Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States (Birkhäuser Advanced Texts Basler Lehrbücher)

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Birkhäuser Advanced Texts Edited by Herbert Amann, Zürich University Steven G. Krantz, Washington University, St. Louis Shrawan Kumar, University of North Carolina at Chapel Hill -DQ1HNRYiĜ8QLYHUVLWp3LHUUHHW0DULH&XULH3DULV

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Superlinear Parabolic Problems

Blow-up, Global Existence and Steady States

Birkhäuser Basel · Boston · Berlin

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix 1

I. MODEL ELLIPTIC PROBLEMS 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classical and weak solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Isolated singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pohozaev’s identity and nonexistence results . . . . . . . . . . . . . . . . . . . . . . . . . . . Homogeneous nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minimax methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liouville-type results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Positive radial solutions of ∆u + up = 0 in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . A priori bounds via the method of Hardy-Sobolev inequalities . . . . . . . . . . A priori bounds via bootstrap in Lpδ -spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A priori bounds via the rescaling method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A priori bounds via moving planes and Pohozaev’s identity . . . . . . . . . . . . .

7 7 12 18 20 29 36 50 55 61 65 68

II. MODEL PARABOLIC PROBLEMS 14. 15. 16. 17. 18. 19.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Well-posedness in Lebesgue spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maximal existence time. Uniform bounds from Lq -estimates . . . . . . . . . . . . Blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fujita-type results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Global existence for the Dirichlet problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Small data global solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Structure of global solutions in bounded domains . . . . . . . . . . . . . . . . . . . 3. Diffusion eliminating blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20. Global existence for the Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Small data global solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Global solutions with exponential spatial decay . . . . . . . . . . . . . . . . . . . . . 3. Asymptotic profiles for small data solutions . . . . . . . . . . . . . . . . . . . . . . . . . 21. Parabolic Liouville-type results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. A priori bounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. A priori bounds in the subcritical case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Boundedness of global solutions in the supercritical case. . . . . . . . . . . . . 3. Global unbounded solutions in the critical case . . . . . . . . . . . . . . . . . . . . . . 4. Estimates for nonglobal solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 75 87 91 100 112 112 120 125 129 129 137 139 150 161 161 166 171 175

vi

23. 24. 25. 26. 27. 28.

Blow-up rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Blow-up set and space profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Self-similar blow-up behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Universal bounds and initial blow-up rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complete blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications of a priori bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. A nonuniqueness result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Existence of periodic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Existence of optimal controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Transition from global existence to blow-up and stationary solutions . 5. Decay of the threshold solution of the Cauchy problem . . . . . . . . . . . . . . 29. Decay and grow-up of threshold solutions in the super-supercritical case

177 190 195 202 218 230 230 234 236 237 239 245

III. SYSTEMS 30. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31. Elliptic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. A priori bounds by the method of moving planes and Pohozaev-type identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Liouville-type results for the Lane-Emden system . . . . . . . . . . . . . . . . . . . 3. A priori bounds by the rescaling method . . . . . . . . . . . . . . . . . . . . . . . . . . . . p 4. A priori bounds by the Lδ alternate bootstrap method . . . . . . . . . . . . . . 32. Parabolic systems coupled by power source terms . . . . . . . . . . . . . . . . . . . . . . 1. Well-posedness and continuation in Lebesgue spaces . . . . . . . . . . . . . . . . . 2. Blow-up and global existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Fujita-type results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Blow-up asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33. The role of diffusion in blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Diffusion preserving global existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Diffusion inducing blow-up. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Diffusion eliminating blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

251 251 253 260 263 266 272 273 278 280 283 287 288 301 311

IV. EQUATIONS WITH GRADIENT TERMS 34. 35. 36. 37. 38. 39.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Well-posedness and gradient bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perturbations of the model problem: blow-up and global existence . . . . . . Fujita-type results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A priori bounds and blow-up rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Blow-up sets and profiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

313 314 319 330 338 348

vii

40. Viscous Hamilton-Jacobi equations and gradient blow-up on the boundary 1. Gradient blow-up and global existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Asymptotic behavior of global solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Space profile of gradient blow-up. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Time rate of gradient blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41. An example of interior gradient blow-up. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

355 355 358 364 367 374

V. NONLOCAL PROBLEMS 42. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43. Problems involving space integrals (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Blow-up and global existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Blow-up rates, sets and profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . q 3. Uniform bounds from L -estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Universal bounds for global solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44. Problems involving space integrals (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Transition from single-point to global blow-up. . . . . . . . . . . . . . . . . . . . . . . 2. A problem with control of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. A problem with variational structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. A problem arising in the modeling of Ohmic heating . . . . . . . . . . . . . . . . 45. Fujita-type results for problems involving space integrals . . . . . . . . . . . . . . . 46. A problem with memory term. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Blow-up and global existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Blow-up rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

377 377 378 381 394 395 398 398 403 411 412 418 421 422 424

APPENDICES 47. Appendix A: Linear elliptic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Elliptic regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p q 2. L -L -estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. An elliptic operator in a weighted Lebesgue space . . . . . . . . . . . . . . . . . . . 48. Appendix B: Linear parabolic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Parabolic regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p q 2. Heat semigroup, L -L -estimates, decay, gradient estimates . . . . . . . . . 3. Weak and integral solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49. Appendix C: Linear theory in Lpδ -spaces and in uniformly local spaces . . p 1. The Laplace equation in Lδ -spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p 2. The heat semigroup in Lδ -spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Some pointwise boundary estimates for the heat equation . . . . . . . . . . . 4. Proof of Theorems 49.2, 49.3 and 49.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. The heat equation in uniformly local Lebesgue spaces . . . . . . . . . . . . . . .

429 429 431 434 438 438 439 443 447 447 450 452 456 460

viii

50. Appendix D: Poincaré, Hardy-Sobolev, and other useful inequalities . . . . 1. Basic inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The Poincar´ e inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Hardy and Hardy-Sobolev inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51. Appendix E: Local existence, regularity and stability for semilinear parabolic problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Analytic semigroups and interpolation spaces . . . . . . . . . . . . . . . . . . . . . . . 2. Local existence and regularity for regular data . . . . . . . . . . . . . . . . . . . . . . 3. Stability of equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Self-adjoint generators with compact resolvent . . . . . . . . . . . . . . . . . . . . . . 5. Singular initial data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . q 6. Uniform bounds from L -estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52. Appendix F: Maximum and comparison principles. Zero number . . . . . . . 1. Maximum principles for the Laplace equation . . . . . . . . . . . . . . . . . . . . . . . 2. Comparison principles for classical and strong solutions . . . . . . . . . . . . . 3. Comparison principles via the Stampacchia method . . . . . . . . . . . . . . . . . 4. Comparison principles via duality arguments . . . . . . . . . . . . . . . . . . . . . . . . 5. Monotonicity of radial solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Monotonicity of solutions in time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Systems and nonlocal problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Zero number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53. Appendix G: Dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54. Appendix H: Methodological notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

462 462 463 465 466 466 470 485 488 495 505 507 507 509 512 515 518 520 522 526 528 532

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579

Introduction This book is devoted to the qualitative study of solutions of superlinear elliptic and parabolic partial differential equations and systems. Here “superlinear” means that the problems involve nondissipative terms which grow faster than linearly for large values of the solutions. This class of problems contains, in particular, a number of reaction-diffusion systems which arise in various mathematical models, especially in chemistry, physics and biology. For parabolic problems of this type it is known that a solution may cease to exist in a finite time as a consequence of its L∞ -norm becoming unbounded: The solution blows up. On the other hand, in many of these problems there exist also global solutions (in particular, stationary solutions). Both global and blowingup solutions may be very unstable and they may exhibit a rather complicated asymptotic behavior. Concerning elliptic problems, we consider questions of existence and nonexistence, multiplicity, regularity, singularities and a priori estimates. Special emphasis is put on those results which are useful in the investigation of the corresponding parabolic problems. As for parabolic problems, we study the questions of local and global existence, a priori estimates and universal bounds, blow-up, asymptotic behavior of global and nonglobal solutions. The study of superlinear parabolic and elliptic equations and systems has attracted the attention of many mathematicians during the past decades. Although a lot of challenging problems have already been solved, there are still many open questions even in the case of the simplest possible model problems. Unfortunately, most of the material, including many of the fundamental ideas, is scattered throughout hundreds of research articles which are not always easily readable for non-specialists. One of the main purposes of this book is thus to give an up-to-date and, as much as possible, self-contained account of the most important results and ideas of the field. In particular we try to find a balance between fundamental ideas and current research. Special effort is made to describe in a pedagogical way the main methods and techniques used in the study of these problems and to clarify the connections between several important results. Moreover, a number of the original proofs have been significantly simplified. In this way, the topic should be accessible to a larger audience of non-specialists. The book contains five chapters. The first two are intended to be an introduction to the field and to enable the reader to get acquainted with the main ideas by studying simple model problems, respectively of elliptic and parabolic type. These model problems are of the form −∆u = f (u), u = 0,

x ∈ Ω, x ∈ ∂Ω,

(0.1)

x

Introduction

and ut − ∆u = f (u), u = 0, u(x, 0) = u0 (x),

x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0, x ∈ Ω,

    

(0.2)

where Ω ⊂ Rn and f is a superlinear function, typically f (u) = |u|p−1 u for some p > 1. The subsequent three chapters are devoted to problems with more complex structure; namely, elliptic and parabolic systems, equations with gradient depending nonlinearities, and nonlocal equations. They include several problems arising in biological or physical contexts. These chapters contain many developments which reflect several aspects of current research. Although the techniques introduced in Chapters I and II provide efficient tools to attack some aspects of these problems, they often display new phenomena and specifically different behaviors, whose study requires new ideas. Many open problems are mentioned and commented. For the reader’s convenience we have collected a number of frequently used results in several appendices. These include estimates of solutions of linear elliptic and parabolic equations, maximum principles, and basic notions from dynamical systems. Also, in one of the appendices, we give an account of the local theory of semilinear parabolic problems based on the abstract framework of interpolationextrapolation spaces. However, this material is not essential for the understanding of the main contents of the book and can be left for a second reading. In particular, for the case of the model problem (0.2), the most useful results on local existenceuniqueness are proved by more elementary methods in the main text. On the other hand, we assume knowledge of the fundamentals of ordinary differential equations, of measure theory, of functional analysis (distributions, self-adjoint and compact operators in Hilbert spaces, Sobolev-Slobodeckii spaces and their embeddings, interpolation, Nemytskii mapping) and of the calculus of variations (minimizing of coercive, weakly lower semicontinuous functionals). Finally, a section of methodological notes and an index are provided. We would like to stress that, due to the broadness of the field of superlinear problems, our list of results and methods is of course not complete and is influenced in part by the interests of the authors. For reasons of space, many interesting topics and results could not be mentioned in this book (and we also apologize for any omission.) In particular, we do not touch degenerate problems with superlinear source (involving for instance porous medium, fast diffusion, or p-Laplace operators), nor higher order equations (where the maximum principle does not generally apply). We do not consider superlinear problems involving nonlinear boundary conditions, nor parabolic systems with convection (chemotaxis, Navier-Stokes). These are very interesting and intensively studied topics, but would require a book on their own. Finally, let us mention that there exist several textbooks and monographs dealing, at least in part, with certain aspects of superlinear problems; see [460], [466], [63], [113], [405], [504], [372], [222], for example.

Introduction

xi

We would like to express our gratitude to several colleagues for their careful and critical reading of (some parts of) the manuscript, particularly H. Amann, M. Balabane, M. Chipot, M. Fila, Ph. Lauren¸cot, P. Poláˇcik, A. Rodr´ıguez-Bernal, J. Rossi, F.B. Weissler and M. Winkler. Our special thanks go to H. Amann for his stimulating encouragements to this project. We also thank T. Hempfling from Birkhäuser for his helpfulness and the first author thanks the Slovak Literary Fund for providing financial support.

1. Preliminaries General We denote by BR (x) or B(x, R) the open ball in Rn with center x and radius R. We set BR := BR (0). The (n − 1)-dimensional unit sphere is denoted by S n−1 . The characteristic function of a given set M is denoted by χM . We write D ⊂⊂ D for D , D ⊂ Rn if the closure of D is a compact subset of D. For any real number s, we set s+ := max(s, 0) and s− := max(−s, 0). We also denote R+ := [0, ∞). Domains Let Ω be a domain, i.e. a nonempty, connected, open subset of Rn and let k ∈ N. We shall say that Ω is uniformly regular of class C k (cf. [13, p. 642]), if either Ω = Rn or there exists a countable family (Uj , ϕj ), j = 1, 2, . . . of coordinate charts with the following properties: (i) Each ϕj is a C k -diffeomorphism of Uj onto the open unit ball B1 in Rn mapping Uj ∩Ω onto the “upper half-ball” B1 ∩(Rn−1 × (0, ∞)) and Uj ∩∂Ω onto the flat part B1 ∩ (Rn−1 × {0}). In addition, the functions ϕj and the up to the order k are uniformly bounded on Uj derivatives of ϕj and ϕ−1 j and B1 , respectively. (ii) The set j ϕ−1 j (B1/2 ) contains an ε-neighborhood of ∂Ω in Ω for some ε > 0. (iii) There exists k0 ∈ N such that any k0 + 1 distinct sets Uj have an empty intersection. In an analogous way we define a uniformly regular domain of class C 2+α (shortly domain of class C 2+α ). Unless explicitly stated otherwise1 , we will always assume that Ω ⊂ Rn is a uniformly regular domain of class C 2+α for some α ∈ (0, 1). On the other hand, we do not assume Ω to be bounded unless this is explicitly mentioned. We denote the distance to the boundary function by δ(x) := dist (x, ∂Ω). The exterior unit normal on ∂Ω at a point x ∈ ∂Ω is denoted by ν(x), and the outer normal derivative by ∂ν or ∂/∂ν. The surface measure (on e.g. ∂Ω or S n−1 ) will be denoted by dσ or dω. For a given domain Ω and 0 < T < ∞, we set QT : = Ω × (0, T ), ST : = ∂Ω × (0, T ) PT : = ST ∪ Ω × {0} 1 In

(lateral boundary), (parabolic boundary).

fact, if we want to allow nonsmooth domains, we will refer to an arbitrary domain.

2

1. Preliminaries

Functions of space and time Let u = u(x, t) be a real function of the space variable x ∈ Ω and the time variable t. Without fearing confusion we will also consider u as a function of a single variable t with values in a space of functions defined in Ω, hence u(t)(x) = u(x, t). By a solution of a PDE being positive we usually mean that u(x) > 0 or u(x, t) > 0 in the domain under consideration. Note that, due to the strong maximum principles in Appendix F, positive is often equivalent to nontrivial nonnegative. Radial functions. We say that a domain Ω ⊂ Rn is symmetric if either Ω = Rn , or Ω = BR = {x ∈ Rn : |x| < R}, or Ω = {x ∈ Rn : R < |x| < R }, where 0 < R < R ≤ ∞ (an annulus if R < ∞). Denote r = |x| and let J ⊂ R be an interval. A function u defined on a symmetric domain Ω (resp., on Ω × J) is said to be radially symmetric, or simply radial, if it can be written in the form u = u(r) (resp., u = u(r, t) for each t ∈ J). The function u is said to be radial nonincreasing if it is radial and if, moreover, u is nonincreasing as a function of r. Banach spaces and linear operators If X is a Banach space and p ≥ 1, then X and p denote the (topological) dual space and dual exponent (1/p + 1/p = 1), respectively. We write X → Y or X → → Y if X is continuously or compactly embedded in Y , respectively. If both X → Y and Y → X (that is X and Y coincide and carry equivalent norms), then . we write X = Y . We denote by L(X, Y ) the space of continuous linear operators A : X → Y , L(X) = L(X, X). If A is a linear operator in X with the domain of definition D(A) and Y ⊂ X, then the operator AY , the Y -realization of A, is defined by AY u = Au, D(AY ) := {u ∈ D(A) ∩ Y : Au ∈ Y }. Function spaces We denote by D(Ω) the space of C ∞ -functions with compact support in Ω. The norms in the Sobolev space W k,p (Ω) (or the Sobolev-Slobodeckii space W k,p (Ω) if k is not an integer) and the Lebesgue space Lp (Ω) will be denoted by · k,p and · p , respectively. We denote by W01,2 (Ω) the closure of D(Ω) in W 1,2 (Ω). The spaces W k,2 (Ω), k ∈ N, and W01,2 (Ω) will also be denoted as H k (Ω) and H01 (Ω), respectively. The functions in these spaces are usually understood to be real valued. If no confusion is likely, we shall use the same notation for similar spaces of functions with values in Rn . Otherwise we shall use the notation Lp (Ω, Rn ), for example. Let Ω be a bounded domain in Rn (not necessarily smooth). The weighted Lebesgue spaces Lpδ (Ω) are defined as follows. Denoting as before δ(x) = dist(x, ∂Ω),

x ∈ Ω,

1. Preliminaries

3

we put, for all 1 ≤ p ≤ ∞, Lpδ = Lpδ (Ω) := Lp (Ω; δ(x) dx). For 1 ≤ p < ∞, Lpδ is endowed with the norm u p,δ =

Ω

1/p |u(x)|p δ(x) dx .

∞ ∞ Remark 1.1. Let us note that L∞ δ (Ω) = L (Ω), with same norm. Indeed, Lδ (Ω) consists, by definition, of those measurable functions that are essentially bounded with respect to the measure δ(x) dx.

For any 1 ≤ p < ∞, the uniformly local Lebesgue space (cf. [297], [253]) Lpul is defined by

Lpul = Lpul (Rn ) = φ ∈ Lploc (Rn ) : φ p,ul < ∞ , where φ p,ul := sup

a∈Rn

|y−a| 0, there exists a compact set K ⊂ M such that u(t) X < ε for all t ∈ M \ K. We also set B(M ) := B(M, R), BC(M ) := BC(M, R), etc. Let M ⊂ Rn . A function u : M → X is said to be locally H¨ older continuous if, for each point t ∈ M , there exist α ∈ (0, 1), C > 0 and a neighborhood V of t, such that u(x) − u(y) X

uα,M∩V := sup < ∞. (1.1) |x − y|α x,y∈M∩V, x=y If α in (1.1) can be chosen independent of t ∈ M , then u is said to be locally α-Hölder continuous. The space of such functions is denoted by C α (M, X) (or C α (M ) if X = R) and endowed with the family of seminorms · ∞,K + ·α,K ,

4

1. Preliminaries

where K runs over all compact subsets of M . By U C α (M, X), α ∈ (0, 1), we denote the set of functions u : M → X such that

uα := uα,M < ∞. The norm in the Banach space BU C α (M, X) = B(M, X) ∩ U C α (M, X) is the sum of the sup-norm and the seminorm ·α . Note that if M is compact, then any locally H¨ older continuous function u : M → X belongs to BU C α (M, X) for some α α and C (M, X) = BU C α (M, X). If Ω is an arbitrary domain in Rn , then BC 1 (Ω) denotes the space of functions u ∈ BC(Ω) whose first derivatives in Ω are bounded, continuous and can be continuously extended to Ω. The norm of a function u in this space is defined as the sum of sup-norms of u and its first-order derivatives. The spaces BC k (Ω) and BU C k (Ω), k ≥ 1 integer, are defined in an obvious way. If no confusion is likely, we shall denote their norms by · BC k . The spaces C k+α (Ω), U C k+α (Ω), BU C k+α (Ω), where k ≥ 1 is an integer and α ∈ (0, 1) are defined similarly. Let Ω be a bounded domain in Rn . Then Ω is compact, hence any function in C(Ω) is bounded and uniformly continuous. On the other hand, the functions in BU C(Ω) can be uniquely extended to functions in C(Ω). Identifying the function u ∈ BU C(Ω) with its extension and endowing the space C(Ω) with the sup-norm, we can write BU C(Ω) = C(Ω). Similarly, BU C α (Ω) = C α (Ω). If Q ⊂ Rn × R is a domain in space and time, then C 2,1 (Q) is the space of functions which are twice continuously differentiable in the spatial variable x and once in the time variable t. The space BC 2,1 (Q) has obvious meaning. If u ∈ Lp (Q), then ut , Dx u and Dx2 u denote the time derivative and first and second spatial derivatives of u in the sense of distributions. Alternatively, we shall also use the notation ∇u, D2 u instead of Dx u, Dx2 u. We denote by W 2,1;p (Q) the space of functions u ∈ Lp (Q) satisfying ut , Dx u, Dx2 u ∈ Lp (Q), endowed with the norm u 2,1;p = u 2,1;p;Q := u p;Q + Dx u p;Q + Dx2 u p;Q + ut p;Q . Let Q = QT = Ω × (0, T ) where Ω is an arbitrary domain in Rn and T > 0. Given α ∈ (0, 1] set |f (x, t) − f (y, s)| [f ]α;Q = sup : x, y ∈ Ω, t, s ∈ (0, T ), (x, t) =

(y, s) . |x − y|α + |t − s|α/2 Let k be a nonnegative integer, α ∈ (0, 1) and a = k + α. Then we put sup |Dxβ Dtj f | + [Dxβ Dtj f ]α;Q |f |a;Q = |β|+2j≤k

Q

|β|+2j=k

and BU C a,a/2 (Q) := {f : |f |a;Q < ∞}. The spaces U C a,a/2 (Q) and C a,a/2 (Q) are defined analogously as in the case of functions defined in Rn . Note that if p > n+2,

1. Preliminaries

5

a < 2 − (n + 2)/p and Ω is smooth enough (for example, if Ω satisfies a uniform interior cone condition), then W 2,1;p (Q) → BU C a,a/2 (Q);

(1.2)

see [320, Lemmas II.3.3, II.3.4], [399, Theorem 6.9] and the references therein for this statement and more general embedding and trace theorems for anisotropic spaces. Embedding (1.2) can also be derived by using the interpolation embedding in Proposition 51.3 and embeddings for isotropic spaces. Eigenvalues and eigenfunctions If Ω is bounded, then we denote by λ1 , λ2 , . . . the eigenvalues of −∆ in W01,2 (Ω) and by ϕ1 , ϕ2 , . . . the corresponding eigenfunctions. Recall that λ1 < λ2 ≤ λ3 ≤ · · · , λk → ∞ as k → ∞, that

1 1,2 2 = sup u dx : u ∈ W0 (Ω), |∇u|2 dx = 1 , (1.3) λ1 Ω Ω and that we can choose ϕ1 > 0. Unless explicitly stated otherwise, we shall assume that ϕ1 is normalized by

Ω

ϕ1 dx = 1.

We shall often use the fact that if Ω is of class C 2 , then there exist constants c1 , c2 > 0 such that c1 δ(x) ≤ ϕ1 (x) ≤ c2 δ(x),

x∈Ω

(1.4)

(this is a consequence of u ∈ C 1 (Ω) and of Hopf’s lemma; cf. Proposition 52.1(iii)). Further frequent notation We denote by G(x, y, t) = GΩ (x, y, t) the Dirichlet heat kernel; Gt (x) = G(x, t) is the Gaussian heat kernel in Rn . The (elliptic) Dirichlet Green kernel is denoted by K(x, y) = KΩ (x, y). We implicitly mean by e−tA the Dirichlet heat semigroup in Ω. The Dirac distribution at point y will be denoted by δy . We shall use the symbols C, C1 , etc. to denote various positive constants. The dependence of these constants will be made precise whenever necessary. Definitions of various critical exponents (pF , pBT , psg , pS , pJL , pL , 2∗ , 2∗ , qc ) and other symbols can be found via the List of Symbols.

Chapter I

Model Elliptic Problems

2. Introduction In Chapter I, we study the problem −∆u = f (x, u), u = 0,

x ∈ Ω, x ∈ ∂Ω,

(2.1)

where f : Ω × R → R is a Carathéodory function (i.e. f (·, u) is measurable for any u ∈ R and f (x, ·) is continuous for a.e. x ∈ Ω). Of course, the boundary condition in (2.1) is not present if Ω = Rn . We will be mainly interested in the model case f (x, u) = |u|p−1 u + λu,

where p > 1 and λ ∈ R.

