , cf. [16, Theorem 2.6]. The condition (A3) can be easily veriﬁed by using the cut-oﬀ functions ϕ± and the corresponding statement for the half-spaces Rn± . We will analyze the resolvent (λ + Aq )−1 near the origin by comparing it with the resolvents of the Stokes operator on Rn± . Hence let Rλ± = (λ + ARn± ,q )−1 , 1 < q < ∞, denote the resolvent of the Stokes operator on Rn± . Then |λ|uq + ∇2 uq ≤ Cq,δ f q ,

λ ∈ Σδ , f ∈ Lq (Rn± ),

(5.1)

cf., e.g., [15, Theorem 1.3]. Moreover, ± Lemma 5.2. Let 1 < q < n, δ ∈ (0, π), and let BM = BM (0) ∩ Rn± . Then for every M >0 (Rλ± f, ∇Rλ± f )Lq (B ± ) ≤ Cq,δ,M f q M

uniformly in λ ∈ Σδ and f ∈ Lq (Rn± )n . Proof. By Poincar´e’s inequality, Rλ± f Lq (B ± ) ≤ Cq,M ∇Rλ± f Lq (B ± ) . Hence it remains to estimate ∇Rλ± . Then, if ∇Rλ± f Lq (B ± ) M

1 q∗

=

1 q

M

− n1 ,

M

≤ CM ∇Rλ± f Lq∗ (Rn± ) ≤ Cq,M ∇2 Rλ f q ≤ Cq,δ,M f q

uniformly in λ ∈ Σδ and f ∈ Lq (Rn± )n by (5.1).

12

H. Abels

In order to analyze (1.2) for A = Aq with Γ replaced by ΓR := Γ ∩ BR (0), R > 0, we consider uλ := (λ + Aq )−1 f − ψ + Rλ+ ϕ+ f − ψ − Rλ− ϕ− f,

f ∈ Lqσ (Ω).

(5.2)

− − −1 Moreover, let pλ = qλ − ψ + p+ f and λ − ψ pλ , where ∇qλ = (I − Pq )∆(λ + Aq ) ± ± ± ± ± ∇pλ = (I − Pq )∆Rλ ϕ f is chosen such that B ± pλ = 0 for M > 0 so large that M ψ ± ≡ 1 on Rn± \ BM (0). Then (uλ , pλ ) solves

(λ − ∆)uλ + ∇pλ = (1 − ϕ+ − ϕ− )f + fλ+ + fλ− =: f˜λ

in Ω,

gλ−

in Ω,

div uλ =

gλ+

+

=: g˜λ

uλ |∂Ω = 0

on ∂Ω,

where gλ± = −∇ψ ± · Rλ± ϕ± f and fλ± = 2(∇ψ ± ) · ∇Rλ± ϕ± f + (∆ψ ± )Rλ± ϕ± f − (∇ψ ± )p± λ. Moreover, φ(uλ ) = φ((λ + Aq )−1 f ) = 0. Lemma 5.3. For every 1 < q < ∞ there is some 0 < a < 1 such that |λ|a uλ q ≤ Cf˜λ q uniformly in λ ∈ Σδ , |λ| ≤ 1. Proof. By [16, Theorem 2.1] for every 1 < q < n |λ|uλ q + ∇2 uλ q ≤ Cq,δ f˜λ q + ∇˜ gλ q + (1 + |λ|)˜ gλ W˙ −1 q,0

uniformly in λ ∈ Σδ . Moreover, gλ± dx = ± BM

± BM

(5.3)

div (1 − ψ ± )Rλ± ϕ± f dx = 0.

Hence gλ± W˙ −1 (Rn ) ≤ CR,q gλ± q by Poincar´e’s inequality. Furthermore, if n < q,0

±

r < ∞ and 1 < q < ∇uλ r

n n−1

≤ ≤

is deﬁned by 1q = 1r + n1 , then by [16, Corollary 2.4] Cr,δ |λ|uλ q + f˜λ q + ∇˜ gλ q + ˜ gλ W˙ −1 q,0 gλ 1,r Cr,δ f˜λ r + ˜

for λ ∈ Σδ , |λ| ≤ 1, since supp f˜λ , supp g˜λ ⊂ BM (0). Moreover, since ∇uλ q ≤ Cq (uλ , ∇2 uλ )q , interpolation of the latter inequality with (5.3) yields that for every 1 < q < ∞ there is some 0 < a < 1 such that λ ∈ Σδ , |λ| ≤ 1. gλ 1,q , |λ|a ∇uλ q ≤ Cq f˜λ q + ˜ n This implies that for n < r < ∞ and 1 < q < n−1 deﬁned by 1q = 1r + n1 there is some 0 < a < 1 such that |λ|a uλ r ≤ Cr |λ|a ∇uλ q ≤ Cr,δ f˜λ q + ˜ gλ 1,q ≤ Cr,δ f˜λ r + ˜ gλ 1,r

for λ ∈ Σδ with |λ| ≤ 1 using Sobolev’s inequality, cf. [16, Lemma 3.1]. Interpolating again ﬁnishes the proof.

Stokes Operator in Unbounded Domains

13

Combining Lemma 5.2 and Lemma 5.3 we obtain: Corollary 5.4. Let uλ be deﬁned as in (5.2). Then for every 1 < q < n there is some 0 < a < 1 such that uλ q ≤ C|λ|−a f q for all f ∈ Lqσ (Ω) uniformly in λ ∈ Σδ with |λ| ≤ 1. Proof of Theorem 1.4. First of all, (λ + Aq )−1 exists for every λ ∈ C \ (−∞, 0] and (Aq ) = Aq , cf. [16, Theorem 2.5]. Hence by duality and interpolation it is n . Moreover, R(Aq ) is dense, cf. suﬃcient to prove the statement for 1 < q < n−1 Franzke [17, Theorem 6]. Therefore Aq is sectorial and it is suﬃcient to prove (1.3) for h ∈ H(δ), cf. Denk, Hieber, and Pr¨ uss [11, Section 2.4]. Moreover, because of Theorem 1.3 it remains to prove h(−λ)(λ + Aq )−1 dλ ≤ Cδ,q h∞ for all h ∈ H(δ). (5.4) q Γ L(Lσ (Ω))

R

−1

Since (λ + Aq ) is bounded on compact subsets of C \ (−∞, 0], it is suﬃcient to consider R = 1. Because of (5.2) and Corollary 5.4, we can replace Aq by Aq,Rn± . Moreover, using 1 1 h(−λ)(λ + ARn± ,q )−1 dλ = h(ARn± ,q ) − h(−λ)(λ + ARn± ,q )−1 dλ 2πi Γ1 2πi Γ1 with Γ1 = Γ \ B1 (0) the estimate (5.4) is a consequence of the bounded H∞ uss [12], and of calculus for the Stokes operator on Rn± , cf. Desch, Hieber, and Pr¨ Theorem 1.3 for Ω = Rn± .

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[8] H. Amann, M. Hieber, and G. Simonett. Bounded H∞ -calculus for elliptic operators. Diﬀ. Int. Eq., Vol. 7, No. 3, 613–653, 1994. [9] W. Borchers and H. Sohr. On the semigrup of the Stokes operator for exterior domains in Lq -spaces. Math. Z. 196, 415–425, 1987. [10] S.K. Chua. Extension theorems on weighted Sobolev space. Indiana Univ. Math. J., 41, 1027–1076, 1992. [11] R. Denk, M. Hieber, and J. Pr¨ uss. R-boundedness, Fourier multipliers and problems of elliptic and parabolic Type. Mem. Am. Math. Soc. 788, 114 p., 2003. [12] W. Desch, M. Hieber, and J. Pr¨ uss. Lp -theory of the Stokes equation in a half-space. J. Evol. Eq. 1, 115–142, 2001. [13] G. Dore and A. Venni. On the closedness of the sum of two closed operators. Math. Z. 196, 189–201, 1987. [14] R. Farwig. Weighted Lq -Helmholtz decompositions in inﬁnite cylinders and in inﬁnite layers. Adv. Diﬀ. Eq. 8, 357–384, 2003. [15] R. Farwig and H. Sohr. Generalized resolvent estimates for the Stokes system in bounded and unbounded domains. J. Math. Soc. Japan 46, No. 4, 607–643, 1994. [16] R. Farwig and H. Sohr. Helmholtz Decomposition and Stokes Resolvent System for Aperture Domains in Lq -Spaces. Analysis 16, 1–26, 1996. [17] M. Franzke. Strong Lq -theory of the Navier-Stokes equations in aperture domains. Ann. Univ. Ferrara, Nuova Ser., Sez. VII 46, 161–173, 2000. [18] A. Fr¨ ohlich. Maximal regularity for the non-stationary Stokes system in an aperture domain. J. Evol. Eq. 2, 471–493, 2002. [19] Y. Giga. Analyticity of the semigroup generated by the Stokes operator in Lr spaces. Math. Z. 178, 297–329, 1981. [20] Y. Giga. Domains of fractional powers of the Stokes operator in Lr Spaces. Arch. Ration. Mech. Anal. 89, 251–265, 1985. [21] Y. Giga and H. Sohr. On the Stokes operator in exterior domains. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36, 103–130, 1989. [22] Y. Giga and H. Sohr. Abstract Lp estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains. J. Funct. Anal. 102, 72–94, 1991. [23] G. Grubb. Functional Calculus of Pseudodiﬀerential Boundary Problems, 2nd Edition. Birkh¨ auser, Basel - Boston - Berlin, 1996. [24] G. Grubb and V.A. Solonnikov. Boundary value problems for the nonstationary Navier-Stokes equations treated by pseudo-diﬀerential methods. Math. Scand. 69, 217–290, 1991. [25] A. McIntosh. Operators which have an H∞ -calculus. In “Miniconference on Operator Theory and Partial Diﬀerential Equations”, B. Jeﬀeries, A. McIntosh, W. Ricker, editors Proc. Center Math. Anal. A.N.U., 14, 210–231, 1986. [26] T. Miyakawa. The Helmholtz decomposition of vector ﬁelds in some unbounded domains. Math. J. Toyama Univ. 17, 115–149, 1994. [27] A. Noll and J. Saal. H ∞ -calculus for the Stokes operator on Lq -spaces. Math. Z. 244, 651–688, 2003.

Stokes Operator in Unbounded Domains

15

[28] C.G. Simader and H. Sohr. A new approach to the Helmholtz decomposition and the Neumann problem in Lq -spaces for bounded and exterior domains. Series on Advances in Mathematics for Applied Sciences, Vol. 11, Singapore, World Scientiﬁc, 1–35, 1992. [29] H. Triebel. Interpolation Theory, Function Spaces, Diﬀerential Operators. NorthHolland Publishing Company, Amsterdam, New York, Oxford, 1978. Helmut Abels Max Planck Institute for Mathematics in the Sciences Inselstr. 22 D-04103 Leipzig, Germany e-mail: [email protected]

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 17–35 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Exact Estimates of Solutions to the Robin Boundary Value Problem for Elliptic Non-divergent Second-order Equations in a Neighborhood of the Boundary Conical Point Mikhail Borsuk and Agneszka Zawadzka Abstract. We investigate the behavior of strong solutions to the Robin boundary value problem for linear and quasilinear elliptic non-divergent secondorder equations in a neighborhood of the boundary conical point. We establish precise exponent of the solution decreasing rate. Mathematics Subject Classiﬁcation (2000). 35J65, 35J70, 35B05, 35B45, 35B65. Keywords. Elliptic equations, the Robin problem, conical points.

1. Introduction Let G ⊂ Rn , n ≥ 2 be a bounded domain with boundary ∂G that is a smooth surface everywhere except at the origin O ∈ ∂G and near the point O it is a convex conical surface with its vertex at O. We consider the elliptic value problems L[u] ≡ aij (x)uxi xj + ai (x)uxi + a(x)u = f (x), x ∈ G, (L) ∂u 1 B[u] ≡ ∂ n + |x| γ(x)u = g(x), x ∈ ∂G \ O. and

aij (x, u, ux )uxi xj + a(x, u, ux ) = 0, aij = aji , x ∈ G, ∂u 1 ∂ n + |x| γ(x)u = g(x), x ∈ ∂G \ O

(QL)

(summation over repeated indices from 1 to n is understood), n denotes the unite outward normal to ∂G \ O. We obtain the best possible estimates of the problems (L) and (QL) strong solutions near a conical boundary point. Analogous results were established in [2, 3, 4, 1] for the Dirichlet problem. Many mathematicians

18

M. Borsuk and A. Zawadzka

have considered the third boundary value problem. For the ﬁrst time, V.A. Kondrat’ev [10] studied general elliptic boundary value problems in domains with conical points. He proved the solvability of such problems with inﬁnitely diﬀerentiable coeﬃcients in weighted and usual L2 -Sobolev spaces. Later V.G. Maz’ya and B.A. Plamenevskiy extended the results of V.A. Kondrat’ev to the Lp -Sobolev weighted spaces (see, e.g., [11, 12]). The oblique derivative problem for elliptic equations in non-smooth domains investigated M. Faierman [6], G. Lieberman [13],[15], M. Garroni, V.A. Solonnikov and M. Vivaldi [7], H. Reisman [18]. G. Lieberman [13, 15] proved the local and global maximum principles for general second-order linear and quasi-linear elliptic equations. He studied the problems of the solutions existence and regularity in Lipschitz domains for equation with the H¨ older continuous coeﬃcients. Recently G. Lieberman [15] proved the H¨older continuity of strong solutions under weak hypotheses on the (L) coeﬃcients. M.Faierman [6] investigated the regularity in the Sobolev space W 2,p (G) of generalized solutions of (L) in a rectangle. H.Reisman [18] investigated such problem in weighted Sobolev spaces for the domain with dihedral edges and the equation coeﬃcients being inﬁnitely smooth. At last, authors of [7] considered the problem for the Poisson equation on the inﬁnite angle. A principal new feature of this article is the consideration of our estimates for equations with minimal smooth coeﬃcients. Our examples demonstrate this fact. Let C be the rotational cone {x1 > r cos ω20 }. For a domain G which has a conical point at O ∈ ∂G we introduce the notations: • Ω := C ∩ S n−1 ; dΩ denotes the (n − 1)-dimensional area element of Ω; • Gba := G ∩ {(r, ω) : 0 ≤ a < r < b, ω ∈ Ω}: a layer in Rn ; • Γba := ∂G ∩ {(r, ω) : 0 ≤ a < r < b, ω ∈ ∂Ω}: the lateral surface of the layer Gba ; • Gd := G \ Gd0 ; Γd := ∂G \ Γd0 ; • Ωρ := Gd0 ∩ ∂B (0), ≤ d. We use the standard function spaces: • C k (G), C0k (G) with the norm |u|k,G , • Lebesgue space Lp (G), p ≥ 1 with the norm up,G , • the Sobolev space W k,p (G). k We deﬁne the weighted Sobolev spaces: Vp,α (G) for integer k ≥ 0 and real α as the closure of C0∞ (G \ O) with respect to the norm ⎛ ⎞ p1 k uVp,α rα+p(|β|−k) |Dβ u|p dx⎠ k (G) = ⎝ G |β|=0

1 k− p

and Vp,α (∂G) as the space of functions ϕ, given on ∂G, with the norm ϕ

k− 1

Vp,α p (∂G)

= inf ΦVp,α k (G) ,

Exact Estimates of Solutions to the Robin Boundary Value Problem

19

where the inﬁmum is taken over all functions Φ such that Φ = ϕ in the sense ∂G of traces. We denote W k (G) ≡ W k,2 (G), ◦ k− 1 W α 2 (∂G)

◦k W α (G)

k ≡ V2,α (G),

k− 1

≡ V2,α 2 (∂G).

2,n Deﬁnition 1. A strong solution of problem (L) is a function u(x) ∈ Wloc (G) ∩ 2 0 W (Gε ) ∩ C (G) that for each ε > 0 satisﬁes an equation for almost all x ∈ Gε and the boundary condition in the sense of traces on Γε .

Deﬁnition 2. A strong solution of problem (QL) is a function u(x) ∈ C 0 (G) ∩ 2,q (G \ O), q ≥ n that satisﬁes the equation for almost all x ∈ G, and W 1 (G) ∩ Wloc the boundary condition in the sense of traces on ∂G \ O. We assume that M0 = max |u(x)| is known (see, e.g., [13, 15]). x∈G

We assume the existence d > 0 such that Gd0 is the convex rotational cone with the vertex at O and the aperture ω0 ∈ ( π2 , π), thus n ω0 π 2 d 2 ω0 2 , ω0 ∈ ( , π) . (1.1) x ; |ω1 | = Γ0 = (r, ω)x1 = cot 2 i=2 i 2 2 Linear problem. Regarding the (L) equation we assume that the following conditions are satisﬁed: (a) the condition of the uniform ellipticity: νξ 2 ≤ aij (x)ξi ξj ≤ µξ 2 ,

∀x ∈ G,

∀ξ ∈ Rn ;

(b) aij ∈ C 0 (G), ai ∈ Lp (G), p > n, 12 n ij ij 2 |a (x) − a (y)| ≤ A(|x − y|);

ν, µ = const > 0, and aij (0) = δij ;

a, f ∈ Ln (G); for these the inequalities 12 n i 2 |x| |a (x)| + |x|2 |a(x)| ≤ A(|x|)

i,j=1

i=1

hold for x, y ∈ G, where A(r) is the monotonically increasing, nonnegative function, continuous at zero, A(0) = 0; (c) there exist numbers f1 ≥ 0, g1 ≥ 0, s > 1, β ≥ s − 2, γ0 > tan ω20 such that |f (x)| ≤ f1 |x|β , |g(x)| ≤ g1 |x|s−1 , γ(x) ≥ γ0 ; (d) a(x) ≤ 0 in G, γ(x) ∈ L∞ (∂G) ∩ C 1 (∂G \ O); → ν Let Ω ⊂ S n−1 be a bounded domain with the smooth boundary ∂Ω. Let − 0 be the exterior normal to ∂Ω. Let γ(x) ∈ C (∂Ω), γ(x) ≥ γ0 > 0. We consider the eigenvalue problem for the Laplace-Beltrami operator ω on the unit sphere: ω u + ϑu = 0, ω ∈ Ω, (EVP ) ∂u + γ(x)u = 0, − ∂→ ν ∂Ω

which consists of the determination of all values ϑ (eigenvalues) for which (EV P ) has a non-zero weak solutions (eigenfunctions).

20

M. Borsuk and A. Zawadzka

Remark 1. The eigenvalue problem (EV P ) was studied in Section VI [5] and in §2.5 [19] (see as well §2.4 [12]). We observe that ϑ = 0 is not an eigenvalue of (EV P ). We denote for the smallest positive eigenvalue ϑ 2 − n + (n − 2)2 + 4ϑ λ= . 2 Our main results are following theorems.

(1.2)

Theorem 1.1. Let u be a strong solution of the problem (L) and assumptions (a)– (d) are satisﬁed with A(r) Dini-continuous at zero. Suppose, in addition, that 1

2 g(x) ∈ W◦ 4−n (∂G)

and

0

and there exist numbers sup −s f ◦ 0 + g ◦ 1/2 W 4−n (G0 )

>0

1

2 a(x) ∈ W◦ 4−n (G), γ(x) ∈ W◦ 2−n (∂G) if u(0) = 0

W 4−n (Γ 0)

+ |u(0)| a ◦ 0

W 4−n (G 0)

+ γ ◦ 1/2

W 2−n (Γ 0)

=: ks , (1.3)

sup 1−s f n,G/2 + |u(0)|an,G/2 =: κs . >0

Then there are d ∈ (0, 1) and a constant C > 0 depending only on ν, µ, d, s,

d d n, λ, γ0 , γC 1 (∂G\O) , meas G and on the quantity A(r) r dr such that ∀x ∈ G0 0

|u(x) − u(0)| ≤ C |u|0,G + ks + κs + f ◦ 0

+ g ◦ 1/2 + g1 W 4−n (∂G) ⎧ ⎪ ⎪|x|λ , ⎨ λ 3/2 1 + |u(0)| 1 + a ◦ 0 + γ ◦ 1/2 |x| ln |x| , W 4−n (G) W 2−n (∂G) ⎪ ⎪ ⎩|x|s , W 4−n (G)

if s > λ, if s = λ, if s < λ.

If, in addition, there is a number sup >0

−s

0 f + u(0)aVp,2p−n (G ) + g /2

(1.4)

1− 1

p Vp,2p−n (Γ ) /2

+ |u(0)|γ

1− 1

p Vp,p−n (Γ ) /2

=: τs , (1.5)

then

|∇u(x)| ≤ C |u|0,G + ks + κs + τs + f ◦ 0

+ g ◦ 1/2 + g1 W 4−n (∂G) ⎧ ⎪ ⎪|x|λ−1 , ⎨ λ−1 3/2 1 + |u(0)| 1 + a ◦ 0 + γ ◦ 1/2 |x| ln |x| , W 4−n (G) W 2−n (∂G) ⎪ ⎪ ⎩|x|s−1 , W 4−n (G)

if s > λ, if s = λ, if s < λ. (1.6)

Exact Estimates of Solutions to the Robin Boundary Value Problem

21

Theorem 1.2. Let u be a strong solution of the problem (L) and assumptions of Theorem 1.1 are satisﬁed with A(r) that is a function continuous at zero but not Dini-continuous at zero. Then there are d ∈ (0, 1) and for each ε > 0 a constant Cε > 0 depending only on ν, µ, d, s, n, λ, γ0 , γC 1 (∂G\O) , ε, meas G and on A(diam G) such that ∀x ∈ Gd0 |u(x) − u(0)| ≤ Cε |u|0,G + f ◦ 0 + g ◦ 1/2 + g1 + ks + κs (1.7) W 4−n (G) W 4−n (∂G) |x|λ−ε , if s > λ, + |u(0)| 1 + a ◦ 0 + γ ◦ 1/2 W 4−n (G) W 2−n (∂G) |x|s−ε , if s ≤ λ and

|∇u(x)| ≤ Cε |u|0,G + f ◦ 0

+ g1 + ks + κs + τs |x|λ−1−ε , if s > λ, +|u(0)| 1 + a ◦ 0 + γ ◦ 1/2 W 4−n (G) W 2−n (∂G) |x|s−1−ε , if s ≤ λ. W 4−n (G)

+ g ◦ 1/2

W 4−n (∂G)

(1.8)

Theorem 1.3. Let u be a strong solution of the problem (L) and assumptions of Theorem 1.1 are satisﬁed with s ≥ λ, A(r) ln 1r ≤ const, r > 0 and A(0) = 0. Then there are d ∈ (0, 1) and the constants C > 0, c > 0 depending only on ν, µ, d, n, λ, γ0 , γC 1 (∂G\O) , meas G and on A(diam G) such that ∀x ∈ Gd0 |u(x) − u(0)| ≤ C |u|0,G + f ◦ 0 + g ◦ 1/2 + ks + κs + g1 W 4−n (G) W 4−n (∂G) (1.9) λ c+1 1 |x| ln + |u(0)| 1 + a ◦ 0 + γ ◦ 1/2 W 4−n (G) W 2−n (∂G) |x| and

+ g ◦ 1/2 + g1 + ks + κs + τs |∇u(x)| ≤ C |u|0,G + f ◦ 0 W 4−n (G) W 4−n (∂G) 1 . |x|λ−1 lnc+1 + |u(0)| 1 + a ◦ 0 + γ ◦ 1/2 W 4−n (G) W 2−n (∂G) |x|

(1.10)

These Theorems are proved in the same way as in [2] for the Dirichlet problem with corresponding modiﬁcations considering the other boundary condition. There the idea of the proof is based on the deduction a new inequality of the Hardy-Friedrichs-Wirtinger type with the exact constant adapted to the Robin problem. The precise exponent of the solution decreasing rate depends on this exact constant. Therefore we restrict oneself here only to the deduction of the similar inequality (see below). Quasilinear problem. Let M = {(x, u, z)|x ∈ G, u ∈ R, z ∈ Rn }. Regarding the problem (QL) we assume that the following conditions are satisﬁed: (A) aij (x, u, z) ∈ W 1,q (M), q > n; γ(x) ∈ L∞ (∂G) ∩ C 1 (∂G \ O); the condition of Carath´eodory: functions a(x, u, z), ∂a(x,u,z) ∈ CAR, that is: ∂u (i) they are measurable on G as functions of variable x for ∀u, z; (ii) they are continuous with respect to u, z for almost all x ∈ G;

22

M. Borsuk and A. Zawadzka

(AA) aij (0, u(0), 0) = δij (i, j = 1, . . . , n) – the Kronecker symbol; (B) the condition of the uniform ellipticity: νξ 2 ≤ aij (x, u, z)ξi ξj ≤ µξ 2 ,

∀ξ ∈ Rn , (x, u, z) ∈ M; ν, µ = const > 0;

≤ 0, ∀(x, u, z) ∈ M; (C) ∂a(x,u,z) ∂u (D) there exist numbers β > −1, γ0 > tan ω20 , γ1 ≥ γ0 , positive constants δ, µ1 , k1 , g0 ≥ 0 and nonnegative functions b(x), f (x) ∈ Lqloc (G \ O), q ≥ n such that on M the inequalities ∂a(x, u, z) |a(x, u, z)| + ≤ µ1 |z|2 + b(x)|z| + f (x), ∂u b(x) + f (x) ≤ k1 |x|β , |g(x)| ≤ g0 |x|δ ; γ0 ≤ γ(x) ≤ γ1 hold; (E) the coeﬃcients of the problem (QL) satisfy such conditions that guarantee u ∈ C 1+ κ (G ) and the existence of a local a priori estimate |u|1+ κ ,G ≤ ∈ (0, 1) for any smooth G ⊂⊂ G \ {O} (see Theorems 13.13, 13.14 M1 , κ [17] and Lemma 2.3 [16]). Our main result about the problem (QL) is the following Theorem 1.4. Let λ > 1 be deﬁne by (1.2) and let u(x) be a solution of problem (QL), q > n and the assumptions (A), (AA), (C) − −(E) are satisﬁed with β > ◦ 1/2

λ − 2 > −1, δ ≥ λ − 1 > 0. Suppose, in addition, that γ(x) ∈ W 2−n (∂G), g(x) ∈ ◦ 1/2

W 4−n (∂G) and for each ε > 0 there is sup ρ−λ−ε g ◦ 1/2 ρ + |u(0)|γ ◦ 1/2 W 4−n (Γ0 )

ρ>0

:= k2 .

W 2−n (Γρ 0)

(1.11)

Then there exist numbers d > 0, C 0 , C 1 not depending on u but only on n, λ, ν, µ, µ1 , β, k1 , q, M0 and the domain G such that 1) |u(x) − u(0)| ≤ C0 |x|λ ;

d/2

|∇u(x)| ≤ C1 |x|λ−1 , x ∈ G0 . 1−1/q

In addition, if g(x) ∈ Vq,α

1−1/q

(∂G) and γ(x) ∈ Vq,α−q (∂G) if u(0) = 0 and

g(x)V 1−1/q (Γ ) + |u(0)| · γ(x)V 1−1/q (Γ ) ≤ Cλ−2+ q,α

0

q,α−q

α+n q

0

, 0 < ρ < d/2, (1.12)

then there exist numbers d > 0, C 2 , not depend on u(x), but only on n, λ, ν, µ, µ1 , β, k1 , q, M0 and the domain G, such that: 2 (G) and 2) if α + q(λ − 2) + n > 0, then u(x) − u(0) ∈ Vq,α λ−2+ u(x) − u(0)Vq,α 2 (Gρ ) ≤ C2 ρ 0

3) if 1 < λ < 2, q >

n 2−λ ,

n+α q

; 0 < ρ < d/2; d/2

then u(x) − u(0) ∈ C λ (G0 ).

For the proof of this Theorem we use the barrier techniques and the comparison principle.

Exact Estimates of Solutions to the Robin Boundary Value Problem

23

2. The Hardy-Friedrichs-Wirtinger inequality Deﬁnition 3. Function u is called a weak solution of the problem (EV P ) provided that u ∈ W 1 (Ω) and satisﬁes the integral identity 1 ∂u ∂η − ϑuη dΩ + γ(x)uηdσ = 0 (II) qi ∂ωi ∂ωi Ω

∂Ω

for all η(x) ∈ W (Ω). 1

Let us introduce functionals on W 1 (Ω): F [u] = |∇ω u|2 dΩ + γ(x)u2 dσ, Ω

∂Ω

!

|∇ω u|2 − ϑu2 dΩ +

H[u] = Ω

u2 dΩ,

G[u] = Ω

γ(x)u2 dσ ∂Ω

and corresponding to them bilinear forms 1 ∂u ∂η dΩ + γ(x)uηdσ, F (u, η) = qi ∂ωi ∂ωi Ω

G(u, η) =

uηdΩ. Ω

∂Ω

We deﬁne the set K = u ∈ W 1 (Ω) G[u] = 1 . Since K ⊂ W 1 (Ω), F [u] is bounded from below for u ∈ K. The greatest lower bound of F [u] for this family we denote by ϑ : inf F [u] = ϑ. We formulate the following statement: u∈K

Theorem 2.1. (see Theorem of Subsection 4 §2.5, p. 123 [19]). Let Ω ⊂ S n−1 be bounded domain with smooth boundary ∂Ω. Let γ(x) ∈ C 0 (∂Ω), γ(x) ≥ γ0 > 0. There exist ϑ > 0 and a function u ∈ K such that F (u, η) − ϑG(u, η) = 0 for arbitrary η ∈ W 1 (Ω). In particular, F [u] = ϑ. In addition, on Ω, u has continuous derivatives of arbitrary order and satisﬁes the equation ω u + ϑu = 0, ω ∈ Ω and the boundary condition of (EV P ) in the weak sense (see Remark on p. 121–122 [19]). Next from the variational principle we obtain the Friedrichs-Wirtinger inequality: Theorem 2.2. Let ϑ be the smallest positive eigenvalue of problem (EV P ) (it there exists according to Theorem 2.1). Let Ω ⊂ S n−1 be bounded domain. Let u ∈ W 1 (Ω) and γ(x) ∈ C 0 (∂Ω), γ(x) ≥ γ0 > 0. Then 2 2 (2.1) ϑ u (ω)dΩ ≤ |∇ω u(ω)| dΩ + γ(ω)u2 (ω)dσ. Ω

Ω

∂Ω

Proof. Consider functionals F [u], G[u], H[u] on W 1 (Ω) described above. We will ﬁnd the element u that gives the minimum of the functional F [u] in the set K. For

24

M. Borsuk and A. Zawadzka

this we investigate the minimization of the quadratic functional H[u] on all functions u(ω), for which the integrals exist and which satisfy the boundary condition from (EV P ). The necessary condition of existence of the functional extremum is δH[u] = 0. By the calculation of the ﬁrst variation δH we have ∂u δudσ + 2 δH[u] = −2 (ω u + ϑu)δudΩ + 2 γ(x)uδudσ. → ∂− ν Ω

∂Ω

∂Ω

Hence we obtain the Euler equation and boundary condition that are our (EV P ). Backwards, let u(ω) be the solution of (EV P ). By Theorem 2.1 u ∈ C 2 (Ω). Therefore we can multiply both sides of equation (EV P ) by u and integrate over Ω using the Gauss-Ostrogradskiy formula: 0 = (uω u + ϑu2 )dΩ = ϑ u2 dΩ − |∇ω u|2 dΩ Ω

Ω

Ω

∂ ∂u J ∂u 2 2 + dσ u dω = ϑ u dΩ − |∇ω u| dΩ + u − ∂ωi qi ∂ωi ∂→ ν Ω Ω Ω ∂Ω 2 2 2 = ϑ u dΩ − |∇ω u| dΩ − γ(x)u dσ = ϑ − F [u] ⇒ ϑ = F [u].

(by K)

Ω

Ω

∂Ω

Consequently the required minimum is the least eigenvalue of (EV P ). The existence a function u ∈ K such that F [u] ≤ F [v] for all v ∈ K was proved in Theorem 2.1. Throughout what follows we work with the value λ deﬁned by (1.2). Therefore the Friedrichs-Wirtinger inequality will be written in the following form 2 2 λ(λ + n − 2) ψ dΩ ≤ |∇ω ψ| dΩ + γ(x)ψ 2 dσ, (2.2) Ω

Ω

∂Ω

∀ψ ∈ W 1 (Ω), γ(x) ∈ C 0 (∂Ω), γ(x) ≥ γ0 > 0. The Hardy-Friedrichs-Wirtinger inequality Let u ∈ C 0 (G) ∩ W 1 (G) and γ(x) ∈ C 0 (∂G \ O), γ(x) ≥ γ0 > 0. Then ⎧ ⎫ ⎪ ⎪ ⎨ ⎬ α−4 2 α−2 2 α−3 2 r u dx ≤ H(λ, n, α) r |∇u| dx + r γ(x)u (x)ds , ⎪ ⎪ ⎩ d ⎭ d d G0

G0

Γ0

1 H(λ, n, α) = , α≤4−n λ(λ + n − 2) + 14 (4 − n − α)2 provided that integrals on the right are ﬁnite.

(2.3)

(2.4)

Exact Estimates of Solutions to the Robin Boundary Value Problem

25

Proof. By Theorem 2.2 the inequality (2.2) holds. Multiplying it by rn−5+α and integrating over r ∈ (0, d) we obtain ⎫ ⎧ ⎪ ⎪ ⎬ ⎨ 1 1 rα−4 u2 dx ≤ rα−2 2 |∇ω u|2 dx + rα−3 γ(x)u2 (x)ds , ⎪ λ(λ + n − 2) ⎪ r ⎭ ⎩d d d G0

Γ0

G0

∀α ≤ 4 − n.

(2.5)

Hence (2.3) follows for α = 4 − n. Now, letα < 4 − n. We show that u(0) = 0. In fact, from the representation u(0) = u(x) − u(x) − u(0) by the Cauchy inequality we have 12 |u(0)|2 ≤ |u(x)|2 + |u(x) − u(0)|2 . Putting v(x) = u(x) − u(0) we obtain 1 |u(0)|2 rα−4 dx ≤ rα−4 u2 (x)dx + rα−4 |v|2 dx < ∞ (2.6) 2 Gd 0

Gd 0

Gd 0

(the ﬁrst integral from the right is ﬁnite by (2.5) and the second integral

α−4is also r dx = ﬁnite, in virtue of the Hardy inequality (see Theorem 330 [9])). Since measΩ

d

Gd 0

rα+n−5 dr = ∞, by α + n − 4 < 0, the assumption u(0) = 0 contradicts

0

(2.6). Thus u(0) = 0. Therefore we can use the Hardy inequality and we obtain 4 rα−4 u2 dx ≤ rα−2 u2r dx. |4 − n − α|2 Gd 0

(2.7)

Gd 0

Adding the inequalities (2.5), (2.7) and using the formula 2 ∂u 1 2 |∇u| = + 2 |∇ω u|2 , ∂r r

we get the desired (2.3).

Lemma 2.3. Let Gd0 be the conical domain and ∇v(, ·) ∈ L2 (Ω) a.e. ∈ (0, d). Let 2−n 2 V (ρ) = r |∇v| dx + r1−n γ(x)v 2 (x)ds < ∞, ∈ (0, d). G o

Then

Γ 0

n − 2 2 ∂v + v v ∂r 2 Ω

dΩ ≤ r=

V (). 2λ

26

M. Borsuk and A. Zawadzka

Proof. Writing V () in spherical coordinates ⎛ ⎞ ⎛ ⎞ V () = r2−n ⎝ |∇v|2 dΩ⎠ rn−1 dr + r1−n ⎝ γ(x)|v|2 dσ ⎠ rn−2 dr 0

=

⎛ r⎝

0

Ω

⎞

|∇v|2 dΩ⎠ dr +

0

Ω

⎛ 1⎝ r

0

∂Ω

⎞

γ(x)|v|2 dσ ⎠ dr

∂Ω

and diﬀerentiating with respect to we obtain 2 ∂v 1 1 2 + |∇ω v| dΩ + γ(, ω)v 2 (, ω)dσ. V () = ∂r r=

Ω

∂Ω

ε 2 1 2 ∂v 2 Moreover, by Cauchy’s inequality, we have for all ε > 0 : ρv ∂v . ∂r ≤ 2 v + 2ε ρ ∂r Thus choosing ε = λ we obtain, by the Friedrichs-Wirtinger inequality (2.2), 2 ∂v ε+n−2 2 ∂v n − 2 2 2 + v dΩ ≤ v dΩ + dΩ v ∂r 2 2 2ε ∂r r= Ω Ω Ω ε+n−2 |∇ω v|2 dΩ ≤ 2λ(λ + n − 2) 2 + 2ε

Ω

∂v ∂r

2

dΩ +

Ω

ε+n−2 2λ(λ + n − 2)

γ(x)v 2 (x)dσ =

V (). 2λ

∂Ω

Proposition 2.4. The comparison principle. Let Gd0 be a convex rotational cone with vertex at O and the aperture ω0 ∈ ( π2 , π). Let L be uniformly elliptic in Gd0 ; d d 0 d d ai (x), a(x) ∈ L∞ loc (G0 ), a(x) ≤ 0 in G0 . Let γ(x) ∈ C (Γ0 ), γ(x) ≥ γ0 > 0 on Γ0 . 2,n Suppose that v and w are functions in Wloc (Gd0 ) ∩ C 0 (Gd0 ) satisfying ⎧ d ⎪ ⎨L[w(x)] ≤ L[v(x)], x ∈ G0 ; d (2.8) B[w(x)] ≥ B[v(x)], x ∈ Γ0 ; ⎪ ⎩ w(x) ≥ v(x), x ∈ Ωd ∪ O. Then v(x) ≤ w(x) in Gd0 . This proposition is the direct consequence of the Lieberman global and strong maximum principles for Lipschitz domains (Lemma 1.1 [13], Proposition 2.1 [15], Corollary 3.2 [15]). Here it should be noted that our condition ω0 ∈ ( π2 , π) guarantees the obliqueness of the operator B at ∀x0 ∈ Γd0 with the modulus of obliqueness lesser than 1.

Exact Estimates of Solutions to the Robin Boundary Value Problem

27

3. The barrier function. The preliminary estimate of the solution modulus Let Gd0 be a convex rotational cone with a solid angle ω0 ∈ (0, π) and the lateral surface Γd0 , such that Gd0 ⊂ {x1 ≥ 0}. Let deﬁne the linear elliptic operator: L0 ≡ 2 aij (x) ∂x∂i ∂xj ; aij (x) = aji (x), x ∈ Gd0 ; νξ 2 ≤ aij (x)ξi ξj ≤ µξ 2 , ∀x ∈ Gd0 , ∀ξ ∈ Rn ; ν, µ = const > 0 and the boundary operator: B ≡ 0, x ∈ Γd0 .

∂ − ∂→ n

+

1 |x| γ(x),

γ(x) ≥ γ0 >

Lemma 3.1. (Existence of the barrier function). Fix the numbers γ0 > tan ω20 , δ > 0, g1 ≥ 0, d ∈ (0, 1). There exist h > 0 depending only on ω0 , the number κ0 ∈ (0, γ0 cot ω20 − 1), a number B > 0 and a function w(x) ∈ C 1 (G0 ) ∩ C 2 (G0 ) that depend only on ω0 , the ellipticity constants ν, µ of the operator L0 and the quantities γ0 , δ, g1 , κ0 , such that for any κ ∈ (0; min(δ, κ0 )) the following hold: L0 [w(x)] ≤ −νh2 |x|κ−1 ; x ∈ Gd0 ; B[w(x)] ≥ g1 |x| ; δ

(3.1)

x ∈ Γd0 ; κ+1

0 ≤ w(x) ≤ C0 (κ0 , B, ω0 )|x|

(3.2)

; x ∈ Gd0 ; x ∈ Gd0 .

κ

|∇w(x)| ≤ C1 (κ0 , B, ω0 )|x| ;

(3.3) (3.4)

Proof. Let (x, y, x ) ∈ Rn , where x = x1 , y = x2 , x = (x3 , . . . , xn ). In {x1 ≥ 0} we consider the cone K with the vertex in O, such that K ⊃ Gd0 (we recall that Gd0 ⊂ {x1 ≥ 0}). Let ∂K be the lateral surface of K and let ∂K ∩ yOx = Γ± be x = ±hy, where h = cot ω20 , 0 < ω0 < π, such that in the interior of K the inequality x > h|y| holds. We shall consider the function: w(x; y, x ) ≡ xκ−1 (x2 − h2 y 2 ) + Bxκ+1 , with some κ ∈ (0; 1), B > 0. Let coeﬃcients of the operator L0 be: a

2,2

1,2

= a, a

= b, a

1,1

(3.5)

= c. Then we have:

L0 w = awyy + 2bwxy + cwxx ;

(3.6)

νη ≤ + 2bη1 η2 + ≤ µη ; η = + ∀η1 , η2 ∈ R. (3.7) y 1 Let us calculate the operator L0 on the function (3.5). For t = x , |t| < h we obtain: L0 w = −h2 xκ−1 φ(κ), where 2

aη12

cη22

2

2

η12

η22 ;

φ(κ) = 2a − 4bt + 4btκ − ch−1 (1 + B)(κ 2 + κ) + ct2 κ 2 − 3ct2 κ + 2ct2 = c(t2 − h−2 (1 + B))κ 2 + (4bt − ch−2 (1 + B) − 3ct2 )κ + 2(ct2 − 2bt + a); 2 y 1+B B 2 −2 c(t − h (1 + B)) = c 2 − ≤ −c 2 < 0. 2 x h h Because of (3.7), we have φ(0) = 2(ct2 − 2bt + a) ≥ 2ν and since φ(κ) is the square function there exists the number κ0 > 0 depending only on ν, µ, h such that φ(κ) ≥ ν for κ ∈ [0; κ0 ]. Therefore we obtain (3.1). Now, let us notice that ω0 Γ± : x = ±hy, h = cot , 0 < ω0 < π. (3.8) 2

28

M. Borsuk and A. Zawadzka

Then we have

on Γ+ : on Γ− : sin

x = r cos ω20 , y = r sin ω20

x y

∠(n, x) = ∠(n, y) =

= r cos ω20 , = −r sin ω20

1 ω0 = √ , 2 1 + h2

ω0 π 2 + 2 , ω0 2

∠(n, x) = π2 + ω20 , ∠(n, y) = π + ω20

cos

h ω0 = √ . 2 1 + h2

Therefore we obtain:

wx = (1 + κ)xκ (1 + B) − (κ − 1)h2 y 2 xκ−2 ⇒ wx Γ± = [2 + B(1 + κ)]xκ , (3.9) wy = −2h2 yxκ−1 ⇒ wy Γ± = ∓2hxκ .

Because of

∂w ∂n

= wx cos ∠(n, x) Γ±

Γ±

+ wy cos ∠(n, y)

Γ±

and (3.9), we get:

∂w ∂n

= −rκ

Γ±

Hence it follows: B[w]

Γ±

hκ (1 + h2 )

κ+1 2

[2(1 + h2 ) + B(1 + κ)].

% & hκ ≥ rκ Bhγ0 − B(1 + κ) − 2(1 + h2 ) . κ+1 1 + h2 2

Since h > γ10 for κ ≤ κ0 we obtain: % & hκ 0 r κ 2 δ B[w] d ≥ κ02+1 B(hγ0 − 1 − κ0 ) − 2(1 + h ) ≥ g1 r , 0 < r < d < 1 Γ± 2 1+h if we take κ ≤ δ and κ +1 ' g 1 + h2 02 ( 1 1 2 B≥ + 2(1 + h ) · hκ 0 hγ0 − 1 − κ0

(3.10)

(it should be pointed out that we can choose (if it is necessary) κ0 so small that κ0 < hγ0 − 1). Now we’ll show (3.3). Let us rewrite the function (3.5) in spherical coordinates. Recalling that h = cot ω20 we obtain: w(x; y, x ) = (1 + B)(r cos ω)1+κ − h2 r2 sin2 ω(r cos ω)κ−1 % ω0 ω0 & χ(ω) , = r1+κ cosκ−1 ω B cos2 ω + ∀ω ∈ − ; 2 ω0 , 2 2 sin 2

Exact Estimates of Solutions to the Robin Boundary Value Problem

29

where χ(ω) = sin ω20 − ω · sin ω20 + ω . We ﬁnd χ (ω) = − sin 2ω and χ (ω) = 0 for ω = 0. Now we see that χ (0) = −2 cos 0 = −2 < 0. In this way we have max χ(ω) = χ(0) = sin2 ω20 and therefore:

ω∈[−ω0 /2,ω0 /2]

w(x; y, x ) ≤ r1+κ cosκ−1 ω(B cos2 ω + 1) 1 1 1+κ κ+1 1+κ cos ω B+ ≤r B+ ≤r . cos2 ω cos2 ω Hence (3.3) follows. Finally, (3.4) follows in virtue of (3.9).

With the help of this Lemma we can estimate |u(x)| for (L) and (QL) in the neighborhood of a conical point. Theorem 3.2. Let u(x) be a strong solution of problem (L) and satisfy assumptions (a)–(d). Then there exist numbers d ∈ (0, 1) and κ > 0 depending only on ν, µ, n, κ0 , ω0 , f1 , β, γ0 , s, g1 , M0 and the domain G such that |u(x) − u(0)| ≤ C0 |x|κ+1 , x ∈ Gd0 ,

(3.11)

where the positive constant C0 depends only on ν, µ, n, f1 , g1 , β, s, γ0 , M0 and the domain G, and does not depend on u(x) Proof. Without loss of generality we may suppose that u(0) ≥ 0. Let us take the barrier function w(x) deﬁned by (3.5) with κ ∈ (0, κ0 ) and the function v(x) = u(x) − u(0). For them we shall prove (2.8). Let us calculate the operator L on this functions. Because of Lemma 3.1 and the assumptions (b), (d), we obtain: Lv(x) = Lu(x) − a(x)u(0) = f (x) − a(x)u(0) ≥ f (x) ≥ −f1 rβ ; Lw(x) ≤ L0 w + ai (x)wxi ≤ −νh2 rκ−1 +

A(r) 1 C1 rκ ≤ − νh2 rκ−1 . r 2

By the continuity of A(r) d > 0 has been chosen so small that C1 A(r) ≤ C1 A(d) ≤

1 2 νh for r ≤ d. 2

(3.12)

Since 0 < κ < κ0 hence it follows that 1 L[Aw(x)] ≤ − Aνh2 rκ0 −1 ≤ Lv(x), x ∈ Gd0 , 2 where numbers κ0 , A are chosen such that κ0 ≤ β + 1, A ≥ From (3.2) we get:

B[Aw]

Γd ±

2f1 . νh2

≥ Ag1 rδ .

(3.13)

(3.14)

30

M. Borsuk and A. Zawadzka

Let us calculate B[v] on Γd± . If A ≥ 1 and 0 < δ ≤ s − 1 then B[v(x)] =

∂u 1 1 + γ(x) (u(x) − u(0)) = g(x) − γ(x)u(0) ≤ g(x) ∂n |x| |x|

≤ g1 r

s−1

≤ g1 r ≤ B[Aw], δ

x∈

(3.15)

Γd± ,

by of (3.14). Now we compare v(x) and w(x) on Ωd . Since x2 ≥ h2 y 2 in K from (3.5) we have ω0 . (3.16) ≥ B|x|1+κ = Bd1+κ cosκ+1 w(x) 2 r=d r=d On the other hand v(x) = (u(x) − u(0)) ≤ M0 (3.17) Ωd

Ωd

and therefore from (3.16)–(3.17), in virtue of (3.10), we obtain: κ +1 ' g 1 + h2 02 ( 1 1+κ κ+1 ω0 2 ≥A Aw(x) ≥ ABd cos + 2(1 + h ) κ 2 h 0 Ωd 1+κ0 1 d1+κ0 h1+κ0 (1 + h2 )− 2 × hγ0 − 1 − κ0 ≥ M0 ≥ v , Ωd

where A is made great enough to satisfy A≥

M0 (hγ0 − 1 − κ0 ) ) 1−κ0 * . hd1+κ0 g1 + 2hκ0 1 + h2 2

(3.18)

Thus, if we choose the small number d > 0 according to (3.12) and large numbers B > 0, A ≥ 1 according to (3.10), (3.13), (3.18), we provide the validity of (2.8). Therefore the functions v(x), Aw(x) satisfy the comparison principle (Proposition 2.4) and we have u(x) − u(0) ≤ Aw(x), x ∈ Gd0 . Similarly, we derive the estimate u(x)−u(0) ≥ −Aw(x) if we consider an auxiliary function v(x) = u(0)−u(x). The theorem is proved, in virtue of (3.3). Theorem 3.3. Let u(x) be a strong solution of the problem (QL) and satisfy assumptions (A)–(D). Then there exist the numbers d > 0 and κ > 0 depending only on ν, µ, µ1 , n, κ0 , ω0 , k1 , β, δ, γ0 , g0 , M0 and the domain G such that |u(x) − u(0)| ≤ C0 |x|κ+1 , x ∈ Gd0 ,

(3.19)

where the positive constant C0 depends only on ν, µ, µ1 , g0 , n, k1 , β, γ0 , M0 and the domain G, and does not depend on u. This theorem is proved in the same way as above theorem if we consider 2 ∂ , x ∈ G, with aij (x) = the linear elliptic operator: L ≡ aij (x) ∂x∂i ∂xj + ai (x) ∂x i aij (x, u(x), ux (x)); ai (x) = b(x)|∇u(x)|−1 uxi (x) and we suppose that ai (x) = 0, i = 1, . . . , n in such points x, where |∇u(x)| = 0. For the barrier function (3.5)

Exact Estimates of Solutions to the Robin Boundary Value Problem

31

and the auxiliary function v(x) = −1 + exp(ν −1 µ1 (u(x) − u(0))) we verify the comparison principle 2.4. Now the proof of Theorems 1.1–1.4 is based on the deduction of the global and local integral weighted estimates of the (L)- and (QL)-solutions with the sharp estimating constants. For this we use the Hardy-Friedrichs-Wirtinger inequality, Lemma 2.3, Theorems 3.2, 3.3 and the following theorem about solutions of the Cauchy problem for a diﬀerential inequality Theorem 3.4. ([2, 1]). Let V () be a monotonically increasing, nonnegative diﬀerentiable function deﬁned on [0, 2d] and satisfy the problem V (ρ) − P()V () + N (ρ)V (2ρ) + Q(ρ) ≥ 0, 0 < ρ < d, (CP) V (d) ≤ V0 , where P(), N (), Q() are nonnegative continuous functions deﬁned on [0, 2d] and V0 is a constant. Then d d d τ V () ≤ exp B(τ )dτ V0 exp − P(τ )dτ + Q(τ ) exp − P(σ)dσ dτ

2

with B() = N () exp P(σ)dσ .

4. Examples We present examples that show that conditions of Theorems 1.1–1.3 (in particular the Dini condition on the function A(r) in condition (b) at the point O in Theorem 1.1) are essential for their validity. Suppose n = 2, the domain G lies inside the corner ω0 ω0 <ω< }, ω0 ∈]0, π[, G0 = {(r, ω) |r > 0; − 2 2 O ∈ ∂G and in some neighborhood of O the boundary ∂G coincides with the sides ω0 ω0 ω0 of the corner ω = − 2 and ω = 2 . We denote Γ± = {(r, ω) | r > 0; ω = ± 2 } and we put γ(x) = γ± = const > 0. ω0 ω=±

2

I. We consider the following problem: ⎧ ⎪ ∈ G0 ; ⎨u = 0, x ∂u 1 ⎪ ⎩ ∂n + r γ± u = 0. Γ±

32

M. Borsuk and A. Zawadzka We verify that the function u(r, ω) = rλ ψ(ω) is a solution of our problem, if λ2 is the least positive eigenvalue of the problem ⎧ 2 ⎪ ω ∈ − ω20 , ω20 ⎨ψ + λ ψ = 0, ⎪ = 0. ⎩(±ψ + γ± ψ) ω0 ω=±

2

and ψ(ω) is regular eigenfunction associated to λ. Precisely λ is deﬁned from transcendence equation tan(λω0 ) =

λ(γ+ + γ− ) . λ2 − γ+ γ−

(4.1)

Then we ﬁnd the eigenfunction % % ω0 & ω0 & − γ+ sin λ ω − . ψ(ω) = λ cos λ ω − 2 2

(4.2)

The existence of the positive solution of (4.1) may be veriﬁed by the graphic method. This example shows that the exponent λ in (1.3) cannot be increased. Remark 2. In order that λ > 1 we show that the condition γ(x) ≥ γ0 > tan ω20 from the assumption (c) of our Theorems is justiﬁed. In fact, we rewrite the equation (4.1) in the equivalent form λ=

1 γ+ γ− arctan + arctan . ω0 λ λ

(4.3)

Hence it follows that 1 (arctan γ+ + arctan γ− ) ⇒ ω0 γ+ + γ− , provided γ+ γ− < 1 ω0 < arctan 1 − γ+ γ− 1<λ<

(4.4)

has to be fulﬁlled. But our condition from the assumption (c) means that γ± ≥ γ0 > tan ω20 . Hence we obtain 2 tan ω20 γ+ + γ− 2γ0 π . ≥ > ω0 = tan ω0 , ω0 < 2 2 1 − γ+ γ− 1 − γ0 2 1 − tan 2 Thus we get (4.4). In the case γ± ≥ γ0 > tan ω20 ≥ 1 for ω0 ∈ [ π2 , π) the inequality λ > 1 is fulﬁlled a fortiori, because of the property of the monotonic increase of the eigenvalues together with the increase of γ(x) (see for example Theorem 6 §2, Chapter VI [5]).

Exact Estimates of Solutions to the Robin Boundary Value Problem II. The function u(r, ω) = r

λ

33

λ−1 λ+1 ln

1 r

ψ(ω) with λ and ψ(ω) deﬁned by (4.1)–

(4.2) is a solution of the problem ⎧ + n ⎪ ⎪ aij (x)uxi xj = 0, x ∈ G0 , ⎨ i,j=1 ⎪ ∂u 1 ⎪ ⎩ ∂n + r γ± u = 0, γ± > 0 Γ±

in the corner G0 , where x2 2 · 2 2 , r > 0; λ + 1 r ln 1/r x1 x2 2 a12 (x) = a21 (x) = · , r > 0; λ + 1 r2 ln 1/r x2 2 a22 (x) = 1 − · 2 1 , r > 0; λ + 1 r ln 1/r a11 (x) = 1 −

aij (0) = δij , (i, j = 1, 2). In the domain Gd0 , d < e−2 the equation is uniformly elliptic with ellipticity 2 2 . Further, A(r) = λ+1 ln−1 1r , i.e., the constants µ = 1 and ν = 1 − ln(1/d) function A(r) does not satisfy the Dini condition at zero. Moreover, aij (x) are continuous at the point O. This example shows that the condition of Theorem 1.1 about Dini-continuity of leading coeﬃcients of (L) are essential, and it illustrates the precision of assumptions of Theorem 1.3 as well. III. The function u(r, ω) = rλ ln 1r ψ(ω) with λ and ψ(ω) deﬁned by (4.1)–(4.2) is a solution of the problem ⎧ ⎪ u = 0, x ∈ G0 , + r22λ ⎨u ln 1r ∂u 1 ⎪ ⎩ ∂n + r γ± u = 0, γ± > 0 Γ±

in the corner G0 . This example shows that assumptions of Theorems 1.1 and 1.3 on lowest coeﬃcients of (L) are precise and essential. IV. The function u(r, ω) = rλ ln 1r ψ(ω) with λ and ψ(ω) deﬁned by (4.1)–(4.2) is a solution of the problem ⎧ λ−2 ⎪ ⎨u = −2λr ψ(ω), x ∈ G0 , ∂u 1 ⎪ ⎩ ∂n + r γ± u = 0, γ± > 0 Γ±

in the corner G0 . All assumptions of Theorem 1.1 are fulﬁlled with s = λ. This example shows the precision of assumptions for the right-hand side of (L) in Theorem 1.1.

34

M. Borsuk and A. Zawadzka

References [1] M. Bochniak, M.Borsuk, Dirichlet problem for linear elliptic equations degenerating at a conical boundary point, Analysis. M¨ unchen, Germany 23, 3 (2003), pp. 225–248. [2] M.V. Borsuk, Best-possible estimates of solutions of the Dirichlet problem for linear elliptic nondivergence equations of second-order in a neighborhood of a conical point on the boundary, Math. USSR Sbornik 74 (1993), 185–201. [3] M.V. Borsuk, Estimates of solutions of Dirichlet problem for elliptic nondivergence second-order equations in a neighbourhood of a conical boundary point, Diﬀer. Uravn. 30, 1 (1994), pp. 104–108. [4] M.V. Borsuk, On the solvability of the ﬁrst boundary value problem for second-order elliptic equations in a domain with a conical point on the boundary, Mat. Fiz. Anal. Geom. 4, 4 (1997), pp. 428–452. [5] R. Courant, D. Hilbert, “Methoden der mathematischen Physik”, Bd. 1, SpringerVerlag, Berlin, 1931. [6] M. Faierman, Regularity of solutions of an elliptic boundary value problem in a rectangle, Comm. in PDE 12 (1987), 285–305. [7] M.G. Garroni, V.A. Solonnikov and Vivaldi M.A., On the oblique derivative problem in an inﬁnite angle, Topological methods in nonlinear analysis 7 (1996), 299–325. [8] D. Gilbarg and N.S. Trudinger, “Elliptic Partial Diﬀerential Equations of Second Order”, Springer-Verlag, Berlin/Heidelberg/New York, 1977. Revised Third Printing, 1998. [9] G.H. Hardy, J.E. Littlewood and G. P´olya, “Inequalities”, University Press, Cambridge, 1952. [10] V.A. Kondrat’ev, Boundary value problem for elliptic equations in domains with conical or angular points, Trudy Moscov. Mat. Obshch. 16 (1967), 209–292. [11] V.A. Kozlov, V.G. Maz’ya and Rossman J., “Elliptic boundary value problem in domains with point singularities”, AMS. Mathematical surveys and monographs, 52 (1997). [12] V.A. Kozlov, V.G. Maz’ya and Rossman J., “Spectral problems associated with corner singularities of solutions to elliptic equations”, AMS. Mathematical surveys and monographs, 85 (2001). [13] G.M. Lieberman, Local estimates for subsolutions and supersolutions of oblique derivative problems for general second-order elliptic equations, Trans. of AMS. 304 (1987), 343–353. [14] G.M. Lieberman, Oblique derivative problems in Lipschitz domains. Continuous boundary data, Bull. Un. Mat. Ital. B (7) 1 (1987), 1185–1210. [15] G.M. Lieberman, Pointwise estimates for oblique derivative problems in nonsmooth domains, J. Diﬀ. Equat. 173 (2001), no. 1, 178–211. [16] Gary M. Lieberman, The nonlinear oblique derivative problem for quasilinear elliptic equations. Nonlinear Analysis. Theory, Methods and Applications, 8 (1984), p. 49– 65. [17] Gary M. Lieberman, Second-order parabolic diﬀerential equations, World Scientiﬁc, Singapore – New Jersey – London – Hong Kong, 1996, 439 p.

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[18] H. Reisman, Second-order elliptic boundary value problem in a domain with edges, Comm. in PDE 6 (1995), 1023–1042. [19] S.L. Sobolev, “Some applications of functional analysis in mathematical physics”, Trans. of Math. Monographs, 90 (1991). AMS, Providence, Rhode Island. Mikhail Borsuk and Agneszka Zawadzka Department of Mathematics and Informatics, University of Warmia and Mazury in Olsztyn, PL-10-957 Olsztyn-Kortowo, Poland e-mail: [email protected]

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 37–61 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Well-posedness of a Two-phase Flow with Soluble Surfactant Dieter Bothe, Jan Pr¨ uss and Gieri Simonett Dedicated to Herbert Amann

Abstract. The presence of surfactants, ubiquitous at most ﬂuid/liquid interfaces, has a pronounced eﬀect on the surface tension, hence on the stress balance at the phase boundary: local variations of the capillary forces induce transport of momentum along the interface – so-called Marangoni eﬀects. The mathematical model governing the dynamics of such systems is studied for the case in which the surfactant is soluble in one of the adjacent bulk phases. This leads to the two-phase balances of mass and momentum, complemented by a species equation for both the interface and the relevant bulk phase. Within the model, the motions of the surfactant and of the adjacent bulk ﬂuids are coupled by means of an interfacial momentum source term that represents Marangoni stresses. Employing Lp -maximal regularity we obtain well-posedness of this model for a certain initial conﬁguration. The proof is based on recent Lp -theory for two-phase ﬂows without surfactant. Mathematics Subject Classiﬁcation (2000). Primary 35R35, Secondary 35Q30, 76D45, 76T10. Keywords. Navier-Stokes equations, surface tension, Marangoni forces, maximal regularity, surface transport theorem.

1. Introduction Surfactants are surface active agents that are adsorbed at gas/liquid or liquid/liquid interfaces, where they lower the surface tension. The presence of surfactants in multiphase systems, either in the form of unavoidable impurities, or as speciﬁc substances like emulsiﬁers to aﬀect the system’s behavior, has a signiﬁcant inﬂuence on the shape and the dynamics of the interfaces. Indeed, resulting variations of the capillary force induce Marangoni stresses which increase the drag of ﬂuid particles and hinder the internal circulation [7]. For example, the terminal rise

38

D. Bothe, J. Pr¨ uss and G. Simonett

velocity of bubbles or droplets can be drastically reduced, which results in larger contact times between the ﬂuid phases and, hence, alters mass transfer. Surfactants are usually soluble in at least one of the adjacent bulk phases, hence there is also exchange of surfactants between the relevant bulk phases and the interface by adsorption and desorption [8]. Along the interface surfactant is transported by convection and diﬀusion. Furthermore, changes of the interfacial area due to compression or stretching cause corresponding changes in the surfactant concentration. Mathematically, the analysis of two-phases ﬂows with surfactant corresponds to the study of free boundary problems for the Navier-Stokes equations, coupled with a diﬀusion equation having a dynamic condition on the free boundary. Existence results for the one-phase Navier-Stokes equations with a free surface were obtained in [2, 5, 31, 32, 33]. In a series of publications, Solonnikov considered the motion of an isolated viscous incompressible drop subject to capillary forces, see [27] and the references contained therein. There is signiﬁcantly less work for the one-phase Navier-Stokes equations with Marangoni eﬀects, which refers to situations where the surface endures tangential stresses caused by variations of surface tension. The latter can be caused by temperature gradients, for instance. This case is considered in [1], where the ﬂow of a viscous liquid with its free surface subjected to a nonuniform thermal ﬂux is studied; see also [19]. Assuming the Boussinesq approximation to account for density variations, stationary solutions are obtained in weighted H¨ older spaces. The instationary situation is considered in [34], where small time existence of the free surface ﬂow of a heat conducting viscous droplet of constant density in three dimensions is proved. Problems involving the two-phase Navier-Stokes equations with surface tension have been investigated in [28, 29, 10, 9]. In [28, 29] small time existence for smooth initial data as well as global existence and convergence to the equilibrium conﬁguration for nearby initial data is proven. Small time existence in weighted H¨older spaces is obtained in [10] for incompressible ﬂuids, while combinations of compressible and incompressible ﬂuids are allowed in [9]. For compressible twophase ﬂow without surface tension but with additional energy balance see [30]. There is a large amount of literature in the Engineering Sciences on two-phase ﬂows in the presence of surface active substances, but a rigorous mathematical analysis is still missing so far. However, interesting ﬁrst numerical results in 2D are obtained in [18], while analytical calculations for bubbles in plane extensional ﬂow of low velocity are given in [25]. In these contributions, the surfactant is considered as insoluble, a common idealization. In the present paper we consider the two-phase Navier-Stokes equations with surfactant dependent surface tension complemented by a convection-diﬀusion equation inside one phase with another convection-diﬀusion equation on the free surface which are nonlinearly coupled. We give a derivation of the mathematical model, for which an appropriate surface transport theorem is employed. To keep the paper self-contained, a proof of the latter together with an introduction of the

Two-phase Flows with Soluble Surfactant

39

notion of surface divergence is included in two appendices. The main result of this paper is the existence of a unique classical solution to this free boundary problem for a certain initial conﬁguration of the phases. The latter corresponds to situations encountered by the usual localization procedure which leads to “deformed half-spaces” for the two phases. The strategy here is to build up the proof based on recent results on two-phase ﬂows without surfactant. This works, since, luckily, the surfactant equations are decoupled from the Navier-Stokes equations in the associated linear problem. Our proof makes use of recent developments in the theory of maximal regularity for parabolic problems. For a general reference to this subject, we refer to the monograph [11]. Of particular importance for this work are the papers [12, 13, 14, 22, 23], where basic notations used throughout this paper can be found.

2. The mathematical model The subsequent mathematical model describes two-phase ﬂows with a free boundary to capture the motion of a ﬂuid particle (droplet or bubble) in an ambient ﬂuid phase (gas or liquid). We consider isothermal ﬂows of two immiscible Newtonian ﬂuids with constant densities. Based on continuum mechanics, balance of mass and momentum inside the phases lead to the Navier-Stokes equations, i.e., ∇ · u = 0, with the stress tensor

∂t (ρ± u) + ∇ · (ρ± u ⊗ u − S) = 0

S = −πI + µ± ∇u + (∇u)T .

These balances hold in Ω+ (t) ∪ Ω− (t), where Ω± (t) are the domains occupied at time t by two ﬂuids with density ρ± and viscosity µ± , respectively. Hence the material parameters depend on the phase. Whenever a distinction between the diﬀerent phases is not necessary, the phase indices + and − are omitted. The phases are separated by a phase boundary Γ(t) at which at least one of the material parameters has a jump-discontinuity. At this interface, the additional jump conditions [ρ(u − uint )] n [ρu ⊗ (u − uint ) − S] n

= 0, = σκn + gradΓ σ

appear, where uint is the interfacial velocity, gradΓ σ denotes the surface gradient of the surface tension σ (cf. Appendix A) and κ = −divΓ n is the sum of the (local) principal curvatures of the interface. More precisely, it should read divΓ(t) , but the dependence on t is dropped whenever this is reasonable. Throughout the paper, n denotes the unit normal at the interface directed into

40

D. Bothe, J. Pr¨ uss and G. Simonett

the phase Ω+ (t), say, in which case κ(t, x) > 0 at x ∈ Γ(t) iﬀ Ω+ (t) ∩ Br (x) is convex for small r > 0. The notation [φ] stands for the jump of a physical quantity φ across the interface, i.e., [φ](t, x) = lim φ(t, x + h n) − φ(t, x − h n) , x ∈ Γ(t). h→0+

Below, it is assumed that the two-ﬂuid system under consideration does not endure phase changes like melting or evaporation. Consequently, there are no convective ﬂuxes across the interface, and hence the normal components of the ﬂuid velocities are continuous at the phase boundary, and the interface itself is advected with the ﬂow, i.e., (u+ · n)|Γ(t) = (u− · n)|Γ(t) = V, where V denotes the normal velocity of the interface. Furthermore, we assume no-slip at the interface. Then the jump conditions simplify to [u] = 0,

[−S] n = σκn + gradΓ σ.

Throughout this paper, we consider the case of a soluble surfactant, i.e., a chemical species which is adsorbed at the interface but also lives in at least one of the bulk phases. For technical simplicity, we concentrate on the case in which this species is only present in phase Ω+ , which is also of practical relevance. The local concentration c of a dilute chemical species is governed by the balance equation ∂t c + ∇ · (cu + J) = 0, where we assume that the molecular ﬂux is given by Fick’s law, J = −D∇c, with constant diﬀusivity D > 0. This leads to the convection-diﬀusion equation ∂t c + u · ∇c − D∆c = 0 inside Ω+ . For more details about the continuum mechanical modeling of twophase ﬂows up to this point see [16, 26]. At the phase boundary, mass balance for a material control area, i.e., M (t) = Γ(t) ∩ V (t) with a material control volume V (t), implies d cΓ dσt = − JΓ · N dst − [J] · n dσt . dt M(t) ∂M(t) M(t) Here cΓ denotes the surface speciﬁc concentration of the surfactant on the interface Γ and JΓ is the interfacial molecular ﬂux of surfactant which will be modeled by Fick’s law again, i.e., JΓ = −DΓ gradΓ cΓ with DΓ > 0. Observe also that [J] · n reduces to −D(∇c · n)|Γ(t) , understood as the Ω+ -sided limit, here, since the species c is only present in Ω+ . Application of the surface divergence theorem (Theorem A in Appendix A), the surface transport theorem (Theorem B in Appendix B) and the usual localization procedure

Two-phase Flows with Soluble Surfactant

41

leads to the diﬀerential formulation of the surfactant balance. A completely Eulerian formulation is problematic since cΓ is only deﬁned on gr(Γ). An intermediate Eulerian/Lagrangian formulation reads as ∂t,n cΓ + divΓ (cΓ uΓ ) − cΓ κV − DΓ ∆Γ cΓ = D(∇c · n)|Γ(t) , where ∂t,n cΓ stands for the derivative of cΓ along a purely normal path, uΓ is the tangential part of u and ∆Γ denotes the Laplace-Beltrami operator on Γ; cf. Appendix B. One more constitutive equation is needed to determine the surfactant distribution. Here we assume that the process of ad- and desorption of surfactant at the interface is instantaneous if viewed on the time scale of convective and diﬀusive transport. This leads to the additional relation cΓ = γ(c|Γ(t) ), where it is reasonable to assume that γ : R+ → R+ is C 2 with γ > 0. In concrete constitutive equations, γ will usually satisfy γ(∞) = c∞ Γ , since the phase boundary has a ﬁnite capacity concerning adsorption of surfactant. This relation gets more complicated in case of liquid/liquid/surfactant-systems in which the surfactant is soluble in both phases; a survey on adsorption and partitioning of surfactants is given in [24]. Finally, the surface tension σ is a function of cΓ and it can be signiﬁcantly lowered due to the adsorption of surfactant. Below, we assume that σ : R+ → R+ is C 2 with σ > 0. In this situation, a nonhomogeneous surfactant distribution leads to nonzero surface gradients of the surface tension. This corresponds to socalled Marangoni forces which cause tangential stress at the phase boundary and can trigger currents along the interface, the so-called Marangoni convection. This back-eﬀect of the surfactant distribution on the hydrodynamics is mediated via the surface gradient of σ within the momentum jump condition. The full model of a two-phase ﬂow with soluble surfactant now reads as Balance equations ∇ · u = 0 for t > 0, x ∈ Ω± (t), ∂t (ρ± u) + ∇ · (ρ± u ⊗ u − S) = 0 for t > 0, x ∈ Ω± (t),

(1)

∂t c + u · ∇c − D∆c = 0 for t > 0, x ∈ Ω+ (t). Free boundary conditions V = u·n

for t > 0, x ∈ Γ(t),

[−S] n = σ(cΓ )κn + σ (cΓ ) gradΓ cΓ

for t > 0, x ∈ Γ(t),

[u] = 0,

D(∇c · n)|Γ(t) = ∂t,n cΓ + divΓ (cΓ uΓ ) − cΓ κV − DΓ ∆Γ cΓ cΓ (t, x) = γ(c(t, x))

for t > 0, x ∈ Γ(t), for t > 0, x ∈ Γ(t).

(2)

42

D. Bothe, J. Pr¨ uss and G. Simonett

R

y

n + 1

7

W

G (t)

W

t +

y = h (t,x )

n W -

x

t W +

W

t

-

t

Figure 1. Conﬁguration of the phases at the interface and localization. Initial conditions

Γ(0) = Γ0 ,

u(0, x) = u0 (x),

for

x ∈ Ω± (0),

c(0, x) = c0 (x)

for

x ∈ Ω+ (0),

c0Γ (x)

for

x ∈ Γ0 .

cΓ (0, x) =

Recall that S is given by S = −πI + S 0

with

S 0 = µ± ∇u + (∇u)T .

(3)

(4)

3. Main result: Small time existence in a deformed half-space The principal aim of this investigation is to obtain local existence of classical solutions for Γ0 being a closed manifold such that the bounded component of Rn+1 \ Γ0 represents a ﬂuid particle like a drop or a bubble. This general situation can be treated by a localization procedure, which will be carried out in a forthcoming paper. Here, we are interested in the model problem which occurs by localization at points on Γ0 ; cf. Figure 1. In this case, Ω± are deformed half-spaces and Γ0 is close to a plane, say Rn , i.e., Γ0 is a graph over Rn given by a function h0 with |∇h0 |∞ small. The main result for this model problem reads as follows. Theorem 1. Let ρ± , µ± , D, DΓ > 0 and γ, σ ∈ C 2 (R+ ) with σ > 0. Fix p > n + 3 and suppose the following conditions to hold. Regularity of initial data: h0 ∈ Wp3−2/p (Rn ),

Γ0 = gr(h0 ),

c0 − c¯ ∈ Wp2−2/p (Ω0+ ),

u0 ∈ Wp2−2/p (Rn+1 \ Γ0 ; Rn+1 ), c0Γ − c¯Γ ∈ Wp2−2/p (Γ0 )

with appropriate c¯, c¯Γ ∈ R and Ω0+ = {(x, y) ∈ Rn+1 : x ∈ Rn , y > h0 (x)}. Compatibility condition: ∇ · u0 = 0,

0 0 [S|t=0 n − n, S|t=0 n n] = −σ (c0Γ ) gradΓ0 c0Γ ,

Two-phase Flows with Soluble Surfactant

43

[u0 ] = 0, c0Γ = γ(c0|Γ0 ), c¯Γ = γ(¯ c). Then there is η > 0 such that the following holds. Given h0 with |∇h0 | ≤ η and c0Γ ≥ 0 with sup c0Γ − inf c0Γ ≤ η, x∈Γ0 0 t0 (u0 , h0 , c0 , cΓ ) and

x∈Γ0

there exist t0 = a classical solution (u, π, c, cΓ ) of (1)–(4) in (0, t0 ). The solution is unique within the class of maximal regularity as described in Theorem 2 below. The proof of Theorem 1 consists of three main steps as given in the subsections to follow. 3.1. Reduction to a half-space Suppose that Γ(t) is a graph over Rn , parameterized as Γ(t) = {(x, h(t, x)) : x ∈ Rn }

for t ∈ J = [0, T ]

with Ω+ (t) lying “above” Γ(t), i.e., Ω± (t) = {(x, y) ∈ Rn × R : y ≷ h(t, x)}

for t ∈ J .

Reduction from deformed into true half-spaces is achieved by means of the transformations ⎤ ⎡ u1 (t, x, h(t, x) + y) ⎥ ⎢ .. v(t, x, y) = ⎣ ⎦ . un (t, x, h(t, x) + y) w(t, x, y)

= un+1 (t, x, h(t, x) + y)

p(t, x, y)

= π(t, x, h(t, x) + y)

where t ∈ J, x ∈ R , y ∈ R and n

c˜(t, x, y)

= c(t, x, h(t, x) + y) − c¯

c˜Γ (t, x)

= cΓ (t, x, h(t, x)) − γ(¯ c) ,

where t ∈ J, x ∈ Rn , y ∈ R+ = [0, ∞). Notice that we passed to concentration diﬀerences w.r. to the “values at inﬁnity” and that now (x, y) instead of x denotes a point in Rn+1 . For the two-phase balance equations for mass and momentum, this conversion has already been carried out in [14]. Therefore, we concentrate on those calculations needed to transform the surfactant balance. Since the orientation of Γ(t) is such that the normal ﬁeld is exterior to Ω− (t), the unit normal at the point (x, h(t, x)) is given by

−∇x h(t, x) −1/2 n(t, x) = β(t, x) with β(t, x) = 1 + |∇x h(t, x)|2 . 1 The normal velocity V of Γ(·) is V (t, x) = β(t, x)−1/2 ∂t h(t, x) .

44

D. Bothe, J. Pr¨ uss and G. Simonett

Together with the kinematic condition V = (u · n)|Γ this yields the transport equation ∂t h + v|y=0 · ∇x h = w|y=0 ,

(5)

which describes the evolution of h. Note that u is continuous across Γ(t), hence v|y=0 and w|y=0 are well deﬁned. According to (24) in Appendix A, the curvature of Γ(t) at x ∈ Γ(t) can be calculated as κ(t, x) = −divΓ(t) n(t, x) = ∇x · (β(t, x)−1/2 ∇x h(t, x)) ,

(6)

hence κ = ∆x h − Gκ (h) with Gκ (h) = (1 − β −1/2 )∆x h + β −3/2 ∇x h , ∇2x h ∇x h .

(7)

Diﬀerentiation of c(t, x, y) = c˜(t, x, y − h(t, x)) + c¯ yields u · ∇c = v · ∇x c˜ + (w − v · ∇x h) ∂y c˜ as well as ∆c = ∆x c˜ + β ∂y2 c˜ − 2 ∇x ∂y c˜ , ∇x h − ∂y c˜ ∆x h , where the corresponding arguments have to be used. Therefore, the surfactant balance in Ω+ (t) transforms into the quasilinear partial diﬀerential equation ∂t c˜ − D ∆x c˜ − D ∂y2 c˜ = Fc (v, c˜, h) ,

t ∈ J, x ∈ Rn , y ∈ R .

Here, the nonlinearity is given by Fc (v, c˜, h) = −v · ∇x c˜ + D (β − 1) ∂y2 c˜ − 2 ∇x ∂y c˜ , ∇x h − ∂y c˜ ∆x h .

(8)

For the transformation of the surfactant balance on Γ(t), we prefer to use the formulation DcΓ + cΓ divΓ uΓ − cΓ κ V − DΓ ∆Γ cΓ = D (∇c · n)|Γ , Dt i.e., the one which employs the full Lagrangian derivative DcΓ d (t, x, h(t, x)) = cΓ (t + s, ξ(t + s), η(t + s))|s=0 Dt ds with (ξ, η) the solution of ξ˙k (s) = uk (s, ξ(s), η(s)) ,

ξk (t) = xk for k = 1, . . . , n ,

η(s) ˙ = un+1 (s, ξ(s), η(s)) ,

η(t) = h(t, x) .

Two-phase Flows with Soluble Surfactant

45

Exploiting (5) it follows that η(s) = h(s, ξ(s)), hence DcΓ (t, x, h(t, x)) Dt

=

d cΓ (t + s, ξ(t + s), h(t + s, ξ(t + s)))|s=0 ds

=

d c˜Γ (t + s, ξ(t + s))|s=0 ds

= ∂t c˜Γ (t, x) + v(t, x) · ∇x c˜Γ (t, x) . We will now compute divΓ u = divΓ uΓ − κV in the new variables. According to (19), the surface divergence divΓ u of the vector ﬁeld u is given by divΓ u(x, h(x)) = =

n

g ij (x)∂i u(x, h(x)), (ej , ∂j h(x))

i,j=1 n

δij − β −1 ∂i h(x)∂j h(x) ∂i vj (x) + ∂i w(x)∂j h(x)

i,j=1

= ∇x · v(x) − β −1 ∇x w(x) − ∇x v(x)∇x h(x), ∇x h(x). In the computations above we have suppressed the variable t. The Laplace-Beltrami operator (see Appendix A) transforms as ∆Γ cΓ = β −1/2 ∇x · β 1/2 ∇x c˜Γ − β −1/2 ∇x h , ∇x c˜Γ ∇x h . A direct calculation shows that ∆Γ cΓ = ∆x c˜Γ − β −1 ∇x h , ∇2x c˜Γ ∇x h + ∆x h ∇x c˜Γ + β −2 ∇2x h ∇x h , ∇x h ∇x h · ∇x c˜Γ . Here, ∇2x c˜Γ and ∇2x h denote the matrix of all (mixed) second-order derivatives of c˜Γ and h, respectively. Finally, exploitation of ∇c(t, x, y) = (∇x c˜ − ∂y c˜ ∇x h , ∂y c˜)(t, x, y − h(t, x)) yields (∇c · n)|Γ = (β 1/2 ∂y c˜ − β −1/2 ∇x c˜ · ∇x h)|y=0 . Altogether, this leads to the quasilinear partial diﬀerential equation ∂t c˜Γ − DΓ ∆x c˜Γ = Gc (v, w, c˜Γ , c˜, h) ,

t ∈ J , x ∈ Rn

with cΓ + c¯Γ )(∇x · v + β −1 ∇x w − ∇x v ∇x h , ∇x h) Gc (v, w, c˜Γ , c˜, h) = −v · ∇x c˜Γ − (˜ − DΓ β −1 ∇x h , ∇2x c˜Γ ∇x h + ∆x h ∇x c˜Γ + DΓ β −2 ∇2x h ∇x h , ∇x h ∇x h · ∇x c˜Γ + D β 1/2 ∂y c˜ |y=0 − D β −1/2 ∇x c˜ |y=0 · ∇x h . (9)

46

D. Bothe, J. Pr¨ uss and G. Simonett

The nonlinear boundary condition for the transformed concentration reads as c + c˜|y=0 ) − γ(¯ c) . c˜Γ = γ(¯ To obtain the linearized version, this is rewritten as c˜Γ − γ0 c˜|y=0 = Gγ (˜ c|y=0 )

c) and with γ0 = γ (¯ c|y=0 ) = γ(¯ c + c˜|y=0 ) − γ(¯ c) − γ (¯ c) (˜ c|y=0 ) . Gγ (˜

(10)

To exploit the results from [14] for constant surface tension, those terms have to be speciﬁed that correspond to the changes in the momentum jump condition in case cΓ ) σ depends on the surfactant concentration cΓ . For this purpose, let σ0 = σ(¯ and rewrite the second equation in (2) as [−S] n − σ0 κn = (σ(cΓ ) − σ0 )κn + σ (cΓ ) gradΓ cΓ . The right-hand side consists of a tangential part corresponding to v and a normal part corresponding to w. After application of the above transformations, this leads to nonlinearities Gsurf and Gsurf which are given by v w cΓ , h) = (σ(˜ cΓ ) − σ0 )(Gκ (h) − ∆x h)∇x h Gsurf v (˜ + σ (˜ cΓ )∇x c˜Γ − β −1 σ (˜ cΓ )(∇x c˜Γ · ∇x h)∇x h, Gsurf cΓ , h) = (σ(˜ cΓ ) − σ0 )(∆x h − Gκ (h)) + β −1 σ (˜ cΓ )∇x c˜Γ · ∇x h. w (˜ Before writing down the full system of transformed equations, we will introduce some more notation. We set ˙ := {(x, y) ∈ Rn × R : x ∈ Rn , y = 0}, Rn × R H n+1 := Rn × (0, ∞), + H n+1 := Rn × (−∞, 0). − ˙ → Rk is some given function. Then Suppose that f : Rn × R f+ := f | H n+1 , +

H n+1 +

f− := f | H n+1 −

H n+1 − ,

denote the restrictions of f to and respectively. It is clear that f completely determines the functions f+ and f− , and vice versa. Moreover, the mapping ˙ Rk ) → W s (H n+1 ; Rk ) × W s (H n+1 ; Rk ) f → (f+ , f− ), W s (Rn × R; p

p

+

p

−

deﬁnes an isometric isomorphism, where Wps are the usual Sobolev-Slobodeskii spaces. Next suppose that we are given a pair of positive numbers, say µ = (µ+ , µ− ), and a function f : Rn × R˙ → Rk . Then µ+ f+ (x, y) if y > 0, µf (x, y) := µ− f− (x, y) if y < 0.

Two-phase Flows with Soluble Surfactant

47

If the function f has one-sided limits at y = 0 these are denoted by f+ (x, 0) and f− (x, 0), respectively. Then we set [µf ](x) := µ+ f+ (x, 0) − µ− f− (x, 0),

x ∈ Rn .

If [f ] = 0 we write f (x) := f (x, 0) := f+ (x, 0) = f− (x, 0). Note that in this case we have [µf ] = [µ]f for any pair µ = (µ+ , µ− ) ∈ R2 . Combining the transformed equations derived above with those from [14], we end up with the following coupled system of nonlinear equations. Diﬀerential Equations ρ ∂t v − µ ∆v + ∇x p = Fv (v, w, p, h)

in J × Rn × R˙

ρ ∂t w − µ ∆w + ∂y p = Fw (v, w, h)

in J × Rn × R˙

∇x · v + ∂y w = Fp (v, h) c = Fc (v, c˜, h) ∂t c˜ − D ∆˜

in J × Rn × R˙ in J × H n+1 +

∂t h − w = Gh (v, h)

on J × Rn

∂t c˜Γ − DΓ ∆x c˜Γ = Gc (v, w, c˜Γ , c˜, h)

on J × Rn

−[µ ∂y v] − [µ] ∇x w = Gv (v, w, [p], h) + Gsurf cΓ , h) on J × Rn v (˜ −2 [µ ∂y w] + [p] − σ0 ∆h = Gw (v, w, h) + Gsurf cΓ , h) v (˜

on J × Rn . (11)

Boundary conditions [v] = 0 ,

[w] = 0 ,

c˜Γ − γ0 c˜|y=0 = Gγ (˜ c|y=0 ) on J × Rn .

(12)

Here Gκ , Fc , Gc and Gγ are given by (7), (8), (9) and (10), respectively. The other nonlinearities have been computed in [14] and are given by Fv (v,w,p,h) = µ − 2(∇x h · ∇x )∂y v + |∇x h|2 ∂y2 v − ∆x h∂y v + ∇x h∂y p +ρ − (v · ∇x )v + (v · ∇x h)∂y v − w∂y v + (w − v · ∇x h)∂y v , Fw (v,w,p,h) = µ − 2(∇x h · ∇x )∂y w + |∇x h|2 ∂y2 w − ∆x h∂y w +ρ − (v · ∇x )w + (v · ∇x h)∂y w − w∂y w + (w − v · ∇x h)∂y w , Fp (v,h) = ∇x h · ∂y v, Gh (v,h) = −∇x h · v, Gv (v,w,[p],h) = σ0 (Gκ (h) − ∆x h)∇x h − [µ∂y w]∇x h + [p]∇x h −[µ] ∇x v + (∇x v)T ∇x h + |∇x h|2 [µ∂y v] + ∇x h · [µ∂y v]∇x h, Gw (v,w,h) = −σ0 Gκ (h) − [µ]∇x h · ∇x w + |∇x h|2 [µ∂y w] − ∇x h · [µ∂y v].

48

D. Bothe, J. Pr¨ uss and G. Simonett

3.2. The associated linear problem According to Section 3.1, the linear system associated with (11), (12) has the following form. Diﬀerential Equations

ρ ∂t w − µ ∆w + ∂y p = fw

in J × Rn × R˙ in J × Rn × R˙

∇x · v + ∂y w = fp

in J × Rn × R˙

ρ ∂t v − µ ∆v + ∇x p = fv

∂t c˜ − D ∆˜ c = fc

in J × H n+1 +

∂t h − w = gh

on J × Rn

∂t c˜Γ − DΓ ∆x c˜Γ = gc

on J × Rn

−[µ ∂y v] − [µ] ∇x w = gv

on J × Rn

−2 [µ ∂y w] + [p] − σ0 ∆h = gw

(13)

on J × Rn .

Boundary Conditions [v] = 0 ,

c˜Γ − γ0 c˜ = gγ

[w] = 0 ,

on J × Rn .

(14)

Initial Conditions v|t=0 = v0 ,

h|t=0 = h0 ,

w|t=0 = w0

˙, on Rn × R

c˜|t=0 = c˜0

on H n+1 , +

c˜Γ |t=0 = c˜Γ0

(15)

on Rn .

Maximal regularity in an appropriate Lp -setting for this problem is given by the following result. Before stating this result we introduce some function spaces. Let n+1 ∞ H˙ p1 (H n+1 ± ) denote the closure of Cc (H ± ) w.r. to the Lp -norm of the gradient, n+1 ∞ factored over the constants, and let 0 H˙ p1 (H n+1 ± ) be the closure of Cc (H ± ) w.r. n+1 n+1 −1 1 to the same norm. Moreover, let H˙ (H ) be the dual of 0 H˙ (H ). p

±

p

±

Theorem 2. Let ρ± , µ± , D, DΓ , γ0 , σ0 > 0. Let 1 < p < ∞ with p = 3/2, 3 and J = [0, T ]. Then problem (13)–(15) admits a unique solution v, w, p, h, c˜, c˜Γ ) such that ˙ Rn+1 ) ∩ Lp J; H 2 (Rn × R; ˙ Rn+1 ) , (v, w) ∈ Hp1 J; Lp (Rn × R; p p± ∈ Lp J; H˙ p1 (H n+1 ± ; R) , [p] ∈ Wp1/2−1/2p J; Lp (Rn ) ∩ Lp J; Wp1−1/p (Rn ) , h ∈ Wp2−1/2p J; Lp (Rn ) ∩ Hp1 J; Wp2−1/p (Rn ) ∩ Lp J; Wp3−1/p (Rn ) , n+1 2 c˜ ∈ Hp1 J; Lp (H n+1 + ) ∩ Lp J; Hp (H + ) , c˜Γ ∈ Hp1 J; Lp (Rn ) ∩ Lp J; Hp2 (Rn )

Two-phase Flows with Soluble Surfactant

49

if and only if the data of the problem satisfy ˙ Rn+1 ) , a) (fv , fw ) ∈ Lp J; Lp (Rn × R; 2−2/p ˙ Rn+1 ) , h0 ∈ Wp3−2/p (Rn ) , b) (v0 , w0 ) ∈ Wp (Rn × R; 2−2/p

c) d) e) f) g) h)

2−2/p

c˜0 ∈ Wp (H n+1 ˜Γ0 ∈ Wp (Rn ) , + ), c n+1 ± 1 −1 ˙ fp ∈ Hp J; Hp (H ± ) ∩ Lp J; Hp1 (H n+1 ± ) , fc ∈ Lp J; Lp (H n+1 + ) , 1/2−1/2p 1−1/p J; Lp (Rn ; Rn+1 ) ∩ Lp J; Wp (Rn ; Rn+1 ) , (gv , gw ) ∈ Wp 1−1/2p 2−1/p gh ∈ Wp (Rn ) , J; Lp (Rn ) ∩ Lp J; Wp gc ∈ Lp J; Lp (Rn ) , 1−1/2p 2−1/p gγ ∈ Wp (Rn ) , J; Lp (Rn ) ∩ Lp J; Wp

and the compatibility conditions i) j) k) l)

[v0 ] = 0, [w0 ] = 0 if p > 3/2 , c˜0 |y=0 − c˜Γ0 = gγ |t=0 if p > 3/2 , ∇x · v0± + ∂y w0± = fp±|t=0 in H˙ p−1 (H n+1 ± ), −[µ ∂y v0 ] − [µ] ∇x w0 = gv |t=0 if p > 3 .

Proof. Concerning the proof, notice ﬁrst that the “only if”-part follows by taking traces; see section 5 in [12]. For the “if”-part observe that problem (13)–(15) consists of two decoupled subsystems. The ﬁrst one involves only (v, w, p, h) and the corresponding result is Theorem 3.3 in [14]. The second one is built by the surfactant equations, i.e., those for c˜ and c˜Γ . Here the desired maximal regularity follows from that of the diﬀusion equation on Rn and on a half-space with Dirichlet boundary condition; cf. [11] and [22]. 3.3. Solution of the quasilinear problem We do not intend to give the detailed proof here, but rather present the main ideas of the proof. In fact, the arguments are very similar to those employed in [12] for the Stefan problem with surface tension, and in [14] for the free boundary value problem of the Navier-Stokes equations. Introduce function spaces XT × YT for the right-hand side of the diﬀerential equations in (13) and ZT for the corresponding solutions as follows. The elements of XT are functions f = (fv , fw , fp , fc ),

˙ deﬁned on [0, T ] × Rn+1 × R,

with regularity given by a), c), d) of Theorem 2. The elements of YT are functions g = (gv , gw , gh , gc , gγ ),

deﬁned on [0, T ] × Rn ,

with regularity deﬁned in e), f), g), h) of Theorem 2. The solution space ZT consists of functions z = (v, w, p, h, c˜, c˜Γ ) with regularity as described in Theorem 2. ZT additionally also contains the ﬁrst of the boundary conditions in (14), i.e., [v] = 0, [w] = 0. Equipped with their natural norms these spaces are Banach

50

D. Bothe, J. Pr¨ uss and G. Simonett

spaces. The corresponding spaces of functions with zero trace at t = 0 (if they exist) will be denoted by a lower left subscript 0. Next it is convenient to introduce an auxiliary function z ∗ ∈ ZT which resolves the initial conditions and the compatibilities j), k) and l) of Theorem 2. By the assumptions of Theorem 1 such a function z ∗ exists, it can be constructed via Theorem 2 applied to special right-hand sides. This way we can now work in the spaces 0 Z T and 0 X T × 0 Y T , which allow for embeddings with embedding constants independent of T . The choice of the exponent p > n + 3 yields (v± , w± ) ∈ C([0, T ]; C01 (H n+1 ± )),

c˜ ∈ C([0, T ]; C01 (H n+1 + )),

c˜Γ ∈ C([0, T ]; C01 (Rn+1 )),

h ∈ C([0, T ]; C02 (Rn )) ∩ C 1 ([0, T ]; C01 (Rn )),

and the spaces for the g’s are Banach algebras, except for that of gc . Introducing the linear operator L : 0 Z T → 0 X T × 0 Y T , deﬁned by the left-hand side of (13) and the last term in (14), Theorem 2 shows that L is linear, bounded, and invertible. The norm of L−1 is bounded by some constant M , which does not depend on T ∈ (0, T0 ], where T0 < ∞ is ﬁxed. We now deﬁne H : ZT → XT × YT by the right-hand side of (11) and by the last boundary condition in (12). Then the nonlinear problem can be rewritten as Lz = H(z + z ∗ ) − Lz ∗ := H0 (z),

z ∈ 0Z T .

Note that H0 maps 0 Z T into 0 X T × 0 Y T by construction. Inverting L, this problem becomes the ﬁxed point problem z = L−1 H0 (z),

z ∈ 0ZT .

¯r (0) into ¯r (0) ⊂ 0 Z T . The aim is to show that L−1 H0 maps B Fix a closed ball B ¯ itself and is a strict contraction on Br (0). Thus we have to show that 1 r ¯r (0). |z−¯ z|0 Z Z , |H0 (z)|0 X T ×0 Y T ≤ , z, z¯ ∈ B 2M M Among the ﬁnitely many terms appearing in the deﬁnition of the F ’s and G’s we encounter three types: |H0 (z)−H0 (¯ z )|0 X T ×0 Y T ≤

• terms containing the factor ∇x h; • terms containing the factor σ(˜ cΓ ) − σ0 ; • lower order terms. Next observe that terms containing derivatives of highest order are linear in these derivatives and are of the ﬁrst two types. They can be made small by choice of η > 0. On the other hand, lower order terms can be made small by the choice of T > 0 and r > 0. These estimates are of the same character as in the two papers cited above. ¯r (0) → In conclusion, we can manage this way to show that L−1 H0 : B ¯ Br (0) is a strict contraction, and hence admits a unique ﬁxed point, thanks to the contraction mapping principle. This yields a unique solution on a possibly small time interval in ZT , the optimal regularity class of type Lp .

Two-phase Flows with Soluble Surfactant

51

Further regularity of the solutions can be obtained by the parameter trick; cf. [23] and [13]. Under the assumptions of Theorem 1 we obtain in this way classical solutions for t ∈ (0, T ], and even real analyticity in case the given functions γ and σ have this property. We refrain from giving more details here.

Appendix A. Surface divergence Let M be an m-dimensional C k -submanifold of Rn . Then, locally around any point p ∈ M , the submanifold is given by a parametrization of class C k . Hence there is an open parameter region B ⊂ Rm and a C k -function g : B → Rn such that g(B) = M ∩ U for some open neighborhood U ⊂ Rn of p. Furthermore, the set {∂1 g(y), . . . , ∂m g(y)} is a linearly independent subset of Rn , and g is a homeomorphism from B onto M ∩ U . The tangent space Tp M at p ∈ M is given by Tp M = {τ ∈ Rn : ∃ C 1 -curve γ : (−, ) → M s.t. γ(0) = p , γ (0) = τ } . Then, Tp M = span{∂1 g(y), . . . , ∂m g(y)} for p ∈ M, y = g −1 (p) . Let f be a vector ﬁeld on M , meaning that f : M → Rn is everywhere tangential to M , i.e., f (p) ∈ Tp M for all p ∈ M . Such an f is a C 1 -vector ﬁeld on M if f ◦ g is continuously diﬀerentiable for every C 1 -parametrization g of M . It then follows that f ◦ γ is continuously diﬀerentiable for every C 1 -curve γ : (−, ) → M . In this case, given p ∈ M and τ ∈ Tp M with τ = 0, the directional derivative of f at p in direction of τ is d ∂f (p) = (f ◦ γ)(0) ∂τ dt for a C 1 -curve γ : (−, ) → M such that γ(0) = p, γ (0) = τ . Correspondingly, a scalar ﬁeld φ : M → R is C 1 if φ ◦ g is C 1 for every C 1 -parametrization g. In this case, the surface gradient gradM φ is given by gradM φ(p) =

m ∂φ τi , ∂τi i=1

where {τ1 , . . . , τm } is an orthonormal basis of Tp M . Given a C 1 -vector ﬁeld f on M , a coordinate-free deﬁnition of the surface divergence divM f of f is 1 f, N dσm−1 , (16) divM f (p) = lim r→0+ |Mr (p)| ∂M (p) r where Mr (p) = M ∩ Vr (p) with a family of open sets Vr (p) ⊂ Rn such that p ∈ Vr (p) , diam(Vr (p)) = O(r)

as r → 0+

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D. Bothe, J. Pr¨ uss and G. Simonett

and ∂Vr (p) is piecewise of class C 2 ; notice that Mr (p) is an m-dimensional submanifold with boundary ∂Mr (p) and the latter is itself an (m − 1)-dimensional submanifold of M without boundary. In (16), · , · is the Euclidean (inner) product on Rn and N denotes the exterior unit normal ﬁeld to ∂Mr (p), i.e., N (q) ∈ Tq M , N (q) ⊥ Tq ∂Mr (p) at q ∈ ∂Mr (p) and N (q) points outward w.r. to Mr (p). Further, |Mr (p)| denotes the surface area of Mr (p). If M has parametrization g : B ⊂ Rm → Rn near p ∈ M , then det G(y) dy |Mr (p)| = Br

with Br = g

−1

(Mr (p)) and the Gramian G(y) = ∂i g(y), ∂j g(y) 1≤i,j≤m . In the following, dσm = det G(y) dy denotes the surface measure on M and dσm−1 is the corresponding surface measure on the respective (m − 1)-dimensional boundary-submanifold. A more common formulation of divM f is obtained as follows. Given p ∈ M ∗ and a basis {τ1 , . . . , τm } of Tp M , let {τ1∗ , . . . , τm } ⊂ Tp M denote the co-basis such that τi , τj∗ = δij , i.e., {τi }, {τj∗ } ⊂ Tp M form a biorthogonal system. Extend {τ1 , . . . , τm } by means of τm+1 , . . . , τn ∈ (Tp M )⊥ to a basis {τ1 , . . . , τn } of Rn . We now exploit the fact that M is locally a graph over Tp M : Let ϕ : M ∩ B (p) → Tp M be given by ϕ(q) =

m

yi τi for q ∈ M with q = p +

i=1

m

yi τi +

i=1

n

tk τk .

k=m+1

For suﬃciently small > 0, the map ϕ : M ∩ B (p) → ϕ(M ∩ B (p)) is a diﬀeomorphism. Now, let Br = (−r, r)m ⊂ Rm with r > 0 and deﬁne the parametrization g : Br → Rn by m yi τi . g(y) = ϕ−1 i=1

Then ∂i g(y) = τi + o(|y|) as y → 0 , since ϕ approximates the identity near the point p. Let n ' ( Vr (p) = p + ti τi : ti ∈ (−r, r) i=1

and Mr (p) = M ∩ Vr (p) = g(Br ) for small r > 0 . Then

2 √ det ∂i g(y) , ∂j g(y) dy = (2r)m det G + o(rm ) as r → 0+

|Mr (p)| = Br

Two-phase Flows with Soluble Surfactant

53

with G = τi , τj 1≤i,j≤m . The boundary of Mr (p) is m 3

∂Mr (p) =

Wrj (p)

with Wrj (p) = Wrj,+ (p) ∪ Wrj,− (p) ,

j=1

where

Wrj,± (p) = g(y) : yi ∈ [−r, r] for i = j , yj = ±r .

On ∂Mr (p) the outer unit normal ﬁeld N satisﬁes N (q) = ± Hence

for q ∈ Wrj,± (p) .

Wrj (p)

f , N dσm−1 =

= [−r,r]m−1

r

= [−r,r]m−1

(2r)m

Wrj,+ (p)

f , N dσm−1 +

Wrj,− (p)

f , N dσm−1

yj =r τj∗ ! (f ◦ g)(y) , ∗ + o(r) det Gj (y) dy j yj =−r |τj |

=

τj∗ + o(r) |τj∗ |

−r

τj∗ ! ∂j (f ◦ g)(0) + o(r), ∗ dyj + o(r) det Gj (y) dy j |τj |

τj∗ ! √ ∂f (p), ∗ det Gj + o(rm ) ∂τj |τj |

(as r → 0+) ,

where dy j = dy1 . . . dyj−1 dyj+1 . . . dym and the matrix Gj is obtained from G by elimination of the jth row and column. Due to the basic geometric fact that “volume = base area × height”, it follows that √ √ τj∗ ! 1 √ det G = det Gj τj , ∗ = ∗ det Gj for j = 1, . . . , m . |τj | |τj | Hence, divM f (p) =

m j=1

! ∂f (p) , τj∗ , ∂τj

(17)

where {τi }, {τj∗ } ⊂ Tp M is any biorthogonal system. In case {τ1 , . . . , τm } is an orthonormal basis of Tp M , (17) resembles the well-known deﬁnition of the divergence in Rm . For concrete computations it is more useful to write divM f (p) in the local coordinates that correspond to a given parametrization, i.e., to use τj = ∂j g(y) with y = g −1 (p) . By deﬁnition of G, it follows that {τj∗ } is then given by ∗ ∗ = τ1 . . . τm G−1 , τ1 . . . τm

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D. Bothe, J. Pr¨ uss and G. Simonett

i.e., τj∗

m

=

in case (g ij ) := G−1 .

(18)

g ij ∂j (f ◦ g)(y) , ∂i g(y) for y = g −1 (p) .

(19)

g ij τi

i=1

Inserting this into (17) yields divM f (p) =

m i,j=1

Using the fact that

1 divM √ τk = 0 for k = 1, . . . , m , det G

(20)

a signiﬁcant simpliﬁcation of (19) can be obtained. Indeed, √ 1 divM f = √ ∂j (v j ◦ g) det G , det G j=1 m

(21)

where the v j denote the components of f w.r. to the basis {τj }, i.e., f (p) =

m

v j (p) τj (p) on M .

j=1

Let us brieﬂy justify equation (20). By (17) we have m 1 1 ! τk = τk , τj∗ divM √ ∂j √ det G det G j=1 m ∂j det G 1 − τk , τj∗ + ∂j τk , τj∗ = √ 2 det G det G j=1 m 1 ∂k det G + ∂j τk , τj∗ . − = √ 2 det G det G j=1

The matrix G = (τi , τj ) satisﬁes the equation ∂k G = QG, with Q = (∂k G)G−1 . According to Liouville’s Theorem, the determinant det G then satisﬁes ∂k det G = (tr Q) det G. The matrix Q is given by Q = (∂k τi , τj )G−1 . For the trace of Q one obtains tr Q =

=

m m ∂k τi , τj + τi , ∂k τj g ij = ∂k τi , g ij τj + g ij τi , ∂k τj i,j=1 m

m

i=1

j=1

∂k τi , τi∗ +

Consequently, ∂k det G = 2

+

i,j=1 m

τj∗ , ∂k τj = 2

∗ j τj , ∂k τj

τj∗ , ∂k τj .

j=1

det G and (20) follows.

Two-phase Flows with Soluble Surfactant

55

The divergence theorem is inherited from the deﬁnition of the surface divergence. It reads as (see Theorem 2.1 in Chapter XII of [20]). Theorem A (Divergence theorem). Let M be a compact m-dimensional C 2 -submanifold of Rn with continuous normal ﬁeld, and let N be the exterior unit normal ﬁeld to its boundary ∂M . Then, for every C 1 -vector ﬁeld f on M , divM f dσm = f, N dσm−1 . (22) M

∂M

By means of (17) or (19) it is possible to deﬁne divM f also for nontangential functions f : M → Rn . One important example is the surface divergence of the unit normal ﬁeld of an (n − 1)-dimensional submanifold (i.e., a hypersurface). In fact, the sum of the principal curvatures of M at p is given by (see, for instance, p. 223 in [4]) (23) κ = −divM nM . Indeed, equation (19) then becomes −divM nM = −

m

g ij ∂j (nM ◦ g) , ∂i g =

i,j=1

m

g ij hji ,

i,j=1

where hji := −∂j (nM ◦ g) , ∂i g = nM ◦ g, ∂j ∂i g is the second form of M (with respect to the parametrization g). The +fundamental ij g h is the trace of the second fundamental form, and it gives expression m ji i,j=1 the sum of the principal curvatures of M . In case M is the graph of a function h : B → R for some open set B ⊂ Rm , that is, M := {(x, h(x)) : x ∈ B}, one easily shows that ∂i h ∂j h g ij = δij − 1 + |∇h|2 and m 1 ∂i h ∂j h 1 m δij − ∂ ∂ h = div ∇h . (24) κ= i j R 1 + |∇h|2 1 + |∇h|2 i,j=1 1 + |∇h|2 Let us note in passing that for n = 3, Gauss called H = 12 κ the mean curvature of M . Notice also that the divergence theorem does not apply to f = nM . For a not necessarily tangent ﬁeld f : M → Rn the decomposition f = fM + f, nM nM , where fM is the tangential part of f , leads to the formula divM f = divM fM − κ f, nM ; observe that

divM f, nM nM = gradM f, nM , nM + f, nM divM nM

(25)

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D. Bothe, J. Pr¨ uss and G. Simonett

and gradM φ(p) ∈ Tp M for any scalar ﬁeld φ. Finally, recall that in local coordinates given by a parametrization g, the Laplace-Beltrami operator ∆M , deﬁned by ∆M φ = divM gradM φ, is given as m 1 ∆M φ(p) = ∂j det G(y) g jk ∂k (φ ◦ g)(y) , y = g −1 (p) , det G(y) j,k=1 for y ∈ B. For more information concerning vector analysis on (sub-)manifolds see, e.g., [3, 4, 20].

Appendix B. Surface transport theorem With the notation from A, we are in a position to prove the following result. Theorem B (Surface transport theorem). Let I ⊂ R open, Ω ⊂ Rn open and the velocity ﬁeld v : J × Ω → Rn be C 1 . Given t0 ∈ I, x0 ∈ Ω let φ(t; t0 , x0 ) denote the solution of the ODE-system ˙ = v(t, φ(t)) , φ(t0 ) = x0 . φ(t) Given a compact C 2 -hypersurface M0 ⊂ Ω, let Mt = {φ(t; t0 , x0 ) : x0 ∈ M0 } denote the advected (material) surface; due to the assumptions on v, Ω and M0 the surface Mt exists for all t ∈ Iδ = (t0 − δ, t0 + δ) for some δ > 0. Let c : gr(Mt ) = {(t, x) : t ∈ Iδ , x ∈ Mt } → R be continuously diﬀerentiable. Then, Dc d + c divM v dσt c dσt = dt Mt Mt Dt Dc + c divM vM − c κ V dσt = Mt Dt

(26)

for all t ∈ Iδ . Here, dσt denotes the surface measure on Mt , κ = κ(t, x) the curvature of Γ(t) at x ∈ Γ(t) according to (23), vM the tangential part of v, V = v , nM the normal component of v, and Dc d (t, x) = c(t + s, φ(t + s; t, x)) |s=0 Dt ds is the Lagrangian derivative of c. Proof. Let m = n − 1, g : B → Rn , with B ⊂ Rm open and bounded, be a parametrization of M0 and Φt (y) = φ(t; t0 , y) be the ﬂow map associated with v. Then, g t : B → Rn with g t := Φt ◦ g

Two-phase Flows with Soluble Surfactant

57

is a parametrization of Mt . Hence, c(t, x) dσt = c(t, Φt (y)) det G(t, y) dy , Mt

B

where G(t, y) is the Gramian

G(t, y) = ∂i g t (y) , ∂j g t (y) 1≤i,j≤m .

Therefore, d c(t, x) dσt dt Mt d c(t, Φt (y)) 1 c(t, Φt (y)) ∂t det G(t, y) dy = det G(t, y) dy + det G(t, y) 2 B dt B (27) and, by the deﬁnition of Dc Dt , the ﬁrst term on the right is d Dc t c(t, Φ (y)) (t, x) dσt . det G(t, y) dy = dt B Mt Dt The derivative of ∂t det G(t, y) can be computed by using Liouville’s Theorem. However, we prefer here to give a self-contained derivation. For this purpose, notice ﬁrst that ∂t det G(t, y)

=

⎛

⎜ ⎜ det ⎜ ⎝ j=1

m

∂1 g t , ∂1 g t

···

∂t ∂1 g t , ∂j g t

.. .

···

∂1 g t , ∂m g t

.. .

∂m g t , ∂1 g t · · ·

∂t ∂m g t , ∂j g t · · ·

.. .

⎞ ⎟ ⎟ ⎟. ⎠

∂m g t , ∂m g t

Using ∂t ∂j g t (y) = ∇x v(t, g t (y)) ∂j g t (y) we obtain 1 ∂t ∂i g t (y) , ∂j g t (y) = D ∂i g t (y) , ∂j g t (y) 2 with

1 (∇x v + (∇x v)T ) , (∇x v)T = ∇x v1 . . . ∇x vn . 2 Consequently, if τi = ∂i g t , then ⎛ τ1 , τ1 · · · Dτ1 , τj · · · τ1 , τm m ⎜ 1 ⎜ .. .. .. ∂t det G(t, y) = det ⎜ . . . ⎝ 2 D=

j=1

τm , τ1 · · ·

Dτm , τj

···

⎞ ⎟ ⎟ ⎟. ⎠

τm , τm

Deﬁne Q ∈ Rm×m by means of

QG = Q τi , τj = Dτi , τj .

(28)

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D. Bothe, J. Pr¨ uss and G. Simonett

Let aij = τi , τj and αi = (αi1 , . . . , αim )T . Then m 1 ∂t det G(t, y) = det α1 . . . Qαj . . . αm . 2 j=1

Due to the properties of the determinant it is now easy to obtain 1 ∂t det G(t, y) = tr Q det α1 . . . αm = tr Q det G(t, y) . 2 By (28), the matrix Q satisﬁes Q = Dτi , τj G(t, y)−1 , hence Qij =

m

if (g ij ) := G−1 .

Dτi , τk g kj

k=1

This yields

6 Qij =

Dτi ,

m

7 g

kj

τk

= Dτi , τj∗

k=1 ∗ with τj∗ according to (18), i.e., {τ1∗ , . . . , τm } ⊂ span{τ1 , . . . , τm } is a co-basis. Therefore m 1 (∇x v) τi , τi∗ + (∇x v) τi∗ , τi , tr Q = 2 i=1

hence

1 ∂t det G(t, y) = divMt v(t, g t (y)) det G(t, y) 2 by (17). Insertion into (27) ﬁnally yields d d c(t, Φt (y)) c(t, x) dσt = det G(t, y) dy dt Mt B dt c(t, Φt (y)) divMt v(t, g t (y)) det G(t, y) dy + B Dc (t, x) + c(t, x) divMt v(t, x) dσt . = Mt Dt

The literature contains several variants of Theorem B (cf. [35]), sometimes with uncommon notations. One such variant of (26) is ) * ∂c d c dσt = + divM (c vM ) − c κ V dσt , (29) dt Mt ∂t n Mt where

) ∂c *

d c(t + s, φn (t + s; t, x)) |s=0 (t, x) = ∂t n ds with φn the purely normal motion. In the sections above, we used the % interface & notation ∂t,n c instead of ∂c to avoid confusion with the interfacial jump of a ∂t n quantity. The function φn is the solution of φ˙ n (t) = vn (t, φn (t)) , φn (t0 ) = x0 , (30)

Two-phase Flows with Soluble Surfactant

59

where vn (t, x) = v(t, x) , nM (x) nM (x)

on gr(Mt ) .

The existence of a solution φn of (30) is not obvious, since vn (t, x) is only deﬁned on gr(Mt ). In such a situation, the tangency condition 1 dist(x0 + hvn (t0 , x0 ), Mt0 +h ) = 0 h h→0+ lim

for all t0 and x0 ∈ Mt0

is necessary and suﬃcient for local existence of solutions, given that Mt enjoys some mild regularity which holds if gr(Mt ) is a C 1 -submanifold of Rn+1 and vn is locally Lipschitz continuous; see, e.g., [6]. Now, given x0 ∈ Mt0 , the C 1 -regularity of Mt0 and the fact that v − vn with v = v(t0 , x0 ) and vn = vn (t0 , x0 ) is tangent to Mt0 at x0 yields x0 − h(v − vn ) + eh ∈ Mt0

for small h > 0

with certain eh ∈ R such that |eh | = o(h) as h → 0+. This implies n

dist(x0 + hvn , Mt0 +h ) ≤ |x0 + hvn − φ(t0 + h; t0 , x0 − h(v − vn ) + eh )| = |x0 + hvn − (x0 − h(v − vn ) + hv(t0 , x0 − h(v − vn )))| + o(h) = h|v(t0 , x0 ) − v(t0 , x0 − h(v − vn ))| + o(h) = o(h)

as h → 0+

if v is continuous. Hence the required tangency condition holds. Application of Theorem A yields the formulation ) * ∂c d c dσt = − c κ V dσt + c v , N dst , dt Mt ∂t n Mt ∂Mt where dst denotes the surface measure on ∂Mt . This form of the surface transport theorem nicely displays the diﬀerent contributions to the change in surfactant mass due to transport of species and geometrical changes. A more formal derivation of Theorem B in the language of Diﬀerential Geometry can be found in [4], p. 228ﬀ, while a more heuristic version for n = 3 is given in [35]. For purely tangential surface motion Theorem B reduces to Theorem 2.11 in Chapter XII of [3], while for purely normal surface motion a generalization to Mt ∩ V with ﬁxed V ⊂ Rn instead of Mt is given in [15]; for extensions cf. Remark 3 in [15] and [17, 21].

References [1] F. Abergel, C. Dupaix: Existence of smooth, stationary interfaces for Marangoni-type ﬂow. Nonl. Anal., Theory Meth. Appl. 27(11), 1329–1350 (1996). [2] G. Allain: Small-time existence for the Navier-Stokes equations with a free surface. Appl. Math. Optim. 16, 37–50 (1987). [3] H. Amann, J. Escher: Analysis III. Birkh¨ auser, 2001. [4] R. Aris: Vectors, tensors, and the basic equations of ﬂuid mechanics. Dover Publications, 1989.

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[5] J.T. Beale: Large-time regularity of viscous surface waves. Arch. Rational Mech. Anal. 84, 307–352 (1984). [6] D. Bothe: Multivalued diﬀerential equations on graphs. Nonl. Anal., Theory Meth. Appl. 18(3), 245–252 (1992). [7] R. Clift, J.R. Grace, M.E. Weber: Bubbles, Drops, and Particles. Academic Press, New York, 1978. [8] R. Defay, I. Prigogine, A. Bellemans: Surface tension and adsorption. Wiley and Sons, New York, 1966. [9] I.V. Denisova: Evolution of compressible and incompressible ﬂuids separated by a closed interface. Interfaces Free Bound. 2(3), 283–312 (2000). [10] I.V. Denisova, V.A. Solonnikov: Classical solvability of the problem on the motion of two viscous incompressible ﬂuids. St. Petersburg Math. J. 7(5), 755–786 (1996); translation from Algebra Anal. 7(5), 101–142 (1995). [11] R. Denk, M. Hieber, J. Pr¨ uss: R-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Amer. Math. Soc. 166(788), 2003. [12] J. Escher, J. Pr¨ uss, G. Simonett: Analytic solutions for a Stefan problem with GibbsThomson correction. J. reine angew. Math. 563, 1–52 (2003). [13] J. Escher, J. Pr¨ uss, G. Simonett: A new approach to the regularity of solutions for parabolic equations. In: Evolution Equations. Lecture Notes in Pure and Appl. Math. 234, 167–190. Dekker, New York, 2003. [14] J. Escher, J. Pr¨ uss, G. Simonett: Analytic solutions of the free boundary value problem for the two-phase Navier-Stokes system. Preprint. [15] M.E. Gurtin, A. Struthers, W.O. Williams: A transport theorem for moving interfaces. Quarterly of Applied Mathematics 47(4), 773–777 (1989). [16] M. Ishii: Thermo-Fluid Dynamic Theory of Two-Phase Flow. Eyrolles, Paris, 1975. [17] J.P. Jaric: On a transport theorem for moving interface. Int. J. Engng. Sci. 30(10), 1535–1542 (1992). [18] A.J. James, J. Lowengrub: A surfactant-conserving volume-of-ﬂuid method for interfacial ﬂows with insoluble surfactant. J. Comp. Phys. 201(2), 685–722 (2004). [19] M.V. Lagunova: On the solvability of a three-dimensional problem of thermocapillary convection. J. Sov. Math. 64(6), 1233–1240 (1993); translation from Probl. Math. Anal. 11, 18–27 (1990). [20] S. Lang: Diﬀerential and Riemannian Manifolds. Springer, New York, 1995. [21] H. Petryk, Z. Mroz: Time derivates of integrals and functionals deﬁned on varying volume and surface domains. Arch. Mech. 38(5-6), 697–724 (1986). [22] J. Pr¨ uss: Maximal regularity for abstract parabolic problems with inhomogeneous boundary data in Lp -spaces. Math. Bohem. 127(2), 311–327 (2002). [23] J. Pr¨ uss: Maximal regularity for evolution equations in Lp -spaces. Conf. Semin. Mat. Univ. Bari (2002) 285, 1–39 (2003). [24] F. Ravera, M. Ferrari, L. Liggieri: Adsorption and partitioning of surfactants in liquid-liquid systems. Adv. Coll. Int. Sci. 88, 129–177 (2000). [25] M. Siegel: Inﬂuence of surfactant on rounded and pointed bubbles in two-dimensional Stokes ﬂow. SIAM J. Appl. Math. 59(6), 1998–2027 (1999).

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[26] J.C. Slattery: Advanced Transport Phenomena. Cambridge University Press, Cambridge 1999. [27] V.A. Solonnikov: On the quasistationary approximation in the problem of motion of a capillary drop. pp. 643–671 in Topics in Nonlinear Analysis. The Hermann Amann Anniversary Volume, (J. Escher, G. Simonett, eds.). Birkh¨ auser, Basel, 1999. [28] N. Tanaka: Global existence of two-phase non-homogeneous viscous incompressible ﬂuid. Commun. Partial Diﬀer. Equations 18(1-2), 41–81 (1993). [29] N. Tanaka: Two-phase free boundary problem for viscous incompressible thermocapillary convection. Jap. J. Math., New Ser. 21(1), 1–42 (1995). [30] A. Tani: Two-phase free boundary problem for compressible viscous ﬂuid motion. J. Math. Kyoto Univ. 24, 243–267 (1984). [31] A. Tani: Small-time existence for the three-dimensional Navier-Stokes equations for an incompressible ﬂuid with a free surface. Arch. Rat. Mech. Anal. 133, 299–331 (1996). [32] A. Tani, N. Tanaka: Large-time existence of surface waves in incompressible viscous ﬂuids with or without surface tension. Arch. Rat. Mech. Anal. 130, 303–314 (1995). [33] Y. Teramato: On the Navier-Stokes ﬂow down an inclined plane. J. Math. Kyoto Univ. 32, 593–619 (1992). [34] A. Wagner: Nonstationary Marangoni convection. Appl. Math. 26(2), 195–220 (1999). [35] H. Wong, D. Rumschitzki, C. Maldarelli: On the surfactant mass balance at a deforming ﬂuid interface. Phys. Fluids 8(11), 3203–3204 (1996). Dieter Bothe Institut f¨ ur Mathematik & Department Chemie, Technische Chemie und Chemische Verfahrenstechnik Universit¨ at Paderborn Warburger Str. 100 D-33098 Paderborn, Germany e-mail: [email protected] Jan Pr¨ uss Fachbereich Mathematik und Informatik Martin-Luther-Universit¨ at Halle-Wittenberg D-60120 Halle, Germany e-mail: [email protected] Gieri Simonett Department of Mathematics Vanderbilt University Nashville, TN 37240, USA e-mail: [email protected]

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 63–70 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Resolvent Diﬀerences for General Obstacles J. Brasche and M. Demuth* Abstract. In case of potential perturbations the second resolvent equation transforms the resolvent diﬀerence into a product of operators. For obstacle perturbations this behavior is maintained due to Dynkin’s formula. In the present article we study generalized obstacle perturbations, e.g., perturbations by measures with inﬁnite weight. It turns out that the resolvent diﬀerence equals a product of two operators, one factor is the free resolvent, the second factor contains all the interaction. This result is applicable to diﬀerential operators of arbitrary order and to a wide class of perturbations.

1. Introduction Resolvent diﬀerences play an essential role in spectral theory of selfadjoint operators. For instance if the diﬀerence is compact the essential spectra are stable (Weyl’s Theorem), or if it is trace class the absolutely continuous spectra are the same (Birman-Kuroda Theorem). For studying resolvent diﬀerences the simplest example is the Schr¨odinger operator L = L0 + MV in L2 (Rd ), where L0 is the selfadjoint realization of the Laplacian and MV is the multiplication operator with a real-valued potential function V (.), such that L0 + MV + 1 is a selfadjoint, positive operator. Let dom(L0 ) = dom(L0 + MV ), set H0 = L0 + 1 and H = L0 + MV + 1. Then the second resolvent equation says H0−1 − H −1 = PV H0−1 ,

(1)

PV = (L0 + MV + 1)−1 MV

(2)

with the denotation deﬁned on dom(H0 ). That means the resolvent diﬀerence is equal to a product of the free resolvent with an operator PV which contains all the information on the perturbation. *Supported by the Volkswagenstiftung.

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This procedure becomes more diﬃcult, if L0 is perturbed by imposing Dirichlet boundary conditions. One possibility comes from stochastic analysis. For simplicity, let L0 be again the Laplacian in L2 (Rd ). It generates the semigroup {e−tL0 , t ≥ 0}. The operators are given by (e−tL0 f )(x) = Ex {f X(t) }, f ∈ L2 (Rd ), where X(.) are the trajectories of the Wiener process and Ex {.} is the associated expectation. For investigating the perturbation we introduce an obstacle region, called Γ. Γ is assumed to be a closed set in Rd with positive volume and smooth boundary. Deﬁne τΓ = inf{s, s > 0, X(s) ∈ Γ} , the ﬁrst hitting time of Γ. Set Σ = Rd Γ. Then the family of operators TΣ (t)f (x) = Ex {f X(t) , τΓ ≥ t} restricted to L2 (Σ) forms a strongly continuous semigroup. We denote its generator by L∞ . In this simple situation L∞ is known to be the Friedrichs extension of L0 |L2 (Σ) , or L∞ is given by L0 with Dirichlet boundary conditions on the boundary of Γ. −1 to Now we set H0 = L0 + 1, H∞ = L∞ + 1. Extending the resolvent H∞ 2 d L (R ) the resolvent diﬀerence is given by −1 ⊕ 0) = PΓ H0−1 . H0−1 − (H∞

(3)

Formula (3) is known as Dynkins’s formula (see, e.g., [FOT] or [DvC]). The operator PΓ contains again all the information about the interaction. Here it is the perturbation by an obstacle on Γ. This can be modelled by a potential barrier with inﬁnite height, because if Hβ = H0 + βχΓ , then −1 s-lim Hβ−1 = H∞ . β→∞

PΓ is given by dom(PΓ ) = dom(H0 ) , ' ( (PΓ f )(x) = Ex e−τΓ f X(τΓ ) . 1/2

It is a projection operator in a new Hilbert space H1 consisting of f ∈ dom(H0 ) having the scalar product 1/2

1/2

f, g1 = H0 f, H0 g . PΓ projects onto the orthogonal complement of the subspace FΣ of H1 , given by 1/2 FΣ = {f, f ∈ dom(H0 ), f˜(x) = 0 quasi everywhere on Γ} . 1/2 Here f˜ is the quasi-continuous version of f ∈ dom(H0 ). The stochastic representation and Dynkin’s formula (3) is restricted to L0 which are generators of Markov processes, i.e., to second-order diﬀerential or

Resolvent Diﬀerences for General Obstacles

65

pseudo-diﬀerential operators. And it is restricted to obstacles which are in some sense regular enough.

2. Resolvent diﬀerences for general free operators In a previous article ([BD], Theorem 2) an essentially more general situation was studied. There L0 was only assumed to be a selfadjoint nonnegative operator in a separable Hilbert space H with the scalar product . , .. Setting H0 = L0 + 1 we 1/2 deﬁne a further Hilbert space H1 with vectors f ∈ dom(L0 ) and with the scalar product 1/2

1/2

f, g1 = L0 f, L0 g + f, g 1/2

1/2

= H0 f, H0 g . Let J be a bounded linear operator with dense range, mapping H1 into another Hilbert space H2 . ( H2 contains the perturbation. It is arbitrary in the present framework. If H = L2 (Rd , |.|), then H2 can be L2 (Rd , µ) with some measure µ. In this case J describes perturbations by measure valued potentials.) Now we deﬁne in H a selfadjoint operator Hβ , Hβ ≥ 1, by 1/2

dom(Hβ ) = H1 ⊂ H 1/2

1/2

||Hβ f ||2 = ||H0 f ||2 + β ||Jf ||2H2 . Then there is a unique nonnegative, invertible selfadjoint operator H∞ given by 1/2 dom(H∞ ) = ker(J) , 1/2

1/2 ||H∞ f || = ||H0 f || .

H∞ models a generalized obstacle perturbation, because (Hβ−1 )g

−1 → H∞ g,

β→ ∞

Hβ−1 h → 0 β→ ∞

,

g ∈ ker(J) h ∈ ker(J)

⊥

,

where ker(J) is the closure of ker(J) in H. It turns out that the resolvent diﬀerence is given by −1 H0−1 − (H∞ ⊕ 0) = PJ H0−1 .

(4) ⊥

PJ is an orthogonal projection operator in H1 with PJ H1 = [ker(J)] , where ⊥ means orthogonal w.r.t. . , .1 . Equation (4) can be considered as a generalization of Dynkin’s formula or as a generalized second resolvent equation. Again PJ contains all the information due to the interaction. For instance if H2 = L2 (R, µ) where µ is some singular measure the selfadjoint operator L0 is perturbed by inﬁnitely high obstacles on the support of this measure µ.

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3. Resolvent diﬀerences for unbounded interaction operators The aim of this article is to extend the last result to densely deﬁned closed operators J mapping H1 into H2 . We will prove the following result. Theorem 1. Let L0 be a nonnegative selfadjoint operator in the separable Hilbert space H endowed with the scalar product . , .. 1/2

Let H1 be the Hilbert space {dom(L0 ), . , .1 } with 1/2

1/2

f, g1 = L0 f, L0 g + f, g . Let J be a densely deﬁned closed operator mapping {H1 , . , .1 } into a further separable Hilbert space {H2 , . , .2 }. Assume ran(J) to be dense in H2 . Deﬁne Hβ , β > 0, by 1/2

dom(Hβ ) = dom(J) 1/2

||Hβ f ||2 = ||(L0 + 1)1/2 f ||2 + β ||Jf ||22 . That means Hβ is the perturbation of L0 given by J with the coupling constant β. Deﬁne a nonnegative selfadjoint operator H∞ by 1/2 ) = ker J , dom(H∞ 1/2 ||H∞ f ||2 = ||(L0 + 1)1/2 f ||2 . 1/2

(In other words: H∞ is deﬁned in ker(J) ⊂ dom(L0 ) via strong resolvent convergence of Hβ as β tends to inﬁnity.) Setting H0 = L0 + 1 the second resolvent equation reads −1 ⊕ 0) = PJ H0−1 . H0−1 − (H∞ ⊥ Decomposing H1 = ker(J) ⊕. , . 1 ker(J) then PJ is the orthogonal pro ⊥ jector onto ker(J) . Before we are able to prove this theorem, we formulate a lemma which will imply ﬁnally the result.

Lemma 2. Assume L0 , J, Hβ , H∞ and H, H1 , H2 are as given in Theorem 1. Then there is a compact operator Jc : H1 → H2 such that dom(Jc ) = H1 , ker(Jc ) = ker(J) , and ran(Jc ) is dense in H2 . Proof of Lemma 2. J ∗ is a closed operator from the separable Hilbert space {H2 , . , .2 } to the Hilbert space {H1 , . , .1 }. Hence {dom(J ∗ ), ||.||J ∗ } is separable where ||g||2J ∗ = ||g||2 + ||J ∗ g||2 . Let {fn }n∈N be a dense set in {dom(J ∗ ), ||.||J ∗ }. From that we can select a set {gn }n∈N , such that gn are linearly independent and span {gn , n ∈ N} = span {fn , n ∈ N} .

Resolvent Diﬀerences for General Obstacles

67

Using the Gram-Schmidt orthogonalisation we get an orthonormal system {en }n∈N in {H2 , . , .2 } with span {en , n ∈ N} = span {gn , n ∈ N} , and span {en , n ∈ N} is dense in {dom(J ∗ ), ||.||J ∗ }. J was assumed to be closed, so dom(J ∗ ) is dense in H2 , i.e., {en , n ∈ N} is an orthonormal basis in H2 . With this basis we are able to deﬁne the compact operator Jc . Set 1 ,k ∈ N . λk = 2−k 1 + ||J ∗ ek ||1 Deﬁne an operator J0 by dom(J0 ) = dom(J) J0 f = Σ λk ek , Jf 2 ek . k∈N

By deﬁnition J0 is densely deﬁned and bounded, mapping H1 into H2 . Its bounded extension is denoted by Jc , i.e., J¯0 = Jc , dom(Jc ) = H1 . Jc is a Hilbert-Schmidt operator. To show that take an orthonormal basis {fj , j ∈ N} in dom(J) ⊂ H1 . Then Σ ||Jc fj ||22

j∈N

= Σ || Σ λk ek , Jfj 2 ek ||22 j∈N

k∈N

Σ λ2k |J ∗ ek , fj 1 |2

= Σ

j∈N k∈N

= Σ λ2k ||J ∗ ek ||21 < ∞ . k∈N

Next we show ker(J) = ker(Jc ). Obviously, ker(J) ker(Jc ), because of the deﬁnition of J0 and Jc , Jf = 0 implies J0 f = Jc f = 0. On the other hand, J is densely deﬁned and closed. Hence ker(J) = [ran(J ∗ )]⊥ . Take an f ∈ ker(Jc ). Then there is a sequence {fn }, fn ∈ dom(J0 ) such that fn → f and J0 fn → Jc f = 0. Let {ek , k ∈ N} be the orthonormal basis in H2 introduced above. Then Jc f, ek 2 = 0 = lim J0 fn , ek 2 n→∞ Σ λm em , Jfn 2 em , ek 2 = lim n→∞

m∈N

= lim λk ek , Jfn 2 n→∞

= λk J ∗ ek , f 1 . Therefore f ∈ ker(Jc ) is orthogonal to any J ∗ ek , k ∈ N, implying f ⊥ ran(J ∗ ) or f ∈ ker(J).

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It remains to prove the density of ran(Jc ) in H2 . The range of J was assumed to be dense. Fix a k0 ∈ N and an ε > 0. Choose f ∈ dom(J) in such a way that ek0 2 ||Jf − || < ε . λk0 2 Then ||Jc f − ek0 ||22 < ε, because ||Jc f − ek0 ||22 2 = Σ λk ek , Jf 2 ek − ek0 k∈N 2 = Σ λ2k |ek , Jf 2 |2 + λ2k0 ek0 , Jf 2 − k∈N

k =k0

1 2 λk0

1 2 λk0

Σ |ek , Jf 2 |2 + ek0 , Jf 2 −

k∈N

k =k0

ek0 2 = Σ ek , Jf 2 ek − k∈N λk0 2 ek0 2 = Jf − < ε. λk0 2 Thus ek0 ∈ ran(Jc ). Because clospan {ek , k ∈ N} = H2 , ran(Jc ) = H2 . This proves Lemma 2.

Proof of Theorem 1. In [BD] Theorem 1 the result was shown for a compact operator Jc given in such a way that −1/2

Jc H0

g = Σ µk vk , g wk k∈N

with an orthonormal basis {vk , k ∈ N} in H and with an orthonormal basis {wk , k ∈ N} in H2 , and with µk → 0 as k → ∞. The result was & % −1 −1 ⊕ 0) g H0 − (H∞ −1/2

= Σ H0 k∈N

−1/2

= Σ H0 k∈N

−1/2

−1/2

vk , g H0

vk −1/2

vk , H0−1 g1 H0

vk .

k ∈ N, is an orthonormal system in ⊥ Deﬁning the projection operator to ker(Jc ) ⊂ H1 by Here H0

vk ,

−1/2

PJc · = Σ H0 k∈N

we get

%

−1/2

vk , · 1 H0

⊥ ker(Jc ) ⊂ H1 .

vk

& −1 ⊕ 0) g = PJc H0−1 g . H0−1 − (H∞

(5)

Resolvent Diﬀerences for General Obstacles

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However, this representation is not unique. In the Lemma 2 above we have shown that for any closed J there is always a compact Jc such that ker(Jc ) = ker(J). Thus PJc = PJ and the deﬁnition of H∞ is unchanged as long as ker(Jc ) = ker(J).

4. Applications The general second resolvent formula (5) can be used to study the spectrum of H∞ in this abstract framework. Possible applications can be explained in analogy to stochastic spectral analysis. Assume L0 to be a diﬀerential operator of second order in L2 (Rd ) = H, generating a Hunt process (for details see [FOT]). Let L∞ be the generator of the strongly continuous semigroup ' ( −tL∞ e f (x) = Ex f X(t) , τΓ t . τΓ is the ﬁrst hitting time of Γ similar to the description in Section 1 for Wiener trajectories. Then Dynkin’s formula (3) can be used in the following way: % −1 & −1 H0 − (H∞ ⊕ 0) ½ = PJc H0−1 ½ . If the process is conservative, i.e., if e−tL0 ½ (x) = ½(x), then (H0−1 ½)(x) = ½(x). On the other hand % −1 & −1 H0 ½ − (H∞ ⊕ 0)½ (x) ∞ % & = dλ e−λ Ex {½} − Ex {τΓ ≥ λ} 0

⎧∞ ⎨

⎫ ⎬ e−λ dλ = Ex ⎩ ⎭ τΓ −τΓ , = Ex e

where Ex {.} is the expectation of the process having in mind. Ex e−τΓ = vΓ (x) is known to be the one-equilibrium potential of Γ. This is related to the capacity of Γ. For a conservative process we have cap (Γ) = vΓ (x)dx . Rd

Sets of ﬁnite capacity play an essential role in spectral theory of obstacle perturbations (see [DvC]). If the capacity is ﬁnite the essential and the absolutely continuous spectra of L0 and L∞ are the same. Certain integral conditions for vΓ guarantee the absence of singularly continuous spectra for H∞ . In the general case of Equation (5) the situation is similar. Consider H = L2 (Rd ) and assume that H0−1 maps L∞ (Rd ) to L∞ (R d ). Then (PJ ½)(x) can be considered as a generalized equilibrium potential and (PJ ½)(x)dx as a general Rd

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−1 notion of a capacity. If H∞ and H0−1 are integral operators it is known that the comparison function −1 D(x) = |H0−1 (x, y) − (H∞ ⊕ 0)(x, y)|dy

= (PJ ½)(x) plays the essential role for the spectral behavior of H∞ (see, e.g., [D]). More detailed applications in H = L2 (R) and for H2 = L2 (R, µ) with diﬀerent measures µ are given in [BD]. Also the large coupling behavior as β → ∞ is studied there in an optimal way.

References [BD]

J. Brasche, M. Demuth, Dynkin’s formula and large coupling convergence, J. Funct. Analysis 219 (2005), 34–69. [D] M. Demuth, Integral conditions for the asymptotic completeness of two-space scattering systems, Helv. Phys. Acta 103 (1993), 333–339. [DvC] M. Demuth, J. van Casteren, Stochastic Spectral Theory for Selfadjoint Feller Operators, Birkh¨ auser Verlag, Basel, 2000. [FOT] M. Fukushima, Y. Oshima, M. Takeda, Dirichlet Forms and Symmetric, Markov Processes, de Gruyter, Berlin, 1994. J. Brasche and M. Demuth Institute of Mathematics TU Clausthal Erzstr. 1 D-38678 Clausthal-Zellerfeld Germany

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 71–97 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Special Finite Time Extinction in Nonlinear Evolution Systems: Dynamic Boundary Conditions and Coulomb Friction Type Problems J.I. D´ıaz Dedicated to Herbert Amann on the occasion of his 65th birthday

1. Introduction The ﬁnite extinction time phenomenon (the solution reaches an equilibrium after a ﬁnite time) is peculiar to certain nonlinear problems which solutions exhibit an asymptotic behavior entirely diﬀerent from the typical behavior of solutions of linear problems. After recalling, in Section 2, some general results and methods on this property, we shall consider several examples of nonlinear systems where ﬁnite extinction time is not an universal property of all the solutions of the problem: a feature very diﬀerent from the case of scalar dissipative equations. For instance, sometimes the vector solution has some components which stabilize in ﬁnite time, and others for which this phenomenon does not occur. This is the case for the linear heat equation with a suitable nonlinear dynamical boundary condition. In Section 3 we present an unpublished result by H. Amann and the author (Madrid, October 1988), the vector solution has two components, one is the solution of the pde in the interior, the other one its trace on the boundary which vanishes after a ﬁnite time. In other nonlinear systems, ﬁnite extinction time is peculiar to a ﬁnite set of orbits. Such behavior arises, for instance, in the case of oscillation under strong friction which is close to Coulomb dry friction for small values of the velocity. This will be presented in Section 4, where, in particular, we shall recall the results by H. Amann and the author on this type of problems.

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2. A survey on ﬁnite extinction time properties In order to ﬁx ideas, let Ω ⊂ RN , N ≥ 1, be a general open set, let Q∞ = Ω × (0, +∞), Σ∞ = ∂Ω × (0, +∞), and consider an evolution boundary value problem formulated as ⎧ ⎨ ut + Au = f (x, t) in Q∞ , Bu = g(x, t) on Σ∞ , (1) ⎩ u(x, 0) = u0 (x) on Ω. Here, Au denotes a nonlinear operator and Bu denotes a boundary operator (we assume, for simplicity, that A and B are autonomous operators). In the study of the stabilization of solutions, as t → +∞, it is usually assumed that f (x, t) → f∞ (x) and g(x, t) → g∞ (x) as t → +∞, in some functional spaces and the main task is to prove that u(x, t) → u∞ (x), as t → +∞, in some topology of a suitable functional space, with u∞ (x) solution of Au∞ = f∞ (x) in Ω, (2) Bu∞ = g∞ (x) on ∂Ω. Here we are interested in a stronger property. We start by assuming that f (x, t) = f∞ (x) g(x, t) = g∞ (x)

for t ≥ Tf , for t ≥ Tg .

(3)

We arrive to the following natural concept Deﬁnition 2.1. Let u be a solution of problem (1). We will say that u(x, t) possesses the property of “stabilization in a ﬁnite time” to the stationary solution u∞ (x) if there exists t∗ < ∞ such that u(x, t) ≡ u∞ (x), on Ω, for any t ≥ t∗ . In many cases f∞ (x) ≡ 0, g∞ (x) ≡ 0, and then u∞ (x) ≡ 0. Then, the above property is known as the ﬁnite time extinction property. Most of the material collected in this Section is devoted to the case of the extinction property. For a diﬀerent survey on this property and other applications, in particular to some problems in Fluid Mechanics, see Chapters 2 and 4 of the monograph [5]. 2.1. Some abstract results for multivalued operators: ﬁnite extinction time and a dichotomy for hyperbolic equations Perhaps, the pioneer abstract result on ﬁnite extinction time of solutions is due to H. Brezis [41]. He proved that if X = H is a Hilbert space, and A : D(A) → P(H) is a maximal monotone operator multivalued at 0 (with 0 ∈ int D(A)) then the solution of the abstract Cauchy problem du dt (t) + Au(t) f (t) in X, u(0) = u0 , possesses the property of ﬁnite extinction in ﬁnite time once we assume f (t) such that B(f (t), ) ⊂ A0, for a.e. t ≥ tf , for some > 0 and tf ≥ 0. (4)

Special Finite Time Extinction in Nonlinear Evolution Systems

73

It turns out that if, for instance, H = L2 (Ω) this property is diﬃcult to be checked for most multivalued operators. This was the motivation of the work [50] in which the property of ﬁnite extinction time was proved for Banach spaces X and A : D(A) → P(X) a multivalued m-accretive operator. Several applications for the special case of X = L∞ (Ω), to some parabolic problems of the type ut − ∆u + β(u) f, with β a maximal monotone graph of R2 (including second-order parabolic obstacle problems) were given in that paper. The application to multivalued nonlinear diﬀusion equations (of the porous media type) ut − ∆β(u) f, was carried out in [54]. The ﬁnite extinction property can be proved also (via this abstract result) for other nonlinear multivalued parabolic problems of the type ⎧ ∇u ⎪ = f (t, x) in Q∞ , ⎨ ut − ν∆u − gdiv |∇u| u=0 on Σ∞ , ⎪ ⎩ u(0, x) = u0 (x) on Ω, for ν ≥ 0, g > 0 and f (t, x) = 0. Such problems arise in several applied contexts (non-Newtonian ﬂuids of Bingham type, image processing, microgranular structures: see references in [2]). The above multivalued operator is also related to some old works in Diﬀerential Geometry ([81]). A problem which looks quite similar to the previous ones but for which the above abstract results does not apply is the multivalued (second-order in time) hyperbolic dry friction type problem as, for instance, ⎧ ⎪ ⎪utt − uxx + β(ut ) 0 in (0, 1) × (0, +∞), ⎪ ⎨u(t, 0) = u(t, 1) = 0 t ≥ 0, (DSP) ⎪u(0, .) = u0 (.) in (0, 1), ⎪ ⎪ ⎩ ut (0, .) = v0 (.) in (0, 1), where now β denotes the maximal monotone graph of R2 given by β(u) = {1} if u > 0, β(0) = [−1, 1] and β(u) = {−1} if u < 0.

(5)

According well-known results (see, e.g., Haraux [70] and also [39]) u(t, x) → ζ(x) in H01 (0, 1) as t → +∞, with ζ verifying −1 ≤ ζxx ≤ 1. Since the beginnings of the seventies, H. Brezis proposed a conjecture: the equilibrium position is reached after a ﬁnite time (stabilization in ﬁnite time). Some partial results in this direction were obtained by H. Cabannes [42], [43] for some special initial data u0 and v0 . The case of arbitrary initial data, u0 and v0 , seems to be still an open problem. Due to the diﬃculty of the above problem (and also suggested by the numerical approach of solutions) some easier formulations were considered in the

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literature, as, for instance, the spatially discretized vibrating string via ﬁnite differences: ⎧ 2 d ui dui ui+1 − 2ui + ui−1 ⎪ ⎪ ⎪ − +β 0 i = 1, . . . , n, ⎪ ⎨ dt2 h2 dt (DDSP) u0 = un+1 = 0, ⎪ ⎪ ⎪ ⎪ ⎩ui (0) = ai and dui (0) = bi . dt This system also arises in the study of the dynamics vibration of N -particles of equal mass m. Indeed, we denote the located positions, along the interval (0, 1) of the x axis, by xi (t) and we assume that each particle is connected to its neighbors by two harmonic springs of strength k. We also assume that the particles are subject to a resultant friction force which is the composition of a Coulomb (or solid) friction and other type of frictions such as, for instance, the one due to the viscosity of an surrounding ﬂuid. Then, the equations of motion for this mechanical system are m¨ xi (t) + k(−xi−1 (t) + 2xi (t) − xi+1 (t)) + µβ β(x˙ i (t)) + µg g(x˙ i (t)) 0 (PN ) xi (0) = u0,i , x˙ i (0) = v0,i i = 1, . . . , N , where we are assuming that x0 (t) = 0, xN +1 (t) = 1

for any t ∈ (0, +∞),

µβ , µg are positive constants, the term µβ β(x˙ i (t)) represents the Coulomb friction, g is a Lipschitz continuous function such that g(0) = 0 and the initial data (u0,i ), (v0,i ) are given in RN . Notice that if we write, for simplicity, k = h12 (with h = 1/(N +1)) and m = 1, then problem (PN ) coincides with the spatial discretization, by ﬁnite diﬀerences, of the damped string equation ⎧ ⎨ utt − uxx + µβ β(ut ) + µg g(ut ) 0 in (0, 1) × (0, +∞), u(0, t) = u(1, t) = 0, t ∈ (0, +∞), (6) (P∞ ) ⎩ x ∈ (0, 1). u(x, 0) = u0 (x), ut (x, 0) = v0 (x) In fact, it was by passing to the limit, N → ∞ in (PN ) (in absence of any friction) how the wave equation was obtained in 1746 by Jean Le Rond D’Alembert. In order to give several criteria to have the stabilization in a ﬁnite time for (PN ) it is useful to start with the study of the special case of a single oscillator, N = 1, without viscous friction m¨ x + 2kx + µβ β(x) ˙ 0.

(7)

The study of this elementary equation can be found in many textbooks (see, for instance, [73]). It is easy to see then that the motion stops deﬁnitively after a ﬁnite µ µ time: i.e., there exists Te < +∞ and x∞ ∈ [− 2kβ , 2kβ ] such that x(t) ≡ x∞ for any t ≥ Te . Concerning the case of N -particles we can mention the work by Bamberger and Cabannes [29] in which they prove the stabilization in a ﬁnite time in absence

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of viscous friction (µg = 0). We point out that this type of friction arises very often in the applications and that its consideration was already proposed by Lord Rayleigh (see, e.g., [80]). Concrete expressions for g can be found also in [73]. System (PN ) can be written, in short, as a vectorial problem ·· · · mx(t) + kAx(t) + µβ B(x(t)) + µβ G(x(t)) 0, (PN ) · x(0) = x0 , x(0) = v0 where x(t) := (x1 (t), x2 (t), . . . , xN (t))T , trix of RN ×N given by ⎛ 2 −1 ⎜ −1 2 ⎜ A =⎜ ⎜ 0 −1 ⎝ ... 0 0 ...

A is the symmetric positive deﬁnite ma0 ... −1 0 2 −1 −1 2 0 −1

0 ... 0 −1 2

⎞ ⎟ ⎟ ⎟, ⎟ ⎠

B :RN → P(RN ) and G :RN → RN denote the (multivalued) maximal monotone operator and the Lipschitz continuous function given by B(y1 , . . . , yN ) = (β(y1 ), . . . , β(yN ))T and G(y1 , . . . , yN ) = (g(y1 ), . . . , g(yN ))T (here hT means the transposed vector of h). As in the case of the damped wave equation, it is not diﬃcult to prove ([63]) that for any (x0 , v0 ) ∈ R2N , problem (PN ) admits a unique weak solution x ∈ C 1 ([0, +∞) : RN ) and that there exists µ µ a unique equilibrium state x∞ ∈ RN (i.e., satisfying that Ax∞ ∈ ([− 2kβ , 2kβ ]N )T ) ·

such that x(t) + x(t) − x∞ → 0 as t → +∞. Sharper results on the asymptotic behavior were obtained in ([63]). It was proved there that the presence of a viscous friction (with a suitable behavior of g near 0) may originate a qualitative distinction among the orbits in the sense that the state of the system may reach an equilibrium state in a ﬁnite time or merely in an asymptotic way (as t → +∞), according the initial data x(0) = x0 · and x(0) = v0 . This dichotomy seems to be new in the literature and contrasts with the phenomena of ﬁnite extinction time for ﬁrst-order (in time) ordinary and parabolic nonlinear equations. More precisely, the following was proved in ([63]): i) if g(r)r ≤ 0 in some neighborhood of√0, then all solutions of (PN ) stabilize in a ﬁnite time, ii) if g(r) = λr with λ ≥ 2 λ1 mk/(µβ µg ), where λ1 denotes the ﬁrst eigenvalue of A then there exist solutions of (PN ) which √ do not stabilize in any ﬁnite time, and iii) if N = 1, A = 1 ∈√R and g (0) < 2 mk/(µβ µg ) any solution stabilize in ﬁnite time but if g (0) ≥ 2 mk/(µβ µg ) there exist solutions which do not stabilize in any ﬁnite time. We point out that the positive results on stabilization in a ﬁnite time remain true under the presence of some impulsive forces f (t) leading to the system ··

·

·

mx(t) + kAx(t) + µβ B(x(t)) + µβ G(x(t)) f (t)

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assuming that their amplitude is small enough: more precisely µβ µβ f (t)T ∈[− + , − ]N for a.e. t ≥ Tf , for some Tf , > 0. 2k 2k Some sharper results are the main goal of some work in progress with G. Hetzer [59]. This behavior contrasts with the case in which the amplitude of f (t) becomes large and g (v) < 0 for any v = 0. It was proved in [45] that, then, the dynamics generates a wide range of events leading to the chaos. As mentioned before, the simultaneous possibility of the occurrence of stabilization in a ﬁnite or inﬁnite time does not hold for solutions of scalar ﬁrst-order in time equations of the form ut − d∆u + β(u) 0

(8)

for β(u) multivalued at u = 0 and d ≥ 0 (see, for instance, [50] and its references). We assume given some Dirichlet and initial conditions. Moreover, if we add an extra term of the form g(u), such that, g(u)u ≥ 0 for any u ∈ R, then the solutions of Ut − d∆U + β(U ) + g(U ) 0

(9)

satisfy that u(t, .)Lp (Ω) ≥ U (t, .)Lp (Ω) and, so, the extinction in a ﬁnite time of u(t, .) implies the same property for U (t, .). The opposed comparison holds when g(u)u ≤ 0. This explain the important diﬀerent behaviors among the solutions of problems of ﬁrst- and second-order in time. Notice that if we assume k = 0 in (P1 ) · then we get that U (t) = x(t) satisﬁes an equation similar to (9) with d = 0. Notice, also, that if m is very small then problem (P1 ) becomes a quasi-static problem (in the terminology of [69]) and then the solutions are closed to the solutions of the ﬁrst-order in time problem 2kx + µβ β(x) ˙ + µg g(x(t)) ˙ 0, (QSP1 ) x(0) = x0 . In that case, g(u)u ≥ 0 implies a comparison opposite to the above mentioned one with respect the solutions with g = 0. Nevertheless, now the multivalued character of β at u = 0 does not imply the stabilization in a ﬁnite time for the solutions of (QSP1 ). The mentioned dichotomy may arise for other dynamical systems, as, for instance, the damped wave equation utt − uxx + β(ut ) + λut 0

in (0, 1) × (0, +∞),

(10)

with Dirichlet boundary conditions u(0, t) = u(1, t) = 0 for t ∈ (0, +∞). If we assume λ ≥ 2π then we can ﬁnd some solutions of (10) which does not stabilize in any ﬁnite time although there are solutions which stabilize in a ﬁnite time. Indeed: we can construct solutions of the ﬁrst type in the form u(x, t) = a(t)sin(πx) + 1 2 x(x − 1), with a(t) such that a ¨ + π 2 a + λa˙ = 0,

(11)

and a(t) ˙ > 0 for all time (which is possible since λ ≥ 2π). By the contrary, if ˙ ˙ we choose b(t) solution of (11) such that b(t) > 0 for all t ∈ [0, 1), b(1) = 0,

Special Finite Time Extinction in Nonlinear Evolution Systems

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b(1) = 1/π 2 and take a(t) = b(t) if t ≤ 1 and a(t) = 1/π 2 for t ≥ 1 then we get a solution which attains the stationary state u∞ ≡ 1 after t = 1. It is possible to obtain some abstract results for the localization in a ﬁnite time of solutions of some second-order in time Cauchy problem of the type d2 u du dt2 (t) + ∂Ψ(u(t)) + ∂Φ( dt (t)) 0 in H, du u(0) = u0 , dt (0) = v0 , in the framework of the subdiﬀerential operators of convex functions Ψ and Φ on an Hilbert space H when we assume Φ(u) ≥ α u for some α > 0, 0 ∈ int ∂Φ(0) and ∂Ψ is single-valued. As before, we assume proved, as a previous step, that there exists u∞ ∈ H such that du (t) + u(t) − u∞ → 0 as t → +∞. (12) dt Then, if −∂Ψ(u∞ ) ∈ int ∂Φ(0) (13) there exists t∗ ≥ 0 such that u(t) = u∞ for any t ≥ t∗ . This abstract theorem ([64]) can be applied, once we assume (13), not only to the ﬁnite-dimensional system of the vibration of N damped particles (already given in [1] and [44]) but also to the damped string equation (6) and to some viscoelastic Bingham materials leading to the multivalued equation utt − ∆u − ∆β(ut ) 0 (this time with the choice H = H −1 (Ω) as in [54]). The last remark of this section concerns the case of periodic solutions of scalar second-order in time equations as (7). Several results on solutions presenting a dead core can be found in [47], [37] and the monograph [75] where many other references can be found. On the other hand, when the friction is replaced by β(x˙ − 8 v ) containing a given excitation v8 it is possible to prove the existence of a periodic solution (a limit cycle) which is attaint in a ﬁnite time by other orbits. This behavior can be also present for damped wave equations of the form (10) [64]. 2.2. Finite extinction time for single-valued operators via semidiscretization One of the possible motivations of a diﬀerent set of results on the extinction in ﬁnite time property for single-valued operators comes from the study of one of the archetype of quasilinear partial diﬀerential operators: the p-Laplacian p−2

∆p u = div(|∇u|

∇u),

1 < p < ∞.