(2.2)

Denote by pS the critical Sobolev exponent, pS :=

∞ if n ≤ 2, (n + 2)/(n − 2) if n > 2.

We shall refer to the cases p < pS , p = pS or p > pS as to (Sobolev) subcritical, critical or supercritical, respectively.

3. Classical and weak solutions ˜ Let u be a solution of (2.1) and f(x) := f (x, u(x) . Then u solves the linear problem −∆u = f˜ in Ω, (3.1) u=0 on ∂Ω. In what follows we define several types of solutions of the linear problem (3.1) (and, consequently, of (2.1)). Definition 3.1. (i) We call u a classical solution of (3.1) if f˜ ∈ C(Ω), u ∈ C 2 (Ω) ∩ C(Ω) and u satisfies the equation and the boundary condition in (3.1) pointwise.

8

I. Model Elliptic Problems

(ii) We call u ∈ W01,2 (Ω) a variational solution of (3.1) if f˜ ∈ W01,2 (Ω) and

∇u · ∇ϕ dx = (3.2) f˜ϕ dx for all ϕ ∈ W01,2 (Ω). Ω

Ω

(iii) Let Ω be bounded, u ∈ L1 (Ω). Set δ(x) := dist (x, ∂Ω)

L1δ (Ω) := L1 (Ω, δ(x)dx).

and

We call u an L1 -solution of (3.1) if f˜ ∈ L1 (Ω) and

f˜ϕ dx for all ϕ ∈ C 2 (Ω), ϕ = 0 on ∂Ω. u(−∆ϕ) dx = Ω

(3.3)

Ω

More generally, we call u an L1δ -solution, or a very weak solution, of (3.1) if f˜ ∈ L1δ (Ω) and (3.3) is satisfied. Note that the definition makes sense since |ϕ| ≤ Cδ hence f˜ϕ ∈ L1 (Ω). Existence-uniqueness and properties of L1δ solutions of the linear problem (3.1) are studied in Appendix C. (iv) If Ω = Rn , then u ∈ L1loc (Ω) is called a distributional solution of (3.1) if the integral identity in (3.3) is true for all ϕ ∈ D(Rn ). Remarks 3.2. (i) If we assume that f˜ is a bounded Radon measure in Ω (instead of f˜ ∈ L1 (Ω)), then the definition of an L1 -solution still makes sense and we refer to [20] and the references therein for properties of such solutions. (ii) If f˜ ∈ L∞ (Ω), then any classical solution of (3.1) satisfies u ∈ W 2,q (K) for any K ⊂⊂ Ω and any q < ∞. This is a consequence of Remark 47.4(iii). If we further assume that f˜ is locally H¨ older continuous in Ω, then u ∈ C 2 (Ω). (iii) Assume Ω bounded. If f˜ ∈ C(Ω), for example, then any classical solution of (3.1) is also a variational solution (this follows from Remark (ii) and integration by parts). If f˜ ∈ L2 (Ω), then any variational solution is an L1 -solution. Some other relations between various types of solutions defined above will be mentioned below (see also Lemma 47.7 in Appendix A). In the following sections we shall often use variational methods in order to prove the solvability of (2.1). Therefore, we derive now a sufficient condition on f which guarantees that any variational solution of (2.1) is classical. If n ≥ 3 we set 2∗ := pS + 1 = 2n/(n − 2), 2∗ := (2∗ ) = 2n/(n + 2). Assume that the Carathéodory function f satisfies the following growth assumption |f (x, u)| ≤ α(x)+Cf (|u|+|u|p ),

α ∈ L(p+1) (Ω)+L2 (Ω), Cf > 0, p ≤ pS . (3.4)

This growth condition can be significantly weakened if n ≤ 2 but (3.4) will be sufficient for our purposes; cf. (2.2). Denote

u F (x, u) := f (x, s) ds 0

3. Classical and weak solutions

9

1 2 E(u) := |∇u(x)| dx − F x, u(x) dx. (3.5) 2 Ω Ω Since p ≤ pS we have W 1,2 (Ω) → Lp+1 (Ω) and the embedding is compact provided p < pS and Ω is bounded. In addition, the energy functional E is C 1 (continuously Fréchet differentiable) in W 1,2 (Ω) and

∇u · ∇ϕ dx − f (·, u)ϕ dx E (u)ϕ = and

Ω

Ω

for all u, ϕ ∈ W 1,2 (Ω). In particular, each critical point of E in W01,2 (Ω) is a variational solution of (2.1). The following proposition is essentially due to [96]; our proof closely follows the proof of [505, Lemma B.3]. Proposition 3.3. Assume (3.4). If n ≥ 3 assume also α ∈ Ln/2 (Ω). Let u be a variational solution of (2.1). Then u ∈ Lq (Ω) for all q ∈ [2, ∞). Proof. Since the assertion is obviously true if n ≤ 2 due to W 1,2 (Ω) → Lq (Ω), we may assume n ≥ 3. Denote f˜(x) := f x, u(x) . Then |f˜| ≤ α + Cf (|u| + |u|p ) ≤ a + b + 2Cf (|u| + |u|pS ), where a := αχ|u|>1 ∈ Ln/2 (Ω), b := αχ|u|≤1 and α can be written in the form α = α1 + α2 with α1 ∈ L(p+1) (Ω), α2 ∈ L2 (Ω). ∗

Choose s ≥ 0 such that u ∈ L2(s+1) (Ω). We shall prove that u ∈ L2 (s+1) (Ω) so that an obvious bootstrap argument proves the assertion. Choose L > 0 and set ϕ := uψ 2 , ΩL := {x ∈ Ω : |u|s ≤ L}. ψ := min |u|s , L , In what follows we denote by C, C1 , C2 various positive constants which may vary from step to step and which may depend on u, s, α, Cf but which are independent of L. We have ∇(uψ) = (1 + sχΩL )(∇u)ψ, ∇ϕ = (1 + 2sχΩL )(∇u)ψ 2 , and ϕ ∈ W01,2 (Ω). Therefore, we obtain

|∇u|2 ψ 2 dx ≤ ∇u · ∇ϕ dx = f˜ϕ dx = f˜uψ 2 dx Ω Ω Ω Ω

∗ (a + b)|u|ψ 2 + u2 ψ 2 + |u|2 ψ 2 dx ≤C

Ω 2 2 ∗ au ψ + b|u| + |u|2s+2 + |u|2 ψ 2 dx ≤C Ω

∗ ≤C 1+ (a + |u|2 −2 )u2 ψ 2 dx , Ω

10


where we have used

b|u| dx ≤ α|u| dx ≤ (|α1 | + |α2 |)|u| dx Ω

Ω

Ω

≤ α1 (p+1) u p+1 + α2 2 u 2 = C. ∗

Consequently, denoting v := a + |u|2 −2 ∈ Ln/2 (Ω), we obtain

2 2 2 |∇(uψ)| dx ≤ C |∇u| ψ dx ≤ C 1 + vu2 ψ 2 dx Ω Ω Ω

2 2 2 u ψ dx + v(uψ) dx ≤C 1+K |v|≤K

|v|>K

2/n

(n−2)/n ∗ |u|2s+2 dx + v n/2 dx |uψ|2 dx Ω |v|>K Ω

2 |∇(uψ)| dx, ≤ C1 (1 + K) + C2 εK

≤C 1+K

Ω

2/n where εK := |v|>K v n/2 dx → 0 as K → +∞. Choosing K such that C2 εK < 1/2 we arrive at

|∇(|u|s+1 )|2 dx = |∇(uψ)|2 dx ≤ 2C1 (1 + K). ΩL

ΩL

∗

Letting L → +∞ we get |u|s+1 ∈ W 1,2 (Ω), hence u ∈ L2

(s+1)

(Ω).

Corollary 3.4. If f has the form (2.2) with p ≤ pS , then any variational solution u of (2.1) is also a classical solution. Moreover, u ∈ C 2 (Ω). Proof. The assertion is a consequence of standard regularity results for linear elliptic equations. More precisely, for any 2 ≤ q < ∞, since f˜ := f (u) ∈ Lq (Ω), u = f˜. Theorem 47.3(i) implies the existence of u ˜ ∈ W 2,q ∩ W01,q (Ω) such that −∆˜ 1 Since u, u ˜ ∈ H0 (Ω), the maximum principle in Proposition 52.3(i) yields u = u˜. Due to the embedding W 2,q (Ω) ⊂ C 1 (Ω) for q > n, we deduce that f˜ ∈ C 1 (Ω). Applying now Theorem 47.3(ii), and Proposition 52.3(i) again, we deduce that u ∈ C 2 (Ω). As for L1 -solutions, we have the following regularity result (we shall see in Remarks 3.6 below that the growth conditions in Propositions 3.3 and 3.5 are optimal). Proposition 3.5. Assume Ω bounded. Let the Carathéodory function f satisfy the growth assumption |f (x, u)| ≤ C(1 + |u|p ),

p < psg ,

(3.6)

3. Classical and weak solutions

11

where psg is defined in (3.8). Let u be an L1 -solution of (2.1). Then u ∈ C0 ∩ W 2,q (Ω) for all finite q. Proof. It is based on a simple bootstrap argument. Fix ρ ∈ (1, n/(n − 2)p) and put f˜(x) = f x, u(x) . Assume that there holds i f˜ ∈ Lρ (Ω)

(3.7)

for some i ≥ 0 (this is true for i = 0 by assumption). Since 1 2 1 1 1 < , 1 − − = ρi pρi+1 ρi pρ n i+1 i+1 by using Proposition 47.5(i), we obtain u ∈ Lpρ (Ω), hence f˜ ∈ Lρ (Ω) due to (3.6). By induction, it follows that (3.7) is true for all integers i. In particular f˜ ∈ Lk (Ω) for some k > n/2 and we may apply Proposition 47.5(i) once more to deduce that u ∈ L∞ (Ω). The conclusion then follows similarly as in the proof of Corollary 3.4 (using the uniqueness part of Theorem 49.1 instead of Proposition 52.3).

Remarks 3.6. (i) Singular solution. Define the exponent ∞ if n ≤ 2, psg := n/(n − 2) if n > 2.

(3.8)

For p > psg (hence n ≥ 3), we let U∗ (r) := cp r−2/(p−1) ,

r > 0,

where cp−1 := p

2 (n − 2)p − n . (3.9) 2 (p − 1)

One can easily check that u∗ (x) := U∗ (|x|) is a positive, radial distributional solution of the equation −∆u = up in Rn . This singular solution (hence the notation psg ) plays an important role in the study of the parabolic problem (0.2) with f (u) = |u|p−1 u (see for example Theorems 20.5, 22.4 and 23.10). On the other hand, if we set u(x) := u∗ (x) − cp for 0 < |x| ≤ 1, Ω := B1 (0) = {x ∈ Rn : |x| < 1}, then it is easy to verify that u is an L1 -solution of (2.1) with f (x, u) = (u + cp )p . Moreover, u is a variational solution of this problem if p > pS . Hence the condition p ≤ pS in Proposition 3.3 is necessary. (ii) Let n ≥ 3 and let Ω be bounded, f ∈ C 1 , |f (x, u)| ≤ C(1 + |u|p ). The example in (i) shows that an L1 -solution need not be classical if p > psg . In fact, it was proved in [44], [394] that the problem −∆u = |u|p−1 u in Ω, (3.10) u=0 on ∂Ω,

12


has a positive unbounded radial L1 -solution u ∈ C 2 (Ω \ {0}) provided p ∈ [psg , pS ) and Ω = B1 (0). See also [404] and the references therein for related nonradial results. (iii) For the case of L1δ -solutions, we shall see in Section 11 that the critical exponent is different, namely (n + 1)/(n − 1). Remark 3.7. Classical vs. very weak solutions for the nonlinear eigenvalue problem. Another type of relations between different notions of solutions appears when one considers the nonlinear eigenvalue problem −∆u = λf (u),

x ∈ Ω,

u = 0,

x ∈ ∂Ω.

(3.11)

Here we assume that f : [0, ∞) → (0, ∞) is a C 1 nondecreasing, convex function, and λ > 0. Namely, it was proved in [94] (see also [233] for earlier related results) that if there exists a very weak solution of (3.11) for some λ0 > 0, then there exists a classical solution for all λ ∈ (0, λ0 ). The proof is based on a perturbation argument relying on a variant of Lemma 27.4 below. As a consequence of this and of results from [305], [142], assuming in addition that limu→∞ f (u)/u = ∞, there exists λ∗ ∈ (0, ∞) such that: (i) for 0 < λ < λ∗ , problem (3.11) has a (unique minimal) classical solution uλ , and the map λ → uλ is increasing; (ii) for λ = λ∗ , problem (3.11) has a very weak solution defined by uλ∗ = limλ↑λ∗ uλ ; (iii) for λ > λ∗ , problem (3.11) has no very weak solution. On the other hand, the solution uλ∗ may be classical or singular, depending on the nonlinearity. For instance, in the case f (u) = (u + 1)p with Ω = BR , (3.11) has a classical solution for λ = λ∗ if and only if p < pJL , where pJL is defined in (9.3); in the case f (u) = eu , the condition is replaced with n ≤ 9 (see [293], [369]). Illustrations of these facts appear on the bifurcation diagram in Remark 6.10(ii) (see Figure 3).

4. Isolated singularities In this section we study the question of isolated singularities of positive classical solutions to the equation −∆u = up . The following result classifies the possible singular behaviors for subcritical or critical p. Theorem 4.1. Let n ≥ 3 and 1 0 such that C1 ψ(x) ≤ u(x) ≤ C2 ψ(x), where

0 < |x| < 1/2,

⎧ 2−n ⎪ ⎨ |x| ψ(x) = |x|2−n (− log |x|)(2−n)/2 ⎪ ⎩ −2/(p−1) |x|

if 1 0. Furthermore, for all p > 1, we have the following result, which explains in what sense the equation can be extended to the whole unit ball. Theorem 4.2. Let p > 1 and n ≥ 3. Assume that u is a positive classical solution of −∆u = up in B1 \ {0}. (i) Then up ∈ L1loc (B1 ) and there exists a ≥ 0 such that u is a solution of −∆u = up + aδ0

in D (B1 ),

˜ with a ˜= where δ0 denotes the Dirac delta distribution. Moreover, we have a ≤ a a ˜(n, p) > 0. (ii) If p < psg and a = 0, then the singularity is removable, i.e. u is bounded near x = 0. (iii) If p ≥ psg , then a = 0. Remarks 4.3. (i) Theorem 4.1 follows from [340], [44], [240] (see also [83]), and [108], for the cases p < psg , p = psg , psg < p < pS and p = pS respectively. Theorem 4.2 follows from [97] and [340]. See also the book [523] for further results and references. (ii) Under the assumptions of Theorem 4.1 with psg 0, as x → 0 (see the proof below). Examples in [340] show that singular solutions do exist for 1 pS , then the upper estimate u(x) ≤ C|x|−2/(p−1) is still true in the radial case (cf. [213], [397]). In fact, as a consequence of −(rn−1 ur )r = rn−1 up > 0, for r > 0 small, we have either ur > 0, hence r u bounded, or ur ≤ 0. In this second case, by integration, we get −ur ≥ r1−n 0 sn−1 up (s) ds ≥ (r/n)up for r > 0 small, hence (u1−p )r ≥ Cr, and the upper estimate follows by a further integration. The estimate is unknown in the nonradial case for p > pS , but related integral estimates of solutions can be found in e.g. [238] and [89].

14


(iv) A result similar to Theorem 4.1 is true for n = 2, with ψ(x) given by the fundamental solution log |x| instead of |x|2−n . These results are related to the fact that the (H 1 –) capacity of a point is 0 when n ≥ 2. When the origin is replaced by a closed subset of 0 capacity, related results can be found in [170]. On the other hand, the upper estimate u(x) ≤ C|x|−2/(p−1) , from the case psg < p < pS , can be generalized to sets other than a single point (see Theorem 8.7 in Section 8). We shall first prove Theorem 4.2. Theorem 4.1 in the case 1 < p < psg will then follow as a consequence of Theorem 4.2 and of a bootstrap argument. In the case psg < p < pS , the upper estimate will be a consequence of the more general result Theorem 8.7 in Section 8. For the cases p = psg , p = pS , and for the lower estimate when psg < p < pS , see the above mentioned references. In view of the proofs, we introduce the following notation. We denote by Γ(x) = cn |x|2−n the fundamental solution of the Laplacian (Newton potential), i.e. −∆Γ = δ0 in D (Rn ). We let ω = {x ∈ Rn : |x| < 1/2} and fix χ ∈ D(B1 ) such that χ = 1 on ω and 0 ≤ χ ≤ 1. For each positive integer j, denote χj (x) = χ(jx). By a straightforward calculation using n ≥ 3, we see that χj → 0 in H 1 (B1 ) as j → ∞. For any ϕ ∈ D(B1 ), we put ϕj := (1 − χj )ϕ. Observe that ϕj → ϕ in H 1 (B1 ). We need the following lemma. Lemma 4.4. Let n ≥ 3. Assume that u ∈ C 2 (B1 \ {0}) satisfies u ≥ 0 and Then u ∈

L1loc (B1 )

−∆u ≥ 0

in B1 \ {0}.

−∆u ≥ 0

in D (B1 ).

and

Proof. For each k > 0, we take a function Gk ∈ C 2 ([0, ∞)) such that Gk (s) = s for 0 ≤ s ≤ k, Gk (s) = k + 1 for s large, Gk ≥ 0 and Gk ≤ 0. Define uk := Gk (u) and note that the sequence {uk }k is monotone increasing and converges to u pointwise in B1 \ {0}. The function uk satisfies −∆uk = −Gk (u)∆u − Gk (u)|∇u|2 ≥ 0

in B1 \ {0}.

(4.2)

Fix α > 0 and ϕ ∈ D(B1 ). Multiplying inequality (4.2) by the test-function ϕ2j (1 + uk )−α and integrating by parts, we obtain

∇uk · ∇(ϕ2j (1 + uk )−α ) 0≤ B1

= −α |∇uk |2 ϕ2j (1 + uk )−1−α + 2 ∇uk · ∇ϕj (1 + uk )−α ϕj . B1

B1

It follows that

|∇uk |2 ϕ2j (1 + uk )−1−α α B1

α ≤ |∇uk |2 ϕ2j (1 + uk )−1−α + C(α) |∇ϕj |2 (1 + uk )1−α , 2 B1 B1

4. Isolated singularities

hence

2

|∇uk |

ϕ2j (1

−1−α

+ uk )

|∇ϕj |2 (1 + uk )1−α .

≤ C(α)

B1

15

B1

Since |∇ϕj |2 → |∇ϕ|2 in L1 (B1 ) and (1 + uk )1−α ∈ L∞ (B1 ), we may pass to the limit j → ∞ (using Fatou’s lemma on the LHS) and we obtain

2 2

−1−α

|∇uk | ϕ (1 + uk )

|∇ϕ|2 (1 + uk )1−α .

≤ C(α)

B1

B1

First taking α = 1 and using 1 + uk ≤ k + 2, we deduce that uk ∈ H 1 (ω), hence 1 uk ∈ Hloc (B1 ). Next take α = 2/n. Consider ϕ such that ϕ = 1 for |x| ≤ 1/4 and with support in ω. Applying the Sobolev and H¨ older inequalities, we get, for any ρ ∈ (0, 1/2),

n−2 2 n−2 n ∇ (1 + uk ) n−2 +C 2n (1 + uk ) ≤C (1 + uk ) n |x| 2 and choose α2 ∈ (0, α1 ) otherwise. Notice that α2 ∈ (0, α1 ) = (0, n − 2) in both cases due to pα1 < n. Since up ≤ C(v1 )p+ + C|x|−pα1 ≤ C(v1 )p+ + C|x|−α2 −2 , there exists C2 > 0 such that v2 := v1 − C2 |x|−α2 satisfies −∆v2 ≤ C(v1 )p+

in D (B1 ).

Since (v1 )p+ ≤ C(v2 )p+ + C|x|−pα2 , we can iterate this procedure and we obtain functions vi (i = 0, 1, . . . ) satisfying vi+1 = vi − Ci+1 |x|−αi+1 , with αi+1 ∈ (0, αi ), and −∆vi+1 ≤ Ci (vi )p+ in D (B1 ). Moreover, due to 0 < a ≤ a ˜(n, p), the constants Ci , Ci may be chosen to depend only on n, p, i. To conclude, we apply a bootstrap argument similar to that in the proof of Proposition 3.5: Fix ρ ∈ (1, n/(n − 2)p), let Ω1 = {|x| < 2/3}, and assume that i (vi )+ ∈ Lpρ loc (Ω1 ) for some i ≥ 0 (this is true for i = 0 in view of (4.6)). Since i

(−∆vi+1 )+ ∈ Lρloc (Ω1 ) and 1 2 1 1 1 < , 1 − − = i i+1 i ρ pρ ρ pρ n i+1

we may apply Proposition 47.6(ii) to deduce that (vi+1 )+ ∈ Lpρ loc (Ω1 ). By iterating, we get (vi )+ ∈ Lkloc (Ω1 ) for some sufficiently large i and some k > n/2. We

18


may then apply Proposition 47.6(ii) once more to deduce that (vi+1 )+ ∈ L∞ (ω). This implies u − aΓ = v1 = vi+1 +

i+1

Cj |x|−αj ≤ C(1 + |x|−α2 ),

|x| < 1/2.

j=2

Moreover, starting from (4.6), it is easy to check that the constant C depends only on n, p. This along with (4.7) yields the conclusion.

5. Pohozaev’s identity and nonexistence results In this section we prove the nonexistence of nontrivial solutions of (2.1) provided f satisfies (2.2) with p ≥ pS , λ ≤ 0 and Ω is a bounded starshaped domain. The following identity [419] plays a crucial role in the proof. Theorem 5.1. Let u be a classical solution of (2.1) with f = f (u) being locally Lipschitz and Ω bounded. Then n−2 2 where F (u) =

u 0

Ω

|∇u|2 dx − n

F (u) dx + Ω

1 2

∂u 2 x · ν dσ = 0, ∂Ω ∂ν

(5.1)

f (s)ds.