During some time, this operator was considered as an academic illustration of nonlinear diﬀusion operators but without any special relevant role in applied frameworks. Perhaps this was the reason (and because the often use of it in the J.L. Lions’ school literature: see, e.g., [78]) why the operator was sometimes called as the “French nonlinear Laplacian”. As we shall mention later, the situation changed

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soon and many important applications were modeled in terms of such quasilinear operator. If, to ﬁx ideas, we consider the problem ⎧ ∂u ⎪ ⎨ in Q∞ , = ∆p u ∂t (P p ) u(t, x) = 0 on Σ∞ , ⎪ ⎩ u(0, x) = u0 (x) on Ω . Very diﬀerent qualitative properties were shown for its solutions according the degenerate or singular nature of the operator. So, for instance, if p > 2 there is ﬁnite speed of propagation (i.e., if supp(u0 ) ⊂ B(0, r) ⊂⊂ Ω, then the solution of problem (P p ) satisﬁes that supp(u(t)) is a compact set for any t > 0 ([57], [58]). In contrast to that, if 1 0 or u(t) = 0 in Ω for all t > 0 ([58]). The ﬁnite time extinction of the solutions of (P p ) when 2N 2N N +2 ≤ p < 2, N ≥ 2 was proved in [28], and, for 1 < p < N +1 , in [72] (see also [5]). Problems of this type are connected to some problems in ﬂuid mechanics as, for instance, the discharge of a turbulent and perfect gas in a pipeline. Indeed, if we assume the pipeline occupying the interval (0, L) and with a section of diameter D very small in comparison with L, the hydraulic approximation leads to a system of equations for the density ρ, velocity u, pressure p and temperature T ⎧ ∂ρ ∂(ρu) ⎪ ⎪ + = 0, ⎪ ⎪ ∂t ∂x ⎪ ⎪ ⎪ ⎪ ∂u ∂p 1 ∂u ⎪ ⎪ ⎨ ρ + ρu =− − ρ |u| u, ∂t ∂x ∂x 2 (14) ⎪ ⎪ ⎪ ( ∂u + u ∂ )( γ T + 1 u2 ) − ∂p = − 1 |u| ( γ T − 1 u2 ), ⎪ ⎪ ∂t ∂x γ − 1 2 ∂t 2 γ −1 2 ⎪ ⎪ ⎪ ⎪ p ⎪ ⎩ = T. ρ Replacing the third equation (the enthalpy equation) by (

∂ γ−1 2 |u| γ ∂u + u )Ln(p/ργ ) = − (T − 1 − u ) ∂t ∂x 2 T 2γ

(15)

and assuming the initial and boundary conditions corresponding to an static initially full pipeline with one closed boundary point and other in which the discharge make take place at the pressure p0 (p0 > pa ) and temperature T0 (for any time) it was shown in ([60]) that when t >> 1/f the second and fourth equation can be simpliﬁed, by neglecting lower-order terms and using some suitable variable scales), to 0=−

∂p 1 − ρ |u| u ∂x 2

and

p = T = 1. ρ

Special Finite Time Extinction in Nonlinear Evolution Systems

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Then, from the ﬁrst equation we deduce that p satisﬁes that ⎧ 2 1/2 2 ⎪ ⎪ ∂p − ∂ ( ∂p sign ∂p ) = 0 t > 0, x ∈ (0, 1), ⎪ ⎪ ⎨ ∂t ∂x ∂x ∂x ∂p ⎪ (0, t) = 0, p(1, t) = pa t > 0, ⎪ ⎪ ⎪ ⎩ ∂x p(x, 0) = 1 x ∈ (0, 1). Notice that since u ≥ 0, making p2 − p2a := w we arrive to the doubly nonlinear parabolic problem ⎧ ∂ψ(w) ⎪ − ∆q w = 0 t > 0, x ∈ (0, 1), ⎪ ⎪ ⎨ ∂t ∂w (16) (0, t) = 0, w(1, t) = 0 t > 0, ⎪ ⎪ ∂x ⎪ ⎩ w(x, 0) = w0 x ∈ (0, 1), 1/m with ψ(w) = (w + pm which is a nondecreasing function of w. The correct a ) exponents here are m = 2 and q = 3/2 nevertheless other interesting cases are m = 7/4 and q = 11/7 (case of very polished pipes) and m = 1 and q = 3/2 (laminar regime). The existence and uniqueness of solutions of a larger class of problems of this type was the main motivation of the paper [68]. The ﬁnite extinction property (here w(x, t) ≡ pa for any t ≥ t0 and x ∈ (0, 1)) was proved in [60]. A family of more general problems, including as special case problem (16), corresponds to the following formulation, also arising in the study of the nonlinear heat equation with absorption ⎧ ∂ ⎨ ∂t u |u|γ−1 − ∆p u + |u|σ−1 u = f + div g in Q∞ , (P) u=0 on Σ∞ , ⎩ u(x, 0) = u0 (x) in Ω,

where Ω ⊂ RN is a bounded open set, γ > 0, σ > 0, 1 ≤ p < ∞, λ ≥ 0 (if p = 1, ∇ u represents the total variation). In order to study the extinction in ﬁnite time of solutions we assume that there exists T0 ≥ 0 such that f (t, .) = 0, g(t, .)= 0 in Ω, if t > T0 . Under suitable conditions, the “solution” (notion to be made precise in each case) satisfy an integral energy inequality leading to the extinction in a ﬁnite time of the function Theorem 1. Let u ∈ L1loc (T0 , +∞ : W01,p (Ω)) for some p > 1 (or u ∈ L1loc (T0 , +∞ : BV0 (Ω)), if p = 1) such that ∃ γ, k, c > 0, λ ≥ 0, σ > k − 1 for which γ+k

|u|

σ+k

, |u|

and

p

k−1

, |∇u| |u|

t

p

k−1

|∇u| |u|

y(t) + c s

∈ L1loc (T0 , +∞ : L1 (Ω)) t

Ω

≤ y(s) a.e. s, t ∈ (T0 , +∞),

σ+k

|u|

+λ tf

(17)

Ω

(18)

80 where y(t) =

J.I. D´ıaz

Ω

γ+k

|u|

dx. Assume that 1≤p<γ+1

and

λ=0

(19)

and

0 < λ,

(20)

or 1 ≤ p,

σ<γ

and let ⎧ ⎪ ⎨1 k = N −p p(N − 1) ⎪ ⎩ 1+γ − >1 p N −p

Np , N −p Np . if 1 (N − p) if N ≤ p or (γ + 1) ≤

0,α Then u ∈ Cloc ([T0 , +∞) : Lγ+k (Ω)) for some α ∈ (0, 1) and there exists a Te ∈ (T0 , +∞) such that u(t, .) ≡ 0 in Ω ∀t ≥ Te .

The proof uses an integral version of the Torricelli-Bernoulli energy inequality found in [5] (Proposition 1.1. or Theorem 2.1, Chapter 2) for p > 1 and [2] for

t p = 1. More precisely y(t) + C s y(τ )µ dt ≤ y(s), for a.e. s, t ∈ (T0 , +∞) for some µ ∈ (0, 1). Some relevant choices of the parameters γ, p, σ which provide the fulﬁllment of the above conditions are: p = 2, γ = 1 and σ < 1; σ = 1, p = 2 and γ > 1; σ = 1, γ = 1 and p < 2. Several notions of solutions are possible (for simplicity, we assume now p > 1). The “variational theory” search for solutions in the “energy space” u ∈ Lp (0, T ; 1,p 2 −1,p W01,p (Ω)), and use that (if p ≥ N2N (Ω). At least +2 ) W0 (Ω) ⊂ L (Ω) ⊂ W for k = 1, u ∈ Lp (0, T ; W01,p (Ω)), ∀T > 0 implies that |∇u|p |u|k−1 ∈ L1loc (0, +∞ : L1 (Ω)). A ﬁrst problem arises with the zero-order term |u|σ−1 u since u ∈ Lp‘ (0, T ; σ+k W01,p (Ω)) ⇒ |u| ∈ L1loc (0, +∞ : L1 (Ω)). Then, if the equation takes place in γ−1 D (Ω) the natural regularity for ut is |ut | ut ∈ Lploc (0, +∞ : W −1,p (Ω)) + L1loc (0, +∞ : L1 (Ω)). In that case the test functions must be taken in Lp (0, T ; W01,p (Ω)) ∩ L∞ (0, T : L∞ (Ω)). The existence of solutions in the above framework is due to many authors (see references in [5]) assumed |u0 |(γ−1) u0 ∈ L2 (Ω) and f, g ∈Lploc (0, +∞ : Lp (Ω)) and the, so-called, weak solution satisﬁes that u ∈ C([0, +∞) : L2 (Ω)). The regularity |u|γ+k , |u|σ+k , |∇u|p |u|k−1 ∈ L1loc (T0 , +∞ : L1 (Ω)) can be obtained by asking some extra regularity to the data (see, e.g., [36]). A nontrivial fact is the justiﬁcation of the time integration by parts formula T ! d γ γ−1 k−1 γ+k (|u| u)t , |u| u = |u(t, .)| dx dt. γ + k 0 dt Ω ∂ It could be easily justiﬁed for the case of strong solutions (i.e., ∂t u |u|γ−1 ∈ L1 (Q)) but it is know that this class of solutions are quite exceptional. More in general (but for γ = k = 1, λ = 0) this was proved in a pioneering paper by J.L. Lions [77]. See other references, in particular for γ = 1 and k = 1, in [46].

γ+k dx may be not well In some cases the extinction energy y(t) = Ω |u| 1,p 2 deﬁned for solutions u(t) ∈ W0 (Ω) ⊂ L (Ω). For instance, this is the case if

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81

γ = 1, λ = 0 and 1 ≤ p < 2N/(N + 2). Due to this diﬃculty, following to [34], it is useful to justify the energy inequality (for k = 1) by working in the space W = W01,p (Ω) ∩ Lγ+1 (Ω) if γ < p − 1 or 1 < σ ≤ p or Ω bounded, otherwise W is deﬁned as the closure of C0∞ (Ω) in the Banach space {u ∈ Lγ+1 (Ω), ∇u ∈ Lp (Ω)}, u = uγ+1 + ∇up . The existence of an energy solution (i.e., u ∈ C([0, +∞) : Lγ+1 (Ω))∩Lp (0, T : W )∩Lσ+1 ((0, T )×Ω)) for any ﬁnite T , satisfying the equation in D and with u(0, .) = u0 (.)) was proved by assuming that u0 ∈ Lγ+1 (Ω), and f + div g ∈ Lp (0, T : W ) + L(σ+1) ((0, T ) × Ω). But the notion of solution can be found out of the energy space W . Among the several types of solutions in this framework we could mention, specially, the so called mild solutions motivated by the numerical analysis and the abstract semigroup theory: given > 0 and a time discretization t0 = 0 < t1 < · · · < tn ≤ T, ti − ti−1 < , T − tn < , and given fi ∈ L∞ (Ω), w0 ∈ L∞ (Ω) we consider the implicit time-discretization, b(wi )−b(wi−1 ) − ∆p wi + λ|wi |σ−1 wi = fi in Ω, ti −ti−1 (DP ) on ∂Ω, wi = 0 where b(u) = |u| u. Notice that wi ∈ W01,p (Ω) ∩ L∞ (Ω). Now, let u0 ∈ Lγ (Ω), 1 1 f ∈ L (0, T : L (Ω)), g = 0. γ−1

Deﬁnition. A mild solution of (P) is a function u such that b(u) ∈ C([0, +∞) : L1 (Ω)), u(0, .) = u0 (.), and, for any > 0 there exists (t0 , t1 , . . . , tn , f0 , f1 , . . . , fn , w0 , w1 , . . . , wn ) satisfying (DP ) with + ti b(u0 ) − b(w0 )1 ≤ , i ti−1 f (t) − fi 1 dt ≤ and b(u(t)) − b(w0 )1 ≤ for any t ∈ (ti−1 , ti ], i = 1, . . . , n. The existence of a mild solution was due to [33]. Moreover, it was proved there that if, in addition, u0 ∈ Lγ+1 (Ω), f ∈ Lp (0, T : W ) + L(σ+1) ((0, T ) × Ω) then the mild solution is also an energy solution. Now we can study the ﬁnite extinction time for the step function w (t) := wi if t ∈ (ti−1 , ti ], i = 1, . . . , n. Deﬁnition. We say that w (t) extincts in a ﬁnite time if there exists T,e = tj , for some j ≤ n such that w (t)∞ > 0 for t ∈ [0, T,e ) and w (t)∞ = 0 for t ∈ [T,e , T ]. Since w (t) satisﬁes the integral energy inequality (17) we get to the following result (due to [32] for p = 2, λ = 0 and [72] for p > 1 and λ = 0) Corollary 1. Assume that there exists T0 = tm , m ≤ n such that f (t, .) = 0, in Ω, if t > T0 ( f (t, .) deﬁned in a similar way to w (t)). Then, under the assumption of Theorem 1 on γ, p, k, and σ, function w (t) extincts in a ﬁnite time T,e . Moreover, if u is a mild solution and assume that there exists T0 ≥ 0 such that f (t, .) = 0, g(t, .) = 0 in Ω, if t > T0 , and that u(T0 ) ∈ Lγ+k (Ω)) then u(t) extincts in a ﬁnite time Te (only dependent on u(T0 )γ+k ).

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Notice that due to the regularizing eﬀects (see [33]), it is possible to have a ﬁnite energy at time T0 (u(T0 ) ∈ Lγ+k (Ω))) even if u0 ∈ Lγ (Ω). An unpleasant fact of mild solutions is the lake of an easy characterization in terms of test functions and the lake of information on their spatial regularity. A diﬀerent notion of solutions corresponds to the so called renormalized solutions (introduced, for second-order equations, in [36]). Since the general integral energy holds, the ﬁnite extinction time phenomenon can be obtained also for such solutions assumed, again, u(T0 ) ∈ Lγ+k (Ω)). The assumption u(T0 ) ∈ Lγ+k (Ω) is, in some sense, necessary. A counterexample can be done in other case: take γ = 1, λ = 0, p = 1, assume that 0 ∈ Ω and u(0, .) = δ0 (the Dirac delta at the origin). Then, it is possible to show ([3]) that there is not any regularizing eﬀect and u(t, .) = C(t)δ0 with C(t) > 0. The extinction time also exists for other time-discretizations (now of semiimplicit type). We write (assuming now w ≥ 0) γ+1−p (wγ+1−p − wi−1 ) γ γ (wp−1 )(wγ+1−p )t ≈ (wip−1 ) i . γ+1−p γ+1−p ti − ti−1 Given > 0, a time discretization t0 = 0 < t1 < · · · < tn ≤ T, ti − ti−1 < , T − tn < , and given fi ∈ L∞ (Ω), w0 ∈ L∞ (Ω) we consider the semi-implicit time-discretization,

(wγ )t =

γ+1−p (wγ+1−p − wi−1 ) γ (wip−1 ) i − ∆p wi + λ|wi |σ−1 wi = fi in Ω. γ+1−p ti − ti−1 When fi and w0 are nonnegative, the existence and uniqueness of a nonnegative wi is consequence of the results of [67]. The convergence of the scheme was given in [76] for p = 2, λ = 0 and in [79] for p = 2 and λ = 0. We have

Corollary 2. Assume that there exists T0 = tm , m ≤ n such that f (t, .) = 0, in Ω, if t > T0 ( f (t, .) deﬁned as done for w (t)). Then, under the assumption of the Theorem 1 on γ, p, k, and σ function w (t) extincts in a ﬁnite time T,e . The existence of a ﬁnite extinction time can be also proved for another type of semi-implicit time-discretization (see [35]) b(wi ) − b(wi−1 ) − div (|∇wi−1 |p−2 ∇wi ) + λ|wi |σ−1 wi = fi in Ω. ti − ti−1 A collection of results on ﬁnite extinction time for other type of schemes in which the discretization is not in time but in space can be found in [53]. We end this section by pointing out that, some times, the ﬁnite extinction time property can be proved by many other methods that the mentioned above. For instance, it is quite usual to get it via comparison principle (see references, e.g., in [5] and the contribution by Ph. Souplet to this volume). The comparison of symmetrical rearrangements allows, sometimes, to improve pointwise criteria (see, e.g., [65]). Spectral arguments can be also used to this end ([31]). Finally, we mention that the ﬁnite extinction time property is clearly connected with the question of the controllability to zero property in control theory (see, for instance, [66]).

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3. Extinction by components In this section we present an unpublished result by H. Amann and the author [26], lost for sometime among the ﬁles of both authors but found by one of them recently (joking, almost as the case of the 1519 Leonardo’s manuscript about the anatomy lost since 1630 discovered, by chance, on 1966 in the archives of the Spanish National Library at Madrid). The main goal of this study was to exhibit some nonlinear system for which only one of the two components satisﬁes the ﬁnite extinction time property. One of such systems can be formulated as follows ⎧ ut − ∆u = f (t, x) in Q∞ , ⎪ ⎪ ⎨ ut + uν + β(u) g(t, x) on Σ∞ , u(0) = uΩ in Ω, ⎪ 0 ⎪ ⎩ u(0) = uΓ0 in Γ. Here Ω denotes a convex bounded domain in RN and β : D(β) → P(R) is the Signorini maximal monotone graph ⎧ φ if r < 0, ⎨ (−∞, 0] if r = 0, β(r) = ⎩ 0 if r > 0, ∞ Γ ∞ uΩ 0 ∈ L (Ω), u0 ∈ L (Γ) and uν is the conormal derivative of u at points of Γ. The existence of solutions is well known in the literature, specially, after the deep contributions by H. Amann and collaborators (a large list of references can be found, for instance, in [30]). The dynamics of the problem becomes quite peculiar when β is not Lipschitz continuous near the origin.

Theorem 2. Assume that f (t, x) ≤ −ε2 near ∂Ω for t ≥ tf , g(t, x) ≤ −ε2 on ∂Ω × (tg , +∞). Then the trace u(t, ·) on Γ vanishes after a ﬁnite time t0 . The proof uses the explicit construction of a local barrier function of the form u (t, x) = Φ(t − t0 ) + C |x − x0 |2 for any x0 ∈ ∂Ω and for some suitable Φ, t0 and C (see some related coincidence results in [55] and [56]). Notice that we can reformulate the problem in terms of a vectorial abstract problem: take the Hilbert spaces V and H, with V = {(u, v) ∈ H 1 (Ω) × H 1/2 (Γ); u|Γ = v} which is a real separable Hilbert space isomorphic to H 1 (Ω) where the latter is endowed with the equivalent norm 1/2 uH 1 (Ω) = ∇u2L2 (Ω) + u|Γ 2L2 (Γ) , H = L2 (Ω) × L2 (Γ) which endowed with the usual inner product (u, v), (˜ u, v˜) = u, u ˜L2 (Ω) + v, v˜L2 (Γ) , is a real Hilbert space. Let us deﬁne A : V → V ∗ by (A(u, v), (ϕ, ψ)) = ∇u · ∇ϕ dx, Ω

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where (·, ·) is the usual pairing between V and V ∗ . We deﬁne the restriction AH : D(AH ) ⊂ H → H of A to H by D(AH ) = {(u, v) ∈ V ; A(u, v) ∈ H} and AH (u, v) = A(u, v), for each (u, v) ∈ D(AH ). It is easy to see that D(AH ) = {(u, v) ∈ L2 (Ω) × L2 (Γ); ∆u ∈ L2 (Ω), uν ∈ L2 (Γ), u|Γ = v} and AH (u, v) = (−∆u, uν ). Then the solution is associated to the vector U (t, .) = (u(t, ·), u(t, ·)|Γ ) ∈ L2 (Ω) × L2 (Γ) can be formulated as d dt U (t, .)

+ AU (t, .) + BU (t, .) F (t, .) U (0, .) = U0

with BU (.) = (0, β(u)) and the above theorem shows that the component u(t, ·)|Γ vanishes in a ﬁnite time although, by the strong maximum principle for the linear heat equation, u(t, ·) cannot vanish in any subset of positive measure of Ω for any time t > 0. Another nonlinear system for which only one of the two components satisﬁes the ﬁnite extinction time property is the hybrid problem ⎧ vtt (t, x) − vxx (t, x) = 0, in Q∞ , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ t > 0, ⎨ v(t, 0) = 0, v(t, 1) = z(t), (21) v(0, ·) = 0, vt (0, ·) = 0, in Ω, ⎪ ⎪ ⎪ ⎪ ˙ 0, z¨(t) + αvx (t, 1) + z(t) + β(z) ⎪ ⎪ ⎩ z(0) = 0, z(0) ˙ = 1, where Ω = (0, 1) and β is given by (5). This system corresponds, for instance, to the modelling of a point block which lies on a rough surface and is connected to a ﬁxed support by a spring and one end of a ﬂexible string is attached to the block, the other end is ﬁxed above the surface so that the string is tense. For small horizontal displacements v(t, x) the dynamics of the string are governed by the wave equation and z(t) denotes the displacement of the block which is assume to be in contact, without any lubrication with a surface, so dry friction occurs (see [38]).

4. Finite extinction time for a ﬁnite set of orbits A very special dynamics arise in the case of the damped oscillator α−1

m¨ x + µ |x| ˙

x˙ + kx = 0,

(22)

when α ∈ (0, 1). Here µ and k > 0 are ﬁxed parameters. In fact we can simplify the above formulation to x¨ + |x| ˙ α−1 x˙ + x = 0, (23) 1/(α−1) x(λt) where by dividing by k and by introducing the rescaling x (t) = β √ √ λ = m/ k and β = µ/(k (2−α)/2 mα/2 ). Notice that the x-rescaling fails for the linear case α = 1 since there is not any deﬁned scale √ for x and the equation is merely reduced to x ¨ + β x˙ + x = 0 with β = µ/( km), a parameter which

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characterizes the dynamics. Notice also that the limit case α → 0 corresponds to the Coulomb friction equation (7). α−1 We recall that, even if the nonlinear term |x| ˙ x˙ is not a Lipschitz continuous function of x, ˙ the existence and uniqueness of solutions of the associate Cauchy problem α−1 x¨ + |x| ˙ x˙ + x = 0 t > 0, (Pα ) ˙ = v0 x(0) = x0 , x(0) is well known in the literature: see, e.g., Brezis [39]. The asymptotic behavior, for t → ∞, of solutions of the Coulomb and linear problems (P0 ) and (P1 ) (limit cases when α → 0 or α → 1) is well known. In the second case the decay is exponential. In the ﬁrst one, as mentioned in Section 2, given x0 and v0 there exist a ﬁnite time T = T (x0 , v0 ) and ζ ∈ [−1, 1] such that x(t) ≡ ζ for any t ≥ T (x0 , v0 ). When ˙ → (0, 0) α ∈ (0, 1) it is well known that the solutions of (Pα ) verify that (x(t), x(t)) as t → ∞ (see, e.g., Haraux [70]). In a series of papers ([61], [62] and [25]) it was shown that the generic asymptotic behavior above described for the limit case (P0 ) is only exceptional for the sublinear case α ∈ (0, 1) since the generic orbits (x(t), x(t)) ˙ decay to (0, 0) in a inﬁnite time and only two one-parameter families of them decay to (0, 0) in a ﬁnite time: in other words, when α → 0 the exceptional behavior becomes generic. In order to present the main arguments of those papers, we can start with some formal results via asymptotic arguments. We can rewrite the equation (23) as the planar system x˙ = y (24) α−1 y y˙ = −x − |y| which, by eliminating the time variable, for y = 0, leads to the diﬀerential equation of the orbits in the phase plane −x − |y|α−1 y (25) y and that allows us to carry out a phase plane description of the dynamics. The plane phase is antisymmetric since if y = σ(x) is a solution of (25) then the function y = −σ(−x) is also solution. So, it is enough to describe a semiplane (for instance x ≥ 0). By multiplying by x and y, respectively, we get that α+1 (x2 + y 2 )t = 2 |y| and (1/x, 1/y) satisfy a system which has the point (0, 0) as a spiral unstable point. For values of x2 + y 2 >> 1 the orbits of the system are given, α−1 y is small compared with in ﬁrst approximation, by x2 + y 2 = C because |y| x. The eﬀect of this term is to decrease slowly C with time giving the trajectory a spiral character. Notice that for α = 1 the character of the trajectories close to the origin depends on the parameter β: for β > βc := 2 the origin is a stable mode and for β < βc is a stable spiral (underdamped oscillations). As we shall see, there are two modes of approach to the origin and so that the origin (0, 0) is a node for the system (24). The lines of zero slope are given by yx =

α−1

−x = |y|

y.

(26)

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So the convergence to (0, 0) is only possible through the regions {(x, y) : x > 0, y < −x1/α } ∪ {(x, y) : x < 0, y > (−x)1/α }. Let us see (formally) that the “ordinary” mode corresponds to orbits that are very close to the ones corresponding to small eﬀects of the inertia. Due to the symmetry it is enough to describe this behavior for the orbits approaching the origin with values of x > 0 and y < 0. Let −y = y > 0. Equation (25) takes the form y y x = −x + y α .

(27)

The line of zero slope is y = x1/α and we search for orbits obeying, for 0 < x << 1, to the expression y = x1/α + z(x) for some function z(x). If we anticipate the 1 condition 0 < z(x) << x1/α , equation (25) takes the “linearized form” α1 x( α −1) z + 1 1 x α zx −αx(1− α ) z = 0. Thus the ﬁrst term can be neglected, compared with the last 2(1−α) one, and then the solution can be written as z(x) ∼ C exp{−[α2 /2(1−α)]x− α } with C an arbitrary constant (which explain the name of “ordinary” orbits). This type of orbits are given, close to the origin, by the approximate equation (26), which for the orbits that reach the origin from below implies that y ∼ x1/α ∼ − dx dt and so, integrating the simpliﬁed equation dx = −x1/α dt

(28)

we get that x(t) ∼ [

α ]α/(1−α) (1 − α)(t + t1 )

(29)

and so that it takes an inﬁnite time to reach the origin. Some diﬀerent orbits approaching the origin can be found by searching among 1/α solutions with large values of |y| compared with |x| . Thus, close to the origin, the orbits with negative y are “very close” to the solutions of the equation found by replacing (27) by the simpliﬁed the equation y y x = y α

(30)

corresponding to a balance of inertia and damping. The solution ending at the origin ( y (0) = 0) is given by y (x) = −{(2 − α)x}1/(2−α) .

(31)

Notice that it involves no arbitrary constant. So this curve is unique (a symmetric curve arises for y > 0 and x < 0) which justiﬁes the term of “extraordinary” orbit. The time evolution of this orbit is given, for x << 1, by integrating the equation − and so x(t) =

dx = [(2 − α)x]1/(2−α) dt

1 (2 − α)(1 − α) [ (t0 − t)+ ](2−α)/(1−α) , (2 − α) 2α

(32)

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where, in general, h(t)+ = max{0, h(t)}. This indicates that the motion (of this approximated solution) ends at a ﬁnite time, t0 , determined by the initial conditions which, by (32) must satisfy that v0 ∼ ±[(2 − α) |x0 |]1/(2−α) . We point out that the two exceptional orbits emanating from the origin spiral around the origin when x2 + y 2 grows toward inﬁnity and so each of them is a separatrix curve in the phase plane. Notice that due to the autonomous nature of the equation, if x(t) is the solution of the Cauchy problem (Pα ) of initial data (x0 , v0 ) then for any parameter τ ≥ 0 the function x (t) := x(t + τ ) coincides with the solution of (Pα ) of initial data (x(τ ), x(τ ˙ )). In this way, the above extraordinary orbits give rise to two curves of initial data for which the corresponding solutions of (Pα ) vanish after a ﬁnite time. In order to make more rigorous the arguments on the extraordinary orbits we can use a ﬁxed point argument to show that there exists two curves Γ+ and Γ− of initial data (x0 , v0 ) for which the solutions x(t) of the corresponding Cauchy problem (Pα ) vanish after a ﬁnite time.(see [61]). Moreover, we can obtain some additional results on these two curves ([60]): (i) Near the origin the curves Γ+ and Γ− can be represented by two functions, y = ψ+ (x) and y = ψ− (x), solutions of the equation (25), where ψ+ : [0, ε] → (−∞, 0] and ψ− : [−ε, 0] → [0, +∞), for some ε > 0. (ii) Functions ψ+ and ψ− , satisfy that ψ± (0) = 0 ε 0 ds ds −∞ < and < +∞. ψ (s) ψ + 0 −ε − (s)

(33)

(x) ↓ −∞ when x ↓ 0 and ψ− (x) ↑ +∞ when x ↑ 0. In particular, ψ+

(iii) We have 1

−Cx 2−α 1 (−x) α

1

≤ ψ+ (x) ≤ −x α for x ∈ [0, ε] and 1 ≤ ψ− (x) ≤ C(−x) 2−α for x ∈ [−ε, 0],

for some C > 0. 1

1

(iv) There exists a xs ∈ (0, ε] such that ψ+ (xs ) = −(xs ) α and (−xs ) α = ψ− (xs ). 1 Moreover the regions D+ := {(x, y): x ∈ [0, xs ] and ψ+ (x) ≤ y ≤ −x α }, 1 D− := {(x, y): x ∈ [−xs , 0] and (−x) α ≤ y ≤ ψ− (x)} are time invariants for equation (25). Curiously, the rigorous proof of the other decay to zero (now in an inﬁnite time) is harder than for the exceptional case. Although an invariant subset M (such that each orbit meeting M needs inﬁnite time to reach the rest) was given in [60], a more careful study of this case was carried out in [25] for an equation slightly more general x ¨ + ϕ(x) ˙ + x = 0, (34) where the friction function ϕ is assumed now to be continuous on R and locally Lipschitz continuous on R{0}, odd, positive for positive arguments and such that

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ϕ(η) = aη α for 0 < η ≤ ε, for some a, ε > 0 and α ∈ (0, 1). In addition we assume that 1 (35) lim ϕ(η)/η < . η→∞ 2 α−1

It is clear that the above case ϕ(η) = |η| η is included as well as some nonmonotone functions such as ⎧ 0 ≤ η ≤ ε, ⎨ aη α , 1 + 1/η, ε < η < ∞, ϕ(η) = ⎩ −ϕ(−η), −∞ < η < 0, where a = ε−1−α + ε−α . This nonlinearity appears in the literature since the pioneering work of Lord Rayleigh (see [73], [80]). The continuous approximations of the Coulomb equation also leads often to such type of formulations. As in the previous case, is easy to see that for any R > 0, the disc ¯ R := (x, y) ∈ R2 ; x2 + y 2 ≤ R2 B is positively invariant and the origin is globally asymptotically stable. In [25] we show, by elementary techniques from the theory of ordinary diﬀerential equations, that problem (34) possesses also at least two orbits converging to the rest state in ﬁnite time. Furthermore, we give a rather detailed analysis of the ﬂow in the neighborhood of the critical point, showing that most orbits reach it in inﬁnite time only. In fact, on the basis of our results we conjecture that there are precisely two orbits converging to zero in ﬁnite time. By the oddness of ϕ, the phase portrait of x˙ = y (36) y˙ = −x − ϕ(y) is, again, invariant under the reﬂection (x, y) → (−x, to study −y). Thus it suﬃces the semiﬂow on the closure of the half-plane H := (x, y) ∈ R2 ; y > 0 , induced by restriction from the ﬂow generated by 36. Consequently, in the rest of this section all assertions on invariant regions etc. pertain to this semiﬂow. We refer to [6] for the elementary facts about semiﬂows and invariant regions. From (36) we see that every orbit meets the real axis oﬀ zero vertically. For 0 < s ≤ 1 we denote by Γs the curve Γs := (x, y) ∈ H ; −x = sϕ(y) , oriented so that y is decreasing. Then every orbit in H meets Γ := Γ1 horizontally, and the direction ﬁeld has a positive x-component and a negative y-component on the right of Γ. Thus no orbit approaches the origin through the ﬁrst quadrant. Furthermore, given any η > 0, (x, y) ∈ H ; −x ≤ ϕ(y), y ≤ η is positively invariant.

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We denote by C(x, y) the solution curve of (36) passing through (x, y) ∈ H and being oriented in positive time direction. It is not diﬃcult to prove C(x, y) meets the positive y-axis whenever (x, y) ∈ Γs for some s ∈ (0, 1]. In order to built a sharper invariant region, for small values of s > 0, we deﬁne the functions k1 (s) := −sϕ(k2 (s)) and k2 (s) := λ(s)β with

1 (1 − α). 2 Then we deﬁne the oriented curve K := Kε in H− by means of the parametrization (0, sε ] → H− , s → k(s) := k1 (s), k2 (s) , λ(s) := αa2 s(1 − s) and β :=

for some suitable sε > 0. Then, by construction k(s) ∈ Γs ,

0 < s ≤ sε ,

and the positive unit tangent vectors to C(x, y) and Γs at (x, y) = k(s) coincide for 0 < s ≤ sε . In the following picture we depict H− in the neighborhood of the origin together with K, Γ, and Γs for three values of s, as well as seven orbits for problem (36) with α = 0.4 and ε = 1.2.

_½

½

|

Then we prove in [4] that

k1 (sε ), η ; 0 < η ≤ k2 (sε ) and K, is a) the region M in H− , bounded by positively invariant, b) there exists a positive semiorbit in H− reaching the origin in ﬁnite time and c) every orbit meeting M needs inﬁnite time to reach the origin.

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The main theorem of [4] gives a rather precise picture of the ﬂow on R2 generated by (36) Theorem 3. The origin is the only critical point of the phase ﬂow of (36), and it ¯ R is is globally asymptotically stable. More precisely, given any R > 0, the disc B positively invariant and every orbit converges to the rest state. The phase portrait is 8 symmetric with respect to the origin. There exist real numbers satisfying 0 < x ≤ x with the following: (i) Every orbit meets [−8 x, x 8 ] × {0}. (ii) If an orbit meets [−8 x, 0) × {0}, resp. (0, x 8 ] × {0}, then it stays in H− , resp. (−H)+ := (x, y) ∈ R2 ; x > 0, y < 0 . If an orbit meets (−∞, −8 x ) × {0}, resp. (8 x, ∞) × {0}, then it leaves H− , + resp. (−H) , after ﬁnite time. (iii) The orbits through (−8 x, 0) and (0, x 8 ) reach the rest point in ﬁnite time. % & (iv) Every orbit meeting (−x, 0) ∪ (0, x) × {0} needs inﬁnite time to reach the critical point. In the next picture a portion of the phase plane of problem (36) is depicted where ϕ is given as in (35) with α = 0.4 and ε = 1.2.

_½

¾ |

In the mentioned paper it was conjectured that there are precisely two orbits reaching the origin in ﬁnite time. This was proved in the paper V´ azquez [82] where other related equations were also considered.

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5. Final conclusions The methods and results of most of the results presented here have many common points with the exposition made by H. Amann ([24]) in occasion of his nomination as Academico Extranjero de la Real Academia de Ciencias de Espa˜ na. At the Introduction of his lecture he says: “In the following, I shall try to give an idea of some of my research interests of the last twenty years. During that period I was predominantly concerned with a functional-analytical approach to parabolic evolution equations. In my opinion, functional analysis, combined with so-called “hard analysis” and many other mathematical subjects, is particularly well suited for providing an abstract, powerful, and suﬃcient general framework for the study of nonlinear partial diﬀerential equations.” Of course, it is well known that there are intimate connections between functional analysis and partial diﬀerential equations. In fact, large parts of linear functional analysis have been developed in order provide the abstract tools for an eﬃcient and uniﬁed study of linear partial diﬀerential equations. The point I want to make is that functional analysis is also very useful for the investigation of nonlinear diﬀerent equations. The central exposition was concerning some nonlinear systems for which he explained some of his many works (he mentioned, in particular, [7], [8], [10], [11], [11], [13], [14], [15], [9], [12], [16], [17], [19], [20], [21], [22], [23] among them. At the end of his paper he included some ﬁnal remarks: a splendid sample of the beauty and depth reached by the Mathematical Analysis and the Applied Mathematics of the last third of the last century and beginnings of the present one thanks to his immense work. “In order to summarize and to give a somewhat broader view I discuss now some of the interrelationships of the theory of parabolic evolution equations with other ﬁelds of analysis, as indicated in the following diagram.

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I have put parabolic evolution equations in the middle, since they are in the center of my present interest, and have grouped around them several other subjects. I did not put arrows on the connecting lines since in many cases the interaction is bilateral. Let us start at the left upper corner. It is well known – and I have taken reaction-diﬀusion systems as an example – that many concrete models for the understanding of phenomena in science lead to parabolic evolution equations and, vice versa, results on parabolic evolution equations have immediate interpretations and consequences for those models. Partial diﬀerential equations are, of course, intimately connected with parabolic evolution equations. However, neither forms a subﬁeld of the other. For example, parabolic evolution equations encompass also other systems like integrodiﬀerential equations or inﬁnite systems of reaction-diﬀusion equations involving even uncountably many unknowns, as they occur in statistical physics (see [15], [23]). The connection between parabolic evolution equations and Fourier analysis lies on a more technical level and can be described adequately by more detailed explanations only. As pointed out earlier, the choice of the correct state space is fundamental when studying partial diﬀerential equations, parabolic evolution equations in particular. The well-developed theory of function spaces provides us with a wide variety of possibilities. Spaces more reﬁned than integer order Sobolev spaces like Besov and Bessel potential spaces have become increasingly important during the last years. This is true, in particular, in the study of the Navier-Stokes equations (cf. [16], [19], [20], and the references therein). Interpolation theory provides us, on a more abstract level, with the right tool for measuring very precisely regularity properties which are the key to a successful approach to nonlinear equations. As I have explained, semigroup theory is precisely what is needed – on the abstract level – to derive the most general local existence theory for parabolic evolution equations. Spectral theory comes in when one starts to study stability questions and the long-time behavior. I could not into detail on the relation between parabolic evolution equations and inﬁnite-dimensional harmonic analysis. Among other things, it has to do with “maximal regularity” questions and has, in particular during the last few years, stimulated much research in Banach space theory. I only want to mention recent results of Weis [83], Kalton [74], and others on Fourier multiplier theorems with operator-valued symbols in vector-valued Lp-spaces. Those results are tied to the theory of UMD spaces, for example. (Also see [48], [49] and, for results valid for arbitrary Banach spaces, [16], [17].) Finally, methods from nonlinear functional analysis, ﬁxed point theorems, bifurcation theory, etc., play an important role in the diﬃcult and fascinating investigation of qualitative properties of the semiﬂows generated by parabolic evolution equations.

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I hope that this enumeration of subjects, which is far from being complete, shows that the ﬁeld of parabolic evolution equations is a fascinating one, invoking a lot of deep and beautiful mathematics.” Acknowledgments Research partially supported by project REN2003-0223-C03 of the DGISGPI (Spain) and RTN HPRN-CT-2002-00274 of the EC.

References [1] S. Adly, H. Attouch and A. Cabot, Finite time stabilization of nonlinear oscillators subject to dry friction, to appear in Progresses in Nonsmooth Mechanics and Analysis (edited by P. Alart, O. Maisonneuve and R.T. Rockafellar), Advances in Mathematics and Mechanics, Kluwer. [2] F. Andreu, V. Caselles, J.I. D´ıaz and J.M. Maz´ on, Some Qualitative properties for the Total Variation, Journal of Functional Analysis, 188, 516–547, 2002. [3] F. Andreu, V. Caselles, J. M. Maz´ on and S. Moll, The Minimizing Total Variation Flow with Measure Initial Conditions, To appear in ARMA. [4] S.N. Antonsev and J.I. Diaz, New results on space and time localization of solutions of nonlinear elliptic or parabolic equations via energy methods, Soviet Math. Dokl. 303 524–528 (1988). [5] S.N. Antontsev, J.I. D´ıaz and S.I. Shmarev, Energy Methods for Free Boundary Problems: Applications to Nonlinear PDEs and Fluid Mechanics, Progress in Nonlinear Diﬀerential Equations and Their Applications, 48, Birkh¨ auser, Boston, 2002. [6] H. Amann, Ordinary Diﬀerential Equations, Walter de Gruyter, Berlin, 1990. [7] H. Amann, Dynamic theory of quasilinear parabolic equations - II. Reaction-diﬀusion systems. Diﬀ. Int. Equ., 3, 1990, 13–75. [8] H. Amann, Hopf bifurcation in quasilinear reaction-diﬀusion systems. In S. Busenberg, M. Martelli, editors, Delay Diﬀerential Equations and Dynamical Systems, Proceedings, Claremont 1990, pages New York, Lecture Notes in Math. #1475. Springer Verlag. 1991, 53–63. [9] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary valu´e problems. In H.J. Schmeisser, H. Triebel, editors, Function Spaces, Diﬀerential Operators and Nonlinear Analysis, Teubner, Stuttgart, Leipzig. 1993, 9–126. [10] H. Amann, Stability of the rest state of a viscous incompressible ﬂuid. Arch. Rat. Mech. Anal, 126, 1994, 231–242. [11] H. Amann, Heat-conducting incompressible viscous ﬂuids. In A. Sequeira, editor, Navier-Stokes Equations and Related Non-Linear Problems, Plenum Press, New York, 1995, 231–243. [12] H. Amann, Linear and Quasilinear Parabolic Problems, Volume I: Abstract Linear Theory. Birkh¨ auser, Basel. 1996. [13] H. Amann, Stability and bifurcation in viscous incompressible ﬂuids. Zapiski Nauchn. Seminar. POMI, 233, 1996, 9–29.

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[14] H. Amann, Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications. Math. Nachr., 186, 1997, 5–56. [15] H. Amann, Coagulation-fragmentation processes. Arch. Rat. Mech. Anal, 151, 2000, 339–366. [16] H. Amann, On the strong solvability of the Navier-Stokes equations. J. Math. Fluid Mech., 2, 2000, 16–98. [17] H. Amann, Elliptic operators with inﬁnite-dimensional state spaces. Evol. Equ., 1, 2001, 143–188 [18] H. Amann, Linear parabolic problems involving measures.Rev. R. Acad. Cien. Serie A Matem. RACSAM, 95, 2001, 85–119 (http://racefyn.insde.es/Publicaciones/ racsam/art´ıculos/racsam%2095 1/Amann.pdf) [19] H. Amann, Nonhomogeneous Navier-Stokes equations in spaces of low regularity. Quaderni di matem´ atica, 12 (2002), 13–31. [20] H. Amann, Nonhomogeneous Navier-Stokes equations with integrable low-regularity data. In Nonlinear Problems in Mathematical Physics and Related Topics II. In Honor of Professor O.A. Ladyzhenskaya (M.S. Birman, St. Hildebrandt, V.A. Solonnikov, N.N. Uraltseva, eds.), Kluwer/Plenum, in print 2002. [21] H. Amann and P. Quittner, Elliptic boundary valu´e problems involving measures: existence, regularity, and multiplicity. Advances in Diﬀ. Equ., 3, 1998, 753–813. [22] H. Amann and P. Quittner, Semilinear parabolic equations involving measures and low-regularity data, Trans. AMS, 356 (2004), 1045–1119. [23] H. Amann, and F. Weber, On a quasilinear coagulation-fragmentation model with diﬀusion. Adv. Math. Sel. Appl., 11, 2001, 227–263. [24] H. Amann, Abstract methods in diﬀerential equations, Rev. R. Acad. Cien. Serie A Matem. RACSAM, 97 (1), 2003, 89–105, (http://racefyn.insde.es/Publicaciones/ racsam/art´ıculos/racsam%2097 1/2002-20-Amann.pdf) [25] H. Amann and J.I. Diaz, A note on the dynamics of an oscillator in the presence of strong friction, Nonlinear Anal. 55, 209–216 (2003) [26] H. Amann and J.I. D´ıaz, Unpublished. Madrid, October 1988. [27] L. Alvarez and J.I. D´ıaz, The waiting time property for parabolic problems trough the nondiﬀusion of the support for the stationary discretization problems. Rev. R. Acad. Cien. Serie A Matem. RACSAM, 97 (1), 2003, 83–88. (http://racefyn.insde.es/ Publicaciones/racsam/art´ıculos/racsam%2097 1/2002-7-Alvarez-Diaz.pdf) [28] A. Bamberger, Etude d’une ´equation doublement non lin´eaire,J. Funct. Anal. 24 148–155 (1977). [29] A. Bamberger and H. Cabannes, Mouvements d’une corde vibrante soumise ` a un frottement solide, C. R. Acad. Sc. Paris, 292, (1981), 699–702. [30] I. Bejenaru, J.I. D´ıaz and I.I. Vrabie, An abstract approximate controllability result and applications to elliptic and parabolic systems with dynamical boundary conditions, Electr. J. Diﬀ. Eqns. 2001 (No. 50), 2001, 1–19. [31] Y. Belaud and J.I. D´ıaz: Finite extinction time for solutions of some parabolic equations with a singular term. Lecture at Nonlinear Partial Diﬀerential Equations describing Front Propagation and other Singular Phenomena, Leiden, 8–12 November, 2004, and paper in preparation.

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[32] Ph. B´enilan and M. Crandall, The continuous dependence on ϕ of solutions of ut − ∆ϕ(u) = 0, Indiana Univ. Math. J., 30, 1981, 161–177. [33] Ph. Benilan and P. Wittbold, On mild and weak solutions of elliptic-parabolic problems, Adv. Diﬀ. Eq., 1, 1996, 1053–1073. [34] F. Bernis, Finite speed of propagation and asymptotic rates for some nonlinear higher-order parabolic equations with absorption, Proc. Roy. Soc. Edinburgh Sect. A, 104, 1986, 1–19. [35] R. Bermejo, J.I. D´ıaz and L. Tello, Mathematical and Numerical Analysis of a Nonlinear Diﬀusive Climatological Energy Balance Model, to appear. [36] L. Boccardo, D. Giachetti, J.I. D´ıaz and F. Murat, Existence of a solution for a weaker form of a nonlinear elliptic equation, in Recent Advances in Nonlinear Elliptic and Parabolic Problems, Pitman Res. Notes Math., Ser. 208, Longman Sc. Tech., Harlow, (1989), 229–246. [37] D. Bothe, Periodic solutions of non-smooth friction oscillations, Z. angew. Math. Phys. 50, 1990, 779–808. [38] D. Bothe and J.I. D´ıaz. Article in progress. [39] H. Br´ezis, Probl`emes unilat´eraux, J. Math. Pures Appl. 51, 1–168 (1972). [40] H. Brezis, Op´erateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert, North-Holland, Amsterdam 1972. [41] H. Brezis, Monotone operators, nonlinear semigroups and applications. Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 2, Canad. Math. Congress, Montreal, Que., 1975, 249–255. [42] H. Cabannes, Mouvement d’une corde vibrante soumise `a un frottement solide, C. R. Acad. Sci. Paris S´ er. A-B 287 (1978), 671–673. [43] H. Cabannes, Study of motions of a vibrating string subject to a solid friction, Math. Meth. in the Appl. Sci., 3, 1981, 287–300. [44] A. Cabot, Stabilization of oscillators subject to dry friction: ﬁnite time convergence versus exponential decay results, to appear. [45] J.M. Carlson and J.S. Langer, Properties of Earthquakes Generated by Fault Dynamics, Physical Review Letters, 62, 1989, 2632–2635. [46] J. Carrillo and P. Wittbold, Uniqueness of renormalized solutions of degenerate elliptic-parabolic problems, J. Diﬀ. Eq., 156, 1999, 93–121. [47] K. Deimling, G. Hetzer and W.X. Shen, Almost periodicity enforced by Coulomb friction, Adv. Diﬀerential Equations, 1, 1996, 265–281. [48] R. Denk, G. Dore, M. Hieber, J. Pr¨ uss and A. Venni, New thoughts on old results of R.T. Seeley, preprint. 2002. [49] R. Denk, G. Dore, M. Hieber and J. Pr¨ uss, R-boundedness, Fourier multipliers and problems of elliptic and parabolic type, preprint. 2001. [50] J.I.D´ıaz, Anulaci´ on de soluciones para operadores acretivos en espacios de Banach. Aplicaciones a ciertos problemas parab´ olicos no lineales. Rev. Real. Acad. Ciencias Exactas, F´ısicas y Naturales de Madrid, Tomo LXXIV, 865–880, 1980. [51] J.I. D´ıaz, Nonlinear Partial Diﬀerential Equations and Free Boundaries. Research Notes in Mathematics no 106, Pitman, London 1985.

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[52] J.I. D´ıaz, Qualitative study of the total variation ﬂow and a related formulation for Bingham ﬂuids. Lecture at the Universit´e Pierre et Marie Curie, June, 13, 2001. [53] J.I. D´ıaz, Energy methods for free boundary problems: new results and some remarks on numerical algorithms. To appear in Esaim Proc. [54] G. D´ıaz and J.I. D´ıaz Finite extinction time for a class of nonlinear parabolic equations. Comm. Part. Diﬀ. Equat. 4, No. 11, 1213–1231, 1979. [55] J.I. D´ıaz and R. Jimenez, Behaviour on the boundary solutions of parabolic equations with nonlinear boundary conditions: the parabolic Signorini problem. Lecture at the VIII CEDYA, Univ. de Santander, 1986. Appeared in An´ alisis no lineal, R.F. Jim´enez and H. Mennickent Eds., Concepci´ on (Chile), 69–80, 1986. [56] J.I. D´ıaz and R. Jim´enez, Boundary behaviour of solutions of the Signorini problem: The elliptic case, Bolletino dell’Unione Matem´ atica Italiana, 7, No. 2 B, 1988, 127– 139. [57] J.I. D´ıaz and M.A. Herrero, Properti´es de support compact pour certaines ´equations elliptiques et paraboliques non lin´eaires, C. R. Acad. Sci. Paris S´er. I Math. 286, 815–817, 1978. [58] J.I. D´ıaz and M.A. Herrero, Estimates on the support of the solutions of some nonlinear elliptic and parabolic problems, Proc. Roy. Soc. Edinburgh, 89A, 249–258, 1981. [59] J.I. D´ıaz and G. Hetzer, Article in preparation. [60] J.I. D´ıaz and A. Li˜ na ´n, Movimiento de descarga de gases en conductos largos: modelizaci´ on y estudio de una ecuaci´ on doblemente no lineal. In the book Reuni´ on Matem´ atica en Honor de A. Dou (J.I. D´ıaz and J.M. Vegas eds.) Universidad Complutense de Madrid, 1989. [61] J.I. D´ıaz and A. Li˜ n´ an, On the asymptotic behavior of solutions of a damped oscillator under a sublinear friction term: from the exceptional to the generic behaviors. In.the Proceedings of the Congress on nonlinear Problems (Fez, May 2000), Lecture Notes in Pure and Applied Mathematics (A. Benkirane and A. Touzani. eds.), Marcel Dekker, New York, 2001, 163–170 . [62] J.I. D´ıaz and A. Li˜ n´ an, On the asymptotic behaviour of solutions of a damped oscillator under a sublinear friction term, Rev. R. Acad. Cien.Serie A Matem. (RACSAM), 95, 2001, 155–160. (http://racefyn.insde.es/Publicaciones/racsam/comunicaciones/ vol95 1/DiazLinan.pdf) [63] J.I. D´ıaz and V. Millot, Coulomb friction and oscillation: stabilization in ﬁnite time for a system of damped oscillators. CD-Rom Actas XVIII CEDYA / VIII CMA, Servicio de Publicaciones de la Univ. de Tarragona 2003. [64] J.I. D´ıaz and V. Millot, Article in preparation. [65] J.I.D´ıaz and J. Mossino, Isoperimetric inequalities in the parabolic obstacle problems. Journal de Math´ematiques Pures et Applique´es, Vol. 71, 233–266. 1992. [66] J.I. D´ıaz, A.M. Ramos, Some results about the approximate controllability property for quasilinear diﬀusion equations, Comptes Rendus Acad. Sci. Paris, 324, S´erie I, 1997, 1243–1248. [67] J.I. D´ıaz and J.E. Saa, Existence et unicit´e de solutions positives pour certaines ´equations elliptiques quasilin´eaires, Comptes Rendus Acad. Sc. Paris, t. 305, S´erie I, 1987, 521–524.

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[68] J.I. D´ıaz and F. de Thelin, On a nonlinear parabolic problems arising in some models related to turbulence ﬂows, SIAM J. Math. Anal. 25, 1994, 1085–1111. [69] G. Duvaut and J. L. Lions, Les In´equations en M´ecanique et en Physique, Dunod, Paris, 1972. [70] A. Haraux , Comportement ` a l’inﬁni pour certains syst`emes dissipatifs non lin´eaires, Proc. Roy. Soc. Edinburgh, Sect. A 84A, 1979, 213–234. [71] A. Haraux, Syst`emes dynamiques et applications, Masson, Paris, 1991. [72] M. Herrero and J.L. V´ azquez, Asymptotic behaviour of the solutions of a strongly nonlinear parabolic problem, Ann. Fac. Sci. Toulouse 3, 1981, 113–127. [73] D.W. Jordan and P. Smith, Nonlinear Ordinary Diﬀerential Equations, (Second Edition), Clarendon Press, Oxford 1979. [74] N. Kalton, Applications of Banach space theory to sectorial operators. In Recent progress in functional analysis (Valencia, 2000), Amsterdam. North-Holland Math. Stud. 189, North-Holland, 2001, 61–74. [75] M. Kunze, Non-Smooth Dynamical Systems, Lecture Notes in Mathematics 1744, Springer, Berlin, 2000. [76] M.N. Le Roux, Semi-discretization in Time of a Fast Diﬀusion Equation, Journal of Mathematical Analysis and Applications, 137, 1989, 354–370. [77] J.L. Lions, Quelques remarques sur les ´equations diﬀerentielles op´erationnelles du 1er ordre, Rend. Sem. Math. Padova, 33, 1963, 213–225. [78] J.L. Lions, Quelques M´ethodes de R´ esolution des Probl`emes aux Limites non Lin´eaires, Dunod, Gauthier-Villars, Paris 1969. [79] E. Melendez, Estudio num´ erico de una ecuaci´ on parab´ olica doblemente no lineal con datos mixtos de contorno: Propiedad de extinci´ on en tiempo ﬁnito de su soluci´ on. T´esis de Licenciatura, Univ. Complutense de Madrid, 1990. [80] J.W. Rayleigh, B. Strutt, The theory of sound, Dover Publications, New York, 2d ed., 1945. ¨ [81] B. Riemann: Uber die Fl¨ ache vom kleinsten Inhalt bei gegebener Begrenzung, Abh. K¨ onigl. Ges. d. Wiss. G¨ ottingen, Mathem. Cl. 13, 3–52, 1867. [82] J.L. V´ azquez, The nonlinearly damped oscillator, ESAIM Control Optim. Calc. Var. 9 231–246 (2003). [83] L. Weis, Operator-valued Fourier multiplier theorems and maximal Lp -regularity. Math. Ann., 319, 2001, 735–758. J.I. D´ıaz Departamento de Matem´ atica Aplicada Facultad de Matem´ aticas Universidad Complutense de Madrid E-28040 Madrid, Spain e-mail: ji [email protected]

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 99–111 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Spectral Projections, Riesz Transforms and H ∞ -calculus for Bisectorial Operators Markus Duelli and Lutz Weis

1. Introduction For a bisectorial operator A with its spectrum in a double cone around the imaginary axis, the spectral projections P1 , P2 corresponding to the upper and lower cone in general cannot be deﬁned by the Dunford calculus and, indeed, they may d be unbounded. E.g., for A = dx on Lp (R) the projection P1 is only bounded for 1 < p < ∞ and projects Lp (R) onto the Hardy space Hp . However, the boundedness of these spectral projections is closely connected to some important inequalities in analysis. We study these questions in the framework of the H ∞ -functional calculus, which allows us to give more direct proofs for some known results, improve on them and also to discover new connections. E.g., it is a result of Dore and Venni that, for 0 ∈ ρ(A), the boundedness of these spectral projections is equivalent to the “symmetry” of the operators A and −A in the sense that for w ∈ C \ Z we have Aw x ≈ (−A)w x

for x ∈ D(Aw ) = D((−A)w ).

In Section 3 we give a diﬀerent proof using the H ∞ -functional calculus that also works for 0 ∈ / ρ(A). More importantly, the boundedness of the spectral projection is equivalent to the estimate (see Section 4) for x ∈ D(A) = D((−A2 )1/2 ), Ax ≈ (−A2 )1/2 x i.e., to the boundedness and invertibility of A(−A2 )1/2 which equals the Hilbert d transform if A = dx on Lp (R). In the case of Hilbert space and operators A with ∞ a bounded H -calculus this was ﬁrst shown in [2]. If A1 , . . . , An is a commuting family of bisectorial operators, the classical example Ai = dxd i , i = 1, . . . , n, on Lp (Rn ) motivates us to call the operators +n Ai (−A)−1/2 , where A = i=1 A2i , Riesz transforms for A1 , . . . , An . If A1 , . . . , An

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have a joint H ∞ -functional calculus one can not only show that these Riesz transforms are bounded but also obtains inequalities of the form n n E E ri Ai x ≈ (−A)1/2 x , r A A x ij i j ≈ Ax i,j=1 i=1 where (ri ), (rij ) are Bernoulli random variables, and which extend the classical inequalities n n 1/2 2 1/2 2 ∂ 2 ≈ ≈ ∆f ∂ f (−∆)1/2 f , Lp ∂xi f Lp i=1 i,j=1 ∂xi ∂xj Lp

Lp

for f ∈ Lp (R ) with 1 < p < ∞. See Section 4. For bisectorial invertible operators that generate a bounded group, it is already shown in [12] that the boundedness of the spectral projection is equivalent to the boundedness of the Hilbert transform of the group. Instead of using the analytic generator of the group as in [12] we give in Section 5 an alternative more direct approach to this result based on the H ∞ -functional calculus. In this way we can avoid the assumption 0 ∈ ρ(A). The H ∞ -functional calculus provides a convenient and unifying framework for all results in this paper. We recall its construction shortly in Section 2. n

2. H ∞ -calculus for bisectorial operators For ω ∈ (0, π) we deﬁne the sector Sω as the set {reiφ : r > 0, φ ∈ (−ω, ω)}. For ω ∈ (0, π/2) we deﬁne the bisector S ω = {reiφ : r > 0, |φ ± π/2| < ω}. Given two sectors (bisectors) Σ1 , Σ2 we write Σ1 < Σ2 if there are 0 < ω1 < ω2 such that Σk = Sωk (Σk = S ωk ), k = 1, 2. In the following X will always denote a complex Banach space. An operator A : D(A) → X having dense domain and range is called sectorial, if there exists ω ∈ (0, π) such that σ(A) ⊂ Sω and sup{zR(z, A) : z ∈ / Sω } < ∞ for all ω ∈ (ω, π). We will write A ∈ Sect(Sω ) in this case. The inﬁmum of all such ω is called the spectral angle of A and denoted by ω(A). We will say that A ∈ Sect(eiη Sω ) if e−iη A ∈ Sect(Sω ). We will call A bisectorial, if there exists ω ∈ (0, π/2) such that both A and −A are sectorial of angle π/2 + ω. The inﬁmum of all such ω is the spectral angle of the bisectorial operator A and is denoted by ω(A). In the following we will sketch the construction of a holomorphic functional calculus for sectorial (bisectorial) operators. Assume that A is sectorial (bisecto˜ where rial) of angle ω. Let Σ = Sω (Σ = S ω ) and Σ < Σ . The contour Γ = ∂ Σ, ˜ Σ < Σ < Σ , is orientated in the positive sense. In a ﬁrst step we deﬁne f (A) for

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holomorphic functions satisfying an estimate s

|f (z)| ≤ C

|z| , (1 + z)2s

z ∈ Σ

(1)

for some s > 0. We denote this class by H0∞ (Σ ). This estimate implies that the integral 1 f (A) = f (z)R(z, A)dz (2) 2πi Γ exists as a Bochner integral in L(X). In a second step we consider functions in H ∞ (Σ ) (or, more generally, polynomially bounded functions). Consider the regularizing function φ(z) = z(1 + z)−2 . Since φ ∈ H0∞ (Σ ) we can deﬁne φ(A) by (1) and check that it is injective with φ(A)−1 = 2I + A + A−1 , and that D(φ(A)−1 ) = R(φ(A)) = D(A) ∩ R(A). To deﬁne f (A) for f ∈ H ∞ (Σ ) we write f (z) = φ(z)−1 (φ(z)f (z)). The function φf is in H0∞ (Σ ); hence we may deﬁne D(f (A)) = {x ∈ X : (φf )(A)x ∈ D(φ(A)−1 )} f (A) = φ(A)−1 (φf )(A). One can show that this deﬁnition is consistent in case f ∈ H0∞ (Σ ) and that the usual properties of a holomorphic functional calculus hold. In particular we have the algebraic properties for f, g ∈ H ∞ (Σ ) • D(f (A)) ∩ D(g(A)) ⊂ D((f + g)(A)) and (f + g)(A)x = f (A)x + g(A)x, for x ∈ D(f (A)) ∩ D(g(A)) • D(f (A)g(A)) = D((f g)(A)) ∩ D(g(A)) and (f g)(A)x = f (A)[g(A)x] for x ∈ D(f (A)g(A)) and the important ‘convergence’ property • If fn ∈ H ∞ (Σ ) with supn fn ∞ < ∞ and fn (z) → f (z) for z ∈ Σ , then for all x ∈ D(A) ∩ R(A) lim fn (A)x = f (A)x.

n→∞

We say that A has a bounded H ∞ (Σ )-calculus, if the operator f (A) is bounded for all f ∈ H ∞ (Σ ). In this case the map f ∈ H0∞ (Σ ) → f (A) ∈ L(X) extends to a bounded algebra homomorphism on H ∞ (Σ ). The inﬁmum of all an gles ω such that A has a bounded H ∞ (Σ )-calculus (Σ = Sω or Σ = S ω ) is denoted by ωH ∞ (A). For details of this construction for sectorial operators, see, e.g., [7], [11, Chapter II], for bisectorial operators [5] and [6]. We call the operator A R-sectorial, if there is a ω ≤ ω ˜ < π/2 such that the set {zR(z, A) : z ∈ C \ Sω˜ } is R-bounded. If the set {zAR(z, A)2 : z ∈ C \ Sω˜ } is R-bounded, we call A almost R-sectorial. In general, the notion of almost Rsectoriality is strictly weaker than R-sectoriality. Following [9], we say that A is R-bisectorial (almost R-bisectorial) of angle ω, if ±A are R-sectorial (almost R-sectorial) of angle π/2 + ω. The inﬁmum of all such angles is denoted by ωr (A). We will write A ∈ RSect(Σ) (A ∈ aRSect(Σ))

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in this case. It is remarkable, that, if A has a bounded H ∞ -calculus, we have ωH ∞ (A) = ωr (A) (cf. [9]). Let A ⊂ L(X) be an algebra of operators that commute with the resolvents of A. We denote by RH ∞ (Σ, A) the set of all A-valued holomorphic functions on Σ that have R-bounded range. For F ∈ RH ∞ (Σ, A) we may deﬁne F (A) by (2). If A has a bounded scalar H ∞ (Σ)-calculus, then the map F → F (A) extends to an operator-valued functional calculus mapping RH ∞ (Σ , A) into L(X) for all Σ > Σ with analogous properties, see [5], [10] and [11] for details. If the Banach space X has Pisier’s property (α), as an application of the operator-valued functional calculus one may construct the joint functional calculus of resolvent commuting operators each having a bounded H ∞ -calculus. If the Banach space has property (α) the joint functional calculus is bounded from H ∞ (Σ1 × · · · × Σn ) into L(X), see [11, Thm. 12.12] and [6] for details.

3. Spectral projections and fractional powers In this section let A always be a bisectorial operator of angle ω. Let Σ = S ω and Σ = S ω for some ω < ω < π/2. We assume that A is densely deﬁned and has dense range. Recall that this implies that A is injective. We will denote by Σ1 and Σ1 the component of Σ and Σ lying in the upper half-plane; similarly, Σ2 and Σ2 will denote the component in the lower half-plane. We say that X = X1 ⊕ X2 is a spectral decomposition of X if Xi is invariant under A and if the restriction Ai of A to Xi is in Sect(Σi ). In particular, σ(Ai ) = σ(A) ∩ Σi . We will call Ai the reduced operators. The following proposition gives uniqueness of such a decomposition and links it to the functional calculus. Clearly, if the operator A would not be injective, there could be no hope to obtain uniqueness, as we could choose to incorporate the kernel of A into either of the two subspaces. 3.1. Proposition. 1. Let X = X1 ⊕ X2 be a spectral decomposition of X and denote by P the canonical projection onto X1 , that is P : x = x1 ⊕ x2 → x1 . Then P = p(A), where p ∈ H ∞ (Σ ) is the indicator function of the set Σ1 . 2. The spectral decomposition exists if and only if the operator P = p(A) is bounded. Proof. 1. Since R(z, Ai ) = R(z, A)|Xi for all z ∈ ρ(A) = ρ(A1 ) ∩ ρ(A2 ), it is easy to see that (3) f (A) = f (A1 ) ⊕ f (A2 ) ∞ with D(f (A)) = D(f (A1 )) ⊕ D(f (A2 )) for all f ∈ H (Σ ). The fact that Ai ∈ Sect(Σi ) together with Cauchy’s integral theorem implies that the H ∞ (Σi )-functional calculus for Ai extends its H ∞ (Σ )-functional calculus. As p is constant on both Σ1 and Σ2 we obtain that p(A) = p(A1 ) ⊕ p(A2 ) = IX1 ⊕ 0 = P .

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2. Assume that p(A) is bounded. It follows readily from elementary properties of the functional calculus that P := p(A) is then a bounded projection commuting with A. Thus X1 = R(P ) and X2 = ker P are A-invariant subspaces of X. We denote by Ai the restriction of A to Xi . It remains to show that Ai ∈ Sect(Σi ). First, we verify that the spectrum of Ai is situated where it should be: σ(Ai ) = σ(A) ∩ Σi . It suﬃces to consider the case i = 1 (i = 2 follows by symmetry, as p(A) is bounded if and only if (1 − p)(A) is bounded) and to show that 0 = µ ∈ Σ2 is in the resolvent set of A1 . Fix λ ∈ C \ Σ . Then the identity (µ − z)−1 = λµ−1 (λ − z)−1 + (λ − µ)µ−1 z(λ − z)−1 (µ − z)−1 plugged into the functional calculus gives (rµ p)(A) = (c1 rλ p + c2 hp)(A) = c1 R(λ, A)P + c2 (hp)(A), −1

−1

−1

−1

(4) −1

where h(z) = z(λ−z) (µ−z) , rc (z) = (c−z) , c1 = λµ and c2 = (λ−µ)µ . As hp ∈ H0∞ (Σ ), the operator (hp)(A) is bounded and hence also (rµ p)(A). The identity (µ − A)(rµ p)(A) = p(A) and the fact that A and (rµ p)(A) commute imply that µ ∈ ρ(A1 ) and R(µ, A1 ) = (rµ p)(A)|X1 . It remains to establish the bound on the resolvent. By (4) we have µR(µ, A1 ) ≤ P µ(rµ p)(A) ≤ P (λR(λ, A) + |λ − µ| (hp)(A)). We will ﬁx the argument of λ and choose λ such that |λ| = |µ|. Then the ﬁrst term is bounded by our sectoriality assumption and as |λ − µ| ≤ 2 |µ|. Thus, we only have to ﬁnd an upper bound for |µ| (hp)(A), 0 = µ ∈ Σ2 . Denote by Γ+ the part of the contour lying in the upper half-plane. A direct estimate gives |µ| |dz| |µ| hp(A) ≤ C Γ+ |λ − z| |µ − z| |dz| 1 |dz| =C = C iρ iν Γ+ λ − z µ − z |µ| Γ+ |e − z| |e − z| |λ| |µ| |µ| |µ| where ρ = arg λ and ν = arg µ, and this integral is uniformly bounded for µ ∈ Σ2 . The resolvent estimate for A2 follows by symmetry, as p(A) is bounded if and only if (1 − p)(A) is bounded. If A has a bounded H ∞ (Σ )-functional calculus it is easy to see that the spectral projection is bounded and induces a spectral decomposition. Conversely, the boundedness of the spectral projection connects the spectral properties of A with those of the reduced operators A1 , A2 . ˜ 3.2. Corollary. Assume that the spectral projections of A are bounded and Σ ⊂ Σ. ˜ ˜ A ∈ aRSect(Σ)] ˜ if and Then A has a bounded H ∞ (Σ)-calculus [or A ∈ RSect(Σ), ˜ i )-calculus [or Ai ∈ RSect(Σ ˜ i ), Ai ∈ aRSect(Σ ˜ i )] only if Ai has a bounded H ∞ (Σ for i = 1, 2. Proof. For the H ∞ -calculus use again the ﬁrst part of the proof of Proposition 3.1. For the (almost) R-sectoriality use in addition Example 2.16 in [11].

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However, the spectral projection is not always bounded, not even in Hilbert space. In [6] is constructed a counterexample using a conditional Schauder basis. See also 4.1 below. Now we will link the notions of fractional powers and the spectral projection. 3.3. Theorem. The following statements are equivalent: 1. P = p(A) is bounded. 2. For one (all) w ∈ C \ Z we have D(Aw ) = D((−A)w ) and the equivalence of norms: Aw x ∼ (−A)w x , x ∈ D(Aw ). 3. For one (all) w ∈ C \ Z the operator Aw (−A)−w admits a bounded extension to X. 4. The operator log A − log(−A) admits a bounded extension to all of X. If A is invertible, we may add the following equivalent statements to the list: 5. There exists w ∈ C \ Z with Re(w) > 0 such that D(Aw ) ⊃ D((−A)w ). 6. For all w ∈ C with Re(w) > 0 we have D(Aw ) = D((−A)w ). 3.4. Remark. For invertible operators this was ﬁrst shown in [5]. Proof. Let w ∈ C. The function h = hw : z → z w (−z)−w is holomorphic on C \ R. As the sum of the exponents is zero, we have, for z ∈ / R, h(z) = eiw(arg(z)−arg(−z)) , where arg : C \ (−∞, 0] → (−π, π] denotes the argument function associated to the principal branch of the logarithm used to deﬁne the complex powers z w . Thus, h is constant on the upper and lower half-planes. It is easy to see that, if w ∈ / Z, the function h takes two diﬀerent values c1 and c2 . p = (c1 − c2 )−1 (h − c2 ). Applying the H ∞ -functional calculus to this identity gives immediately P = p(A) = (c1 − c2 )−1 (h − c2 )(A) = (c1 − c2 )−1 h(A) − c2 (c1 − c2 )−1 ,

(5)

that is, the spectral projection is bounded if and only if h(A) is bounded. The equivalence of 1. and 2. is immediate from the identity (5) and the fact that the densely deﬁned operator Aw (−A)−w is a restriction of h(A) (the operator is densely deﬁned, as its domain contains the range of φ(A)). The equivalence of 2. and 3. is clear. The other statements follow analogously. We give some details concerning the last two statements. The implication 7. ⇒ 1. is true, as the relation R((−A)−w ) = D((−A)w ) and the fact that the invertibility of A implies the boundedness of (−A)−w tell us D(h(A)) ⊃ D(Aw (−A)−w ) = X. Finally, we show 1. ⇒ 8.: Let P be bounded and denote by Ak ∈ Sect(Σk ) the reduced operators, then D((±A)w ) = D((±A1 )w ) ⊕ D((±A2 )w ) (compare (3)). Now it suﬃces to observe that, as h is constant on Σk , we have h(Ak ) = ck = 0, hence A−w = k ck (−Ak )−w . In particular, the ranges agree, which proves the claim.

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4. Boundedness of Riesz transforms Our results are motivated by the following well-known example. d on Lp (R, X) with domain 4.1. Example. Let p ∈ [1, ∞) and deﬁne A = dt 1,p D(A) = W (R, X). As A is the generator of the bounded group of translations on Lp (R, X), it follows that A ∈ Sect(Σ) for any bisector Σ. In order to decide whether the spectral projection corresponding to the upper complex half-plane is bounded we compute the operator A1/2 (−A)−1/2 using Fourier multipliers. Since A is the Fourier multiplier associated to the symbol m(x) = ix, the operators A1/2 and (−A)−1/2 have the symbols

(ix)1/2 = e1/2(log|x|+i arg(ix)) and (−ix)−1/2 = e−1/2(log|x|+i arg(−ix)) . Hence A1/2 (−A)−1/2 has the symbol (ix)1/2 (−ix)−1/2 = ei/2(arg(ix)−arg(−ix)) = i sign (x). Thus, up to a constant A1/2 (−A)−1/2 is the Hilbert transform. On the other hand, the spectral projection P1 corresponding to the upper half-plane, has symbol χR+ . As a consequence of Theorem 3.3 we obtain the equivalence of the following statements. 1. The P1(the Poisson projection) is bounded; spectral projection d 1/2 d 1/2 d 1/2 d 1/2 ∀f ∈ D( dx 2. dx f ≈ − dx f ) = D( − dx ). d2 1/2 d −1 3. the operator − dx2 , the Hilbert transform, is bounded; dx d2 1/2 d ∀f ∈ W 1,p (R, X). f ≈ dx f 4. − dx2 It is well known that the Hilbert transform is bounded, if and only if p ∈ (1, ∞) and if X is a UMD space. With the same arguments as in the proof of Theorem 3.3 we can generalize the equivalence of 1. ⇔ 4. Note that for a bisectorial operator A ∈ Sect(S ω ), the operator −A2 is sectorial in Sect(S2ω ). 4.2. Theorem. Let A be a bisectorial operator on X. Then the spectral projection P corresponding to the upper half-plane is bounded if and only if A(−A2 )−1/2 extends to a bounded operator, i.e., (−A)2 x ≈ Ax for x ∈ D(A) = D((−A2 )1/2 ). For the Hilbert space case it was noted in [2] that the boundedness of the H ∞ -calculus implies the above estimate. Let A1 , . . . , An now be a family of commuting bisectorial operators on X. Motivated by the example Ai = dxd i on Lp (Rn ) and the classical Riesz trans+n forms dxd i (−∆)−1/2 we call the operators Ai (−A)−1/2 with A = i=1 A2i Riesz transforms of the family A1 , . . . , An . By means of the H ∞ -calculus we obtain the boundedness of such Riesz-transforms and related inequalities.

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4.3. Theorem. Let A1 , . . . , An , be commuting bisectorial operators with a bounded joint H ∞ (Σn )-calculus, Σ = S ω , ω ∈ (0, π/4). Then we have the inequalities n n 1/2 E E ri Ai x ≈ (−A) x , rij Ai Aj x ≈ Ax i=1

i,j=1

where (ri ), (rij ) are iid Bernoulli random variables.

+n +n Proof. Consider h(z) = ( i=1 i zi )(− i=1 zi2 )−1/2 +nthe holomorphic +n functions and g(z) = ( i,j=1 ij zi zj )( i=1 zi2 )−1 . It is easy to see that g, h ∈ H ∞ (Σn ). Moreover, the functions are bounded uniformly with respect the choice of signs i , ij ∈ {−1, 1}. Hence, by the boundedness of the joint functional calculus, we obtain the inequalities n n i Ai x ≤ C h∞ (−A)1/2 x , A A x ij i j ≤ C g∞ Ax i,j=1 i=1 where C denotes the norm of the joint calculus. Since this estimate is uniform in the choice of signs, we obtain one of the two desired inequalities by integration. We will obtain the reverse inequalities combining randomization and a duality argument. We will give the details for the ﬁrst equivalence, the argument for the second is similar. We denote by X # the moon-dual of X, that is the set D(A∗ ) ∩ R(A∗ ) where the closure is taken in the norm of X ∗ . The moon-dual operator A# of A is the part of A∗ in X # ; if A is sectorial or has a bounded H ∞ -calculus then so does A# . If X is reﬂexive we have A# = A∗ . We note that X # is norming for X (see [11, Section 15A] for details). Let x ∈ D((−A)1/2 ), y ∈ D((−A# )1/2 ), then A2i x, y = Ai x, A# A2i )1/2 x, ((− A2i )1/2 )# y = (− i y 2 1/2 2 1/2 # ≤ E ri Ai x ≤ cE ri Ai x (−A# )1/2 y ; E ri Ai y in the last step we used the Khinchine-Kahane inequality and made use the inequality proved above (substituting A# for A). Therefore, since the set R((−A# )1/2 ) is norming, we ﬁnd ri Ai x (−A)1/2 x = sup (−A)1/2 x, z ≤ cE z=1

which completes the proof. n 4.4. Example. Let X = · · · dx xn ). For i = 1, . . . , n, deﬁne on t semigroup Ti (t)f (x) = f (x1 , . . . , xi−1 , e xi , xi+1 , . . . , xn ). As the measure 1 Lp (Rn , dx x1

X the of the Lp space X is invariant under scaling, we ﬁnd that Ti are contraction semigroups. Moreover, it is easy to see that they are positive and commute. Their generators, d , have thus a bounded bisectorial functional calculus (apply given by Ai = xi dx i

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the result of Hieber and Pr¨ uß [8]). Since the Lp space X has property (α), we ﬁnd that (A1 , . . . , An ) have a bounded joint bisectorial functional calculus. d2 The operator A2i is just xi dxd i (xi dxd i ) = xi dxd i + x2i dx 2 . Therefore, the precedi ing theorem gives the equivalence ⎛ ⎞ 2 1/2 n n 2 d d d d ⎝ xi xj f ⎠ ≈ x2i 2 f + xi f . dx dx dx dx i j i i i=1 i,j=1 Making use of [11, Cor. 10.15] we may extend this result to the vector-valued setting, allowing for functions taking values in a UMD space.

5. Spectral projection and Hilbert transform of a bounded C0 -group Let p denote the indicator function of the upper complex half-plane. The example d A = dt shows that the spectral projection P = p(A) is bounded if and only if the Hilbert transform is a bounded operator on Lp . In general, let (U (t))t∈R be a bounded C0 -group on the Banach space X with generator A. For 0 < < T < ∞ consider the operators U (s)x i U ds; H,T x := π <|s|
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By iteration and using the resolvent equation this implies that, for x ∈ D(A2 ), the function s → R(d + is, A)x − R(−d + is, A)x is majorized by s → −2 −1 Cd |d + is| , with C = C(x) ≥ 0, which is integrable on R. 5.2. Lemma. Let d > 0 and x ∈ D(A2 ), then 1 U sign s (R(−d + is, A)x − R(d + is, A)x) ds. (6) Hd x = 2π R

∞ Proof. We have r≤|s|≤R e−d|s| U(s)x ds = −∞ hr,R (s)e−d|s| U (s)x ds, where hr,R s denotes the function s → 1s 1{r≤|u|≤R} (s) ∈ Lp (R) for all p ∈ [1, ∞]. As the resolvent is given by the Laplace transform of its semigroup, we obtain F (e−d|·| U (·))(t) = e−ist e−d|s| (s)U (s) ds = R(d + it, A) − R(−d + it, A). (7) R

Recall that, given two functions f ∈ L1 (R), g ∈ L1 (R, Y ) for some

Banach space Y , we have as a consequence of Fubini’s theorem the relation R fˇ(t)g(t) dt =

f (t)ˇ g (t) dt, where fˇ denotes the inverse Fourier transform of f . By the inversion R formula, applying this relation to the identity (7) we ﬁnd ∞ U (s)x ˇ r,R (s)((R(d + is, A)x − R(−d + is, A)x)) ds. ds = e−d|s| h s −∞ r≤|s|≤R We compute the inverse Fourier transform of hr,R : for s ∈ R \ {0} we have R R/s sin u −1 −1 ist −ist −1 ˇ hr,R (s) = (2π) du; t (e − e ) dt = iπ u r r/s

∞ it is well known from calculus that the integral 0 u−1 sin u du converges and has ˇ r,R are uniformly bounded on R and conas value π/2. Therefore, the functions h verge pointwise (as r → 0 and R → ∞) to the limit i2−1 sign (·). Hence, Lebesgue’s theorem implies 1 HdU x = sign s (R(−d + is, A)x − R(d + is, A)x) ds, 2π R which proves the claim.

Using this representation, we will compare the Hilbert transform H U and the spectral projection P . Denote by Σ the bisector S ω with ω ∈ (π/4, π/2). Denote by p ∈ H ∞ (C \ R) the indicator function of the upper complex halfplane and deﬁne the signum-function s by s(z) = p(z)−p(−z) = 2p(z)−1; then p = (1+s)/2. Thus, the projection P = p(A) is bounded if and only if the operator s(A) is bounded, where both operators are deﬁned by means of the H ∞ (Σ)-functional calculus. We choose the following approximations of the identity ϕn (z) = eis(z)z/n ; the functions ϕn are in H ∞ (Σ) with norms uniformly bounded by 1 and converge pointwise to the function which is identically 1. The operator s(A) is approximated

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by the bounded operators Sn = (sϕn )(A): for x ∈ D(A)∩R(A) ⊂ D(P ) = D(s(A)) we have s(A)x = lim Sn x (8) n→∞

by the Convergence Lemma. The vector Sn x is given by the Cauchy integral 1 Sn x = s(z)ϕn (z)R(z, A)x dz. 2πi Γ ˜ d as {w ∈ C : |Re(w)| = d} with For d > 0 we deﬁne also the contour Γ positive orientation, and we consider the operators 1 s(z)ϕn (z)R(z, A)x dz. Qd,n x = 2πi Γ˜ d Let x ∈ D(A2 ) ∩ R(A). Since x ∈ R(A), say x = Ay, we have R(z, A)x = −y + zR(z, A)y. Consequently, the function R(·, A)x is bounded on the complement of S θ for all θ > 0. Hence estimating the diﬀerence of the two integrals implies the existence of a constant Cx > 0, independent of n, such that Qd,n x − Sn x ≤ Cx d.

(9)

As n tends to inﬁnity, Sn x converges to s(A)x = (2P − I)x. Hence, it remains to study the behavior of the expression Qd,n x: ∞ −1 Qd,n x = (2π) s(is)ϕn (d + is)R(d + is, A)x −∞

−s(−is)ϕn (−d − is)R(−d − is, A)x ds. As s(−is) = sign(−s) = −s(is) and as ϕn (−z) = ϕn (z) we obtain ∞ % & −1 Qd,n x = (2π) s(is)ϕn (d + is) R(d + is, A)x − R(−d + is, A)x ds (10) −∞ ∞ % & s(is)ϕn (d + is) R(−d + is, A)x + R(−d − is, A)x ds. + (2π)−1 −∞

We consider the ﬁrst integral in (10): as x ∈ D(A2 ), the function in the parentheses is integrable; hence we may apply Lebesgue’s theorem, which tells us that the ﬁrst term converges to −HdU x as n tends to inﬁnity. In order to ﬁnd an estimate for the second integral in (10) note that by substituting s by −s we have ∞ % & s(is)ϕn (d + is) R(−d + is, A)x + R(−d − is, A)x ds (2π)−1 −∞ ∞ % & = (2π)−1 s(is) ϕn (d + is) − ϕn (−d + is) R(−d + is, A)x ds. −∞

Rewriting the term in the squared brackets

ϕn (d + is) − ϕn (−d + is) = ϕn (−d + is) e2is(is)d/n − 1

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the second integral in (10) can be rewritten as 0 −2id/n −1 −(e − 1)(2π) ϕn (−d + is)R(−d + is, A)x ds −∞ ∞ 2id/n −1 + (e − 1)(2π) ϕn (−d + is)R(−d + is, A)x ds. 0

By Cauchy’s theorem we may deform the path of integration i(−∞, 0) − d into the path γ = (−1 − i)(0, ∞) − d without changing the value of the integral, and similarly we may deform the path i(0, ∞) − d into (−1 + i)(0, ∞) − d. Thus, we obtain by the sectoriality of A and as x ∈ R(A) a constant Cx independent of d such that 0 = (11) ϕ (−d + is)R(−d + is, A)x ds ϕ (z)R(z, A)x dz n n −∞ γ ≤ Cx |ϕn (z)| |dz| = Cx n. γ

The same estimate also holds for the second term. Hence, the second integral in (10) is estimated by 2id/n − 1) + (e−2id/n − 1) (2π)−1 Cx n = 2 sin(d/n)π −1 Cx n, (12) (e which implies

lim sup Qd,n x + HdU x ≤ 2π −1 Cx d.

(13)

n→∞

Putting all this together, we ﬁnd for x ∈ D(A2 ) ∩ R(A) and for all d > 0 and n∈N s(A)x + H U x ≤ s(A)x − Sn x + Sn x − Qd,n x + Qd,n x + H U x + H U x − H U x ; d

d

taking the limes superior with respect to n, we obtain by (8), (9) and (13) s(A)x + H U x ≤ Cx d + 2Cx π −1 d + HdU x − H U x , from which we conclude applying Lemma 5.1, by letting d → 0, that s(A)x + H U x = 0. Hence, the operators H U and I − 2P agree on the dense set D(A2 ) ∩ R(A). We summarize these considerations in the following theorem. 5.3. Theorem. Let A be the generator of the bounded C0 -group U . Assume that A has dense range. Then the following assertions are equivalent: 1. The spectral projection P = p(A) is bounded; 2. The Hilbert transform H U has a bounded extension to X; more precisely H U = I − 2P .

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It was shown in [12] that the spectral projection is bounded if the principal value integral deﬁning the Hilbert transform converges strongly. The converse was established for invertible operators.

References [1] W. Arendt and S. Bu, Sums of Bisectorial Operators and applications, Integral Equations and Operator Theory, submitted 2004. [2] P. Auscher, A. McIntosh and A. Nahmod, Holomorphic functional calculi of operators, quadratic estimates and interpolation, Indiana Univ. Math. J. 46 (1997), no. 2, 375–403. [3] G. Da Prato and P. Grisvard, Sommes d’op´ erateurs lin´eaires et ´equations diff´ erentielles op´erationnelles, J. Math. Pures Appl. (9) 54 (1975), no. 3, 305–387. [4] G. Dore and A. Venni, Separation of two (possibly unbounded) components of the spectrum of a linear operator, Integral Equations and Operator Theory, Vol. 12 (1989), 470–485. [5] G. Dore and A. Venni, H ∞ functional calculus for sectorial and bisectorial operators, Preprint 2001. [6] M. Duelli, Spectral calculus for bisectorial operators and applications to linear and non-linear evolution equations, doctoral dissertation, Universit¨ at Ulm, in preparation. [7] M. Haase, The functional calculus for sectorial operators and similarity methods, doctoral dissertation, Universit¨ at Ulm, 2003. [8] M. Hieber and J. Pr¨ uss, Functional calculi for linear operators in vector-valued Lp spaces via the transference principle, Adv. Diﬀerential Equations 3 (1998), no. 6, 847–872. [9] N.J. Kalton, P.C. Kunstmann and L. Weis, Perturbation and Interpolation Theorems for the H ∞ -calculus with Applications to Diﬀerential Operators, submitted. [10] N.J. Kalton and L. Weis, The H ∞ -calculus and sums of closed operators, Math. Ann. 321 (2001), no. 2, 319–345. [11] P.C. Kunstmann and L. Weis, Maximal Lp -regularity for Parabolic equations, Fourier multiplier theorems and H ∞ -functional calculus, in: Functional Analytic Methods for Evolution Equations, M. Ianelli, R. Nagel and S. Piazzera, eds., Lecture Notes for Mathematics, Springer, 2004. [12] S. Monniaux, G´en´erateur analytique et r´egularite maximale, Th`ese, Universit´e de Franche-Compt´e, Besan¸con, 1995. [13] J. Pr¨ uss, Evolutionary Integral Equations and Applications, Birkh¨ auser Verlag, Monographs in Mathematics, 87, 1993. Markus Duelli and Lutz Weis Mathematisches Institut I Universit¨ at Karlsruhe Englerstraße 2 D-76128 Karlsruhe e-mail: [email protected] e-mail: [email protected]

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 113–136 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Very Weak Solutions of Stationary and Instationary Navier-Stokes Equations with Nonhomogeneous Data Reinhard Farwig, Giovanni P. Galdi and Hermann Sohr Dedicated to Professor Dr. H. Amann on the Occasion of his 65th Birthday

Abstract. We investigate several aspects of very weak solutions u to stationary and nonstationary Navier-Stokes equations in a bounded domain Ω ⊆ R3 . This notion was introduced by Amann [3], [4] for the nonstationary case = g leading to a new and very with nonhomogeneous boundary data u| ∂Ω large solution class. Here we are mainly interested to investigate the “largest possible” class for the more general problem with arbitrary divergence k = div u, boundary data g = u| and an external force f , as weak as possible. ∂Ω In principle, we will follow Amann’s approach. Mathematics Subject Classiﬁcation (2000). Primary 35Q30, 76D05; Secondary 35J25, 35J65, 35K60. Keywords. Stokes and Navier-Stokes equations, very weak solutions, nonhomogeneous data.

1. Introduction Throughout this paper Ω ⊂ R3 is a bounded domain with boundary ∂Ω of class C 2,1 and N = N (x) ∈ R3 denotes the unit outer normal at x = (x1 , x2 , x3 ) ∈ ∂Ω. In Ω×[0, T ), where 0 < T ≤ ∞, we consider the system of Navier-Stokes equations in the very general form ut − ∆u + u · ∇u + ∇p = f,

div u = k,

u|

∂Ω

= g,

u|

t=0

= u0

with nonhomogeneous data f = div F and k, g, u0 satisfying k ∈ Ls 0, T, Lr (Ω) , F = (Fij )i,j=1,2,3 ∈ Ls 0, T ; Lr (Ω) , 1 g = (g1 , g2 , g3 ) ∈ Ls 0, T ; W − q ,q (∂Ω) , u0 ∈ J q,s (Ω),

(1.1)

(1.2)

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where 3 < q < ∞, 2 < s < ∞, 1 < r < q such that 13 + 1q = 1r , 2s + 3q = 1. For simplicity, we assume that the coeﬃcient of viscosity equals 1. See (2.19) concerning the space of initial values J q,s (Ω) as a space of functionals. Further we suppose the compatibility condition k(t)dx = N · g(t)dS for a.a. t ∈ [0, T ), (1.3)

Ω

∂Ω

where ∂Ω N · g(t)dS := N · g, 1∂Ω must be understood as the functional N · g ∈ 1 W − q ,q (∂Ω) applied to the test function 1 ∈ W 1−1/q ,q (∂Ω). The largest possible class of very weak solutions seems to be Serrin’s uniqueness class in the context Ls 0, T ; Lq (Ω) for u deﬁned by 2s + 3q = 1. Indeed, we cannot expect, up to now, to obtain the desired uniqueness and regularity properties for any larger class. To obtain the relation which deﬁnes a very weak solution u of (1.1), we follow Amann [3], [4] in principle and apply formally to (1.1) the test function 1 2 2 w ∈ C0 [0, T ); C0,σ (Ω) , where C0,σ (Ω) = {v ∈ C 2 (Ω)3 : div v = 0 in Ω, v | = 0}. ∂Ω Then an integration by parts yields the relation T − u, wt Ω − u, ∆wΩ + g, N · ∇w∂Ω − uu, ∇wΩ − ku, wΩ dt (1.4) 0

T

= u0 , w(0)Ω −

F, ∇wΩ dt,

2 w ∈ C01 [0, T ); C0,σ (Ω) .

0

Here ·, ·Ω is the usual Lq − Lq -pairing in Ω and g(t), N · ∇w(t)∂Ω denotes 1 the value of the boundary distribution g(t) ∈ W − q ,q (∂Ω) at the normal derivative wN (t) = N · ∇w(t)| ; furthermore, u0 , w(0)Ω means the value of ∂Ω

2 ∈ C0,σ (Ω), see (2.19), and the functional u0 ∈ J q,s (Ω) at w(0) = w| t=0 uu = u ⊗ u = (ui uj )i,j=1,2,3 for u = (u1 , u2 , u3 ). We also use the relation +3 u · ∇u = (u · ∇)u = div (uu) − ku. Note that div F = i=1 (∂/∂xi )Fij j=1,2,3 . 2 Using in (1.4) in particular the test function w ∈ C01 [0, T ); C0,σ (Ω) with 2 (Ω) = {v ∈ C02 (Ω)3 ; div v = 0}, we obtain, together with some appropriate C0,σ distribution p, the validity of the ﬁrst equation in (1.1) in the sense of distributions. The second equation div u = k in (1.1) must be supposed additionally to (1.4). To explain the boundary condition u| = g in (1.1) we observe that the normal ∂Ω derivative wN (t) of w(t) at ∂Ω has the form wN (t) = N · ∇w(t) = curl w(t) × N, (1.5)

and therefore, the relation (1.4) contains only the tangential component N × g of u at ∂Ω. Indeed, we will show that the tangential component of u is well deﬁned by (1.4) as a distribution on ∂Ω, and we will derive an explicit formula, see (4.5). The (well-deﬁned) condition N · u(t)| = N · g(t) for the normal component of ∂Ω u at ∂Ω must be supposed additionally. This leads to a precise formulation of

Very Weak Solutions of Navier-Stokes Equations

115

the boundary condition u| = g, see Remarks 3(2) in Section 4 below. Thus we ∂Ω suppose, additionally to (1.4), the conditions div u(t) = k(t),

N · u(t)|

∂Ω

= N · g(t) for a.a. t ∈ [0, T ).

(1.6)

This leads to the following Deﬁnition 1. Assume that the data F, k, g and u0 satisfy (1.2) and (1.3). Then s q u ∈ L 0, T ; L (Ω) is called a very weak solution of the Navier-Stokes system (1.1) if the conditions (1.4) and (1.6) are satisﬁed. Note that a very weak solution u need not have any diﬀerentiability property in space and time. In particular, u need not satisfy any energy inequality with ﬁnite energy u2L∞ (0,T ;L2 (Ω)) + ∇u2L2(0,T ;L2 (Ω)) < ∞ like weak solutions in the sense of Hopf. This justiﬁes the notation “very weak solution”. On the other hand, a very weak solution is unique – a fact which is not known in general for weak solutions in the sense of Hopf. The notion of very weak solutions is not new for homogeneous data k = 0 and g = 0, see [3], [4], [15], and the literature therein. However, Amann’s notion of very weak solutions in [3], [4] for k = 0 and boundary values g = 0 introduces a completely new aspect. It leads to new solution classes of very low smoothness in space such that the boundary condition u| = g is not deﬁned by usual trace the∂Ω orems but more generally by the conditions (1.3), (1.4). This will have interesting applications. Following Amann’s approach our aim is to extend the solution classes for (1.1) 1 to div u = k = 0 and to the “weakest” possible case u| = g ∈ W − q ,q (∂Ω) for a.a. ∂Ω t ∈ [0, T ], leading to the solution class in Deﬁnition 1 without any smoothness in space and only satisfying Serrin’s condition, see Theorem 1. We will develop such a theory also for the linear nonstationary Stokes system, see Theorem 4. Further we improve the results on very weak solutions of stationary Stokes and Navier-Stokes systems developed in [15] to the more general case that k ∈ Lr (Ω), 13 + 1q = 1r , see Theorems 2 and 3. Concerning the initial condition u| = u0 in (1.1) we note, see Theorem t=0

1, that A−1 q Pq u is well deﬁned as a continuous function on [0, T ) with values in Lqσ (Ω); here, Aq means the Stokes operator and Pq the Helmholtz projection. We obtain the well-deﬁned condition A−1 q Pq u|

t=0

= A−1 q Pq u0 ,

(1.7)

which can be understood as the precise meaning of u| = u0 , see (2.19) for t=0 −1 Aq Pq u0 and the proof of Theorem 1. Thus each condition in the system (1.1) has a well-deﬁned direct meaning for a very weak solution u. The ﬁrst two equations hold in the sense of distributions on Ω × (0, T ), u| = g holds in the sense of distributions in ∂Ω, and u| = u0 holds ∂Ω t=0

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in the sense of (1.7). Moreover, if the data are suﬃciently smooth, u coincides with the usual strong solution. Our main theorem on the system (1.1) reads as follows. Theorem 1. Suppose the data F, k, g and u0 satisfy (1.2) and (1.3) with 3 < q < ∞, 2 < s < ∞, 1 < r < q, 13 + 1q = 1r , 2s + 3q = 1. Then there exists some T = T (F, k, g, u0 ) > 0, 0 < T ≤ T , and a uniquely determined very weak solution s/2 u ∈ Ls 0, T ; Lq (Ω) of the system (1.1) satisfying A−1 0, T ; Lqσ (Ω) q Pq ut ∈ L −1 q for all 0 < T < T , and Aq Pq u ∈ C [0, T ); Lσ (Ω) . The existence interval [0, T ) is determined by the condition (4.23), depending on the data, and includes the case T = T = ∞ if the data are suﬃciently small. Up to now we cannot prove that there exists an open maximal existence interval as in [3], [4] for the case k = 0. The reasons are the very weak assumptions on g and k in (1.2). In the linearized case u · ∇u ≡ 0 we have to omit the term u · ∇u, wΩ = −uu, ∇wΩ − ku, wΩ in (1.4), and we may omit the restriction 2s + 3q = 1 in (1.2), which is caused by the nonlinear term. Then we can show the existence and uniqueness of a very weak solution u ∈ Ls 0, T ; Lq (Ω) of the linearized system (1.1) satisfying the estimate A−1 q Pq ut Ls (0,T ;Lq (Ω)) + uLs (0,T ;Lq (Ω)) ≤ C u0 J q,s (Ω) + F Ls (0,T ;Lr (Ω)) + kLs (0,T ;Lr (Ω)) + g

(1.8)

Ls (0,T ;W

− 1 ,q q (∂Ω))

with C = C(Ω, q, s) > 0, see Theorem 4 in Section 4 below. In the stationary case we consider the system −∆u + u · ∇u + ∇p = f,

div u = k,

with data f = div F and k, g satisfying 1

F = (Fij ) ∈ Lr (Ω), k ∈ Lr (Ω), g ∈ W − q ,q (∂Ω),

u|

∂Ω

=g

(1.9)

k dx = N · g, 1∂Ω , (1.10) Ω

with 3 < q < ∞, 1 < r < q,

1 3

+

1 q

= 1r .

An obvious modiﬁcation of the nonstationary case yields the following Deﬁnition 2. Assume that the data F, k and g satisfy (1.10). Then u ∈ Lq (Ω) is called a very weak solution of the stationary Navier-Stokes system (1.9) if the relation −u, ∆wΩ + g, N · ∇w∂Ω − uu, ∇wΩ − ku, wΩ = −F, ∇wΩ ,

(1.11)

2 (Ω), and the conditions holds for all w ∈ C0,σ

div u = k, are satisﬁed.

N · u| = N · g ∂Ω

(1.12)

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117

In this case we obtain the following result. Theorem 2. Suppose the data F, k and g satisfy (1.10) with 3 < q < ∞, 1 < r < q, 1 1 1 3 + q = r . There exists a constant K = K(Ω, q) > 0 such that if F Lr (Ω) + kLr (Ω) + g

W

− 1 ,q q (∂Ω)

≤ K,

(1.13)

then we obtain a uniquely determined very weak solution u ∈ Lq (Ω) to the stationary Navier-Stokes system (1.9). This solution satisﬁes the estimate (1.14) uLq (Ω) ≤ C F Lr (Ω) + kLr (Ω) + g − q1 ,q (∂Ω)

W

with C = C(Ω, q) > 0. Similarly as in (1.5) we obtain for x ∈ ∂Ω the identity wN = N · ∇w = (curl w) × N,

2 w ∈ C0,σ (Ω).

(1.15)

2 Setting w ∈ C0,σ (Ω) in (1.11) we obtain that −∆u + u · ∇u + ∇p = f holds in the sense of distributions with some distribution p. In the stationary case we can also prove that each very weak solution u has a well-deﬁned trace u| even in ∂Ω

1

the space W − q ,q (∂Ω), and there is an explicit representation formula for u|

∂Ω

, see

(3.6). Note that Theorem 2 improves the result in [15] where kLr (Ω) is replaced by the stronger norm kLq (Ω) . In the linearized case u · ∇u ≡ 0, we omit uu, ∇wΩ , ku, wΩ in relation (1.11), and the existence result together with estimate (1.14) holds without any smallness condition for every 1 < q < ∞, 1 < r < q, 13 + 1q ≥ 1r , see Theorem 3 in Section 3 below. The improvement concerning kLr (Ω) leads to a certain scaling invariance in the following sense. Let λ > 0, consider some ball Ba (x0 ) ⊆ R3 with radius a > 0 and center x0 ∈ R3 , and let F, k, g be data as in (1.10) with Ω = Ba (x0 ), 3 < q < ∞, 1 < r < q, 13 + 1q = 1r . Then it is easy to show that u ∈ Lq Ba (x0 ) ˜ is a very weak solution to the system (1.9) on Ω with data F, k, g iﬀ u ˜ ∈ Lq (Ω), ˜ Ω = Ba/λ (x0 ), is a very weak solution of the system −∆˜ u + u˜ · ∇˜ u + ∇˜ p = div F˜ , ˜ g˜ are deﬁned by ˜ where u on Ω, ˜, p˜, F˜ , k, u ˜(x) = λu λ(x − x0 ) + x0 , F˜ (x) = λ2 F λ(x − x0 ) + x0 , g˜(x) = λg λ(x − x0 ) + x0 ).

˜ div u ˜ = k,

p˜(x) ˜ k(x)

= =

u ˜| ˜ = g˜ ∂Ω

(1.16)

λ2 p λ(x − x0 ) + x0 , λ2 k λ(x − x0 ) + x0 ),

Then we conclude that if Ka,q = K Ba (x0 ), q > 0 is the constant in 3 (1.13) for Ba (x0 ), then λ1− q Ka,q can be chosen as the corresponding constant for Ba/λ (x0 ). We also conclude that (1.14) holds for u, F, k, g in Ω = Ba (x0 ) with

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˜ g˜ in Ω ˜ = Ba/λ (x0 ) with the same conC = C Ba (x0 ), q iﬀ (1.14) holds for u ˜, F˜ , k, stant C. These properties have several applications in the local regularity theory. Similar results hold in the nonstationary theory. The proofs of these theorems are organized as follows. First we consider the linearized stationary and nonstationary system and prove explicit representation formulas. Applying these formulas to the nonlinear system leads to a ﬁxed point problem which can be solved by Banach’s ﬁxed point theorem for suﬃciently small data. In Section 2 we prepare several preliminaries.

2. Notations and preliminaries q Classical Function Spaces. Let 1 < q < ∞ and q = q−1 such that 1q + q1 = α,q q α,q 1. We need the usual spaces L (Ω) and W (Ω), W0 (Ω), 0 ≤ α ≤ 2, with norms ·Lq (Ω) = ·q,Ω and ·W α,q (Ω) = ·α;q,Ω , resp. The space W −α,q (Ω) =

W0α,q (Ω) is the dual space of W0α,q (Ω) with the natural duality pairing ·, ·Ω and the norm ·W −α,q (Ω) = ·−α;q,Ω . Thus, e.g., f, hΩ means the value of

the functional f ∈ W −α,q (Ω) at h ∈ W0α,q (Ω). Similarly, for functions on the boundary, Lq (∂Ω) and W α,q (∂Ω), W −α,q (∂Ω), with norms ·Lq (∂Ω) = ·q,∂Ω and ·W α,q (∂Ω) = ·α;q,∂Ω , ·W −α,q (∂Ω) = ·−α;q,∂Ω , resp., and the duality pairing ·, ·∂Ω are the corresponding notions for ∂Ω. In particular, the pairing between Lq (∂Ω) and its dual space Lq (∂Ω) = Lq (∂Ω) is given by f, g∂Ω = f · g dS

∂Ω

where ∂Ω . . . dS means the surface integral on ∂Ω, see [25] and [27], p. 33, p. 40. For more details on these spaces cf. [1], [12], [13], [27], [29]. In general, we use the same symbol for scalar, vector, and tensor-valued spaces. By C0ν (Ω), C ν (Ω), C ν (Ω), ν = 0, 1, . . . and ν = ∞, we denote the usual spaces of smooth functions. We set C0ν (Ω) = {v ∈ C ν (Ω); v | = 0}. The space of distri∂Ω butions C0∞ (Ω) is the dual space of the test space C0∞ (Ω) with the usual topology, the duality pairing is again denoted by ·, ·Ω . Similarly, the space C 1 (∂Ω) of boundary distributions is the dual space of the test space C 1 (∂Ω) with the duality pairing ·, ·∂Ω . This test space has the form C 1 (∂Ω) since ∂Ω has no boundary. Spaces of solenoidal vector-valued functions are denoted by appending “σ”. ν ν (Ω) = {v ∈ C0ν (Ω); div v = 0} and C0,σ (Ω) = {v ∈ C0ν (Ω); Thus we have C0,σ div v = 0}. The corresponding functional space for solenoidal test functions ∞ ∞ C0,σ (Ω) is the dual space C0,σ (Ω) , again with pairing ·, ·Ω . By a theorem of de Rham, [28], Chapter I, Proposition 1.1, a distribution d = (d1 , d2 , d3 ) ∈ C0∞ (Ω) ∞ with d, vΩ = 0 for all v ∈ C0,σ (Ω) has the form d = ∇h with some scalar distribution h.

Very Weak Solutions of Navier-Stokes Equations

119

∞ Let Lqσ (Ω) be the closure of C0,σ (Ω) in the norm ·q,Ω . Then Lqσ (Ω) = Lqσ (Ω) q is the dual space of Lσ (Ω) with pairing ·, ·Ω .

Traces and Extensions. Let α = 1, 2. Then the trace map f → f | is a well∂Ω 1 deﬁned bounded operator from W α,q (Ω) onto W α− q ,q (∂Ω). Conversely, there ex1 ists a linear and bounded extension operator E: h → Eh from W α− q ,q (∂Ω) into W α,q (Ω) satisfying Eh | = h. Thus it holds Eh α;q,Ω ≤ Chα− q1 ;q,∂Ω with ∂Ω C = C(Ω, α, q) > 0. Let 1 < r ≤ q, 13 + 1q ≥ 1r , and let f = (f1 , f2 , f3 ) ∈ Lq (Ω) with div f ∈ Lr (Ω). Then we use Green’s identity div f, Eh Ω = N · f, h∂Ω − f, ∇Eh Ω 1−

1

,q

1

for h ∈ W q (∂Ω) = W q ,q (∂Ω) and with the extension operator E : 1 W q ,q (∂Ω) → W 1,q (Ω). This leads, using the embedding property Eh r ,Ω ≤ C(Eh q ,Ω + ∇Eh q ,Ω ), C = C(Ω, q, r) > 0, to the estimate |N · f, h∂Ω | ≤ C(f q,Ω + div f r,Ω )h q1 ;q ,∂Ω

(2.1)

1

with C = C(Ω, q, r) > 0. Hence the trace N · f | ∈ W − q ,q (∂Ω) of the normal ∂Ω component of f at ∂Ω is well deﬁned and it holds the estimate N · f − 1q ;q,∂Ω ≤ C(f q,Ω + div f r,Ω ).

(2.2)

Using the corresponding identity curl f, Eh Ω = N × f, Eh ∂Ω + f, curl Eh Ω 1−

1

,q

now for h = (h1 , h2 , h3 ) ∈ W q (∂Ω), we obtain the following trace property: 1 If f = (f1 , f2 , f3 ) ∈ Lq (Ω), curl f ∈ Lr (Ω), then the trace N × f | ∈ W − q ,q (∂Ω) ∂Ω of the tangential component of f at ∂Ω is well deﬁned and it holds the estimate N × f − 1q ;q,∂Ω ≤ C(f q,Ω + curl f r,Ω )

(2.3)

with C = C(Ω, q, r) > 0. The identity f = (N · f )N + (N × f ) × N at ∂Ω shows that it is justiﬁed to call N × f | the tangential component of f at ∂Ω. ∂Ω

Let f ∈ Lq (Ω) with Ω f dx = 0. Then there exists some b = bf ∈ W01,q (Ω) with div bf = f such that f → bf is a linear mapping satisfying bf 1;q,Ω ≤ Cf q,Ω ,

C = C(Ω, q) > 0.

(2.4)

If moreover f ∈ W01,q (Ω), then bf ∈ W02,q (Ω) and bf 2;q,Ω ≤ C∇f q,Ω , see [6], [12], Theorem III.3.2 and [27], p. 68.

C = C(Ω, q) > 0,

(2.5)

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R. Farwig, G.P. Galdi and H. Sohr

Using properties of the weak Neumann problem in Lq (Ω), see [24], we ﬁnd 1 for each h ∈ W − q ,q (∂Ω) some E h = (E1h , E2h , E3h ) ∈ Lq (Ω) with div E h ∈ Lr (Ω), N · E h | = h, such that h → E h is a linear map satisfying ∂Ω

E h q,Ω + div E h r,Ω ≤ Ch− q1 ;q,∂Ω

(2.6)

with C = C(Ω, q, r) > 0. 1 Let h = (h1 , h2 , h3 ) ∈ W 1− q ,q (∂Ω). Then we ﬁnd an extension wh ∈ = h depending linearly on h; moreW 2,q (Ω) ∩ W01,q (Ω) such that N · ∇wh | ∂Ω over, wh 2;q,Ω ≤ Ch1−1/q;q,∂Ω (2.7) with C = C(Ω, q) > 0, see [23], Theorem 5.8, p. 104, or [29], 5.4.4, p. 385. If additionally N · h| = 0, then we can show that div wh | = 0 and ∂Ω ∂Ω

h h h N · ∇w | = −N × curl w | = h, see [15]. This yields Ω div w dx = 0, ∂Ω ∂Ω 1,q h div w ∈ W0 (Ω), and we ﬁnd b = b(wh ) ∈ W02,q (Ω) satisfying div b = div wh and (2.4), (2.5). Setting wˆh = wh − b(wh ) we see that wˆh ∈ W 2,q (Ω) satisﬁes w ˆh |

∂Ω

= 0,

N · ∇w ˆh |

∂Ω h

= −N × curl w ˆh |

∂Ω

= h,

div w ˆh = 0.

(2.8)

Moreover, the mapping h → w ˆ is linear and w ˆh 2;q,Ω ≤ Ch1−1/q;q,∂Ω ,

C = C(Ω, q) > 0,

(2.9)

see [15], (2.14). The Stokes Operator. Let f = (f1 , f2 , f3 ) ∈ Lq (Ω). Then, the weak Neumann problem ∆H = div f , N · (∇H − f )| = 0, has a unique solution ∇H ∈ Lq (Ω) ∂Ω satisfying the estimate ∇Hq,Ω ≤ Cf q,Ω (2.10) with C = C(Ω, q) > 0, see [24]. Setting Pq f = f − ∇H we deﬁne the Helmholtz projection operator Pq as a bounded operator from Lq (Ω) onto Lqσ (Ω), satisfying Pq2 = Pq and Pq f, gΩ = f, Pq gΩ ,

for all f ∈ Lq (Ω), g ∈ Lq (Ω) .

Hence Pq = Pq holds for the dual operator Pq of Pq . The Stokes operator Aq , with domain D(Aq ) = Lqσ (Ω) ∩ W01,q (Ω) ∩ W 2,q (Ω) ⊆ Lqσ (Ω) and range R(Aq ) = Lqσ (Ω), is deﬁned by Aq u = −Pq ∆u, u ∈ D(Aq ). The fractional α q α α q power Aα q : D(Aq ) → Lσ (Ω) = R(Aq ) with D(Aq ) ⊆ D(Aq ) ⊆ Lσ (Ω), 0 ≤ α ≤ 1, α −1 −α = Aq is a bounded operator is well deﬁned, bijective and its inverse (Aq ) α ) = D(A from Lqσ (Ω) into Lqσ (Ω) with range R(A−α q q ); furthermore, the operator α α α (Aq ) = Aq is the dual operator of Aq . The norms u2;q,Ω and Aq uq,Ω are 1/2

equivalent for u ∈ D(Aq ); analogously, the norms u1;q,Ω and Aq uq,Ω are 1 2

equivalent for u ∈ D(Aq ) = Lqσ (Ω)∩W01,q (Ω). Note that the space D(Aα q ) endowed

Very Weak Solutions of Navier-Stokes Equations

121

α with the graph norm Aα q uq,Ω , u ∈ D(Aq ), is a Banach space. Furthermore, we mention the important embedding property

uq,Ω ≤ CAα γ uγ,Ω ,

u ∈ D(Aα γ ),

1 < γ ≤ q,

2α +

3 3 = , q γ

(2.11)

with C = C(Ω, q) > 0. See [2], [10], [14], [18], [19], [22], [26], [27], [29] concerning proofs and further properties of the Stokes operator. Finally we observe that Aq u = Aρ u holds if u ∈ D(Aq ) ∩ D(Aρ ), 1 < q < ∞, 1 < ρ < ∞. It is well known that −Aq generates a bounded analytic semigroup {e−tAq : t ≥ 0}, see [2], [17], [20], [26], [27], and that −tAq vq,Ω ≤ Ce−δt t−α vq,Ω , Aα qe

v ∈ Lqσ (Ω), t > 0,

(2.12)

with constants C = C(Ω, q) > 0, δ = δ(Ω, q) > 0. Let 0 < α ≤ 1, 1 < q < ∞, and let d = (d1 , d2 , d3 ) ∈ C0∞ (Ω) be a distribution. Assume that d, vΩ is well deﬁned for all v ∈ D(Aα q ) and is con tinuous in the norm Aα v , i.e., there exists a constant C > 0 such that q ,Ω q |d, vΩ | ≤ CAα v . In other words, the functional d, v , v ∈ D(Aα q ,Ω Ω q q ), is α α a well-deﬁned element of the dual space D(Aq ) of D(Aq ). Writing formally d, vΩ = d, Pq vΩ = Pq d, vΩ , we call Pq d = Pq d, ·Ω the restriction of the functional d to test functions v ∈ D(Aα q ), giving Pq a generalized meaning; in ) . short, we write Pq d ∈ D(Aα q

α q Let d ∈ C0∞ (Ω) with Pq d ∈ D(Aα q ) . Since R(Aq ) = Lσ (Ω), there ex∗ q ists a uniquely determined element d ∈ Lσ (Ω) satisfying the relation d, vΩ = ∗ −α −α a generalized meaning. d∗ , Aα q vΩ . We set d = Aq Pq d, giving the operator Aq −α q Thus Aq Pq d ∈ Lσ (Ω) is well deﬁned by the relation α −α α d, vΩ = Pq d, vΩ = Pq d, A−α q Aq vΩ = Aq Pq d, Aq vΩ ,

(2.13)

v ∈ D(Aα q ), similarly as in the theory of distributions. We conclude that the q ∞ operation A−α q Pq d ∈ Lσ (Ω) is well deﬁned by (2.13) if d ∈ C0 (Ω) and Pq d ∈ α D(Aq ) . To obtain examples, we consider a functional f in the (vector-valued) Bessel potential space Hq−2α (Ω) for 0 < α ≤ 12 , see [1], [2], [29]. Then Pq f ∈ D(Aα q ) , 1 ∞ α since C0,σ (Ω) is dense in the Banach space D(Aq ) because of α ≤ 2 . Therefore, q A−α q Pq f ∈ Lσ (Ω) is well deﬁned in this case. Let 1 < r ≤ q, 13 + 1q ≥ 1r , and let u = (u1 , u2 , u3 ) ∈ Lq (Ω). Assume that 1

the distribution d = ∆u is continuous in the norm Ar2 · r ,Ω . Then the element −1/2 Pr ∆u ∈ Lrσ (Ω) is well deﬁned by the relation Ar −1

1

Ar 2 Pr ∆u, Ar2 vΩ = u, ∆vΩ ,

∞ v ∈ C0,σ (Ω),

(2.14)

∞ (Ω) is dense in according to (2.13); see also [15], Section 2. Here we use that C0,σ 1

the Banach space D(Ar2 ).

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Let F = (Fij ) ∈ Lr (Ω) where 1 < r ≤ q, using (2.11) and the estimate

1 3

+

1

1 q

≥ 1r , and set d = div F . Then

∇vr ,Ω ≤ C1 Ar2 vr ,Ω ≤ C2 Aq vq ,Ω ,

v ∈ D(Aq ) ,

Ci = Ci (Ω, q, r) > 0, i = 1, 2, we see that the distribution v → −F, ∇vΩ is q contained in D(Aq ) . Therefore, the element Fˆ = −A−1 q Pq div F ∈ Lσ (Ω) is well deﬁned by the relation Fˆ , Aq vΩ = −A−1 q Pq div F, Aq vΩ = −div F, vΩ

(2.15)

Fˆ q,Ω = A−1 q Pq div F q,Ω ≤ CF r,Ω

(2.16)

and it holds with C = C(Ω, q, r) > 0; see [27], III, Lemma 2.6.1 concerning similar operations. Spaces Lq (0, T ; X). Let 1 < q, s < ∞. Then we introduce the usual Bochner space T 1/s s Ls (0, T ; X) with norm ·Ls (0,T ;X) = where X is any Banach 0 ·X dt space with norm ·X . In the case X = W α,q (Ω), −1 ≤ α ≤ 1, we set ·Ls (0,T ;W α,q (Ω)) = ·α;q,s,Ω , and for X = W α,q (∂Ω) let ·Ls (0,T ;W α,q (∂Ω)) = ·α;q,s,∂Ω . Finally, if X = Lq (Ω) or X = Lq (∂Ω), we set ·Ls (0,T ;Lq (Ω)) = ·q,s,Ω , and ·Ls (0,T ;Lq (∂Ω)) = ·q,s,∂Ω , resp. As duality pairing we deﬁne T f, gΩ,T = f, gΩ dt with f, gΩ = f · g dx (2.17) 0

Ω s

for f = (f1 , f2 , f3 ) ∈ L (0, T ; L (Ω)), g = (g1 , g2 , g3 ) ∈ L (0, T ; Lq (Ω)). Similarly, we use the notation T f, g∂Ω,T = f, g∂Ω dt with f, g∂Ω = f · g dS. s

q

0

∂Ω

We also need the spaces C [0, T ); X , ν = 0, 1, 2, . . ., of X-valued functions ν v(t), such that v, (d/dt)v, . . . , (d/dt) v are continuous on [0, T ). 0 1 We set C [0, T ); X = C [0, T ); X . The space C [0, T ); X is the subspace 0 of functions v with compact support contained in [0, T ), of C 1 [0, T ); X consisting whereas C01 (0, T ); X is the subspace of C 1 [0, T ); X -functions v with compact support contained in (0, T ). Let f ∈ Ls 0, T ; Lqσ (Ω) . Then there exists a unique function v ∈ Ls 0, T ; D(Aq ) with vt ∈ Ls 0, T ; Lqσ (Ω) and v ∈ C [0, T ); Lqσ (Ω) , ν

satisfying the evolution system vt + Aq v = f, To be more precise,

v(t) = 0

t

0 ≤ t < T,

e−(t−τ )Aq f (τ )dτ,

v(0) = 0.

0 ≤ t < T,

Very Weak Solutions of Navier-Stokes Equations

123

and it holds the ‘maximal regularity’ estimate vt q,s,Ω + Aq vq,s,Ω ≤ Cf q,s,Ω ,

C = C(Ω, q, s) > 0,

(2.18)

see [20], [26]. The space of initial values. Let 1 < q, s < ∞. The space of initial values J q,s (Ω) q consists of distributions u0 satisfying A−1 q Pq u0 ∈ Lσ (Ω), see (2.13), and an additional integrability condition in time. To be more precise, let ∞ q,s ∞ −1 q s Aq e−tAq A−1 J (Ω) = {u0 ∈ C0 (Ω) : Aq Pq u0 ∈ Lσ (Ω), q Pq u0 q dt < ∞} 0

(2.19) and

u0 J q,s (Ω) = A−1 P u + ( q 0 q,Ω q

∞

1

s s Aq e−tAq A−1 q Pq u0 q dt) .

0

Obviously, · J q,s (Ω) deﬁnes a seminorm in J q,s (Ω) which becomes a norm if we identify two elements u0 , v0 ∈ J q,s (Ω) satisfying A−1 q Pq (u0 − v0 )J q,s (Ω) = 0, i.e., 2 u0 − v0 is a gradient. Of course, since w(0) ∈ C0,σ (Ω) in (1.4) is solenoidal, initial values can be prescribed only modulo gradients. − 1 +ε As an example, let u0 ∈ C0∞ (Ω) and assume that Aq s Pq u0 ∈ Lqσ (Ω) where 0 < ε < 1s . Then by (2.12) u0 ∈ J q,s (Ω). Using similar calculations as in [4] we −2/s can show that u0 ∈ Bq,s (Ω) is suﬃcient for u0 ∈ J q,s (Ω) in the case s ≥ 2; here −2/s Bq,s (Ω) denotes a Besov space, see [2], [3], [4], [29]. Consider u ∈ Ls 0, T ; Lq (Ω) such that its time derivative (A−1 q Pq u)t = s q 0, T ; L A−1 P u ∈ L (Ω) in the sense of distributions. Then, after redeﬁning on q t q σ −1 q −1 q a null set of [0, T ), we get Aq Pq u ∈ C [0, T ); Lσ (Ω) . Thus Aq Pq u(t) ∈ Lσ (Ω) −1 is well deﬁned for each t ∈ [0, T ), and therefore A−1 q Pq u(0) = Aq Pq u0 in (1.7) is well deﬁned.

∞ Let v0 ∈ Lqσ (Ω) such that 0 Aq e−tAq v0 sq dt < ∞, let f ∈ Ls 0, T ; Lqσ (Ω) and consider the general system vt + Aq v = f, v(0) = v0 . Then we apply (2.18) to vˆ(t) = v(t) − e−tAq v0 , obtain the estimate T 1s −tAq s Aq e v0 q dt + f q,s,Ω (2.20) vt q,s,Ω + Aq vq,s,Ω ≤ C 0

with C = C(Ω, q, s) > 0, and the representation formula t −tAq v0 + e−(t−τ )Aq f (τ ) dτ, 0 ≤ t < T. v(t) = e

(2.21)

0

3. Stationary very weak solutions First we consider the linearized stationary system −∆u + ∇p = f,

div u = k,

u|

∂Ω

=g

(3.1)

124

R. Farwig, G.P. Galdi and H. Sohr

with data f = div F and k, g satisfying F ∈ L (Ω), k ∈ L (Ω), g ∈ W r

r

− q1 ,q

k dx = N · g∂Ω ,

(∂Ω),

(3.2)

Ω

where 1 < q < ∞, 1 < r ≤ q, 13 + 1q ≥ 1r . Here we follow [15] in principle, but in [15] the stronger condition k ∈ Lq (Ω) is supposed. Modifying Deﬁnition 2 in an obvious way for the linearized case, u ∈ Lq (Ω) is called a very weak solution of the system (3.1) with data (3.2) if −u, ∆wΩ + g, N · ∇w∂Ω = −F, ∇wΩ ,

2 w ∈ C0,σ (Ω),

(3.3)

and additionally the conditions (1.12) are satisﬁed, i.e., div u = k and N ·u = N ·g on ∂Ω. Our main result on (3.1), improving [15], Theorem 3, reads as follows. Theorem 3. Suppose the data F, k, g satisfy (3.2) with 1 < q < ∞, 1 < r ≤ q, 1 1 1 q 3 + q ≥ r . Then there exists a unique very weak solution u ∈ L (Ω) of the system (3.1) satisfying the estimate uq,Ω ≤ C(F r,Ω + kr,Ω + g−1/q;q,∂Ω )

(3.4)

with C = C(Ω, q, r) > 0. ∞ Remarks 1. (1) Setting w ∈ C0,σ (Ω) in (3.3) and using de Rham’s argument we −1,q (Ω) such that −∆u + ∇p = f holds in the sense ﬁnd some pressure p ∈ W of distributions, and that p−1;q,Ω satisﬁes the same estimate as uq,Ω in (3.4). Moreover we conclude from (3.3) that −1

−1

Ar 2 Pr ∆u = −Ar 2 Pr div F, −1 where Ar 2 Pr div F is deﬁned by the ∞ v ∈ C0,σ (Ω), see 2.14 and 2.15 .

(

)

(

relation

−1 Ar 2 Pr

(3.5) 1 2 r

div F, A vΩ = −F, ∇vΩ ,

)

(2) Assume for a moment that u is suﬃciently smooth. Then, inserting w = w ˆ h from (2.8), (2.9) in the expression u, ∆wΩ , and using integration by parts, 1 as a functional in W − q ,q (∂Ω). To we obtain an explicit trace formula for u| ∂Ω

1

be more precise, the map h → u| , h∂Ω for h ∈ W 1−1/q ,q (∂Ω) = W q ,q (∂Ω), ∂Ω N · h| = 0 is deﬁned by ∂Ω

u|

∂Ω

−1

1

, h∂Ω = u, ∆w ˆ h Ω − Ar 2 Pr ∆u, Ar2 w ˆh Ω ,

1

h ∈ W − q ,q (∂Ω).

(3.6)

Using (2.9) with q replaced by q we then obtain the estimate |u|

∂Ω

−1

, h∂Ω | ≤ C(uq,Ω + Ar 2 Pr ∆ur,Ω )h1/q;q ;∂Ω

(3.7)

with C = C(Ω, q, r) > 0. Formula (3.6) is well deﬁned for each very weak solution u ∈ Lq (Ω) and yields an explicit formula for the tangential component of u| . ∂Ω

The normal component N · u| is well deﬁned by (1.12), (2.1). This shows that ∂Ω

Very Weak Solutions of Navier-Stokes Equations

125

1

the trace u| ∈ W − q ,q (∂Ω) is well deﬁned for a very weak solution u, and (2.1), ∂Ω (3.7) yield the estimate u|

−1

∂Ω

− 1q ;q,∂Ω ≤ C(uq,Ω + Ar 2 Pr ∆ur,Ω + div ur,Ω )

(3.8)

with C = C(Ω, q, r) > 0. (3) We conclude that a very weak solution u ∈ Lq (Ω) of (3.1) satisﬁes the 1 conditions (1.12), (3.5) and the condition u| = g as elements of W − q ,q (∂Ω). ∂Ω

Conversely, if u ∈ Lq (Ω) satisﬁes (1.12), (3.5), and the (well-deﬁned) trace u| ∂Ω is equal to g, then u is a very weak solution of (3.1). Proof of Theorem 3. Following in principle [15], we ﬁrst assume that u ∈ Lq (Ω) is a given very weak solution u of (3.1), and prepare some estimates. Using the trace map W 1,q (Ω) → W 1−1/q ,q (∂Ω) and the embedding estimate (2.11) we obtain that |g, N · ∇w∂Ω | ≤ Cg− 1q ;q,∂Ω vq ,Ω ,

2 w ∈ C0,σ (Ω), v = Aq w

(3.9)

with C = C(Ω, q) > 0. Therefore, the functional v → g, N · ∇A−1 q v∂Ω is con−1 tinuous in vq ,Ω , and it holds g, N · ∇Aq v∂Ω = G, vΩ with some unique G ∈ Lqσ (Ω) satisfying Gq,Ω ≤ Cg− 1q ;q,∂Ω .

Similarly, v → F, ∇A−1 q vΩ , v = Aq w, is continuous in vq ,Ω , and with ˆ ˆ Fˆ = −A−1 P div F , see ( 2.15 ), we get that F, ∇A−1 q q q vΩ = F , vΩ and F q,Ω ≤ CF r,Ω with C = C(Ω, q, r) > 0 by (2.16). Using E h for h = N · g, cf. (2.6), the compatibility condition in (3.2) yields

(div E h − k)dx = 0. Hence there exists b ∈ W01,r (Ω) satisfying div b = div E h − k Ω and, due to (2.6), (2.4), bq,Ω ≤ C1 ∇br,Ω ≤ C2 (div E h r,Ω + kr,Ω )

(3.10)

where Ci = Ci (Ω, q, r) > 0, i = 1, 2. Then we use the solution ∇H ∈ L (Ω) of the weak Neumann problem q

∆H = div (E h − b) = k = div u,

N · ∇H |

∂Ω

= N · (E h − b)|

∂Ω

,

and applying (2.10), (2.6), (3.10) leads to the estimate ∇Hq,Ω ≤ C1 E h − bq,Ω ≤ C2 (g− q1 ;q,∂Ω + kr,Ω )

(3.11)

with Ci = Ci (Ω, q, r) > 0. Further we get from (2.10) that ∇Hq,Ω ≤ Cuq,Ω with C = C(Ω, q) > 0. Obviously, ∇H only depends on the data k, g. ∈ Using (2.2) and (2.3) with curl ∇H = 0 we conclude that ∇H | 1

W − q ,q (∂Ω) is well deﬁned, and that

∇H− 1q ;q,∂Ω ≤ C(g− 1q ;q,∂Ω + kr,Ω ) with C = C(Ω, q, r) > 0.

∂Ω

(3.12)

126

R. Farwig, G.P. Galdi and H. Sohr Set u ˆ = Pq u = u − ∇H ∈ Lqσ (Ω) and gˆ = g − ∇H |

∂Ω

1

∈ W − q ,q (∂Ω). Then

|ˆ g , N · ∇w∂Ω | ≤ C(g− q1 ;q,∂Ω + kr,Ω )vq ,Ω ,

2 w ∈ C0,σ (Ω), v = Aq w, (3.13) −1 q ˆ ˆ cf. (3.9). As above, we construct G ∈ Lσ (Ω) satisfying ˆ g , N ·∇Aq v∂Ω = G, vΩ , and the estimate ˆ q,Ω ≤ C(g 1 G (3.14) − q ;q,∂Ω + kr,Ω )

holds with C = C(Ω, q, r) > 0. In the next step we use the relation ∇H, ∆wΩ = ∇H, N · ∇w∂Ω , w as in (3.13), which follows using an approximation of H by smooth functions and g , N · ∇w∂Ω + an integration by parts. Then by (3.3) a calculation leads to ˆ ˆ Fˆ yields G, ˆ vΩ + Fˆ , vΩ = −ˆ F, ∇wΩ = −ˆ u, vΩ , and inserting G, u, vΩ . From the regularity properties of the Stokes operator Aq , see [26], we know that 2 (Ω) is dense in Lqσ (Ω). Thus we may use the the set of all v = Aq w with w ∈ C0,σ last relation for all v ∈ Lqσ (Ω), and we get the representation formula ˆ − Fˆ u = ∇H − G

(3.15)

for the given very weak solution u. Since the right-hand side of (3.15) depends only on the data F, k, g, we can use (3.15) to construct u ∈ Lq (Ω). Then the same arguments as above show that u satisﬁes (3.3) and (1.12). Thus u deﬁned by (3.15) is a very weak solution of (3.1). Since each given very weak solution of (3.1) has the form (3.15), we obtain the uniqueness assertion. The estimate (3.4) follows from (3.12), (3.14), and (2.16). This proves Theorem 3. Remarks 2. (1) Suppose that the data F, k, g in (3.1), (3.2) satisfy the stronger conditions F ∈ Lq (Ω), k ∈ Lq (Ω), g ∈ W 1−1/q,q (∂Ω), 1 < q < ∞. Then the very weak solution u in Theorem 3 satisﬁes u ∈ W 1,q (Ω) and estimate u1;q,Ω ≤ C(F q,Ω + kq,Ω + g1−1/q;q,∂Ω )

(3.16)

with C = C(Ω, q) > 0. The existence of such a solution u of (3.1) is well known, see [10], [12], [14]. Since u is obviously also a very weak solution which is unique, we conclude this regularity property; see also [15], Lemma 4. (2) In the same way we conclude that if the data in (3.1), (3.2) satisfy the conditions f = div F ∈ Lq (Ω), k ∈ W 1,q (Ω), g ∈ W 2−1/q,q (∂Ω), then this solution satisﬁes (3.17) u2;q,Ω ≤ C(f q,Ω + k1;q,Ω + g2−1/q;q,∂Ω ) with C = C(Ω, q) > 0. Thus (3.16) and (3.17) are regularity properties of the very weak solution u if the data are suﬃciently smooth. Proof of Theorem 2. Following [15] we ﬁrst consider a given very weak solution u ∈ Lq (Ω) of the system (1.9). Using similar arguments as in the previous proof

Very Weak Solutions of Navier-Stokes Equations we obtain, since u · ∇u = div (uu) − ku and

1 3

+

1 q

127

= 1r , that

|u · ∇u, wΩ | ≤ C(u2q,Ω + kr,Ω uq,Ω )∇wr ,Ω ,

2 w ∈ C0,σ (Ω).

(3.18)

Hence we ﬁnd W (u) ∈ Lr (Ω) satisfying u · ∇u, wΩ = div (uu) − ku, wΩ = div W (u), wΩ = −W (u), ∇wΩ , (3.19) and W (u)r,Ω ≤ C(u2q,Ω + kr,Ω uq,Ω )

(3.20)

with C = C(Ω, q, r) > 0, similarly as in (2.15), (2.16). We see that u is a very weak solution of the linear system −∆u + ∇p = div F − W (u) , div u = k, u| = g. (3.21) ∂Ω Then (3.15) leads to the formula ˆ − Fˆ − W ˆ (u) u = ∇H − G

(3.22)

ˆ Fˆ , W ˆ (u) are determined by where ∇H, G, ∆H

= k,

N · (∇H − g)|

ˆ vΩ G,

= g − ∇H |

Fˆ , vΩ

=

ˆ (u), vΩ W

,N·

∂Ω F, ∇A−1 q vΩ ,

= 0,

∂Ω ∇A−1 q v∂Ω ,

−1 = −uu, ∇A−1 q vΩ − ku, Aq vΩ

2 for all w ∈ C0,σ (Ω), v = Aq w.

ˆ−W ˆ (ˆ Setting uˆ = u − ∇H, F (ˆ u) = −Fˆ − G u + ∇H), we obtain from (3.22) the equation uˆ = F (ˆ u), which can be solved by Banach’s ﬁxed point theorem. This leads to the desired solution u = u ˆ + ∇H. For this purpose we use similar estimates as in the previous proof, and obtain F (ˆ u)q,Ω

≤ C(ˆ uq,Ω + g− q1 ;q,∂Ω + kr,Ω )2 +Ckr,Ω ˆ uq,Ω + g− 1q ;q,∂Ω + kr,Ω +C F r,Ω + g− q1 ;q,∂Ω + kr,Ω

(3.23)

with C = C(Ω, q) > 0. Setting a = C, β = g− q1 ;q,∂Ω + kr,Ω , γ = C(F r,Ω + β) + β, δ = Ckr,Ω , we obtain the estimate F (ˆ u)q,Ω + β ≤ a(ˆ uq,Ω + β)2 + δ(ˆ uq,Ω + β) + γ.

(3.24)

Then we consider the closed ball B = {ˆ u ∈ Lqσ (Ω); ˆ uq,Ω + β ≤ y1 } where y1 > 0 means the smallest root of the equation y = ay 2 + δy + γ. Supposing the smallness condition 4aγ + 2δ < 1 we get y1 > β and F (ˆ u) − F(ˆ v )q,Ω ≤ a ˆˆ u − vˆq,Ω with some 0 < a ˆ < 1. Now Banach’s ﬁxed point theorem yields a unique solution uˆ ∈ B with u ˆ = F (ˆ u), see [27], V.4.2, for details. Then u = u ˆ + ∇H solves (3.22) and is a very weak solution of (1.9). The smallness condition 4aγ + 2δ < 1 can be written in the form (1.13).

128

R. Farwig, G.P. Galdi and H. Sohr

To prove uniqueness we follow [15] and assume that there exists another very weak solution v ∈ Lq (Ω) of the system (1.9) with the same data F, k, g as for u. Setting U = u − v we can show that the equation U, ∆w + v · ∇w + u · (∇w)T + kwΩ = 0

(3.25)

2 is satisﬁed for all w ∈ C0,σ (Ω), where “T ” denotes the transpose. Note that v need not satisfy any smallness condition, but that uq is small. Then, by standard arguments, we solve for each G ∈ C0∞ (Ω) the modiﬁed Stokes system

−∆w − v · ∇w − u · (∇w)T − kw + ∇π = G,

div w = 0, w| = 0 ∂Ω

1,3/2 2 to get a solution (w, ∇π) ∈ W 2,3/2 (Ω) ∩ W0 (Ω) × L3/2 (Ω). Since C0,σ (Ω) is 2,3/2 1,3/2 3/2 dense in W (Ω) ∩ W0 (Ω) ∩ Lσ (Ω), we may insert this w in (3.25) which ﬁnally shows that U = 0, u = v. This proves Theorem 2. The estimate (1.14) is an easy consequence.

4. Nonstationary very weak solutions Consider the linearized nonstationary system ut − ∆u + ∇p = f,

div u = k,

u|

∂Ω

= g,

u|

t=0

= u0

(4.1)

with data f = div F and k, g, u0 satisfying 1 F ∈ Ls 0, T ; Lr (Ω) , k ∈ Ls 0, T ; Lr (Ω) , g ∈ Ls 0, T ; W − q ,q (∂Ω)), k dx = N · g, 1∂Ω for a.a. t ∈ [0, T ), (4.2) u0 ∈ J q,s (Ω), Ω

where 1 < r ≤ q < ∞, 13 + 1q ≥ 1r , 1 < s < ∞. Modifying Deﬁnition 1 in an obvious way for the linearized case, a vector ﬁeld u ∈ Ls 0, T ; Lq (Ω) is called a very weak solution of the system (4.1) with data (4.2) if −u, wt Ω,T − u, ∆wΩ,T + g, N · ∇w∂Ω,T = u0 , w(0)Ω − F, ∇wΩ,T , (4.3) 2 w ∈ C01 [0, T ); C0,σ (Ω) , and additionally the conditions (1.6) are satisﬁed, i.e., div u = k and N · u| = N · g a.e. in (0, T ). ∂Ω

Our main result on this system reads as follows. Theorem 4. Suppose that the data F, k, g and u0 satisfy (4.2) with 1 < r ≤ q < ∞, 1 1 1 3 + q ≥ r , 1 < s < ∞. Then there exists a unique very weak solution u ∈ Ls 0, T ; Lq (Ω) of the system (4.1), satisfying −1 −1 s q q = A−1 A−1 q Pq ut ∈ L 0, T ; Lσ (Ω) , Aq Pq u ∈ C [0, T ); Lσ (Ω) , Aq Pq u| q Pq u0 t=0

Very Weak Solutions of Navier-Stokes Equations

129

and

A−1 q Pq ut q,s,Ω + uq,s,Ω ≤ C u0 J q,s (Ω) + F r,s,Ω + kr,s,Ω + g− q1 ;q,s,∂Ω (4.4) with C = C(Ω, q, r, s) > 0. ∞ (Ω) in (4.3), we obtain, Remarks 3. (1) Setting in particular w ∈ C0∞ (0, T ); C0,σ see [27], p. 248, p. 202, [28], the existence of a distribution p such that ut − ∆u + ∇p = f holds in Ω × (0, T ) in the sense of distributions. (2) Let h = (h1 , h2 , h3 ) ∈ C01 (0, T ); W 1−1/q ,q (∂Ω) with N ·h| = 0. Then ∂Ω

h(t) → w ˆh(t) , see (2.8), (2.9), is a linear mapping satisfying (wˆh )t = w ˆht . We may h insert w = w ˆ in (4.3) and obtain the formula g, h∂Ω,T = u, w ˆ ht Ω,T + u, ∆w ˆ h Ω,T − F, ∇wˆh Ω,T .

(4.5)

Since the normal component N · h of the test function h is zero, this formula yields a well-deﬁned expression for the tangential component N ×g of the boundary values. It is easy to see using integration by parts that N × g coincides with the usual trace N × u| if u is suﬃciently smooth. Therefore, we may call the right∂Ω hand side of (4.5) the trace N × u| of the tangential component of u at ∂Ω ∂Ω in the sense of distributions. Since the normal component N · u| of u at ∂Ω ∂Ω

is well deﬁned by (1.6), we get an explicit trace formula for u| in the sense of ∂Ω distributions at ∂Ω which coincides with the usual trace of u at ∂Ω if u is smooth. This yields a precise meaning of the general boundary condition u| = g in the ∂Ω sense of boundary distributions. (3) Since w(0) in (4.3) is solenoidal we expect that the initial condi= u0 only makes sense “modulo gradients”. Therefore, the condition tion u| t=0 −1 = A−1 Aq Pq u| q Pq u0 , see (1.7), seems to be the adequate precise formulation t=0 of the initial condition u| = u0 . If u is suﬃciently smooth, we need additional t=0

(necessary) compatibility conditions in order to reach that u(0) = u0 , see (4.14). Proof of Theorem 4. Let E(t) = E k(t),g(t) ∈ Lq (Ω) be the very weak solution of the stationary system −∆E(t) + ∇p(t) = 0,

div E(t) = k(t), E(t)| = g(t) for a.a. t ∈ [0, T ]. (4.6) ∂Ω

Then from the relation g, N · ∇w∂Ω,T = E, ∆wΩ,T for every (3.3) 2we obtain 1 w ∈ C0 [0, T ); C0,σ (Ω) . Therefore, given a very weak solution u, (4.3) can be written in the form −1 −1 −1 −A−1 q Pq u, vt Ω,T − u − E, ∆Aq vΩ,T = Aq Pq u0 , v(0)Ω − F, ∇Aq vΩ,T (4.7) 2 q (Ω) , v = Aq w. Since A−1 P u ∈ L (Ω), see ( 2.19 ), where w ∈ C01 [0, T ); C0,σ q 0 q σ and since div (u − E) = 0, N · (u − E)| = 0 yielding u − E = Pq (u − E) we ∂Ω

130

R. Farwig, G.P. Galdi and H. Sohr

obtain that u − E, ∆A−1 q vΩ,T

=

Pq (u − E), ∆A−1 q vΩ,T

=

u − E, Pq ∆A−1 q vΩ,T = −u − E, vΩ,T .

Further we use (2.15) for a.a. t ∈ [0, T ) and get a unique Fˆ = −A−1 q Pq div F ∈ s q L 0, T ; Lσ (Ω) satisfying the relation 2 ˆ for all v = Aq w, w ∈ C01 [0, T ); C0,σ (Ω) . (4.8) F, ∇A−1 q vΩ,T = F , vΩ,T This leads to the relation −A−1 q Pq u, vt Ω,T

= A−1 q Pq u0 , v(0)Ω − Pq u, vΩ,T +Pq E, vΩ,T +

A−1 q Pq

(4.9)

div F, vΩ,T .

Then a standard argument shows, see [28], III, 1.1 or [27], IV, 1.3, that A−1 q Pq ut ∈ Ls 0, T ; Lqσ (Ω) is well deﬁned, that q −1 A−1 A−1 q Pq u ∈ C [0, T ); Lσ (Ω) , q Pq u(0) = Aq Pq u0 , and that the evolution system −1 −1 k,g (A−1 , q Pq u)t + Aq (Aq Pq u) = Aq Pq div F + Pq E

−1 (A−1 q Pq u)(0) = Aq Pq u0 (4.10) is satisﬁed. From (2.21) we now obtain the representation formula

u ˆ(t) ≡ Pq u(t) = Aq e−tAq A−1 q Pq u0 +

(4.11) t

k,g dτ Aq e−(t−τ )Aq A−1 q Pq div F + Pq E

0

for the very weak solution u. As in (3.22) we get u ˆ(t) = Pq u(t) = u(t) − ∇H(t) where ∇H(t) is determined by ∆H(t) = k(t), N · ∇H(t) − g(t) | = 0 for a.a. ∂Ω

t ∈ [0, T ). Since ∇H(t) only depends on g(t), k(t), see (3.11), we obtain by (4.11) a formula for u = u ˆ + ∇H which determines u uniquely by the data F, k, g and u0 . Now use (4.11) to construct a very weak solution u. Using the same calculation as above we obtain the existence assertion of u, the uniqueness of which follows from the representation (4.11). The estimate (4.4) is based on (2.20) and the estimates of E, Fˆ , see (3.4) and (2.16). Using (2.20) we conclude that the term u0 J q,s (Ω) in (4.4) can be replaced by the weaker norm T 1s s q,s Aq e−tAq A−1 . (4.12) u0 JT (Ω) ≡ q Pq u0 q,Ω dt 0

Now the proof of Theorem 4 is complete.

Very Weak Solutions of Navier-Stokes Equations

131

Next we consider some regularity properties. Suppose the data f = div F and k, g, u0 of the system (4.1) satisfy the stronger conditions F ∈ Ls 0, T ; W 1,q (Ω) , k ∈ Ls 0, T ; W 1,q (Ω) , kt ∈ Ls 0, T ; Lr (Ω) , 1 g ∈ Ls 0, T ; W 2−1/q,q (∂Ω) , gt ∈ Ls 0, T ; W − q ,q (∂Ω) , u0 ∈ W 2,q (Ω) (4.13) with 1 < r ≤ q < ∞, 13 + 1q ≥ 1r , 1 < s < ∞, and the compatibility conditions k dx = N · g, 1∂Ω for t ∈ [0, T ), u0 | = g(0), div u0 = k(0). (4.14) ∂Ω

Ω

Observe that g(0) and k(0) are well deﬁned because of the assumptions on kt , gt . For simplicity the assumption on u0 is not optimally chosen. Then we will show the existence of a unique solution u ∈ Ls 0, T ; W 2,q (Ω) with ut ∈ Ls 0, T ; Lq (Ω) of the system (4.1), together with a pressure function p such that ∇p ∈ Ls 0, T ; Lq (Ω) , satisfying the estimate ut q,s,Ω + u2;q,s,Ω + ∇pq,s,Ω ≤ C u0 2;q,Ω + f q,s,Ω + k1;q,s,Ω + kt r,s,Ω + g2−1/q;q,s,∂Ω + gt −1/q;q,s,∂Ω

(4.15)

with C = C(Ω, q, s) > 0. The equations ut − ∆u + ∇p = f , div u = k and u| = g in (4.1) are satisﬁed in the strong sense for a.a. t ∈ [0, T ); hence u ∈ ∂Ω C [0, T ); Lq (Ω) and u(0) = u0 is well deﬁned. The initial value u0 ∈ W 2,q (Ω) can be treated as a functional from J q,s (Ω), see (2.19). Thus we see that (4.13), (4.14) are stronger than the conditions (4.2). Therefore, Theorem 4 yields a unique very weak solution u ∈ Ls 0, T ; Lq (Ω) to (4.1) which coincides with each more regular solution by the uniqueness property. To show the existence of a regular solution u satisfying (4.15) we ﬁrst suppose that such a solution is given. Let E = E k,g be chosen as in (4.6). Then E(0) = E k(0),g(0) satisﬁes the system −∆E(0) + ∇p(0) = 0,

div E(0) = k(0),

E(0)| = g(0) ∂Ω

and (3.17) shows that E(0) ∈ W 2,q (Ω). Using (4.14) we see that u0 − E(0)| = 0, ∂Ω div u0 − E(0) = 0 which leads to u0 − E(0) ∈ D(Aq ). Further, using Et = (E k,g )t = E kt ,gt , the assumptions on k, kt , g, gt , estimate (3.4) with u, F, k, g replaced by Et , 0, kt , gt , and the estimate (3.17) with u, f, k, g replaced by E, 0, k, g, we obtain the estimate Et q,s,Ω + E2;q,s,Ω ≤ C k1;q,s,Ω + kt r,s,Ω + g2−1/q;q,s,Ω + gt − q1 ;q,s,∂Ω with C = C(Ω, q, s) > 0.

(4.16)

132

R. Farwig, G.P. Galdi and H. Sohr Setting u ˜(t) = u(t) − E(t) we obtain the evolution system u˜t + Aq u ˜ = Pq f − Pq Et , div u ˜ = 0, u ˜| = 0, u˜| = u0 − E(0). ∂Ω t=0

Then (2.21) yields the representation formula t −tAq u0 −E(0) + u(t) = E(t)+e e−(t−τ )Aq (Pq f −Pq Eτ )dτ,

0 ≤ t < T, (4.17)

0

for the given regular solution u. In the next step we use (4.17) to construct the desired solution u, and we use (2.20), (4.16). Further we apply (3.17) to E(0), and use that k(0) = div u0 , g(0) = u0 | . This yields the regularity properties of u, the estimate (4.15) for u, ∂Ω and its uniqueness. The pressure term ∇p, constructed by de Rham’s argument, can be written in the form ∇p = f − ut + ∆u proving (4.15) for p. Proof of Theorem 1. First let u be a given solution of (1.1) for some 0 < T ≤ T with the properties of this theorem. Further we consider the solution E = E F,k,g,u0 of the corresponding linearized system Et − ∆E + ∇ˆ p = div F,

div E = k,

E|

∂Ω

= g,

E|

t=0

= u0

according to Theorem 4. Setting u ˜ = u − E, the calculation as in (3.19) shows that u ˜ is a very weak solution of the linear system u ˜t − ∆˜ u + ∇˜ p = −div W (u),

div u ˜ = 0,

u˜|

∂Ω

= 0,

u˜|

t=0

= 0,

(4.18)

where W (u) is deﬁned as in (3.19); in particular, div W (u) = div (uu) − ku. Using (3.20) we conclude that W (u) ∈ Ls/2 0, T ; Lr (Ω) for 0 < T < T . If T < ∞ we set T = T . Thus we may use the representation formula (4.11) with k = 0, g = 0, u0 = 0, F = −W (u) = −W (˜ u + E) and with s replaced by s/2. Hence t u ˜(t) = (F (˜ u)(t) := − Aq e−(t−τ )Aq A−1 0 ≤ t < T . (4.19) q Pq div W (u) dτ, 0

To solve

(4.19) by Banach’s ﬁxed point theorem we have to estimate

T 1 3 F (˜ u)q,s,Ω where ·q,s,Ω = ·q,s,Ω,T = ·sq,Ω dt s . Let α = 2q yield0 1 1 1 3 3 ing 2 − α + s = s/2 and 2α + q = q/2 . Using (2.12) and (2.11) with α = α , γ = 2q , we get that t 1 −1/2 Aq/2 Pq/2 div W (u)q/2 dτ . F (˜ u)(t)q,Ω ≤ C 1/2+α 0 (t − τ ) Looking at the integrand, we apply the estimate −1/2

Aq/2 Pq/2 (ku)q/2 ≤ CPγ (ku)γ ≤ Ckr uq

Very Weak Solutions of Navier-Stokes Equations which is based on (2.11) with α = inequality. Furthermore,

−1/2 Aq/2 Pq/2

1 2

and

1 γ

=

1 3

2 q

+

=

1 r

+

1 q

133 and on H¨ older’s

div (uu)q/2 ≤ Cuuq/2 ≤ Cu2q since

−1/2

−1/2

|Aq/2 Pq/2 div (uu), ϕΩ | = | − uu, ∇A(q/2) P(q/2) ϕΩ | ≤ Cuuq/2 ϕ(q/2)

for all ϕ ∈ L(q/2) (Ω). Summarizing we conclude that t 1 F (˜ u)(t)q,Ω ≤ C u2q + kr uq dτ . 1/2+α 0 (t − τ ) Then the Hardy-Littlewood inequality, see [27], p. 103, [29], 1s s1˜ T t T | (t − τ )α−1 h(τ ) dτ |s dt ≤C |h(t)|s˜ dt 0

0

0

− α and s˜ = 2s yields the estimate 2 (4.20) F (˜ u)q,s,Ω ≤ C ˜ uq,s,Ω + Eq,s,Ω uq,s,Ω + Eq,s,Ω + kr,s,Ω ˜

with α =

1 2

with C = C(Ω, q, s) > 0. Setting a = C, β = Eq,s,Ω , δ = Ckr,s,Ω and γ = β, (4.20) is equivalent to the estimate F (˜ u)q,s,Ω + β ≤ a(˜ uq,s,Ω + β)2 + δ(˜ uq,s,Ω + β) + γ;

(4.21)

cf. (3.24) in the proof of Theorem 2 for the stationary case. Thus, in the same way as in that proof, we obtain a solution u˜ ∈ Ls 0, T ; Lq (Ω) of the ﬁxed point equation u ˜ = F (˜ u) if the condition 4aγ + 2δ < 1, i.e., T 1s T 1s 4C Esq,Ω dt + 2C ksr,Ω dt < 1. (4.22) 0

0

is satisﬁed. Using (4.4) and (4.12) we may use also the (weaker) smallness condition T 1s T 1s −tAq −1 s Aq e Aq Pq u0 q,Ω dt + F sr,Ω dt (4.23) 0

T

ksr,Ω dt

+ 0

1s

+ 0

0 T

gs− 1 ;q,∂Ω dt

1s

q

with c = c(Ω, q, s) > 0. This condition is always satisﬁed if T > 0 is suﬃciently small; note that the case T = ∞ is possible. Writing (4.19) in the form −1 A−1 ˜(t) = A−1 q u q Pq u(t) − Aq Pq E(t) t =− e−(t−τ )Aq A−1 q Pq div W (u)dτ, 0

(4.24) 0 ≤ t < T ,

134

R. Farwig, G.P. Galdi and H. Sohr

we conclude using (2.20), (2.16), (3.20), together with H¨older’s inequality, that (A−1 ˜)t q,s/2,Ω q u

≤ ≤

C1 A−1 q Pq div W (u)q,s/2,Ω ≤ C2 W (u)r,s/2,Ω C3 u2q,s,Ω + kr,s,Ω uq,s,Ω < ∞

with Ci , i = 1, 2, 3, depending on Ω, q, s. Furthermore, we obtain from (4.4) that s q s/2 0, T ; Lqσ (Ω) , 0 < T < T . A−1 q Pq Et ∈ L 0, T ; Lσ (Ω) ⊆ L s/2 0, T ; Lqσ (Ω) for all T with 0 < T < T , and This proves that A−1 q Pq ut ∈ L all 0 < T < ∞ if T = ∞. A calculation shows that u deﬁned by u = u ˜ + E is a very weaksolution of (1.1). To prove the uniqueness of u we assume that v ∈ Ls 0, T ; Lq (Ω) is another very weak solution of (1.1). Setting U = u − v we obtain in the same way as in (4.18) that U is a very weak solution of the system Ut − ∆U + ∇P = −div (U u) − div (vU ) + kU, div U = 0,

U|

∂Ω

= 0,

U|

t=0

= 0.

(4.25)

The same method as used for (4.18) and (4.19) then leads to the estimate U q,s,Ω ≤ C(uq,s,Ω + vq,s,Ω + kr,s,Ω ) U q,s,Ω

(4.26)

with C = C(Ω, q, s) > 0. Since · q,s,Ω = · q,s,Ω,T , we observe that C does not depend on T . Thus we can choose T ∈ (0, T ) such that 1 . 2C This leads to U q,s,Ω,T ≤ 0, hence U = 0 and u = v on the interval [0, T ]. If T < T we can continue this procedure and get u = v on [0, T ) in ﬁnitely many steps. The proof of Theorem 1 is complete. uq,s,Ω,T + vq,s,Ω,T + kr,s,Ω,T ≤

References [1] Adams, R.A.: Sobolev Spaces, Academic Press, New York 1975 [2] Amann, H.: Linear and Quasilinear Parabolic Equations, Birkh¨auser Verlag, Basel 1995 [3] Amann, H.: Navier-Stokes Equations with Nonhomogeneous Dirichlet Data, Journal Nonlinear Math. Phys., 10, Suppl. 1 (2003), 1–11 [4] Amann, H.: Nonhomogeneous Navier-Stokes Equations with Integrable Low-Regularity Data, Int. Math. Ser., Kluwer Academic/Plenum Publishing, New York (2002), 1–26 [5] Berselli, L.C., and Galdi, G.P.: On the space-time regularity of C(0, T ; Ln )-very weak solutions to the Navier-Stokes equations. Nonlinear Anal., 58 (2004), 703–717 [6] Bogowski, M.E.: Solution of the First Boundary Value Problem for the Equation of Continuity of an Incompressible Medium, Soviet Math. Dokl., 20 (1979), 1094–1098

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[7] Cannone, M.: Viscous ﬂows in Besov Spaces, Advances in Mathematical Fluid Mechanics, Springer-Verlag, Berlin (2002), 1–34 [8] Fabes, E.B., Jones, B.F., and Rivi`ere, N.M.: The Initial Value Problem for the Navier-Stokes Equations with Data in Lp , Arch. Rational Mech. Anal., 45 (1972), 222–240 [9] Farwig, R., and Sohr, H.: The stationary and nonstationary Stokes system in exterior domains with non-zero divergence and non-zero boundary values, Math. Methods Appl. Sci., 17 (1994), 269–291 [10] Farwig, R., and Sohr, H.: Generalized Resolvent Estimates for the Stokes System in Bounded and Unbounded Domains, J. Math. Soc. Japan, 46 (1994), 607–643 [11] Fujiwara, D., and Marimoto, H.: An Lr -theorem of the Helmholtz Decomposition of Vector Fields, J. Fac. Sci. Univ. Tokyo (1A), 24 (1977), 685–700 [12] Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations; Linearized Steady Problems, Springer Tracts in Natural Philosophy, Vol. 38, Springer-Verlag, New York 1998 [13] Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations; Nonlinear Steady Problems, Springer Tracts in Natural Philosophy, New York 1998 [14] Galdi, G.P., Simader, C.G., and Sohr, H.: On the Stokes Problem in Lip-schitz Domains, Ann. Mat. Pura Appl., 167 (1994), 147–163 [15] Galdi, G.P., Simader, C.G., and Sohr, H.: A Class of Solutions to Stationary Stokes 1 and Navier-Stokes Equations with Boundary Data in W − q ,q (∂Ω), Math. Ann., 331 (2005), 41–74 [16] Galdi, G.P., and Simader, C.G.: Existence, uniqueness and Lq -estimates for the Stokes problem in exterior domains, Arch. Rational Mech. Anal., 112 (1990), 147– 163 [17] Giga, Y.: Analyticity of the Semigroup Generated by the Stokes Operator in Lr spaces, Math. Z., 178 (1981), 287–329 [18] Giga, Y.: Domains of Fractional Powers of the Stokes Operator in Lr -spaces, Arch. Rational Mech. Anal., 89 (1985), 251–265 [19] Giga, Y., and Sohr, H.: On the Stokes Operator in Exterior Domains, J. Fac. Sci. Univ. Tokyo, Sec. IA, 36 (1989), 103–130 [20] Giga, Y., and Sohr, H.: Abstract Lq -estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, J. Funct. Anal., 102 (1991), 72–94 [21] Kato, T.: Strong Lp -solutions to the Navier-Stokes Equations in Rm , with Applications to Weak Solutions, Math. Z., 187 (1984), 471–480 [22] Kozono, H., and Yamazaki, M.: Local and Global Solvability of the Navier-Stokes Exterior Problem with Cauchy Data in the Space Ln,∞ , Houston J. Math., 21 (1995), 755–799 ´ [23] Neˇcas, J.: Les M´ethodes Directes en Th´eorie des Equations Elliptiques, Academia, Prag 1967

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[24] Simader, C.G., and Sohr, H.: A new approach to the Helmholtz decomposition and the Neumann problem in Lq -spaces for bounded and exterior domains, Adv. Math. Appl. Sci., 11, World Scientiﬁc (1992), 1–35 [25] Simader, C.G., and Sohr, H.: The Dirichlet Problem for the Laplacian in Bounded and Unbounded Domains, Pitman, Longman Scientiﬁc, Vol. 360, 1997 [26] Solonnikov, V.A.: Estimates for Solutions of Nonstationary Navier-Stokes Equations, J. Soviet Math., 8 (1977), 467–528 [27] Sohr, H.: The Navier-Stokes Equations. An Elementary Functional Analytic Approach, Birkh¨ auser Advanced Texts, Birkh¨ auser Verlag, Basel 2001 [28] Temam, R.: Navier-Stokes Equations, North-Holland, Amsterdam, New York, Tokyo 1977 [29] Triebel, H.: Interpolation theory, function spaces, diﬀerential operators, NorthHolland, Amsterdam 1978 Reinhard Farwig Fachbereich Mathematik Darmstadt University of Technology D-64283 Darmstadt, Germany e-mail: [email protected] Giovanni P. Galdi Department of Mechanical Engineering University of Pittsburgh 15261 Pittsburgh, USA e-mail: [email protected] Hermann Sohr Fakult¨ at f¨ ur Elektrotechnik Informatik und Mathematik University of Paderborn D-33098 Paderborn, Germany e-mail: [email protected]

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 137–150 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Grow-up on the Boundary for a Semilinear Parabolic Problem Marek Fila, Juan J.L. Vel´azquez and Michael Winkler Dedicated to Herbert Amann on the occasion of his 65th birthday

Abstract. We study the asymptotic behavior of positive solutions of a semilinear parabolic equation with a nonlinear boundary condition. This problem admits a unique stationary solution which is not bounded and attracts all positive solutions. We ﬁnd their growth rate at the singular point on the boundary. Mathematics Subject Classiﬁcation (2000). 35K60, 35B40. Keywords. Semilinear parabolic equation, nonlinear boundary condition, asymptotic behavior.

1. Introduction Consider the problem ut = ∆u − aup , ∂u = uq , ∂ν u(x, 0) = u0 (x) ≥ 0,

x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0, x ∈ Ω,

where a > 0, p, q > 1 and Ω is a smoothly bounded domain in RN . The study of this problem was initiated in [5] (and independently in [17]) and continued in [2], [3], [4], [6], [7], [8], [10], [11], [13], [16], [18], [19], [20], [21], [22], for example. Earlier results on more general problems can be found in [1] and [12], for instance. Concerning blow-up and global existence, the results of [5], [21] and [2] can be summarized as follows. If p < 2q − 1

or

a < q and p = 2q − 1

138

M. Fila, J.J.L. Vel´ azquez and M. Winkler

then there are solutions which blow up in ﬁnite time, while for p > 2q − 1

a > q and p = 2q − 1

or

all solutions are global and bounded. It was also shown in [5] that for a = q,

p = 2q − 1,

N = 1 and Ω = (−1, 1)

all nontrivial solutions are global and converge pointwise (as t → ∞) to the unique solution w of the problem wxx − qw2q−1 = w(±1) =

|x| < 1,

0, ∞.

In this paper we study the growth rate of these global unbounded solutions as t → ∞. We consider the problem ut −ux

= =

uxx − pu2p−1 , up ,

ux

= =

0, u0 ≥ 0,

u|t=0

0 < x < 1, t > 0, x = 0, x = 1, 0 ≤ x ≤ 1,

(1.1)

where p > 1. Our main result is: Theorem 1.1. i) There is C1 > 0 (depending on p and u0 L∞ ((0,1)) only) such that 1

u(·, t)L∞ ((0,1)) ≤ C1 (t + 1) 2 ,

t > 0.

ii) For any ν > 0 there is C2 > 0 (depending on p, u0 and ν) such that 1

u(·, t)L∞ ((0,1)) ≥ C2 (t + 1) 2 −ν ,

t > 0.

This implies in particular that the large-time behavior is not selfsimilar. Non-selfsimilar behavior of global unbounded solutions was discussed in [9] for the problems ut = ∆u + up , 1

p−1 2 2 u= , N −2− p−1 p−1 4 √ , p≥1+ N −4−2 N −1

x ∈ B1 = {x ∈ RN | |x| < 1}, t > 0, x ∈ ∂B1 , t > 0, N > 10,

and ut = ∆u + eu ,

x ∈ BR = {x ∈ RN | |x| < R}, t > 0,

u = 0, R = 2(N − 2),

x ∈ ∂BR , t > 0, N > 10.

Grow-up on the Boundary for a Semilinear Parabolic Problem

139

These problems possess stationary solutions which are singular at the origin and stable from below. The rate at which the solutions starting below the singular stationary solution tend to inﬁnity at the origin was found in [9]. For the Cauchy problem for the equation ut

x ∈ RN , t > 0,

= ∆u + up , p≥1+

4 √ , N −4−2 N −1

N > 10,

this was done in [14] but there the growth rate depends on the spatial decay rate of initial data. The growth rate of global unbounded solutions of the problem ut

∆u + up ,

=

u =

x ∈ B1 , t > 0, x ∈ ∂B1 , t > 0,

0, p=

N +2 , N −2

N ≥ 3,

was found in [15]. In this case the global unbounded solutions do not converge to a singular stationary solution but, again, their behavior is not of selfsimilar nature. In Section 2 we ﬁnd the growth rate of solutions of (1.1) formally by matching an asymptotic development of the solution (obtained by a suitable rescaling and linearization) with the asymptotics of the singular steady state w. This matching procedure is diﬀerent from those in [9], [14] and [15]. Section 3 contains a rigorous proof of Theorem 1.1. The proof is based on comparison with suitable super- and subsolutions. Their form is suggested by the formal matched asymptotics from Section 2.

2. Matched asymptotics We begin with an analysis of the solution of the steady-state problem wxx − pw2p−1 w wx

= 0, = ∞,

0 < x < 1, x = 0,

= 0,

x = 1.

By Lemma 4.1 in [5], it is unique and we can write it as ∞ du √ , m = w(1). x= 2p u − m2p w(x) Since

∞

1= m

du √ , 2p u − m2p

(2.2)

(2.3)

140

M. Fila, J.J.L. Vel´ azquez and M. Winkler

an obvious change of variables yields m

p−1

∞

= 1

dz √ . 2p z −1

Taylor’s expansion gives √

1 1 = p u u2p − m2p

1 m2p + · · · 1+ 2 u2p

as u → ∞.

From (2.3) it then follows that x=

1 m2p 1 1 + + ··· p−1 p−1w (x) 2(3p − 1) w3p−1 (x)

as x → 0;

(2.4)

therefore, 1

w(x) ≈ αp x− p−1

1

αp = (p − 1)− p−1 .

as x → 0,

Now, (2.4) implies 3p−1 3p−1 1 1 m2p (p − 1) p−1 x p−1 + · · · = x − p − 1 wp−1 2(3p − 1)

as x → 0.

Hence, 2p−1

1

w(x) = αp x− p−1 + Kp m2p x p−1 + · · ·

as x → 0,

(2.5)

where Kp =

(p − 1)2p−1 . 2(3p − 1)

Now we make a local analysis in a region near x = 0. We introduce the following change of variables: 1

u(x, t) = (ε(t))− p−1 v(y, t),

y=

x , ε(t)

where ε(t) is deﬁned by 1

u(0, t) = αp (ε(t))− p−1 . Then v satisﬁes −

1 εεv ˙ − εεyv ˙ y + ε2 vt p−1 v

=

vyy − pv 2p−1 ,

=

αp ,

−vy

=

vp ,

vy

=

0,

1 , t > 0, ε(t) y = 0, t > 0,

0
y = 0, t > 0, 1 , t > 0. y= ε(t)

Grow-up on the Boundary for a Semilinear Parabolic Problem

141

Since ε(t) → 0 as t → ∞ and we expect that yvy and vt are bounded, we may assume that v(y, t) → v˜(y) as t → ∞, where v˜ satisﬁes v 2p−1 v˜yy − p˜ −˜ vy

= =

0, v˜p ,

y > 0, y = 0,

v˜ =

αp ,

y = 0.

Note that v˜ is given explicitly by 1

v˜(y) = αp (y + 1)− p−1 ,

y > 0.

Writing v(y, t) = v˜(y) + z(y, t)

(2.6)

and linearizing formally, we obtain that z satisﬁes the equation 1 v˜ + y˜ vy + ε2 zt = zyy − p(2p − 1)˜ −εε˙ v 2(p−1) z, p−1 that suggests the asymptotics z(y, t) ≈ εεG(y), ˙ where G solves the problem 1 − = v˜ + y˜ vy p−1 G = −Gy =

(2.7)

Gyy − p(2p − 1)˜ v 2(p−1) G, 0, p˜ v p−1 G,

y > 0, y = 0, y = 0,

which can be written as Gyy −

p p(2p − 1) G = −[(p − 1)(y + 1)]− p−1 , 2 2 (p − 1) (y + 1)

y > 0,

(2.8)

with initial conditions G(0) = Gy (0) = 0.

(2.9)

The general solution of (2.8) is 2p−1

p

G(y) = −Ap (y + 1) p−1 − Bp (y + 1)− p−1 +

p−2 1 αp (y + 1) p−1 . 2(p + 1)

The constants Ap and Bp for which (2.9) holds are Ap =

p−2 1 (p − 1) p−1 , (p + 1)(3p − 1)

Bp =

1 αp . 2(3p − 1)

It follows then that G(y) ≈ −Ap y

2p−1 p−1

as y → ∞.

(2.10)

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M. Fila, J.J.L. Vel´ azquez and M. Winkler

From (2.10) we now obtain (as y → ∞) u(x, t) =

1

ε− p−1 v(y, t) 1

≈

ε− p−1 (˜ v (y) + εεG(y)) ˙ 2p−1 1 1 ε− p−1 αp y − p−1 − εεA ˙ p y p−1

=

αp x− p−1 − Ap εε ˙ − p−1 x p−1 .

≈

1

p+1

2p−1

Since we want this asymptotics to match with (2.5), we require ε˙ = −

p+1 Kp m2p p−1 ε . Ap

Therefore, ε(t) ≈ λp t−

p−1 2

,

λp =

2K m2p − p−1 2 p . (p − 1)Ap

3. The rigorous proof of the blow-up rate In this section we prove Theorem 1.1. The main idea is to construct sub- and supersolutions of the form (2.6), (2.7) that is suggested by matched asymptotics. To obtain a subsolution we modify the function G slightly and because of this modiﬁcation we are not able to reach the power 1/2 in Theorem 1.1 ii). i) As ux satisﬁes a linear parabolic equation with smooth coeﬃcients and ux ≤ 0 in {0, 1} × (0, ∞), the maximum principle shows that ux ≤ 0 if u0 is nonincreasing and smooth. Replacing u0 by u0 L∞ ((0,1)) if necessary, we thus may assume ux ≤ 0 in (0, 1) × (0, ∞). With a diﬀerentiable function ε : [0, ∞) → (0, ∞) to be speciﬁed below, we deﬁne ' 1 ( (3.11) Q := (y, t) ∈ (0, ∞)2 | 0 < y < ε(t) and the function v : Q → R by 1

u(x, t) =: ε− p−1 (t)v

x ,t . ε(t)

(3.12)

Then P v := vt −

ε˙ 1 1 ε˙ v − yvy = 0, (vyy − pv 2p−1 ) − ε2 p−1ε ε −vy = v p , vy = 0, 1

v(y, 0) = ε− p−1 (0)u0 (ε(0)y),

(y, t) ∈ Q, y = 0, 1 , y= ε(t) 0
1 . ε(0)

Grow-up on the Boundary for a Semilinear Parabolic Problem

143

We ﬁrst claim that 1 ˙ v(y, t) ≤ w(y, t) := v˜(y) + ε(t)ε(t)G(y) 2

in Q

if we choose ε(t) := λ(t + 1)−

p−1 2

with λ > 0 small enough (see below). Indeed, we have * 1) 1 1 1 2 (ε˙ + ε¨ ˙ yy − p(˜ ˙ 2p−1 ε)G − 2 v˜yy + εεG v + εεG) Pw = 2 ε 2 2 1 ε˙ 1 1 − v˜ + y˜ vy + ε˙2 − G − yGy , ε p−1 2(p − 1) 2 where p p 1 v˜ + y˜ vy = (p − 1)− p−1 (y + 1)− p−1 . p−1

(3.13)

˙ ≥ 0, we have Assume ﬁrst that p ≥ 32 . Since εG 2p−1 1 p(2p − 1) p(2p − 1)(p − 1) 2 2 2p−3 2 p v˜ + εεG ˙ εε˜ ˙v 2p−2 G + ε ε˙ v˜ ≥ p˜ v 2p−1 + G 2 2 4 (3.14) v 2p−1 and due to the equation deﬁning G, and hence, since v˜yy = p˜ * p p p(2p − 1) 1 ε˙ ) − p−1 − p−1 −2 Gyy − (y + 1) G + 2(p − 1) (y + 1) Pw ≥ − 2ε (p − 1)2 ) * p−2 2p−3 1 p − 2 ε¨ ε 1 + ε˙2 + 2 G − yGy + p(2p − 1)(p − 1)− p−1 (y + 1)− p−1 G2 2 p − 1 ε˙ 2 p−2 2p−3 ε 1 1 2 ) p − 2 ε¨ G − yGy + p(2p − 1)(p − 1)− p−1 (y + 1)− p−1 G2 ε˙ + = 2 p − 1 ε˙2 2 * p p 1 − (p − 1)− p−1 (y + 1)− p−1 εε˙ * 2p−3 p 1 2) = ε˙ c1 G − yGy + c2 (y + 1)− p−1 G2 + c3 λ−2 (t + 1)p (y + 1)− p−1 2 * 2p−3 p 1 2) ≥ (3.15) ε˙ c1 G + c2 (y + 1)− p−1 G2 + c3 λ−2 (y + 1)− p−1 2 with c1 =

2p−1 p−1 ,

2p−1

p−2

c2 = 12 p(2p − 1)(p − 1)− p−1 > 0 and c3 = 2(p − 1)− p−1 > 0. As 2p−1

|G(y)| ≤ c(y + 1) p−1 and 2p−3

2p+1

(y + 1)− p−1 G2 (y) ≥ c(y + 1) p−1

2p−3

for all y ≥ 0, we can ﬁnd y0 > 0 such that c1 G(y) + c2 (y + 1)− p−1 G2 (y) ≥ 0 for y ≥ y0 and hence (3.16) P w ≥ 0 whenever y ≥ y0 .

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M. Fila, J.J.L. Vel´ azquez and M. Winkler

On the other hand, if λ2 ≤

−1 p c3 · min G(y) (y0 + 1)− p−1 c1 y∈[0,y0 ] p

then c1 G(y)+c3 λ−2 (y+1)− p−1 ≥ 0 for y ∈ (0, y0 ), implying P w ≥ 0 for y ∈ (0, y0 ). We thus have proved P w ≥ 0 in Q. (3.17) In the case 1 0 and all ξ ≥ 0. Accordingly, in the set ' ( 1 ˙ v −1 (y)G(y) ≤ 1 , Q1 := (y, t) ∈ Q | ε(t)ε(t)˜ 2 we have an estimate similar to (3.14) – except for a smaller but still positive constant factor in the last term – and we conclude as before that if λ is small enough then P w ≥ 0 in Q1 . (3.18) If (y, t) ∈ Q \ Q1 , however, (3.14) is replaced by 2p−1 p(p − 1) 1 ˙ εε˜ ˙v 2p−2 G + pc4 |εε| p v˜ + εεG ≥ p˜ v 2p−1 + ˙ 2p−1 |G|2p−1 , 2 2 so that instead of (3.15) we ﬁnd * p p p(2p − 1) 1 ε˙ ) − p−1 − p−1 −2 Gyy − (y + 1) G + 2(p − 1) (y + 1) Pw ≥ − 2ε (p − 1)2 ) * 1 p − 2 ε¨ ε + ε˙2 + 2 G − yGy + 2pc4 |εε| ˙ 2p−3 |G|2p−1 . 2 p − 1 ε˙ 2 3−2p −2(3−2p) 2 3−2p Using that |εε| ˙ 2p−3 = ( p−1 ) λ (t + 1)p(3−2p) ≥ ( p−1 ) for λ ≤ 1 and t ≥ 0, we arrive at * p 1 ) P w ≥ ε˙2 c1 G + c˜2 |G|2p−1 + c3 λ−2 (y + 1)− p−1 2 2 3−2p with c1 , c3 as above and c˜2 = 2p( p−1 ) c4 > 0. As 2p − 1 > 1, we again have p

c1 G+ c˜2 |G|2p−1 ≥ 0 for y ≥ y˜0 and some y˜0 > 0, while c1 G+ c3 λ−2 (y + 1)− p−1 ≥ 0 for y ∈ (0, y˜0 ) if λ is small enough. Hence, Pw ≥ 0

in Q \ Q1 ,

(3.19)

so that in fact (3.17) continues to hold for p ∈ (1, 32 ). To check the corresponding inequalities on the boundary, we ﬁrst note that −wy = −˜ vy = v˜p = wp

for y = 0,

(3.20)

Grow-up on the Boundary for a Semilinear Parabolic Problem while at y =

145

1 ε(t) ,

1 wy = v˜y + εεG ˙ y. 2 Due to the asymptotics of G, we can ﬁnd y1 large and c5 > 0 such that Gy (y) ≤ p 1 −c5 (y + 1) p−1 for all y > y1 . Thus, if λ < y11 then y = ε(t) > y1 and ) * p p 2p wy ≥ (p − 1)− p−1 (y + 1)− p−1 −1 − c6 εε(y ˙ + 1) p−1 ) * p p p+1 ≥ (p − 1)− p−1 (y + 1)− p−1 −1 − c6 ε− p−1 ε˙ ) * p p 2 p−1 c6 λ− p−1 ≥ (p − 1)− p−1 (y + 1)− p−1 −1 + 2 ≥ 0, (3.21) provided that λ is small enough. Finally, at t = 0 we have 1

v(y, 0) =

λ p−1 u0 (λy) 1

≤

λ p−1 u0 L∞ ((0,1)) ,

and, since by interpolation G(y) ≤ − w(y, 0)

Ap 2 (y

2p−1

1

+ 1) p−1 + c7 (y + 1)− p−1 ,

1 1 p−1 2 λ G(y) = (p − 1)− p−1 (y + 1)− p−1 − 4 * ) 2p−1 1 1 p−1 (p − 1)Ap 2 ≥ (p − 1)− p−1 − c7 λ2 (y + 1)− p−1 + λ (y + 1) p−1 . 4 4

Hence, for λ small enough we have v≤w

at t = 0

(3.22)

if u0 L∞ ((0,1))

≤

1

1

2p−3

2p−1

c8 λ− p−1 (y + 1)− p−1 + c9 λ p−1 (y + 1) p−1

=: ϕ(y). Minimizing with respect to y ∈ [0, ∞), we obtain that 1

) 1 with c10 = (2p − 1) 2p

ϕ(y) ≥ c10 λ− p(p−1) * 2p−1 1 2p−1 + (2p − 1)− 2p c8 2p c92p and thus (3.22) is satisﬁed if λ

lies below some positive constant depending on p and u0 L∞ ((0,1)) only. Altogether, we gain from (3.17), (3.20)–(3.22) and the comparison principle that x 1 u(x, t) ≤ ε− p−1 (t)w ,t in (0, 1) × (0, ∞), ε(t)

146

M. Fila, J.J.L. Vel´ azquez and M. Winkler

whence in accordance with our assumption on the monotonicity of u with respect to x, 1

1

1

u(t)L∞ ((0,1)) = u(0, t) ≤ ε− p−1 (t) = λ− p−1 (t + 1) 2 , which completes the proof of part i). ii) We ﬁrst note that for small positive δ, the solutions Gδ of Gδyy − (1 + δ)

p p p(2p − 1) (y + 1)−2 Gδ + (p − 1)− p−1 (y + 1)− p−1 = 0 in (0, ∞), 2 (p − 1) Gδ (0) = Gδy (0) = 0,

have the form 2p−1

p

p−2

Gδ (y) = −Apδ (y + 1) p−1 +ηδ − Bpδ (y + 1)− p−1 −ηδ + Cpδ (y + 1) p−1 , 2 2 p(2p−1) 1 − where ηδ = 14 + (1 + δ) p(2p−1) 2 (p−1) 4 + (p−1)2 # 0 and Apδ , Bpδ and Cpδ converge to Ap , Bp and Cp , respectively, as δ # 0. Therefore, setting β := p−1 2 − Apδ Bpδ Cpδ 1 (p − 1)ν < p−1 , we can ﬁx δ > 0 small such that ≤ , , ≤ 2 and 2 2 Ap Bp Cp 1 2 ηδ < β − p−1 . Using ε(t) := λ(t + λκ )−β ,

t ≥ 0,

2 with λ suﬃciently large and κ ∈ ( 2β+1 , β1 ) arbitrary but ﬁxed, we shall this time compare v in Q (see (3.11) and (3.12)) with

w(y, t) := v˜(y) + 2ε(t)ε(t)G ˙ δ (y). Then Pw

* 1) ε)Gδ − 2 v˜yy + 2εεG ˙ δyy − p(˜ v + 2εεG ˙ δ )2p−1 = 2(ε˙2 + ε¨ ε ε˙ 1 2 − v˜ + y˜ vy + ε˙2 − Gδ − 2yGδy . ε p−1 p−1

(3.23) 1

Since for γ > 1 we have (1 + ξ)γ ≤ 1 + (1 + δ)γξ for 0 ≤ ξ ≤ ξδ := (1 + δ) γ−1 − 1, the estimate v 2p−1 + 2(1 + δ) · p(2p − 1)εε˜ ˙v 2p−2 Gδ p(˜ v + 2εεG ˙ δ )2p−1 ≤ p˜

(3.24)

holds whenever |2εε˜ ˙v −1 Gδ | ≤ ξδ . But 1

|2εε˜ ˙v −1 Gδ | ≤ c11 λ− β ε

2β+1 β

2p

(y + 1) p−1 +ηδ ,

so that if (y, t) ∈ Q fulﬁls ε(t) ≤ 1 then, since y + 1 ≤ |2εε˜ ˙v −1 Gδ | ≤ ≤ ≤

1

c12 λ− β · ε

1 ε(t)

+1≤

(3.25) 2 ε(t)

in this case,

2β+1 2p β − p−1 −ηδ

1

c12 λ− β ξδ

(3.26)

Grow-up on the Boundary for a Semilinear Parabolic Problem

147

2p for λ suﬃciently large, where we have used that 2β+1 − p−1 − ηδ > 0 due to our β restriction on δ (and thus of ηδ ). If, conversely, (y, t) ∈ Q with ε(t) > 1 then y + 1 ≤ 2 and therefore

|2εε˜ ˙v −1 Gδ |

1

≤ c13 λ− β · ε

2β+1 β

= c13 λ2 (t + λκ )−2β−1 ≤ c13 λ2−(2β+1)κ ≤ ξδ

(3.27)

2 for large λ, since κ > 2β+1 . Having thus proved (3.24) in all of Q, we use it in (3.23) to infer that * p p p(2p − 1) 1 ε˙ ) − p−1 − p−1 −2 (p − 1) (y + 1) G + (y + 1) P w ≤ −2 Gδyy − (1 + δ) δ ε (p − 1)2 2 ) p − 2 ε¨ * ε 2 Gδ − yGδy + +2ε˙ p − 1 ε˙2 ) p − 2 ε¨ * p p ε 1 + 2 Gδ − yGδy + (p − 1)− p−1 (y + 1)− p−1 . = 2ε˙2 p − 1 ε˙ 2εε˙

Since p − 2 ε¨ ε yGδy + − p − 1 ε˙2 Gδ

→ =

p − 2 β + 1 2p − 1 + − − ηδ p−1 β p−1 2 1 − − ηδ > 0 as y → ∞, β p−1

we can ﬁnd y2 > 0 such that P w ≤ 0 whenever y ≥ y2 .

(3.28)

But −εε˙ = βλ2 (t + λκ )−2β−1 ≤ βλ2−(2β+1)κ for t ≥ 0, so that, since κ > ) * p 1 P w ≤ 2ε˙2 max −yGδy (y) − (p − 1)− p−1 λ(2β+1)κ−2 y∈[0,y2 ] 2β ≤ 0 for y < y2 , provided that λ is large enough. Evidently, −wy = wp

for y = 0,

1 then we use the estimate Gδy (y) ≥ −c14 (y + 1) while if y = ε(t) any y > 0, in deriving

wy

= ≤

2 2β+1 ,

(3.29)

(3.30) p p−1 +ηδ

, valid for

v˜y + 2εεG ˙ δy

) * p p 2β+1 2p 1 (p − 1)− p−1 (y + 1)− p−1 −1 + c15 λ− β ε β (y + 1) p−1 +ηδ .

The positive term in the large brackets is a multiple of the right-hand side in (3.25), so that the same ideas which lead to (3.26) and (3.27) also apply here to

148

M. Fila, J.J.L. Vel´ azquez and M. Winkler

yield for large λ 1 . (3.31) ε(t) To achieve the ‘suitable’ inequality also at t = 0, we observe that if λ ≥ 1 and 1 (y, 0) ∈ Q then y + 1 ≤ ε(0) + 1 = λβκ−1 + 1 ≤ 2 due to κ < β1 , so that wy ≤ 0 if y =

w(y, 0)

= v˜(y) + 2ε(0)ε(0)G ˙ δ (y) 1

2p−1

1

≤ (p − 1)− p−1 (y + 1)− p−1 + c16 λ2−(2β+1)κ (y + 1) p−1 +ηδ 2p−1

1

≤ (p − 1)− p−1 + c16 · 2 p−1 +ηδ λ2−(2β+1)κ ≤ c17 , 2 . On the other hand, the strong maxwhere we have once more used κ > 2β+1 imum principle guarantees that without loss of generality we may assume m := min u0 (x) > 0 (otherwise we take t + 1 as initial time) and hence obtain

x∈[0,1]

λ

1−βκ p−1

≥

λ

1−βκ p−1

≥

c17

v(y, 0) =

u0 (λ1−βκ y) m

for any (y, 0) ∈ Q and λ suﬃciently large, because κ < β1 . Thus, v ≥ w at t = 0 and accordingly, due to (3.28)–(3.31) and the comparison principle, v ≥ w in Q. In particular, 1

1

β

u(t)L∞ ((0,1)) ≥ ε− p−1 (t)w(0) = λ− p−1 (t + λκ ) p−1 , which yields the assertion, since

β p−1

=

1 2

− ν.

Acknowledgement Part of this work was done while the third author was a postdoc at the Comenius University supported by the European Community’s Human Potential Programme under contract HPRN-CT-2002-00274, Fronts - Singularities. The ﬁrst author was partially supported by VEGA Grant 1/0259/03. The authors thank Victor A. Galaktionov for a stimulating discussion.

References [1] H. Amann, Parabolic evolution equations and nonlinear boundary conditions, J. Differ. Equations 72 (1988), 201–269. [2] F. Andreu, J.M. Maz´ on, J. Toledo and J.D. Rossi, Porous medium equation with absorption and a nonlinear boundary condition, Nonlinear Anal. TMA 49 (2002), 541–563. [3] J.M. Arrieta and A. Rodr´ıguez-Bernal, Localisation near the boundary of blow-up for reaction-diﬀusion equations with nonlinear boundary conditions, Comm. Part. Diﬀ. Equations 29 (2004), 1127–1148.

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[4] T.K. Boni, Sur l’explosion et le comportement asymptotique de la solution d’une ´equation parabolique semi-lin´ eaire du second ordre, C. R. Acad. Sci. Paris, S´erie I 326 (1998), 317–322. [5] M. Chipot, M. Fila and P. Quittner, Stationary solutions, blow-up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions, Acta Math. Comenianae 60 (1991), 35–103. [6] M. Chipot and P. Quittner, Equilibria, connecting orbits and a priori bounds for semilinear parabolic equations with nonlinear boundary conditions, J. Dynamics Differ. Equations 16 (2004), 91–138. [7] M. Chipot and M. Ramaswamy, Semilinear elliptic problems with nonlinear boundary conditions, Diﬀer. Equ. Dyn. Syst 6 (1998), 51–75. [8] M. Chipot and F. Voirol, On a class of nonlinear elliptic problems with Neumann boundary conditions growing like a power, Z. Anal. Anwend. 14 (1995), 853–868. [9] J.W. Dold, V.A. Galaktionov, A.A. Lacey and J.L. V´azquez, Rate of approach to a singular steady state in quasilinear reaction-diﬀusion equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci 26 (1998), 663–687. [10] Y.V. Egorov and V.A. Kondratiev, Two theorems on blow-up solutions for semilinear parabolic equations of second order, C. R. Acad. Sci. Paris, S´erie I 327 (1998), 47–52. [11] Y.V. Egorov and V.A. Kondratiev, On blowing-up solutions to second-order parabolic equations, Doklady Math. 59 (1999), 90–92. [12] J. Escher, Global existence and nonexistence for semilinear parabolic systems with nonlinear boundary conditions, Math. Ann. 284 (1989), 285–305. [13] M. Fila and B. Kawohl, Large time behavior of solutions to a quasilinear parabolic equation with a nonlinear boundary condition, Adv. Math. Sci. Appl. 11 (2001), 113–126. [14] M. Fila, M. Winkler and E. Yanagida, Grow-up rate of solutions for a supercritical semilinear diﬀusion equation, J. Diﬀer. Equations 205 (2004), 365–389. [15] V.A. Galaktionov and J.R. King, Composite structure of global unbounded solutions of nonlinear heat equations with critical Sobolev exponents, J. Diﬀer. Equations 189 (2003), 199–233. [16] A.A. Lacey, J.R. Ockendon, J. Sabina and D. Salazar, Perturbation analysis of a semilinear parabolic problem with nonlinear boundary conditions, Rocky Mt. J. of Math. 26 (1996), 195–212. [17] J. L´ opez G´ omez, V. M´ arquez and N. Wolanski, Dynamic behavior of positive solutions to reaction-diﬀusion problems with nonlinear absorption through the boundary, Rev. Uni´ on Mat. Argent. 38 (1993), 196–209. [18] K. Pﬂ¨ uger, On indeﬁnite nonlinear Neumann problems, Partial diﬀerential and integral equations (H.G.W. Begehr et al, eds.), Kluwer Academic Publishers: Dordrecht (1999), 335–346. [19] P. Quittner, On global existence and stationary solutions for two classes of semilinear parabolic problems, Comment. Math. Univ. Carolinae 34 (1993), 105–124. [20] A. Rodr´ıguez-Bernal, Attractors for parabolic equations with nonlinear boundary conditions, critical exponents, and singular initial data, J. Diﬀer. Equations 181 (2002), 165–196.

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[21] A. Rodr´ıguez-Bernal and A. Tajdine, Nonlinear balance for reaction-diﬀusion equations under nonlinear boundary conditions: dissipativity and blow-up, J. Diﬀer. Equations 169 (2001), 332–372. [22] J.D. Rossi, The blow-up rate for a semilinear parabolic equation with a nonlinear boundary condition, Acta Math. Univ. Comenianae 67 (1998), 343–350. Marek Fila Institute of Applied Mathematics Comenius University, Mlynsk´ a dolina 84248 Bratislava, Slovakia Juan J.L. Vel´ azquez Departamento de Matem´ atica Aplicada Facultad de Ciencias Matem´ aticas Universidad Complutense E-28040 Madrid, Spain Michael Winkler Department of Mathematics I RWTH Aachen D-52056 Aachen, Germany

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 151–188 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Existence and Uniqueness Results for Reaction-diﬀusion Processes of Electrically Charged Species H. Gajewski and I.V. Skrypnik Dedicated to Prof. Herbert Amann on the occasion of his 65th birthday

Abstract. We study initial-boundary value problems for elliptic-parabolic systems of nonlinear partial diﬀerential equations describing drift-diﬀusion- reaction processes of electrically charged species in N -dimensional bounded Lipschitzian domains. We include Fermi-Dirac statistics and admit nonsmooth material coeﬃcients. We prove existence and uniqueness of bounded global solutions. Mathematics Subject Classiﬁcation (2000). 35B45, 35K15, 35K20, 35K65. Keywords. Nonlinear elliptic-parabolic systems, nonlocal drift, global bounded solutions, uniqueness, nonstandard assumptions, degenerate type.

1. Introduction We study a mathematical model describing drift-diﬀusion-reaction processes of electrically charged species. Such processes play an important role in many branches of modern technology (see [4], [13], [14], [17]). The classical drift-diﬀusion model of charged carrier transport in semiconductors was established by van Roosbroeck [16]. It consists of a Poisson equation for the electrostatic potential v0 and continuity equations for the densities u1 , u2 of electrons and holes. The classical van Roosbroeck model describes processes in homogeneous semiconductor materials (e.g., silicon). Modern devices are often heterostructures where complex reactions take place. By this reason we admit nonsmooth data and n (≥ 2) species with densities ui and speciﬁc charges qi . The mathematical model is formulated below in Section 2. It consists of a Poisson equation (2.1) and n continuity equations (2.2). The equation (2.1) expresses the Gauss law, the system (2.2) means local carrier conservation. The

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system (2.1), (2.2) is completed by current relations (2.3), which suppose the antigradients of the electrochemical potentials ζi from (2.5) to be driving forces for carrier transport. We consider the system (2.1), (2.2) in a bounded Lipschitzian domain Ω ⊂ RN , N ≥ 3, completed by boundary and initial conditions (2.7)–(2.9). The initial-boundary value problem (2.1), (2.2), (2.7)–(2.9) was formulated and studied in [8]. In that paper the free energy was recovered to be Lyapunov functional of the system and was used for proving a priori estimates, existence and uniqueness results. However, for space dimension N ≥ 3 a gap remained between existence and uniqueness results in particular for the physically most relevant case that the state relation (2.4) is according to Fermi-Dirac statistics. Actually for N ≥ 3 the paper [8] rests on following restricting hypotheses: The existence result holds for dielectric permittivity from (2.1) and conductivity functions di from (2.3) such that = constant ,

di (t, x, z, ξ) = σi (z) ξ ,

(1.1)

that means, Ji = −σi (vi ) ∇ζi . Moreover, except for the special case of Boltzmann statistics (i.e., ei in (2.4) is speciﬁed as exponential function), the uniqueness result in [8] supposes the electrostatic potential v0 to satisfy the regularity condition ∇v0 ∈ L∞ (0, T ); Lp (Ω) for some p > N . (1.2) (As to the validity of (1.2) in some nonsmooth situations comp. [5].) The present paper mainly aims to ﬁll that gap by proving global existence and uniqueness results without the restricting hypotheses (1.1), (1.2). To this end we apply to problem (2.1), (2.2), (2.7)–(2.9) an approach developed for model situations in our papers [9], [10], [11], [12]. The key role play sophisticated test functions in integral identities for proving a priori estimates and the uniqueness result. The paper is organized as follows. Formulations of all hypotheses and main results are contained in Section 2. Integral estimates for the chemical potentials vi and the electrostatical potential v0 are proved in Section 3. In Section 4 we study the boundedness of the potentials vi and v0 . A proof of the existence result is sketched in Section 5, where also a detailed proof of our main result, the uniqueness theorem, is given. Finally, Section 6 is devoted to the special case of functions di being linear with respect to ζ. This case is studied without growth conditions for the reaction terms.

2. Formulation of model and main results The drift-diﬀusion model describing n species with densities ui and speciﬁc charges qi was formulated in [6], [7], [8] and reads as follows −∇ · (∇v0 ) = f +

n

qi ui

in QT = Ω × (0, T ) ,

(2.1)

i = 1, . . . , n ,

(2.2)

i=1

∂ui + ∇ · Ji + Ri = 0 in QT , ∂t

Existence and Uniqueness Results for Reaction-diﬀusion Processes

153

where T is a ﬁnite time and Ω is a bounded Lipschitzian domain in RN . We suppose later on that N ≥ 3. In (2.1) v0 is the electrostatical potential, is the dielectric permittivity, f describes external sources (impurities). The currents Ji are given in the form Ji = −di (·, vi , ∇ζi ), i = 1, . . . , n , (2.3) where vi are chemical potentials related to the densities ui by the state equations ui = u∗i ei (vi ),

i = 1, . . . , n ,

with given, strictly positive state densities tentials ζi from (2.3) are deﬁned by ζi = qi v0 + vi ,

u∗i

(2.4)

∞

∈ L (Ω). The electrochemical po-

i = 1, . . . , n.

(2.5)

Remark 2.1. The state equations 2.4 are chosen for simplicity. The results of the paper remain true for state equations like ui = u∗i ei (vi + qi gi ),

i = 1, . . . , n ,

with given band edges gi ∈ H (Ω), q > N . This can be seen by replacing the argument vi = ζi − qi v0 of ei and its derivatives by vi = ζi − qi v¯0 , where v¯0 = v0 − gi . However, an extension of our results to piece-wise constant gi ’s in view of heterostructures is not trivial. That makes a diﬀerence to the peculiar situation of Boltzmann statistics, i.e., ei = exp, where even the case gi ∈ L∞ (Ω) can be simply included by setting u∗i := u∗i exp(qi gi ) ∈ L∞ (Ω). 1,q

The reaction terms Ri in (2.2) have the form: % & Ri (·, v, ζ) = rαβ (·, v, α · ζ) − rαβ (·, v, β · ζ) (αi − βi ) ,

(2.6)

(α,β)∈R

where α = (α1 , . . . , αn ), β = (β1 , . . . , βn ) ∈ R ⊂ Rn are vectors of stoichiometric coeﬃcients and the ﬁnite set R denotes the reactions actually taking place in the volume Ω occupied by the species. Remark 2.2. There are modiﬁed drift-diﬀusion models of charged species. So in the papers [3] and [4] Poisson’s equation (2.1) is replaced by the neutrality condition n qi ui = 0 on QT = Ω × (0, T ) . f+ i=1

We complete system (2.1), (2.2) by boundary and initial conditions : ν · Ji + RiΓ = 0 on ΓT = (0, T ) × ∂Ω,

(2.7)

ν · (∇v0 ) + κv0 = f on ΓT , (2.8) (2.9) ui (0, ·) = hi in Ω , i = 1, . . . , n, where ν(x ) is the outer unit normal at x ∈ ∂Ω , RiΓ represents reactions taking place on the boundary ∂Ω of Ω. We assume that Γ Γ rαβ (·, v, α · ζ) − rαβ (·, v, β · ζ) (αi − βi ) , (2.10) RiΓ = Γ

(α,β)∈RΓ

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H. Gajewski and I.V. Skrypnik

where RΓ is a ﬁnite set of vector pairs of stoichiometric coeﬃcients and the funcΓ tions rαβ model surface reaction rates. Remark 2.3. As a special feature the boundary condition (2.8) with (2.10) allows thermal equilibria, i.e., steady states with vanishing driving forces ∇ζi . However, the results of the paper remain true for other kinds of boundary conditions, for example ν · Ji + κi (ζi − fiΓ ) = 0

on

ΓT = (0, T ) × ∂Ω,

with κi , fiΓ ∈ L∞ (∂Ω), κi ≥ 0. The system (2.1), (2.2), (2.7)–(2.9) will be solved for the unknown vector v = (v0 , v1 , . . . , vn ) taking into account the relations (2.3)–(2.5) between v and J = (J1 , . . . , Jn ), u = (u1 , . . . , un ), ζ = (ζ1 , . . . , ζn ), respectively. We assume the data of problem (2.1), (2.2), (2.7)–(2.9) to satisfy following hypotheses: Γ i) dij (t, x, z, ξ), i = 1, . . . , n, j = 1, . . . , N, rαβ (t, x, v, y), rγδ (t, x , v, y), Γ (α, β) ∈ R, (γ, δ) ∈ R , are measurable functions of (t, x) ∈ QT , (t, x ) ∈ ΓT with respect to Lebesgue and surface measures respectively for every z, y ∈ R1 , ξ ∈ RN , v ∈ Rn+1 and continuous functions with respect to y, z, ξ, v for almost every (t, x) ∈ QT , (t, x ) ∈ ΓT , di (t, x, z, 0) = 0 for i = 1, . . . , n; and u∗i are measurable functions on Ω; κ ∈ L∞ (∂Ω); κ ≥ 0, κ = 0; qi ∈ R1 ; ii) ei ∈ (R1 → R1 ) is continuously diﬀerentiable such that ei (z) > 0, z ∈ R1 ;

0 lim ei (z) = 0, lim ei (z) = + ∞, −∞ ei (z) dz < ∞, i = 1, . . . , n; z→−∞

z→+∞

iii) there exist positive constants ν1 , ν2 such that for arbitrary ξ, ξ , ξ ∈ RN , (t, x) ∈ QT , z ∈ R1 N % & dij (t, x, z, ξ ) − dij (t, x, z, ξ ) (ξj − ξj ) ≥ ν1 ei (z)|ξ − ξ |2 , j=1

dij (t, x, z, ξ) ≤ ν2 (1 + |ξ|)e (z); e (z) ≤ ν2 ei (z) for i i ν1 ≤ (x) ≤ ν2 , ν1 ≤ u∗i (x) ≤ ν2 ,

i = 1, . . . , n;

z < 0,

j = 1, . . . , N ;

Γ iv) the functions rαβ (t, x, v, y), rγδ (t, x , v, y), (α, β) ∈ R, (γ, δ) ∈ RΓ , are 1 increasing in y ∈ R for (t, x) ∈ QT , (t, x ) ∈ ΓT , v ∈ Rn+1 and there exist convex functions M : R1 → R1+ = {z ∈ R1 : z > 0}, M Γ : R1 → R1+ such that

[rαβ (t, x, v, α · ζ) − rαβ (t, x, v, β · ζ)](α − β) · ζ ≤ M (|v|), Γ Γ [rγδ (t, x , v, γ · ζ) − rγδ (t, x , v, δ · ζ)](γ − δ) · ζ ≤ M Γ (|v|),

ζi = vi + qi v0 .

Existence and Uniqueness Results for Reaction-diﬀusion Processes

155

Finally, we assume the data f, f Γ , hi to satisfy: ∂f N ∈ L2 0, T ; [W 1,2(Ω)]∗ , p1 > , f ∈ C [0, T ]; Lp1 (Ω) , ∂t 2 Γ 1 ∂f (2.11) f Γ ∈ C [0, T ]; Lp2 (∂Ω) , ∈ L2 0, T ; [W 2 ,2 (∂Ω)]∗ , p2 > N − 1 , ∂t log(hi ) ∈ L∞ (Ω), i = 1, . . . , n. Deﬁnition 2.1. A vector v = (v0 , . . . , vn ) is called solution of problem (2.1), (2.2), (2.7)–(2.9), if for i = 1, . . . , n: , i) v0 ∈ C [0, T ]; W 1,2 (Ω) , vi ∈ L2 0, T ; W 1,2 (Ω) % &∗ ∂ ∗ 2 2 , ui = ui ei (vi ) ∈ C [0, T ]; L (Ω) , ∂t ui ∈ L 0, T ; W 1,2 (Ω) where the time derivative is to be understood in the sense of distributions, ' ( ) ∂v 2 ∂v 2 * i 0 + M (|v|) dx dt < ∞ , ei (vi ) + ∂x ∂x QT

(2.12) M (|v|) dx dt < ∞ ; Γ

ΓT

ii) for arbitrary test functions ϕ ∈ C ∞ (QT ), ψ ∈ C ∞ (Ω), almost every τ ∈ (0, T ) and i = 1, . . . , n the following integral identities hold: τ N ∂(vi + qi v0 ) ∂ϕ ∂ui , ϕ + dij t, x, vi , ∂t ∂x ∂xj 0 Ω j=1 (2.13) Γ + Ri (t, x, v, ζ) ϕ dx + Ri (t, x, v, ζ) ϕ ds dt = 0 , ∂Ω

N n ∂v0 ∂ψ − qi ui + f (t, x) ψ dx (x) ∂xj ∂xj Ω j=1 i=1 (2.14) + κ(x)v0 − f Γ ψ ds = 0 , ∂Ω where ζ(t, x) = ζ1 (t, x), . . . , ζn (t, x) , ζi (t, x) = vi (t, x) + qi v0 (t, x); iii) for test functions ϕ ∈ C ∞ QT with ϕ(τ, x) = 0, x ∈ Ω, the integral identity τ τ ∂ui ∂ϕ , ϕdt + dx dt = 0 (2.15) [ui − hi ] ∂t ∂t 0 0 Ω holds for τ ∈ (0, T ),

i = 1, . . . , n.

Besides of (2.2), (2.7) we shall consider for δ ∈ [0, 1] the regularized equations ∂ui (δ) + ∇ · Ji + Ri = 0 in ∂t (δ)

ν · Ji

+ RiΓ = 0 ,

QT ,

on ΓT ,

(2.16) (2.17)

156

H. Gajewski and I.V. Skrypnik (δ)

Ji

(δ) = −di ·, vi , ∇ζi ,

(δ)

vi

' 1( (0) = max vi , − , Ji = Ji , δ

i = 1, . . . , n.

(2.18) Solutions of problem (2.1), (2.16), (2.17), (2.8), (2.9) are deﬁned as in Deﬁnition 2.1. In what follows we understand as known parameters the numbers ν1 , ν2 , n, N , T , vectors in R, RΓ , norms of the data f , f Γ , hi in respective spaces and numbers that depend only on Ω, M , M Γ and κ. Moreover, we denote by ck , k = 1, . . ., constants depending only on known parameters. Theorem 2.1. Let the conditions i)–iv), (2.11) be satisﬁed. Then there exists a constant K1 depending only on known parameters and independent of δ ∈ [0, 1] such that each solution v of problem (2.1), (2.16), (2.17), (2.8), (2.9) satisﬁes

∂v0 (t, x) 2 2 dx + ess supt∈(0,T ) κ(x)v (t, x) ds Λi vi (t, x) + 0 ∂x Ω ∂Ω 2 ∂(vi + qi v0 ) dx dt ≤ K1 , + ei (vi ) ∂x QT

(2.19) where

v

Λi (v) =

s ei (s) ds,

i = 1, . . . , n.

(2.20)

0

For establishing further integral estimates we need growth conditions for the functions ei , Ri , RiΓ , i = 1, . . . , n: ν3 (v γi + 1) ≤ ei (v) ≤ ν4 (v γi + 1), Ri (t, x, v, ζ) ≥ −ν4

n

v > 0,

[vj ]p+3

+ |v0 |

− α1 (t, x),

p3

4 , N −2

(2.21)

for vi > 0 ,

(2.22)

0 ≤ γi <

j=1

RiΓ (t, x , v, ζ) ≥ −ν4

n

[vj ]p+4 + |v0 |p4

− α2 (t, x ),

for vi > 0 ,

(2.23)

j=1

where ν3 , ν4 are positive constants and 2 1 (γ∗ + 2), p4 < γ∗ + 1 + (γ∗ + 2), γ∗ = min(γ1 , . . . , γn ), N N N +2 r1 r2 α1 ∈ L (QT ), r1 > , α2 ∈ L (ΓT ), r2 > N + 1. 2

p3 < γ∗ + 1 +

Remark 2.4. The growth condition (2.21) is satisﬁed by functions ei according to Fermi-Dirac statistics, i.e., by Fermi Integrals: ∞ 1 sγi ds . ei (v) = Fγi (v) = Γ(γi + 1) 0 1 + exp(s − v)

Existence and Uniqueness Results for Reaction-diﬀusion Processes

157

Note that the exponential function (Boltzmann statistics) violates (2.21). Standard reaction terms like Shockley-Read and Auger recombination/generation [8] satisfy (2.22)–(2.23). We understand numbers from conditions (2.21)–(2.23) and norms of the functions α1 , α2 as known parameters too. Theorem 2.2. Let the assumptions of Theorem 2.1 and the conditions (2.21)–(2.23) be satisﬁed. Then there exists a constant K2 depending only on known parameters and independent of δ ∈ [0, 1] such that each solution v of problem (2.1), (2.16), (2.17), (2.8), (2.9) satisﬁes ∂vi 2 ∂v0 2 dx dt ≤ K2 , i = 1, . . . , n. ei (vi ) + (2.24) ∂x ∂x QT

Theorem 2.3. Let the assumptions of Theorem 2.2 be satisﬁed. Then there exist constants K3 and η ∈ (0, 1) depending only on known parameters and independent of δ such that for arbitrary t ∈ [0, T ], x, y ∈ Ω v0 L∞ (QT ) ≤ K3 ,

|v0 (t, x) − v0 (t, y)| ≤ K3 |x − y|η .

(2.25)

In view of controlling vi (t, x) from below we suppose additionally to (2.22) and (2.23) that for vi < 0, i = 1, . . . , n: % & (2.26) Ri (t, x, v, ζ) ≤ ν4 ei (vi ) F v0 , e(v) + α1 (t, x) , % & RiΓ (t, x , v, ζ) ≤ ν4 ei (vi ) F v0 , e(v) + α2 (t, x ) (2.27) with ν4 , α1 (t, x), α2 (t, x ) as in (2.21), (2.22), e(v) = (e1 (v1 ), . . . , en (vn )) and some continuous function F : Rn+1 → R1 . Theorem 2.4. Let the assumptions of Theorem 2.2 and the conditions (2.26), (2.27) be satisﬁed. Then there exists a constant K4 depending only on known parameters and independent of δ ∈ [0, 1] such that for each solution v = (v0 , v1 , . . . , vn ) of problem (2.1), (2.16), (2.17), (2.8), (2.9) (2.28) ess sup |vi (t, x)| : (t, x) ∈ QT ≤ K4 , i = 0, . . . , n . Theorem 2.5. Let the conditions i)–iv), (2.11), (2.21)–(2.23), (2.26), (2.27) be satisﬁed. Then the initial-boundary value problem (2.1), (2.2), (2.7)–(2.9) has at least one solution in the sense of the Deﬁnition 2.1. Theorem 2.6. Let the conditions of Theorem 2.5 be satisﬁed. Assume additionally that for i = 1, . . . , n, j = 1, . . . , N : (i) the functions dij (t, x, z, ξ) have the special structure dij (t, x, z, ξ) = ei (z)γij (t, x, ξ) ei

e−1 i

(2.29)

where ◦ : (0, ∞) → (0, ∞) is piece-wise diﬀerentiable and concave; Γ (ii) the functions ei , γij (t, x, ξ), rαβ (t, x, v, y), rγδ (t, x , v, y) are locally Lipschitzian with respect to ξ, v, y.

158

H. Gajewski and I.V. Skrypnik

Then the initial-boundary value problem (2.1), (2.2), (2.7)–(2.9) has a unique solution in the sense of the Deﬁnition 2.1. Remark 2.5. The Fermi integrals from Remark 2.4 satisfy the respective assumptions of Theorem 2.6 [1]. Corollary 2.1. Let the conditions of Theorem 2.6 be satisﬁed and assume additionΓ are Lipschitzian with respect to t. ally that the functions f, f Γ , dij , rαβ , rαβ Then the solution v of problem (2.1), (2.2), (2.7)–(2.9) is regular in the sense that t→t

∂vi ∈ L∞ (0, T ; L2 (Ω)) ∩ L2 (0, T ; W 1,2(Ω)), i = 1, . . . , n. ∂t

Remark 2.6. Corollary 2.1 and Theorem 2.4 imply that t→t

∂ui ∈ L∞ (0, T ; L2(Ω)). ∂t

Consequently, (2.2) can be understood not only in the sense of distributions, but even as an equation in L2 (0, T ; L2 (Ω)). We conclude this section considering the special case that the currents Ji are linear with respect to the gradients of the electrochemical potentials ζi . This case is interesting in so far as we do not need the growth restrictions (2.22), (2.23) for the reaction terms. Theorem 2.7. Let the conditions i)–iv), (2.11), (2.21), (2.26), (2.27) be satisﬁed. Suppose that the reference densities from (2.4) and the exponents γi from (2.21) satisfy u∗i = u∗ ,

γi = γ ,

i = 1, . . . , n

(2.30)

and that for (α, β) ∈ R, (γ, δ) ∈ RΓ α · q = β · q,

γ·q = δ·q .

(2.31)

Moreover, assume the functions dij to have the structure dij (t, x, z, ξ) =

N

ei (z)akj (t, x)ξk ,

i = 1, . . . , n,

j = 1, . . . , N.

(2.32)

k=1

Then all assertions of the Theorems 2.2–2.6 are valid. Remark 2.7. We assumed the coincidence of the γi ’s for simplicity. It is possible to replace it by some restriction on max{|γi −γj |, 1 ≤ i, j ≤ n}. Analogously to [11] it is possible to prove Theorem 2.7 for γi satisfying only 0 ≤ γi < N 2−2 , i = 1, . . . , n. We shall prove the Theorems 2.1, 2.2 in Section 3, the Theorems 2.3, 2.4 in Section 4 and the Theorems 2.5, 2.6 in Section 5. Finally we shall make some comments with respect to the proof of Theorem 2.7 in Section 6.

Existence and Uniqueness Results for Reaction-diﬀusion Processes

159

3. Proof of integral estimates The proof of a priori estimates in this section rests on testing the integral identities (2.13), (2.14) by suitable functions. For that purpose the following remark is useful: Remark 3.1. Let F : Rn+1 → R1 be an arbitrary piece-wise diﬀerentiable function with bounded gradient and let v(t, x) be a solution of problem (2.1), (2.16), (2.17), (2.8), (2.9). Then the equality (2.13) holds for ϕ(t, x) = F (v(t, x)). Moreover, (2.14) holds for arbitrary functions ψ ∈ W 1,2 (Ω). That follows from (2.12) after approximating v(t, x) by smooth functions. Proof of Theorem 2.1. Let v be a solution of problem (2.1), (2.16), (2.17), (2.8), (2.9). Denote by g0 (x) a solution of the problem −∇ · (∇g0 ) = f (0, x) +

n

qi hi (x)

in Ω ,

(3.1)

i=1

ν · (∇g0 ) + κg0 = f Γ (0, x ) on ∂Ω .

(3.2)

We extend vi(t, x) for t < 0, x ∈ Ω by setting vi (t, x) = gi (x), where hi (x) u∗ (x) , i = 1, . . . , n. In an analogous way we extend f (t, x) and

gi (x) = e−1 i

i

f Γ (t, x ). Testing the integral identity (2.14) with ψ(x) = v0 (t + s, x) − v0 (t, x) and integrating on t, we obtain for τ ∈ (0, T ), s ∈ (0, T − τ ), τ N * ∂ % & ∂ ) v0 (t + s, x) + v0 (t, x) v0 (t + s, x) − v0 (t, x) (x) ∂x ∂x j j −s Ω j=1

n qi ui (t + s, x) + ui (t, x) + f (t + s, x) + f (t, x) − (3.3) i=1 τ % & & % × v0 (t + s, x) − v0 (t, x) dx dt + κ(x) v0 (t + s, x) + v0 (t, x) −s ∂Ω % & − f Γ (t + s, x) − f Γ (t, x) v0 (t + s, x) − v0 (t, x) ds dt = 0 . Arguing as in the proof of Theorem 2.3 in [11], we infer from (3.3) τ n ∂v0 (τ, x) 2 ∂ui 2 , v0 dt (x) dx + κ(x)v0 (τ, x) ds − qi ∂x ∂t 0 Ω ∂Ω i=1 τ τ ∂v0 (τ, x) 2 2 dx dt + (x) κ(x)v (t, x) ds dt . ≤ c1 1 + 0 ∂x 0 0 Ω ∂Ω Note that W 1,2 (Ω) can be normed equivalently by 12 ∂u(x) 2 2 dx + κ(x)u (x) ds . ∂x Ω ∂Ω

(3.4)

160

H. Gajewski and I.V. Skrypnik

Remark 3.1 allows us to test the regularized version of 2.13 with ϕ = vi + qi v0 :

τ

0

N ∂(vi + qi v0 ) ∂ϕ ∂ui , ϕ + dij t, x, viδ , ∂t ∂x ∂xj Ω j=1 + Ri (t, x, v, ζ)ϕ dx dt τ + RiΓ (t, x, v, ζ)ϕ ds dt = 0. 0

(3.5)

∂Ω

So, using (3.4), we get n τ ∂v0 (τ, x) 2 ∂ui 2 , vi (x) κ(x)v0 (τ, x) ds + dx + ∂x ∂t Ω ∂Ω i=1 0 N ∂ (δ) ∂(vi + qi v0 ) + dij t, x, vi , (vi + qi v0 ) ∂x ∂x j Ω j=1 + Ri (t, x, v, ζ)(vi + qi v0 ) dx dt τ RiΓ (t, x, v, ζ)(vi + qi v0 ) ds dt +

0

∂Ω τ

≤ c2 1 + 0

Ω

∂v0 (t, x) 2 (x) dx dt + ∂x

0

τ

(3.6)

κ(x)v02 (t, x)

ds dt .

∂Ω

i We transform the integral with ∂u ∂t by means of Lemma 1 and Lemma 3 from [10] and obtain τ % & ∂ui , vi > dt = < u∗i (x) Λi vi (τ, x) − Λi gi (x) dx. (3.7) ∂t 0 Ω

Estimating terms with Ri , RiΓ by means of condition iv), we get n i=1

=

Ri (t, x, v, ζ)(vi + qi v0 ) %

& rα,β (t, x, v, α · ζ) − rαβ (t, x, v, β · ζ) · (α − β) · ζ ≥ 0,

(α,β)∈R n i=1

=

(3.8) RiΓ (t, x , v, ζ)(vi

(α,β)∈RΓ

+ qi v0 )

% Γ & Γ rαβ (t, x , v, α · ζ) − rαβ (t, x , v, β · ζ) (α − β) · ζ ≥ 0.

Existence and Uniqueness Results for Reaction-diﬀusion Processes By condition iii) we obtain from (3.6)–(3.8) ∂v0 (τ, x) 2 dx + (x) κ(x)v02 (τ, x) dx ∂x Ω ∂Ω 2 n τ ∂(v + q i i v0 ) + ei (vi ) dx dt ∂x i=1

0

Ω

≤ c2 1 +

τ

0

161

(3.9)

τ ∂v0 (t, x) 2 2 dx dt + (x) κ(x)v (t, x) ds dt . 0 ∂x 0 Ω ∂Ω

The last inequality and Gronwall’s lemma imply (2.19) and the proof of Theorem 2.1 is complete. Lemma 3.1. Let the conditions of Theorem 2.1 be satisﬁed. Suppose that ess supt∈(0,T ) uri (t, x) dx ≤ L1 for i = 1, . . . , n , (3.10) Ω N with numbers r ∈ N2N and L1 depending only on known parameters. Then +2 , 2 2 NpN ∂v (t, x) p−2 0 v0 (t, x) −2 + v0 (t, x) ess supt∈(0,T ) dx ∂x Ω (3.11) −1) p(N N −2 v0 (t, x) + ds ≤ L2 , ∂Ω

where the constant L2 depends only on known parameters and p is deﬁned by r N = (p − 1) . (3.12) p· N −2 r−1 Proof. For arbitrary functions w we deﬁne wk (t, x) = min w(t, x), k , k ∈ R1 , (t, x) ∈ QT . (3.13) Testing the integral identity (2.14) with ψ(t, x) = |v0 (t, x)|p−1 sign v0 (t, x), k > 0, k using the conditions iii), (2.11), (3.10) and H¨ older’s inequality, we obtain ∂|v0 |k 2 dx + |v0 |p−2 κ(x)|v0 |pk ds k ∂x Ω ∂Ω (r−1) r−1 N −1) r r(N −1) r (p−1) r(N (p−1) r−1 N (r−1) ≤ c4 |v0 |k dx + c4 |v0 |k ds . Ω

∂Ω

(3.14) Hence Sobolev’s embedding theorem yields −2 NN−2 N N −1 p N p N −1 |v0 |k N −2 dx + |v0 |k N −2 ds Ω

≤ c5 Ω

∂Ω

r (p−1) r−1

|v0 |k

r−1 r dx + c5 ∂Ω

−1) (p−1) r(N N (r−1) |v0 |k

(3.15)

(r−1) N r(N −1)

ds

.

162

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In view of the restriction on r and (3.12) we infer (3.11) from (3.14) and (3.15) letting k → ∞. The proof of Lemma 3.1 is completed. In what follows we suppose the conditions (2.21)–(2.23) to be satisﬁed. We ﬁx a ∆ ∈ (0, 1) such that 2 (γ∗ + 2) − p3 , N 1 ∆ ≤ 1 + γ∗ + (γ∗ + 2) − p4 , N ∆N 4 γ ∗ = max{γ1 , . . . , γn } ≤ − N −2 N −2 ∆ ≤ 1 + γ∗ +

(3.16)

and deﬁne r(m) = ∆m ,

m = 0, 1, 2 . . . .

(3.17)

Lemma 3.2. Let the conditions of Theorem 2.2 be satisﬁed. Suppose that for some nonnegative integer m

2 2+r(m) v0 (t, x)r(m) ∂v0 dx dt + κ(x)v0 (t, x) ds dt ≤ L3 , ∂x ΓT

QT

&r(m) % vi (t, x) +

∂vi 2 ∂x dx dt ≤ L3 ,

(3.18) i = 1, . . . , n ,

QT

with [vi (t, x)]+ = max{vi (t, x), 0} and a constant L3 depending only on known parameters and m. Then there exists a constant L4 depending only on known parameters and m such that

2 r(m+1)+2 v0 (t, x)r(m+1) ∂v0 dx dt + κ(x)v0 (t, x) ds dt ≤ L4 . (3.19) ∂x ΓT

QT

Proof. Remark that by condition (2.21) ei (v) ≤ c6 v γi +1 , v ≥ 1,

Λi (v) ≥ c6 v γi +2 ,

i = 1, . . . , n,

(3.20)

where the function Λi is deﬁned by (2.20). From (2.19), (3.20) we have ess supt∈(0,T ) Ω

&γi +2 % vi (t, x) + dx ≤ c7 ,

i = 1, . . . , n.

(3.21)

Existence and Uniqueness Results for Reaction-diﬀusion Processes

163

r(m+1)+1 Testing the integral identity (2.14) with ψ(t, x) = v0 (t, x)k sign v0 (t, x) and using condition iii) and (2.11) we have 2 r(m+1) ∂|v0 |k r(m+1)+2 |v0 |k dx dt + κ(x)|v0 |k ds dt ∂x QT

≤ c8

ΓT

n

r(m+1)+1

ui |v0 |k

i=1 Q T

T

T

0

Ω

v0 (t, x)[r(m+1)+1]p2 ds k

+ 0

with pi =

pi pi −1 ,

(3.22)

1 p v0 (t, x)[r(m+1)+1]p1 dx 1 dt k

+

dx dt

∂Ω

1 p 2

dt

i = 1, 2. The embedding theorem and (3.18) imply

T

[r(m)+2] NN −2

|v0 | 0

Ω

T

−1 [r(m)+2] N N −2

|v0 |

+ 0

NN−2 dx dt −2 N N −1

ds

(3.23) dt ≤ c9 .

∂Ω

Hence we can estimate the second and the third integral on the right-hand side of (3.22) by a constant depending only on known parameters. In order to estimate the ﬁrst integral on the right-hand side of (3.22) we derive ﬁrstly an auxiliary estimate for vi (t, x). By H¨ older’s inequality, the embedding theorem, (3.18) and (3.21) we obtain with an arbitrary number q ∈ 0, NN−2 : 0

T

%

Ω

T

1q &(γi +2)[1−q NN−2 ]+[r(m)+2]q vi (t, x) + dx dt

%

≤ 0

Ω

1q − NN−2 &γi +2 vi (t, x) + dx

%

·

T

Ω

%

≤ 0

Ω

NN−2 & [r(m)+2]N N −2 vi (t, x) + dx dt

(3.24)

1q − NN−2 &γi +2 vi (t, x) + dx

· Ω

%

&r(m) ∂vi 2 dx dt ≤ c10 . vi (t, x) + ∂x

Let us choose the number q∗ such that % & % & N , r(m + 1) + 1 q∗ = r(m) + 2 N −2

q∗ =

q∗ . q∗ − 1

(3.25)

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Since ∆ ∈ (0, 1) and r(m) = m∆, we have q∗ > (2.4) and (2.21), we get r(m+1)+1 ui |v0 |k dx dt QT

T

[r(m)+2] NN −2

≤ c11 0

Ω

T

|v0 |k

×

0 T

Ω

Using H¨ older’s inequality,

1 q1 q∗ ∗ (γ +1)q∗ dx dx dt 1 + [vi ]+ i Ω

[r(m)+2] NN −2

≤ c11

N N −2 .

|v0 |k

NN−2 N q∗ (N −2) dx dt

(3.26)

1 1− N N (q∗ −1) q∗ (N −2) q∗ − (γi +1)q∗ N −2 dx dt . 1 + [vi ]+

0

Ω

Let q = q∗ − NN−2 (q∗ − 1) = NN−2 − N2−2 q∗ ∈ 0, NN−2 with q∗ deﬁned by (3.25). Since % & % & 1 N · = q∗ r(m)−r(m+1)+1 = q∗ (1−∆) , r(m)+2 q = q∗ r(m)+2 1− N − 2 q∗ we have by (3.16) ) & N − 2* % + r(m) + 2 q − (γi + 1)q∗ (γi + 2) 1 − q N )2 * & q∗ % (γi + 2) − γi − ∆ q∗ = 4 − N ∆ − γi (N − 2) ≥ 0 . = N N The inequalities (3.18), (3.21), (3.24), (3.26), (3.27) imply r(m+1)+1 ui |v0 |k dx dt ≤ c12 .

(3.27)

(3.28)

QT

So we obtain the desired estimate (3.19) from (3.22), (3.23) and (3.28). This ends the proof of Lemma 3.2. Lemma 3.3. Suppose that the assumptions of Theorem 2.2 and the inequalities (3.18) are satisﬁed for a nonnegative integer m such that γi0 ≥ r(m + 1) ,

i0 ∈ {1, . . . , n} .

(3.29)

Then there exists a constant L5 depending only on known parameters such that % &r(m+1) ∂vi0 (t, x) 2 dx dt ≤ L5 . vi0 (t, x) + (3.30) ∂x QT

Proof. For arbitrary functions w1 (t, x), w2 (t, x) deﬁned on QT we deﬁne the set w1 ≤ w2 = (t, x) ∈ QT : w1 (t, x) ≤ w2 (t, x) .

Existence and Uniqueness Results for Reaction-diﬀusion Processes By (2.19) and (3.19) we have % &r(m+1) ∂vi0 (t, x) 2 dx dt ≤ c13 . vi0 (t, x) + ∂x {[vi0 ]+ ≤2|v0 |}

165

(3.31)

To complete the proof we need an analogous estimate with respect to {[vi0 ]+ > 2|v0 |}. Testing the identity (2.14) with '% (r(m)+ε & ψ = |v0 |k [vi0 − |v0 |k ]+ k + |v0 |k sign v0 , ε ∈ (0, ∆], k > 1, and using condition iii) and (2.11), we obtain % & I1 + I2 ≤ c14 I3 (1) + I4 + I5 where

'%

I1 =

[vi0 − |v0 |k ]+

{|v0 |

κ(x)|v0 |2k

I2 = ΓT

(r(m)+ε ∂v 2 0 dx dt, + |v0 |k ∂x k

'% (r(m)+ε & ds dt, [vi0 − |v0 |k ]+ k + |v0 |k

|v0 |lk

I3 (l) =

&

(3.32)

'%

[vi0 − |v0 |k ]+

&

(r(m)+ε−l ∂v ∂v i0 0 dx dt, + |v0 |k ∂x ∂x k

{|v0 |k

I5 =

i=1

'% (r(m)+ε Γ & f (t, x)|v0 |k [vi0 − |v0 |k ]+ + |v0 |k ds dt. k

ΓT

Up to the end of Lemma 3.3 we choose ε = ∆. We estimate I3 (l) for natural numbers l < r(m + 1) by Young’s inequality '% (r(m+1)−2l ∂v 2 & c15 i0 dx dt [v |v0 |2l − |v | ] + |v | I3 (l) ≤ ε1 I1 + i0 0 k + k 0 k k ∂x ε1 {|v0 |k
(3.33) where ε1 is an arbitrary positive number. Using the simple inequality 2 ∂vi0 2 ≤ c16 ∂(vi0 + qi0 v0 ) + ∂vi0 ∂v0 , ∂x ∂x ∂x ∂x

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H. Gajewski and I.V. Skrypnik

we have from (3.33), (2.19), (3.19) and (3.29) 2 c17 r(m+1) ∂v0 I3 (l) ≤ ε1 I1 + |v0 | dx dt 1 + I3 (2l) + ε1 ∂x +

QT

∂(vi0 + qi0 v0 ) 2 dx dt ≤ ε1 I1 + c18 1 + I3 (2l) . [vi0 ]γ+i ∂x ε1

(3.34)

QT

The inequalities (2.19), (3.19) imply also I3 (l) ≤ c19 for l > r(m + 1). Therefore, iterating (3.34), we get 1 I1 + c19 . (3.35) I3 (1) < 2c14 Next we estimate the term I4 by H¨ older’s inequality and condition (2.11): 1 T n p1 I4 ≤c20 |vi |γi +2+r(m+1) dx dt + |vi |[r(m+1)+1]p1 dx dt 0

i=1 Q T

+

n

r(m+1)+1 |v0 |k ui

i=1 Q T

T

Ω

dx dt + 0

Ω

[r(m+1)+1]p1 |v0 |k

1 p1 dx dt + 1 . (3.36)

Now all integrals in (3.36) can be estimated from above by a constant depending only on known parameters. Indeed, since by (3.16) % & 2 (γi + 2) + r(m) + 2 − γi + 2 + r(m + 1) = N & 1% 2 4 − (N − 2)γi − N ∆ ≥ 0, = (γi + 2) − γi − ∆ = N N an estimate of the ﬁrst integral in (3.36) follows from (3.24) with q = 1. In analogous way the second integral in (3.36) can be estimated by means of (3.24) with q = p1 . Estimates for the third and the fourth integral in (3.36) follow from (3.28) and (3.23), respectively. So we have shown that I4 ≤ c21 .

(3.37)

Further, condition (2.11), (3.18), (3.23), (3.24) and the embedding theorem yield −2 N T N −1 −1 % &[r(m)+2] N N −2 |vi0 | + |v0 |k I5 ≤ c22 ds dt + 1 0

∂Ω

2

r(m) ∂vi0 2 ≤ c23 + |v | |vi0 | i 0 ∂x QT

+ |v0 |

r(m)

∂v0 2 + |v0 |2 dx dt + 1 ≤ c24 . ∂x

(3.38)

Existence and Uniqueness Results for Reaction-diﬀusion Processes Now (3.32), (3.35), (3.37), (3.38) and (3.19) imply 2 r(m+1) ∂v0 [vi0 ] ∂x dx dt ≤ c25 .

167

(3.39)

{vi0 >|v0 |}

Finally, the desired inequality (3.30) follows from (2.19), (3.31), (3.39) and the proof of Lemma 3.3 is completed. Lemma 3.4. Let the assumptions of Theorem 2.2 be satisﬁed and suppose that the inequalities (3.18) hold for some nonnegative integer m. Moreover, let m and i0 be such that (3.40) γi0 < r(m + 1) and suppose that % % &r(m)+2 &r(m) ∂v0 2 dx dt ≤ L6 , (3.41) vi (t, x) + vi (t, x) + ess supt∈(0,T ) dx + ∂x Ω QT

with a constant L6 depending only on m and known parameters. Then there exists a constant L7 depending only on the same parameters such that % &r(m+1)+2 vi0 (t, x) + ess supt∈(0,T ) dx +

Ω

%

&r(m+1) ∂v0 2 ∂vi0 2 vi0 (t, x) + dx dt ≤ L7 . ∂x + ∂x

(3.42)

QT

Proof. We start by proving that &r(m)+ε ∂v0 2 % vi (t, x) + ∂x dx dt ≤ c26

(3.43)

QT ∆ . We want to apply (3.32) with this ε. Since in the for i = i0 and ε = ε1 = r(m)+2 proof of Lemma 3.3 I4 , I5 have been estimated without using assumption (3.29), we can suppose (3.38) to be hold. Further, (3.19) holds also true. To estimate I3 (l) with the chosen ε we apply Young’s inequality, (3.18), (3.41) and (3.19): 2 ∂v0 2 r(m) ∂vi0 I3 (1) ≤ c27 + [vi0 ]+ ∂x ∂x QT

+

r(m)+1 |v0 |k

'% ([r(m)+1]ε−1 ∂v 2 & 0 dx dt ≤ c28 . [vi0 − |v0 |k ]+ k + |v0 |k ∂x (3.44)

Now the inequalities (3.31), (3.32), (3.37), (3.38) and (3.44) imply (3.43) for i = i0 . Note that this estimate follows in the same way for i = 1, . . . , n.

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The key for continuing our previous discussions is following estimate &r(m)+ε+2 &r(m)+ε ∂vi 2 % % ess supt∈(0,T ) dx + vi (t, x) + vi (t, x) ∂x dx dt ≤ c29 . Ω QT

(3.45) Indeed, it can be seen from the proof of the Lemma 3.2, that (3.19) and (3.45) imply (3.19) with r(m+1)+ε instead of r(m+1). This ensures that (3.43) remains 2∆ and even for further steps. true for ε = ε2 = r(m)+2 The estimate (3.45) follows immediately from (3.43) and (2.19) provided r(m) + ε ≤ γi . So it remains to prove (3.45) for the case that r(m) + ε > γi . To this end we test integral identity (3.5) with ρ )% )% & * & *2 1 , ρ>− , ϕ = ei (vi ) − ei (m0 ) + a(ρ) + ei (vi ) − ei (m0 ) + 2 k(i) k(i) where m0 z+

−1 hi (x) ; x ∈ Ω, = ess sup ei u∗i (x) = max(z, 0), [s]k(i) = min(s, k(i)),

k(i) = ei (k) − ei (m0 ) for k > m0 ;

i = 1, . . . , n ,

a(ρ) = 1 for ρ ≤ 1, a(ρ) = 0 for ρ > 1 .

Then, using Lemma 2 from [10], we can evaluate the ﬁrst term: τ ∂ui (ρ) , ϕ > dt = < u∗ (x)Λk,i ei vi (τ, x) − ei (m0 ) dx , ∂t 0 Ω where (ρ) Λk,i (z)

=

z

' (ρ 2 [s]k(i) a(ρ) + [s]k(i) ds ≥

0

+

1 [z+ ]2ρ+2 k(i) . 2(ρ + 1)

(3.46)

(3.47)

We write the space derivative of ϕ in the form ∂vi ∂ϕ (ρ) = Φk,i (vi ) χ(m0 < vi < k) , ∂xj ∂xj

(3.48)

where χ(m0 < vi < k) is the characteristic function of the set {m0 < vi < k} and (ρ) the function Φk,i (vi ) satisﬁes for ρ > − 12 the estimate ' % &2 ( ρ (ρ) ≤ Φk,i (vi ) ρ∗ ei (vi ) a(ρ) + ei (vi ) − ei (m0 ) ' % &2 ( ρ ≤ c31 (1 + ρ) ei (vi ) a(ρ) + ei (vi ) − ei (m0 ) (3.49) for m0 < vi < k with ρ∗ = min(1; 1 + 2ρ).

Existence and Uniqueness Results for Reaction-diﬀusion Processes

169

Using (3.46)–(3.49) and the conditions iii), (2.20)–(2.22), we obtain from (3.5) with the chosen test function ϕ ) % Ω

& *2(γi +1)(ρ+1) vi (τ, x) + dx

) %

τ

+

Ω

0

≤ c32

ρ+1 ρ∗

k

& *2γi +2(γi +1)ρ ∂vi 2 vi (t, x) + ∂x χ(m0 < vi < k) dx dt k

2 0

τ

) % & *2γi +2(γi +1)ρ ∂v0 2 vi (t, x) + ∂x k Ω

× (1 + ρ) χ(m0 < vi < k) dx dt 2

+

n 0

j=1

τ

+ 0

%

τ

Ω

) % Ω

τ

0

τ

∂Ω

1 r1

k

∂Ω j=0

+ 0

& *(γi +1)(1+2ρ)r1 vi (t, x) + dx dt

n %

+

(γi +1)(1+2ρ)+p3 &(γi +1)(1+2ρ)+p3 + v0 (t, x) vj (t, x) + dx dt

)%

&(γi +1)(1+2ρ)+p4 vj (t, x) + ds dt

& *(γi +1)(1+2ρ)r2 vi (t, x) + ds dt

1 r2

+1 .

(3.50)

k

To continue the proof of the inequality (3.45) we choose ρ such that 2γi + 2(γi + 1)ρ = r(m) + ε

(3.51)

and estimate the right-hand side of (3.50) integral by integral. An estimation of the ﬁrst one follows from (3.43). Note that by (3.51) and (3.16) ) * 2 (γi + 1)(1 + 2ρ) + p3 = r(m) + 2 + (γ∗ + 2) N * ) 2 + p3 − 1 − γi − (γ∗ + 2) + ε N < r(m) + 2 +

2 (γ∗ + 2). N

Hence estimates for the vj terms, j = 1, . . . , n, of the second integral on the righthand side of (3.50) follow from (3.24) with q = 1. Taking into account (3.11), the v0 term can be estimated by the same arguments.

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In order to estimate the third integral we use the next inequality that follows analogously to the inequality (3.24): % &(r(m)+2)(1+ N2 ) vi (t, x) dx dt QT

N2 NN−2 % &r(m)+2 &(r(m)+2) NN−2 ≤ vi (t, x) + vi (t, x) + dx dx dt 0 Ω Ω N2 (3.52) &r(m)+2 % &2 % ≤ c33 r(m) + 2 ess supt∈(0,T ) dx vi (t, x) + Ω % &r(m) ∂vi 2 % &r(m)+2 vi (t, x) + × + vi (t, x) + dx dt. ∂x

T

%

QT

It is simple to check that (γi + 1)(1 + 2ρ)r1 < (r(m) + 2 (1 + N2 ), such that the third integral can be estimated by means of (3.52), (3.41), (3.18). To estimate the last integrals in (3.50) we note ﬁrstly following auxiliary inequality that follows analogously to the inequality (3.52): % % &2 &(r(m)+2)(1+ N1 ) vi (t, x) + ds dt ≤ c34 r(m) + 2 NN+1 N2N +1 &r(m)+ N2+1 ∂vi % × dx dt vi (t, x) + ∂x + [vi ]+ 0 Ω N1 % &2 % &r(m)+2 vi (t, x) + dx ≤ c35 r(m) + 2 ess supt∈(0,T ) 2 Ω % &r(m) ∂vi % &2 vi (t, x) + × ∂x + vi (t, x) + dx dt.

ΓT

T

(3.53)

QT

Then analogous arguments as used for proving (3.24) and (3.53) lead to % &r(m)+2+ N1 (γi +2) vi (t, x) + ds dt ≤ c36 .

(3.54)

ΓT

Since by (3.51) and (3.16) 1 (γ∗ + 2) , N (3.54) implies an estimation for the fourth integral on the right-hand side of (3.50). Finally, (3.53) implies an estimate for the last integral in (3.50). With (3.50) the key estimate (3.45) is fully proved. This ends the proof of Lemma 3.4. (γi + 1)(1 + 2ρ) + p4 < r(m) + 2 +

Proof of Theorem 2.2. Remark that for m = 0 the conditions (3.18), (3.41) follow from Lemma 3.1 and Theorem 2.1. Starting from m = 0, we can iterate the application of the Lemmas 3.2–3.4. After M + 1 steps we arrive at the inequalities (3.19) and (3.42) with m = M . Taking M so large that ∆ · (M + 1) ≥ γ ∗ , we get Theorem 2.2.

Existence and Uniqueness Results for Reaction-diﬀusion Processes

171

4. Proof of L∞ -estimates Proof of Theorem 2.3. We apply Lemma 3.4 with m = M and M such that r(M + 1) + 2 > N2 (γ ∗ + 1), γ ∗ = max(γ1 , . . . , γn ). Then Theorem 2.3 follows immediately from (3.42), conditions i), iii), (2.11) and well-known results on the regularity of solutions of linear elliptic equations (see, for example [15]) to Poisson’s equation (2.1). In what follows we assume the conditions of Theorem 2.4 to be satisﬁed. We shall estimate for vi , i = 1, . . . , n, separately on the sets {vi > 0} (Lemma 4.1) and {vi < 0} (Lemma 4.2). Lemma 4.1. Let the condition of Theorem 2.4 be satisﬁed. Then there exists a constant L8 depending only on known parameters such that for i = 1, . . . , n, (4.1) ess sup vi (t, x) : (t, x) ∈ QT ≤ L8 . Proof. Using Lemma 3.4 and (3.50) we get for r ≥ r∗ = 2 + 4 max(γ1 , . . . , γn ) τ % % &r+2 &r ∂vi 2 dx dt vi (τ, x) + dx + vi (t, x) + ∂x 0 Ω Ω τ % &r ∂v0 2 2 dx dt ≤ c37 r 1 + vi (t, x) + ∂x 0 Ω

τ 1

τ 1 r r % % &(r+1)r1 &(r+1)r2 1 2 + vi (t, x) + vi (t, x) + dx dt + ds dt . 0

0

Ω

∂Ω

(4.2) Remark only that Lemma 3.4 gives us the estimate of [vj ]p+3 in Lr1 (QT ), j = 1, . . . , n. We start estimating the ﬁrst integral on the right-hand side of (4.2). Let {ϕj ∈ C ∞ (RN ), j = 1, . . . , J, } be a partition of unity such that J

ϕ2j (x)

j=1

JR

N

= 1,

∂ϕj c0 ∂x ≤ R

% &N ≤ c0 d(Ω) ,

for x ∈ Ω,

R < 1,

supp ϕj ⊂ B(xj , R),

J χ B(xj , R) ≤ c0 ,

(4.3)

j=1

where B(xj , R) is the ball of radius R with center xj ∈ Ω, c0 is a constant depending only on N , d(Ω) is the diameter of Ω, χ(B(xj , R)) is the characteristic function of B(xj , R). The radius R will be ﬁxed later.

172

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We test the integral identity (2.14) with % &r % & Ψ(t, x) = vi (t, x) + · v0 (t, x) − v0,l (t) ϕ2l (x), v0,l (t) = v0 (t, xl ).

(4.4)

Integration with respect to t and summing up on l yield τ % &r ∂v0 2 vi (t, x) + dx dt ≤ c38 r I1 (r) + I2 (r) ∂x Ω 0 1 1 τ τ p1 p2 % % &rp1 &rp2 vi (t, x) + dx vi (t, x) + ds dt + dt , + 0

Ω

0

∂Ω

(4.5) where

∂vi ∂v0 &r−1 dx dt , vi (t, x) + v0 (t, x) − v0,l (t)ϕ2l (x) ∂x ∂x Ω l=1 0 % &r ∂v0 1 τ dx dt. I2 (r) = vi (t, x) + R 0 Ω ∂x I1 (r) =

J

Since by (2.25)

τ

%

v0 (t, x) − v0,l (t) ≤ K3 Rη

we obtain

for x ∈ B(xl , R) ,

(4.6)

% &r ∂v0 2 dx dt vi (t, x) + ∂x 0 Ω

τ % &r ∂vi 2 dx dt + 1 . + c39 r2 R2η vi (t, x) + ∂x 0 Ω

1 c38 rI1 (r) ≤ 2

τ

(4.7)

We ﬁx R such that 4 c39 c37 r4 R2η = 1. Estimating I2 (r) by Cauchy’s inequality and using (4.5), (4.7), we deduce from (4.2) τ % % &r+2 &r ∂vi 2 dx dt vi (τ, x) + dx + vi (t, x) + ∂x 0 Ω Ω 1

τ r1 % &(r+1)r1 2+ η4 ≤ c40 r vi (t, x) + dx dt 1+

τ

τ

+

0

0

%

Ω

+ 0

∂Ω

Ω

1

p1 &rp1 vi (t, x) + dx dt + %

& vi (t, x) + ds rp2

1 p2

0

τ

%

&(r+1)r2 vi (t, x) ds dt

1 r2

∂Ω

dt .

Hence Sobolev’s embedding theorem and standard Moser iteration lead to (4.1) and the proof of Lemma 4.1 is completed. For ε > 0 and arbitrary functions g deﬁned on QT we use the notations g (ε) (t, x) = max g(t, x), ε , g− (t, x) = min g(t, x), 0 . (4.8)

Existence and Uniqueness Results for Reaction-diﬀusion Processes

173

Lemma 4.2. Let the conditions of the Theorem 2.4 be satisﬁed. Then there exists a constant L9 depending only on known parameters such that ess inf vi (t, x) : (t, x) ∈ QT ≥ −L9 ,

i = 1, . . . , n.

(4.9)

Proof. Denote hi (x) m0 = ess sup e−1 : x ∈ Ω, i u∗ (x)

i = 1, . . . , n

,

i

ei (v) , ei (−m0 ) 1 r Ψ(r) (z) = − 2 |ln z|r + 2 |ln z|r−1 e2i (−m0 ) , z > 0 . z z e i (v) =

We test the integral identity (3.5) with ϕ=

1 (ε) ei (vi )

ln− e (ε) (vi )r , i

0 < ε < 1,

r≥1,

to get

τ

0

n ∂ui (δ) ∂(vi + qi v0 ) , ϕdt + dij t, x, vi , Ψ(r) e i (vi ) ∂t ∂x j=1 Qτ

∂vi −1 × ei (vi ) χ e i (ε) < vi < −m0 dx dt ∂xj r 1 (ε) ln− e i (vi ) dx dt + Ri (t, x, v, ζ) (ε) ei (vi ) Qτ 1 ln− e (ε) (vi )r ds dt = 0 . + RiΓ (t, x, v, ζ) (ε) i ei (vi )

(4.10)

Γτ

Evaluating the ﬁrst integral in (4.10) analogously to equality (40) in [10] yields

τ

0

∂ui 1 , ϕ dt = ∂t r+1

ln− e (ε) vi (τ, x) r+1 u∗i (x) dx i

{ ei (vi )>ε}

(ε) |ln ε|r ln− e i vi (τ, x) u∗i (x) dx

− { ei (vi )≤ε}

1 ≤− r+1

ln− e (ε) vi (τ, x) r+1 u∗i (x) dx . i

Ω

(4.11)

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H. Gajewski and I.V. Skrypnik

We estimate the second integral in (4.10) by using the condition iii) to obtain n j=1 Q τ

∂vi (δ) ∂(vi + qi v0 ) dij t,x, vi , ei (vi )Ψ(r) e i (vi ) ∂x ∂xj × χ e −1 i (ε) < vi < −m0 dx dt 2 ln− e (ε) (vi )r−1 ∂ ln− e (ε) (vi ) dx dt ≤ −c41 r i i ∂x Qτ

+ c42 r

(4.12)

r ∂v0 2 (ε) dx dt . 1 + ln− e i (vi ) ∂x

Qτ

Estimating the two last integrals in (4.10) by using (2.26), (2.27) and Lemma 4.1, we get 1 ln− e (ε) (vi )r dx dt Ri (t, x, v, ζ) (ε) i ei (vi ) Qτ 1 ln− e (ε) (vi )r ds dt + RiΓ (t, x, v, ζ) (ε) i ei (vi ) Γτ (4.13) r % & (ε) 1 + α1 (t, x) ln− e i (vi ) dx dt ≤ c43 +

Qτ

r & % (ε) 1 + α2 (t, x) ln− e i (vi ) ds dt .

Γτ

(ε) By (4.10), (4.11) and (4.13) we ﬁnd for w(ε) (t, x) = ln− e i (vi )(t, x)

%

w

(ε)

&r+1 (τ, x) dx +

Ω

%

w

(ε)

&r−1 ∂wε 2 (t, x) ∂x dx dt

Qτ

% &r ∂v0 2 + α1 (t, x) + 1 dx dt 1 + w(ε) (t, x) ≤ c44 r2 ∂x +

Qτ

%

&r w(ε) (t, x) α2 (t, x) + 1 ds dt

(4.14)

.

Γτ

To estimate that term in (4.14) with the derivative of v0 , we test the integral identity (2.14) with % &2 % & Ψ(t, x) = w(ε) (t, x) v0 (t, x) − v0,l (t) ϕ2l (x) ,

(4.15)

Existence and Uniqueness Results for Reaction-diﬀusion Processes

175

where v0,l (t), ϕl (x) are the functions from (4.4). By integration on t and taking the sum on l we get % (ε) &r ∂v0 2 % ε &r dx dt ≤ c45 r I(r) + 1 w (t, x) w (t, x) dx dt 2 ∂x R Qτ

1 p % (ε) &rp1 1 w (t, x) dx dt +

τ

+ 0

τ

0

Ω

Qτ

%

w

(ε)

&rp (t, x) 2 ds

1 p 2

dt

,

∂Ω

(4.16) where I(r) =

J %

w

(ε)

l=1 Q τ

2 ∂w(ε) ∂v0 &r−1 dx dt. v0 (t, x) − v0,l (t) ϕl (x) (t, x) ∂x ∂x

Using (4.7), we can estimate the last integral &r ∂v0 2 % (ε) 1 dx dt c45 r I(r) ≤ w (t, x) 2 ∂x Qτ

(ε) 2 (ε) w (t, x)r ∂w dx dt . + c46 1 + r2 R2η ∂x

(4.17)

Qτ

Fixing the number R such that 4 c44 c46 r4 R2η = 1, we obtain from (4.14), (4.16) and (4.17) % (ε) % (ε) &r+1 &r−1 ∂w(ε) 2 w (τ, x) w (t, x) dx + ∂x dx dt Ω Qτ

1

r1 % (ε) &r·r1 2+ η4 ≤ c47 r w (t, x) dx dt 1+

+

Γτ τ

Qτ

&r r % (ε) w (t, x) 2 ds dt

1 r2

τ

%

&r p w(ε) (t, x) 2 ds

1 p1 &r p1 % (ε) dx dt w (t, x)

+ 0

+ 0

1 p2

(4.18)

Ω

dt

.

∂Ω

Remark also that (4.18) implies % (ε) &2 w (τ, x) dx + Ω

(ε) ∂w (t, x) 2 dx dt ≤ c48 ∂x

(4.19)

Qτ

with a constant c48 depending only on known parameters. To verify (4.19) we have to estimate the integrals on the right-hand side of (4.18) with r = 1 and then to apply Gronwall’s Lemma. As an example we consider the third integral. Deﬁne p

176

H. Gajewski and I.V. Skrypnik

by p1 =

N N −p .

Then p < 2 and we can assume p > 1. Using Sobolev’s embedding

theorem we have with p∗ =

τ

1 p1 % (ε) &p1 w (t, x) dx dt

c47 0

Np N −p

Ω

≤ c49 1 +

τ

0

pp∗ % (ε) &p∗ w (t, x) dx dt

Ω

(ε) p ∂w (ε) p w dx dt ≤ c50 1 + ∂x

(4.20)

Qτ

(ε) 2 ∂w (ε) 2 1 ≤ w (t, x) dx dt . ∂x dx dt + c51 1 + 8 Qτ

Qτ

Now (4.18), (4.19) and standard Moser iterations give w(ε) (t, x) ≤ c52 ,

(4.21)

with a constant c52 depending only on known parameters and independent of ε. The estimate (4.21) implies that the measure of the set { ei (vi (t, x)) < ε} is equal to zero if |ln ε| > c52 , i.e., −c52 −1 vi (t, x) > e −1 ) e i

and the proof of Lemma 4.2 is complete.

Proof of Theorem 2.4. Theorem 2.4 follows immediately from the inequalities (4.1) and (4.9).

5. Proof of existence and uniqueness Proof of Theorem 2.5. We modify the functions ei (z), dij (t, x, z, ξ), r(t, x, v, y), Γ rγδ (t, x , v, y) in following way: z ei (z) = ei (min[s, K4 ]) ds , −∞

d ij (t, x, z, ξ) = dij t, x, min[z, K4 ], ξ , r αβ (t, x, v, y) = rαβ t, x, min[v, K3 + K4 ], min[y, Kαβ ] , Γ Γ r γδ (t, x , v, y) = rγδ (t, x , min[v, K3 + K4 ], min[y, Kγδ ] ,

(5.1)

where K3 , K4 are the constants from Theorem 2.3, 2.4 and min[v, K3 + K4 ] = min [vj , K3 + K4 ], j=0,...,n

Kαβ =

n |αi | + |βi | (K3 + K4 ). i=1

Existence and Uniqueness Results for Reaction-diﬀusion Processes

177

Now we consider the system −∇ · (∇v0 ) = f +

qi u i

in

QT ,

(5.2)

i=1

∂ ui (δ) i = 0, + ∇ · J i + R ∂t

i = 1, . . . , n in QT ,

(5.3)

i are deﬁned by (2.4), (2.18), (2.6) with e i , d i , r αβ instead of where u i , J iδ , R ei , di , rαβ . We assume further that δ = K14 . In an analogous way we modify the boundary condition (2.17): (δ) Γ = 0, ν · J i + R i

i = 1, . . . , n on ΓT .

(5.4)

The solvability of the nondegenerate problem (5.2)–(5.4), (2.8), (2.9) can be simply shown by using backward time discretization (see, for example [2]). By Theorems 2.3, 2.4 each solution v = (v0 , v1 , . . . , vn ) of that nondegenerate problem, satisﬁes the a priori estimates (2.25), (2.28). But, because of (5.1), v is automatically solution of the original problem (2.1), (2.2), (2.7)–(2.9). So theorem 2.5 is proved. We want now to prepare the proof of the uniqueness result. Let us to this aim suppose contradictionarily the existence of two solutions (1) (1) (2) (2) v (1) = v0 , v1 , . . . , vn(1) , v (2) = v0 , v1 , . . . , vn(2) of problem (2.1), (2.2), (2.7)–(2.9). Remark that both solutions necessarily fulﬁll the a priori estimates (2.25), (2.28). We shall show that v (1) = v (2) . We start by proving auxiliary Lemmas. Lemma 5.1. Let the assumptions of Theorem 2.6 be satisﬁed. Then there exists a constant L10 depending only on known parameters such that for arbitrary τ ∈ (0, T ] n ∂(vi(1) − vi(2) ) 2 (1) dx dt v (τ, x) − v (2) (τ, x)2 dx + i i ∂x Ω i=1 Qτ

n ∂(v0(1) − v0(2) ) 2 (1) v − v (2) 2 ≤ L10 + l l ∂x +

(5.5)

l=0

Qτ

n (1) (1) (1) (1) v − v (2) 2 1 + ∂(vi + qi v0 ) ∂v0 dx dt . i i ∂x ∂x i=1

Proof. We test the integral identity (2.13) for the solution v (k) , k = 1, 2, with ϕ(k) , where ϕ(1) =

1 (1) ei (vi )

(2) & % (1) − ei vi , ei vi

(2)

ϕ(2) = vi

(1)

− vi .

178

H. Gajewski and I.V. Skrypnik

Taking the sum of the obtained equalities, we get n 2

N (k) (k) ∂ui ∂ζ ∂ϕ(k) (k) , ϕ(k) > dt + ei (vi )γij t, x, i ∂t ∂x ∂xj 0 i=1 k=1 j=1 Qτ + Ri t, x, v (k) , ζ (k) ϕ(k) dx dt + RiΓ t, x, v (k) , ζ (k) ϕ(k) ds dt = 0. τ

<

Γτ

(5.6) We evaluate the ﬁrst integral in (5.6) analogously to Lemma 2 from [10] and obtain 2 k=1

τ

0

(k)

∂ui , ϕ(k) > dt = < ∂t

Ω

u∗i (x)

≥ c53

(1)

vi (τ,x) % (2) vi (τ,x)

& (1) vi (τ, x) − z ei (z) dz dx

(1) v (τ, x) − v (2) (τ, x)2 dx. i i

(5.7)

Ω

The second one can be estimated by the assumptions (i), (ii) of Theorem 2.6: z1 & ei (s) e (z1 ) % ei (z1 ) − ei (z2 ) ≥ ei (z1 ) − ei (z1 ) − i (5.8) (s) ei (s) ds = ei (z2 ) , ei (z1 ) e z2 i ei (z1 ) − ei (z2 ) −

& ei (z1 ) % ei (z1 ) − ei (z2 ) ≤ c54 |z1 − z2 |2 . ei (z1 )

(5.9)

The last inequalities, conditions (2.29), iii) and the local Lipschitz continuity of γij imply 2 N (k) (k) ∂ζ ∂ϕ(k) ei vi γij t, x, i ∂x ∂xj j=1

k=1

(1) N

(2) (1) (2) (2) ∂ ζi − ζi ∂ζi ∂ζi vi ≥ γij t, x, − γij t, x, ∂x ∂x ∂xj j=1 (1) (1) (1) ∂ζ ∂v (2) 2 − c55 vi − vi 1 + i 0 ∂x ∂x (1) (2) (1) (2) ∂ ζi − ζi ∂ v0 − v0 − c55 ∂x ∂x (1) ∂ vi − vi(2) 2 ≥ c56 ∂x (1) (1) (1) (1) ∂ζi ∂v0 ∂ v0 − v0(2) 2 (2) 2 . − c57 vi − vi 1+ + ∂x ∂x ∂x ei

(5.10)

Existence and Uniqueness Results for Reaction-diﬀusion Processes By the local Lipschitz continuity of Ri , RiΓ we get 2 n (1) (k) (k) (k) v − v (2) 2 , ≤ c R , ζ t, x, v ϕ i 58 l l k=1

179

(5.11)

l=0

n 2 Γ (1) (k) (k) (k) v − v (2) 2 . t, x, v ϕ R , ζ ≤ c 58 i l l k=1

(5.12)

l=0

Using the interpolation inequality 2 ∂v 2 2 v (t, x) ds dt ≤ ε + cε |v| dx dt ∂x Γτ (1)

Qτ (2)

for functions vl − vl , l = 0, 1, . . . , n, and suitable ε > 0, we obtain (5.5) from (5.6), (5.7), (5.10), (5.12) and the proof of Lemma 5.1 is completed. Lemma 5.2. Let the conditions of Theorem 2.6 be satisﬁed. Then a constant L11 depending only on known parameters exists such that n (1) ∂(v0(1) − v0(2) ) 2 (1) + v − v (2) 2 dx ≤ L11 v − v (2) 2 dx . (5.13) 0 0 i i ∂x Ω i=1 Ω Proof. We test the integral identity (2.14) associated with the solutions v (k) , (1) (2) (2) (1) k = 1, 2, with ψ (1) = v0 − v0 , ψ (2) = v0 − v0 . The sum of the obtained equalities reads: (1) n 2 ∂ v0 − v0(2) 2 (1) (2) 2 (k) (x) κ(x) v0 − v0 ds = qi ui ψ (k) dx. dx + ∂x Ω ∂Ω Ω k=1 i=1 (5.14) Now (5.13) follows from (5.14), Cauchy and embedding inequalities. Lemma 5.3. Let the conditions of Theorem 2.6 be satisﬁed. Then a constant L12 depending only on known parameters exists such that for arbitrary τ ∈ (0, T ]

n (1) (1) (1) 2 v (τ, x) − v (2) (τ, x)2 dx + v − v (2) 2 ∂vi i i i i ∂x Ω i=1 Qτ

(2) 2 n (1) ∂v ∂ vi − vi(2) 2 + i dx dt ≤ L12 ∂x ∂x i=1 Qτ

(1) (2) 2 ∂v ∂v (1) (2) 2 + 1 + 0 + 0 vi − vi ∂x ∂x 2 (1) ∂ (1) (2) (2) 2 v − v0 + v0 − v0 + dx dt . ∂x 0

(5.15)

180

H. Gajewski and I.V. Skrypnik

Proof. We test the integral identity (2.13) for the solution v (k) , k = 1, 2, with ϕ

(1)

ϕ (2)

=

(1) (2) exp A ei (vi ) − exp Aei (vi ) (1)

ei (vi ) & % (2) (1) (2) = A vi − vi exp A ei (vi ) ,

,

where A is a positive number, depending only on known parameters, such that &2 % A ei (s) + 2 ei (s) ≥ 1 for

|s| ≤ K4 ,

i = 1, . . . , n ,

(5.16)

with K4 from (2.28). Taking the sum of the obtained equalities, we get n 2

N (k) (k) ∂ϕ (k) ∂ui ∂ζ (k) ,ϕ (k) > dt + ei (vi )γij t, x, i ∂t ∂x ∂xj 0 i=1 k=1 j=1 Qτ (k) (k) + Ri t, x, v (k) , ζ (k) ϕ RiΓ t, x, v (k) , ζ (k) ϕ ds dt = 0. dx dt + τ

<

Γτ

(5.17) We transform the ﬁrst integral in (5.17) analogously to the inequality (5.7) to obtain 2 i=1

0

τ

(k)

∂ui ,ϕ (k) > dt = A < ∂t ×

ei (z) exp Aei (z) dz dt

u∗i (x)

Ω

%

≥ c59

(1)

vi (τ,x) %

(1)

vi (τ, x) − z

(2) vi (τ,x)

(1) vi (τ, x)

−

&2 (2) vi (τ, x)

& (5.18) dx.

Ω

To estimate the second term in (5.17) we use the inequality z1 & ei (z1 ) % − exp[Aei (z1 )] − exp[Aei (z2 )] ≥ −A ei (z) exp[Aei (z)] dz ei (z1 ) z2 z1 % % &2 & 2 = A ei (z2 ) exp[Aei (z2 )] − ei (z1 ) exp[Aei (z1 )] + A ei (z) exp[Aei (z)] dz , z2

(5.19)

that follows for z1 , z2 ∈ R1 from condition 1) of Theorem 2.6. So we obtain N j=1 Q τ

ei

(1) (1) ∂ϕ (1) ∂ζi dx dt ≥ I (1) + I (2) + I (3) , vi γij t, x, ∂x ∂xj

(5.20)

Existence and Uniqueness Results for Reaction-diﬀusion Processes where I (1) = A2

I

(2)

=A

N j=1 Q τ

I

(3)

(1) (1) vi(1) % &2 ∂ζi ∂ζ γij t, x, i ei (z) exp[Aei (z)] dz dx dt, ∂x ∂xj vi(2)

N j=1 Q τ

= A qi

181

(2) (1) (1) (2) (2) ∂ vi − vi ∂ζi γij t, x, ei vi exp[Aei vi ] dx dt, ∂x ∂xj

N

γij

j=1 Q τ

(1)

∂ζ t, x, i ∂x

(1) (1) ei (vi ) (1) ei (vi )

∂v0 ∂xj

(1)

vi

(2)

ei (z) exp[A ei (z)] dz

vi

(1) (1) (2) (2) − ei vi exp[Aei vi ] + ei vi exp[Aei vi ] dx dt. We rewrite the second term in (5.17) with k = 2 as follows N (2) (2) ∂ϕ (2) ∂ζ ei vi γij t, x, i dx dt = I (4) + I (5) + I (6) + I (7) , ∂x ∂x j j=1 Qτ

I (4) = −A2

N j=1 Q

τ

I (5) = qi A2

N j=1 Q

I (6) = A2

N

τ

j=1 Q

(5.21)

(1) (1) (2) (1) ∂ζi % (2) &2 ∂ζ (2) ei vi γij t,x, i exp[Aei vi ] vi − vi dxdt, ∂x ∂xj (1) (1) (2) (1) ∂v0 % (2) &2 ∂ζ (2) ei vi γij t,x, i exp[Aei vi ] v1 − vi dxdt, ∂x ∂xj (1) (1) (2) (2) ∂vi ∂vi ∂ζ ∂ζ − γij t,x, i γij t,x, i ∂x ∂xj ∂x ∂xj

τ

% (2) &2 (2) (1) (2) × ei vi exp[Aei vi ] vi − vi dxdt, N (2) (2) (2) ∂ vi(1) − vi(2) ∂ζi (7) γij t,x, dxdt. I = −A ei vi exp[Aei vi ] ∂x ∂xj j=1 Qτ

We want to estimate sums of terms from (5.20) and (5.21). Note that by (5.16) z1 % &2 % &2 ei (z) exp[Aei (z)] dz − ei (z2 ) exp[Aei (z2 )](z1 − z2 ) z2 z1 z % 2 ei (θ) + A ei (θ)]2 ei (θ) exp[Aei (θ)] dθ dz = z2

z2

≥ c60 |z1 − z2 |2

for

and hence

I

(1)

+I

(4)

≥ c61 Qτ

|z1 |, |z2 | ≤ K4 (1) (1) 2 v − v (2) 2 ∂ζi dx dt . i i ∂x

(5.22)

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H. Gajewski and I.V. Skrypnik

The next estimate follows from conditions iii) and (2.29) I

(2)

+I

(7)

(1) ∂ ζi − ζi(2) 2 dx dt − c63 ≥ c62 ∂x Qτ

(1) ∂ v0 − v0(2) 2 dx dt . ∂x

Qτ

(5.23) The local Lipschitz continuity of the function ei implies % &2 ei (z1 ) z1 e (z) exp[Aei (z)] dz A ei (z2 ) exp[Aei (z2 )](z1 − z2 ) + ei (z1 ) z2 i − ei (z1 ) exp[Aei (z1 )] + ei (z2 ) exp[Aei (z2 )] ≤ c64 |z1 − z2 |2 for arbitrary numbers z1 , z2 ∈ [−K4 , K4 ] and consequently (3) I + I (5) ≤ c65

(1) (1) ∂ζ ∂v (1) (2) 2 1 + i 0 · vi − vi dx dt. ∂x ∂x

(5.24)

Qτ

Further, the local Lipschitz condition for γij yields: (6) I ≤c66

(1) ∂ vi − vi(2) ∂vi(1) ∂ζi(2) + ∂x ∂x ∂x Qτ

(1) (1) ∂vi ∂ v0 − v0(2) (1) v − v (2) dx dt. + · i i ∂x ∂x

(5.25)

Finally, we obtain from (5.17), (5.18), (5.22)–(5.25) with view of (5.11), (5.12) n % i=1

(1) vi (τ, x)

−

&2 (2) vi (τ, x)

dx +

Ω

(1) (1) 2 v − v (2) 2 ∂vi dx dt i i ∂x

Qτ

(1) 2

n (1) ∂ vi − vi(2) 2 + 1 + ∂v0 v (1) − v (2) 2 ≤ c67 i i ∂x ∂x Qτ

i=1

(1) ∂ vi − vi(2) ∂ζi(2) (1) (v − v (2) + i ∂x i ∂x ∂ v (1) − v (2) 2 (1) (2) 2 0 0 + dx dt. + v0 − v0 ∂x (1)

(2)

(5.26)

Changing the places of vi and vi in (5.26) and applying Cauchy’s inequality, we arrive at the desired estimate (5.15) and the proof of Lemma 5.3 is complete.

Existence and Uniqueness Results for Reaction-diﬀusion Processes

183

Lemma 5.4. Under the conditions of Theorem 2.6 a constant L13 depending only on known parameters exists such that for all τ ∈ (0, T ], R ∈ (0, 1],

(1) (1) (2) 2 v − v (2) 2 ∂v0 + ∂v0 dx dt i i ∂x ∂x Qτ

≤ L13 R

2η Qτ

(1) 1 R2 |vi

τ

+ 0

−

(2) vi |2

(1) ∂ vi − vi(2) 2 dx dt + ∂x

1 p (1) v − v (2) 2p1 dx 1 i i

Ω

(1) v − v (2) 2p2 ds i i

+

1 p2

dt

,

(5.27)

∂Ω

where η is the H¨ older exponent from Theorem 2.3. Proof. Let {ϕj (x)}, j = 1, . . . , J, be a partition of unity satisfying (4.3) with a number R to be chosen later on. We test the integral identity (2.14) associated with the solution v (k) , k = 1, 2, with Ψ(k) (t, x) =

J %

(1) 2 & (k) (k) (2) v0 (t, x) − v0,l (t) ϕ2l (x)vi (t, x) − vi (t, x) ,

l=1 (k)

(k)

v0,l (t) = v0 (xl ). We obtain after integration with respect to t and using the H¨ older inequality (k) 2 (1) (2) 2 ∂v (x)vi − vi 0 dx dt ∂x Qτ

≤ I (8) (k) + I (9) (k) 1 τ p (1) v − v (2) 2p1 dx 1 + c68 i i 0

τ

+ 0

I

(8)

(k) = −2

N J

(x)

l=1 j=1 Q

τ

I (9) (k) = −2

N J l=1 j=1 Q

(x)

Ω

(1) v − v (2) 2p2 ds i i

1 p2

(5.28)

dt ,

∂Ω (k) % (k) ∂v0 ∂ϕl (k) & (1) (2) 2 ϕl · v0 − v0,l vi − vi dx dt, ∂xj ∂xj

(1) (2) (k) % (k) ∂ vi − vi ∂v0 (k) &% (1) (2) & · ϕ2l v0 − v0,l vi − vi dxdt. ∂xj ∂xj

τ

(8)

(9)

Estimating I (k), I (k) by Cauchy’s inequality and using (4.6), we obtain (5.27) immediately from (5.28) and the proof of Lemma 5.4 is complete.

184

H. Gajewski and I.V. Skrypnik

Proof of Theorem 2.6. From (5.5) we get by applying Cauchy’s inequality to the second term and using (5.13), (5.15), (5.27) with suitable R n ∂(vi(1) − vi(2) ) 2 (1) dx dt v (τ, x) − v (2) (τ, x)2 dx + i i ∂x Ω

i=1

≤ c69

n i=1

τ

(1) v − v (2) 2 dx dt + i i

τ

(1) v − v (2) 2p2 ds i i

1 p 2

1 p (1) v − v (2) 2p1 dx 1 dt (5.29) i i

0

Qτ

+ 0

Qτ

Ω

dt .

∂Ω

Here the second integral on the right-hand side can by estimated by the interpolation inequality 1 p (1) v − v (2) 2p1 dx 1 dt i i 0 Ω (1) v (θ, x) − v (2) (θ, x)2 dx ≤ ε ess sup0<θ<τ i i

τ

(5.30)

Ω

(1) ∂ vi − vi(2) 2 (1) v − v (2) 2 dx dt , + i i dx dt + c(ε) ∂x Qτ

where p1 = (5.29)

Qτ p1 p1 −1

<

N N −2

and ε > 0. Analogously we get for the last integral in

n n (1) (1) v (τ, x) − v (2) (τ, x)2 dx ≤ c70 v − v (2) 2 dx dt. i i i i i=1

Ω

(5.31)

i=1 Q τ (1)

(2)

(1)

Hence Gronwall’s lemma yields vi = vi for i = 1, . . . , n. Finally, v0 follows from (5.13) and Theorem 2.6 is proved.

Proof of Corollary 2.1. Let v be the solution of (2.1), (2.2), (2.7)–(2.9). Set (1)

vi

= vi (t, x),

(2)

vi (t, x) = vi (t + δ, x),

i = 0, . . . , n , δ ∈ (0, T − t).

We test integral identity (3.5) with the functions ϕ(1) = t2

1 (1) ei (vi )

(2) & % (1) − ei vi , ei vi

(2)

ϕ(2) = t2 (vi

(1)

− vi ).

(2)

= v0

Existence and Uniqueness Results for Reaction-diﬀusion Processes

185

Then, arguing essentially as in the proof of (5.29) and (5.31), we obtain n (1) (2) 2 (1) − vi ) 2 ∂(vi v (τ, x) − v (2) (τ, x)2 dx + t dx dt τ2 i i ∂x i=1

≤ c71

Ω

n

Qτ

(2) 2 2 (1) 2 t vi − vi dx dt + δ .

i=1 Q τ

Now dividing by δ 2 , applying Gronwall’s lemma and taking the limit δ → 0, the corollary follows.

6. Proof of Theorem 2.7 We start from the proof of (2.24) making some changes in the proof of Theorem 2.2. In the proof of Lemmas 3.1, 3.2, 3.3 we did not use the conditions (2.22), (2.23). Consequently, the results of these Lemmas remain valid. We replace the test function ϕ in the proof of Lemma 3.4 by ) & *ρ % ϕ = a(ρ) + Λ(v) − M0 + vi ,

Λ(v) =

k

n

Λi (vi ),

i = 1, . . . , n ,

(6.1)

i=1

+ where ρ > 0, k > 1, the function Λi (z) is deﬁned by (2.20), M0 = ni=1 Λi (m0 ), m0 and a(ρ) are the same numbers as in the proofs of the Lemmas 4.2 and 3.4, respectively. Using the equalities α · q = β · q, γ · q = δ · q for (α, β) ∈ R, (γ, δ) ∈ RΓ and that Γ the functions rαβ , rγδ are increasing increasing in the last argument, we ﬁnd n

Ri (·, v, ζ)vi =

i=1

n %

& rαβ (·, v, α · ζ) − rαβ (·, v, β · ζ) (αi − βi )vi

i=1 (α,β)∈R

=

%

& rαβ (·, v, α · ζ) − rαβ (·, v, β · ζ) (α − β) · ζ ≥ 0 , (6.2)

(α,β)∈R n

RiΓ (·, v, ζ)vi ≥ 0 ,

ζ = (ζ1 , . . . , ζn ),

ζi = qi v0 + vi .

i=1

Using the test function from (6.1) and the inequality (6.2), we obtain from (2.13) n τ N % & *ρ ∂ui ) , a(ρ) + Λ(v) − M0 + vi > dt + < ei (vi )ajk (t, x) ∂t k 0 i=1 Qτ j,k=1 ) % & *ρ ∂ ∂ a(ρ) + Λ(v) − M0 + vi dx dt ≤ 0. × (vi + qi v0 ) ∂xk ∂xj k (6.3)

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We evaluate the ﬁrst integral in (6.3) following Lemma 2 in [10] to get n τ % & *ρ ∂ui ) , a(ρ) + Λ(v) − M0 + vi > dt < ∂t k i=1 0 Λ(v(τ,x)) ) % & *ρ (6.4) a(ρ) + z − M0 + dz dx u∗ = k 0 Ω ) % & *ρ+1 1 ∗ ≥ u a(ρ) + Λ(v(τ, x)) − M0 + dx − c71 a(ρ) . ρ+1 k Ω To estimate the second integral in (6.3) we note that: n N % & *ρ ∂vi ∂ ) a(ρ) + Λ(v) − M0 + vi ei (vi )ajk ∂xk ∂xj k i=1 j,k=1

N n ) & *ρ % ∂vi ∂vi = a(ρ) + Λ(v) − M0 + ei (vi )ajk ∂x k k ∂xj i=1 j,k=1

n N ) & *ρ−1 % ∂vi + ρ a(ρ) + Λ(v) − M0 + ajk ei (vi )vi ∂xk k i=1 j,k=1 n ∂vl × el (vl )vl χ 0 < Λ(v) − M0 < k − a(ρ) ∂xj l=1 n ) ∂vi 2 % & *ρ . ei (vi ) ≥ c72 a(ρ) + Λ(v) − M0 + ∂x k i=1 (6.5) Now (6.3)–(6.5) and (2.21) imply ) % & *ρ+1 u∗ a(ρ) + Λ(v(τ, x)) − M0 + dx Ω

) +

k

%

a(ρ) + Λ(v) − M0

QT

n & *ρ + k

i=1

2

∂vi ei (vi ) ∂x

dx dt

') ( % & *ρ a(ρ) + Λ(v) − M0 + + M0ρ ≤ c73 (ρ + 1)

(6.6)

2

k

Qτ

×

∂v0 2 dx dt + a(ρ) . ei (vi ) ∂x i=1

n

With view of the proof of Lemma 3.4 we want to reestablish (3.45) as a consequence of (3.43) (and (3.41)). Note that by assumption γi = γ and let us assume that r(m)+ε > γ. Choosing ρ in (6.6) such that (2+γ)ρ+γ = r(m)+ε we can estimate the right-hand side of (6.6) by (3.43) for i = 1, . . . , n. Hence (6.6) implies (3.45). Repeating all another discussions from the proofs of Lemma 3.4 and Theorem 2.2,

Existence and Uniqueness Results for Reaction-diﬀusion Processes

187

we obtain the inequality (2.24). The proof of (2.25) in the considered case coincides with that one in the proof of Theorem 2.3. In order to prove (2.28) we need only to check (4.1). We have from (6.6) with ρ ≥ 1 ) n ) *ρ+1 *ρ ∂vi 2 dx dt Λ(v) − M0 u∗ Λ v(τ, x) − M0 dx + ei (vi ) ∂x + + Ω i=1 Qτ

) (6.7) n *ρ ∂v0 2 ρ 2 Λ(v) − M0 ei (vi ) dx dt + M0 . ≤ c73 (ρ + 1) ∂x + i=1 Qτ

To estimate the last integral we test the identity (2.14) with n *ρ * ) γ ) 1 + [vi ]+ v0 (t, x) − v0,l (t) ϕ2l (x) , ψ(t, x) = Λ v(t, x) − M0 +

i=1

where the notations from (4.4) are used. Integration on t and summing up on l give J ) n n *ρ ∂v0 2 dx dt ≤ c74 ρ Λ(v) − M0 ei (vi ) ei (vi ) ∂x + i=1 i=1 l=1 Q Qτ τ *ρ ) 1 2 ∂vi ∂v0 + 2 dx dt |v0 − v0,l |ϕl × Λ(v) − M0 ∂x ∂x R + 1 τ ) *ρ p1 p1 + Λ(v) − M0 ei (vi ) dx dt + 0 Ω τ ) *ρ p2 p12 ρ + Λ(v) − M0 ei (vi ) ds dt + M0 . 0

+

∂Ω

(6.8) Using (4.6), we obtain from (6.7), (6.8) ) ) n *ρ+1 *ρ ∂vi 2 dx dt Λ v(τ, x) −M0 Λ(v) − M0 dx + ei (vi ) ∂x + + Ω i=1

4

Qτ

≤ c75 ρ2+ η M0ρ +

τ

) *(ρ+1)p1 p11 dx dt (6.9) Λ(v) − M0

0

τ

+ 0

∂Ω

)

Λ(v) − M0

+

Ω

*(ρ+1)p2

ds

1 p2

dt .

+

The last inequality implies (4.1) by standard Moser iteration. The proofs of the existence and uniqueness (Theorems 2.5, 2.6) remain valid under the assumptions of Theorem 2.7. This ends the proof of Theorem 2.7

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References [1] G. Albinus, Convex analysis of the energy model of semiconductor devices, WIAS Preprint 285, Berlin (1996). [2] H.W. Alt, S. Luckhaus, Quasilinear elliptic-parabolic diﬀerential equations, Math. Z., 183 (1983), 311–341. [3] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in: Function spaces, Diﬀerential Operators and Nonlinear Analysis (H.J. Scmeisser and H. Triebel, eds.), Teubner-Texte zur Mathematik, vol. 133, Teubner, Stuttgart - Leipzig, 1993, pp. 9–116. [4] H. Amann, M. Renardy, Reaction-diﬀusion problems in electrolysis, Nonlinear Diﬀer. Equ. Appl., 1 (1994), 91–117. [5] C. Ebmeyer, J. Frehse, Mixed boundary value problems for nonlinear elliptic equations in multidimensional non-smooth domains, Math. Nachr. 203 (1999) 47–74 . [6] H. Gajewski, On the uniqueness of solutions to the drift-diﬀusion model of semiconductor devices, Math. Models Methods Appl. Sci., 4 (1994), 121–133. [7] H. Gajewski, K. Gr¨ oger, Semiconductor equations for variable mobilities based on Boltzmann statistics or Fermi-Dirac statistics, Math. Nach., 140 (1989), 7–36. [8] H. Gajewski, K. Gr¨ oger, Reaction-diﬀusion processes of electrically charged species, Math. Nachr., 177 (1996), 109–130. [9] H. Gajewski, I.V. Skrypnik, To the uniqueness problem for nonlinear elliptic equations, Nonlinear Anal., Theory Methods Appl., 52 (2003), 291–304. [10] H. Gajewski, I.V. Skrypnik, On the uniqueness of solutions for nonlinear parabolic equations, Discrete Contin. Dyn. Syst., 10 (2003), 315–336. [11] H. Gajewski, I.V. Skrypnik, On the uniqueness of solutions for nonlinear ellipticparabolic equations, J. Evol. Equ., 3 (2003), 247–281. [12] H. Gajewski, I.V. Skrypnik, On the unique solvability of nonlocal drift-diﬀusion-type problems, Nonlinear Anal. 56 (2004), 803–830. [13] A. Glitzky, R. H¨ unlich, Electro-reaction-diﬀusion systems including cluster reactions of higher order, Math. Nachr. 216 (2000), 95–118. [14] J.W. Jerome, Analysis of Charge Transport, Springer-Verlag, Berlin, (1996) xii+167 pp. [15] O.A. Ladyzhenskaya, N.N. Uraltseva, Linear and quasilinear elliptic equations, Nauka, Moscow (1973) (Russian), 576 pp. [16] W. van Roosbroeck, Theory of ﬂow of electrons and holes in germanium and other semiconductors, Bell System Tech. J., 29 (1950), 560–607. [17] S. Selberherr, Analysis and simulation of semiconductor devices, Springer-Verlag, Wien, New York (1984). H. Gajewski Weierstraß-Institute for Applied Analysis and Stochastics, Mohrenstr. 39 D-10117 Berlin, Germany I.V. Skrypnik Institute for Applied Mathematics and Mechanics, Rosa Luxemburg Str. 74 340114, Donetsk, Ukraine

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 189–210 c 2005 Birkh¨ auser Verlag Basel/Switzerland

An Inverse Problem for a Phase-ﬁeld Model in Sobolev Spaces Fabrizio Colombo and Davide Guidetti Dedicated to Herbert Amann, in occasion of his retirement

Abstract. We prove an existence and uniqueness result for an inverse problem arising from a phase-ﬁeld model with two memory kernels. More precisely, we identify the convolution memory kernels and the diﬀusion coeﬃcient besides the temperature and the phase-ﬁeld parameter. We prove our results in the framework of Sobolev spaces. Our fundamental tools are an optimal regularity result in the Lp spaces and ﬁxed point arguments. Mathematics Subject Classiﬁcation (2000). primary 35R30, 45K05; secondary 45N05, 80A20. Keywords. Phase-ﬁeld system with memory, heat equation, Cahn-Hilliard type equation, inverse problem, optimal regularity in Lp .

1. Introduction Phase-ﬁeld models are described by systems of equations ruling the evolution of the temperature θ and the phase-ﬁeld χ, that may stand for the local proportion of one of the two phases, solid and liquid, in a three-dimensional body Ω. If we consider phase-ﬁeld models with memory, the thermal memory is taken into account modifying the Fourier’s heat conduction law by introducing a memory convolution kernel (see (1.4)), denoted in the sequel by b. Unfortunately, b cannot be measured directly, to determine b additional measurements on the temperature are required, so that we have to face the problem of identifying θ, χ and b simultaneously. The model we have introduced and studied in [15] is the same we consider in this paper (it has two unknown convolution kernels), but the inverse problem differs, since the heat source is unknown in [15], but the diﬀusion coeﬃcient is given, while here we have the opposite situation. More precisely, in [15], we have proved an existence and uniqueness result for the problem dealing with the determination

190

F. Colombo and D. Guidetti

of the state variables (temperature and phase parameter), the convolution kernels and the time dependence of the heat source. The setting in the space C(Ω), used in [15], has the advantage that it allows to make additional boundary measurements on the temperature, to determine the convolutions kernels, while the Sobolev setting adopted here has the good property of requiring less compatibility conditions on the data. We point out that only in the recent literature we can ﬁnd contributions related to inverse problems for phase-ﬁelds models. We refer the reader to [14] for more details. Many important direct problems in the phase-ﬁeld theory have been proposed and solved. For sake of completeness we have to mention some of the contributions in this research area, given by several authors. Among them, without claim of completeness, we mention [2],[3],[6], [7], [8], [9], [10],[11], [12], [13] and, for phaseﬁeld models with memory, the papers [16], [17] and the literature there in. We recall here in the following the model, introduced in [17], that we are going to study. 1.1. The model Let Ω be an open bounded set in R3 with suﬃciently regular boundary ∂Ω occupied by an isotropic, rigid and homogeneous heat conductor. We consider only small variations of the absolute temperature and its gradient. The materials which exhibit phase transitions, due to the temperature variations, are described by two state variables: the absolute temperature Θ and the phase-ﬁeld χ at each point x ∈ Ω and t ∈ [0, T ] for T > 0, where χ takes approximately value −1 in the liquid and +1 in the solid. In our model we prefer to work with the temperature variational ﬁeld θ deﬁned by: Θ − Θc , (1.1) θ= Θc where Θc is the reference temperature at which the transition occurs. The energy balance equation is (1.2) ∂t e + ∇ · J = F , where e is the internal energy, J is the heat ﬂux and F is the external heat supply. Taking into account a linearized version of the Coleman-Gurtin theory, we assume (as in (Coleman-Gurtin [11] and in Giorgi, Grasselli and Pata [16]) the constitutive equations: t ˜ a(s)θ(t − s) ds + Θc λ(χ(t, x)), (1.3) e(t, x) = ec + cv Θc θ(t, x) + 0

t

b(s)∇θ(t − s) ds,

J = −k∇θ(t, x) −

(1.4)

0

where a and b account for the memory eﬀects, ec , cv and k are the internal energy ˜ is at equilibrium, the speciﬁc heat and the conductivity, respectively. Moreover, λ

An Inverse Problem for a Phase-ﬁeld Model in Sobolev Spaces a suitable regular given function. By (1.2), (1.3) and (1.4) we get

t ˜ ∂t ec + cv Θc θ(t, x) + 0 a(s)θ(t − s) ds + Θc λ(χ(t, x))

t −∇ · k∇θ(t, x) + 0 b(s)∇θ(t − s) ds = F (t, x), t ∈ [0, T ], x ∈ Ω.

191

(1.5)

Following Novick-Cohen [23], we couple the evolution equation (1.5) with the Cahn-Hilliard type equation, which rules the phase evolution (see Brokate and Sprekels [4], Caginalp [8] and Novick-Cohen [22] for its justiﬁcation): ˜ (χ)θ(t, x)], ε∂t χ(t, x) = ∆[−∆χ(t, x) + ϕ(χ(t, x)) − λ

t ∈ [0, T ], x ∈ Ω, (1.6)

where ϕ is a smooth given function and ε > 0 is a parameter. We will assume ˜ is a linear function of its argument. Associated to the evolution equations, that λ we will consider also the initial and the Neumann boundary conditions, to be introduced just below. As we have already observed, the kernels a and b cannot be measured directly and the physical observable that can be easily measured is the temperature, so a and b have to be indirectly determined by additional measurements on θ, made on suitable parts of the material. We consider additional measurements on the temperature which can be represented as Φj (θ)(t) := ϕj (x)θ(t, x) dx = gj (t), ∀t ∈ [0, T ], j = 1, 2, (1.7) Ω

where, for j ∈ {1, 2}, ϕj : Ω → R is a given function depending on the type thermometer used for the additional measurements on θ and gj : [0, T ] → R represents the result of the corresponding additional measurement on θ. We have given two conditions because we have to identify the two unknown kernels a and b. 1.2. The identiﬁcation problem Before we can precisely state the problem, we need some notations: • R+ denotes positive real numbers. • If Ω is an open subset of Rn , lying on one side of its boundary ∂Ω, which is a submanifold of Rn of class at least C 1 , we denote by ν(x) the unit vector normal to ∂Ω in x, pointing outside Ω, and by Dν the normal derivative. • If X and Y are Banach spaces and k ∈ N, we shall indicate with Lk (X, Y ) the Banach space of k-linear and continuous mappings from X k to Y . In case k = 1, we shall simply write L(X, Y ). • If s ∈ N0 , p ∈ [1, +∞], W s,p (Ω) is the usual Sobolev space. • • • •

If s ∈ Z, s ≥ 2, WBs,p (Ω) := {f ∈ W s,p (Ω) : Dν f ≡ 0}. s,p If s ∈ Z, s ≥ 4, WBB (Ω) := {f ∈ W s,p (Ω) : Dν f ≡ Dν ∆f ≡ 0}. s Bp,q (Ω) (s > 0, 1 ≤ p, q ≤ +∞) is the Besov space. (., .)θ,p is the real interpolation functor (0 < θ < 1, 1 ≤ p ≤ +∞).

192

F. Colombo and D. Guidetti

• If a ∈ L1 (0, T ) and f : (0, T ) → X, where X is a Banach space, we set t (a ∗ f )(t) := a(t − s)f (s)ds, 0

whenever the integral has a meaning. • If p ∈ [1, +∞), T ∈ R+ , m ∈ N0 , X is a Banach space, f ∈ W m,p (0, T ; X), we set m−1 f (j) (0)X + f (m) Lp (0,T ;X) . f W m,p (0,T ;X) := j=0

• If T ∈ R , p ∈ [1, +∞], we set +

W 1,p (0, T ; Lp (Ω)) ∩ Lp (0, T ; W 2,p(Ω)).

X(T, p) :=

If u ∈ X(T, p), we deﬁne uX(T,p)

:= uW 1,p (0,T ;Lp (Ω)) + uLp(0,T ;W 2,p (Ω)) .

• If T ∈ R , p ∈ [1, +∞], we set +

Y (T, p) :=

W 1,p (0, T ; Lp(Ω)) ∩ Lp (0, T ; W 4,p (Ω)),

If v ∈ Y (T, p),we deﬁne vY (T,p)

:=

vW 1,p (0,T ;Lp (Ω)) + vLp (0,T ;W 4,p (Ω)) .

• C(α, β, . . . ) will indicate a positive constant depending on α, β,. . . . To distinguish diﬀerent constants depending on α, β, etc. in a sequence of estimates, we shall write C1 (α, β, . . . ), C2 (α, β, . . . ), etc. With the above notations we can give the deﬁnition of the inverse problem we are investigating in the sequel. Deﬁnition 1.1. Let p ∈ (1, +∞). Determine θ, χ, a, b, and k with θ ∈ W 2,p (0, τ ; Lp (Ω)) ∩ W 1,p (0, τ ; W 2,p (Ω)), χ ∈ W 2,p (0, τ ; Lp (Ω)) ∩ W 1,p (0, τ ; W 4,p (Ω)), a ∈ W 1,p (0, τ ), b ∈ Lp (0, τ ), k ∈ R, satisfying the system ⎧ Dt (θ + λχ + a ∗ θ)(t, x) ⎪ ⎪ ⎪ ⎪ −∆[kθ(t, x) + b ∗ θ](t, x) = f (t, x), (t, x) ∈ [0, τ ] × Ω, ⎪ ⎪ ⎪ ⎪ εDt χ(t, x) = ∆[−∆χ + ϕ(χ) − λθ](t, x), (t, x) ∈ [0, τ ] × Ω, ⎨ Dν θ(t, x ) = Dν χ(t, x ) = Dν ∆χ(t, x ) = 0, (t, x ) ∈ [0, τ ] × ∂Ω, ⎪ ⎪ θ(0, x) = θ0 (x), x ∈ Ω, ⎪ ⎪ ⎪ ⎪ χ(0, x) = χ0 (x), x ∈ Ω, ⎪ ⎪ ⎩ Φj [θ(t)] = gj (t), j ∈ {1, 2}, t ∈ [0, τ ], under suitable regularity and compatibility conditions on the data.

(1.8) (1.9) (1.10) (1.11) (1.12)

(1.13)

An Inverse Problem for a Phase-ﬁeld Model in Sobolev Spaces

193

1.3. The main result The main result of this paper states that, under suitable regularity and compatibility conditions on the data (see the list (C1)–(C14) is Section 2) the inverse problem in Deﬁnition 1.1 has a unique solution in the Sobolev setting (see Theorem 4.1). The proof of Theorem 4.1 is nontrivial an it is spread over three sections. More precisely: • Section 2 contains an equivalent reformulation (see Theorem 2.1), in suitable Sobolev spaces, of the inverse problem in Deﬁnition 1.1. • In Section 3 we have some optimal regularity results in the Lp spaces for the linear version of the problem (see Theorem 3.2), together with are technical facts that we shall use afterwards. • In Section 4 thanks to the results of the previous sections, by ﬁxed point arguments, we state and prove our main result.

2. Equivalent reformulation of the inverse problem The main goal, when dealing with parabolic integro-diﬀerential inverse problems, is to ﬁnd for them equivalent reformulations which lead to a set of regularity and compatibility conditions on the data that make the inverse problem well posed. The equivalent reformulation is also an essential tool to prove, by optimal regularity results and ﬁxed point arguments, the existence and the uniqueness of a solution to the inverse problem. Let us introduce the set of conditions that allow us to make this reformulation: (C1) Ω is an open bounded subset of Rn , lying on one side of its boundary ∂Ω, which is a submanifold of Rn of class C 4 ; (C2) λ ∈ R, ε ∈ R+ ; (C3) p ∈ (1, +∞), n ∈ N, n < 4p; (C4) ϕ ∈ C ∞ (R); 4,p (Ω); (C5) χ0 ∈ WBB

(C6) θ0 ∈ WB2,p (Ω); (C7) for some T ∈ R+ , f ∈ W 1,p (0, T ; Lp(Ω));

(C8) for j ∈ {1, 2}, u ∈ Lp (Ω), Φj [u] = Ω ϕj (x)u(x)dx, with ϕj ∈ Lp (Ω); (C9) for j ∈ {1, 2}, gj ∈ W 2,p (0, T ); 4,p (C10) v0 := ε−1 ∆[−∆χ0 + ϕ(χ0 ) − λθ0 ] ∈ (Lp (Ω), WBB (Ω))1−1/p,p ;

(C11) χ−1 := Φ2 [θ0 ]Φ1 [∆θ0 ] − Φ1 [θ0 ]Φ2 [∆θ0 ] = 0; (C12) k0 := χ[Φ1 [θ0 ]{Φ2 [f (0) − λv0 ] − g2 (0)} − Φ2 [θ0 ]{Φ1 [f (0) − λv0 ] − g1 (0)}] ∈ R+ ; (C13) Φj (θ0 ) = gj (0), j ∈ {1, 2}; (C14) u0 := f (0)−λv0 −a0 θ0 +k0 ∆θ0 ∈ (Lp (Ω), WB2,p (Ω))1−1/p,p , with a0 deﬁned in (2.5);

194

F. Colombo and D. Guidetti We can state and prove the following equivalence theorem.

Theorem 2.1. Let the assumptions (C1)–(C14) hold. Let θ, χ, a, b, k verify the conditions (1.8)–(1.12) and solve the system (1.13). We set u := Dt θ, v := Dt χ, h := Dt a. Then k = k0 (deﬁned in (C12)), u, v and h satisfy the following conditions (2.8)–(2.10) and solve the system (2.14)–(2.15). On the other hand,if u, v, h satisfy the conditions (2.8)–(2.10) and solve the system (2.14)–(2.15), if we set θ := θ0 + 1 ∗ u, χ := χ0 + 1 ∗ v, a := a0 + 1 ∗ h, k = k0 , then θ, χ, a, b, k verify the conditions (1.8)–(1.12) and solve the system (1.13). Proof. Assume that a solution (θ, χ, a, b, k) exists, satisfying the conditions (1.8)– (1.12). From the second equation in (1.13), we obtain immediately Dt χ(0) = v0 .

(2.1)

We observe that Dt (a ∗ θ) = a(0)θ + Dt a ∗ θ,

(2.2)

Dt a ∗ θ ∈ C([0, τ ]; L (Ω)) and b ∗ ∆θ ∈ C([0, τ ]; L (Ω)). So, for t = 0, the ﬁrst equation in (1.13) becomes p

p

Dt θ(0) + λv0 + a(0)θ0 − k∆θ0 = f (0).

(2.3)

Applying Φj (for j ∈ {1, 2}) to the ﬁrst equation in (1.13) and using the last condition (always in (1.13)), we obtain a(0)Φj [θ0 ] − kΦj [∆θ0 ] = Φj [f (0) − λv0 ] − gj (0), j ∈ {1, 2}.

(2.4)

From assumption (C11), it follows a(0) = a0 := χ[{Φ2 [f (0) − λv0 ] − g2 (0)}Φ1 [θ0 ] − {Φ1 [f (0) − λv0 ] − g1 (0)}Φ2 [θ0 ]]. (2.5) k = k0 , (2.6) Dt θ(0) = u0 .

(2.7)

u := Dt θ,

(2.8)

v := Dt χ,

(2.9)

h := Dt a.

(2.10)

u ∈ W 1,p (0, τ ; Lp (Ω)) ∩ Lp (0, τ ; W 2,p (Ω)),

(2.11)

v ∈ W 1,p (0, τ ; Lp (Ω)) ∩ Lp (0, τ ; W 4,p (Ω)),

(2.12)

h ∈ Lp (0, τ ).

(2.13)

Now we set

We have that

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Diﬀerentiating the two ﬁrst equations in (1.13), we see that (u, v, h, b) solves the system: ⎧ Dt u(t, x) + λDt v(t, x) = k0 ∆u(t, x) − a0 u(t, x) + Dt f (t, x) ⎪ ⎪ ⎪ ⎪ −h(t)θ ⎪ 0 (x) + b(t)∆θ0 (x) − (h ∗ u)(t, x) + (b ∗ ∆u)(t, x), ⎪ ⎪ ⎪ (t, x) ∈ [0, τ ] × Ω, ⎪ ⎪ ⎪ ⎪ 2 ⎨ εDt v(t, x) = −∆ v(t, x) + ∆[ϕ (χ0 + 1 ∗ v)v](t, x) − λ∆u(t, x), (t, x) ∈ [0, τ ] × Ω, (2.14) ⎪ ⎪ ⎪ ⎪ Dν u(t, x ) = Dν v(t, x ) = Dν ∆v(t, x ) = 0, (t, x ) ∈ [0, τ ] × ∂Ω, ⎪ ⎪ ⎪ ⎪ ⎪ u(0, x) = u0 (x), x ∈ Ω, ⎪ ⎪ ⎪ ⎩ v(0, x) = v0 (x), x ∈ Ω. Applying now Φj to the ﬁrst equation in (2.14), using the fact that Φ[u(t)] = gj (t), Φ[Dt u(t)] = gj (t) and replacing Dt v with −1 {−∆2 v + ∆[ϕ (χ0 + 1 ∗ v)v] − λ∆u}, we can couple (2.14) with the two further conditions h(t)Φj (θ0 ) − b(t)Φj (∆θ0 ) = Φj [Dt f (t)] − gj (t) − a0 gj (t) − (h ∗ Φj (u))(t) + k0 Φj [∆u(t)] + (b ∗ Φj (∆u))(t) + −1 λΦj {∆2 v(t) − ∆[ϕ (χ0 + 1 ∗ v)(t)v(t) + λu(t)]},

(2.15)

with t ∈ [0, τ ], j ∈ {1, 2}. Assume now that u, v, h satisfy the regularity conditions (2.11), (2.12), (2.13). We set θ := θ0 + 1 ∗ u, χ := χ0 + 1 ∗ v, a := a0 + 1 ∗ h. We remark that the ﬁrst equation in (2.14) can be written in the form Dt [Dt θ + λDt χ + a0 θ + Dt a ∗ θ − k0 ∆θ − b ∗ ∆θ] = Dt f. From (C14), it follows Dt θ + λDt χ + a0 θ + Dt a ∗ θ − k0 ∆θ − b ∗ ∆θ = f, so that the ﬁrst equation in (1.13) is satisﬁed, if k = k0 . The second equation in (2.14) can be written in the form Dt {εDt χ − ∆[−∆χ + ϕ(χ) − λθ]} = 0. From (C10), it follows εDt χ − ∆[−∆χ + ϕ(χ) − λθ] = 0. The boundary and the initial conditions in (1.13) are clearly satisﬁed by θ and χ. It remains only to verify that Φj [θ(t)] = gj (t), ∀t ∈ [0, τ ], j ∈ {1, 2}. To this aim, we ﬁrst observe that, by using the second equation in (2.14), we can write (2.15) in the form h(t)Φj (θ0 ) − b(t)Φj (∆θ0 ) =

Φj [Dt f (t)] − gj (t) − a0 gj (t) − (h ∗ Φj (u))(t) +k0 Φj [∆u(t)] + (b ∗ Φj (∆u))(t) − λΦj (Dt v(t)). (2.16)

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Applying Φj to the ﬁrst equation in (2.14) and comparing the result with (2.16), we obtain (2.17) Dt2 Φj [θ] + a0 Dt Φj [θ] = gj + a0 gj . Moreover, Dt {Φj [θ]}(0) = Φj [u0 ] = Φj [f (0) − λv0 ] − a0 Φj [θ0 ] + k0 Φj [∆θ0 ] = gj (0). (2.18) So (2.17), (C13), (2.18) imply Φj [θ(t)] = gj (t), ∀t ∈ [0, τ ] and for j ∈ {1, 2}.

3. Some preparatory results We start by studying the following system ⎧ Dt u(t, x) + λDt v(t, x) = k0 ∆u(t, x) − a0 u(t, x) + f (t, x), ⎪ ⎪ ⎪ ⎪ ⎨ εDt v(t, x) = −∆2 v(t, x) − λ∆u(t, x) + g(t, x), Dν u(t, x ) = Dν v(t, x ) = Dν ∆χ(t, x ) = 0, ⎪ ⎪ u(0, x) = u0 (x), ⎪ ⎪ ⎩ v(0, x) = v0 (x),

(t, x) ∈ [0, T ] × Ω, (t, x) ∈ [0, T ] × Ω, (t, x ) ∈ [0, T ] × ∂Ω, x ∈ Ω, x ∈ Ω. (3.1)

under the following assumptions: (A1) assumption (C1) holds; (A2) λ ∈ R, k0 ∈ R+ , ε ∈ R+ , a0 ∈ R. We shall use the following result, which is a consequence of Theorem 8.2 in [18]: Theorem 3.1. (I) Consider the problem ⎧ ⎨ Dt u(t, x) = k0 ∆u(t, x) − a0 u(t, x) + f (t, x), (t, x) ∈ [0, T ] × Ω, (t, x ) ∈ [0, T ] × ∂Ω, Dν u(t, x ) = 0, ⎩ u(0, x) = u0 (x), x ∈ Ω.

(3.2)

with the assumption (A1), k0 ∈ R+ , a0 ∈ R. Then, if p ∈ (1, +∞), f ∈ Lp (0, T ; Lp (Ω)) and u0 ∈ (Lp (Ω), WB2,p (Ω))1−1/p,p , (3.2) has a unique solution u ∈ X(T, p). Moreover, ∀ T0 ∈ R+ , there exists C(T0 ) ∈ R+ , such that, if 0 < T ≤ T0 , uX(T,p) ≤ C(T0 )(f Lp(0,T ;Lp (Ω)) + u0 (Lp (Ω),W 2,p (Ω))1−1/p,p ).

(3.3)

B

(II) Consider the problem ⎧ ⎨ εDt v(t, x) = −∆2 v(t, x) + λ2 ∆v(t, x) + g(t, x), (t, x) ∈ [0, T ] × Ω, Dν v(t, x ) = 0, Dν ∆v(t, x ) = 0 (t, x ) ∈ [0, T ] × ∂Ω, ⎩ v(0, x) = v0 (x), x ∈ Ω.

(3.4)

with the assumption (A1) and ε ∈ R+ . Then, if p ∈ (1, +∞),g ∈ Lp (0, T ; Lp (Ω)) 4,p and v0 ∈ (Lp (Ω), WBB (Ω))1−1/p,p , (3.4) has a unique solution v ∈ Y (T, p). More+ over, ∀ T0 ∈ R , there exists C(T0 ) ∈ R+ , such that, if 0 < T ≤ T0 , vY (T,p) ≤ C(T0 )(f Lp (0,T ;Lp (Ω)) + v0 (Lp(Ω),W 4,p (Ω))1−1/p,p ). BB

(3.5)

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Proof. We show only (I). The proof of (II) is similar. By Proposition 1.2.10 in [21], there exists z ∈ X(T, p) such that z(0) = u0 . Subtracting z to u, one is reduced to the case u0 = 0. The existence and uniqueness of a solution in X(T, p) in case u0 = 0 is a particular case of Theorem 8.2 in [18]. The estimate of the solution can be obtained employing the following argument: let 0 < T ≤ T0 . Extend f to an element f˜ ∈ Lp (0, T0 ; Lp (Ω)) setting f˜(t, .) ≡ 0 if t ∈]T, T0 ]. Then the solution u˜ of (3.2), replacing T with T0 and f with f˜, is an extension of u. So uX(T0 ,p) uX(T,p) ≤ ˜ ≤ C(T0 )(f˜Lp (0,T0 ;Lp (Ω)) + u0 (Lp (Ω),W 2,p (Ω))1−1/p,p ) (3.6) B = C(T0 )(f Lp(0,T ;Lp (Ω)) + u0 (Lp (Ω),W 2,p (Ω))1−1/p,p ). B

Remark 3.1. From Theorem 3.5 in [19], we have that, for p ∈ (1, +∞), (Lp (Ω), WB2,p (Ω))1−1/p,p 2(1−1/p) Bp,p (Ω) = 2(1−1/p) {f ∈ Bp,p (Ω) : Dν f ≡ 0}

if if

1 < p < 3, 3 < p < +∞.

(3.7)

Moreover, 4,p (Ω))1−1/p,p (Lp (Ω), WBB ⎧ 4(1−1/p) ⎪ (Ω) if ⎨ Bp,p 4(1−1/p) = {f ∈ Bp,p (Ω) : Dν f ≡ 0} if ⎪ ⎩ 4(1−1/p) (Ω) : Dν f ≡ Dν ∆f ≡ 0} if {f ∈ Bp,p

1 < p < 5/3, 5/3 < p < 5, 5 < p < +∞.

(3.8)

Corollary 3.1. Consider the solution v of the problem (3.4) in case v0 = 0. Then, if 0 < T ≤ T0 and s ∈ Z, 0 ≤ s ≤ 4, s

vLp (0,T ;W s,p (Ω)) ≤ C(T0 , s)T 1− 4 gLp(0,T ;Lp (Ω)) . Proof. If s = 0, we have, owing to Young’s inequality, vLp (0,T ;Lp (Ω))

= ≤ ≤

1 ∗ Dt vLp (0,T ;Lp (Ω)) 1L1(0,T ) Dt vLp (0,T ;Lp (Ω)) C(T0 )T gLp(0,T ;Lp (Ω)) .

In general, there exists C(s) > 0 such that, ∀v ∈ W 4,p (Ω)), 1− s

s

4 4 vW vW s,p (Ω) ≤ C(s)vLp (Ω) 4,p (Ω) .

So, employing H¨ older’s inequality,

T vLp(0,T ;W s,p (Ω)) = ( 0 v(t)pW s,p (Ω) dt)1/p

T (1−s/4)p sp/4 ≤ C(s)( 0 v(t)Lp (Ω) v(t)W 4,p (Ω) dt)1/p 1−s/4

s/4

≤ C(s)vLp (0,T ;Lp (Ω)) vLp (0,T ;W 4,p (Ω)) ≤ C(T0 , s)T 1−s/4 gLp(0,T ;Lp (Ω)) .

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Theorem 3.2. Let p ∈ (1, +∞), T ∈ R+ . Then the following conditions are necessary and suﬃcient in order that (3.1) have a solution (u, v) ∈ X(T, p) × Y (T, p): (I) f, g ∈ Lp (0, T ; Lp (Ω)); (II) u0 ∈ (Lp (Ω), WB2,p (Ω))1−1/p,p ; 4,p (III) v0 ∈ (Lp (Ω), WBB (Ω))1−1/p,p . If (I)–(III) are satisﬁed, the solution is unique. Moreover, ∀T0 ∈ R+ there exists C(T0 ) ∈ R+ , such that, if 0 < T ≤ T0 , uX(T,p) + vY (T,p) ≤ C(T0 )[f Lp(0,T ;Lp (Ω)) + gLp(0,T ;Lp (Ω)) + u0 (Lp (Ω),W 2,p (Ω))1−1/p,p

(3.9)

B

+ v0 (Lp (Ω),W 4,p (Ω))1−1/p,p ]. BB

Proof. The condition (I) is obviously necessary. From the trace method to deﬁne real interpolation spaces (see [21], Section 1.2.2), we know that {u(0) : u ∈ X(T, p))} and {v(0) : v ∈ Y (T, p)} coincide with the interpolation spaces (Lp (Ω), WB2,p (Ω))1−1/p,p

4,p and (Lp (Ω), WBB (Ω))1−1/p,p

respectively. So, even conditions (II) and (III) are necessary. To show that (I)–(III) are also suﬃcient, we set z := u + λv and consider the new problem ⎧ Dt z(t, x) = k0 ∆z(t, x) − a0 z(t, x) ⎪ ⎪ ⎪ ⎪ +(λa0 − k0 λ∆)v(t, x) + f (t, x), (t, x) ∈ [0, T ] × Ω, ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎨ εDt v(t, x) = −∆ v(t, x) + λ ∆v(t, x) − λ∆z(t, x) + g(t, x), (t, x) ∈ [0, T ] × Ω, ⎪ ⎪ Dν z(t, x ) = Dν v(t, x ) = Dν ∆v(t, x ) = 0, (t, x ) ∈ [0, T ] × ∂Ω, ⎪ ⎪ ⎪ ⎪ ⎪ z(0, x) = z0 (x), x ∈ Ω, ⎪ ⎪ ⎪ ⎩ v(0, x) = v0 (x), x ∈ Ω.

(3.10)

(3.11)

with z0 := u0 + λv0 . (Ω), WB2,p (Ω))1−1/p,p

(3.12)

and that to show that (3.1) has a We observe that z0 ∈ (L unique solution (u, v) in X(T, p) × Y (T, p) is equivalent to show that (3.11) has a unique solution with the same regularity. To this aim, we indicate with TT (u0 , f ) the solution in X(T, p) of (3.2), with ST (v0 , g) the solution in Y (T, p) of (3.4). We start by looking for a solution in some interval [0, τ ], with 0 < τ ≤ T . p

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Let X := {z ∈ W 1,p (0, τ ; Lp (Ω)) ∩ Lp (0, T ; WB2,p(Ω)) : z(0) = z0 }, which is a closed subspace of X(τ, p). It is easily seen that, if a solution (z, v) exists in some interval [0, τ ], then v = Sτ (v0 , g − λ∆z), (3.13) = Tτ [u0 , f + (λa0 − k0 λ∆)Sτ (v0 , g − λ∆z)] = Tτ [u0 , f + (λa0 − k0 λ∆)Sτ (v0 , g)] − λ2 Tτ [0, (a0 − k0 ∆)Sτ (0, ∆z)] := Uτ (z). (3.14) The operator Uτ maps X into itself. If z1 and z2 are elements of X, we have, employing Theorem 3.1 and Corollary 3.1, z

Uτ (z1 ) − Uτ (z2 )X(τ,p)

= λ2 Tτ (0, (a0 − k0 ∆)Sτ (0, ∆(z1 − z2 ))X(τ,p) ≤ C1 (T )(a0 − k0 ∆)Sτ (0, ∆(z1 − z2 ))Lp (0,τ ;Lp (Ω)) ≤ C2 (T )Sτ (0, ∆(z1 − z2 ))Lp (0,τ ;W 2,p (Ω)) ≤ C3 (T )τ 1/2 ∆(z1 − z2 ))Lp (0,τ ;Lp(Ω)) ≤ C4 (T )τ 1/2 z1 − z2 Lp (0,τ ;W 2,p (Ω)) .

So, if C4 (T )τ 1/2 < 1, the operator Uτ has a unique ﬁxed point z in X. We observe that the length τ of the interval does not depend on the data of our problem. From (3.13), we can determine also v. Starting from the new data z(τ ) and v(τ ) (again 4,p (Ω))1−1/p,p respectively, we belonging to (Lp (Ω), WB2,p (Ω))1−1/p,p and (Lp (Ω), WBB can extend the solution to the interval [0, 2τ ]. Proceeding in this way, we construct a solution in [0, T ]. The proof of the estimate (3.9) can be obtained with the same argument used to show (3.3) and (3.5). Lemma 3.1. Let X be a Banach space, p ∈ (1, +∞), τ ∈ R+ , h ∈ Lp (0, τ ), f ∈ Lp (0, τ ; X). Then h ∗ f ∈ Lp (0, τ ; X) and h ∗ f Lp (0,τ ;X) ≤ τ 1−1/p hLp(0,τ ) f Lp(0,τ ;X) . Proof. Using Young’s inequality and H¨ older’s inequality, we have h ∗ f Lp (0,τ ;X) ≤ hL1(0,τ ) f Lp(0,τ ;X) ≤ τ 1−1/p hLp(0,τ ) f Lp(0,τ ;X) .

Lemma 3.2. Under the conditions (C1) and (C3), W 4,p (Ω) is continuously embedded in C(Ω) and is a space of pointwise multipliers for W s,p (Ω), ∀s ∈ Z, 0 ≤ s ≤ 4. Proof. Analogous to the proof given in [20], Lemma 4.1 in the case p = 2.

Lemma 3.3. Under the assumptions (C1) and (C3), if S ∈ C ∞ (R), the map v → S ◦ v is of class C ∞ from W 4,p (Ω) into itself. Moreover, ∀k ∈ N0 , (S ◦.)(k) is bounded with values in Lk (W 4,p (Ω), W 4,p (Ω)) in every bounded subset of W 4,p (Ω). Proof. Analogous to the proof given in [20], Lemma 4.2 in the case p = 2.

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Lemma 3.4. Assume that (C1) and (C3) are satisﬁed, S ∈ C ∞ (R), χ0 ∈ W 4,p (Ω). Let R ∈ R+ , 0 < τ ≤ T , V1 and V2 be elements of Y (τ, p), max Vj Lp (0,τ ;W 4,p (Ω)) ≤ R.

j∈{1,2}

Then S(χ0 + 1 ∗ V1 )V1 − S(χ0 + 1 ∗ V2 )V2 Lp (0,τ ;W 2,p (Ω)) ≤ C(R, T )τ (1−1/p)∧(1/(2p)) V1 − V2 Y (τ,p) . Proof. We have S(χ0 + 1 ∗ V1 )V1 − S(χ0 + 1 ∗ V2 )V2 Lp(0,τ ;W 2,p (Ω)) ≤ [S(χ0 + 1 ∗ V1 ) − S(χ0 + 1 ∗ V2 )]V1 Lp (0,τ ;W 2,p (Ω)) + S(χ0 + 1 ∗ V2 )(V1 − V2 )Lp (0,τ ;W 2,p (Ω)) := I1 + I2 . By Lemma 3.2, I1

≤ ≤

CS(χ0 + 1 ∗ V1 ) − S(χ0 + 1 ∗ V2 )L∞ (0,τ ;W 4,p (Ω)) V1 Lp (0,τ ;W 2,p (Ω)) C1 (R)S(χ0 + 1 ∗ V1 ) − S(χ0 + 1 ∗ V2 )L∞ (0,τ ;W 4,p (Ω))

For j ∈ {1, 2}, by virtue of H¨ older’s inequality, χ0 + 1 ∗ Vj L∞ (0,τ ;W 4,p (Ω))

≤ χ0 W 4,p (Ω) + τ 1−1/p Vj Lp (0,τ ;W 4,p (Ω)) ≤ χ0 W 4,p (Ω) + T 1−1/p R.

So, by Lemma 3.3 and H¨older’s inequality, I1

≤ ≤

C2 (R)1 ∗ (V1 − V2 )L∞ (0,τ ;W 4,p (Ω)) C2 (R)τ 1−1/p V1 − V2 Lp (0,τ ;W 4,p (Ω)) .

With analogous arguments, we obtain I2

≤

C3 (R)V1 − V2 Lp (0,τ ;W 2,p (Ω)) .

As W 2,p (Ω) ∈ J 12 (Lp (Ω), W 4,p (Ω)), by H¨older’s inequality, τ 1 V1 − V2 )(t)pW 2,p (Ω) dt) p ( 0 τ p p 1 2 p ≤ C( (V1 − V2 )(t)L2 p (Ω) (V1 − V2 )(t)W 4,p (Ω) dt) 0

1

1

≤ CV1 − V2 L2 p (0,τ ;Lp (Ω)) V1 − V2 L2 p (0,τ ;W 4,p (Ω)) . We have, employing Young’s inequality, V1 − V2 Lp (0,τ ;Lp(Ω)) = V1 (0) − V2 (0) + 1 ∗ Dt (V1 − V2 )Lp (0,τ ;Lp (Ω)) ≤ τ 1/p V1 (0) − V2 (0)Lp (Ω) + τ Dt (V1 − V2 )Lp (0,τ ;Lp (Ω)) ≤ (τ 1/p ∨ τ )V1 − V2 W 1,p (0,τ ;Lp (Ω)) .