Proof. First notice that u ∈ C 2 (Ω) (see Remark 3.2(ii)). Using integration by parts we obtain

Ω

∇u · ∇(x · ∇u) − |∇u|2 dx

∂u ∂ ∂u ∂u ∂ ∂u dx = dx = xj xj ∂xi ∂xj ∂xj ∂xi Ω i,j ∂xi Ω i,j ∂xi

∂u ∂u ∂ ∂u ∂u xj = xj νj dσ − n |∇u|2 dx − dx, ∂xi ∂xi ∂Ω i,j ∂xi Ω Ω i,j ∂xj ∂xi

hence

2

∂u ∂ ∂u 1 ∂u dx = xj |∇u|2 dx x · ν dσ − n ∂xj ∂xi 2 ∂Ω ∂ν Ω i,j ∂xi Ω and

Ω

∇u · ∇(x · ∇u) dx =

1 2

∂u 2 n−2 |∇u|2 dx. x · ν dσ − 2 ∂Ω ∂ν Ω

5. Pohozaev’s identity and nonexistence results

19

Multiplying the equation in (2.1) by x·∇u, integrating over Ω and denoting the leftor the right-hand side of the resulting equation by (LHS) or (RHS), respectively, we obtain

∂u (x · ∇u) dσ − −(LHS) = ∆u(x · ∇u) dx = ∇u · ∇(x · ∇u) dx Ω ∂Ω ∂ν Ω

2

1 n−2 ∂u = |∇u|2 dx, x · ν dσ + 2 ∂Ω ∂ν 2 Ω

(RHS) = f (u)(x · ∇u) dx

Ω F (u)(x · ν) dσ − n F (u) dx = −n F (u) dx, = Ω

∂Ω

Ω

where we have used that, on ∂Ω, ∇u = Cν for suitable C ∈ R and F (u) = F (0) = 0. The comparison of (LHS) and (RHS) yields now the assertion. Corollary 5.2. Assume Ω bounded and starshaped with respect to some point x0 ∈ Ω (i.e. the segment [x0 , x] is a subset of Ω for any x ∈ Ω), n ≥ 3. Assume that n−2 F (u) ≤ f (u)u for all u. (5.2) 2n Then (2.1) does not possess classical positive solutions. If, in addition, f (0) = 0, then (2.1) does not possess classical nontrivial solutions. Condition (5.2) is satisfied if, for example, f (u) = |u|p−1 u + λu, p ≥ pS and λ ≤ 0. Proof. We proceed by contradiction. We can assume that Ω is starshaped with respect to x0 = 0. Then x · ν ≥ 0 on ∂Ω and

x2 dx > 0, x · ν dσ = ∆ 2 ∂Ω Ω hence x · ν > 0 on a set of positive surface measure in ∂Ω. If u is a positive solution of (2.1), then ∂u/∂ν < 0 on ∂Ω by the maximum principle and we obtain

2 1 ∂u (5.3) x · ν dσ > 0. 2 ∂Ω ∂ν Multiplication of the equation in (2.1) by u and integration by parts yields

2 |∇u| dx = f (u)u dx (5.4). Ω

Ω

Using (5.1), (5.3), (5.4) we arrive at

n−2 f (u)u − F (u) dx < 0, 2n Ω

20


which contradicts (5.2). If f (0) = 0 and u is a sign-changing solution of (2.1), then the assertion follows from the unique continuation property. In fact, let x1 ∈ ∂Ω be such that x · ν > 0 in a neighborhood Γ1 of x1 in ∂Ω (recall that ∂Ω is smooth). Then the above arguments guarantee ∂u/∂ν = 0 on Γ1 . Since u = 0 and ∆u = f (u) = 0 on ∂Ω, / Ω. Then all the second derivatives of u have to vanish on Γ1 . Set u(x) := 0 for x ∈ u is a solution of (2.1) in a neighborhood of Γ1 , hence u ≡ 0 in this neighborhood due to [272, Satz 2]. Using the same result one can easily show u ≡ 0 in Ω. Remark 5.3. The idea of considering the multiplier x · ∇u was used before in [455] in the linear case f (u) = µu (for a different purpose, namely an integral representation of the eigenvalues of the Laplacian). Identities similar to (5.1) (see Lemma 31.4 for the case of systems and see also [431]) are sometimes called RellichPohozaev type identities in the literature.

6. Homogeneous nonlinearities In this section we use variational methods in order to study the problem −∆u = |u|p−1 u + λu, u = 0,

x ∈ Ω,

(6.1)

x ∈ ∂Ω.

The energy functional E has the form E(u) = Ψ(u) − Φ(u), where Ψ(u) :=

1 2

Ω

|∇u(x)|2 − λu2 dx and Φ(u) :=

1 p+1

Ω

|u|p+1 dx.

(6.2)

Notice that Ψ is quadratic and Φ is positively homogeneous of order p + 1 = 2. Therefore, if Ψ (w) = µΦ (w) (6.3) for some µ > 0, then, setting t := µ1/(p−1) , we get E (tw) = Ψ (tw) − Φ (tw) = t Ψ (w) − tp−1 Φ (w) = 0.

(6.4)

Consequently, tw is a critical point of E, hence a classical solution of (6.1) if p ≤ pS . A nontrivial function w satisfying (6.3) will be found by minimizing the functional Ψ with respect to the set M := {u : Φ(u) = 1} and using the following well-known Lagrange multiplier rule.

6. Homogeneous nonlinearities

21

Theorem 6.1. Let X be a real Banach space, w ∈ X and let Ψ, Φ1 , . . . , Φk : X → R be C 1 in a neighborhood of w. Denote M := {u ∈ X : Φi (u) = Φi (w) for i = 1, . . . , k} and assume that w is a local minimizer of Ψ with respect to the set M . If Φ1 (w), . . . , Φk (w) are linearly independent, then there exist µ1 , . . . , µk ∈ R such that k Ψ (w) = µi Φi (w). i=1

Our proofs of the main results of this section (Theorem 6.2 and Theorem 6.7(i)) follow those in [505, Theorem I.2.1 and Lemma III.2.2]. Let us first consider the subcritical case. Theorem 6.2. Assume Ω bounded. Let 1 0, λ1 Ω the functional Ψ is convex and coercive. Let uk ∈ M := {u ∈ X : Φ(u) = 1}, uk u in X. Then uk → u in Lp+1 (Ω) due to X → → Lp+1 (Ω), hence u ∈ M . Consequently, the set M is weakly sequentially closed in the reflexive space X and there exists w ∈ M such that Ψ(w) = inf M Ψ. Since |w| ∈ M and Ψ(|w|) = Ψ(w), we may assume w ≥ 0. Moreover, Φ (w)w = (p + 1)Φ(w) = p + 1, hence Φ (w) = 0. Theorem 6.1 guarantees the existence of µ ∈ R such that Ψ (w) = µΦ (w), hence 0 < 2Ψ(w) = Ψ (w)w = µΦ (w)w = µ(p + 1)Φ(w) = µ(p + 1). Consequently, µ > 0 and we deduce from (6.4) that u := µ1/(p−1) w is a nonnegative variational solution of (6.1), u ≡ 0. Corollary 3.4 guarantees that u is a classical solution and the strong maximum principle shows u > 0 in Ω. Remarks 6.3. (i) Annulus. Assume that Ω = {x ∈ Rn : 1 < |x| < 2}, λ < λ1 and let X denote the space all of radial functions in W01,2 (Ω). It is easily seen that X is compactly embedded into the space Y of all radial functions in Lp+1 (Ω) 1,2 p+1 (1, 2) , for any p > 1 (in fact, X and Y are isomorphic to W0 (1, 2) and L respectively). Moreover, any critical point of E in X is obviously a classical solution of (6.1). Hence the proof of Theorem 6.2 guarantees the existence of a positive classical solution of (6.1) for all p > 1. (ii) Nonexistence for λ ≥ λ1 . If Ω is bounded, λ ≥ λ1 and p > 1 is arbitrary, then (6.1) does not have positive stationary solutions. To see this, it is sufficient to multiply the equation in (6.1) by the first eigenfunction ϕ1 to obtain

0= |u|p−1 uϕ1 dx + (λ − λ1 ) uϕ1 dx > 0 Ω

provided u is a positive solution.

Ω

22


Remark 6.4. Unbounded domains. Let Ω = Rn , 1 < p < pS and λ < 0 (notice that 0 is the minimum of the spectrum of −∆ in W 1,2 (Rn )). Let X and Y denote the space of radial functions in W 1,2 (Rn ) and Lp+1 (Rn ), respectively. If n ≥ 2, then X is compactly embedded in Y (see [76, Theorem A.I’]) so that we may use the approach above in order to get a positive solution of (6.1). Moreover, using symmetrization it is easy to see that the minimizer of Ψ(u) = Schwarz 1 1 2 2 |∇u| dx in M − λu := {u ∈ X : p+1 |u|p+1 dx = 1} is also a minimizer X 2 Ω Ω p+1 1 1,2 n in the larger set M := {u ∈ W (R ) : p+1 Ω |u| dx = 1}. In the case Ω = Rn one can use a similar approach to that used in Theorem 6.2 for functions f = f (u) (or f = f (|x|, u)) which need not be homogeneous. In fact, 1 2 if one is able to find a minimizer u of |∇u| dx in the set N := {u ∈ X : 2 Ω F (u) dx = 1}, then there exists σ > 0 such that the function uσ (x) := u(x/σ) Ω solves (6.1). This idea was used in [76], for example. For more recent results on existence and uniqueness of positive solutions of this problem with f = f (u) we refer to [235], [420] and the references therein. If f depends on x (and not only on |x|) or if Ω is unbounded and not symmetric, then the situation is more delicate. In some cases, one can use the concentration compactness arguments in order to get a solution (see [50] and the references therein). Let us now turn to the critical case p = pS . In view of Corollary 5.2 and the proof of Theorem 6.2, the functional Ψ cannot attain its infimum over the set M if Ω is starshaped and λ = 0. In other words, denoting 2 2 Ω |∇u| − λ|u| dx Sλ (u, Ω) := , u 22∗ Sλ (Ω) := inf{Sλ (u, Ω) : u ∈ W01,2 (Ω), u = 0}

= inf{Sλ (u, Ω) : u ∈ W01,2 (Ω), u 2∗ = 1}, the value S0 (Ω) cannot be attained if Ω is starshaped. The following proposition shows that the same is true for any Ω = Rn . In particular, this means that the solution from Remark 6.3(i) (for p = pS and λ = 0) is not a minimizer of S0 (·, Ω). Proposition 6.5. We have S0 (Ω1 ) = S0 (Ω2 ) for any open sets Ω1 , Ω2 ⊂ Rn . If Ω = Rn , then S0 (Ω) is not attained. Proof. Let Ω1 , Ω2 ⊂ Rn be open. Since S0 (Ω) = S0 (x + Ω) for any x ∈ Rn , we may assume 0 ∈ Ω1 ∩ Ω2 . Denote wR (x) := w(Rx). Let ε > 0 and 0 = u1 ∈ W01,2 (Ω1 ), S0 (u1 , Ω1 ) < S0 (Ω1 ) + ε. Setting u˜1 (x) := ˜1 (x) = 0 if x ∈ / Ω1 , we have u ˜1 ∈ W01,2 (Rn ) = W 1,2 (Rn ) and u(x) if x ∈ Ω1 , u supp (˜ uR ˜R 1 ) ⊂ Ω2 if R is sufficiently large. Let u2 be the restriction of u 1 to Ω2 . 1,2 Then u2 ∈ W0 (Ω2 ), u2 = 0, and n uR u 1 , Rn ) S0 (Ω2 ) ≤ S0 (u2 , Ω2 ) = S0 (˜ 1 , R ) = S0 (˜

= S0 (u1 , Ω1 ) < S0 (Ω1 ) + ε.


23

Letting ε → 0 we obtain S0 (Ω2 ) ≤ S0 (Ω1 ). Exchanging the role of Ω1 and Ω2 we obtain the reversed inequality. Now assume Ω = Rn , u ∈ W01,2 (Ω) and S0 (u, Ω) = S0 (Ω). We may assume u ≥ 0, u = 0. Set u ˜(x) := u(x) for x ∈ Ω, u ˜(x) := 0 otherwise. Then S0 (˜ u , Rn ) = n n S0 (Ω) = S0 (R ), hence u ˜ is a minimizer of S0 (·, R ) and the proof of Theorem 6.2 shows the existence of µ > 0 such that u ˜ is a classical positive solution of the equation −∆u = µ|u|p−1 u in Rn . But this is a contradiction with u = 0 outside Ω. Remark 6.6. Best constant in Sobolev’s inequality. The function S0 (·, Rn ) 1/n −1/2 Γ(n)/Γ(n/2) at any attains its minimum S := S0 (Rn ) = n(n − 2)π function of the form uε (x − x0 ), where ε > 0, x0 ∈ Rn and uε (x) := (ε2 + |x|2 )−(n−2)/2 . This was proved by symmetrization techniques in [43] and [508] (for more general results of this kind see [111] and the references therein). If we set (n−2)/4 Cε := n(n − 2)ε2 , then the functions Cε uε (·−x0 ) are the only positive classical solutions of (6.1) with Ω = Rn , p = pS and λ = 0: This follows from Theorems 8.1 and 9.1 below. Theorem 6.7. Let n ≥ 3 and p = pS . Assume Ω bounded, 0 < λ < λ1 . Let S be the constant from Remark 6.6. (i) If Sλ (Ω) < S, then there exists u ∈ W01,2 (Ω) such that u > 0 in Ω and Sλ (Ω) = Sλ (u, Ω). (ii) If λ is close to λ1 , then Sλ (Ω) < S. Proof. (i) Let {uk } be a minimizing sequence for Sλ (·, Ω), uk 2∗ = 1. Replacing uk by |uk | we may assume uk ≥ 0. Since

λ 2 1− |∇uk |2 − λu2k dx = Sλ (uk , Ω) → Sλ (Ω), |∇uk | dx ≤ λ1 Ω Ω the sequence {uk } is bounded in W01,2 (Ω) and we may assume uk u in W01,2 (Ω). ∗ Due to the embeddings W01,2 (Ω) → L2 (Ω) and W01,2 (Ω) → → L2 (Ω) we obtain ∗ uk u in L2 (Ω) and uk → u in L2 (Ω). Passing to a subsequence we may assume uk (x) → u(x) a.e. Given t ∈ [0, 1], denote 2∗ −2 , ψk = ψk (t) := 2∗ uk + (t − 1)u uk + (t − 1)u

∗

ψ = ψ(t) := 2∗ tu|tu|2

−2

.

24


Then ψk → ψ a.e. in Ω and ψk , ψ are uniformly bounded in L2∗ (Ω), where 2∗ := (2∗ ) = 2n/(n + 2). Using Vitali’s convergence theorem we obtain

Ω

2∗ d uk + (t − 1)u dt dx Ω 0 dt

1

1

∗ ψk u dx dt → ψu dx dt = |u|2 dx as k → ∞, = 2∗

|uk |

0

2∗

− |uk − u|

Ω

0

hence

1

dx =

Ω

Ω

∗

∗

u 22∗ = 1 − uk − u 22∗ + o(1),

where o(1) → 0 as k → ∞. The weak convergence uk u in W01,2 (Ω) implies

Ω

2

|∇uk | dx =

Ω

2

|∇(uk − u)| dx +

Ω

|∇u|2 dx + o(1),

hence Sλ (Ω) = Sλ (uk , Ω) + o(1)

2 = |∇u|2 − λu2 dx + o(1) |∇(uk − u)| dx + Ω

Ω

≥ S uk − u 22∗ + Sλ (Ω) u 22∗ + o(1) ∗

∗

≥ S uk − u 22∗ + Sλ (Ω) u 22∗ + o(1) ∗ = S − Sλ (Ω) uk − u 22∗ + Sλ (Ω) + o(1). ∗

Now S > Sλ (Ω) implies uk → u in L2 (Ω), hence u 2∗ = 1. The weak lower semi-continuity of the norm in W01,2 (Ω) guarantees Sλ (u, Ω) ≤ lim inf Sλ (uk , Ω) = Sλ (Ω), k→∞

thus Sλ (u, Ω) = Sλ (Ω). Similarly as in the proof of Theorem 6.2, a suitable positive multiple of u is a classical positive solution of (6.1) with p = pS , hence u > 0 in Ω. (ii) Let ϕ1 be the first eigenfunction, ϕ1 2∗ = 1. Then

Sλ (Ω) ≤ Sλ (ϕ1 , Ω) = (λ1 − λ)

Ω

ϕ21 dx < S

if λ is close to λ1 . Corollary 6.8. Let n ≥ 3 and p = pS . Assume Ω bounded, 0 < λ < λ1 . If λ is close to λ1 , then problem (6.1) has a classical positive solution.


25

Remarks 6.9. (i) Positive solutions in the critical case [98]. Let Ω be bounded, p = pS , λ∗ := inf{λ ∈ (0, λ1 ) : Sλ (Ω) < S}. Set uε (x) := (ε + |x|2 )−(n−2)/2 (cf. Remark 6.6) and assume 0 ∈ Ω. If n ≥ 4 and λ > 0, then careful estimates show Sλ (uε ϕ, Ω) < S provided ϕ ∈ D(Ω) is nonnegative, ϕ = 1 in a neighborhood of 0 and ε is small enough. Consequently, λ∗ = 0 in this case and problem (6.1) possesses a positive solution for any λ ∈ (0, λ1 ). Now let n = 3 and Ω = B1 (0). If λ > λ1 /4, then Sλ (uε ϕ, Ω) < S provided ϕ(x) = cos(π|x|/2) and ε is small enough. On the other hand, one can use a Pohozaev-type identity for radial functions in order to prove that (6.1) does not have positive radial solutions if λ ≤ λ1 /4. Since any positive solution of (6.1) is symmetric due to [239] we have λ∗ = λ1 /4 in this case and the problem possesses positive solutions if and only if λ ∈ (λ1 /4, λ1 ). Another proof of the above results for Ω = B1 (0) based on the ODE techniques can be found in [40]. The authors use the symmetry of positive solutions u = u(|x|) of (6.1) and the substitution y(t) = u(|x|), t = (n − 2)n−2 |x|−(n−2) , which transforms the problem into the ODE y +t−k (λy+y pS ) = 0 with k := 2(n − 1)/(n − 2). (ii) Uniqueness for p ≤ pS . Uniqueness of positive solutions of (6.1) in the case Ω = B1 (0), p ≤ pS , was established in [239] (if λ = 0), [393] (if λ ≥ 0, p ≤ psg ), [310] (if λ < 0, p < pS ) and [544], [503] (if λ > 0, p ≤ pS ). Some of these articles contain also uniqueness results for more general functions f (|x|, u) and for Ω being an annulus. Uniqueness fails for general bounded domains (see (iii) and (iv) below). On the other hand if Ω satisfies some convexity and symmetry properties, then uniqueness (and non-degeneracy) for positive solutions of (6.1) is true, at least for some values of p and/or λ (see [147], [118], [256], for example). (iii) Nonradial minimizers. Let Ω = {x : 1 < |x| < 2}, n ≥ 3, λ = 0 and p > 1. Set S(u, Ω, p) :=

|∇u|2 dx , u 2p+1

Ω

S(Ω, p) := inf{S(u, Ω, p) : u ∈ W01,2 (Ω) u = 0}, S r (Ω, p) := inf{S(u, Ω, p) : u ∈ W01,2 (Ω) u = 0, u is radial}. By Remark 6.3(i), problem (6.1) with λ = 0 has a positive radially symmetric solution u which minimizes S(·, Ω, p) in the class of radial functions. Since S(Ω, pS ) is not attained (see Proposition 6.5), we have S(Ω, pS ) < S r (Ω, pS ). It is easy to see that the functions p → S(Ω, p) and p → S r (Ω, p) are continuous. Consequently, S(Ω, p) < S r (Ω, p) also for p < pS , p close to pS . Since S(Ω, p) is attained in the subcritical case, the corresponding (positive) minimizer is not radially symmetric.

26


(iv) Effect of the topology of domain. Let Ω be bounded, n ≥ 3, p = pS and λ = 0. The above considerations show that (6.1) has a positive solution if Ω is an annulus but it does not possess positive solutions if Ω is starshaped. It was proved in [47] that this problem has positive solutions whenever the homology of dimension d of Ω with Z2 coefficients is nontrivial for some positive integer d. In particular, this is true when n = 3 and Ω is not contractible. On the other hand, there are several examples showing that positive solutions do exist even if Ω is contractible (see [149], for example). Let Ω be bounded and let its Ljusternik-Schnirelman category be bigger than 1. If p < pS , then problem (6.1) admits multiple positive solutions whenever p is close to pS or λ < 0 and |λ| is large enough (see [68]); the same is true if p = pS , λ > 0 is small and n ≥ 4 (see [456], [323]). Again, this topological condition on Ω is not necessary (see [148], where multiple positive solutions are constructed for any p < pS , λ = 0 and Ω being starshaped, and see [408] for the critical case). (v) Critical case in the unit ball. Let Ω = B1 (0), n ≥ 3, p = pS and consider radial (classical) solutions of (6.1). Due to Corollary 5.2, nontrivial solutions do not exist if λ ≤ 0. Denote by X the space of all radial functions in W01,2 (Ω) and let λrk denote the k-th eigenvalue of −∆ in X (λrk = k 2 π 2 if n = 3). The corresponding radial eigenfunction ϕrk (considered as a function of r := |x|) has (k −1) zeros in (0, 1) and each point (0, λrk ) ∈ X ×R is a bifurcation point for (6.1) (see [451]). The corresponding bifurcation branch Bk of nontrivial solutions is an unbounded continuous curve and u has (k − 1) zeros for any (u, λ) ∈ Bk . Moreover, there exists µk := lim{λ : (u, λ) ∈ Bk , u X → ∞}, k = 1, 2, . . . , and we have µk = (k − 12 )2 π 2 if n = 3, µ1 = 0 if n ≥ 4, µk+1 = λrk if n = 4, 5, µk+1 ∈ (0, λrk ) if n = 6, µk = 0 if n ≥ 7 (see Figure 1 and [40], [41], [42], [39]). Denote µ ˜k := inf{λ : (u, λ) ∈ Bk }. The results mentioned in (i) and (ii) imply µ ˜1 = µ1 = λ1 /4 if n = 3, µ ˜1 = µ1 = 0 if n ≥ 4. Similarly, [34] and [234] imply µ ˜2 = µ2 if n = 4, µ ˜2 < µ2 if n = 5 but the relation between µ ˜2 and µ2 for n ∈ {3, 6} seems to be an open problem. Denote also λ∗ := inf{˜ µk : k ≥ 2}. Then λ∗ > 0 provided n ≤ 6 (see [39]). On the other hand, problem (6.1) with Ω = B1 (0), n ≥ 4, p = pS and λ > 0 has infinitely many nontrivial solutions in W01,2 (Ω) (see [212]). Consequently, if n ∈ {4, 5, 6} and λ < λ∗ , then all these solutions (except for ±u1 where u1 denotes the unique positive solution) have to be nonradial. The existence of (nonradial sign-changing) solutions for Ω = B1 (0), n = 3 and λ ∈ (0, λ1 /4] seems to be open. Many interesting results on singular radial solutions of (6.1) for Ω = B1 (0) and p > 1 can be found in [73]. Remarks 6.10. Supercritical case. Let n ≥ 3, p > pS . (i) If λ = 0, then the analogue of the result of [47] mentioned in Remark 6.9(iv) does not hold (see [406], [407]).


27

u 6 B1

0 µ1

B2

λr2

λ1 µ2 n=3

6

6

µ1 = 0

λ1 = µ2

λr2

µ1 = 0

n=4

λr2

n=5

6

µ1 = 0

λ1 = µ2

6

µ2

λ1 n=6

λr2

µk = 0

λ1

λr2

n≥7

Figure 1: Bifurcation diagrams for radial solutions of (6.1) with p = pS and Ω = B1 (0).

(ii) Let Ω = B1 (0). Then the points (0, λrk ) from Remark 6.9(v) are bifurcation points for (6.1) also in this case. Let Bk (p) denote the corresponding bifurcation branch and let µk (p), µ ˜k (p) have similar 6.9(v). If as in Remark √ meaning √ n > 6, assume also p < pZZ := n + 1 − 2n − 3 /(n − 3 − 2n − 3 . Then

28


u 6

0 p < pS

6

λ1

6

6

λ1

0 p = pS , n > 3

λ1 0 λ1 /4 p = pS , n = 3

0

λ1 p > pS , n ≤ 6

Figure 2: Bifurcation diagrams for positive solutions of (6.1) with Ω = B1 (0).

0 < µ ˜1 (p) < µ1 (p) < λ1 and problem (6.1) has infinitely many radial positive (classical) solutions if λ = µ1 (p) (see Figure 2 and [546], [104], [365]). It is not clear whether the condition p < pZZ for n > 6 is optimal, but some restrictions on n or p for this behavior of B1 (p) may be expected. In fact, bifurcation diagrams for positive solutions of the related problem −∆u = λ(1 + u)p , x ∈ B1 (0), (6.5) u = 0, x ∈ ∂B1 (0), in the supercritical case are completely different for p < pJL and p ≥ pJL , where pJL is defined in (9.3) (see Figure 3 and [293]). Note also that the same diagrams as in Figure 3 are true for the problem x ∈ B1 (0), −∆u = λeu , (6.6) u = 0, x ∈ ∂B1 (0), and the three cases I, II and III correspond to n ≤ 2, 3 ≤ n ≤ 9 and n ≥ 10, respectively.

7. Minimax methods

u 6

29

6

λ

0

0

I. p ≤ pS

6

λ II. pS < p < pJL

λ

0 III. p ≥ pJL

Figure 3: Bifurcation diagrams for positive solutions of (6.5).

7. Minimax methods In this section we look for saddle points of the energy functional E defined in (3.5) by minimax methods. Throughout this section we assume that f satisfies the growth assumption (3.4) so that E is a C 1 -functional in the Hilbert space W01,2 (Ω) and its critical points correspond to (variational) solutions of (2.1). Even if we considered a finite-dimensional space X = R2 and a smooth functional E : X → R, then (looking at the graph of E as the earth’s surface) existence of a saddle (mountain pass) on a mountain range between two valleys is not clear, in general. For example, if E : R2 → R : (x, y) → ex − y 2 , A0 = (0, −2), A1 = (0, 2), then any path from A1 to A2 in R2 has to cross the line {y = 0} where E > 0 > max{E(A1 ), E(A2 )}, but the functional E does not possess critical points at all. If one looks for a point with a minimal height on the “mountain range” described by the graph of E on {(x, y) : y = 0}, then any minimizing sequence has the form (xk , 0), where xk → −∞. In particular, it is not compact and we cannot choose a subsequence converging to the desired saddle point. Therefore, dealing with abstract functionals E in a real Banach space X, we shall need additional information on E which will prevent the problem mentioned above. Definition 7.1. A sequence {uk } in X is called a Palais-Smale sequence if the sequence {E(uk )} is bounded and E (uk ) → 0. We say that E satisfies condition (PS) if any Palais-Smale sequence is relatively compact. We say that E satisfies condition (PS)β (Palais-Smale condition at level β) if any sequence {uk } satisfying E(uk ) → β, E (uk ) → 0, is relatively compact. A real number β is called a critical value of E if there exists u ∈ X with E (u) = 0 and E(u) = β. The following mountain pass theorem is due to [23]. Our proofs of this theorem and Theorems 7.4, 7.8 below closely follow those in [505, Chapter II].

30


Theorem 7.2. Suppose that E ∈ C 1 (X) satisfies (PS). Let u0 , u1 ∈ X, M := max{E(u0 ), E(u1 )}, P := {p ∈ C([0, 1], X) : p(0) = u0 , p(1) = u1 },

(7.1)

β := inf max E(p(t)). p∈P t∈[0,1]

If β > M , then β is a critical value of E. Given β ∈ R and δ > 0, denote Nδ = Nδ (β) := {u ∈ X : |E(u) − β| ≤ δ, E (u) ≤ δ} and Eβ := {u ∈ X : E(u) < β}. In the proof of Theorem 7.2 we shall need the following deformation lemma. Lemma 7.3. Suppose that E ∈ C 1 (X) and let Nδ (β) = ∅ for some δ < 1. Choose ε = δ 2 /2. Then there exists a continuous mapping Φ : X × [0, 1] → X such that (i) Φ(u, t) = u whenever t = 0 or |E(u) − β| ≥ 2ε, (ii) t → E(Φ(u, t)) is nonincreasing for all u, (iii) Φ(Eβ+ε , 1) ⊂ Eβ−ε . In addition, Φ(·, t) is odd if E is even. Proof. In order to avoid all technicalities we shall assume, in addition, that E ∈ C 2 (X) and X is a Hilbert space. Notice that these assumptions are satisfied in our applications if f has the form (2.2), for example (and see e.g. [505] for the proof in the general case). Choose functions ϕ, ψ : R → [0, 1] such that ϕ is smooth, ϕ(t) = 1 for |t−β| ≤ ε, ϕ(t) = 0 for |t − β| ≥ 2ε, ψ(t) = 1 for t ≤ 1 and ψ(t) = 1/t for t > 1. The vector field F : X → X : u → −ϕ E(u) ψ E (u) ∇E(u) is bounded and locally Lipschitz. Consequently, the initial value problem Φt (u, t) = F Φ(u, t) ,

for t ∈ [0, 1],

Φ(u, 0) = u has a unique solution for any u ∈ X. The function Φ defined in this way is obviously continuous and satisfies (i). Denoting v := Φ(u, t) we have d d E Φ(u, t) = E(v) = E (v)F (v) = −ϕ E(v) ψ E (v) E (v) 2 ≤ 0, dt dt thus (ii) is true.

7. Minimax methods

31

Assertion (iii) will be proved by a contradiction argument. Let u ∈ Eβ+ε and (ii) implies |E Φ(u, t) −β| ≤ ε < δ for t ∈ [0, 1], assume Φ(u, 1) ∈ / Eβ−ε . Then hence Nδ = ∅ implies E Φ(u, t) ≥ δ for t ∈ [0, 1]. Using this estimate and the properties of the functions ϕ, ψ we get

E Φ(u, 1) = E(u) +

= E(u) −

0

1

d E Φ(u, t) dt dt

1

ϕ(. . . ) ψ(. . . ) E Φ(u, t) 2 dt

0

=1

≥δ 2

2

< β + ε − δ ≤ β − ε, a contradiction. Proof of Theorem 7.2. Assume that β is not a critical value of E. Then it is easy to use condition (PS) in order to find δ > 0 such that Nδ (β) = ∅. We may assume δ < 1, δ 2 < β − M . Let ε := 12 δ 2 be β there from Lemma 7.3. By the definition of 2 ) ≤ M < β−δ = β−2ε exists p ∈ P such that maxt∈[0,1] E p(t) < β+ε. Since E(u i for i = 0, 1, Lemma 7.3(i) guarantees that p1 : t → Φ p(t), 1 is an element of P . Now Lemma 7.3(iii) implies maxt∈[0,1] E p1 (t) ≤ β − ε, which contradicts the definition of β. The next theorem is again due to [23]. It represents a symmetric variant of Theorem 7.2 and we will use it for the proof of existence of infinitely many solutions of problem (2.1). Theorem 7.4. Suppose that E ∈ C 1 (X) is even and satisfies (PS). Let X + , X − be closed subspaces of X with dim X − = codim X + + 1 < ∞. Let E(0) = 0 and let there exist α, ρ, R > 0 such that E(u) ≥ α for all u ∈ Sρ+ := {u ∈ X + : u = ρ} and E(u) ≤ 0 for all u ∈ X − , u ≥ R. Set Γ := {h ∈ C(X, X) : h is odd, h(u) = u if E(u) ≤ 0}, β := inf max E h(u) . h∈Γ u∈X −

Then β is a critical value of E, β ≥ α. The proof of the above theorem will be almost the same as the proof of Theorem 7.2 provided we prove the following Intersection Lemma. Lemma 7.5. If ρ > 0 and h ∈ Γ, then h(X − ) ∩ Sρ+ = ∅. Proof of Theorem 7.4. Lemma 7.5 implies β ≥ α. Assume that β is not a critical value of E. Then Nδ (β) = ∅ for some δ > 0 and we may assume δ < 1, δ 2 < α. Let ε := δ 2 /2 and Φ be from Lemma 7.3 and choose h ∈ Γ such that

32


− E h(u) < β + ε for allu ∈ X . Set h1 (u) := Φ h(u), 1 . Then h1 ∈ Γ and E h1 (u) = E Φ h(u), 1 < β − ε, due to Lemma 7.3(iii). But this contradicts the definition of β. In the proof of Lemma 7.5 we shall need the notion of Krasnoselskii genus. Definition 7.6. Let A be the set of all closed subsets of X satisfying A = −A. If A ∈ A, then we set γ(A) := 0 if A = ∅ and γ(A) := inf{m : ∃h ∈ C(A, Rm \ {0}), h odd} otherwise. The following proposition is proved in [505, Propositions II.5.2 and II.5.4]: Proposition 7.7. Suppose that A, A1 , A2 ∈ A and h ∈ C(X, X) is odd. Then the following is true: (1) γ(A) ≥ 0, γ(A) = 0 if and only if A = ∅, (2) if A1 ⊂ A2 , then γ(A1 ) ≤ γ(A2 ), (3) γ(A1 ∪ A2 ) ≤ γ(A 1 ) + γ(A2 ), (4) γ(A) ≤ γ h(A) , (5) if A is compact and 0 ∈ / A, then γ(A) < ∞ and there exists a symmetric neighborhood U of A such that U ∈ A and γ(A) = γ(U). (6) Let D be a bounded symmetric neighborhood of zero in Y , where Y is a subspace of X with m := dim(Y ) < ∞, and let ∂Y D denote the boundary of D in Y . Then γ(∂Y D) = m. − := Proof of Lemma 7.5. Let ρ > 0 and h ∈ Γ. Set R1 := max{R, ρ}, BR 1 − {u ∈ X : u < R1 } and Sρ := {u ∈ X : u = ρ}. Since E(u) ≤ 0 for u ∈ X − , u ≥ R, we have h(u) = u > ρ for all u ∈ X − , u > R1 , hence − h(X − ) ∩ Sρ = h BR ∩Sρ is compact. In particular, A := h(X − ) ∩ Sρ+ fulfills the 1 assumptions of Proposition 7.7(5), thus there exists its symmetric neighborhood U with γ(U ) = γ(A). By (2) and (3) in Proposition 7.7 we obtain

γ(A) = γ(U ) ≥ γ h(X − ) ∩ Sρ ∩ U ≥ γ h(X − ) ∩ Sρ −γ(B),

(7.2)

where Sρ := {u ∈ X : u = ρ} and B := h(X − ) ∩ Sρ \ U . Let Z be a direct complement of X + in X and let π : X → Z denote the projection along X + . Since U is a neighborhood of h(X − ) ∩ Sρ+ , we get B ∩ X + = ∅, hence 0 ∈ / π(B) and the definition of γ implies γ(B) ≤ dim Z = codim X + . (7.3) −1 − Now (2) and (4) in Proposition 7.7 guarantee γ h(X ) ∩ Sρ ) ≥ γ h (Sρ ) ∩ X − . Since h(0) = 0 and h(u) = u for u ∈ X − , u > R, the set h−1 (Sρ ) ∩ X − contains the relative boundary of {u ∈ X − : h(u) < ρ} which is a symmetric bounded

7. Minimax methods

33

neighborhood of zero in X − . Consequently, using (2) and (6) in Proposition 7.7 we arrive at γ h(X − ) ∩ Sρ ≥ dim X − = codim X + + 1. (7.4) Now (7.2)–(7.4) imply γ(A) ≥ 1, hence A = ∅. Theorems 7.2 and 7.4 guarantee the following solvability result. Theorem 7.8. Assume Ω bounded. Let f be a Carathéodory function, and let there exist p < pS , R > 0 and µ > 2 such that |f (x, u)| ≤ C(1 + |u|p ) for all x ∈ Ω, u ∈ R and f (x, u)u ≥ µF (x, u) > 0 for all x ∈ Ω and |u| > R. (i) If there exist c < λ1 and ρ ∈ (0, 1) such that f (x, u)/u ≤ c for all x ∈ Ω and |u| < ρ, then there exists a positive solution of (2.1). (ii) If f (x, −u) = −f (x, u) for all x ∈ Ω and u ∈ R, then there exists a sequence {uk } of solutions of (2.1) with E(uk ) → ∞ as k → ∞. Proof. The energy functional E associated with (2.1) is C 1 . Let us first verify that E satisfies condition (PS). Let {uk } be a Palais-Smale sequence. Denote |u|1,2 := 1/2 2 and notice that this is an equivalent norm in X := W01,2 (Ω). Then Ω |∇u| dx

f (x, uk )uk dx o(1 + |uk |1,2 ) = −E (uk )uk = −|uk |21,2 + Ω

µ − 1 |uk |21,2 + f (x, uk )uk − µF (x, uk ) dx − µE(uk ) = 2 Ω µ 2 − 1 |uk |1,2 − C1 , ≥ 2 where C1 > 0 is independent of k. Consequently, the sequence {uk } is bounded in X. We have ∇E(u) = u + F1 (u), where F1 is compact.2 Since {uk } is bounded in X, we may assume (passing to a subsequence if necessary) F1 (uk ) → w in X for some w ∈ X. Since o(1) = ∇E(uk ) = uk − F1 (uk ), we obtain uk → w, hence {uk } is relatively compact. (i) We will use Theorem 7.2. In order to get a positive solution, let us define u ˜ ˜ ˜ ˜ ˜ := f (x, u) := f (x, u) if u ≥ 0, f (x, u) = 0 otherwise, F (x, u) := 0 f (x, s) ds, E(u) 1 2 1 ˜ ˜ |∇u| dx − and satisfies condition (PS). F (x, u) dx, and notice that E is C 2 Ω Ω ˜ 0 ) = 0. The assumption f (x, u)/u ≤ c for |u| < ρ guarantees Set u0 := 0, then E(u 2 ˜ |F (x, u)| ≤ (c/2)u for |u| < ρ. If |u| ≥ ρ, then the growth assumption |f (x, u)| ≤ C(1 + |u|p ) implies |F˜ (x, u)| ≤ C(|u| + |u|p+1 ) ≤ (c/2)u2 + C2 |u|p+1 , 2 The

Nemytskii mapping F : Lp+1 (Ω) → L(p+1) (Ω) : u → f (·, u) is continuous. The embed ding Ip : X → Lp+1 (Ω) is compact, hence the dual mapping Ip : Lp+1 (Ω) → X is compact as well. Let R : X → X denote the Riesz isomorphism in the Hilbert space X (thus RE (u) = ∇E(u)) and let J : L(p+1) (Ω) → Lp+1 (Ω) be the isomorphism defined by (Jw)u = Ω uw dx for u ∈ Lp+1 (Ω). Then ∇E(u) = u + F1 (u), where F1 : X → X : u → RIp JF Ip (u) is compact.

34


where C2 := C(1 + ρ−p ). Consequently, if Cp denotes the norm of the embedding X → Lp+1 (Ω), then

1 c ˜ E(u) ≥ |∇u|2 dx − u2 dx − C2 |u|p+1 dx 2 Ω 2 Ω Ω 1 c 2 − − C2 Cpp+1 |u|p−1 ≥ 1,2 |u|1,2 ≥ α > 0 2 2λ1 provided |u| = δ is small enough. Now the assumption f (x, u)u ≥ µF (x, u) > 0 1,2 d u−µ F (x, u) ≥ 0 for u > R, hence F (x, u) ≥ b(x)uµ for u > R, where implies du b(x) := R−µ F (x, R) > 0. Hence, fixing u ∈ X, u > 0 in Ω, denoting

1 A(u) := |∇u|2 dx, B(u) := b(x)uµ dx > 0, (7.5) 2 Ω Ω and taking t > 0, we obtain ˜ E(tu) = E(tu) ≤ t2 A(u) − tµ B(u) + C3 → −∞ where we used the estimate

0 0 and Theorem 7.2 guarantees the existence of a solution u with ˜ E(u) ≥ α. Since f˜(x, u) = 0 for u ≤ 0, the maximum principle implies u ≥ 0. Now ˜ E(u) = E(u) > 0, hence u = 0 and using the maximum principle again we obtain u > 0 in Ω. (ii) Choose a positive integer k. Let X − denote the linear hull of ϕ1 , ϕ2 , . . . , ϕk , and X + be the closure of the linear hull of ϕk , ϕk+1 , . . . . The growth condition on f guarantees |F (x, u)| ≤ CF (1 + |u|p+1 ) for suitable CF > 0. Set q := pS if n ≥ 3 and choose q > p otherwise. Let C4 := CF Cqp+1−r and C5 := CF |Ω|, where Cq denotes the norm of the embedding Iq : W01,2 (Ω) → Lq+1 (Ω), r ∈ (0, p + 1) is defined by r/2 + (p + 1 − r)/q = 1 and |Ω| denotes the measure of Ω. If u ∈ X + r/2 1/(p−1) and u = ρ := ρk := λk /(4C4 ) , then

1 |∇u|2 dx − CF |u|p+1 dx − C5 E(u) ≥ 2 Ω Ω 1 ≥ |u|21,2 − CF u r2 u p+1−r − C5 q+1 2 1 −r/2 2 − C4 λk |u|p−1 ≥ 1,2 |u|1,2 − C5 2 1 −r/2 r/(p−1) − C4 λk ρp−1 ρ2 − C5 = C6 λk = − C5 , 2

7. Minimax methods

35

where C6 = (4p C4 )−1/(p−1) . Denote α = αk := inf{E(u) : u ∈ X + , |u|1,2 = ρ}. Since λk → ∞ as k → ∞, we have αk → ∞. On the other hand, estimates in (i) show E(tu) ≤ t2 A(u) − tµ B(u) − C3 , where A, B are defined in (7.5). Since A(u) = 1/2 for |u|1,2 = 1 and C7 := inf{B(u) : u ∈ X − , |u|1,2 = 1} > 0, we have E(u) ≤

1 2 |u| − C7 |u|µ1,2 + C3 2 1,2

for all u ∈ X − ,

hence the assumptions of Theorem 7.4 are satisfied for any k large enough and we obtain a sequence of critical points uk of E satisfying E(uk ) ≥ αk . (In fact, a more careful choice of ρ above enables one to use Theorem 7.4 for any k.) Remarks 7.9. (i) Linking. Let f be differentiable in u, f (x, 0) = 0, f (x, u)/u ≥ fu (x, 0) for all x ∈ Ω and u ∈ R. If the assumption f (x, u)/u ≤ c < λ1 for u small in Theorem 7.8(i) fails, then one can use a modification of the mountain pass theorem, so called “linking”, in order to prove the existence of a nontrivial solution of (2.1) (see [505, Section II.8] and the references therein). (ii) Perturbation results. Consider the problem −∆u = |u|p−1 u + ϕ, x ∈ Ω, (7.6) u = 0, x ∈ ∂Ω, where Ω ⊂ Rn is bounded, 1 < p < pS and ϕ ∈ W −1,2 (Ω) := W01,2 (Ω) . Theorem 7.8(ii) guarantees existence of infinitely many solutions of (7.6) provided ϕ = 0. The same result is known to be true for ϕ belonging to a residual set in W −1,2 (Ω) (see [45]) and for all ϕ ∈ W −1,2 (Ω) provided p(n − 2) < n (see [300, Théorème V.4.6.]; see also [506], [46], [452] and [48]). On the other hand, if n > 2, p ∈ [psg , pS ) and ϕ is a general (smooth) function, then even the solvability of (7.6) seems to be open. (iii) Unbounded domains. If Ω = Rn , then the existence of infinitely many solutions of (2.1) is known in many cases as well. We refer to [76], [140], [139], [9] and the references therein. (iv) Critical case. Let Ω ⊂ Rn be bounded, p = pS and λ > 0. If n ≥ 7, then problem (6.1) possesses infinitely many solutions, see [162]. Such a result is known for any n ≥ 4 if the domain Ω exhibits suitable symmetries (see [212]) but not for general domains (cf. also the results for Ω being a ball mentioned in Remark 6.9(v)). If n = 6 and λ ∈ (0, λ1 ), then (6.1) has at least two (pairs of) solutions for any bounded Ω, see [117]. Recall also that if λ ≤ 0, p ≥ pS and Ω is starshaped, then (6.1) does not possess nontrivial classical solutions due to Corollary 5.2.

36


8. Liouville-type results In order to prove a priori bounds for positive solutions of (2.1) with f (x, u) ∼ up as u → +∞, 1 < p < pS (see the rescaling method in Section 12), it will be important to know that the problems −∆u = up , and

−∆u = up , u = 0,

x ∈ Rn x ∈ Rn+ ,

(8.1)

x ∈ ∂Rn+ ,

(8.2)

do not possess positive bounded (classical) solutions. Here Rn+ denotes the halfspace {x ∈ Rn : xn > 0}. In fact, we shall see in Chapter II that these Liouvilletype results have important applications for parabolic problems as well. In this section we even prove that these problems do not possess any positive classical solution. The following two results are due to [240], [241], except for Theorem 8.1(ii) which was proved in [108]. Theorem 8.1. Let Ω = Rn and p > 1. (i) If p < pS , then equation (8.1) does not possess any positive classical solution. (ii) If p = pS , then any positive classical solution of (8.1) is radially symmetric with respect to some point. Theorem 8.2. Let 1 < p ≤ pS . Then problem (8.2) does not possess any positive classical solution. We will see in the next section that the condition p < pS is optimal for nonexistence in Rn . However, in the case of a half-space and if we consider only bounded positive solutions, nonexistence is known for a larger range of exponents, namely p < pS := (n + 1)/(n − 3)+ (note that pS is the Sobolev exponent in (n − 1) dimensions). This result is due to [150]. Theorem 8.3. Let 1 3.

Then problem (8.2) does not possess any positive, bounded classical solution. On the other hand, under a stronger assumption on p, one can extend the nonexistence result in Rn to elliptic inequalities. The following result is due to [238].

8. Liouville-type results

37

Theorem 8.4. Let 1 < p ≤ psg . Then the inequality −∆u ≥ up ,

x ∈ Rn

(8.3)

does not possess any positive classical solution. Remarks 8.5. (i) It seems unknown if the condition p ≤ pS is optimal for the nonexistence of positive solutions of (8.2). In the case of positive bounded solutions, the results recently announced in [178] indicate that the condition p < pS can be improved, but the optimal exponent seems to remain unknown. (ii) The condition p ≤ psg in Theorem 8.4 is optimal, as shown by the explicit example u(x) = k(1 + |x|2 )−1/(p−1) with n ≥ 3, p > psg and k > 0 small enough. (iii) Consider the inequality −∆u ≥ up in the half-space Rn+ (no boundary conditions required). Then nonexistence of positive solutions holds whenever p ≤ (n + 1)/(n − 1) (see [74]). (iv) Consider “quasi-solutions” of (8.1), i.e. (nonnegative) functions satisfying the double inequality aup ≤ −∆u ≤ up , x ∈ Rn , (8.4) for some a ∈ (0, 1). It is shown in [509] that if 1 psg and a ∈ (0, 1) is small enough, then (8.4) possesses positive solutions u ∈ C 2 (Rn ). Note that a simple example is provided by the function u(x) = k(1 + |x|2 )−1/(p−1) with k > 0 large enough. We start by proving Theorem 8.4, which is much easier than Theorems 8.1 and 8.2. The following proof (cf. [74], [501], [372]) is based on a rescaled test-function argument, and it is different and simpler than the original proof of [238]. Proof of Theorem 8.4. Take ξ ∈ D(B1 ), 0 ≤ ξ ≤ 1, with ξ = 1 for |x| ≤ 1/2, and let m = 2p/(p − 1). Fix R > 0 and define ϕR (x) = ξ m (x/R). We observe that ∆ϕR = mR−2 ξ m−1 ∆ξ + (m − 1)ξ m−2 |∇ξ|2 (x/R) hence

1/p

|∆ϕR | ≤ CR−2 ξ m−2 (x/R)χ{|x|>R/2} = CR−2 ϕR χ{|x|>R/2} . Multiplying (8.3) by ϕR , integrating by parts, and using H¨ older’s inequality, we obtain

1/p up ϕR ≤ − u ∆ϕR ≤ CR−2 u ϕR Rn

Rn

R/2 n/2. We may then apply Proposition 47.6(ii) once more to deduce that v(0) ≤ C(n, p). (8.14) Now assume that u is a solution of (8.9), fix x0 ∈ Ω and let R := dist(x0 , ∂Ω). Then v(x) := R2/(p−1) u(x0 + Rx) solves (8.12) and the conclusion follows from (8.14).