(3.15)

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So we obtain, if 0 < τ ≤ T , 1/2

≤ ≤

1/2

C4 (R)(τ 1/(2p) ∨ τ 1/2 )V1 − V2 Lp (0,τ ;W 4,p (Ω)) V1 − V2 W 1,p (0,τ ;Lp(Ω)) C4 (R, T )τ 1/(2p) V1 − V2 Y (τ,p) . (3.16) From (3.15) and (3.16), we get the result. I2

Lemma 3.5. Let p ∈ (1, +∞), Ω satisfying (C1), ψ ∈ Lp (Ω), τ ∈ R+ . We deﬁne in Lp (0, τ ; Lp (Ω)) the operator ψ(x)f (t, x)dx. Φ[f ](t) := Ω

Then, if u ∈ X(τ, p), Φ[∆u] ∈ L (0, τ ) and p

Φ[∆u]Lp(0,τ ) ≤ ω(τ )uX(τ,p) ,

(3.17)

with ω(τ ) > 0, independent of u, and lim ω(τ ) = 0; if v ∈ Y (τ, p), Φ[∆2 v] ∈ τ →0

Lp (0, τ ) and

Φ[∆2 v]Lp (0,τ ) ≤ ω(τ )vY (τ,p) ,

(3.18)

with ω(τ ) > 0, independent of u, and lim ω(τ ) = 0. τ →0

Proof. We prove only the ﬁrst statement; the second can be shown similarly. It is easily seen that Φ[∆u]Lp(0,τ ) ≤ CψLp (Ω) uLp(0,τ ;W 2,p (Ω)) ≤ CψLp (Ω) uX(τ,p),

(3.19)

with C > 0 independent of ψ, τ , u. If ψ ∈ D(Ω), we have ∆ψ(x)u(t, x)dx, Φ[∆u](t) = Ω

so that Φ[∆u]Lp(0,τ )

≤ = ≤ ≤ ≤

∆ψLp (Ω) uLp(0,τ ;Lp(Ω)) ∆ψLp (Ω) u(0) + 1 ∗ Dt uLp(0,τ ;Lp(Ω)) ∆ψLp (Ω) (τ 1/p u(0)Lp(Ω) + τ Dt uLp(0,τ ;Lp(Ω)) ) ∆ψLp (Ω) (τ 1/p ∨ τ )uW 1,p (0,τ ;Lp(Ω)) ∆ψLp (Ω) (τ 1/p ∨ τ )uX(τ,p) . (3.20) The conclusion is a simple consequence of (3.19), (3.20) and the density of D(Ω) in Lp (Ω).

4. The main result Theorem 4.1. Assume that the assumptions (C1)–(C14) are satisﬁed. Then there exists τ ∈ (0, T ] such that the problem (1.13) has a unique solution (θ, χ, a, b, k) satisfying the conditions (1.8)–(1.12).

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Proof. We start by showing the existence of a solution with the required properties. We already know from Lemma 2.1 that, if a solution exists, necessarily k = k0 and a(0) = a0 . Moreover, if u, v and h are the functions deﬁned in (2.8)–(2.10), they satisfy the system (2.14)–(2.15) and vice versa. So we focalize our attention on the system (2.14)–(2.15) and look for a solution (u, v, h, b) satisfying the regularity conditions (2.11)–(2.13) and (1.11). We start by ﬁxing an element ˜ ∈ W 1,p (0, T ; Lp(Ω)) ∩ Lp (0, T ; W 2,p (Ω)) U B ˜ (0) = u0 and an element such that U 4,p V˜ ∈ W 1,p (0, T ; Lp (Ω)) ∩ Lp (0, T ; WBB (Ω))

such that V˜ (0) = v0 . Next, we indicate with (H, B) the solution in Lp (0, T )×Lp(0, T ) of the system H(t)Φj (θ0 ) − B(t)Φj (∆θ0 ) = Φj [Dt f (t)] − gj (t) − a0 gj (t) ˜ (t)] + −1 λΦj {∆2 V˜ (t) + k0 Φj [∆U

(4.21)

− λ∆U˜ (t)]}, t ∈ [0, T ], j ∈ {1, 2}. The existence and uniqueness of (H, B) follows immediately from (C11). Now we consider the problem ⎧ Dt U (t, x) + λDt V (t, x) = k0 ∆U (t, x) − a0 U (t, x) + Dt f (t, x) ⎪ ⎪ ⎪ ⎪ ⎪ − H(t)θ0 (x) + B(t)∆θ0 (x), (t, x) ∈ [0, τ ] × Ω, ⎪ ⎪ ⎪ ⎪ ⎨ εDt V (t, x) = −∆2 V (t, x) − λ∆U (t, x), (t, x) ∈ [0, T ] × Ω, (4.22) ⎪ ⎪ Dν U (t, x ) = Dν V (t, x ) = Dν ∆V (t, x ) = 0, (t, x ) ∈ [0, T ] × ∂Ω, ⎪ ⎪ ⎪ ⎪ U (0, x) = u0 (x), x ∈ Ω, ⎪ ⎪ ⎪ ⎩ V (0, x) = v0 (x), x ∈ Ω. By Theorem 3.2, the problem (4.22) has a unique solution (U, V ) in X(T, p) × Y (T, p). We set C1 := U X(T,p) + V Y (T,p) + HLp(0,T ) + BLp(0,T ) .

(4.23)

Let ρ ∈ R+ . We put Z(τ, ρ) := {(u, v, h, b) ∈ X(τ, p) × Y (τ, p) × Lp (0, τ ) × Lp (0, τ ) : u(0) = u0 , v(0) = v0 , Dν u ≡ Dν v ≡ Dν ∆v ≡ 0, u − U X(τ,p) + v − V Y (τ,p) + h − HLp (0,τ ) + b − BLp (0,τ ) ≤ ρ}. Z(τ, ρ) is a closed subset of X(τ, ρ) × Y (τ, p) × Lp (0, τ ) × Lp (0, τ ).

(4.24)

An Inverse Problem for a Phase-ﬁeld Model in Sobolev Spaces

203

If (U1 , V1 , H1 , B1 ) ∈ Z(τ, ρ), we consider the problem ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Dt u(t, x) + λDt v(t, x) = k0 ∆u(t, x) − a0 u(t, x) + Dt f (t, x) −h(t)θ0 (x) + b(t)∆θ0 (x) − (h ∗ U1 )(t, x) + (b ∗ ∆U1 )(t, x), (t, x) ∈ [0, τ ] × Ω, εDt v(t, x) = −∆2 v(t, x) + ∆[ϕ (χ0 + 1 ∗ V1 )V1 ](t, x) − λ∆u(t, x), (t, x) ∈ [0, τ ] × Ω, Dν u(t, x ) = Dν v(t, x ) = Dν ∆v(t, x ) = 0, (t, x ) ∈ [0, τ ] × ∂Ω, u(0, x) = u0 (x),

x ∈ Ω,

v(0, x) = v0 (x),

x ∈ Ω,

(4.25)

h(t)Φj (θ0 ) − b(t)Φj (∆θ0 ) = Φj [Dt f (t)] − gj (t) − a0 gj (t) −(H1 ∗ Φj (U1 ))(t) + k0 Φj [∆U1 (t)] + (B1 ∗ Φj (∆U1 ))(t) +−1 λΦj {∆2 V1 (t) − ∆[ϕ (χ0 + 1 ∗ V1 )(t)V1 (t) + λU1 (t)]}, t ∈ (0, τ ), j ∈ {1, 2}.

It is easily seen that the system (4.25) has a unique solution S(U1 , V1 , H1 , B1 ) belonging to X(τ, p) × Y (τ, p) × Lp (0, τ ) × Lp (0, τ ). In fact, one can determine ﬁrst h and b using (C11). Next, one can ﬁnd u and v, using Theorem 3.2 and the knowledge of h and b. Clearly, we have to look for a ﬁxed point of the operator S. Let now (U1 , V1 , H1 , B1 ) ∈ Z(τ, ρ) for some τ ∈ (0, T ], ρ ∈ R+ . We put (u, v, h, b) := S(U1 , V1 , H1 , B1 ). We estimate u − U X(τ,p) + v − V Y (τ,p) + h − HLp (0,τ ) + b − BLp (0,τ ) . To this aim, we observe that (u − U, v − V, h − H, b − B) solves the system ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

Dt (u − U )(t, x) + λDt (v − V )(t, x) = k0 ∆(u − U )(t, x) −a0 (u − U )(t, x) − (h(t) − H(t))θ0 (x) + (b(t) − B(t))∆θ0 (x) −(h ∗ U1 )(t, x) + (b ∗ ∆U1 )(t, x), (t, x) ∈ [0, τ ] × Ω, εDt (v − V )(t, x) = −∆2 (v − V )(t, x) + ∆[ϕ (χ0 + 1 ∗ V1 )V1 ](t, x) −λ∆(u − U )(t, x), (t, x) ∈ [0, τ ] × Ω,

Dν (u − U )(t, x ) = Dν (v − V )(t, x ) = Dν ∆(v − V )(t, x ) = 0, (t, x ) ∈ [0, τ ] × ∂Ω, ⎪ ⎪ ⎪ ⎪ ⎪ (u − U )(0, x) = 0, x ∈ Ω, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (v − V )(0, x) = 0, x ∈ Ω, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (h(t) − H(t))Φj (θ0 ) − (b(t) − B(t))Φj (∆θ0 ) = −(H1 ∗ Φj (U1 ))(t)+ ⎪ ⎪ ⎪ k Φ [∆(U − U)(t)] ˜ ⎪ + (B1 ∗ Φj (∆U1 ))(t) + −1 λΦj {∆2 (V1 − V˜ )(t) ⎪ 0 j 1 ⎪ ⎩ ˜ )(t)]}, t ∈ (0, τ ), j ∈ {1, 2}. −∆[ϕ (χ0 + 1 ∗ V1 )(t)V1 (t) + λ(U1 − U

(4.26)

204

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We have h − HLp(0,τ ) + b − BLp (0,τ ) ≤ C2

2

{H1 ∗ Φj (U1 )Lp (0,τ )

j=1

+ B1 ∗ Φj (∆U1 ))Lp (0,τ ) ˜ + −1 λ∆2 (V1 − V˜ ) − −1 λ2 ∆(U1 − U ˜ )}Lp (0,τ ) + Φj {k0 ∆(U1 − U) + Φj {∆[ϕ (χ0 + 1 ∗ V1 )V1 ]}Lp(0,τ ) }. By Lemma 3.1, it follows H1 ∗ Φj (U1 )Lp (0,τ ) + B1 ∗ Φj (∆U1 ))Lp (0,τ ) ≤ τ 1−1/p (H1 Lp (0,τ ) Φj (U1 )Lp (0,τ ) + B1 Lp (0,τ ) Φj (∆U1 )Lp (0,τ ) ) ≤ C3 (C1 + ρ)2 τ 1−1/p . By Lemma 3.5, we get ˜ ) + −1 λ∆2 (V1 − V˜ ) − −1 λ2 ∆(U1 − U ˜ )}Lp(0,τ ) Φj {k0 ∆(U1 − U ˜ ˜ ≤ ω1 (τ )(U1 − U X(τ,p) + V1 − V )Y (τ,p) ). The letter ω will be used to indicate that the constant tends to 0 as τ goes to 0. Finally, from Lemma 3.4, we have (setting V2 = 0) Φj {∆[ϕ (χ0 + 1 ∗ V1 )V1 ]}Lp (0,τ ) } ≤ ≤

C4 ϕ (χ0 + 1 ∗ V1 )V1 Lp (0,τ ;W 2,p (Ω)) C1 (T, ρ)τ (1−1/p)∧1/(2p) .

Summarizing, we have obtained an estimate of the form h − HLp(0,τ ) + b − BLp (0,τ )

≤ C2 (T, ρ)ω2 (τ ).

(4.27)

So, from Theorem 3.2, u − U X(τ,p) + v − V Y (τ,p) ≤ C5 {h − HLp (0,τ ) + b − BLp (0,τ ) + h ∗ U1 Lp(0,τ ;Lp (Ω)) + b ∗ ∆U1 Lp (0,τ ;Lp(Ω)) + ϕ (χ0 + 1 ∗ V1 )V1 Lp (0,τ ;W 2,p (Ω)) }. Then we consider h ∗ U1 Lp (0,τ ;Lp(Ω)) ≤ τ 1−1/p hLp(0,τ ) U1 Lp (0,τ ;Lp(Ω)) ≤ τ 1−1/p [HLp(0,T ) + C2 (T, ρ)ω2 (τ )][U X(T,p) + ρ]. and analogously b ∗ ∆U1 Lp (0,τ ;Lp (Ω)) ≤ C6 τ 1−1/p bLp(0,τ ) U1 Lp (0,τ ;W 2,p (Ω)) ≤ C6 τ 1−1/p [BLp (0,T ) + C2 (T, ρ)ω2 (τ )][U X(T,p) + ρ], so, employing also Lemma 3.4, we obtain u − U X(τ,p) + v − V Y (τ,p)

≤ C3 (T, ρ)ω3 (τ ).

(4.28)

An Inverse Problem for a Phase-ﬁeld Model in Sobolev Spaces

205

In conclusion, from (4.27) and (4.28), we have u − U X(τ,p) + v − V Y (τ,p) + h − HLp(0,τ ) + b − BLp (0,τ ) ≤ C4 (T, ρ)ω4 (τ ).

(4.29)

(4.29) implies that, if τ ∈ (0, T ] and C4 (T, ρ)ω4 (τ ) ≤ ρ,

(4.30)

then S maps Z(τ, ρ) into itself. Now we consider two elements (U1 , V1 , H1 , B1 ) and (U2 , V2 , H2 , B2 ) of Z(τ, ρ). Writing (uj , vj , hj , bj ) instead of S(Uj , Vj , Hj , Bj ), we estimate u1 − u2 X(τ,p) + v1 − v2 Y (τ,p) + h1 − h2 Lp (0,τ ) + b1 − b2 Lp (0,τ ) . The diﬀerence (u1 − u2 , v1 − v2 , h1 − h2 , b1 − b2 ) satisﬁes the system ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Dt (u1 − u2 )(t, x) + λDt (v1 − v2 )(t, x) = k0 ∆(u1 − u2 )(t, x) −a0 (u1 − u2 )(t, x) − [h1 (t) − h2 (t)]θ0 (x) + [b1 (t) − b2 (t)]∆θ0 (x) −[(h1 ∗ U1 ) − (h2 ∗ U2 )](t, x) + [(b1 ∗ ∆U1 ) − (b2 ∗ ∆U2 )](t, x), (t, x) ∈ [0, τ ] × Ω, εDt (v1 − v2 )(t, x) = −∆2 (v1 − v2 )(t, x) + ∆[ϕ (χ0 + 1 ∗ V1 )V1 −ϕ (χ0 + 1 ∗ V2 )V2 ](t, x) − λ∆(u1 − u2 )(t, x), (t, x) ∈ [0, τ ] × Ω, Dν (u1 − u2 )(t, x ) = Dν (v1 − v2 )(t, x ) = 0, (t, x ) ∈ [0, τ ] × ∂Ω, Dν ∆(v1 − v2 )(t, x ) = 0, (t, x ) ∈ [0, τ ] × ∂Ω, (u1 − u2 )(0, x) = 0, x ∈ Ω, (v1 − v2 )(0, x) = 0, x ∈ Ω, (h1 (t) − h2 (t))Φj (θ0 ) − (b1 (t) − b2 (t))Φj (∆θ0 ) = −[(H1 ∗ Φj (U1 )) − (H2 ∗ Φj (U2 ))](t) + k0 Φj [∆(U1 − U2 )(t)] +[(B1 ∗ Φj (∆U1 )) − (B2 ∗ Φj (∆U2 ))](t) + −1 λΦj {∆2 (V1 − V2 )(t) −∆[ϕ (χ0 + 1 ∗ V1 )(t)V1 (t) − ϕ (χ0 + 1 ∗ V2 )(t)V2 (t) +λ(U1 − U2 )(t)]}, t ∈ (0, τ ), j ∈ {1, 2}.

(4.31)

So we have h1 − h2 Lp (0,τ ) + b1 − b2 Lp (0,τ ) ≤ C7

2

{H1 ∗ Φj (U1 ) − H2 ∗ Φj (U2 )Lp (0,τ )

j=1

+ B1 ∗ Φj (∆U1 ) − B2 ∗ Φj (∆U2 )Lp (0,τ ) + Φj {k0 ∆(U1 − U2 ) + −1 λ∆2 (V1 − V2 ) − −1 λ2 ∆(U1 − U2 )}Lp(0,τ ) + Φj {∆[ϕ (χ0 + 1 ∗ V1 )V1 − ϕ (χ0 + 1 ∗ V2 )V2 ]}Lp(0,τ ) }.

206

F. Colombo and D. Guidetti H1 ∗ Φj (U1 ) − H2 ∗ Φj (U2 )Lp (0,τ ) ≤ (H1 − H2 ) ∗ Φj (U1 )Lp (0,τ ) + H2 ∗ Φj (U1 − U2 )Lp (0,τ ) ≤ C8 τ 1−1/p [H1 − H2 Lp (0,τ ) U1 Lp (0,τ ;Lp(Ω) + H2 Lp (0,τ ) U1 − U2 Lp (0,τ ;Lp (Ω)) ] ≤ C8 τ 1−1/p [H1 − H2 Lp (0,τ ) (U X(T,p) + ρ) + (HLp(0,T ) + ρ)U1 − U2 X(τ,p) ].

Analogously we get B1 ∗ Φj (∆U1 ) − B2 ∗ Φj (∆U2 )Lp (0,τ ) ≤ C9 τ 1−1/p [B1 − B2 Lp (0,τ ) (U X(T,p) + ρ) + (BLp (0,T ) + ρ)U1 − U2 X(τ,p) ]. By Lemma 3.5, we have Φj {k0 ∆(U1 − U2 ) + −1 λ∆2 (V1 − V2 ) − −1 λ2 ∆(U1 − U2 )}Lp(0,τ ) ≤ ω5 (τ )(U1 − U2 X(τ,p) + V1 − V2 )Y (τ,p) ). Finally, from Lemma 3.4, we have Φj {∆[ϕ (χ0 + 1 ∗ V1 )V1 − ϕ (χ0 + 1 ∗ V2 )V2 ]}Lp(0,τ ) ≤ C4 ϕ (χ0 + 1 ∗ V1 )V1 − ϕ (χ0 + 1 ∗ V2 )V2 Lp (0,τ ;W 2,p (Ω)) ≤ C1 (T, ρ)τ (1−1/p)∧1/(2p) V1 − V2 Y (τ,ρ) . Summarizing, we have obtained an estimate of the form h1 − h2 Lp (0,τ ) + b1 − b2 Lp (0,τ ) ≤ C5 (T, ρ)ω6 (τ )[U1 − U2 X(τ,ρ)

(4.32)

+ V1 − V2 Y (τ,ρ) + H1 − H2 Lp (0,τ ) + B1 − B2 Lp (0,τ ) ]. So, from Theorem 3.2, u1 − u2 X(τ,p) + v1 − v2 Y (τ,p) ≤ C5 {h1 − h2 Lp (0,τ ) + b1 − b2 Lp(0,τ ) + h1 ∗ U1 − h2 ∗ U2 Lp (0,τ ;Lp(Ω)) + b1 ∗ ∆U1 − b2 ∗ ∆U2 Lp(0,τ ;Lp (Ω))

(4.33)

+ ϕ (χ0 + 1 ∗ V1 )V1 − ϕ (χ0 + 1 ∗ V2 )V2 Lp (0,τ ;W 2,p (Ω)) } ≤ C6 (T, ρ)ω4 (τ )[U1 − U2 X(τ,ρ) + V1 − V2 Y (τ,ρ) + H1 − H2 Lp (0,τ ) + B1 − B2 Lp (0,τ ) ]. In conclusion, from (4.32) and (4.33), we have u1 − u2 X(τ,p) + v1 − v1 Y (τ,p) + h1 − h2 Lp (0,τ ) + b1 − b2 Lp (0,τ ) ≤ C7 (T, ρ)ω7 (τ )[U1 − U2 X(τ,p) + V1 − V2 Y (τ,p) + h1 − h2 Lp (0,τ ) + b1 − b2 Lp (0,τ ) ].

(4.34)

An Inverse Problem for a Phase-ﬁeld Model in Sobolev Spaces

207

If τ ∈ (0, T ] is such that (4.30) holds together with C7 (T, ρ)ω7 (τ ) < 1, then S maps Z(τ, ρ) into itself and is a contraction in it. It follows that the system (4.25) has a unique solution (u, v, h, b) in Z(τ, ρ). Now we show that, if (θ1 , χ1 , a1 , b1 , k1 ) and (θ2 , χ2 , a2 , b2 , k2 ) are solutions of (1.13) both belonging to X(τ, p) × Y (τ, p) × W 1,p (0, τ ) × Lp (0, τ ) × R, for some τ ∈ (0, T ], then they coincide. We have already seen that, necessarily, k1 = k2 = k0 . Setting, for i ∈ {1, 2}, ui := Dt θi ,

(4.35)

vi := Dt χi ,

(4.36)

hi := Dt ai ,

(4.37)

then (u1 , v1 , h1 , b1 ) and (u2 , v2 , h2 , b2 ) both satisfy the system (2.14)–(2.15). As, for a ﬁxed ρ ∈ R+ , the system (2.14)–(2.15) has a unique solution in Z(σ, ρ) if σ is suﬃciently small, necessarily (u1 , v1 , h1 , b1 ) and (u2 , v2 , h2 , b2 ) coincide almost everywhere in some right neighborhood of 0. So we set τ1 := inf{t ∈ [0, τ ] :u1 − u2 X(t,p) + v1 − v2 Y (t,p) +h1 − h2 Lp (0,t) + b1 − b2 Lp (0,t) > 0}.

(4.38)

Of course, we assume that the set in the second term of (4.38) is not empty, so that τ1 ∈ (0, τ ). We observe that u1 −u2 X(τ1 ,p) +v1 −v2 Y (τ1 ,p) +h1 −h2 Lp (0,τ1 ) +b1 −b2 Lp (0,τ1 ) = 0, (4.39) which implies also u1 (τ1 ) = u2 (τ1 ),

v1 (τ1 ) = v2 (τ1 ).

(4.40)

For t ∈ [0, τ − τ1 ], we set U (t) := u1 (τ1 + t) − u2 (τ1 + t),

(4.41)

V (t) := v1 (τ1 + t) − v2 (τ1 + t),

(4.42)

H(t) := h1 (τ1 + t) − h2 (τ1 + t),

(4.43)

B(t) := b1 (τ1 + t) − b2 (τ1 + t).

(4.44)

If t ∈ [0, τ − τ1 ] we have (h1 ∗ u1 )(τ1 + t) − (h2 ∗ u2 )(τ1 + t) = [(h1 − h2 ) ∗ u1 ](τ1 + t) + [h2 ∗ (u1 − u2 )](τ1 + t) t [h1 (τ1 + t − s) − h2 (τ1 + t − s)]u1 (s)ds = 0 t h2 (t − s)[u1 (τ1 + s) − u2 (τ1 + s)]ds + 0

= (H ∗ u1 )(t) + (h2 ∗ U )(t).

(4.45)

208

F. Colombo and D. Guidetti

Moreover, for i ∈ {1, 2}, t ∈ [0, τ − τ1 ],

τ

t χ0 + (1 ∗ vi )(τ1 + t) = χ0 + 0 1 vi (s)ds + 0 vi (s + τ1 )ds = χ1 + (1 ∗ vi (. + τ1 )),

with

τ1

χ1 := χ0 +

v1 (s)ds = χ0 + 0

(4.46)

τ1

v2 (s)ds.

(4.47)

0

4,p (Ω). Consequently, we have We observe that χ1 ∈ WBB ⎧ ⎪ ⎪ Dt U (t, x) + λDt V (t, x) = k0 ∆U (t, x) − a0 U (t, x) ⎪ ⎪ −H(t)θ0 (x) + B(t)∆θ0 (x) − (H ∗ u1 )(t, x) ⎪ ⎪ ⎪ ⎪ −(h ⎪ 2 ∗ U )(t, x) + (B ∗ ∆u1 )(t, x) + (b2 ∗ ∆U )(t, x), ⎪ ⎪ ⎪ (t, x) ∈ [0, τ − τ1 ] × Ω, ⎪ ⎪ ⎪ ⎪ 2 ⎪ V (t, x) = −∆ V (t, x) + ∆[ϕ (χ + 1 ∗ v1 (. + τ1 )) εD ⎪ t 1 ⎨ (t, x) ∈ [0, τ − τ1 ] × Ω, ⎪ ⎪ v1 (. + τ1 ) − ϕ (χ1 + 1 ∗ v2 (. + τ1 ))v2 (. + τ1 )](t, x) − λ∆U (t, x), ⎪ ⎪ ⎪ ⎪ Dν U (t, x ) = Dν V (t, x ) = Dν ∆V (t, x ) = 0, ⎪ ⎪ ⎪ ⎪ (t, x ) ∈ [0, τ − τ1 ] × ∂Ω, ⎪ ⎪ ⎪ ⎪ ⎪ U (0, x) = 0, x ∈ Ω, ⎪ ⎪ ⎪ ⎩ V (0, x) = 0, x ∈ Ω

(4.48)

H(t)Φj (θ0 ) − B(t)Φj (∆θ0 ) = −(H ∗ Φj (u1 ))(t) − (h2 ∗ Φj (U ))(t) + k0 Φj [∆U (t)] + (B ∗ Φj (∆u1 ))(t) + (b2 ∗ Φj (∆U ))(t) + −1 λΦj {∆2 V (t) − ∆[ϕ (χ1 + 1 ∗ v1 (. + τ1 ))(t)v1 (t + τ1 ) − ϕ (χ1 + 1 ∗ v2 (. + τ1 ))(t)v2 (t + τ1 ) + λU (t)]}, (4.49) with t ∈ [0, τ − τ1 ], j ∈ {1, 2}. If we set R := u1 X(τ,p) + v1 (. + τ1 )Y (τ −τ1 ,p) + v2 (. + τ1 )Y (τ −τ1 ,p) + h2 Lp (0,τ ) + b2 Lp (0,τ ) ,

(4.50)

then, from (4.49), employing Lemmata 3.1, 3.4, 3.5, we get, for σ ∈ (0, τ − τ1 ], HLp (0,σ) + BLp(0,σ) ≤ C5 (R)ω7 (σ)[HLp (0,σ) + BLp (0,σ) + U X(σ,p) + V Y (σ,p) ].

(4.51)

The estimate (4.51) implies that, for some σ1 ∈ (0, τ − τ1 ], if 0 < σ ≤ σ1 , HLp (0,σ) + BLp (0,σ) ≤ C6 (R)ω7 (σ)[U X(σ,p) + V Y (σ,p) ]

(4.52)

From (4.48), using Theorem 3.2 and again Lemmata 3.1 and 3.4, we obtain U X(σ,p) + V Y (σ,p) ≤ C7 (R)[HLp (0,σ) + BLp(0,σ) + ω8 (σ)(U X(σ,p) + V Y (σ,p) + HLp(0,σ) + BLp (0,σ) )]. (4.53)

An Inverse Problem for a Phase-ﬁeld Model in Sobolev Spaces

209

(4.52) and (4.53) together imply U X(σ,p) + V Y (σ,p)

≤

C8 (R)ω9 (σ)[U X(σ,p) + V Y (σ,p) ].

(4.54)

From (4.54), we have that U and V vanish in some right neighborhood of 0. From (4.52) it follows that even H and B vanish in some right neighborhood of 0. But this is in contradiction with the deﬁnition of τ1 . With this the proof is complete.

References [1] R. Adams, Sobolev spaces, Pure and Applied Mathematics, vol. 65, Plenum Press (1975). [2] G. Bonfanti, F. Luterotti, Global solution to a phase-ﬁeld model with memory and quadratic nonlinearity, Adv. Math. Sci. Appl., 9 (1999), 523–538. [3] G. Bonfanti, F. Luterotti, Regularity and convergence results for a phase-ﬁeld model with memory, Math. Meth. Appl. Sci., 21 (1998), 1085–1105. [4] M. Brokate, J. Sprekel, Hysteresis and Phase Transition, Springer, New York, 1996. [5] D. Brandon, W.J. Hrusa, Construction of a class of integral models for heat ﬂow in materials with memory, J. Integral Equations and Appl., 1 (1998), 175–201. [6] G. Caginalp, An analysis of a phase ﬁeld model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205–245. [7] G. Caginalp, Stefan and Hele-Shaw type models as asymptotic limits of the phase ﬁeld equations, Phys. Rev. A, 39 (1989), 5887–5896. [8] G. Caginalp, The dynamics of a conserved phase-ﬁeld system: Stefan-like, Hele-Shaw, and Cahn-Hilliard models as asymptotic limits, IMA J. Appl. Math., 44 (1990), 77– 94. [9] G. Caginalp, X. Chen, Convergence of the phase ﬁeld model to its sharp interface limits, European Journal of Applied Mathematics, 9 (1998), 417–445. [10] G. Caginalp, X. Chen, Phase ﬁeld equations in the singular limit of sharp interface problems, in: M. Gurtin and G.B. McFadden (eds.), On the Evolution of Phase Boundaries, IMA Volume of Mathematics and Its Applications, 43 (1992), 1-28, Springer-Verlag. [11] B.D. Coleman, M.E. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys., 18 (1967), 199–208. [12] P. Colli, G. Gilardi, M. Grasselli, Global smooth solution to the standard phase ﬁeld models with memory, Adv. Diﬀerential Equations, 2 (1997), 453–486. [13] P. Colli, G. Gilardi, M. Grasselli, Well-posedness of the weak formulation for the phase ﬁeld models with memory, Adv. Diﬀerential Equations, 2 (1997), 487–508. [14] F. Colombo, Direct and inverse problems for a phase-ﬁeld model with memory, J. Math. Anal. Appl., 260 (2001), 517–545. [15] F. Colombo, D. Guidetti, V. Vespri, An inverse parabolic problem for a phase-ﬁeld model for boundary measurements of the temperature, preprint. [16] C. Giorgi, M. Grasselli, V. Pata, Uniform attractors for a phase-ﬁeld model with memory and quadratic nonlinearity, Indiana Univ. Math. J., 48 (1999), 1395–1445.

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[17] M. Grasselli, V. Pata, F.M. Vegni, Longterm dynamics of a conserved phase-ﬁeld system with memory, Asymp. Analysis, 33 (2003), 261–320. [18] R. Denk, M. Hieber, J. Pr¨ uss, R-boundedness, Fourier multipliers and problems of elliptic and parabolic type, preprint: Technische Universit¨ at Darmstadt, n. 2156 (2001). [19] D. Guidetti, On interpolation with boundary conditions, Math. Zeit., 207 (1991), 439–460. [20] D. Guidetti, A. Lorenzi, A mixed identiﬁcation problem related to a phase-ﬁeld model with memory, preprint. [21] A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Diﬀerential Equations and Their Applications, Birkh¨auser (1995). [22] A. Novick-Cohen, The Cahn-Hilliard equation: mathematical and modeling perspectives, Adv. Math. Sci. Appl., 8 (1998), 965–985. [23] A. Novick-Cohen, Conserved phase–ﬁeld equations with memory, in Curvature Flows and related Topics, 1994, GAKUTO Internat. Ser. Math. Sci. Appl., Vol 5, Gakk¯ otosho, Tokyo, 1995, pp. 179–197. [24] A. Pazy, Semigroups of linear operators and applications to partial diﬀerential equations, Springer Verlag, New York, Berlin, Heidelberg, Tokyo, 1983. [25] E. Sinestrari, On the Cauchy problem in spaces of continuous functions, J. Math. Anal. Appl., 107 (1985), 16–66. [26] H.B. Stewart, Generation of analytic semigroups by strongly elliptic operators, Trans. Amer. Math. Soc., 199 (1974), 141–162. [27] H. Triebel, Theory of functions spaces, Monogra. Math., Birkh¨ auser (1983). [28] H. Triebel: Interpolation theory, function spaces, diﬀerential operators, North Holland, Amsterdam, New York, Oxford, 1978. Fabrizio Colombo Dipartimento di Matematica Politecnico di Milano Via Bonardi 9 I-20133 Milano, Italy Davide Guidetti Dipartimento di Matematica Universit` a di Bologna Piazza di Porta S. Donato 5 I-40126 Bologna, Italy

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 211–225 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Numerical Analysis of Microstructures: The Inﬂuence of Incompatibility O. Anza Hafsa and M. Chipot Abstract. We are concerned with the appearance of microstructures in some problems of Calculus of Variations experiencing “wells” or minimum of energy. Using piecewise linear ﬁnite elements, we give energy estimates and analyze their dependence on the incompatibility of the wells.

1. Introduction We denote by Mn×N the space of n × N matrices with real coeﬃcients. Let K be a compact subset of matrices in Mn×N and let ϕ be a nonnegative continuous function vanishing only on K, i.e., such that ϕ(M ) > 0 ∀M ∈ K , ϕ(M ) = 0 ∀M ∈ K. It is well known that the quasiconvex envelope of ϕ is given by 1 inf Qϕ(W ) = ϕ(W + ∇u(x))dx, |Ω| u∈W01,∞ (Ω;RN ) Ω

(1)

(2)

where |Ω| is the Lebesgue measure of Ω, ∇u denotes the gradient of u, i.e., the Jacobian matrix of the mapping x → u(x) W01,∞ (Ω; RN )

from Ω ⊂ Rn into RN , is the space of RN -valued functions such ∞ nN that ∇u ∈ (L (Ω)) and which vanish on the boundary Γ = ∂Ω of Ω. We refer the reader to [4] for these notions. The compatibility is described by the following deﬁnition. Deﬁnition 1.1. Two matrices A, B ∈ Mn×N are said to be rank-one compatible if and only if Rk(A − B) ≤ 1. Let us introduce the class of problems that we would like to address.

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1.1. The continuous problem We are interested in the following minimization problem inf ϕ(∇u(x))dx. 1,∞ (Ω;RN )

u∈W0

(3)

Ω

Then we have: Proposition 1.1. Suppose that ϕ is a function satisfying (1). Then, if 0 ∈ K c where K c denotes the laminate convex hull of K, the inﬁmum (3) is 0. Proof. It follows from (2) that the inﬁmum (3) is equal to 0 if and only if Qϕ(0) = 0.

(4)

From (1), (2), we have Qϕ(W ) = 0 ∀W ∈ K. The set K c is the set of matrices obtained by making convex combinations of rank one compatible matrices starting with matrices in K (see [5]). Since Qϕ is rank-one convex, (see [4]), for any matrices W1 , W2 rank-one compatible it holds that 0 ≤ Qϕ(αW1 + (1 − α)W2 ) ≤ αQϕ(W1 ) + (1 − α)Qϕ(W2 ). If in particular Qϕ(W1 ) = Qϕ(W2 ) = 0, then it follows that Qϕ(αW1 + (1 − α)W2 ) = 0. The proposition follows then by iteration.

1.2. The discrete problem We will consider below an approximation to (3). Ω will be a polygonal domain of RN , {τh }h a family of regular triangulations. The mesh size h is deﬁned by h = max hK . K∈τh

The approximation problem is

ϕ(∇v(x))dx,

inf

v∈V0h

(5)

Ω

where V0h = {v ∈ V h : v = 0 on ∂Ω} and V h = {v = (v1 , . . . , vN )|vi : Ω → R is continuous, vi |K is aﬃne for i = 1, 2 and K ∈ τh }. We would like to obtain, in simple cases of the proposition 1.1 diﬀerent estimates of (5) in terms of the mesh size. We will restrict ourselves in this note to the case of 2 × 2 matrices and show somehow how the incompatibility between the wells slows down, for a given mesh size, the speed of convergence of the minimizing sequences of (3).

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2. An example with three levels of laminations A generic matrix M ∈ M2×2 is denoted by x11 x12 . M= x21 x22 In the x11 , x22 plane, we consider the following “wells” (element of K): 1/2 0 W1 = 0 2

1 0 W2 = 0 −1

−1 0 W3 = 0 −1

−1 0 W4 = . 0 1

From the above deﬁnition, it is easy to see that W1 is not rank-one compatible with any other Wi , but (Wi , Wi+1 ) are rank-one compatible for i = 2, 3. Let ϕ : M2×2 → [0; +∞[ be a nonnegative continuous function which satisﬁes (1) with K = {W1 , W2 , W3 , W4 }.

(6)

Let us consider two intermediate matrices W1 =

1/2 0 0 1

W2 =

1/2 0 , 0 −1

which are represented below. W1 x22 W1’

W4

x11

0

W3

W2’

W2

Figure 1 It is easy to see that 0 ∈ K c and thus in this case the inﬁmum (3) is 0. In addition we have: Proposition 2.1. Under the above assumptions the inﬁmum of (3) is not achieved.

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Proof. If u is a minimizer it holds that ∇u( · ) ∈ K a.e. in Ω. If u = (u1 , u2 ) this implies that ∂x2 u1 = ∂x1 u2 = 0 and since u vanishes at the boundary u = 0. This is impossible since 0 ∈ K and the proof is complete. In view of Proposition 2.1 we have to direct our attention on the minimizing sequences of (3). Regarding the discrete problem we have: Theorem 2.1. Under the above assumptions, there exists a constant C > 0 which does not depend on h & 1 such that 1 ϕ(∇v(x))dx ≤ Ch 4 . inf v∈V0h

Ω

Proof. Let α ∈]0; 1[. Let vh : [0; 2hα ]2 → R2 be a vector-valued function deﬁned by vh = v = (v1 , v2 ) where x1 /2 if x1 ∈ [0; 43 hα ], (7) v1 (x1 , x2 ) = α 2h − x1 if x1 ∈ [ 43 hα ; 2hα ], x2 if x2 ∈ [0; hα ], v2 (x1 , x2 ) = (8) α 2h − x2 if x2 ∈ [hα ; 2hα ]. ' ( ∇v(x) ∈ W1 , W2 , W3 , W4 a.e. in Ω,

We have

the repartition of the gradients being depicted on the Figure 2. x2 2h α

W2’

W3

W1’

W4

hα

4 α 3h

0

2h α

x1

Figure 2 We remark that W2 =

3 1 W2 + W3 4 4

and

W1 =

2 1 W1 + W2 . 3 3

(9)

Numerical Analysis of Microstructures Our goal is to ﬁnd a function w deﬁned ⎧ ⎪ ⎨ (w1 (x), w2 (x)) w(x) = (u1 (x), v2 (x)) ⎪ ⎩ v(x)

215

by for x ∈ [0; 43 hα ] × [0; hα ], for x ∈ [0; 43 hα ] × [hα ; 2hα ], otherwise,

satisfying

' ( ∇w(x) ∈ W1 , W2 , W3 , W4 except on a small set of [0; 2hα ]2 .

Set

4 4 Q1 = [0; hα ] × [0; hα ] and Q2 = [0; hα ] × [hα ; 2hα ]. 3 3 Let k, l ∈ N∗ to be chosen later on. We proceed in three steps. Step 1. We modify the function v on Q2 . From (9) we have that it holds 3 1 W2 = W2 + W3 . 4 4 Then a lamination with W2 , W3 could produce W2 . Let χ be the periodic function of period 1 and deﬁned on ]0; 1[ by 1 if t ∈]0; β], χβ (t) = 0 if t ∈]β; 1[, with β ∈]0; 1[. Let ω1 : Q2 → R be deﬁned by x1 3kt 2χ 34 ( α ) − 1 dt. ω1 (x) = ω1 (x1 ) = 4h 0 It is easy to check that ∇ω1 (x) = (−1, 0) or (1, 0). Let ∂2 Q2 ⊂ ∂Q2 be deﬁned by ∂2 Q2 = {(x1 , x2 ) ∈ Q2 : x2 = hα or x2 = 2hα }. One sets u1 (x) = ω1 (x) ∧ (v1 (x) + dist(x, ∂2 Q2 ))

for x ∈ Q2 and

w(x) = (u1 (x), v2 (x)) for x ∈ Q2 , then ∇w ∈ {W2 , W3 } a.e. in Q2 except on a small neighborhood of ∂2 Q2 , when 1 hα . dist(x, ∂2 Q2 ) < 2 k Step 2. We modify the function v on Q1 . From (9) we have that 2 1 W1 = W1 + W2 . 3 3 Similarly to the ﬁrst step, it is easy to ﬁnd a function w2 : Q1 → R such that the function w = (v1 , w2 ) satisﬁes ∇w(x) ∈ {W1 , W2 } a.e. on Q1

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except a small neighborhood of 4 α 2 hα ∂1 Q1 = (x1 , x2 ) ∈ Q1 : x1 = 0 or x1 = h , i.e., when dist(x, ∂1 Q1 ) < . 3 3 k We can deﬁne w2 : Q1 → R as w2 (x) = ω2 (x) ∧ (v2 (x) + dist(x, ∂1 Q1 )), where

kt ) − 1 dt. α h 0 Step 3. Now, it remains to modify w on the strips

) α α * ' α ( k−1 3 2h h h 4 α ; + p Σp where Σp = 0; h × Q1 = . 3 3 k k k p=0 ω2 (x) = ω2 (x2 ) =

x2

3χ 32 (

We choose the function ω1 of the ﬁrst step but with k replaced by l, i.e., x1 3lt 2χ 34 ( α ) − 1 dt. ω1 (x) = ω1 (x1 ) = 4h 0 We set on each strips Σp up1 = ω1 ∧ (v1 + dist(., ∂2 Σp )). with

' hα hα hα ( 2 hα ∂2 Σp = (x1 , x2 ) ∈ Q1 : x2 = +p or +p . 3 k k k k Then we set p (u1 (x), w2 (x)) if x ∈ Σp , w(x) = (v1 (x), w2 (x)) if x ∈ Q1 \ Q1 . We have ∇w(x) ∈ {W3 , W2 } a.e. on Q1 except on a small neighborhood of ∪{∂2 Σp : 0 ≤ p ≤ k − 1} when 1 hα dist(x, ∂2 Σp ) < 2 l for each p = 0, . . . , l − 1. We suppose that the following inequality holds 1 hα 1 hα < , 2 l 6 k i.e., we choose l > 3k. Thus, we have ∇w(x) ∈ {W3 , W2 , W1 } a.e. on Q1 except a small neighborhood of ∂2 Q1 and ∪{∂2 Σp : 0 ≤ p ≤ k − 1}. Now, suppose that w is extended by periodicity of period 2hα in both directions x1 , x2 (note that w is periodic on [0, 2hα ]2 ). In order to match the 0 boundary conditions one sets w ˆ = (w1 ∧ dist(., Γ), w2 ∧ dist(., Γ))

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217

where Γ is the boundary of Ω and ﬁnally one deﬁnes w ˜=w ˜h the interpolant of w ˆ on τh . Since ∇w˜ is bounded – due to the regularity of the mesh – independently of h, we have ϕ(∇w)dx ˜ ≤ C|{x ∈ Ω : ∇w(x) ˜ ∈ / K}| (10) Ω

where | | is the Lebesgue measure of sets. Let us evaluate in terms of h the righthand side of (10). The gradient of w ˜h does not possibly belong to K on diﬀerent subsets. • On a tubular neighborhood S1 of Γ with thickness of the order of hα (see [1] p. 336-337). • On a neighborhood S2 of thickness 2h of the lines 4 + 2p hα and x2 = phα with p ∈ Z. x1 = 2phα , 3 (This is due to interpolation.) • On a neighborhood S3 of thickness 1 hα 2 hα 1 hα of ∂2 Q2 , of ∂1 Q1 , and of ∂2 Σp for p = 0, . . . , k − 1. 2 k 3 k 2 l • On a neighborhood S4 of thickness α2h of the lines 4h 4p hα , (1 + p) with p ∈ {0, . . . , k − 1} in Q2 ; • x1 = 1 + 3 k 3 k α α h 2 + 3p h • x2 = ,p with p ∈ {0, . . . , k − 1} in Q1 \ Q1 ; k k 3 4 hα 4p hα , (1 + p) with p ∈ {0, . . . , l − 1} in Q1 . • x1 = 1 + 3 l 3 l We have the following estimates: i) |S1 | ≤ Chα , |S2 | ≤ Ch1−α ; 1 k C 2 hα α 1 hα 4 α 1 hα 4 α h + h +k h ≤C + ; ii) |S3 | ≤ 2α h 3 k 2 k 3 2 l 3 k l C 4 1 hα iii) |S4 | ≤ 2α k2hhα + k2h hα + k(l2h ) ≤ C(k + l)h1−α . h 3 3 k (C/h2α is the number of squares of size 2hα × 2hα covering Ω.) We deduce that

' 1 k( . ϕ(∇w)dx ˜ ≤ C hα + h1−α + (k + l)h1−α + + k l Ω 1

1−α

The function l → kl + lh1−α reaches its minimum for l = k 2 h− 2 . We obtain ' 1 1( . ϕ(∇w)dx ˜ ≤ C hα + h1−α + (kh1−α ) 2 + kh1−α + k Ω

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O. Anza Hafsa and M. Chipot 1

Since h & 1 then (kh1−α ) 2 > kh1−α (see below) we get that ' 1 1( . ϕ(∇w)dx ˜ ≤ C hα + h1−α + (kh1−α ) 2 + k Ω The function k →

1 k

1

1−α

2

1−α

+ k 2 h 2 is minimum for k = 2− 3 h− 3 . It follows: ( ' 1−α ϕ(∇w)dx ˜ ≤ C hα + h1−α + h 3 . Ω

It follows that the best possible α is α=

1 , 4

which completes the proof.

Remark 2.1. The minimizing sequence that we constructed uses each of the wells with a ﬁxed probability. It generates – as every bounded minimizing sequence in this case – a unique Young measure given by 2 1 5 1 δW1 + δW2 + δW3 + δW4 , 9 3 18 6 denotes the Dirac mass at Wi . νx =

where δWi

(11)

3. An example with four levels of laminations In this section we consider the following four wells (see the ﬁgure below): 1/2 0 1 0 −1 0 −2 0 W1 = W2 = W3 = W4 = . 0 2 0 −1 0 −1 0 1/2 We have: Proposition 3.1. The inﬁmum (3) is equal to 0, moreover it is not achieved (i.e., the minimisation problem (3) does not have solutions). Proof. The proof is similar to the proof of Propositions 1.1 and 2.1. To see that 0 ∈ K c one has to consider the intermediate matrices 1/2 0 1/2 0 −1 0 W2 = W4 = , W1 = 0 1/2 0 −1 0 1/2 which are represented in Figure 3.

Numerical Analysis of Microstructures

219

W1 x22

W1’

W4’

W4

x11

0

W3

W2’

W2

Figure 3 Regarding the discrete problem we have: Theorem 3.1. Under the above assumptions, there exists a constant C > 0 which does not depend on h & 1 such that 1 inf ϕ(∇v(x))dx ≤ Ch 5 . v∈V0h

Ω

Proof. Let vh : [0; 2hα ]2 → R2 be a vector-valued function deﬁned by (7) and v2 (x1 , x2 ) = v1 (x2 , x1 ). We have that

( ' ∇v(x) ∈ W1 , W2 , W3 , W4 a.e. in Ω.

Let us denote 4 4 4 4 4 4 Q1 = [0; hα ]×[0; hα ], Q2 = [0; hα ]×[ hα ; 2hα ] and Q4 = [ hα ; 2hα ]×[0; hα ] 3 3 3 3 3 3 We proceed in several steps. The following equalities give us the successive steps of modiﬁcations of the aﬃne function v: (i) W2 = 34 W2 + 14 W3 ; (ii) W1 = 12 W1 + 12 W2 ; (iii) W4 = 25 W1 + 35 W4 . Then one proceeds as follows: Step 1: Second lamination On Q2 a lamination with W2 , W3 replaces W2 ; On Q1 a lamination with W1 , W2 replaces W1 ; On Q4 a lamination with W1 , W4 replaces W4 ;

220

O. Anza Hafsa and M. Chipot

Step 2: Third lamination On Q1 a lamination with W2 , W3 replaces W2 ; On Q4 a lamination with W1 , W2 replaces W1 ; Step 3: Fourth lamination On Q4 a lamination with W2 , W3 replaces W2 ; For the reader convenience the scheme is illustrated by the pictures below. x2

x2

2h α

2h α W2’

W3

4 α 3h

( W3 ,W2 )

W3

(W1, W2’ )

W1’ W4

4 α 3h

W4’

W1’

2h α

4 α 3h

0

x1

x2

x2

2h α

2h α ( W3 ,W2 )

4 α 3h

0

( W3 ,W2 )

W3

4 α 3h

2h α

x1

2h α

x1

W3

4 α 3h

W1 W1

(W1, W2 ,W3 )

0

(W1, W2 ,W3 )

W2’ W4

4 α 3h

W2 W3 W4

2h α

x1

0

Figure 4 Let k, l, m ∈ N∗ to be chosen later on. Step 1: Second lamination. 1) On Q2 a lamination with W2 , W3 replaces W2 . Set x1 3kt 2χ 34 ( α ) − 1 dt. ω1 (x) = ω1 (x1 ) = 4h 0

4 α 3h

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221

Let ∂2 Q2 ⊂ ∂Q2 be deﬁned by ' ( 4 ∂2 Q2 = (x1 , x2 ) ∈ Q2 : x2 = hα or x2 = 2hα . 3 One sets for every x ∈ Q2 u1 (x) = ω1 (x) ∧ (v1 (x) + dist(x, ∂2 Q2 )) and, w(x) = (u1 (x), v2 (x)). Then ∇w ∈ {W2 , W3 } a.e. in Q2 except on a small neighborhood of ∂2 Q2 , when 1 hα . dist(x, ∂2 Q2 ) < 2 k 2) On Q1 a lamination with W1 , W2 replaces W1 . Set x2 3kt 3χ 12 ( α ) − 1 dt. ω2 (x) = ω2 (x2 ) = 4h 0 Let ∂1 Q1 ⊂ ∂Q1 be deﬁned by ' 4 ( ∂1 Q1 = (x1 , x2 ) ∈ Q1 : x1 = 0 or x1 = hα . 3 One sets for every x ∈ Q1 u2 (x) = ω2 (x) ∧ (v2 (x) + dist(x, ∂1 Q1 )) and, w(x) = (v1 (x), u2 (x)). Then w = (v1 , u2 ) satisﬁes ∇w(x) ∈ {W1 , W2 } a.e. on Q1 except on a small neighborhood of ∂1 Q1 , when 3 hα dist(x, ∂1 Q1 ) < . 4 k 3) On Q4 a lamination with W1 , W4 replaces W4 . Set 32 hxα1 −2 5 2 α χ 25 (kt) − 2dt . ω1,4 (x) = ω1,4 (x1 ) = h 1 + 3 2 0 Let ∂2 Q4 ⊂ ∂Q4 be deﬁned by ' 4 ( ∂2 Q4 = (x1 , x2 ) ∈ Q4 : x2 = 0 or x2 = hα . 3 One sets for every x ∈ Q4 u1 (x) = ω1,4 (x) ∧ (v1 (x) + dist(x, ∂2 Q4 )) and, w(x) = (u1 (x), v2 (x)). Then w = (u1 , v2 ) satisﬁes ∇w ∈ {W4 , W1 } a.e. in Q4 except on a small neighborhood of ∂2 Q4 , when 3 hα . dist(x, ∂2 Q4 ) < 4 k

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O. Anza Hafsa and M. Chipot

Step 2: Third lamination. 1) On Q1 a lamination with W2 , W3 replaces W2 ; We change the function w on the strips Q1 deﬁned by Q1 =

' 4 hα 4 hα 4 hα ( 4 ; ] + {p } . Σp where Σp = [0; hα ] × [ 3 6 k 3 k 3 k p=0

k−1 3

Set

x1

ω1,1 (x) = 0

2χ 43 (

3lt ) − 1 dt. α 4h

For each p ∈ {0, . . . , k − 1} one sets ∂2 Σp ⊂ ∂Q1 given by ' 4 hα 4 hα ( 4 hα 4 hα ∂2 Σp = (x1 , x2 ) ∈ Q1 : x2 = +p or x2 = +p . 6 k 3 k 3 k 3 k We deﬁne for every p = 0, . . . , k − 1 up1 (x) = ω1,1 (x) ∧ (v1 (x) + dist(x, ∂2 Σp ) and we change w like w(x) =

(up1 (x), w2 (x)) (v1 (x), w2 (x))

for x ∈ Σp ,

if x ∈ Σp if x ∈ Q1 \ Q1 .

Then ∇w ∈ {W2 , W3 } a.e. in Q2 except on a small neighborhood of ∂2 Σp dist(x, ∂2 Σp ) <

1 hα 2 l

for each p = 0, . . . , l − 1. 2) On Q4 a lamination with W1 , W2 replaces W1 ; We change the function w on the strips Q4 deﬁned by Q4 =

'4 2 hα ( 4 4 4 hα ] + {p } × [ hα ; 2hα ]. ∆p where ∆p = [ hα ; hα + 3 3 15 k 3 k 3 p=0

k−1 3

We do l laminations on each strips ∆p with a similar function as Step 1. 2) to produce a function ω2,4 which satisﬁes ∇ω2,4 = (0, −1) or (0, 2) on ∆p . For each p ∈ {0, . . . , k − 1} one sets ∂1 ∆p ⊂ ∂Q1 given by ' 2 hα ( 4 2 hα 4 4 hα or x1 = hα + +p . ∂1 ∆p = (x1 , x2 ) ∈ Q4 : x1 = hα + p 3 3 k 3 15 k 3 k We deﬁne for every p = 0, . . . , k − 1 up2 (x) = ω2,4 (x) ∧ (v2 (x) + dist(x, ∂1 ∆p )

for x ∈ ∆p ,

Numerical Analysis of Microstructures and we change w like

w(x) =

(u1 (x), up2 (x)) (u1 (x), v2 (x))

223

if x ∈ ∆p , if x ∈ Q4 \ Q4 .

Then ∇w ∈ {W1 , W2 , W4 } a.e. in Q4 except on a small neighborhood of ∂1 ∆p dist(x, ∂1 ∆p ) <

3 hα 4 l

for each p = 0, . . . , k − 1. Step 3: Fourth lamination On Q4 a lamination with W2 , W3 replaces W2 ; We change the function w on the strips of the form ) ) α 4 α 4 α 2 h 4 hα * ' 4 hα ( 4 hα * ' 2 hα ( Ip,r = h ; h + + p ; + r × ⊂ ∆p , 3 3 15 k 3 k 3 l 3 l 3 k where p = 0, . . . , k − 1 and r = 0, . . . , l − 1. We do m laminations on each rectangle p,r which satisﬁes Ip,r with a similar function as in Step 1. 1) to obtain a function ω1,4 p,r = (−1, 0) or (1, 0) on Ip,r ∇ω1,4

For each p ∈ {0, . . . , k − 1} and r ∈ {0, . . . , l − 1} one sets ∂2 Ip,r ⊂ ∂∆p given by ' 4 hα 4 hα ( 2 hα 4 hα +r or x2 = +r . ∂2 Ip,r = (x1 , x2 ) ∈ ∂Ip,r : x2 = 3 l 3 k 3 l 3 k We deﬁne for every p = 0, . . . , k − 1 and r = 0, . . . , l − 1 p,r up,r 1 (x) = ω1,4 (x) ∧ (u1 (x) + dist(x, ∂2 Ip,r )

and we change w like

w(x) =

p (up,r 1 (x), u2 (x)) (u1 (x), up2 (x))

for x ∈ Ip,r ,

if x ∈ Ip,r , otherwise in ∆p .

Then ∇w ∈ {W1 , W2 , W3 } a.e. in ∆p except on a small neighborhood of ∂2 Ip,r for which it holds 1 hα dist(x, ∂2 Ip,r ) < 2 mk for each p = 0, . . . , k − 1 and r = 0, . . . , l − 1 . We extend then w by periodicity of period 2hα in both directions x1 , x2 . In order to match the 0 boundary conditions we set w ˆ = (w1 ∧ dist(., Γ), w2 ∧ dist(., Γ)) and ﬁnally one deﬁnes w ˜=w ˜h the interpolant of w ˆ on τh .

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Since ∇w ˜ is bounded independently of h, then ϕ(∇w)dx ˜ ≤ C|{x ∈ Ω : ∇w(x) ˜ ∈ / K}|.

(12)

Ω

Let us evaluate in terms of h the right-hand side of (12). The gradient of w ˜h does possibly not belong to K on diﬀerent subsets (see the previous section). As before we have: |S1 | ≤ Chα , |S2 | ≤ Ch1−α . Let us now examine more closely our lamination process. Suppose that we are laminating a cell of size a × b, say [x0 ; x0 + a] × [y0 ; y0 + b], in the x1 direction. If we split this cell in k the gradient of the laminate does not belong to the good set: • on a domain near the lateral border of size a. ak , • on strips of width h, i.e., on a domain of size khb. The size of the total domain is of the order of magnitude: a a. + khb × # of cells k (i.e., # of cells that we are laminating). In the process above we have laminated cells of size hα × hα (in fact c1 hα × c2 hα but these constants are playing no rˆ ole here.) Thus the set of bad gradients which will always remain after the ﬁrst lamination is of size 1 C α hα h . + khhα = C + kh1−α . 2α h k k α

We then laminate in l, k cells of size hα × hk . This generates an unremovable set of bad gradients of size: k hα C α hα 1−α + lh = C + lh k h . . h2α l k l α

α

Finally we laminate lk cells of size hk × hl in m and this produces a set of bad gradients of size: l C hα hα hα 1−α . . + mh = C + mkh lk h2α k km l km We deduce that ( ' l 1 k + (k + l + mk)h1−α . ϕ(∇w)dx ˜ ≤ C hα + h1−α + + + k l mk Ω 1

l The function m → mkh1−α + mk reaches its minimum for m = l 2 k −1 h− follows that ( ' 1 1 k ϕ(∇w)dx ˜ ≤ C hα + + + kh1−α + lh1−α + (lh1−α ) 2 . k l Ω 1

Since h & 1 if l is chosen such that lh1−α < (lh1−α ) 2 (see below) we get ( ' 1 1 k ϕ(∇w)dx ˜ ≤ C hα + + + kh1−α + (lh1−α ) 2 . k l Ω

1−α 2

. It

Numerical Analysis of Microstructures 1

2

The function l → (lh1−α ) 2 + kl is minimum for l = (2k) 3 h− ( ' 1−α 1 1 ϕ(∇w)dx ˜ ≤ C hα + + k 3 h 3 . k Ω 1

The function k → k 3 h

1−α 3

3

is minimum for k = 3 4 h− ( ' 1−α ϕ(∇w)dx ˜ ≤ C hα + h 4 . +

1 k

1−α 4

225 1−α 3

. It follows

and we obtain

Ω

The best constant α is α =

1−α 4

i.e., α = 15 . This completes the proof.

Remark 3.1. It would be interesting to check the sharpness of these estimates. For results in this direction see [3], [6]. Remark 3.2. Our estimates give direct proofs of the Propositions 2.1 and 3.1. Acknowledgements This paper was written while the ﬁrst author was visiting the Institute of Mathematics of the University of Z¨ urich. The ﬁnancial support provided by the Institute, as well as the support of the programme HMS2000, is gratefully acknowledged. The second author is supported by the Swiss National Science Foundation under the contracts # 20-67618.02 and 20-103300/1.

References [1] M. Chipot, The appearance of microstructures in problems with incompatible wells and their numerical approach, Numerische Math. 83 (1999): 325–352. [2] M. Chipot, Elements of nonlinear analysis, Birkh¨ auser Advanced Texts 2000. ¨ [3] M. Chipot, S. Muller, Sharp Energy Estimates For Finite Element Approximations of Non-convex Problems. Proceedings of a IUTAM Symposium on Variations of Domains and Free-Boundary Problems in Solid Mechanics, P. Argoul, M. Fr´emond, Q.S. Nguyen Eds., Kluwer, (1999), p. 317–325. [4] B. Dacorogna, Direct methods in the calculus of variations, Springer, 1989. [5] G. Dolzmann, Variational Methods for Crystalline Microstructure-Analysis and Computation, Springer Lecture Notes in Mathematics # 1803, Springer Berlin, 2003. [6] A. Lorent, An optimal scaling law for ﬁnite element approximations of a variational problem with non-trivial microstructure, M2AN. 5 (2001): 921–934. O. Anza Hafsa and M. Chipot University of Zurich Angewandte Mathematik Winterthurerstr. 190 CH-8057 Zurich

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 227–242 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Nonlinear Singular Elliptic Problems: Recent Results and Open Problems Jes´ us Hern´andez, Francisco J. Mancebo and Jos´e M. Vega To Herbert Amann, with admiration, acknowledgement and friendship

1. Introduction Reaction-diﬀusion equations and systems have been studied very extensively in the last years as models for many problems arising in applications as, e.g., population dynamics (Lotka-Volterra systems) combustion, chemically reacting systems, etc. A typical example is ∂u − ∆u = f (x, u) in Ω×]0, +∞[ ∂t

(1.1)

where Ω is a bounded domain in Rn , ∆ is the Laplacian, which models diﬀusion and the nonlinearity f represents the reaction term. This evolution equation must be supplemented with appropriate boundary and initial conditions. Of course, this equation admits much more general versions, but it still retains most of the interesting features arising along this theory. The reader may consult the references in [32]. Stationary solutions of this parabolic equation are very important, since they correspond to equilibria solutions, and its existence and multiplicity have been considered in detail as well. Very often only positive solutions are the relevant ones, a fact which is very closely related with the meaning of the unknowns (concentrations, populations, etc.). Usually the nonlinearity is a smooth (actually C 1 or at least locally Lipschitz) function such that f (0) ≥ 0 but functions like f (u) = uα with 0 < α < 1 are considered in the literature, since they arise in fractional order chemical reactions or nonlinear diﬀusion models, see the book [15] and the references therein. Also nonlinearities deﬁned for u ∈]0, +∞[ but going to inﬁnity when u goes to zero are of interest in situations as chemical heterogeneous catalysts or non-Newtonian

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ﬂuids, besides its intrinsic mathematical interest. A model example of this kind is λ uα u=0

−∆u =

in Ω

(1.2)

on ∂Ω,

where α > 0 and λ > 0 is a real parameter. Problems of this type, for monotone decreasing nonlinearities, were treated by Fulks and Maybee in the pioneering paper [18], a fact which is really remarkable if one takes into account that the considerable development in the study of reaction-diﬀusion equations and systems only started in the late sixties. They proved existence and uniqueness for the parabolic problem by using a ﬁxed point argument for the associated integral equation and also existence and uniqueness for the stationary problem; moreover solutions of the parabolic problem tend to the unique solution of the elliptic one. As usual, the one-dimensional case was studied by using ordinary diﬀerential equations methods that cannot be extended to the case of a general bounded domain (see, e.g., [19]). It is clear that, due to the singular character of the problem, only positive solutions are to be considered. The general case was studied later by diﬀerent authors by replacing the original problem by an approximate one that is regular, where the usual techniques (sub- and super-solutions, bifurcation) can be applied and then go to the limit to obtain the solution to the singular problem. This was done ﬁrst by Stuart in [33] by replacing the homogeneous Dirichlet boundary condition by (positive) nonhomogeneous ones. Then Crandall, Rabinowitz and Tartar in [12] proved several interesting results. First, they obtain very general existence results (valid for any α > 0 in the case of the model example (1.2) and for diﬀerential operators not necessarily in divergence form) by an approximation argument using either sub- and supersolutions in the monotone case or global bifurcation (and a more sophisticated convergence argument for continua) in the general one. They only look for classical solutions in the space C 2 (Ω) ∩ C(Ω) and do not try to get optimal regularity results. However it is shown that if α > 1 the solutions are H¨older continuous 2 (more precisely in C 1+α (Ω)) and a necessary and suﬃcient condition for solutions to be Lipschitz up to the boundary (which amounts to 0 < α < 1 in the case model) is obtained in [12] Theorem 3.25. Simpler proofs of some of these results were obtained by Lazer and McKenna in [28], where they studied the behavior of solutions close to the boundary and also showed that classical solutions are in the Sobolev space H01 (Ω) if and only if 0 < α < 3. Several improvements concerning both existence and regularity were obtained by Gomes [21] and del Pino [14]. And some general regularity results were obtained by Gui and Hua-Lin [22]. We also mention existence results by using approximation and then sub- and super-solutions due to Coclite ( [9]– [11]), see also [30], [31]. Existence of weak solutions in the Sobolev space H01 (Ω) has been studied in [35]. Finally Ghergu and Radulescu [20] and Aranda and Lami-Dozo [3] use asymptotic bifurcation to prove existence, and the latter get some (partial) uniqueness as well.

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Other existence results may be found in [34]. Multiplicity results can be found in [27], [35] and [23]. In a recent interesting paper D´ avila and Montenegro [13] study both positive and dead core solutions for the singular problem in Application 4 below. Apparently the techniques used there can be applied elsewhere. Recently the authors of this paper have extended the method of sub- and super-solutions to these problems (see [24]). First, they proved a general existence theorem for maximal and minimal positive solutions in the interval between ordered sub- and super-solutions. We intended to get a theorem as similar as possible to the well-known theorems in the regular case (see, e.g., [1]) which could be applied directly avoiding any kind of approximation procedure. To obtain this generalization, we have needed some additional conditions are imposed to sub- and super-solutions (see Remark 2.1). We allow operators not necessarily in divergence form, where even some coeﬃcients may be singular. In order to implement this program it was necessary to have suitable existence, uniqueness and regularity results for linear problems with singular data (regularity is important in order to get compactness of the solution operator) and to extend the Strong Maximum Principle when the coeﬃcients are singular. These results are still useful for studying the existence of principal eigenvalues for singular linear eigenvalue problems (now in the sense of having unbounded coeﬃcients close to the boundary) extending to the case of general operators the results proved by Berstsch and Rostamian in [5] in the variational case by using variational arguments, weighted Sobolev spaces and the Hardy-Sobolev inequality. We also obtained as corollaries of the same general theorems the diﬀerentiability of the solution (or Green’s) operator in the interior ∂u < 0 on ∂Ω) and a linearized of the positive cone (i.e., functions u > 0 in Ω and ∂n stability theorem again as in the regular case. In a subsequent paper we apply the results in [24] to many examples, extending and unifying in this way most of the results in the literature. Most of the above-mentioned results will be reviewed in the survey paper [26]. The paper is organized as follows. After this introduction, we collect in Section 2 the main results in [24], which will be applied in Section 3. Full proofs and many related results and comments can be found in [24], [26] and its references. In Section 3 we consider a few selected applications exhibiting some interesting features (see [25], [26] for more details).

2. Main general theorems In this section we collect, usually without proof, the main general results which will be applied to the problems considered in Section 3. Most of them are taken from [24], where the reader will also ﬁnd complements and comments. We have also included here several results which are more or less easy corollaries of the main theorems in [24] we consider of independent interest in order to be used in Section 3 and elsewhere.

230

J. Hern´ andez, F.J. Mancebo and J.M. Vega We deal with the semilinear elliptic problem n ∂2u ∂u Lu ≡ − aij (x) + bi (x) = f (x, u) in Ω ∂x ∂x ∂x i j i i,j=1 u=0

on ∂Ω

(2.1) (2.2)

under the following assumptions, which hold for some ﬁxed α such that −1 < α < 1. (H.1) Ω ⊂ Rn is a bounded domain, with a C 3,γ boundary, for some γ > 0 if n > 1. Note that the distance from x ∈ Ω to ∂Ω, d(x), deﬁnes a function ¯ 1 ), with Ω1 = {x ∈ Ω : d(x) < ρ1 } for some ρ1 > 0. d ∈ C 2,γ (Ω (H.2) The second-order part of the operator −L is uniformly, strongly elliptic in ¯ bi ∈ C 2 (Ω), Ω. Also, for all i, j, k = 1, . . . , n, aij = aji ∈ C 3 (Ω) ∩ C(Ω), and there is a constant K such that |∂aij /∂xk | + |bi | < K[1 + d(x)α ] and |∂ 2 aij /∂xi ∂xj | + |∂bi /∂xj | < Kd(x)α−1 for all x ∈ Ω. As a consequence, ¯ the functions aij , x → d(x)∂aij (x)/∂xk and x → d(x)bi (x) are in C 0,δ (Ω) whenever 0 < δ < min{α + 1, γ}. (H.3) There is an integer m > 0 such that f , ∂ j f /∂uj , ∂ j f /∂uj−1 ∂xk ∈ C(Ω×]0, ∞[) for all k = 1, . . . , n and all j = 1, . . . , m+1. And if u : Ω → R is such that 0 < k1 d(x) < u(x) < k2 d(x) for all x ∈ Ω and some constants k1 and k2 , then |f (x, u(x))| < K0 [1 + d(x)α ] and |∂ j f (x, u(x))/∂uj | + n + |∂ j f (x, u(x))/∂uj−1 ∂xk | < Kj d(x)α−j for all x ∈ Ω, all k = 1, . . . , n k=1

and all j = 1, . . . , m + 1, where Kj (can depend on k1 and k2 but) is independent of u . As the reader can see, (H.2) allows some singularity for the coeﬃcients bi at the boundary. We also require the coeﬃcients in L to be such as the formal adjoint operator L∗ deﬁned as

∂ ∂aij ∂ ∂u ∗ L u≡− aij − bi − u ∂xi ∂xj ∂xi ∂xj also satisﬁes (H.2). Since all sums apply to the values 1, . . . , n of+the involved indexes, the limits 1 and n are omitted hereafter in the symbol . Note that assumption (H.3) is satisﬁed, for example, by the usual power-law nonlinearities ¯ or, more generally, for g ∈ C 1 (Ω) f (x, u) = g(x)uα1 if α1 > −1 and g ∈ C 1 (Ω) satisfying |g(x)| ≤ kd(x)α2 for some k > 0 and some α2 such that |α1 + α2 | < 1. Our main existence theorem is the following (Theorem 4.1 in [24]). Theorem 2.1. Under the assumptions (H.1)–(H.3), let us assume that (2.1), (2.2) ¯ has a sub-solution u0 and a super-solution u0 such that u0 , u0 ∈ C 2 (Ω) ∩ C01,δ (Ω) for some δ > 0 and 0 < kd(x) < u0 (x) ≤ u0 (x) for all x ∈ Ω. Then (2.1), (2.2) possesses a minimal and a maximal solution in the interval [u0 , u0 ], u∗ and u∗ , ¯ whenever 0 < δ < δ0 = min{γ, α+ 1} which are such that u∗ , u∗ ∈ C 2 (Ω)∩C01,δ (Ω) ∗ 0 and u0 ≤ u∗ ≤ u ≤ u in Ω. Also, u∗ (resp. u∗ ) is the C01,δ (Ω)-limit from below

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(resp. from above) of a monotone sequence of sub-solutions (resp. super-solutions) ∂u∗ ∂u∗ of (2.1) (2.2). Moreover, we have < 0, < 0 on ∂Ω. ∂n ∂n As it was pointed out after Theorem 4.1 in [24], the condition u0 > kd(x) > 0 is often too stringent for applications. The Lemma 4.2 in [24] is instrumental in order to extend the applicability of Theorem 2.1 above. Remark 2.1. Theorem 2.1 provides an existence result for positive solutions in the interval between ordered sub- and super-solutions. In this sense it is similar to wellknown results for nonsingular elliptic problems (see, e.g., [1] and its references). However, the singularity forces us to deal only with positive solutions. Moreover, the additional assumption u0 = u0 = 0 on ∂Ω is made here, and also we impose u0 > kd(x) > 0 in Ω. Remark 2.2. Concerning the regularity of solutions given by Theorem 2.1, we have ¯ with δ < δ0 = min{γ, α + 1}, a result that coinproved that they are in C01,δ (Ω) cides with the one obtained in [22] for a particular nonlinearity. Other (sometimes nonoptimal) regularity results were given in [12] (where the case α < −1 was also studied), [21], [28], [14] and [9], [10], [35] and [34]. Notice that we work from the beginning in these spaces, instead of getting some kind of weak solutions and then improve their smoothness by using some regularity techniques. In particular by the ∂u < 0 on ∂Ω. But examples Strong Maximum principle in Appendix B in [24] ∂n ∂u =0 due to Ouyang, Shi and Yao [29] show that there are solutions such that ∂n on ∂Ω. The above references provide more information concerning the boundary behavior of solutions. The proof of Theorem 2.1 follows the usual pattern for results of this type whose main ingredients are: i) an existence and uniqueness result for the associated linear problem giving the iterates converging to the (maximal and minimal) solutions; ii) the compactness of the solution operator for the linear problem in i); iii) a comparison result (following from a Maximum Principle) for the same problem. In our case, we have stated an existence and uniqueness theorem for the linear problem Lu = M (x)v in Ω (2.3) u=0 on ∂Ω, where M (x)is a weight function satisfying the natural assumption (H.3 ) M ∈ C 1 (Ω) and for all k = 1, . . . , n , d(x)2−α |∂M (x)/∂xk | is bounded in ¯ whenever Ω. As a consequence, the function x → d(x)2 M (x) is in C 0,δ (Ω) 1−α M (x) is bounded in Ω. 0 < δ < min{γ, 1 + α} and d(x)

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J. Hern´ andez, F.J. Mancebo and J.M. Vega

The natural character of assumption (H.3 ) comes out from the fact that if we consider the linearized problem along the positive solution u we obtain LU − M (x)U = λU U =0

in Ω on ∂Ω,

(2.4)

where M (x) = fu (x, u(x)). Proposition 2.1. (Proposition 2.3 in [24]) Let Ω, L and M satisfy assumptions ¯ then (2.3) has a unique solution u ∈ C 2 (Ω)∩ (H.1)–(H.2) and (H.3 ). If v ∈ C00,1 (Ω) 1,δ ¯ C (Ω) for all δ such that 0 < δ < δ0 = min{γ, α + 1} and there is a constant K > 0, which (can depend on δ but) is independent of v, such that uC 1,δ (Ω) ¯ ≤ KvC 0,1 (Ω) ¯ .

(2.5)

Proof. See Appendix A in [24]. We also add a special instance of Proposition 2.1 which will be useful in Section 3. Corollary 2.1. Under the assumptions of Proposition 2.1, if 0 < β < 1, there exists ¯ for any 0 < δ < δ0 = min{γ, 1 − β} of the a unique solution u ∈ C 2 (Ω) ∩ C 1,δ (Ω) problem 1 in Ω Lu = d(x)β (2.6) u=0 on ∂Ω. ∂u < 0 on ∂Ω. ∂n The necessary extension of the Strong Maximum Principle to our singular problem was obtained in the Appendix B in [24]; we will come back to this point below. And the compactness of the solution operator to (2.3) follows readily from estimate (2.5) (See the proof of Theorem 2.6 in [24].) This property is also instrumental for the extension of classical results concerning existence and properties of principal eigenvalues to the case of singular coeﬃcients. We actually consider the more general linear eigenvalue problem Moreover, we have

LU − M (x)U = λN (x)U U =0

in Ω on ∂Ω,

(2.7)

where N satisﬁes (H.4) N is (strictly) positive in Ω and satisﬁes (H.3 ). Theorem 2.2. (Theorem 2.6 in [24]) Under the assumptions (H.1)–(H.2), (H.3 ) and (H.4), the spectrum of the linear eigenvalue problem (2.7), with U ∈ C 2 (Ω) ∩ ¯ is such that C 0,1 (Ω), (i) It consists, at most, of an inﬁnite, countable set of eigenvalues which are ¯ for all δ such that 0 < δ < isolated, and the eigenfunctions are in C 1,δ (Ω) δ0 = min{γ, α + 1}.

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(ii) It contains a unique principal eigenvalue (i.e., a real eigenvalue with an as¯ namely, sociated eigenfunction in the interior of the positive cone of C01 (Ω), such that U > 0 in Ω and ∂U/∂n < 0 on ∂Ω), which is simple, and it is the only eigenvalue with this property. Uniqueness of the positive solution of (2.1)–(2.2) can be proved if the nonlinearity f (x, u) satisﬁes a concavity assumption. This is well known in the regular case (see, e.g., [1]) and we extend the result to the present situation. The idea in the proof goes back as far to Krasnoselskii and we have adapted the version given by Brezis and Kamin in Method III in the Appendix in [6] to obtain the following theorem. Theorem 2.3. Under the assumptions of Theorem 2.1 and moreover (H.5)

f (x, u) is (strictly) decreasing for u > 0, u there is a unique positive solution to problem (2.1)–(2.2).

For any x ∈ Ω, the mapping

Proof. See [6] and [25]. For more uniqueness results see [30] (where an argument involving integration by parts in [2] or [6] is used) or [31]. We have also studied in [24] the linearization of the semilinear problem (2.1)– ∂u < 0 on ∂Ω. With this (2.2) around a smooth solution such that u > 0 in Ω, ∂n aim we rewrite the problem in the integral form F (u) ≡ u − G(f (·, u)) = 0,

(2.8)

¯ → C 1 (Ω) ¯ is the Green operator corresponding to the linear where G : C01 (Ω) 0 equation LU = V

in Ω,

U = 0 on ∂Ω,

(2.9)

(deﬁned as G(V ) = U ). Then we have Theorem 2.4. ( [24], Theorem 3.1) Under the assumptions (H.1)–(H.3), the oper¯ → C 1 (Ω) ¯ deﬁned in (2.8) is of class C m in the positive cone of ator F : C01 (Ω) 0 1 ¯ ¯ that satisfy u > 0 in Ω and C0 (Ω) (that is, the set of those functions of C01 (Ω) ∂u/∂n < 0 on ∂Ω), where m ≥ 1 is as deﬁned in Assumption (H.3) and the linear ¯ → C 1 (Ω) ¯ is given by operator F (u) : C01 (Ω) 0 F (u)v = v − G(fu (·, u)v).

(2.10)

If m > 1 and 1 < j ≤ m then the j-linear operator , ∂ j F (u)/∂uj ≡ F (j) (u) : ¯ ]j → C 1 (Ω), ¯ is given by [ C01 (Ω) 0 F (j) (u)(v1 , . . . , vj ) = −G((∂ j f (·, u)/∂uj )v1 · · · vj ).

(2.11)

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J. Hern´ andez, F.J. Mancebo and J.M. Vega

˜ > 0 such that for any Lemma 2.1. If 0 < α < 1 and 0 0 ﬁxed, the associated problem δ Lu + α = λup in Ω u u=0 on ∂Ω,

(2.13)

and the associated equation u−G(− uδα +λup ) = 0 where G is the Green’s function. For δ = 0 (2.13) has a unique positive solution u ¯ and it is easy to see by using up−1 ) > Lemma 2.2 below with f (u) = up that H(u) = (1−p)up > 0 and λ1 (L−λp¯ 0. Hence by the Implicit Function Theorem for (δ, u) = (0, u ¯), there exists δ0 > 0 such that for any 0 < δ < δ0 (2.13) has a positive solution. Now we write u = µv and hence δ Lv + α+1 α = λµp−1 v p in Ω µ v (2.14) v=0 on ∂Ω. If we pick δ = µ1+α (δ → 0 if and only if µ → 0) we obtain that Lv +

1 λ = 1−p v p α v µ v=0

in Ω

(2.15)

on ∂Ω.

has a positive solution for λ > 0 and µ > 0 “small”, and this ends the proof. (Notice that 1 − p > 0). In [24] we have also extended standard linearized stability results for regular ¯ semilinear parabolic problems to solutions that are in the positive cone of C01 (Ω) as steady states of ∂u + Lu = f (x, u) in Ω, u = 0 on ∂Ω, u(·, 0) = u0 in Ω, (2.16) ∂t 1 by using the fact that for α > − the operator L is sectorial in Lq (Ω) for any n q > n. Notice that if the linear diﬀerential operator has, e.g., constant coeﬃcients, 1 then we can get rid of the limitation α > − . n Theorem 2.5. ( [24], Theorem 4.6) In addition to the assumptions (H.1)–(H.3), let ¯ be a us assume that α > −1/n, let us ∈ C 2 (Ω) ∩ C(= the positive cone of C01 (Ω)) solution of (2.1)–(2.2), and let M ≡ fu (·, us ). If the principal eigenvalue of (2.4) is strictly positive (resp., strictly negative) then us is an exponentially stable (resp., ¯ unstable) steady state of (2.16) in the Lyapunov sense, with the norm of C 1 (Ω). Also, if u0 ∈ C then the problem (2.16) has a unique solution, t → u(·, t) ∈ C, in a

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maximal existence interval, 0 ≤ t < T ≤ ∞, and if T < ∞ then there is a sequence {tm } such that tm ( T and either max{u(x, tm ) : x ∈ Ω} → ∞, or u(x, tm ) # 0 for some x ∈ Ω, or min{∂u(x, tm )/∂n : x ∈ ∂Ω} → 0 as m → ∞. The following lemma provides a simple, rather general and useful, result in order to check that the principal eigenvalue of (2.4) is strictly positive. This property will be useful when applying the Implicit Function Theorem. ¯ is a solution of (2.1)–(2.2) such Lemma 2.2. Suppose that u ∈ C 2 (Ω) ∩ C01 (Ω) ∂u that u > 0 in Ω and ∂n < 0 on ∂Ω and let µ1 be the principal eigenvalue of the linearized problem Lw − fu (x, u)w = µ w

in

Ω,

w=0

on

∂Ω,

(2.17)

around the solution u. Then if f (x, u) satisﬁes the assumption (H.6)

H(x, u) = f (x, u) − ufu (x, u) > 0,

(2.18)

we have µ1 > 0.

3. Applications In this section we consider a few selected applications, already studied in the literature, which exhibit some interesting features or specially relevant ones in one way or another. Additional applications with detailed proofs can be found in [25] or in [26] and its references. We assume all along this section that Ω is a bounded domain in Rn with the smoothness given by (H.1) and the diﬀerential operator L satisﬁes assumptions (H.2). Application 1: The model case For the model problem λK(x) uα u=0

Lu =

in Ω

(3.1)

on ∂Ω,

we prove (see [24]) the following Theorem 3.1. If 0 < α < 1, and K(x) is a function such that 0 < K(x) ≤ k1 d(x)−β where β > 0 is such that α + β < 1 and f (x, u) = K(x)u−α satisﬁes (H.3), then for any λ > 0 there exists a unique positive solution z(λ) > 0 of (3.1) with z(λ) ∈ C 2 (Ω) ∩ C01,δ (Ω), where δ < δ0 = min{γ, α + 1}. Moreover, the mapping λ → z(λ) from (0, ∞) into C01,δ (Ω) is C ∞ and z(λ) is asymptotically stable. Proof. As a sub-solution we try u0 ≡ ce, where e is the unique solution of Le = K(x) in Ω e=0

on ∂Ω,

(3.2)

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given by Corollary 2.1 . We have Lu0 −

λK(x) λK(x) (c1+α eα − λ)K(x) = cK(x) − α α = α (u0 ) c e cα e α

and u0 will be a sub-solution if, for λ > 0 given, c1+α eα L∞ ≤ λ, which holds for c > 0 small enough. In order to ﬁnd a super-solution, we start by recalling that it follows from Theorem 2.2 that there exist λ1 > 0 eigenvalue and ϕ1 > 0 eigenfunction to Lϕ1 = λ1 ϕ1 ϕ1 = 0

in Ω on ∂Ω,

(3.3)

normalized by ϕ1 L∞ = 1. Then there is (again by Corollary 2.1, notice that 0 < α + β < 1) a unique solution u1 > 0 to K(x) Lu1 = in Ω ϕα (3.4) 1 on ∂Ω, u1 = 0 and we look for a super-solution u0 ≡ µu1 for a suitable µ. We have for λ > 0 given α λK(x) µK(x) λK(x) (µα+1 uα 1 − λϕ1 )K(x) Lu0 − = − = α (u0 )α ϕα µα uα µα ϕα 1 1 1 u1 and it follows from the smoothness of u1 and ϕ1 that µu1 > ϕ1 for µ > 0 large enough, and hence Lu0 −

λK(x) (µ − λ)K(x)ϕα (µ − λ)K(x) 1 ≥ = >0 α α 0 α α (u ) µ ϕ1 u1 µα uα 1

for µ large enough. It is clear that for µ large enough we have u0 ≤ u0 and then Theorem 2.1 gives existence. Uniqueness follows immediately from Theorem 2.3 and the rest of the statement from Theorem 2.4 for f (x, u) = K(x)u−α , H(x, u) = K(x)u−α + αK(x)u−α = (1 + α) K(x)u−α > 0, Theorem 2.5 and Lemma 2.2. Theorem 3.1 extends in diﬀerent ways existence and uniqueness results, with a simpler proof, in [12], [28]. The smoothness of the branch z(λ) and its stability seem to be new in the literature. Application 2: A “mixed” nonlinearity We consider now the problem λ + up in Ω uα u=0 on ∂Ω.

Lu =

(3.5)

¯ > 0 such that for 0 < λ < λ ¯ Theorem 3.2. If 0 < α < 1 and p > 1, then there is a λ there is a positive solution to (3.5).

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Proof. It is clear that u0 ≡ z(λ) > 0, z(λ) being the unique solution of 3.1 for K(x) ≡ 1 is a sub-solution. Now we look for a super-solution of the form u0 ≡ Az(λ), z(λ) > 0 solution to (3.1). We have λ λA λ − (u0 )p = α − α α − Ap z p 0 α (u ) z A z λ(Aα+1 − 1) − Ap+α z p+α = Aα z α which will be positive if Lu0 −

λ(A1+α − 1) ≥ Ap+α z(λ)p+α ∞ . 1

But since z(λ) = λ 1+α z(1), it is enough to have p+α

1+α . λ(A1+α − 1) ≥ Ap+α z(1)p+α ∞ λ

For A > 1 ﬁxed, this inequality can be written as p+α

λ ≥ C(A)λ 1+α and since

p+α > 1 it turns out that u0 will be a super-solution for λ > 0 small 1+α

enough. We always have u0 ≤ u0 and existence follows from Theorem 2.1. Problem (3.5) for L = −∆ was studied by Coclite and Palmieri [11] using a much more complicated approximation argument involving sub- and supersolutions. Later the case −1 < α < 0, corresponding to the sum of a concave and a convex nonlinearities was considered in [2], where the existence of a minimal positive solution was shown using sub- and super-solutions; the authors also proved that the solutions small enough (in the L∞ (Ω) norm) are unique. Existence of a second positive solution was then proved by an application of the Mountain Pass Theorem. A sharp result showing that there are exactly two positive solutions for N = 1 was obtained in [7]. In the singular case this second solution was obtained by using diﬀerent (and rather more complicated) variational arguments (see [23] [35] and [27]). Application 3: Open multiplicity problems We analyze now the problem Lu +

1 = λf (x) uα u=0

in Ω

(3.6)

on ∂Ω,

where 0 < α < 1, f ≥ 0 and λ > 0 is a real parameter. Theorem 3.3. If 0 < α < 1, c is a strictly positive constant and f is such that c ¯ > 0 such that if λ > λ ¯ there exists at 0 ≤ f (x) ≤ in Ω, then there exists λ d(x)α least a positive solution to (3.6), where d(x) = dist(x, ∂Ω).

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Proof. We pick as a super-solution the only solution to Lw = λf (x) in Ω, w = 0 on ∂Ω. By the Maximum Principle, w > 0 on Ω. We obtain a sub-solution by reasoning as in the proof of Lemma 2.1. Consider the auxiliary problem δ Lu + α = f (x) in Ω (3.7) u u=0 on ∂Ω. Proceeding as in the proof of Lemma 2.1 it is obtained that the problem Lv +

1 1 = f (x) vα µ

1 large enough. µ Deﬁne A = {λ > 0/∃ solution u > 0}. Again A is nonempty and 0 ∈ / A. If ¯ = inf A and λ > λ ¯ then there exists λ ˜ ∈ A such that λ ¯<λ ˜ < λ. Then if u λ ˜ is a ˜ we get solution of (3.6) for λ = λ

has one solution for

L˜ u+

1 ˜ − λ)f (x) ≤ 0 − λf (x) = (λ u ˜α

and we have a sub-solution. Remark 3.1. D´ıaz, Morel and Oswald [17] proved the same result in Theorem 3.3 for L = −∆ and f ∈ L∞ (Ω). Uniqueness raises interesting questions, see the results and examples given in [13] and [8] for n = 1. Application 4: “Stopping” branches Let us consider the problem Lu +

1 = λup uα u=0

in Ω

(3.8)

on ∂Ω

with 0 < α < 1 and 0 0 such that for any Theorem 3.4. If 0 < α < 1 and 0 λ there exists at least a positive solution of (3.8). Let ε˜ > 0 be a ﬁxed number small enough, then for λ large enough there is a unique solution u of (3.8) such 1 that λ 1−p ε˜ϕ1 < u, where ϕ1 is deﬁned by (3.3). Proof. It is not diﬃcult to ﬁnd a super-solution. Indeed, it suﬃces to pick the unique positive solution of problem −∆u = λup in Ω, u = 0 on ∂Ω. In order to ﬁnd a sub-solution, we use Lemma 2.1 which provides existence of a solution for a λ > 0 large enough. If we deﬁne A = {λ ∈ R/∃u > 0 solution to (3.8)} then A is nonempty and 0 ∈ / A. It is easy to see that if a ∈ A and b ≥ a, then b ∈ A. Indeed, if u is a solution for λ = a then it is a sub-solution for λ = b, b ∈ A.

Nonlinear Singular Elliptic Problems

239

˜ < λ with λ ˜ ∈ A and if u Now, let λ∗ = inf A. If λ > λ∗ , there exists λ∗ < λ ˜ > 0 is ˜ we have a solution for λ = λ, L˜ u+

1 ˜ − λ)˜ − λ(˜ u)p = (λ up ≤ 0 u ˜α

and u˜ is a sub-solution. Remark 3.2. The above result extends Theorem 1.2 in [29] and Lemma 4.11 in [13]. The stability of the maximal positive solution is also proved in [13] for L = −∆. Moreover, the radial case is studied in [29] where the exact number of positive solutions is obtained; besides the maximal (stable) positive solution for λ > λ0 we obtain in the general case by using sub- and super-solutions, existence of a second “lower” positive solution u(λ) is proved for a bounded interval ]λ0 , λ1 [ of λ; ∂u(λ1 ) = 0 on ∂B, moreover, these solutions form a smooth branch and u(λ1 ) = ∂n where B is the unit ball. This situation strongly recalls the transition from positive to nonnegative dead core solutions described in detail for the problem −∆u + uβ = λuα in Ω, u = 0 on ∂Ω for 0 < β < α < 1, in the case Ω =]0, 1[ in [16]. This suggests to deal with the singular problem by using a similar approach improving in this way the results in [29] and looking for an answer for the conjecture on the behavior near the boundary made there. The analogous singular problem −1 < β < α < 0 could be treated by similar arguments. Application 5: Positive versus dead core solutions in mathematical biology Our last example is the problem Lu = λM (x)uα u=0

in

Ω,

on ∂Ω.

(3.9)

This problem arises in population dynamics when looking for stationary solutions to the usual logistic equation with nonlinear diﬀusion. After a change of unknown we get the problem (3.9) with 0 < α < 1. It is well known (and easy to show) that if M ≡ 1 for each λ > 0 there is a nontrivial positive solution which is asymptotically stable. If, however, M (x) changes sign over Ω, the situation becomes more delicate. A very careful and detailed study was carried out in [4]. Since 0 < α < 1, the nonlinearity is not locally Lipschitz near the origin and “dead core” solutions (i.e., solutions identically zero on subdomains of Ω with positive measure) may appear, see [15]. Existence of these dead core solutions was obtained in [4] by using sub- and super-solutions and uniqueness of solutions satisfying some additional conditions was proved as well. All these results were closely related with the structure and properties of the subsets where M (x) is positive. But no result concerning existence of positive solutions was given in [4]. In [24] the authors gave a suﬃcient condition on M (x) for existence of positive solutions. More precisely, they actually prove the following result, under the condition that

240

J. Hern´ andez, F.J. Mancebo and J.M. Vega

the unique solution to the linear problem LU = M (x) U =0

in Ω on ∂Ω,

(3.10)

∂U < 0 on ∂Ω. This condition is satisﬁed if M (x) ≥ 0 ∂n but there are also changing sign functions M (x) that satisfy this property.

is such that U > 0 on Ω and

Theorem 3.5. If the operator L and the domain Ω satisfy assumptions (H.1) and (H.2), let α be such that 0 < α < 1 and M ∈ C 1 (Ω) be such that |M (x)| ≤ K1 d(x)α2 for some K1 and some α2 such that −1 < α2 < 1 − α. Then there is a unique positive solution u(λ) of (3.9) for any λ > 0. Moreover the mapping ¯ is C ∞ and u(λ) is asymptotically stable. λ → u(λ) from (0, +∞) into C01,δ (Ω) Proof. We only sketch it, details can be found in [24]. Let Ψ be the unique solution of LΨ = |M (x)| in Ω, Ψ = 0 on ∂Ω then u0 = CΨ is a super-solution of (3.9) for C 1 large enough. Reasoning as in [24] it is obtained that u0 = (λ(1−α)Ψ) 1−α is a subsolution. This gives existence. Uniqueness follows from an argument which seems due to Spruck (see [24], [4], [6]). The last part is proved by means of Theorem 2.5 and Lemma 2.2. Acknowledgement Research partially supported by the Spanish Ministry of Education Grant MTM2004-03808 and REN 2003-0223 and by the European Community HPRNCT-2002-00274.

References [1] H. Amann. Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Review, 18:620–709, 1976. [2] A. Ambrosetti, H. Brezis, and G. Cerami. Combined eﬀects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal, 122:519–543, 1994. [3] C. Aranda and E. Lami-Dozo. Multiple solutions to a singular nonlinear Dirichlet problem. to appear, 00:1–2, 2002. [4] C. Bandle, A. Pozio, and A. Tesei. The asymptotic behavior of the solutions of degenerate parabolic equations. Trans. Amer. Math. Soc, 303:487–501, 1987. [5] M. Bertsch and R. Rostamian. The principle of linearized stability for a class of degenerate diﬀusion equations. J. Diﬀer. Equat, 57:373–405, 1985. [6] H. Brezis and S. Kamin. Sublinear elliptic equations in Rn . Manus. Math., 74:87–106, 1992. [7] T. Cazenave, M. Escobedo, and A. Pozio. Some stability properties for minimal solutions of −∆u = λg(u). Portugaliae Mathematica, 59:373–391, 2002. [8] Y.S. Choi, A.C. Lazer, and P.J. McKenna. Some remarks on a singular elliptic boundary value problem. Nonlinear Anal., TMA, 32:305–314, 1998.

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[9] M.M. Coclite. On a singular nonlinear Dirichlet problem II. Boll. U. Mat. Ital, 5(B):955–975, 1991. [10] M.M. Coclite. On a singular nonlinear Dirichlet problem IV. Nonlinear Anal., TMA, 23:925–936, 1994. [11] M.M. Coclite and G. Palmieri. On a singular nonlinear Dirichlet problem. Comm. Partial Diﬀer. Equat, 14:1315–1327, 1989. [12] M.G. Crandall, P.H. Rabinowitz, and L. Tartar. On a Dirichlet problem with singular nonlinearity. Comm. Part. Diﬀ. Equat, 2:193–222, 1977. [13] J. D´ avila and M. Montenegro. Positive versus free boundary solutions to a singular equation. Jour. d’Analyse Math., 90:303–305, 2003. [14] M. del Pino. A global estimate for the gradient in a singular elliptic boundary value problem. Proc. Royal Soc. Edinburgh, 122(A):341–352, 1992. [15] J.I. D´ıaz. Nonlinear Partial Diﬀerential Equations and Free Boundaries. Pitman, 1985. [16] J.I. D´ıaz and J. Hern´ andez. Global bifurcation and continua of nonnegative solutions for a quasilinear elliptic problem. C.R. Acad. Sci. Paris, Serie I, 329:587–592, 1999. [17] J.I. D´ıaz, J.M. Morel, and L. Oswald. An elliptic equation with singular nonlinearity. Comm. Part. Diﬀ. Equat, 12:1333–1344, 1987. [18] W. Fulks and J.S. Maybee. A singular nonlinear equation. Osaka J. Math, 12:1–19, 1960. [19] J.A. Gatica, V. Oliker, and P. Waltman. Singular nonlinear boundary value problems for second-order ordinary diﬀerential equations. J. Diﬀ. Equat., 79:62–78, 1989. [20] M. Ghergu and V.D. Radulescu. Bifurcation and asymptotics for the Lane-Emden equation. C.R. Acad. Sci. Paris, Serie I, 337:259–264, 2003. [21] S.N. Gomes. On a singular nonlinear elliptic problem. SIAM J. Math. Anal, 17:1359– 1369, 1986. [22] C. Gui and F. Hua Lin. Regularity of an elliptic problem with a singular nonlinearity. Proc. Roy. Soc. Edinburgh, 123(A):1021–1029, 1993. [23] Y. Haitao. Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem. J. Diﬀer. Equat., 189:487–512, 2003. [24] J. Hern´ andez, F.J. Mancebo, and J.M. Vega. On the linearization of some singular nonlinear elliptic problems and applications. Annales I.H.P., Anal. non lin`eaire, 19(6):777–813, 2002. [25] J. Hern´ andez, F.J. Mancebo, and J.M. Vega. Positive solutions for singular nonlinear elliptic equations. Preprint, 2005. [26] J. Hern´ andez, F.J. Mancebo, and J.M. Vega. Survey in preparation. In M. Chipot and P. Quittner, editors, Handbook of Diﬀerential Equations, Elsevier, Amsterdam, 2006. [27] N. Hirano, C. Saccon, and N. Shioji. Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities. Adv. Diﬀer. Eqs., 9:197–220, 2004. [28] A.C. Lazer and P.J. McKenna. On a singular nonlinear elliptic boundary value problem. Proc. Amer. Math. Soc, 111:721–730, 1991.

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[29] T. Ouyang, J. Shi, and M. Yao. Exact Multiplicity and bifurcation of solutions of a singular equation. Preprint, 1996. [30] J. Shi and M. Yao. Positive solutions of elliptic equations with singular nonlinearity. Preprint, 1996. [31] J. Shi and M. Yao. On a singular nonlinear semilinear elliptic problem. Proc. Royal Soc. Edinburgh, 138(A):1389–1401, 1998. [32] J. Smoller. Shock Waves and Reaction-Diﬀusion Equations. Springer-Verlag, 1985. [33] C.A. Stuart. Existence and approximation of solutions of nonlinear elliptic equations. Math. Z., 147:53–63, 1976. [34] M. Wiegner. A Degenerate Diﬀusion Equation with a Nonlinear Source Term. Nonlinear Anal., TMA, 28:1977–1995, 1997. [35] S. Yijing, W. Shaoping, and L. Yiming. Combined eﬀects of singular and superlinear nonlinearities in some singular boundary value problems. J. Diﬀer. Equat., 176:511– 531, 2001. Jes´ us Hern´ andez Departamento de Matem´ aticas Universidad Aut´ onoma de Madrid E-28049 Madrid Francisco J. Mancebo and Jos´e M. Vega E.T.S.I. Aeron´ auticos Universidad Polit´ecnica de Madrid Plaza del Cardenal Cisneros 3 E-28040 Madrid

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 243–251 c 2005 Birkh¨ auser Verlag Basel/Switzerland

The Navier-Stokes Flow in the Exterior of Rotating Obstacles Matthias Hieber Dedicated to Prof. H. Amann on the Occasion of his 65th Birthday

Abstract. In this note we describe recent results on the equations of NavierStokes in the exterior of a rotating domain. After rewriting the problem on a ﬁxed exterior domain Ω in Rn , it is shown that for initial data u0 ∈ Lpσ (Ω) with p ≥ n and which are satisfying a certain compatibility condition there exists a unique local mild solution to the Navier-Stokes problem. In the case of the whole space of Rn , this local mild solution is even analytic in the space variable x.

1. Introduction In this paper we describe recent developments in the mathematical analysis of the Navier-Stokes ﬂow past rotation obstacles. Observe that whereas the description of the Navier-Stokes ﬂow in the exterior of a ﬁxed obstacle can be regarded as fairly well understood (see, e.g., [26], [31], [12], [28], [30], [1], [8], [19], [18]), this is not the case for rotating or moving obstacles. It is the aim of this paper to discuss and present recent results in the latter direction. Let us start with the following situation: we denote by O denotes a threedimensional rigid body, the obstacle, and assume that the exterior domain Ω = R3 \O is ﬁlled with a viscous, incompressible ﬂuid. Assume that the obstacle is rotating about the x3 -axis with constant angular velocity ω = (0, 0, 1)T . We set Ω(t) := {y(t) = etR x, x ∈ Ω} and Γ(t) := {y(t) = etR x, x ∈ Γ} ⎛ ⎞ 1 −1 0 where R = ⎝ 1 1 0 ⎠ is the 3 × 3-matrix describing the above rotation. 0 0 1 Then the motion past the rotating obstacle O is governed by the equations of

244

M. Hieber

Navier-Stokes given by ∂t w − ∆w + w · ∇w + ∇q ∇·w w w(y, 0)

= = = =

0, 0, Ry, w0 (y),

y y y y

∈ Ω(t), ∈ Ω(t), ∈ Γ(t), ∈ Ω.

t > 0, t > 0, t > 0,

(1.1)

Here w = w(y, t) and q(y, t) denote the velocity and the pressure of the ﬂuid, respectively. The boundary condition on Γ(t) is the usual no-slip boundary condition. By a suitable change of variables we are led to the following set of equations deﬁned on the ﬁxed domain Ω: ∂t u − ∆u + u · ∇u − Rx · ∇u + Ru + ∇p = 0, x ∈ Ω, t > 0, ∇ · u = 0, u = Rx, u(x, 0) = w0 (x),

x ∈ Ω, t > 0, x ∈ Γ, t > 0,

(1.2)

x ∈ Ω.

T

Observe ﬁrst that the term Ru = (−u2 , u1 , 0) , the Coriolis force, can be considered as a lower-order perturbation of the Laplacian. On the other hand, the ﬁrst-order term Rx · ∇u = −x2 ∂1 u + x1 ∂2 u cannot be regarded as subordinate to the Laplacian. The above equation (1.2) was ﬁrst considered by Borchers [6] and by Chen and Miyakawa [9] in the framework of weak solutions. Later, Hishida [23], [24] proved, that the solution of the linearized equation is governed by a strongly continuous semigroup on L2σ (Ω). Moreover, he proved the existence of a unique local mild solution in L2σ (Ω) to problem (1.2) provided the initial data w0 belongs to suitable subset of L2 (Ω). For the case of Ω = Rn and 1 < p < ∞, Babin, Mahalov and Nicolaenko [3], [4] proved the existence of a local mild solution to (1.2) provided w0 lies in Lpσ (Rn ) or w0 is a periodic function satisfying certain properties. The stationary problem, i.e., the time independent problem, was considered before by Galdi [13] and by Farwig, Hishida and M¨ uller [11] on R3 . They proved p a priori L -estimates for the elliptic problem using Littlewood-Paley theory. Recently, the original time dependent problem (1.2) was considered for Ω = Rn and for arbitrary n × n matrices M by Sawada and Hieber in [22] and for exterior domains by Geissert, Heck and Hieber in [14]. The main technique in [22] and [14] was to show that the operator −A deﬁned as Au = P (−∆ − M x · ∇)u + M u and equipped with a suitable domain generates a C0 -semigroup T on Lpσ (Rn ) or Lpσ (Ω), respectively. The semigroup T satisﬁes in addition Lp -Lq smoothing estimates as well as gradient estimates similar to the ones known for the usual Stokes operator. By adjusting Kato’s iteration procedure (see [25], [17], [19], [21]) to the given situation one obtains a local mild solution to the problem on Rn .

Navier-Stokes Flow

245

Note that in contrast to the situation of the classical Navier-Stokes equation, the semigroup T on Lpσ (Ω) is not expected to be analytic, even for p = 2.

2. The Navier-Stokes ﬂow with undounded drift terms: the case of Rn We start this section by considering the equation ∂t u − ∆u + u · ∇u − M x · ∇u + M u + ∇p = ∇·u = u(0) =

0 0 u0

in Rn × (0, T ), in Rn × (0, T ), in Rn ,

(2.1)

where u0 ∈ Lpσ (Rn )n for some p satisfying 1 < p < ∞. Here M denotes an n × nmatrix with real coeﬃcients. We deﬁne the operator A in Lpσ (Rn )n as Au := −∆u − M x · ∇u + M u

(2.2)

· ∇u ∈ L (R ) }. with domain D(A) := {u ∈ W (R ) ∩ Modifying techniques developed in [10], [27] and [29] it was proved in [22] n that −A generates a C0 -semigroup e−tA on Lpσ (Rn ) , which is not analytic. Applying the Helmholtz projection P to (2.1) we may rewrite (2.1) as u (t) + Au + Pu · ∇u − 2PM u = 0 (2.3) u(0) = u0 . 2,p

n n

Lpσ (Rn )n ; M x

p

n n

In the given situation of Rn , the Helmholtz projection P can be expressed explicitly by P := (δij + Ri Rj )1≤i,j≤n , where δij denotes the Kronecker’s delta, and Ri is the ith Riesz transform on Rn deﬁned by Ri := ∂i (−∆)−1/2 , i = 1, . . . , n. Note that A and P commute, since ∇ · Au = 0 if ∇ · u = 0. For T > 0 we call a function u ∈ C([0, T ); Lpσ (Rn )) a mild solution of (2.3) if u satisﬁes the integral equation t t u(t) = e−tA u0 − e−(t−s)A Pu(s) · ∇u(s)ds + 2 e−(t−s)A PM u(s)ds, t > 0. 0

0

(2.4) and u(0) = u0 . We now state the local existence and uniqueness result for mild solutions of (2.3) in Lp spaces which was proved recently in [22]. Note that the underlying semigroup e−tA is not analytic; hence it is a priori not obvious that the classical iteration procedure due to Kato is applicable in the given situation. Theorem 2.1 ([22]). (Local Existence and Uniqueness) Let n ≥ 2, p ∈ [n, ∞) and q ∈ [p, ∞]. Let M be an n × n-matrix with real coeﬃcients and assume that u0 ∈ Lpσ (Rn ). Then there exist T0 > 0 and a unique mild solution u of (2.3) such that 1

1

t 2 ( p − q ) u ∈ C([0, T0 ); Lqσ (Rn )) n

1

1

1

t 2 ( p − q )+ 2 ∇u ∈ C([0, T0 ); Lq (Rn )). n

246

M. Hieber

Regularity questions of mild solution of (2.3) are of particular interest. The estimates for higher-order derivatives given in the following result imply that the mild solution to (2.3) is even analytic in the space variable x. In fact, the following result holds true. Theorem 2.2 ([22]). Let n ≥ 2, u0 ∈ Lnσ (Rn ), r ∈ (n, ∞) and q ∈ [n, ∞]. Assume that etM ≤ 1 for all t ≥ 0. Let u be the local mild solution of (2.3) for some T > 0. Assume further that there exist constants M1 , M2 ≥ 0 such that 1

1

and sup t 2 ( n − r ) u(t)r ≤ M2 < ∞. n

sup u(t)n ≤ M1 < ∞

0
0
Then there exist constants K1 and K2 such that 1

1

∇m u(t)q ≤ K1 (K2 m)m t− 2 − 2 ( n − q ) , m

n

t ∈ (0, T ), m ∈ N0 .

(2.5)

It follows from the above Theorem 2.2 that the mild solution u of (2.3) is analytic in x. More precisely, we have the following estimate on the radius of analyticity of u. Corollary 2.3. Let u be a mild solution of (2.3) satisfying the assumptions of Theorem 2.2. Then u is analytic in x. Moreover, there exists a constant C > 0 such that the radius ρ(t) of analyticity is given by m −1/m √ ∇ u(t)∞ ≥ C t. ρ(t) ≥ lim sup m! m→∞ In the following section we deal with the corresponding situation in exterior domains.

3. The Stokes semigroup for unbounded drift terms: exterior domains Let Ω ⊂ Rn be a domain in Rn with C 1,1 -boundary and let 1 < p < ∞. We then set Lu := (∆ + M x · ∇)u − M u and deﬁne its realization LΩ in Lp (Ω)n by LΩ u := D(LΩ ) :=

Lu {u ∈ W 2,p (Ω)n ∩ W01,p (Ω)n : M x · ∇u ∈ Lp (Ω)n }.

Of course, if Ω is bounded, then D(LΩ ) = W 2,p (Ω)n ∩ W01,p (Ω)n . In the following we will not distinguish between Lp (Ω)n and Lp (Ω) and often write LΩ u := ∆u + M x · ∇u − M u. Moreover, we deﬁne the operator AΩ in Lpσ (Ω)n by AΩ u := PΩ LΩ u. with domain D(AΩ ) := {u ∈ W (Ω)n ∩W01,p (Ω)n ∩Lpσ (Ω)n : M x·∇u ∈ Lp (Ω)n }. Here, PΩ : Lp (Ω) → Lpσ (Ω) denotes the Helmholtz projection. For the proof of the 2,p

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local existence result in exterior domains (see Theorem 4.1) we need to consider perturbations of LΩ and AΩ of the following form: for b ∈ Cc∞ (Ω)n deﬁne Lb by Lb u := Lu + b · ∇u + u · ∇b and set LΩ,b u := Lb u,

D(LΩ,b ) := D(LΩ )

as well as AΩ,b u := AΩ u + PΩ (b · ∇u + u · ∇b),

D(AΩ,b ) := D(AΩ ).

Here u = (u1 , . . . , un ) ∈ Lp (Rn )n . The following result states that the operator AΩ,b generates a C0 -semigroup TΩ,b on Lpσ (Ω) which satisﬁes so-called Lp -Lq smoothing properties. They will be the key ingredient in the proof of the local existence Theorem 4.1. Proposition 3.1 ([14]). Let 1 < p < ∞ and let Ω ⊂ Rn be an exterior domain with C 1,1 -boundary. Then the operator AΩ,b generates a C0 -semigroup TΩ,b on Lpσ (Ω). In particular, AΩ generates a C0 -semigroup TΩ on Lpσ (Ω). Moreover, let 1 0, ω ≥ 0 such that for f ∈ Lpσ (Ω) n 1 1 (a) TΩ,b (t)f Lqσ (Ω) ≤ Ct− 2 p − q eωt f Lpσ (Ω) , t > 0, 1 (b) ∇TΩ,b (t)f Lp (Ω) ≤ Ct− 2 eωt f Lpσ (Ω) , t > 0. Moreover, for f ∈ Lpσ (Ω) n 1 1 1 t 2 p − q TΩ,b (t)f Lqσ (Ω) + t 2 ∇TΩ,b (t)f Lp (Ω) → 0,

for

t → 0.

Recall that the ﬁrst assertion of the above proposition was proved ﬁrst by Hishida [23], [24] for the special case p = 2. Observe also that the corresponding results for the classical Stokes operator can be found in [16], [7] and [20]. The proof of Proposition 3.1 relies on techniques on Laplace Transforms [2], sums of convolution operators and on properties of solutions of the equation div u = f in suitable function spaces. For Sobolev spaces of nonnegative order there is a well-known answer to the latter problem due to Bogovski˘ı [5]. Proposition 3.2 (Bogovski˘ı, [5]). Let Ω ⊂ Rn , n ≥ 2, be a bounded Lipschitz domain, 1 < p < ∞ and k ∈ N0 . Then there exists a continuous operator V : W0k,p (Ω) → (W0k+1,p (Ω))n such that div V f = f for all f ∈ W0k,p (Ω) satisfying

Ω

(3.1)

f = 0.

For an extension of this result to Sobolev spaces of negative order, see the monograph of Galdi [12] or [15]. The result on iterated convolutions (for a proof see [14]) needed for the proof of Proposition 3.1 reads as follows.

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Lemma 3.3. Let X, Y be Banach spaces and let T : (0, ∞) → L(Y, X) and S : (0, ∞) → L(Y ) be strongly continuous functions. Assume that T (t)L(Y,X) ≤ C0 tα eωt ,

S(t)L(Y ) ≤ C0 tβ eωt ,

t > 0,

for some C0 , ω > 0 and α, β > −1. For f ∈ Y set T0 (t)f := T (t)f and t Tn−1 (t − s)S(s)f ds,

Tn (t)f :=

n ∈ N, t > 0.

0

Then there exist C, ω ˜ > 0 such that ∞ Tn (t)f X ≤ Ctα eω˜ t f Y ,

t > 0.

n=0

In the following section we modify the well-known iteration procedure due to Kato [25] to our situation. Of central importance are the Lp − Lq -smoothing properties as well as the gradient estimates for TΩ,b developed in Proposition 3.1.

4. The Navier-Stokes ﬂow in exterior domains In this section we are considering the Navier-Stokes equations in the exterior domain Ω(t) as deﬁned in Section 1. Recall that after rewriting the problem to a ﬁxed domain, the Navier-Stokes equations read as follows ∂t u − ∆u + u · ∇u − M x · ∇u + M u + ∇p ∇·u u u(x, 0)

= = = =

0, 0, M x, u0 (x),

x ∈ Ω, x ∈ Ω, x ∈ Γ, x ∈ Ω,

t > 0, t > 0, t > 0,

(4.1) where div u0 = 0 and the matrix M satisﬁes tr M = 0. Furthermore, assume that the initial value u0 satisﬁes the compatibility condition u0 · n = M x · n

on ∂Ω,

where n denotes the outer normal. Let ζ ∈ Cc∞ (Rn ) be a cut-oﬀ function with 0 ≤ ζ ≤ 1 and ζ = 1 near Γ. Further let supp ∇ζ =: K. We then deﬁne b : Rn → Rn by b(x) := ζM x − BK ((∇ζ)M x), where B is the operator deﬁned in Proposition 3.2. Then div b = 0 and b(x) = M x on Γ. Setting u ˜ := u − b it follows that u ˜ satisﬁes ∂t u − ∆u + u · ∇u − M x · ∇u +M u + b · ∇u + u · ∇b + ∇p = F2 ∇·u=0 u|Γ = 0, u(0) = u0 − b,

in Ω × (0, T ), in Ω × (0, T ),

(4.2)

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249

with ∇ · (u0 − b) = 0 in Ω and F2 = −∆b − M x · ∇b + M b + b · ∇b, provided u satisﬁes (4.1). Applying the Helmholtz projection PΩ to (4.2), we may rewrite (4.2) as an evolution equation in Lpσ (Ω) as u − AΩ,b u + PΩ (u · ∇u) = u(0) =

PΩ F2 , u0 − b.

(4.3)

Note that the compatibility condition on u0 implies u0 − b ∈ Lpσ (Ω). In the following, given 0 < T < ∞, we call a function u ∈ C([0, T ); Lpσ (Ω)) a mild solution of (4.3) if u satisﬁes the integral equation t u(t) = TΩ,b (t)(u0 − b) −

TΩ,b (t − s)PΩ (u · ∇u)(s) ds 0

t TΩ,b (t − s)PΩ F2 (s) ds,

+

0 < t < T.

0

Our main result concerning the Navier-Stokes ﬂow past rotating obstacles is then the following theorem. Theorem 4.1 ([14]). Let n ≥ 2, n ≤ p ≤ q < ∞ and u0 − b ∈ Lpσ (Ω). Then there exists T0 > 0 and a unique mild solution u of (4.3) such that n 1 1 t → t 2 ( p − q ) u(t) ∈ C ([0, T0 ] ; Lqσ (Ω)) , n 1 1 1 t → t 2 ( p − q )+ 2 ∇u(t) ∈ C ([0, T0 ] ; Lq (Ω)) .

References [1] H. Amann, On the strong solvability of the Navier-Stokes equations. J. Math. Fluid Mech., 2 (2000), 16–98. [2] W. Arendt, Ch. Batty, M. Hieber, and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems. Birkh¨ auser, Basel, 2001. [3] A. Banin, A. Mahalov, and B. Nicolaenko, Global regularity of 3D rotating NavierStokes equations for resonant domains. Indiana Univ. Math. J. 48 (1999), 1133–1176. [4] A. Banin, A. Mahalov and B. Nicolaenko, 3D Navier-Stokes and Euler equations with initial data characterized by uniformly large vorticity. Indiana Univ. Math. J. 50 (2001), 1–35. [5] M.E. Bogovski˘ı, Solution of the ﬁrst boundary value problem for an equation of continuity of an incompressible medium. Dokl. Akad. Nauk SSSR 248 (1979), 1037–1040 [6] W. Borchers, Zur Stabilit¨ at und Faktorisierungsmethode f¨ ur die Navier-StokesGleichungen inkompressibler viskoser Fl¨ ussigkeiten, Habilitationschrift Universit¨ at Paderborn, 1992. [7] W. Borchers, and H. Sohr, On the semigroup of the Stokes operator for exterior domains in Lq -spaces. Math. Z. 196, (1987), 415–425.

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[8] M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations. Handbook of Mathematical Fluid Dynamics, vol 3 (eds. S. Friedlander, D. Serre), Elsevier, 2003. [9] Z. Chen, and T. Miyakawa, Decay properties of weak solutions to a perturbed NavierStokes system in Rn . Adv. Math. Sci. Appl. 7 (1997), 741–770. [10] G. Da Prato, and A. Lunardi, On the Ornstein-Uhlenbeck operator in spaces of continuous functions. J. Func. Anal. 131 (1995), 94–114. [11] R. Farwig, T. Hishida, and D. M¨ uller, Lq -theory of a singular ‘winding’ integral operator arising from ﬂuid dynamics, Preprint, TU Darmstadt, 2003. [12] G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes equations. Vol. I, Springer Tracts in Natural Philosophy, Vol. 38, Springer 1998. [13] G.P. Galdi, Steady ﬂow of a Navier-Stokes ﬂuid around a rotating obstacle, J. Elasticity 71 (2003), no. 1-3, 1–31. [14] M. Geissert, H. Heck, and M. Hieber, Lp -Theory of the Navier-Stokes ﬂow past rotating or moving obstacles, Preprint (2004). [15] M. Geissert, H. Heck, and M. Hieber, Remarks on Bogovskii’s operator in Sobolev spaces of negative order, Preprint (2004). [16] Y. Giga, Analyticity of the semigroup generated by the Stokes operator in Lr spaces, Math. Z. 178 (1981), no. 3, 297–329. [17] Y. Giga, Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes system. J. Diﬀerential Equations, 62 (1986), 186–212. [18] Y. Giga, S. Matsui and O. Sawada, Global existence of two-dimensional Navier-Stokes ﬂow with nondecaying initial velocity. J. Math. Fluid Mech., 3 (2001), 302–315. [19] Y. Giga, and T. Miyakawa, Solutions in Lr of the Navier-Stokes initial value problem. Arch. Rational Mech. Anal., 89 (1985), 267–281. [20] Y. Giga, H. Sohr, On the Stokes operator in exterior domains. J. Fac. Sci. Univ. Tokyo, 36, (1989), 313–333. [21] Y. Giga and O. Sawada, On regularizing-decay rate estimate for solutions to the Navier-Stokes initial value problem. Nonlinear Anal. and Appl., (2003), to appear. [22] M. Hieber, and O. Sawada, The Navier-Stokes equations in Rn with linearly growing initial data. Arch. Rational Mech. Anal., to appear 2004. [23] T. Hishida, An existence theorem for the Navier-Stokes ﬂow in the exterior of a rotating obstacle. Arch. Rat. Mech. Anal., 150 (1999), 307–348. [24] T. Hishida, The Stokes operator with rotation eﬀect in exterior domains. Analysis, 19 (1999), 51–67. [25] T. Kato, Strong Lp -solutions of Navier-Stokes equations in Rn with applications to weak solutions. Math. Z., 187 (1984), 471–480. [26] O. Layzhenskaya, The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York, 1969. [27] A. Lunardi, On the Ornstein-Uhlenbeck operator in L2 -spaces with respect to invariant measures. Trans. Amer. Math. Soc. 349 (1997), 155–169. [28] A. Majda, Vorticity and the mathematical theory of incompressible ﬂuid ﬂow. Comm. Pure Appl. Math., 34 (1986), 187–220.

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[29] G. Metafune, J. Pr¨ uss, A. Rhandi, and R. Schnaubelt, The domain of the OrnsteinUhlenbeck operator on an Lp -space with invariant measure. Ann. Sc. Norm. Super. Pisa Cl. Sci., 1 (2002), 471–485. [30] H. Sohr, The Navier-Stokes equations, Birkh¨ auser Advanced Texts: Birkh¨ auser, Basel, 2001. [31] V.A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations. J. Soviet Math., 8 (1977), 467–529. Matthias Hieber Technische Universit¨ at Darmstadt Fachbereich Mathematik Schlossgartenstr. 7 D-64289 Darmstadt, Germany e-mail: [email protected]

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 253–264 c 2005 Birkh¨ auser Verlag Basel/Switzerland

A Global Bifurcation Result for Variational Inequalities Milan Kuˇcera, Jan Eisner and Lutz Recke Dedicated to Professor Herbert Amann in the occasion of his 65th birthday

1. Introduction Let H be a real Hilbert space with the inner product ·, · and the norm · , let K be a closed convex cone with the vertex at the origin in H. We will consider variational inequalities λ ∈ R, u ∈ K :

u − λLu − G(λ, u), ϕ − u ≥ 0 for all ϕ ∈ K

(1.1)

with a real bifurcation parameter λ. We assume that L : H → H is a linear compact mapping, G : R × H → H is a continuous nonlinear compact mapping, limu→0 G(λ,u) = 0 uniformly on compact λ-intervals. u

(1.2)

Denote by CI := {(λ, u) ∈ R × H : (1.1) is fulﬁlled, u = 0} the closure of the set of all nontrivial solutions to (1.1). The aim of this paper is to show that for a given couple (λ0 , u0 ) satisfying the “linearized variational inequality” and certain particular assumptions, there is a component of CI bifurcating at (λ0 , 0) from the set of trivial solutions in the direction u0 . This component is global in the sense that it is unbounded or reaches a point (λ, u) satisfying certain conditions determined by u0 . An application is given to a simple reaction-diﬀusion system with a unilateral nonlocal boundary condition. The variational inequality (1.1) is equivalent to the equation u = PK (λLu + G(λ, u)), where PK is the projection onto the cone K. If the Leray-Schauder index of the mapping I − PK (λL) changes between two values λ1 and λ2 then there is a global bifurcation between λ1 and λ2 (see [7]). However, I − PK (λL) is a nonlinear The research has been supported by the grant A 1019202 of the Grant Agency of the Academy of Sciences of the Czech Republic and by Deutsche Forschungsgemeinschaft (Sfb 555).

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nonsmooth mapping and its index can be calculated only in particular situations (see [7, 9]). We consider here a situation when there need not be a jump of the index for this nonlinear mapping but there is a jump of the index of the linear mapping I − λP L, where P is the orthogonal projection onto a certain subspace determined by u0 . Using Dancer’s global bifurcation theorem [1] for the equation u = P (λLu + G(λ, u))

(1.3)

and some relationship between the sets of solutions to (1.1) and (1.3), we obtain our result.

2. Main result It is well known that the cone K can be written in the form K = {u ∈ H : u, vα ≥ 0 for all α ∈ A},

(2.1)

where A is a certain set of indices and {vα }α∈A is a family of vectors in H, vα = 1 (see [2]). We assume that A = ∅, i.e., K = H. We will always consider a ﬁxed solution (λ0 , u0 ) ∈ R× H of the “linearized” variational inequality (more precisely, the variational inequality with the linearized operator) λ ∈ R, u ∈ K : u − λLu, ϕ − u ≥ 0 for all ϕ ∈ K.

(2.2)

We assume that u0 = 1 and A0 := {α ∈ A : u0 , vα = 0} = ∅.

(2.3)

Let us introduce the closed subspace H0 := {u ∈ H : u, vα = 0 for all α ∈ A0 } and the orthogonal projection P of H onto H0 . We assume that there is c0 > 0 such that u0 , vα ≥ c0 for all α ∈ A \ A0 ,

(2.4)

λ0 Lu0 , (I − P )ϕ ≤ −c0 (I − P )ϕ for all ϕ ∈ K,

(2.5)

λ0 is an (algebraically) simple characteristic value of the operator P L,

(2.6)

i.e., (λ0 )−1 is a simple eigenvalue of P L. In what follows, solutions (λ, u) to (1.1) satisfying the following conditions will play a basic role: u, vα > 0 for all α ∈ A \ A0 ,

(2.7)

λLu + G(λ, u), (I − P )ϕ < 0 for all ϕ ∈ K with (I − P )ϕ = 0. (2.8) We denote CI (u0 ) := the component of {(λ, u) ∈ CI : u ∈ H0 , (2.7), (2.8) hold } containing (λ0 , 0).

A Global Bifurcation Result for Variational Inequalities

255

Remark 2.1. The assumptions (2.4), (2.5) ensure that the couple (λ0 , u0 ) satisﬁes (2.2) if and only if it satisﬁes u0 = P Lu0 (see Lemma 2.5 and Remark 2.9 below). The assumption (2.6) guaranties that u0 is the only normalized solution to (2.2) satisfying u ∈ H0 , (2.7). Hence, the continuum CI (u0 ) (if it is nonempty) bifurcates at (λ0 , 0) in the direction u0 (see Remark 2.10 for details). Let us remark that there could exist solutions (λ0 , u1 ) to (2.2) with u1 = 1 and u1 = u0 which satisfy (2.3)–(2.6), but with another set A0 , another subspace H0 and another projection P and, hence, another correspondingly deﬁned component CI (u1 ) of solutions to (1.1). Remark 2.2. In many applications the variational inequality (1.1) can be interpreted as a principle of virtual work. Then, for a solution (λ, u) of (1.1) with u ∈ H0 , the condition (2.8) says the following. If there is a nonactive constraint for a given “virtual displacement” ϕ−u among those active for u0 (i.e., ϕ, vα > 0 for a certain α ∈ A0 ), then the “virtual work”, carried out by the generalized force on such a displacement, should be negative (i.e., λLu + G(λ, u), ϕ − u < 0), cf. [5, Remark 2.2] for more details. In the application discussed in Section 3, the condition (2.8) is equivalent to (3.9), which means a nontrivial ﬂux through an “active” part of the boundary. Theorem 2.3. Let (λ0 , u0 ) satisfy (2.4)–(2.6). Then at least one of the following conditions is fulﬁlled: (i) CI (u0 ) is unbounded; (ii) CI (u0 ) contains a point (λ, u) = (λ0 , 0) for which u, vα = 0 for some α ∈ A \ A0 or λLu + G(λ, u), (I − P )ϕ = 0 for some ϕ ∈ K with (I − P )ϕ = 0. Let us note that the point (λ, u) from (ii) can be of the type (λ, 0), λ = λ0 . Of course, in this case (2.2) with this λ has a nontrivial solution. Further properties of the set CI (u0 ) will be mentioned on the end of this section. Remark 2.4. Let CI0 be the component of CI containing CI (u0 ). Our considerations do not exclude the case when CI0 (starting in the direction u0 ) can return to (λ0 , 0) in the direction of an other solution u ˜0 = u0 of (2.2) (see Remark 2.1), e.g., that CI0 “contains a closed loop”. Lemma 2.5. If (λ, u) satisﬁes (1.3) and the weakened conditions (2.7), (2.8) where the sharp sign > is replaced by ≥, then (λ, u) fulﬁlls also (1.1). Proof. We have u ∈ H0 for any solution of (1.3) and this together with the weakened condition (2.7) gives u ∈ K. Further, u−λLu−G(λ, u), P (ϕ−u) = P (u−λLu− G(λ, u)), ϕ−u = 0 and u−λLu−G(λ, u), (I −P )(ϕ−u) = −λLu+G(λ, u), (I − P )ϕ ≥ 0 for all ϕ ∈ K by the weakened condition (2.8). Adding the expressions with P and I − P we get (1.1). For ε > 0, R > 0, we denote Kε (u0 ) = (λ, u) ∈ R × H

u −u0 : <ε , u

BR (λ0 , 0) = {(λ, u) ∈ R × H : |λ − λ0 | + u ≤ R}.

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Lemma 2.6. Let (λ0 , u0 ) satisfy (2.4), (2.5). If (λ, u) ∈ Kc0 (u0 ) with c0 from (2.4) u then u, vα ≥ c(u) := u(c0 − u − u0 ) > 0 for all α ∈ A \ A0 . There is ε > 0 such that if (λ, u) ∈ Bε (λ0 , 0) ∩ Kε (u0 ) then the condition (2.8) holds. Proof. The ﬁrst assertion follows from a direct calculus. Assume that there is no ε > 0 such that (2.8) holds for all (λ, u) considered. Then there are (λn , un ) such that (λn , un ) → (λ0 , 0), uunn → u0 and λn Lun + G(λn , un ), (I − P )ϕn ≥ 0 for some ϕn ∈ K, (I − P )ϕn = 0. Due to the assumptions (2.5), (1.2), we have 9 : λn Lun + G(λn , un ) (I − P )ϕn c0 ≤ − λ0 Lu0 , → 0, un (I − P )ϕn

which is a contradiction.

Remark 2.7. Let us recall the Dancer’s global bifurcation theorem [1]. Consider a ˜ : E → E and N : R × E → E real Banach space E with the norm · . Let L be a linear compact mapping and a continuous nonlinear compact mapping such (λ,u) that limu→0 Nu = 0 uniformly on compact λ-intervals. Let λ0 be a simple ˜ let u0 ∈ E and u∗ ∈ E ∗ be the corresponding characteristic value of the operator L, 0 ˜ and of the adjoint operator L ˜ ∗ , respectively, such that u∗0 , u0 = 1 eigenvector of L (here ·, · denotes the duality pairing between E and E ∗ ). Set ˜ − N (λ, u) = 0} C = {(λ, u) ∈ R × E : u = 0, u − λLu and let C0 be the component of C containing (λ0 , 0). Following [1], let us deﬁne subcontinua C0+ and C0− of C0 starting at (λ0 , 0) in the direction of u0 and −u0 , respectively. For η ∈ (0, 1), set Kη = {(λ, u) ∈ R × E : |u, u∗0 | > ηu} Kη+ = {(λ, u) ∈ Kη : u, u∗0 > 0}, Kη− = {(λ, u) ∈ Kη : u, u∗0 < 0}. By Lemma 1.24 in [10], there exists R > 0 such that (C \ {(λ0 , 0)}) ∩ BR (λ0 , 0) ⊂ Kη .

(2.9)

For each r ∈ (0, R] denote by Dr± the components of the sets {(λ0 , 0)} ∪ (C ∩ + − Br (λ0 , 0) ∩ Kη± ), respectively, containing (λ0 , 0). Denote by C0,r and C0,r the components of C0 \ Dr− and C0 \ Dr+ , respectively, containing (λ0 , 0). Set 3 3 + − C0+ = C0,r , C0− = C0,r . 0
0
and C0− are independent of the It follows from (2.9) that the deﬁnitions of + − choice of η ∈ (0, 1), both C0 and C0 are connected and C0 = C0+ ∪ C0− (see also [1]). The Dancer’s global bifurcation theorem [1] states that C0+

either C0+ ∩ C0− = {(λ0 , 0)} or both C0+ and C0− are unbounded.

(2.10)

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˜ = P Lu, N (λ, u) = P G(λ, u) and consider Proof of Theorem 2.3. Set E = H, Lu the sets C, C0+ and C0− from Remark 2.7 corresponding to this choice. Hence, C0+ and C0− is the half-branch of nontrivial solutions to (1.3) bifurcating at (λ0 , 0) in the direction u0 and −u0 , respectively. Let us denote + CA – the component of the set {(λ, u) ∈ C0+ : (2.7), (2.8) hold} containing (λ0 , 0).

Let ε be from Lemma 2.6. Standard considerations using (1.2) imply that there is R > 0 such that (C ∩ BR (λ0 , 0)) \ {(λ0 , 0)} ⊂ Kε (u0 ) ∪ (−Kε (u0 ))

(2.11)

(see [10], Lemma 1.24). Let us choose R such that (2.11) holds, 0 < R < ε. We know from Lemma 2.6 that (2.7), (2.8) hold for all (λ, u) ∈ Bε (λ0 , 0) ∩ Kε (u0 ), and if (λ, u) ∈ −Kε (u0 ) then clearly (2.7) is not fulﬁlled. It follows by using (2.11) + and then realizing the deﬁnition of CA and (2.10), that + CA ∩ BR (λ0 , 0) \ {(λ0 , 0)} = C0+ ∩ BR (λ0 , 0) ∩ Kε (u0 ) = ∅. (2.12) + Any (λ, u) ∈ CA satisﬁes (1.3) and the weakened conditions (2.7), (2.8) where + the sharp sign > is replaced by ≥. Hence, Lemma 2.5 implies that any (λ, u) ∈ CA + satisﬁes (1.1). The set CA is connected and contains (λ0 , 0) and therefore it follows + CA ⊂ CI (u0 ).

(2.13)

Now, let us assume that (ii) does not hold. First, let us show that then + + CA = C0+ . If CA

= C0+ then there is (λ, u) ∈ C0+ such that at least one of the conditions u, vα ≤ 0 for some α ∈ A \ A0 or λLu + G(λ, u), (I − P )ϕ ≥ 0 for some ϕ ∈ K with (I − P )ϕ = 0 holds. We know from (2.12) and Lemma 2.6 + that CA \ {(λ0 , 0)} contains points (λ, u) satisfying (2.7), (2.8), and it follows by + using the connectedness of C0+ , (2.13) and the deﬁnition of CA that one of the + cases mentioned in (ii) must occur. Further, let us show that C0 ∩ C0− = {(λ0 , 0)}. + Indeed, if C0+ ∩ C0− = {(λ0 , 0)} then CA contains a closed loop and, according + − to the deﬁnition of C0 , C0 and the assumption (2.4) (i.e., −u0 , vα < 0 for all + α ∈ A \ A0 ), CA must contain a point (λ, u) with u, vα < 0 for some α ∈ A \ A0 , + = C0+ must be unbounded by (2.10). Now, (i) is a which is not true. Hence, CA consequence of (2.13). Remark 2.8. (see [5, Lemma 2.3]) Let us suppose that A0 is a ﬁnite set and assume that the vectors {vα }α∈A0 are linearly independent. Then there exists a basis span {vα : α ∈ A0+ } such that vα∗ , vβ = δαβ for α, β ∈ A0 and {vα∗ }α∈A0 in+ ∗ (I − P )u = α∈A0 u, vα vα = α∈A0 u, vα vα∗ for u ∈ H. Then the condition (2.5) holds if and only if λ0 Lu0 , vα∗ < 0 for all α ∈ A0 . Moreover, we have λLu + G(λ, u), (I − P )ϕ = 0 for some ϕ ∈ K with (I − P )ϕ = 0 (the condition from (ii) in Theorem 2.3) if and only if λLu + G(λ, u), vα∗ = 0 for some α ∈ A0 . Remark 2.9. If (λ, u) ∈ R × H0 is a solution to (1.1) such that there is c > 0 with u, vα ≥ c for all α ∈ A \ A0 , then (λ, u) is also a solution to (1.3) (see [5, Lemma

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2.2]). Consequently, Lemmas 2.5 and 2.6 imply that the problems (1.1) and (1.3) are equivalent in BR (λ0 , 0) ∩ Kε (u0 ) for suﬃciently small R and ε. Analogously, if (λ, u) ∈ R × H0 is a solution to (2.2) with u, vα ≥ c for all α ∈ A \ A0 , then u = λP Lu. Hence, (2.6) implies that u0 is an isolated normalized solution to (2.2) for λ = λ0 . Remark 2.10. In the proof of Theorem 2.3, ε could be taken arbitrarily small and R + and Remark was such that (2.11) holds. Hence, the deﬁnitions of CI (u0 ) and CA 2.9 imply that for any ε > 0 there is R > 0 such that (C (u ) ∩ B I 0 R (λ0 , 0)) \ + {(λ0 , 0)} = CA ∩ BR (λ0 , 0) \ {(λ0 , 0)}. Therefore (2.12) ensures that (CI (u0 ) ∩ BR (λ0 , 0)) \ {(λ0 , 0)} ⊂ Kε (u0 ) and in this sense one can say that CI (u0 ) bifurcates in the direction u0 .

3. Application Let Ω be a bounded domain in Rn , n ≥ 2, with a Lipschitzian boundary ∂Ω. Let ΓD and ΓN be open subsets of ∂Ω, meas ΓD > 0, meas (∂Ω \ (ΓD ∪ ΓN )) = 0. Furthermore, let ΓU be a subset of ΓN . We will consider the problem d01 ∆u1 + λ(b11 u1 + b12 u2 + g(u1 , u2 )u1 ) = 0 d02 ∆u2 + λ(b21 u1 + b22 u2 ) = 0

in Ω,

∂u2 1 ΓD , ∂u ∂n = 0 on ΓN , ∂n = 0 on ΓN \ ΓU ,

u1 = u2 = 0 on ∂u ∂u2 ∂u2 2 u dΓ ≥ 0, ≥ 0, is constant, 2 ∂n ∂n ∂n · ΓU u2 dΓ = 0 on ΓU . ΓU

(3.1) (3.2)

Here d0 := (d01 , d02 ) is a couple of positive numbers (diﬀusion coeﬃcients), bij ∈ R (i, j = 1, 2), g is a real diﬀerentiable function on R2 , g(0, 0) = 0, λ ∈ R is the bifurcation parameter. Set HD := {ϕ ∈ W 1,2 (Ω) : ϕ = 0 on ΓD in the sense of traces}, H := HD × HD . The elements of H will be denoted by u = (u1 , u2 ). Introduce the inner product in HD by ϕ, ψ := ∇ϕ · ∇ψ dx for all ϕ, ψ ∈ HD Ω

and denote by · the corresponding norm in HD . The same symbols will be used for the corresponding inner product and norm in H. Further, consider the closed convex cone K2 :=

ϕ ∈ HD :

ϕ dΓ ≥ 0 ΓU

with the vertex at the origin in HD . A weak solution to (3.1), (3.2) is a couple u = (u1 , u2 ) ∈ H satisfying the variational inequality u1 ∈ HD : d01 u1 − λ(b11 Au1 + b12 Au2 + G1 (u1 , u2 )u1 ) = 0, u2 ∈ K2 : d02 u2 − λ(b21 Au1 + b22 Au2 ), ψ − u2 ≥ 0 for all ψ ∈ K2

(3.3)

where the operators A : HD → HD , G1 : H → L(HD ) (the space of all linear mappings in HD ) are deﬁned by Aϕ, ψ = Ω ϕψ dx, G1 (u1 , u2 )ϕ, ψ =

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g(u1 , u2 )ϕψ dx for all u1 , u2 , ϕ, ψ ∈ HD . Clearly, A is a linear, completely conΩ tinuous, positive symmetric operator. We will assume g(ξ, η) ≤ c(1 + |ξ|q−2 + 2n |η|q−2 ), q arbitrary for n = 2, q < n−2 for n > 2, so that G1 is well deﬁned, that the map (u1 , u2 ) ∈ H → G1 (u1 , u2 )u1 ∈ HD is completely continuous and G1 (u1 , u2 )u1 /(u1, u2 ) → 0 for (u1 , u2 ) → 0. We need also the “linearized” variational inequality (which is a nonlinear problem again) u1 ∈ HD : d01 u1 − λ(b11 Au1 + b12 Au2 ) = 0 u2 ∈ K2 : d02 u2 − λ(b21 Au1 + b22 Au2 ), ψ − u2 ≥ 0 for all ψ ∈ K2 .

(3.4)

This is a weak formulation of the problem d01 ∆u1 + λ(b11 u1 + b12 u2 ) = 0 d02 ∆u2 + λ(b21 u1 + b22 u2 ) = 0

in Ω

(3.5)

with (3.2). The problem (3.3) can be written in the form (1.1) with the cone K := HD × K2 , which is of the form (2.1) with A := {0}, v0 = (0, v20 ) ∈ H deﬁned

by v0 , u = ΓU u2 dΓ for all u = (u1 , u2 ) ∈ H, and with the operators Lu := ((b11 Au1 + b12 Au2 )/d01 , (b21 Au1 + b22 Au2 )/d02 ), G(λ, u) := λG1 (u1 , u2 )u1 /d01 , 0 for all λ ∈ R, u = (u1 , u2 ) ∈ H. The conditions (1.2) follow from our growth assumptions. ∂v can be deRemark 3.1. If v ∈ HD , ∆v ∈ L2 (Ω) then the normal derivative ∂n ∂v

ﬁned as a linear bounded functional on the space HD by ∂n , ϕ = Ω ∆vϕ + ∇v∇ϕdx for all ϕ ∈ HD where (·, ·) denotes the dual pairing. All equalities and inequalities for the normal derivative used can be understood in the sense of this functional (see [5, Remark 5.2] for details). It is easy to see that u = (u1 , u2 ) ∈ H is a solution of (3.3) or of (3.4) (i.e., a weak solution of (3.1) or (3.5) with (3.2)) if and only if ∆u1 , ∆u2 ∈ L2 (Ω), (3.1) or (3.5), respectively, holds a.e. in Ω and the boundary conditions (3.2) are fulﬁlled in the sense mentioned. Analogously for the other problems considered. See [5, Observation 5.2].

By a solution of any boundary value problem we will always mean its weak solution. Further, we will assume that b11 > 0, b12 < 0, b21 > 0, b22 < 0, b11 + b22 < 0, det bij > 0.

(3.6)

We will need the following notation (see also Fig. 1A): ( ' b12 b21 /κ2 hj := d = [d1 , d2 ] ∈ R2+ : d2 = d1 −b11 /κjj + bκ22j , j = 1, 2, . . . , DS := d ∈ R2+ : d is to the right from all hj , j = 1, 2, . . . , DU := d ∈ R2+ : d is to the left from at least one hj , ω := DS ∩ DU , ∂v where κj are eigenvalues of ∆v + κv = 0 on Ω with v = 0 on ΓD , ∂n = 0 on ΓN . Under the assumption (3.6), this notation has the following interpretation related

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to the problem d1 ∆u1 + b11 u1 + b12 u2 = 0 in Ω (3.7) d2 ∆u2 + b21 u1 + b22 u2 = 0 ∂u2 ∂u1 = = 0 on ΓN (3.8) u1 = u2 = 0 on ΓD , ∂n ∂n with parameters (d1 , d2 ) ∈ R2+ . The trivial solution of the parabolic system corresponding to (3.7), (3.8) is stable for d ∈ DS (the domain of stability) and unstable for d ∈ DU (the domain of instability). Simultaneously, the trivial solution of ODE obtained for d1 = d2 = 0 is stable. Hence, the “diﬀusion driven instability” occurs, which plays an essential role in many mathematical models in biology (see, e.g., [6]). Remark 3.2. Let (3.6) hold. Then the problem (3.7), (3.8) has a nontrivial solution if and only if d ∈ hj for some j = 1, 2, . . . . See, e.g., [8] for details. Lemma 3.3. Let (3.6) hold. Let (λ, u) satisfy (3.4) with d0 /λ ∈ DS , u = 0. Then ∂u2 > 0 on ΓU . (3.9) ∂n 2 Proof. If ∂u ∂n = 0 on ΓU then u is simultaneously a weak solution to the problem (3.5), (3.8) (see also Remark 3.1). Hence, it cannot be d0 /λ ∈ DS due to Remark 3.2.

Now we will distinguish two cases: d0 /λ ∈ / h1 for all λ > 0 d /λc ∈ h1 for some λc > 0, 0

d /λ ∈ DS for all 0 < λ < λc . 0

(3.10) (3.11)

Remark 3.4. Since hj have the joint tangent intersecting the origin, (3.10) means that the straight line σ := {d0 /λ : λ > 0} does not intersect the border ω between DS and DU , while (3.11) occurs always if σ intersects ω, d0 /λc being the ﬁrst intersection point. Further, CI (u0 ) will be the set from Section 1 for our particular H, K, L, G, H0 , P . We will assume that g(u1 , u2 ) ≤ 0 for all (u1 , u2 ) ∈ R2 .

(3.12)

Theorem 3.5. Let (3.6), (3.12) hold. Let us assume that (λ0 , u0 ) = (λ0 , u01 , u02 ) satisﬁes (3.4), d0 /λ0 ∈ DS , u0 > 0 and u01 2 +

d02 b12 0 2 u = 0. d01 b21 2

(3.13)

(a) If (3.10) holds and there is no (λ, 0) ∈ CI (u0 ) with λ = λ0 then CI (u0 ) is unbounded. (b) If (3.11) holds and there is no (λ, 0) ∈ CI (u0 ) with λ = λ0 , λ < λc , then CI (u0 ) is unbounded or for any λ ∈ (λ0 , λc ) there is u ∈ H such that (λ, u) ∈ CI (u0 ), u = 0.

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The last assertion in (b) means that the set CI (u0 ) emanating at (λ0 , 0) reaches at least λc , i.e., d0 /λ reaches the border between DS and DU (see Fig. 1). The existence of the points (λ0 , u0 ) satisfying our assumptions will be discussed on the end of the paper.

d2 . . .

h3

h2

h1

σ d0 /λ0 0 d /λc

DU ω

ω

DS CI (u0 )

ω d1

0

λ0

λc

Figure 1. Left: The system of hyperbolas hj , their envelope ω, domain of stability DS (to the right from ω) and instability DU (to the left from ω), the straight line σ = {d0 /λ : λ > 0}. Right: A possible bifurcation continuum CI (u0 ) emanating from (λ0 , 0) in the case (3.11). Remark 3.6. Let (3.6) hold. The weak formulation of (3.7), (3.8) is d1 u1 − b11 Au1 − b12 Au2 = 0 d2 u2 − b21 Au1 − b22 Au2 = 0.

(3.14)

Expressing u2 from the second equation and substituting to the ﬁrst one, we see that (3.14) is equivalent to u2 = b21 (d2 I − b22 A)−1 Au1 d1 u1 − b11 Au1 − b12 b21 A(d2 I − b22 A)−1 Au1 = 0.

(3.15)

It follows from Remark 3.2 that for any ﬁxed d2 > 0, the greatest eigenvalue dmax (d2 ) of the linear operator Sd2 := b11 A + b12 b21 A(d2 I − b22 A)−1 A coincides with the maximal d1 such that (d1 , d2 ) ∈ ω, and therefore it is positive. The operator Sd2 is compact and symmetric, and it follows from the variational characterization of its greatest positive eigenvalue that ; < b11 Aw + b12 b21 A(d2 I − b22 A)−1 Aw, w . max dmax (d2 ) = w∈HD ,w=1

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Further, it follows from the deﬁnition of DU and DS that (d1 , d2 ) ∈ DU if and only if d1 < dmax (d2 ), and (d1 , d2 ) ∈ DS if and only if d1 > dmax (d2 ). For the proof of Theorem 3.5, we will compare the problems (3.7), (3.8) and d1 ∆u1 + b(x)u1 + b12 u2 = 0 in Ω d2 ∆u2 + b21 u1 + b22 u2 = 0

(3.16)

with (3.8), the parameters d1 , d2 ∈ R+ and b ∈ Lr (Ω) with r arbitrary for n = 2, r < n2 for n > 2. The weak formulation of (3.16), (3.8) is ˜ 1 − b12 Au2 = 0, d1 u1 − Au d2 u2 − b21 Au1 − b22 Au2 = 0 ˜ ϕ = where the operator A˜ : HD → HD is deﬁned by Av, v, ϕ ∈ HD .

(3.17)

Ω

b(x)vϕ dx for all

Lemma 3.7. Let (3.6) hold, b ∈ Lr (Ω) with r as above, b(x) ≤ b11 . If (3.16), (3.8) has a nontrivial solution for some (d1 , d2 ) ∈ R2+ then (d1 , d2 ) ∈ DU . If the nonlinear problem (3.1), (3.8) has a nontrivial solution for some λ > 0 and (3.12) holds then d0 /λ ∈ DU . Proof. Applying considerations from Remark 3.6 to (3.17) and using the assumption b(x) ≤ b11 , for any solution (u1 , u2 ) to (3.16) and (3.8) we obtain ! ˜ 1 + b12 b21 A(d2 I − b22 A)−1 Au1 , u1 u1 −2 ≤ dmax (d2 ). d1 = Au Hence, Remark 3.6 yields (d1 , d2 ) ∈ DU . If (u1 , u2 ) is a solution to (3.1), (3.8) then it is a solution to (3.16), (3.8) with di := d0i /λ, i = 1, 2, and b(x) := b11 + g(u1 (x), u2 (x)) ≤ b11 by (3.12). Hence, the last assertion of our Lemma is a consequence of the ﬁrst one. Proof of Theorem 3.5. We will apply Theorem 2.3 with A0 = {0}, H0 = {(ϕ, ψ) ∈ H : ΓU ψ dΓ = 0}. Lemma 3.3 and (3.2) yield that ∂u02 > 0 on ΓU , u02 dΓ = 0 (3.18) ∂n ΓU under our assumptions. Hence, u0 ∈ H0 and (2.4) is clearly fulﬁlled. In the notation of Remark 2.8 we have v0∗ = v0 and we obtain λ0 Lu0 , v0∗ = Ω λd00 (b21 u01 + 2

∂u0 b22 u02 )v20 dx = − Ω ∆u02 v20 dx = u02 , v20 − ΓU ∂n2 v20 dΓ = u0 , v0 −cv0 , v0 = −c, ∂u0

where c := ∂n2 on ΓU . Hence, (2.5) follows from (3.18) by using Remark 2.8. Due to Remark 2.1, for the proof of (2.6), we need to show that ker(I − λ0 P L) = span {(u0 )}, u0 , u∗0 = 0

(3.19)

where u∗0 is an eigenvector corresponding to λ0 of the adjoint operator L∗ P . The couple u = (u1 , u2 ) ∈ H satisﬁes u = λP Lu if and only if ∆u1 , ∆u2 ∈ L2 (Ω), (3.5)

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holds a.e. in Ω and the boundary conditions 1 ΓD , ∂u u ∂n = 0 on ΓN ,

1 = u2 = 0 on ∂u 2 ΓU u2 dΓ = 0, ∂n is constant on ΓU

∂u2 ∂n

= 0 on ΓN \ ΓU ,

(3.20)

are fulﬁlled (cf. Remark 3.1). Clearly, (λ0 , u0 ) satisﬁes (3.5), (3.20). If there were an other normalized solution u = (u1 , u2 ) = ±u0 to (3.5), (3.20) then we could u ˜2 ˜2 ) such that ∂∂n = 0 on ΓU . Hence, u ˜ ﬁnd their linear combination u ˜ := (˜ u1 , u would be a solution of (3.5), (3.8), which would contradict Remark 3.2 because we have d0 /λ0 ∈ DS . Hence, the ﬁrst condition in (3.19) is proved. The second condition in (3.19) follows from (3.13) because we can choose u∗0 = (u∗1 , u∗2 ), u∗1 = d0 b u01 , u∗2 = d20 b12 u02 . 1 21 If (ii) from Theorem 2.3 does not hold then CI (u0 ) is unbounded. Further, let the condition (ii) from Theorem 2.3 be fulﬁlled. Assume that there is no (λ, 0) ∈ CI (u0 ), λ = λ0 , as in the assumptions of (a), (b) in Theorem 3.5. We have A \ A0 = ∅ and therefore there is (λ, u) = (λ, u1 , u2 ) ∈ CI (u0 ) such that u = 0, λLu + G(λ, u), (I − P )ϕ = 0 for some ϕ ∈ K with (I − P )ϕ = 0. Because of Remark 2.8 and the fact A = A0 = {0}, we λLu+G(λ, u), v0∗ = 0.

have λ ∗ Similarly as above we obtain λLu + G(λ, u), v0 = Ω d0 (b21 u1 + b22 u2 )v20 dx = −c 2

∂u2 2 where c := ∂u ∂n on ΓU . This implies ∂n = 0 on ΓU . Hence (u1 , u2 ) is a solution of (3.16), (3.8) with b(x) = b11 + g(u1 , u2 ), d = d0 /λ. Lemma 3.7 implies that d0 /λ ∈ DU . In the case (3.10), this is excluded. In the case (3.11), d0 /λ ∈ DU implies λ ≥ λc and the connectedness of CI (u0 ) implies that for any λ ∈ (λ0 , λc ) there is u ∈ H, u = 0, (λ, u) ∈ CI (u0 ).

Let us note that in the case (3.11), the existence of at least one (λ0 , u0 ) satisfying all assumptions of Theorem 3.5 with the exception of (3.13) is guaranteed by the following Theorem 3.8, which follows from [4, Theorem 2.1]. It can be shown by the methods used in [3, 4, 9] that such points exist also in the case (3.10) when the straight line σ does not intersect ω but is suﬃciently close to it. The assumption (3.13) must be checked in concrete examples. Theorem 3.8. Let (3.6) and (3.11) hold. Then there is a bifurcation point λI < λc of the problem (3.1), (3.2). In particular, dI := (d01 /λI , d02 /λI ) ∈ DS and there is a nontrivial solution to (3.3). (See Fig. 1.)

References [1] E.N. Dancer, On the structure of solutions of non-linear eigenvalue problems. Indiana Univ. Math. J., Vol. 23, No. 11 (1974), 1069–1076. [2] E. DiBenedetto, Real Analysis, Birkh¨ auser Advanced Texts, Birkh¨ auser, Boston, 2002. [3] P. Dr´ abek, M. Kuˇcera, M. M´ıkov´ a, Bifurcation points of reaction-diﬀusion systems with unilateral conditions. Czechoslovak Math. J., 35 (1985), 639–660.

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[4] J. Eisner, M. Kuˇcera, Spatial patterning in reaction-diﬀusion systems with nonstandard boundary conditions, Fields Institute Communications 25 (2000), 239–256. [5] J. Eisner, M. Kuˇcera, L. Recke, Smooth continuation of solutions and eigenvalues for variational inequalities based on the implicit function theorem. J. Math. Anal. Appl. 274, No. 1 (2002), 159–180. [6] J.D. Murray, Mathematical Biology, 2nd ed., Biomathematics Texts, Vol. 19, (1993), Springer-Verlag Berlin, Heidelberg. [7] V.K. Le, K. Schmitt, Global Bifurcation in Variational Inequalities. Application to Obstacle and Unilateral Problems. Springer-Verlag, New York, 1997. [8] M. Mimura, Y. Nishiura, M. Yamaguti, Some diﬀusive prey and predator systems and their bifurcation problems. Ann. N.Y. Acad. Sci. 316 (1979), 490–521. [9] P. Quittner, Bifurcation points and eigenvalues of inequalities of reaction-diﬀusion type. J. Reine Angew. Math., 380 (1987), 1–13. [10] P.H. Rabinowitz, Some global results for non-linear eigenvalue problems. J. Functional Analysis 7 (1971), 487–513. [11] L. Recke, J. Eisner, M. Kuˇcera, Smooth bifurcation for variational inequalities based on the implicit function theorem. J. Math. Anal. Appl. 275 No. 2 (2002), 615–641. Milan Kuˇcera and Jan Eisner Mathematical Institute of the Academy of Sciences of the Czech Republic ˇ a 25 Zitn´ 115 67 Prague 1 Lutz Recke Humboldt University Unter den Linden 6 D-10099 Berlin

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 265–272 c 2005 Birkh¨ auser Verlag Basel/Switzerland

On Elliptic Non-divergence Operators with Measurable Coeﬃcients Peer Christian Kunstmann Abstract. We study properties of the coeﬃcient matrices of non-divergence operators on Rn aiming at sectoriality and R-sectoriality of these operators. In particular, we present results on approximation, scaling, and the behavior in the Lp -scale.

1. Introduction Let m ∈ N. In this paper we study sectoriality and R-sectoriality of elliptic operators of order 2m in Lp (Rn , C), p ∈ (1, ∞), which have the following form A= aα (x)Dα , D(A) = Wp2m (Rn , C), (1) |α|=2m

where aα : R → C are bounded measurable functions. Here we use the usual multi-index notation α = (α1 , . . . , αn ) ∈ Nn0 , |α| = α1 + · · · + αn , and Dα = D1α1 . . . Dnαn where Dj := −i∂/∂xj for j = 1, . . . , n. The classical three step approach to show sectoriality of non-divergence operators can be described as follows: First study operators with constant coeﬃcients. Second study small perturbations of constant coeﬃcient operators. In the last step use localization and patch together the localized operators (we refer in particular to the localization scheme in [1] for this last step). The approach has also been successfully used to establish a number of further properties such as maximal Lp regularity, R-sectoriality, boundedness of imaginary powers (BIP), or boundedness of an H ∞ -functional calculus. For the second and third step, it seems to be unavoidable to assume some kind of smoothness for the highest-order coeﬃcients, e.g., uniform continuity ([1], [9], [13], [14], [5]) or VMO ([10]) for the properties of sectoriality and R-sectoriality, and H¨ older continuity for BIP ([16]) or a bounded H ∞ -calculus ([1], [6]) (at least, H¨older continuity is needed for boundary value problems; on Rn , however, it is known that uniform continuity ([7]) or even VMO n

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([8]) is suﬃcient). It is also very well known that for all these properties lower+ order terms |β|<2m aβ (x)Dβ with bounded and measurable aβ may be studied by perturbation methods. Hence we may neglect lower-order terms in our study. In this paper we somehow change the point of view. We make no assumptions on the smoothness of the coeﬃcients and we do not use localization. Instead, we investigate what can be said in the general case by looking not on a particular operator in a particular Lp , but on sets of operators in a particular Lp or on a particular operator in a scale of Lp -spaces. On the technical side we limit ourselves to abstract nonsense and hand-waving. Consequently, our observations are not of an absolute (certain regularities imply certain properties) but rather of a relative nature. Nevertheless, they give some useful information. The paper is organized as follows: in Section 2 we study approximation of coeﬃcients and prove a result (Theorem 1) on approximation in the strong operator topology (not in uniform operator topology, the latter being, of course, very well known and usually used for bounded uniformly continuous coeﬃcients). This result on sectoriality also holds for R-sectoriality and makes clear how a precise control on the constants may be exploited. The result on R-sectoriality is new, and even the result on sectoriality has, to our knowledge, not been explicitly stated before. We recall that, for operators in divergence form, scaling may sometimes be used to improve constants in certain estimates (cf. [2]). In Section 3 we concentrate on non-divergence operators with coeﬃcients that are invariant under scaling. On the one hand, this additional assumption makes life more easy and, in fact, the results are somehow analogous to those for operators in divergence form. On the other hand, however, famous counterexamples in elliptic theory belong to this class. In Section 4, the last one, we use a result from abstract interpolation theory to investigate the behavior of a given operator in a scale of Lp -spaces. These results are also new. Notation: We denote tuples (aα )|α|=2m of functions aα ∈ L∞ (Rn , C) by a and the set of all such a by A 2m . For a given tuple of coeﬃcients a ∈ A 2m and p ∈ (1, ∞) ﬁxed we deﬁne the associated operator in Lp (Rn , C) by (Op(a)u)(x) := aα (x)(Dα u)(x), x ∈ Rn . |α|=2m

where u ∈ D(Op(a)) :=

Wp2m (Rn , C).