40


Remarks 8.8. (i) More general nonlinearities. Results similar to Theorem 8.7 for more general nonlinearities can be found in [240], [83], [473], [424]. In particular, universal singularity estimates of the type of (8.10) are established in [424] when the nonlinearity up is replaced by any f (x, u) such that f (x, u) ∼ up , as u → ∞, with 1 < p < pS . The method of proof is different: The estimate is directly deduced from the Liouville-type Theorem 8.1(i) by using rescaling and doubling arguments (see Theorem 26.8 and Lemma 26.11 below for a similar approach in the parabolic case). (ii) Singularities of quasi-solutions. For “quasi-solutions” of (8.1) (cf. Remark 8.5(iv)), the local behavior near an isolated singularity was studied in [509]. Let Ω = B(0, 1) \ {0}. If psg psg and a ∈ (0, 1) is small enough, then there exist solutions of (8.15) with arbitrarily large growth rates as x → 0. On the other hand, by a straightforward modification of the proof of [424, Theorem 2.1], one can show the following uniform and global property: For each p ∈ (1, pS ), there exist a = a(n, p) ∈ (0, 1) and C(n, p) > 0 such that, for any domain Ω ⊂ Rn , estimate (8.10) is true for any positive solution u ∈ C 2 (Ω) of (8.15). Note that, as a consequence of this estimate, one recovers the nonexistence result in Remark 8.5(iv). (iii) Radial supercritical case. When p ≥ pS , Ω = BR and u is a radial positive classical solution of (8.9), a similar argument as in Remark 4.3(iii) shows that u(r) ≤ C(R − r)−2/(p−1) , 0 ≤ r < R, for some C > 0. However the constant C cannot depend only on n, p, since otherwise this would imply nonexistence of radial positive classical solutions of (8.9) for Ω = Rn and p ≥ pS , hence contradicting Theorem 9.1 below. We now turn to the proof of Proposition 8.6. It is based on a key gradient estimate for local solutions of (8.7) (see (8.22) below). To establish this estimate, we prepare the following lemma, which provides a family of integral estimates relating any C 2 -function with its gradient and its Laplacian. The proof relies on the Bochner identity (8.18), on the change of variable v = uk+1 , and on testfunctions of the form ϕv m . In the rest of this section, we use the notation = Ω for simplicity. Lemma 8.9. Let Ω be an arbitrary domain in Rn , 0 ≤ ϕ ∈ D(Ω), and 0 < u ∈ C 2 (Ω). Fix q ∈ R and denote

I = ϕ uq−2 |∇u|4 , J = ϕ uq−1 |∇u|2 ∆u, K = ϕ uq (∆u)2 .


41

Then, for any k ∈ R with k = −1, there holds

1 uq |∇u|2 ∆ϕ + uq ∆u + (q − k)u−1 |∇u|2 ∇u · ∇ϕ, (8.16) αI + βJ + γK ≤ 2 where α=−

n−1 2 q(q − 1) k + (q − 1)k − , n 2

β=

n+2 3q k− , n 2

γ=−

n−1 . n

Proof. Step 1. We first claim that for all v ∈ C 2 (Ω), v > 0 and any m ∈ R, there holds

m(1 − m) 3m n−1 m−2 4 m−1 2 ϕv ϕv ϕ v m (∆v)2 |∇v| − |∇v| ∆v − 2 2 n

m 1 v m |∇v|2 ∆ϕ + ≤ v ∆v + mv m−1 |∇v|2 ∇v · ∇ϕ. 2 (8.17) First note that, by density, it suffices to prove (8.17) for v ∈ C 3 (Ω). To prove the claim, we start from the identity 1 ∆|∇v|2 = ∇(∆v) · ∇v + |D2 v|2 , 2 where |D2 u|2 =

(8.18)

(uxi xj )2 . Multiplying by ϕ v m and integrating over Ω, we

1≤i,j≤n

obtain T1 + T2 :=

ϕ v m ∇(∆v) · ∇v +

ϕ v m |D2 v|2 =

1 2

ϕ v m ∆|∇v|2 =: T3 . (8.19)

Integrating by parts and using ϕ ∈ D(Ω), we can rewrite the first and third terms as follows:

T1 = − (∆v)∇ · (ϕ v m ∇v)

m m−1 2 |∇v| ∆v − ϕ v m (∆v)2 = − v (∆v) ∇v · ∇ϕ − m ϕ v and

1 m |∇v|2 v m ∆ϕ + mv m−1 ∇v · ∇ϕ + ϕ(v m−1 ∆v + (m − 1)v m−2 |∇v|2 ) 2 2

1 v m |∇v|2 ∆ϕ + m v m−1 |∇v|2 ∇v · ∇ϕ = 2

m m(m − 1) ϕ v m−1 |∇v|2 ∆v + ϕ v m−2 |∇v|4 . + 2 2

T3 =

42


Moving the first term of T1 to the right of (8.19) and the last two terms of T3 to the left, it follows that

m(1 − m) 3m ϕ v m−2 |∇v|4 − ϕ v m−1 |∇v|2 ∆v + ϕ v m |D2 v|2 2 2

m 1 m 2 m v |∇v|2 ∆ϕ + = ϕ v (∆v) + v ∆v + mv m−1 |∇v|2 ∇v · ∇ϕ. 2 (8.20) By Cauchy-Schwarz’ inequality (applied with the inner product (A, B) = tr(AB ∗ ) on matrices), we have 2 (∆v)2 = tr(D2 v) ≤ tr (D2 v)(D2 v)∗ tr (In ) = n|D2 v|2 .

(8.21)

Due to ϕ ≥ 0, Claim (8.17) follows by combining (8.20) and (8.21). Step 2. We set v = uk+1 , m = (k + 1)−1 (q − 2k), that is q = (k + 1)m + 2k, and we compute

m−2 4 4 ϕv ϕ u(k+1)(m−2)+4k |∇u|4 = (k + 1)4 I, |∇v| = (k + 1)

ϕ v m−1 |∇v|2 ∆v = (k + 1)3

ϕ u(k+1)(m−1)+3k |∇u|2 (∆u + ku−1 |∇u|2 )

= (k + 1)3 (kI + J), ϕ v m (∆v)2 = (k + 1)2

ϕ u(k+1)m+2k (∆u)2 + 2k(∆u)u−1 |∇u|2 + k 2 u−2 |∇u|4

= (k + 1)2 (k 2 I + 2kJ + K),

m 2 v (∆v)∇v · ∇ϕ = (k + 1) u(k+1)m+2k ∆u + ku−1 |∇u|2 ∇u · ∇ϕ, and

v m−1 |∇v|2 ∇v · ∇ϕ = (k + 1)3

u(k+1)m+2k−1 |∇u|2 ∇u · ∇ϕ.

Substituting in (8.17) and dividing by (k + 1)2 , we get 3m n − 1 2 n − 1 3m (k + 1)2 − k(k + 1) − k I+ − (k + 1) − 2k J 2 2 n 2 n

n−1 1 − K≤ uq |∇u|2 ∆ϕ + uq ∆u + (k + m(k + 1))u−1 |∇u|2 ∇u · ∇ϕ, n 2

m(1 − m)

which readily implies the lemma.


43

Lemma 8.10. (i) Let Ω be an arbitrary domain in Rn , and 0 ≤ ϕ ∈ D(Ω). Let 0 −p, k = −1 and denote

I = ϕ uq−2 |∇u|4 , K = ϕ u2p+q . Then there holds αI + δK ≤

1 2

uq |∇u|2 ∆ϕ + C

up+q + uq−1 |∇u|2 |∇u · ∇ϕ|,

(8.22)

where C = C(n, p, q, k) > 0 and α=−

n−1 2 q(q − 1) k + (q − 1)k − , n 2

δ=

1 3q n + 2 n − 1 − k − . (8.23) p+q 2 n n

(ii) Assume that 1 0,

2p + q > n(p − 1)/2.

(8.24)

Proof. (i) Since −∆u = up , we have

p+q−1 2 (p + q)J = − ϕ (p + q)u |∇u| = − ϕ ∇u · ∇(up+q )

= ϕ (∆u)up+q + (∇ϕ · ∇u)up+q , where J is defined in Lemma 8.9, hence

(p + q)J = − ϕ u2p+q + (∇ϕ · ∇u)up+q . Substituting in (8.16), we obtain (8.22). (ii) A simple computation shows that δ > 0 and 2p+q > n(p−1)/2 is equivalent to (n − 1)p q (n − 4)p − n and q > q0 (p) := . k < k0 (q) := − 2 n+2 2 For k = k0 (q), we have (n − 1)pq (n − 1)2 p2 n − 1 q2 − + − α = α(k0 (q)) = n 4 n+2 (n + 2)2 q (n − 1)p q(q − 1) + (q − 1) − − 2 n+2 2 (n − 1)pq (n − 1)2 p2 n − 1 q2 np(q − 1) − + − = − n 4 n+2 (n + 2)2 n+2 q2 pq n(n + 2)p − (n − 1)2 p2 n−1 − − + . = n 4 n+2 (n + 2)2

44


The discriminant of the above polynomial in q is given by p2 + n(n + 2)p − (n − 1)2 p2 np[(n + 2) − (n − 2)p] = > 0. (n + 2)2 (n + 2)2 √ √ 2p 2p Therefore we have α(k0 (q)) > 0 for q ∈ (− n+2 − 2 D, − n+2 + 2 D). Moreover, 2p − n+2 > q0 (p) is equivalent to n(n + 2) > (n2 − 2n − 4)p, which is true due to p < pS . Choosing D=

q=−

2p n+2

and

we see that (8.24) is fulfilled.

np − k = k0 (q)− = − n+2

(with k = −1),

Proof of Proposition 8.6. Take q, k as in Lemma 8.10(ii) and Ω = B1 . We shall estimate the terms on the RHS of (8.22). Let ξ ∈ D(Ω), be such that ξ = 1 for |x| ≤ 1/2 and 0 ≤ ξ ≤ 1. Put θ = (3p + 1 + 2q)/2(2p + q) ∈ (0, 1). By taking ϕ = ξ m with m = 2/(1 − θ), we have |∇ϕ| ≤ Cξ m−1 ≤ Cϕθ ,

|∆ϕ| ≤ Cξ m−2 ≤ Cϕθ .

(8.25)

Fix ε > 0. Using Young’s inequality under the form xyz ≤ εxa + εy b + C(ε)z c ,

a−1 + b−1 + c−1 = 1,

and (8.25), we obtain

q 2 u |∇u| ∆ϕ = ϕ1/2 u(q−2)/2 |∇u|2 ϕ(q+2)/2(2p+q) u(q+2)/2

× ϕ−(p+1+q)/(2p+q) ∆ϕ ≤ ε ϕ uq−2 |∇u|4 + ε ϕ u2p+q + C(ε),

C

ϕ1/4 u(q−2)/4 |∇u| ϕ(4p+3q+2)/4(2p+q) u(4p+3q+2)/4

× ϕ−(3p+1+2q)/2(2p+q) |∇ϕ| ≤ ε ϕ uq−2 |∇u|4 + ε ϕ u2p+q + C(ε),

up+q |∇u · ∇ϕ| ≤

and

q−1 2 ϕ3/4 u3(q−2)/4 |∇u|3 ϕ(q+2)/4(2p+q) u(q+2)/4 C u |∇u| |∇u · ∇ϕ| ≤

× ϕ−(3p+1+2q)/2(2p+q) |∇ϕ| ≤ ε ϕ uq−2 |∇u|4 + ε ϕ uq+2p + C(ε). Combining this with (8.22), we obtain αI + δK ≤ C(n, p, q, k)ε(I + K) + C(ε).


45

Since α, δ > 0, by choosing ε sufficiently small, we conclude that I, K ≤ C, hence (8.8) with r = 2p + q > max(n(p − 1)/2, p). Proof of Theorem 8.2. Let u be a positive solution of (8.2). Assume n ≥ 2 and denote x = (x1 , . . . , xn−1 ). Choose x ¯, x ˜ ∈ Rn+ with x ¯n = xñ . We will show u(¯ x) = u(˜ x) so that u depends only on xn . x Choose the origin to be the point x¯+˜ , 0 . Given x ∈ Rn+ , set 2 z=

x + en , |x + en |2

v(z) = |x + en |n−2 u(x) =

u(x) . |z|n−2

The function v is the Kelvin transform of u. It solves the problem ∆v + |z|γ v p = 0

in D, on ∂D \ {0},

v=0

(8.26)

where D := B1/2 (en /2) and γ := (n − 2)p − (n + 2) ≤ 0. We want to show that v is axisymmetric about the zn axis, i.e. v = v(|z |, zn ). Choose any direction e perpendicular to the zn -axis. Without loss of generality we may assume e = e1 . 6 zn D en 2

λ

Σ(λ) z1

Figure 4: Moving planes.

We shall apply the moving planes method to problem (8.26). Given λ ∈ [0, 1/2), set Σ(λ) := {z ∈ D : z1 > λ}, z λ := (2λ − z1 , z2 , . . . , zn ). The point z λ is the reflection of z with respect to the hyperplane {z1 = λ} and Σ(λ) is called a cap. We next define w(z; λ) := v(z λ ) − v(z)

for z ∈ Σ(λ)

46


(the parameter λ will be omitted in w when no risk of confusion arises). Then ∆w = ∆v(z λ ) − ∆v(z) = −|z λ |γ v p (z λ ) + |z|γ v p (z) = |z|γ − |z λ |γ v p (z λ ) − |z|γ v p (z λ ) − v p (z) . Since v p (z λ ) − v p (z) = pξ p−1 w(z; λ) for some ξ = ξ(z, λ) lying between v(z λ ) and v(z), we obtain ∆w + |z|γ pξ p−1 w = |z|γ − |z λ |γ v p (z λ ) ≥ 0 in Σ(λ) . The maximum principle (see Proposition 52.1) implies v > 0 in D and ∂v/∂ν < 0 on ∂D \ {0}, hence w ≥ 0 on Σ(λ) for λ close to 1/2. Set µ ¯ := inf{µ > 0 : w ≥ 0 in Σ(λ) for all λ ≥ µ} and assume µ ¯ > 0. Then w ≥ 0 on Σ(¯ µ) and there exist λi ∈ (0, µ ¯), λi → µ ¯, such that inf{w(z; λi ) : z ∈ Σ(λi )} < 0. Since w(·; λi ) ≥ 0 on ∂Σ(λi ), this infimum is attained at some qi ∈ Σ(λi ) and ∇w(qi , λi ) = 0. Since w(·; λi ) ≥ 0 in an ε-neighborhood of ∂D ∩ Σ(λi ) (with ε being independent of i), we may µ) \ ∂D. Continuity arguments and w ≥ 0 on Σ(¯ µ) guarantee assume qi → q¯ ∈ Σ(¯ w(¯ q; µ ¯) = 0 and ∇w(¯ q; µ ¯) = 0, hence w(·; µ ¯) ≡ 0 by the maximum principle. This contradicts w > 0 on {z ∈ ∂Σ(¯ µ) : z1 > µ ¯ }. Consequently, µ ¯ = 0 and w(·; 0) ≥ 0 on Σ(0). A symmetric argument shows w(·; 0) ≤ 0 on Σ(0), hence v is symmetric with respect to the hyperplane {e1 = 0}. Since this holds for any hyperplane containing the zn -axis, v is axially symmetric. Hence, u = u(|x |, xn ) and, consequently, u(¯ x) = u(˜ x). Thus we have reduced the problem to the case n = 1. Assume that u is a positive solution of u (t) + up (t) = 0,

t > 0,

u(0) = 0. Since u is concave and positive for t > 0, it must fulfill u ≥ 0. Fix t1 > 0 and consider t > t1 . Then

t u(t) = u(t1 ) + (t − t1 )u (t1 ) + (t − s)u (s) ds. t1

Since u (s) = −up (s) ≤ −up (t1 ), we obtain 1 0 < u(t) ≤ u(t1 ) + (t − t1 )u (t1 ) − (t − t1 )2 up (t1 ), 2 hence up (t1 )
0, v(z) ≤ C|z|2−n

as |z| → ∞,

(8.27)

∆v + |z|γ v p = 0 in Rn \ {0},

(8.28)

and v solves the equation

where γ := (n − 2)p − (n + 2) ≤ 0. Due to (8.28) and n ≥ 3, we infer from Lemma 4.4 that ∆v ≤ 0 in D (Rn ). It follows from the maximum principle in Proposition 52.3(ii) that, for each R > 0, v ≥ η(R) := min v > 0 |x|=R

in BR (0) \ {0}.

(8.29)

Given λ ≤ 0, set z λ := (2λ − z1 , z2 , . . . , zn ), Σ(λ) := {z ∈ Rn : z1 < λ}, Σ (λ) := Σ(λ) \ {0λ } and

w(z; λ) := v(z λ ) − v(z),

z ∈ Σ(λ) \ {0λ }

(the parameter λ will be omitted in w when no risk of confusion arises). As in the preceding theorem we obtain ∆w + |z|γ pξ p−1 w ≤ 0 in Σ (λ),

(8.30)

where ξ = ξ(z, λ) lies between v(z λ ) and v(z). Set α := (n − 2)/2 and w(z; ˜ λ) = |z|α w(z; λ). Then ∆w ˜−

n−2 z · ∇w˜ + c(z, λ)w ˜≤0 |z|2

where c(z, λ) := −

in Σ (λ),

(8.31)

(n − 2)2 + |z|γ pξ p−1 (z, λ). 4|z|2

Let us first show that w ˜ ≥ 0 in Σ (λ),

for λ −1.

(8.32)

48


We shall argue by contradiction. Assume that λi → −∞ and inf Σ (λi ) w(·; ˜ λi ) < 0. By (8.27) and (8.29) with R = 1, we have w(z; ˜ λi ) ≥ 0 if |z − 0λi | < 1 and i is large enough. Since also, for each i, w(z; ˜ λi ) = 0 on ∂Σ(λi )

and

w(z; ˜ λi ) → 0, |z| → ∞,

(8.33)

we see that the infimum of w(·; ˜ λi ) over Σ (λi ) is attained at some qi ∈ Σ (λi ) and |qi − 0λi | ≥ 1. We have |qi | → ∞, thus v(qi ) → 0. If the sequence {qiλi } were bounded, then (8.29) would imply v(qiλi ) ≥ c1 > 0, hence w(qi ) > 0 for i large, a contradiction. Therefore |qiλi | → ∞. Now the definition of v implies v(z)|z|n−2 → u(0) if |z| → ∞, so that we cannot have |qiλi |/|qi | → 0 (otherwise w(qi ) > 0 for large i). Thus both v(qi ) and v(qiλi ) can be estimated above by Cqi2−n for some fixed C > 0 and the same is true for ξ(qi , λi ). Hence, c(qi , λi ) ≤ −

(n − 2)2 Cp + 4 µ ¯, λi → µ ¯ , such that inf{w(z; ˜ λi ) : z ∈ Σ (λi )} < 0. Assume that w(·, ˜ µ ¯) is not identically zero. Since ∆w(·, µ ¯) ≤ 0 in Σ (¯ µ), the maximum principle (see Proposition 52.1) implies w(·, µ ¯) > 0 in Σ (¯ µ). Arguing similarly as for (8.29), we deduce that w(·, µ ¯) ≥ c2 > 0 in U := Bµ¯/2 (0µ¯ ) \ {0µ¯ }. Due to the continuity of v in U and w(z; λi ) = w z − 2(λi − µ e1 := (1, 0, . . . , 0), ¯)e1 ; µ ¯ +v z − 2(λi − µ ¯ )e1 −v(z), ˜ λi ) ≥ 0) in Bµ¯/4 (0λi ) \ {0λi } for i large. Consewe obtain w(·; λi ) ≥ 0 (hence w(·; quently, in view of (8.33), the infimum of w(·; ˜ λi ) over Σ (λi ) has to be attained at λi ¯/4. Assume |qi | → ∞. Then |qiλi |/|qi | → 1 and some qi ∈ Σ (λi ), with |qi − 0 | ≥ µ we obtain a contradiction as above (cf. (8.34)). Therefore we may assume that {qi } µ) \ {0µ¯ }. By continuity and w(·, ˜ µ ¯ ) ≥ 0, we obtain is bounded and qi → q¯ ∈ Σ(¯ w(¯ ˜ q, µ ¯) = 0 and ∇w(¯ ˜ q, µ ¯ ) = 0, hence w(¯ q, µ ¯) = 0 and ∇w(¯ q, µ ¯ ) = 0. Applying the maximum principle in Proposition 52.1(ii) and (iii) to equation (8.30), it follows that w(·, µ ¯) ≡ 0, hence w(·, ˜ µ ¯) ≡ 0, a contradiction. Consequently, w(·, ˜ µ ¯) ≡ 0, which means that v is symmetric with respect to {z1 = µ ¯}. Now using (8.28) we see that (−∆v)/v p = |z|γ has the same symmetry, which is not possible unless p = pS .


49

If p < pS , then we get µ ¯ = 0, so that w(·, 0) ≥ 0 and v(z 0 ) ≥ v(z) provided z1 ≤ 0. Considering the function v˜(z) := v(z 0 ) instead of v we obtain the reversed inequality, hence v(z1 , z2 , . . . , zn ) = v(−z1 , z2 , . . . , zn ). Repeating this procedure with any given direction instead of e1 we see that v, hence u, are radially symmetric (about zero). If we repeat this procedure with u ˜(x) = u(x − x0 ) for a given x0 = 0 instead of u, we show that u is radially symmetric about the point x0 . Since this is true for any x0 , the function u has to be constant. But the only constant solution of (8.1) is the trivial solution. If p = pS and µ ¯ < 0, then v is symmetric with respect to {z1 = µ ¯}. If µ ¯ = 0, then we can repeat the procedure with v˜(z) := v(z 0 ) and in any case we obtain the symmetry of v with respect to {z1 = µ ˜} for suitable µ ˜. Now we can repeat the above proof with directions e2 , e3 , . . . , en instead of e1 and we obtain the existence of z¯ ∈ Rn such that v is symmetric with respect to {zk = z¯k } for k = 1, 2, . . . , n, hence v(¯ z + z) = v(¯ z − z) for all z. Rotating the coordinate system and repeating the procedure we find z˜ ∈ Rn such that v(˜ z + z) = v(˜ z − z) for all z. Assume z¯ = z˜. Without loss of generality, we may assume z¯ = 0. The symmetry relations for v imply v(¯ z ) = v(2˜ z − z¯) = v(3¯ z − 2˜ z ) = v(4˜ z − 3¯ z ) = · · · → 0, hence v(¯ z ) = 0, a contradiction. Consequently, z¯ = z˜ and we obtain the rotational symmetry of v (hence of u) about z¯. Proof of Theorem 8.3. Assume that (8.2) admits a positive, bounded classical solution u. As a special case of Theorem 21.10 below (which we shall prove by using moving planes arguments), it follows that u is nondecreasing in xn : ∂xn u(x) ≥ 0,

x ∈ Rn+ .

Therefore, for each x ∈ Rn−1 , U (x ) := lim u(x , xn ) xn →∞

is well defined and is a bounded positive function. Take now ϕ ∈ D(Rn−1 ) and 1 ψ ∈ D(R), with supp ψ ⊂ (0, 1) and 0 ψ = 1. Let k > 0. Testing the equation with ϕ(x )ψ(xn − k), we have

p u ϕ(x ) ψ(xn − k) dxn dx = ϕ(x ) ψ(xn − k) ∆u dxn dx − Rn−1 R Rn−1 R

= u ∆ ϕ(x ) ψ(xn − k) dxn dx , Rn−1

hence −

Rn−1

R

R

up (x , s + k) ϕ(x ) ψ(s) ds dx

= u(x , s + k) ∆ ϕ(x ) ψ(s) ds dx . Rn−1

R

50


By dominated convergence, letting k → ∞, it follows that

1

U p (x ) ϕ(x ) dx = − U p (x ) ϕ(x ) ψ(s) ds dx − Rn−1 Rn−1 0

= U (x ) ∆ ϕ(x ) ψ(s) ds dx . Rn−1

But the RHS is equal to

U (x ) ∆x ϕ(x ) dx Rn−1

R

1

U (x ) ϕ(x ) dx

ψ(s) ds + Rn−1

0

=

Rn−1

1

ψ (s) ds

0

U (x ) ∆x ϕ(x ) dx .

It follows that U solves (8.1) in Rn−1 in the distribution sense, hence in the classical sense (this is a consequence of the boundedness of U and of Remark 47.4). The result is then a consequence of Theorem 8.1(i).

9. Positive radial solutions of ∆u + up = 0 in Rn In this section we study positive radial classical solutions of the equation −∆u = up ,

x ∈ Rn .

(9.1)

Since this problem does not possess positive classical solutions if 1 < p < pS due to Theorem 8.1, we restrict ourselves to the case p ≥ pS . Consequently, n ≥ 3. Positive radial classical solutions of (9.1) can be written in the form u(x) = U (r), where r = |x| and U ∈ C 2 ([0, ∞)) is a positive classical solution of U +

n−1 U + U p = 0, r

r ∈ (0, ∞),

U (0) = 0.

(9.2)

It is easily seen that prescribing initial values U (0) = α > 0, U (0) = 0, the equation in (9.2) has a unique solution for r small enough. In fact, this equation can be written in the form (rn−1 U ) = −rn−1 U p and, by integration we obtain the equivalent integral equation

r s n−1 t U (r) = α − U p (t) dt ds, s 0 0 which can be solved by the Banach fixed point theorem. Let U∗ (r) = cp r−2/(p−1) be the singular solution defined in (3.9) and set pJL :=

+∞

if n ≤ 10,

1+

if n > 10.

√ n−1 4 n−4+2 (n−2)(n−10)

The main result of this section is the following theorem.

(9.3)

9. Positive radial solutions of ∆u+up =0 in Rn

51

Theorem 9.1. Let p ≥ pS . Given α > 0, problem (9.2) possesses a unique positive solution Uα ∈ C 2 ([0, ∞)) satisfying Uα (0) = α. This solution is decreasing and we have Uα (r) = αU1 (α(p−1)/2 r). (9.4) If p > pS , then r2/(p−1) Uα (r) → cp as r → ∞. If p = pS , then U1 (r) =

n(n − 2) (n−2)/2 . n(n − 2) + r2

(9.5)

Let α1 > α2 > 0. If p ≥ pJL , then U∗ (r) > Uα1 (r) > Uα2 (r) for all r > 0. If pS < p < pJL , then Uα1 and Uα2 intersect infinitely many times and Uα1 , U∗ intersect infinitely many times as well. If p = pS , then Uα1 , Uα2 intersect once and Uα1 , U∗ intersect twice. Proof. Using the transformation w(s) = r2/(p−1) U (r),

s = log r,

(9.6)

problem (9.2) becomes w + βw + wp − γw = 0,

s ∈ R,

(9.7)

where β :=

1 (n − 2)p − (n + 2) ≥ 0, p−1

γ := cp−1 = p

2 (n − 2)p − n > 0, 2 (p − 1)

and we are looking for solutions w satisfying w(s), w (s) → 0 as s → −∞. Set E(w) = E(w, w ) :=

1 2 γ 2 1 |w | − w + wp+1 . 2 2 p+1

Then E is a Lyapunov functional for (9.7); more precisely, 2 d E w(s) = −β w (s) ≤ 0. ds

(9.8)

Denoting x := w and y := w , problem (9.7) can be written in the form x y

=

y =: F (x, y) −βy − xp + γx

(9.9)

where x > 0 and (x, y) → (0, 0) as s → −∞. Problem (9.9) possesses two equilibria, (0, 0) and (cp , 0) lying in the half-space {(x, y) : x ≥ 0}. Denote A1 := ∇F (0, 0) =

0 γ

1 , −β

A2 := ∇F (cp , 0) =

0 1 . −γ(p − 1) −β

52


y 6 PP q

y = ν2 x

1

-

B BN 1 ) 0 BM B

T

6

j cp

PP q

? @@ I

x

y = 0

Figure 5: The flow generated by (9.9) for pS !pS . Then β > 0 and the matrix A1 has two real eigenvalues ν1,2 := − 12 β ± β 2 + 4γ with ν1 < 0 < ν2 = 2/(p − 1). The corresponding (xi , yi) satisfy yi = νi xi , i = 1, 2. The eigenvalues ! eigenvectors ν˜1,2 := − 21 β ± β 2 − 4γ(p − 1) of A2 are real iff β 2 ≥ 4γ(p − 1), that is iff p ≥ pJL . Assume pS < p < pJL . In this case, the eigenvalues ν˜1 , ν˜2 are complex and their real parts are negative so that the critical point (cp , 0) is a stable spiral. The flow for the planar system (9.9) is illustrated in Figure 5. We are interested in the trajectory T emanating from the origin to the right half-space, since it represents the graph of any positive solution of (9.7) in the w-w plane. This trajectory cannot hit the axis x = 0 again since the energy functional E is nonnegative on this axis, E(0, 0) = 0, β > 0 and (9.8) is true. Moreover, the corresponding solutions w exists for all s ∈ R and w, w remain bounded for all s ∈ R due to (9.8). Consequently, T has to converge to the critical point (cp , 0) which corresponds to the singular solution w∗ (s) = r2/(p−1) U∗ (r) ≡ cp . Thus, if Uα is the unique local solution of (9.2) such that Uα (0) = α > 0, then its transform wα (s) = r2/(p−1) Uα (r) exists globally and satisfies wα (s) → cp as s → ∞. Consequently, Uα exists globally and r2/(p−1) Uα (r) → cp as r → ∞. It ˜α (r) := αU1 (α(p−1)/2 r) is a solution of (9.2) is easily verified that the function U ˜α = Uα by uniqueness. The graphs of wα and w1 ˜α (0) = α, hence U satisfying U in the w-w plane are identical, so that there exists sα ∈ R such that Uα (es ) = wα (s) = w1 (s − sα ) for all s ∈ R. Hence, given α1 > α2 > 0, Uα1 (r) = Uα2 (r) for some r > 0 iff w1 (s − sα1 ) = w1 (s − sα2 ) for some s ∈ R. This happens for infinitely many s since T spirals around the point (cp , 0). Similarly, wα1 (s) = cp

9. Positive radial solutions of ∆u+up =0 in Rn

53

for infinitely many s, hence Uα1 and U∗ intersect infinitely many times. Next consider the case p ≥ pJL . On the halfline y = − β2 (x − cp ), x < cp , we have for suitable xθ ∈ (x, cp ): 2x(xp−1 − cp−1 ) y x p−1 p (x = −β − − γ) = −β + x y β(x − cp ) β 2 2 < −β + (p − 1)γ ≤ − . = −β + x(p − 1)xp−2 θ β β 2 y 6 PP q

y = ν2 x

1

B BN 1 ) 0 BM B

T 6

-

-

y = − β2 (x − cp ) ? ?

cp

x

y = 0 Figure 6: The flow generated by (9.9) for p ≥ pJL .

Consequently, the trajectory T ends up at (cp , 0) again but the x-coordinate is increasing along T (see Figure 6). Hence, the solutions U of (9.2) are ordered according to their values at r = 0, U∗ > Uα1 > Uα2 if α1 > α2 . Finally consider the case p = pS . Then β = 0 and the energy functional E is constant along any solution. Since E(cp , 0) < 0 and E(0, y) > 0 for y = 0, the trajectory T is a homoclinic orbit (see Figure 7). Let wα , sα have the same meaning as above. Given α1 = α2 , there exists a unique s ∈ R such that w1 (s − sα1 ) = w1 (s − sα2 ). Hence, the corresponding solutions Uα1 , Uα2 of (9.2) intersect exactly once. Similarly, given α > 0, we have wα (s) = cp for two values of s, so that Uα and U∗ intersect twice. One can easily check that the function U1 defined by (9.5) is a solution of (9.2) satisfying the initial condition U1 (0) = 1. Remarks 9.2. (i) The exponent pJL appeared for the first time in [293] where the authors studied mainly problems with the nonlinearities f (u) = λ(1 + au)p and f (u) = λeu , λ, a > 0.

54


-

y 6

R @

T R @ @ @ I @

?

cp

x

I @

Figure 7: The flow generated by (9.9) for p = pS .

(ii) The intersection properties of the solutions U in Theorem 9.1 play an important role in the study of stability and asymptotic behavior of solutions of the corresponding parabolic problem, see Sections 22, 23. Remark 9.3. Let p = pS and a > 0. For all α ≥ M0 (a) with M0 (a) > 0 large enough, if V is a positive classical solution of V +

n−1 V + V p = 0, r

0 < r < a,

such that V (a) = Uα (a) and limr→0 V (r) = ∞, then V has to intersect Uα in (0, a). In fact, denoting wα (s) := r2/(p−1) Uα (r), s = log r, the rescaled function from the last proof, it suffices to chose M0 (a) such that wM (log a) < 0 0 (a)

(9.10)

(hence wα (log a) < 0 for all α ≥ M0 (a)). Indeed the trajectory of W (s) := r2/(p−1) V (r), s ∈ (−∞, log a), has to be a subset of a periodic orbit lying inside the trajectory T (see Figure 7). Due to (9.10) there exists s0 ∈ (−∞, log a) such that wα (s0 ) = W (s0 ), hence Uα (es0 ) = V (es0 ). Note also that there exist infinitely many periodic orbits of (9.7) for p = pS , p corresponding to positive singular solutions of u + n−1 r u + u = 0 for r > 0. Remark 9.4. Let p > pJL . Since the trajectory T approaches the limit point (cp , 0) below the dotted line with slope −β/2 and ν˜2 < −β/2 < ν˜1 < 0, it has to converge along the eigenvector (1, ν˜1 ) corresponding to the eigenvalue ν˜1 , hence y(s) → ν˜1 x(s) − cp

as

s → ∞.

10. A priori bounds via the method of Hardy-Sobolev inequalities

55

Returning to the original variables and denoting V (r) := U (r) − U∗ (r) we obtain rV (r) = ν˜1 − m, r→∞ V (r)

(9.11)

lim

where m := 2/(p − 1). Assuming that V (r) = cr−α + h.o.t. for some c = 0 and α > m, (9.11) guarantees c < 0 and α = m + λ− , where ! 1 β − β 2 − 4γ(p − 1) 2 ! 1 = n − 2 − 2m − (n − 2 − 2m)2 − 8(n − 2 − m) . 2

λ− := −˜ ν1 =

This expansion is indeed true: In fact, a more precise asymptotic expansion of V was established in [260] and [334].

10. A priori bounds via the method of Hardy-Sobolev inequalities A priori estimates of solutions can be used for the proof of existence and multiplicity results. Unlike the variational methods in sections 6 and 7, this approach does not require any variational structure of the problem and enables one to prove the existence of continuous branches of solutions. Due to Theorem 7.8(ii) one cannot hope for a priori estimates of all solutions. The bifurcation diagrams in Figure 2 suggest that there is some hope for such estimates if we restrict ourselves to positive solutions and to the subcritical case.3 In the present and the following three sections we introduce four different methods which are often used in the proofs of a priori bounds for positive solutions of superlinear elliptic problems. We will study mainly the scalar problem −∆u = f (x, u, ∇u), u = 0,

x ∈ Ω, x ∈ ∂Ω,

(10.1)

where Ω is bounded and f is a sufficiently smooth function with superlinear growth in the u-variable. Some of the possible generalizations and modifications will be mentioned as remarks, others can be found in the subsequent chapters. This section is devoted to the method of [99], which is based on a Hardytype inequality and enables one to treat rather general nonlinearities f . On the 3 In fact, in the subcritical case one can get a priori estimates of all solutions with bounded Morse indices (without the positivity assumption), see [49], [539], [32].

56


other hand, it requires an upper growth restriction corresponding to the limiting exponent ∞ if n = 1, pBT := (n + 1)/(n − 1) if n > 1, which is stronger than what is imposed by the methods in Sections 12 and 13 (for instance, in the particular case f (x, u, ∇u) = up , we have to assume p < pBT ). However, the exponent pBT is not technical and its role will be clarified in the next section. Theorem 10.1. Let Ω ⊂ Rn be bounded, n ≥ 3, β := pBT . Let f : Ω× R+ × Rn → R+ be continuous and bounded on Ω × M × Rn for M ⊂ R+ bounded. Let lim inf u→∞

f (x, u, s) > λ1 , u

lim

u→∞

f (x, u, s) = 0, uβ

uniformly for (x, s) ∈ Ω × Rn .

(10.2) Then there exists C > 0 with the following property: If t ≥ 0 and u ∈ H01 ∩ L∞ (Ω) is a positive variational solution of −∆u = f (x, u, ∇u) + tϕ1 , u = 0,

x ∈ Ω, x ∈ ∂Ω,

(10.3)

then u ∞ + t ≤ C.

(10.4)

Proof. We shall denote by C various positive constants which may vary from step to step but which are independent of u and t. Let t ≥ 0 and u be a positive solution of (10.3). The proof of (10.4) will consist of the following three steps: 1. Ω uδ dx ≤ C, t ≤ C and Ω f (x, u, ∇u)δ dx ≤ C, 2. ∇u 2 ≤ C, 3. u ∞ ≤ C. Step 1. Due to (10.2) there exist C1 > λ1 and C2 > 0 such that f (x, u, s) ≥ C1 u − C2 for all (x, u, s). Multiplying the equation in (10.3) by ϕ1 yields

λ1

Ω

u(−∆ϕ1 ) dx = (−∆u)ϕ1 dx = (f ϕ1 + tϕ21 ) dx Ω Ω

uϕ1 dx − C2 ϕ1 dx + t ϕ21 dx, ≥ C1

uϕ1 dx =

Ω

Ω

Ω

Ω

where f = f (x, u(x), ∇u(x)). This estimate can be written in the form

(C1 − λ1 )

Ω

uϕ1 dx + t

Ω

ϕ21 dx ≤ C,

(10.5)


57

hence

Ω

uϕ1 dx ≤ C

and

t ≤ C.

(10.6)

Now (10.5) and δ ≤ Cϕ1 guarantee

Ω

f δ dx ≤ C

Ω

f ϕ1 dx = Cλ1

Ω

uϕ1 dx − Ct

ϕ21 dx ≤ C.

Ω

(10.7)

Step 2. Multiplying the equation in (10.3) by u yields ∇u 22

2

= Ω

|∇u| dx =

f u dx + t Ω

Ω

ϕ1 u dx ≤

f u dx + C.

(10.8)

Ω

Denoting α := 2/(n + 1) ∈ (0, 1) we have β + 1/(1 − α) = 2/(1 − α). Given ε > 0 there exists Cε > 1 such that f (x, u, s) ≤ εuβ + Cε .

(10.9)

Using Hölder’s inequality, Step 1, (10.9) and Lemma 50.4 we obtain

α

u u1/(1−α) 1−α (f α δ α ) f 1−α α dx ≤ f δ dx f α/(1−α) dx δ Ω Ω Ω δ uβ+1/(1−α) 1−α u1/(1−α) 1−α ≤ ε1−α dx + Cε dx α/(1−α) α/(1−α) Ω δ Ω δ " u "2 "u" " " " " = ε1−α " α/2 " + Cε " α " ≤ ε1−α C ∇u 22 + CCε ∇u 2 . δ 1/(1−α) δ 2/(1−α)

f u dx = Ω

This estimate and (10.8) guarantee ∇u 2 ≤ C.

(10.10)

Step 3. Choose p ∈ (n/2, n). Then W 2,p (Ω) → L∞ (Ω)

and W 1,2 (Ω) → Lp(β−1) (Ω)

due to n(β − 1) < 2∗ . These embeddings, Lp -estimates (see Appendix A), (10.9), Step 1 and (10.10) imply u ∞ ≤ C u 2,p ≤ C f + tϕ1 p ≤ ε uβ p + C(Cε + 1) ≤ ε u β−1 u ∞ + C˜ε ≤ ε ∇u β−1 u ∞ + C˜ε ≤ εC u ∞ + C˜ε . p(β−1)

2

Now choosing ε > 0 small enough yields u ∞ < C.

58


Remarks 10.2. (i) The proof of Theorem 10.1 can be easily modified for more general second-order elliptic differential operators. In the case of a nonsymmetric operator one has to work with the first eigenfunction of the adjoint operator, of course. One could also allow more general nonlinearities (nonlocal, for example). The boundedness assumption on f could be relaxed as well. (ii) The term tϕ1 in (10.3) is needed for the proof of existence of a positive solution of (10.3) with t = 0 (see Corollary 10.3 below). This lower order term does not play any significant role in a priori estimates in the following sections provided t ≤ C. Since this bound for t was proved in Step 1 of the proof of Theorem 10.1 by using only the lower bound for f in (10.2), in the following sections we shall restrict ourselves to the case t = 0 only. (iii) A priori estimates of solutions of problems like (10.3) appeared first in [400] and [517]. The assumptions on the growth of f or the dimension n in these articles are more restrictive than those in Theorem 10.1 which is due to [99]. Corollary 10.3. Let Ω and f be as in Theorem 10.1 and let lim sup u→0+

f (x, u, s) < λ1 u

uniformly for (x, s) ∈ Ω × Rn .

(10.11)

Then problem (10.3) with t = 0 possesses at least one positive solution u, with u ∈ W 2,q ∩ C0 (Ω) for all finite q. Proof. Set X := C 1 (Ω). Given u ∈ X and t ≥ 0, let Ft (u) = w be the unique solution of the linear problem −∆w = f (x, u, ∇u) + tϕ1 ,

x ∈ Ω, x ∈ ∂Ω

w = 0,

(10.12)

(cf. Theorem 47.3(i)). Note that, since f (·, u, ∇u) ∈ L∞ (Ω), we have u ∈ W 2,q ∩ C0 (Ω) for all finite q. Then Ft : X → X is compact and we are looking for a positive fixed point of F0 . Let u X = r 1, τ ∈ [0, 1] and assume τ F0 (u) = u. Multiplying the equation in (10.12) by u and applying (10.11) yield

|∇u|2 dx = τ f u dx ≤ (λ1 − ε) u2 dx, Ω

Ω

Ω

which contradicts (1.3). Hence τ F0 (u) = u and the homotopy invariance of the topological degree implies deg I − F0 , 0, Br = deg I, 0, Br = 1, (10.13) where I denotes the identity and Br := {u ∈ X : u X < r}.


59

Let u X = R. If R is large enough, then Theorem 10.1 and Lp -estimates (see Appendix A) imply Ft (u) = u for any t ≥ 0. The same theorem implies also FT (u) = u provided T is large enough. Consequently, deg I − F0 , 0, BR = deg I − FT , 0, BR = 0. (10.14) ¯r = −1, hence there Now (10.13) and (10.14) guarantee deg I − F0 , 0, BR \ B ¯r such that F0 (u) = u. The positivity of u is a consequence of exists u ∈ BR \ B the maximum principle. In what follows we present an alternative proof of Theorem 10.1 in the special case f (x, u, s) = |u|p−1 u, 1 < p < pBT , n ≥ 1. Instead of Hardy’s inequality we shall use the following lemma (see [89], [450], and cf. also [143] and the references in [450, Remark 4.1]). It provides a useful singular test-function and will also be used later in Section 26. Lemma 10.4. Assume Ω bounded and 0 < α < 1. Then the problem −∆ξ = ϕ−α x ∈ Ω, 1 , ξ = 0,

x ∈ ∂Ω

(10.15)

∈ admits a unique classical solution ξ ∈ C(Ω) ∩ C 2 (Ω). Moreover, we have ϕ−α 1 L1 (Ω), ξ ∈ H01 (Ω), and ξ(x) ≤ C(Ω, α)δ(x),

x ∈ Ω.

(10.16)

Proof. Define h(s) = 3s − s2−α , s ≥ 0. The function h ∈ C 1 ([0, ∞)) ∩ C 2 ((0, ∞)) satisfies h = 3 − (2 − α)s1−α , and h(s) ≤ 3s,

−h = (2 − α)(1 − α)s−α ,

h (s) ≥ 1,

s>0

for all s ∈ [0, 1].

Let ϕ = ϕ1 −1 ∞ ϕ1 , and set v(x) = h(ϕ(x)). Simple computation yields −∆v = −h (ϕ)|∇ϕ|2 − h (ϕ)∆ϕ = C1 ϕ−α |∇ϕ|2 + λ1 h (ϕ)ϕ ≥ C1 ϕ−α |∇ϕ|2 + λ1 ϕ. Now, for δ(x) ≤ ε small enough, we have |∇ϕ|2 ≥ η > 0, hence −∆v ≥ C1 ηϕ−α . On the other hand, for δ(x) ≥ ε, we have ϕ ≥ c > 0, hence −∆v ≥ λ1 c ≥ C2 ϕ−α . We conclude that for some c > 0, w := cv satisfies −∆w ≥ ϕ−α 1

and w(x) ≤ C3 δ(x),

for all x ∈ Ω.

(10.17)

60


Next, for all ε > 0, let ξε be the (classical) solution of −∆ξε = (ϕ1 + ε)−α in Ω, with ξε = 0 on ∂Ω. By (10.17) and the maximum principle, we have ξε (x) ≤ w(x) ≤ C3 δ(x) ≤ C4 ,

x∈Ω

(10.18)

and ξε is increasing as ε decreases to 0. Denote by ξ the (pointwise) limit of ξε . Elliptic estimates along with (10.18) imply that ξ ∈ C(Ω) ∩ C 2 (Ω), that ξ satisfies (10.16) and is a classical solution of (10.15). The uniqueness follows immediately from the maximum principle. The fact that ϕ−α ∈ L1 (Ω) can be easily deduced from the inequality ϕ1 ≥ cδ, 1 by flattening the boundary and using a partition of unity (see e.g. [485] for details). Finally, to show that ξ ∈ H01 (Ω), it suffices to note that, since α < 1,

|∇ξε |2 = −

Ω

ξε ∆ξε =

Ω

Ω

ξε (ϕ1 + ε)−α ≤ C4

ϕ−α < ∞. 1

Ω

Alternative proof of Theorem 10.1 for f = up , t = 0. Let ε > 0 be small and α := r /r, where r is defined by 1/r = 1/2 − ε/(p − 1). Let ξ be the solution of (10.15). As in Step 1 of the proof of Theorem 10.1 we obtain Ω up δ dx ≤ C. Testing the equation with ξ, we obtain

Ω

uϕ−α 1 dx =

Ω

∇u · ∇ξ dx =

(−∆u)ξ dx = Ω

Ω

up ξ dx ≤ C

(where we used ϕ−α ∈ L1 (Ω) and ξ ∈ H01 (Ω)). Denoting pε := (p + 1)/2 − ε, we 1 get

p/r 1/r 1/r −1/r pε u ϕ1 u dx u dx = ϕ1 Ω Ω

1/r 1/r ≤ up ϕ1 dx uϕ−α ≤ C. 1 dx Ω

Ω

∗

Define θ ∈ (0, p + 1) by θ/pε + (p + 1 − θ)/2 = 1. Then p + 1 − θ < 2 provided ε is small enough and the interpolation inequality yields

Ω

2

|∇u| dx =

Ω

p+1−θ θ up+1 dx = u p+1 ≤ C ∇u p+1−θ , p+1 ≤ u pε u 2∗ 2

which guarantees a bound for u in W 1,2 (Ω). The rest of the proof is the same as in the proof of Theorem 10.1 (Step 3).

11. A priori bounds via bootstrap in Lpδ -spaces

61

11. A priori bounds via bootstrap in Lpδ -spaces This section is devoted to the Lpδ bootstrap method, which, in the scalar case, was developed independently in [85], [449]. It applies to problem (10.1) under essentially the same assumptions on the nonlinearities f as in the method of the previous section, with a growth restriction still given by the exponent pBT of Section 10. However, unlike that method (and those in the next two sections), it applies to very weak solutions. The optimality of the Lpδ bootstrap method was studied in [489] and it turns out that the exponent pBT is optimal for the regularity of very weak solutions, thus showing the critical role played by this exponent for problems of the form (10.1). Let us point out that in the case of systems, studied in [449], the growth restrictions of the Lpδ bootstrap method become much weaker than those imposed by the (generalization of the) method of Hardy-Sobolev inequalities (see Section 31). In this section, by a solution u of (10.1), we understand a very weak (or L1δ -) solution, cf. Definition 3.1. Namely, if f does not depend on ∇u, this means that u ∈ L1 (Ω),

and −

u∆ϕ = Ω

f (·, u)ϕ, Ω

f (·, u) ∈ L1δ (Ω),

(11.1)

for all ϕ ∈ C 2 (Ω), ϕ|∂Ω = 0.