We deﬁne

Sp2m := {a ∈ A 2m : Op(a) is quasi-sectorial in Lp (Rn , C) }. Recall that an operator A in a Banach space X is called sectorial if, for some θ0 ∈ (0, π), the spectrum σ(A) of A is contained in the closure of the open sector Σθ0 := {z ∈ C \ {0} : |arg z| < θ0 } and, for all θ ∈ (0, π − θ0 ), the set {λ(λ + A)−1 : λ ∈ Σθ } is bounded. The operator A is called quasi-sectorial if ν + A is sectorial for some ν ≥ 0. If, in these deﬁnitions, the set {λ(λ + A)−1 : λ ∈ Σθ } is even R-bounded, then the operator A is called R-sectorial or quasi-R-sectorial, respectively. We denote the set of coeﬃcients a such that Op(a) is quasi-R-sectorial in Lp (Rn , C) by RS 2m p . Recall that a set T of linear operators X → Y is called

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R-bounded, if there is a constant C > 0 such that, for all k ∈ N, T1 , . . . , Tk ∈ T, and x1 , . . . , xk ∈ X, one has E

k j=1

εj Tj xj Y ≤ CE

k

εj xj X ,

j=1

where (εj ) denotes a sequence of independent and symmetric, {±1}-valued random variables on a probability space, e.g., the Rademachers. The inﬁmum of all such C is called the R-bound of T. For the notions of R-boundedness, R-sectoriality, and their connection with maximal Lp -regularity we refer to [14]. Since all functions are complex-valued on Rn we drop Rn and C in notation and write only Lp and Wp2m .

2. Sets of coeﬃcients and approximation

+ On A 2m we consider the norm a∞ := |α|=2m aα ∞ . For any q ∈ [1, ∞] we denote by τq the topology induced by the strong operator topology when considering the elements aα of a as pointwise multipliers on Lq (Rn ). Then τ1 ⊂ τq ⊂ τr ⊂ τ∞ for 1 ≤ q ≤ r ≤ ∞ and τ∞ is induced by ·∞ . Note that ak → a (k → ∞) in τq for q ∈ [1, ∞) if (ak ) is a sequence such that supk ak ∞ < ∞, and ak → a (k → ∞) point- and componentwise a.e. In the following p ∈ (1, ∞) is ﬁxed. For a tuple a of coeﬃcients and A = Op(a) we introduce the following notation for the constants appearing in a priori estimates, where ν ≥ 0 and θ ∈ (0, π): Pθ,ν (a) := inf{M > 0 : ∀λ ∈ Σθ , u ∈ Wp2m : λup + (−∆)m up ≤ M (λ + ν + A)up }. We use the convention inf ∅ := ∞. For ν = 0 we just write Pθ (a) = Pθ,0 (a). We also shall use the constants in resolvent estimates: Mθ,ν (a) := sup{λ(λ + ν + A)−1 p→p + (−∆)m (λ + ν + A)−1 p→p : λ ∈ Σθ }. Again we let Mθ (a) := Mθ,0 (a) and understand Mθ,ν (a) = ∞ if −Σθ ⊂ ρ(ν + A). Since we are also interested in R-sectoriality, we introduce the corresponding RR boundedness counterparts. We denote by Pθ,ν (a) the inﬁmum of all constants C > 0 such that, for all k ∈ N, λ1 , . . . , λk ∈ Σθ , and u1 , . . . , uk ∈ Wp2m we have E

k j=1

εj λj uj p + E

k

εj (−∆)m uj p ≤ CE

j=1

k

εj (λj + ν + A)uj p ,

(2)

j=1

and we denote by Rθ,ν (a) the R-bound of the set {(λ(λ + ν + A)−1 , (−∆)m (λ + ν + A)−1 ) : λ ∈ Σθ }, considered as operators from Lp into Lp × Lp . Recall that p was ﬁxed. If we want to indicate the special Lp -space to which we refer then we use p as an additional subscript, e.g., Mθ,ν,p (a) etc.

268

P.C. Kunstmann We have the following theorem on approximation of coeﬃcients.

Theorem 1. Let p ∈ (1, ∞) and K ⊂ Sp2m be convex and · ∞ -bounded. Suppose τp that M, ν ≥ 0 and θ ∈ (0, π) are such that Mθ,ν ≤ M on K . Then K ⊂ Sp2m τp and Mθ,ν ≤ M on K . τp and Rθ,ν ≤ M on K , then K ⊂ RS 2m and If, in addition, K ⊂ RS 2m p p τp Rθ,ν ≤ M on K . Proof. First we notice that Mθ,ν ≤ M on K implies Pθ,ν ≤ M on K . Then we observe that Pθ,ν is lower semi-continuous for τp on A 2m which is immediate from τp τp the deﬁnitions. Hence Pθ,ν ≤ M on K . For a ∈ K we choose a0 ∈ K and τp consider aµ := (1 − µ)a0 + µa for µ ∈ [0, 1]. Since K is again convex, we have Pθ,ν (aµ ) ≤ M for all µ ∈ [0, 1]. Since a0 ∈ K ⊂ Sp2m , the continuity method yields the ﬁrst assertion. τp τp R Since K ⊂ Sp2m , the assertion Rθ,ν ≤ M on K is equivalent to Pθ,ν ≤M τp on K . Now we notice that E

k

εj fj p =

j=1

k 1 σj fj p , 2k σ =±1 j=1

(3)

j

and the RHS hereof is just an average over choices of signs in the sum. Hence, if we let am → a in τp , then (3) shows that (2) for Op(am ) in place of A will converge τp R R to (2) for Op(a) in place of A. Thus Pθ,ν ≤ M on K implies Pθ,ν ≤ M on K , τp which in turn implies Rθ,ν ≤ M on K . Remark 2. The proof shows that the convexity assumption may be weakened. It τp is suﬃcient that K is arc-wise connected for τ∞ . For applicability of the above results we want to mention the following. Remark 3. For a tuple a = (aα )|α|=2m ∈ A 2m we denote the associated symbol by Sy(a, x, ξ) := aα (x)ξ α , x, ξ ∈ Rn , |α|=2m

where, as usual, ξ := for ξ = (ξ1 , . . . , ξn ) ∈ Rn and α = (α1 , . . . , αn ) ∈ n N0 . If K ⊂ {z ∈ C : Re z > 0} is compact and convex then α

ξ1α1 ·. . .·ξnαn

AK2m := {a ∈ A 2m : for a.e. x ∈ Rn : ∀|ξ| = 1 : Sy(a, x, ξ) ∈ K} is convex. Let a = (aα ) ∈ AK2m . Taking a standard molliﬁer sequence (ρk ) = (k n ρ(k·)) we have that all ak := (ρk ∗ aα )|α|=2m , k ∈ N, belong to AK2m , and that ak → a (k → ∞) in τq for any q ∈ [1, ∞). Hence via Theorem 1 a good control on sectoriality [R-sectoriality] constants for smooth coeﬃcients yields sectoriality [R-sectoriality] also for non-smooth coeﬃcients.

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269

3. Scaling and scaling invariant coeﬃcients We deﬁne, for µ > 0, the dilation operator Sµ f := f (µ·) and extend it to tuples a = (aα )|α|=m ∈ A 2m via Sµ a = (Sµ aα )|α|=2m . Observe that Sµ f p = µ−n/p f p for f ∈ Lp (Rn ) and p ∈ (1, ∞). For µ > 0 and a ∈ A 2m we have S1/µ Op(a)Sµ u = µ2m Op(S1/µ a)u

(4)

for u ∈ Wp2m . Sine Sµ is an isomorphism of Lp with inverse S1/µ , we conclude that Sµ leaves also Sp2m invariant. Lemma 4. Let p ∈ (1, ∞), a ∈ A 2m and θ ∈ (0, π), ν ≥ 0. For any µ > 0 we have Kθ,ν (a) = Kθ,νµ−2m (S1/µ a)

where K ∈ {P, M, R, P R }.

(5)

Proof. The other cases being similar we only study K = P . In λup + (−∆)m up ≤ M (λ + ν + Op(a))up we write u = Sµ v and S1/µ · p in place of · p . By (4) we obtain λup + µ2m (−∆)m up ≤ M (λ + ν + µ2m Op(S1/µ a))up . Dividing by µ2m we obtain the assertion since µ−2m Σθ = Σθ .

Remark 5. If we have ν = 0 in the preceding lemma we see that the constants cannot depend on, e.g., the modulus of continuity or the H¨ older norm of the aα in a. This makes ﬁniteness of the constants for ν = 0 the most interesting case. We call an a ∈ A 2m scaling-invariant if Sµ a = a for all µ > 0. For scalinginvariant coeﬃcients we obtain the following. Proposition 6. Let p ∈ (1, ∞) and a ∈ A 2m be scaling-invariant. For any θ ∈ (0, π) and ν > 0 we have Kθ,ν (a) = Kθ (a)

where K ∈ {P, M, R, P R }.

(6)

In particular, Pθ,ν (a) < ∞ for some θ, ν implies the G˚ arding inequality Op(a)up ≥ (Pθ (a))−1 up ,

u ∈ Wp2m .

(7)

Moreover, if (0, ∞) ∩ ρ(−Op(a)) = ∅ then Op(a) ∈ Sp2m . Proof. For (6) we use (5) in Lemma 4 and let µ → ∞ (where we may use Theorem 1). We obtain (7) by letting λ → 0 in the a priori estimate. To prove the last assertion let λ0 > 0 be in ρ(−Op(a)) and M := (λ0 + Op(a))−1 p→p . By (4) and scaling-invariance we have, for any µ > 0 and A = Op(a), M = S1/mu (λ0 + A)−1 Sµ p→p = (λ0 + µ2m A)−1 p→p = µ−2m (λ0 µ−2m + A)−1 p→p , which means λ(λ + A)−1 p→p = M λ0 for any λ > 0.

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P.C. Kunstmann

For general a ∈ S 2m we would obtain (7) only for ν + Op(a) and some ν > 0 in place of Op(a). Examples of scaling-invariant operators are, of course, constant coeﬃcient operators, for which the technique is well known. Other examples are given by operators n xj xk ∂j ∂k ), c ≥ 0. −(∆ + c |x|2 j,k=1

For n ≥ 3, operators of this type provide well-known counterexamples in the theory of elliptic boundary value problems (we refer to [15, Sect. 1.1]). It seems, however, not to be clear, if they also serve as counterexamples to sectoriality on Rn .

4. Extrapolation via interpolation In this section we exploit Sneiberg’s Lemma (cf. [2, Lem. 23, p. 53], cf. also [12]): Lemma 7 (Sneiberg’s Lemma). Let (Xθ ) and (Yθ ) complex interpolation scales of reﬂexive spaces where θ ∈ [0, 1]. If S is a linear operator which is bounded Xθ → Yθ for each θ ∈ [0, 1]. Then the following subsets of (0, 1) are open: {θ : ∃η > 0 : SxYθ ≥ ηxXθ },

{θ : S : Xθ → Yθ is an isomorphism}.

The second statement follows from the ﬁrst if we also consider the dual operator S ∗ in the dual scales (Yθ∗ ) and (Xθ∗ ). Theorem 8. For ﬁxed a ∈ A 2m , θ ∈ (0, π), the following subsets of (1, ∞) are open: {p ∈ (1, ∞) : Kθ,0,p < ∞} where K ∈ {P, M, R, P R }. For ν > 0 also the subsets {p ∈ (1, ∞) : Kθ,ν,p < ∞ and ν + Op(a) is invertible in Lp } where K ∈ {P, M, R, P R } are open. Proof. We choose a dense sequence (λj ) in Σθ and let A := Op(a). We ﬁrst treat the case ν = 0 and start with K ∈ {P, M }. We shall apply Lemma 7 to Xp := {(uj ) ∈ (Wp2m )N : (uj )Xp := ( (λj uj p + (−∆)m uj p )p )1/p < ∞} j

and Yp := lp (Lp ) and the operator S : (uj ) → ((λj + A)uj ). Observe that (λj + A)uj p ≤ λj uj p + Auj p ≤ λj uj p + C(−∆)m uj p where C depends on p and a∞ . This yields boundedness of S : Xp → Yp for any p ∈ (1, ∞). Now Lemma 7 proves the claim. For K ∈ {P R , R} we use the same operator S but we use Xp := {(uj ) ∈ (Wp2m )N : (uj )Xp := E εj λj uj p +E εj (−∆)m uj p < ∞} j

j

On Elliptic Non-divergence Operators

271

+ and Yp := Rad(Lp ) with norm (fj )Yp := E j εj fj p . For elementary properties of Rad(X) we refer to [14]. Since Lp -spaces have non-trivial type we have [Rad(Lp ), Rad(Lq )]θ = Rad(Lr ) for 1/r = (1 − θ)/p + θ/q (cf. [11]), and (Yp ) is a complex interpolation scale. A similar argument shows that (Xp ) is a complex interpolation scale. Now we observe εj (λj + A)uj p ≤ E εj λj uj p + EA εj u j p E j

j

≤ E

j

εj λj uj p + CE

j

(−∆)m uj p

j

where C is the same constant as before. Thus Lemma 7 yields the claim for K ∈ {P R , R}. The case ν > 0 is treated very similar. We just replace A by ν +A and (−∆)m by 1 + (−∆)m . Observe, e.g., that Pθ,ν,p (a) < ∞ and ν + A has a bounded inverse Lp → Lp if and only if there is a C > 0 such that λup + (1 + (−∆)m )up ≤ C(λ + ν + A)up , For k ∈ {M, R, P R } the situation is similar.

u ∈ Wp2m .

References [1] H. Amann, M. Hieber, G. Simonett, Bounded H∞ calculus for elliptic operators, Diﬀerential Integral Equations 7, 613–653 (1994). [2] P. Auscher, Ph. Tchamitchian, Square root problem for divergence operators and related topics, Ast´erisque 249, Paris, Soci´et´e Math´ematique de France (1998). [3] S. Blunck, P.C. Kunstmann, Weighted norm estimates and maximal regularity, Adv. Diﬀer. Equ. 7, 1513–1532 (2002). [4] E.B. Davies, Uniformly elliptic operators with measurable coeﬃcients, J. Funct. Anal. 132, 141–169 (1995). [5] R. Denk, M. Hieber, J. Pr¨ uss, R-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Am. Math. Soc. 788 (2003). [6] R. Denk, G. Dore, M. Hieber, J. Pr¨ uss, A. Venni, New thoughts on old results of R. T. Seeley, Math. Ann. 328, 545–583 (2004). [7] X.T. Duong, G. Simonett, H∞ -calculus for elliptic operators with non-smooth coefﬁcients, Diﬀer. Integral Equ. 10, 201–217 (1997). [8] X.T. Duong, Li Xin Yan, Bounded holomorphic functional calculus for nondivergence form diﬀerential operators, Diﬀer. Integral Equ. 15, 709–730 (2002). [9] R. Haller, H. Heck, M. Hieber, Muckenhoupt weights and maximal Lp -regularity, Arch. Math. 81, 422–430 (2003). [10] H. Heck, M. Hieber, Maximal Lp -regularity for elliptic operators with VMOcoeﬃcients, J. Evol. Equ. 3, 332–359 (2003). [11] N.J. Kalton, P.C. Kunstmann, L. Weis, Perturbation and interpolation theorems for the H ∞ -calculus with applications to diﬀerential operators, to appear in Math. Ann.

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[12] N.J. Kalton, M. Mitrea, Stability results on interpolation scales of quasi-Banach spaces and applications, Trans. Am. Math. Soc. 350, 3903–3922 (1998). [13] P.C. Kunstmann, Maximal Lp -regularity for second-order elliptic operators with uniformly continuous coeﬃcients on domains, in Iannelli, Mimmo (ed.) et al., Evolution equations: applications to physics, industry, life sciences and economics, Basel, Birkh¨ auser. Prog. Nonlinear Diﬀer. Equ. Appl. 55, 293–305 (2003). [14] P.C. Kunstmann, L. Weis, Maximal Lp -regularity for parabolic equations, Fourier multiplier theorems and H ∞ -functional calculus, in M. Iannelli, R. Nagel, S. Piazzera (eds.), Functional Analytic Methods for Evolution Equations, Springer Lecture Notes Math. 1855, 65–311 (2004). [15] A. Maugeri, D.K. Palagachev, L.G. Softova, Elliptic and Parabolic Equations with Discontinuous Coeﬃcients, Mathematical Research Vol. 109, Wiley-VCH, Berlin, 2000. [16] J. Pr¨ uss, H. Sohr, Imaginary powers of elliptic second-order diﬀerential operators in Lp -spaces, Hiroshima Math. J. 23, 161–192 (1993). Peer Christian Kunstmann Mathematisches Institut I Universit¨ at Karlsruhe Englerstr. 2 D-76128 Karlsruhe, Germany e-mail: [email protected]

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 273–290 c 2005 Birkh¨ auser Verlag Basel/Switzerland

A Chemotaxis Model with Threshold Density and Degenerate Diﬀusion Philippe Lauren¸cot and Dariusz Wrzosek* Abstract. A quasilinear degenerate parabolic system modelling the chemotactic movement of cells is studied. The system under consideration has a similar structure as the classical Keller-Segel model, but with the following features: there is a threshold value which the density of cells cannot exceed and the ﬂux of cells vanishes when the density of cells reaches this threshold value. Existence and uniqueness of weak solutions are proved. In the one-dimensional case, ﬂat-hump-shaped stationary solutions are constructed. Mathematics Subject Classiﬁcation (2000). 35K55, 35K65, 34B15, 34C25. Keywords. Chemotaxis equations, degenerate diﬀusion, quasilinear parabolic equation, compactness method.

1. Introduction We investigate the well-posedness and the properties of steady states for a degenerate parabolic system describing the aggregation phase of the chemosensitive motion of cells. This system is actually a particular case of the general system introduced by Patlak [7] and Keller and Segel [4] and describes the space and time evolution of the cell density u and the chemoattractant density v (see also [3] and the references therein). More precisely, it reads ∂t u = ∇ · (D(u) ∇u − h(u) ∇v)

in

(0, +∞) × Ω ,

(1)

∂t v = ∆v + g(u, v) (D(u) ∇u − h(u) ∇v) ν˙ = 0

in on

(0, +∞) × Ω , (0, +∞) × ∂Ω ,

(2) (3)

(u, v)(0) = (u0 , v0 )

in

Ω,

(4)

where Ω is a bounded open subset of R with smooth boundary ∂Ω and ν denotes the outward normal unit vector ﬁeld to ∂Ω. Here, D(u) denotes the density-dependent diﬀusion coeﬃcient and h(u) is usually written in the form N

* The second author was supported by Polish KBN grant 2 P03A 03022.

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Ph. Lauren¸cot and D. Wrzosek

h(u) = u h1 (u) where h1 is commonly referred to as the chemotactic sensitivity function. The function g describes the rates of production and degradation of the chemoattractant v, and the classical Keller-Segel model corresponds to the case where D and h1 are constants. The basic assumption made here is the existence of a threshold value for the cell density which corresponds to a tight packing state. In other words, cells cannot accumulate without bound at a given point of Ω and the corresponding mathematical assumption is that the function h vanishes identically when u crosses a given value, say u ≥ 1. The eﬀect of a threshold cell density or a volume-ﬁlling eﬀect has been taken into account in the modelling of chemotaxis phenomenon in [2, 6], and the well-posedness and large time behavior of solutions to chemotaxis systems incorporating the volume-ﬁlling eﬀect were studied in [9, 10]. A particular feature of the models studied in the above-mentioned papers is that there is a relationship between D and h: more precisely, there is a non-increasing function q such that q(0) = 1, q(1) = 0, D(r) = q(r) − r q (r) and h(r) = r q(r) ,

r ∈ [0, 1] .

(5)

Furthermore, q is assumed to be such that D is bounded from below by a positive constant, so that (1)–(4) is a non-degenerate parabolic system. Now, since a threshold density for u has been introduced in the model, it is natural to wonder whether this threshold density is attained by u during the evolution. Numerical experiments performed in [6] suggest that the threshold density may be reached by the u-component of stationary states. To the contrary, it turns out that the u-component of steady states to (1)–(3) never attains the threshold value 1 when (5) and the non-degeneracy assumption on D are fulﬁlled (unless u ≡ 1). This was shown in [10, Proposition 1.4] and we extend it in Proposition 8 below. Furthermore, as we will see below, removing the non-degeneracy assumption on D is not suﬃcient to warrant that the u-component of steady states could take the value 1. The purpose of this paper is thus twofold: on the one hand, we establish the well-posedness of (1)–(4) without the condition (5) and allow the diﬀusion coefﬁcient to vanish at the threshold value (D(1) = 0). From a biological viewpoint, this means that the diﬀusivity of cells decreases gradually to zero as u approaches values close to the threshold. On the other hand, we exhibit a condition on D and h which guarantees that, at least in the one-dimensional case, there are stationary states (u, v) to (1)–(3) such that u reaches the threshold density either on large sets (ﬂat-hump-shaped solutions) or at one point (spike-shaped solutions), see Theorem 10 and Remark 11 below. Let us now state the assumptions on the data we will use in the sequel, together with the main results we obtain in this paper. We assume that D ∈ C 2 (R), h ∈ C 2 (R) satisfy D(r) > 0 for r ∈ [0, 1) , D(1) = h(0) = h(1) = 0 and (6) h(r) > 0 for r ∈ (0, 1) ,

Chemotaxis with Threshold Density and Degenerate Diﬀusion

275

and that the reaction term g ∈ C 2 (R2 ) satisﬁes the (one-sided) growth condition g(r, 0) ≥ 0 and ∂s g(r, s) ≤ κ for (r, s) ∈ [0, 1] × [0, +∞) ,

(7)

for some κ > 0. As for the initial data, we assume that (u0 , v0 ) ∈ L∞ (Ω; R2 ) with 0 ≤ u0 ≤ 1 and 0 ≤ v0 a.e. in Ω .

(8)

Further assumptions on D, h and g will be needed to ensure the uniqueness of a solution to (1)–(4) and read: there are C0 > 0, C1 > 0 and functions g1 ∈ C 2 (R) and g2 ∈ C 2 (R) such that g1 (0) ≥ 0, g2 (0) ≥ 0, 2

(h(r) − h(s)) ≤ C0 (r − s) (D(r) − D(s)) , g(r, s) = g1 (s) + r g2 (s) for (r, s) ∈ [0, 1] × [0, +∞), where D(r) :=

and

(9)

max {g1 (s), g2 (s)}

≤ C1 ,

(10)

r

D(s) ds ,

r ∈ [0, 1] .

0

Deﬁnition 1. Let (u0 , v0 ) ∈ L∞ (Ω; R2 ) be such that (8) holds true. A weak solution to (1)–(4) is a couple (u, v) of functions such that, for each T > 0, 0 ≤ u(t, x) ≤ 1 ,

0 ≤ v(t, x)

a.e. in

(0, T ) × Ω ,

(11)

u ∈ L∞ ((0, T ) × Ω) ∩ Cw ([0, T ]; L2 (Ω)) , D(u) ∈ L2 (0, T ; H 1 (Ω)) , u(0) = u0 , v ∈ L∞ ((0, T ) × Ω) ∩ H 1 (0, T ; L2(Ω)) ∩ L2 (0, T ; H 2(Ω)) , and (u, v) satisfy T ∂t u, ψ dt

T

v(0) = v0 ,

(13)

(∇D(u) − h(u) ∇v) (t, x) · ∇ψ(t, x) dxdt = 0

+

0

(12)

0

Ω

∂t v

= ∆v + g(u, v) a.e. in (0, T ) × Ω ,

∂ν v

= 0 a.e. on (0, T ) × ∂Ω ,

for each t ∈ [0, T ] and ψ ∈ L2 (0, T ; H 1(Ω)). Here, ., . denotes the duality pairing between H 1 (Ω) and H 1 (Ω) . Here, Cw ([0, T ]; L2 (Ω)) denotes the set of functions from [0, T ] in L2 (Ω) which are continuous for the weak topology of L2 (Ω). Theorem 2. Consider (u0 , v0 ) ∈ L∞ (Ω; R2 ) such that (8) holds true. Under the assumptions (6) and (7) on D, h and g, there exists at least a weak solution (u, v) to (1)–(4) in the sense of Deﬁnition 1 such that u(t, x) dx = u0 (x) dx for each t ≥ 0 . (14) Ω

Ω

In addition, if D, h and g also fulﬁl (9) and (10), then (1)–(4) has a unique weak solution.

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As already mentioned, the well-posedness of (1)–(4) has been established in [2, 9] when D and h are related by (5) and D(r) ≥ δ > 0 for r ∈ [0, 1]. In particular, the case where q(1) = q (1) = 0 is not covered by the analysis performed in [9] and Theorem 2 thus provides an existence theorem for that case. As for uniqueness, we may use Proposition 4 to check that D and h fulﬁl the condition (9) if q (r) ∼ −(1 − r)k as r → 1 for some k > 0. Indeed, in that case, we have h (r) ∼ −(1 − r)k and D(r) ∼ (1 − r)k as r → 1. The proof of Theorem 2 is given in Section 2.1. In the last section, we state some properties of steady states to (1)–(3) and, in the one-dimensional case, construct stationary solutions (u, v) to (1)–(3) such that u ≡ 1 on a non-empty subinterval of Ω.

2. Well-posedness 2.1. Existence The existence statement in Theorem 2 is proved by a compactness method. Owing to the degeneracy of D for u = 1, we ﬁrst study a non-degenerate regularized problem for which the well-posedness is deduced from the abstract theory developed by Amann [1]. More precisely, we have the following result: Theorem 3. Let p > N and consider (u0 , v0 ) ∈ W 1,p (Ω; R2 ) such that (8) holds true. Let D, h and g be functions satisfying (6) and (7), and assume further that there is η > 0 such that D(r) ≥ η

for

r ∈ [0, 1] .

(15)

Then there is a unique classical solution ¯ R2 ) ∩ C 1,2 ((0, +∞) × Ω; ¯ R2 ) (u, v) ∈ C([0, +∞) × Ω; ¯ to (1)–(4) such that (u(t, x), v(t, x)) ∈ [0, 1]×[0, +∞) for each (t, x) ∈ [0, +∞)× Ω. In addition, u(t, x) dx = u0 (x) dx for each t ≥ 0 . (16) Ω

Ω

Proof. Owing to (15) and the continuity of D, there exists δ > 0 such that D ≥ η/2

on

(−δ, 1 + δ) .

We put D0 := (−δ, +∞) × (−δ, 1 + δ) and deﬁne a ∈ C 2 (D0 ; M2 (R)) by 1 0 m,n for y = (y1 , y2 ) ∈ D0 . a(y) = (a )1≤m,n≤2 := −h(y2 ) D(y2 )

(17)

Chemotaxis with Threshold Density and Degenerate Diﬀusion

277

We next set aj,k (y) := a(y) δj,k for 1 ≤ j, k ≤ N and y ∈ D0 and introduce the operators A(y)z

:= −

N

∂j (aj,k (y) ∂k z) ,

j,k=1

B(y)z

:=

N

νj · aj,k (y) ∂k z ,

j,k=1

g(y2 , y1 ) , 0 with z = (z1 , z2 ). With these notations, an abstract formulation of (1)–(4) reads and the function

f (y) = (f m )1≤m≤2 := ∂t z + A(z)z

=

B(z)z = z(0) =

f (z) , 0, (v0 , u0 ) ,

with z = (v, u). Owing to (17), the eigenvalues of the matrix a(y) are positive for each y ∈ D0 , and the boundary-value operator (A, B) is of separated divergence form in the sense of [1, Example 4.3 (e)]. Consequently, the boundary-value operator (A, B) is normally elliptic by [1, Section 4]. We are then in a position to apply [1, Theorems 14.4 & 14.6] to conclude that (1)–(4) has a unique maximal classical solution ¯ R2 ) ∩ C 1,2 ((0, T ) × Ω; ¯ R2 ) , z := (v, u) ∈ C([0, T ) × Ω; where T ∈ (0, +∞]. In addition, thanks to (6) and (7), we have a1,2 (0, y2 ) = 0, a2,1 (y1 , 0) = −h(0) = 0 and f 1 (0, y2 ) = g(y2 , 0) ≥ 0. These properties along with (8) and [1, Theorem 15.1] ensure that v(t, x) ≥ 0 and u(t, x) ≥ 0 for (t, x) ∈ ¯ We next observe that z˜ := (v, 1 − u) is a solution to a similar initial[0, T ) × Ω. boundary value problem with 1 0 for y = (y1 , y2 ) ∈ D0 . a ˜(y) = (˜ am,n )1≤m,n≤2 := −h(1 − y2 ) D(1 − y2 ) Since a ˜2,1 (y1 , 0) = −h(1) = 0, a further application of [1, Theorem 15.1] and (8) ¯ So far, we have thus proved that yield that 1 − u(t, x) ≥ 0 for (t, x) ∈ [0, T ) × Ω. 0 ≤ u(t, x) ≤ 1

and

0 ≤ v(t, x)

(18)

¯ Also, integrating (1) over (0, t) × Ω implies that (16) holds for (t, x) ∈ [0, T ) × Ω. true for t ∈ [0, T ). We ﬁnally infer from (2), (7) and (18) that ∂t v − ∆v ≤ g(u, 0) + κ v ≤ C (1 + v) with C := g(., 0)L∞ (0,1) + κ, from which we conclude that ¯. v(t, x) ≤ (1 + v0 L∞ ) eCt , (t, x) ∈ [0, T ) × Ω We are then in a position to use [1, Theorem 15.5] to conclude that T = +∞. Recalling (18), the proof of Theorem 3 is complete.

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Proof of Theorem 2 (Existence). For ε ∈ (0, 1) and r ∈ R, we put Dε (r) := D(r)+ε. Also, there exist (uε0 , v0ε ) ∈ W 1,N +1 (Ω; R2 ) satisfying (8) and v0ε L∞ ≤ 2 v0 L∞

and

uε0 − u0 L2 + v0ε − v0 L2 ≤ ε .

(19)

By Theorem 3, the initial-boundary value problem (1)–(4) with Dε instead of D and (uε0 , v0ε ) instead of (u0 , v0 ) has a unique classical solution (uε , v ε ) such that 0 ≤ uε (t, x) ≤ 1

and

0 ≤ v ε (t, x) ,

¯. (t, x) ∈ [0, +∞) × Ω

(20)

In the following we denote by Ci , i ≥ 1, any positive constant which does not depend on ε ∈ (0, 1). The dependence of the Ci ’s upon additional parameters will be indicated explicitly. We next proceed as in the last step of the proof of Theorem 3 to show that, for each T > 0, ¯. 0 ≤ v ε (t, x) ≤ C1 (T ) , (t, x) ∈ [0, T ] × Ω

(21)

Consequently, the right-hand side of (2) is bounded on ﬁnite time intervals and classical parabolic estimates warrant that, for each T > 0 and p ∈ (1, ∞), there is a positive constant C2 (p, T ) such that v ε Lp (0,T ;W 2,p (Ω)) + ∂t v ε Lp ((0,T )×Ω) ≤ C2 (p, T ) .

(22)

˜ ε deﬁned by We next introduce the functions Dε and D ˜ε d2 D dDε ˜ ε (0) = Dε (0) = 0 . = Dε and D = 2 dr dr We multiply (1) by Dε (uε ) and integrate over Ω to obtain d ε ε ˜ D (u ) dx = − ∇Dε (uε ) · [∇Dε (uε ) − h(uε ) ∇v ε ] dx dt Ω Ω

whence

2

≤

− ∇Dε (uε )L2 + hL∞ (0,1) ∇Dε (uε )L2 ∇v ε L2

≤

−

h2L∞ (0,1) 1 2 2 ∇Dε (uε )L2 + ∇v ε L2 , 2 2

T

2

|∇Dε (uε )| dxdt ≤ C3 (T ) 0

(23)

Ω

by (22). Recalling (1), we deduce from (6), (20), (22) and (23) that T 2 ∂t uε H 1 (Ω) dt ≤ C4 (T ) .

(24)

0

Also, the non-negativity of D implies that (dDε /dr) ≥ D, which, together with (23), implies that T 2 |∇D(uε )| dxdt ≤ C3 (T ) . (25) 0

Ω

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We now deﬁne the function P ∈ C 2 (R) by P (0) = 0 and dP/dr := D2 and claim that (P (uε )) is bounded in w ∈ L2 (0, T ; H 1 (Ω)) , ∂t w ∈ L1 (0, T ; W 1,N +1 (Ω) ) (26) for each T > 0. Indeed, on the one hand, it follows from (6), (20) and (25) that P (uε )L∞ ≤ P L∞ (0,1) , and

T 0

∇P (uε )2L2 dt ≤ D2L∞ (0,1)

T 0

∇D(uε )2L2 dt ≤ C5 (T ) .

On the other hand, we infer from (6) and the continuous embedding of W 1,N +1 (Ω) in L∞ (Ω) that, for ϕ ∈ W 1,N +1 (Ω) and t ∈ (0, T ), ϕ ∂t P (uε )(t) dx ≤ ∂t uε (t) 1 D(uε (t))2 ϕ 1 H (Ω) H (Ω) Ω

≤ ∂t uε (t)H 1 (Ω) D2L∞ (0,1) ϕH 1 dD ε + 2 ∂t u (t)H 1 (Ω) ∇D(uε (t))L2 ϕL∞ dr ∞ L

(0,1)

≤ C6 ∂t uε (t)H 1 (Ω) (1 + ∇D(uε (t))L2 ) ϕW 1,N +1 . and the right-hand side of the above estimate belongs to L1 (0, T ) by (24) and (25). Thanks to (26), we may apply a classical compactness result [5, Corollary 4] to conclude that (P (uε )) is relatively compact in L2 ((0, T ) × Ω) for each T > 0. Since P is increasing, the relative compactness of (P (uε )) in L2 ((0, T ) × Ω) and the bound (20) allow us to deduce that (uε ) is relatively compact in Lp ((0, T ) × Ω) for each p ∈ [1, ∞) and T > 0. Also, (20), (24) and a classical compactness result [5, Th´eor`eme 1.12.1] ensure that (uε ) is relatively compact in C([0, T ]; H 1 (Ω) ) for each T > 0. Recalling (22), we conclude from the above analysis that there are functions (u, v) ∈ L∞ ((0, T ) × Ω; R2 ) and a subsequence of (uε , v ε ) (not relabeled) such that (uε , v ε ) −→ (u, v) in Lp ((0, T ) × Ω; R2 ) ,

(27)

(uε , v ε ) −→ (u, v) in C([0, T ]; H 1(Ω) ) × C([0, T ]; Lp (Ω)) ,

(28)

and for each p ∈ [1, ∞) and T > 0. Furthermore, (20) and (27) warrant that (u(t, x), v(t, x)) ∈ [0, 1] × [0, +∞) for almost every (t, x) ∈ [0, +∞) × Ω, while (25) ensures that D(u) belongs to L2 (0, T ; H 1(Ω)) for each T > 0. Recalling (19) and the deﬁnition of Dε , the convergences (27), (28) and classical arguments allow us to show that (u, v) is actually a solution to (1)–(4) in the sense of Deﬁnition 1 and complete the proof of the existence part of Theorem 2.

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2.2. Uniqueness In this section, we prove the uniqueness statement of Theorem 2 under the additional assumptions (9) and (10) on D, h and g. The proof relies on a classical duality technique: introducing the subset L20 (Ω) of L2 (Ω) deﬁned by L20 (Ω) := w ∈ L2 (Ω) , w(x) dx = 0 , Ω

we denote by N w the unique solution to −∆N w

= w in Ω ,

(29)

∂ν N w

= 0 on ∂Ω ,

(30)

in H 2 (Ω) ∩ L20 (Ω). Proof of Theorem 2 (Uniqueness). Let (u, v) and (ˆ u, vˆ) be two solutions to (1)–(4) in the sense of Deﬁnition 1 and ﬁx T > 0. We put U (t, x) := u(t, x) − u ˆ(t, x) ,

V (t, x) := v(t, x) − vˆ(t, x) ,

for (t, x) ∈ [0, T ] × Ω. Then U (t) ∈ L20 (Ω) for each t ∈ [0, T ] and it follows from Deﬁnition 1 and (14) that t ∂t U, ψ ds (31) 0 t [∇ (D(u) − D(ˆ u)) − (h(u) − h(ˆ u)) ∇v − h(ˆ u) ∇V ] · ∇ψ dxds =− 0

Ω

for each ψ ∈ L (0, t; H 1 (Ω)). We may then take ψ = N U in (31) and obtain, for each δ ∈ (0, 1), t |∇N U (t)|2 dx = |∇N U (0)|2 dx + 2 ∂t U, N U ds 0 Ω Ω t ∆N U (D(u) − D(ˆ u)) dxds =2 0 Ω t (h(u) − h(ˆ u)) ∇N U · ∇v dxds +2 0 Ω t h(ˆ u) ∇N U · ∇V dxds +2 2

0

t ≤ −2

Ω

(u − u ˆ) (D(u) − D(ˆ u)) dxds 0

Ω t

∇vL∞ ∇N U L2 h(u) − h(ˆ u)L2 ds t ∇N U L2 ∇V L2 ds + 2 hL∞ (0,1) +2

0

0

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t (u − u ˆ) (D(u) − D(ˆ u)) dxds t 1 2 (h(u) − h(ˆ u)) dxds + ∇v2L∞ ∇N U 2L2 ds +δ δ 0 Ω 0 t t C 2 ∇V L2 ds + ∇N U 2L2 ds . +δ δ 0 0

≤ −2

0

Ω

t

It next follows from (2), (10) and the boundedness of u ˆ, v and vˆ on (0, T ) × Ω that 1 d V 2 2 + ∇V 2L2 2 dt L = V [g1 (v) − g1 (ˆ v) + u ˆ (g2 (v) − g2 (ˆ v ))] dx + g2 (v) U V dx Ω Ω ≤ C(T ) V 2L2 + ∇N U · ∇ (g2 (v) V ) dx Ω ≤ C(T ) V 2L2 + ∇V L2 ∇N U L2 + ∇vL∞ V L2 ∇N U L2 1 ≤ ∇V 2L2 + C(T ) V 2L2 + C(T ) 1 + ∇v2L∞ ∇N U 2L2 , 2 whence

t

V (t)2L2 +

∇V 2L2 ds 0 t t 1 + ∇v2L∞ ∇N U 2L2 ds . V 2L2 ds + C(T ) ≤ C(T ) 0

0

We now sum the previous two inequalities with δ ≤ min {1/4, 1/C0} ﬁxed and use (9) to deduce that V

(t)2L2

+

∇N U (t)2L2

t

≤ C(T )

1 + ∇v2L∞ V 2L2 + ∇N U 2L2 ds .

0

(32) As u and v are bounded in (0, T ) × Ω, classical parabolic regularity results and (2) imply that v ∈ Lp (0, T ; W 2,p (Ω)) for each p ∈ (1, ∞). By the Sobolev embedding, there is p ≥ 2 large enough such that ∇v ∈ L2 (0, T ; L∞ (Ω)). The Gronwall lemma and (32) then entail that V (t) = ∇N U (t) = 0 for every t ∈ [0, T ], which completes the proof. In the ﬁnal result of this section, we provide a suﬃcient condition on h and D which guarantees that (9) holds true. Proposition 4. If h D−1/2 ∈ L∞ (0, 1), then (9) holds true.

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Proof. For 0 ≤ r ≤ r ≤ 1, we have r |h (s)| ds ≤ C |h(r ) − h(r)| ≤ r

≤C

r

D1/2 (s) ds

r 1/2

r

D(s) ds

|r − r |1/2

r

≤ C |r − r |1/2 |D(r) − D(r )|

1/2

,

whence the claim.

In particular, D(r) = (1 − r)m and h(r) = r (1 − r)n fulﬁl (6) and (9) for m ≥ (n + 2)/2 and n ≥ 2.

3. Stationary states In this section, we assume that g(r, s) := γ r − β s

(33)

for some β > 0 and γ > 0. Deﬁnition 5. (i) A stationary solution to (1)–(3) is a couple (u, v) of functions in L∞ (Ω; R2 ) such that D(u) ∈ H 1 (Ω), v ∈ H 2 (Ω), (u(x), v(x)) ∈ [0, 1] × [0, +∞) a.e. in ∇D(u) − h(u) ∇v = 0 a.e. in Ω , −∆v + β v = γ u a.e. in Ω , ∂ν v = 0 a.e. on ∂Ω .

Ω,

(34) (35) (36) (37)

(ii) If M ∈ [0, |Ω|], a stationary solution to (1)–(3) with mass (M, γM/β) is a stationary solution (u, v) to (1)–(3) satisfying in addition β u(x) dx = v(x) dx = M . (38) γ Ω Ω Obviously, (u, v) = (0, 0) and (u, v) = (1, γ/β) are the only stationary solutions to (1)–(3) with mass (0, 0) and (|Ω|, γ |Ω|/β), respectively. Also, if m ∈ (0, 1), (m, γm/β) is a stationary solution to (1)–(3) with mass (m|Ω|, γm|Ω|/β). We next describe some properties of stationary solutions to (1)–(3) with mass (M, γM/β) for M ∈ (0, |Ω|). For further use, we introduce the auxiliary functions r r D(s) j(r) := ds and J(r) := j(s) ds , r ∈ (0, 1) , 1/2 h(s) 1/2 which enjoy the following property.

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Lemma 6. The function j is an increasing function from (0, 1) onto (−∞, j1 ) where 1 D(s) j1 := ds ∈ (0, +∞] . 1/2 h(s) Proof. The strict monotonicity of j follows at once from (6). We also infer from (6) that, for s ∈ (0, 1/2], min D D(s) C 1 [0,1/2] ≥ ≥ h(s) h L∞ (0,1) s s for some C > 0, so that D/h does not belong to L1 (0, 1/2).

Proposition 7. Let M ∈ (0, |Ω|) and consider a stationary solution (u, v) to (1)– (3) with mass (M, γM/β). Then Ωu := {x ∈ Ω , u(x) ∈ [0, 1)} is a non-empty open subset of Ω, j(u) − v is constant on each connected component of Ωu and ¯ ∩ C 1 (Ωu ). In addition, u(x) > 0 for x ∈ Ω ¯ and u ∈ C(Ω)

γ ¯. v(x) ∈ 0, , x∈Ω (39) β Proof. Since u ∈ L∞ (Ω), we infer from (36) and (37) that v ∈ W 2,p (Ω) for every ¯ which, together p ∈ (1, ∞). Choosing p large enough, we realize that ∇v ∈ C(Ω), 1,∞ ¯ with (34) and (35), implies that D(u) ∈ W (Ω). Consequently, D(u) ∈ C(Ω) ¯ and, since D is increasing, we have also u ∈ C(Ω). A further use of (35) then yields ¯ Therefore, recalling that u ≥ 0, we deduce that Ωu is an open that D(u) ∈ C 1 (Ω). subset of Ω (which is non-empty by (34) since M < |Ω|) and (6) ensures that u ∈ C 1 (Ωu ) with ∇D(u) = D(u) ∇u on Ωu . Now, let Λ be a connected component of {x ∈ Ω , u(x) ∈ (0, 1)}. For x ∈ Λ, (35) also reads ∇ (j(u) − v) (x) = 0. Therefore, there is λ ∈ R such that j(u(x)) = v(x) + λ for each x ∈ Λ. Thanks to the monotonicity of j, we have in fact u(x) = j −1 (v(x) + λ) ≥ j −1 (λ) > 0 for x ∈ Λ. The continuity of u then entails that ¯ Consequently, u(x) > 0 for x ∈ Ω ¯ and Ωu = {x ∈ u(x) ≥ j −1 (λ) > 0 for x ∈ Λ. Ω , u(x) ∈ (0, 1)}. It then follows from (35) that ∇ (j(u) − v) = 0 on Ωu . Finally, the bounds (39) are straightforward consequence of (36) and (37) by the comparison principle. 3.1. The non-degenerate case: D/h ∈ L1 (1/2, 1) We next derive some upper and lower bounds for stationary solutions which generalize [10, Proposition 1.4]. Proposition 8. Let M ∈ (0, |Ω|) and consider a stationary solution (u, v) to (1)–(3) with mass (M, γM/β). Assume further that j1 = +∞. Then M M γ γ ¯. j −1 j − ≤ u(x) ≤ j −1 j + , x∈Ω |Ω| β |Ω| β ¯ In particular, u(x) ∈ (0, 1) for x ∈ Ω.

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Proof. Let Λ be a connected component of Ωu . By Proposition 7, there is λ ∈ R such that j(u(x)) = v(x) + λ for x ∈ Λ . (40) Since j1 = +∞, we infer from (39) that u(x) ≤ j −1 ((γ/β) + λ) < 1, from which we conclude that Λ = Ωu = Ω by the continuity of u. Therefore,

γ +λ , M= u(x) dx = j −1 (v(x) + λ) dx ∈ |Ω| j −1 (λ), |Ω| j −1 β Ω Ω whence

M M γ λ∈ j − ,j . |Ω| β |Ω| The claimed bounds then follow from (39), (40) and the above estimates for λ. Remark 9. When D and h are related by (5), we have j(r) = ln (2r) − ln (q(r)) + ln (q(1/2)) and D/h ∈ L1 (1/2, 1). 3.2. The degenerate case: D/h ∈ L1 (1/2, 1) In that case, j1 < +∞. In contrast to the previous case where we have shown that the ﬁrst component u of any steady state to (1)–(3) cannot reach the threshold value 1 (unless u ≡ 1), we shall prove in this subsection that there are stationary solutions (u, v) to (1)–(3) such that u = 1 on some subset of Ω with non-empty interior, so that the graph of u is ﬂat-hump-shaped. To this end, we recall that, by Proposition 7, j(u) − v is constant on each connected component of Ωu = {x ∈ Ω , u(x) ∈ [0, 1)} and look for a stationary solution (u, v) such that the constant does not depend on the connected component of Ωu . Introducing f := j −1 and fλ (.) := f (. + λ) for λ ∈ R, we thus look for a stationary solution (u, v) to (1)–(3) such that there is λ ∈ R for which u(x) = fλ (v(x)) for x ∈ Ωu . Since u(x) = 1 for x ∈ Ω \ Ωu , we actually have u(x) = Fλ (v(x)) where

Fλ (y) =

fλ (y) 1

for for

¯, for x ∈ Ω

(41)

y ≤ vλ := j1 − λ , y > vλ .

(42)

Next, (36) and (41) yield that −∆v = γFλ (v) − βv

in Ω

(43)

with no-ﬂux boundary conditions on ∂Ω. Observe that the continuity of v established in Proposition 7 and (43) imply that −∆v v

= γ − βv in Ω0 := Ω \ Ω¯u , = vλ on ∂Ω0 .

(44) (45)

We ﬁnally recall that (39) warrants that vλ ≤ γ/β .

(46)

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From now on, we restrict ourselves to the one-dimensional case Ω = (0, l) and provide a suﬃcient condition for the existence of a ﬂat-hump-shaped stationary solution to (1)–(3). To this end, we say that the triple (λ , γ , β) ∈ R × (0, +∞)2 satisﬁes the condition (C) if ⎧ There are −1 , 0 , 1 = γ/β such that −1 < 0 < vλ < 1 and ⎪ ⎪ ⎪ ⎪ ⎪ β ⎪ ⎪ i , i ∈ {−1 , 0 , 1} , Fλ (i ) = ⎪ ⎪ γ ⎨ (C) β ⎪ y , y ∈ (0 , 1 ) , Fλ (y) > ⎪ ⎪ γ ⎪ ⎪ ⎪ ⎪ β ⎪ ⎪ ⎩ y , y ∈ (−1 , 0 ) , Fλ (y) < γ Theorem 10. Suppose that there is a triple (λ , γ , β) which satisﬁes the condition (C) together with vλ 1 β vλ + −1 . (47) fλ (y) dy < vλ − −1 −1 γ 2 Then, for l large enough, there exists a ﬂat-hump-shaped stationary solution (u, v) to (1)–(3) in Ω = (0 , l). More precisely, there exists x1 ∈ (0, l/2) such that u(x) ∈ [0, 1) for x ∈ [0, x1 ) ∪ (l − x1 , l], u(x) = 1 for x ∈ [x1 , l − x1 ] and v(x) ≥ vλ for x ∈ [x1 , l − x1 ]. If additionally f ∈ L2 (j1 − δ , j1 )

for some δ > 0 ,

(48)

then u ∈ H (0, l). 1

Proof. Step 1. Let us ﬁrst consider the one-dimensional boundary-value problem v Φ(v)

= Φ(v) in (0 , l) , v (0) = v (l) = 0 , = −γ Fλ (v) + β v .

(49) (50)

Notice that the previous second-order ordinary diﬀerential equation is equivalent to the hamiltonian system v = w ,

w = Φ(v) ,

with “energy” E and “potential” U given by v 1 Φ(y)dy . E(v , w) = w2 + U(v) , U(v) = − 2 0

(51)

(52)

We denote by x −→ ϕ(x; v0 , w0 ) = (ϕ1 (x; v0 , w0 ) , ϕ2 (x; v0 , w0 )) the solution to (51) starting from (v0 , w0 ) at x = 0 and restrict the analysis of the phase portrait of (51) to the strip P = {(v , w) : v ∈ [−1 , 1 ]}. It follows from (C) that there are three equilibria Ai := (i , 0), i ∈ {−1, 0, 1} in P . Since Φ(v) > 0 for v ∈ (−1 , 0 ) and Φ(v) < 0 for v ∈ (0 , 1 ) by (C), we realize that U has a local minimum at 0 and U(0 ) = 0. Consequently, for c ∈ I := (0, min {U(−1 ) , U(1 )}), the level sets Ec = {(v , w) : E(v, w) = c} of E are closed curves, from which we deduce that

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any orbit starting from (v0 , 0) such that U(v0 ) ∈ I is periodic and surrounds the equilibrium A0 . Step 2. In order to satisfy the boundary conditions we seek a periodic solution (v, w) to (51) such that v(0) ∈ (−1 , 0 ), w(0) = 0, and which crosses the line lλ := {(v , w) : v = vλ }. Suppose that U(−1 ) > U(vλ ) ,

(53)

and notice that U(vλ ) < U(1 ) since vλ ∈ (0 , 1 ) and U is increasing on (0 , 1 ). Therefore, U(vλ ) ∈ I and there exists v0 ∈ (−1 , 0 ) such that U(v0 ) ∈ I and U(v0 ) > U(vλ ). Owing to Step 1, the solution ϕ(.; v0 , 0) to (51) is periodic with period T (v0 ) and reaches lλ for some x = x1 . We then set l = T (v0 ) and deﬁne v(x)

=

ϕ1 (x; v0 , 0) for x ∈ [0 , l] ,

(54)

u(x) u(x)

= =

fλ (v(x)) for x ∈ [0 , x1 ) ∪ (l − x1 , l] , 1 for x ∈ [x1 , l − x1 ] .

(55) (56)

It is then straightforward to check that the couple (u, v) thus constructed is a stationary solution to (1)–(3) in the sense of Deﬁnition 5 and that v ∈ C 2 ([0 , l]) and u ∈ C([0 , l]). These two facts actually imply that D(u) ∈ C 1 ([0 , l]). Notice further that v satisﬁes vxx (x) = v(x1 ) =

−γ + βv(x) for x ∈ (x1 , l − x1 ) , v(l − x1 ) = vλ .

It remains to check (53). It follows from (50) that (53) also reads −1 vλ β 2 β 2 (vλ − 20 ) < ( − 20 ) , Fλ (y)dy − Fλ (y)dy − 2γ 2γ −1 0 0 which is equivalent to (47). We ﬁnally assume that f enjoys the additional property (48). Owing to (54), (55) and (56), to show that ux ∈ L2 (0 , l) it is suﬃcient to prove that ux ∈ L2 (x1 − ε , x1 ) and ux ∈ L2 (l − x1 − ε , l − x1 )

(57)

for ε small enough. Since vx (x) > 0 for x ∈ [x1 − δ , x1 ] we infer from (55) that x1 x1 |ux (x)|2 dx = |f (v(x) + λ)|2 (vx (x))2 dx x1 −ε

x1 −ε v(x1 )+λ

=

v(x1 −ε)+λ

|f (z)|2 vx (v −1 (z − λ)) dz ,

and the ﬁrst assertion of (57) readily follows from (48). A similar argument provides the second assertion of (57) and completes the proof.

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Remark 11. (i) It clearly follows from the proof that, for λ , γ , β satisfying the assumptions of Theorem 10, there exists a unique (singular) solution such that u is spikeshaped (it suﬃces to choose the unique v0 ∈ (−1 , 0 ) such that U(v0 ) = U(vλ ) in the second step of the proof of Theorem 10). In that case Ωu = (0, l)\{l/2}. (ii) It is possible to merge a ﬁnite number of ﬂat-hump-shaped stationary solutions to obtain a multi-hump solution on a suitably enlarged interval. (iii) It is possible to modify the proof of Theorem 10 to construct a stationary hump-shaped solution which is not symmetric with respect to l/2. To this end one should choose two diﬀerent values of λ on each connected component of the set Ωu . Proceeding as in the proof of Theorem 10 one can construct separately a part of the solution deﬁned on a subinterval containing 0 and another part on a subinterval containing l. Then one can merge both in such a way that, at a merging point x0 , we have lim vx (x) = 0 = lim vx (x)

x→x− 0

x→x+ 0

and there is a possible jump discontinuity of vxx at x0 . We now specify conditions under which Theorem 10 can be applied. We ﬁrst show that the condition (C) is valid under a simple condition on j or f near j1 . Proposition 12. If lim f (y) >

y→j1−

β , γ

(58)

then there is ε > 0 such that the triple (λ , β , γ) satisﬁes the condition (C) for any λ ∈ (j1 − γ/β , j1 + ε − γ/β). Proof. We put λ = j1 − γ/β, so that vλ = γ/β. Since lim f (y) = 0 ,

y→−∞

f (0) =

1 , 2

and fλ (vλ ) = 1, the graph of y → Fλ (y) has at least one intersection point S = (λ , Fλ (λ )) satisfying λ < vλ with the straight line y → (β/γ)y. Suppose that S is the closest intersection point with respect to R = (γ/β , 1) . If S is a transversal intersection point it is suﬃcient to shift the graph of y → Fλ (y) to the left by a small λ ∈ (0 , ε) and then (λ , γ , β) satisﬁes (C) with λ ∈ (λ , λ + ε) . If S is not a transversal intersection point, using the fact that fλ ∈ C 1 (−∞ , j1 ), a small shift of the graph to the right creates a transversal intersection in the vicinity of S. Then one proceeds as in the previous case. Proposition 13. Consider (β, γ) ∈ (0, +∞) × (0, +∞). Assume that D/h has a limit j2 in [0, +∞] as r → 1 such that γ (59) j2 > , β

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and that (D/h) (r) < 0 for every r ∈ [0, 1) such that (D/h) (r) = 0. Introducing γ r0 := sup r ∈ (0, 1) , j (r) = , (60) β the triple (λ, γ, β) satisﬁes the condition (C) if γ γ j1 − < λ < j(r0 ) − r0 . β β

(61)

Proof. For r ∈ (0, 1) and λ ∈ R, we put γ ψ(r) := r − j(r) and ψλ (r) := ψ(r) + λ . β Since ψ (r) = (γ/β) − (D/h)(r), the assumptions (6) and (59) guarantee that γ lim ψ (r) = − j2 > 0 and lim ψ (r) = −∞ . r→1 r→0 β Consequently, ψ has a least one zero in (0, 1) and we denote by r0 the largest zero of ψ . Owing to (59), we clearly have r0 < 1 and ψ (r) > 0 for r ∈ (r0 , 1), which implies that ψ (r0 ) ≥ 0. Either ψ (r0 ) > 0 and there is ε > 0 such that ψ (r) < 0 for r ∈ (r0 − ε, r0 ). Or ψ (r0 ) = 0 and the assumption in Proposition 13 ensures that ψ (r0 ) > 0. We therefore conclude that, in this case also, there is ε > 0 such that ψ (r) < 0 for r ∈ (r0 − ε, r0 ). Consider now λ ∈ (−ψ(1), −ψ(r0 )). Then ψλ (1) > 0 > ψλ (r0 ). Recalling that ψλ = ψ is positive in (r0 , 1), there is a unique s0 ∈ (r0 , 1) such that ψλ (s0 ) = 0, ψλ (r) > 0 for r ∈ (s0 , 1) and ψλ (r) < 0 for r ∈ [r0 , s0 ). Next, ψλ (r) → +∞ as r → 0 by Lemma 6. Consequently, ψλ has at least one zero in (0, r0 ) and we denote by s−1 ∈ (0, r0 ) the largest zero of ψλ in (0, r0 ). Then, ψλ (r) < 0 for r ∈ (s−1 , s0 ). Introducing i := (γ si )/β, i = −1, 0, and observing that β β r +j r , fλ (y) = f ψλ γ γ we easily see that the condition (C) is fulﬁlled.

We now check that there are situations in which Theorem 10 can be applied. More precisely, we assume that h(r) = r D(r) for r ∈ [0, 1]. In that particular case, we have j(r) = ln 2r for r ∈ (0, 1), j1 = ln 2, j2 = 1 and r0 = β/γ. On the one hand, by Proposition 13, the triple (λ, γ, β) fulﬁls the condition (C) if γ γ γ > 1 and ln 2 − < λ < ln 2 − ln −1. (62) β β β In addition, when (λ, γ, β) fulﬁls (62), a further study of the function y −→ f (y + λ) − (βy)/γ on (−∞, vλ ) shows that γ 0 < −1 < ln 2 − ln − λ. (63) β

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On the other hand, since h(r) = r D(r) for r ∈ [0, 1], we have vλ 1 β fλ (y) dy = r j (r) dr = 1 − −1 . γ −1 (β−1 )/γ Now assume that γ γ > 2 and λ = ln 2 − + δ , β β for some δ ∈ (0, (γ/β) − 2). Then vλ > 2 and we infer from (63) that vλ β 2 2 vλ − −1 − ν = fλ (y) dy 2γ −1

β vλ2 − 2−1 γ = − + −1 γ 2 β

β vλ − 2 γ (vλ − −1 ) − + vλ ≥ γ 2 β

γ β vλ − 2 γ ≥ vλ − ln 2 + ln + λ − + vλ γ 2 β β

γ β 1 γ ≥ − δ − 2 ln −δ γ 2 β β > 0 for δ small enough. Observe that the larger γ/β is, the larger δ can be. As a ﬁnal remark, let us point out that, in the case h(r) = r D(r), a ﬂat-humpshaped stationary solution (u, v) to (1)–(3) coincides with a stationary solution to the so-called minimal chemotaxis model in which D ≡ 1 and h(r) = r (cf. [3]) on the set {x : u(x) < 1}. Acknowledgments Partial support from the EU ﬁnanced network HYKE no. HPRN-CT-2002-00282 and the Universit´e de Toulouse I is gratefully acknowledged. Also, part of this work was done while both authors enjoyed the friendly atmosphere of the workshop “Nonlinear Elliptic and Parabolic Problems: A Special Tribute to the Work of Herbert Amann” held in Z¨ urich from June 28 to June 30, 2004.

References [1] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in “Function Spaces, Diﬀerential Operators and Nonlinear Analysis”, H. Triebel, H.J. Schmeisser (eds.), Teubner-Texte Math. 133, Teubner, Stuttgart, 1993, pp. 9–126. [2] T. Hillen and K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math. 26 (2001), 280–301.

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[3] D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I., Jahresber. Deutsch. Math.-Verein. 105 (2003) 103–165. [4] E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399–415. [5] J.-L. Lions, Quelques M´ethodes de R´ esolution des Probl` emes aux Limites non Lin´eaires, Dunod, Paris, 1969. [6] K. Painter and T. Hillen, Volume-ﬁlling and quorum-sensing in models for chemosensitive movement, Canadian Appl. Math. Q. 10 (2002), 501–543. [7] C.S. Patlak, Random walk with persistence and external bias, Bull. Math. Biol. Biophys. 15 (1953), 311–338. [8] J. Simon, Compact sets in the space Lp (0, T ; B), Ann. Mat. Pura Appl. 146 (1987), 65–96. [9] D. Wrzosek, Global attractor for a chemotaxis model with prevention of overcrowding, Nonlinear Anal. 59 (2004), 1293–1310. [10] D. Wrzosek, Long time behaviour of solutions to a chemotaxis model with volume ﬁlling eﬀect, Proc. Roy. Soc. Edinburgh Sect. A, to appear. Philippe Lauren¸cot Math´ematiques pour l’Industrie et la Physique CNRS UMR 5640 Universit´e Paul Sabatier–Toulouse 3 118 route de Narbonne F–31062 Toulouse cedex 4, France e-mail: [email protected] Dariusz Wrzosek Institute of Applied Mathematics and Mechanics Warsaw University Banacha 2 02-097 Warszawa, Poland e-mail: [email protected]

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 291–327 c 2005 Birkh¨ auser Verlag Basel/Switzerland

In the Blink of an Eye Juli´an L´opez-G´omez and Marcela Molina-Meyer Dedicated to H. Amann, with our deepest admiration, at the occasion of his retirement from Z¨ urich University

Abstract. The mathematical analysis of this paper shows how the eﬀects of strategic local symbiosis provide an exceptional mechanism to increase productivity in highly competitive environments. The most striking consequence from our analysis is that productivity can blow-up in cooperation areas, though some of the species might become extinct elsewhere, as a result of the aggressions received from competitors. As a by-product, it is realized why strategic symbiosis eﬀects help to avoid massive extinctions of populations, or industrial and ﬁnancial companies. Going beyond, it has been numerically observed that, in the presence of local strategic symbiosis, high level aggressions might provoke a substantial increment of the complexity of the system; a mechanism that might explain the extraordinary bio-diversity of Earth’s biosphere, as well as the complexity of Global Economy. Mathematics Subject Classiﬁcation (2000). 34A34, 34C23, 92B05. Keywords. Symbiosis in competitive environments, coexistence states, complexity, biodiversity.

1. Introduction 544 millions years ago there were 3 animal phyla with their corresponding variety of external forms, though at 538 millions years ago there were 38, the same number that exists today, except for one or two extinctions, Parker [44]. Consequently, the vast existing diversity of body architectures has emerged during a ﬁve-million-year interval beginning 543 millions year ago. Such a time interval is referred to in the specialized literature as the Cambrian explosion – the ﬁrst fossils from that period of time where found in the Cambrian Hills in Wales by the recognized geologist Sedgwick, who named it as the Cambrian. Cambrian explosion supports that the history of life has been governed by long time intervals of micro-evolution – gradual evolution – combined with short macro-evolution periods which have been the most proliﬁc generators of biodiversity, in contrast with the classical paradigm of

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J. L´opez-G´omez and M. Molina-Meyer

Darwin and Wallace, according with it evolution occurred gradually. Parker [44] defends the theory that Cambrian explosion took place as a result of a revolutionary event occurred in between 544 and 543 million-years-ago: in one-million-year interval most of skin energy receptors specialized into eyes; in the blink of an eye vision was born. Consequently, it seems that Cambrian explosion was extraordinarily facilitated by the explosion of all kind of mutual interactions between Precambrian animals as a natural consequence from vision. Quite strikingly, it seems that, indeed, predating and competition mechanisms are in the origin of the extraordinary diversity of the Earth biosphere. A reminiscent phenomenology has been observed as well in studying the diversity and productivity of grass-lands, Tilman [46]. Quite surprisingly, some species in competition have better productivity than any individual species in monoculture; a phenomenon usually referred to as overyielding which is far from well understood. Cambrian explosion might be regarded as a genuine overyielding phenomenon originated by the extraordinary increment of animal interactions as a result of eyes formation. Consequently, understanding overyielding seems to be imperative in order to explain biodiversity, complexity and productivity. This paper provides a possible mathematical approach to the study of these crucial problems by analyzing the eﬀects of local strategic symbiosis in the dynamics of competing species systems; a mechanism also adopted by commercial and industrial companies to avoid extinction and improve productivity, simultaneously generating complex commercial networks and global market diversity. Competition, as most ecologists employ the word, means the active demand by a number of individuals of the same species – intraspeciﬁc competition – or members of a number of species at the same trophic level – interspeciﬁc competition – for a common resource or requirement that is actually, or potentially, limiting, Wilson [47], Begon et al. [4]. It is commonly agreed that this deﬁnition is consistent with the general assumptions of the Lotka-Volterra equations, which still conform the basis of the mathematical theory of competition. Though non-spatial models are far from realistic, the spatial ones allowed to establish the range of validity of the competitive exclusion principle, [25], [8], one of the greatest paradigms of evolution, not well understood yet. Rather than extinction, in nature one observes segregation when the level of the aggressions blows-up, and, after appropriate periods of time, species diﬀerentiation, as a result of spatial and temporal heterogeneities. Our mathematical analysis reveals that overyielding is a direct consequence from the super-linear character of competition in the presence of strategic symbiosis, which might explain diversity through the existence of multiple stable coexistence states. These multiplicities might allow the species to evolve adopting diﬀerent proﬁles according to their inhabiting territories, though a most rigorous analysis of this issue would require a strong additional multidisciplinary eﬀort. Focusing attention into the mathematical contains of this paper, more than in the eventual scientiﬁc signiﬁcance of the underlying ideas, this paper is distributed as follows. In Section 2 we introduce the mathematical model. It consists

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of a generalized Lotka-Volterra system where the interaction coeﬃcients are assumed to change sign; so, it is far from being classiﬁed within the classical families of Lotka-Volterra models, where the interactions have constant sign. Section 3 analyzes the qualitative behavior of the semi-trivial positive solutions of the model. Section 4 ascertains the shape of the curves of neutral stability of the semi-trivial positive solutions. Section 5 recalls a series of fundamental results for the classical competition Lotka-Volterra model. Collecting those results is imperative to realize the signiﬁcance of the new mathematical results of this paper. Section 6 gives a general existence result concerning the coexistence states of the competition model in the presence of local symbiosis. Some of these results are reminiscent of well-known results for general superlinear indeﬁnite problems, while others are completely new. It is worth-mentioning that our model is a sort of superlinear indeﬁnite system and, hence, all known properties for those problems have a counterpart for the system. Section 7 analyzes the behavior of one of the species, say u, when the intrinsic growth rate of the competitor, say v, blows-up. Its main result shows that the species u must entirely segregate into the regions where it is free from the aggressions of v, arbitrarily increasing within the cooperation areas. As the classical Lotka-Volterra competition model with diﬀusion entails the extinction of u, our result shows how strategic symbiosis is an extraordinary mechanism to avoid extinction, increasing simultaneously productivity, even when the level of the aggressions within competition areas is arbitrarily high. This result might be a breakthrough from the point of view of the applications of the underlying theory to Population Dynamics and Macro-Economy. Finally, in Section 8 we provide an example where our numerical experiments show the existence of multiple stable coexistence states. Such kind of phenomenology is rather striking, since the single equation admits, at most, a unique stable positive solution. Consequently, it seems that competition is an extraordinary mechanism to generate multiple stable positive solutions. This feature should position to many multidisciplinary scientists in the right direction to realize why predating and competition are on the basis of the great diversity of Earth Biosphere, as well as on the foundations of the great complexity of Global Economy. Although we are dealing with general systems, without any kind of variational structure, where no global comparison techniques are available, the maximum principle and the nice properties derived from it, whose study was systematically carried out from the seminal results of Amann (cf. [1] and the list of references therein), still are applicable to make local comparisons which are on the basis of the mathematical analysis carried out overall this honoring paper. Consequently, comparison techniques might be imperative to explain why, in the blink of an eye, Biosphere diversity blown-up, and why, Markets globalization, facilitated by the development of new communications and technologies, is originating a drastic increment of Markets complexity, as well in the blink of an eye, or course!

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2. The mathematical model Throughout this paper, we will work with two species, u and v, interacting in a certain habitat Ω, which is modeled by a smooth bounded domain of RN , N ≥ 1, which is assumed to be divided into three regions, Ω+ , Ω− and Ω0 , in such a way that the species u and v compete within Ω+ , cooperate within Ω− , and are free from the action of the other within Ω0 . Figure 1 shows one among the multiple admissible conﬁgurations. Subsequently, to simplify the exposition, we shall always assume that, like in Figure 1, ¯0 ∪ Ω ¯ − ⊂ Ω and Ω ¯0 ∩ Ω ¯− = ∅ , Ω thought this condition is far from necessary for the validity of the analysis carried out in this paper. Also, though it is everywhere assumed that Ω− = ∅ = Ω0 , some of these regions might be empty.

Ω

Ω+ −

Ω

0

Figure 1. An admissible habitat. Then, our spatial heterogeneous model for analyzing the dynamics of the species u and v is the following: ⎧ ∂u 2 ⎪ ∂t − ∆u = λu − u − buv ⎪ in Ω × R+ , ⎨ ∂v 2 ∂t − ∆v = µv − v − cuv (2.1) u=v=0 on ∂Ω × R+ , ⎪ ⎪ ⎩ u(·, 0) = u0 ≥ 0 , v(·, 0) = v0 ≥ 0 , in Ω , where Ω is a bounded domain of RN , N ≥ 1, with boundary ∂Ω, of class C 2+ν for ¯ some ν ∈ (0, 1), ∆ is the Laplacian of RN , R+ := (0, ∞), λ, µ ∈ R, b, c ∈ C ν (Ω) 2,p are two functions changing sign in Ω, and u0 , v0 ∈ W0 (Ω), for some p > N/2. Typically, Ω represents the inhabiting area, u and v measure the densities of the species at x ∈ Ω after time t > 0, the Laplacian measures the dispersion rates of u

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and v overall Ω – they have been normalized to unity –, λ and µ are the intrinsic growth rates of u and v, respectively, and b(x), c(x) model the interaction patterns, and other spatial eﬀects, between u and v. The regions where b are c are positive (resp. negative) are the areas where competition (resp. cooperation) takes place, the regions where bc < 0 are the areas where some of the species preys on the other, and the regions where some of these coeﬃcients vanishes provide us with the areas where the corresponding species is free from the action of the other. Finally, u0 and v0 are the initial population densities. Our boundary conditions entail the habitat Ω to be surrounded by completely hostile areas. As we will think of (2.1) as a model for competition in the presence of strategic symbiosis, we will suppose Ω0 = Int b−1 (0) = Int c−1 (0) , Ω+ = b−1 (R+ ) = c−1 (R+ ) , Ω− = b−1 (−R+ ) = c−1 (−R+ ) . Consequently, Ω− is the region where cooperation takes place, Ω+ is the region where competition takes place, and Ω0 is the region where both species are free from the action of the other. It should be noted that, under our general assumptions, (2.1) is not a Lotka-Volterra model. Now, setting h+ := max{h, 0} ,

h− := h+ − h ,

¯ , h ∈ C(Ω)

we have that b = b+ − b− , ¯ + = supp b+ = supp c+ , Ω

c = c+ − c− , ¯ − = supp b− = supp c− , Ω

and Ω0 = Int [Ω \ (Ω+ ∪ Ω− )]. Moreover, b+ > 0, b− > 0, c+ > 0 and c− > 0. ¯ it is said that h > 0 if h ≥ 0 and h = 0. Throughout this paper, for any h ∈ C(Ω) ¯ it is said that h ) 0 if it lies in the interior of the cone of Also, given h ∈ C 1 (Ω), ¯ Under these assumptions, for each p > N/2 and u0 , positive functions of C 1 (Ω). 2,p v0 ∈ W0 (Ω), there exists a maximal existence time T = T (u0 , v0 , λ, µ, b, c) ∈ (0, ∞] such that (2.1) possesses a unique solution (u, v) := (u(x, t; u0 , v0 , λ, µ, b, c), v(x, t; u0 , v0 , λ, µ, b, c)) , satisfying ¯ × [0, T )) ∩ C 2+ν,1+ ν2 (Ω ¯ × (0, T )) u , v ∈ C(Ω and

lim u(·, t)C(Ω) =∞ ¯ + v(·, t)C(Ω) ¯ t↑T

if T < ∞ .

Moreover, by the parabolic maximum principle, for each t ∈ (0, T ), u(·, t) ) 0 if u0 > 0, while u = 0 if u0 = 0. By symmetry, v satisﬁes the same property. In

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general, T < ∞, since b and c change sign in Ω. Consequently, in the absence of the other, each species grows according to the logistic low ⎧ ∂w in Ω × (0, ∞) , ⎨ ∂t − ∆w = γw − w2 (2.2) w=0 on ∂Ω × (0, ∞) , ⎩ in Ω , w(·, 0) = w0 ≥ 0 , and, hence, limt↑∞ (u, v) is determined by the non-negative steady states of (2.1) with a vanishing component. Note that the steady states of (2.1) are the nonnegative solutions (u, v) of the system ⎧ ⎨ −∆u = λu − u2 − b(x)uv in Ω , −∆v = µv − v 2 − c(x)uv (2.3) ⎩ u=v=0 on ∂Ω . whose structure might be extraordinarily involved, since it depends upon the geometry of the habitat Ω and the spatial dimension, as well as on b and c, because, in a number of regimes, (2.3) exhibits a genuine super-linear behavior. As in the classical spatially homogeneous Lotka-Volterra models, u = λu − u2 − buv , (2.4) v = µv − v 2 − cuv , d where = dt and b, c ∈ R, in order to ascertain the dynamics of (2.1), it is imperative to analyze the existence, multiplicity and stability properties of all nonnegative solutions of (2.3). Consequently, at our present early stage, most of our eﬀort will be focused to analyze (2.3). Although under some appropriate circumstances the limiting proﬁles of the solutions of (2.1) are given by the metasolutions of (2.3) (e.g., [30]), here we restrict ourselves to study the classical solutions of (2.3). Besides (0, 0), Problem (2.3) exhibits three types of non-negative solutions. Those of the form (u, 0), or (0, v), often refereed to as the semi-trivial positive solutions, and the so called coexistence states, which are the solutions of the form (u, v) with both components positive. In any circumstance, as a consequence from the strong maximum principle, u ) 0 if u > 0, and v ) 0 if v > 0.

3. The curves of change of stability of the semi-trivial states Subsequently, for each smooth subdomain D ⊂ Ω and V ∈ L∞ (D), we shall denote by σ[−∆ + V ; D] the lowest eigenvalue of the linear boundary value problem (−∆ + V )w = τ w in D , (3.1) w|∂D = 0 , referred to as the principal eigenvalue of (3.1); it is algebraically simple, and its associated eigenfunction, ϕ, can be chosen to satisfy ϕ ) 0 in D – the principal

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eigenfunction –; σ[−∆ + V ; D] is the unique eigenvalue of (3.1) to a positive eigenfunction (cf. Amann [2]). Subsequently, we set σ := σ[−∆; Ω] . It is folklore that (2.3) possesses a semi-trivial positive solution of the form (u, 0) if, and only if, λ > σ. Moreover, if λ > σ and we denote by θγ the unique positive solution of −∆w = γw − w2 in Ω , (3.2) w|∂Ω = 0 , then, (θλ , 0) is the unique semi-trivial positive solution of the form (u, 0). Similarly, (2.3) possesses a semi-trivial positive solution of the form (0, v) if, and only if, µ > σ, and, in such case, (0, θµ ) is the unique one. Condition λ > σ measures the facilities (among them, the size and natural resources) that Ω should oﬀer to each of the species to maintain it in the absence of the other; e.g., in the absence of v, u is driven to extinction if λ ≤ σ. In this section we study the stability of the semi-trivial positive solutions of (2.3). To show the signiﬁcance of this analysis we will ﬁrst concentrate into the spatially homogeneous model (2.4) with λ, µ, b, c ∈ R+ , which is the classical competition Lotka-Volterra model. In such case, the semi-trivial positive solutions are (λ, 0) and (0, µ) and the coexistence state should be given by λ − bµ µ − cλ , (u, v) = (3.3) 1 − bc 1 − bc if it exists and, in addition, bc = 1. Regarding applications, the case bc = 1 is not of interest, since, generically, bc = 1. So, suppose bc = 1. Then, two diﬀerent situations must be distinguished, according to the size of bc: • 0 < bc < 1, referred to as the low level competition case. • bc > 1, refereed to as the high level competition case. In case bc < 1, (3.3) provides us with a coexistence state of (2.4) if, and only if, λ > bµ and µ > cλ ,

(3.4)

while, in case bc > 1, (3.3) is a coexistence state if, and only if, λ < bµ and µ < cλ .

(3.5)

As the matrices of the linearizations of (2.4) at (λ, 0) and (0, µ) are −λ −bλ λ − bµ 0 and , 0 µ − cλ −cµ −µ respectively, it is obvious that the straight lines µ = cλ

and λ = bµ

(3.6)

provide us with the curves of change of stability of the semi-trivial states (λ, 0) and (0, µ) in the (λ, µ)-parameter space. Indeed, (λ, 0) is a stable node if µ < cλ, while it is a saddle point if µ > cλ. Similarly, (0, µ) is stable if λ < bµ, while it is unstable if λ > bµ.

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Therefore: • In case bc < 1, (2.4) has a coexistence state (unique) if, and only if, (λ, 0) and (0, µ) are saddle points. • In case bc > 1, (2.4) has a coexistence state (unique) if, and only if, (λ, 0) and (0, µ) are stable nodes. Actually, it is folklore that in case bc < 1 the coexistence state is a global attractor for all positive solutions of (2.4), while, in case bc > 1, the coexistence state is a saddle point and, generically, there is extinction of some of the species, according to the relative size of the initial populations. Moreover, when one of the semi-trivial states is linearly stable and the other linearly unstable, then one of the species becomes extinct. This elementary analysis makes apparent the signiﬁcance of the curves of neutral stability of the semi-trivial positive solutions in ascertaining the dynamics of the positive solutions of (2.4). Now, we analyze the linearized stability of the semi-trivial positive solutions (θλ , 0) and (0, θµ ) of (2.3) as steady-states of (2.1). Suppose λ > σ. Then, the local attractive character of (θλ , 0) is given through the sign of the real parts of the eigenvalues τ of the linear eigenvalue problem ⎧ ⎨ −∆u = λu − 2θλ u − bθλ v + τ u in Ω , −∆v = µv − cθλ v + τ v (3.7) ⎩ u=v=0 on ∂Ω . By the linearized stability principle, (θλ , 0) is exponentially asymptotically stable if all those real parts are positive, while it is unstable if it has an eigenvalue with negative real part. Suppose τ is an eigenvalue of (3.7) with associated eigenfunction (u, v) = (0, 0). If v = 0, then (−∆ + 2θλ − λ)u = τ u

in Ω ,

u|∂Ω = 0 ,

and, since σ[−∆+2θλ −λ; Ω] > σ[−∆+θλ −λ; Ω] = 0, necessarily τ > 0. Therefore, such eigenvalues do not aﬀect the local behavior of (θλ , 0). If v = 0, then τ must be an eigenvalue of −∆ + cθλ − µ, and, hence, τ ∈ R and τ ≥ σ[−∆ + cθλ − µ; Ω] = σ[−∆ + cθλ ; Ω] − µ . Moreover, since σ[−∆ + cθλ ; Ω] − µ is the lowest one among these eigenvalues, the curve µ = σ[−∆ + cθλ ; Ω] [in the (λ, µ)-plane] provides us with the geometrical locus of the set of values (λ, µ) where (θλ , 0) is neutrally stable. Most precisely, (θλ , 0) is linearly stable (l.s.) if, and only if, µ < σ[−∆ + cθλ ; Ω], while (θλ , 0) is linearly unstable (l.u.) if, and only if, µ > σ[−∆ + cθλ ; Ω]. By symmetry, (0, θµ ) is l.s. if, and only if, λ < σ[−∆ + bθµ ; Ω], while it is l.u. if, and only if, λ > σ[−∆ + bθµ ; Ω]. Consequently, the curves µ = σ[−∆ + cθλ ; Ω] and λ = σ[−∆ + bθµ ; Ω]

(3.8)

will play a similar role, with respect to (2.1), as the curves µ = cλ and λ = bµ for its spatially homogeneous counterpart (2.4). These curves will be refereed to as the curves of change of stability, or neutral stability, of the semi-trivial solutions.

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4. Limiting behaviors of the curves of change of stability The following result will provide us with the limiting behavior at (λ, µ) = (σ, σ) and at inﬁnity of the curves (3.8). ¯ and set, for each γ > σ, Proposition 4.1. Suppose a ∈ C ν (Ω) f (γ) := σ[−∆ + a(x)θγ ; Ω]. Then, f is real analytic in (σ, ∞), and it admits the asymptotic expansion aϕ3 + O(|γ − σ|2 ) as γ ↓ σ , f (γ) = σ + (γ − σ) Ω

where ϕ ) 0 is the unique eigenfunction of σ satisfying

Ω

(4.1)

ϕ3 = 1. Moreover,

f (γ) = min a . ¯ γ↑∞ γ Ω lim

(4.2)

In particular, if we deﬁne f (σ) := σ, then f ∈ C([σ, ∞)). Furthermore, setting Ωa0 := Int a−1 (0) ,

Ωa− := Int a−1 (−R+ ) ,

one has that f (γ) ≤ σ[−∆; Ωa0 ∪ Ωa− ] ,

γ ≥ σ,

(4.3)

and, actually, lim f (γ) = σ[−∆; Ωa0 ]

if a ≥ 0 (i.e., Ωa− = ∅) and

γ↑∞ Ωa0 is

(4.4)

a nice smooth open subset of Ω.

¯ γ → θγ , is real analytic follows from Proof. The fact that the map (σ, ∞) → C ν (Ω), the implicit function theorem, since θγ is the unique positive solution of (3.2) and it is non-degenerate. By the simplicity of the principal eigenvalue it readily follows ¯ that f is real analytic in (σ, ∞). Moreover, since θγ → 0 as γ ↓ σ in C ν (Ω), limγ↓σ f (σ) = σ, and, hence, f ∈ C([σ, ∞)) if we deﬁne f (σ) := σ. The asymptotic expansion (4.1) follows from the main theorem of Crandall & Rabinowitz [9] by adapting the argument of the proof of [25, Lemma 3.6] (cf. [15, Lemma 4.3] as well). Now, note that for each γ > σ, 1 f (γ) = σ[− ∆ + a(x)ϑγ ; Ω] , γ γ where ϑγ := θγ /γ; ϑγ is the unique positive solution of 1 − ∆w = w − w2 γ

in Ω ,

w|∂Ω = 0 ,

and, hence, thanks to, e.g., [20, Theorem 3.4], limγ↑∞ ϑγ = 1, uniformly in compact subsets of Ω. Therefore, adapting the proof of [20, Lemma 3.1], (4.2) holds. Estimate (4.3) follows from the fact that a ≤ 0 in Ωa0 ∪ Ωa− . Indeed, for each γ ≥ σ, f (γ) = σ[−∆ + aθγ ; Ω] ≤ σ[−∆ + aθγ ; Ωa0 ∪ Ωa− ] ≤ σ[−∆; Ωa0 ∪ Ωa− ] . Finally, by [25, Theorem 3.4], (4.4) holds if a ≥ 0.

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In analyzing the behavior of the curves (3.8) by means of Proposition 4.1 one must take a ∈ {b, c}. In such case, a changes sign, and, hence, minΩ¯ a < 0, and the sign of the limiting slope limγ↓σ f (σ) = Ω aϕ3 at γ = σ depends upon the relative sizes of a+ and a− versus ϕ3 . In fact, there exists ε > 0 such that f (γ) is increasing in [σ, σ + ε] if Ω aϕ3 > 0, while it is decreasing if Ω aϕ3 < 0. Thus, the behavior of f (γ) at γ = σ depends, among other things, upon the sign of

local 3 aϕ . Also, by (4.2), the map γ → f (γ) is asymptotic, at γ = ∞, to the straight Ω line γ → minΩ¯ a · γ. Moreover, due to the estimate (4.3), within the quadrant min{λ, µ} > σ1 := σ[−∆; Ω0 ∪ Ω− ] ,

(4.5)

(θλ , 0) and (0, θµ ) are linearly unstable.

5. The classical competition model It is model (2.1) under the very special requirement that ¯. b(x) > 0 and c(x) > 0 for each x ∈ Ω

(5.1)

In particular, Ω− = Ω0 = ∅. This model has been extensively analyzed in [5], [10], [11], [39], [40], [12], [23] [41], [17], [19], [43], [27], [20], [13], [29], and the list of references therein. This section gives a short summary of some of the most important ﬁndings of these references; to illustrate the most striking diﬀerences between the classical model and (2.1). Suppose (5.1) and a ∈ {b, c}. Then, the function f (λ), γ ≥ σ, deﬁned in Proposition 4.1, in increasing in γ. Moreover, aϕ3 and γ → min a · γ (5.2) γ → σ + (γ − σ) Ω

¯ Ω

provide us with the tangents of f (γ) at γ = σ and at inﬁnity, respectively. The shape of the regions enclosed by the curves (3.8), where the attractiveness character of (θλ , 0) and (0, θµ ) is the same, is strongly dependent on the intensity of the competition, measured by the function bc. 5.1. Low level of aggressions According to the theory developed in Furter & L´ opez-G´omez [19] (going back to [39] and [41]), there exists a constant K1 ∈ (0, 1) such that the curves (3.8) cannot meet within the quadrant min{λ, µ} > σ if b∞ < 1 ,

c∞ < 1 ,

bc∞ < K1 .

(5.3)

Moreover, the region enclosed by the curves (3.8) consists of the values of (λ, µ) for which (θλ , 0) and (0, θµ ) are linearly unstable. In Figure 2 we have represented the curves (3.8) in the special case when b and c are positive constants. Except for the values of the slopes of the limiting tangents (5.2), these curves, in the general case when b and c are arbitrary functions satisfying (5.3), look very similar to those shown in Figure 2. As illustrated by Figure 2, the curves (3.8) divide the quadrant min{λ, µ} > σ in three regions. In Region 1, (θλ , 0) is l.u. and (0, θµ ) is l.s.; in Region 2, (θλ , 0)

In the Blink of an Eye µ

301

λ=σ+ b (µ−σ)

λ= b µ

1 λ=σ[−∆+ b θ ;Ω] µ

µ=σ+ c (λ−σ) 2

µ=c λ µ=σ[−∆+ c θ ;Ω] λ 3

(σ,σ) λ (0,0)

Figure 2. The curves (3.8) under condition (5.3).

and (0, θµ ) are, simultaneously, l.u.; and, in Region 3, (θλ , 0) is l.s. and (0, θµ ) is linearly unstable. By a result of Blat & Brown [5], (2.3) possesses a coexistence state for each (λ, µ) in Region 2. Actually, by a result of Hess & Lazer [23], (2.1) is compressive, or permanent, in Region 2, i.e., the ω-limit of any positive solution of (2.1) lies in the interior of the positive quadrant; with both components bounded away from zero. Further, by a result of L´ opez-G´omez & Sabina [43], for each (λ, µ) in Region 2, (2.3) possesses a unique coexistence state if bc∞ is suﬃciently small, which is a global attractor for all positive solutions of (2.1). As an eﬀect from dispersion, in strong contrast with the behavior exhibited by the spatially homogeneous model (2.4), (2.3) can exhibit a coexistence state within Regions 1 and 3. Actually, the set of values of (λ, µ) for which (2.3) possesses a coexistence can be strictly larger than the region enclosed by the curves (3.8). Figure 3 shows an admissible conﬁguration where such a situation occurs (see Eilbeck et al. [17] and Furter & L´ opez-G´omez [19], [20], for further details). In Figure 3, Region 1 is the region where (θλ , 0) is l.u., (0, θµ ) is l.s., and (2.3) does not exhibit a coexistence state. Similarly, Region 5 is the region where (θλ , 0) is l.s., (0, θµ ) is l.u., and (2.3) does not exhibit a coexistence state. Thanks to a result attributable to Dancer [12], in Region 1 (0, θµ ) is a global attractor for all positive solutions of (2.1), while in Region 5 (θλ , 0) attracts to all positive solutions of (2.1). For each pair of values (λ, µ) in the interior of Regions 2 and 4, (2.3) possesses at least two coexistence states. Actually, as a result of the theory of monotone dynamical systems, (2.3) possesses a minimal and a maximal coexistence

302

J. L´opez-G´omez and M. Molina-Meyer λ=σ[−∆+ b θµ ;Ω]

µ

1

3 2

µ=σ[−∆+ c θλ;Ω]

Permanence wedge

4 5

(σ,σ) λ (0,0)

Figure 3. Most relevant regions of the quadrant min{λ, µ} > σ.

state; one of them stable and the other unstable. (cf. Dancer [12] and L´opez-G´omez [30]). Finally, Region 3 in Figure 3 stands for the region where (2.1) is permanent. Further, by the a priori bounds of Furter & L´ opez-G´omez [19] and L´ opezG´ omez & Sabina [43], for each λ > σ, there exists µ2 = µ2 (λ) such that (2.3) does not admit a coexistence state if µ > µ2 . Therefore, (0, θµ ) attracts to all positive solutions of (2.1) if µ > µ2 .

(5.4)

By symmetry, for each µ > σ, there exists λ2 = λ2 (µ) such that (2.3) does not admit a coexistence state if λ > λ2 , and, hence, (θλ , 0) attracts to all positive solutions of (2.1) if λ > λ2 .

(5.5)

Consequently, u (resp. v) is driven to extinction by v (resp. u) if µ (resp. λ) is suﬃciently large, which is a rather natural result from the point of view of Population Dynamics, in complete agreement with the behavior exhibited by the classical non-spatial model (2.4). Figure 4 shows a genuine bifurcation diagram of non-negative solutions of (2.3), where λ > σ has been ﬁxed and µ has been regarded as the main bifurcation parameter. By the celebrated local bifurcation theorem of Crandall & Rabinowitz [9], the curves (3.8) provide us with the set of values of (λ, µ) for which a curve of coexistence states bifurcates from the semi-trivial positive solutions.

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max u + max v (0,θ µ )

(θ ,0) λ

σ

µ

1

µ(λ)

µ∗

µ

µ 2

Figure 4. A genuine bifurcation diagram of non-negative steady-states. In Figure 4, we have denoted µ(λ) := σ[−∆ + cθλ ; Ω] ,

(5.6) ∗

the value of µ where the stability of (θλ , 0) is lost, while µ stands for the unique value of µ for which (5.7) λ = σ[−∆ + bθµ∗ ; Ω] , where (0, θµ ) becomes stable. Thanks to L´opez-G´omez [29, Theorem 7.2.2], (2.3) possesses a continuum of coexistence states linking the states (µ(λ), θλ , 0) and (µ∗ , 0, θµ∗ ). Incidentally, it should be noted that the unilateral global theorems of Rabinowitz [45] cannot be applied to get these results as they are wrong in the way they are stated (cf. the recent counterexample of Dancer [14]; prompted from the discussion carried out in L´ opez-G´omez [29]). In Figure 4, dashed lines represent curves ﬁlled in by unstable solutions. If we assume that Figure 4 shows all the non-negative solutions of (2.3), then, it is apparent that the following occurs: • (θλ , 0) is a global attractor for all positive solutions of (2.1) if µ < µ1 , where µ1 ≤ µ(λ) is the minimal value of µ for which (2.3) has a coexistence state. • For each µ ∈ (µ1 , µ(λ)) ∪ (µ∗ , µ2 ) – it might be empty –, (2.3) possesses two coexistence states, at least, one of them stable and another unstable, though the ﬁne topological structure of the continuum of coexistence states might be quite involved. • For each µ ∈ (µ(λ), µ∗ ), (2.3) has a coexistence state and (2.1) is permanent. • For each µ > µ2 , (0, θµ ) attracts to all positive solutions of (2.1). 5.2. Intermediate level of aggressions As b and c grow and bc crosses the critical value 1, then the problem of analyzing the ﬁne structure of the region enclosed by the curves (3.8), and, in particular,

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ﬁnding out the cardinal of its components, becomes extraordinarily more involved, since these curves meet in the interior of the quadrant min{λ, µ} > σ. Indeed, Eilbeck et al. [17] found that, if b and c are positive constants such that bc > 1

and bc − 1 ∼ 0 ,

(5.8)

then the curves (3.8) cross, in dramatic contrast with the situation exhibited by the model under low level of aggressions. Figure 5 shows one admissible situation. The crossing point near (σ, σ) perturbs from (σ, σ) as bc − 1 > 0 separates from 0. The other crossing point perturbs from inﬁnity as a result of the change of the relative positions between the curves (3.8) (cf. [27]). µ=σ[−∆+ cθ λ ;Ω]

µ

1

5 λ=σ[−∆+ b θµ;Ω] Permanence

4 3

Unstable coexistence 6

2 (σ,σ) λ (0,0)

Figure 5. Relevant (λ, µ)-regions for an intermediate level of aggressions. Now, in Region 1 (0, θµ ) is a global attractor for the positive solutions of (2.1). In Regions 2 and 5, (θλ , 0) and (0, θµ ) are simultaneously l.s. and (2.1) admits an unstable coexistence state. In Region 3, (0, θµ ) is l.s., (θλ , 0) is l.u., and the model exhibits, at least, two coexistence states, one stable and another unstable. In Region 4, (2.3) has a coexistence state and (2.1) is permanent. Finally, in Region 6, (θλ , 0) is a global attractor for the positive solutions of (2.1). At the crossing points of the curves (3.8), where both semi-trivial positive solutions are linearly neutrally stable, (2.3) might not admit a coexistence state as it has been proven by Dancer [13], by sharpening the theory developed by Eilbeck et al. [17] and Furter & L´ opez-G´omez [19]. Dancer’s example has shown that a

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number of published results in the literature are completely false, as a consequence from a wrong use of the topological degree (cf. [13] for further details). In the presence of spatial heterogeneities, i.e., in the general case when b and c are arbitrary positive functions satisfying (5.8), Furter & L´opez-G´omez [19] developed a number of strategies to construct examples where the curves (3.8) cross at an arbitrarily large number of points, by using singularity theory techniques at the co-dimension two singularity (σ, σ). 5.3. Severe aggressions This is the case when bc is suﬃciently large, separated away from 1. µ=σ[−∆+ c θ ;Ω] λ

µ

1 Unstable coexistence state 3 2 λ=σ[−∆+ b θµ;Ω]

4 5

(σ,σ) λ (0,0)

Figure 6. Relevant (λ, µ)-regions for severe aggressions. Now, the qualitative behavior of (2.1) changes drastically, as in Region 3, instead of permanence, (2.1) may exhibit extinction of some of the species, u or v, according to the relative size of each of the initial populations. Note that, in all circumstances analyzed in this section, (5.4) and (5.5) remain valid. Therefore, for any λ, u is driven to extinction by v if µ – the intrinsic growth rate of v – is suﬃciently large, while, for any µ, v is driven o extinction by u if λ – the intrinsic growth rate of u – is suﬃciently large. As a result of local symbiosis eﬀects, our model does not always exhibit such a behavior, as it will become apparent later.

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6. The general competitive-symbiotic prototype model Throughout this section we work under the general assumptions of Section 2. In particular, Ω+ , Ω− and Ω0 are non-empty and b, c satisfy the sign requirements ¯ we denote of Section 2. Subsequently, for any a ∈ C(Ω) aL := min a , ¯ Ω

aM := max a . ¯ Ω

(6.1)

Then, thanks to Proposition 4.1, the curves (3.8) look like shows Figure 7. Although they might cross as many times as we wish in a neighborhood of (σ, σ), this is far from being important in our subsequent analysis. Note that if b− ↓ 0 and c− ↓ 0, then, on any compact subset of the quadrant min{λ, µ} ≥ σ, (3.8) must approximate uniformly to the curves of change of stability of the semi-trivial positive solutions of the associated competition model ⎧ ⎨ −∆u = λu − u2 − b+ (x)uv in Ω , −∆v = µv − v 2 − c+ (x)uv (6.2) ⎩ u=v=0 on ∂Ω , as a result of the continuous dependence of the principal eigenvalues with respect to their potentials. This suggests that the symbiotic eﬀects of (2.1) should play their most signiﬁcant role for suﬃciently large intrinsic growth rates. Indeed, by (4.2), µ = cL λ and λ = bL µ are the tangents at inﬁnity of the curves µ = σ[−∆ + cθλ ; Ω] and λ = σ[−∆ + bθµ ; Ω], respectively, and, due to (4.3), the whole quadrant min{λ, µ} > σ1 := σ[−∆; Ω0 ∪ Ω− ]

(6.3)

entirely consists of values of (λ, µ) where (θλ , 0) and (0, θµ ) are l.u.; this is a completely new phenomenology, inherent to the model we are dealing with, which cannot be covered by the classical competition model revisited in Section 5. Note that limmax{|Ω0 |,|Ω− |}↓0 σ1 = ∞ (cf. [26]), where | · | stands for the Lebesgue measure of RN . Subsequently, we set λs := max σ[−∆ + bθµ ; Ω] , µ>σ

µs := max σ[−∆ + cθλ ; Ω] . λ>σ

(6.4)

Note that max{λs , µs } < σ1 . As it will become clear later, (2.3) exhibits a genuine competitive behavior within the box delimited by (σ, σ) and (λs , µs ), while it inherits a symbiotic behavior within λ > λs , µ > µs . Indeed, thanks to [29, Theorem 7.2.2], if we ﬁx λ > λs and regard to µ as the main bifurcation parameter, then there is a component C(µ,θλ ,0) of the set of coexistence states of (2.3) that emanates from the semi-trivial state (θλ , 0) at µ = σ[−∆ + cθλ ; Ω]. Moreover, C(µ,θλ ,0) satisﬁes some of the following alternatives: either C(µ,θλ ,0) is ¯ × C0 (Ω), ¯ or (µ∗ , 0, θµ∗ ) ∈ C ¯ (µ,θ ,0) , for some µ∗ such that unbounded in R × C0 (Ω) λ λ = σ[−∆ + bθµ∗ ; Ω]. Obviously, the last alternative cannot occur if λ > λs , and, ¯ × C0 (Ω). ¯ Now, we will hence, in this case, C(µ,θλ ,0) is unbounded in R × C0 (Ω) study some numerical one-dimensional examples to illustrate the diﬀerent global

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307

λ=σ[−∆+ b θµ ;Ω] µ

λ= b µ L (σ1 ,σ 1)

(σ,σ) λ (0,0) µ=σ[−∆+ c θ λ ;Ω]

µ= c λ L

Figure 7. Relevant (λ, µ)-regions for the general model. behaviors that the component C(µ,θλ ,0) can have. Precisely, we consider ⎧ ⎨ −u = λu − u2 − b(x)uv in (0, 1) , −v = µv − v 2 − b(x)uv ⎩ u(0) = u(1) = v(0) = v(1) = 0 , where

⎧ + πx ⎨ b sin 0.4 %π & −b− sin (x − 0.4) b(x) := 0.2 % & ⎩ + π b sin 0.4 (x − 0.6)

if 0.0 ≤ x ≤ 0.4 , if 0.4 ≤ x ≤ 0.6 , if 0.6 ≤ x ≤ 1.0 .

(6.5)

(6.6)

for some positive constants b+ > 0 and b− > 0. In this example, Ω0 = ∅, Ω− = (0.4, 0.6), and Ω+ = (0.0, 0.4) ∪ (0.6, 1.0). Figure 8 shows a plot of the curves (3.8) for the special choice b− = 0.4 . (6.7) b+ = 2 , They have been computed by combining a path-following code for approximating θλ and θµ , through centered divided diﬀerences, with the inverse power method to compute the associated principal eigenvalues. Note that, thanks to Proposition 4.1, 2

2

d d σ[− dx σ[− dx 2 + bθλ ; (0, 1)] 2 + bθµ ; (0, 1)] lim = lim = bL = −0.4 , λ↑∞ µ↑∞ λ µ

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J. L´opez-G´omez and M. Molina-Meyer 250

200

µ

150

100

50

0

0

50

100

λ

150

200

250

Figure 8. The curves (3.8) for the model (6.5), (6.6), (6.7). provide us with the slopes at inﬁnity of the curves (3.8). Figure 9 shows a plot of the component C(µ,θλ ,0) for λ = 48.98 > λs . To compute it we have ﬁxed λ = 48.98 and used µ as the main bifurcation parameter by coupling a pure spectral method with collocation and a path global continuation solver (cf. [24], [35], [8], and [38] for any further detail required). In Figure 9 the component C(µ,θ48.98 ,0) emanates from (θ48.98 , 0) at the critical value of the parameter d2 + bθ48.98 ; (0, 1)] ∼ 23.51 dx2 and it is actually deﬁned for all further value of µ for which we tried to compute it. All computed coexistence states are exponentially asymptotically stable and, hence, they are local attractors for the positive solutions of (2.1). Figure 9 shows four signiﬁcant curves; stable solutions are indicated by solid lines, unstable by dashed lines. We are plotting the value of µ against the norm µ(48.98) := σ[−

|u| + |v| := max u(x) + max v(x) . 0≤x≤1

0≤x≤1

The two horizontal lines describe the trivial state (0, 0), and the semi-trivial state (θ48.98 , 0); |θ48.98 | ∼ 44.02. Then, Figure 9 shows the curve of semi-trivial positive solutions (0, θµ ). This curve emanates from (0, 0) at µ = π 2 and it grows with µ for any further value of the parameter. Finally, Figure 9 shows the curve of coexistence states. It is the curve emanating super-critically from the state (θ48.98 , 0)

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250

200

|u|+|v|

150

100

50

0

0

50

100

150

200

250

mu

Figure 9. A plot of the component C(µ,θ48.98 ,0) of (6.5), (6.6), (6.7). at the value of µ where (θλ , 0) loses stability. Strongly motivated by our numerical computations, we conjecture that in this example Pµ C(µ,θ48.98 ,0) = (π 2 , ∞), where Pµ stands for the µ–projection operator. If, instead of (6.7), we make the choice b+ = 2 ,

b− = 1.2 ,

(6.8)

then, bL = −1.2 and, thanks again to Proposition 4.1, 2

2

d d σ[− dx σ[− dx 2 + bθλ ; (0, 1)] 2 + bθµ ; (0, 1)] = lim = −1.2 , λ↑∞ µ↑∞ λ µ

lim

which provides us with the slopes at inﬁnity of the curves (3.8), whose plots are shown in Figure 10. Rather naturally, as bigger is b− as wider is the region where both semi-trivial states are unstable, as a result of strengthening symbiotic eﬀects if b− grows, which explains why the region enclosed by the curves (3.8) is larger in Figure 10 than in Figure 9. Figure 11 shows a plot of C(µ,θ48.98 ,0) for the special choice (6.8). Now, the component bifurcates from (θ48.98 , 0) at µ ∼ 8.46 and it exhibits a sub-critical turning point at µ ∼ 165.1353, where it turns backward. The coexistence states along the lower-half curve are exponentially asymptotically stable, while the solutions along the upper-half curve are linearly unstable. In this example, based upon our numerical experiments, a reasonable conjecture is Pµ C(µ,θ48.98 ,0) = (−∞, 165.1353], though a proof of this fact will not be given here in.

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J. L´opez-G´omez and M. Molina-Meyer 250

200

150

µ

100

50

0

−50

−100

−150 −150

−100

−50

0

50 λ

100

150

200

250

Figure 10. The curves (3.8) for the model (6.5), (6.6), (6.8). In general, the following result is satisﬁed. Theorem 6.1. Suppose λ > λs , b− M < 1,

c− M < 1,

(6.9)

and (2.3) possesses a coexistence state. Then, − − λ > (1 − b− M cM ) σ − b M µ ,

− − µ > (1 − b− M cM ) σ − cM λ .

(6.10)

Moreover, there exists µ0 (λ) ∈ R such that − − (1 − b− M cM )σ − cM λ < µ0 (λ) ≤ σ[−∆ + cθλ ; Ω]

(6.11)

and (6.12) Pµ C(µ,θλ ,0) ∈ { [µ0 (λ), ∞) , (σ[−∆ + cθλ ; Ω], ∞) } . In particular, (2.3) possesses a coexistence state for each µ > σ[−∆ + cθλ ; Ω]. − Note that b− M = cM in example (6.5), (6.6). Thus, (6.9) holds for the choice (6.7), while it fails for (6.8). Therefore, by simply looking at Figures 9 and 11, it seems that some kind of condition (6.9) is really necessary for the validity of the theorem. Most crucially, thanks to Theorem 6.1, it is obvious that the principle of competitive exclusion fails to be true for model (2.1), because if Ω0 ∪ Ω− can maintain to u in the absence of v, i.e., if λ > σ1 , then λ > λs and, for any µ > µ0 (λ), (2.3) possesses a coexistence state. Consequently, u can avoid extinction, even in the most dramatic case when the intrinsic growth rate of its competitor v, measured

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5500 5000 4500 4000

|u|+|v|

3500 3000 2500 2000 1500 1000 500 0 0

20

40

60

80

100 mu

120

140

160

180

200

Figure 11. The component C(µ,θ48.98 ,0) of (6.5), (6.6), (6.8). by µ, blows-up. In Section 7 we shall show that, rather naturally, u must pay the price of segregation within Ω0 ∪Ω− if µ ↑ ∞. Therefore, strategic symbiosis seem to be imperative to avid extinction in the presence of good competitors; an strategy adopted by most of world-leading companies, of course ! Proof of Theorem 6.1. Suppose (u0 , v0 ) is a coexistence state of (2.3). Then, −∆u0 = λu0 − u20 − bu0 v0 ≤ λu0 − u20 + b− u0 v0 ≤ λu0 − u20 + b− M u0 v0 , −∆v0 = µv0 − v02 − cu0 v0 ≤ µv0 − v02 + c− u0 v0 ≤ µv0 − v02 + c− M u0 v0 , in Ω, while u0 = v0 = 0 on ∂Ω. Thus, (u0 , v0 ) is a positive sub-solution of the pure symbiotic model ⎧ ⎨ −∆u = λu − u2 + b− M uv in Ω , (6.13) −∆v = µv − v 2 + c− M uv ⎩ u = v = 0, on ∂Ω . ' ( µ λ Due to (6.9), for each constant N ≥ max 1−b , the pair (N, N ) is a − , − 1−c M

M

super-solution of (6.13), and – if necessary – enlarging N one has that (u0 , v0 ) ≤ (N, N ). Consequently, thanks to [15, Theorem 8.7], (6.13) has a coexistence state (ˆ u0 , vˆ0 ) such that u0 ≤ u ˆ0 ≤ N , v0 ≤ vˆ0 ≤ N . (6.14)

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Therefore, due [15, Lemma 6.1], (λ, µ) satisﬁes (6.10). Moreover, u ˆ0 ≤

λ + µb− M − , 1 − b− c M M

vˆ0 ≤

µ + λc− M − . 1 − b− c M M

(6.15)

v0 ≤

µ + λc− M − , 1 − b− M cM

(6.16)

In particular, (6.14) and (6.15) imply u0 ≤

λ + µb− M − , 1 − b− M cM

for any coexistence state (u0 , v0 ) of (2.3). Note that, due to (6.10), λ+µb− M > 0 and > 0. Consequently, the coexistence states of (2.3) possess uniform a priori µ + λc− M bounds in L∞ on any compact subset of (λ, µ) ∈ R2 . The remaining assertions of the theorem follow straight ahead from the fact that C(µ,θλ ,0) is an unbounded closed and connected set. When, instead of (6.9), the following conditions are satisﬁed b− M > 1,

c− M > 1,

(6.17)

then (2.3) might not exhibit coexistence states for suﬃciently large µ, as strongly supported by Figure 11. But, from this fact, it should not concluded that the principle of competitive exclusion will hold as in the classical competition model. Actually, as larger are the symbiotic eﬀects between u and v, measured by b− and c− , as greater should be the possibilities of avoiding extinction for each of the species. Consequently, the absence of coexistence states when b− and c− grow should entail some kind of Malthusian growth for the species. To realize what’s going on, we will focus our attention into the simplest case when λ = µ and b = c .

(6.18)

Then, all pairs of the form (u, v) = (w, w), where w is a positive solution of in Ω , −∆w = λw − (1 + b)w2 (6.19) w|∂Ω = 0 , are coexistence states of (2.3). The solutions of (6.19) exhibit a number of diﬀerent ¯ behaviors according to the size of b− . If b− M < 1, then 1 + b(x) > 0 for each x ∈ Ω and, hence, (6.19) has a positive solution (unique) if, and only if, λ > σ; denoted by wλ . Moreover, if λ > σ, then wλ is a global attractor for ⎧ ∂w in Ω × (0, ∞) , ⎨ ∂t − ∆w = λw − (1 + b)w2 (6.20) w=0 on ∂Ω × (0, ∞) , ⎩ w(·, 0) = w0 > 0 in Ω . Consequently, (wλ , wλ ) attracts to all solutions of (2.1) with u0 = v0 > 0. If b− M = 1 in the closure of some appropriate smooth subdomain D0 ⊂ Ω− , ¯ 0 and 1 + b(x) > 0 for each x ∈ Ω \ D ¯ 0 . Thus, thanks to the then 1 + b = 0 in D theory of Fraile et al. [18], (6.19) has a positive solution (unique) if, and only if, σ < λ < σ0 := σ[−∆; D0 ], say wλ . Moreover, as in the previous case, wλ is a global attractor for (6.20), and, therefore, (wλ , wλ ) is a global attractor for all solutions of (2.1) with u0 = v0 > 0 if σ < λ < σ0 . Furthermore, thanks to the theory of

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313

Garc´ıa-Meli´an et al. [21], Lσ0 := limλ↑σ0 wλ provides us with the minimal positive solution of the singular problem ¯0 , −∆w = σ0 w − (1 + b)w2 in Ω \ D (6.21) w|∂D0 = ∞ , ¯ 0 . For each λ ≥ σ0 , the dynamics of (6.20) ¯ 0 , while limλ↑σ0 wλ = ∞ in D in Ω \ D ¯ 0 . By a is regulated by the minimal metasolutions of (6.19) supported in Ω \ D metasolution it is meant the extension by inﬁnity of the solutions of (6.21) (cf. [28], [31] and [33]). Therefore, the limit as t ↑ ∞ of the solutions of (2.1) with λ ≥ σ0 and u0 = v0 > 0 is regulated by the metasolutions of (6.19), like in [30]. In such case, (2.3) cannot admit a coexistence state. If b− M > 1, then the function 1 + b(x) changes sign in Ω and, in such case, the dynamics of (6.20) becomes much more involved, since (6.20) is a superlinear indeﬁnite problem. Among the main results available in this case, it is known that there exists λ∗ > σ such that (6.19) cannot have a positive solution if λ > λ∗ (cf. Amann & L´ opez-G´omez [3]). Moreover, if λ∗ > σ and σ < λ < λ∗ , then the minimal positive solution is the unique non-negative stable solution of (6.20) (cf. G´ omez-Re˜ nasco and L´opez-G´omez [22]), though, quite strikingly, as a result of the superlinear character of the model, (6.19) might exhibit an arbitrarily large number of positive – multiple spike – solutions; all of them necessarily unstable; the uniqueness of the stable solution in this general class of superlinear indeﬁnite problems showing the extraordinary depth beneath the celebrated theorem of Amann concerning the stability of the minimal positive solution, [1]. As in the previous case, if λ > λ∗ , then the dynamics of (6.20) are regulated by the metasolutions of (6.19), which can be reached in a ﬁnite, or inﬁnity, time according to the size of b− M , but this – rather sharp – analysis is outside the general scope of this work (cf. L´opez-G´omez and Quittner [42] and [34] for complete details). The previous discussion, besides showing the necessity of condition (6.9) for the validity of Theorem 6.1, supports the thesis that, in the absence of coexistence states, the species will exhibit a genuine Malthusian growth within cooperation areas, up to blow-up in a ﬁnite time if b− , or c− , are suﬃciently large. A model that might avoid the previous blow-up phenomenology, simultaneously maintaining arbitrarily large the population levels as time passes by is the following fast dispersion model ⎧ ⎪ ⎪ ⎪ ⎨

∂u ∂t ∂v ∂t

− ∆(um ) = λu − u2 − b(x)uv − ∆(v n ) = µv − v 2 − c(x)uv

⎪ u=v=0 ⎪ ⎪ ⎩ u(·, 0) = u0 ≥ 0 ,

v(·, 0) = v0 ≥ 0 ,

in Ω × R+ , on ∂Ω × R+ , in Ω ,

where n > 1 and m > 1 are suﬃciently close to 1, but we refrain of giving more details here (cf. M. Delgado et al. [16] for the case of one single equation).

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7. Segregation within cooperation areas The next result provides us with the proﬁles of the coexistence states of (2.3) when the intrinsic growth rate of the species v, µ, reaches high levels. It explains the ultimate reasons why the principle of competition exclusion fails in our model (2.1). As the level of the aggressions from the competitor blows-up, the species segregates within regions where cooperation takes place, so avoiding extinction, and facilitating diversity through further diﬀerentiation. Theorem 7.1. Suppose λ > λs (see (6.4)), b− M < 1,

c− M < 1,

c+ M <

− 1 − b− M cM , b− M

(7.1)

and let {(µn , un , vn )}n≥1 be a sequence of coexistence states of (2.3) such that µn ↑ ∞ as n ↑ ∞. Then, lim vn = ∞

n→∞

and

where θ[λ,Ω0 ]

uniformly in compact subsets of Ω ,

⎧ in Ω+ ⎨ 0 θ[λ,Ω0 ] in Ω0 lim un = n→∞ ⎩ ∞ in Ω− is the maximal non-negative solution of −∆u = λu − u2 in Ω0 , u=0 on ∂Ω0 .

(7.2)

(7.3)

Note that θ[λ,Ω0 ] = 0 if λ ≤ σ[−∆; Ω0 ], while θ[λ,Ω0 ] ) 0 if λ > σ[−∆; Ω0 ]. Also, note that the third condition of (7.1) imposes a control on the size of c+ M, which measures the stress suﬀered by v from its interaction with u. So, (7.1) seems rather natural for getting (7.2). By Theorem 6.1, the ﬁrst two estimates of (7.1) guarantee the existence of a coexistence state of (2.3) for any suﬃciently large µ. Note that the model (6.5), (6.6), (6.7) ﬁts within the setting of Theorem 7.1, since 2 = c+ M <

− 1 − b− 1 − (0.4)2 M cM = 2.1 . = − 0.4 bM

Figure 12 shows a series of plots of un and vn for a sequence of coexistence states (µn , un , vn ) of (6.5), (6.6), (6.7), with λ = 48.98 and µn increasing. Actually, it shows the components of a series of coexistence states along the curve of coexistence states of Figure 9. Their point-wise behaviors, as µn increases, completely agrees with the predictions made from Theorem 7.1. Indeed, vn grows to inﬁnity all over (0, 1) as µn ↑ ∞, while un ↑ ∞ as µn ↑ ∞ in Ω− = (0.4, 0.6), and un ↓ 0 as µn ↑ ∞ in Ω+ = (0.0, 0.4) ∪ (0.6, 1.0). Rather naturally, vn ↑ ∞ at a higher rate within Ω+ than in Ω− , as an eﬀect of symbiosis with u within Ω+ . The proﬁles of un and vn clearly illustrate the permanence of both species as µ ↑ ∞ at the cost of the segregation of u within the cooperation area. A rather new phenomenology that cannot occur in the classical competing species

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315

1500

400

350

300

1000

v

u

250

200

150

500

100

50

0

0

0.2

0.4

0.6

0.8

0

1

0

0.2

x

0.4

0.6

0.8

1

x

Figure 12. A series of coexistence states along C(µ,θ48.98 ,0) for (6.5), (6.6), (6.7). models revisited in Section 5, where the principle of competitive exclusion predicts extinction of u. Proof of Theorem 7.1: Fix λ > λs and a sequence {(µn , un , vn )}n≥1 of coexistence states of (2.3) with limn→∞ µn = ∞, whose existence is guaranteed by Theorem 6.1. Without lost of generality we can assume that µn ≥ 0 for each n ≥ 1. Thanks to (6.16), we already know that un ≤

λ + µn b− M − , 1 − b− M cM

n ≥ 1,

and, hence, −∆vn = µn vn − vn2 − c+ un vn + c− un vn ≥ µn vn − vn2 − c+ un vn λ + µn b− M ≥ µn − c+ vn − vn2 − 1 − b− c M M

+ c b− λ c+ M M M ≥ 1− µn − vn − vn2 . − − 1 − b− 1 − b− M cM M cM Thus, setting γn :=

1−

+ b− M cM − − 1 − b M cM

µn −

λ c+ M − , 1 − b− M cM

n ≥ 1,

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J. L´opez-G´omez and M. Molina-Meyer

we ﬁnd that, for each n ≥ 1, vn is a positive super-solution of −∆w = γn w − w2 in Ω , w=0 on ∂Ω .

(7.4)

Due to (7.1), limn→∞ γn = ∞ and, in particular, there is n0 ≥ 1 such that γn > σ for each n ≥ n0 . For such values of n, (7.4) has a (unique) positive solution, say wn . Thus, vn ≥ wn if n ≥ n0 . Now, by [20, Theorem 3.4], limn→∞ wn = ∞ uniformly in compact subsets of Ω. Therefore, limn→∞ vn = ∞. Once proven (7.2), the validity of (7.3) will be obtained from the un -problem (−∆ + bvn ) un = λun − u2n in Ω , n ≥ 1. on ∂Ω , un = 0 Subsequently, we will prove each of the parts of (7.3) separately. Part 1: Demographic explosion in Ω− . It suﬃces to show that for each x0 ∈ Ω− ¯ρ (x0 ) ⊂ Ω− and limn→∞ un = ∞ uniformly in there exists ρ > 0 such that B ¯ ¯ R (x0 ) ⊂ Ω− , and consider the Bρ (x0 ). Indeed, let x0 ∈ Ω− and R > 0 with B auxiliary problems ⎧ ⎨ − in BR (x0 ) , −∆ − b min vn U = λU − U 2 ¯ R (x0 ) (7.5) B ⎩ U =0 on ∂BR (x0 ) . For each n ≥ 1, we have that σ[−∆ − b− min vn ; BR (x0 )] ≤ σ[−∆; BR (x0 )] − min b− · min vn ¯ R (x0 ) B

¯ R (x0 ) B

¯ R (x0 ) B

and, hence, lim σ[−∆ − b− min vn ; BR (x0 )] = −∞ ,

n→∞

¯ R (x0 ) B

¯R (x0 ). In particular, there exists n0 ≥ 1 such that, since vn → ∞ uniformly in B for each n ≥ n0 , λ > σ[−∆ − b− min vn ; BR (x0 )] . ¯ R (x0 ) B

Consequently, (7.5) possesses a unique positive solution, say Un . As un is a positive super-solution of (7.5), for each n ≥ n0 , un |BR (x0 ) ≥ Un in BR (x0 ) and, hence, to obtain the limiting behavior of un in Ω− it suﬃces to prove that ¯ R (x0 ) . lim Un = ∞ uniformly in B (7.6) n→∞

2

−1 To show (7.6), we perform the change of variable U = ε−1 := n Z where εn minB¯R (x0 ) vn , to transform (7.5) into −εn ∆Z = (λεn + b− ) Z − Z 2 in BR (x0 ) , (7.7) Z=0 on ∂BR (x0 ) .

¯ρ (x0 ), Thanks to [20, Theorem 3.6], one has that limn→∞ Zn = b− uniformly in B for each ρ ∈ (0, R), where Zn is the unique positive solution of (7.7). This shows

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317

(7.6). Consequently, limn→∞ un = ∞ uniformly in compact subsets of Ω− , which concludes the proof of this part. ¯R (x0 ) ⊂ Part 2: Demographic extinction in Ω+ . Let x0 ∈ Ω+ and R > 0 such that B Ω+ , and consider the singular auxiliary problems in BR (x0 ) , (−∆ + αn ) U = λU − U 2 (7.8) U =∞ on ∂BR (x0 ) , where αn := min b+ · min vn . ¯ R (x0 ) B

¯ R (x0 ) B

We already know that limn→∞ αn = ∞. Moreover, for each n ≥ 1, (7.8) possesses a unique solution, denoted by Ln (cf., e.g., [33]), and, since un is a positive subsolution of (7.8), un ≤ Ln for each n ≥ 1. Therefore, to show that un → 0 uniformly on compact subsets of Ω+ it suﬃces to prove that limn→∞ Ln = 0 uniformly in ¯ R (x0 ). Indeed, set D := BR/2 (x0 ) and let ϕ ) 0 be a principal eigenfunction B 2 associated to σ[−∆; D]. Then, multiplying the diﬀerential equation of Ln by ϕ, integrating in D and applying the formula of integration by parts, gives ∂ϕ ∂ϕ − , Ln ϕ = − Ln L2n ϕ ≤ − Ln (αn − λ + σ[−∆; D]) ∂ν ∂ν D ∂D D ∂D where ν stands for the outward unit normal of D. Let L0 be the unique large solution of −∆U = λU − U 2 in D. As, for each n ≥ 1, λ − αn ≤ λ, we have that Ln ≤ L0 , n ≥ 1, and, hence, since ϕ has negative ﬂux on ∂D, we obtain that ∂ϕ (αn − λ + σ[−∆; D]) , n ≥ 1. Ln ϕ ≤ − L0 ∂ν D ∂D

Thus, since limn→∞ (αn − λ + σ[−∆; D]) = ∞, necessarily limn→∞ D Ln ϕ = 0. On the other hand, thanks to the results of [28], for each n ≥ 1, Ln is radially symmetric and, for n suﬃciently large, minD¯ Ln = Ln (x0 ), since −∆Ln ≤ 0. Necessarily, limn→∞ Ln (x0 ) = 0, and, therefore, limn→∞ un (x0 ) = 0, since un ≤ Ln . As this holds for each x0 ∈ Ω+ , the functions un approximate to zero in Ω+ ¯ one can argue as as n → ∞. To show the uniformity of that convergence in D x follows. For each x ∈ D = B R (x0 ), let Ln denote the large solution of 2

(−∆ + αn ) U = λU − U 2

in B R (x) , 2

U |∂B R (x) = ∞ . 2

Since Ln ≤ Lxn = Lxn0 (x0 − x + ·) in B R (x), we have that Ln (x) ≤ Lxn0 (x0 ) for each 2 ¯ and concludes the second x ∈ D, which implies limn→∞ Ln = 0 uniformly in D part of the proof. Part 3: Demographic stabilization in Ω0 . First, suppose that λ > σ[−∆; Ω0 ]

(7.9)

and denote by θλ the unique positive solution of −∆u = λu − u2

in Ω0 ,

u|∂Ω0 = 0 .

(7.10)

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J. L´opez-G´omez and M. Molina-Meyer

As, for each n ≥ 1, un is a positive super-solution of (7.10), we ﬁnd that ¯0 , u n ≥ θλ in Ω n ≥ 1.

(7.11)

Now, for each suﬃciently small δ > 0, consider the δ-neighborhood of Ω0 Ωδ := { x ∈ Ω : dist (x, Ω0 ) < δ } ; δ is assumed to be chosen so that ∂Ωδ ⊂ Ω+ (see Figure 1). By monotonicity, (7.9) implies λ > σ[−∆; Ω0 ] > σ[−∆; Ωδ ] , (7.12) ¯ 0 ⊂ Ωδ . Thus, the problem since Ω −∆u = λu − u2

in Ωδ ,

u|∂Ωδ = 0 ,

(7.13)

possesses a unique positive solution. Let denote it by θ[λ,δ] and consider a smooth function of the following type in Ω δ θ[λ,δ] 2 Φδ := ¯δ \ Ωδ ψδ in Ω 2

where ψδ is any positive extension (bounded away from zero) of θ[λ,δ] |Ω δ to the 2

totality of Ωδ . We now show that for suﬃciently large n the function Φδ provides us with a positive super-solution of the following problem in Ωδ , (−∆ + bvn ) U = λU − U 2 (7.14) U = un on ∂Ωδ , whose unique positive solution is un . Since ψδ is bounded away from zero in ∂Ωδ and, thanks to the result already obtained in Part 2, we have that lim un = 0

n→∞

uniformly in ∂Ωδ ⊂ Ω+ .

Thus, there exists n0 ≥ 1 such that for each n ≥ n0 one has that Φδ ≥ un in ∂Ωδ . Moreover, in Ω δ one has that Φδ = θ[λ,δ] and, hence, 2

2 , (−∆ + bvn ) Φδ = (−∆ + bvn ) θ[λ,δ] > −∆θ[λ,δ] = λθ[λ,δ] − θ[λ,δ]

whereas in Ωδ \ Ω δ , we have Φδ = ψδ and, so, 2

(−∆ + bvn ) Φδ = (−∆ + bvn ) ψδ ≥ λψδ − ψδ2 , if n is suﬃciently large, since limn→∞ minΩ¯ δ \Ωδ/2 (bvn ) = ∞. Summarizing, there exists n0 ≥ 1 such that, for each n ≥ n0 , Φδ is a positive strict super-solution of (7.14), whose unique solution is un . Consequently, Φδ ≥ un , n ≥ n0 . In particular, lim sup un |Ω0 ≤ θ[λ,δ] for each suﬃciently small δ > 0. Therefore, θλ ≤ lim inf un |Ω0 ≤ lim sup un |Ω0 ≤ θ[λ,δ] . n→∞

n→∞

Now, thanks to theory of Cano-Casanova & L´ opez-G´omez [7], limδ↓0 θ[λ,δ] = θλ ¯ 0 , and, consequently, limn→∞ un |Ω¯ = θλ uniformly in Ω ¯ 0 , which uniformly in Ω 0 concludes the proof of the theorem under condition (7.9).

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319

Now, suppose that λ ≤ σ[−∆; Ω0 ]. Then, for each ε > 0 the function un is a positive sub-solution of (−∆ + bvn ) U = (σ[−∆; Ω0 ] + ε)U − U 2 in Ω , (7.15) U =0 on ∂Ω , and, hence, in Ω , (7.16) un ≤ u[n,ε] where u[n,ε] stands for the unique positive solution of (7.15); it exists for suﬃciently large n. As ρε := σ[−∆; Ω0 ] + ε > σ[−∆; Ω0 ], by the result that we have just obtained, we know that ¯0 , uniformly in Ω (7.17) lim u[n,ε] = θρε n→∞

where θρε is the unique positive solution of −∆U = ρε U − U 2 U =0

in Ω0 , on ∂Ω0 .

Therefore, thanks to (7.16) and (7.17), it is apparent that for each ε > 0 lim sup un ≤ θρε .

(7.18)

n→∞

On the other hand, as a consequence from the celebrated theorem of Crandall & ¯ 0 . Consequently, Rabinowitz [9], it is rather clear that limε↓0 θρε = 0, uniformly in Ω ¯ 0 , which concludes the proof. (7.18) implies limn→∞ un = 0 uniformly in Ω

8. Generating biological and economical complexity In this section we consider the prototype model ⎧ −0.4 sin (5πx) , ⎪ ⎪ ⎪ ⎪ ⎨ 20 sin [5π(x − 0.2)] , b(x) := −0.4 sin [5π(x − 0.4)] , ⎪ ⎪ 40 sin [5π(x − 0.6)] , ⎪ ⎪ ⎩ −0.4 sin [5π(x − 0.8)] ,

(6.5), (6.6) with 0.0 ≤ x ≤ 0.2, 0.2 < x ≤ 0.4, 0.4 < x ≤ 0.6, 0.6 < x ≤ 0.8, 0.8 < x ≤ 1.0.

(8.1)

For this choice, u and v compete in Ω+ = (0.2, 0.4)∪(0.6, 0.8), while they cooperate in Ω− = (0, 0.2)∪(0.4, 0.6)∪(0.8, 1). Note that the level of the aggressions between u and v has been chosen relatively high, 20 and 40, in comparison with the level of the cooperative eﬀects within symbiotic areas, 0.4. Consequently, at ﬁrst glance, symbiotic interactions are relatively weak. For this choice, σ1 = 25π 2 ∼ 246.74 and, hence, in the quadrant min{λ, µ} > 246.74 both semi-trivial states are linearly unstable, which is clearly illustrated by Figure 13. Thanks to Proposition 4.1, the tangents at inﬁnity of the curves (3.8) are given by µ = −0.4 × λ and λ = −0.4 × µ, respectively, which complete agrees with the numerical computations. Figure 14 shows the bifurcation diagram of nonnegative solutions for a λ ∼ 140; it shows three signiﬁcant curves for describing

320

J. L´opez-G´omez and M. Molina-Meyer 500

450

400

350

µ

300

250

200

150

100

50

0

0

50

100

150

200

250 λ

300

350

400

450

500

Figure 13. The curves (3.8) for the choice (6.5), (6.6) and (8.1). the dynamics of (2.1); stable solutions are indicated by solid lines, unstable by dashed lines. As (0, 0) has been represented in the µ–axis, its character cannot be distinguished in the plot, though it is always unstable. The horizontal line describes the semi-trivial state (θ140 , 0); |θ140 | ∼ 116.57. Then, Figure 14 shows the curve of semi-trivial solutions (0, θµ ), which emanates from (0, 0) at µ = π 2 and it grows for all further values of µ. All solutions along this curve are unstable. The component of coexistence states C(µ,θ140 ,0) emanates, super-critically, from (θ140 , 0) at d2 + bθ140 ; (0, 1)] ∼ 93.40 , dx2 where the previous stability of (θ140 , 0) is lost. All coexistence states are stable until C(µ,θ140 ,0) reaches its ﬁrst turning point, at µ1 ∼ 171.0, where it turns backwards. Once crossed the ﬁrst turning point all coexistence states become unstable, with Morse index one, until C(µ,θ140 ,0) reached its second turning point, super-critical, at µ2 = 126.47. The Morse index counts the dimension of the unstable manifold of the steady-states. And so on. . . until the eight turning point exhibited by C(µ,θ140 ,0) is crossed, super-critically, at µ8 := 100.7, where all solutions become stable for all further values of µ. Table 1 collects the parameter values of each of the turning points exhibited by C(µ,θ140 ,0) as well as the Morse indices of the coexistence states along each of the associated arcs of curve, c0 , c1 , . . . , c8 ; c0 stands for the arc of curve of coexistence states emanating from the semitrivial positive solution (θ140 , 0) and, for each j ≥ 1, µ = σ[−

In the Blink of an Eye

321

200 180 160 140

|u|+|v|

120 100 80 60 40 20 0

0

20

40

60

80

100 mu

120

140

160

180

200

Figure 14. The component C(µ,θ140 ,0) of (6.6), (6.7), (8.1). cj stands for the arc of curve in between the turning points corresponding to µj and µj+1 . Consequently, for each µ ∈ (126.47, 153.93), (6.5), (6.6), (8.1) possesses 9 coexistence states: 4 among them are asymptotically stable, and the remaining 5 are unstable. The Morse index changed by ±1 whenever crossed a turning point, as predicted by well-known results of classical bifurcation theory. Figure 15 shows a series of plots of representative coexistence states along c0 . We have plotted the proﬁles of the u and the v-components of the coexistence states for a series of values of µ in between the bifurcation point from (θ140 , 0) and the ﬁrst turning point at µ1 = 171.0. arc c0 c1 c2 c3 c4 c5 c6 c7 c8

Morse index 0 1 0 1 2 1 0 1 0

Turning point 171.0 126.47 165.9 109.0 166.2 118.97 153.93 100.7 none

Table 1. The turning points and the Morse index along the arcs cj .

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As the value of µ separates from the bifurcation value µ = 93.40, u decreases from θ140 , whereas v increases from zero. In Figure 15 and in all subsequent plots of coexistence states, the arrows in each of the components of (0, 1) where b has a constant sign, indicate (very roughly) if the corresponding species grows, ↑, or, conversely, decays, ↓, as µ moves between two consecutive turning points. 160

120

↓ ↓ ↓ ↓ ↓

↑ ↑ ↑ ↑ ↑

140 100 120 80

80

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Figure 15. Representative coexistence states along c0 . Rather naturally, as the intrinsic growth rate of v, measured by µ, increases, the species v grows. Moreover, it grows at the fastest rate within the symbiotic regions, b < 0, where cooperation with u takes place, than in the competitive domains, b > 0, where u and v compete. Not surprisingly, u decays as µ increases, and its fastest decay rates occur within regions where competition takes place. Similarly, Figure 16 shows the plots of a series of representative coexistence states along the arc of curve c2 . As in Figure 15, we have plotted the proﬁles of the u and the v-components of the computed coexistence states for a series of values of µ in between the turning points µ2 = 126.47 and µ3 = 165.9. As along c0 , the coexistence states along c2 exhibit a genuine sub-linear behavior, in the sense that u is point-wise decreasing and v is increasing in (0, 1) for all values of µ. Such a nice monotonicity seems to be closely related to the attractive character of these coexistence states, as it is lost along arcs where solutions are unstable. Once crossed the last turning point exhibited by C(µ,θ140 ,0) , at µ8 = 100.7, the coexistence states become stable for all further value of µ. In Figure 17 we have represented the plots of a series of representative coexistence states along c8 .

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160

↑ ↑ ↑ ↑ ↑

↓ ↓ ↓ ↓ ↓

120

140

100

120

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u

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Figure 16. Representative coexistence states along c2 .

As illustrated by Figure 17, the distribution of the population v grows everywhere in (0, 1) as µ increases, reaching arbitrarily large values, while the species u grows arbitrarily in the regions where cooperation with v takes place, (0, 0.2), (0.4, 0.6) and (0.8, 1), though it naturally decays up to extinction within the regions where u is stressed by v, (0.2, 0.4) and (0.6, 0.8), in complete agreement with Theorem 7.1. It should be noted that, for the choice (8.1), one has − b− M = cM = 0.4 < 1 ,

c+ M = 40 >

− 1 − b− 1 − (0.4)2 M cM = 2.1 , = 0.4 b− M

and, consequently, the last estimate of (7.1) fails to be true in this example, which suggests that the range of validity of Theorem 7.1 might be substantially wider. This example shows that, although (6.19) possesses at most a unique stable positive solution (by the main theorem of G´ omez-Re˜ nasco & L´opez-G´omez [22]), (2.3) might have an arbitrarily large number of stable coexistence states for some appropriate range of values of the parameters involved in the setting of the model, so, strongly suggesting that competition combined with some strategic symbiotic eﬀects might result into a substantial growth of the dynamical complexity of the system, a sort of principle corroborating that vision indeed extraordinarily facilitated the Cambrian explosion.

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Figure 17. Representative coexistence states along c8 . Acknowledgements The authors thank to the Spanish Ministry of Education and Science under grant REN2003-00707.

References [1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), 620–709. [2] H. Amann, Maximum principles and principal eigenvalues, in Ten Mathematical Essays on Approximation in Analysis and Topology (J. Ferrera, J. L´ opez-G´ omez and F.R. Ru´ız del Portal, eds.), pages 1–60, Elsevier, Amsterdam, 2005. [3] H. Amann and J. L´ opez-G´ omez, A priori bounds and multiple solutions for superlinear indeﬁnite elliptic problems, J. Diﬀ. Eqns. 146 (1998), 336–374. [4] M. Begon, J.L. Harper, and C.R. Townsend, Ecology, Individual, Populations and Communities, Blackwell Scientiﬁc Publications, Cambridge, Massachusetts, 1990. [5] J. Blat and K.J. Brown, Bifurcation of steady-state solutions in predator-prey and competition systems, Proc. Roy. Soc. Edinburgh 97A (1984), 21–34. [6] S. Cano-Casanova and J. L´ opez-G´ omez, Permanence under strong aggression is possible, An. Inst. H. Poincar´e, Anal. non Lin. 20 (2003), 999–1041. [7] S. Cano-Casanova and J. L´ opez-G´ omez, Varying domains in a general class of sublinear elliptic problems, Elec. J. Diﬀ. Eqns. 74 (2004), 1–41.

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[8] S. Cano-Casanova, J. L´ opez-G´ omez, and M. Molina-Meyer, Permanence through spatial segregation in heterogeneous competition, in Proc. 9th IEEE International Conference on Methods and Models in Automation and Robotics (R. Kaszy´ nski, Ed.), pages 123–130, Szczecin, 2003. [9] M.G. Crandall and P.H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321–340. [10] E.N. Dancer, On positive solutions of some pairs of diﬀerential equations, I, Trans. Amer. Math. Soc. 284 (1984), 729–743. [11] E.N. Dancer, On positive solutions of some pairs of diﬀerential equations, II, J. Diﬀ. Eqns. 60 (1985), 236–258. [12] E.N. Dancer, The existence and the uniqueness of positive solutions for competing species models with diﬀusion, Trans. Amer. Math. Soc. 326 (1991), 829–859. [13] E.N. Dancer, A counterexample of competing species equations, Diﬀ. Int. Eqns. 9 (1996), 239–246. [14] E.N. Dancer, Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one, Bull. London Math. Soc. 34 (2002), 533–538. [15] M. Delgado, J. L´ opez-G´ omez and A. Su´ arez, On the symbiotic Lotka-Volterra model with diﬀusion and transport eﬀects, J. Diﬀ. Eqns. 160 (2000), 175–262. [16] M. Delgado, J. L´ opez-G´ omez and A. Su´ arez, Non-linear versus linear diﬀusion from classical solutions to metasolutions, Adv. Diﬀ. Eqns. 7 (2002), 1101–1124. [17] J.C. Eilbeck, J.E. Furter and J. L´ opez-G´ omez, Coexistence in the competition model with diﬀusion, J. Diﬀ. Eqns. 107 (1994), 96–139. [18] J.M. Fraile, P. Koch, J. L´ opez-G´ omez and S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a nonlinear elliptic problem, J. Diﬀ. Eqns. 127 (1996), 295–319. [19] J.E. Furter and J. L´ opez-G´ omez, On the existence and uniqueness of coexistence states for the Lotka-Volterra competition model with diﬀusion and spatially dependent coeﬃcients, Nonl. Anal. TMA 25 (1995), 363–398. [20] J.E. Furter and J. L´ opez-G´ omez, Diﬀusion-mediated permanence problem for a heterogeneous Lotka-Volterra competition model, Proc. Roy. Soc. Edinburgh 127A (1997), 281–336. [21] J. Garc´ıa-Meli´ an, R. G´ omez-Re˜ nasco, J. L´ opez-G´ omez and J.C. Sabina de Lis, Pointwise growth and uniqueness of positive solutions for a class of sublinear elliptic problems where bifurcation from inﬁnity occurs, Arch. Rat. Mech. Anal. 145 (1998), 261–289. [22] R. G´ omez-Re˜ nasco and J. L´ opez-G´ omez, The eﬀects of varying coeﬃcients on the dynamics of a class of superlinear indeﬁnite reaction-diﬀusion equations, J. Diﬀ. Eqns. 167 (2000), 36–72. [23] P. Hess and A. Lazer, On an abstract competition model and applications, Nonl. Anal. Th. Meth. Appns. 16 (1991), 917–940. [24] J. L´ opez-G´ omez, Estabilidad y Bifurcaci´ on Est´ atica. Aplicaciones y M´etodos Num´ ericos, Cuadernos de Matem´ atica y Mec´ anica, Cursos y Seminarios 4, CONICET, Santa Fe, 1988

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[25] J. L´ opez-G´ omez, Permanence under strong competition, in Dynamical Systems and Applications, pages 473–488, World Scientiﬁc Series in Applied Analysis 4, Singapore, 1995. [26] J. L´ opez-G´ omez, The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems, J. Diﬀ. Eqns. 127 (1996), 263– 294. [27] J. L´ opez-G´ omez, On the structure of the permanence region for competing species models with general diﬀusivities and transport eﬀects, Disc. Cont. Dyn. Sys. 2 (1996), 525–542. [28] J. L´ opez-G´ omez, Large solutions, metasolutions, and asymptotic behavior of the regular positive solutions of a class of sublinear parabolic problems, Elec. J. Diﬀ. Eqns. Conf. 05 (2000), 135–171. [29] J. L´ opez-G´ omez, Spectral Theory and Nonlinear Functional Analysis, Research Notes in Mathematics 426, CRC Press, Boca Raton (FL), 2001. [30] J. L´ opez-G´ omez, Coexistence and meta-coexistence for competing species, Houston J. of Maths., 29 (2003), 483–536. [31] J. L´ opez-G´ omez, Dynamics of parabolic equations. From classical solutions to metasolutions, Diﬀ. Int. Eqns. 16 (2003), 813–828. [32] J. L´ opez-G´ omez, The boundary blow-up rate of large solutions, J. Diﬀ. Eqns. 195 (2003), 25–45. [33] J. L´ opez-G´ omez, Metasolutions: Malthus versus Verhulst in population dynamics. A dream of Volterra, in Handbook of Diﬀerential Equations “Stationary Partial Diﬀerential Equations” (M. Chipot and P. Quittner, Eds.), Elsevier Science B. V., North Holland, to appear. [34] J. L´ opez-G´ omez, Global existence versus blow-up in superlinear indeﬁnite parabolic problems, Sci. Mat. Japonica 61 (2005), 493–517. [35] J. L´ opez-G´ omez, J.C. Eilbeck, M. Molina and K.N. Duncan, Structure of solution manifolds in a strongly coupled elliptic system, IMA J. Numer. Anal. 12 (1992), 405–428. [36] J. L´ opez-G´ omez and M. Molina-Meyer, Symbiosis in competitive environments, in Variational Methods and Related Topics, Vol. 12, pages 118–141, RIMS KYOTO 1347, Kyoto, 2003. [37] J. L´ opez-G´ omez and M. Molina-Meyer, Singular perturbations in Economy and Ecology. The eﬀect of strategic symbiosis in random competitive environments, Adv. Math. Sci. Appns. 14 (2004), 87–107. [38] J. L´ opez-G´ omez and M. Molina-Meyer, Biodiversity through competition, in Proc. 10th IEEE International Conference on Methods and Models in Automation and Robotics (R. Kaszy´ nski, Ed.), pages 39–44, Szczecin, 2004. omez, M. Molina-Meyer and M. Villarreal, Existence and numerical sim[39] J. L´ opez G´ ulation of coexistence states for some elliptic Lotka-Volterra models, Quadernos del IAC Mauro Picone 27 (1989), 119–133. [40] J. L´ opez G´ omez, M. Molina-Meyer and M. Villarreal, Numerical simulation of coexistence states, SIAM J. Numer. Anal. 29 (1992), 1074–1092. [41] J. L´ opez-G´ omez and R.M. Pardo, Coexistence regions in Lotka-Volterra models with Diﬀusion, Nonl. Anal. Th. Meth. Appns. 19 (1992), 11–28.

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[42] J. L´ opez-G´ omez and P. Quittner, Complete and energy blow-up in indeﬁnite superlinear parabolic problems, Disc. Cont. Dyn. Syst., to appear. [43] J. L´ opez-G´ omez and J.C. Sabina de Lis, Coexistence states and global attractivity for some convective diﬀusive competing species models, Trans. Amer. Math. Soc. 347 (1995), 3797–3833. [44] A. Parker, In the blink of an eye: the cause of the most dramatic event in the history of life, Free Press, London, 2003. [45] P.H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487–513. [46] D. Tilman, Diversity and Production in European Grasslands, Science 285 (1999), 1099–1100. [47] E.O. Wilson, Sociobiology, Harvard University Press, Cambridge, Massachusetts, 1980. Juli´ an L´ opez-G´ omez Dpto. de Matem´ atica Aplicada Univ. Complutense de Madrid E-28040 Madrid, Spain e-mail: Lopez [email protected] Marcela Molina-Meyer Dpto. de Matem´ aticas Univ. Carlos III de Madrid E-28911 Legan´es, Madrid, Spain e-mail: [email protected]

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 329–349 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Generalized Minimal Cardinal of the λ-slices of the Semi-bounded Components Arising in Global Bifurcation Theory Juli´an L´opez-G´omez and Carlos Mora-Corral Dedicated to H. Amann, with our deepest admiration

Abstract. We use the degree introduced by P. Benevieri and M. Furi [3] to obtain the generalized minimal cardinal of the number of solutions of the λsections of those components of the set of nontrivial solutions of an abstract equation of the form F(λ, x) = 0 that are compact in one direction of the parameter; here F is a C 1 Fredholm map of index 1 such that F(λ, 0) = 0 for all λ ∈ R. These bounds are given in terms of the parity of the linearized Fredholm family D2 F(·, 0). The parity is a local invariant measuring the change of the orientation of D2 F(λ, 0) as λ crosses an interval. The set of eigenvalues of D2 F(·, 0) is not assumed to be discrete. Therefore, as regards applications, the theory developed in this paper should be extremely versatile. Mathematics Subject Classiﬁcation (2000). 47J15, 58C30, 47H11, 58C40. Keywords. Bifurcation theory. Counting the number of solutions of λ-sections.

1. Introduction In this paper, if E and F are two real Banach spaces, the set L(E, F ) denotes the space of linear bounded operators from E to F , and Φ0 (E, F ) stands for the subset of L(E, F ) consisting of all Fredholm operators of index zero. Throughout this work, F: R×E →F is a continuous map satisfying the following properties: F1. For each λ ∈ R, the partial map F(λ, ·) ∈ C 1 (E, F ) and D2 F(λ, x) ∈ Φ0 (E, F ) for all x ∈ E, where D2 stands for derivation with respect to x ∈ E. F2. The map D2 F : R × E → Φ0 (E, F ) is continuous. F3. F(λ, 0) = 0 for each λ ∈ R.

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Our main goal is to analyze the global topological behavior of the compact components of the set of nontrivial solutions of the equation F(λ, x) = 0 ,

λ∈J,

(1.1)

where, for some ﬁxed Λ ∈ R, J ∈ {R, (−∞, Λ], [Λ, ∞)} . As usual, by a component it is meant a maximal (for the inclusion) connected subset. Under assumption F3, R × {0} provides us with a solution curve of (1.1) in R × E. A bifurcation value of F = 0 from R × {0} is any λ0 ∈ R for which (λ0 , 0) belongs to the closure of S := F−1 (0) ∩ [R × (E \ {0})]. The set of nontrivial solutions of F = 0 is then deﬁned as the union of S with B × {0}, where B ⊂ R is a set of possible bifurcation values; the precise concept will be given in Section 4. In P. Benevieri and M. Furi [5] some bifurcation results concerning (1.1) for the special case where J = R were derived: a local one, [5, Theorem 3.1], in the spirit of M.A. Krasnosel’skii [16]; and two global ones, [5, Theorems 3.2 and 3.6], in the spirit of P.H. Rabinowitz [25]. The aim of this paper is twofold: • To show that the sign jump (a key concept ensuring bifurcation, going back within our setting to [5, Theorem 3.1] and reminiscent of the concept of crossing number attributable to H. Poincar´e) can be detected by means of the algebraic multiplicity of D2 F(·, 0) (cf. [18], [23], and the list of references therein). • To ascertain the generalized minimal number of solutions of the λ-slices of the compact components of the set of nontrivial solutions of (1.1), for an unbounded interval J, through the signature of the component; this shares the spirit of [20] and [21], where the special case when the map (λ, x) → F(λ, x)−x is completely continuous was dealt with. Although in this paper F is slightly more regular than in [20] and [21], where F was imposed to be completely continuous, it should be noted that here F is allowed to be a general Fredholm operator of index 1, not necessary a compact perturbation of the identity; essentially, to avoid compactness, one must impose C 1 regularity. Consequently, we are substantially enlarging the range of applicability of the abstract theory developed therein. Fig. 3 in [21] illustrates how a compact component of (1.1) in case J = R might be a subset of a larger unbounded component of the set of nontrivial solutions of F in the whole space R × E, for which no available result in the literature applies, except those coming from [20] and [21]. Consequently, the methodology adopted here might result into a breakthrough in global bifurcation theory. Actually, the classic results by P.H. Rabinowitz [25], E.N. Dancer [7], R.J. Magnus [22] and J. Ize [13] exclusively treated the case J = R, as well as most of the work originated from them, obtaining or reﬁning the so-called global alternative of

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P.H. Rabinowitz, which reads as follows. If J = R and C is a component of (1.1) emanating from R× {0} at a nonlinear eigenvalue λ0 ∈ R – bifurcation value based on the linear part –, then some of the following (non-exclusive) options occurs: 1. The component C is unbounded in R × E. 2. There is another nonlinear eigenvalue λ1 ∈ R \ {λ0 } such that (λ1 , 0) ∈ C. This result is certainly seminal. Its importance and popularity is due to its huge number of applications in wide areas of nonlinear diﬀerential equations, as it provides a universal property (very appealing and easily remembered) satisﬁed by the compact components of the set of non-trivial solutions; in many circumstances, it can be used to conclude that C is unbounded. Thus, that result provides one of the most powerful techniques to get general existence results in nonlinear eigenvalue problems. This is the case, for instance, if C consists of non-negative solutions of a nonlinear equation satisfying a generalized maximum principle in an ordered Banach space (cf. [19] and the list of references therein). Almost simultaneously with the publication of the seminal theorems of P.H. Rabinowitz [25] and the sharpened versions of E.N. Dancer [7], H. Amann and S.A. Weiss obtained their celebrated uniqueness theorem of the Leray-Schauder topological degree, which appeared in [2] after a rather lengthy elaboration process. The theorem of Amann and Weiss axiomatized the topological degree, which has shown to be the most powerful topological tool for nonlinear analysis developed in the 20th century. At that time, H. Amann had already familiarized himself with the use of the topological degree in obtaining a number of multiplicity results within the context of abstract ﬁxed point equations in ordered Banach spaces and its applications to the search of positive solutions in nonlinear elliptic eigenvalues problems. Undoubtedly, those results, collected by himself in one of the most inﬂuential references on nonlinear analysis of the second half of the 20th century, [1], have shown to be a milestone in the development of modern nonlinear analysis. The spirit of H. Amann [1] has also impregnated the present honoring paper, where the topological degree for Fredholm mappings is used to count the minimal number of solutions of the λ-slices of all bounded components of (1.1), so giving the topological degree the original use for which it was introduced by L.E.J. Brouwer [6] and further extended by J. Leray and J. Schauder [17]. Consequently, the spirit of this paper is extremely classic, strongly inﬂuenced by P.H. Rabinowitz and H. Amann, where some new mathematical challenges are addressed. Among them, the development, or eventual adaptation, of techniques from topological shape theory for ascertaining the minimal “topological complexity” of the compact components of (1.1) – a rather natural question once one knows the minimal number of solutions of the λ-slices of C in terms of its signature, which concept we will introduce in Section 7. The distribution of this paper is as follows. In Section 2 we recall the concepts and properties of orientation and topological degree for Fredholm maps. In Section 3 it is shown that the algebraic multiplicity detects every change of orientation of any Φ0 -valued family of real variable at an algebraic eigenvalue; as a by-product, it

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provides us with a suﬃcient condition for a real number to be a bifurcation value from R×{0}. The change of orientation is, therefore, the counterpart of the change of the Leray-Schauder index. In Section 4 we introduce the concepts of admissible families and associated parities. Roughly speaking, an admissible family is a locally ﬁnite family of disjoint open intervals such that D2 F(λ, 0) is an isomorphism for each λ in each of those intervals, and the parity is a local invariant measuring the change of orientation of D2 F(λ, 0) as λ crosses one of those intervals. Section 5 gives a formula for the degree of F in an appropriate open set containing a compact component of the set of nontrivial solutions of (1.1) by means of the associated parities; this is the main result of this paper. Section 6 applies the main theorem to establish some (optimal) lower bounds for the number of solutions of the λ-sections (obtained by ﬁxing the value of the parameter λ) of any compact component of the set of nontrivial solutions of (1.1). Finally, in Section 7 the concepts of signature and generalized minimal cardinal are introduced. Then, the reader is sent to the ﬁnal sections of [21] for a detailed discussion illustrating the profoundness of the abstract theory developed here. Throughout this paper, the subset of L(E, F ) consisting of the topological isomorphisms is denoted by Iso(E, F ), and IE stands for the identity map in E.

2. Topological degree for Fredholm maps This section collects the concepts of orientation and topological degree for C 1 Fredholm maps given in P. Benevieri and M. Furi [3, 4, 5], which sharpen those going back to P.M. Fitzpatrick and J. Pejsachowicz [9] (cf. [4] for a comparison with other related constructions). Let E, F, G be three real Banach spaces, and, for each L ∈ Φ0 (E, F ), let C(L) denote the (non-empty) set of ﬁnite-rank operators A ∈ L(E, F ) for which L + A ∈ Iso(E, F ). In C(L) an equivalence relation can be deﬁned by declaring that A, B ∈ C(L) are equivalent, A ∼L B, if & % det (L + B)−1 (L + A) > 0 . Note that (L + B)−1 (L + A) = IE + (L + B)−1 (A − B) is a ﬁnite-rank perturbation of the identity and, hence, its determinant is well deﬁned (see, e.g., T. Kato [14, Section III.§4.3]). The relation ∼L has exactly two equivalence classes. Each of these equivalence classes is called an orientation of L. The operator L is said to be oriented once an orientation has been chosen. In such a case, the orientation given to L will be denoted by C+ (L), and we will call C− (L) := C(L) \ C+ (L). Once an operator L ∈ Iso(E, F ) has been oriented, its sign, denoted by sgn L, is deﬁned through 1, if 0 ∈ C+ (L) , sgn L := −1 , if 0 ∈ C− (L) .

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This notation should not be confused with the concept of sign of a real number sign a = a/|a| ,

a ∈ R \ {0} .

For each pair of oriented operators L1 ∈ Φ0 (E, F ) and L2 ∈ Φ0 (F, G), their oriented composition is deﬁned as the operator L2 L1 equipped with the product orientation C+ (L2 L1 ), which is taken as the orientation containing L2 A1 + A2 A1 + A2 L1 , where A1 ∈ C+ (L1 ) and A2 ∈ C+ (L2 ). This concept is consistent, as C+ (L2 L1 ) is independent of the choice of A1 and A2 . The following result is satisﬁed. Lemma 2.1. Let L1 ∈ Iso(E, F ) and L2 ∈ Iso(F, G) be two oriented isomorphisms and consider the oriented composition L2 L1 . Then, sgn(L2 L1 ) = sgn L2 sgn L1 . Proof. For each i ∈ {1, 2}, let Ai ∈ C+ (Li ). Then, thanks to the properties of the determinant (cf., e.g., T. Kato [14]), & det [(L2 L1 + L2 A1 +A2 A1 + A2 L1 )−1 (L2 L1 ) & % = det (L1 + A1 )−1 (L2 + A2 )−1 L2 L1 % & −1 L2 L1 = det (L1 + A1 )−1 L1 L−1 1 (L2 + A2 ) & % & % −1 L2 L1 = det (L1 + A1 )−1 L1 det L−1 1 (L2 + A2 ) & % & % = det (L1 + A1 )−1 L1 det (L2 + A2 )−1 L2 ,

which readily concludes the proof.