(11.2)

If f depends on ∇u, we assume in addition that ∇u is a function, i.e. ∇u ∈ L1loc (Ω) and we replace f (·, u) by f (·, u, ∇u) in (11.1)–(11.2). Remark 11.1. If u ∈ L1 (Ω) and ∆u ∈ L1δ (Ω) (where ∆u is understood in the distribution sense), we say that u = 0 on ∂Ω in the weak sense if

u ∆ϕ =

Ω

for all ϕ ∈ C 2 (Ω), ϕ|∂Ω = 0.

ϕ ∆u Ω

If (11.1) is satisfied (and ∇u ∈ L1loc (Ω) in case f depends on ∇u), then u is a very weak solution of (10.1) if and only if it solves the differential equations in (10.1) in the distribution sense and the boundary conditions in the weak sense. Theorem 11.2. Assume Ω bounded and 1 λ1 . (11.4)

62


There exists C > 0 such that if u is a nonnegative very weak solution of (10.1), then u ∈ L∞ (Ω) and u ∞ ≤ C. Condition (11.4) can be weakened or replaced by other conditions of different form. For instance, by applying the same method, we obtain regularity and a priori estimates for the following simple equation: −∆u = a(x)up , u = 0,

x ∈ Ω, x ∈ ∂Ω.

(11.5)

Theorem 11.3. Assume Ω bounded and a ∈ L∞ (Ω), a ≥ 0, a ≡ 0 and 1 < p < pBT . Then the conclusions of Theorem 11.2 remain valid for problem (11.5). Remarks 11.4. (i) The growth condition (11.3) in Theorem 11.2 is slightly stronger than that in Theorem 10.1 (where (10.2) allows some “almost critical” f ’s). (ii) Under the assumptions of Theorems 11.2 and 11.3, as a consequence of standard regularity results for linear elliptic equations, we moreover obtain u ∈ C0 ∩ W 2,q (Ω) for all finite q (argue similarly as in the proof of Corollary 3.4, using the uniqueness part of Theorem 49.1 instead of Proposition 52.3). The optimality of the exponent pBT in Theorems 11.2 and 11.3 is shown by the following result from [489]. Theorem 11.5. Assume Ω bounded and p > pBT . Then there exists a function a ∈ L∞ (Ω), a ≥ 0, a ≡ 0, such that problem (11.5) admits a positive very weak solution u such that u ∈ L∞ (Ω). The method of proof of Theorems 11.2–11.3 is based on bootstrap and uses the Lpδ regularity theory of the Laplacian (cf. Theorem 49.2 and Proposition 49.5 in Appendix C). Proof of Theorem 11.2. Step 1. Initialization. By (10.6), (10.7) in the proof of Theorem 10.1, we know that u 1,δ ≤ C,

f (·, u, ∇u) 1,δ ≤ C.

Since p < pBT , we may fix ρ > 1 and k0 such that n + 1 1 n+1 max p, p− < k0 < . 2 ρ n−1 By (11.6) and Proposition 49.5, it follows that u k0 ,δ ≤ C.

(11.6)

11. A priori bounds via bootstrap in Lpδ -spaces

63

Step 2. Bootstrap. Put ki = k0 ρi , i = 1, 2, . . . . Assume that there holds u ki ,δ ≤ C(i)

(11.7)

for some i ≥ 0 (this is true for i = 0 by Step 1). Since p 2 1 1 1 < , p − − = i ki ki+1 k0 ρ ρ n+1 by using Theorem 49.2(i) and (11.3), we obtain u ki+1 ,δ ≤ C ∆u ki /p,δ = C f ki /p,δ ≤ C(1 + v p ki /p,δ ) = C(1 + v pki ,δ ) ≤ C. By induction, it follows that (11.7) is true for all integers i. Taking i large enough, we thus have (11.7) for some ki > (n+ 1)p/2. Applying Theorem 49.2(i) and (11.3) once more, and Remark 1.1, we obtain u ∞ ≤ C. Proof of Theorem 11.3. We only need to modify Step 1, the bootstrap step being then unchanged. Assume that u is a nonnegative (very weak) solution of (11.5). It follows from the quantitative version of Hopf’s lemma (see Remark 49.12(i) in Appendix C) that

u≥c aup δ dy δ ≥ c1 aup ϕ1 dy ϕ1 , Ω

Ω

for some constant c1 > 0 depending only on Ω. We deduce that

p

aup ϕ1 dx ≥ cp1 aup ϕ1 dx aϕp+1 dx ≥ 2 aup ϕ1 dx − C, 1 Ω

Ω

hence λ1

Ω

Ω

Ω

uϕ1 dx =

Ω

aup ϕ1 dx ≤ C.

We now turn to the proof of Theorem 11.5. It is based on Lemma 49.13 from Appendix C, where a singular solution of the linear Laplace equation with an appropriate right-hand side belonging to L1δ is constructed. The right-hand side has to possess suitable boundary singularities, supported in a conical subdomain of Ω. In order to re-construct a posteriori the coefficient a(x), the key point is the lower estimate (11.8) for the solution in the same cone. Proof of Theorem 11.5. Assume that 0 ∈ ∂Ω without loss of generality. Let α = 2/(p − 1). By assumption, we have α < n − 1. By Lemma 49.13, there exist R > 0 and a revolution cone Σ1 of vertex 0, with Σ := Σ1 ∩ B2R ⊂ Ω, such that the function φ := |x|−(α+2) χΣ

64


belongs to L1δ and such that the (very weak) solution u > 0 of −∆u = φ, x ∈ Ω, u = 0, x ∈ ∂Ω satisfies

u ≥ C|x|−α χΣ .

Therefore, we have u ∈ L

∞

(11.8)

and

up ≥ C|x|−αp χΣ = C|x|−(α+2) χΣ = Cφ. Setting a(x) = φ/up ≥ 0, we get −∆u = φ = a(x)up and a(x) ≤ 1/C, hence a ∈ L∞ . The proof is complete. Remarks 11.6. Localization of singularities. (a) In Theorem 11.5, it is to be noted that, in spite of the imposed homogeneous Dirichlet boundary condition, the singularity of the solution occurs at a boundary point, actually a single point. The boundary conditions continue to be satisfied not only in the weak sense but also in the sense of traces (see Remark 49.4(c) in Appendix C). (b) If we assume that p < psg and that a given weak solution of (11.5) is bounded near the boundary, then one can use usual Lebesgue spaces instead of Lpδ -spaces in the proof of Theorem 11.2, to show that the solution is bounded in Ω. Therefore, the occurrence of boundary singularities is necessary if pBT psg , the situation is different and much easier, since it is then not difficult to construct examples of similar equations with only an interior singularity (see Remarks 3.6). (c) The support of a in Theorem 11.5 can be localized in an arbitrarily small neighborhood of a boundary point. However, it is also possible to construct an example where the function a is positive in Ω, uniformly away from ∂Ω (see [489] for details). Remarks 11.7. (a) The cases f (u) = up and p = pBT . Similar counterexamples as in Theorem 11.5 have been constructed recently in [155] for the model problem (3.10) (a(x) ≡ 1) when p > pBT is close to pBT . Moreover the critical case p = pBT was shown to belong to the singular case. Related results have also been announced in [81]. (b) Variable critical exponents in nonsmooth domains. The notion of very weak solution has been recently extended in [362] to the case of some nonsmooth domains, namely Lipschitz domains, and generalizations of Theorems 11.2 and 11.5 have been obtained. For suitable cone-shaped domains, the analogue of the exponent pBT was computed. Interestingly, it was found to depend on the domain and to be smaller than (n + 1)/(n − 1).

12. A priori bounds via the rescaling method

65

12. A priori bounds via the rescaling method In this section we present a priori estimates of solutions of (10.3) based on rescaling and Liouville-type theorems. In this context, this method was first used in [241]. In comparison to the method of Section 10, it requires a rather precise asymptotic behavior for f as u → ∞ (f has to behave like up for u large) but the growth condition on f is optimal (p < pS ). The method also works for general second-order elliptic operators but for simplicity we restrict ourselves to the Laplace operator. As explained in Remark 10.2(ii) we consider the case t = 0 only. Theorem 12.1. Assume Ω bounded, 1 0 for all x ∈ Ω, g ∈ C(Ω × R × Rn ), and |g(x, u, s)| ≤ C 1 + |u|q + |s|r ,

where q < p, r
0 such that any positive strong solution u ∈ C 1 (Ω) of −∆u = a(x)up + g(x, u, ∇u),

x ∈ Ω,

x ∈ ∂Ω

u = 0,

(12.2)

satisfies u ∞ ≤ C. 2,1 Remark 12.2. Here, u being a strong solution means that u ∈ Wloc (Ω) and u satisfies the differential equation a.e. in Ω. Since we also assume u ∈ C 1 (Ω), Remarks 47.4(i) and (iii), actually imply u ∈ W 2,q (Ω) for all finite q.

Proof of Theorem 12.1. Assume the contrary. Then there exist positive solutions uj of (12.2) such that uj ∞ → ∞ as j → ∞. Let xj ∈ Ω be such that uj (xj ) + |∇uj (xj )|2/(p+1) = sup uj + |∇uj |2/(p+1) =: Mj Ω

and let dj := dist (xj , ∂Ω). Since Ω is compact, we may assume xj → x0 for some −(p−1)/2 x0 ∈ Ω. Set κj := Mj . The sequence dj /κj is either unbounded or bounded. In the former case we may assume dj /κj → ∞, in the latter dj /κj → c ≥ 0. Case 1. Let dj /κj → ∞. Set vj (y) :=

1 uj (x), Mj

y :=

x − xj , κj

and Ωj := {y ∈ Rn : |y| < dj /κj }. Then vj + |∇vj |2/(p+1) ≤ vj (0) + |∇vj (0)|2/(p+1) = 1

(12.3)

66


and −∆vj (y) = a(κj y + xj )vjp (y) + gj (y),

y ∈ Ωj ,

(12.4)

where 2p/(p−1)

gj (y) := κj

−2/(p−1) (p+1)/(p−1) g κj y + xj , κj vj (y), κj ∇vj (y)

satisfies |gj | ≤ Cκεj ,

ε := min 2(p − q), 2p − (p + 1)r /(p − 1).

(12.5)

Interior elliptic Lp -estimates (see Appendix A) guarantee that vj are locally bounded in W 2,z for any z > 1 (uniformly with respect to j). Let α ∈ (0, 1), R > 0 and BR := {y ∈ Rn : |y| < R}. There exists z = z(α) > 1 such that W 2,z (BR ) is compactly embedded into BU C 1+α (BR ). Consequently, we may assume vj → v in C 1+α . Passing to the limit in (12.4) and (12.3) we see that v is a positive (classical) solution of −∆v = a(x0 )v p in Rn , which contradicts Theorem 8.1. ˜j ∈ ∂Ω be such that dj = |xj − x ˜j |. For any j we Case 2. Let dj /κj → c ≥ 0. Let x can choose a local coordinate z = z(j) = (z 1 , z 2 , . . . , z n ) in an ε-neighborhood Uj of x˜j such that the image of the boundary ∂Ω will be contained in the hyperplane z 1 = 0, x ˜j becomes 0, xj becomes zj := (dj , 0, 0, . . . , 0), and the image of Uj will contain the set {z : |z| < ε } for some ε > 0. We may assume that ε, ε are independent of j and the local charts are uniformly bounded in C 2 . In these new coordinates, the equation for w = wj (z) = uj (x) becomes −

i,k

⎫ ∂2w ∂w i p 1 ⎬ a (z) i k − b (z) i = a(x(z))w + g˜(z), |z| < ε, z > 0, ⎪ ∂z ∂z ∂z (12.6) i ⎪ ⎭ 1 w = 0, |z| < ε, z = 0, ik

where g˜(z) := g x(z), w(z), D(z)∇z w(z) , D = D(j) = (∂z i /∂xk )i,k , bi = bi(j) = # ∂zi ∂zk ik t ∆z i , aik = aik

∂x ∂x , hence A = A(j) := (a(j) )i,k = D · D, and the A(j) (j) = 2 are uniformly elliptic. Also, since ∂Ω is uniformly C , it follows that the aik (j) are 1 i ∞ uniformly bounded in C and the b(j) in L . Moreover, since D(0) is a Euclidean transformation, it follows that A(j) (0) = D(0) · tD(0) = Id. Set vj (y, s) := where

1 wj (κj y + zj ), Mj

ε zj dj . , y1 > − y ∈ Ωj := y : y − < κj κj κj

12. A priori bounds via the rescaling method

67

Then vj is a solution of −

aik (κj y + zj )

i,k

∂ 2v ∂v − κ bi (κj y + zj ) i j ∂y i ∂y k ∂y i = a(x(κj y + zj ))v p + gj

in Ωj , on {y ∈ ∂Ωj : y 1 = −dj /κj },

v=0 where 2p/(p−1)

gj (y) := κj

−2/(p−1) −(p+1)/(p−1) g x(κj y + zj ), κj v(y), κj D(κj y + zj )∇v(y)

satisfies (12.5). Interior-boundary Lp -estimates (see Appendix A) and the bounds i on the coefficients aik (j) , b(j) again yield a subsequence of {vj } converging to a positive (classical) solution v of ∆v = a(x0 )v p , v = 0, which contradicts Theorem 8.2.

y1 > −c, y1 = −c,

Remarks 12.3. (i) If g is independent of the gradient variable, then it is sufficient to choose Mk := sup uk in the proof of Theorem 12.1. (ii) Indefinite coefficients. Assume that the function a in problem (12.2) changes sign. Under suitable assumptions on a, g and p one can still use the method of [241] in order to get a priori bounds for positive solutions (see [74], [19] and [167], for example). In addition to the limiting problems in the proof of Theorem 12.1 one has to deal with problems of the form −∆u = h(y)up ,

y ∈ Rn ,

where typically h(y) = |y1 |α y1 for some α ≥ 0. In some cases, a combination of the above approach with other arguments (moving planes, energy, . . . ) yields the a priori bounds, see [124], [453], [237] and the references therein. Of course, if the problem has variational structure, then the existence of nontrivial solutions can often be proved by variational or dynamical methods, see [8], [75], [7], [257], [121], [3] and the references therein. (iii) The rescaling method is sometimes referred to as the “blow-up method”, because one performs a zoom of the microscopic scales of the solution. Here we shall not use this terminology, in order to avoid confusion with the blow-up phenomenon in the parabolic problem.

68


13. A priori bounds via moving planes and Pohozaev’s identity In this section we describe the method of a priori estimates of solutions of (2.1) due to [183]. Similarly as in the preceding section, the growth condition for function f will be optimal. The advantage of this method consists in the fact that it does neither require precise asymptotic behavior of f for u large nor Liouville-type theorems. On the other hand, the symmetry of the Laplace operator plays an important role, f cannot depend on ∇u in a general way and we also have to assume that either Ω is convex or f satisfies a restrictive monotonicity condition, see (13.3) below. The assumptions for a general function f = f (x, u) are rather complicated (see [183, Remark 1.5]) and therefore we restrict ourselves to the case f = f (u). Hence, we shall study positive solutions of the problem −∆u = f (u), x ∈ Ω, (13.1) u = 0, x ∈ ∂Ω. Theorem 13.1. Assume n ≥ 2 and Ω bounded. Let f : R+ → R be locally Lipschitz continuous and assume f (u) f (u) lim = 0, > λ1 , u→∞ uσ u where σ = pS if n ≥ 3, σ < ∞ is arbitrary if n = 2. Let one of the following assumptions be satisfied: (i) Ω is convex and lim inf u→∞

lim sup u→∞

uf (u) − θF (u) ≤ 0, u2 f (u)κ

θ ∈ [0, 2∗ ),

(13.2)

where κ = 2/n. (ii) Condition (13.2) is satisfied with κ = 2/n and, in the case n ≥ 3, the function u → f (u)u−pS is nonincreasing on (0, ∞).

(13.3)

(iii) Condition (13.2) is satisfied with κ = 2/(n + 1), n ≥ 3, ∂Ω = Γ1 ∪ Γ2 , where Γ1 , Γ2 are closed and satisfy (1) at every point of Γ1 , all sectional curvatures of Γ1 are bounded away from 0 by a positive constant a; (2) there exists x0 ∈ Rn such that (x − x0 , ν(x)) ≤ 0 for all x ∈ Γ2 . Then there exists C > 0 such that u ∞ < C for any positive classical solution u of (13.1). In view of the proof we set some notation. For each ε > 0, let Ωε := {z ∈ Ω : δ(z) < ε}.

13. A priori bounds via moving planes and Pohozaev’s identity

69

For y ∈ ∂Ω and λ > 0, we define T (y, λ) := {x ∈ Rn : (y − x, ν(y)) = λ}, Σ(y, λ) := {x ∈ Ω : (y − x, ν(y)) ≤ λ}, we denote by R(y, λ) the reflection with respect to the hyperplane T (y, λ) and we set Σ (y, λ) := R(y, λ)Σ(y, λ). We need the following lemma. Lemma 13.2. Assume Ω bounded and convex, λ0 > 0, and 0 ≤ u ∈ C(Ω)∩C 1 (Ω). Assume that (∇u(x), ν(y)) ≤ 0, y ∈ ∂Ω, x ∈ Σ(y, λ0 ). (13.4)

Then sup u ≤ C Ωε

Ω

uϕ1 dx,

where ε, C > 0 depend only on Ω and λ0 . Proof. Let us first recall that ν(∂Ω) = S n−1 .

(13.5)

This follows from a standard degree argument. We give the proof for completeness. Assume without loss of generality that 0 ∈ Ω and select ν˜, a continuous extension of ν to Ω. The homotopy H1 (t, x) := t˜ ν (x) + (1 − t)x has no zero on ∂Ω, since (x, ν(x)) ≥ 0 on ∂Ω due to the convexity of Ω. Therefore d(˜ ν , 0, Ω) = d(id, 0, Ω) = 1, where d denotes the Brouwer degree. Assume for contradiction that η ∈ ν(∂Ω) for some η ∈ S n−1 . Then the homotopy H2 (t, x) = t˜ ν (x) − (1 − t)η has no zero on ∂Ω. Consequently d(˜ ν , 0, Ω) = d(−η, 0, Ω) = 0, a contradiction which proves (13.5). Next, by decreasing λ0 if necessary, we may assume that {y − λν(y) ∈ Rn : λ ∈ (0, λ0 ]} ⊂ Ω,

y ∈ ∂Ω.

(13.6)

˜ ∈ ∂Ω satisfy |x − x ˜| = δ(x). Notice that x ˜ is Let ε ∈ (0, λ0 /4], x ∈ Ωε , and let x uniquely determined and (˜ x − x)/|˜ x − x| = ν(˜ x) if ε is small. Let α ∈ (0, 1) and let η ∈ S n−1 be such that (η, ν(˜ x)) ≥ α. Using the fact that Ω is contained in the half-space {z ∈ Rn : (z − x, ν(˜ x)) ≤ |˜ x − x|} (due to the convexity of Ω), we obtain ! (y(η)−x, η) ≤ (y(η)−x, ν(˜ x))+|y(η)−x||η−ν(˜ x)| ≤ ε+diam(Ω) 2(1 − α) ≤ λ0 /2, provided α is close to 1 and ε is small enough, say 1 − α + ε < ε0 = ε0 (Ω, λ0 ). This along with (13.6) implies {x − λη ∈ Rn : λ ∈ [0, λ0 ]} ⊂ Σ(y(η), λ0 ).

70


It then follows from (13.4) that [0, ε] λ → u(x − λη) is nondecreasing for any η ∈ S n−1 satisfying (η, ν(˜ x)) ≥ α. This property guarantees the existence of γ = γ(Ω, λ0 ) > 0 such that for all x ∈ Ωε there exists a measurable set Ix ⊂ Ω \ Ωε (13.7) satisfying meas Ix ≥ γ and u(ξ) ≥ u(x) for all ξ ∈ Ix . Indeed (decreasing the value of ε if necessary), it is sufficient to take a conical piece Ix = Ωcε ∩ {x − λη : η ∈ S n−1 , (η, ν(˜ x)) ≥ α, λ ∈ [0, λ0 ]}. Since ϕ1 ≥ Cε on Ω \ Ωε for some Cε > 0, we deduce from (13.7) that

Cε γu(x) ≤ Cε u(ξ) dξ ≤ u(ξ)ϕ1 (ξ) dξ ≤ u(ξ)ϕ1 (ξ) dξ Ix

Ω

Ix

and the lemma is proved. Proof of Theorem 13.1. First assume (i). The proof will consist of the following four steps: 1. Ω uδ dx ≤ C, Ω |f (u)|δ dx ≤ C, where δ(x) = dist (x, ∂Ω), 2. u + |∇u| ≤ C in a neighborhood of ∂Ω, 3. ∇u 2 ≤ C, 4. u ∞ ≤ C. Step 1. This step is almost the same as Step 1 in the proof of Theorem 10.1 and we leave the detailed proof to the reader. Step 2. Since Ω is convex and smooth, we can find λ0 , c0 > 0 such that Σ (y, λ) ⊂ Ω,

λ ≤ λ0

and

(ν(x), ν(y)) > c0 ,

x ∈ ∂Σ(y, λ0 ) ∩ ∂Ω.

We shall now apply the moving planes method (cf. [239], [183]) to show that u(R(y, λ)x) ≥ u(x),

y ∈ ∂Ω, x ∈ Σ(y, λ), λ ≤ λ0 .

(13.8)

Without loss of generality, we may assume that y = 0 and that ν(0) = −e1 (in particular, Ω lies entirely in the upper half-space {x1 > 0}). For each x = (x1 , x ), we denote xλ := R(0, λ)x = (2λ − x1 , x ), Σλ := Σ(0, λ) = Ω ∩ {x1 < λ}, and Σλ := Σ (0, λ). Define wλ (x) = u(xλ ) − u(x),

for x ∈ Σλ , 0 < λ ≤ λ0 ,

and set

E := µ ∈ (0, λ0 ] : wλ (x) ≥ 0 for all x ∈ Σλ and λ ∈ (0, µ) .


71

∂u Since ∂x (0) > 0 by Hopf’s lemma, we have λ ∈ E for λ > 0 small. Assume for 1 ¯ := sup E < λ0 . We have contradiction that λ

wλ ≥ 0,

¯ for all x ∈ Σλ and λ ∈ (0, λ],

(13.9)

¯ with λ ¯ < λi < λ0 , such that min wλi < 0. and there exists a sequence λi → λ, Σλi

Since w = 0 on {x1 = λ} ∩ Ω and λ

wλ > 0 on {x1 < λ} ∩ ∂Ω,

for all λ < λ0 ,

(13.10)

it follows that this minimum is attained at a point qi ∈ Σλi . Therefore ∇wλi (qi ) = ∂u = (e1 · ν) ∂u 0. On the other hand, since ∂x ∂ν ≥ c > 0 on {x1 ≤ λ0 } ∩ ∂Ω and 1 wλ (x) = u(2λ − x1 , x ) − u(x1 , x ) = 2(λ − x1 )

∂u (ξ(x)), ∂x1

with |ξ(x) − x| ≤ 2(λ − x1 ), we see that wλ (x) ≥ 0 for x in an ε-neighborhood of {x1 = λ} ∩ ∂Ω, with ε > 0 independent of λ ∈ (0, λ0 ]. Therefore, we may assume ¯ ∩ ∂Ω, and by continuity we get / {x1 = λ} that qi → q¯ ∈ Σλ¯ , q¯ ∈ ¯

q) = 0 wλ (¯

and

¯

∇wλ (¯ q ) = 0.