Now, let X be a topological space and L ∈ C(X, Φ0 (E, F )). An orientation of L is a map X x → α(x) such that α(x) is an orientation of L(x) for all x ∈ X, and, for each x0 ∈ X and A ∈ α(x0 ), there exists a neighborhood U of x0 in X such that = α(x). A∈ x∈U

Although L might not admit an orientation, the following result holds from P. Benevieri and M. Furi [3, 4, 5], where we send for further details. Proposition 2.2. Suppose X is a simply-connected topological space. Then, every L ∈ C(X, Φ0 (E, F )) possesses exactly two orientations, say C+ (L) and C− (L), and each of them is uniquely determined by the orientation of L(x) for an arbitrary x ∈ X. More precisely, for every x ∈ X and every orientation α(x) of L(x), there exists a unique ε ∈ {+, −} such that Cε (L)(x) = α(x). In this paper the space X will always be simply-connected, since X ∈ {I,E,I × E} for some interval I ⊂ R. Consequently, throughout the remaining of this paper X will be assumed to be simply connected, and every family L ∈ C(X, Φ0 (E, F ))

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will be thought of as oriented by C+ (L). Moreover, if g ∈ C 1 (E, F ) satisﬁes Dg(x) ∈ Φ0 (E, F ) for each x ∈ E, then it will be said that g is oriented once an orientation C+ (Dg) for Dg has been chosen. Similarly, any operator F ∈ C(R×E, F ) satisfying F1 and F2 of Section 1 will be assumed to be oriented, and this means that an orientation C+ (D2 F) has been chosen for D2 F. Subsequently, A stands for the set of pairs (g, U) consisting of an oriented function g ∈ C 1 (E, F ) satisfying Dg(x) ∈ Φ0 (E, F ) for all x ∈ E, and an open set U ⊂ E for which g −1 (0) ∩ U is compact. According to P. Benevieri and M. Furi [3], a Z-valued topological degree is deﬁned in A and satisﬁes the usual properties of normalization, additivity and homotopy invariance. In this paper, we denote by Deg(g, U) the degree of g in U with respect to the origin 0, according to [3]. The following result collects some of the main properties of this topological degree (cf. P. Benevieri and M. Furi [3, 5] for a proof). Throughout the remaining of this paper, given any subset S ⊂ R × E and λ ∈ R we will deﬁne Sλ := { x ∈ E : (λ, x) ∈ S } . Proposition 2.3. Suppose g ∈ C 1 (E, F ). Then, the following properties hold: 1. (Additivity) If U1 , U2 are two disjoint open subsets of an open set W ⊂ E such that (g, W ) ∈ A and g −1 (0) ∩ W ⊂ U1 ∪ U2 , then Deg(g, W ) = Deg(g, U1 ) + Deg(g, U2 ) . 2. (Regular-value formula) If (g, U) ∈ A, and Dg(x) ∈ Iso(E, F ) for every x ∈ g −1 (0) ∩ U, then sgn Dg(x). Deg(g, U) = x∈g−1 (0)∩U

3. (Homotopy invariance) Let Ω be an open subset of R × E, two real numbers a ≤ b, and F : R × E → F an oriented continuous function satisfying F1 and F2 of Section 1 such that {(λ, x) ∈ Ω : F(λ, x) = 0, λ ∈ [a, b]}

is compact.

Then, Deg(F(λ, ·), Ωλ ) is constant for each λ ∈ [a, b]. Let g ∈ C 1 (E, F ) and suppose that for some open subsets U1 , U2 of E and some x ∈ E, g −1 (0) ∩ [U1 ∪ U2 ] ⊂ {x} . Then, by the additivity property of Proposition 2.3, Deg(g, U1 ) = Deg(g, U2 ) . This common value will be denoted by Ind(x, g) and called the index of g at x.

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3. Change of orientation through the algebraic multiplicity In many circumstances, a generalized concept of algebraic multiplicity has been deﬁned for Φ0 -valued families of a real or complex variable (e.g., I.C. Gohberg and E.I. Sigal [12], P. Sarreither [26], R.J. Magnus [22], J. Ize [13], P.J. Rabier [24], or [18], and the references therein). Actually, it was shown in [23] that there is a unique possible concept of algebraic multiplicity. In this section we show that this algebraic multiplicity detects every change of orientation of an operator family at any algebraic eigenvalue, a concept going back to [18, Deﬁnition 4.3.1]. Our result extremely sharpens the corresponding theorems of P.M. Fitzpatrick and J. Pejsachowicz [8]. The following deﬁnition ﬁxes the concept of algebraic eigenvalue. Deﬁnition 3.1. Let Ω be an open interval of R, and L ∈ C r (Ω, Φ0 (E, F )), for some integer r ≥ 0. A value λ0 ∈ Ω is said to be an algebraic eigenvalue of L if there exist C, δ > 0, and an integer 0 ≤ k ≤ r such that L(λ) ∈ Iso(E, F ) for 0 < |λ − λ0 | < δ and 0 < |λ − λ0 | < δ . (3.1) L(λ)−1 L(F,E) ≤ C|λ − λ0 |−k , If k is the minimum integer number for which (3.1) holds for some C > 0 and δ > 0, then λ0 is said to be a k-algebraic eigenvalue. Note that λ0 is a 0-algebraic eigenvalue of L if and only if L(λ0 ) ∈ Iso(E, F ). If λ0 is an algebraic eigenvalue of L, then it was shown in [18] that the algebraic multiplicity χ[L; λ0 ] of L at λ0 can be deﬁned and is a natural number. In [18, Chapter 5] the theories of [18, Chapter 4] and R.J. Magnus [22] are combined to obtain the following result. Proposition 3.2. Let Ω be an open interval, and λ0 ∈ Ω an algebraic eigenvalue of L ∈ C r (Ω, Φ0 (E, F )). Then, one of the following exclusive alternatives occurs: (a) L(λ0 ) ∈ Iso(E, F ) and χ[L; λ0 ] = 0. (b) λ0 is a k-algebraic eigenvalue of L for some 1 ≤ k ≤ r, and there exist k ﬁnite-rank projections P1 , . . . , Pk ∈ L(E, E) \ {0} such that L(λ) = M(λ) [(λ − λ0 )P1 + IE − P1 ] · · · [(λ − λ0 )Pk + IE − Pk ] ,

λ ∈ Ω,

for a certain M ∈ C r−k (Ω, Φ0 (E, F )) satisfying M(λ0 ) ∈ Iso(E, F ). Moreover, k rank Pi , χ[L; λ0 ] = i=1

independently of the projections Pi chosen. The main theorem in this section establishes that the algebraic multiplicity χ[L; λ0 ] detects every sign jump (as discussed by P. Benevieri and M. Furi [5]) of L at an algebraic eigenvalue λ0 . It reads as follows.

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Theorem 3.3. Let Ω be an open interval, and λ0 ∈ Ω a k-algebraic eigenvalue of the oriented family L ∈ C r (Ω, Φ0 (E, F )), for some 0 ≤ k ≤ r. Then, sgn L(λ) changes as λ crosses λ0 if and only if χ[L; λ0 ] is odd. Proof. Suppose L(λ0 ) ∈ Iso(E, F ) and, for example, sgn L(λ0 ) = 1. Then, by deﬁnition of orientation, 0 ∈ C+ (L(λ)) for λ * λ0 , and, hence, sgn L(λ) = 1 for λ * λ0 . Suppose L(λ0 ) is not invertible. Then, due to Proposition 3.2(b), we can decompose L in the form L(λ) = M(λ)f1 (λ) · · · fk (λ),

λ ∈ Ω,

(3.2)

where the family M ∈ C(Ω, Φ0 (E, F )) satisﬁes M(λ0 ) ∈ Iso(E, F ), and fi (λ) = (λ − λ0 )Pi + IE − Pi ,

λ ∈ Ω,

1 ≤ i ≤ k,

for some ﬁnite-rank projections P1 , . . . , Pk ∈ L(E, E) \ {0}. Clearly, the statement of the proposition is independent of the orientation chosen for L. We choose the following orientations: for each 1 ≤ i ≤ k, the orientation of fi (λ0 ) is deﬁned as Pi ∈ C+ (fi (λ0 )), and 0 ∈ C+ (M(λ0 )). This determines the orientations C+ (fi ) for 1 ≤ i ≤ k, and C+ (M). The orientation of L is deﬁned as the product orientation given by (3.2). Now ﬁx 1 ≤ i ≤ k and λ * λ0 with λ = λ0 . Then, Pi ∈ C+ (fi (λ)) and rank Pi % & det fi (λ)−1 (fi (λ) + Pi ) = det IE + (λ − λ0 )−1 Pi = 1 + (λ − λ0 )−1 . Consequently, sgn fi (λ) = sign(λ − λ0 )rank Pi ,

λ * λ0 ,

λ = λ0 .

Therefore, by Lemma 2.1 and Proposition 3.2(b), it is apparent that sgn L(λ) = sign(λ − λ0 )χ[L;λ0 ] , which concludes the proof.

λ * λ0 ,

λ = λ0 ,

The relevance of Theorem 3.3 comes from the fact that, according to P. Benevieri and M. Furi [5, Theorem 3.1], a change of sgn L(λ) as λ crosses λ0 entails λ0 being a bifurcation value of F = 0 from R × {0} for every continuous function F : R × E → F vanishing at Ω × {0} and such that D2 F(·, 0) = L. Therefore, combining [5, Theorem 3.1] with Theorem 3.3 gives rise to the next result. Theorem 3.4. If λ0 is an algebraic eigenvalue of L with χ[L; λ0 ] ∈ 2N + 1, then λ0 is a bifurcation value from R × {0} for every continuous function F : R × E → F satisfying F1, F2 and F3 of Section 1 such that D2 F(·, 0) = L. Actually, F needs only to be deﬁned in a neighborhood of (λ0 , 0). Also, condition χ[L; λ0 ] ∈ 2N + 1 is optimal even in the classic setting where the map (λ, x) → F(λ, x) − x is compact; see [18, Theorem 4.3.4] for the precise statement and proof of this fact.

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4. Admissible families and associated parities To introduce the concepts of admissible families and associated parities, it is convenient to recall some elementary facts about locally ﬁnite families. A family A whose elements are subsets of a topological space X is said to be locally ﬁnite if for each x ∈ X there exists a neighborhood Ω of x such that the set {A ∈ A : A ∩ Ω = ∅} is ﬁnite. An elementary property of locally ﬁnite families is the fact that for every compact K of X and every locally ﬁnite family A, the set {A ∈ A : A ∩ K = ∅} is ﬁnite. The next lemma shows that every locally ﬁnite family of disjoint open intervals can be ordered ; these are the families with which we will work. Given A, B two non-empty subsets of R, we will write A < B when a < b for all a ∈ A and b ∈ B. If A = {a} (respectively, B = {b}) the notation a < B (respectively, A < b) will also be used. Lemma 4.1. Let A be a non-empty, locally ﬁnite family of disjoint, non-empty open intervals of R. Then, A is ﬁnite or countable, and there exist r, s ∈ Z ∪ {−∞, ∞} with r ≤ s such that A can be expressed as A = {Ai }si=r , where Ai−1 < Ai for all i ∈ Z ∩ [r + 1, s]. Proof. For each n ∈ N deﬁne An := > {A ∈ A : [−n, n]∩A = ∅}, which is a ﬁnite set ∞ because A is locally ﬁnite. As A = n=1 An , then A is countable. Fix n ∈ N such that An is non-empty; as it is formed by a ﬁnite number of disjoint, non-empty open intervals, then there exist p0 , q0 ∈ Z with p0 ≤ q0 such that An = {Ap0 , . . . , Aq0 } with Ap0 < · · · < Aq0 . It is apparent that An+1 = {Ap1 , . . . , Aq1 } for some integers p1 , q1 with p1 ≤ p0 ≤ q0 ≤ q1 , and Ap1 < · · · < Aq1 . An inductive process concludes the proof. Deﬁnition 4.2. Let r, s ∈ Z ∪ {−∞, ∞} be with r ≤ s. The family {Ji }si=r is said to be an admissible family for a given set Σ ⊂ R (or just an admissible family, if Σ is clear from the context) when it is a locally ﬁnite family of open and disjoint intervals Ji ⊂ R such that Ji = ∅ and Ji ∩ Σ = ∅ for all i ∈ Z ∩ [r, s]. In such a case, according to Lemma 4.1, we can (and will) suppose that Ji−1 < Ji for all i ∈ Z ∩ [r + 1, s]. Now, let L ∈ C(R, Φ0 (E, F )) be an oriented family, consider the closed set Σ := {λ ∈ R : L(λ) ∈ / Iso(E, F )} and an admissible

family {Ji }si=r s 3

(4.1)

for Σ. The map

Ji λ → sgn L(λ) ∈ {−1, 1}

i=r

is continuous, and the proof of this fact is essentially the ﬁrst paragraph of the proof of Theorem 3.3. Therefore, for each i ∈ Z ∩ [r, s] there exists ai ∈ {−1, 1} such that sgn L(λ) = ai for every λ ∈ Ji . We deﬁne the parity map P associated with L and with the admissible family {Ji }si=r as ai − ai−1 . (4.2) P : Z ∩ [r + 1, s] → {−1, 0, 1} , P (i) = 2

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Note that, when one sets Γ0 := {i ∈ Z ∩ [r + 1, s] : ai−1 = ai } ,

Γ1 := {i ∈ Z ∩ [r + 1, s] : ai−1 = ai } ,

the parity P satisﬁes the following properties: • P (i) = 0 if i ∈ Γ0 . • P (i) ∈ {−1, 1} if i ∈ Γ1 . • P (i)P (j) = −1 if i, j ∈ Γ1 with i < j and (i, j) ∩ Γ1 = ∅. Moreover, every map deﬁned in Z ∩ [r + 1, s] satisfying these three properties must equal P or −P . Thus, either set, Γ0 and Γ1 , determines P up to a sign switch. The knowledge of P up to a sign switch suﬃces in order to apply the results in the next sections. This fact also increases the relevance of Theorem 3.3, since it helps to compute the sets Γ0 and Γ1 . The family of compact intervals {Ii }si=r+1 associated with the admissible family {Ji }si=r is deﬁned by Ii := [sup Ji−1 , inf Ji ],

i ∈ Z ∩ [r + 1, s].

(4.3)

Each of these Ii is non-empty, but it may consist of a single point. Also, note that Ii < Ii+1 for i ∈ Z ∩ [r + 1, s − 1]. The proof of following lemma, being elementary, is omitted here. Lemma 4.3. The family {Ii }si=r+1 is locally ﬁnite. As a consequence of Lemma 4.3, if K is a compact subset of R × E, then the set {i ∈ Z ∩ [r + 1, s] : (Ii × {0}) ∩ K = ∅} is ﬁnite, since {Ii × {0}}si=r+1 is locally ﬁnite in R × E. Throughout the remaining of this paper, we consider a continuous map F : R × E → F satisfying F1, F2 and F3 of Section 1 with D2 F(·, 0) = L, and also an admissible family {Ji }si=r for the set Σ deﬁned in (4.1); we then deﬁne

s 3 % & −1 (4.4) S := S ∪ R \ Ji × {0} . S := F (0) ∩ R × (E \ {0}) , i=r

The set S will be subsequently referred to as the set of nontrivial solutions of F = 0 with respect to the admissible family {Ji }si=r . According to the next lemma, S is closed. Lemma 4.4. The set S is closed in R × E. Proof. It suﬃces to show that S¯ ⊂ S. Let {(λn , xn )}n∈N be a sequence in S with limit (λ, x) ∈ R×E. By continuity, we have F(λ, x) = 0. If x = 0, then (λ, x) ∈ S. If x = 0, then λ is a bifurcation value from >s R × {0} and, due to the Implicit Function Theorem, λ ∈ Σ; thus, (λ, 0) ∈ (R \ i=r Ji ) × {0}. Although F is not of class C 1 in its two variables, it has enough regularity to apply the Implicit Function Theorem (e.g., S.G. Krantz and H.R. Parks [15, Theorem 3.4.10]).

Generalized Minimal Cardinal

339

The most common situation arising in the literature occurs when Σ is discrete. Then, the optimal choice of an admissible family consists in taking the components (open intervals) of R \ Σ. In this case, every Ii consists of a single point, and s 3 Σ= Ii . i=r+1

5. The topological degree of F in terms of the parity map Subsequently, we consider an oriented continuous map F : R × E → F satisfying F1, F2 and F3 of Section 1 with L := D2 F(·, 0), an admissible family {Ji }si=r for the set Σ deﬁned in (4.1), the parity P associated with L and with {Ji }si=r , and the associated family {Ii }si=r+1 deﬁned through (4.3). Also, this section supposes that, for some Λ ∈ R, J ∈ R , (−∞, Λ] , [Λ, ∞) , (5.1) and that C is a compact component of S ∩ (J × E). Moreover, we deﬁne B := {i ∈ Z ∩ [r + 1, s] : (Ii × {0}) ∩ C = ∅} . The set B is ﬁnite, since C is compact and the family {Ii × {0}}si=r+1 is locally ﬁnite. Note that 3 Ii ) × {0} ⊂ C, (J ∩ i∈B

because C is a>component of S∩(J ×E). Also, every λ ∈ R\ s satisﬁes λ ∈ i=r+1 Ii , since C is bounded. Therefore, 3 Ii × {0} . C ∩ R × {0} = J ∩

>s i=r

Ji with (λ, 0) ∈ C (5.2)

i∈B

The proof of the following topological property can be found in M. Furi and M.P. Pera [10, 11]. This lemma will be very useful in the proof of Proposition 5.3. Lemma 5.1. Suppose Y is a locally compact Hausdorﬀ topological space and Y0 ⊂ Y is compact. Then, some of the following alternatives occurs: 1. There exists a compact set in Y containing Y0 and having empty boundary. 2. There exists a connected set C in Y contained in Y \ Y0 such that C¯ is not compact and C¯ ∩ Y0 = ∅. The following concept will play a crucial role in the subsequent analysis. The ball of radius R > 0 centred at (λ, x) ∈ R × E is denoted by BR (λ, x). Deﬁnition 5.2. Let β > 0. An open set Ω ⊂ R × E is said to be an open β-isolating neighborhood of C in J × E if C ⊂ Ω ⊂ C + Bβ (0, 0) and the set

Ω∩ S∪

3 i∈B

Ii × {0}

∩ (J × E)

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J. L´opez-G´omez and C. Mora-Corral

is compact, where S is the set deﬁned in (4.4). An open set Ω ⊂ R × E is said to be an open isolating neighborhood of C in J × E if it is an open β-isolating neighborhood of C in J × E for some β > 0. The following result establishes the existence of open β-isolating neighborhoods. Proposition 5.3. Suppose (5.1) and C is a compact component of S ∩ (J × E). Then, for each β > 0, the set C possesses an open β-isolating neighborhood in J × E. Proof. Let β > 0 and consider 3 D := S ∪ Ii × {0} , Y0 := C,

% & Y := D ∩ C + Bβ (0, 0) ∩ (J × E),

i∈B

where S is deﬁned in (4.4). Let us prove that D is locally compact. Let (λ, x) ∈ D; in order to construct a compact neighborhood of (λ, x) in D we distinguish two cases. First, we suppose that x = 0; let r > 0 be such that Br (λ, x) ∩ (R × {0}) = ∅. As P. Benevieri and M. Furi [5, Lemma 3.5] showed, F is locally proper and, hence, F−1 (0) is locally compact. Thus, there exists a compact neighborhood K of (λ, x) in F−1 (0) such that K ⊂ Br (λ, x). Then, K is a compact neighborhood of (λ, x) in D. >s > > Secondly, suppose that x = 0. Then, λ ∈ i∈B Ii ; as i=r Ji ∪ i∈B Ii is open in R, there exists r > 0 such that s 3 3 (λ − r, λ + r) ⊂ Ji ∪ Ii . i=r

i∈B

−1

As F (0) is locally compact, there exists a compact neighborhood K of (λ, 0) in F−1 (0) such that K ⊂ Br (λ, 0). Then, 3 K := K \ [(R \ Ii ) × {0}] i∈B

is a compact neighborhood of (λ, 0) in D. Indeed, the only not immediate fact is that K is compact, and this will we proved as soon as we show that K is closed in>K. So let {(λn , xn )}n∈N be a sequence in K with limit (λ0 , 0) such that λ0 ∈ R \ i∈B Ii . We shall derive a contradiction. By the election of r we have that >s value. Then, there exists n0 ∈ N such that λ0 ∈ i=r Ji , so λ0 is not a bifurcation> xn = 0 for all n ≥ n0 . Therefore, λn ∈ i∈B Ii for all n ≥ n0 , a contradiction. So D is locally compact. Therefore, Y is a locally closed subset of the locally compact space D, so Y is locally compact. The set Y0 is compact and, due to (5.2), it is contained in Y . Now we prove that Alternative 2 of Lemma 5.1 cannot happen. Let C be a connected subset of Y \ Y0 such that C¯ ∩ Y0 = ∅, where C¯ stands for the closure of C in R × E. Then C¯ is a connected set of S ∩ (J × E) with non-empty intersection with C, and, hence, C¯ ⊂ C. Thus, C¯ ∩ Y = C¯ is compact. Therefore, Alternative 1 of Lemma 5.1 must occur. According to it, there exists a

Generalized Minimal Cardinal

341

compact set K with empty boundary in Y and such that Y0 ⊂ K ⊂ Y . Necessarily, K is open in Y and consequently K = U ∩ Y for some U open set in R × E. The set U ∩ [C + Bβ (0, 0)] constitutes an open β-isolating neighborhood of C in J × E. Subsequently, BR denotes the ball of radius R centred at the origin in E. Also, recall that for a subset S ⊂ R × E and λ ∈ R, the set Sλ is formed by all x ∈ E such that (λ, x) ∈ E, and, as a usual convention, a summation over the empty set is taken as zero. The main result of this section reads as follows. Theorem 5.4. Suppose (5.1) and C is a compact component of S ∩ (J × E).>Let Ω be an open isolating neighborhood of C in J × E. Then, for each λ∗ ∈ J \ i∈B Ii there exists ρ∗ > 0 such that for all 0 < ρ ≤ ρ∗ , ¯ρ = 2 sign J ∗ Deg F(λ∗ , ·), Ωλ∗ \ B P (i) , (5.3) i∈J ∗

where ∗

J := ∗

sign J :=

{i ∈ B : Ii > λ∗ } if J = [Λ, ∞), {i ∈ B : Ii < λ∗ } if J = (−∞, Λ], 1 if J ∗ = {i ∈ B : Ii > λ∗ }, −1 if J ∗ = {i ∈ B : Ii < λ∗ },

and J ∗ ∈ {{i ∈ B : Ii > λ∗ }, {i ∈ B : Ii < λ∗ }} if J = R. Proof. Suppose J = [Λ, ∞). Fix an orientation for F : R × E → F . By Deﬁnition 5.2, the set

3 Ii × {0} ∩ (J × E) K := Ω ∩ S ∪ i∈B

is compact. Now, we distinguish two diﬀerent cases. Suppose J∗ = ∅ .

(5.4)

Consider b > λ∗ such that Ω ⊂ (−∞, b)×E. The set K is disjoint from [λ∗ , b]×{0}, so there exists ρ∗ > 0 such that ¯ρ∗ ) = ∅. K ∩ ([λ∗ , b] × B This implies that for each 0 < ρ ≤ ρ∗ , ¯ρ ) : F(λ, x) = 0, λ ∈ [λ∗ , b] = K ∩ [λ∗ , b] × (E \ Bρ ) , (λ, x) ∈ Ω \ (R × B which is compact. Therefore, the homotopy invariance of Proposition 2.3 shows that ¯ρ = Deg F(b, ·), Ωb \ B ¯ρ = 0, Deg F(λ∗ , ·), Ωλ∗ \ B because Ωb = ∅. This concludes the proof of the theorem under condition (5.4). Now, suppose J ∗ = {i1 , . . . , iM } , (5.5)

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instead of (5.4), where i1 < · · · < iM . Take b ∈ R such that Ω ⊂ (−∞, b) × E. Let δ > 0 be such that 3% & (J ∩ Ii ) × {0} + Bδ (0, 0) ⊂ Ω, i∈B

the intervals Ii + (−δ, δ) are disjoint for i ∈ B, and 3 / Ii + (−δ/2, δ/2). λ∗ ∈ i∈B

Also, δ > 0 can be chosen so that, when we denote ∗ s+ 0 := λ ,

s− M+1 := b,

s− j := inf Iij −δ/2,

s+ j := sup Iij +δ/2,

1≤j≤M,

+ we have that s− j ∈ Jij −1 and sj ∈ Jij for 1 ≤ j ≤ M . We observe that for all 1 ≤ j ≤ M, % − + & − + + (λ, x) ∈ Ω : F(λ, x) = 0, λ ∈ [s− ∪ [sj , sj ] × {0} j , sj ] = K ∩ [sj , sj ] × E

is compact. Therefore, by the homotopy invariance of Proposition 2.3, + Deg(F(s− j , ·), Ωs− ) = Deg(F(sj , ·), Ωs+ ), j

Now, since K is disjoint from K∩

>M

+ − j=0 ([sj , sj+1 ] × {0}),

M 3

1 ≤ j ≤ M.

j

(5.6)

there exists ρ∗ > 0 such that

− ¯ [s+ j , sj+1 ] × Bρ∗ = ∅.

j=0

This implies that for each 0 < ρ ≤ ρ∗ and 0 ≤ j ≤ M , the set ¯ρ ) : F(λ, x) = 0, λ ∈ [s+ , s− ] = K ∩ [s+ , s− ] × (E \ Bρ ) (λ, x) ∈ Ω \ (R × B j j+1 j j+1 is compact. By the homotopy invariance of Proposition 2.3, there exist d0 , . . . , dM ∈ Z such that ¯ρ = dj , λ ∈ [s+ , s− ], 0 ≤ j ≤ M. Deg F(λ, ·), Ωλ \ B j

j+1

Moreover, dM = 0, because Ωb = ∅. By the additivity and the regular-value formula of Proposition 2.3, for each ρ > 0 small enough and each 1 ≤ j ≤ M , we have Deg F(s− = dj−1 + sgn D2 F(s− j , ·), Ωs− j , 0), j + = dj + sgn D2 F(s+ Deg F(sj , ·), Ωs+ j , 0). j

These equalities together with (5.6) and (4.2) give dj−1 − dj = 2P (ij ),

1 ≤ j ≤ M.

Thus, adding them and taking into account that dM = 0, we get d0 = 2

M j=1

P (ij )

Generalized Minimal Cardinal

343

and, therefore, M ¯ρ = 2 Deg F(λ∗ , ·), Ωλ∗ \ B P (ij ) .

(5.7)

j=1

Thanks to (5.5), equation (5.7) provides us with (5.3) when J = [Λ, ∞). The case when J = (−∞, Λ] can be proved similarly. Consequently, we omit the details of its proof. Finally, suppose J = R, let C be a compact component > of S and consider an open isolating neighborhood Ω of C in R × E, and λ∗ ∈ R \ i∈B Ii . Take a Λ ≤ λ∗ such that Ω ⊂ [Λ, ∞) × E. Then, C is a component of S ∩ ([Λ, ∞) × E), and Ω is an open isolating neighborhood of C in [Λ, ∞) × E. Therefore, thanks to the ﬁrst part of the proof, ¯ρ = 2 Deg F(λ∗ , ·), Ωλ∗ \ B P (i). i∈B, Ii >λ∗

Similarly,

¯ρ = −2 Deg F(λ∗ , ·), Ωλ∗ \ B

P (i) ,

i∈B, Ii <λ∗

which completes the proof.

Theorem 5.4 provides us with a valuable information about the structure of the compact components of S. The following result should be compared with R. J. Magnus [22, Theorem 2.10], E. N. Dancer [7, Corollary 2] and P. Benevieri and M. Furi [5, Theorem 3.2]. It constitutes a substantial extension of the main results in these references. Actually, it is the result from which the global alternative of P.H. Rabinowitz [25] readily follows. Corollary 5.5. Suppose C is a compact component of S. Then, P (i) = 0 . i∈B

6. Lower estimates of Card Cλ In this section we use Theorem 5.4 to estimate the number of elements of Cλ . The notation J ∗ introduced in the statement of Theorem 5.4 will be subsequently maintained. Theorem 6.1. Suppose (5.1) and C is a>compact component of S∩(J ×E). Suppose, in addition, that there exist λ∗ ∈ J \ i∈B Ii and n ∈ N diﬀerent isolated zeros of F(λ∗ , ·), say x1 , . . . , xn ∈ Cλ∗ , such that Ind(xj , F(λ∗ , ·)) ∈ {−1, 0, 1} , Set

1 ≤ j ≤ n.

n± := Card j ∈ {1, . . . , n} : Ind xj , F(λ∗ , ·) = ±1 .

(6.1) (6.2)

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Then, Card Cλ∗ ≥ n + 1

whenever

2 P (i) = |n+ − n− | .

(6.3)

i∈J ∗

In particular, Card Cλ∗ ≥ n + 1 if n+ + n− ∈ 2N + 1. Proof. Suppose Cλ∗ = {x1 , . . . , xn }. Necessarily, xj = 0 for each 1 ≤ j ≤ n, since > ¯ρ = ∅ for suﬃciently small ρ > 0. The λ∗ ∈ J \ i∈B Ii and (5.2). Thus, Cλ∗ ∩ B set C is compact and has empty intersection with Z := {λ∗ } × x ∈ E \ Cλ∗ : F(λ∗ , x) = 0 , which is closed since xj is an isolated zero of F(λ∗ , ·) for each 1 ≤ j ≤ n. Therefore, there exists β > 0 such that (C + Bβ (0, 0)) ∩ Z = ∅. By Proposition 5.3, there exists Ω an open isolating neighborhood of C in J × E such that Ω ∩ Z = ∅. Then, ¯ρ : F(λ∗ , x) = 0 Cλ∗ = x ∈ Ωλ∗ \ B and, according to the additivity property of Proposition 2.3, we ﬁnd that n ¯ρ = Deg F(λ∗ , ·), Ωλ∗ \ B Ind xj , F(λ∗ , ·) . j=1

Therefore, it follows from Theorem 5.4 that P (i) = |n+ − n− | , 2 i∈J ∗

which completes the proof of (6.3). Finally, if n+ + n− is odd, then n+ − n− is odd as well, which concludes the proof. Theorem > 6.2. Suppose (5.1), C is a compact component of S ∩ (J × E), and λ∗ ∈ J \ i∈B Ii is a parameter value for which Cλ∗ consists of exactly n ∈ N isolated zeros of F(λ∗ , ·), say x1 , . . . , xn , satisfying (6.1). Then, (6.4) P (i) . Card Cλ∗ ≥ 2 i∈J ∗

Moreover, Card Cλ∗ ∈ 2N if n − n+ − n− ∈ 2N, where n+ and n− are deﬁned in (6.2). Proof. We have 0 ∈ Cλ∗ , because of (5.2). Moreover, an adaptation of the argument given in the proof of Theorem 6.1 gives n 2 P (i) = Ind xj , F(λ∗ , ·) i∈J ∗

≤

j=1 n

| Ind(xj , F(λ∗ , ·))| = n+ + n− ≤ Card Cλ∗

j=1

and, therefore, (6.4) holds.

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Further, suppose n − n+ − n− ∈ 2N. Then, n ∗ P (i) = Ind xj , F(λ , ·) = |n+ − n− | , 2 i∈J ∗

j=1

and, hence, n+ + n− and n are even. This concludes the proof.

7. Concepts of Signature and Generalized Minimal Cardinal Strongly motivated by Theorem 6.2, the following concepts are rather natural. Deﬁnition 7.1. Suppose (5.1) and C is a compact component of S∩(J ×E); assume > / i∈B Ii if J ∈ {(−∞, Λ], [Λ; ∞)}. Then: CΛ = ∅ and Λ ∈ (a) When B = {i1 , . . . , iN } with i1 < · · · < iN , the signature of C in J × E is deﬁned by i1 ··· iN Signature[C; J × E] := , P (i1 ) · · · P (iN ) whereas the signature of C in J × E is said to be empty if B = ∅. (b) When B = {i1 , . . . , iN } with i1 < · · · < iN , the generalized minimal cardinal of C in J × E is the map 3 Ii → N GMC := GMC[C;J×E] : J \ i∈B

deﬁned by GMC(λ) := ⎧ ⎧ ⎫ ⎬ N ⎨ ⎪ ⎪ ⎪ , ⎪ 2 max 1, P (i ) if λ ∈ [Λ, inf Ii1 ) , ⎪ j ⎪ ⎩ ⎪ ⎪ ⎭ j=1 ⎨ ⎫ ⎧ ⎬ N ⎨ ⎪ ⎪ , if λ ∈ (sup Iik , inf Iik+1 ) , 1 ≤ k ≤ N − 1 , P (i ) 2 max 1, ⎪ j ⎪ ⎪ ⎩ ⎪ ⎭ j=k+1 ⎪ ⎪ ⎩ 0, if λ ∈ (sup IiN , ∞) , if J = [Λ, ∞), by GMC(λ) := ⎧ ⎧ ⎫ ⎬ N ⎨ ⎪ ⎪ ⎪ ⎪ , if λ ∈ (sup IiN , Λ] , 2 max 1, P (i ) ⎪ j ⎪ ⎭ ⎩ ⎪ ⎪ ⎨ ⎫ ⎧ j=1 ⎬ k ⎨ ⎪ ⎪ P (ij ) , if λ ∈ (sup Iik , inf Iik+1 ) , 1 ≤ k ≤ N − 1 , 2 max 1, ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ j=1 ⎪ ⎪ ⎩ 0, if λ ∈ (−∞, inf Ii1 ) ,

346

J. L´opez-G´omez and C. Mora-Corral if J = (−∞, Λ], and by GMC(λ) := ⎧ 0, ⎪ ⎧ ⎫ if λ ∈ (−∞, inf Ii1 ) ∪ (sup IiN , ∞) , ⎪ ⎨ ⎬ k ⎨ , if λ ∈ (sup Ii , inf Ii ) , 1 ≤ k ≤ N − 1 , 2 max 1, P (i ) ⎪ j k k+1 ⎪ ⎩ ⎩ ⎭ j=1 if J = R, whereas we take GMC := 0

if

B = ∅.

Note that GMC[C;J×E] is uniquely determined from the Signature of C in J × E, once the admissible family A for Σ is ﬁxed. With these concepts in mind, Theorem 6.2 may be rephrased as follows: GMC >[C;J×E] (λ) is a lower bound of Card Cλ for every value of λ ∈ J \ i∈B Ii for which Cλ consists of a ﬁnite number of isolated zeros of F(λ, ·) satisfying (6.1). Subsequently we will assume that J = [Λ, ∞). In the special case that the signature of C in J × E is empty we have deﬁned GMC[C;J×E] = 0 and it should be noted that, although Cλ might have an arbitrarily large number of solutions for some set of values of the parameter λ > Λ, it might have none for each λ > Λ. Actually, this occurs if Pλ C = {Λ}, where Pλ stands for the λ-projection operator. In the case when i1 i2 i1 i2 , , (7.1) Signature[C; J × E] ∈ −1 −1 1 1 according to Deﬁnition 7.1 we have that ⎧ ⎨ 4, 2, GMC[C;J×E] (λ) = ⎩ 0,

if λ ∈ [Λ, inf Ii1 ) , if λ ∈ (sup Ii1 , inf Ii2 ) , if λ > sup Ii2 .

Figure 1 shows some admissible components with signature (7.1) with the following additional assumptions: every λ ∈ [Λ, ∞)\(Ii1 ∪Ii2 ) is a strongly regular parameter value of C, Card CΛ = 4, Ii1 = {σ1 }, and Ii2 = {σ2 }. By a strongly regular parameter value it is meant any value of λ for which Cλ consists of exactly n ∈ N isolated zeros of F(λ, ·), say x1 , . . . , xn , satisfying Ind(xj , F(λ, ·)) ∈ {−1, 1} ,

1 ≤ j ≤ n.

It should be noted that, thanks to Theorem 6.1, Cλ must possess an even number of solutions for each λ ∈ [Λ, ∞) \ (Ii1 ∪ Ii2 ). The analysis carried out in the last sections of [21] showed that all situations illustrated in Figure 1 are admissible. The ﬁve cases shown by Figure 1 correspond with each of the possible cases accordingly to the number of arcs of C connecting {Λ} × CΛ to Ii1 × {0}.

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u

u

Λ

σ1

σ

u

Λ

λ

λ σ

2

(a)

σ

1

Λ

λ σ

2

σ

1

2

(c)

(b)

u

u

Λ σ

σ 1

λ 2

Λ σ

σ 1

λ 2

(e)

(d)

Figure 1. Some admissible components with Card CΛ = 4. Now, suppose i1 i2 i1 i2 i1 i2 i1 i2 , , , . ±1 ∓1 ±1 0 0 ±1 0 0

Signature[C; J × E] ∈

(7.2)

Then, GMC[C;J×E] (λ) =

2, 0,

if λ ∈ [Λ, inf Ii1 ) ∪ (sup Ii1 , inf Ii2 ) , if λ > sup Ii2 .

(7.3)

Figure 2 shows some admissible components with signature in the set (7.2) in the special case that every λ ∈ [Λ, ∞) \ (Ii1 ∪ Ii2 ) is a strong regular parameter value of C, Card CΛ = 2, Ii1 = {σ1 } and Ii2 = {σ2 }. Again, (7.3) provides us with the minimal number of solutions in all these cases. When some degenerate points appear along any of these curves of C, the number of solutions might drastically increase, or decrease, of course, but it seems the previous diagrams provide us with the minimal topological patterns that C should contain. The analysis already carried out in [21, Sections 6,7], where we send for further details, justiﬁes why GMC(λ) is indeed a generalized minimal cardinal of C.

348

J. L´opez-G´omez and C. Mora-Corral u

u

Λ

σ1

σ

(a)

λ 2

u

Λ

λ σ

σ

1

(b)

2

Λ

λ σ

σ

1

2

(c)

Figure 2. Some admissible components with Card CΛ = 2.

Acknowledgements The authors thank the Ministry of Education and Science of Spain for research support under Grant REN2003-00707. The second author also thanks the Spanish MECD for partial support, through a predoctoral fellowship under the supervision of J. L´ opez-G´omez.

References [1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), 620–709. [2] H. Amann and S.A. Weiss, On the uniqueness of the topological degree, Math. Z. 130 (1973), 39–54. [3] P. Benevieri and M. Furi, A simple notion of orientability for Fredholm maps of index zero between Banach manifolds and degree theory, Ann. Sci. Math. Qu´ebec 22 (1998), 131–148. [4] P. Benevieri and M. Furi, On the concept of orientability for Fredholm maps between real Banach manifolds, Topol. Meth. Nonl. Anal. 16 (2000), 279–306. [5] P. Benevieri and M. Furi, Bifurcation results for families of Fredholm maps of index zero between Banach spaces. Nonlinear analysis and its applications (St. John’s, NF, 1999), Nonlinear Anal. Forum 6 (2001), 35–47. ¨ [6] L.E.J. Brouwer, Uber Abbildung von Mannigfaltigkeiten, Math. Ann. 71 (1912), 97– 115. [7] E.N. Dancer, On the structure of solutions of non-linear eigenvalue problems, Ind. Univ. Math. J. 23 (1974), 1069–1076. [8] P.M. Fitzpatrick and J. Pejsachowicz, Parity and generalized multiplicity, Trans. Amer. Math. Soc. 326 (1991), 281–305. [9] P.M. Fitzpatrick and J. Pejsachowicz, Orientation and the Leray-Schauder theory for fully nonlinear elliptic boundary value problems, Memoirs of the American Mathematical Society 483, Providence, R. I., 1993.

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[10] M. Furi and M.P. Pera, A continuation principle for periodic solutions of forced motion equations on manifolds and applications to bifurcation theory, Paciﬁc J. Math. 160 (1993), 219–244. [11] M. Furi and M.P. Pera, Global Bifurcation of Fixed Points and the Poincar´e Translation Operator on Manifolds, Annali di Mat. Pura ed Appl. (IV) 173 (1997), 313–331. [12] I.C. Gohberg and E.I. Sigal, An Operator Generalization of the Logarithmic Residue Theorem and the Theorem of Rouch´e, Math. Sbornik 84(126) (1971), 607–629. English Trans.: Math. USSR Sbor. 13 (1971), 603–625. [13] J. Ize, Bifurcation theory for Fredholm operators, Memoirs of the American Mathematical Society 174, Providence, R. I., 1976. [14] T. Kato, Perturbation Theory for Linear Operators, Grundlehren der mathematischen Wissenschaften 132, Springer, Berlin 1980. [15] S.G. Krantz and H.R. Parks, The implicit function theorem. History, theory, and applications, Birkh¨ auser, Boston, MA, 2002. [16] M.A. Krasnosel’skii, Topological Methods in the Theory of Nonlinear Integral Equations, Gos. Izdat. Tehn.-Teor. Lit., Moscow, 1956. ´ [17] J. Leray and J. Schauder, Topologie et ´equations functionelles, Ann. Sci. Ecole Norm. Sup. S´er. 3 51 (1934), 45–78. [18] J. L´ opez-G´ omez, Spectral Theory and Nonlinear Functional Analysis, Chapman & Hall / CRC. Research Notes in Mathematics 426, Boca Raton, 2001. [19] J. L´ opez-G´ omez and M. Molina-Meyer, Bounded components of positive solutions of abstract ﬁxed point equations: mushrooms, loops and isolas, J. Diﬀ. Eqns. 209 (2005), 416–441. [20] J. L´ opez-G´ omez and C. Mora-Corral, Counting solutions of nonlinear abstract equations, Top. Meth. Nonl. Anal. 24 (2004), 307–335. [21] J. L´ opez-G´ omez and C. Mora-Corral, Minimal complexity of semi-bounded components in bifurcation theory, Nonl. Anal. T.M.A. 58 (2004), 749–777. [22] R.J. Magnus, A Generalization of Multiplicity and the Problem of Bifurcation, Proc. London Math. Soc. 32 (1976), 251–278. [23] C. Mora-Corral, On the Uniqueness of the Algebraic Multiplicity, J. London Math. Soc. 69 (2004), 231–242. [24] P.J. Rabier, Generalized Jordan chains and two bifurcation theorems of Krasnoselskii, Nonl. Anal. T.M.A. 13 (1989), 903–934. [25] P.H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487–513. [26] P. Sarreither, Transformationseigenschaften endlicher Ketten und allgemeine Verzweigungsaussagen, Math. Scand. 35 (1974), 115–128. Juli´ an L´ opez-G´ omez and Carlos Mora-Corral Facultad de Ciencias Matem´ aticas Departamento de Matem´ atica Aplicada Universidad Complutense de Madrid E-28040 Madrid, Spain e-mail: Lopez [email protected] e-mail: Carlos [email protected]

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 351–356 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Blow-up of Solutions of a Semilinear Heat Equation with a Visco-elastic Term Salim A. Messaoudi Abstract. In this work we consider an initial boundary value problem related to the equation t ut − ∆u + g(t − s)∆u(x, s)ds = |u|p−2 u 0

and prove, under suitable conditions on g and p, a blow-up result for solutions with negative or vanishing initial energy. This result improves an earlier one by the author. Mathematics Subject Classiﬁcation (2000). 35K05 - 35K65. Keywords. Blow-up, ﬁnite time, viscoelastic, relaxation function, vanishing energy.

1. Introduction In this work we study the ﬁnite-time blow-up of solutions for the following initial boundary value problem t g(t − s)∆u(x, s)ds = |u|p−2 u, x ∈ Ω, t>0 ut − ∆u + 0 (1.1) u(x, t) = 0, x ∈ ∂Ω, t ≥ 0 u(x, 0) = u0 (x),

x ∈ Ω,

where g : R+ → R+ is a bounded C 1 function, p > 2, and Ω is a bounded domain of Rn (n ≥ 1), with a smooth boundary ∂Ω. This type of equations arises from a variety of mathematical models in engineering and physical sciences. For example, in the study of heat conduction in materials with memory, the classical Fourier’s law of heat ﬂux is replaced by the following form t q = −d∇u − ∇ [k(x, t)u(x, τ )] dτ, (1.2) −∞

352

S.A. Messaoudi

where u is the temperature, d the diﬀusion coeﬃcient and the integral term represents the memory eﬀect in the material. The study of this type of equations has drawn a considerable attention see [3], [4], [12] , [14] [15]. From a mathematical point of view, one would expect the integral term to be dominated by the leading term in the equation. Therefore, the theory of parabolic equations applies to this type of equations. In the absence of the memory term (g = 0) problem (1.1) has been studied by various authors and several results concerning global and nonglobal existence have been established. For instance, in the early 1970’s Levine [8] introduced the concavity method and showed that solutions with negative energy blow up in ﬁnite time. Later, this method was improved by Kalantarov and Ladyzhenskaya [7] to accommodate more general situations. Ball [2] also studied (1.1) with f (u, ∇u) instead of |u|p−2 u and established a nonglobal existence result in bounded domains. This result had been extended to unbounded domains by Alfonsi and Weissler [1]. For the quasilinear case, Junning [6] studied ut − div(|∇u|m−2 ∇u) = f (u),

x ∈ Ω,

t>0

u(x, t) = 0,

x ∈ ∂Ω

t≥0

u(x, 0) = u0 (x),

x ∈ Ω,

(1.3)

and established a global existence result. He also proved a nonglobal existence result under the condition 1 4(m − 1) m |∇u0 (x)| dx − F (u0 (x))dx ≤ − u2 (x)dx (1.4) m Ω mT (m − 2)2 Ω 0 Ω

u where F (u) = 0 f (s)ds. More precisely he showed that if there exists T > 0, for which (1.4) holds then the solution blows up in a time less than T. This type of results have been extensively generalized and improved by Levine, Park, and Serrin in a paper [9], where the authors proved some global, as well as nonglobal, existence theorems. Their result, when applied to problem (1.3), requires that 1 |∇u0 (x)|m dx − F (u0 (x))dx < 0. (1.5) m Ω Ω We note that the inequality (1.5) implies (1.4). In a note, Messaoudi [10] extended the blow-up result to solution with initial datum satisfying 1 |∇u0 (x)|m dx − F (u0 (x))dx ≤ 0. (1.6) m Ω Ω In the present work, we consider (1.1) and prove, for suitable conditions on p and g, a blow-up result for solutions with negative or vanishing initial energy. This result improves an earlier one in [10].

Solutions of a Semilinear Heat Equation

353

2. Blow-up In order to state and prove our result we introduce the “modiﬁed” energy functional t 1 1 1 2 p g(s)ds) ∇u(t)2 − u(t)p (2.1) E(t) = (g + ∇u)(t) + (1 − 2 2 p 0 where

(g + v)(t) = 0

t

2

g(t − τ ) v(t) − v(τ )2 dτ.

For the relaxation function g and p, we assume that ∞ 1− g(s)ds = l > 0 g(s) ≥ 0, g (s) ≤ 0,

(2.2)

(2.3)

0

and

2(n − 1) , n > 2, p > 2, n = 1, 2. (2.4) n−2 By multiplying the equation in (1.1) by ut , integrating over Ω we get, after some manipulations, see [11], 1 d 1 2 2 E(t) = − g(t) ∇u(t)2 − (g + ∇u)(t) + |ut | ut dx ≤ 0, (2.5) dt 2 2 Ω 2
for regular solutions. The same result can be established, for weaker solutions and for almost every t, by a simple density argument. Similarly to [13], we give a deﬁnition for a strong solution of (1.1). Deﬁnition: A strong solution of (1.1) is a function u ∈ C([0, T ); H01 (Ω))∩C 1 ([0, T ); L2 (Ω)), satisfying (2.5) and t s p−2 ∇u(τ ).∇φ(s)dτ + ut φ − |u| uφ dxds = 0, ∇u.∇φ − 0

Ω

0

for all t in [0, T ) and all φ in C([0, T ), H01 (Ω)). Remark 2.1: Condition (2.4) is needed so that |u|p−2 u ∈ L2 (Ω); hence

∞ p−2 |u| uφdx makes sense. Condition 1 − 0 g(s)ds = l > 0 is necessary to Ω guarantee the parabolicity of the system (1.1). Theorem. Assume that (2.3) and (2.4) hold. Given u0 ∈ H01 (Ω) satisfying E (0) ≤ 0. If ∞ p−2 (2.6) g(s)ds < p − 3/2 0 then any strong solution of (1.1) blows up in ﬁnite time. Proof. We deﬁne 1 L(t) = 2

u2 (x, t)dx Ω

354

S.A. Messaoudi

and diﬀerentiate with respect to t to get L (t) = uut (x, t)dx Ω t u∆udx − u(x, t) g(t − s)∆u(x, s)dsdx + |u|p dx = 0 Ω Ω Ω t |∇u|2 dx + g(t − s)∇u(x, t).∇u(x, s)dsdx + |u|p dx = − Ω Ω 0 Ω t 2 2 p |∇u| dx + g(t − s)||∇u(t)||2 dτ + |u| dx (2.7) ≥ − Ω 0 Ω t g(t − s) |∇u(t).[∇u(s) − ∇u(t)]|dxdτ. − 0

Ω

By using the Schwarz inequality, (2.7) takes the form t p L (t) ≥ |u| dx − (1 − g(s)ds)||∇u(t)||22 0 Ω t g(t − τ )||∇u(t)||2 ||∇u(τ ) − ∇u(t)||2 dτ −

(2.8)

0

By applying Young’s inequality to the last term of (2.8) we arrive at

3 t L (t) ≥ |u|p dx − 1 − g(s)ds ||∇u(t)||22 − (g + ∇u)(t) 4 0 Ω We then substitute for ||∇u(t)||22 from (2.1); hence (2.9) becomes ) *

t 1 − 34 0 g(s)ds |u|p dx + 2 H(t) L (t) ≥

t (1 − 0 g(s)ds) Ω

t 1 − 34 0 g(s)ds + − 1 (g + ∇u)(t)

t (1 − 0 g(s)ds)

t 2 1 − 34 0 g(s)ds − |u|p dx ≥ γ |u|p dx

t p (1 − g(s)ds) Ω Ω 0 where

(2.9)

(2.10)

∞ 2 1 − 34 0 g(s)ds

γ =1− >0 p (1 − 0∞ g(s)ds)

because of (2.6). Next we have, by the embedding of the Lq spaces, Lp/2 (t) ≤ C||u||pp .

(2.11)

By combining (2.10) and (2.11) we get L (t) ≥ ΓLp/2 (t),

(2.12)

Solutions of a Semilinear Heat Equation

355

A direct integration of (2.12) then yields Lp/2−1 (t) ≥

1 L1−p/2 (0)

Therefore L blows up in a time t∗ ≤

1 ΓL(p/2)−1 (0)

.=

1 Γ

− Γt

.

1−p/2 1 u20 (x)dx 2 Ω

Acknowledgment The author would like to express their sincere thanks to King Fahd University of Petroleum and Minerals for its support. This work has been funded by KFUPM under Project # MS/VISCO ELASTIC/270.

References [1] Alfonsi L. and F. Weissler, Blow-up in Rn for a parabolic equation with a damping nonlinear gradient term, Progress in nonlinear diﬀerential equations and their applications 7 (1992), 1–20. [2] Ball J., Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford Ser. 28 (1977), 473–486 [3] Da Prato G. and M. Iannelli, Existence and regularity for a class of integrodiﬀerential equations of parabolic type, J. Math. Anal. Appl., 112 (1985), 36–55. [4] A. Friedman, Mathematics in Industrial Problems, Springer-Verlag, New York, 1992. [5] Friedman A., Partial diﬀerential equations of parabolic type, Prentice-Hall Englewood N. J. 1964. [6] Junning Z., Existence and nonexistence of solutions for ut = div(|∇u|p−2 ∇u) + f (∇u, u, x, t), J. Math. Anal. Appl. 172 (1993), 130–146. [7] Kalantarov V. K. and O. A. Ladyzhenskaya, The occurrence of collapse for quasilinear equations of parabolic and hyperbolic type, J. Soviet Math. 10 (1978), 53–70. [8] Levine H., Some nonexistence and instability theorems for solutions of formally parabolic equations of the form P ut = −Au+F (u), Archive Rat. Mech. Anal. 51 (1973), 371–386. [9] Levine H., S. Park, and J. Serrin, Global existence and nonexistence theorems for quasilinear evolution equations of formally parabolic type J. Diﬀ. Eqns 142 (1998), 212–229. [10] Messaoudi S.A., A note on blow-up of solutions of a quasilinear heat equation with vanishing initial energy, J. Math. Anal. Appl. 273 (2002), 243–247. [11] Messaoudi S.A., Blow-up and global existence in a nonlinear viscoelastic equation, Mathematische Nachrichten vol. 260 (2003), 58–66. [12] Nohel J.A., Nonlinear Volterra equations for the heat ﬂow in materials with memory, Integral and functional diﬀerential equations, Lecture notes in pure and applied mathematics, edited by T.L. Herdman, S.M. Rankin, III, H.W. Stech, Marcel Dekker Inc., 1981.

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[13] Pucci P. and J. Serrin, asymptotic stability for nonlinear parabolic systems, Energy methods in continuum mechanics, (Oviedo, 1994), 66–74, Kluwer Acad. Publ., Dordrecht, 1996. [14] Yin H.M., On parabolic Volterra equations in several space dimensions, SIAM J. Mathematical Analysis 22 (1991), 1723–1737. [15] Yin H. M., Weak and classical solutions of some Volterra integro-diﬀerential equations, Communication in P.D.E 17 (1992), 1369–1385. Salim A. Messaoudi 1. Mathematical Sciences Department KFUPM, Dhahran 31261 Saudi Arabia e-mail: [email protected]

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 357–371 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Finite Element Methods for Investigating the Moving Boundary Problem in Biological Development Cornel Marius Murea and George Hentschel Abstract. We describe two ﬁnite element algorithms which can be used to study organogenesis or organ development during biological development. Such growth can often be reduced to a free boundary problem with similarities to two-ﬂuid ﬂow in the presence of surface tension, though material is added at a constant growth rate to the developing organ. We use the speciﬁc case of avian limb development to discuss our algorithms.

1. Introduction Biological development involves both growth and changes of form which can often involves free moving boundaries [22]. Such moving boundary problems are similar in some respects to two ﬂuid ﬂow interfaces such as the Hele-Shaw problem with surface tension also called Mullins-Sekerka problem. In general, however, in contrast to incompressible ﬂows, growth due to mitosis and nutrients ensure that material is constantly being added (and sometimes removed when cell death or apoptosis occurs). In addition, speciﬁc boundary conditions (in general diﬀerent for each organ or cell type considered) will result in a more complex boundary value problem than those studied in Hele Shaw cells. In this paper to be speciﬁc we shall consider avian limb development, though we believe that similar ﬁnite element algorithms described here will be useful for other problems of organogenesis or organ morphogenesis and biological development. In this paper we will consider the evolution of two-dimensional moving domains. The more realistic but more complex case of three-dimensional domains separated by two-dimensional interfaces will be described in a future publication. In the case of the avian limb, the ventral-dorsal length scale (back of limb to palm) is normally small compared to the proximal-distal (tip of ﬁnger to point at which the limb joins the main body of the organism) or the posterior-anterior distance

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(from thumb to little ﬁnger) and therefore two-dimensional simulations are quite informative. In addition, at the developmental stage we are interested in, namely the embryo the whole limb which is only of a millimeter in scale, has approximately the shape of an ellipse with boundary Γ1 (t) and a boundary Γ2 (t) grafted to the trunk of the organism. In general only the growth velocity of Γ1 (t) parallel to the gradient of a pressure, while the growth of Γ2 (t) can be described by the motion of the joining vertex with the main trunck (in the more complex three-dimensional case this single point becomes a closed one-dimensional contour). The pressure in the limb whose gradient describes the rate of growth of the limb is the solution of a Poisson problem with Dirichlet boundary conditions depending on the curvature of the boundary. In Section 2 we give a brief description of some relevant aspects of avian limb development. Then in Section 3 we present a mathematical formulation of the resulting free boundary problem. In Section 4 two algorithms are described to solve numerically integrate the resulting equations of motion and ﬁnd the dynamical evolution of the interface. In the ﬁrst algorithm the boundary of the domain is approached by a polygon and the pressure is computed by a Finite Element Method. The computed pressure is a piecewise linear function, globally continuous. The curvature is computed as the inverse of the ray of the circle passing through three consecutive vertices of the boundary. The gradient of the pressure is then computed in a vertex of the mesh, as a weighted mean of the gradients in the neighborhood triangles. For the time discretization, we use the forward ﬁnite diﬀerence Euler’s scheme. A dynamic mesh technique is used in order to generate a triangular mesh at each time step. Starting from the mesh at the precedent time step and knowing the boundary at the current time step, we generate a mesh by redistributing the interior vertices using an optimization algorithm. The number of the interior vertices are constant. Also the connections of all meshes are the same, i.e., if i, j, k are the vertices of a triangle in the mesh at the precedent time step, these points are the vertices of a triangle in the current mesh. While in the second algorithm, the boundary is approached by cubic spline interpolation which gives a curve twice continuously diﬀerentiable. The curvature is computed using the parametrization of the splines. Again at each time step, a new mesh is generated, but this time, the generation of the current mesh is independent from the previous one. In Section 5 we describe the numerical tests of our algorithms. Finally in Section 6, we give a brief discussion of the potential of these approaches for studies of organogenesis and biological development.

2. Early avian limb development Early avian limb development presents a beautiful example of organogenesis and biological pattern formation: Well-deﬁned developmental axes exist which need

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to be understood. Limb growth changes the size and shape of internal domains in which biochemical processes occur. Cartilege formation via mesenchymal cell condensation occurs which will later diﬀerentiate into bone and form the skeletal limb. Many of these features appear to be robust: if comparisons are made anatomically with such an apparently diﬀerent vertabrates as chicken and mouse, it is remarkable the extent to which the gross features of patterning observed during development are conserved by evolution. All of which suggests that universal physical mechanisms controlling development exist. The embryo produces the raw materials (e.g., proteins, polysaccharides, RNAs) for its development from the available nutrients, according to rules embodied in the genetic code; diﬀusion, spreading, diﬀerential adhesion and chemotaxis transport these materials to speciﬁc regions of the organism. The mechanical or chemical changes which may take place in the course of the transport (change of concentration, cell shape, adhesiveness and cohesiveness) are signals that often aﬀect the production of the building material itself, that is, gene activity. In the course of these events cells diﬀerentiate and become more specialized. Diﬀerentiation involves regulated gene expression, but elaborate interactions among cells determine where and when new genes are expressed. In addition, morphogenetic changes require coordinated cell movement. The formation of the avian limb requires the establishment of proximal-distal positional gradients and transverse periodic modulations of morphogens to control the formation of individual and multiple parallel skeletal elements. These morphogen patterns act on limb mesenchyme to promote the formation of precartilage condensations, and ultimately the chondrocytes that will give rise to the cartilaginous primordia of the limb skeleton, which ultimately are replaced by bone. In this paper we wish to investigate only one aspect of this morphogenesis. Namely what type of overall shape is to be expected as a result of growth of the developing embryonic limb. To investigate this problem we consider a minimal model which incorporates the key features of this biological growth. Key is the addition of material at a rate S to the extracellular matrix in which the cells move (more generally the rate of growth will be S(x, t) as it could be both spatially varying and have a temporal dependence due to genetic switching mechanisms). This means that that the material ﬂow in the limb will obey ∇ · v = S.

(1)

We treat growth of the limb as due to a creeping ﬂow because of the very low Reynolds numbers involved [8]. Therefore we can expect the ﬂow to obey Darcy’s Law v = −α∇p

(2)

where p is a pseudo pressure ﬁeld, which obeys ∆p = −S/α in the limb domain. Finally we need biologically reasonable boundary conditions. As it appear there is no ﬂow of material into the main body of the organism at Γ2 (t) we shall

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take slip boundary condition here v · ν = 0,

(3)

where ν is the outer unit normal vector to the boundary, while the elastic properties of the epithelial layer of cells forming the skin layer at Γ1 (t) will result a pressure at this boundary obeying p = γκ, (4) where γ is the eﬀective surface tension of the limb [9] while κ is the limb curvature. The equation of the normal velocity of the boundary Γ1 (t) is Vν = v · ν.

(5)

The above condition requires that the boundary Γ1 (t) moves with the ﬂuid. It is the integration of this free boundary value problem that we study below. This mathematical model agrees favorably with the analysis presented in [3] based on biological experiments where the limb is considered as a homogeneous and highly hydrated core embedded in an dense envelope.

3. The free boundary problem We study the evolution of a bounded connected open domain Ω(t) of R2 with boundary ∂Ω(t) = Γ1 (t) ∪ Γ2 (t), where Γ1 (t) and Γ2 (t) are two non-empty subsets of ∂Ω(t). Here t ≥ 0 is the time. We assume that Γ1 (t) is a non-closed curve of class C 2 and its ends evolve on the Oy ax. The boundary Γ2 (t) is the segment which has the same ends as Γ1 (t). Let ν denote the outer unit normal vector to the boundary.

y

Γ1 (t) Γ2 (t)

Ω (t)

x ν

V

Figure 1. Schematic illustration of the free boundary problem

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From the equations (1)–(5), we can eliminate v and we obtain a system in p only. In the moving domain Ω(t), we have to ﬁnd the pressure p(x, y, t) : Ω(t) → R, such that ⎧ ⎪ ⎨ −∆p = S/α in Ω(t) p = γκ on Γ1 (t) (6) ⎪ ⎩ ∂p = 0 on Γ2 (t) ∂ν where α, γ, S are positive real constants and κ is the curvature of Γ1 (t). We use the sign convention that convex domains have positive curvature of the boundary. The boundary Γ1 (t) evolves according to the law ∂p ∂ν

(7)

Ω(0) = Ω0 .

(8)

Vν = −α where Vν is the normal velocity of Γ1 (t). We know the initial domain

We consider the problem (6)–(8) of determining the evolution of Ω(t) and to ﬁnd the pressure p(x, y, t) for t ∈ [0, T ], where T > 0 is a given real constant. This problem is similar to the Hele-Shaw problem with surface tension, but in our case the pressure is no longer harmonic (∆p = 0). S Let p(x, y, t) = 4α x2 + y 2 . We set P = p + p and we obtain from (6)–(7) the following problem: in the moving domain Ω(t), we have to ﬁnd the pressure P (x, y, t) : Ω(t) → R, such that ⎧ in Ω(t) ⎪ ⎨ ∆P = 0 P = γκ + p on Γ1 (t) (9) ⎪ ⎩ ∂P = 0 on Γ2 (t) ∂ν and the normal velocity of the boundary Γ1 (t) is ∂P ∂p − Vν = −α . (10) ∂ν ∂ν Though we cannot prove the existence and uniqueness of our solution, the existence and uniqueness of classical solution for the Hele-Shaw with surface tension problem was proved in [7], suggests that this is the case here also. It is possible that, in order to obtain the existence and uniqueness of solution, we have to prescribe the angles between the boundaries Γ1 (t) and Γ2 (t). The problem of existence and uniqueness is now under active investigation by G. Simonett. As it was shown in [12], the shape of the moving domain is determined solely by its normal velocity. In other words, if the velocity of Γ1 (t) has the form V = Vν · ν + Vτ · τ where τ is the unit tangent vector to the boundary and Vν is given by (7), then the movement of the domain and the pressure are the same as in the case Vτ = 0. We set Vτ = −α (∇p · τ ) and then V = −α∇p. The advantage of this choice is that

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we do not need to evaluate the normal vector to the boundary when we compute the velocity of the boundary. It is convenient to describe the curve Γ1 (t) by the parametric coordinates x = y =

r1 (θ, t), r2 (θ, t),

θ ∈ [a, b] .

Let us introduce the following generalized cylinder: 3 ΩT = (Ω(t) × {t}) . t∈]0,T [

The problem (6), (7) and (8) is equivalent to the following: ﬁnd r = (r1 , r2 ) : [a, b] × [0, T ] → R2 and p : ΩT → R, such that ∂r1 ∂p (θ, t) = −α (r1 (θ, t) , r2 (θ, t) , t) , ∀θ ∈ [a, b] , ∀t ∈]0, T [ ∂t ∂x ∂p ∂r2 (θ, t) = −α (r1 (θ, t) , r2 (θ, t) , t) , ∀θ ∈ [a, b] , ∀t ∈]0, T [ ∂t ∂y 0 r (θ, 0) = r1 (θ) , r20 (θ) , ∀θ ∈ [a, b] where r0 = r10 , r20 is a parametric representation of Γ1 (0) and p (x, y, t) is the solution of (6). ∂p ∂p = 0 and ν = (−1, 0)T on Γ2 (t), we have ∂x = 0 on Γ2 (t). If we Since ∂ν ∂p 2 1 suppose that p ∈ C (Ω(t)) ∩ C Ω(t) , we obtain that ∂x = 0 at the ends of Γ2 (t), also ∂p ∂x (r1 (a, t) , r2 (a, t) , t) = 0, ∀t ∈]0, T [ ∂p ∂x (r1 (b, t) , r2 (b, t) , t) = 0, ∀t ∈]0, T [. ∂r1 1 Then ∂r ∂t (a, t) = ∂t (b, t) = 0 and consequently r1 (a, t) = r1 (b, t) = 0, ∀t ∈]0, T [. The boundary could be parametrized in multiple ways, but the solution must be independent of parametrization.

4. Numerical methods The free boundary problem (6)–(8) is similar to the Hele-Shaw problem with surface tension also called Mullins-Sekerka problem. To solve numerically the Hele-Shaw problem with surface tension, there exists an eﬃcient approach named θ − L introduced in [12]. The variables are the tangent angle θ to the moving boundary and its arc length L. This framework makes the application of an implicit method for time integration easy and it permits to study the problem in a long time interval [13], [4]. In this approach a Fredholm like boundary integral has solved and the integral representation of a harmonic function is used. This is speciﬁc to some linear problems with constant coeﬃcients. This method is not appropriate if we replace the linear Darcy’s law (2) by the nonlinear Navier-Stokes equation with a volume source located at a certain point in

Investigating the Moving Boundary Problem the domain

ρ0

363

µ ∂v + (v · ∇) v − µ∆v + ∇p = f + ∇ (∇ · v) ∂t 3

where ρ0 > 0 is the density of the ﬂuid and µ > 0 its viscosity. A frequented framework used for Navier-Stokes equation in moving domain is Arbitrary Lagrangian Eulerian together with the Finite Element Method [14]. Other approaches are Time-Space Finite Elements [1], Level Set Method [21] and Immersed Boundary Methods [17]. The last one was employed to study the avian limb development in [6]. One of the disadvantage of the continuum models is the complex implementation required to handle the moving boundary of the domain where we have to solve a system of PDEs. In [15] the software CompuCell is presented, where a purely continuum approach for morphogenesis is used in combination with a discrete cellular automata. One of the part of CompuCell is based on the cellular Potts model (CMP). A criticism of this formalism is that it neglects simple force balance between cells. In this paper we present two algorithms which belong to the general framework called “front-tracking methods” [5]. The numerical results were produced for the Darcy’s law, but these algorithms could be used for the steady Navier-Stokes equation also. 4.1. The ﬁrst algorithm For the time discretization, we use the forward ﬁnite diﬀerences Euler’s scheme. We denote by ∆t the time step and by N = T /∆t the number of time steps. We approximate Γ1 (n∆t) by a polygonal line Γn1 of vertices (xni , yin ) for i = 0, . . . , M . We have xn0 = xnM = 0, for all n. We denote by Ωn the polygonal domain bounded by Γn1 and the Oy ax. For each vertex (xi , yi ) of Γn1 , we compute the discrete curvature κn (xi , yi ) as the inverse of the ray of the circle passing through the n n three points (xni−1 , yi−1 ), (xni , yin ) and (xni+1 , yi+1 ). The problem (6) is solved numerically by the Finite Element Method. The computed pressure pn is approached by P1 function, globally continuous. We follow [18] for computing the discrete gradient of pn . Let A be a vertex of Γn1 . We denote by star(A) the set of all triangles T of the mesh such that A is a vertex of T . We compute the discrete gradient of pn at the point A as following: + n T ∈star(A) Area(T ) · ∇p | T + , T ∈star(A) Area(T ) where pn | T is the linear function representing the restriction of the function pn on the triangle T . Algorithm 1 Generate a triangular mesh for Ω0 using freefem+ [2]. for each n from 0 to N − 1 do Step 1: Compute the discrete curvature κn (xi , yi ) at each vertex (xi , yi ) of Γn1 .

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Step 2: Compute pn by the Finite Element ⎧ −∆pn (x, y) = S/α, ⎪ ⎪ ⎪ ⎪ ⎨ pn (xi , yi ) = γ κn (xi , yi ) , ⎪ ⎪ n ⎪ ⎪ ⎩ ∂p (x, y) = 0, ∂ν

Method in Ωn ∀ (xi , yi ) vertex of Γn1 on Γn2 .

Step 3: Compute the discrete gradient of pn at each vertex of Γn1 . Step 4: Compute the vertices of Γn+1 1 xn+1 i yin+1

∂pn n n (x , y ) ∂x i i ∂pn n n (xi , yi ) = yin − α∆t ∂y = xni − α∆t

for each vertex (xni , yin ) ∈ Γn1 , i = 0, 1, . . . , M , then set xn+1 = xn+1 = 0. 0 M n+1 and is the segment of ends xn+1 , yM Step 5: The boundary Γn+1 2 M n+1 n+1 x0 , y0 . Step 6: Compute a triangular dynamic mesh for Ωn+1 , where ∂Ωn+1 = Γn+1 ∪ Γn+1 using the algorithm [16]. 1 2 endfor; At the Step 6, we start from the mesh at the precedent time step and knowing the boundary at the current time step, we generate a mesh by redistributing the interior vertices using an optimization algorithm. The number of the interior vertices are constant. Also, the connections of all meshes are the same, i.e., if i, j, k are the vertices of a triangle in the mesh at the precedent time step, these points are the vertices of a triangle in the current mesh. 4.2. The second algorithm Let (xni , yin ) for i = 0, . . . , M be points on Γ1 (n∆t). Let {a = s0 < s1 < · · · < sM = b} be a partition of an interval [a, b]. We will compute the interpolating cubic spline functions Γn1 = {(x(s), y(s)) , s ∈ [a, b]} with the properties: • x(s) and y(s) are twice continuously diﬀerentiable on [a, b], • x(s) and y(s) coincide on every subinterval [si , si+1 ], i = 0, . . . , M − 1 with polynomials of degree three, • x(si ) = xni and y(si ) = yin for i = 0, . . . , M , • x (a) = x (b) = 0 and y (a) = y (b) = 0. For the numerical tests, we have chosen si = i, for i = 0, . . . , M . In order to prevent the oscillations, we could choose a = 0 and 2 2 2 n xni − xni+1 + yin − yi+1 . si+1 − si =

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The curve (x(s), y(s)), s ∈ [a, b] has a continuous curvature given by x (s)y (s) − x (s)y (s) k(s) = 3/2 . 2 2 (x (s)) + (y (s))

(11)

Algorithm 2 Let (x0i , yi0 ) for i = 0, . . . , M be points on Γ1 (0). for each n from 0 to N − 1 do Step 1: Compute the cubic spline functions Γn1 = {(x(s), y(s)) , s ∈ [a, b]} Step 2: Compute the curvature κn (xi , yi ) at each vertex (xni , yin ) using (11). Step 3: Generate a triangular mesh for Ωn using freefem+ [2], where n ) and ∂Ωn = Γn1 ∪ Γn2 and Γn2 is the segment of ends (xnM , yM n n (x0 , y0 ). Step 4: Step 5: Step 6: Compute pn , ∇pn and (xn+1 , yin+1 ) as in the i Algorithm 1. endfor; At he step Step 3, the generation of the current mesh is independent from the previous one. We shall now describe some numerical tests of the eﬃcacy of these algorithms in studies of organogenesis.

5. Numerical tests 5.1. The initial domain is a semicircle First let consider the case where the initial domain is a semicircle of ray R0 . Then, if we set the parametric representation of Γ1 (0) as ) π π* r10 (θ) = R0 cos(θ), θ∈ − , 0 r2 (θ) = R0 sin(θ), 2 2 the evolution of the boundary Γ1 (t) is described by St ) π π* r1 (θ, t) = R0 e 2 cos(θ), , θ∈ − , St 2 2 r2 (θ, t) = R0 e 2 sin(θ),

t > 0.

The pressure has the form γ S 2 St R0 e − x2 − y 2 + St . 4α R0 e 2 The algorithms have been implemented using the programming language C++ and the Finite Element classes of F. Hecht [10]. The numerical results were displayed using gnuplot. p (x, y, t) =

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The ﬁrst simulation was performed using the Algorithm 1 for R0 = 1, S = 2, γ = 1, α = 0.5, ∆t = 0.05. The number of time steps is N = 10. A dynamic mesh technique is used in order to generate a triangular mesh at each time step. We have used the algorithm described in [16] for the mesh generation. The initial mesh has: 208 vertices, 362 triangles, M = 32 (the number of vertices on the boundary Γ1 ), h = 0.155178 (the mesh size). 2

2

1.5

1.5

1

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5

-2

-2 0

0.5

1

1.5

2

0

0.5

1

1.5

2

Figure 2. The initial mesh (left) and the mesh after 10 time steps (right) The number of vertices, triangles, boundary edges are the same for the ﬁrst 10 meshes. In the below table, we see the evolution of the mesh size h. Time step (n) Mesh size (h)

1 2 3 4 5 0.155 0.171 0.187 0.202 0.216

Time step (n) Mesh size (h)

6 7 8 9 10 0.231 0.244 0.256 0.266 0.276

After 10 time steps, the domain might be a semicircle of ray e0.5 ≈ 1.648721. 2

1.5

1

0.5

0

-0.5

-1

-1.5

-2 0

1

2

3

4

5

Figure 3. The evolution of the moving boundary (0–10 time steps)

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The second simulation was performed using the Algorithm 2 for ∆t = 0.0005 and N = 1000 (the number of time steps). The others parameters are the same as in the ﬁrst simulation. The software freefem+ [2] was used to generate a triangular mesh at each time step. The numbers of the vertices and of the triangles are not the same for the all meshes. For example, the mesh after 1000 time steps has 205 vertices and 356 triangles. After 1000 time steps, the domain might be a semicircle of ray e0.5 ≈ 1.648721. 2

1.5

1

0.5

0

-0.5

-1

-1.5

-2 0

1

2

3

4

Figure 4. The boundary after 0, 100, . . . , 1000 time steps

5.2. A non-convex initial domain Let now consider a case when the initial domain is non-convex as in Figure 5. The boundary Γ1 (0) has two ﬂat parts on the bottom, on the top and three semicircles of rays r1 = 0.6, r2 = 0.2 and r3 = 0.2 respectively. The simulations ware performed for: S = 2, α = 0.5, γ = 0.5, ∆t = 0.0001 and N = 180 (the number of time steps). 2

1.5

1

0.5

0

-0.5

-1

-1.5

-2 0

1

2

3

4

5

Figure 5. The initial domain

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3 2 0 -2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0

1

1

2

3

4

368

Figure 6. The initial (continuous) and the ﬁnal (dashed) boundaries. Algorithms 1 (left) and 2 (right)

We have observed that the pressure is almost constant near the two ﬂat parts of Γ1 (t) and near the largest semicircle. Consequently, these parts of boundary don’t move. We obtain boundary with self-intersection (like the 8) after 182 time steps using the Algorithm 2 and after 230 time steps using the Algorithm 1. 5.3. Concluding remarks The second algorithm is superior to the ﬁrst one due in principal to a better approximation of the curvature and a better mesh. We can improve the results by moving the boundary along the normal velocity which preserves a reasonable distribution of the vertices on the boundary. The velocity of the boundary could be computed more accurate by using P2 Finite Element for the pressure. Also adapting mesh techniques could be employed for improving the quality of the mesh. In the ﬁrst numerical test, we have solved the free boundary problem until time t = 0.5 and in the second, until t = 0.018. We have to use implicit in time algorithms in order to study this kind of free boundary problem in a long time interval. The Arbitrary Lagrangian Eulerian framework together with the Finite Element Method will be employed in a future paper. These methods could be employed for the Navier-Stokes equation with surface tension.

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6. Discussion The algorithms described above could form the basis for many important investigations of organogenesis and biological development in general. Of course they will need to be extended in several directions to give a quantitative picture of how growth and form develops. Obviously highly computationally intensive threedimensional simulations are necessary. While genetic switching mechanisms will need to be incorporated in order to understand the temporal properties of biological development. But in addition to these questions many other lines of investigation need to be developed. For example we know that cell condensation and bone development depend on reaction diﬀusion mechanisms in a progress zone of undiﬀerentiated cells at the tip of the limb. The size of this progress zone can be expected to have a signiﬁcant impact on the resulting prepattern created in the limb [19, 20, 11]. This is because even at the most basic level the number of standing waves of a heterogeneous distribution of a chemical species formed by a reaction-diﬀusion mechanism depends both on the scale of the basic pattern (set by the magnitude of the biological parameters) and on the space available for this pattern to develop (set by the domain size). Thus a very question is how the size of this progress zone changes with time? In order to answer this question studies of limb growth will form a vital ingredient. Another question involves how the skeletal elements themselves, once formed, would inﬂuence growth of the developing limb? Fairly signiﬁcant changes in internal domain organisation occur between early development when the stylopod and zeugopod are created, and later on when the digits appear. This question will require the development of algorithms for complex connected domains in which the skeletal elements create internal boundaries to growth. In addition, the existence of such internal domains will in turn inﬂuence reaction-diﬀusion mechanisms in the interdigital regions. Such reaction-diﬀusion mechanisms might be relevant to properly controlled cell death leading ultimately to digit formation. The inﬂuence will be very dependent on the relative scale of the patterning compared to the size of the interdigital domains. Clearly these algorithms need to be developed in several directions for studies of biological development and these will be reported in future publications. Acknowledgments The authors are very grateful to Professor Stuart A. Newman from the New York Medical College whose questions and comments have contributed to improve the contents of this paper.

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References [1] Bansch E., Finite element discretization of the Navier-Stokes equations with a free capillary surface. Numer. Math. 88 (2001) no. 2, 203–235. [2] Bernardi D., Hecht F., Ohtsuka K., Pironneau O., A ﬁnite element software for PDE: freefem+. http://www-rocq.inria.fr/Frederic.Hecht [3] Borkhvardt V.G., Growth and Shaping of the Fin and Limb Buds. Russian Journal of Developmental Biology, 31 (2000) no. 3, 154–161. [4] Ceniceros H.D., Hou T.Y., Si H., Numerical study of Hele-Shaw ﬂow with suction. Phys. Fluids 11 (1999) no. 9, 2471–2486. [5] Crank J., Free and moving boundary problems. The Clarendon Press, Oxford University Press, New York, 1987. [6] Dillon R., Othmer H.G., A mathematical Model for outgrowth and spatial pattering of the vertebrate limb bud. J. Theor. Biol., 197 (1999) 295–330. [7] Escher J., Simonett G., Classical solutions for Hele-Shaw models with surface tension. Adv. Diﬀerential Equations 2 (1997) no. 4, 619–642. [8] Forgacs G., Foty R.A., Shafrir Y., Steinberg M.S., Viscoelastic properties of living tissues: a quantitative study. Biophysical Journal 74 (1998) 2227–2234. [9] Foty R.A., Pfleger C.M., Forgacs G., Steinberg M.S., Surface tensions of embryonic tissues predict their mutual envelopment behavior. Development 122 (1996) 1611–1620. [10] Hecht F., C++ et El´ements Finis, Cours DEA, Universit´e Paris VI, http://www-rocq.inria.fr/Frederic.Hecht [11] Hentschel H.G.E., Glimm T., Glazier J.A., Newman S.A., Dynamical Mechanisms for Skeletal Pattern Formation in the Avian Limb, Proc. Royal Soc. B. 271 (2004) 1713–1722. [12] Hou T.Y., Lowengrub J.S., Shelley M.J., Removing the stiﬀness from interfacial ﬂows with surface tension. J. Comput. Phys. 114 (1994) 312–328. [13] Hou T. Y., Lowengrub J. S., Shelley M. J., The long-time motion of vortex sheets with surface tension. Phys. Fluids 9 (1997) no. 7, 1933–1954. [14] Hughes T.J.R., Liu W.K., Zimmermann T.K., Lagrangian-Eulerian ﬁnite element formulation for incompressible viscous ﬂows. Comput. Methods Appl. Mech. Engrg. 29 (1981) no. 3, 329–349. [15] Izaguirre J.A., Chaturvedi R., Huang C., Cickovski T., Cof J., Thomas G., Forgac G., Alber M., Hentschel G., Newman S.A., Glazier J.A., COMPUCELL, a multi-model framework for simulation of morphogenesis, Bioinformatics 20 (2004) no. 7, 1129–1137. [16] Murea C.M., Dynamic meshes generation using the relaxation method with applications to ﬂuid-structure interaction problems. An. Univ. Bucuresti Mat. 47 (1998) no. 2, 177–186. [17] Peskin C.S., The immersed boundary method. Acta Numer. 11 (2002), 479–517. [18] Pironneau O., Hecht F., Introduction au Calcul Scientiﬁque en C++, Cours Matrˆıse de Math´ematiques, Ing´enierie Math´ematique. Universit´e Paris VI, 2000. http://www-rocq.inria.fr/Frederic.Hecht

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[19] Newman S.A., Frisch H.L., Dynamics of skeletal pattern formation in developing chick limb. Science 205 (1979), 662–668. [20] Newman S.A., Frisch H.L., Percus J.K. On the stationary state analysis of reaction-diﬀusion mechanisms for biological pattern formation. J. Theor. Biol. 134, (1988) 183–197. [21] Sethian J. A., Level set methods: evolving interfaces in geometry, ﬂuid mechanics, computer vision and material science, Cambridge University Press, 1996. [22] Wolpert L., Beddington R., Brockes J., Jessell T., Lawrence P., Meyerowitz E., Principles of Development. Oxford, New York, Tokyo. Oxford University Press, 1998. Cornel Marius Murea Laboratoire de Math´ematiques et Applications Universit´e de Haute-Alsace 4, rue des Fr`eres Lumi`ere F-68093 Mulhouse Cedex, France e-mail: [email protected] George Hentschel Emory University Department of Physics Atlanta, GA 30322 USA e-mail: [email protected]

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 373–390 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Existence of Weak Solutions to the Equations of Stationary Motion of Heat-conducting Incompressible Viscous Fluids J. Naumann Dedicated to Professor H. Amann on the occasion of his 65th birthday

1. Introduction The stationary motion of a heat-conducting incompressible viscous ﬂuid in a domain Ω ⊂ Rn (n = 2 resp. n = 3) is governed by the following system of conservation laws: ∇·u= 0 (u · ∇)u = ∇ · S − ∇p + f γ(θ)(u · ∇θ) = tr(SD) − ∇ · q

mass,

(1.1)

momentum,

(1.2)

energy,

(1.3)

where u = (u1 , . . . , un ) velocity, S = {Sij }

extra stress1) , p = pressure,

f = (f1 , . . . , fn ) external force,

θ = temperature, γ(θ) = speciﬁc heat, 1 ∂ui ∂uj + D = {Dij }, Dij = Dij (u) = rate of strain 2 ∂xj ∂xi tr(SD) = Sij Dij

trace of SD,

q = heat ﬂux

(cf., e.g., Landau/Lifschitz [9], M¨ uller [15], Serrin [19], Spurk [21]). The ﬁrst term on the right-hand side of (1.3) represents the production of heat due to viscosity, while the second term represents the heat due to the heat ﬂux. 1)

In what follows, Latin indices take the values 1, . . . , n (n = 2 resp. n = 3). Repeated indices imply summation over 1, . . . , n.

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J. Naumann

We specify the constitutive assumptions for the extra stress tensor S and the vector of heat ﬂux q: S = µ(θ)D Newton’s law, (1.4) q = −κ(θ)∇θ Fourier’s law where µ(θ) = viscosity , κ(θ) = heat capacity (cf. [15], [21] for a discussion of these constitutive assumptions).

We consider system (1.1)–(1.4) in a bounded domain Ω ⊂ Rn (n = 2 resp. n = 3) whose boundary ∂Ω is assumed to satisfy the following conditions: ∂Ω is Lipschitzian, ∂Ω = Γ0 ∪ Γ1 (Γ0 ∩ Γ1 = ∅, Γ0 relatively open). With these assumptions, the boundary conditions upon u and θ we are going to consider, are as follows: u=0

on ∂Ω,

(1.5)

θ = θ0

on Γ0 ,

(1.6)

on Γ1 .

(1.7)

q·ν =0

Here θ0 is a given function, while ν = (ν1 , . . . , νn ) denotes the unit outward normal with respect to Ω along ∂Ω. Boundary condition (1.5) means that the ﬂuid adheres to the boundary ∂Ω; conditions (1.6), (1.7) model the situation where the temperature of the ﬂuid is prescribed on Γ0 , while there is no heat ﬂux through Γ1 . Various modiﬁcations of problem (1.1)–(1.7) have been studied in recent time. First, Morimoto [12], [13] proved the existence of weak solutions to the stationary Boussinesq equations (with µ = const > 0, κ = const > 0). These equations can be formally obtained by replacing f in (1.2) by a term depending linearly on θ, and by neglecting tr(SD) = µDij (u)Dij (u) in (1.3). This modiﬁcation of (1.3) simpliﬁes considerably proving the existence of weak solutions to (1.1)–(1.7). General outﬂow boundary conditions for the stationary Boussinesq equations are studied by the same author in [14]. A uniqueness result for weak solutions to the stationary Boussinesq equations with temperature dependent viscosity µ = µ(θ) has been proved by Goncharova [8]. With µ = const > 0, κ = const > 0, f in (1.2) replaced by a nonlinear term ukaszewicz [11] proved the existence of a strong solution involving θ, and Γ1 = ∅, L to (1.1)–(1.6). The proof is based on estimates for strong solutions to the Stokes system. In [2], Baranger and Mikeliˇc studied (1.1)–(1.3) with constitutive assumptions S = µ(θ, |D|)D,

q = −κ∇θ

(κ = const > 0),

Existence of Weak Solutions to the Equations of Stationary Motion

375

where the conditions upon µ(θ, ·) make S to include the well-known power-law model of non-Newtonian ﬂuid mechanics as special case. The authors present a proof of existence of a weak solution to (1.1)–(1.3), (1.5), (1.6) (i.e., Γ1 = ∅) for the case when the term (u · ∇)u is neglected in (1.2). An important step of their proof consists of solving (1.3), (1.6) by decomposing θ = θ1 + θ2 where θ1 is determined by a linear elliptic problem, whereas θ2 is determined by a weakly nonlinear elliptic problem. The constitutive assumptions S = τ (D),

q = −κ∇θ

(κ = const > 0)

are considered in Roub´ıˇcek [18], where the conditions upon the tensor-valued function τ are motivated by the power law model. In that paper, source terms depending on θ, are added to the right-hand side of (1.2) and (1.3), respectively. An existence theorem for weak solutions to (1.1)–(1.3) under rather general constitutive assumptions on S and q has been established by Consiglieri [5]. In [2], [11] and [18] the Schauder ﬁxed point theorem is used for proving the existence of a weak solution to the boundary value problem under consideration. All the papers mentioned above, are concerned with the special case γ ≡ 1. Under the constitutive assumptions S = µ(θ)a(D),

q = −κ(θ)b(∇θ),

Consiglieri and Shilkin [6] proved the existence of a weak solution {u, θ} to (1.1)– (1.3) where u possesses locally integrable second-order derivatives. From this result the local H¨ older continuity of ∇u and ∇θ is then deduced in case n = 2 (note that [12] contains an interior diﬀerentiability result for weak solutions to the stationary Boussinesq equations). Finally, with S = µ(θ)D,

q = −∇θ,

Shilkin [20] proved a result on the partial H¨ older continuity of weak solutions to (1.1)–(1.3) for the case n = 3. The aim of the present paper is to prove the existence of a weak solution {u, θ} to (1.1)–(1.7) for variable κ and under very mild conditions upon the speciﬁc heat γ. Our method of proof is based on an approximation argument which does not require any uniqueness result for weak solutions to auxiliary problems. In addition, we obtain an estimate on the integral |∇θ|2 dx (δ > 0) (1 + θ − θ0 )1+δ Ω

and a result on the higher integrability of ∇u and ∇θ. The paper is arranged as follows: 2. Statement of the main result 3. Proof of the main result 3.1 Existence of an approximate solution {uε , θε }

376

J. Naumann

3.2 A priori-estimates on ∇uε 3.3 A priori-estimates on ∇θε 3.4 Passage to limit ε → 0. 4. Higher integrability 4.1 Higher integrability of ∇uε 4.2 Higher integrability of ∇θε References

2. Statement of the main result We consider the following conditions upon the data in (1.4), (1.3) and (1.2): µ ∈ C(R), 0 < µ1 ≤ µ(ξ) ≤ µ2 < +∞ ∀ξ ∈ R

(µ1 , µ2 = const),

(2.1)

κ ∈ C(R), 0 < κ1 ≤ κ(ξ) ≤ κ2 < +∞ ∀ξ ∈ R

(κ1 , κ2 = const),

(2.2)

(γ0 = const),

(2.3)

γ ∈ C(R), |γ(ξ)| ≤ γ0 < +∞ ∀ξ ∈ R

6 if n = 3). (2.4) f ∈ [Ls (Ω)]n (s > 1 if n = 2, s = 5 Next, let W 1,p (Ω) (1 ≤ p < +∞) denote the usual Sobolev space. Deﬁne W01,p (Ω) := {ϕ ∈ W 1,p (Ω) ϕ = 0 a.e. on ∂Ω}, D1,p (Ω) := {v ∈ [W 1,p (Ω)]n div v = 0 a.e. in Ω} 0

0

and WΓ1,p (Ω) := {ϕ ∈ W 1,p (Ω) ϕ = 0 a.e. on Γ0 }. 0 Finally, for u, v ∈ [W 1,2 (Ω)]n we deﬁne 1 ∂ui ∂uj ∂vi ∂vj D(u, v) := Dij (u)Dij (v) = + + 4 ∂xj ∂xi ∂xj ∂xi Deﬁnition. Assume (2.1)–(2.4), and θ0 ∈ W 1,2 (Ω). The pair {u, θ} ∈ D1,2 0 (Ω) × W 1,p (Ω) 3 6 (1 < p < 2 if n = 2, ≤ p < if n = 3) is called a weak solution to (1.1) - (1.7) if 5 2 ∂uj µ(θ)D(u, v)dx + ui vj dx = fi vi dx ∀ v ∈ D1,2 (2.5) 0 (Ω), ∂xi Ω

Ω

Ω

⎧ ∂θ ⎪ ⎨ κ(θ)∇θ · ∇ϕdx + γ(θ)ui ϕdx = µ(θ)D(u, u)ϕdx ∂xi Ω Ω ⎪ ⎩ Ω 1,q ∀ ϕ ∈ WΓ0 (Ω), θ = θ0

a.e. on Γ0 .

(2.6) (2.7)

Existence of Weak Solutions to the Equations of Stationary Motion

377

The main result of our paper is the following Theorem 1. Assume (2.1)–(2.4), and θ0 = const 1) . ? 1,p (Ω) × W (Ω) which satisﬁes (2.5)– Then there exists {u, θ} ∈ D1,2 0 n p∈[1, n−1 [

(2.7). In addition, there holds

θ ≥ θ0

a.e. in Ω,

|∇u|2 dx ≤ c1 f 2[Ls(Ω)]n ,

(2.8) (2.9)

Ω

c2 |∇θ|2 dx ≤ f 2[Ls(Ω)]n ∀δ > 0, 1+δ (1 + θ − θ0 ) δ Ω ⎧ 2p/(1−δ) ⎪ ⎪ |∇θ|p dx ≤ f p[Ls(Ω)]n + f [Ls(Ω)]n ⎪ ⎪ ⎪ ⎪ ⎨Ω ∀ p ∈ [1, 2[, ∀δ ∈]0, 1[ if n = 2, ⎪ ⎪ ⎪ ⎪ 3 3 − 2p ⎪ ⎪ if n = 3 ⎩ ∀ p ∈ [1, [, δ = 2 3−p

(2.10)

(2.11)

(ci = const; i = 1, 2, 3). Remark 1. The existence of weak solutions in = W 1,p (Ω) (n ≥ 2) n p∈[1, n−1 [

to linear, uniformly elliptic equations with L∞ -coeﬃcients, right-hand side in L1 and zero boundary conditions has been proved in Stampacchia [22]. A very general theory of linear and semi-linear elliptic boundary value problems involving measures as right-hand sides, has been developed by Amann and Quittner [1]. Existence results for very large classes of nonlinear elliptic equations with right-hand side in L1 , are proved by Rakotoson [17] and Boccardo and Gallou¨et [3], [4]. ∂θ The coeﬃcient κ(θ) and the presence of the term γ(θ)ui in (1.3) prevent ∂xi proving the existence of a weak solution to this equation (cf. (2.6)) by appealing to the existence results in [22], [1] and [3], [4], [17]. Nevertheless, using techniques developed in [3], [4], [17] we are able to establish integral estimates on ∇θ. For this, the key observation is that the integral ∂θ γ(θ)ui ϕdx ∂xi Ω

vanishes for appropriate functions ϕ. 1) This

assumption we made for the sake of technical simplicity of the calculations in Sect. 3.3.

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J. Naumann

3. Proof of the main result 3.1. Existence of an Approximate Solution {uε , θε } We begin with proving the following existence result. Proposition 1. Suppose that the conditions of Theorem 1 hold. For every ε > 0, 1,2 there exists {uε , θε } ∈ D1,2 (Ω) such that 0 (Ω) × W ∂uεj µ(θε )D(uε , v)dx + uεi vj dx = fi vi dx ∀ v ∈ D1,2 (3.1) 0 (Ω), ∂xi Ω

Ω

Ω

⎧ ∂θε D(uε , uε ) ⎪ ⎪ ϕdx κ(θε )∇θε · ∇ϕdx + γ(θε )uεi ϕdx = µ(θε ) ⎨ ∂xi 1 + εD(uε , uε ) (3.2) Ω Ω Ω ⎪ ⎪ ⎩ ∀ ϕ ∈ WΓ1,2 (Ω), 0 θε = θ0

a.e. on Γ0 .

(3.3)

Proof. By Korn’s inequality, [W01,2 (Ω)]n is a Hilbert space with respect to the scalar product u, v →

D(u, v)dx. Ω

1,2 We deﬁne H := D1,2 0 (Ω) × WΓ0 (Ω). By {u, θ}, {v, ϕ}, . . . we denote the elements of H. Clearly, H is a Hilbert space with respect to the scalar product ({u, θ}, {v, ϕ}) := (D(u, v) + ∇θ · ∇ϕ)dx. Ω

The associated norm is

{u, θ} =

1/2 (D(u, u) + |∇θ| )dx 2

.

Ω

We furnish

∗ (D1,2 0 (Ω))

×

(WΓ1,2 (Ω))∗ 2) 0

u∗ 2(D1,2 (Ω))∗ 0

with the norm

+

1/2 θ∗ 2(W 1,2 (Ω))∗ Γ

.

0

1,2 ∗ ∗ Then H∗ is linearly isometric to (D1,2 0 (Ω)) × (WΓ0 (Ω)) . Identifying both spaces, then for {u∗ , θ∗ } ∈ H∗ and {u, θ} ∈ H we may write

{u∗ , θ∗ }, {u, θ} = u∗ , u + θ∗ , θ. 2) Let

X be a normed linear space. We denote by X ∗ the dual of X, by · X ∗ the dual norm on X ∗ and by x∗ , x the value of x∗ ∈ X ∗ at x ∈ X.

Existence of Weak Solutions to the Equations of Stationary Motion

379

Let ε > 0. The assumptions (2.1)–(2.3) permit to deﬁne a nonlinear mapping A : H → H∗ by ∂uj µ(θ + θ0 )D(u, v)dx + ui vj dx A({u, θ}), {v, ϕ} := ∂xi Ω Ω ∂θ + κ(θ + θ0 )∇θ · ∇ϕdx + γ(θ + θ0 )ui ϕdx ∂xi Ω Ω D(u, u) ϕdx, {u, θ}, {v, φ} ∈ H. − µ(θ + θ0 ) 1 + εD(u, u) Ω

We show that A is coercive, i.e., 1 A({u, θ}), {u, θ} → +∞ as {u, θ} → ∞. {u, θ} Indeed, we have

ui

(3.4)

∂uj uj dx = 0 ∀u ∈ D1,2 0 (Ω), ∂xi

(3.51 )

Ω

γ(θ + θ0 )ui

∂θ θdx = 0 ∀u ∈ D1,2 0 (Ω), ∂xi

∀ θ ∈ W 1,2 (Ω).

Ω

Here (3.51 ) is readily seen by integration by parts. To verify (3.52 ), deﬁne t Φ(t) :=

t ∈ R.

sγ(s + θ0 )ds, 0

Then

Φ ∈ C 1 (R), |Φ (t)| ≤ γ0 |t| ∀t ∈ R (cf. (2.3)), and for every θ ∈ W 1,2 (Ω) ∂ ∂θ Φ(θ) = θγ(θ + θ0 ) a.e. in Ω ∂xi ∂xi

(i = 1, . . . , n),

Φ(θ) ∈ W 1,r (Ω) (r ∈ [1, 2[ if n = 2, r =

3 if n = 3). 2

Thus, by integration by parts, ∂θ ∂ γ(θ + θ0 )ui θdx = ui Φ(θ)dx = 0. ∂xi ∂xi Ω

We obtain

Ω

A({u, θ}), {u, θ} ≥ µ1

D(u, u)dx + κ1

Ω

for all {u, θ} ∈ H. Whence (3.4).

µ2 |∇θ| dx − ε

|θ|dx

2

Ω

Ω

(3.52 )

380

J. Naumann Next, we show that A is pseudo-monotone, i.e.,

1 A maps bounded sets in H into bounded sets in H∗ , o

2o for every sequence ({uk , θk }) ⊂ H with {uk , θk } → {u, θ} weakly in H, lim supA({uk , θk }), {uk , θk } − {u, θ} ≤ 0 there exists a subsequence (not relabelled) such that lim infA({uk , θk }), {uk , θk } − {v, ϕ} ≥ A({u, θ}), {u, θ} − {v, ϕ} for all {v, ϕ} ∈ H. Indeed, 1o is easily deduced from the deﬁnition of A. To establish 2o , we note that there exists a subsequence of ({uk , θk }) (not relabelled) such that uki → ui , θk → θ strongly in L4 (Ω), θk → θ a.e. in Ω as k → +∞ (i = 1, . . . , n). Observing that µ(θk + θ0 )D(uk − u, uk − u)dx + κ(θk + θ0 )|∇(θk − θ)|2 dx Ω

Ω

= A({uk , θk }), {uk , θk } − {u, θ} − µ(θk + θ0 )D(u, uk − u)dx − κ(θk + θ0 )∇θ · ∇(θk − θ)dx Ω

−

uki

∂ukj ∂xi

ukj − uj dx −

Ω

+

Ω

γ(θk + θ0 )uki

∂θk (θk − θ)dx ∂xi

Ω

D(uk , uk ) (θk − θ)dx, µ(θk + θ0 ) 1 + εD(uk , uk )

Ω

we ﬁnd

min{µ1 , κ1 } lim sup

(D(uk − u, uk − u) + |∇(θk − θ)|2 )dx Ω

≤ lim supA({uk , θk }), {uk , θk } − {u, θ} ≤ 0. Hence uk → u strongly in [W 1,2 (Ω)]n , θk → θ strongly in W 1,2 (Ω), and (again by passing to a subsequence if necessary) D(uk , uk ) → D(u, u)

a.e. in

Ω

Existence of Weak Solutions to the Equations of Stationary Motion

381

as k → ∞. Thus, for every ϕ ∈ W 1,2 (Ω), D(uk , uk ) D(u, u) lim µ(θk + θ0 ) (θk − ϕ)dx = µ(θ + θ0 ) (θ − ϕ)dx. 1 + εD(uk , uk ) 1 + εD(u, u) Ω

Ω

It follows lim infA({uk , θk }), {uk , θk } − {v, ϕ} ≥ A({u, θ}), {u, θ} − {v, ϕ} for all {v, ϕ} ∈ H. Whence 2o . From the theory of pseudo-monotone operators we obtain: for every ε > 0 there exists {uε , θˆε } ∈ H such that A({uε , θˆε }) = {f, 0} (cf., e.g., Lions [10]). The pair {uε , θε }, θε := θˆε + θ0 , satisﬁes (3.1)–(3.3).

3.2. A priori-estimates on ∇uε We insert v = uε in (3.1). By (3.51 ), µ(θε )D(uε , uε )dx = fi uεi dx. Ω

Ω

Thus |∇uε |2 dx ≤ c1 f 2[Ls(Ω)]n

∀ε > 0.

(3.6)

Ω

3.3. A priori-estimates on ∇θε For notational simplicity, throughout this section we write u and θ in place of uε and θε , respectively. Inserting ϕ = (θε − θ0 )− (ξ − := min{ξ, 0}, ξ ∈ R) in (3.2) gives ∂θ − 2 κ(θ)|∇(θ − θ0 ) | dx + γ(θ)ui (θ − θ0 )− dx ∂xi Ω Ω D(u, u) = µ(θ) (θ − θ0 )− dx 1 + εD(u, u) Ω

≤ 0.

382

J. Naumann Recalling that θ = θˆ + θ0 , we have ∂θ γ(θ)ui (θ − θ0 )− dx = ∂xi Ω

γ(θˆ + θ0 )ui

∂ θˆ ˆ θdx ∂xi

ˆ {x|θ(x)<0}

=

ui

∂ ˆ Φ(θ)dx ∂xi

ˆ {x|θ(x)<0}

=

ui

∂ Φ(θˆ− )dx ∂xi

Ω

=

0

(cf. the proof of (3.52 )). It follows |∇(θ − θ0 )− |2 dx = 0. Ω

Thus θ ≥ θ0 a.e. in Ω.

(3.7)

To prove estimates on ∇θ, we make use of arguments of Rakotoson [17]. Similar techniques are developed in Boccardo and Gallou¨et [3], [4]. 1 is admissible in (3.2) (observe Let δ > 0. The function ϕ = 1 − (1 + θ − θ0 )δ (3.7)). We obtain ∂θ |∇θ|2 1 1− dx dx + γ(θ)ui δ κ(θ) (1 + θ − θ0 )1+δ ∂xi (1 + θ − θ0 )δ Ω

Ω

= Ω

1 D(u, u) 1− µ(θ) dx. 1 + εD(u, u) (1 + θ − θ0 )δ

Here the second integral on the left-hand side vanishes. Indeed, deﬁning t 1 ds, t ∈ [0, +∞[, Ψ(t) := γ(s + θ0 ) 1 − (1 + s)δ 0

we ﬁnd

Ω

∂θ ∂ 1 γ(θ)ui 1− dx = ui Ψ(θ − θ0 )dx = 0 δ ∂xi (1 + θ − θ0 ) ∂xi Ω

Existence of Weak Solutions to the Equations of Stationary Motion (cf. the proof of (3.52 )). Observing (3.6) it follows |∇θ|2 c2 dx ≤ f 2[Ls(Ω)]n , (1 + θ − θ0 )1+δ δ

383

(3.8)

Ω

where the constant c2 is independent of both ε and δ. Now, let p ∈]1, 2[, δ ∈]0, 1[ 3 − 2p 3 p ∈]1, [, δ = 2 3−p

if

n = 2,

if

n = 3.

Then p(1 + δ) ∈]1, +∞[ 2−p p(1 + δ) 3p = 2−p 3−p

if

n = 2,

if

n = 3.

Combining H¨ older’s inequality, estimate (3.8) and Sobolev’s embedding theorem we obtain p/2 (2−p)/2 2 |∇θ| |∇θ|p dx ≤ dx (1 + θ − θ0 )p(1+δ)/(2−p) dx (1 + θ − θ0 )1+δ Ω

Ω

≤

c p/2 2

δ

Ω

f p[Ls(Ω)]n

c p/2 2

≤ 2(2−p)/2 δ

(2−p)/2

(1 + θ − θ0 )p(1+δ)/(2−p) dx Ω

f p[Ls(Ω)]n ×

× (meas Ω)

(2−p)/2

(1+δ)/2 |∇θ| dx p

+ C0 Ω

(here C0 denotes the constant of the embedding WΓ1,p (Ω) ⊂ Lp(1+δ)/(2−p) (Ω); note 0 1+δ < 1, we ﬁnally obtain by the that if n = 2, then C0 → +∞ as p → 2). Since 2 aid of Young’s inequality 2p/(1−δ)

|∇θ|p dx ≤ c3 f p[L2(Ω)]n + f [Ls(Ω)]n

(3.9)

Ω

with c3 = const independent of ε.

3.4. Passage to limit ε → 0 By passing to a subsequence, from (3.6) and (3.9) we conclude uε → u weakly in [W 1,2 (Ω)]n , θε → θ weakly in W 1,p (Ω),

(3.10)

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J. Naumann

and (again by passing to a subsequence if necessary) uε → u strongly in [L4 (Ω)]n , θε → θ strongly in L1 (Ω),

(3.11)

θε → θ a.e. in Ω

(3.12)

as ε → 0. Clearly, u ∈ D1,2 0 (Ω), θ = θ0 a.e. on Γ0 and θ ≥ θ0 a.e. in Ω. Using (3.10), (3.11) and (3.12), the passage to limit ε → 0 in (3.1) easily gives (2.5). Estimates (2.9) and (2.11) are readily deduced from (3.6) and (3.9), respectively. To carry out the passage to limit ε → 0 in (3.2), we ﬁrst note that from (3.1) it follows that µ(θε )D(uε − u, uε − u)dx = − µ(θε )D(u, uε − u)dx Ω

Ω

−

uεi

∂uεj (uεj − uj )dx + ∂xi

Ω

fi (uεi − ui )dx. Ω

Hence, by (3.10), (3.11) and (3.12), lim µ(θε )D(uε − u, uε − u)dx = 0.

(3.13)

ε→0

Ω

We obtain uε → u strongly in [W 1,2 (Ω)]n , and then (for a subsequence) D(uε , uε ) → D(u, u) a.e. in Ω

(3.14)

as ε → 0. Taking into account the elementary identity (a − b, a + b) ε|a|2 |b|2 |a|2 − |b|2 = − , a, b ∈ Rn , 2 2 1 + ε|a| 1 + ε|a| 1 + ε|a|2 we obtain by the aid of (3.12), (3.13) and (3.14) D(uε , uε ) lim µ(θε ) − µ(θ)D(u, u)dx = 0. ε→0 1 + εD(uε , uε ) Ω

Now the passage to limit ε → 0 in (3.2) with ϕ ∈ WΓ1,p (Ω) 0

1
readily carried out. This gives (2.6). Finally, from (3.8), (3.10) and (3.12) we conclude ∂θε ∂θ ∂xi ∂xi −→ (1 + θε − θ0 )(1+δ)/2 (1 + θ − θ0 )(1+δ)/2

weakly in L2 (Ω)

n n−1

is

Existence of Weak Solutions to the Equations of Stationary Motion

385

as ε → 0 (i = 1, . . . , n). It follows |∇θ|2 |∇θε |2 dx ≤ lim inf dx (1 + θ − θ0 )1+δ (1 + θε − θ0 )1+δ Ω

Ω

c2 ≤ f 2[Ls(Ω)]n . δ

4. Higher integrability In this section, we consider system (1.1)–(1.7) for the case n = 3. For what follows, let 6 s> (4.1) f ∈ [Ls (Ω)]3 5 (cf. (2.4)). 4.1. Higher integrability of ∇uε Let (2.1)–(2.3) and (4.1) be satisﬁed, and let θ0 ∈ W 1,2 (Ω). Let {u, θ} ∈ D1,2 (Ω)× 6 3 ≤p< be a weak solution to (1.1)–(1.7) (cf. Sect. 2). W 1,p (Ω) 5 2 From Wolﬀ [23] it follows: ∃ t > 2 : ∇u ∈ [Lt (Ω)]9 , ∇u[Lt (Ω)]9 ≤ K0 f [Ls(Ω)]3 ,

(4.2)

where K0 is a continuous increasing function deﬁned on [0, +∞[. This result which holds for dimensions n = 2 and n = 3, is proved by the method of higher integrability by reverse H¨ older inequality (cf. Giaquinta [7] for a presentation of this method for weak solutions to nonlinear elliptic systems). We note that this method does not provide an information about the size of t. Let (2.1)–(2.3) and (4.1) be satisﬁed, and let θ0 = const. Let {uε , θε } ∈ (Ω) × W 1,2 (Ω) (ε > 0) be as in Prop. 1 (cf. Sect. 3.1). D1,2 0 Again appealing to [23] we obtain: ∃ t > 2 : ∇uε ∈ [Lt (Ω)]9 , ∇uε [Lt (Ω)]9 ≤ K0 f [Ls(Ω)]3 , (4.2 ) with the same function K0 as above in (4.2). Clearly, neither t nor K0 depend on ε3) . We shall use (4.2 ) to establish a result on the higher integrability of ∇θε . 3) Note

that the estimates on the integrals providing (4.2 ) depend on µ1 and µ2 (cf. (2.1)), but not on µ(θε ) itself.

386

J. Naumann

4.2. Higher integrability of ∇θε We continue to assume (2.1)–(2.3), (4.1) and θ0 = const. Let {uε , θε } ∈ D1,2 0 (Ω) × W 1,2 (Ω) (ε > 0) be as in Prop. 1 (cf. Sect. 3.1). The function θε represents a weak solution to the nonlinear elliptic boundary value problem ∂ D(uε , uε ) ∂θ ∂θ = 0 in Ω, − − µ(θ) κ(θ) + γ(θ)uεi ∂xi ∂xi ∂xi 1 + εD(uε , uε ) ∂θ = 0 on Γ1 , θ = θ0 on Γ0 , ∂ν where uε can be viewed as a given function (cf. (3.2)). From (4.2 ) it follows t/2 D(uε , uε ) t/2 dx ≤ µ2 (D(uε , uε ))t/2 dx (4.3) µ(θε ) 1 + εD(uε , uε ) Ω Ω ˆ 0 f [Ls(Ω)]3 , ≤ K ˆ 0 is a continuous increasing function deﬁned on [0, +∞[. where K Proving the higher integrability of ∇θε by the method of reverse H¨older t 6 inequality requires > in (4.3) (cf. [7] for local gradient estimates, and Na2 5 umann and Wolﬀ [16] for global gradient estimates). However, this method does 12 in (4.2 ). Moreover, establishing a global higher not provide the inequality t > 5 ¯ 0 ∩Γ1 (cf. [18]). integrability result for ∇θε requires additional assumptions on Γ We now prove a higher integrability result for ∇θε by a variant of the gradient estimate technique developed in Sect. 3.3. Let t be as in (4.2 ). We deﬁne p0 :=

6(t − 1) . 3t − 2

Clearly, 3 < p0 < 2. 2 We then have the following Proposition 2. There holds ∇θε ∈ [Lp0 (Ω)]3 , ∇θε [Lp0 (Ω)]3 ≤

2 c0 t K0 f [L2(Ω)]3 , t−2

where c0 = const independent of ε and t, K0 according to (4.2’).

Existence of Weak Solutions to the Equations of Stationary Motion

387

Proof. Let τ > 0. The function ϕ :=

θε − θ0 τ + (θε − θ0 )2/t

4)

is admissible in (3.2). We obtain

κ(θε ) 2 (θε − θ0 )2/t ∇(θε − θ0 ) dx ∇θε · τ ∇(θε − θ0 ) + 1 − 2/t 2 t (τ + (θε − θ0 ) ) Ω ∂θε θε − θ0 + γ(θε )uεi dx ∂xi τ + (θε − θ0 )2/t Ω θε − θ0 D(uε , uε ) = µ(θε ) dx. (4.4) 1 + εD(uε , uε ) τ + (θε − θ0 )2/t Ω

Deﬁne σ Ψ0 (σ) :=

γ(η + θ0 )

η dη, τ + η 2/t

σ ∈ [0, +∞[

0

(cf. (3.52 ) and Sect. 3.3). Then, by integration by parts, Ω

∂θε θε − θ0 γ(θε )uεi dx = ∂xi τ + (θε − θ0 )2/t =

uεi

∂ Ψ0 (θε − θ0 )dx ∂xi

Ω

0.

With the help of (4.2 ) from (4.4) it follows 2 θε − θ0 |∇θε |2 D(uε , uε ) κ1 1 − dx ≤ µ(θε ) dx t 1 + εD(uε , uε ) τ + (θε − θ0 )2/t τ + (θε − θ0 )2/t Ω Ω ≤ C1 |∇uε |2 (θε − θ0 )(t−2)/t dx Ω

(t−2)/t 2 (θε − θ0 )dx ≤ C1 K0 f [L2(Ω)]3

Ω

4) The

use of this function in (3.2) has been suggested by J. Wolf.

388

J. Naumann

with C1 = const independent of ε. Thus, by H¨ older’s inequality, |∇θε |p0 dx ≤ Ω

≤ Ω

p0 /2 (2−p0 )/2 |∇θε |2 2/t p0 /(2−p0 ) dx (τ + (θε − θ0 ) ) dx τ + (θε − θ0 )2/t Ω

C1 t ≤2 κ1 (t − 2)

p2 /2

p0 (t−2)/2t 2 K0 f [Ls(Ω)]3 (θε − θ0 )dx Ω

× τ (mes Ω)

(2−p0 )/2

(2−p0 )/2

(θε − θ0 )

2p0 /t(2−p0 )

+

(4.5)

.

dx

Ω

By the deﬁnition of p0 , 2p0 3p0 = = p∗0 t(2 − p0 ) 3 − p0

(Sobolev embedding exponent).

We obtain

(θε − θ0 )dx ≤

|∇θε | dx

C2

Ω

1/p0 p0

Ω

p∗ 0

(θε − θ0 ) dx ≤ Ω

,

p∗0 /p0 |∇θε |p0 dx

C3 Ω

(C1 , C2 = const). Inserting both estimates into (4.5) and carrying out the passage to limit τ → 0 gives |∇θε |p0 dx Ω

≤ C4

t p0 /2 p0 (t−2)/2t+p∗0 (2−p0 )/2p0 K0 f Ls(Ω)]3 |∇θε |p0 dx . t−2 Ω

Finally, observing that t − 2 p∗0 (2 − p0 ) 1 = , + 2t 2p0 2 we obtain the result claimed.

The estimates on ∇uε and ∇θε obtained in Sect. 3.2 and 3.3, and those in (4.2 ) and Prop. 2 provide the following

Existence of Weak Solutions to the Equations of Stationary Motion

389

Theorem 2. Assume (2.1)–(2.3), (4.1) and θ0 = const. Then there exists t > 2 such 6(t − 1) 1,p0 (Ω) p0 = that: There exists a pair {u, θ} ∈ D1,t satisfying 0 (Ω) × W 3t − 2 (2.5)–(2.11), and ∇u[Lt (Ω)]9 ≤ K0 f [Ls(Ω)]3 , 2 c0 t K0 f [Ls(Ω)]3 ∇θ[Lp0 (Ω)]3 ≤ . t−2 Remark 2. The higher integrability of ∇θ in the case n = 2 will be discussed in a forthcoming paper.

Literatur [1] Amann, H.; Quittner, P., Elliptic boundary value problems involving measures: existence, regularity, and multiplicity. Advances Diﬀ. Equs. 3 (1998), 753–813. ˇ, A., Stationary solutions to a quasi-Newtonian ﬂow with [2] Baranger, J.; Mikelic viscous heating. Math. Models Meth. Appl. Sci. 5 (1995), 725–738. ¨t, T., Nonlinear elliptic equations with right-hand side [3] Boccardo, L.; Galloue measures. Commun. Partial Diﬀ. Equs. 17 (1992), 641–655. [4] —— ; —— , Strongly nonlinear elliptic equations having natural growth terms and L1 data. Nonl. Analysis, Theory, Meth. & Appl. 19 (1992), 572–579. [5] Consiglieri, L., Stationary weak solutions for a class of non-Newtonian ﬂuids with energy transfer. Int. J. Non-Linear Mech. 32 (1997), 961–972. [6] Consiglieri, L.; Shilkin, T., Regularity to stationary weak solutions in the theory of generalized Newtonian ﬂuids with energy transfer. Zapiski Nauchn. Seminarov POMI 271 (2000), 122–150. [7] Giaquinta, M., Multiple integrals in the calculus of variations and nonlinear elliptic systems. Ann. Math. Studies 105, Princeton Univ. Press, Princeton, New Jersey 1983. [8] Goncharova, O.N., On the uniqueness of the solution of a stationary problem for equations of free convection (Russian). Dinamika Sploshn. Sredy. Sibirsk. Otdel. AN SSSR, Novosibirsk, no. 97 (1990), 22–28. [9] Landau, L.D.; Lifschitz, E.M., Lehrbuch der theoretischen Physik; Band VI: Hydrodynamik. 3. Auﬂ., Akademie-Verlag, Berlin 1978. [10] Lions, J.L., Quelques m´ethodes de r´esolution des probl`emes aux limites non lin´eaires. Dunod, Gauthier-Villars, Paris 1969. [11] L ukaszewicz, G., On the stationary ﬂows of viscous, incompressible and heatconducting ﬂuids. Math. Methods Appl. Sci. 10 (1988), 329–337. [12] Morimoto, H., On the existence of weak solutions of equation of natural convection. J. Fac. Sci. Univ. Tokyo, Sect. IA, Math. 36 ( 1989), 87–102. [13] —— , On the existence and uniqueness of the stationary solution to the equations of natural convection. Tokyo J. Math. 14 (1991), 217–226.

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[14] —— , A note on the existence of solutions to stationary Boussinesq equations under general outﬂow conditions. In: H. Amann et al. (eds.): Navier-Stokes equations and related nonlinear problems. Utrecht, VSP/TEV (1998); pp. 183–193. [15] M¨ uller, I., Grundz¨ uge der Thermodynamik. 3. Auﬂ., Springer-Verlag, Berlin 2001. [16] Naumann, J.; Wolff, M., On a global Lq -gradient estimate on weak solutions of nonlinear elliptic systems. Dip. Mat. Univ. Catania (rapporto interno, 18. 02. 1991) (19 pp.). [17] Rakotoson, J.M., Quasilinear elliptic problems with measures as data. Diﬀ. Integral Equs. 4 (1991), 449–457. ˇek, T., Steady-state bouyancy-driven viscous ﬂow with measure data. Math. [18] Roub´ıc Bohemica 126 (2001), 493–504. [19] Serrin, J., Mathematical principles of classical ﬂuid mechanics. In: Handbuch der Physik (Hrsg. S. Fl¨ ugge), Bd. VIII/1: Str¨ omungsmechanik I, Springer-Verlag, Berlin 1959; pp. 125–263. [20] Shilkin, T., Partial regularity of weak solutions of the stationary 3D-Boussinesq system. Zapiski Nauchn. Seminarov POMI 288 (2002), 256–270. [21] Spurk, J.H., Str¨omungslehre. 4. Auﬂ., Springer-Verlag, Berlin 1996. [22] Stampacchia, G., Le probl`eme de Dirichlet pour les ´equations elliptiques du second ordre a ` coeﬃcients discontinus. Ann. Inst. Fourier Grenoble 15 (1965), 189–258. [23] Wolff, M., A global Lκ -gradient estimate on weak solutions to nonlinear stationary Navier-Stokes equations under mixed boundary conditions. Inst. f. Math., HumboldtUniv. Berlin, preprint nr. 96-3 (1996). J. Naumann Institut f¨ ur Mathematik (Sitz: Rudower Chaussee 25) Humboldt-Universit¨ at zu Berlin Unter den Linden 6 D-10099 Berlin, BRD

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 391–402 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Liouville Type Theorems and Complete Blow-up for Indeﬁnite Superlinear Parabolic Equations Peter Pol´aˇcik and Pavol Quittner Dedicated to Herbert Amann

1. Introduction In this paper we study nonnegative bounded solutions of parabolic equations of the form ut − ∆u = h(x1 )f (u),

x = (x1 , x2 , . . . , xN ) ∈ RN , t ∈ R,

(1.1)

where h, f are nondecreasing continuous functions, h(0) = 0,

h is strictly increasing for x1 > 0,

lim h(x1 ) = +∞.

x1 →+∞

(1.2)

We are particularly interested in solutions deﬁned for all t ∈ R. Below we refer to such solutions as entire solutions. Our main goal is to prove a Liouville type result to the eﬀect that there are no positive bounded entire solutions, see Theorem 1.1 below. If, for example, f (u) = up , p > 1, the theorem guarantees that (1.1) does not possess any such solutions. Using this fact we obtain a new result on complete blow-up for the problem ⎫ ut − ∆u = λu + a(x)up , x ∈ Ω, t > 0, ⎪ ⎬ x ∈ ∂Ω, t > 0,

u = 0,

u(x, 0) = u0 (x) ≥ 0,

x ∈ Ω,

⎪ ⎭

(1.3)

¯ changes sign and where Ω is a smooth bounded domain in R , λ ∈ R, a ∈ C(Ω) p > 1 is subcritical, see Theorem 1.2 below. More speciﬁcally, our nonexistence result for (1.1) together with known nonexistence results for the equation N

ut − ∆u = up ,

x ∈ RN , t ∈ R,

The ﬁrst author was supported in part by NSF grant DMS-0400702. The second author was supported in part by VEGA grant 1/0259/03.

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guarantee suitable a priori bounds for solutions of (1.3) and these bounds can be eﬃciently used in the study of blow-up. Let us explain the relation between a priori bounds and Liouville type theorems in more detail. First consider steady states of (1.3) with a > 0. In order to show an a priori bound, we proceed by contradiction assuming that there exist positive equilibria uk , k = 1, 2, . . . , of (1.3) satisfying Mk := max uk (x) = uk (xk ) → ∞. x∈Ω

Set

−(p−1)/2

2/(p−1)

uk (xk + λk y), λk := Mk . vk (y) := λk Then a subsequence of (suitable transforms of) vk converges to a positive bounded solution of one of the following limit problem −∆v = v p where H

(1.4)

−∆v = v in H , v = 0 on ∂H , N := {x ∈ R : x1 > 0}. However, if N +2 p < pS := (N − 2)+ p

N

in RN ,

N

N

(1.5)

(1.6)

then problems (1.4) and (1.5) do not possess such solutions, see [6] and [11], respectively. Consequently, (1.6) guarantees a uniform bound for positive equilibria of (1.3) if a > 0. If a changes sign then in addition to the limit problems mentioned above one has to deal with equations of the form −∆v = h(x1 )v p

in RN ,

(1.7)

(or the corresponding problem in a half-space), where h depends on the behavior of a close to its zero set (for example, h(x1 ) = x1 if zero is a regular value of a). The solvability of such equations and related a priori estimates have been intensively studied, see [4, 2, 7, 8] and references therein. A priori bounds play an important role in the study of equilibria. For example, they appear as crucial ingredients in proofs of the existence of positive equilibria. Similarly, in the study of time dependent solutions of (1.3), suitable a priori bounds are often crucial tools of analysis. These bounds have either the form u(·, t)∞ ≤ C(u0 ∞ , δ, τ )

for any t ∈ (0, min(τ, T (u0 ) − δ)),

(1.8)

∞

where · ∞ denotes the norm in L (Ω), T (u0 ) denotes the maximal existence time of the solution of (1.3) and δ, τ > 0 are arbitrary, or u(·, t)∞ ≤ C(u0 ∞ )

for any t > 0 provided T (u0 ) = ∞.

(1.9)

In fact, estimate (1.8) can be used in the study of blow-up and estimate (1.9) is useful in the study of equilibria and connecting orbits. A connection between Liouville theorems and the a priori bounds is furnished, similarly as in the case the equilibria, by a scaling argument. Assume that estimates

Liouville Type Theorems and Complete Blow-up

393

(1.8) or (1.9) fail and consider solutions uk of (1.3) with uniformly bounded initial data u0,k such that supt∈Ik u(·, t)∞ → ∞, where Ik = [0, min(τ, T (u0,k ) − δ)) in the case of (1.8), Ik = [0, ∞) in the case of (1.9). Then, for a suitable subsequence of uk , we can ﬁnd xk ∈ Ω and tk ∈ Ik such that Mk := uk (xk , tk ) → ∞ and the functions uk remain bounded by 2Mk on the time intervals (tk − kMk1−p , tk + kMk1−p ), see [21, 20]. Set 2/(p−1)

vk (y, s) := λk

uk (xk + λk y, tk + λ2k s),

−(p−1)/2

λk := Mk

.

Standard parabolic estimates imply that, along a subsequence, vk converges to a positive bounded entire solution of a parabolic problem on the whole space or on a half-space. In case a > 0, the limit problem has one of the following forms vt − ∆v = v p ,

x ∈ RN , t ∈ R,

(1.10)

vt − ∆v = v , x ∈ H , t ∈ R, v = 0, x ∈ ∂H , t ∈ R. (1.11) In addition, still with a > 0, one can obtain a uniform bound for the corresponding energy functionals 1 λ 1 2 2 E(uk (·, t)) := |∇uk | dx − u dx − aup+1 dx, (1.12) 2 Ω 2 Ω k p+1 Ω k for t ∈ Ik and this bound guarantees a uniform estimate for the time derivatives of uk in L2 (Ω × Ik ). Using this estimate it is easy to see that the limit function v does not depend on t, hence the limit problems (1.10) and (1.11) reduce to (1.4) and (1.5), respectively. Consequently, the nonexistence results mentioned above imply a priori bounds (1.8) and (1.9) for solutions of (1.3) if (1.6) holds and a > 0. If, on the other hand, a changes sign (or if we considered problem (1.3) with some lower-order perturbations destroying the variational structure, see [21]) then the energy bound for (1.12) is not known and one has to guarantee the nonexistence of positive bounded solutions for the time-dependent problems (1.10), (1.11). In addition, for sign-changing functions a we also have to show nonexistence results for limit problems of the form p

N

N

vt − ∆v = h(x1 )v p ,

x ∈ RN , t ∈ R.

(1.13)

Unlike the nonexistence results for the elliptic problems (1.4), (1.5) and (1.7), Liouville type theorems for the parabolic counterparts (1.10), (1.11) and (1.13) do not seem to be known in the full subcritical range. Optimal nonexistence results are well known for these parabolic problems if we replace the time interval t ∈ R with t ∈ (0, ∞). However, these Fujita type results provide just suﬃcient conditions for the nonexistence of entire solutions. For example, the problem vt − ∆v = v p ,

x ∈ RN , t ∈ (0, ∞),

does not possess positive global solutions if and only if p ≤ 1+2/N . This restriction on p is certainly not optimal for positive entire solutions of problem (1.10). In fact, the nonexistence of such solutions follows from [5] for any p < N (N + 2)/(N − 1)2 and the nonexistence of radial positive bounded entire solutions of (1.10) for any p < pS is proved in [17] (see also [16] for such result on radially decreasing solutions

394

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in dimensions N ≤ 3 or [15, 18] for related results mentioned below). Similarly, the Fujita type result for (1.13) with h(x1 ) = x1 requires p ≤ 1 + 3/(N + 1) (see [20]), but the nonexistence result for entire positive bounded solutions of this problem is true for any p > 1. This follows from the following theorem which is the main result of this paper. Theorem 1.1. Suppose that h ∈ C(R) and f ∈ C 1 (R+ , R+ ) are nondecreasing functions satisfying (1.2) and f (0) = f (0) = 0, f (u) > 0 for u > 0. Then the only nonnegative bounded classical solution of the equation ut − ∆u = h(x1 )f (u),

x ∈ RN , t ∈ R,

(1.14)

is the trivial solution u ≡ 0. The proof of this theorem is based on the method of moving planes and it was inspired by the proof of the corresponding elliptic Liouville theorem in [8]. The elliptic result now becomes a special case of Theorem 1.1 dealing with steady states. We remark that our assumptions on the functions h and f are weaker than those in [8], so we have a slight improvement of the theorem in [8] even in this special case. Nonexistence results for (1.13) and (1.10) were used in [20] in order to prove a priori estimates and complete blow-up results for (1.3). Since the nonexistence for (1.13) was proved only for p ≤ 1 + 3/(N + 1) in [20], the main result in [20] required this restriction on the exponent p. Here we can use our new nonexistence result for (1.13) (valid for any p > 1) and the nonexistence result for (1.10) from [5] to justify that the condition p ≤ 1 + 3/(N + 1) in [20] can be replaced with p < N (N + 2)/(N − 1)2 . In order to formulate the corresponding statement, let us ﬁrst recall the notion of complete blow-up. Assume T (u0 ) < ∞ and let uk be solutions of the approximation problems ⎫ ut − ∆u = fk (x, u), x ∈ Ω, t > 0, ⎪ ⎬ u = 0, x ∈ ∂Ω, t > 0, (1.15) ⎪ ⎭ u(x, 0) = u0 (x), x ∈ Ω, where

fk (x, u) :=

λu + a(x) min(up , k), if a(x) ≥ 0, λu + a(x)up , if a(x) < 0,

k = 1, 2, 3, . . . .

The solutions uk exist globally and uk+1 ≥ uk , k = 1, 2, . . . . Set u ¯(x, t) := lim uk (x, t) ∈ [0, ∞], k→∞

x ∈ Ω, t ∈ [0, ∞).

¯ remains ﬁnite in Then u ¯(x, t) = u(x, t) for any x ∈ Ω and t < T (u0 ) and u Ω− := {x ∈ Ω : a(x) < 0} for any t > 0. Let D ⊂ Ω. We say that u blows up completely in D at t = T if u ¯(x, t) = ∞ for any x ∈ D and any t > T . The following theorem is an immediate consequence of results in [20] and Theorem 1.1.

Liouville Type Theorems and Complete Blow-up

395

Theorem 1.2. Assume that p < N (N + 2)/(N − 1)2 , Ω+ := {x ∈ Ω : a(x) > 0} is connected and a ∈ C 2 (Ω),

a < 0 on ∂Ω,

∇a(x) = 0 whenever a(x) = 0.

(1.16)

If T (u0 ) < ∞ then the solution u of (1.3) blows up completely in Ω at t = T (u0 ). In addition, the function +

T : {u0 ∈ L∞ (Ω) : u0 ≥ 0} → (0, ∞] : u0 → T (u0 ) is continuous. If a ≡ 1 then a complete blow-up result was shown in [3] for the full subcritical range p ∈ (1, pS ). On the other hand, it is known that solutions of (1.3) need not blow up completely if p > pS , see [10] or [9] and the references therein. Similarly, the function T need not be continuous if p > pS . Liouville type theorems have many other applications than those discussed above. As an illustration we mention two related results on the problem x ∈ RN , t > 0, ut − ∆u = up , (1.17) u(x, 0) = u0 (x) ≥ 0, x ∈ RN , where u0 ∈ L∞ (RN ). The ﬁrst one concerns positive solutions of the equation ut − ∆u = up ,

x ∈ RN , t ∈ (−∞, T ),

(1.18)

satisfying the estimate 1

u(x, t) ≤ C(T − t)− p−1

(x ∈ RN , t < T ),

(1.19)

where C is a constant. Note that there is a “trivial” family of such solutions: u(x, t) = κ(T1 − t)−1/(p−1) , T1 ≥ T , where κ = (1/(p − 1))1/(p−1) . A theorem of [15] asserts that this family contains all positive solutions of (1.18) satisfying (1.19) if p < pS . This result is used in a theorem on the blow-up rate of positive unbounded solutions of (1.17). The second result concerns a diﬀerent class of solutions of (1.18), namely solutions satisfying ϕ1 (x) ≤ u(x, t) ≤ ϕ2 (x)

(x ∈ RN , t < T ),

(1.20)

where ϕ1 < ϕ2 are positive radial steady states of (1.17) (ϕ2 can be singular). It is well known (see [14]) that the related steady states exist if and only if N > 10 and p ≥ pJL , where √ (N − 2)2 − 4N + 8 N − 1 . pJL := (N − 2)(N − 10) In fact, there is an increasing curve of steady states, which constitutes a “trivial” set of solutions satisfying (1.20). According to a theorem of [18], each solution of (1.18) satisfying (1.20) is a steady state if p > pJL . This result is used in the proof of quasiconvergence (convergence to a set of steady states) of positive solutions of (1.17) bounded above by the singular steady state.

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Finally let us mention that a priori bounds (1.8) and (1.9) can also be obtained for sign-changing solutions. In that case we cannot use rescaling but one can use energy and interpolation estimates, see [19, 1, 13] and the references therein.

2. Proof of Theorem 1.1 Assume on the contrary that (1.14) possesses a positive bounded classical solution u. Denote Lu := ut − ∆u, M := sup u. Given λ ∈ R, set Σλ := {x ∈ RN : x1 < λ}, xλ = (xλ1 , x2 , . . . , xN ) := (2λ − x1 , x2 , . . . , xN ), ¯ λ × R, wλ (x, t) := u(xλ , t) − u(x, t), (x, t) ∈ Σ ˜ : wλ (x, t) ≥ 0 for all x ∈ Σλ and λ < λ}, ˜ λ(t) := sup{λ λ∗ := inf{λ(t) : t ∈ R}. We will prove the following assertions. Claim 1. If x ∈ Σmin(λ,0) , t ∈ R and wλ (x, t) ≤ 0 then Lwλ (x, t) ≥ 0. Claim 2. λ(t) ≥ 0 for all t ∈ R. Claim 3. The mapping t → λ(t) is nondecreasing. Claim 4. λ∗ = ∞. Claim 5. u ≡ 0. Proof of Claim 1. Let x1 < min(λ, 0) and wλ (x, t) ≤ 0. If xλ1 ≥ 0 then h(xλ1 ) ≥ 0 ≥ h(x1 ) and Lwλ (x, t) = h(xλ1 )f (u(xλ , t)) − h(x1 )f (u(x, t)) ≥ 0. If xλ1 < 0 then h(x1 ) ≤ h(xλ1 ) ≤ 0 and, since f is nondecreasing, Lwλ (x, t) = h(xλ1 )f (u(xλ , t)) − h(x1 )f (u(x, t)) ≥ h(xλ1 ) f (u(xλ , t)) − f (u(x, t)) ≥ 0. Proof of Claim 2. Let λ ≤ 0. We prove wλ (x, t) ≥ 0 for all x ∈ Σλ and t ∈ R. Assume on the contrary that inf wλ (x, t0 ) < 0 for some t0 ∈ R.

(2.1)

x∈Σλ

Set g(y) := ln(λ + 1 − y) + 1, y ≤ λ,

and

w ¯λ (x, t) :=

wλ (x, t) . g(x1 )

Notice that g(y) ≥ g(λ) = 1, g (y) = −(λ + 1 − y)−2 < 0 and g(x1 )Lw ¯λ (x, t) = Lwλ (x, t) + 2(∂x1 w ¯λ (x, t))g (x1 ) + w ¯λ (x, t)g (x1 ).

(2.2)

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397

Assume x ∈ Σλ and wλ (x, t) ≤ 0. Then Lwλ (x, t) ≥ 0 due to Claim 1, hence Lw ¯λ (x, t) ≥ 2(∂x1 w ¯λ )

g g +w ¯λ . g g

We compare w ¯λ with the solution of the problem zt − zyy = F (y, z, zy ), y ∈ R, t > 0, where

z(y, 0) = −M, y ∈ R,

⎧ ⎪ y < λ − 1, ⎨2zy g /g, F (y, z, zy ) := 2zy g /g − az, y ∈ [λ − 1, λ], ⎪ ⎩ 0, y > λ,

and a := −g (λ−1)/g(λ−1) > 0. Note that, by the maximum principle, z(y, t) < 0 for all y ∈ R, t ≥ 0, and, since F (y, −M, 0)) ≥ 0 with the strict inequality in (λ − 1, λ), z is increasing in t. In view of the obvious inequalities g g +z g g w ¯λ (x, t0 − T ) ≥ z(x1 , 0) F (x1 , z, zx1 ) ≤ 2zx1

wλ (x, t) = 0 > z(x1 , t)

(x1 < λ), (x ∈ Σλ ), (x ∈ ∂Σλ ),

the maximum principle implies w ¯λ (x, t0 ) ≥ z(x1 , T ) for any T > 0 and x ∈ Σλ . We now show that z(x1 , T ) → 0 as T → ∞, which yields a contradiction to (2.1). First oﬀ, z being increasing in t and negative implies that the limit z¯(y) := limt→+∞ z(y, t) is a solution of −¯ z = F (y, z¯, z¯ ) satisfying −M ≤ z¯ ≤ 0. Since F (y, z¯, z¯ ) = 0 for y > λ, the function z¯ is constant on [λ, ∞). One can also easily z g /g on (−∞, λ − 1) are constant, show that all bounded solutions of −¯ z = 2¯ hence z¯ is constant on (−∞, λ − 1]. Finally, on [λ − 1, λ], z¯ satisﬁes the following equation, with matching conditions at the boundary points: −¯ z = 2¯ z g /g − a¯ z,

z¯ (λ − 1) = z¯ (λ) = 0.

Assume that z¯ is not identically zero in [λ − 1, λ] and let z attain its negative minimum in this interval at y0 ∈ [λ − 1, λ]. Then z¯ (y0 ) = 0, hence z¯ (y0 ) = a¯ z (y0 ) < 0 which yields a contradiction. Consequently, z¯ ≡ 0 in R, as claimed. Proof of Claim 3. Let t0 ∈ R and λ ≤ λ(t0 ). Then wλ (x, t0 ) ≥ 0 for any x ∈ Σλ . If x ∈ Σλ then xλ1 > x1 and Lwλ (x, t) = h(xλ1 )f (u(xλ , t)) − h(x1 )f (u(x, t)) ≥ h(xλ1 ) f (u(xλ , t)) − f (u(x, t)) = h(xλ1 )f (θ(x, t))wλ (x, t) for some θ(x, t) between u(x, t) and u(xλ , t). Since wλ (x, t) = 0 for x ∈ ∂Σλ , the maximum principle implies wλ (x, t) ≥ 0, hence λ(t) ≥ λ(t0 ), for all t ≥ t0 . This proves that the function t → λ(t) is nondecreasing.

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Proof of Claim 4. Claim 2 guarantees λ∗ ≥ 0. Assume λ∗ < ∞ and choose tk → −∞. Then λ(tk ) → λ∗ hence there exist λk # λ∗ such that inf wλ (x, tk ) < 0.

(2.3)

x∈Σλk

There exists Cf > 0 such that f ≤ Cf on [0, M ]. Set b2 := h(λ∗ + 1)Cf , ﬁx δ > 0 such that 2δ −2 ≥ 33 (2b2 + 1)

(2.4)

and η > 0 such that f ≤

δ on [0, η]. h(λ∗ + 1)(λ∗ + 1)3

(2.5)

Let ε ∈ (0, δ) be suﬃciently small (as speciﬁed below). Fix k such that λk < λ∗ + ε and set λ := λk , g(y) := 2 −

δ , y ∈ (−∞, λ], δ+λ−y

w ¯λ (x, t) :=

wλ (x, t) . g(x1 )

We next derive an inequality satisﬁed by w ¯λ in the set where it is negative. Let x ∈ Σλ and wλ (x, t) ≤ 0. First we treat the case x1 < λ∗ − δ. If Lwλ (x, t) ≥ 0, then (2.2) gives Lw ¯λ (x, t) ≥ 2(∂x1 w ¯λ )

g . g

(2.6)

Assume Lwλ (x, t) < 0. Then Claim 1 guarantees that x1 > 0. Standard local parabolic estimates give |∇u(x, t)| ≤ Cλ

whenever x1 ∈ [0, 2λ].

Since ∗

u(xλ , t) ≥ u(x, t) ≥ u(xλ , t),

(2.7)

h(x1 ) ≥ 0, and ∗

∗

|xλ − xλ | = |xλ1 − xλ1 | = 2(λ − λ∗ ) ≤ 2ε, we have 0 > Lwλ (x, t) = h(xλ1 )f (u(xλ , t)) − h(x1 )f (u(x, t)) ∗ ∗ ≥ h(xλ1 ) − h(x1 ) f (u(xλ , t)) + h(xλ1 ) f (u(xλ , t)) − f (u(xλ , t)) ∗ ∗ ≥ h(λ∗ ) − h(λ∗ − δ) f (u(xλ , t)) − h(2λ∗ + 1)Cf u(xλ , t) − u(xλ , t) ∗ ≥ h(λ∗ ) − h(λ∗ − δ) f (u(xλ , t)) − 2h(2λ∗ + 1)Cf Cλ ε. Consequently, ∗

f (u(xλ , t)) ≤

2h(2λ∗ + 1)Cf Cλ ε h(λ∗ ) − h(λ∗ − δ)

Liouville Type Theorems and Complete Blow-up

399

∗

and this bound guarantees u(xλ , t) ≤ η if ε is small enough. Now (2.7) and (2.5) imply Lwλ (x, t) ≥ h(x1 ) f (u(xλ , t)) − f (u(x, t)) δ ≥ h(λ∗ ) wλ (x, t) ≥ b1 wλ (x, t), ∗ h(λ + 1)(λ∗ + 1)3 where g (y) δ δ g (y) ≤− for y ∈ [0, λ∗ ]. ≤ =− b1 := ∗ 3 3 (λ + 1) (δ + λ − y) 2 g(y) Hence we infer from (2.2) that (2.6) holds in this case as well. We conclude that Lw ¯λ (x, t) ≥ 2(∂x1 w ¯λ )

g g

whenever wλ (x, t) ≤ 0 and x1 < λ∗ − δ.

Next assume x1 ∈ [λ∗ − δ, λ]. Then Lwλ (x, t) ≥ h(x1 ) f (u(xλ , t)) − f (u(x, t)) ≥ h(λ∗ + 1)Cf wλ (x, t) = b2 wλ (x, t). Notice that −g (y) =

2δ ≥ 2b2 + 1 ≥ g(y)b2 + 1 for y ∈ [λ∗ − δ, λ], (δ + λ − y)3

due to (2.4), hence Lw ¯λ (x, t) ≥ 2(∂x1 w ¯λ )

w ¯λ g − g g

whenever wλ (x, t) ≤ 0 and x1 ∈ [λ∗ − δ, λ].

We employ the above inequalities in a comparison argument, similarly as in the proof of Claim 2. By the maximum principle, we have w ¯λ (x, tk ) ≥ z(x1 , T ) for any T > 0 and x ∈ Σλ , where z is the solution of the problem zt − zyy = F (y, z, zy ), y ∈ R, t > 0, where

z(y, 0) = −M, y ∈ R,

⎧ ⎪ y < λ∗ − δ, ⎨2zy g /g, F (y, z, zy ) := 2zy g /g − az, y ∈ (λ∗ − δ, λ), ⎪ ⎩ 0, y > λ,

and a := inf [λ∗ −δ,λ] 1/g = 1/g(λ∗ − δ). One shows, by the same arguments as in the proof of Claim 2, that z(x1 , T ) → 0 as T → ∞, hence w ¯λ (x, tk ) ≥ 0, which contradicts (2.3). Proof of Claim 5. Set ˜) ∈ R × RN −1 : σ < x1 < σ + 1, |˜ x| < Ωσ := {x = (x1 , x Let µ0 be the ﬁrst eigenvalue of the problem −∆ϕ = µϕ in Ω0 ,

ϕ = 0 on ∂Ω0 ,

1 }, 2

σ ≥ 0.

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and let ϕ0 be the corresponding positive eigenfunction satisfying maxΩ0 ϕ0 = 1. Set ϕσ (x) := ϕ0 (x1 − σ, x˜) and ψσ (t) := u(x, t)ϕσ (x) dx. Ωσ

Then ψσ (0) ≥ c0 := ψ0 (0) since u is nondecreasing in x1 . Denote Ωσ (t) := {x ∈ Ωσ : u(x, t)ϕσ (x) ≥ c0 /2}. If t ≥ 0 is such that ψσ (t) ≥ c0 then |Ωσ (t)| ≥ ξ := c0 /(2M ). Indeed, c0 c0 ≤ u(x, t)ϕσ (x) dx ≤ |Ωσ (t)| · M + , 2 Ωσ and the stated inequality follows. Notice that ξ ∈ (0, 1) and c0 u(x, t)ϕσ (x) dx ≥ ξ ≥ ξ u(x, t)ϕσ (x) dx, 2 Ωσ (t) Ωσ \Ωσ (t) hence

ξ u(x, t)ϕσ (x) dx ≥ 1+ξ Ωσ (t)

u(x, t)ϕσ (x) dx = Ωσ

c0 ψσ (t). 2M + c0

We conclude that, as long as ψσ (t) ≥ c0 , we have d ψσ (t) = ∆u(x, t)ϕσ (x) dx + h(x1 )f (u(x, t))ϕσ (x) dx dt Ωσ Ωσ ≥ u(x, t)∆ϕσ (x) dx + h(σ) f (u(x, t))ϕσ (x) dx Ωσ

Ωσ (t)

f (u(x, t)) u(x, t)ϕσ (x) dx u(x, t)∆ϕσ (x) dx + h(σ) M Ωσ Ωσ (t) c 1 0 u(x, t)ϕσ (x) dx ≥ −µ0 ψσ (t) + h(σ)f 2 M Ωσ (t) * ) c 1 c0 0 . ≥ ψσ (t) −µ0 + h(σ)f 2 M 2M + c0 Taking σ large, the expression in the square brackets becomes greater than one, hence ψσ (t) ≥ ψσ (t). Since ψσ (0) ≥ c0 , we deduce that the inequality is valid for all t ≥ t0 . Consequently, ψσ (t) → +∞ as t → ∞ which contradicts the boundedness of u. ≥

Acknowledgment This research was done while the second author was visiting University of Minnesota. He thanks the School of Mathematics for the hospitality. The authors also thank Yihong Du for correcting a choice of the function g in the proof of Claim 4.

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References [1] N. Ackermann, T. Bartsch, P. Kaplick´ y and P. Quittner, A priori bounds, nodal equilibria and connecting orbits in indeﬁnite superlinear parabolic problems, preprint. [2] H. Amann, J. L´ opez-G´ omez, A priori bounds and multiple solutions for superlinear indeﬁnite elliptic problems, J. Diﬀerential Equations 146 (1998), 336–374. [3] P. Baras and L. Cohen, Complete blow-up ater Tmax for the solution of a semilinear heat equation, J. Funct. Anal. 71 (1987), 142–174. [4] H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Superlinear indeﬁnite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonlinear Anal. 4 (1994), 59–78. [5] M.-F. Bidaut-V´eron, Initial blow-up for the solutions of a semilinear parabolic equation with source term, In: Equations aux d´eriv´ees partielles et applications, articles d´edi´es ` a Jacques-Louis Lions, Gauthier-Villars, Paris (1998), 189–198. [6] W. Chen and C. Li, Classiﬁcation of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991), 615–622. [7] W. Chen and C. Li, Indeﬁnite elliptic problems in a domain, Discrete Contin. Dyn. Syst. 3 (1997), 333–340. [8] Y. Du and S. Li, Nonlinear Liouville theorems and a priori estimates for indeﬁnite superlinear elliptic equations, Adv. Diﬀerential Equations 10 (2005), 841–860. [9] M. Fila, H. Matano and P. Pol´ aˇcik, Immediate regularization after blow-up, SIAM J. Math. Analysis, to appear. [10] V. Galaktionov and J.L. V´ azquez, Continuation of blow-up solutions of nonlinear heat equations in several space dimensions, Comm. Pure Appl. Math. 50 (1997), 1–67. [11] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Diﬀerential Equations 6 (1981), 883–901. [12] Y. Giga, A bound for global solutions of semilinear heat equations, Comm. Math. Phys. 103 (1986), 415–421. [13] Y. Giga, S. Matsui and S. Sasayama, Blow up rate for semilinear heat equation with subcritical nonlinearity, Indiana Univ. Math. J. 53 (2004), 483–514. [14] C. Gui, W.-M. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in Rn , Comm. Pure Appl. Math. 45 (1992), 1153–1181. [15] F. Merle and H. Zaag, Optimal estimates for blowup rate and behavior for nonlinear heat equations, Comm. Pure Appl. Math. 51 (1998), 139–196. [16] J. Matos and Ph. Souplet, Universal blow-up rates for a semilinear heat equation and applications, Adv. Diﬀerential Equations 8 (2003), 615–639. [17] P. Pol´ aˇcik and P. Quittner, A Liouville-type theorem and the decay of radial solutions of a semilinear heat equation, Nonlinear Analysis TMA, to appear. [18] P. Pol´ aˇcik and E. Yanagida, A Liouville property and quasiconvergence for a semilinear heat equation, J. Diﬀerential Equations 208 (2005), 194–214. [19] P. Quittner, Continuity of the blow-up time and a priori bounds for solutions in superlinear parabolic problems, Houston J. Math. 29 (2003), 757–799.

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[20] P. Quittner and F. Simondon, A priori bounds and complete blow-up of positive solutions of indeﬁnite superlinear parabolic problems, J. Math. Anal. Appl. 304 (2005), 614–631. [21] P. Quittner and Ph. Souplet, A priori estimates of global solutions of superlinear parabolic problems without variational structure, Discrete Contin. Dyn. Syst. 9 (2003), 1277–1292. Peter Pol´ aˇcik School of Mathematics University of Minnesota Minneapolis MN 55455, USA Pavol Quittner Department of Applied Mathematics and Statistics Comenius University Mlynsk´ a dolina 84248 Bratislava, Slovakia

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 403–411 c 2005 Birkh¨ auser Verlag Basel/Switzerland

On Reducing the 2d Navier-Stokes Equations to a System of Delayed ODEs Dalibor Praˇz´ak Abstract. We give a simple proof that projecting the 2d Navier-Stokes equations to suﬃciently many eigenfunctions of the Stokes operator leads to a system of delayed ODEs. The proof is based on the repeated use of the so-called squeezing property. The reduced system is uniquely solvable and dissipative. Moreover, the solutions on the attractor to the full NSEs are in one-to-one correspondence to the solutions on a compact, invariant subset to a global attractor of the reduced system. Mathematics Subject Classiﬁcation (2000). 37D45, 75D05. Keywords. Navier-Stokes equations, global attractor, equations with delay.

0. Introduction Is the behavior of a given dissipative PDE ﬁnite-dimensional for large times? There is vast literature dealing with this question, see [10], [9], [1] and the references therein. Generally speaking, the answer is yes, yet it comes in a variety of results. The “simplest” possible type of theorems state the existence of a global attractor with ﬁnite (fractal or Hausdorﬀ) dimension. This means, all solutions are attracted to a set that is ﬁnite-dimensional. However, the convergence to the attractor can be very slow, and the distance is still measured in a inﬁnite-dimensional space. Hence, we would like to be more speciﬁc in expressing the ultimate ﬁnitedimensionality. One possible approach is the following. Assume that our PDE has an abstract form (1) ut + Au = F (u) where u = u(t) lives in a suitable (inﬁnite-dimensional) space, A is a linear operator and F is the lower-order nonlinearity. Let P be a projection to ﬁrst N eigenfunctions of A and set p = P u. We would like to show that for N large enough, the behavior of u is dominated by p. Research supported by GACR grant 201/01/D054 and MSM grant 1132 00007.

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Here come the theorems on the number of determining modes. In many situations one indeed proves that if p1 − p2 → 0, then u1 − u2 → 0, meaning for t → ∞. See [8], for example. But let us proceed even further. The projection P gives an equation for p pt + Ap = P F (u)

(2)

and one could ask whether we can ‘decouple’ this equation from (1) so that it would be possible to solve for p independently. More precisely, we seek a function Φ : R(P ) → R(I − P ) such that if p solves the reduced equation pt + Ap = P F (p + Φ(p))

(3)

then p + Φ(p) solves (1). Such a decoupling function (suitably smooth) is of course nothing else than the so-called inertial manifold. There is a well-known method of Lyapunov-Perron for constructing such a manifold, see, e.g., [6]. In this approach, the function Φ is sought by means of a Banach contraction theorem in a suitable space of functions from R(P ) to R(I−P ). The key assumption around which the contraction argument evolves is that the gap between the spectra of AP and A(I − P ) is suﬃciently large to dominate the nonlinearity F . Unfortunately, there are important examples of PDEs for which this ‘gap condition’ fails. The 2d Navier-Stokes equations are probably the most prominent example. At any rate, the spectral gap condition is rather restrictive. Several attempts were made to extend the validity of Lyapunov-Perron construction to more general situations. In [3] Debussche and Temam showed that, regardless the spectral gap, one can use the ﬁxed point argument to obtain a function Φ : R(P )×R(I −P ) → R(I −P ) such that Φ(p(t), q(t − T )) = q(t), where q = (I − P )u. That means, one can ﬁnd q(t) provided we know p(t) and also q at some earlier time t − T . Thus Φ is, as the authors called it, an inertial manifold with delay. Another generalization of Lyapunov-Perron method was presented by Hale and Raugel in [7]. Here the authors proved that the ﬁxed point argument will work – once again without the gap condition – to obtain a function Φ from the space of R(P )-valued functions to the space of R(I − P )-valued functions. In other words, one can ﬁnd the trajectory {q(s)}s∈R provided we know the trajectory {p(s)}s∈R . We would also like to mention a diﬀerent approach of [4]. Here the authors started directly from the fact that the attractor has ﬁnite dimension. Hence by Ma˜ n´e’s theorem there always exists a projection P that is – when restricted to the attractor – invertible. Thus one can write pt + Ap = P F (P −1 (p)) .

(4)

There are certain disadvantages to this approach, namely (i) one doesn’t know P explicitly and (ii) in general, P −1 is only H¨older continuous, hence (4) is not uniquely solvable.

On Reducing the 2d Navier-Stokes Equations

405

But let us return to the papers [3] and [7]. Both of them obtain an interesting corollary. Although the function Φ would not reduce (1) to a system of ODEs, it can still reduce it to a system of ODEs with an inﬁnite delay. Moreover, the properties of Φ show that the delay has at least Lipschitz dependence with exponentially fading memory. The aim of our paper is to give an alternative and perhaps simpler proof of this reduction. Without recalling to a ﬁxed point argument we show in an elementary way (see Lemma 2.1) that the value u(t) can be computed from the past values of p(t). Moreover, this functional dependence is Lipschitz continuous. This reduces the equation to a system of delayed ODEs, ﬁrst only on the global attractor. However, since the domain of a Lipschitz function can easily be extended to the whole space, (cf. Lemma 1.4), we ﬁnally embed the original equation into a full system of delayed ODEs (12). We then brieﬂy study the properties of the reduced system. We prove existence, uniqueness (Theorem 3.1), dissipativity and existence of global attractor (Theorem 3.2). Finally, in Theorem 3.3 we summarize the correspondence between the solutions of the original equation (5) and the reduced system (12). Our paper deals exclusively with the 2d Navier-Stokes system on torus. Yet the argument depends mainly on the squeezing property (9) that holds quite generally, cf. [4]. Hence a similar construction certainly can be carried out for other dissipative equations. Last, but not least, one is also lead to several questions of a more general character: • How larger is Ar (the attractor to the reduced system) than A (the attractor to the full Navier-Stokes system)? In particular, has Ar ﬁnite dimension? • Does the reduced system describe also the behavior outside the attractor? For example, does it have some kind of tracking property? • Can the inﬁnite delay be replaced by the bounded one? We would like to address these issues in the future research. Notation. If p is a function on (−∞, t] then [pt ](s) = p(t + s) is a function on (−∞, 0]. Less formally we use a superscript p0 to emphasize that also the past values are taken into account, e.g., A0 denotes the set of all backward trajectories in A.

1. Preliminaries Consider the Navier-Stokes system ut + (u · ∇) u − νu + ∇p = f div u = 0 u|t=0 = u0

(5)

406

D. Praˇz´ak

where u(x, t) : Ω × [0, ∞) → Rn and p(x, t) : Ω × [0, ∞) → Rn are the velocity and pressure and f (x) : Ω → Rn are given external forces. We restrict ourselves to the two-dimensional spatially-periodic setting, i.e., Ω = (0, ω)2 and u, p are with zero mean and periodic boundary conditions. We set V = u : R2 → R2 : div u = 0, u = 0, u is periodic , Ω

H k = clW k,2 (Ω) V

with uH k = ∇k u2 .

Fix f ∈ H 1 . It is well known (see, e.g., [2]) that for any u0 ∈ H 0 , there exists unique u ∈ C([0, T ]; H 0 ) ∩ L2 (0, T ; H 1 ) a weak solution to (5). Hence one can deﬁne S(t) : H 0 → H 0 the solution operators to (5). The dynamical system (S(t), H 0 ) is dissipative - there exists a bounded B ⊂ 0 H which is uniformly absorbing and positively invariant for S(t). Due to the regularizing eﬀect one can have additional smoothness of B. We will need that B is bounded in H 2 and closed in H 0 . See, e.g., [4] for the proof. The set = 3 A= clH 0 S(s)B t>0

s>t

is the so-called global attractor. By deﬁnition it is a compact set that is uniformly attracting and invariant for the solutions, cf. [10]. Note that A ⊂ B. The following standard results will be used later. Lemma 1.1. The convective term R(u) = (u · ∇) u is on B Lipschitz continuous from H 1 to H 0 . Proof. Set w = u − v. The embedding W 2,2 ⊂ W 1,4 ⊂ L∞ gives (u · ∇) u − (v · ∇) v2 ≤ u · ∇w2 + w · ∇v2 ≤ c1 u∞ ∇w2 + w4 ∇v4 ≤ c2 ∇w2 u2,2 + v2,2

Since B is bounded in H 2 , the conclusion follows.

Lemma 1.2. For ﬁxed τ > 0, the mapping S(τ ) is on B Lipschitz continuous from H 0 to H 1 . Proof. Subtract equations for u, v and test by w = u − v to get 1 d 2 2 [R(u) − R(v)] · w = I . w2 + ν∇w2 ≤ 2 dt Ω By previous lemma I ≤ R(u) − R(v)2 w2 ≤ c1 ∇w2 w2 ≤

ν c2 ∇w22 + w22 . 2 2

On Reducing the 2d Navier-Stokes Equations

407

d 2 2 2 w2 + ν∇w2 ≤ c2 w2 . dt

(6)

Hence By Gronwall’s lemma

2

2

w(t)2 ≤ ec2 t w(0)2 ,

t ≥ 0.

(7)

Further, integrating (6) over t ∈ (0, τ ) and using (7) gives τ τ 2 2 2 2 ν ∇w2 ≤ c2 w2 + w(0)2 ≤ w(0)2 1 + c2 τ ec2 τ . 0

(8)

0

Next, we test the equation for w by −w to get 1 d 2 2 ∇w2 + ν∇2 w2 ≤ [R(u) − R(v)] · w = J . 2 dt Ω

Here 2

J ≤ R(u) − R(v)2 w2 ≤ c3 ∇w2 ∇2 w2 ≤ ν∇2 w2 +

c4 2 ∇w2 . 2

Hence

d 2 2 ∇w2 ≤ c4 ∇w2 . dt Multiply by a smooth function ξ such that ξ(0) = 0 and ξ(τ ) = 1. Integrating over t ∈ (0, τ ) (by parts on the left) gives τ 2 2 ∇w(τ )2 ≤ c4 |ξ| + |ξ | ∇w2 . 0

Combining with (8) gives the conclusion.

Lemma 1.3. Let P be a projection to the ﬁrst N eigenfunctions of the Stokes operator (i.e., − restricted to H 0 .) For N large enough there exist t∗ > 0 and θ ∈ (0, 1/4) such that S = S(t∗ ) satisﬁes on for any u, v ∈ B the so-called squeezing property: √ (9) Su − Sv2 ≥ 2P (Su − Sv)2 =⇒ Su − Sv2 < θu − v2 . Proof. For the proof including the estimate N = rank P ≤ c G2 ln G4 ν 2 λ1 + 1

(10)

see [4, pp. 57–60]. Here λ1 the ﬁrst eigenvalue of the Stokes operator and G = f 2 /ν 2 λ1 is the Grashof number. We will need a weaker form of squeezing property: √ (11) Su − Sv2 ≤ 2P (Su − Sv)2 + θu − v2 .

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Remark. One can obtain better Lipschitz constants in Lemmas 1.1, 1.2 using more delicate estimates of the convective term. On the other hand, (9) or (11) can be deduced from the ‘smoothing property’ of Lemma 1.2. However, the estimate of N (= rank P ) would be worse. Finally, we will need the following abstract result. Lemma 1.4. Let X be a normed space. Let B ⊂ X be bounded and F8 : B → R be a Lipschitz mapping. Then there exists F : X → R such that F |B = F8 and F is Lipschitz, bounded and has a bounded support. Proof. See, e.g., [5].

2. The reduced system Set

A0 = u(t) : (−∞, 0] → H 0 is a solution to (5) with u(0) ∈ A T = P u : u ∈ A0

Observe that A0 is bounded in C((−∞, 0]; H 0 )∩L∞ (−∞, 0; H 2 ) and T is bounded in C((−∞, 0]; RN ) and compact in X = Cρ ((−∞, 0]; RN ) with ρ = eγt , γ > 0 where χX = sup |χ(t)|eγt . t≤0

To prove the compactness: given χn ∈ T one ﬁnds corresponding un ∈ A0 . Using the regularity of A and the equation, one obtains a subsequence converging to u in (say) L2loc (−∞, 0; H 1 ). This is enough to conclude that u ∈ A0 and χn = P un → P u in X . The following lemma is a key step of our construction. Lemma 2.1. There exists a mapping E : T → H 1 such that E{P u0 } = u(0) for ∀u ∈ A0 . Moreover, E is Lipschitz continuous from X to H 1 provided that γ > 0 is small enough. Proof. Set w = u − v where u, v ∈ A0 . Using Lemma 1.2 and then (11) repeatedly one has √ c−1 2P w(−t∗ )2 + θw(−2t∗ )2 1 ∇w(0)2 ≤ w(−t∗ )2 ≤ √ √ ≤ 2P w(−t∗ )2 + 2θP w(−2t∗ )2 + θ2 w(−3t∗ )2 ... ≤

n √ 2 θk−1 P w(−kt∗ )2 + θn w(−(n + 1)t∗ )2 . k=1

On Reducing the 2d Navier-Stokes Equations

409

Letting n → ∞ and using the boundedness of w(t) one ﬁnds ∞ ∞ √ ∇w(0)2 ≤ c1 2 θk−1 P w(−kt∗ )2 ≤ c2 P w0 X θk eγkt∗ . k=1

k=1

In the second inequality we estimated P w(t) ≤ P w X e the last sum converges and we are done. 0

−γt

. If ln θ + γt∗ < 0,

Now we can deal with the projection of the nonlinear term. Lemma 2.2. There exists a mapping B : X → RN such that if u ∈ A0 then B(P u0 ) = P R(u(0)). Moreover, B is Lipschitz, bounded and has bounded support. 8 : T → RN by B(χ) 8 8 is Proof. We deﬁne B = P R(E(χ)). By Lemmas 2.1, 1.1, B 0 0 8 Lipschitz and B(P u ) = P R(u(0)) if u ∈ A . Applying Lemma 1.4 to each of the 8i , i = 1, . . . , N gives the desired mapping B. components B Let now u be a solution to (5) with values in A. We can assume that u is deﬁned for all t ∈ R. Applying P to (5) gives (denoting p(t) = P u(t)) pt + Ap + B(pt ) = g .

(12)

Here A = ν diag (λ1 , . . . , λN ), g = P f and B comes from Lemma 2.2.

3. Properties of the reduced system We ﬁrst establish the existence and uniqueness of solutions. Theorem 3.1. For any χ ∈ X and T > 0 there exist unique p : (−∞, T ] → RN a solution to (12) with p0 = χ. Proof. Assume p1 , p2 are two solutions to (12). Multiplying the equation for z = p1 − p2 by z gives 1 d 2 |z| + νλ1 |z|2 ≤ |B(pt1 ) − B(pt2 )| |z| = I . 2 dt By Lemma 2.2 c2 2 |I| ≤ c1 z t X |z| ≤ νλ1 |z|2 + z t X . 2 Hence d 2 |z| ≤ c2 sup |z(s)|2 e2γ(s−t) . dt s≤t Integrating over s ∈ (0, τ ) (where τ ∈ (0, T ) is ﬁxed) gives |z(τ )|2 ≤ |z(0)|2 + c2 T sup |z(s)|e2γs + c2 T s≤0

sup |z(s)|2 . s∈[0,T ] 2

Coming to supτ ∈[0,T ] on the left gives (as |z(0)|2 ≤ z 0 X ) 2

2

2

zC([0,T ]) ≤ (c2 T + 1)z 0 X + c2 T zC([0,T ]) .

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Thus if c2 T < 1 one has a local existence and uniqueness by a ﬁxed point argument. As c2 is independent of the data, the results extend globally. We omit further details. Next we show that the reduced system remains dissipative. Theorem 3.2. The system (12) has a global attractor Ar ⊂ X and Ar is bounded in C((−∞, 0]; RN ). Proof. Multiply (12) by p to obtain 1 d 2 |p| + νλ1 |p|2 + B(pt ) p ≤ |f | |p| . 2 dt Let R0 > 0 be such that B(χ) = 0 if χX ≥ R0 . We can assume |f |−νλ1 R0 ≤ −1. Since pt X ≥ |p(t)|, on has % & 1 d 2 |p| ≤ |p| |f | − νλ1 |p| ≤ −|p| 2 dt whenever |p(t)| ≥ R0 . One concludes that |p(t)| ≤ R0 for t ≥ t0 . Set Br = {χ ∈ X : |χ(t)| ≤ R0 for ∀t ≤ 0}. The auxiliary function q = q(t) is deﬁned as p(t) if t ≥ t0 and p(t0 ) if t < t0 . Clearly q t ∈ Br , hence distX (pt , Br ) ≤ q t − pt X = sup |p(s) − q(s)|eγ(s−t) s≤t0

≤ R0 e

γ(t0 −t)

+ pt0 X eγ(t0 −t) → 0

as t → ∞ .

Thus the set Br is uniformly attracting. Hence it is enough to ﬁnd an attractor to solutions starting from Br . Due to [10, Remark 1.4] it suﬃces to verify the asymptotic compactness: if χn = p0n ∈ Br and tn → ∞, the sequence ptnn has a converging subsequence in X (cf. .) However, ptnn are uniformly bounded on (−∞, 0] and its derivatives are uniformly bounded on [−tn , 0] from the equation. Thus there is a subsequence converging locally uniformly in (−∞, 0] and hence in X . Now we can prove our main theorem that relates the large time behavior of (5) and (12). Recall that A, Ar are global attractors to (5), (12) respectively, A0 are backward trajectories on A and T = P A0 . Here P is a projection to ﬁrst N eigenfunctions of the Stokes operator and N is estimated in (10). The mapping E was introduced in Lemma 1.2. Theorem 3.3. (i) Let u be a solution to (5) with an initial condition u0 ∈ A. Then p(t) = P u(t) is a solution to (12) with an initial condition χ = P u0 ∈ T . (ii) Conversely, let p be a solution to (12) with an initial condition χ ∈ T . Then u(t) = E(pt ) is a solution to (5) with an initial condition u0 = E(χ) ∈ A. (iii) The set T ⊂ Ar is compact and positively invariant with respect to (12).

On Reducing the 2d Navier-Stokes Equations

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Proof. Ad (i). Assume u is deﬁned for all t ∈ R. (One can prove that the backward continuation is unique, but we do not need that.) Then apply P to ﬁnd that p = P u solves (12), see after Lemma 2.2. Ad (ii). Take u0 ∈ A0 such that P u0 = χ and extend u to all t ≥ 0. By (i), q = P u solves (12) with an initial condition χ. However, due to the uniqueness (Theorem 3.1), p = q. In particular, by Lemma 2.1, u(t) = E(pt ) for all t ≥ 0. Ad (iii). For the compactness of T see before Lemma 2.1. The invariance follows easily from (i), (ii) and the invariance of A with respect to (5).

References [1] Babin A.V., Vishik, M.I., “Attractors of evolution equations,” Studies in Mathematics and its Applications 25 North-Holland, Amsterdam, 1992. [2] Constantin P., Foias C., “Navier-Stokes equations,” University of Chicago Press, Chicago, 1988. [3] Debussche A., Temam R., Some new generalizations of inertial manifolds, Discrete Contin. Dyn. Syst. (4) 2 (1996), 543–558. [4] Eden A., Foias C., Nicolaenko B., Temam R., “Exponential attractors for dissipative evolution equations,” Masson, Paris, 1994. [5] Engelking R., “General Topology,” Heldermann Verlag, Berlin, 1989. [6] Foias C., Sell G.R., Temam R., Inertial manifolds for nonlinear evolutionary equations, J. Diﬀer. Equations (2) 73 (1988), 309–353. [7] Hale J.K., Raugel G., Regularity, determining modes and Galerkin methods, J. Math. Pures Appl. (9) 82 (2003), 1075–1136. [8] Jones D.A., Titi E.S., Upper bounds on the number of determining modes, nodes, and volume elements for the Navier-Stokes equations, Indiana Univ. Math. J. (3) 42 (1993), 875–887. [9] Sell G.R., You Y., “Dynamics of evolutionary equations,” Applied Mathematical Sciences 143, Springer, New York, 2002. [10] Temam R., “Inﬁnite-dimensional dynamical systems in mechanics and physics,” Springer Verlag, New York, 1997. Dalibor Praˇza ´k Department of Mathematical Analysis Charles University Prague Sokolovska 83 186 75 Prague 8, Czech Republic e-mail: [email protected]

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 413–419 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Quasilinear Parabolic Equations in Lp Joachim Rehberg Abstract. The paper contains a local existence and uniqueness result for quasilinear parabolic equations on a three-dimensional domain including mixed boundary conditions and discontinuous coeﬃcients. Mathematics Subject Classiﬁcation (2000). 35B45, 35D05/10, 35J05/10/20/ 60/65, 35Q40, 47A55/60/75, 47H05/10/15. Keywords. Partial diﬀerential equations, nonsmooth domains, mixed boundary conditions, discontinuous coeﬃcients.

1. Introduction In modelling of real world problems one often is confronted with partial diﬀerential equations which act on nonsmooth domains, possess discontinuous coeﬃcients or/and are complemented by mixed boundary conditions [1], [25] [4] [17], [2], [7], [12], [9], [5]. In particular, in simulation of the current ﬂow in real semiconductor devices this is unavoidable, or the model is meaningless [25]. In this sense, the focus of this paper is on quasilinear equations of the (formal) type ∂u − div F (t, u)µ grad u = G(t, u, grad u), u(T0 ) = u0 (1) ∂t including Neumannn, Dirichlet or mixed type boundary conditions. The aim is to regard these equations in a suitably chosen Lp space; only this justiﬁes the application of Gauss’ theorem which allows to re-establish the ﬁrst principle formulation (cf. [26]) ∂ u dx − ν · j dσ = G(t, u, grad u) dx (2) ∂t Ξ ∂Ξ Ξ (j being the species current belonging to the particle species represented by u and Ξ any suitable subdomain of Ω). In fact, only the knowledge that div j is a (locally integrable) function allows to deﬁne the normal component of the current across boundaries of subdomains. Let us mention that this also has important consequences for the numerical treatment of the equations, see [8] and [21]. It

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turns out that – among other things – an isomorphism property for the associated elliptic operators − div µ grad +1 : HΓ1,q (Ω) −→ HΓ−1,q (Ω)

(3)

(Γ being the Neumann boundary part) is of great use if q is greater than the dimension of the domain Ω (see [18] or [23]). The problem is that for arbitrary (elliptic, L∞ ) coeﬃcients the occurring q s do exceed 2 (cf. [15]), but in general only arbitrarily little (cf.[24]). This suﬃces for the treatment of two-dimensional problems (cf. [18] or [19]), while in the meanwhile the necessity grows to regard not only two-dimensional problems (mostly as cuts of the originally three-dimensional ones) but the three-dimensional models themselves. In particular, this is true in device modelling (cf. [10], [22]) and in many of the above-mentioned applications. This makes it desirable to ﬁnd a class of (three-dimensional) domains Ω, coeﬃcient functions µ and Neumann parts Γ of the boundary, such that the operator (3) provides a topological isomorphism for a q > 3. This we will do ﬁrst within this paper. Afterwards we establish some functional analytic tools with which we ﬁnally solve (1).

2. Notations/General assumptions 1. Assumption. We assume Ω ⊂ R3 to be a bounded three-dimensional Lipschitz domain. Γ is an open boundary part satisfying the following condition: For any x ∈ ∂Ω there is an open neighborhood U of x and a bi-Lipschitz mapping Lx from U onto an open set V such that Lx (U ∩ (Ω ∪ Γ)) is either • the open unit cube • the open unit cube combined with its ground plate • the open unit cube combined with half of its ground plate. For s ∈ [0, 1] and q ∈]1, ∞[ we denote by H s,q the space of Bessel potentials with diﬀerentiability index s and integrability index q on Ω. By HΓs,q we denote the closure of ψ|Ω : ψ ∈ C0∞ (R3 ), suppψ ∩ (∂Ω \ Γ) = ∅

in H s,q . If s ∈ [−1, 0[ then HΓs,q shall denote the dual of HΓ−s,q , provided 1q + q1 = 1. 2. Assumption. We will always assume that µ is an L∞ function on Ω taking its values in the set of real, symmetric 3 × 3 matrices. Additionally, µ shall satisfy the usual ellipticity condition µ := ess inf

(µ(x)y) · y > 0.

inf

x∈Ω yR3 =1

The operator

− div µ grad : HΓ1,2 −→ HΓ−1,2

is deﬁned by

(4)

(5)

− div µ grad ψ1 , ψ2 =

(µ grad ψ1 ) · grad ψ2 dx. Ω

(6)

Quasilinear Parabolic Equations in Lp

415

3. The elliptic regularity result 3. Proposition. If the domain Ω, the Neumann boundary part Γ and the coeﬃcient function µ satisfy one of the following conditions I–V then there is a number qˆ = qˆ(Ω, Γ, µ) > 3 such that (3) provides a topological isomorphism for every q ∈ [2, qˆ]. Condition I Ω is Lipschitzian, Γ = ∅, µ ≡ 1 cf. [16]. Condition II Ω is Lipschitzian, Γ = ∂Ω, µ ≡ 1, cf. [27]. Condition III Ω is a polyhedral Lipschitz domain, Γ = ∅, µ is constant on (ﬁnitely many) polyhedral subdomains and the resulting analytic operator pencils satisfy a speciﬁc condition on the edges, cf. [23]. ¯ ∩ (∂Ω \ Γ) is a ﬁnite union of line Condition IV Ω is a convex polyhedron, Γ segments, µ ≡ 1, cf. [3]. ¯ 1 ), Ω1 is C 1 , µ is uniformly Condition V Ω is Lipschitzian, Ω = Ω1 ∪ ∂Ω1 ∪ (Ω \ Ω ¯ continuous on Ω1 and on Ω \ Ω1 , cf. [6]. 4. Proposition. (Gr¨ oger’s localization principle, cf. [15] and [6]) The assertion of the preceding proposition remains true if there is a (ﬁnite) localization of Ω and Γ such that the resulting local objects satisfy one of the Conditions I–V .

4. Functional analytic instruments Having the goal in mind to treat (1) in an Lp space, we give the following 5. Deﬁnition. If p ∈ [ 65 , ∞[ we denote by −∇ · µ∇ the maximal restriction of the operator − div µ grad to the space Lp . The corresponding domain is denoted by D. 6. Proposition. Let U be a positive, essentially bounded function on Ω with an ˆ and a strictly positive lower bound U . Then the operator −U ∇ · µ∇ upper bound U is densely deﬁned and satisﬁes the resolvent estimate for z with ,z ≥ 0: (−U ∇ · µ∇ + 1 + z)−1 B(Lp ) ≤ Λ

µ

L∞

µ

,p

U ˆ U

·

1 . 1 + |z|

(7)

Hereby, the function Λ : [1, ∞[×]1, ∞[ −→ ]0, ∞[ is continuous. 7. Proposition. Assume θ ∈]0, 1[, q ∈]1, ∞[, s0 , s1 ∈ [−1, 1] \ {1/q, −1/q } and s = (1 − θ)s0 + θs1 . Then [HΓs0 ,q , HΓs1 ,q ]θ = HΓs,q .

(8)

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5. Result At ﬁrst we will now ﬁx the Banach space, namely we put p := 2q . 8. Remark. Generically, q cannot be expected to exceed 4 and, hence, p ≤ 2. 3 , 1] the space [Lp , D]α continuously 9. Theorem. Assume 3 < q ≤ 4. For α ∈] 12 + 2q

embeds into HΓ1,q . 10. Lemma. Assume s ∈ [0, 1[ and τ ∈]s, 1] and set = τ − s. Then the operator (− div µ grad +1)/2 maps HΓ−s,2 isomorphically onto HΓ−τ,2 . Proof. The operator − div µ grad +1 is positive and generates analytic semigroups on both, HΓ−1,2 and L2 . Thus, powers of − div µ grad +1 and its restriction to the range space L2 are well deﬁned and the usual rules for calculus hold. In this sense, we consider the operators def

B = (− div µ grad +1)1/2 : HΓ1,2 → L2 , C = (− div µ grad +1)1/2 : L2 → HΓ−1,2 . def

Clearly, we have (B ∗ ) = (B )∗ = C . By a well-known theorem B maps the domain of B τ isomorphically onto the domain of B s . On the other hand, B is positive and selfadjoint, thus dom(B γ ) = [L2 , dom(B)]γ = [L2 , HΓ1,2 ]γ , γ ∈]0, 1[ holds. Because these interpolation spaces are identical with HΓγ,2 , B provides a topological isomorphism between HΓτ,2 and HΓs,2 . Hence, by duality, C = (−divµgrad)/2 maps HΓ−s,2 isomorphically onto HΓ−τ,2 . We now give the proof of Theorem 9: Assume s = 3( p1 − 12 ) and τ = 3q − 12 and set = τ − s. The deﬁnition of s and τ yields the (continuous) embeddings Lp → HΓ−s,2

and HΓ−τ,2 → HΓ−1,q .

= 1 − ρ/2. Thus, one gets [Lp , D]α → dom (−∇ · µ∇ + 1)1−ρ/2 .

By deﬁnition, we have α >

1 2

+

3 2q

Hence, it suﬃces to prove dom (−∇ · µ∇ + 1)1−ρ/2 → HΓ1,q , or, what is the same, (−∇ · µ∇ + 1)ρ/2−1 ∈ B(Lp ; HΓ1,q ).

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Denoting by ι and κ the embedding constants between the corresponding spaces, we may estimate (−∇ · µ∇ + 1)/2−1 B(Lp ;H 1,q ) Γ

≤ (− div µ grad +1)−1 B(H −τ,2 ;H 1,q ) (− div µ grad +1)/2 B(Lp ;H −τ,2 ) Γ

Γ

Γ

≤ ικ(− div µ grad +1)−1 B(H −1,q ;H 1,q ) (− div µ grad +1)/2 B(H −s,2 ;H −τ,2 ) . Γ

Γ

Γ

Γ

11. Theorem. Assume that Ω, Γ, µ form a constellation which is described in Section 3. Suppose, additionally, that the function F in (1) depends H¨ older continuously from t and locally Lipschitzian from the second argument. Finally, let the dependence of G in (1) from t also be H¨ older continuous, from the second argument locally Lipschitzian and from ∇u not be stronger than quadratic. Then, if the (Dirichlet homogenized) initial value belongs to the space (Lp , D)0α,∞ with 3 , 1], then the equation (1) has locally in time a unique solution which α ∈] 12 + 2q belongs to C 1 (]T0 , T ]; Lp) ∩ C(]T0 , T ]; D) ∩ C([T0 , T ]; (Lp , D)0α,∞ ). Proof. The proof results by applying the local existence result from [20]. The suppositions of that theorem are assured by Proposition 6 and Theorem 9 on one hand and our suppositions made on the mappings F and G on the other. Acknowledgement The author thanks J. Elschner, J.A. Griepentrog, H.-Chr. Kaiser, and G. Schmidt from the Weierstrass Institute for Applied Analysis and Stochastics, Berlin, for scientiﬁc collaboration on the topic.

References [1] H. Amann, Nonhomogeneous Linear and Quasilinear Elliptic and Parabolic Boundary Value Problems, Schmeisser, Hans-J¨ urgen et al (Eds.), Function spaces, diﬀerential operators and nonlinear analysis. Survey articles and communications of the international conference held in Friedrichsroda, Germany, September 20–26, 1992. Stuttgart: B. G. Teubner Verlagsgesellschaft. [2] S.N. Antontsev, M. Chipot, The thermistor problem: Existence, smoothness, uniqueness, blowup, SIAM J. Math. Anal. 25 (1994), 1128–1156. [3] M. Dauge, Neumann and mixed problems on curvilinear polyhedra, Integral Equations Oper. Theory 15, (1992), 227–261 (1992). [4] P. Degond, S. Genieys, A. J¨ ungel, A steady-state system in non-equilibrium thermodynamics including thermal and electrical eﬀects, Math. Meth. Appl. Sci 21 XV (1998) 1399–1413. [5] F. Duderstadt, D. H¨ omberg, A.M. Kludnev, A mathematical model for impulsive resistance welding, Math. Meth. Appl. Sci. 26 717–737.

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[6] J. Elschner, H.-Chr. Kaiser, J. Rehberg, G. Schmidt, Optimal regularity for elliptic operators occurring in real world problems, in preparation. [7] P. Fabrie, T. Gallouet, Modelling wells in porous media ﬂow, Math. Mod. Meth. Appl. Sci. 10 V (2000) 673–70. [8] J. Fuhrmann, H. Langmach, Stability and existence of solutions of time-implicit ﬁnite volume schemes for viscous nonlinear conservation laws, Appl. Num. Math. 37 (2001) 201–230. [9] H. Gajewski, K. Gr¨ oger, Initial boundary value problems modelling heterogeneous semiconductor devices, in: Surveys on Analysis, Geometry and Mathematical Physics, B. W. Schulze, H. Triebel (Eds.) Teubner-Texte zur Mathematik, Leipzig, 1990. [10] H. Gajewski, H.-Chr. Kaiser, H. Langmach, R. N¨ urnberg, R.H. Richter, Mathematical Modelling and Numerical Simulation of Semiconductor Detectors, in: MATHEMATICS: Key Technology for the Future, W. J¨ ager, H.-J. Krebs (Eds.) Springer (2002) 355–364. [11] H. Gajewski, K. Gr¨ oger, Reaction-diﬀusion processes of electrically charged species, Math. Nachr. 177 (1996), 109–130. [12] A. Glitzky, R. H¨ unlich, Global estimates and asymptotics for electro-reactiondiﬀusion equations, Appl. Anal. 66 (1997) 205–226. [13] J.A. Griepentrog, H.-Chr. Kaiser, J. Rehberg, Heat kernel and resolvent properties for second-order elliptic diﬀerential operators with general boundary conditions on Lp , Adv. Math. Sci. Appl. 11 (1), (2001) 87–112. [14] J.A. Griepentrog, K. Gr¨ oger, H.-Chr. Kaiser, J. Rehberg, Interpolation for function spaces related to mixed boundary value problems, Math. Nachr. 241 (2002) 110–120. [15] K. Gr¨ oger, A W 1,p -estimate for solutions to mixed boundary value problems for second-order elliptic diﬀerential equations, Math. Ann. 283 (1989), 679–687. [16] D. Jerison, C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal. 130, (1995), 161–219. [17] A. J¨ ungel, Regularity and uniqueness of solutions to a parabolic system in nonequilibrium thermodynamics, Nonlinear analysis: theor. appl. 41 V-VI, (2000) 669–688. [18] H.-Chr. Kaiser, H. Neidhardt, J. Rehberg, Classical solutions of quasilinear parabolic systems on two-dimensional domains, to appear in NoDEA. [19] H.-Chr. Kaiser, H. Neidhardt, J. Rehberg, Classical solutions of quasilinear parabolic systems on two-dimensional domains, WIAS-Preprint No. 765 (2002). [20] A. Lunardi, Abstract quasilinear parabolic equations, Math. Ann. 267 (1984) 395–415. [21] R. Luce, S. Perez, A ﬁnite volume scheme for an elliptic equation with heterogeneous coeﬃcients. Application to a homogenization problem Appl. Num. Math. 38 (2001) 427–444. [22] G. Lutz, R.H. Richter, L. Str¨ uder, DEPMOS-Arrays for X-ray imaging, XRAY Optics, Instruments and Missions III, J. Tr¨ umper, B. Aschenbach (Eds.), Proceedings of SPIE, vol. 4012, (2000), 249–256. [23] W. Maz’ya, J. Elschner, J. Rehberg, G. Schmidt, Solutions for quasilinear evolution systems in Lp , Arch. Rat. Mech. Anal. 171 (2004) 219–262.

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[24] N.G. Meyers, An Lp -estimate for the gradient of solutions of second-elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa 17, (1963), 189–206. [25] S. Selberherr, Analysis and simulation of semiconductor devices, Springer-Verlag, Wien-New York, 1984. [26] A. Sommerfeld, Thermodynamics and statistical mechanics, Lectures on theoretical physics, vol. V, Academic Press, New York, 1956. [27] D. Zanger, The inhomogeneous Neumann problem in Lipschitz domains, Commun. Partial Diﬀer. Equations 25 (2000), 1771–1808. Joachim Rehberg Weierstrass Institute for Applied Analysis and Stochastics Mohrenstr. 39 D-10117 Berlin, Germany e-mail: [email protected]

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 421–432 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Parabolic Equations in Locally Uniform Spaces An´ıbal Rodr´ıguez-Bernal “Nonlinear Elliptic and Parabolic Problems: A Special Tribute to the Work of Herbert Amann”, Z¨ urich, June 2004

1. The problem In this talk we present some recent results on linear parabolic equations of the form ut − ∆u = V (x)u in RN , t > 0 (1.1) u(x, 0) = u0 (x) where both the potential V (x) and the initial data belong to the so-called locally uniform spaces, [2]. These spaces, deﬁned below, allow for local singularities and impose no growth restrictions at inﬁnity. Moreover they enjoy suitable nesting and embedding properties similar as the Lebesgue spaces in a bounded domain. Also, we consider more general parabolic equations of the form ⎧ N N ⎪ ⎪ ⎨u − akl (x)∂k ∂l u + bj (x)∂j u + c(x)u = 0, x ∈ RN t (1.2) j=1 k,l=1 ⎪ ⎪ ⎩u(x, 0) = u (x), x ∈ RN 0

with complex-valued coeﬃcients and initial data in uniform spaces. In this context our goal is to extend the results in [3] to uniform spaces. Finally some results on nonlinear equations will be also presented, see [4].

2. Uniform spaces Recall that the uniform spaces, [9, 1, 2] consists of functions φ ∈ Lqloc (RN ), such that |φ(y)|q dy < ∞ (2.1) sup x∈RN

B(x,1)

Partially supported by BFM2003-03810, Spain.

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with norm φLqU (RN ) = sup φLq (B(x,1)) . x∈RN

Moreover we also consider the space L˙ qU (RN ) which is the subspace of LqU (RN ) such that τy φ − φLqU (RN ) → 0 as |y| → 0 where {τy , y ∈ RN } are the translations. Analogously one can deﬁne the uniform Sobolev spaces WUk,q (RN ) consisting k,q of functions φ ∈ Wloc (RN ) such that φW k,q (RN ) = sup φW k,q (B(x,1)) < ∞ U

x∈RN

˙ k,q (RN ) is the subspace of W k,q (RN ) for k ∈ N and W U U τy φ − φW k,q (RN ) → 0 as |y| → 0 U

where {τy , y ∈ R } are the translations. Note that embeddings can be sketched in the following diagram N

L∞ (RN ) = BU C(RN ) =

N L∞ U (R ) ⊂ ∪ N L˙ ∞ (R )⊂ U

LqU (RN ) ⊂ ∪ L˙ qU (RN ) ⊂ ∪ Lq (RN ) ⊂

··· ···

··· ···

L1U (RN ) ∪ L˙ 1U (RN ) ∪

⊃ L1 (RN )

3. The heat equation We ﬁrst consider the heat equation in LqU (RN ) ut − ∆u = 0 in RN , t > 0 u(x, 0) = u0 (x) ∈ LqU (RN )

(3.1)

whose solution is given by the convolution with the heat kernel |x−y|2 e− 4t u0 (y) dy. u(x, t) = T (t)u0 = (4πt)−N/2 Then we can show the following LqU −

RN p LU estimates,

[2]

Theorem 3.1. i) There exists M0 = M0 (N ) such that for 1 ≤ q ≤ p ≤ ∞ N 1 1 T (t)u0LpU (RN ) ≤ M0 t− 2 ( q − p ) + 1 u0 LqU (RN ) , 1

Dβ T (t)u0 LpU (RN ) ≤ M0 t− 2 (k− p + q ) u0 LqU (RN ) , for any 1 ≤ |β| = k.

N

N

(3.2)

Parabolic Equations in Locally Uniform Spaces far Qi

423

11111 00000 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 000000 111111 00 11 000000 111111 11111 00000 00 11 000000 111111 00 11 000000 111111 000000 111111 000000 111111

near Qi

Qi

Figure 1. The ith cube, the near and far neighbors ii) For each bounded set B ⊂ RN and u0 ∈ LqU (RN ) T (t)u0 − u0 Lq (B) → 0, L˙ qU (RN ),

If u0 ∈ 1 ≤ p ≤ ∞ and

as t → 0.

1 ≤ q ≤ ∞, then T (t)u0 ∈ L˙ pU (RN ) for t > 0 and any

T (t)u0 − u0 LqU (RN ) → 0,

as t → 0.

Proof. We sketch the proof of i): Let i ∈ ZN , Qi the open cube in RN centered at i and sides parallel to the axes. Hence Qi ∩ Qj = ∅, i = j, RN = ∪i∈ZN Qi . Given i ∈ ZN , we denote j ∈ N (i) if Qi ∩ Qj = ∅ and then Qnear = ∪j∈N (i) Qi , i

N near Qfar i = R \ Qi

For ﬁxed u0 ∈ LqU (RN ) and i, decompose near far T (t)u0 = T (t)u0 + T (t)u0 i

with

near T (t)u0

i

= T (t)(u0 XQnear ), i

i

far T (t)u0 = T (t)(u0 XQfar ) i i

Now the result follows from: near N 1 1 T (t)u0 Lp (Qi ) ≤ (4πt)− 2 ( q − p ) u0 Lq (Qnear ) i

t > 0,

(3.3)

i

and

far L∞ (Qi ) ≤ c(t)u0 L1U (Qfar , T (t)u0 i ) i −N/2 −α/t

with 0 ≤ c(t) ≤ Ct

e

t≥0

(3.4)

as t → 0 and C and α > 0 depending only on N .

424

A. Rodr´ıguez-Bernal

In fact, using the inclusions L∞ (Qi ) → Lp (Qi ) and restrictions LqU (RN ) → Lq (Qnear ), LqU (RN ) → L1U (Qfar i i ) from (3.4),(3.3) we get 1

1

T (t)u0 Lp (Qi ) ≤ C(t− 2 ( q − p ) + c(t))u0 LqU (RN ) . N

Taking the sup in i we get the result. Now we prove (3.3) using standard estimates for the heat equation in Lebesgue spaces near N 1 1 T (t)u0 Lp (Qi ) ≤ T (t)(u0 XQnear )Lp (RN ) ≤ (4πt)− 2 ( q − p ) u0 Lq (Qnear ). i i i

On the other hand, to prove (3.4) we write u0 XQfar = i

uj0 = u0 XQj . Then for each j

T (t)uj0 (x) = (4πt)−N/2

e−

|x−y|2 4t

+ j∈ZN \N (i)

uj0 with

uj0 (y)dy

Qj

and denoting dij := inf{dist(x, y), x ∈ Qi , y ∈ Qj } we get far L∞ (Qi ) ≤ T (t)uj0 L∞ (Qi ) T (t)u0 i

j∈ZN \N (i)

≤ (4πt)−N/2 u0 L1U (Qfar i )

e

−d2 ij 4t

.

j∈ZN \N (i)

Since #{j ∈ Z, dij = k} ≤ Ck N −1 j∈ZN \N (i)

e

−d2 ij 4t

≤C

∞

k N −1 e

−k2 4t

= tN/2 c(t)

k=1

with c(t) as in (3.4).

Moreover we can also prove that the heat equation deﬁnes an analytic semigroup which is not continuous at t = 0, as follows Theorem 3.2. The heat equation deﬁnes an order preserving analytic semigroup in LqU (RN ), for 1 ≤ q ≤ ∞, which is moreover continuous at t = 0 if u0 ∈ L˙ qU (RN ). However the domain of ∆ is not dense and satisﬁes ⎧ 1,p ⎪ for any p < NN−1 if q = 1 ⎨⊂ WU D(∆) = WU2,q for 1 < q < ∞ ⎪ ⎩ ⊂ WU2,p for any p < ∞ if q = ∞ Moreover, for u0 ∈ LqU (RN ), 1 ≤ q ≤ ∞ N N (0, ∞) t −→ T (t)u0 ∈ L˙ ∞ U (R ) = BU C(R )

is analytic.

Parabolic Equations in Locally Uniform Spaces

425

Proof. The ﬁrst part is proved by direct manipulation of (3.2). For the second part note that in particular we have proved the Calderon–Zigmud estimate: for f ∈ LqU (RN ), 1 < q < ∞, we show u0 (x) = G(x − y)f (y) dy ∈ WU2,q (RN ) RN

where G(x) Green’s function for −∆ + I in RN . For this, ﬁx the cube Qi and write finear = f XQnear , fifar = f XQfar . Then i i near near u0 i W 2,q (Qi ) ≤ u0 i W 2,q (RN ) ≤ Cf Lq (Qnear ) i with C independent of i. Using the cubes Qj with j ∈ N (i), since G ∈ C ∞ (RN ) and decays exponentially, for any k ∈ N far u0 i W k,∞ (Qi ) ≤ Cf L1U (Qfar i ) C independent of i. The analyticity part of the theorem is achieved by means of the following interesting Lemma, which allows to transfer analyticity to other spaces. Lemma 3.3. Assume {S(t)}t≥0 analytic semigroup in a Banach space X. Assume that for some Banach space Y and for t > 0, S(t) : X → Y is continuous. Then for each u0 ∈ X, (0, ∞) t → u(t) = S(t)u0 ∈ Y

is analytic.

In particular if Y ⊂ X, continuous, then {S(t)}t≥0 is an analytic semigroup in Y . Proof. For t0 > 0, the Taylor series in X: ∞ 1 m) u(t) = S(t)u0 = u (t0 )(t − t0 )m , m! m=0

with

um) (t0 ) = S m) (t0 )u0 .

For small enough > 0, um) (t0 ) = Am S(t0 )u0 = S()Am S(t0 − )u0 , where −A is the inﬁnitesimal generator in X. Hence, the series converges in Y since um) (t0 )Y ≤ C()Am S(t0 − )u0 X = C()um) (t0 − )X If Y ⊂ X, continuous, and u0 ∈ Y , S m) (t0 )u0 Y ≤ C()um) (t0 − )X ≤ ≤ C()S m) (t0 − )L(X) u0 X ≤ C()S m) (t0 − )L(X) u0 Y . Since the Taylor series converges in L(X) it also converges in L(Y ).

As the proofs above depend crucially on the Gaussian structure of the heat kernel, using [6], we can prove

426

A. Rodr´ıguez-Bernal

Theorem 3.4. Consider ut + L(u) = 0 in RN for L(u) = −

N

∂i ai,j (x)∂j u + ai (x)u + bi (x)∂i u + c0 (x)u

i=1

with real coeﬃcients ai,j , ai , bi , c0 ∈ L∞ (RN ) and the ellipticity condition N

ai,j (x)ξi ξj ≥ α0 |ξ|2 ,

ξ ∈ RN ,

α0 > 0.

i,j=1

Then the fundamental solution of the parabolic problem ut + Lu = 0 in RN satisﬁes a Gaussian bound 0 ≤ k(x, y, t, s) ≤ C(t − s)−N/2 eω(t−s) e−c

|x−y|2 (t−s)

for t > s and x, y ∈ RN where C, c, ω depend on the L∞ norm of the coeﬃcients. Hence, the results above apply to the semigroup u(t, x) = TL (t)u0 = k(x, y, t, 0)u0 (y) dy RN

4. Schr¨ odinger semigroups We consider now Schr¨ odinger semigroups ut − ∆u = V (x)u, x ∈ RN u(0) = u0 ∈ LqU (RN )

t > 0,

(4.1)

where V is in a locally uniform space. By treating this equation as a perturbation of the heat equation we get, [2] Proposition 4.1. Assume V ∈ LσU (RN ), σ > N/2, then i) For 1 ≤ q ≤ p ≤ ∞ 1

1

e(∆+V )t u0 LpU (RN ) ≤ M eat t− 2 ( q − p ) u0 LqU (RN ) , N

a and M depend only on N , σ and V LσU (RN ) . ii) For any bounded B ⊂ RN , u0 ∈ LqU (RN ), 1 ≤ q < ∞, e(∆+V )t u0 − u0 Lq (B) → 0, If moreover u0 ∈ e

L˙ qU (RN ), (∆+V )t

as t → 0.

1 ≤ q ≤ ∞, then

u0 − u0 LqU (RN ) → 0,

as t → 0.

Theorem 4.2. If V ∈ LσU (RN ), σ > N/2, then ∆+V generates an order preserving analytic semigroup in LqU (RN ), for 1 ≤ q ≤ ∞ which is continuous at t = 0 if u0 ∈ L˙ qU (RN ).

Parabolic Equations in Locally Uniform Spaces

427

Moreover for u0 ∈ LqU (RN ), 1 ≤ q ≤ ∞ N N (0, ∞) t −→ e(∆+V )t u0 ∈ L˙ ∞ U (R ) = BU C(R )

is analytic. As for the decay of solutions in locally uniform spaces, we have Lemma 4.3. The semigroup e(∆+V )t decays in LqU (RN ) iﬀ decays in Lp (RN ) for 1 ≤ p, q ≤ ∞

5. General second-order parabolic equations Now we consider general second-order parabolic equations ⎧ N N ⎪ ⎪ ⎨u − a (x)∂ ∂ u + bj (x)∂j u + c(x)u = 0, x ∈ RN t kl k l j=1 k,l=1 ⎪ ⎪ ⎩u(x, 0) = u (x), x ∈ RN 0

(5.1)

with complex-valued coeﬃcients and initial data u0 ∈ LqU (RN ) 1 < q < ∞. Note that systems can be also considered, but we stick to the scalar case for the sake of simplicity. Hereafter we will make the following assumptions for the coeﬃcients of the second-order diﬀerential operator A in (5.1) • For some 0 < M , 0 < θ0 < |A0 (x, ξ)| = |

N

π 2

and for all x, ξ ∈ RN with |ξ| = 1

ai,j (x)ξi ξj | ≥

i,j=1

1 , M

|arg(A0 (x, ξ))| ≤ θ0

• akl ∈ BU C(RN ), with modulus of continuity ω(s). p

• bj ∈ LUj (RN ) where pj > N , pj ≥ q • c ∈ LpU0 (RN ) where p0 > N/2, p0 ≥ q +N +N p p • k,l=1 akl BUC(RN ,ω) + j=1 bj L j (RN ) + cLU0 (RN ) ≤ M. U

Under these assumptions and considering solutions in Lq (RN ) the following results were proven, among other things, in [3] Theorem 5.1. Given 0 < M , 0 < θ0 < θ < π, A in Lq (RN ) with domain W 2,q (RN ), is sectorial and for some µ ∈ R A + µI ∈ Iso(W 2,q (RN ), Lq (RN )) A + µIL(W 2,q (RN ),Lq (RN )) + (A + µI)−1 L(Lq (RN ),W 2,q (RN )) ≤ c.

428

A. Rodr´ıguez-Bernal Therefore −A, generates a C 0 analytic semigroup {S(t)} in Lq (RN ) and +N +N ut − k,l=1 akl (x)∂k ∂l u + j=1 bj (x)∂j u + c(x)u = 0 u(0) = u0 ∈ Lq (RN )

has a unique solution u(t) = S(t)u0 for t ≥ 0. Concerning bounded imaginary powers it was shown in [3]: Proposition 5.2. With 0 < M , 0 < θ0 < θ < π as before if 1 1 ω 3 (t) dt < ∞, t 0 then, Λ = A + µI ∈ BIP Lq (RN ), N , θ , i.e.,

(5.2)

Λit L(Lq (RN )) ≤ N eθ|t| , t ∈ R. In particular, the fractional powers spaces satisfy Xqα = [Lq (RN ), W 2,q (RN )]α = Hq2α (RN ) which are Bessel potential spaces and [·, ·]α denotes complex interpolation. Note that a crucial remark is that in the results above all constants µ, c, N , depend only on 0 < M , θ0 , θ, ω. In order to extend these results to locally uniform spaces we ﬁrst obtain an alternative description of these spaces in terms of some weighted norms. For this consider a weight 0 < ρ ∈ C 2 (RN ) ∩ L1 (RN ) with |Dα ρ(x)| ≤ Cα ρ(x), and the weighted space

x ∈ RN ,

φ ∈ Lqloc (RN ),

Lqρ (RN ) =

RN

|α| ≤ 2

|φ(x)|q ρ(x)dx < ∞ ,

1 with norm φLpρ (RN ) = RN |φ(x)|p ρ(x)dx p . If ρ does not decay too fast, see [2], then LqU (RN )

=

φ∈

Lqloc (RN ),

sup φLqτy ρ (RN ) < ∞

y∈RN

where {τy , y ∈ RN } are the translations. Now we ﬁrst obtain solutions of (5.1) in weighted spaces, [2] Theorem 5.3. With 0 < M , 0 < θ0 < θ < π as before, A in Lqρ (RN ) with domain Wρ2,q (RN ) is sectorial A + µI ∈ Iso(Wρ2,q (RN ), Lqρ (RN )) A + µIL(Wρ2,q (RN ),Lqρ (RN )) + (A + µI)−1 L(Lqρ (RN ),Wρ2,q (RN )) ≤ c.

Parabolic Equations in Locally Uniform Spaces

429

Therefore −A, generates a C 0 analytic semigroup {S(t)} in Lqρ (RN ) and + +N ut − N k,l=1 akl (x)∂k ∂l u + j=1 bj (x)∂j u + c(x)u = 0 q N u(0) = u0 ∈ Lρ (R ) has a unique solution u(t) = S(t)u0 for t ≥ 0. Moreover, if (5.2) holds, then Λ = A + µI ∈ BIP Lqρ (RN ), N , θ and the fractional powers spaces satisfy Wρ2α,q = [Lqρ (RN ), Wρ2,q (RN )]α Note that again all constants, µ, c, N , depend only on 0 < M , θ0 , θ, ω and ρ, but can be taken uniform for all translations τy ρ. Now we are ready to obtain solutions in uniform spaces Theorem 5.4. With 0 < M , 0 < θ0 < θ < π and ρ as before, A in LqU (RN ) with (not dense) domain WU2,q (RN ) is sectorial and A + µI ∈ Iso(WU2,q (RN ), LqU (RN )) A + µIL(W 2,q (RN ),Lq (RN )) + (A + µI)−1 L(Lq (RN ),W 2,q (RN )) ≤ c. U

U

U

U

Therefore −A, generates a (not C ) analytic semigroup {S(t)} in LqU (RN ) 0

and

+ +N ut − N k,l=1 akl (x)∂k ∂l u + j=1 bj (x)∂j u + c(x)u = 0 u(0) = u0 ∈ LqU (RN )

has a unique solution u(t) = S(t)u0 for t ≥ 0. Combining some results above we get p

Corollary 5.5. If akl ∈ BU C(RN ), bj ∈ LUj (RN ) and c ∈ LpU0 (RN ) where pj > N and p0 > N/2, for any 1 < r ≤ ∞ +N +N ut − k,l=1 akl (x)∂k ∂l u + j=1 bj (x)∂j u + c(x)u = 0 u(0) = u0 ∈ LrU (RN ) has a unique solution u(t) = S(t)u0 for t ≥ 0 and (0, ∞) t → S(t)u0 ∈ BU C(RN ) is analytic. For any p, q, r the semigroup is order preserving in Lp (RN ) iﬀ it is order preserving in LqU (RN ) iﬀ it is order preserving in Lrρ (RN ).

430

A. Rodr´ıguez-Bernal

The next result shows that under additional assumptions on the coeﬃcients the semigroup above is continuous up to t = 0 in the dotted uniform spaces Theorem 5.6. Assume 0 < M , 0 < θ0 < θ < π, ρ as before, akl ∈ BU C(RN ), p bj ∈ L˙ Uj (RN )

c ∈ L˙ pU0 (RN )

where pj > N , pj ≥ q and p0 > N/2, p0 ≥ q. ˙ 2,q (RN ) is sectorial, generates a C 0 anThen A in L˙ qU (RN ) with domain W U alytic semigroup {S(t)} in L˙ qU (RN ) and +N +N ut − k,l=1 akl (x)∂k ∂l u + j=1 bj (x)∂j u + c(x)u = 0 u(0) = u0 ∈ L˙ qU (RN ) has a unique solution u(t) = S(t)u0 for t ≥ 0. Moreover, if (5.2) holds, then Λ = A + µI ∈ BIP L˙ q (RN ), N , θ U

and the fractional powers spaces satisfy ˙ 2,q (RN )]α ˙ 2α,q := [L˙ q (RN ), W W U U U

6. Nonlinear equations Now we present some results on nonlinear equations, see [4] ut = ∆u + f (x, u), x ∈ RN , t > 0, q N u(0, x) = u0 (x) ∈ LU (R )

(6.1)

where the nonlinear term satisﬁes f (x, s) = g(x) + m(x)s + f0 (x, s) ∂ f0 (x, 0) = 0. ∂s Concerning local existence, we have f0 (x, 0) = 0,

Theorem 6.1. Assume 1 < q < ∞ and g ∈ LqU (RN ),

m ∈ LpU (RN )

p > N/2.

Then (6.1) is well posed for u0 ∈ LqU (RN ) if ∂ 2q f0 (x, s) ≤ c(1 + |s|ρ−1 ), 1 ≤ ρ ≤ ρC = 1 + . ∂s N

Note that solutions are constructed through the variations of constants formula and that the result cover the case of the critical exponent, ρC , for the space LqU (RN ).

Parabolic Equations in Locally Uniform Spaces

431

The next result ensures global existence of solutions, see [1], for similar conditions in a standard Lebesgue spaces setting, Theorem 6.2. Assume f (x, s)s ≤ C(x)|s|2 + D(x)|s| with C, D ∈ LpU (RN ),

p > N/2,

D ≥ 0.

Then solutions of (6.1) are global for u0 ∈ LqU (RN ), and are bounded in L (RN ) on [ε, T ]. Moreover if u0 ∈ B ⊂ LqU (RN ), a bounded set, then ∞

u(t, B) ⊂ L∞ (RN )

is bounded

Proof. Using comparison and maximum principles we get |u(t, x)| ≤ U (t, x)

x ∈ RN

where Ut − ∆U = C(x)U + D(x), U (0) = |u0 |. With this, D ∈ LpU (RN ) and p > N/2 implies U is bounded in L∞ (RN ) on [ε, T ]. Hence, f0 (·, u)LqU (RN ) ≤ C0 (ε, T ), t ∈ [ε, T ] and from the variation of constants formula, the solution is global.

The next step is obtaining asymptotic bounds on the solutions Theorem 6.3. Assume the solutions of zt − ∆z = C(x)z decay exponentially. Let 0 ≤ φ ∈ WU2,p (RN ) ⊂ CUα (RN ) be the unique solution of −∆φ = C(x)φ + D(x)

in

RN .

Then lim supt→∞ |u(t, x, u0 )| ≤ φ(x), uniformly in x ∈ RN . If |u0 (x)| ≤ φ(x), then |u(t, x, u0 )| ≤ φ(x) for all t > 0. Proof. Now |u(t, x)| ≤ U (t, x) → φ(x), and we use the variation of constants formula.

as t → ∞

With this we can prove Theorem 6.4. With the assumptions above there exists bounded invariant set A ⊂ BU C(RN ) such that for every bounded set B ⊂ LqU (RN ), distCloc (RN ) u(t, B), A → 0, as t → ∞. Example: Assume m ∈ LpU (RN ), with p > N/2 and f (x, s) = m(x)s − s3 ,

x ∈ RN .

432

A. Rodr´ıguez-Bernal

Since ρ = 3 for local existence we take initial data in LqU (RN ) with q > N . On the other hand, global existence follows by taking C(x) = m(x) and D(x) = 0. Assume now that m(x) = m0 (x) − m1 (x),

m0 , m1 ∈ LpU (RN )

such that e(∆−m1 (x)I)t decays exponentially, e.g., m1 can be taken as a large positive constant. Hence using Young’s inequality we get f (x, s)s ≤ C(x)|s|2 + D(x)|s| with D(x) ≈ |m0 |3/2 (x) C = −m1 ∈ LpU (RN ), p > N/2, and the existence of the attractor in the local compact topology follows if m0 ∈ LrU (RN ) with r > 3N 4 . See [5] for related results in weighted spaces, [8] for the case of BU C(RN ) setting and [7] for a diﬀerent construction of locally compact attractors.

References [1] J.M. Arrieta, J.W. Cholewa, T. Dlotko, A. Rodr´ıguez-Bernal, “Asymptotic behavior and attractors for reaction diﬀusion equations in unbounded domains”. Nonlinear Analysis TMA. 56, 515–554 (2004). [2] J.M. Arrieta, J.W. Cholewa, T. Dlotko, A. Rodr´ıguez-Bernal, “Linear parabolic equations in locally uniform spaces”. Math. Mod. Meth. Appl. Sci. 14, 253–293 (2004). [3] H. Amann, M. Hieber, G. Simonett, Bounded H∞ -calculus for elliptic operators, Diﬀ. Int. Eqns 3, 613–653 (1994). [4] J.M. Arrieta, A.Rodr´ıguez–Bernal, in preparation [5] A.V. Babin, M.I. Vishik, Attractors of partial diﬀerential evolution equations in unbounded domain, Proc. Roy. Soc. Edinburgh 116A, 221–243 (1990). [6] D. Daners, Heat kernel estimates for operators with boundary conditions, Math. Nachr. 217, 13–41 (2000). [7] M.A. Efendiev, S.V. Zelik, The attractor for a nonlinear reaction-diﬀusion system in an unbounded domain, Comm. Pure Appl. Math. 54, 625–688 (2001). [8] S. Merino, On the existence of the compact global attractor for semilinear reaction diﬀusion systems on RN , J. Diﬀ. Eqns 132, 87–106 (1996). [9] B. Simon, Schr¨ odinger semigroups, Bull. Amer. Math. Soc. 7, 447–526 (1982). An´ıbal Rodr´ıguez-Bernal Departamento de Matem´ atica Aplicada Universidad Complutense de Madrid E-28040 Madrid, Spain

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 433–462 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Bifurcation of Traveling Waves Related to the B´enard Equations with an Exterior Force B. Scarpellini Abstract. We consider a time-independent B´enard equation (*) −Aw + (λ0 + ε)PM w + N (u, w) + (a∂x + b∂y )w + τ f = 0 on the inﬁnite layer R2 × [− 21 , 12 ]. Here f = f (z) is an exterior force depending only on z ∈ [− 12 , 12 ], w satisﬁes Dirichlet conditions in the z-direction and L1 , L2 -periodic conditions in the x, y-direction, while a, b satisfy a diophantine condition. λ0 is the critical Rayleigh parameter. It is shown that for generic f and small τ, ε, a, b, (*) has solutions w such that (a∂x + b∂y )w = 0. These solutions give rise to periodic traveling wave solutions of the time-dependent version of (*) (with a = b = 0). The proof is via bifurcation methods related to Hopf bifurcation. Mathematics Subject Classiﬁcation (2000). 35B32, 37K50, 35Q35. Keywords. Inhomogenic B´enard equation, traveling waves, Hopf bifurcation.

0. Introduction Below we study the inhomogeneous B´enard equations with a drift term: ν∆v + λkθ + a∂x v + b∂y v − (v∇)v + f − ∇p = 0

(0)

µ∆θ + λv3 + a∂x θ + b∂y θ − (v∇)θ + f4 = 0 div v = 0, k = (0, 0, 1)t , f = (f , f4 ), f = (f1 , f2 , f3 ). Here v = (v1 , v2 , v3 ) is the velocity ﬁeld, θ the temperature, p the pressure, √ a, b constants with ab = 0, ν the viscosity, µ the thermal conductivity and λ = R with R the Rayleigh number; the density has been set = 1. We consider (0) on the inﬁnite layer Ω = R2 × (− 21 , 12 ) with v, θ subject to Dirichlet boundary conditions at ∂Ω and to L1 , L2 -periodic boundary conditions in the unbounded space directions. For a = b = 0, Eq. (0) reduces to the slightly rescaled original B´enard equations (see [11], Sec. (0)). The drift term T = a∂x + b∂y in (0) is included since solutions v, θ of (0) give rise to traveling wave solutions v(x − at, y − bt, z),

θ(x − at, y − bt, z)

434

B. Scarpellini

of the original, time-dependent B´enard equations provided with the time-dependent force f (x − at, y − bt, z). The exterior force, while justiﬁed on mathematical grounds, admits also a physical interpretation. In fact, disregarding the drift term, we may look at (0) as an approximation to the magnetic B´enard problem ([1], [8], [10]) in which the induced magnetic ﬁeld h = (h1 , h2 , h3 ) is neglected in view of its smallness and in which f = ((H0 ∇)A,

(H0 ∇)B,

(H0 ∇)C, 0),

H0 = (A, B, C)

is given in terms of the impressed magnetic ﬁeld H0 ; for a justiﬁcation of this argument, see [8], introduction. In [10] the problem was to prove existence of solutions of (a) via LeraySchauder degree arguments. Our interest here is in the following situation. Let the force f = f (z) depend on z ∈ [− 21 , 12 ] only. In this case there is a unique solution v0 (z), θ0 (z), p0 (z) of Eq. (0), depending on z only. The question to be treated here is: (*) given f = f (z), do there exist solutions v, θ, p of (0) depending nontrivially on x, y and such that T v,T θ are ≡ 0, with T = a∂x + b∂y ? Since degree arguments do not contain suﬃcient information to answer this question, we are forced to proceed by bifurcation arguments. Now if ab = 0 it does not seem possible to apply symmetry arguments in order to achieve reduction to the case of a simple eigenvalue. This diﬃculty is avoided by showing (Sect. 3, 4) that one is in a situation which is analogous to that encountered in case of Hopf bifurcation. In sect. 3 we show that if a certain linear operator L0 , arising from the linearisation of (0) with respect to the “trivial” equilibrium solution v0 (z), θ0 (z), p0 (z), satisﬁes σ(L0 ) ∩ R = {0}, then only solutions v, θ, p can bifurcate which generate trivial traveling waves, i.e., such that ∂t v(x − at, y − bt, z) = −(T v)(x − at, y − bt, z) = 0 and likewise with θ, p; see Sect. 3 for details. If however σ(L0 ) ∩ R contains a unique pair ±iσ, σ = 0, σ ∈ R of simple eigenvalues, then bifurcation of solutions v, θ with T v = 0, T θ = 0 is indeed possible. It is shown in Sect. 4 that under this spectral condition there are a number of conditions such that if any one of these is satisﬁed, bifurcation of nontrivial traveling waves occurs. A diﬃculty with the results in Sect. 4 is that the veriﬁcation of these conditions requires a detailed knowledge of the eigenfunctions of a certain generalised Orr-Sommerfeld eigenvalue problem which is in general diﬃcult to obtain. There is however an instance of Eq. (0) considered which, although it does not satisfy the spectral assumption of Sect. 4, is nevertheless accessible to the techniques of Sect. 4. In fact, an instance of Eq. (0) is considered in which f = f (z) is small, λ close to the critical Rayleigh parameter λ0 and where the constants a, b satisfy some diophantine inequalities, for which the existence of bifurcating solutions v, θ, p such that T v = 0, T θ = 0 can eﬀectively be proved, and which thus give rise to genuine traveling wave solutions of the original, time-dependent B´enard equations with exterior force f = f (z). As to related literature we refer to [14] for an extended list of references; the papers cited in [14] consider one-dimensional traveling waves in contrast to ours which are

Bifurcation of Traveling Waves

435

in 3D. In [15], traveling waves of the 3D-Navier Stokes equations are constructed; however this construction is not based on the standard semigroup setting and does not interfere with ours.

1. Notation R, C denote real and complex numbers, with ζ the complex conjugate of ζ ∈ C. We α set ∂j = ∂xj , x1 = x, x2 = y, x3 = z and set Dα = ∂1α1 . . . ∂p p for any multiindex 3 α = (α1 , . . . , αp ). For Ω ⊆ R a domain with segment property, f ∈ L2 (Ω) is in H p (Ω) iﬀ there are fn ∈ C p (Ω), n ≥ 1 such that fn − f L2 → 0 as n → ∞, and such that fn , n ≥ 1 is Cauchy in H p (Ω); if fn ∈ C0p (Ω), n ≥ 1 then f ∈ H0p (Ω). The scalar product on H p (Ω) is given by (u, v)p = (Dα u, Dα v)0 , (u, v)0 = uv dxn . (1) Ω

|α|≤p

We have H 0 (Ω) = L2 (Ω) and set H p = H p (Ω) for Ω ﬁxed. For vector ﬁelds u, v ∈ H p (Ω)m we set + (2) u, vp = j (uj , vj )p , u = (u1 , . . . , um ), v = (v1 , . . . , vm ) + u2 = u2L2 = uj 2 , u2H p = u, up . Thus u = uL2 and uH p are interpreted according to the context. For u = (u1 , u2 , u3 ), v = (v1 , . . . , vm ) vector ﬁelds and θ a scalar we set + (3) (u∇)θ = uj ∂j θ, (u∇)v = ((u∇)v1 , . . . , (u∇)vm ). With ∆ the Laplacian, ∂ a derivative and f = (f1 , . . . , f4 ) we put ∆f = (∆f1 , . . . , ∆f4 ), ∂f = (∂f1 , . . . , ∂f4 ) and f = (f1 , f2 , f3 ). For u = (u1 , . . . , u4 ), v = (v1 , . . . , vm ) we thus stipulate (u∇)v = ( u∇)v

(4)

in accordance with (1.3).

2. A functional setting In order to handle the problem raised in the introduction we rephrase the system (0) in functional analytic terms along established lines. With L1 , L2 > 0 ﬁxed periods we set QL = (0, L1 ) × (0, L2 ), I = [− 21 , 12 ], Q = QL × I, Ω = R2 × I p and stipulate: (a) f ∈ Cper (Q) iﬀ f ∈ C p (Ω) 1 1 periodic in y, (b) f ∈ Cper,0 (Q) iﬀ f ∈ Cper (Q) 2 p then stipulate that f ∈ L (Q) is in Hper (Q) if p

(0)

and if f is L1 -periodic in x, L2 and f (x, y, ± 21 ) = 0, x, y ∈ R. We p there is a sequence fn ∈ Cper (Q), n ≥ 1 which is Cauchy in H (Q) and such that f − fn L2 (Q) → 0 as n → ∞; the 1 space Hper,0 (Q) is deﬁned in an analogous way. These notions readily extend to

436

B. Scarpellini

vector ﬁelds in an obvious way; we thereby use the same notation, interpreting it according to the context. We stipulate: 1 1 E ⊆ L2 (Q)3 is the L2 -closure of ﬁelds f ∈ Hper (Q)2 × Hper,0 (Q)

(1)

such that div f = 0, and P is the orthogonal projection from L (Q) to E. 2

3

The phase space E of Eq. (0) is now given by E = E × I, and P is the orthogonal projection from L2 (Q)4 onto E : if f = (f , f4 ) ∈ L2 (Q)4 then Pf = (P f , f4 ).

(2)

A Stokes operator related to the ﬁrst three equations of (0) is deﬁned via 2 1 f ∈ dom(AS ) iﬀ f ∈ (Hper (Q) ∩ Hper,0 (Q))3 and div f = 0,

(3)

and AS f = νP ∆f for f ∈ dom(AS ) The Stokes operator BS , associated with the system (0) is given by: 2 1 dom(BS ) = dom(AS ) × (Hper (Q) ∩ Hper,0 (Q)); for

(4)

v ∈ dom(AS ), (v, θ) ∈ dom(BS ), we set BS (v, θ) = (AS v, µ∆θ). One then has ([11],[12]): Proposition 2.1: (a) AS , BS are selfadjoint on E, E respectively, (b) AS < 0, BS < 0, (c) AS , BS have compact resolvents. For convenience we introduce the operator A and the matrix M via A = −BS ;

M = (ajk ),

j, k ≤ 4 with a34 = a43 = 1

(5)

and ajk = 0 otherwise. Notation: For w ∈ dom(A) we set wA = Aw and recall that wH 2 , wA deﬁne equivalent norms on dom(A). In order to handle the nonlinearity in (0) we recall (1.3), (1.4). For wj ∈ dom(A), j = 1, 2 we set N (w1 , w2 ) = −P(w1 ∇)w2 = −P(w 1 ∇)w2 .

(6)

N ( , ) admits a number of estimates in terms of fractional powers A . Here we 2 1 only need the fact that if wj ∈ Hper (Q)4 , j = 1, 2, then (w1 ∇)w2 ∈ Hper (Q)4 and γ

(w1 ∇)w2 H 1 ≤ cw1 H 2 w2 H 2 .

(7)

System (0) in sect. 0 can now be put into abstract form. The pressure p is thereby eliminated with the aid of the projection P. We assume f ∈ E, i.e., Pf = f . By these stipulations, Eq. (0) assumed the forms Aw = λPM w + PT w + N (w, w) + f,

T = a∂x + b∂y .

(8)

To solve (0) now amounts to seek w ∈ dom(A) such that (2.8) holds. A solution w ∈ dom(A) such that T w = 0 is called a nontrivial traveling wave; otherwise it is called trivial.

Bifurcation of Traveling Waves

437

3. Nonexistence of bifurcating traveling waves As noted in the introduction, to ﬁnd nontrivial traveling wave solutions of Eq. (0) leads to bifurcation problems whose solvability depends on certain spectral assumptions. Prior to formulate them we need some preparatory steps. Proposition 3.1: For f = (f1 , . . . , f4 ) ∈ E we have f3 = 0. Proof. Since E = E × I and f = (f , f4 ), f = (f1 , f2 , f3 ) we have f ∈ E by assumption. Now we recall: g ∈ E iﬀ g, ∇p = 0 for all p ∈ H 1 (Q);

(1)

see [5] or [12], proof of Lemma 7.2 and Remarks (9.62). For f = g, (3.1) entails

f (z)∂z p dz = 0, for p ∈ H 1 (I), I = (− 21 , 12 ) I 3

whence f3 = 0.

With f ∈ E and by Prop. 3.1, Eq. (0) admits a unique solution v = (v1 , . . . , v , v4 ), depending on z only: v4 ) = ( ν∂z2 v = fj ,

j = 1, 2,

vj ∈ H (I) ∩

H01 (I),

2

µ∂z2 v4 = f4 ,

v3 = 0,

λ∂z p + v4 = 0,

(2)

p ∈ H (I). 1

From (3.2) we read oﬀ: (a) τ v is the solution associated to τ f , (b) N (v, v) = 0, (c) v does not depend on λ. We stipulate: if ϕ ∈ H 1 (Q) then T ϕ = a∂x ϕ + b∂y ϕ,

(3)

if w = (w1 , . . . , wn ) ∈ H (Q) then T w = (Tw1 , . . . , Twn ). 1

n

Proposition 3.2: If w ∈ dom(A) then T w ∈ E Proof. By periodicity we have g∂s f dx3 = − f ∂s g dx3 , Q

1 f, g ∈ Hper (Q),

s ∈ {x, y}

(4)

Q

whence T f, g = −f, Tg ,

1 f, g ∈ Hper (Q)n .

∈ dom(AS ) ⊆ E. Since Let w = (w, w4 ) whence w and by (3.1) it suﬃces to show: T w, ∇p = 0,

2 Hper (Q)

(5) 1

is H -dense in

2 p ∈ Hper (Q).

1 Hper (Q)

(6)

(3.6) follows via (3.1), (3.5) from the equalities: T w, ∇p = −w, T ∇p = −w, ∇T p, p ∈ H 2 (Q).

By these remarks we have that τ v, with v = v(z) as in (3.2), is the unique x, y-independent solution of the following variant of equation (2.10): −Aw0 + λPM w0 + (1 + χ)T w0 + N (w0 , w0 ) = τ f, w0 ∈ dom(A).

(7)

438

B. Scarpellini

The extra parameters χ, τ are needed later. In order to ﬁnd solutions w0 giving rise to nontrivial traveling waves we set w0 = τ v + w in (3.7) and obtain the following equation for w: (−A + λPM − τ G + (1 + χ)T )w + N (w, w) = 0, w ∈ dom(A)

(8)

where G is the operator given by Gw = −(N (w, v) + N (v, w)),

2 (Q)4 . dom(G) = Hper

(9)

In order to handle (3.9) as a bifurcation problem we pick λ0 > 0, τ0 ≥ 0 arbitrarily for the moment, set λ = λ0 + ε, τ = τ0 + δ in (3.8) so as to obtain (−A + λ0 PM − τ0 G + T )w + εPM w − δGw + χT w + N (w, w) = 0.

(10)

Given ε, δ, χ small we look for solutions w ∈ dom(A) of (3.10) with wH 2 small such that τ v + w is a nontrivial traveling wave. We note that a solution w = 0 of (3.10) depends nontrivially on x, y since otherwise (τ0 + δ)v + w and (τ0 + δ)v would be two diﬀerent solutions of (3.8) for λ = λ0 + ε, τ = τ0 + δ, depending on z only, what is not possible. But in order to guarantee that τ v + w is a nontrivial traveling wave, the stronger condition T w = 0 must hold. The above bifurcation L0 via: problem depends on λ0 , τ0 which deﬁne operators B, = −A + λ0 PM − τ0 G, L0 = B + T, B = dom(L0 ) = dom(A). dom(B)

(11)

and L0 are holomorphic semigroup generators with compact Proposition 3.3: B resolvents. Proof. See, e.g., [12] pg. 69, 139 for comparable situations, see also [7], Thm. 2.1, pg. 80, [4], Thm. 2.4, pg. 4.99. As a consequence we have Lemma 3.1: Let 0 ∈ / σ(L0 ). There are ε0 , δ0 , χ0 , d0 > 0 as follows: if |ε| ≤ ε0 , |δ| ≤ δ0 , |χ| ≤ χ0 and if w ∈ dom(A) is a solution of (3.10) with wH 2 ≤ d0 then w = 0. maps E boundedly onto dom(A), Proof. By Prop. 3.3 and the assumption, L−1 0 H 2 , thus (3.10) is equivalent to −1 w + L−1 0 (εPM w − δGw + χT w) + L0 N (w, w) = 0.

(12)

The left-hand side of (3.12) is a real analytic mapping F (w, ε, δ, χ) from dom(A), H 2 × R3 into dom(A), H 2 , such that F (0, ε, δ, χ) = 0 for all ε, δ, χ. Since (∂w F (0, 0, 0, 0) = I the claim follows from the implicit function theorem. The situation is more involved if the assumption of the Lemma does not hold. Theorem 1: Assume: (a) 0 ∈ σ(L0 ),

Bifurcation of Traveling Waves

439

∩ iR then σ = 0. Then there are ε0 , χ0 , δ0 , d0 > 0 as follows: if (b) if σ ∈ σ(B) |ε| ≤ ε0 , |δ| ≤ δ0 , |χ| ≤ χ0 and if w ∈ dom(A) is a solution of (3.10) with wH 2 ≤ d0 then T w = 0. Remarks: (a), (b) do not exclude the possibility of a small solution w = 0 of (3.10); but w0 = τ v + w is then already a solution of (−A + λPM )w0 + N (w0 , w0 )τ f = 0,

λ = λ0 + ε, τ = τ0 + δ,

(13)

that is, (∂t w0 )(x − at, y − bt, z) = −(T w0 )(x − at, y − bt, z) = 0, (14) whence no nontrivial traveling wave is generated by w0 = τ v + w. For the proof of Thm. 1 we need some facts about Fourier series which underlie the Orr-Sommerfeld method. We recall the periods L1 , L2 and set 2πβ 2πα 8 , β8 = , eαβ = ei(8αx+βy) , α, β ∈ Z. (15) α 8= L1 L2 An element w ∈ L2 (Q) is then characterised by its Fourier series: 2 w= eαβ wαβ , wαβ ∈ L (I), |wαβ |2 dz < ∞. αβ

αβ

1 We then have that w ∈ Hper (Q) iﬀ wαβ ∈ H 1 (I), α, β ∈ Z and in addition (1 + α2 + β 2 ) |wα β|2 dz + |∂z wαβ |2 dz < ∞. I

αβ

αβ

(16)

I

(17)

I

Likewise we have 1 w ∈ Hper,0 (Q) iﬀ wαβ ∈ H01 (I),

α, β ∈ Z and (3.17) holds.

(18)

p Similar characterizations hold for elements w ∈ Hper (Q), p ≥ 2; we refer to [13] for a detailed treatment of such expansions, see also [11], sec. 2. Here it is convenient to rephrase these facts in terms of suitable subspaces of L2 (Q), L2 (Q)n . We deﬁne p 1 subspaces Lαβ , Hαβ , Hαβ0 of L2 (Q) via p Hαβ = {eαβ f /f ∈ H p (I)},

Lαβ = {eαβ f /f ∈ L2 (I)}, 1 Hαβ0

= {eαβ f /f ∈

(19)

H01 (I)}.

Fourier expansion arguments show that one has p p Hαβ = Hper (Q) ∩ Lαβ ,

1 1 Hαβ0 = Hper,0 (Q) ∩ Lαβ .

(20)

These notions and facts extend readily to ﬁelds and product spaces such as p Hper (Q)n , etc. We deﬁne Eαβ = E ∩ L3αβ ,

Eαβ = E ∩ L4αβ

(21)

and note that Eαβ admits the equivalent deﬁnition: Eαβ is the L2 -closure of ﬁelds eαβ f, with f = (f1 , f2 , f3 ) 8 2 + ∂z f3 = 0. αf1 + iβf in H 1 (I)2 × H 1 (I) such that i8 0

(22)

440

B. Scarpellini

From the expansion properties (3.16)–(3.18) and the stipulations (3.21) we infer the orthogonal sum relations: L2 (Q)n = ⊕Lnαβ , E = ⊕Eαβ , E = ⊕Eαβ . (23) αβ

αβ

αβ

Now let p8αβ be the orthogonal projection from L (Q) onto Lαβ , likewise p αβ from L2 (Q)3 onto L3αβ and pαβ from L2 (Q)4 onto L4αβ . These projections satisfy commutation relations: 2

Lemma 3.2: (1) P p αβ = p αβ P , (2) Ppαβ = pαβ P, (3) p αβ As ⊂ As p αβ , (4) pαβ A ⊆ Apαβ , (5) pαβ G ⊆ Gpαβ , (6) pαβ (PM ) = (PM )pαβ . Proof. As to (1) we ﬁrst note: if f ∈ L3αβ then P f ∈ L3αβ .

(24)

We ﬁrst prove (3.24) for the dense subset f = eαβ f ,

f = (f1 , f2 , f3 ) ∈ H 1 (I)2 × H01 (I).

(25)

For f in (3.25) we solve the boundary value problem: 8 2 + ∂z f3 , (∂z2 − γ 2 )p = i8 αf1 + iβf

p ∈ H 2 (I), ∂z p ∈ H01 (I)

(26)

where γ 2 = α 82 + β82 . Now (3.26) has a unique solution if γ 2 > 0 and unique up to an additive constant if γ = 0 (since I ∂z f3 dz = 0). We thus have P f = eαβ f − ∇(eαβ p) ∈ L3αβ ∩ E = Eαβ .

(27)

With (3.24) proved for f ’s of type (3.25), (3.24) follows by a density and approximation argument. Now let f ∈ L2 (Q)3 From (3.23), (3.24) and the boundedness of P we infer P εδ f, P f = f= P p εδ f = P p εδ f εδ

whence p αβ P f =

εδ

εδ

p αβ p εδ P p εδ f = p αβ P p αβ f = P p αβ f

εδ

proving (1). Claim (2) now follows from (1) and (2.2). In order to prove (3), (4) we consider the Dirichlet Laplacian ∆: 2 1 dom(∆) = Hper (Q) ∩ Hper,0 (Q),

∆f = (∂x2 + ∂y2 + ∂z2 )f

(28)

and show pαβ , that is ∆−1 p8αβ = p8αβ ∆−1 p8αβ ∆ ⊆ ∆8

(29)

Bifurcation of Traveling Waves

441

what in turn is a consequence of: if f ∈ Lαβ then ∆−1 f ∈ Lαβ ,

α, β ∈ Z.

(30)

Thus let f = eαβ g, g ∈ L2 (I) and h the solution of: (∂z2 − γ 2 )h = g,

h ∈ H 2 (I) ∩ H01 (I),

γ2 = α 82 + β82 .

(31)

We then have eαβ h ∈ dom(∆),

∆−1 f ∈ Lαβ ,

∆(eαβ h) = eαβ g = f,

(32) −1

proving (3.30). Now let f ∈ L (Q). From (3.30) and the boundedness of ∆ infer: p8εδ f, ∆−1 f = p8εδ ∆−1 p8εδ f whence f= 2

εδ −1

p8αβ ∆

f=

we

εδ −1

p8αβ p8εδ ∆

p8εδ f = p8αβ ∆−1 p8αβ f = ∆−1 p8αβ f

εδ

what proves (3.29). Clauses (3), (4) now follow from (2.3)–(2.5), (3.29) and clauses (1), (2) already proved. The proofs of (5), (6) are straightforward and omitted. Remarks: In order to ﬁnd suitable subspaces which are also invariant under N ( , ) we list a few remarks. 8 (1) If f ∈ (H 1 )4 then T f = Proposition 3.4: Set D(α, β) = +i(a8 α + bβ). 1 (Q)4 then T pαβ f = +iD(α, β)pαβ f . +iD(α, β)f , (2) if f ∈ Hper

αβ

Proof. Based on (3.19) we have in case (1) that f = eαβ g with g ∈ H 1 (I)4 and eαβ as in (3.15) whence 8 αβ g = +iD(α, β)f. T f = ((a∂x + b∂y )eαβ )g = i(a8 α + bβ)e 1 4 ) . (2) follows from (1) pαβ f ∈ (Hαβ

Given α0 , β0 ∈ Z we consider the additive group: m{α0 , β0 } = {(nα0 , nβ0 )/n ∈ Z}

(33)

and note Proposition 3.5: Let a, b ∈ R, a · b = 0. Then there are α0 , β0 ∈ Z such that α + bβ8 = 0, α, β ∈ Z} m{α0 , β0 } = {(α, β)/a8

•

We omit the elementary proof. Based on (3.19), (3.23), we now associate with a pair (α, β) ∈ Z 2 the orthogonal sum [α, β]k = ⊕Lkn(α,β) , (n(α, β) = (nα, nβ)). (34) n

Thus [α, β]k is the space of elements w ∈ L2 (Q)k whose Fourier series assumes the form 2 k en(α,β) wn , wn ∈ L (I) , |wn |2 dz < ∞ (35) w= n

I

442

B. Scarpellini

or likewise (for k = 4) the space of w ∈ L2 (Q)4 such that pεδ w = 0 if (ε, δ) ∈ / + m{α, β}, that is such that w = n pn(αβ) w. Now let Qαβ and Qαβ be the orthogonal projections from L2 (Q)3 onto [α, β]3 and from L2 (Q)4 onto [α, β]4 respectively. We have the commutation properties. Lemma 3.3: εδ = Q εδ p αβ and pαβ Qεδ = Qεδ pαβ , (a) p αβ Q αβ and Qαβ P = PQαβ , αβ P = P Q (b) Q αβ AS =⊆ AS Q αβ and Qαβ A ⊆ AQαβ , (c) Q (d) Qαβ commutes with PM , G, T . For reasons of space we omit the proof which is very similar to that of Lemma 3.2. By Lemma 3.3 the space [α, β]4 is invariant under the linear operators appearing in (3.11). But [α, β]4 is also invariant under the nonlinearity N ( , ) as asserted by Lemma 3.4: 2 2 1 (Q)3 ∩[α, β]3 , g ∈ Hper (Q)∩[α, β]1 then (f ∇)g ∈ Hper (Q)∩[α, β]1 , (a) if f ∈ Hper (b) if f, g ∈ dom(A) ∩ [α, β]4 then N (f, g) ∈ E ∩ [α, β]4 . Proof. Let ﬁrst f = eαβ f , g = eεδ g , f , g ∈ H 2 (I). We then have 1 (Q). f ∂x g, f ∂y g, f ∂z g ∈ Lα+ε,β+δ ∩ Hper

(36)

∩ [α, β]3 , gN ∈ ∩ [α, β]1 be ﬁnite sums: Next let fN ∈ en(α,β) f n , gN = en(α,β) g n , f n ∈ H 2 (I)3 , g n ∈ H 2 (I). fN = 2 Hper (Q)3

2 Hper (Q)

|n|≤N

|n|≤N

From (3.36) we infer 1 (Q) ∩ [α, β]1 . (fN ∇)gN ∈ Hper

Now assume f ∈ f= en(α,β) f n ,

2 Hper (Q)3

∩ [α, β]3 , g ∈ gN = en(α,β) gn ,

2 Hper (Q)

n

(37)

∩ [α, β]1 , with Fourier series

f n ∈ H 2 (I)3 ,

g n ∈ H 2 (I)

n

and set also fN =

en(α,β) f n ,

gN =

|n|≤N

en(α,β) gn .

n≤N

From standard Fourier theory we infer f − fN H2 → 0,

g − gN H 2 → 0

as N → ∞.

(38)

2 2 1 (Q)3 , v ∈ Hper (Q) then (u∇)v ∈ Hper (Q) and We also note: if u ∈ Hper

(u∇)vH 1 ≤ cuH 2 vH 2 , this follows from the continuous embedding From (3.38), (3.39) we infer:

some c;

2 Hper (Q)

(39)

→ Cper (Q).

(f ∇)g − (fN ∇)gN H 1 → 0 as N → ∞.

(40)

Bifurcation of Traveling Waves

443

From (3.37), (3.40) and since [α, β]1 is L2 -closed we obtain clause (a) of the Lemma. From (a) we immediately obtain the vector version 2 if f ∈ Hper (Q)3 ∩ [α, β]3 ,

(f ∇)g ∈

1 (Q)4 Hper

2 g ∈ Hper (Q)4 ∩ [α, β]4 then

(41)

∩ [α, β]4 .

Clause (b) of the Lemma now follows from (b) of Lemma 3.3 and from (3.41). Prior to prove Thm. 1 we list some consequences of Prop. 3.3. With L0 given by (3.11) we set Lj = {ϕ/Lj0 ϕ = 0},

Nj = rg(Lj0 ),

j ≥ 1.

Obviously L1 and N1 are eigenspace and range of L0 . By Prop. 3.3 and RieszSchauder theory there is ν ≥ 1 such that L1 ⊆ L2 · · · ⊆ Lν = Lν+1 = . . . ,

N1 ⊇ N2 ⊇ · · · ⊇ Nν = Nν+1 . . .

we set Lν = L, Nν = N . The properties of L, N are summarised by Proposition 3.6: (A) There are bounded projections S, Q from E onto L, N respectively such that S + Q = 1 and in addition: (a) QL0 ⊆ L0 Q, (b) the mapping QL0 : N ∩ dom(L0 ), H 2 → N , L2 is onto, 1-1 and boundedly invertible. (B) There is a basis e1 , . . . , eN of L and a dual basis e∗1 , . . . , e∗N such that ej , e∗k = δjk and f ∈ N iﬀ f, e∗j = 0, j, k ≤ N . Remarks: Riesz-Schauder theory as summarised above follows readily from RieszSchauder theory for compact operators ([3], Chpt. 5) and from Prop. 3.3. With Lj , L, Q, S as above and Qαβ as in Lemma 3.3 we have Lemma 3.5: (A) There+are (αk , βk ) ∈ Z 2 , k ≤ p such that L1 ∩ Eαk βk = 0, k ≤ p and Lj ⊆ p1 ⊕Eαk βk , j ≤ ν, (B) QQαβ = Qαβ Q. Proof. As to (A), let m ⊆ Z 2 be the set of (α, β) with L1 ∩Eαβ = 0. Then m is ﬁnite. Otherwise one could ﬁnd a sequence (αk , βk ), k < ∞ and elements gk ∈ L1 ∩Eαk βk , gk = 0 with L0 gk = 0. Since the gk are pairwise orthogonal, dom(L1 ) = ∞ would follow, contradicting Prop. 3.6. Thus m = {(αk , βk )/k ≤ p}, some p; we set E = ⊕Eαk βk . (42) k≤p

If L1 ⊆ E an e ∈ L , e ∈ / E could be found. Now + e = pαβ e, pαβ e, pεδ e = 0 if (α, β) = (ε, δ) 1

444

B. Scarpellini

by virtue of (3.23). Since e ∈ / E , pαβ e = 0 would hold for some (α, β) ∈ m. By Lemma 3.2 we have L0 pαβ e = pαβ L0 e = 0, i.e., pαβ e ∈ L1 whence (α, β) ∈ m, contradicting the deﬁnition of m. Thus L1 ⊆ E . Now let Lj ⊆ E be true; let f ∈ Lj+1 . Then L0 f = ϕ for some ϕ ∈ Lj . Now pick (α, β) ∈ / m. Since Lj ⊆ E we have pαβ ϕ = 0 whence L0 pαβ f = pαβ L0 f = pαβ ϕ = 0,

i.e., pαβ f ∈ L1 ∩ Eαβ ,

thus pαβ f = 0 by deﬁnition of m. That is, f ∈ E whence Lj+1 ⊆ E , completing the induction and thus the proof of (A). As to (B) we recall [4], Thm. 6.17 according to which P = −(2πi)−1 (L0 − ζ)−1 dζ, (43) Γ

Γ a small circle centered at ζ, is a bounded projection onto the generalised eigenspace of L0 at ζ = 0, i.e., onto L = Lν . Since S in Prop. 3.5 is also a bounded projection onto L, we have P S = S,

SP = P .

(44)

−1

Now S commutes with (L0 − ζ) by Prop. 3.5 and hence with P in (3.44), i.e., P S = SP . By (3.43), (3.44) we get: S = P = −(2πi)−1 (L0 − ζ)−1 dζ. (45) Γ

By Lemma 3.3, Qαβ commutes with L0 , hence with (L0 − ζ)−1 , ﬁnally with S and thus with Q, what proves (B). Proof of Theorem 1. By (3.11), we can rewrite (3.10) as follows L0 w + εPM w − δGw + χT w + N (w, w) = 0,

w ∈ dom(L0 ).

(46)

Based on the spaces L, N and on Prop. 3.6 we apply the ﬁrst half of the LyapunovSchmidt method to (3.46), i.e., we make the “Ansatz” w = ζe + h,

h ∈ N,

ζe =

N

ζj ej .

(47)

1

We insert (3.47) into (3.46) and apply the projection Q in Prop. 3.5 to the resulting equation so as to get: QL0 h

+ εQPM (ζe + h) − δQG(ζe + h) + + χQT (ζe + h) + QN (ζe + h, ζe + h) = 0.

By Prop. 3.5, (A) we have that K = (QL0 )−1 : N , L2 → N ∩ dom(L0 ), H 2

(48)

Bifurcation of Traveling Waves

445

is a bounded operator. Application of K to (3.48) gives: h + εKQPM (ζe + h) − δKQG(ζe + h) + χKQT (ζe + h)

(49)

+ KQN ζe + h, ζe + h) = 0. We deﬁne the space X = N ∩ dom(L0 ), H2 , i.e., X = H 2 and denote the left-hand side of (3.49) by F (h, ζ, ε, δ, χ). Evidently ((2.8), (2.9)) we have (50) F maps X × RN +3 holomorphically into X. Moreover: (1) F (0, . . . , 0) = 0, (2) (∂h F )(0, . . . , 0) = I. By the the implicit function theorem there are ζ1 , ε1 , δ1 , χ1 , d1 > 0 and a ϕ1 mapping the neighborhood U1 = {(ζ, ε, δ, χ)/|ζ| ≤ ζ1 , |ε| ≤ ε1 , |δ| ≤ δ1 , |χ| ≤ χ1 } ⊆ RN +3 holomorphically into X such that: (a) F (ϕ1 (ζ, ε, δ, χ), ζ, ε, δ, χ) = 0 on U1 , (b) ϕ1 (0, . . . , 0) = 0, (c) if h ∈ X and (ζ, ε, δ, χ) ∈ U1 satisfy F (h, ζ, ε, δ, χ) = 0 and hH 2 ≤ d1 then h = ϕ1 (ζ, ε, δ, χ). Next we deﬁne a smaller space Y ⊆ X. With (αj , βj ) ∈ Z 2 , j ≤ p as in Lemma 3.5, (A) we claim: a8 αj + bβ8j = −D(αj , βj ) = 0,

j ≤ p.

(51)

In fact, by the deﬁnition of the set (αj , βj ), j ≤ p in the proof of Lemma 3.5, (A) and by Prop. 3.4 we ﬁnd ϕj = 0 in L1 ∩ Eαj βj such that: + T )ϕj = Bϕ j − iD(αj , βj )ϕj = 0 L0 ϕj = (B

(52)

(3.51) now follows from (3.52) and assumption (b) of Thm. 1. By Prop. 3.4 there is (α0 , β0 ) in Z 2 with {(nα0 , nβ0 )/n ∈ Z} = {(α, β)/D(α, β) = 0,

(53)

α, β ∈ Z} = m(α0 , β0 ). Thus (αj , βj ) ∈ m(α0 , β0 ), j ≤ p by (3.51) whence (a) ej ∈ dom(L0 ) ∩ [α0 , β0 ]4 ,

(b) T h = 0, h ∈ dom(L0 ) ∩ [α0 , β0 ]4 .

(54)

The space Y is now deﬁned via Y = N ∩ dom(L0 ) ∩ [α0 , β0 ]4 ,

Y = H 2 .

(55)

Y is an H 2 -closed subspace of X. By Lemmas 3.2–3.4, and (3.54) the mapping F in (3.52) leaves Y invariant. We now apply the implicit function theorem to F as a holomorphic mapping from Y × RN +3 to Y so as to ﬁnd ζ2 , ε2 , δ2 , χ2 , d2 > 0 and a ϕ2 mapping the neighborhood U2 = {(ζ, ε, δ, χ)/|ζ| ≤ ζ2 ,

|ε| ≤ ε2 ,

|δ| ≤ δ2 ,

|χ| ≤ χ2 } ⊆ RN +3

holomorphically into Y such that: (α) F (ϕ2 (ζ, ε, δ, χ), ζ, ε, δ, χ) = 0 on U2 , (β) ϕ2 (0, . . . , 0) = 0, (γ) if (ζ, ε, δ, χ) ∈ U2 , h ∈ Y satisfy hH 2 ≤ d2 and F (h, ζ, ε, δ, χ) = 0 then ϕ2 (ζ, ε, δ, χ) = h.

446

B. Scarpellini

By the unicity properties (c) of ϕ1 , (γ) of ϕ2 we ﬁnd ε0 ≤ min(ε1 , ε2 ), δ0 ≤ min(δ1 , δ2 ), χ0 ≤ min(χ1 , χ2 ), ζ0 ≤ min(ζ1 , ζ2 ) such that: ϕ1 (ζ, ε, δ, χ) = ϕ2 (ζ, ε, δ, χ),

|ζ| ≤ ζ0 ,

|ε| ≤ ε0 ,

|δ| ≤ δ0 , |χ| ≤ χ0 .

(56)

To conclude the proof, pick d0 > 0 so small that if w ∈ dom(L0 ), wH2 ≤ d0 then w = ζe + h with |ζ| ≤ ζ0 ,

hH 2 ≤ d1 where h ∈ X.

(57)

Now let |ε| ≤ ε0 , |δ| ≤ δ0 , |χ| ≤ χ0 and w ∈ dom(L0 ) a solution of (3.46) such that wH 2 ≤ d0 . By (3.57), w = ζe + h holds with |ζ| ≤ ζ0 , hH 2 ≤ d1 and F (h, ζ, ε, δ, χ) = 0. Since |ζ| ≤ ζ1 , . . . , |χ| ≤ χ1 and hH 2 ≤ d1 we have h = ϕ1 (ζ, ε, δ, z) and since also |ζ| ≤ ζ0 , . . . , |χ| ≤ χ0 we have by (3.56) h = ϕ1 (ζ, ε, δ, χ) = ϕ2 (ζ, ε, δ, χ) ∈ dom(L0 ) ∩ [α, β]4 + whence T w = ζj T ej + T h = 0 by (3.52).

Corollary to Theorem 1: If τ0 = 0 in (3.10) then the conclusion of Thm. 1 holds. = −A + λ0 PM is selfadjoint and has only real eigenvalues. Proof. In this case, B in (3.11) but stipulate Theorem 1 admits a minor generalization. We retain B + χ0 T, L0 = B

χ0 ∈ R,

χ0 = 0

(58)

and consider instead of (3.10) the equation L0 w + εPM w − δGw + χT + N (w, w) = 0,

L0 via (3.58).

(59)

Theorem 1 : With L0 as in (3.58) and under the assumptions of Thm. 1 there are ε0 , δ0 , χ0 , d0 > 0 as follows: if |ε| ≤ ε0 , |χ| ≤ χ0 and if w ∈ dom(L0 ) is a solution of (3.10) with wH 2 ≤ d0 then T w = 0. Remarks: Theorem 1 follows from Theorem 1 via the substitutions χ0 T → T , χχ−1 0 → χ.

4. Existence of traveling waves Theorem 1, 1 assert nonexistence of small traveling wave solutions w = 0 of (3.10) resp. (3.59) under assumptions (a), (b) which in turn imply existence of −1 (α, β) ∈ Z 2 , (α, β) = 0 with D(α, β) = 0 what means that aL−1 are 1 , bL2 rationally dependent. In order to remedy the situation we drop (b) while (a) has L0 as in (3.11) we assume: H: (α) to be retained in view of Lemma 1. With B, ∩ (iR − {0}) = {±iσ}, and iσ (and There is precisely one σ > 0 such that σ(B) and bL−1 are hence −iσ) has algebraic multiplicity 1, (β) 0 ∈ σ(L0 ), (γ) aL−1 1 2 rationally independent. Remarks: H(α), which implies τ0 = 0 in (3.10), is reminiscent of Hopf bifurcation ([9], sect. V, [2]). In order to exploit this similarity we need some preparatory

Bifurcation of Traveling Waves

447

steps. By H(α) there is a unique pair (α, β) ∈ Z 2 and a w ∈ (H 2 (I) ∩ H01 (I))4 with 8 2 + ∂z w3 = 0, w = (w1 , . . . , w4 ). αβ w) = iσeαβ w, i8 B(e αw1 + iβw (1) One easily shows that α2 + β 2 > 0 must hold. We assume αβ = 0

(2)

but the arguments below apply apart from minor changes also to the case αβ = 0. We note: Proposition 4.1: D(α, β) = σ or D(α, β) = −σ. + T ) by H(β) there is by Lemma 3.2 a pair (γ, δ) ∈ Z 2 and Proof. Since 0 ∈ σ(B 2 1 u ∈ (H (I) ∩ H0 (I))4 with u = 0 and γδ u) + iD(γ, δ)eγδ u = 0, B(e

8 2 + ∂z u3 = 0. i8 γ u1 + δu

(3)

Now γ = δ = 0 is excluded on elementary grounds. But D(γ, δ) = 0, (γ, δ) = 0 cannot hold either in view of H(γ). Thus D(γ, δ) = 0 whence D(γ, δ) = ±σ by whence H(α); let, e.g., D(γ, δ) = σ. Thus eγδ u is an eigenfunction of −iσ ∈ σ(B) 8

eγδ u = ce−i(8αx+βy) w, some c = 0. This implies γ = −α, δ = −β and thus D(α, β) = −σ. Likewise in case D(γ, δ) = −σ.

Remark: Without loss of generality we assume D(α, β) = −σ,

(α, β as in 4.1).

(4)

L0 resp.; recalling eαβ , Eαβ in (3.15), (3.21) ∗ , L∗ are the adjoints of B, Below B 0 we use the shorthand e−αβ = e−α,−β , E−αβ = E−α,−β and likewise with the projections αβ , Qαβ . B ∗ respectively and Proposition 4.2: ±iσ are algebraically simple eigenvalues of B, ∗ with: there are ϕ, ϕ ∈ dom(B) = iσϕ, Bϕ = −iσϕ, B ∗ ϕ∗ = −iσϕ∗ , B ∗ ϕ∗ = iσϕ∗ , Bϕ ∗

∗

∗

(5)

∗

ϕ, ϕ = 1, ϕ, ϕ = 1, ϕ, ϕ = 0, ϕ, ϕ = 0, ϕ, ϕ∗ ∈ Eαβ , ϕ, ϕ∗ ∈ E−αβ . Proof. With ϕ = eαβ w as in (4.1) the claim follows from H(α) by standard func B ∗ tional analysis and the fact that the spaces Eαβ , E−αβ are invariant under B, (Lemma 3.3). Proposition 4.3: (a) 0 ∈ σ(L0 ) has algebraic and geometric multiplicity m = 2; its eigenspace is spanned by ϕ, ϕ, (b) 0 ∈ σ(L∗0 ) has algebraic and geometric multiplicity m = 2; its eigenspace is spanned by ϕ∗ , ϕ∗ .

448

B. Scarpellini

Proof. We ﬁrst claim L0 ϕ = L0 ϕ = L∗0 ϕ∗ = L∗0 ϕ∗ = 0.

(6)

This follows from Prop. 3.3, 3.4 and (4.4); e.g., ∗ ϕ∗ − T ϕ∗ = iσϕ∗ − iD(−α, −β)ϕ∗ = iσϕ∗ − iσϕ∗ = 0. L∗0 ϕ∗ = B The other cases in (4.6) follows in the same way. Next assume L0 ψ = 0, some ψ ∈ dom(L0 ),

ψ = 0.

(7)

Now pick (γ, δ) ∈ Z such that 2

(γ, δ) ∈ / {(α, β), (−α, −β)}.

(8)

We apply pγδ to (4.7); by Lemma 3.2 we get γδ ψ + T pγδ ψ = Bp γδ ψ + iD(γ, δ)pγδ ψ = 0. Bp

(9)

= 0 if w ∈ (H 2 (I) ∩ H 1 (I))4 , div w = 0, w = 0, clause (4.9) implies Since Bw 0 pγδ ψ = 0 if (γ, δ) = 0. If (γ, δ) = 0 then D(γ, δ) = 0 by H(γ). If now pγδ ψ = 0 then D(γ, δ) ∈ {±σ} by H(α) and (4.9) whence D(α + γ, β + δ) = 0 or D(γ − α, δ − β) = 0

(10)

by Prop. 4.1. By H(γ) and (4.10) we thus have (γ, δ) = (α, β) or (γ, δ) = (−α, −β), contradicting (4.8). We thus have pγδ ψ = 0 if (γ, δ) ∈ / {(α, β), (−α, −β)}.

(11)

ψ = pαβ ψ + p−αβ ψ.

(12)

From (4.11) we infer: By setting (γ, δ) = (α, β) and (γ, δ) = (−α, −β) respectively in (4.9) we obtain: pαβ ψ = cϕ, p−αβ ψ = c ϕ, some c, c .

(13)

From (4.7), (4.12), (4.13) it now follows that span (ϕ, ϕ) is the eigenspace of 0 ∈ σ(L0 ). In a last step we assume L0 f = aϕ + bϕ, some f ∈ dom(L0 ), a, b. ∗

(14)

∗

We multiply (4.14) scalarly with ϕ , and ϕ respectively. From (4.5) we infer a = b = 0, i.e., L0 f = 0, what concludes the proof. Remark: By Prop. 4.3, 0 ∈ σ(L0 ) has a generalised eigenspace of dimension m = 2. Due to the lack of a suitable symmetry, the method of bifurcation from a simple eigenvalue is not applicable. As in case of Hopf bifurcation this diﬃculty is overcome by the introduction of an amplitude parameter µ, i.e., instead of (3.10) we consider: (15) L0 w + εPM w − δGw + χT w + µN (w, w) = 0;

Bifurcation of Traveling Waves

449

if w ∈ dom(L0 ) is a solution of (4.15) then µw is a solution of (3.10). Since we seek real solutions we introduce real eigenvectors: φ1 = φ∗1

=

√1 (ϕ + ϕ), 2 √1 (ϕ∗ + ϕ∗ ), 2

φ2 = φ∗2

=

√i (ϕ − ϕ), 2 √i (ϕ∗ − ϕ∗ ). 2

(16)

Proposition 4.4: (a) span(ϕ, ϕ) = span(φ1 , φ2 ), span(ϕ∗ , ϕ∗ ) = span(φ∗1 , φ∗2 ) and φj , φ∗k = δjk , (b) f ∈ rg(L0 ) iﬀ f, φ∗j = 0, j = 1, 2. Proof. (a) follows from Prop. 4.1 and (4.16). (b) follows from the fact that 0 ∈ σ(L0 ) has span(φ1 , φ2 ) as generalised eigenspace, while 0 ∈ σ(L∗0 ) has span(φ∗1 , φ∗2 ) as generalised eigenspace. We add some observations. From Prop. 3.4, 4.2 and from (4.4), (4.16) we infer T φ1 = σφ2 , T φ2 = −σφ1 , T φ∗1 = σφ∗2 , T φ∗2 = −σφ∗1 .

(17)

In order to apply Prop. 3.6 to L0 let Lj , Nj , L = Lν , N = Nν be the spaces prior to Prop. 3.6. By Prop. 3.5, 4.3, 4.4 we have that L = L1 = Lν = span(φ1 , φ2 ), N = N1 = Nν = rg(L0 ).

(18)

The projections S, Q in Prop. 3.5 are now given by Sf = f, φ∗1 φ1 + f, φ∗2 φ2 , Qf = f − Sf, f ∈ E.

(19)

Based on (3.5), (4.17), (4.19) a simple computation shows that S, Q commute with T , i.e.: ST ⊆ T S, QT ⊆ T Q. (20) Finally, based on Prop. 3.6 and (4.18) we deﬁne the bounded operator K via K = (QL0 )−1 : N L2 → N ∩ dom(L0 ), H 2

(21)

and introduce the space X = dom(L0 ) ∩ rg(L0 ), H 2 = N ∩ dom(L0 ), H 2 .

(22)

We now apply the Ljapounov-Schmidt (L-S) method to (4.15). To this eﬀect we insert the “Ansatz” w = ζφ + h,

ζφ = ζ1 φ1 + ζ2 φ2 ,

h∈X

into (4.15), multiply the resulting equation with KQ from the left, obtaining the ﬁrst L-S equation. The other two are the orthogonality conditions which arise from the fact that L0 h, φ∗j = 0, j = 1, 2. The L-S equations so obtained are: h + εKQPM (ζφ + h) − δKQG(ζφ + h) + χT h +µKQN (ζφ + h, ζφ + h) = 0 εPM (ζφ + h), φ∗j − δG(ζφ + h), φ∗j + χT (ζφ), φ∗j +µN (ζφ, h) + N (h, ζφ) + N (h, h), φ∗j = 0, j = 1, 2.

(23)

450

B. Scarpellini

In (4.23) some simpliﬁcations have been taken care of which follow from the relations: QT h = T h ∈ N , QT ζφ = 0, N (ζφ, ζφ), φ∗j = 0, j = 1, 2.

(24)

The ﬁrst two follow from (4.17), (4.20), the third is proved in the appendix. We denote the left-hand sides of (4.23) one after the other by F (h, ζ, ε, δ, χ, µ) and Gj (h, ζ, ε, δ, χ, µ), j = 1, 2 (ζ = (ζ1 , ζ2 )). Clearly F is a holomorphic mapping from X × R6 to X while the Gj are holomorphic mappings from X × R6 to R. Since F (0, . . . , 0) = 0, δh F (0, . . . , 0) = I we have by the implicit function theorem that there are ζ0 , ε0 , δ0 , χ0 , µ0 > 0 and a mapping h which maps the neighborhood (25) U = {(ζ, . . . , µ) |ζ| ≤ ζ0 , |ε| ≤ ε0 , . . . , |µ| ≤ µ0 } ⊆ R6 holomorphically into X such that F (h(ζ, ε, δ, χ, µ), ζ, ε, δ, z, µ) = 0 on U.

(26)

Since F (0, ζ, 0, . . . , 0) = 0 we have by the unicity part of the implicit function theorem: (27) h(ζ, 0, . . . , 0) = 0, |ζ| ≤ ζ0 . We now stipulate: Hj (ζ, ε, δ, χ, µ) = Gj (h(ζ, ε, δ, χ, µ)ζ, ε, δ, χ, µ), j = 1, 2

(28)

H(ζ, . . . , µ) = (H1 (ζ, ε, . . . , µ), H2 (ζ, ε, . . . , µ)). Clearly, H is a holomorphic mapping from U to R2 which by (4.27) satisﬁes H(ζ, 0, . . . , 0) = 0, |ζ| ≤ ζ0 .

(29)

Based on (4.27) we would like to solve H(ζ, ε, δ, χ, µ) = 0,

ζ = 0 ﬁxed, ε, δ, χ, µ small.

This amounts to investigate the matrices ∂Hj (ζ, 0, 0, 0, 0) , j, k = 1, 2, ζ = 0 D(γ1 , γ2 , ζ) = ∂γk

(30)

(31)

for pairs (γ1 , γ2 ), γj ∈ {ε, δ, χ, µ}, γ1 = γ2 what leads to the study of the determinants (32) d(γ1 , γ2 , ζ) = det(D(γ1 , γ2 , ζ)), ζ = 0, γj ∈ {ε, δ, χ, µ} for γ1 = γ2 . Since d(γ1 , γ2 , ζ) = 0 if γ1 = µ or γ2 = µ, it remains to investigate three cases: d(ε, χ, ζ), d(ε, δ, ζ), d(δ, χ, ζ), ζ = (ζ1 , ζ2 ) = 0. (33) The computation of these determinants is completely straightforward. We content us to summarize the result of this computations. We set ajk = PM φj , φ∗k = M φj , φ∗k , bjk = Gφj , φ∗k , djk = aj1 bk2 − aj2 bk1

(34)

Bifurcation of Traveling Waves

451

The determinants in (4.34) now assume the form: d(ε, χ, ζ) = σζ12 a11 − σζ1 ζ2 (a12 + a21 ) + σζ22 a22

(35)

d(δ, χ, ζ) = −σζ12 b11 + σζ1 ζ2 (b12 + b21 ) − σζ22 b22 d(ε, δ, ζ) = −ζ12 d11 + ζ1 ζ2 (d12 + d21 ) − ζ22 d22 . Theorem 2: (A) If one of a11 , a12 + a21 , a22 is = 0 then for almost all ζ ∈ R, |ζ| ≤ ζ0 , ζ = 0 there is a neighborhood Uζ of 0 ∈ R2 and holomorphic functions ε(δ, µ), χ(δ, µ) on Uζ such that ε(0, 0) = χ(0, 0) = 0 and H(ζ, ε(δ, µ), δ, χ(δ, µ), µ) = 0, (δ, µ) ∈ Uζ .

(36)

(B) Likewise with δ, χ and b11 , b12 + b21 , b22 , (C) Likewise with ε, δ and d11 , d12 + d21 , d22 . Proof. Consider (A). If one of a11 , a12 + a21 , a22 is = 0 then for almost all ζ ∈ R2 , |ζ| ≤ ζ0 , ζ = 0 the determinant d(ε, χ, ζ) in (4.35) is = 0. For such ζ the 2×2-matrix D(ε, χ, ζ) in (4.31) is invertible by (4.32), γ1 = ε, γ2 = χ. Since D(ε, χ, ζ) is the Fr´echet derivative of H(ζ, . . . , µ) with respect to (ε, χ) at ε = χ = δ = µ = 0 and by (4.29) we can apply the implicit function theorem so as to ﬁnd a neighborhood U5 and holomorphic functions E(δ, µ), χ(δ, µ) with the required properties. (B), (C) follows likewise. Remarks: Since (4.15) is translation invariant in the x, y-direction, ε(δ, µ), χ(δ, µ) in (A) depend on ζ only via |ζ|; however the smaller |ζ| = 0 the smaller the neighborhood Uρ . Likewise in cases (B), (C). Corollary: Under the assumptions H(α)–(γ) and one of the assumptions in (A), (B) or (C) of Thm. 2 there are arbitrary small values ε, δ, χ, µ, not all = 0, and an of (4.15) such that T w = 0. associated solution w = w(ε, δ, χ, µ) ∈ dom(B) Proof. Under the assumption in (A), e.g., ε(δ, µ), χ(δ, µ) and h = h(ζ, ε(δ, µ), δ, χ(δ, µ), µ) satisfy the L-S equations (4.24). Hence w = ζφ + h is a solution of (4.15). Since we are free to pick µ = 0, δ = 0, not all of ε, χ, δ, µ are = 0. Moreover ST w = ST (ζφ + h) = T S(ζφ + h) = T ζφ = ζ1 σφ2 − ζ2 σφ1 = 0.

Theorem 3: The solution w = 0 of (4.15) associated with ε, δ, χ, µ under the assumptions of Thm. 2 give rise to a time periodic traveling wave, i.e., w(t, x, y, z) = bβ −1 2 w(x − at, y − bt, z) has time period d = ( aα + ) where (α, β) ∈ Z is given by L1 L2 (4.1). Proof. By H(α), a pair (α, β) ∈ Z 2 , unique up to sign, is determined via (4.1). Recalling [α, β]4 in (3.34) we deﬁne the space Y = dom(L0 ) ∩ N ∩ [α, β]4 , H 2

(37)

which is an H 2 -closed subspace of X. Based on Lemmas 3.2–3.4 we conclude as in the proof of Thm. 1 that the mapping F given by the ﬁrst equation in (4.23) is a holomorphic mapping from Y × R6 to Y . As in the proof of Thm. 1 we may

452

B. Scarpellini

replace X by Y in the proof of Thm. 2 in order to infer from the unicity part of the implicit function theorem that the solution h(ζ, ε, . . . , µ) of (4.26) is already in Y ; an eventual shrinking of the neighborhood Uζ in (4.25) is thereby taken into account. That is, w = ζφ + h is in Y and has the representation 8 w= ein(8αx+βy) wn (z), wn ∈ (H 2 (I) ∩ H01 (I))4 . (38) n

This entails w =

8

8

ein(8αx+βy) wn (z)e−in(8αa+βb)t .

(39)

n

Recalling α 8, β8 in (3.15), it follows from (4.39) that d = ( aα L1 + period for w.

bβ −1 L2 )

is a time

Remarks: (1) By Thm. 3, our bifurcating traveling waves are necessarily time periodic, a fact not immediately evident from the outset. (2) While it may be diﬃcult to ﬁnd conditions on the vj in (3.2) which secure that H(α) and at least one of the assumptions in (A)–(C) are satisﬁed, there is a case, which although not subject to H(α), can nevertheless be handled by the arguments in the proof of Thm. 2. This case arises if χ0 = τ0 = 0 in (3.11), (3.58), that is (4.15) now assumes the form: + εPM w − δGw + χT + µN (w, w) = 0, w ∈ dom(B), Bw = −A + λ0 PM. L0 = B

(40)

is now selfadjoint, condition H(α) cannot hold and is thus omitted. ConSince B dition H(γ) is retained while condition H(β) is replaced by the stronger one: H(δ) :

0 ∈ σ(−A + λ0 PM ) and − A + λPM < 0 if λ < λ0

By H(δ), λ0 is the critical Rayleigh parameter of the B´enard problem ([5],[11]). As to the next assumption we invoke a result of V.I. Judovitch ([5],[6]) which states that there is a set J ⊆ R2 with R2 − J of Lebesgue measure zero such that is as (L1 , L2 ) ∈ J (Lj the periods) implies that the eigenspace L of 0 ∈ σ(B) 2 follows. There is a pair (α, β) ∈ Z , (α, β) = 0 and smooth functions w, ϑ = 0 on [− 12 , 12 ] such that ν(∂z2 − γ 2 )2 w = λ0 γ 2 θ, µ(∂z2 − γ 2 )ϑ + λ0 w = 0, ϑ = w = ∂z w = 0 (41) at z = ± 12 and such that L is spanned by the four vectors: iδ iε ∂z w, 2 ∂z w, w, ϑ , |ε| = |α|, |δ| = |β| ϕεδ = eεδ γ2 γ

(42)

where (ε, δ) ∈ Z 2 , γ 2 = α2 + β 2 . We now assume H(ε)

(L1 , L2 ) ∈ J and αβ = 0.

The cases α = 0 or β = 0 subsume under the arguments below. Our basic as has sumptions are now H(γ), (δ), (ε). By H(δ), (ε) the eigenspace L of 0 ∈ σ(B)

Bifurcation of Traveling Waves

453

dimension 4. Since one has only four parameters ε, δ, χ, µ, the L-S method cannot be applied straightforwardly. To avoid this diﬃculty we ﬁx (α, β) ∈ Z 2 given by H(δ), (ε) and recall the space [α, β]4 in (3.34). By Lemmas 3.2–3.4, the left-hand side of (4.40) leaves the space E ∩[α, β]4 invariant. We thus can apply L-S to (4.40) B restricted to E ∩ [α, β]4 , restricted to E ∩ [α, β]4 . The eigenspace L of 0s ∈ σ(B), has dimension d = 2 and is given by spaces (ϕαβ , ϕ−αβ ), with ϕαβ and ϕ−αβ as in and hence its restriction to E ∩ [α, β]4 (4.42). We set ϕ = ϕαβ , ϕ = ϕ−αβ . Since B ∗ is selfadjoint, we have ϕ = ϕ. A real basis of L is again given by (4.16); the orthogonality relations in Prop. 4.4 remain of course valid in the present setting. restricted to E ∩ [α, β]4 is now The projection Q associated by Prop. 3.6 with B given by Qf = f − f, φ1 φ1 − f, φ2 φ2 , f ∈ E ∩ [α, β]4 ; (43) it maps E ∩ [α, β]4 onto rg(B) ∩ [α, β]4 . By Prop. 3.6 the bounded operator K = to E ∩ [α, β]4 : −1 is associated with the restriction of B (QB) ∩ [α, β]4 , L2 → dom(B) ∩ rg(B) ∩ [α, β]4 , H 2 . K : rg(B)

(44)

With the L-S equations given by (4.23) and reinterpreted as above, we let as before F (h, ζ, . . . , µ), Gj (h, ζ, . . . , µ), j = 1, 2 be the left-hand sides of (4.23). F , Gj now map X × R6 holomorphically into X and R respectively, where X is the space ∩ rg(B) ∩ [α, β]4 , H 2 . X = dom(B)

(45)

Since F (0) = 0, ∂h F (0) = I, there is by the implicit function theorem a neighborhood U of type (4.26) (some ζ0 , . . . , µ0 ) and a holomorphic mapping h from U to X which satisﬁes (4.26), (4.27). In terms of h, Gj , j = 1, 2 we deﬁne mappings Hj , j = 1, 2, and H via (4.28). H is now a holomorphic mapping from U to R2 which satisﬁes (4.25). Since solving the L-S equations (4.23) is reduced to solve (4.30) via the implicit function theorem we are again led to study the regularity of the matrices D(γ1 , γ2 , ζ) in (4.32) and hence the determinants in (4.33). Since the eigenfunctions φj (= φ∗j ) are known via (4.16), (4.41), (4.42) these determinants can be computed in some cases. Theorem 4: (A) For ζ ∈ R2 , ζ = 0 suﬃciently small there are a neighborhood Uζ of 0 ∈ R2 and holomorphic functions ε(δ, µ), χ(δ, µ) on Uζ with; (a) ε(0, 0) = χ(0, 0) = 0, (b) H(ζ, ε(δ, µ), δ, χ(δ, µ), µ) = 0, (δ, µ) ∈ Uζ .

(B) Let v4 in (3.2) satisfy I v4 ∂z (wV)dz = 0; for ζ ∈ R2 , ζ = 0 suﬃciently small there is a neighborhood Uζ of v ∈ R2 and holomorphic functions δ(ε, µ), χ(ε, µ) on Uζ with: (a) δ(0, 0) = χ(0, 0) = 0, (b) H(ζ, ε, δ(ε, µ), χ(ε, µ), µ) = 0, (ε, µ) ∈ Uζ . Proof. As to (A) we evaluate d(ε, χ, ζ) via (4.34), (4.35).

454

B. Scarpellini

By H(γ) we have Since φj =

φ∗j

D(α, β) = a8 α + bβ8 = −σ = 0.

(46)

ajj = PM φj , φj , a21 = a12 = PM φ1 , φ2 .

(47)

we have

Since φj , j = 1, 2 are eigenfunctions of 0 ∈ σ(−A + λ0 PM ) we have: λ0 PM φj φj = Aφj , φj > 0,

PM φj , φj > 0.

(48)

As shown in the appendix it holds that PM φj , φk = 0, j = k,

PM φ1 , φ1 = PM φ2 , φ2 .

(49)

By (4.34), (4.35) and (4.46)–(4.49) we thus get: d(ε, χ, ζ) = σ(ζ12 + ζ22 )PM φ1 , φ1 ,

ζ = (ζ1 , ζ2 ) = 0,

(50)

hence D(ε, χ, ζ) is invertible whence (A) follows by the implicit function theorem. As to (B) we compute d(δ, χ, ζ) via (4.35) where bjk = Gφj , φk . Now we need some results proved in the appendix, namely Gφ1 , φ2 = −Gφ2 , φ1 , Gφ1 , φ1 = Gφ2 , φ2 , Gφ1 , φ1 = −A v4 ∂z (wV) dz, some A > 0

(51)

I

with v4 given by (3.2). Based on (4.51), the computation of d(δ, χ, ζ) via (4.35) gives & % (52) d(δ, χ, ζ) = Aσ(ζ12 + ζ22 ) v4 ∂z (wV) dz, I = − 21 , 12 . I

By the assumption of (B) the claim now follows from (4.31), (4.32), (4.46) and (4.52) via the implicit function theorem Remarks: As to (A), (B) it would be desirable to know, e.g., at least the linear terms in the expansion of ε(δ, µ), χ(δ, µ). Below, (4.23)j ; j = 1, 2, 3 denotes the jth equation of (4.23). We ﬁrst consider the expression h(ζ, ε, δ, χ, µ) = h1 ε + h2 δ + h3 χh + h4 µ + . . . ,

hj = hj (ζ).

Inserting this into (4.23)1 a comparison of coeﬃcients yields: h1 = −KQPM ζφ, h2 = KQGζφ, h4 = −KQN (ζφ, ζφ),

(53)

h3 = 0. We now multiply (4.23)2 with ζ1 , and add (4.23)3 multiplied with ζ2 . By taking care of (4.46), of Prop. 4.4 and (4.49) we get ε|ζ|2 PM φ1 , φ1 − δG(ζφ + h), ζφ + εPM h, ζφ +µN (ζφ + h, ζφ + h), ζφ = 0,

|ζ| = 2

ζ12

+

(54)

ζ22 .

We insert the expansion ε = ε0 µ + ε1 δ + · · · into (4.54). Comparison of coeﬃcients yields: ε0 = 0,

ε1 |ζ|2 PM φ1 , φ1 = Gζφ, ζφ.

(55)

Bifurcation of Traveling Waves

455

Next we multiply (4.23)3 with ζ1 and subtract (4.23)2 multiplied with ζ2 so as to obtain: χ|ζ|2 σ − G(ζφ + h), ζ1 φ2 − ζ2 φ1 δ + εPM h, ζ1 φ2 − ζ2 φ1

(56)

+µN (ζφ + h, ζφ + h), ζ1 φ2 − ζ2 φ1 = 0. We insert the expansion χ = χ0 µ+χ1 δ+· · · into (4.51); a comparison of coeﬃcients yields: (57) χ0 = 0, χ1 |ζ|2 σ = Gζφ, ζ1 φ2 − ζ2 φ1 . Crucial for the computation of ε1 , χ1 is Prop. 4.5, to be proved in the appendix: Proposition 4.5: There is a holomorphic function F (z), z ∈ I, not ≡ 0, depending on w, V, α, β in (4.41), (4.42), such that 8 2 )dz (vj via (3.2)). Gφ1 , φ2 = F (z)(8 αv1 + βv (58) I

Theorem 5: Assume (a) ∂z (wV)v4 dz = 0, I

(b)

8 2 ) dz = 0. F (z)(8 αv1 + βv

(59)

I

Then: (A) ε1 = 0, (B) χ1 = 0, (C) for ζ ∈ R2 , ζ = 0 suﬃciently small there is a neighborhood U5 of O ∈ R2 and holomorphic functions ε(χ, µ), δ(χ, µ) on Uζ with (a) ε(0, 0) = δ(0, 0) = 0, (b) H(ζ, ε(χ, µ), δ(χ, µ), χ, µ) = 0 (H via (4.28)). Proof. From (4.55) we infer δε1 |ζ|2 PM φ1 φ1 = ζ12 Gφ1 φ1 + ζ22 Gφ2 , φ2

(60)

+ ζ1 ζ2 (Gφ1 , φ2 + Gφ2 , φ1 ). Claim (A) near follows from (4.46), (4.51), from Ass. (4.59)(a) and from (4.60). In order to prove (B) we recall (4.57); computation yields Gζφ, ζ1 φ2 − ζ2 φ2 = ζ12 Gφ1 , φ2 − ζ22 Gφ2 , φ1 + ζ1 ζ2 (Gφ2 , φ2 − Gφ1 , φ1 ). By (4.51) we thus obtain Gζφ, ζ1 φ2 − ζ2 φ1 = (ζ12 + ζ22 )Gφ1 , φ2 .

(61)

(B) now follows from (4.57), (4.61), Ass. (4.59)(b) and Prop. 4.5. In order to prove (C) we compute the the determined d(ε, δ, ζ) via (4.34), (4.35). By a straightforward computation based on (4.34), (4.35), (4.49), (4.51) and by using Prop. 4.5 and Ass. (4.59)(b) we get d(ε, δ, ζ) = |ζ|2 PM φ1 , φ1 Gφ1 , φ2 = 0. Claim (C) now follows from (4.62) by the implicit function theorem.

(62)

456

B. Scarpellini

Remarks: As with Thm. 2 we can rephrase Thms. 4, 5 so as to apply to equation (4.40) and ultimately to equation (0) in the introduction. We content us to summarize the facts for (4.40). Corollary 1: Let vj , j ≤ 4 in (3.2) satisfy assumptions (4.59). Then there are ε, δ, χ, µ arbitrarily small, all = 0, and an associated solution w = w(ζ, ε, δ, χ, µ) of (4.40) such that T w = 0. Proof. With the assumptions satisﬁed, we have by Thm. 5 that ε(δ, µ), χ(δ, µ) in Thm. 4 have the form: ε = ε1 δ + O(δ 2 + µ2 ),

χ = χ1 δ + O(δ 2 + µ2 ),

ε1 = 0, χ1 = 0.

Thus for µ, δ both = 0 and small, ε, χ are both = 0 and by Thm. 4, (A) we have that w(ζ, δ, µ) = ζφ + h(ζ, ε(δ, µ), δ, χ(δ, µ), µ) = ζφ + h, ζ = 0

(63)

is a solution of (4.40). As in the proof of the corollary to Thm. 2 we conclude that T w = 0. Thus with w as in (4.63) we have that w = δv + µw is a genuine traveling wave solution of −Aw + (λ0 + ε)PM w + χPT w + N (w, w) + δf = 0,

(64)

i.e., such that − χat, y − χbt, z) = −χ(T w)(x − χat, y − χbt, z) = 0. ∂t w(x Since the construction of solutions in the proof of Thm. 4 is in the space E ∩[α, β]4 , Thm. 3 retains its validity here, that is, the traveling waves constructed via Thm. 4.5, Cor. 1 are time periodic with period given by Thm. 3. As for the functions ε(χ, µ), δ(χ, µ) it is easy to see that they admit an expansion of the form: ε(χ, µ) = ε1 χ + O(χ2 + µ2 ),

δ(χ, µ) = δ1 χ + O(χ2 + µ2 ).

(65)

Based on the assumptions of Thm. 5 one can perform computations similar to the above in order to compute ε1 , δ1 . The result, stated without proof, is δ1 = σGφ1 , φ2 −1 , ε1 = σPM φ1 , φ1 −1 Gφ1 , φ2 −1 Gφ1 , φ1 ,

(66)

with Gφ1 , φ2 , Gφ1 , φ1 given by (4.51), (4.58) resp. Remarks: Our construction of traveling wave solutions to (4.40) resp. (0) gives rise to two problems. The ﬁrst is whether the construction extends to other translation invariant domains such as channels. The second is of course the question whether this waves are stable in some sense, and whether they can be seen either numerically or experimentally. These topics will be treated elsewhere.

Bifurcation of Traveling Waves

457

Appendix Below we prove some relations which appear in the proofs of Thm. 4, 5. We recall (4.41), (4.42): i8 α iβ8 8 ϕ = eαβ ∂z w, 2 ∂z w, w, V , eαβ = eiτ , τ = α 8x + βx, (A.1) γ2 γ α 8 =

2πα 8 2πβ , β= , α, β ∈ Z, α · β = 0, γ 2 = α 82 + β82 , L1 L2

α, β given by H(δ), (ε). We have ϕ = a + ib, a, b real, with β8 α 8 a = − 2 ∂z w sin τ, − 2 ∂z w sin τ, w cos τ, V cos τ γ γ α 8 β8 b = − 2 ∂z w cos τ, 2 ∂z w cos τ, w sin τ, V sin τ . γ γ

(A.2)

For computational reasons we replace (4.16) by a rescaled deﬁnition φ1 , φ2 , without aﬀecting the assertions to be proved: φ1 = ϕ + ϕ = 2a,

φ2 = i(ϕ − ϕ) = 2b.

(A.3)

Let d1 = L1 (4α)−1 , d2 = L2 (4β)−1 ; one easily veriﬁes φ1 (x + d1 , y, z) = −φ2 (x, y, z),

φ2 (x + d1 , y, z) = φ1 (x, y, z)

(A.4)

likewise with d2 and y. We also need the Euclidean scalar product u·v =

4

uj vj ,

u = (u1 , . . . , u4 ), v = (v1 , . . . , v4 )

1

and the real scalar product

[u, v] = u, v =

u · v dx3 ,

dx3 = dx dy dz.

Q

Proposition A1: (a) PM φ1 , φ1 = PM φ2 , φ2 , (b) PM φj , φk = 0, j = k. Proof. As to (a) we have by periodicity L1 L2 PM φ1 , φ1 = M φ1 · φ1 dx3 = 0

0

I

L1 +d1

d1

0

L2

M φ1 · φ1 dx3 . I

On the last integral we perform the substitution x + d1 → x; by (A.4) we get PM φ1 , φ1 = [M (−φ2 ), (−φ2 )] = [M φ2 , φ2 ] = PM φ2 , φ2 .

458

B. Scarpellini

In case of (b) the same substitution yields: PM φ1 , φ2 = [M φ1 , φ2 ] = [M (−φ2 ), φ1 ] = −PM φ1 , φ2 , whence PM φ1 , φ2 = PM φ2 , φ1 = 0.

Proposition A2: N (φj , φk ), φl = 0, j, k, l = 1, 2. Proof. Since the φj are real we have τ = N (φj , φk ), φl = −[(φj ∇)φk , φl ]. Recalling (A.1)–(A.3) we have that I is a sum + I = c [(ψp ∇)ψq , ψr ], ψp , ψq , ψr ∈ {ϕ, ϕ} with ϕ, ϕ via (1) and c a constant; due to the structure of ϕ, ϕ, each term [(ψp ∇)ψq , ψr ] is a product of two integrals π1 , π2 say, which have the form: L1 L1 π1 = eα1 β1 eα2 β2 eα3 β3 dx dy, π2 = P(z) dz, (A.5) 0

0

I

and where (A.6) (αj , βj ) ∈ {(α, β), (−α, −β)}, α, β ∈ Z, α · β = 0 with α, β ﬁxed by H(δ), (ε) in Sec. 4. With the exponential factors eαj βj given by (1), and by (A.6), the factors π1 in (A.5) vanish, whence the claim follows. We now come to the claims regarding G deﬁned via (3.2), (3.9). Proposition A3: (a) Gφ1 φ1 = Gφ2 , φ2 , (b) Gφ1 , φ2 = −Gφ2 , φ1 . Proof. As to (b) we recall (2.8), (3.9) according to which Gφ1 , φ2 = −N (v, φ1 , φ2 − N (φ1 , v), φ2 = = [(v∇)φ1 , φ2 ] + [(φ1 ∇)v, φ2 ] = I1 + I2 . By periodicity we have

L1 +d1

L2

(v∇)φ1 · φ2 dx3 .

I1 = d1

0

I

Since the substitution x + d1 → x commutes with diﬀerentiation we have by (A.4): L1 L2 I1 = ((v∇)(−φ2 )) · φ1 dx3 = [N (v, φ2 ), φ1 ]. 0

0

I

Likewise we ﬁnd I2 = [N (φ2 , v), φ1 ] whence Gφ1 , φ2 = [N (v, φ2 ), φ1 ] + [N (φ2 , v), φ1 ] = −Gφ2 , φ1 , proving (b); the proof of (a) is similar and omitted.

We now come to the proof of Prop. 4.5 which requires some preparatory steps, labeled by S1, S2,. . .

Bifurcation of Traveling Waves S1:

459

Based on (3.9) and (A.1)–(A.3), we have Gφ1 , φ2 = [(v∇)]φ1 , φ2 ] + [(φ1 ∇)v, φ2 ] = 4[(v∇)a, b] + 4[(a∇)v, b] = I + II.

(A.7)

In order to evaluation I, II we note that they depend linearly on v = (v1 , v2 , 0, v4 ) in (3.2). This allows us to split the evaluation of I, II into three cases: (a) v = (r, 0, 0, 0),

(b) v = (0, r, 0, 0),

(c) v = (0, 0, 0, r),

with r ∈ H 2 (I) ∩ H01 (I). We ﬁrst compute I under Ass. (A.8)(a). We ﬁrst note: L1 L2 L1 L2 L1 L2 8 , τ =α 8x + βy. sin2 τ dx dy = cos2 τ dx dy = 2 0 0 0 0

(A.8)

(A.9)

Based on (2) we compute the term (v∇)a under the assumption (A.8)(a) and obtain that (v∇)a is equal to α 82 α 8β8 − 2 r∂z w cos τ, − 2 r∂z w cos τ, −wr8 α sin τ, −Vr8 α sin τ . (A.10) γ γ The Euclidean scalar product of (A.10) with b gives for (v∇)a · b the expression: −

α 8β82 α 83 2 2 r(∂ w) cos τ − r(∂z w)2 cos2 τ z γ4 γ4 −rw2 α 8 sin2 τ − rV 2 α 8 sin2 τ.

(A.11)

After a rearrangement of terms in (A.11), integration over Q based on (A.9) yields: (∂z w)2 2 2 + w + V α 8r dz. (A.12) I = 4[(v∇)a, b] = −2L1L2 γ2 I In order to compute II in (A.7) under Ass. (A.8)(a) we note that II = −4[(a∇)b, v]; by (A.8)(a) it suﬃces to compute only the ﬁrst component of (a∇)b, denoted by (a∇)b1 . Computation based on (A.2) yields: (a∇)b1 =

α 83 α 8β82 α 8 2 2 (∂ w) sin τ + (∂z w)2 sin2 τ + 2 w∂z2 w cos2 τ z γ4 γ4 γ

which, after rearrangements of terms, reduces to α 8 (a∇)b1 = 2 (∂z w)2 sin2 τ + w∂z2 w cos2 τ . γ Based on (A.9), (A.13), computation yields α 8 II = −4 (a∇)b1 r dx3 = −2L1 L2 ((∂z w)2 + w∂z2 w) 2 r dz. γ Q I

(A.13)

(A.14)

460

B. Scarpellini

By combining (A.7), (A.12), (A.14) we infer that under the assumption of case (A.8)(a), Gφ1 , φ2 has the form Gφ1 , φ2 = −2L1 L2 (P1 (z) + P2 (z))8 αr dz, (A.15) I

P1 (z) =

1 ((∂z w)2 + w∂z2 w), γ2

P2 (z) =

∂z w + w2 + V 2 . γ2

S2: While (A.15) gives an expression of Gφ1 , φ2 under Ass. (A.8)(a), a completely analogous computation describes Gφ1 , φ2 under Ass. (A.8)(b). Thus with v as in (A.8)(b) we have 8 dz. Gφ1 , φ2 = −2L1 L2 (P1 (z) + P2 (z))βr (A.16) I

S3: It remains to evaluate Gφ1 , φ2 under Ass. (A.8)(c). Since v = (v1 , v2 , v3 ) = 0 under Ass. (A.8)(c) and according (1.4) we have that I deﬁned via (A.7) vanishes: I = 0. By (A.7) we thus have Gφ1 , φ2 = 4[(a∇)v, b] under Ass. (A.8)(c) which also implies that the ﬁrst three components (a∇)vj , j = 1, 2, 3 of (a∇)v vanish. As to the fourth component (a∇)v4 of (a∇)v, computation based on (A.2), (A.8)(c) yields (a∇)v4 = w∂z r cos τ whence L1 L2 [(a∇)v, b] = wV∂z r dz · sin τ cos τ dx dy. (A.17) I

0

0

Since the trigonometric integral in (A.17) vanishes, we have Gφ1 , φ2 = 0 under Assumption (A.8)(c).

(A.18)

S4: It remains to compute Gφ1 , φ1 for v = (v1 , v2 , 0, v4 ) in (3.2), (3.9) arbitrary. By (3.9) and (A.3) we have Gφ1 , φ1 = −N (φ1 , v), φ1 = [(φ1 ∇)v, φ1 ] = 4[(a∇)v, a]. According to (A.2) we have that (a∇)v = (w∂z v1 cos τ, w∂z v2 cos τ, 0, w∂z v4 cos τ ). The Euclidean scalar product of (A.19) with a in (A.2) gives: w 8 z v2 ) sin τ cos τ (a∇)v · a = − 2 ∂z w(8 α∂z v1 + β∂ γ +wV + wV∂z v4 cos τ.

L1 L2 Since 0 0 sin τ cos τ dxdy = 0, we infer from (A.20): L1 L2 Gφ1 , Gφ1 = 4 (a∇)v · a dx3 = 2L1 L2 wV∂z v4 dz 0 0 I = −2L1L2 ∂z (wV)v4 dz I

(A.19)

(A.20)

(A.21)

Bifurcation of Traveling Waves

461

S5: Proof of Prop. 4.5. We invoke (A.15), (A.16), (A.18) in S1, S2, S3 respectively and recall that G in (3.9) depends linearly on v. Since v = (v1 , 0, 0, 0) + (0, v2 , 0, 0) + (0, 0, 0, v4 ) we infer from (A.15), (A.16), (A.18): 8 2 )dz Gφ1 , φ2 = −2L1 L2 (P1 (z) + P2 (z))(8 αv0 + βv

(A.22)

I

with Pj (z), j = 1, 2 given by (A.15). We claim: P1 (z) + P2 (z) ≡ 0. In fact, in case of the contrary we would have: (P1 (z) + P2 (z)) dz = 0.

(A.23)

(A.24)

I

But according to the deﬁnition of the Pj in (15) and since w(± 12 ) = 0 it holds that

P2 (z) dz > 0, P1 (z) dz = γ12 I ∂z (w∂z w) dz = 0. (A.25) I

I

Thus (A.24) cannot hold, whence (A.23) follows. Prop. 4.5 now follows by setting F (z) = c(P1 (z) + P2 (z)), c = 0 a suitable constant.

References [1] S. Chandrasekhar, Hydrodynamic and hydromagnetic stability. Dover Publ., New York 1961. [2] G. Crandall, P. Rabinowitz: The Hopf bifurcation theorem in inﬁnite dimensions. Arch. Rat. Mech. Analysis 67, no. 1, 53–72, 1977. [3] A. Friedman, Foundations of modern analysis. Dover Publ., New York, 1982 [4] T. Kato, Perturbation theory for linear operators. Springer, Berlin, New York, 1980. [5] K. Kirchg¨ assner, H. Kielh¨ ofer, Stability and bifurcation in ﬂuid dynamics. Rocky Mountains J. of Math., vol. 3, no. 2, 275–318, 1973. [6] K. Kirchg¨ assner, Bifurcation in nonlinear hydrodynamic stability. SIAM Review, vol. 17, no. 4, 452–483 (1975). [7] A. Pazy, Semigroups of linear operators and applications to PDE’s. Appl. Math. Sc. 44, Springer, New York, 1983. [8] P. Roberts, An introduction to magneto-hydrodynamics, Longmans, 1974. [9] D. Sattinger, Topics in stability and bifurcation theory. Lecture Notes in Math. 309, Springer, Berlin, New York, 1973. [10] B. Scarpellini, Equilibrium solutions of the B´enard equations with an exterior force. J. Diﬀ. & Integral Eq. 16, no. 2, 129–1958, 2003. [11] B. Scarpellini, W. von Wahl, Stability properties of the Boussinesq equations. ZAMP 49, 294–321, 1998. [12] B. Scarpellini, Stability, instability and direct integrals. Chapman & Hall/CRC, Boca Raton, 1999.

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[13] W. von Wahl, The Boussinesq equations in terms of poloidal and toroidal ﬁelds and the mean ﬂow. Bayreuther Math. Schriften 40, 203–290, 1992. [14] B. Sandstede: Stability of traveling waves. Handbook of dynamical systems, Vol. 2, (2002) 983–1055, Elsevier. [15] D. Chae, P. Dubovski: Traveling wave-like solutions of the Navier-Stokes and related equations. J. Math. Anal. Appl. 204 (1996), 930–939. B. Scarpellini Mathematisches Institut Universit¨ at Basel Rheinsprung 21 CH-4051 Basel, Switzerland

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 463–472 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Vector-valued Sobolev Spaces and Gagliardo-Nirenberg Inequalities Hans-J¨ urgen Schmeißer and Winfried Sickel Abstract. We prove Gagliardo-Nirenberg inequalities for vector-valued Lizorkin-Triebel spaces. Therefrom we derive the corresponding assertions for the vector-valued Sobolev spaces Wpm (Rn , E) . Here we do not assume the UMD property for E. Mathematics Subject Classiﬁcation (2000). 46E35. Keywords. Vector-valued Sobolev spaces, vector-valued Lizorkin-Triebel spaces, Littlewood-Paley theory, Gagliardo-Nirenberg inequalities, embeddings.

1. Introduction Vector-valued Sobolev and Besov spaces are now widely used in abstract evolution equations, cf., e.g., Amann [Am 1, Am 2] and Pr¨ uß [Pr], and in the theory of integral operators, cf., e.g., K¨onig [Ko] and Pietsch [Pi]. For certain applications it is quite inconvenient to assume that the Banach space E has the UMD property which is equivalent to the boundedness of the Hilbert transform in L2 (R, E) by Bourgain [Bo]. There are simple examples of Banach spaces which do not have this property. The UMD property implies reﬂexivity of E. So it rules out H¨ older spaces and L1 and variants of those, cf. Amann [Am 1]. On the other hand the UMD property is essential for the validity of Michlin-H¨ ormander type Fourier multiplier assertions in corresponding Lp -spaces, cf. McConnell [Mc] and Zimmermann [Zi]. So, the question is which properties of distribution spaces really depend on the UMD property and which do not. Here we are concerned with Gagliardo-Nirenberg inequalities which state that (in the scalar-valued case) |β|/m Dα f |Lp Dβ f |Lr ≤ C f |Lq 1−|β|/m |α|=m

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holds for all f ∈ Lp (Rn ) belonging to the Sobolev spaces Wpm (Rn ), provided that 1 − |β|/m |β|/m 1 = + . r q p Further, the constant C does not depend on f . To prove its vector-valued variant one has to avoid techniques which require the UMD property of the Banach space under consideration. Hence one has to avoid techniques which are based on such multipliers. A way out is the use of vector-valued spaces of Lizorkin-Triebel type, where multiplier theorems are available and their Fourier-analytic deﬁnition uses Littlewood-Paley type decompositions. This scale of spaces seems to be of interest also for themselves. However, in contrast to the scalar case they do not contain (fractional) Sobolev spaces as special cases. Nevertheless, it turns out that some assertions as the Gagliardo-Nirenberg inequalities, can be derived via embeddings. Agreement: Throughout the paper E denotes a Banach space and · E its norm without any further restriction. 1 ≤ p, q ≤ ∞,

0 < |β| < m,

and

2. Vector-valued function spaces 2.1. Preliminaries Our main tool will be Fourier analysis. There is not much information about the Fourier transform in the literature in the context of vector-valued functions. We refer to Amann [Am 1, Am 2, Am 3] and [Sc, SS] for more detailed reports. A linear mapping f : S(Rn ) → E is called a tempered distribution with values in E if there exist a constant c and nonnegative integers k and such that f (ϕ) E ≤ c sup sup (1 + |x|)k |Dα ϕ(x)| x∈Rn |α|≤

holds for all ϕ ∈ S(R ). The collection of all such mappings f is denoted by S (Rn , E). In this paper all derivatives have to be understood in the sense of distributions (either in S (Rn , E) or in S (Rn )). As usual we deﬁne the vectorvalued Fourier transform for f ∈ S (Rn , E) and ϕ ∈ S(Rn ) by n

F f (ϕ) := f (F ϕ) , where F ϕ(ξ) = (2π)−n/2

Rn

e−ixξ ϕ(x) dx ,

xξ =

n

xi ξi ,

i=1

denotes the scalar-valued Fourier transform on Rn . As in the preceding two lines we use the same symbol for both versions, hoping that the meaning will be clear from the context. Finally, let us mention that, as in the scalar case, the Fourier transform and its inverse transform are continuous one-to-one mappings of S(Rn , E) onto itself and of S (Rn , E) onto itself. We shall introduce the scale of Lizorkin-Triebel spaces by Fourier-analytic means. To this end we use smooth dyadic decompositions of unity. Let ψ be an inﬁnitely

Sobolev Spaces and Gagliardo-Nirenberg Inequalities

465

diﬀerentiable function such that 0 ≤ ψ(x) ≤ 1, ψ(x) = 1 if |x| ≤ A, and ψ(x) = 0 if |x| > B for some numbers 0 < A < B < ∞. Then we put ϕ0 (x) = ψ(x) ,

ϕ1 (x) = ϕ0 (x/2) − ϕ0 (x) ,

Obviously

∞

ϕj (x) = 1 ,

ϕj (x) = ϕ1 (2−j+1 x) , j = 2, 3, . . . . (1) x ∈ Rn .

j=0

We denote the set of all such smooth dyadic decompositions of the unity by Φ. For any f ∈ S (Rn , E) it holds f=

∞

F −1 [ϕj (ξ) F f (ξ)](x)

j=0

(convergence in S (Rn , E)), see, e.g., [Sc] and [Am 3]. In what follows we work quite often with such decompositions. For the sake of brevity we put fj (x) = F −1 [ϕj (ξ) F f (ξ)](x),

j = 0, 1, . . . ,

f ∈ S (Rn , E).

(2)

2.2. Deﬁnition and basic properties For us it is most convenient to introduce Lizorkin-Triebel classes by means of the so-called decomposition method. It has the disadvantage of some non-transparency but it is nearest to Fourier analysis. Deﬁnition 1. Let {ϕj }∞ j=0 ∈ Φ. Let −∞ < s < ∞ and 1 ≤ q ≤ ∞. s (Rn , E) is the set of all (i) Let 1 ≤ p < ∞. The Lizorkin-Triebel space Fp,q n f ∈ S (R , E) such that ⎛ ⎞1/q ∞ % & q ⎝ s n jsq −1 n Lp (R ) F f |Fp,q (R , E) = ϕj (ξ) F f (ξ) (·) E ⎠ 2 < ∞, j=0

if q < ∞ and f

s |Fp,∞ (Rn , E)

% & jsq −1 n F ϕj (ξ) F f (ξ) (·) E Lp (R ) < ∞ , = sup 2 j=0,1,...

if q = ∞. s (Rn , E) is the set of all (ii) Let 1 ≤ p ≤ ∞. The Besov-Nikol’skij space Bp,∞ n f ∈ S (R , E) such that % & s f |Bp,∞ (Rn , E) = sup 2js F −1 ϕj (ξ) F f (ξ) (·) |Lp (Rn , E) < ∞ . j=0,1,...

Remark 1. In the scalar case the above deﬁnitions go back to papers by Peetre, Nikol’skij, Lizorkin, and Triebel, for the vector-valued case we refer to [Tr 2]. Vector-valued Besov spaces have been introduced in this way in Sparr [Sp], see also Schmeisser [Sc] and Amann [Am 1, Am 2, Am 3]. The classical approach to

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Besov spaces uses moduli of continuity, diﬀerences and derivatives or interpolation of Sobolev spaces. We refer to Amann [Am 1], Grisvard [Gr], Muramatu [Mu], K¨ onig [Ko], Pietsch [Pi] and Pr¨ uß [Pr]. Proposition 1. The spaces deﬁned above are Banach spaces. They are independent of the chosen system {ϕj }∞ j=0 ∈ Φ in the sense of equivalent norms. We have the chain of continuous embeddings s s S(Rn , E) → Fp,q (Rn , E) → Bp,∞ (Rn , E) → S (Rn , E) .

Remark 2. Proposition 1 is a consequence of a Fourier multiplier assertion for entire analytic E-valued functions, cf. Triebel [Tr 2, 15.4]. Based on this tool the proof in the vector-valued situation is the same as in the scalar case, cf., e.g., [Tr 1, 2.3.2]. For the embeddings see [Tr 1, 2.3.3] and [Sc]. Remark 3. We do not distinguish between spaces equipped with equivalent norms. Deﬁnition 2. (i) Let 1 ≤ p ≤ ∞ and let m be a natural number. Then we deﬁne m n n α n Wp (R , E) = f ∈ Lp (R , E) : D f ∈ Lp (R , E) for all |α| ≤ m and put f |Wpm (Rn , E) =

(3)

Dα f |Lp (Rn , E) .

|α|≤m

(ii) Let 1 < p < ∞ and let s ∈ R. Then we deﬁne Hps (Rn , E) = f ∈ S (Rn , E) : F −1 [(1 + |ξ|2 )s/2 F f (ξ)](x) ∈ Lp (Rn , E)

(4)

and put f |Hps (Rn , E) = F −1 [(1 + |ξ|2 )s/2 F f (ξ)](x) |Lp (Rn , E) . Remark 4. (UMD spaces). It is well known (cf. McConnell [Mc], Zimmermann [Zi]), that 1 < p < ∞, Hpm (Rn , E) = Wpm (Rn , E) , if the Banach space E has the UMD property (unconditionality of martingale differences). A survey of characterizations and properties of UMD spaces can be found in Amann [Am 1] and in Pietsch and Wenzel [PW]. In general we cannot expect the above coincidence. For example, it is shown in Bukhvalov [Bu] that in the case n = 1 and if m is odd the UMD property of E is necessary. Remark 5. (Littlewood-Paley theory). In the scalar case we have (equivalent norms) s (Rn ) , Hps (Rn ) = Fp,2

1 < p < ∞.

This is based on the Littlewood-Paley-Theorem ∞ 1/2 −1 2 n L f |Lp (Rn ) ∼ | F [ϕ F f ](x) | (R ) j p , j=0

(5)

1 < p < ∞,

Sobolev Spaces and Gagliardo-Nirenberg Inequalities

467

s and the lift property of the spaces Hps (Rn ) and Fp,2 (Rn ), see, e.g., Lizorkin [Li] and Triebel [Tr 1]. The arguments in Rubio de Francia and Torrea [RT, p. 283] (due to Pisier [Pis]) show in the E-valued case ∞ 1/2 n −1 2 n F [ϕj F f ](x) E f |Lp (R , E) ∼ Lp (R ) , 1 < p < ∞ , j=0

0 (Rn , E)) if and only if E can be renormed as (or equivalently Lp (R , E) = Fp,2 a Hilbert space. This extends to the E-valued version of (5) by the lift property. m (Rn , E) = Wpm (Rn , E) , 1 < p < ∞, if and only if E is Analogously, it holds Fp,2 Hilbertian. n

2.3. Some embeddings Here we collect a few embedding relations between these diﬀerent types of spaces. Proposition 2. (i) Suppose 1 < p < ∞ and m ∈ N0 . Then we have the chain of continuous embeddings m m (Rn , E) → Wpm (Rn , E) → Fp,∞ (Rn , E) . Fp,1

(6)

(ii) Suppose 1 < p < ∞ and s ∈ R. Then we have the continuous embeddings s s (Rn , E) → Hps (Rn , E) → Fp,∞ (Rn , E) . Fp,1

(7)

(iii) Suppose m ∈ N0 . Then we have m m F1,1 (Rn , E) → W1m (Rn , E) → B1,∞ (Rn , E) .

(8)

(iv) Suppose m ∈ N0 . Then we have m m (Rn , E) → B∞,∞ (Rn , E) . W∞

(9)

Proof. Observe that in [Tr 2, Sect. 15] one can ﬁnd the tools needed to reduce the proof of all these embeddings to the case m = 0 (or s = 0) as in the scalar case. For the latter one see, e.g., [Tr 1, 2.5]. Let 1 ≤ p < ∞. Then ∞ f |Lp (Rn , E) ≤ fj E Lp (Rn ) j=0

is a consequence of the triangle inequality. This proves the ﬁrst embeddings in (6) and (8). It follows from the construction of our smooth decomposition of unity that fj (x)E ≤ 2 sup |F −1 [ϕ0 (2−j ξ)](z)| f (x − z)E dz . j=0,1,...

The right-hand side can be estimated by c (M f ( · )E )(x), see Stein [St, p. 56/57]. Applying the Hardy-Littlewood maximal inequality we get 0 (Rn , E) ≤ c M f ( · )E (x) Lp (Rn ) f |Fp,∞

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H.-J. Schmeißer and W. Sickel

for all f ∈ Lp (Rn , E), where 1 < p < ∞. If p = 1 or if p = ∞ we argue that fj |Lp (Rn , E) ≤ c |F −1 ϕj (z)| f (x − z) E dz Lp (Rn ) ≤

c F −1 ϕ0 |L1 (Rn ) f |Lp (Rn , E) .

This proves 0 f |Bp,∞ (Rn , E) ≤ c f |Lp (Rn , E) .

The proof is complete.

3. Gagliardo-Nirenberg inequalities The aim of this section consists in determining upper bounds for the Lp (Rn , E)norms of intermediate derivatives Dβ f, |β| < m in terms of the Lp (Rn , E)-norms of f and its derivatives Dα f, |α| = m. Such interpolation inequalities have been obtained by many authors, e.g., Ehrling, Il’yin, Nirenberg, Browder, and Gagliardo. A proof which does not make use of Fourier analytical techniques may be found in Adams [Ad, IV] (there also exact references are given, but see also Besov, Kudryavtsev, Lizorkin and Nikol’skij [BKLN] for this). Recently, Brezis and Mironescu [BM] published a version based on Littlewood-Paley theory. Surprisingly these inequalities are very simple to prove (even in the vector-valued situation) for the most complicated spaces, namely the Lizorkin-Triebel classes. 3.1. Gagliardo-Nirenberg inequalities for Lizorkin-Triebel classes We follow Brezis and Mironescu [BM]. Proposition 3. Let 1 ≤ q, q0 , q1 ≤ ∞, −∞ < s0 < s1 < ∞, and 0 < Θ < 1. (i) Let 1 ≤ p0 , p1 < ∞. We put 1−Θ 1 Θ = + p p0 p1

and

s = (1 − Θ) s0 + Θ s1 .

Then there exists a positive constant C such that s (Rn , E) ≤ C f |Fps00,q0 (Rn , E)1−Θ f |Fps11,q1 (Rn , E)Θ f |Fp,q

(10)

holds for all f ∈ S (Rn , E). (ii) Let 1 ≤ p0 < ∞. We put 1−Θ 1 = p p0

and

s = (1 − Θ) s0 + Θ s1 .

Then there exists a positive constant C such that s s1 f |Fp,q (Rn , E) ≤ C f |Fps00,q0 (Rn , E)1−Θ f |B∞,∞ (Rn , E)Θ

holds for all f ∈ S (Rn , E).

(11)

Sobolev Spaces and Gagliardo-Nirenberg Inequalities

469

Proof. The proof is based on the following elementary inequality, cf. [BM]. For any sequence {a } of complex numbers it holds 2sj aj |q ≤ C 2s0 j aj |∞ 1−Θ 2s1 j aj |∞ Θ , where s = (1 − Θ) s0 + Θ s1 and 1 ≤ q ≤ ∞. This can be applied with aj = older’s inequality yields fj (x) E . Then H¨ s f |Fp,q (Rn , E) ≤ C f |Fps00,∞ (Rn , E)1−Θ f |Fps11,∞ (Rn , E)Θ ,

as long as p1 < ∞. In the case p1 = ∞ we obtain f

s |Fp,q (Rn , E)

≤ C f

|Fps00,∞ (Rn , E)1−Θ

Θ

sup

sup

x∈Rn

j=0,1,...

2

s1 j

fj (x) E

.

The monotonicity with respect to the third parameter complements the proof. Remark 6. The proof given above is due to Brezis and Mironescu [BM] for E = C. The surprising fact is the independence of the microscopic parameters q, q0 , q1 . 3.2. The vector-valued version of the Gagliardo-Nirenberg inequalities As a consequence of the preceding proposition and Proposition 2 we now obtain the Gagliardo-Nirenberg inequalities in the vector-valued setting. Theorem 1. Let m ∈ N, 1 < r ≤ ∞, 1 < p, q ≤ ∞, and 0 < |β| < m. For Θ = |β|/m

1 1−Θ Θ = + r q p

and

there exists a constant c such that D f |Lr (R , E) ≤ c f |Lq (R , E) β

n

n

1−Θ

Θ D f |Lp (R , E) α

n

(12)

|α|=m

holds for all f ∈ Wpm (Rn , E) ∩ Lq (Rn , E). Proof. We choose s0 = 0, s1 = m and Θ = |β|/m. Step 1. We assume 1 < p, q < ∞. Applying Proposition 3 with q = 1 and q0 = q1 = ∞ it follows |β|

0 m f |Fr,1 (Rn , E) ≤ C f |Fq,∞ (Rn , E)1−Θ f |Fp,∞ (Rn , E)Θ

Now we use Proposition 2 and obtain f |Wr|β| (Rn , E) ≤ c f |Lq (Rn , E)1−Θ f |Wpm (Rn , E)Θ .

(13)

To achieve at (12) we replace the complete Sobolev norm on the left-hand side by Dβ f |Lp (Rn , E) and consider the family f (λx), λ > 0. Then (13) yields λ|β|−n/r Dβ f |Lr (Rn , E) ≤ c λ−n(1−Θ)/q f |Lq (Rn , E)1−Θ

|α|≤m

λ|α|−n/p Dα f |Lp (Rn , E)

Θ

(14)

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H.-J. Schmeißer and W. Sickel

with some c independent of λ. Dividing both sides by λ|β|−n/r we end up with Θ Dβ f |Lr (Rn , E) ≤ c f |Lq (Rn , E)1−Θ λ|α|−m Dα f |Lp (Rn , E) . |α|≤m

For λ → ∞ the desired result follows. Step 2. We consider p = ∞ and 1 < q < ∞. We proceed as above. It follows from (11) that |β|

0 m (Rn , E)1−Θ f |B∞,∞ (Rn , E)Θ f |Fr,1 (Rn , E) ≤ C f |Fq,∞

Now we use Proposition 2 and obtain m (Rn , E)Θ . f |Wr|β| (Rn , E) ≤ c f |Lq (Rn , E)1−Θ f |W∞

The rest is the same as in Step 1. Step 3. Let q = ∞ and 1 < p < ∞. It follows |β|

0 m (Rn , E)1−Θ f |Fp,∞ (Rn , E)Θ f |Fr,1 (Rn , E) ≤ C f |B∞,∞

Now we use Proposition 2 and obtain f |Wr|β| (Rn , E) ≤ c f |L∞ (Rn , E)1−Θ f |Wpm (Rn , E)Θ . The rest is the same as in Step 1. Step 4. It remains to consider the case r = p = q = ∞. Let m ∈ N and 0 < Θ < 1. In [SS] we proved the following continuous embedding for the real interpolation of Nikol’skij-Besov spaces 0 m mΘ (Rn , E), B∞,∞ (Rn , E) → W∞ (Rn , E) . B∞,∞ Θ,1

By standard properties of the real interpolation method it follows mΘ 0 m (Rn , E) ≤ c f |B∞,∞ (Rn , E)1−Θ f |B∞,∞ (Rn , E)Θ f |W∞

with some constant c independent of f . Applying Proposition 2 and a scaling argument as above we have ﬁnished. Remark 7. In case p = 1 and/or q = 1 our knowledge is less complete. Only in case p = q = 1 we have a deﬁnite answer. Observe, if m ∈ N and 0 < Θ < 1 then 0 m mΘ B1,∞ (Rn , E), B1,∞ (Rn , E) → F1,1 (Rn , E) , Θ,1

cf. [SS]. The same type of arguments as used in Step 4 of the proof of the preceding theorem leads to the inequality Θ Dβ f |L1 (Rn , E) ≤ c f |L1 (Rn , E)1−Θ Dα f |L1 (Rn , E) , |α|=m

where 0 < Θ < 1, |β| = mΘ, and f ∈

W1m (Rn , E).

By using simply the sandwich argument, cf. Proposition 2(ii), one obtains the following result with respect to the classes Hps (Rn , E).

Sobolev Spaces and Gagliardo-Nirenberg Inequalities

471

Theorem 2. Let −∞ < s0 < s1 < ∞, and 0 < Θ < 1. (i) Let 1 < p0 , p1 < ∞. We put 1−Θ Θ 1 = + and s = (1 − Θ) s0 + Θ s1 . p p0 p1 Then there exists a positive constant c such that f |Hps (Rn , E) ≤ c f |Hps00 (Rn , E)1−Θ f |Hps11 (Rn , E)Θ

(15)

holds for all f ∈ S (R , E). (ii) Let 1 < p0 < ∞. We put n

1 1−Θ = and s = (1 − Θ) s0 + Θ s1 . p p0 Then there exists a positive constant c such that s1 f |Hps (Rn , E) ≤ c f |Hps00 (Rn , E)1−Θ f |F∞,∞ (Rn , E)Θ

(16)

holds for all f ∈ S (Rn , E). Remark 8. Based on the above strategy one can deal with the Sobolev embeddings and its reﬁned versions, the Jawerth-Franke embeddings (comparing Lizorkin-Triebel classes with Besov classes). We refer to [SS] for details. Let us mention in this context also the paper of Pelczynski and Wojciechowski [PeWo], where this problem is treated for W11 (Rn , E). Moreover, the sandwich strategy to prove a result for the more complicated Lizorkin-Triebel spaces and afterwards to use the independence of the microlocal parameter q to deduce results for the Sobolev spaces is also applicable for the trace problem. We again refer to [SS].

References [Ad] R.A. Adams, Sobolev spaces, Academic Press, New York, 1975. [Am 1] H. Amann, Linear and quasilinear parabolic problems, Vol. 1, Birkh¨ auser, Basel, 1995. [Am 2] H. Amann, Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications, Math. Nachr. 186 (1997), 5–56. [Am 3] H. Amann, Vector-valued distributions and Fourier multipliers, Preprint, Z¨ urich, 1997. [BKLN] O.V. Besov, L.D. Kudrayavtsev, P.I. Lizorkin and S.M. Nikol’skij, Investigations in the theory of spaces of diﬀerentiable functions of several variables, Proc. Steklov Inst. 182 (1990), 73–139. [Bo] J. Bourgain, Some remarks on Banach spaces in which martingale diﬀerence sequences are unconditional, Ark. Mat. 21 (1983), 163–168. [BM] H. Brezis and P. Mironescu, Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces, J. Evol. Equ. 1 (2001), 387–404. [Bu] A.V. Bukhvalov, Sobolev spaces of vector-valued functions, J. Math. Sciences 71 (1994), 2173–2179.

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[Gr]

P. Grisvard, Commutativit´e de deux foncteurs d’interpolation at applications, J. Math. Pures Appl. 45 (1966), 143–290. [Ko] H. K¨ onig, Eigenvalue distribution of compact operators, Birkh¨ auser, Basel, 1986. [Li] P.I. Lizorkin, On multipliers of Fourier integrals in the spaces LpΘ , Trudy Math. Inst. Steklov 89 (1967), 231–248. [Mc] T.R. McConnell, On Fourier multiplier transforms of Banach-valued functions, Trans. Amer. Math. Soc. 285 (1984), 739–757. [Mu] T. Muramatu, Besov spaces and Sobolev spaces of generalized functions deﬁned on a general region, Publ. R.I.M.S. Kyoto Univ. 9 (1974), 325–396. [Pe] J. Peetre, New thoughts on Besov spaces, Duke Univ. Press, Durham, 1976. [PeWo] A. Pelczynski and M. Wojciechowski, Molecular decompositions and embedding theorems for vector-valued Sobolev spaces with gradient norm, Studia Math 107 (1993), 61–100. [Pi] A. Pietsch, Eigenvalues and s-numbers. Cambridge Univ. Press, Cambridge, 1987. [PW] A. Pietsch and J. Wenzel, Orthonormal systems and Banach space geometry. Cambridge Univ. Press, Cambridge, 1998. [Pis] G. Pisier, Type des espaces norm´es. Seminaire Maurey-Schwartz 1973–1974, Ecole Polytechnique, Palaiseau. [Pr] J. Pr¨ uss, Evolutionary integral equations and applications, Birkh¨ auser, Basel, 1993. [RT] J.L. Rubio de Francia and J.L. Torrea, Some Banach techniques in vector-valued Fourier analysis, Coll. Math. 54 (1987), 273–284. [Sc] H.-J. Schmeisser, Vector-valued Sobolev and Besov spaces, In: Seminar Analysis of the Karl-Weierstraß-Institute, 4-44, Teubner-Texte Math. 96, Teubner, Leipzig, 1987. [SS] H.-J. Schmeißer and W. Sickel, Traces, Gagliardo-Nirenberg inequalities and Sobolev type embeddings for vector-valued function spaces, Jenaer Schriften zur Mathematik und Informatik Math/Inf/24/01, Jena, 2001. [Sp] G. Sparr, Interpolation of several Banach spaces, Ann. Mat. Pura Appl. 99 (1974), 247–316. [St] E.M. Stein, Harmonic analysis. Real-variable methods, orthogonality, and oscillatory integrals. Princeton Univ. Press, Princeton, 1993. [Tr 1] H. Triebel, Theory of function spaces, Birkh¨ auser, Basel, 1983. [Tr 2] H. Triebel, Fractals and spectra related to Fourier analysis and function spaces, Birkh¨ auser, Basel, 1997. [Zi] F. Zimmermann, On vector-valued Fourier multiplier theorems, Stud. Math. 93 (1989), 201–222. Hans-J¨ urgen Schmeißer and Winfried Sickel Mathematisches Institut Friedrich-Schiller-Universit¨ at Jena Ernst-Abbe-Platz 1–2 D-07743 Jena, Germany

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 473–495 c 2005 Birkh¨ auser Verlag Basel/Switzerland

The Inﬂuence of Gradient Perturbations on Blow-up Asymptotics in Semilinear Parabolic Problems: A Survey Philippe Souplet Dedicated to Professor Herbert Amann

Abstract. In this paper, we give a survey of results concerning semilinear parabolic problems of the form ut − ∆u = up + g(x, t, u, ∇u). Our goal is to examine the eﬀect of the (gradient) perturbation term g on the asymptotic behavior of blow-up solutions. It turns out that, if the perturbation becomes critical or supercritical in a scaling sense, then the blow-up rate as well as the blow-up proﬁles may become notably diﬀerent from those known in the unperturbed case. In some cases, we give precise asymptotic estimates on blow-up solutions. Mathematics Subject Classiﬁcation (2000). 35K55, 35B40, 35B33.

1. Introduction Consider the semilinear heat ⎧ ⎨ut − ∆u u ⎩ u(x, 0)

equation = |u|p−1 u, = 0, = u0 (x),

x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0, x ∈ Ω,

(1.1)

where p > 1, Ω is a regular domain of RN and u0 ∈ C0 (Ω). It is well known that for large initial data (in a suitable sense), solutions of (1.1) do not exist globally. Namely, they blow up in ﬁnite time T = T (u0 ) in sup norm: lim u(t)∞ = ∞.

t→T

The description of the asymptotic behavior of blowing-up solutions of (1.1) has attracted considerable interest in the past two decades (see, e.g., [24, 14] for recent

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surveys). Main elements in the description are the blow-up rate, set, proﬁles and the possible continuation after blow-up. First of all, it is well known that any blow-up solution of (1.1) satisﬁes the lower blow-up rate estimate u(t)∞ ≥ κ(T − t)−1/(p−1) ,

0 ≤ t < T,

κ = (p − 1)−1/(p−1) .

(1.2)

On the other hand, under certain assumptions, the upper blow-up rate estimate u(t)∞ ≤ C(T − t)−1/(p−1) ,

0 < t < T,

(1.3)

is also satisﬁed for some C > 0. In this case, the order of growth is the same as for the related ODE y = y p , and (1.3) is referred to as type I blow-up. This holds for instance if u, ut ≥ 0, with Ω bounded [20], or if N +2 if N ≥ 3 p < pS := N −2 ∞ if N ≤ 2, and Ω = RN or Ω is convex bounded [27, 29, 30]. Actually, for radial positive solutions (when Ω is the ball BR or Ω = RN ), the critical exponent for the blowup rate is known to be if N ≥ 11 1 + N −4−24√N −1 p∗ := ∞ if N ≤ 10. More precisely, if p < p∗ and u0 ≥ 0 is radial (and satisﬁes some additional assumptions when Ω = RN ), then (1.3) is satisﬁed [39], whereas if p > p∗ and Ω = RN , then there are examples of radial positive solutions such that lim (T − t)1/(p−1) u(t)∞ = ∞

(1.4)

t→T

(see [32]). Property (1.4) is referred to as fast or type II blow-up. On the other hand, for radial sign-changing solutions, formal arguments suggest that type II blow-up can occur when p = pS [18]. In the nonradial case, the question is still open for p ≥ pS . Finally, let us mention that less is known about the blow-up rate for the Neumann problem (see [42] and the references therein). As for the blow-up set and proﬁles, let us mention a few of the many known results. First, if Ω is convex bounded and u0 ≥ 0, then the blow-up set is a compact subset of Ω, i.e., u remains bounded near ∂Ω [20]. Next, if Ω = BR and u0 ≥ 0 is radial, with u0,r ≤ 0, r = |x|, then u blows up only at the origin and satisﬁes u(x, t) ≤ Cε |x|−2/(p−1)−ε ,

0 < |x| < R,

0 < t < T,

(1.5)

for each ε > 0 (see [20] and cf. also [43, 21]). Actually, if in addition p < pS , then the ﬁnal space proﬁle satisﬁes the precise estimate 1/(p−1) , |x| → 0 (1.6) u(x, T ) ∼ cp |x|−2/(p−1) log |x| (see [56, 40]). On the other hand, the asymptotic behavior can be described by a diﬀerent notion of proﬁle, based on self-similar variables. Namely, if p < pS , and

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475

Ω = RN or Ω is convex bounded, then for any blow-up point a ∈ Ω, it holds √ lim (T − t)1/(p−1) u(a + y T − t, t) = ± κ, (1.7) t→T

uniformly for y bounded (see [26–30] and cf. also [21]). Finally, it is known [3] that if, e.g., p < pS or ut ≥ 0, then solutions cannot be continued after t = T , even in a weak sense. On the contrary, for pS < p < p∗ , there are examples of blow-up solutions which can be continued and become immediately regular again after T [15]. We refer to [14, 47] for more details. The aim of this paper is to examine the eﬀect of a perturbation, especially by a gradient term, on the asymptotic behavior of blow-up solutions. For simplicity, we shall restrict our attention to positive solutions. Thus we shall consider perturbed problems of the form ⎧ p x ∈ Ω, t > 0, ⎨ut − ∆u = u + g(x, t, u, ∇u), u = 0, x ∈ ∂Ω, t > 0, (1.8) ⎩ u(x, 0) = u0 (x) ≥ 0, x ∈ Ω. It will turn out that whether or not estimates such as (1.3), (1.5)–(1.7) are preserved for problem (1.8) strongly depends on the size of the perturbation term. Unless otherwise stated, we shall assume that g is a C 1 function, with g(x, t, 0, 0) ≥ 0, and that u0 ∈ X, where X := 0 ≤ w ∈ C 1 (Ω) : w, ∇w ∈ L∞ (Ω), w = 0 on ∂Ω , wX := w∞ + ∇w∞ . Problem (1.8) then admits a unique, maximal solution u ≥ 0, classical for t > 0 (see, e.g., [48] and the references therein). Its maximal existence time will be denoted by T = T (u0 ). Also, when Ω = BR or RN , a solution will be said to be radial (or symmetric when N = 1) if u(x,t) = u(|x|,t), and we shall always denote r = |x|. In some respects, the present article constitutes a sequel to our survey paper [49], incorporating some more recent developments and focusing on the special question of blow-up asymptotics. The outline of the rest of the paper is as follows. In Section 2, we introduce a notion of sub-/supercritical perturbation, based on scaling considerations, which will turn out to be useful for a heuristic interpretation of many results. In Section 3, we present results which show that blow-up remains type I if the perturbation satisﬁes a suitable subcriticality condition. A closely related property, concerning bounds of global solutions, will be also discussed. In Section 4, we report on some recent results that go in the opposite direction: for certain (critical) perturbations, even in one space dimension, fast or type II blow-up occurs. Then in Section 5, we present various results concerning blowup proﬁles in space. In particular, the blow-up proﬁle estimate (1.5) ceases to be true if the perturbation becomes supercritical and we give precise estimates of the diﬀerent proﬁles that occur. In Section 6, we treat similar questions concerning blow-up proﬁles in self-similar variables. Finally in Section 7, we brieﬂy discuss the questions of blow-up sets and continuation after blow-up. In each section, we have

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intended to present proofs for some of the main results (or at least the principal ideas thereof).

2. Sub-/supercritical perturbations and examples It is a well-known fact that scaling and self-similarity play an important role in the analysis of nonlinear PDE’s. For our problem (1.8), the optimal or critical conditions that will appear below will be often interpretable heuristically via scaling considerations. Namely, assume for simplicity that g = g(u, ∇u) and consider the dilations uα (x, t) = α−1 u(α(1−p)/2 x, α1−p t),

gα (z, ξ) = α−p g(αz, α(p+1)/2 ξ),

α > 0.

Observe that for any solution of (1.8)1 , the function uα solves the same equation with g replaced by gα . Let us agree to say roughly that the perturbation g is subcritical (resp., critical or supercritical), if g(αz, α(p+1)/2 ξ) = o(αp ) (resp., ∼ αp or ) αp ),

as α → ∞

(for ﬁxed z, ξ, say). Then it is reasonable to expect that for subcritical perturbations, asymptotic properties of blow-up solutions to (1.8) should be close to those of the unperturbed equation (1.1). Likewise, one can expect that the behavior might be diﬀerent in some respect for supercritical (or even critical) perturbations. Although certain open questions remain, such expectations will turn out to be true. Let us give some typical examples of problem (1.8) which have been studied in the literature. In the following equation: ut − ∆u = up − µ|∇u|q ,

(2.1)

with homogeneous Dirichlet boundary conditions, where p, q > 1, µ > 0, the gradient term has a dissipative eﬀect which may prevent blow-up. Equation (2.1) has been ﬁrst considered in [9] and has next been studied by several authors (see [49] for a survey and list of references). As a second example, let us mention the equation (2.2) ut − ∆u = up + a · ∇(uq ), N where p > 1, q ≥ 1, 0 = a ∈ R , which contains a convective gradient term. For results on blow-up for equation (2.2), we refer to, e.g., [1, 38]. Note that equation (2.1) (resp., (2.2)) is (sub-, super-) critical in the above sense if q (<, >) = 2p/(p + 1) (resp., (p + 1)/2). It is known that, in a bounded domain for instance, ﬁnite time blow-up for (2.1) with µ > 0 may happen if and only if p > q, and that it occurs in L∞ norm (see [54, 13] and cf. also the references in [49]). The same is true for (2.2). Observe that the critical blow-up exponent q = p for (2.1) does not coincide with the scaling exponent q = 2p/(p + 1) of the perturbation, but is rather related to its “size” as compared with that of the reaction term. The situation will be diﬀerent for blow-up asymptotics.

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As a generalization of (2.1), one may also consider the equation ut − ∆u = up − λus |∇u|q ,

(2.3)

with q > 1 and s, λ ∈ R. In the case s < 0, the homogeneous boundary conditions will be replaced by u = b > 0 on ∂Ω and initial conditions will be assumed to verify u0 (x) ≥ b in Ω (and u0 (x) = b on ∂Ω). Note that these assumptions guarantee u ≥ b for t > 0, so that the gradient term is nonsingular. (Although the boundary conditions are diﬀerent from those in (1.8), this change is unessential: (2.3) can be recast in the form (1.8) by setting v = u−b and taking g(x, t, v, ∇v) = (v + b)p − v p − λ(v + b)s |∇v|q .) If λ < 0, then blow-up for (2.3) is of course always possible, but when, e.g., q > max(2, p) and s = 0, gradient blow-up, instead of L∞ blow-up, may occur (see [50]), and this is out of the scope of the present paper. Let us thus assume λ > 0. If p > q + s, then we have T < ∞ for large initial data (this follows from [54, Theorem 1] if s > 0 and from simple modiﬁcations of its proof if s < 0). Moreover, blow-up has to occur in sup norm whenever q ≤ 2 [36] or s ≥ 0 [5]. Finally, if p ≤ q + s then solutions cannot blow up, at least in L∞ (this follows from the proofs in [13, 4]). We shall devote particular attention to the following special, one-dimensional case of (2.3): ⎧ |u |2 ⎪ ⎪ut − uxx = up − λ xm , −1 < x < 1, t > 0, ⎨ u (2.4) u(±1, t) = b, t > 0, ⎪ ⎪ ⎩ −1 < x < 1, u(x, 0) = u0 (x) ≥ b, with m, λ, b > 0, on which several interesting phenomena can be observed. Let us note that (2.3) is (sub-, super-) critical in the above sense for s + q(p + 1)/2 (<, > ) = p. In particular for (2.4), the conditions become just m (>, <) = 1.

3. Type I blow-up rates In this section, we shall see that if the perturbation term in (1.8) satisﬁes a suitable subcriticality condition and p is not too large, then blow-up is type I. A property closely related to type I blow-up is the boundedness of global solutions, i.e.: sup u(t)∞ ≤ C(u0 ) < ∞, t≥0

whenever T (u0 ) = ∞.

(3.1)

This property is true for (1.1) when p < pS , actually in the stronger version (usually called a priori bound) where C(u0 ) = C(u0 ∞ ) (see [25, 46] and the references therein). Likewise, it will be preserved under subcritical perturbation. Although the two properties concern diﬀerent classes of solutions, there are amenable to a uniﬁed approach, based on scaling and Fujita- or Liouville-type theorems (see Lemma 3.1) and on a doubling property (see Lemma 3.2); they will

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be given a single proof. Note that in the classical proofs of (1.3) or (3.1) for problem (1.1) with p < pS , essential use is made of the energy functional E associated to the problem. Here (at the expense of further restriction on p, though), we do not use any energy functional (and indeed, problem (1.8) is not known to possess a variational structure in general). The idea that we use was ﬁrst introduced in [33] to establish blow-up rates for problems with nonlinear boundary conditions, using Fujita theorems. It was adapted in [10, 16] to other problems, e.g., equations of the form (1.8) or parabolic systems, and it was modiﬁed in [48] to prove a priori bounds for global solutions. Next, in [41], parabolic Liouville theorems were used instead of Fujita results, allowing to increase the range of p, and a uniﬁed proof, covering blow-up rates and a priori bounds at the same time, was given. The present result is an extension of [41, Theorem 3.2]. Theorem 3.1.Let p > 1 satisfy p < pB := N (N + 2)/(N − 1)2 p ≤ (N + 3)/(N + 1)

if Ω = RN , otherwise.

(3.2)

Assume that g satisﬁes the growth assumption |g(x, t, u, ξ)| ≤ C0 (1 + |u|p1 + |ξ|q ), for some 1 ≤ p1 0 such that that all solutions u of (1.8) verify the following. (i) if T < ∞, then

1 ≤ C 1 + (T − t)− p−1 , u(t)∞ + ∇u(t)2/(p+1) ∞

0 < t < T;

(3.4)

(ii) if T = ∞, then u(t)∞ + ∇u(t)∞ ≤ C,

t ≥ 0.

(3.5)

Remarks 3.1. (i) Theorem 3.1 applies for instance to (2.1) and (2.2) in the subcritical case (q < 2p/(p + 1) or q < (p + 1)/2, respectively). Although condition (3.3) is relatively general, it is however not satisﬁed by all subcritical perturbations in the sense deﬁned in Section 2. In particular, it does not hold for (2.3) when q ≥ 2p/(p + 1), although the perturbation may be subcritical for certain s < 0. For a type I blow-up result under a diﬀerent subcriticality condition, see Theorem 4.2 below. (ii) If we assume g(x, t, u, 0) ≤ 0, then any blow-up solution of (1.8) satisﬁes the lower estimate (1.2). This follows immediately from the proof of [20, Theorem 4.5]. (iii) In some cases, the assumption (3.2) on p can be weakened. For instance, if Ω = RN or BR , and u is radial, then the conclusions of Theorem 3.1 are true for all p < pS .

Inﬂuence of Gradient Perturbations on Blow-up Asymptotics

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(iv) If we allow the constants in (3.4), (3.5) to depend on the solution u, then Theorem 3.1 remains true for all p < pB if Ω is convex and bounded (in case T = ∞, we thus have boundedness of global solutions). (v) In the case of equation (2.1) with Ω convex and bounded, it was proved in [11] that blow-up is type I for all p > 1 and 1 < q < 2p/(p + 1), under the assumption ut ≥ 0. The proof is based on a modiﬁcation of maximum principle arguments from [20]. Let us turn to the proof of Theorem 3.1. For a given solution u of (1.8), let us denote , t ∈ [0, T (u0)). Mu (t) := sup u(s)∞ + ∇u(s)2/(p+1) ∞ s∈[0,t]

Clearly, Mu is a nonnegative, continuous and nondecreasing function on [0, T (u0)). For any given t ∈ [0, T (u0 )) we can deﬁne t+ u by + tu = sup s ∈ [t, T (u0 )); Mu (s) ≤ 2Mu (t) ≤ T (u0 ) ≤ ∞. We shall ﬁrst prove the following Lemma. Lemma 3.1. Assume (3.2), (3.3) and let K > 0. There exists CK > 0 such that for all solutions of (1.8) with u0 X ≤ K and all t ∈ [0, T (u0 )), it holds either or

Mu (t) ≤ CK ,

(3.6)

p−1 (t+ (t) ≤ CK . u − t)Mu

(3.7)

Proof. Suppose that the Lemma is false. Then there exists a sequence (uj ) of solutions and tj ∈ [0, Tj ), with Tj := T (uj,0 ), such that for all j, uj,0 X ≤ K,

(3.8)

Mj ≥ j, (t+ j

−

tj )Mjp−1

(3.9)

≥ j,

(3.10)

+ where Mj := Muj (tj ) and t+ j := tj,uj . By the well-posedness of (1.8) in X (see, e.g., [48]), (3.8) implies the existence of δ > 0 such that, for all j,

uj (t)X ≤ 2K,

Tj > δ,

0 ≤ t ≤ δ.

For each j, choose tˆj ∈ (0, tj ] and xj ∈ Ω such that 1 2 Mj

≤ u(xj , tˆj ) + |∇u(xj , tˆj )|2/(p+1) ≤ Mj .

(3.11)

Then we have tˆj ≥ δ Put λj :=

−(p−1)/2 Mj .

for large j.

(3.12)

By (3.9) and (3.10), we have + λ−2 j (tj

λj → 0 ˆ − tj ) → ∞

as j → ∞,

(3.13)

as j → ∞.

(3.14)

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Next we rescale the solution u about (xj , tˆj ) with the scaling factor λj as follows: 2/(p−1) uj (λj y + xj , λ2j s + tˆj ), (y, s) ∈ Dj ϕj (y, s) := λj N ˆ −2 ˆ where Dj := Ωj × [−λ−2 j tj , λj (Tj − tj )), Ωj = {y ∈ R ; λj y + xj ∈ Ω}. Then ϕj is a solution of the equation ϕs = ∆ϕ + ϕp + gj

in Dj

with ϕj (0, 0) + |∇ϕj (0, 0)| ≥ 1/2, where 2p/(p−1) −2/(p−1) −(p+1)/(p−1) gj (y, s) := λj g λj y+xj , λ2j s+ tˆj , λj ϕj (y, s), λj ∇ϕj (y, s) . 2/(p+1)

As a consequence of the deﬁnition of t+ j and (3.12), we have j := Ωj × (−δλ−2 , λ−2 (t+ − tˆj )). in D j j j ε p1 The growth assumption (3.3) implies that |gj | ≤ Cλj 1 + |ϕj | + |∇ϕj |q , where 2p 1 ) p+1 ε := min 2(p−p > 0. Consequently, p−1 , p−1 p+1 − q 0 ≤ ϕj + |∇ϕj |2/(p+1) ≤ 2,

|gj | ≤ Cλεj

j. in D

(3.15)

Now, we distinguish the cases (i) dj /λj → ∞ and (ii) dj /λj → c ≥ 0, where dj = dist(xj , ∂Ω). Using parabolic estimates similarly as in [25] (see also [48]), we obtain a subsequence converging to a (bounded) solution of either ϕs = ∆ϕ + ϕp , or

ϕs = ϕ =

∆ϕ + ϕp , 0,

y ∈ RN ,

s ∈ R,

y ∈ Hc , s ∈ R, y ∈ ∂Hc , s ∈ R,

(3.16)

(3.17)

for some c ∈ R, where Hc = {y ∈ RN ; yN > c}, and such that ϕ(0, 0) + |∇ϕ(0, 0)|2/(p+1) ≥ 1/2. But, as a consequence of [7, Th´eor`eme 0.1], (3.16) does not admit any nontrivial solution if p < pB . As for (3.17), it does not admit any nontrivial solution (even just for s > 0) whenever p ≤ (N + 3)/(N + 1) (see, e.g., [37]). The Lemma follows. Since Mu (t) is a nondecreasing function, then either: (i) (3.6) holds on the whole [0, T (u0 )), hence in particular T (u0 ) = ∞, or: (ii) there exists τ ∈ [0, T (u0 )) such that we have (3.6) on [0, τ ] and (3.7) on [τ, T (u0 )). Theorem 3.1 is then an immediate consequence of the following Lemma, which is essentially contained in [33]. Lemma 3.2. Assume that (ii) above occurs. Then T = T (u0 ) < ∞ and 1 u(t)∞ ≤ C1 1 + (T − t)− p−1 , 0 < t < T, where C1 = C1 (CK ) > 0.

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Proof. Fix t ∈ [τ, T ) and deﬁne an increasing sequence by t0 := t and ti+1 := t+ i,u , i = 0, 1, 2, . . . Then i Mi := Mu (ti ) = 2i M (t) and ti+1 − ti ≤ CK Mi1−p = 21−p CK M 1−p (t). (3.18) It follows that ti converges to a limit t∞ < ∞, necessarily equal to T , since Mu (ti ) → ∞. Formula (3.18) then implies that T − t ≤ C0 M 1−p (t),

τ ≤ t < T,

∞ 1−p i where C0 := CK . Since u(t)∞ ≤ CK on the remaining interval [0, τ ], 2 i=0

the Lemma follows.

Remarks 3.2. (i) For p < pS , (3.16) does not admit any bounded, positive radial solution (see [58]). Suitable modiﬁcations of the proof of Lemma 3.1 then yield the result mentioned in Remark 3.1 (iii). It seems to be open whether problem (3.16) can possess positive (bounded) solutions if pB ≤ p < pS . The analogous question for (3.17) is also open for p > (N + 3)/(N + 1). For a given p, negative answers to those questions would imply the conclusions of Theorem 3.1 for any domain Ω. Let us note that even for equation (1.1), type I blow-up is still an open question for p > (N + 3)/(N + 1) if Ω is nonconvex (and u does not satisfy u, ut ≥ 0). (ii) The result mentioned in Remark 3.1 (iv) follows from the proof of Lemma 3.1 and the fact that case (ii) in that proof does not occur, due to a moving plane argument from [11, Theorem 3.1].

4. Fast blow-up rates In this section, we shall show that for certain critical perturbations, namely problem (2.4) with m = 1, blow-up may become type II. Theorem 4.1. [31] Let m = 1 and λ > p. Assume that u0 is symmetric, with u0,r ≤ 0, and let the solution u of (2.4) satisfy T < ∞. Then lim (T − t)1/(p−1) u(t)∞ = ∞.

t→T

(4.1)

It was moreover proved in [31] that the assumption on λ in Theorem 4.1 is optimal: if m = 1 and 0 < λ ≤ p, then blow-up is type I (at least if ut ≥ 0). This shows a phenomenon of strong sensitivity to gradient perturbations. By using results in [52], it is actually possible to conﬁrm the optimality of Theorem 4.1 by extending the type I blow-up result to a class of parabolic inequalities which contains the sub-critical case m > 1.

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Theorem 4.2. Assume that u is symmetric, with ur ≤ 0 and ut ≥ 0 in D = (−1, 1) × (0, T ). Let λ < 1, K > 0 and assume that |ux |2 , for all (x, t) ∈ D such that u(x, t) ≥ K, u Then (1.3) holds. In particular, this applies to problem (2.4) for any m > 1 and λ > 0. ut − uxx ≥ up − λ

Remarks 4.1. (i) The exact type II blow-up rate in Theorem 4.1 is presently unknown. However, we conjecture that there should not exist a single rate for all solutions. Such a situation occurs for problem (1.1) with p > p∗ [32]. (ii) We have mentioned in Section 3 that there is a connection between type I blow-up and property (3.1) for global solutions. Actually, it is known that for stronger perturbations, property (3.1) may fail, too. In case of equation (2.1), an example of unbounded global solution was constructed in [12] for any p > q = 2, with N = 1, Ω = (−1, 1) and some µ = µ(p) > 0. In that example, the solution stabilizes (monotonically) in inﬁnite time to a stationary solution singular at x = 0. On the other hand, for any q ≥ p and µ > 0, and, e.g. Ω = RN , it was shown in [55] that all suﬃciently large, compactly supported initial data give rise to a growing-up global solution, such that u(x, t) → ∞ as t → ∞ for each x ∈ RN . (iii) The blow-up rate for problem (2.4) with m = 1 is actually quite unstable. Consider the following mild modiﬁcation: ⎧ |ux |2 p ⎪ ⎪ −1 < x < 1, t > 0, ⎨ut − uxx = a(x, t)u − λ u , (4.2) u(±1, t) = b(t), t > 0, ⎪ ⎪ ⎩ −1 < x < 1, u(x, 0) = u0 (x) ≥ 1, with p > 1 and any λ ∈ R. Then there exist continuous functions a = a(|x|, t) on [−1, 1] × [0, T ), b(t) on [0, T ) satisfying 0 < c1 ≤ a(|x|, t), b(t) ≤ c2 ,

|x| < 1,

0 < t < T,

and such that for some symmetric u0 , with u0,r ≤ 0, problem (4.2) admits a solution with type I blow-up. An explicit example can be easily found under the form u(x, t) = [x2 + C(T − t)]−1/(p−1) (cf. [45, 47] and [31] for similar examples related to complete blow-up and dead-core rates, respectively). The rest of this section will be essentially devoted to present (the sketch of) a proof of Theorem 4.1 (the short proof of Theorem 4.2 will be given afterwards). Actually, Theorem 4.1 is a consequence of a recent result of [31] on dead-cores for the absorption problem ⎧ γ −1 < x < 1, t > 0, ⎪ ⎨wt − wxx = −w , w(±1, t) = k, t > 0, (4.3) ⎪ ⎩ w(x, 0) = w0 (x), −1 < x < 1,

Inﬂuence of Gradient Perturbations on Blow-up Asymptotics

483

where 0 < γ < 1 and k > 0. Indeed, similarly as in [34], we notice that (2.4) with m = 1 is transformed into (4.3) by the change of unknown λ−p , k = ab1−λ . with a = (λ − 1)(1−λ)/λ , γ = (4.4) w = au1−λ , λ−1 Now w is nondecreasing in |x| and blow-up of u at t = T is equivalent to appearance of a dead-core for w, i.e., w(0, T ) = 0. Note that w ≥ 0 exists for all times t > 0. The fast blow-up estimate (4.1) becomes equivalent to 1 . (4.5) α= lim (T − t)−α w(0, t) = 0, t→T 1−γ The proof of (4.5) relies on the use of self-similar variables (and will actually provide additional information – cf. Theorems 5.4 and 6.3 below). Namely, following [26, 21] and [19], we set √ T − t = e−s , y = x/ T − t and w(x, t) = (T − t)α v(y, s). Then v satisﬁes the equation y vs = vyy − vy + αv − v γ in D, (4.6) 2 where D := {(y, s); − log T < s < ∞, |y| < es/2 }. Under the assumptions of Theorem 4.1, we shall actually show the more precise convergence statement % 2 &α lim v(y, s) = V1 (y) := kγ |y|2α , kγ = (1−γ) , (4.7) 2(1+γ) s→∞

uniformly on {|y| < R} for each R > 0, from which (4.5) (and hence Theorem 4.1) readily follows. A quick check reveals that the right-hand side V1 (y) of (4.7) provides an (unbounded) stationary solution of (4.6), more precisely a solution of y (4.8) Vyy − Vy + αV − V γ = 0, y ∈ R. 2 Note that each solution of (4.8) corresponds to a self-similar solution of wt = √ wxx − wγ in (−∞, T ) × R given by w(x, t) = (T − t)1/(1−γ) V (x/ T − t). On the other hand V1 (x), restricted to [−1, 1], is also a stationary solution of (4.3) with k = kγ . The proof of (4.7) will then be carried out in 3 steps. (i) identify the stationary solutions of (4.6) (in a suitable set); (ii) prove that all the global solutions of (4.6) are attracted by the set of stationary solutions of (4.6) (in the locally compact topology); (iii) discard all the possible limits other than the stationary solution V1 . We shall only brieﬂy outline the main ideas of each step. We refer to [31] for details. We need three lemmas. We start with a lower estimate which is the key ingredient in step (iii). Lemma 4.1. There exist c1 > 0 (depending on w) such that % &α w(x, t) ≥ w1−γ (0, t) + c1 x2 , |x| ≤ 1, T /2 ≤ t ≤ T.

(4.9)

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P. Souplet

Sketch of proof. We consider the auxiliary function J = wx − εxwγ . By a suitable modiﬁcation of an argument of [20], and using a nonlocal version of the maximum principle, it can be proved that for ε > 0 small enough, it holds J ≥0

in [0, 1] × (T /2, T ).

(4.10)

−γ

This yields (w )x = (1 − γ)w wx ≥ ε(1 − γ)x in [0, 1] × (T /2, T ) and the estimate then immediately follows by integrating in space between 0 and x. 1−γ

We next derive similar upper estimates on w which yield polynomial upper bounds for v. Lemma 4.2. There exist c2 > 0 (depending on w) such that % 1−γ &2α w(x, t) ≤ w 2 (0, t) + c2 |x| , |x| ≤ 1, T /2 ≤ t ≤ T.

(4.11)

Moreover, the corresponding global solution v of (4.6) satisﬁes 2

v(y, s) ≤ C(1 + |y|) 1−γ

and

1+γ

|vy (y, s)| ≤ C(1 + |y|) 1−γ

(4.12)

for all − log(T /2) =: s0 < s < ∞, |y| < es/2 . Sketch of proof. We consider the auxiliary function J(x, t) := 12 wx2 − Cwγ+1 , where C > 0 is a suitably large constant. By using the maximum principle, it can be proved that J ≤ 0 in Q := [0, 1] × (T /2, T ). This implies (4.11) and we get in turn the estimate |x| ≤ 1, T /2 ≤ t ≤ T. (4.13) |wx | ≤ Cw(γ+1)/2 , For the proof of (4.12), we ﬁrst notice that, since wxx (0, t) ≥ 0, we have wt (0, t) ≥ −wγ (0, t), and by integrating between t and T we get w(0, t) ≤ C(T − t)α . By combining this with (4.11), we obtain √ √ % 1−γ & 2 v(y, s) = (T − t)−α w(y T − t, t) ≤ (T − t)−α w 2 (0, t) + c2 |y| T − t 1−γ √ %√ & 2 2 ≤ C(T − t)−α T − t + |y| T − t 1−γ = C(1 + |y|) 1−γ .

The estimate of vy then follows from (4.13). Next, for step (i), we have the following. Lemma 4.3. Let V ∈ C 2 (R) by a solution of (4.8) such that V = V (|y|),

with

V ≥ 0,

V >0

for all y > 0,

and such that V is polynomially bounded. Then V = V1 = kγ |y|2α

or

V = V2 := κγ := (1 − γ)1/(1−γ) .

Sketch of proof. We make use of the auxiliary function 1

Z := |W − y2 W | 1−γ ,

W = V 1−γ .

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By some tedious calculation, one can show that at each y such that Z(y) > 0, it holds y2 Z − Z ≤ 0, i.e., 2

(4.14) (e−y /4 Z ) ≥ 0. It then follows that the function Z ≥ 0 is nonincreasing in (0, ∞). Indeed, other2 wise, there would exist y0 such that Z(y0 ) > 0 and Z (y0 ) > 0, hence Z ≥ Cey /4 2 for y ≥ y0 by (4.14). Due to |(y −2 W ) | = 2y −3 Z 1−γ , we would get W ≥ eηy as y → ∞, for some η > 0, contradicting the polynomial bound assumed on V . Now assume for contradiction that Z is nonconstant on (0, ∞). Then there is R > 0 such that Z(0) > Z(R) and we may choose ε > 0 so small that f := 2 Z + εey /2 satisﬁes f (0) > Z(0) > f (R). It follows that f has a local maximum at some y1 ∈ (−R, R) and that Z(y1 ) > 0. Therefore, we get 0 ≤ (y/2)f − f ≤ 2 ε((y 2 /2) − 1 − y 2 )ey /2 < 0, a contradiction. We deduce that W − (y/2)W = C on (0, ∞). By integration, we ﬁnally get W = A + By 2 and the conclusion follows easily by substituting into the equation (4.8). Now, the rest of the proof of (4.7), in particular Step (ii), will be a consequence of energy arguments close to those from Giga and Kohn [26] for blow-up problems. A diﬀerence with [26] is that here v is not uniformly bounded; and indeed it will be proved that, unlike in those works, v converges to an unbounded self-similar proﬁle. End of proof of Theorem 4.1 (sketch). By using the energy functional R(s) 2 vy v γ+1 αv 2 + − (y, s)ρ(y) dy, E(s) = 2 γ+1 2 0 y2

where ρ(y) = e− 4 and R(s) = e 2 , along with estimate (4.12), one can show

∞ R(s) 2 ρvs dy ds < ∞. Therefore, by standard arguments, using (4.12) and that s0 0 p parabolic L estimates, we deduce that, for each sequence sn → ∞, there exists a subsequence sn such that v(·, sn ) converges to a solution V of (4.8), uniformly on {|y| < R} for each R > 0. But on the other hand, by the lower bound in Lemma 4.1, for each y ∈ R and s > 2 log |y|, we have √ √ 2α v(y, s) = (T − t)α w(y T − t, t) ≥ (T − t)α (c1 |y T − t|2 )α = cα 1 |y| . s

In view of Lemma 4.3 and the polynomial bound (4.12), this shows that necessarily V = V1 . The convergence assertion (4.7) follows, hence Theorem 4.1. Proof of Theorem 4.2. Set p−λ > 1. (4.15) 1−λ Then v satisﬁes vt − vxx ≥ (1 − λ)v γ at each (x, t) ∈ D such that v ≥ K 1−λ . Since moreover vr ≤ 0 and vt ≥ 0, we may apply [52, Theorem 5] to deduce that v(x, t) ≤ C(T − t)−1/(γ−1) in D, that is, u(x, t) ≤ C(T − t)−1/(p−1) . v = u1−λ ,

γ=

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5. Blow-up proﬁles in space The results of this section show that the inﬂuence of gradient perturbations on spatial blow-up proﬁles is connected in a precise way with the sub- or super-critical nature of the perturbation (in the sense deﬁned in Section 2). In particular, for equation (2.1), the absorbing gradient term has no signiﬁcant eﬀect on the spatial blow-up proﬁle if q < q0 := 2p/(p + 1), whereas its inﬂuence becomes quite strong for q > q0 , making it more and more singular as q increases to p (recall that blow-up for (2.1) occurs if and only if p > q). Theorem 5.1. [11] Assume Ω = BR , 1 < q < p, µ > 0. Let u0 be radial, with u0,r ≤ 0, and let the solution of (2.1) satisfy T < ∞. Then 0 is the only blow-up point. Moreover, for all α > α0 , it holds u(x, t) ≤ Cα |x|−α , with

α0 =

0 < |x| < R,

0 < t < T,

2/(p − 1)

if 1 < q ≤ 2p/(p + 1)

q/(p − q)

if 2p/(p + 1) < q 2/(p − 1) if 2p/(p + 1) < q < p). As an indication of the optimality of α0 in Theorem 5.1, it was shown in [11] that (5.1) can hold for no α < α0 if in addition N = 1 and ut ≥ 0. Actually, we have the following lower bound on the proﬁle, which shows that Theorem 5.1 is sharp. Theorem 5.2. [51] Under the hypotheses of Theorem 5.1, assume in addition that ut ≥ 0. Then it holds u(x, T ) ≥ C|x|−α0 ,

0 < |x| < η,

for some C, η > 0.

(5.2)

Here and in what follows, we have denoted u(x, T ) = limt→T u(x, t). Recall that if 0 is the only blow-up point (without even assuming ut ≥ 0), then this limit exists and ﬁnite for any x = 0, due to standard parabolic estimates. Remarks 5.1. (i) The proof of Theorem 5.2 actually shows that the lower blow-up proﬁle estimate (5.2) with α0 = 2/(p− 1) is true for the general perturbed problem (1.8) in Ω = BR under the assumption g ≤ 0, whenever u is radial, ur < 0 for r > 0, ut ≥ 0, and u blows up in sup norm. Moreover, no boundary conditions are actually required. (ii) Consider equation (2.3)1 with q ≥ 1, λ > 0, s ∈ R and q + s < p. Assume Ω = BR , u radial, ur ≤ 0, ut ≥ 0 (no boundary conditions are required). Then the proof of Theorem 5.2 yields the lower blow-up proﬁle estimate (5.2) with α0 = q/(p − q − s), whenever blow-up occurs in sup norm. We observe that α0 > 2/(p − 1) when q > 2(p − s)/(p + 1). The argument of the proof can be easily extended to obtain estimates of this kind for more general g’s and/or nonlinearities other than up , with nonlinear diﬀusion as well (see [51]).

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(iii) In Theorem 5.2, we may take C = C(p) if q ≤ 2p/(p + 1) and C = C(p, q, µ) if 2p/(p + 1) < q < p. The proof of Theorem 5.1 was based on nontrivial modiﬁcations of the argument of [20] (which consists in applying the maximum principle to a function of the form J = ur + cε (r)F (u)). Moreover, in order to reach the exponent α0 , the authors had to incorporate an additional bootstrap argument (on the value of γ for F (u) = uγ ). We shall not reproduce this proof here for reasons of space. Proof of Theorem 5.2. First assume 1 < q ≤ 2p/(p + 1). Since ut ≥ 0 and ur ≤ 0, we have 1 2 1 p+1 = (urr + up )ur = ut + µ|ur |q − N r−1 ur ur ≤ 0, 2 ur + p+1 u r 1 1 hence 12 u2r + p+1 up+1 (r, t) ≤ p+1 up+1 (0, t). Therefore, we get ur (t)∞ ≤ C1 u(p+1)/2 (0, t).

(5.3)

1 2 u(0, t).

For 0 < t < T , let r0 (t) be such that u(r0 (t), t) = Note that, since ur < 0 for 0 < t < T and 0 < r ≤ R, the implicit function theorem implies that r0 (t) is unique and is a continuous function of t. Since u is nonglobal, it is known that u cannot remain bounded in sup norm (see Section 2). Consequently, u(0, t) → ∞ as t → T , due to ur ≤ 0 and ut ≥ 0. We may also assume that 0 is the only blow-up point, since otherwise the result is trivial. This implies that r0 (t) → 0 as t → T . Now we have 0 ≤ r ≤ r0 (t). −ur ≤ C2 u(p+1)/2 , Integrating, we get u−(p−1)/2 (r0 (t), t) ≤ u−(p−1)/2 (0, t) + C3 r0 (t) = 2−(p−1)/2 u−(p−1)/2 (r0 (t), t) + C3 r0 (t) hence u(r0 (t), t) ≥ C4 (r0 (t))−2/(p−1) . Using ut ≥ 0, it follows that u(r0 (t), T ) ≥ C4 (r0 (t))−2/(p−1) ,

0 < t < T.

Since r0 is continuous and r0 (t) → 0 as t → T , we deduce that the range r0 ((0, T )) contains an interval of the form (0, η) and the conclusion follows. Let us now consider 2p/(p + 1) < q 1. For each t ∈ (0, T ), at a point r ∈ (0, R] where |ur (·, t)| achieves its maximum, we have µ|ur |q = up + urr − ut +

N −1 r ur

≤ up ,

due to ut ≥ 0, ur ≤ 0 and urr (r, t) ≤ 0. It follows that ur (t)∞ ≤ C1 uγ (0, t).

(5.4)

Using (5.4) in place of (5.3), the conclusion follows exactly as in the case q ≤ 2p/(p + 1). In some special cases, it is possible to obtain sharper estimates or even exact proﬁles. Let us ﬁrst consider the critical case q = 2p/(p + 1) of equation (2.1).

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Theorem 5.3. [53] Let Ω = RN , q = 2p/(p + 1) and µ ∈ (0, 2). There exists p0 (N, µ) > 1 such that for all p ∈ (1, p0 ) and some radial initial data u0 , u0,r ≤ 0, the equation (2.1) admits a solution with single-point blow-up and the ﬁnal proﬁle u(x, T ) = C|x|−2/(p−1) ,

x ∈ RN ,

for some C > 0.

Theorem 5.3 is a consequence of Theorem 6.2 below. As for the critical problem (2.4) with m = 1, considered in Section 4, we have the following result: Theorem 5.4. [31] Let m = 1 and λ > p. Assume that u0 is symmetric, with u0,r ≤ 0, and let the solution u of (2.4) satisfy T < ∞. Then we have single-point blow-up and the ﬁnal proﬁle satisﬁes C1 ≤ |x|2/(p−1) u(x, T ) ≤ C2 ,

0 < |x| < 1, x = 0,

for some C1 , C2 > 0.

Proof of Theorem 5.4. The (globally deﬁned) solution w ≥ 0 of the transformed problem (4.3) (cf. (4.4)) satisﬁes estimates (4.9) and (4.11). Since w(0, T ) = 0, we have in particular (c1 |x|2 )α ≤ w(x, T ) ≤ (c2 |x|)2α , for |x| < 1, and the assertion concerning u(x, T ) follows immediately. Remarks 5.2. (i) Let us note that for the critical problems in Theorems 5.3, 5.4, we have the same order of growth for the space proﬁles near x = 0 (slightly less singular than the usual proﬁle (1.5) for the model problem (1.1)), but the time rates are diﬀerent: type I or type II, respectively. On the other hand, comparison of these results with Theorems 5.1, 5.2 yields the interesting observation that the dissipative gradient term can have opposite eﬀects on the blow-up proﬁle: slightly less singular or strongly more singular proﬁle. (ii) In view of the results of Sections 3–5, it is an interesting open question whether blow-up is type I or type II for equation (2.1) in the supercritical case 2p/(p + 1) < q < p. For the critical case q = 2p/(p + 1), see Remark 6.1 (ii).

6. Blow-up proﬁles in self-similar variables This section concerns the description of blow-up in self-similar variables for problem (1.8). There again, the results can be interpreted in terms of criticality of the perturbations. For instance, as long as q < q0 = 2p/(p + 1) in equation (2.1), the classical Giga-Kohn proﬁle (1.7) is preserved (under suitable assumptions), while a new, nonﬂat proﬁle appears for q = q0 (compare formulas (6.1) and (6.2)). Theorem 6.1. Let 1 < p ≤ pS , Ω = BR or Ω = RN . Assume that g = g(|x|,t,u,|∇u|) satisﬁes g(0, t, u, 0) ≤ 0, along with the subcritical growth assumption (3.3). Moreover, assume that the solution u of (1.8) is radial, with ur ≤ 0, that it blows up in L∞ , and satisﬁes the type I estimate (1.3). Then it holds, √ (6.1) lim (T − t)1/(p−1) u(y T − t, t) = κ = (p − 1)−1/(p−1) , t→T

uniformly for y bounded.

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For q = 2p/(p + 1), equation (2.1) is actually scale-invariant, which allows for the possibility of existence of self-similar solutions. Theorem 6.2. [53] Let Ω = RN , q = 2p/(p + 1) and µ ∈ (0, 2). There exists p0 (N, µ) > 1 such that for all p ∈ (1, p0 ), (2.1) admits a positive self-similar solution of the form √ u(x, t) = (T − t)−1/(p−1) W (x/ T − t), x ∈ RN , −∞ < t < T, equivalently

√ (6.2) (T − t)1/(p−1) u(y T − t, t) = W (y), 2 N where W ∈ C (R ) is radial, with Wr ≤ 0. Moreover, for all such solution, W satisﬁes lim |y|2/(p−1) W (y) = C > 0. |y|→∞

As for the critical problem (2.4) with m = 1, it leads to a still distinctly diﬀerent self-similar proﬁle, namely a singular one. Theorem 6.3. [31] Let m = 1 and λ > p. Assume that u0 is symmetric, with u0,r ≤ 0, and let the solution u of (2.4) satisfy T < ∞. Then it holds √ lim (T − t)1/(p−1) u(y T − t, t) = cp |y|−2/(p−1) , t→T

uniformly for y bounded and bounded away from 0. Theorem 6.2 was proved in [53] by using ODE methods. As for Theorem 6.3, its proof was given in Section 4 above (cf. Formula (4.7)). Theorem 6.1 seems to be new and we now give a proof. Proof of Theorem 6.1. Following [26, 21], we rescale equation (1.8) by similarity variables, by setting √ T − t = e−s , y = x/ T − t and v(y, s) = (T − t)β u(x, t), with β = 1/(p − 1). Then v satisﬁes the equation vs = ∆v − y2 .∇v + v p − βv + h(y, s) in D,

(6.3)

where h(y, s) = e−pβs g(x, t, u(x, t), |∇u(x, t)|) and D = {(y, s) ∈ RN × R, s > s0 := − log T, |y| ≤ Res/2 }. It was proved in [41, Lemma 5.1] that under assumption (3.3) (even with q ≤ 2p/(p + 1)), it holds ∇u(t)∞ ≤ C sup u(τ )(p+1)/2 ∞ τ ∈[0,τ ]

for t close enough to T . Therefore, the assumption (1.3) implies that u(t)∞ ≤ C(T − t)−β = Ceβs ,

∇u(t)∞ ≤ C(T − t)−(p+1)β/2 = Ce

Letting γ = β min(p − p1 , p − q(p + 1)/2) > 0, it follows from (3.3) that h(s)∞ ≤ Ce−pβs (1 + Ceβp1 s + Ce

(p+1)q 2

βs

) ≤ Ce−γs .

p+1 2 βs

.

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Moreover, (1.3) implies that v is bounded in D. Since the perturbation in equation (6.3) is decaying and integrable in time, the energy arguments in [26] (derived there for h = 0) carry over, and we deduce that v(y, s) converges to 0 or κ, as s → ∞, uniformly for y bounded. Finally, since g(0, t, u, 0) ≤ 0 and u blows up in L∞ , we easily obtain (T − t)β u(0, t) ≥ C1 > 0, hence v(0, s) ≥ C1 . Consequently, the 0 limit cannot occur and the result is proved. Remarks 6.1. (i) On a general domain, under the assumptions of Theorem 6.1 (but for radial symmetry), the proof shows that, for any blow-up point a ∈ Ω (i.e., a such that u is not locally bounded nearby (a, T )), it still holds √ lim (T − t)1/(p−1) u(a + y T − t, t) = κ or 0, t→T

uniformly for y bounded. In the case g = 0, the nondegeneracy results in [28] (based, among other things, on weighted energy arguments) show that the 0 limit cannot occur. With more work, it might be possible to extend this to our case, but this extension does not seem immediate. (ii) Consider equation (2.1) in the critical case q = 2p/(p+1). It is known [52] that blow-up is type I if µ is small enough (under suitable assumptions), but the question is open for µ > 0 large. However, the gradient term may already have a (mild) inﬂuence on the blow-up rate for µ > 0 small. Namely, the solution u from Theorem 6.2 satisﬁes limt→T (T −t)1/(p−1) u(t)∞ = W (0) > κ (see [53]), whereas, for q < 2p/(p + 1) and, e.g., p < pB , any radially decreasing blow-up solution in RN satisﬁes limt→T (T − t)1/(p−1) u(t)∞ = κ by Theorems 6.1 and 3.1. (iii) As in Theorem 4.1, the assumption λ > p in Theorems 5.2 and 6.3 is optimal, for there is a bifurcation in the spatial and self-similar proﬁles at λ = p. Indeed, for λ ∈ (1, p), the transformation (4.4) converts equation (2.4) with m = 1 into the quenching problem ut − uxx = −u−γ for some γ > 0. Assume for instance that u is symmetric with ur ≤ 0, ut ≥ 0. By using the results in [17] (and references therein) on the quenching problem, and transforming back to (2.4), we ﬁnd that the space proﬁle at a = 0 is given by (1.6) and the self-similar proﬁle by (1.7). (iv) For the model equation (1.1), more precise self-similar proﬁles based on the larger time-scale (T − t)| log(T − t)| have been studied by several authors, see [56, 8] and the references therein. Some of the results remain valid for equation (1.8) under assumption (3.3) (see [8, Remark 3 p. 541]). (v) For equation (2.1) but now with µ < 0, self-similar proﬁles have been studied in [23] for q = 2 and Ω = RN . It was proved that radial blow-up solutions behave asymptotically like a self-similar solution w of the Hamilton-Jacobi equation wt = wp + |∇w|2 , with w having the form w(x, t) = (T − t)−1/(p−1) θ(x(T − t)−σ ) and σ = (2 − p)/2(p − 1). Note that this kind of self-similar behavior is quite diﬀerent from that in Theorem 6.2; indeed, σ describes the range (−∞, 1/2) for p ∈ (1, ∞).

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7. Blow-up sets and continuation after blow-up We have already seen several results which ensure single point blow-up for equation (1.8) in the radial case (cf. Theorems 5.1, 5.3 and 5.4). In particular, unlike blowup rates and proﬁles, the property of single-point blow-up in the radial case does not seem to be aﬀected by the presence of a dissipative gradient term, even if it is critical or supercritical. On the other hand, it was proved in [11] that if Ω is convex bounded then, as for (1.1) with u0 ≥ 0, the blow-up set of any solution of (2.1) is a compact subset of Ω when µ > 0 and q < 2p/(p + 1). The situation is quite diﬀerent when µ < 0 (see [35, 34, 23]). Namely, one has global blow-up if 1 q = 2, Ω = BR and u0 is radial, with u0,r ≤ 0, then there is still single point blow-up. The proof involves the transformation v = eu − 1, which changes (2.1) into the equation with mildly superlinear source vt − ∆v = (1 + v) logp (1 + v). The authors of [34] interpret this result in the following way. While the term up alone would force the solution to develop a spike at the maximum point, hence causing single point blow-up, the gradient term now has a positive sign and tends to push up the steeper parts of the graph of u(., t). This enhances regional or even global blow-up, the inﬂuence of the gradient term becoming more important as the value of p decreases. We note that the change of behavior is here no longer related to scaling properties of the perturbation, but rather to its size as compared with that of the reaction term (see the end of paragraph 2 of Section 2 for a related phenomenon). As for continuation after blow-up, it can be seen that some solutions of the critical problem (2.4) for m = 1 and λ > p admit a nontrivial continuation after blow-up, in sharp contrast with the unperturbed problem (1.1) in the onedimensional case. Moreover, this continuation is unique can be described in a precise way. It is to be noted that the continuation does not enjoy the property of immediate regularization mentioned in Introduction for (1.1) with pS b0 (λ, p) large enough (but this may be extended to more general initial data). Then we have T < ∞ and, by using the transformation (4.4) and known results [2, 57] on the dead-core problem (4.3), one obtains the following. There exists a blow-up interface σ : [T, ∞) → [0, 1), monotonically increasing, locally Lipschitz, with σ(T ) = 0 < L = σ(∞) < 1, and an extension u of u, with u ∈ C(D; (0, ∞]), D = [−1, 1] × [0, ∞), such that u = ∞ in D∞ := (x, t); T ≤ t < ∞, |x| ≤ σ(t) , u is a classical solution to (2.4)1 in D \ D∞ . Moreover, for x ∈ (L, 1), u(x, t) converges monotonically as t → ∞ to the unique stationary solution v of (2.4)1 in (L, 1) such that v(1) = b and v(L) = ∞. We refer to [6] for similar results on a diﬀerent blow-up problem with gradient perturbation. Remark 7.1. In the opposite direction, note that if λ < 1 in problem (2.4) for m = 1, then blow-up is complete, due to the transformation (4.15) and the known results

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on (1.1). In fact, the threshold is given by λ = (p + 1)/2. Indeed, for λ ∈ (1, p), one can use the results in [22, 44] on the quenching problem (cf. Remark 6.1 (iii)) to deduce that incomplete blow-up may occur if and only if λ > (p + 1)/2.

References [1] J. Aguirre, M. Escobedo, On the blow up of solutions for a convective reaction diﬀusion equation, Proc. Roy. Soc. Edinburgh., 123A, 3 (1993), 433–460. [2] C. Bandle, I. Stakgold, The formation of the dead core in parabolic reactiondiﬀusion problems, Trans. Amer. Math. Soc., 286 (1984), 275–293. [3] P. Baras, L. Cohen, Complete blow-up after Tmax for the solution of a semilinear heat equation, J. Funct. Anal., 71 (1987), 142–174. [4] J.-Ph. Bartier, Global behavior of solutions of a reaction-diﬀusion equation with gradient absorption in unbounded domains, to appear. [5] J.-Ph. Bartier, Ph. Souplet, Gradient bounds for solutions of semilinear parabolic equations without Bernstein’s quadratic condition, C.R. Acad. Sc. Paris, 338 (2004), 533–538. [6] J. Bebernes, D. Eberly, Characterization of blow-up for a semilinear heat equation with a convection term, Quart. J. Mech. Appl. Mech., 42 (1989) 447–456. ´ron, Initial blow-up for the solutions of a semilinear parabolic [7] M.-F. Bidaut-Ve equation with source term, in: Equations aux d´eriv´ees partielles et applications, articles d´edi´es ` a Jacques-Louis Lions. Gauthier-Villars, Paris, 1998, pp. 189–198. [8] J. Bricmont, A. Kupiainen, Universality in blow-up for nonlinear heat equations, Nonlinearity, 7 (1994), 539–575. [9] M. Chipot, F.B. Weissler, Some blow up results for a nonlinear parabolic problem with a gradient term, SIAM J. Math. Anal., 20 (1989), 886–907. [10] M. Chlebik, M. Fila, From critical exponents to blowup rates for parabolic problems, Rend. Mat. Appl., Ser. VII, 19 (1999), 449–470. [11] M. Chlebik, M. Fila, P. Quittner, Blowup of positive solutions of a semilinear parabolic equation with a gradient term, Dyn. Contin. Discrete Impulsive Syst. Ser. A, 10 (2003), 525–537. [12] K. Deng, Stabilization of solutions of a nonlinear parabolic equation with a gradient term, Math. Z., 216 (1994), 147–155. [13] M. Fila, Remarks on blow up for a nonlinear parabolic equation with a gradient term, Proc. Amer. Math. Soc., 111 (1991), 795–801. [14] M. Fila, Blow-up of solutions of supercritical parabolic problems, preprint (2004). [15] M. Fila, H. Matano, P. Polacik, Immediate regularization after blow-up, SIAM J. Math. Anal., to appear. [16] M. Fila, Ph. Souplet, The blow-up rate for semilinear parabolic problems on general domains, Nonlinear Diﬀer. Eq. and Appl. NoDEA, 8 (2001), 473–480. [17] S. Filippas, J.-S. Guo, Quenching proﬁles for one-dimensional semilinear heat equations, Quart. Appl. Math., 51 (1993), 713–729.

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´ zquez, Fast blow-up mechanisms for sign[18] S. Filippas, M.A. Herrero, J.J.L. Vela changing solutions of a semilinear parabolic equation with critical nonlinearity, Proc. R. Soc. Lond. A, 456 (2000), 2957–2982. [19] A. Friedman, M.A. Herrero, Extinction properties of semilinear heat equations with strong absorption, J. Math. Anal. Appl., 124 (1987), 530–546. [20] A. Friedman, J.B. McLeod, Blowup of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34 (1985), 425–447. [21] V.A. Galaktionov, S. Posashkov, Application of new comparison theorems to the investigation of unbounded solutions of nonlinear parabolic equations (Russian), Diﬀer. Uravn., 22 (1986), 1165–1173. (Engl. transl.: Diﬀer. Equations, 22 (1986), 809–815.) ´ zquez, Necessary and suﬃcient conditions for com[22] V.A. Galaktionov, J.L. Va plete blow-up and extinction for one-dimensional quasilinear heat equations, Arch. Rational Mech. Anal., 129 (1995), 225–244. ´ zquez, Blowup for quasilinear heat equations de[23] V.A. Galaktionov, J.L. Va scribed by means of nonlinear Hamilton-Jacobi equations, J. Diﬀer. Equations, 127 (1996), 1–40. ´ zquez, The problem of blow-up in nonlinear parabolic [24] V.A. Galaktionov, J.L. Va equations, Discrete Contin. Dyn. Syst., 8 (2002), 399–433. [25] Y. Giga, A bound for global solutions of semilinear heat equations, Comm. Math. Phys., 103 (1986), 415–421. [26] Y. Giga, R.V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38 (1985), 297–319. [27] Y. Giga, R.V. Kohn, Characterizing blowup using similarity variables, Indiana Univ. Math. J., 36 (1987), 1–40. [28] Y. Giga, R.V. Kohn, Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math., 42 (1989), 845–884. [29] Y. Giga, S. Matsui, S. Sasayama, Blow up rate for semilinear heat equations with subcritical nonlinearity, Indiana Univ. Math. J., 53 (2004), 483–514. [30] Y. Giga, S. Matsui, S. Sasayama, On blow-up rate for sign-changing solutions in a convex domain, Math. Methods Appl. Sci., 27 (2004), 1771–1782. [31] J.-S. Guo, Ph. Souplet, Fast rate of formation of dead-core for the heat equation with strong absorption and applications to fast blow-up, Math. Ann., to appear. [32] M.A. Herrero, J.J.L. Velazquez, Explosion de solutions des ´equations paraboliques semilin´eaires supercritiques, C. R. Acad. Sc. Paris S´ erie I, 319 (1994), 141– 145. [33] B. Hu, Remarks on the blowup estimate for solution of the heat equation with a nonlinear boundary condition, Diﬀer. Integral Equations, 9 (1996), 891–901. [34] B. Kawohl, L. Peletier, Remarks on blowup and dead cores for nonlinear parabolic equations, Math. Z., 202 (1989), 207–217. [35] A.A. Lacey, Global blowup of a nonlinear heat equation, Proc. Royal Soc. Edinburgh, 104 (1986), 161–167. [36] O. Ladyzenskaya, V.A. Solonnikov, N.N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc. Translations, Providence, RI, 1967.

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[37] H.A. Levine, P. Meier, A blowup result for the critical exponent in cones, Israel J. Math., 67 (1989), 129–136. [38] H.A. Levine, L.N. Payne, P.E. Sacks, B. Straughan, Analysis of convective reaction-diﬀusion equation (II), SIAM J. Math. Anal., 20 (1989), 133–147. [39] H. Matano, F. Merle, On nonexistence of type II blow-up for a supercritical nonlinear heat equation, Commun. Pure Appl. Math., 57 (2004), 1494–1541. [40] J. Matos, Self-similar blow up patterns in supercritical semilinear heat equations, Commun. Appl. Anal., 5 (2001), 455–483. [41] J. Matos, Ph. Souplet, Universal blow-up rates for a semilinear heat equation and applications, Advances Diﬀer. Equations, 8 (2003), 615–639. [42] N. Mizoguchi, Blowup rate of solutions for a semilinear heat equation with the Neumann boundary condition, J. Diﬀer. Equations, 193 (2003), 212–238. [43] C. Mueller, F.B. Weissler, Single point blow-up for a general semilinear heat equation, Indiana Univ. Math. J., 34 (1985), 881–913. [44] D. Phillips, Existence of solutions of quenching problems, Appl. Anal., 24 (1987), 253–264. [45] M. Pierre, unpublished manuscript (2003). [46] P. Quittner, Continuity of the blow-up time and a priori bounds for solutions in superlinear parabolic problems, Houston J. Math., 29 (2003), 757–799. [47] P. Quittner, Complete and energy blow-up in superlinear parabolic problems, Proc. NCTS Intern. Conf. on Elliptic and Parabolic Problems, Hsinchu 2004, World Scientiﬁc 2005. [48] P. Quittner, Ph. Souplet, A priori estimates of global solutions of superlinear parabolic problems without variational structure, Discrete Contin. Dyn. Syst., 9 (2003), 1277–1292. [49] Ph. Souplet, Recent results and open problems on parabolic equations with gradient nonlinearities, Electronic J. Diﬀer. Equations, Vol. 2001 (2001), No. 20, pp. 1–19. [50] Ph. Souplet, Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions, Diﬀer. Integral Equations, 15 (2002), 237–256. [51] Ph. Souplet, Blow-up proﬁle estimates for nonlinear parabolic equations, in preparation. [52] Ph. Souplet, S. Tayachi, Blow-up rates for nonlinear heat equations with gradient terms and for parabolic inequalities, Colloq. Math., 88 (2001), 135–154. [53] Ph. Souplet, S. Tayachi, F.B. Weissler, Exact self-similar blow-up of solutions of a semilinear parabolic equation with a nonlinear gradient term, Indiana Univ. Math. J., 48 (1996), 655–682. [54] Ph. Souplet, F.B. Weissler, Self-similar sub-solutions and blow-up for nonlinear parabolic equations, J. Math. Anal. Appl., 212 (1997), 60–74. [55] Ph. Souplet, F.B. Weissler, Poincar´e’s inequality and global solutions of a nonlinear parabolic equation, Ann. Inst. H. Poincar´e, Analyse non lin´eaire, 16 (1999), 337–373.

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´ zquez, Blow up for semilinear parabolic equations, in: Research in ap[56] J.J.L. Vela plied mathematics, Recent advances in partial diﬀerential equations, M.A. Herrero and E. Zuazua (eds.), John Wiley & Sons, 1994, pp. 131–145. [57] H.-M. Yin, The Lipschitz continuity of the interface in the heat equation with strong absorption, Nonlinear Anal., 20 (1993), 413–416. ´c ˇik, P. Quittner, A Liouville-type theorem and the decay of radial solu[58] P. Pola tions of a semilinear heat equation, preprint (2005). Philippe Souplet D´epartement de Math´ematiques INSSET Universit´e de Picardie F-02109 St-Quentin, France and Laboratoire de Math´ematiques Appliqu´ees UMR CNRS 7641 Universit´e de Versailles 45 avenue des Etats-Unis F-78035 Versailles, France e-mail: [email protected]

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 497–507 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Non-existence of Positive Solutions for Diﬀusive Logistic Equations with Nonlinear Boundary Conditions Kenichiro Umezu Dedicated to Professor Herbert Amann on the occasion of his 65th birthday

Abstract. In this paper a diﬀusive logistic equation with large diﬀusion is considered under nonlinear boundary conditions. Non-existence of the corresponding stationary positive solutions is discussed by use of variational techniques.

1. Introduction This paper is devoted to the study of the following semilinear elliptic problem with a nonlinear boundary condition arising in population dynamics (cf. [3], [4], [5], [6], [8], [10]): ⎧ ⎨−∆u = λ(m(x) − u)u in Ω, (1.1) ∂u ⎩ = λb(x)up on ∂Ω. ∂n Here Ω is a bounded domain of RN , N ≥ 2, with smooth boundary ∂Ω, λ > 0 is a parameter, p > 1 is a constant, m ∈ C θ (Ω) satisﬁes that m(x0 ) > 0 for some x0 ∈ Ω and may change sign, b ∈ C 1+θ (∂Ω) is non-negative, and n is the unit exterior normal to ∂Ω. Problem (1.1) denotes steady states of solutions of the parabolic problem ⎧ ∂u ⎪ ⎪ ⎨ ∂t (t, x) = ∇ · (1/λ)∇u(t, x) + (m(x) − u(t, x))u(t, x) in (0, ∞) × Ω, u(0, x) = u0 (x) in Ω, ⎪ ⎪ ⎩ p (1/λ)∇u(t, x) · n = b(x)u(t, x) on (0, ∞) × ∂Ω. From the viewpoint of population dynamics, the unknown function u denotes the population density of some species with diﬀusion rate 1/λ where m represents

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the birth or decay rate, and the nonlinear boundary condition means that the law of the population ﬂux (1/λ)∇u · n to the direction n on the boundary ∂Ω is determined nonlinearly on u, having the ﬂux rate b. If u ∈ C 2 (Ω) is a solution of (1.1) for some λ > 0, then we call (λ, u) a solution of (1.1). A solution (λ, u) of (1.1) is called positive if u > 0 in Ω. The existence and multiplicity of positive

have been studied

solutions of (1.1) in the previous work [10], in which, if p = 2, Ω m dx ≥ 0 and ∂Ω b dσ < |Ω|, then it is proved that there exists a positive solution of (1.1) for any λ > 0 small,

converging to the constant c∗ (b) = Ω m dx/(|Ω| − ∂Ω b dσ) in the topology of C 2 (Ω) as λ ↓ 0, see [10, Theorems 1.1 and 1.6]. Here dσ denotes the surface element of ∂Ω and |Ω| the volume of Ω. Under the same assumption a multiplicity result can be obtained for such λ > 0 small by use of a variational technique, if additionally the space dimension N is equal to two or three, see [10, Theorem 1.8], where the second positive solution tends to +∞ in C(Ω), equivalently in the usual Sobolev space W21 (Ω), as λ ↓ 0 (in fact, only the case Ω m dx = 0 is mentioned in [10, Theorem 1.8], but one can check easily that the assertion of the theorem is also true for the case Ω m dx > 0). The purpose of this paper is to discuss suﬃcient conditions for the non-existence of positive solutions in the case when

p = 2 and Ω m dx ≥ 0. It should be remarked that Hess [8] treats the linear case b = 0 and proves that a unique positive solution for each λ > 0 converges to c∗ (0) as λ ↓ 0, see [8, Example 28.6]. Now our main result is the following. Theorem 1.1. Let the space dimension N equals two or three. Assume that p = 2

and Ω m dx ≥ 0. If ∂Ω b dσ > |Ω|, then problem (1.1) has no positive solutions for any λ > 0 small (simply for 0 < λ & 1). For constants a > 0 and p, q > 1, Chipot, Fila and Quittner [7] consider a semilinear elliptic problem with a nonlinear boundary condition of the form ⎧ ⎨−∆u = −aup in Ω, (1.2) ∂u ⎩ = uq on ∂Ω. ∂n The existence and multiplicity of positive solutions and also their stability are studied there. Our argument is variational, inspired by them. It will be noted that the smallness of parameter λ > 0 is crucial for applying techniques in [7] in our setting. At the end of Introduction it should be mentioned that the space dimension N is assumed to be two or three throughout this paper. This implies that the following three compact embeddings hold true from Sobolev’s embedding theorem: W21 (Ω) ⊂ L2 (Ω), W21 (Ω) ⊂ L3 (Ω) and W21 (Ω) ⊂ L3 (∂Ω). The next section is devoted to the proof of Theorem 1.1.

(1.3)

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499

2. Proof of Theorem 1.1 2.1. Weak super- and subsolutions In this section we consider (1.1) with p = 2: ⎧ ⎨−∆u = λ(m(x) − u)u in Ω, (2.1) ∂u ⎩ = λb(x)u2 on ∂Ω, ∂n

under the assumption that N = 2 or 3, Ω m dx ≥ 0 and ∂Ω b dσ > |Ω|. It has been discussed in [7] that the weak super- and subsolution method constructs a weak solution of (1.2) to the Sobolev space X := W21 (Ω) such that it is a local minimizer of the corresponding energy functional. Our proof starts with its formulation in our setting. A function u ∈ X is called a weak subsolution of (2.1) if we have 2 ∇u∇ϕ dx − λ muϕ dx + λ u ϕ dx − bu2 ϕ dσ ≤ 0 Ω

Ω

Ω

∂Ω

for any non-negative ϕ ∈ X. Similarly a weak supersolution of (2.1) is deﬁned by the reverse of the inequality. Usual pointwise sub- and supersolutions of (2.1) to class C 2 (Ω) are called strong (see [2]). Strong sub- and supersolutions of (2.1) are called strict if they are not solutions of (2.1). It is easy to check that strong suband supersolutions imply weak ones, respectively. Now we prove the following (cf. [7, Lemma 2.4]). Lemma 2.1. Let u, u ∈ C 2 (Ω) be strict, strong sub- and supersolutions of (2.1), respectively, such that 0 ≤ u ≤ u in Ω. Then there exists a positive solution u ∈ C 2 (Ω) of (2.1) such that u
in Ω.

(2.2)

Moreover, the positive solution u is a local minimizer of the corresponding energy functional 1 λ λ λ Eλ (v) = |∇v|2 dx − mv 2 dx + v 3 dx − bv 3 dσ, v ∈ X, 2 Ω 2 Ω 3 Ω 3 ∂Ω meaning that there exists a constant ε > 0 such that v − uX ≤ ε implies Eλ (u) ≤ Eλ (v). Proof. The assertion (2.2) is a direct consequence of applying the strong maximum principle and boundary point lemma (cf. [9]). The remainder assertion will be veriﬁed in accordance with the argument in [7, Lemma 2.4]. (I) First let us show how to use the weak sub- and supersolution method to provide us with the existence of a positive solution u. Let M = {u ∈ X : u ≤ u ≤ u a.e. both in Ω and on ∂Ω}. One can check easily that Eλ is coercive and weakly lower semi-continuous on M . Hence, by the standard argument, minv∈M Eλ (v) is attained by some u ∈ M . We claim here that the function u is a weak solution, and consequently a solution of

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(2.1). Indeed, the assertion that u is a solution is obtained by elliptic regularity with the aid of (1.3). Let ϕ ∈ C 1 (Ω) and ε > 0, and we put uε (x) = min{u(x), max{u(x), u(x) + εϕ(x)}},

x ∈ Ω.

It follows that uε ∈ M . Since M is convex, we have u + τ (uε − u) ∈ M for all τ ∈ [0, 1]. Since u = minv∈M Eλ (v), it follows that d Eλ (u + τ (uε − u))|τ =0 . (2.3) 0≤ dτ Letting uε = (u + εϕ) − ϕε + ϕε , where ϕε = max{0, u + εϕ − u} ≥ 0,

ϕε := − min{0, u + εϕ − u} ≥ 0,

we have by a direct computation d Eλ (u + τ (uε − u))|τ =0 dτ ∇u∇ϕ dx − λ

u ϕ dx − λ bu ϕ dσ ∂Ω ε ε 2 ε 2 ε − ∇u∇ϕ dx − λ muϕ dx + λ u ϕ dx − λ bu ϕ dσ Ω Ω ∂Ω Ω 2 2 + ∇u∇ϕε dx − λ muϕε dx + λ u ϕε dx − λ bu ϕε dσ .

=ε

Ω

muϕ dx + λ

Ω

2

Ω

∂Ω

Ω

Ω

2

Ω

It follows from (2.3) that 2 ∇u∇ϕ dx − λ muϕ dx + λ u ϕ dx − λ Ω

Ω

where

Ω

∇u∇ϕ dx − λ

I1,ε =

∇u∇ϕε dx − λ Ω

bu2 ϕε dσ,

Ω

∂Ω

u ϕε dx − λ 2

muϕε dx + λ Ω

u ϕ dx − λ

muϕ dx + λ Ω

1 (I1,ε − I2,ε ), ε (2.4)

2 ε

ε

Ω

I2,ε =

∂Ω

ε

bu2 ϕ dσ ≥

Ω

bu2 ϕε dσ. ∂Ω

Since u is a weak supersolution, it follows that I1,ε = ∇u∇ϕε dx − λ muϕε dx + λ u2 ϕε dx − λ bu2 ϕε dσ Ω Ω Ω ∂Ω ε ε + ∇(u − u)∇ϕ dx − λ m(u − u)ϕ dx + λ (u2 − u2 )ϕε dx Ω Ω Ω 2 2 ε −λ b(u − u )ϕ dσ ∂Ω ≥ ∇(u − u)∇ϕε dx − λ m(u − u)ϕε dx + λ (u2 − u2 )ϕε dx Ω Ω Ω −λ b(u2 − u2 )ϕε dσ. (2.5) ∂Ω

Non-existence of Positive Solutions for Diﬀusive Logistic Equations

501

Now introduce Ωε = {x ∈ Ω : u + εϕ ≥ u > u},

Γε := {x ∈ ∂Ω : u + εϕ ≥ u > u},

and then ϕ = u + εϕ − u for x ∈ Ωε and for x ∈ Γε . Meanwhile, ϕε = 0 for x ∈ Ωε and for x ∈ Γε . It follows from (2.5) that I1,ε ≥ J1,ε + εJ2,ε , where 2 2 J1,ε = |∇(u − u)| dx − λ m(u − u) dx + λ (u2 − u2 )(u − u) dx Ωε Ωε Ωε b(u2 − u2 )(u − u) dσ −λ Γε J2,ε = ∇(u − u)∇ϕ dx − λ m(u − u)ϕ dx + λ (u2 − u2 )ϕ dx Ωε Ωε Ωε 2 2 b(u − u )ϕ dσ. −λ ε

Γε

Since |Ωε |, |Γε | → 0 as ε ↓ 0, J2,ε converges to zero as ε ↓ 0 by use of Lebesgue’s convergence theorem. On the other hand, either if x ∈ Ωε , or if x ∈ Γε , then |u − u| ≤ εϕC(Ω) , so that J1,ε is estimated below by −C1 ε2 for some constant C1 > 0. The following assertion has been deduced. I1,ε ≥ −C1 ε2 + ε o(1),

ε ↓ 0.

(2.6)

Similarly, since u is a weak subsolution, there exists a constant C2 > 0 such that I2,ε ≤ C2 ε2 + εo(1),

ε ↓ 0.

(2.7)

Combining (2.4) with (2.6) and (2.7), it follows that ∇u∇ϕ dx − λ muϕ dx + λ u2 ϕ dx − λ Ω

Ω

≥ − (C1 + C2 )ε + o(1),

Ω

∂Ω

ε ↓ 0,

which implies ∇u∇ϕ dx − λ muϕ dx + λ u2 ϕ dx − λ Ω

bu2 ϕ dσ

Ω

Ω

bu2 ϕ dσ ≥ 0.

∂Ω

By replacing ϕ by ψ = −ϕ and carrying out the same argument, it is proved that the reverse inequality holds. The claim has been now veriﬁed. (II) Next let us show how to argue by contradiction to obtain that the solution (λ, u) is a local minimizer of Eλ for each λ > 0. Assume to the contrary that there exist uk ∈ X, k = 1, 2, 3, . . . , satisfying 1 and Eλ (u) > Eλ (uk ). uk − uX ≤ (2.8) k Since Eλ is weakly lower semi-continuous in X, there exists wk ∈ X such that Eλ (wk ) = minv−uX ≤1/k Eλ (v). Hence we may assume that Eλ (u) > Eλ (uk ) =

min

v−uX ≤1/k

Eλ (v).

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Kenichiro Umezu

Then there exists a Lagrange multiplier Λk ∈ R such that for any ϕ ∈ X Eλ (uk )ϕ + Λk ∇(uk − u)∇ϕ dx + (uk − u)ϕ dx = 0. Ω

(2.9)

Ω

We assert here that lim inf Λk ≥ 0.

(2.10)

k→∞

To verify (2.10), we let ϕ = uk − u and then Λk uk − u2X = − ∇uk ∇(uk − u) dx − λ muk (uk − u) dx Ω Ω u2k (uk − u) dx − λ bu2k (uk − u) dσ . +λ Ω

∂Ω

Since u is a weak solution of (2.1), it follows that Λk uk − u2X = − |∇(uk − u)|2 dx − λ m(uk − u)2 dx Ω Ω b(uk + u)(uk − u)2 dσ . +λ (uk + u)(uk − u)2 dx − λ Ω

∂Ω

(2.11) Taylor’s formula (cf. [11]) gives us that 1 Eλ (uk ) − Eλ (u) = Eλ (u)(uk − u) + Eλ (u)(uk − u)2 + o(uk − u2X ), 2

k→∞ (2.12)

By deﬁnition, Eλ (u)(uk − u)2 =

|∇(uk − u)|2 dx − λ m(uk − u)2 dx Ω Ω 2 + 2λ u(uk − u) dx − 2λ u(uk − u)2 dσ. Ω

(2.13)

∂Ω

Substituting (2.12) and (2.13) into (2.11), and having in mind u is a weak solution of (2.1), it follows that Λk uk − u2X = −2(Eλ (uk ) − Eλ (u)) − λ (uk − u)3 dx Ω 3 +λ b(uk − u) dσ + o(uk − u2X ), k → ∞. ∂Ω

This implies (2.10), in view of (2.8) and (1.3).

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By the standard argument, it follows from (2.9), (2.10) and (1.3) that uk is a solution to C 2 (Ω) of the problem ⎧ Λk Λk + λm λ Λk ⎪ ⎪ u2k + u = uk − u in Ω, ⎨−∆ uk − 1 + Λ 1 + Λ 1 + Λ 1 + Λk k k ∂ Λk λ ⎪ ⎪ u = bu2 on ∂Ω. uk − ⎩ ∂n 1 + Λk 1 + Λk k By Sobolev’s embedding theorem and elliptic regularity we have sup uk − uWr1 (Ω) < ∞ for any r > N. k≥1

By compactness arguments there exists v ∈ C(Ω) such that some sub-sequence of {uk − u}, still denoted by {uk − u}, converges to v in C(Ω), and then we note vL2 (Ω) ≤ v − (uk − u)L2 (Ω) + uk − uL2 (Ω) ≤ |Ω|1/2 v − (uk − u)C(Ω) + uk − uL2 (Ω) . Hence we get v = 0 in L2 (Ω), since uk → u in X. By the continuity of v it follows that v = 0 in Ω, so that uk → u in C(Ω). In view of (2.2) this implies that u < uk < u in Ω for k ≥ 1 large, and thus that uk ∈ M for such k. Since Eλ (u) = minv∈M Eλ (v), it follows that Eλ (u) ≤ Eλ (uk ), contradictory for (2.8). The proof of Lemma 2.1 is now complete. 2.2. Proof of Theorem 1.1 We show here how to use Lemma 2.1 to prove Theorem 1.1. The proof is due to a contradiction argument. Assume to the contrary that there exists a sequence (λj , uj ), λj ↓ 0, of positive solutions of (2.1). Since ∂Ω b dσ > |Ω|, we can choose a function b1 ∈ C 1+θ (∂Ω) independent of (λj , uj ) such that 0 ≤ b1 ≤ b, b1 ≡ b, j ) is a strict strong supersolution of (2.1) with b = b1 . ∂Ω b1 dσ > |Ω| and (λj , u On the other hand, Since Ω m dx ≥ 0, the eigenvalue problem ⎧ ⎨−∆φ − λmφ = µφ in Ω, ∂φ ⎩ =0 in ∂Ω ∂n has a unique principal eigenvalue µ1 < 0 for λ > 0 (cf. [1]). Let φ1 ∈ C 2 (Ω) be the corresponding eigenfunction, which can be chosen as strictly positive in Ω. It is easy to check that (λj , εj φ) is a strict, strong subsolution of (2.1) with b = b1 and satisﬁes εj φ ≤ uj in Ω if constant εj > 0 is so small. Lemma 2.1 provides us with the existence of a positive solution (λj , vj ) of (2.1) with b = b1 such that εj φ < vj < uj in Ω and vj is a local minimizer of the corresponding energy functional. So, without loss of generality we may assume that there exists a sequence (λj , uj ), λj ↓ 0, of positive solutions of (2.1) which are local minimizers of Eλj . Then we prove the following.

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Proposition 2.2. For some j0 ≥ 1 there exists a constant ηj0 > 0 such that Eλj0 (uj0 + ηuj0 ) − Eλj0 (uj0 ) < 0

for any 0 < η < ηj0 .

(2.14)

Proof. Denoting the positive solution (λj , uj ) simply by (λ, u), it follows that 2 2 2 3 |∇u| dx − λ mu + 2λ u dx − 2λ bu3 dx. Eλ (u)u = Ω

Ω

Ω

∂Ω

Since (λ, u) is a solution of (2.1), we have 2 3 3 u dx − bu dσ Eλ (u)u = λ ∂Ω Ω 2 2 =− |∇u| dx − λ mu dx . Ω

(2.15) (2.16)

Ω

For u ∈ X we set u = ξ + w where ξ = |Ω| −1 Ω u dx. The function w is an orthogonal complement to constants in X, or Ω w dx = 0. From Poincar´e’s inequality

(see (2.21) below), wX is equivalent to ( Ω |∇w|2 dx)1/2 . It is well known that uX is equivalent to |ξ| + wX . We note that if u ∈ C 1 (Ω) is positive in D, then ξ > 0. Let δ0 > 0 be a constant. For the positive solution (λ, u) with u = ξ + w as above, if we consider the case w/ξX ≤ δ0 , then assertion (2.15) gives us that 3 3 w w 2 3 dx − b 1+ dσ . (2.17) Eλ (u)u = ξ λ 1+ ξ ξ Ω ∂Ω Since u is a weak solution of (2.1), we have by Taylor’s formula Eλ (u + ηu) − Eλ (u) = where

1

R3 = 0

(1 − s)2 Eλ (u + sηu)(ηu)3 ds. 2

By deﬁnition, Eλ (u)[ϕ, ψ, φ]

η 2 E (u)u2 + R3 , 2 λ

ϕψφ dx −

= 2λ Ω

bϕψφ dσ ,

∂Ω

so that condition (1.3) gives us that there exists a constant C3 > 0 independent of (λ, u) and 0 < η < 1 both, satisfying Eλ (u + ηu) − Eλ (u) ≤ From (2.17) it follows that λη 2 ξ 3 Eλ (u + ηu) − Eλ (u) ≤ 2

η 2 E (u)u2 + C3 λη 3 u3X . 2 λ

(2.18)

3 3 w w dx − b 1+ dσ + C4 η 1+ ξ ξ Ω ∂Ω (2.19)

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for some constant C4 > 0 depending on δ0 . The compact embeddings (1.3) enable us to have 3 3 w w 3 |Ω| − 1+ dx − b 1+ dσ 0 is chosen small enough. So, from (2.19) it follows that if w/ξX ≤ δ0 , then

3( ∂Ω b dσ − |Ω|) (2.20) Eλ (u + ηu) − Eλ (u) < 0 for any 0 < η < 4C4 Next we consider the case w/ξX ≥ δ0 , where δ0 has been determined in (2.20). Let α0 > 0 be a constant from Poincar´e’s inequality, that is, α0 w2X ≤ |∇w|2 dx, ∀w ∈ {w ∈ X : w dx = 0}. (2.21) Ω

Ω

Then Schwarz’s inequality shows 2 |∇u| dx − λ mu2 dx Ω Ω α w 2 w 0 2 ≥ λξ − m+ C(Ω) m dx , ξ − 2mL2 (Ω) ξ − λ Ω X X where m+ = max{m, 0}. Let K > 4 be chosen so large that 2 m dx Ω . δ0 > (K − 4)mL2(Ω)

(2.22)

(2.23)

If we assume that 0<λ≤

α0 , m+ C(Ω) + (KmL2 (Ω) )/(2δ0 )

(2.24)

then we have KmL2 (Ω) α0 − m+ C(Ω) ≥ , λ 2δ0 by which (2.22) implies 2 |∇u| dx − λ mu2 dx Ω Ω 2 w KmL2(Ω) w 2 − 2mL2(Ω) − m dx . ≥ λξ ξ ξ 2δ0 Ω X X Hence it follows from (2.23) that 2 |∇u| dx − λ mu2 dx ≥ C5 λξ 2 , Ω

Ω

(2.25)

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where C5 is a constant given as K −4 C5 = m dx > 0. mL2 (Ω) δ0 − 2 Ω From (2.16) and (2.25) we see that Eλ (u)u2 ≤ −C5 λξ 2 . From (2.18) it follows that C5 ξ 2 C5 ξ 2 + C3 ηu3X < 0 if 0 < η < Eλ (u + ηu) − Eλ (u) ≤ λη 2 − . 2 2C3 u3X (2.26) In view of (2.20), (2.24) and (2.26), such λj as in (2.24) veriﬁes (2.14). The proposition has been proved. Now assertion (2.14) is contradictory for uj0 being a local minimizer of Eλj0 . The proof of Theorem 1.1 is now complete. Acknowledgements This research was partly supported by the Grant-in-Aid for Scientiﬁc Research (C)(No. 16540165), Japan Society for the Promotion of Science.

References [1] G.A. Afrouzi and K.J. Brown, On principal eigenvalues for boundary value problems with indeﬁnite weight and Robin boundary conditions, Proc. Amer. Math. Soc. 127 (1999), 125–130. [2] H. Amann, Nonlinear elliptic equations with nonlinear boundary conditions, In : New Developments in diﬀerential equations (W. Eckhaus ed.), Math. Studies Vol. 21, North–Holland, Amsterdam, 1976, pp. 43–63. [3] R.S. Cantrell and C. Cosner, Diﬀusive logistic equations with indeﬁnite weights: population models in disrupted environments, Proc. Roy. Soc. Edinburgh 112A (1989), 293–318. [4] R.S. Cantrell and C. Cosner, Diﬀusive logistic equations with indeﬁnite weights: population models in disrupted environments II, SIAM J. Math. Anal. 22 (1991), 1043–1064. [5] R.S. Cantrell and C. Cosner, The eﬀects of spatial heterogeneity in population dynamics, J. Math. Biology 29 (1991), 315–338. [6] R.S. Cantrell, C. Cosner and V. Hutson, Ecological models, permanence and spatial heterogeneity, Rocky Mountain J. Math. 26 (1996), 1–35. [7] M. Chipot, M. Fila and P. Quittner, Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions, Acta. Math. Univ. Comenianae 60 (1991), 35–103. [8] P. Hess, Periodic-parabolic boundary value problems and positivity, Pitman Research Notes in Math. Series Vol. 247, Longman Scientiﬁc & Technical, Harlow, Essex, 1991. [9] M.H. Protter and H.F. Weinberger, Maximum principles in diﬀerential equations, Prentice–Hall, Englewood Cliﬀs New Jersey, 1967.

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[10] K. Umezu, Multiplicity of positive solutions under nonlinear boundary conditions for diﬀusive logistic equations, Proc. Edinburgh Math. Soc. 47 (2004), 495-512. [11] E. Zeidler, Nonlinear functional analysis and its applications I: Fixed-point theorems, Springer-Verlag, New York Berlin Heidelberg Tokyo, 1993. Kenichiro Umezu Faculty of Engineering Maebashi Institute of Technology Maebashi 371-0816, Japan

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 509–516 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Extremal Equilibria and Asymptotic Behavior of Parabolic Nonlinear Reaction-diﬀusion Equations An´ıbal Rodr´ıguez-Bernal and Alejandro Vidal-L´opez “Nonlinear Elliptic and Parabolic Problems: A Special Tribute to the Work of Herbert Amann”, Z¨ urich, June 2004

1. Introduction We give some conditions on the nonlinear term that ensure some “nice” properties for the asymptotic behavior of parabolic nonlinear reaction diﬀusion equations. Namely, the existence of two ordered extremal equilibria such that the order interval they deﬁne attracts the dynamics of the problem uniformly in space for bounded sets of initial data. The proof relies in the use of a parabolic version of the sub-supersolution method as well as the comparison principle that parabolic equations of this type satisfy. As a consequence of our results we can recover, among others, some results of a very well-known paper of H. Amann “Fixed points equations in ordered Banach spaces and nonlinear eigenvalue problem” where he studies elliptic equations by mean of topological techniques such as the topological degree or Leray-Schauder index. It is also in this paper that H. Amann introduces the sub-supersolution method for elliptic equations.

2. Our main result We consider the following problem ⎧ ⎨ ut − ∆u = u(0) = ⎩ Bu =

f (x, u) in Ω u0 0 on ∂Ω

(1)

where Ω ⊂ RN is a bounded domain, u0 ∈ X (e.g., X = C(Ω)), B is a boundary operator (Dirichlet, Robin or Neumann) and f : Ω × R → R is a continuous

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function, locally Lipschitz in u and satisﬁes the structure condition s f (x, s) ≤ C(x)s2 + D(x)|s|

∀s ∈ R

(2)

with C ∈ Lp (Ω), p > N/2, and 0 ≤ D ∈ Lq (Ω), q > N/2. Remark 2.1. The above condition on f means that is bounded above by a linear function for s positive and bounded below by another for s negative. Remark 2.2. For the sake of simplicity we deal with the Laplacian operator but due to the methods we will use the results remains true for more general operators. Under the above conditions, problem (1) has a unique solution for all time t ≥ 0 that we denote by uf (t, x; u0 ) (see, e.g., Henry [5]). Therefore, the solutions of the problem (1) deﬁne a dynamical system, S(t), that maps every initial data u0 to the solution at time t of problem (1) starting at u0 . As a consequence of the parabolic maximum principle, the following comparison principle holds for (1) (see Lunardi [7]). Theorem 2.3. Let f, g(x, u) : Ω × R → R be continuous functions, locally Lipschitz in u. Suppose that for all x ∈ Ω and u ∈ R we have f (x, u) ≤ g(x, u). Then, uf (t, x; u0 ) ≤ ug (t, x; u0 ) while both solutions exist, where we have denoted by uf (t, x; u0 ) the solution at time t of problem (1) starting at u0 . Moreover, if u0 ≤ v0 are two ordered initial conditions then uf (t, x; u0 ) ≤ uf (t, x; v0 ) while both solutions exist. Finally, the solutions of (1) are regular because of the smoothing property of the equation (see, e.g., Lunardi [7]). We state now our main result. Theorem 2.4. Assume f satisﬁes (2). Suppose that S∆+C (t) has exponential decay, i.e., solutions of problem wt − ∆w = C(x)w Bw = 0. decay exponentially to zero as t goes to ∞. Then, there exist maximal and minimal equilibrium points of (1), ϕm ≤ ϕM , such that (3) ϕm ≤ lim inf u(t, x; u0 ) ≤ lim sup u(t, x; u0 ) ≤ ϕM t→∞

t→∞

uniformly for x ∈ Ω and for u0 in bounded sets of X. In particular, the attractor for the problem (1) exists and is contained between the extremal equilibria and, of course, these equilibria belong to the attractor.

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Moreover, the maximal equilibrium is stable from above and the minimal one from below. Remark 2.5. Notice that the extremal equilibria do not have to be sign-constant. Remark 2.6. The property (3) implies, in particular, that any equilibrium ψ satisﬁes ϕm ≤ ψ ≤ ϕM . For the proof of Theorem 2.4 we need the following lemma that give us an estimate concerning the asymptotic behavior of the solutions of the nonlinear problem in terms of the linear one coming from the structure condition (2). Lemma 2.7. Let φ be the unique solution of −∆φ = C(x)φ + D(x) Bφ = 0

in on

Ω ∂Ω.

Then lim sup |u(t, x; u0 )| ≤ φ(x) t→∞

uniformly for x ∈ Ω and for u0 in bounded sets of X. Remark 2.8. The key in the proof of this lemma is the exponential decay of the semigroup associated to ∆ + C. Proof of Theorem 2.4 (sketch). First, we set η = φ + δ with δ a positive constant. Now, from Lemma 2.7, we obtain that S(T )η ≤ η at a certain time T large enough. Making the semigroup act on both sides of this inequality, we have that the sequence S(nT )η, is decreasing by the monotonicity property of the solutions. Moreover, it is bounded from below by −η. So it converges to a function ϕM pointwise. It turns out that by the Ascoli-Arzel`a theorem, this convergence is uniform. In fact, the solution starting at η converges to ϕM as t goes to ∞. Furthermore, by the continuity of S(t), ϕM is an equilibrium point. Finally, we prove that the solutions of (1) satisﬁes (3) uniformly for x ∈ Ω and for u0 in bounded sets of initial data. In particular, this implies that ϕM is maximal. The rest of the statements of the theorem follow in a straightforward way from the above arguments.

3. Some consequences for positive solutions We state now some consequences of Theorem 2.4 dealing with nonnegative and positive equilibria of problem (1). The ﬁrst result gives a condition for the existence of a minimal nonnegative equilibrium which is easy to check. Corollary 3.1. Suppose that f (·, 0) ≥ 0, f (·, 0) ≡ 0, and there exist C ∈ Lp (Ω), p > N/2 and 0 ≤ D ∈ Lq (Ω) such that f (x, s) ≤ C(x)s + D(x)

(4)

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for almost x ∈ Ω and for all s ≥ 0. Then (1) has two ordered extremal nonnegative equilibria provided that the semigroup associated to ∆ + C, S∆+C , has exponential decay. Proof. The existence of the maximal equilibrium ϕ follows from Theorem 2.4 (notice that f satisﬁes (2) and S∆+C has exponential decay). Moreover it is nonnegative since f (·, 0) ≥ 0. Now f (·, 0) ≥ 0, f (·, 0) ≡ 0, implies that 0 is a strict subsolution. So, S(t)0 is increasing. Furthermore, S(t)0 is bounded by the maximal solution ϕM whose existence follows from Theorem 2.4 . Thus, S(t)0 converges to a certain function + ϕ+ m that is positive (by the strong maximum principle). By continuity of S(t), ϕm is an equilibrium. Finally, notice that from the comparison principle, if ψ is a nonnegative equilibrium then ψ = S(t)ψ ≥ S(t)0 ≥ 0. Thus, taking limits as t goes to ∞ we have that ϕ+ m is the minimal nonnegative equilibrium for (1). Remark 3.2. If 0 is solution, that is, f (·, 0) ≡ 0, then it is the minimal nonnegative equilibrium. Another interesting question is when the problem (1) has positive equilibria even with f (x, 0) = 0. The next result goes in that direction when the former assumption holds (i.e., problem (1) has zero as an equilibrium solution). Essentially, what is needed for the existence of positive equilibria is the instability of 0 and the global boundedness of solutions starting near of 0. This holds, in particular, if all the solutions are globally bounded. Corollary 3.3. Let f (·, 0) ≡ 0. Suppose that there exist C ∈ Lp (Ω), p > N/2 and 0 ≤ D ∈ Lq (Ω) such that f (x, s) ≤ C(x)s + D(x)

(5)

for almost x ∈ Ω and for all s ≥ 0. Then (1) has at least one positive solution provided that 0 is unstable and the semigroup generated by ∆ + C has exponential decay. Proof. From Theorem 2.4 there exists a maximal equilibrium ϕM which is stable from above. We recall that we obtain the existence of the maximum equilibrium as limits of the solution of (1) starting at φ ≥ 0. Since f (·, 0) ≡ 0 we have that ϕM is nonnegative, by the comparison principle. Now, since 0 is unstable it follows that ϕM is positive. Otherwise, ϕM ≡ 0 (by the maximum principle ϕM is either zero or positive). Thus, 0 would be stable that is a contradiction. Remark 3.4. Notice that the positive equilibrium which existence we have obtained is the maximal equilibrium of (1). Remark 3.5. In the last case, there exists an attractor of nonnegative solutions for problem (1) which is contained in the order interval [0, ϕM ]. We recall that given two ordered elements of X, u ≤ v, the order interval is deﬁned by [u, v] = {x ∈ X : u ≤ x ≤ v}.

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4. Amann’s result revisited In [1], H. Amann studies the existence of positive solutions of elliptic problems of the form Lu = f (x, u) in Ω (6) Bu = 0 on ∂Ω where Ω ⊂ RN is a bounded domain of RN ; L is a uniformly elliptic operator of the form N N Lu = − aij Di Dk u + ai Di u + au i,j=1

i=1

with aij , ai , a ∈ C µ (Ω), (aij ) is symmetric and a is nonnegative; B is a boundary operator of the form Bu = b(x)u + δ

∂u , ∂ν

δ = 0, 1

with 0 ≤ b ∈ C 1+µ (∂Ω); and f ∈ C µ (Ω × R+ ) such that there exists ω ≥ 0 for which f (x, ξ) − f (x, η) > −ω(ξ − η) (7) for all x ∈ Ω, ξ > η ≥ 0. Remark 4.1. We consider more general functions f . Notice that we do not require (7) holds for f nor f to be H¨ older continuous. We just need f continuous and locally Lipschitz in u. Theorem 4.2 (Theorem 9.6 in Amann [1], p. 649). Assume that f (·, 0) ≥ 0 and there exist fˆ ∈ C(Ω) and 0 < m ∈ C µ (Ω) such that f (x, ξ) ≤ fˆ(x) + λm(x)ξ

(x, ξ) ∈ Ω × R+

(8)

with λ > 0. Then (6) has a minimal nonnegative solution provided that λ < λ0 (m), where λ0 (m) is the principal eigenvalue associated to m. This result can be obtained as a consequence of Corollary 3.1. Namely, notice that (8) implies that (2) holds with C(x) = λm(x) and D(x) = fˆ(x) and λ < λ0 (m) implies the exponential decay of the semigroup generated by ∆+λm(x). The result follows now from Theorem 3.1. Remark 4.3. Note that in our proof we do not need to assume the positivity of m or λ, nor such regularity in the coeﬃcients. In fact, what we need is that C(x) = λm(x) belongs to some Lq (Ω) space with q > N/2 without restriction over its sign. Theorem 4.4 (Theorem 9.8 in Amann [1], p. 650). Assume that f (·, 0) ≡ 0. Suppose that ∂u f exists, is continuous in a neighborhood of zero, and satisﬁes 0 < ∂u f (·, 0) ∈ C µ (Ω).

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If ψ > 0 is a supersolution for (6), then there exists a maximal positive solution in the order interval [0, ψ], provided that λ0 (∂u f (·, 0)) < 1.

(9)

This result is easy to obtain following the argument in the proof of Corollary 3.3 since condition (9) implies the instability of the zero solution. Namely, since ψ is a positive supersolution, the solution of (1) starting at ψ is nonincreasing. Furthermore, it is bounded from below. Thus, it converges to some equilibrium ψM ≥ 0. Now, the proof follows from the proof of Corollary 3.3 with ψM instead of ϕM . Notice that to obtain the result we do not need as much regularity for the coeﬃcients nor the positivity of ∂u f (·, 0). In fact, it is enough for f to be continuous and locally Lipschitz in u, and such that there exists a ∈ Lq (Ω) for some q > N/2 with a(x) ≤ lim inf s→0+ f (x, u)/u and 0 is unstable for the semigroup generated by ∆ + a (i.e., the ﬁrst eigenvalue of ∆ + a is negative). Theorem 4.5 (Theorem 9.9 in Amann [1], p. 650). Assume that f (·, 0) ≡ 0. Suppose that ∂u f exists, is continuous in a neighborhood of zero, and satisﬁes 0 < ∂u f (·, 0) ∈ C µ (Ω). Moreover, suppose that there exist fˆ ∈ C(Ω) and 0 < m ∈ C µ (Ω) such that f (x, ξ) ≤ fˆ(x) + λm(x)ξ

(x, ξ) ∈ Ω × R+

with λ ≥ 0. Then (6) has at least one positive solution provided that λ0 (∂u f (·, 0)) < 1

and

λ < λ0 (m).

Note again that with our techniques we can prove the result above with no sign and weaker regularity assumptions on ∂u f , m(x) and λ, the following result Remark 4.6. Some other related results that can be obtained as a consequence of our results can be found in Berestycki and P.L. Lions [2], P.L. Lions [6] or de Figueiredo [4], for instance.

5. An example: the logistic equation We consider now the logistic equation ⎧ ⎨ ut − ∆u = a(x)u − b(x)u3 u(0) = u0 ⎩ u = 0

in

Ω (10)

on ∂Ω.

in X + . Suppose that a(x) = a0 (x)+ a1 (x) and S∆+a0 has exponential decay. Using Young’s Inequality, for u ≥ 0, we have 3 1/2 a1 (x) 3 f (x, u) = a(x)u − b(x)u ≤ a0 (x)u + M b(x)

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with a similar computation for u ≤ 0. Thus, setting C(x) = a0 (x), property (2) is satisﬁed provided that a31 ∈ Lr (Ω) (11) b for some r > N/4. From the results on Section 3 it is possible to obtain the following result concerning the existence of a positive solution of problem (10) and its stability. Theorem 5.1. Suppose that there exists a nontrivial decomposition of a(x) = a0 (x) + a1 (x) (i.e., a0 and a1 are not identically zero) such that S∆+a0 has exponential decay and (11) holds. Moreover, suppose that 0 is unstable for the problem ⎧ Ω ⎨ ut − ∆u = a(x)u in u(0) = u0 (12) ⎩ u = 0 on ∂Ω. Then, there exist a unique positive solution for the problem (10). Moreover, this solution is globally asymptotically stable for positive solutions. Remark 5.2. Notice that if 0 is stable for the problem (12) then it is the unique solution of the problem (10) and it is asymptotically globally stable. Proof. The existence is an straightforward application of Corollary 3.3. Actually, what we prove is the existence of the maximal positive equilibrium ϕ (see Remark 3.4). For the uniqueness, let ψ ≤ ϕ another equilibrium. Then, −∆ϕ = f (x, ϕ)

and

− ∆ψ = f (x, ψ)

where f (x, u) = a(x)u − b(x)u . Multiplying the ﬁrst equation by ψ, the second one by ϕ, subtracting and integrating in Ω, we have f (x, ϕ) f (x, ψ) f (x, ϕ) f (x, ψ) ϕψ − ϕψ = − 0= ϕψ ϕ ψ ϕ ψ Ω Ω Ω 3

Now, notice that f is such that f (x, s)/s is strictly decreasing. Thus, f (x, ϕ) f (x, ψ) −

= 0. ϕ ψ Therefore, must be ψ ≡ 0. The stability of the positive solution is clear.

Remark 5.3. For the uniqueness argument in the proof above we follow the argument in Stackgold and Payne [8] (see also Ex. 7, p. 127, in Henry [5]).

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References [1] H. Amann. Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Review, 18(4):620–709, 1976. [2] H. Berestycki and P.L. Lions. Some applications of the method of super and subsolutions. In Bifurcation and nonlinear eigenvalue problems (Proc., Session, Univ. Paris XIII, Villetaneuse, 1978), volume 782 of Lecture Notes in Math., pages 16–41. Springer, Berlin, 1980. [3] H. Br´ezis and Luc Oswald. Remarks on sublinear elliptic equations. Nonlinear Anal., Theory Methods Appl., 10:55–64, 1986. [4] D.G. de Figueiredo. Positive solutions of semilinear elliptic problems. In Diﬀerential equations (S˜ ao Paulo, 1981), volume 957 of Lecture Notes in Math., pages 34–87. Springer, Berlin, 1982. [5] D. Henry. Geometric Theory of Semilinear Parabolic Equations. Number 840 in Lecture Notes in Mathematics. Springer-Verlag, 1981. [6] P.L. Lions. On the existence of positive solutions of semilinear elliptic equations. SIAM Review, 24(4):441–467, 1982. [7] A. Lunardi. Analytic semigroups and optimal regularity in parabolic problems. Progress in Nonlinear Diﬀerential Equations and their Applications, 16. Birkh¨auser Verlag, Basel, 1995. [8] I. Stakgold and L.E. Payne. Nonlinear problems in nuclear reactor analysis. In Nonlinear problems in the physical sciences and biology (Proc. Battelle Summer Inst., Seattle, Wash., 1972), volume 322 of Lecture Notes in Math., pages 298–307. Springer, Berlin, 1979. An´ıbal Rodr´ıguez-Bernal and Alejandro Vidal-L´ opez Departamento de Matem´ atica Aplicada Universidad Complutense de Madrid E-28040 Madrid, Spain

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 517–528 c 2005 Birkh¨ auser Verlag Basel/Switzerland

A Remark on Continuous Coagulation-Fragmentation Equations with Unbounded Diﬀusion Coeﬃcients Christoph Walker Abstract. Continuous coagulation-fragmentation processes with diﬀusion are studied. It is shown that the parameter dependent diﬀusion term d(y)∆ generates an analytic semigroup in suitable state spaces even for unbounded diﬀusion coeﬃcients d(y). This yields existence and uniqueness of local-intime smooth solutions that are global for small initial values in the absence of fragmentation.

1. Introduction In the present paper we demonstrate how to extend a recent result of H. Amann and the author on diﬀusive continuous coagulation-fragmentation equations [4] to the case of unbounded diﬀusion coeﬃcients. Recall that these equations describe the time evolution of a system of a large number of particles that may change size due to coalescence or breakage. The mechanism leading to aggregation is assumed to be governed merely by Brownian motion. Applications of these processes can be found in various scientiﬁc and industrial disciplines, such as biology, physics, chemistry, or oil industry (e.g., see [9] and the references therein). More precisely, denoting by y the particle size and by u = u(y) = u(t, y, x) the particle size distribution function at time t and position x, the continuous version of the diﬀusive coagulation-fragmentation equations reads as t>0, y ∈ (0, y0 ) , ∂t u(y) − d(y)∆x u(y) = L[u](y) in Ω , on ∂Ω , t > 0 , y ∈ (0, y0 ) , ∂ν u(y) = 0 u(0, y, ·) = u0 (y) in Ω , y ∈ (0, y0 ) .

(1)

Here Ω ⊂ Rn is a bounded domain with smooth boundary ∂Ω and u0 = u0 (y, x) is a given initial distribution. The right-hand side L[u] := Lb [u] + Lc [u, u] + Ls [u, u]

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C. Walker

consists of the integral operators y0 γ(y , y) u(y ) dy − u(y) Lb [u](y) := 0

y

Lc [u, v](y) :=

1 2

y

0

y0

− u(y)

2y0

y0

− u(y)

y

K(y , y − y ) Q(y , y − y )

0

y

1 Ls [u, v](y) := 2

y γ(y, y ) dy , y

K(y , y − y ) P (y , y − y ) u(y − y ) v(y ) dy

1 + 2

y

× βc (y , y) u(y ) v(y − y ) dy dy y0 −y K(y, y ) P (y, y ) + Q(y, y ) v(y ) dy ,

0 y0

y −y y0 0

K(y , y − y ) βs (y , y) u(y ) v(y − y ) dy dy K(y, y ) v(y ) dy ,

y0 −y

for y ∈ (0, y0 ), where Lb , Lc , and Ls account for the formation and depletion of particles due to spontaneous breakage, coalescence and collisional breakage, and due to scattering, respectively. We refer to [4], [13] for a precise deﬁnition of the kernels and a more detailed interpretation of the above terms. By Y := (0, y0 ) we denote the admissible range for particle sizes, which is either unbounded if particles are allowed to become arbitrarily large, that is, if y0 = ∞, or bounded if y0 ∈ (0, ∞). In the former case of the classical coagulation-fragmentation model, the scattering operator Ls is identically zero. As in [4] we treat both cases simultaneously. We refrain from recalling the present state of research on continuous coagulation-fragmentation processes with diﬀusion, but refer instead to [2], [4], [5], [8], [13] and the references therein. It has been shown in [4] that the above system of equations possesses a unique smooth solution locally in time, which preserves the total mass. Moreover, this solution exists globally for small initial values provided that the linear fragmentation terms are neglected. These results rely on the fact that the analytic semigroup generated by the size-dependent diﬀusion term d(y)∆x on the state spaces L1 (Y, Lp (Ω)), 1 ≤ p < ∞, has smoothing properties. The basic assumption in [4] for the generation result and the regularizing eﬀects is that the diﬀusion coeﬃcients satisfy a bound of the form 0 < d ≤ d(y) ≤ d < ∞ ,

y∈Y .

The main motivation for the present paper is to relax this restriction in order to include unbounded coeﬃcients. We will show that neither the upper nor the lower bound for the diﬀusion coeﬃcients is necessary to obtain the generation result. Nevertheless, we point out that the lower bound d > 0 is crucial in our analysis to guarantee existence of smooth solutions. This bound provides a suitable control for the time singularities arising from the regularizing eﬀects.

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2. The diﬀusion semigroup We brieﬂy recall the most important notations and abbreviations already used in [4] and refer to [4] for more details. The abbreviation Lp := Lp (Ω) stands for the µ µ := Hp,B (Ω) for the Bessel potential spaces including Lebesgue spaces and Hp,B Neumann boundary conditions (if meaningful). For p ∈ [1, ∞) we denote by ∆p the (well-deﬁned) closure in Lp of the linear operator ∆ |C 2 (Ω) ¯ subject to Neumann boundary conditions, which generates a positive analytic semigroup {et∆p ; t ≥ 0} 2 of contractions on Lp . The domain of ∆p equals Hp,B provided p > 1. Since the t∆1 t∆p restriction of e to Lp coincides with e , we may put ∆ := ∆1 and obtain the estimates 1 1 α −n 2 ( p − q )− 2 , α ) ≤ c (1 ∧ t) et∆ L(Lp,Hq,B t>0, (2) for α ∈ [0, 2] \ {1 + 1/q} and 1 ≤ p ≤ q ≤ ∞, where q ∈ (1, ∞) if α > 0, and also − α ,H µ ) ≤ c (1 ∧ t) et∆ L(Hp,B p,B

µ−α 2

,

t>0,

(3)

for 1 < p < ∞ and 0 ≤ α ≤ µ ≤ 2 with α, µ = 1 + 1/p. We then set Lp := L1 Y, Lp , (1 + y)dy , 1 ≤ p ≤ ∞ , and

α Hα p,B := L1 Y, Hp,B , (1 + y)dy ,

with the convention of Lp .

H0p,B

1
:= Lp for p ∈ {1, ∞}. By

L+ p

α ∈ [0, 2] \ {1 + 1/p} , we denote the positive cone

Throughout this paper we assume for the diﬀusion coeﬃcients that d ∈ L1,loc (Y, (0, ∞))

(4)

and that there exists d with d(y) ≥ d > 0 ,

y∈Y .

(5)

If d satisﬁes (4), we deﬁne for each p ∈ [1, ∞) an operator Ap by virtue of (Ap u)(y) := −d(y)∆p u(y) ,

a.e. y ∈ Y ,

for u ∈ D(Ap ) := {u ∈ Lp ; u(y) ∈ D(∆p ) for a.e. y ∈ Y , d(·)∆p u ∈ Lp } . Observe then that, provided d additionally satisﬁes (5), the continuous injections 2 , d(y)(1 + y)dy → D(Ap ) → H2p,B , p ∈ (1, ∞) , (6) L1 Y, Hp,B hold. Finally, let the projection P ∈ L(Lp ) be deﬁned by 1 Pu := u(·, x) dx , u ∈ Lp , 1 ≤ p < ∞ , |Ω| Ω so that the space Lp has the direct sum decomposition Lp = P(Lp ) ⊕ (1 − P)(Lp ) .

(7)

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Note that P(Lp ) = L1 (Y, (1 + y)dy) ⊂ Lp . We put L•p := (1 − P)(Lp ) to shorten notation. Now we can prove analogous statements to [4, Thm. 2, Prop. 3] for diﬀusion coeﬃcients obeying (4) and (5). Theorem 1. Suppose d satisﬁes (4) and (5). Then −Ap generates a positive strongly continuous analytic semigroup of contractions on Lp for each p ∈ [1, ∞). It is given by (e−tAp u)(y) = etd(y)∆p u(y) , a.e. y ∈ Y , t ≥ 0 , u ∈ Lp , (8) and it holds that Ap ⊃ Aq for 1 ≤ p < q < ∞. Furthermore, the estimates 1

1

e−tAp L(Lp ,Hαq,B ) ≤ c(T ) t− 2 ( p − q )− 2 , n

α

0
(9)

for α ∈ [0, 2] \ {1 + 1/q} and 1 ≤ p ≤ q ≤ ∞, where q ∈ (1, ∞) if α > 0, and e−tAp L(Hαp,B ,Hµp,B ) ≤ c(T ) t−

µ−α 2

0
,

(10)

for 1 0 such that, for 1 0, 1

1

e−tAp |L•p L(L•p ,L•q ) ≤ M e−ω0 t t− 2 ( p − q ) , n

t>0.

(11)

Proof. Fix 1 ≤ p < ∞. Analogously to [4, Lem. 9] one shows that the tensor product D(Y ) ⊗ D(Ω) is dense in Lp , where D(X) denotes the test functions on an open subset X ⊂ Rm . Due to (4), D(Y ) ⊗ D(Ω) is contained in the domain of Ap . Therefore, Ap is densely deﬁned. That it is a closed operator in Lp follows from the fact that d(y)∆p is closed in Lp for a.e. y ∈ Y . Next we write Lp = R · 1 ⊕ L•p with L•p := (1 − P)(Lp ) as in the proof of [4, Prop. 3], which then decomposes ∆p into ∆p = 0 ⊕ ∆•p . Hereby, ∆•p := ∆p |D(∆p )∩L•p generates an analytic semigroup •

{et∆p = et∆p |L•p ; t ≥ 0} on L•p and

[Rez ≥ −ω] ⊂ (∆•p ) = (∆p ) ∪ {0} for some ω > 0, where (∆•p ) and (∆p ) denote the resolvent sets of the operators ∆•p and ∆p , respectively. Therefore, there exist N ≥ 1 and a number α ∈ (π/2, π) such that Σα := {z ∈ C \ {0} ; | arg z| < α} ⊂ (∆•p ) and N , λ ∈ Σα . (λ − ∆•p )−1 L(L•p ) ≤ |λ| Observing the resolvent decomposition @ −1 −1 A λ λ −1 −1 −1 d(y) • (λ − d(y)∆p ) = d(y) − ∆p ⊕ − ∆p = d(y) d(y) λ d(y)

Continuous Coagulation-Fragmentation Equations

521

for λ ∈ Σα and a.e. y ∈ Y , we derive (λ − d(y)∆p )−1 L(Lp ) ≤

N , |λ|

λ ∈ Σα ,

a.e. y ∈ Y,

for some N ≥ 1. Clearly, this implies that Σα belongs to the resolvent set of the operator −Ap and that the resolvent is given by (λ + Ap )−1 u (y) = (λ − d(y)∆p )−1 u(y) , a.e. y ∈ Y , u ∈ Lp , λ ∈ Σα . Hence we deduce that (λ + Ap )−1 L(Lp ) ≤

N , |λ|

λ ∈ Σα ,

so that well-known generation results (e.g., [6, Thm. 4.2.1]) then ensure that −Ap is the generator of a strongly continuous analytic semigroup on Lp for 1 ≤ p < ∞. Moreover, from [7, Thm. 11.6.6] we infer (8). Consequently, the semigroup generated by −Ap is a positive semigroup of contractions and satisﬁes the estimates (9) and (10) in view of (2), (3) and assumption (5). Finally, that e−tAp leaves both of the the spaces P(Lp ) and L•p invariant and that the estimate (11) holds can be shown exactly as in the proof of [4, Prop. 3]. Remark 2. Observe that (5) is required for the estimates (9)–(11) but not for the analyticity of the semigroup {e−tAp ; t ≥ 0}.

3. Well-posedness In the sequel, for ϑ ≥ 0 given, we say that hypothesis H(ϑ) is satisﬁed provided that (H1 ) K is a non-negative symmetric function deﬁned on Y × Y and there is k > 0 such that K(y, y ) d(y + y )ϑ + d(y)ϑ ≤ k , (y, y ) ∈ Y × Y . P and Q are non-negative and symmetric functions belonging to L∞ (Ξ), where Ξ := {(y, y ) ∈ Y × Y ; y + y ∈ Y }, such that 0 ≤ P (y, y ) + Q(y, y ) ≤ 1

for a.e. (y, y ) ∈ Ξ .

(H2 ) γ is a measurable function from {(y, y ) ; 0 < y < y < y0 } into R+ such that there exists mγ > 0 with y y y γ(y, y ) dy ≤ mγ for a.e. y ∈ Y . d(y )ϑ γ(y, y ) dy + d(y)ϑ 0 0 y (H3 ) βc is a non-negative measurable function on {(y, y ) ; 0 < y < y < y0 } such that y+y y βc (y + y , y ) dy − y − y = 0 for a.e. (y, y ) ∈ Ξ , Q(y, y ) 0

522

C. Walker and there exists mc > 0 with y+y Q(y, y ) d(y )ϑ βc (y + y , y ) dy ≤ mc

for a.e. (y, y ) ∈ Ξ .

0

(H4 ) βs is a measurable function from (y0 , 2y0 ) × (0, y0 ) into R+ such that y0 y βs (y + y , y ) dy = y + y for a.e. y + y ∈ (y0 , 2y0 ) , 0

and there exists ms > 0 with y0 d(y )ϑ βs (y + y , y ) dy ≤ ms

for a.e. y + y ∈ (y0 , 2y0 ) .

0

Clearly, hypothesis H(ϑ) with ϑ ≥ 0 implies hypothesis H(0) in view of (5). Let us mention right now that H(ϑ) ensures ’the gain of integrability’ (see [12, Lem. 2.6]) (12) L[u] ∈ L1 Y, d(y)ϑ (1 + y)dy for u ∈ L1 (Y, (1 + y)dy) . Also note that hypothesis H(0) coincides with the assumptions made in [4]. We therefore refer to [4, Ex. 1] for kernels obeying H(0). If, in the case y0 < ∞, the kernels are of the form 0 < y < y < y0 , γ(y, y ) ∝ y α−ξ−1 (y )ξ , −1−ζ ζ βc (y, y ) := (ζ + 2) y (y ) , 0 < y < y < y0 , −2−ν ν βs (y, y ) := (ν + 2) y0 y (y ) , 0 < y < y0 ≤ y < 2y0 , τ (y, y ) ∈ Ξ , Q(y, y ) ∝ (y + y ) , µ K(y, y ) ∝ (yy ) , y, y ∈ Y , with 0 ≥ ξ, ζ, ν > −1 and α, µ, τ > 0, we can choose ϑ > 0 suﬃciently small such that hypothesis H(ϑ) is satisﬁed for diﬀusion coeﬃcients d(y) ∼ y −λ , λ > 0. In order to show that problem (1) is well posed, we assume in the sequel that (4), (5), and at least hypothesis H(0) are satisﬁed. Then it follows, for 1 ≤ p < ∞, that (13) Lb ∈ L(Lp ) and G := Lc + Ls ∈ L2 (L2p , Lp ) since the pointwise product L2p × L2p → Lp is a multiplication. If J denotes an interval in R+ containing 0, we put J˙ := J \ {0}. We then call u ∈ C(J, Lp ) a mild Lp -solution to the re-written problem u˙ + Ap u = L[u] ,

t>0,

u(0) = u0 ,

(14)

provided it solves the ﬁxed point equation u(t) = U (t)u0 + U L[u](t) in Lp , where U (t) := e

−tA

, t ≥ 0, with A := A1 and t U v(t) := U (t − s) v(s) ds ,

t∈J ,

t∈J .

0

˙ D(Ap )). A mild solution is a strong Lp -solution if u ∈ C 1 (J˙, Lp ) ∩ C(J,

(15)

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523

Given a Banach space E and µ ∈ R, we denote by BCµ (J˙, E) the Banach space of all functions u : J˙ → E such that t → tµ u(t) is bounded and continuous from J˙ into E, equipped with the norm := sup tµ u(t)E . u → uBCµ (J,E) ˙ t∈J˙

We write Cµ (J˙, E) for the closed linear subspace thereof consisting of all u satisfying tµ u(t) → 0 in E as t → 0. Due to Theorem 1 we have (see [4, Prop. 4]) Proposition 3. Let 1 ≤ p ≤ q ≤ ∞ and α ∈ [0, 2) \ {1 + 1/q} be such that n(1/p − 1/q)/2 + α/2 < 1 and either q ∈ (1, ∞) or α = 0. Then, for µ < 1, ˙ Hα u → U Lb [u] ∈ L Cµ (J˙, Lp ), Cµ+ n2 ( p1 − 1q )+ α2 −1 (J, (16) q,B ) and

˙ Hα ) . u → U G[u, u] ∈ L2 Cµ/2 (J˙, L2p ), Cµ+ n2 ( p1 − 1q )+ α2 −1 (J, q,B

(17)

Moreover, the analogue of [4, Prop. 6]) is still valid: Proposition 4. Let 1 0 and q ∈ (1, ∞). Then, for u0 ∈ Lp , ˙ Hα U u0 := t → U (t)u0 ∈ C n2 ( p1 − q1 )+ α2 (J, q,B ) . Proof. Owing to (6), n(1/p − 1/q)/2 + α/2 < 1, and classical embedding arguments we have D(Ap ) → H2p,B → Hα q,B . Hence, since U (t)u0 ∈ D(Ap ), t > 0, due to Theorem 1, we deduce 2 e(t+h)d(y)∆u0 (y) − etd(y)∆ u0 (y)Hp,B (1 + y) dy U (t + h)u0 − U (t)u0 Hαq,B ≤ c Y

for 0 ≤ |h| < t. Taking (2) and (5) into account and that {eτ ∆ ; τ ≥ 0} is a 2 in view of [1, V. Thm. 2.1.3], bounded and strongly continuous semigroup on Hp,B we infer from Lebesgue’s theorem that the right-hand side of the above inequality tends to zero as h → 0, and thus U u0 ∈ C(J˙, Hα q,B ). Theorem 1 then yields that the ζ 0 ˙ α map t → t U (t)u Hq,B remains bounded on J for ζ := n(1/p − 1/q)/2 + α/2 < 1. The fact that Hα is dense in Lp implies U u0 ∈ Cζ (J˙, Hα ) as in the proof q,B

of [4, Prop. 6].

q,B

Under hypothesis H(0) we can prove now existence and uniqueness of mild solutions to problem (14). Subsequently, we will derive more regularity (with respect to time) for these solutions.

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C. Walker

Our global-in-time statement requires that collisional breakage is dominated by coalescence, i.e., that y+y Q(y, y ) βc (y + y , y ) dy − 2 ≤ P (y, y ) , y + y ∈ Y , (18) 0

and that scattering is a binary processes, meaning that βs (y, y ) = βs (y, y − y ) ,

(y, y ) ∈ Ξ ,

(19)

0 < y < y − y0 < y0 .

(20)

and βs (y, y ) = 0 ,

Then we have the following existence and uniqueness result. Theorem 5. Let (4), (5), and hypothesis H(0) be satisﬁed and assume that (n/2 ∨ 1) < p < ∞. Then, given any non-negative initial value u0 ∈ Lp , problem (14) possesses a unique maximal non-negative mild Lp -solution u := u(·; u0 ) on an open interval J(u0 ) ⊂ R+ such that tn/4p u(t)L2p → 0 as t → 0+ . In addition, u has the regularity ˙ 0 ), H2 ) , u ∈ C(J(u q,B

q ∈ (1, ∞) .

(21)

If t+ := sup J(u0 ) < ∞, then sup t+ /2
u(t)Lq = ∞ ,

(n/2 ∨ 1) < q < ∞ .

The solution u preserves the total mass, that is, y u(t, y, x) dydx = y u0 (y, x) dydx , Ω

Y

Ω

t ∈ J(u0 ) .

(22)

(23)

Y

Finally, if γ ≡ 0 and (18)–(20) are satisﬁed, then, for p ∈ (n/2, ∞) and p ≥ 2, the solution remains bounded in Lp whenever u0 Lp is small. In particular, u exists globally in time in this case. Proof. The proofs of the existence and uniqueness parts of this theorem including the blow-up behavior (22) are, verbatim, the same as the ones of [4, Thm. 7] (see step (i) and (iii) of the proof of [4, Thm. 7] and Propositions 3 and 4). For the ˙ 0 ) and ε > 0 regularity result (21) we may focus on q > p. We choose T ∈ J(u suﬃciently small and derive L[u(· + ε)] ∈ C([0, T ], Hνq,B ) and u ∈ C((0, T ], Hνq,B ) for some ν > 0. This can be done as in the second step of the proof of [4, Thm. 7] in view of Propositions 3 and 4. Since U is strongly continuous on Hνq,B due to Lebesgue’s theorem (see the proof of Proposition 4), estimate (10) implies U L[u(· + ε)] ∈ C([0, T ], H2q,B ) and U u(ε) ∈ C((0, T ], H2q,B ) .

Continuous Coagulation-Fragmentation Equations

525

Due to t ∈ [0, T ] ,

u(t + ε) = U (t)u(ε) + U L[u(· + ε)](t) ,

(24)

assertion (21) follows since ε > 0 was arbitrary. ˙ 0 ), the mild solution u = u(·; u0 ) is bounded on Next observe that, for T ∈ J(u α α [0, T ] with values in Hq,B provided u0 ∈ Hα q,B . Since Hq,B embeds continuously in L∞ for q suﬃciently large and since u solves the ﬁxed point equation u(t) = Uω (t)u0 + Uω (L[u] + ωu)(t) ,

t ∈ J(u0 ) ,

we deduce that u is non-negative for non-negative initial values u0 ∈ Hα q,B as in [4, Thm. 10]. Here, the constant ω > 0 suitably depends on sup0≤t≤T u(t)L∞ and Uω is given by Uω (t) := e−ωt U (t). The continuous dependence of u(·; u0 ) on + + u0 and the density of Hα q,B ∩ Lp in Lp (see [4, Lem. 9]) yields then the positivity assertion. The conservation of mass formula (23) follows from (15) by taking into account the equality y [U (t)v](y) dx dy = y etd(y)∆ v(x, y) dx dy Y Ω Y Ω y v(x, y) dx dy , t > 0 , v ∈ Lp , = Y

Ω

which is due to the Neumann boundary conditions, and observing that y L[v](y) dy = 0 , v ∈ L1 (Y, ydy) , Y

(see hypotheses (H1 )–(H4 ) and [12, Lem. 2.6]). Finally, based on estimate (11), the global-in-time existence result is obtained as in [4, Thm. 16] noticing that the proof of the latter merely requires mild solutions. This proves the assertion. Before we prove that hypothesis H(ϑ) with ϑ > 0 provides strong solutions, let us show that also the weaker assumption H(0) implies that the original equation (1) is satisﬁed pointwise almost everywhere. More precisely, deﬁning Lq := L1 (Y, Lq , dµ(y)) with

dµ(y) := d(y)−1 (1 + y)dy ,

we have Theorem 6. The mild solution u = u(·; u0 ) provided by Theorem 5 belongs to ˙ 0 ), Lq ) for each q ∈ [1, ∞) and it holds that C 1 (J(u u(t) ˙ = d∆u(t) + L[u(t)] in Lq ,

˙ 0) . t ∈ J(u

In particular, for almost every y ∈ Y , it holds that u(t, ˙ y) − d(y)∆u(t, y) = L[u(t)](y) in Lq ,

˙ 0) . t ∈ J(u

˙ 0 ) and ε > 0 suﬃciently small and set Proof. Fix T ∈ J(u uε := u(· + ε) ∈ C([0, T ], H2q,B ) ,

q ∈ (1, ∞) .

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C. Walker

Then there is ν > 0 such that fε := L[uε ] ∈ C([0, T ], Hνq,B ) (see [10, Cor. 4.5.2]). Given t ∈ [0, T ] and h > 0 suﬃciently small, we deduce −1 h U (t + h)u(ε) − U (t)u(ε) + A U (t)u(ε)L q t+h ∆ eτ d(y)∆ − etd(y)∆ uε (y) dτ dµ(y) ≤ h−1 d(y) Lq Y

≤ h−1

t t+h

t

Y

eτ d(y)∆ − etd(y)∆ uε (y) 2 H

dτ (1 + y)dy .

q,B

Since {eτ d(y)∆ ; τ ≥ 0} is for a.e. y ∈ Y a bounded and strongly continuous 2 semigroup on Hq,B , we may apply Lebesgue’s theorem to deduce that the righthand side of the above inequality tends to zero as h does. Hence ∂+ U (t)u(ε) = −A U (t)u(ε) in Lq ,

t ∈ [0, T ] ,

(25)

and, since by the same arguments ∂+ U u(ε) ∈ C([0, T ], Lq ), we obtain that U u(ε) belongs to C 1 ([0, T ], Lq ) with derivative given by (25). Next, for h > 0 small and t ∈ (0, T ], we write h−1 U fε (t + h) − U fε (t) t −1 −1 U (t + h − s) − U (t − s) fε (s) ds + h =h 0

t+h

U (t + h − s)fε (s) ds t

=: Mh + Nh . Then, as above, we compute Mh + A (U fε )(t)Lq t t+h (τ −s)d(y)∆ e ≤ ch−1 − e(t−s)d(y)∆ fε (s, y)H 2 dτ ds (1 + y)dy . Y

0

q,B

t

According to (3) and (5), for h ≤ 1 and 0 < s < t ≤ T , it holds that h

−1

t

t+h

(τ −s)d(y)∆ e − e(t−s)d(y)∆ fε (s, y)H 2

dτ

q,B

≤c

sup t≤τ ≤t+1

−1+ν/2 ν 1 ∧ (τ − s)d(y) fε (s, y)Hq,B

ν ≤ c(T ) (t − s)−1+ν/2 fε (s, y)Hq,B

so that we may again apply Lebesgue’s theorem to deduce Mh −→ −A (U fε )(t) in

Lq ,

h → 0+ .

(26)

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527

On the other hand, we have Nh − fε (t)Lq t+h −1 ≤h e(t+h−s)d(y)∆L(Lq ) fε (s, y) − fε (t, y)Lq ds dµ(y) Y

+ h−1

t

Y

≤ d−1

sup

Lq

t

t≤s≤t+h

+ d−1

e(t+h−s)d(y)∆fε (t, y) − fε (t, y)

t+h

ds dµ(y)

fε (s) − fε (t)Lq sup

Y 0<τ
eτ ∆fε (t, y) − fε (t, y)Lq (1 + y) dy .

Lebesgue’s theorem shows that the right side converges towards zero as h → 0+ , whence Nh → fε (t) in Lq . Taking (26) into account we obtain ∂+ (U fε )(t) = −A U fε (t) + fε (t) in Due to (3), (5), and the strong continuity of U on

Lq ,

Hνq,B ,

t ∈ (0, T ] .

(27)

we derive

∂+ (U fε ) ∈ C((0, T ], Lq ) and U fε ∈ C((0, T ], Lq ) . Thus U fε ∈ C 1 ((0, T ], Lq ). Recalling (25), we infer from equation (24) that uε ∈ C 1 ((0, T ], Lq ). Since ε > 0 was arbitrary, the assertion follows. Stronger assumptions on the kernels ensure that the mild solutions actually are strong solutions as the concluding theorem shows. Theorem 7. Suppose that H(ϑ) holds with ϑ > 0. Then, given the assumptions of Theorem 5, the mild solution u = u(·; u0 ) is a strong Lp -solution to problem (14) and has the additional regularity ˙ 0 ), Lq ) ∩ C(J˙(u0 ), D(Aq )) , q ∈ (1, ∞) . u ∈ C 1 (J(u Proof. We may assume that q ∈ (1, ∞) is large. By [·, ·]θ we denote the complex interpolation functor of exponent θ ∈ (0, 1). Then, in view of [11, Thm. 1.18.4], we may follow the lines of the second step of the proof of [11, Thm. 1.18.5] to show that % & 2 2θ L1 (Y, Lq , (1 + y)dy), L1 (Y, Hq,B , d(y)(1 + y)dy) θ = L1 Y, Hq,B , d(y)θ (1 + y)dy provided 2θ ∈ (0, 1) \ {1 + 1/q}. Consequently, (6) implies, for 2θ = 1 + 1/q, % & 2θ L1 Y, Hq,B , d(y)θ (1 + y)dy → Lq , D(Aq ) θ =: Eθ . (28) Let now u ∈ C(J(u0 ), Lp ) ∩ C(J˙(u0 ), H2q,B ) be the mild solution provided by Theorem 5. Since q is large, we infer from [10, Cor. 4.5.2] that there is some ν > 0 2 2 2ν small such that the pointwise product Hq,B × Hq,B → Hq,B is continuous. By making ϑ smaller if necessary, we may assume that ν = ϑ. Therefore, hypothesis ˙ 0 ) with values in H(ϑ) guarantees that the map t → L[u(t)] is continuous on J(u 2ϑ ϑ L1 (Y, Hq,B , d(y) (1 + y)dy) (see (12)). Taking (28) into account we thus obtain

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L[u] ∈ C(J˙(u0 ), Eϑ ). As in part (ii) of the proof of [4, Thm. 7] we can now shift the equation (14) and apply [1, Thm. IV.1.5.1] in order to deduce that u(· + ε) is a strong Lq -solution for each ε > 0 small. This proves the claim. Acknowledgement I would like to thank G. Simonett for fruitful discussions on the subject.

References [1] H. Amann. Linear and quasilinear parabolic problems. Volume I: Abstract linear theory. Birkh¨ auser. Basel, Boston, Berlin 1995. [2] H. Amann. Coagulation-fragmentation processes. Arch. Rat. Mech. Anal. 151 (2000), 339–366. [3] H. Amann, J. Escher. Strongly continuous dual semigroups. Ann. Mat. Pura Appl. CLXXI (1996), 41–62. 151 (2000), 339-366. [4] H. Amann, C. Walker. Local and global strong solutions to continuous coagulationfragmentation equations with diﬀusion. To appear in: J. Diﬀ. Eq. [5] H. Amann, F. Weber. On a quasilinear coagulation-fragmentation model with diﬀusion. Adv. Math. Sci. Appl. 11 (2001), 227–263. [6] H.O. Fattorini. The Cauchy problem. Addison-Wesley Publishing Company. London, Amsterdam 1983. [7] E. Hille, R.S. Phillips. Functional Analysis and Semigroups. American Mathematical Society Colloquium Publications. Volume XXXI (Revised edition). Providence, Rhode Island, 1957. [8] P. Lauren¸cot, S. Mischler. The continuous coagulation-fragmentation equations with diﬀusion. Arch. Rat. Mech. Anal. 162, No. 1 (2002), 45–99. [9] P. Lauren¸cot, S. Mischler. On coalescence equations and related models. In: P. Degond, L. Pareschi, G. Russo (eds.), Modeling and computational methods for kinetic equations. Birkh¨ auser. Boston 2004. [10] T. Runst, W. Sickel. Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial diﬀerential equations. De Gruyter Series in Nonlinear Analysis and Applications 3. De Gruyter. Berlin, New York 1996. [11] H. Triebel. Interpolation theory, function spaces, diﬀerential operators. Second edition. Johann Ambrosius Barth. Heidelberg, Leipzig 1995. [12] C. Walker. Coalescence and breakage processes. Math. Meth. Appl. Sci. 25 (2002), 729–748. [13] C. Walker. On a new model for continuous coalescence and breakage processes with diﬀusion. Adv. Diﬀ. Equ. 10, no. 2 (2005), 121–152. Christoph Walker Vanderbilt University Department of Mathematics 1326 Stevenson Center Nashville, TN 37240, USA e-mail: [email protected]

Progress in Nonlinear Diﬀerential Equations and Their Applications, Vol. 64, 529–536 c 2005 Birkh¨ auser Verlag Basel/Switzerland

On Lp-Estimates of Optimal Type for the Parabolic Oblique Derivative Problem with VMO-Coeﬃcients – A Reﬁned Version Peter Weidemaier Dedicated to Prof. Dr. Herbert Amann

Abstract. We prove Wp2,1 (ΩT )-estimates (1 < p < ∞) for parabolic operators with a second-order elliptic part in non-divergence with essentially +form n b (ξ, t)∂i u(ξ, t) = bounded VMO-coeﬃcients. The boundary condition i i=1 g(ξ, t) on ∂ΩT is considered in the non-degenerate case, and the bi are only assumed to be in space-time Sobolev-spaces (see condition (B)).

1. Introduction As in [7] we consider the initial-boundary value problem (IBVP) ut − aij (x, t)∂i ∂j u = u(x, 0) =

f (x, t) u0 (x)

in ΩT in Ω

(1) (2)

b(ξ, t), ∇x u =

g(ξ, t)

on ΓT ,

(3)

+n

where a, b := i=1 ai bi and Ω ⊂ Rn . The focus is on weak assumptions for the coeﬃcients, which still allow estimates of the solution in Wp2,1 spaces. As in [7] we assume that aij ∈ L∞ ∩ V M O(ΩT ). While in [7] we assumed bi ∈ C α, α/2 (∂ΩT ) for some α > 1 − 1/p, in the present work the bi are assumed to be in space-time Sobolev spaces (see condition (B)). The method of proof is similar to that in [7]. Let us introduce some notation : Ω denotes an open subset of Rn with boundary Γ := ∂Ω, while ΩT stands for Ω × (0, T ). The conjugate exponent of p is denoted p , p1 + p1 = 1. The Sobolev-Slobodeˇcki spaces are denoted Wps , s > 0 with norm · s,p . They coincide with the usual Sobolev spaces for integer s and are

530

P. Weidemaier

deﬁned for non-integer s and X-valued functions, X a Banach space, as Wps (Ω, X) = {f ∈ Wp[s] (Ω, X) | |∂ α f |W s−[s] ( Ω,X) < ∞ ∀ |α| = [s] } , p

where [s] is the biggest integer smaller s and for θ ∈ (0, 1) ⎛ ⎞1/p p f (x) − f (y) X dxdy ⎠ . |f |Wpθ (Ω,X) := ⎝ |x − y|n+θp Ω Ω

+ The norm in is given by f W [s] (Ω,X) + |α|=[s] |∂ α f |W s−[s] (Ω,X) . The p p Sobolev-Slobodeˇcki spaces on manifolds are deﬁned by local charts. In particular, αi Γ, i = 1, . . . , N are local charts whose images cover Γ and if Rn−1 ⊃ Vi −→ (φi )i=1,...,N is a smooth partition of unity on Γ with supp φi ⊂ αi (Vi ), then Wps (Ω, X)

Wps (Γ) = { f ∈ Lp (Γ) | α∗i (φi f ) ∈ Wps (Vi ) all i }, f Wps (Γ) :=

N

α∗i (φi f )Wps (Vi ) ,

i=1

α∗i

where is the pull-back deﬁned by αi , i.e., α∗i (g) = g ◦ αi . The spaces Lp (Γ) are also deﬁned by local charts in a similar way. We write · p,Γ for · Lp (Γ) . Furα,β k,m (ΓT ) := Lq ( 0, T ; Wpα (Γ)) ∩ Wqβ ( 0, T ; Lp (Γ)) and Wpk,m = Wp,p . thermore, Wp,q Finally we denote 1/q T q |f |Wp,q α,β |f (·, t)|W α (Γ) dt + |f |Wqβ ( 0,T ; Lp (Γ)) . (ΓT ) := 0

p

The H¨older spaces are denoted C α with seminorm [·]α , resp. C α,β for space-time H¨older spaces. Furthermore, Lip(ΓT ) denotes the globally Lipschitz-continuous functions on ΓT . Finally, c denotes a constant which may vary from line to line and the notation c∗ is used to emphasize the uniform boundedness of the constant c(T ) for T ≤ 1.

2. Main result We formulate condition (B). 1− 1 +δ,(1− 1 +δ)/2

p bi ∈ Wq1 ,qp2 (ΓT ), where q1 , q2 can be any of the following numbers parametrized by θ:

n−1 q1 = θ−1/p q1 > p q1 = p

and q2 =

2p 1−θ

and δ > 0 and δ > 0 and δ = 0 in each case.

if θ ∈ ( p1 , min(1, np ) ) if θ = np < 1 if θ ∈ ( np , 1 )

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(B)

On Lp -Estimates for the Parabolic Oblique Derivative Problem

531

Theorem 2.1. Assume that Ω is a bounded C 1,1 -domain (see Kufner et al. [3], 6.2.2. Deﬁnition) in Rn with ν denoting the vector ﬁeld of exterior unit normals on Γ. Assume further that (A) A(x, t) := (aij (x, t))1≤i,j≤n is symmetric and positive deﬁnite uniformly in Ω × [ 0, T ], say A(x, t) ≥ λIwith λ > 0, aij (x, t) ∈ L∞ ∩ V M O(ΩT ) (E) 1 0 on ΓT (“nondegeneracy condition”) and the bi satisfy assumption (B). (F) f ∈ Lp (ΩT ) 1−1/p,(1−1/p)/2 (G) g ∈ Wp (ΓT ) 2−2/p (Iv) u0 ∈ Wp (Ω) (C) b(·, 0), ∇u0 (·) = g(·, 0) on Γ for p > 3 . Then (1) → (3) has a unique solution u ∈ Wp2,1 (ΩT ) and there is a constant c∗ (p, n, λ, ∂Ω, T, b 1− p1 +δ,(1− p1 +δ)/2 , aij L∞ ∩V MO ) such that (ΓT )

Wq1 ,q2

∗

uWp2,1 (ΩT ) ≤ c · (u0 W 2−2/p (Ω) + f Lp (ΩT ) + gW 1−1/p,(1−1/p)/2 (ΓT ) ) . p

p

(4)

Remark 2.2. i.) The theorem is a reﬁnement of previous results of Softova ([6], [5]) who assumed boundary coeﬃcients bi ∈ Lip (∂ΩT ), and of the present author ([7]) who assumed bi ∈ C α, α/2 (∂ΩT ) for some α > 1 − 1/p. ii) Condition (B) implicitly implies that the coeﬃcients are H¨older-continuous in space-time: Indeed an easy calculation based on the interpolation inequality θ f σ, q1 , Ω ≤ c · f 1−θ q1 , Ω f s1 ,q1 , Ω ,

∀f ∈ Wqs11 (Ω) ,

where σ = θs1 ∈ / N, θ ∈ [ 0, 1 ] and q1 > 1 (cf. [2], 4.1 Proposition), together with H¨ older’s inequality shows that α σ + < 1, σ ∈ / N. Wqs11,q,s22 (ΓT ) → Wqα2 (0, T ; Wqσ1 (Γ)) if s2 s1 But the last space embeds continuously into C β2 (0, T ; C β1 (Γ)), if α − q12 > β2 and σ − n−1 q1 > β1 , by standard Sobolev embeddings (cf. [1]). Altogether such positive β s can be chosen iﬀ n−1 1 + < 1. s2 q2 s1 q1 This condition is satisﬁed in all cases in (B). However, for δ and q1 small, the H¨ older exponents β1 , β2 thus obtained are smaller than 1 − 1p , (1 − 1p )/2 respectively. Proof. The proof is by localizing the problem on small domains as detailed in [7], i.e., we consider the boundary value problems fulﬁlled by uk = uφk with φk (k ≥ 1) being a smooth function from a partition of unity on Γ with supp φk ⊂ BR (ξk ), ξk ∈ Γ. We may assume ∇φk ∞ ≤ C/R. The parameter R ∈ ( 0, 1 ) will be chosen small later. Then uk satisﬁes the boundary condition b(ξk , 0), ∇uk = gk + g¯

with

g¯ := b(ξk , 0) − b(ξ, t), ∇uk + b, ∇φk u .

532

P. Weidemaier

The key estimate in [7] for the perturbation terms in g¯ was that for all ε > 0 there exist R(ε), T (ε, R) such that 1 ¯ g W 1−1/p,(1−1/p)/2 (ΓT ) ≤ ε uk Wp2,1 (ΩT ) + uWp2,1 (ΩT ) + c( )u0 p, Ω . (5) p ε This estimate then implies (4) in the sense of an a priori estimate (see [7]), and the rest of the proof is along the lines of [7]. Estimate (5) follows from Lemma 3.1 applied to each of the two terms of the sum in g¯. In this lemma we choose ψ such that ψ BR (ξk ) ≡ 1, supp ψ ⊂ B2R (ξk ), ∇ψ∞ ≤ C/R, [ψ]α ≤ C/Rα . As for the ﬁrst term of g¯ we observe that, by the uniform continuity of b on ΓT , (b(ξk , 0) − b)ψ∞, ΓT ≤ ε provided R(ε), T (ε) are suﬃciently small. From now on R will be kept ﬁx. As for the second term in g¯ we apply Lemma 3.1 with m2 (ξ, t) := b(ξ, t), ∇φk (ξ) and v = u. Then we estimate b∞, ΓT R uW 1−1/p,(1−1/p)/2 (ΓT ) ≤ c∗ uLp (0,T ; Wp1 (Ω) + T 1/2 uWp2,1 (ΩT ) , m2 ψ∞, ΓT ≤ c∗ p

(6)

where in the second estimate we used uLp (0,T ; W 1−1/p (Γ) ≤ c∗ uLp (0,T ; Wp1 (Ω) , p the estimate |u|Wpβ1 (0,T ; Lp (Γ)) ≤ T β2 −β1 |u|Wpβ2 (0,T ; Lp (Γ)) ,

(7)

1−1/(2p)

for 0 < β1 < β2 < 1 and the embedding Wp2,1 (ΩT ) → Wp (0, T ; Lp (Γ)) (cf. [4], Chapter II, Lemma 3.4.). Estimate (7) follows directly from the deﬁnition of the seminorms and is applied with β1 = (1 − 1/p)/2, β2 = 1 − 1/(2p). Using in (6) that uLp (0,T ; Wp1 (Ω) ≤ εuWp2,1 (ΩT ) + T 1/p c(1/ε)u0p, Ω for T ≤ T0 (ε) (cf. Lemma 3.4), we see that m2 ψ∞, ΓT uW 1−1/p,(1−1/p)/2 (ΓT ) can p be estimated by the right-hand side in (5).

3. Some tools from calculus First we prove a result on estimating products of functions in the Sobolev spaces on the boundary. Lemma 3.1. Assume that u ∈ Wp2,1 (ΩT ), p ∈ ( 1, ∞ ) and that m = m(ξ, t) belongs to the class described in (B). Let ψ be a smooth function on Γ with supp ψ ≡ 1 on supp u(·, t) for a.e. t ∈ ( 0, T ), 0 ≤ ψ ≤ 1. Then it holds for

On Lp -Estimates for the Parabolic Oblique Derivative Problem

533

v ∈ {u, ∂j u | j = 1, . . . , n:

mvW 1−1/p,(1−1/p)/2 (ΓT ) ≤ c mψ∞, ΓT vW 1−1/p,(1−1/p)/2 (ΓT ) p p

1 1 +|m| 1− p1 +δ,(1− p1 +δ)/2 (ε + c( )T 1/p )uWp2,1 (ΩT ) + c( )u0 p, Ω , ε ε Wq1 ,q2 (ΓT )

where ε > 0 is arbitrary and c( 1ε ) is a constant that increases with decreasing ε. Proof. The Lemma follows from Lemma 3.2, applied with s = 1 − 1/p, v = ∂j u and v = u, and Lemma 3.3. Lemma 3.2. Let ψ be as in the preceding Lemma and let s ∈ ( 0, 1 ), p, r ∈ ( 1, ∞ ), q ∈ ( 1, ∞ ]. Then it holds |mv|W s, s2 (Γ

T)

≤ c mψ∞, ΓT |v|W s, 2s (Γ

T)

+ |m|

s+δ, s+δ

Wpq ,pr2 (ΓT )

vLpr (0,T ; Lpq (Γ))

.

Here δ > 0 is arbitrary, if q < ∞ and δ = 0, if q = ∞. Proof. We give the proof for q < ∞. The case q = ∞ follows by obvious modiﬁcations. Suppressing the t-argument of v for convenience, we have ⎛ ⎞1/p ∗ ∗ p |αi (φi mv)(x) − αi (φi mv)(y)| |α∗i (φi mv)|Wps (Vi ) = ⎝ dx dy ⎠ . |x − y|n−1+sp Vi Vi

Since vψ ≡ v on Γ, we replace v by vψ in the last identity. Using f g(x) − f g(y) = [f (x) − f (y)] g(x) + [g(x) − g(y)] f (y) with f = α∗i (φi v), g = α∗i (m ψ) the last expression is majorized by ⎛ ⎞1/p ∗ ∗ p |α (m ψ)(x) − α (m ψ)(y)| i i mψ∞, Γ |v|Wps (Γ) + ⎝ |α∗i (φi v)(y)|p dx dy ⎠ . |x − y|n−1+sp Vi Vi

We continue to estimate the last integral. Splitting n−1 n−1 − δp] + (s + δ)p] + [ q q 1 1 with + = 1, q > 1 and δ > 0 q q

n − 1 + sp = [

and applying H¨ older’s inequality with q, q this integral is estimated by ⎛ ⎞1/q ⎛ ⎞1/q ∗ ∗ pq ∗ pq |α (m ψ)(x) − α (m ψ)(y)| |α (φ v)(y)| i i i i ⎝ ⎝ dx dy ⎠ dx dy ⎠ . |x − y|n−1−δpq |x − y|n−1+(s+δ)pq Vi Vi

Vi Vi

534

P. Weidemaier

The second factor can be estimated by ⎞1/q ⎛ ⎝ |α∗i (φi v)(y)|pq dy ⎠ , since sup y∈Vi

Vi

dx < ∞. |x − y|n−1−δpq

Vi

By the estimates derived so far, we get for functions depending on t the estimate 1/p 1/p T T p p |m(t)v(t)|W s (Γ) dt ≤ mψ∞, ΓT |v(t)|W s (Γ) dt p

0

+c 0

p

0

1/p

T

|m(t)ψ(t)|pW s+δ (Γ) v(t)ppq, Γ dt pq

.

After applying H¨ older’s inequality with r, r in the t-integral, the last term can be estimated by cmψL

pr

s+δ vLpr (0,T ; Lpq (Γ)) (0,T ; Wpq (Γ))

.

Next we treat the terms corresponding to |mv|W s/2 (0,T ; Lp (Γ)) analogously. We p have to consider 1/p T T [m(t) − m(τ )]v(t)p p, Γ dτ dt . (8) s |t − τ |1+ 2 p 0 0 After application of H¨ older’s inequality with q , q in the integral over Γ, splitting 1 the term with |t−τ according to | 1 1 s s 1 + p = ( + ( + δ)p) + ( − δp) with δ > 0 2 r 2 r and applying H¨ older’s inequality with r , r in the τ − t-integral, the expression in (8) can be estimated by

T

T

s

0

0

1/p r

m(t) − m(τ )pr pq , Γ |t − τ |1+( 2 +δ)p r

T

dτ dt 0

0

T

v(t)pr pq, Γ |t − τ |1−δpr

1/p r dτ dt

.

Carrying out the integration with respect to τ in the last integral, we arrive at the desired result. Lemma 3.3. For u ∈ Wp2,1 (ΩT ) the inequality uLpr (0,T ;Lpq (Γ)) + ∇x uLpr (0,T ; Lpq (Γ)) 1 ≤ εuWp2,1 (ΩT ) + c( ) u0 p, Ω + T 1/p ∂t up, ΩT ε

On Lp -Estimates for the Parabolic Oblique Derivative Problem

535

holds, where ε > 0 is arbitrary, c( 1ε ) is a constant that increases with decreasing ε and r, q are as follows: if θ ∈ ( p1 , min(1, np ) ) if θ = np < 1 if θ ∈ ( np , 1 )

n−1 q = n−θp q<∞ q=∞

and r =

2 1+θ

in each case.

Proof. By standard embedding results (cf. [1], 5.14 Lemma, 5.16 Corollary) Wpθ (Ω) → Wpθ−1/p (Γ) → Ls (Γ) for

1 1 1 θ − 1/p 1 ≥ ∗ := − , if θ > and s∗ > 0 . s s p n−1 p

The last condition is equivalent with θp < n. If s∗ = 0, s arbitrary large is allowed, and if s∗ < 0, s = ∞ is allowed. Putting s = pq, we thus have 1+θ

1−θ

u(t)pq, Γ + ∂j u(t)pq, Γ ≤ c · u(t)1+θ, p, Ω ≤ c · u(t)2,2p, Ω u(t)p,2Ω for q as given in the Lemma and θ < 1. The last estimate is the interpolation 2 inequality (cf. [2], 4.1 Proposition). From the last estimate we get for r = 1+θ and v ∈ {u, ∂j u} T T p(r−1) pr v(t)pq, Γ d t ≤ c u(t)p2, p, Ω u(t)p, Ω d t, 0

0

since

1−θ 1 pr = , = p(r − 1) . 2 r r

Inserting the estimate

sup u(t)p, Ω ≤ u0 p, Ω + T 1/p ∂t u(t)p, ΩT ,

t∈[ 0,T ]

which follows easily by H¨older’s inequality from t ∂t u(·, s)d s , u(·, t) = u0 (·) + 0

we get 1/r

vLpr (0,T ; Lpq (Γ)) ≤ uW 2,1 (Ω p

T

1/r 1/p u + T ∂ u . 0 p, Ω t p, ΩT )

H¨older’s inequality “with ε” with r, r yields the assertion of the Lemma.

Lemma 3.4. Let 1 ≤ p < ∞ and 0 < s < 2. For every ε > 0 there exists T0 (ε) such that for T ≤ T0 uLp (0,T ; Wps (Ω) ≤ εuWp2,1 (ΩT ) + T 1/p c(1/ε)u0 p, Ω . Proof. See [7], Lemma 3.4.

536

P. Weidemaier

References [1] Adams, R.A., Sobolev Spaces, Academic Press, New York-San Francisco-London, 1975. [2] Amann, H., Global existence for semilinear parabolic systems, J. Reine Angew. Math. 360 (1985), 47–83. [3] Kufner, A., John, O., and Fuˇcik, S., Function Spaces, Noordhoﬀ Int. Publ., Leyden, 1977. [4] Ladyzhenskaya, O.A., Solonnikov, V.A., and Uralceva, N.N., Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr., 23, Amer. Math. Soc., Providence, RI, 1968. [5] Maugeri, A., Palagachev, D.K., and Softova, L.G., Elliptic and Parabolic Equations with Discontinuous Coeﬃcients, Wiley-VCH, Berlin, 2000. [6] Softova, L.G., Oblique derivative problem for parabolic operators with VMO coeﬃcients, Manuscripta Math. 103 (2000), 203–220. [7] Weidemaier P., On Lp -estimates of optimal type for the parabolic oblique derivative problem with V M O-coeﬃcients, Diﬀerential Integral Equations 18 (2005), 935–946. Peter Weidemaier Fraunhofer-Institut Kurzzeitdynamik Eckerstr. 4 D-79104 Freiburg, Germany e-mail: [email protected]