(13.11)

But (13.9) implies ¯ ¯ ¯ −∆wλ (x) = f u(xλ ) − f (u(x)) ≥ −cwλ (x)

and

¯

wλ (x) ≥ 0,

x ∈ Σλ¯ ,

for some constant c > 0 (depending on u). By Hopf’s lemma (cf. Proposition 52.1 ¯ and Remark 52.2), this along with (13.11) implies wλ = 0 in Σλ¯ , contradicting ¯ = λ0 , which proves (13.8). This guarantees that u satisfies (13.10). Consequently, λ (13.4). By Lemma 13.2 and Step 1, we deduce that u ≤ C on Ωε for some ε, C > 0 depending only on Ω. Now the bound for ∇u in Ωε/2 follows from interior-boundary elliptic Lp -estimates (see Appendix A) and the embedding W 2,p → C 1 for p > n. In particular, we have shown that ∂u ≤ C, ∂ν

x ∈ ∂Ω.

(13.12)

Step 3. Notice that Steps 1 and 2 imply f (u) 1 ≤ C.

(13.13)

First consider the case n ≥ 3. The H¨ older and Sobolev inequalities and (13.13) guarantee

2/n

Ω

u2 |f (u)|2/n dx ≤ u 22∗ f (u) 1

≤ C ∇u 22 .

72


Pohozaev’s identity (5.1) and (13.12) yield

2 ∗ |∇u| dx − 2 F (u) dx ≤ C. Since 2∗

Ω

uf (u) dx, the last two estimates and (13.2) imply

F (u) dx ≤ uf (u) dx + C ≤ θ F (u) dx + ε u2 |f (u)|2/n dx + Cε Ω Ω Ω Ω

≤ (θ + εC) F (u) dx + C˜ε .

Ω

|∇u|2 dx =

Ω

Ω

Ω

Choosing ε < (2∗ − θ)/C we obtain Ω F (u) dx ≤ C, hence ∇u 2 ≤ C. Next let n = 2. Set γ := 1 − 1/σ. Given ε > 0, the assumption limu→∞ f (u)/uσ = 0 guarantees the existence of Cε > 0 such that uf (u) ≤ εu2 f (u)γ + Cε . Similarly as above we obtain

∇u 22 = uf (u) dx ≤ ε u2 |f (u)|γ dx + Cε Ω

Ω

≤ ε u 22/(1−γ) f (u) γ1 ≤ εC ∇u 22 + Cε , which proves the assertion. Step 4. If f (u) ≤ C(1 + up )

for some p < pS

(13.14)

(which is always true if n = 2), then one can use standard bootstrap estimates based on Lq -estimates (see Appendix A) to show that the W 1,2 -bound from Step 3 guarantees an L∞ -bound. If n ≥ 3 and (13.14) is not true, then we use the following estimates (see [96] and cf. the proof of Proposition 3.3). Let p > 1, ap := (p + 1)2 /4 and q := (p + 1)n/(n − 2). Then

(n−2)/n " "2 (p+1)/2 2 ∇u dx = Cap uq dx = "u(p+1)/2 "2∗ ≤ C |∇u|2 up−1 dx Ω Ω Ω

ap p =C f (u)u dx ≤ ε up+σ dx + Cε , p Ω Ω where σ = (n + 2)/(n − 2). Next H¨ older’s inequality and Step 3 yield

(n−2)/n

2/n ∗ up+σ dx = uq(n−2)/n+4/(n−2) dx ≤ uq dx u2 dx Ω Ω Ω Ω

(n−2)/n q ≤C u dx . Ω


73

These estimates imply u q ≤ C, hence f (u) q/σ ≤ C. Since q can be made arbitrarily large, the Lp -estimates (see Appendix A) conclude the proof in case (i). Next consider assumption (ii). Instead of Ω being convex we now assume (13.3). Since the convexity assumption was used only in the proof of Step 2, it is sufficient to modify the proof of this step. Choose x0 ∈ ∂Ω. Then there exists a ball Br ⊂ Rn \Ω of radius r such that x0 ∈ ∂Br . The radius r can be chosen independent of x0 and, without loss of generality, we may assume r = 1. Choose a coordinate system such that Br is centered at the origin and x0 = (1, 0, . . . , 0). Set y = J(x) := x/|x|2 and w(y) = |x|n−2 u(x). Then −∆w(y) = g(y, w) n−2

in O := J(Ω),

n+2

w)/|y| is nonincreasing in y due to (13.3). Since where g(y, w) := f (|y| O ⊂ Br is smooth and x0 ∈ ∂O ∩ ∂Br we can use the moving planes method in order to get the existence of εx0 , γx0 > 0 with the following property: for any y ∈ O, |y−x0 | < εx0 , there exists a set Ky ⊂ {z ∈ O : dist (z, ∂O) > εx0 } satisfying meas Ky ≥ γx0 and w(ξ) ≥ w(y) for all ξ ∈ Ky . Going back to the original variables and using the compactness of ∂Ω we get the existence of ε, γ, c > 0 such that (13.7) is true, with u(ξ) ≥ u(x) replaced by u(ξ) ≥ cu(x). The rest of the proof of Step 2 is the same as in case (i). Finally consider case (iii). Then Steps 1 and 4 can be proved in the same way as in case (i). Repeating the arguments in the proof of Step 2 of case (i) we obtain a uniform bound for u and |∇u| in a neighborhood of Γ1 . Without loss of generality we may assume x0 = 0, hence x·ν(x) ≤ 0 for all x ∈ Γ2 . These facts and Pohozaev’s identity (5.1) imply

∗ 2 F (u) dx − uf (u) dx ≤ C. (13.15) Ω

Ω

Next using Lemma 50.4 with τ := 1/(n + 1) and q := 2(n + 1)/(n − 1), Step 1 and Hölder’s inequality, we obtain

" u "2 " u "2 " " " " 1−2/q uf (u) dx = ∇u 22 ≥ c1 " τ " ≥ c2 " τ " f (u)δ 1 δ q δ q Ω

1−2/q u2 |f (u)|δ ≥ c2 dx = c2 u2 |f (u)|2/(n+1) dx. 2τ Ω δ Ω Now (13.2) with κ = 2/(n + 1), (13.15) and the last estimate imply

uf (u) dx ≤ θ F (u) dx + ε u2 |f (u)|2/(n+1) dx + Cε Ω Ω Ω

∗ ≤ (θ/2 + εC) uf (u) dx + Cε Ω

and the choice of ε small enough concludes the proof. The following corollary can be proved in the same way as Corollary 10.3.

74


Corollary 13.3. Let f : R+ → R+ satisfy the assumptions in Theorem 13.1 and lim supu→0+ f (u)/u < λ1 . Then problem (2.1) possesses at least one positive classical solution. Remark 13.4. If one is interested only in the existence of positive solutions of (2.1) without knowing their a priori bounds, then the technical assumption (13.2) can be omitted, see [183]. The proof is based on an approximation of the function f , on the mountain pass theorem (including uniform bounds for the energy of approximating solutions) and Pohozaev’s identity.

Chapter II

Model Parabolic Problems 14. Introduction In Chapter II, we mainly consider semilinear parabolic problems of the form ⎫ x ∈ Ω, t > 0, ut − ∆u = f (u), ⎪ ⎬ u = 0, x ∈ ∂Ω, t > 0, (14.1) ⎪ ⎭ x ∈ Ω, u(x, 0) = u0 (x), where f is a C 1 -function with a superlinear growth. For simplicity, we formulate most of our assertions for the model case f (u) = |u|p−1 u with p > 1, but the methods of our proofs can be applied to more general parabolic problems (not necessarily of the form (14.1)). Some of possible generalizations and modifications will be mentioned as remarks, other can be found in the subsequent chapters.

15. Well-posedness in Lebesgue spaces Definition 15.1. Given a Banach space X of functions defined in Ω, u0 ∈ X and T ∈ (0, ∞], we say that the function u ∈ C([0, T ), X) is a solution (more precisely, a classical X-solution) of (14.1) in [0, T ) if u ∈ C 2,1 (Ω × (0, T )) ∩ C(Ω × (0, T )), u(0) = u0 and u is a classical solution of (14.1) for t ∈ (0, T ). If Ω is unbounded, ∞ then we also require u ∈ L∞ loc ((0, T ), L (Ω)). ∞ If X = L (Ω), then, instead of the condition u ∈ C([0, T ), X), we require u ∈ C((0, T ), X) and u(t) − e−tA u0 ∞ → 0 as t → 0, where e−tA is the Dirichlet heat semigroup in Ω (cf. Appendix B). We say that (14.1) is well-posed in X if, given u0 ∈ X, there exist T > 0 and a unique classical X-solution of (14.1) in [0, T ]. It is well known that (14.1) is well-posed in X = W01,q (Ω) for any q > n if Ω is bounded, or in X = L∞ (Ω) for any Ω (see Example 51.9 and Remark 51.11). In this section we study the well-posedness of the model problem ⎫ ut − ∆u = |u|p−1 u, x ∈ Ω, t > 0, ⎪ ⎬ u = 0, x ∈ ∂Ω, t > 0, (15.1) ⎪ ⎭ x ∈ Ω, u(x, 0) = u0 (x),

76

II. Model Parabolic Problems

in the Lebesgue spaces Lq (Ω), 1 ≤ q < ∞, since this well-posedness will play a crucial role in many subsequent sections. The following two results show that the exponent qc := n(p − 1)/2 is critical for the well-posedness. The existence/nonexistence part of these results is due to [528], [529] (where uniqueness was proved in a more restrictive class of solutions). The uniqueness and nonuniqueness parts were proved in [93] and [53], [425], respectively. An alternative proof of the existence-uniqueness part of Theorem 15.2 based on interpolation and extrapolation spaces can be found in Appendix E (see Theorem 51.25 and Example 51.27). In what follows we write shortly Lq -solution instead of Lq (Ω)-solution. Theorem 15.2. Let p > 1, u0 ∈ Lq (Ω), 1 ≤ q < ∞, q > qc . Then there exists T = T ( u0 q ) > 0 such that problem (15.1) possesses a unique classical Lq -solution in [0, T ) and the following smoothing estimate is true: u(t) r ≤ C u0 q t−αr ,

αr :=

n 1 1 − , 2 q r

(15.2)

for all t ∈ (0, T ) and r ∈ [q, ∞], with C = C(n, p, q) > 0. In addition, u ≥ 0 provided u0 ≥ 0. Theorem 15.3. Let p > 1 + 2/n and 1 ≤ q < qc . (i) There exists a nonnegative function u0 ∈ Lq (Ω), such that (15.1) does not admit any nonnegative classical Lq -solution in [0, T ) for any T > 0. (ii) Assume p < pS , Ω = BR , and let u0 ∈ L∞ (Ω), u0 ≥ 0, be radial nonincreasing. Then there exists a time T > 0 such that (15.1) possesses infinitely many positive radial nonincreasing classical Lq -solutions in [0, T ). Remarks 15.4. (i) The critical case. It was also proved in [529], [93] that when u0 ∈ Lq (Ω) and q = qc > 1, then there exists T = T (u0 ) > 0 such that (15.1) possesses a unique classical Lq -solution in [0, T ) (see Example 51.27 and cf. Remark 20.24(i)). The same arguments as in Remark 51.26(vi) guarantee that this solution satisfies (15.2) in (0, T ). In addition, it is nonnegative if u0 ≥ 0. Unlike in the case q > qc , T cannot be chosen uniform for all u0 lying in a bounded subset of Lq (Ω). Indeed assume without loss of generality that Ω ⊃ B(0, 1) and choose 0 ≤ u0 ∈ Lq ∩ L∞ (Ω) such that T0 := Tmax (u0 ) < ∞ (see Section 16 for the definition of the maximal existence time Tmax (u0 ) and Section 17 for the existence of such solution). For each j ≥ 1, set ωj = B(0, 1/j) and define u0,j (x) :=

j 2/(p−1) u0 (jx),

x ∈ ωj ,

0,

x ∈ Ω \ ωj .

(15.3)

15. Well-posedness in Lebesgue spaces

77

By direct computation, we see that u ˜j (x, t) := j 2/(p−1) u(jx, j 2 t) solves (15.1) in −2 ωj × (0, j T0 ) with initial data u0,j |ωj . Let uj be the solution of problem (15.1) (in Ω) with initial data u0,j . Since uj ≥ 0 on ∂ωj , it follows from the comparison principle that uj ≥ u ˜j in ωj as long as uj exists. Consequently, Tmax (u0,j ) ≤ j −2 T0 → 0, as j → ∞, while u0,j qc = u0 qc due to qc = n(p − 1)/2. See Remark 22.10(iii), Remark 27.8(g) and [36] for further results in that direction. If q = qc = 1 (i.e. q = 1, p = 1 + 2/n), then there exists a positive function u0 ∈ L1 (Ω) for which (15.1) does not possess any nonnegative classical L1 -solution in [0, T ) for any T > 0; see [93, Theorem 11]) and see also [116] for a similar example with the weaker notion of integral solution. (ii) Nonuniqueness in Rn . Assume Ω = Rn , u0 = 0, 1 + 2/n 0 and which is a global classical Lq -solution of (15.1) for any q < qc (and a W 1,q -solution for any q < n(p−1)/(p+1), see [269]). Moreover, there exist infinitely many nontrivial functions which are global classical Lq -solutions of (15.1) for any q < qc (see [532]). All these solutions are (forward) self-similar, that is u(x, t) = λ2/(p−1) u(λx, λ2 t),

λ > 0.

√ Such solutions can be found in the form u(x, t) = t−1/(p−1) w(x/ t), where w = w(y) solves the problem ∆w +

1 y · ∇w + w + |w|p−1 w = 0, 2 p−1

y ∈ Rn .

(15.4)

In [269] and [532], radial positive and infinitely many radial nontrivial solutions of (15.4) (with a rapid decay at infinity) were found by ODE techniques (see also [533]). Variational methods for solving (15.4) were used in [174]. (iii) Uniqueness and nonuniqueness in the class of mild solutions. If u is a classical Lq -solution of (14.1) in [0, T ), then it satisfies the variation-ofconstants formula

t u(t) = e−(t−τ )A u(τ ) + e−(t−s)A f (u(s)) ds, 0 < τ < t < T. (15.5) τ

Indeed, applying the operator e−(t−s)A to the equation ut (s) + Au(s) = f (u(s)), d (e−(t−s)A u(s)) = e−(t−s)A (ut (s) + Au(s)) we integrating in s ∈ (τ, t) and using ds obtain (15.5). Any function u ∈ C([0, T ), Lq (Ω)) satisfying f (u) ∈ L1loc ((0, T ), L1 + L∞ (Ω)), u(0) = u0 and (15.5) is called a mild Lq -solution of (15.1). (If q = ∞, then we modify this definition in the same way as in the case of classical solutions.) Now assume q ≥ p and let u be a mild Lq -solution of (15.1). Then we can pass to the limit in (15.5) as τ → 0 to get

t u(t) = e−tA u0 + e−(t−s)A |u(s)|p−1 u(s) ds. (15.6) 0

78


On the other hand, any solution of (15.6) in C [0, T ), Lq (Ω) is obviously a mild Lq -solution. If, in addition, q ≥ qc (and q > p if q = qc ), then each mild Lq -solution is a classical Lq -solution so that the uniqueness in Theorem 15.2 and (i) holds in the class of mild Lq -solutions (see [93], [536]). This is not true for the limiting case q = qc = p = n/(n − 2). In fact, if Ω is the unit ball and q = p = n/(n − 2), then there exists a singular stationary solution us ∈ Lq (Ω) \ C(Ω) of (15.1) (see [394] and cf. Remark 3.6(ii)). The function u(t) := us is a mild Lq -solution of (15.1) with u0 := us which is not classical for t > 0. On the other hand, (i) guarantees the existence of a classical Lq -solution. A similar example for Ω = Rn was constructed in [511]. (iv) Integral solutions. Consider problem (14.1) with f nonnegative and u0 ≥ 0. We say that u is an integral solution of (14.1) in [0, T ) if u : Ω×[0, T ) → [0, ∞] is measurable, finite a.e. and

t

u(x, t) = Ω

G(x, y, t)u0 (y) dy +

0

Ω

G(x, y, t − s)f (u(y, s)) dy ds

(15.7)

for a.e. (x, t) ∈ QT , where G is the Dirichlet heat kernel in Ω (cf. Appendix B). If u0 ∈ Lq (Ω) is nonnegative and u is a mild Lq -solution of (14.1), then u is also an integral solution of (14.1). In fact, since e

−tA

w(x) =

G(x, y, t)w(y) dy, Ω

the functions u, f are nonnegative and u : [0, T ) → Lq (Ω) is continuous, it is easy to pass to the limit in (15.5) as τ → 0 in order to obtain (15.7). Let us mention that the nonexistence statement in Theorem 15.3 is true in the class of integral solutions. (v) Weak solutions. Assume that Ω is bounded and u0 ∈ L1δ (Ω). A function u ∈ C([0, T ), L1δ (Ω)) is called a weak solution (more precisely weak L1δ solution) of (14.1) in [0, T ) if the functions u, δf (u) belong to L1loc ((0, T ), L1 (Ω)), u(0) = u0 and

t

τ

Ω

t

f (u)ϕ = − τ

Ω

u(ϕt + ∆ϕ) −

u(τ )ϕ(τ ) Ω

for any 0 < τ < t < T and any ϕ ∈ C 2 (Ω × [τ, t]) such that ϕ = 0 on ∂Ω × [τ, t] and ϕ(t) = 0. One can prove that any mild Lq -solution (hence any classical Lq solution) is a weak L1δ -solution for any q ≥ 1 (see Corollary 48.11) and that the linear problem ut − ∆u = f, x ∈ Ω, t ∈ [0, T ), u = 0, u(x, 0) = u0 (x),

x ∈ ∂Ω, t ∈ [0, T ), x∈Ω

(15.8)


79

possesses a unique weak L1δ -solution in [0, T ) for any f ∈ L1loc ([0, T ), L1δ (Ω)) and u0 ∈ L1δ (Ω) (see Proposition 48.9). It is also easy to see that the notions of integral solution and weak L1δ -solution coincide if Ω is bounded, f is nonnegative and we consider nonnegative, locally integrable solutions only (see Corollary 48.10). Notice that uniqueness of weak solutions need not be true for the nonlinear problem (see (iii)). (vi) Initial traces. In view of Theorems 15.2 and 15.3, it is a natural question to ask what should be the most general admissible initial data for local existence in problem (15.1). This question can be formulated as a problem of initial traces and has been studied in [56], [28]. In the case Ω = Rn , it is known that for any nonnegative classical solution u of ut − ∆u = up ,

x ∈ Rn , 0 < t < T

(15.9)

with p > 1, there exists a unique nonnegative Radon measure µ such that u(t) → µ in the sense of measures,

as t → 0.

(15.10)

The measure µ is called the initial trace of u. If, moreover, p < 1 + 2/n, then µ is uniformly locally finite, i.e.:

sup dµ < ∞, (15.11) x∈Rn

B(x,1)

and the result is optimal. Namely, if p < 1 + 2/n and µ is any nonnegative Radon measure verifying (15.11), then there exists a nonnegative classical solution of (15.9) which satisfies (15.10), and it is unique in a suitable class. Actually these results remain valid for properly defined weak solutions. On the other hand, in the range p ≥ 1 + 2/n, the initial trace of a given solution u has to satisfy conditions stronger than (15.11) (in particular the Dirac measure µ = δ0 is not admissible), but the necessary and sufficient condition on initial traces seems to be unknown. See also Theorem 15.11 and Remark 15.12 below for related results. Remark 15.5. Independence of the local solution with respect to q. If u0 ∈ Lq1 ∩ Lq2 (Ω) for some 1 ≤ q1 , q2 ≤ ∞, with q1 , q2 > qc or q1 , q2 ≥ qc > 1, then the corresponding solutions ui on [0, T i ), given by Theorem 15.2 (or Remark 15.4(i)), coincide for t < min(T 1 , T 2 ). This is a consequence of the following general argument. By decreasing one of the Ti ’s, we may assume T1 = T2 . The solution ui is obtained as the unique fixed point of a contraction Φiu0 : X i → X i , where X i is a complete metric space (of functions of t ∈ [0, T1 )). For u0 as above, Φ1u0 coincides with Φ2u0 on X := X 1 ∩ X 2 , and it is a contraction on the complete metric space X (with norm · X = max( · X1 , · X2 ). It thus has a unique fixed point u. By uniqueness in each X i , we immediately deduce that u1 = u = u2 .

80


Proof of Theorem 15.2. It is divided into several steps. Step 1. Fixed-point argument. To handle the singularity of the initial data, the idea is to introduce a Banach space of functions with a temporal weight which has a suitable decay as t → 0. We may assume that u0 q > 0. Let T > 0 be small and consider the Banach space pq YT := {u ∈ L∞ u YT := sup tα u(t) pq , loc (0, T ), L (Ω) : u YT < ∞}, 0 0 small and set κ1 := pq. Then u ∈ L∞ [ε, T ], Lκ1 (Ω) and u(t + ε) = e−tA u(ε) +

t

0

e−(t−s)A |u(s + ε)|p−1 u(s + ε) ds.

Choose κ2 > κ1 such that β1 := n2 κp1 − (15.17) and the Lp -Lq -estimates we get u(t + ε) κ2 ≤ t−β2 u(ε) κ1 + ∞

0

1 κ2

t

< 1 and set β2 :=

n 1 2 κ1

(15.17) −

1 κ2

. Using

(t − s)−β1 u(s + ε) pκ1 ds ≤ C(ε)

for t ∈ [ε, T − ε]. Hence u ∈ L [2ε, T ], Lκ2 (Ω) and an obvious bootstrap argu ∞ ment shows u ∈ L∞ loc (0, T ], L (Ω) . Now standard existence and regularity results for linear parabolic equations (see Appendix B) guarantee that u is a classical solution for t > 0, hence a classical Lq -solution. Let us explain this in more detail in the case of bounded domains; in the general case one can use smooth cut-off functions and use localized versions of the regularity statements in Appendix B. Fix δ > 0 small and let ψ : R → [0, 1] be a smooth function satisfying ψ(t) = 0 for t ≤ δ and ψ(t) = 1 for t ≥ 2δ. Since u is a mild solution, it is also a weak (L1δ -) solution (see Corollary 48.11). Consequently, ψu is a weak solution of the linear problem (15.8) with f := ψt u + ψ|u|p−1 u ∈ L∞ (Q), where Q := QT . Now Theorem 48.1(iii) guarantees that this linear problem has a strong solution v ∈ W 2,1;q (Q) for any q ∈ (1, ∞). This strong solution is obviously a weak solution and the uniqueness of weak solutions (see Proposition 48.9) guarantees ψu = v, consequently u ∈ W 2,1;q (Ω × (2δ, T )). Now fixing q > n + 2 we see that f (u) is

82


Hölder continuous in Ω × (2δ, T ). Next consider the function ψ(t − 2δ)u(t) and use Theorem 48.2(ii) to see that u is a classical solution for t > 4δ. Step 3. Continuous dependence. Let us denote by U (t)u0 the solution u(t) constructed above. The existence proof shows that U (·)v0 is defined and belongs to BM,T for any v0 satisfying v0 q < M and any T ≤ T0 . In addition, (15.13) guarantees U (·)u0 − U (·)v0 YT ≤ u0 − v0 q + C(p, α)M p−1 T 1−pα U (·)u0 − U (·)v0 YT , hence the choice of T0 implies U (·)u0 − U (·)v0 YT ≤ 2 u0 − v0 q .

(15.18)

It follows that U (t)u0 − U (t)v0 q ≤ u0 − v0 q +

0

t

|U (s)u0 |p−1 U (s)u0 − |U (s)v0 |p−1 U (s)v0 q ds

≤ u0 − v0 q + C(p)M p−1 T01−α U (·)u0 − V (·)v0 YT ≤ 2 u0 − v0 q (15.19) whenever t ≤ T0 . Consequently, the map Lq (Ω) → Lq (Ω) : v0 → U (t)v0 is Lipschitz continuous in a neighborhood of u0 . Step 4. Uniqueness. a classical Lq -solution of (15.1) in an interval Let v be q ∞ [0, T1 ), that is v ∈ C [0, T1 ), L (Ω) ∩L∞ loc ((0, T1 ), L (Ω)), v(0) = u0 and v is a classical solution of (15.1) for t ∈ (0, T1 ). Due to the uniqueness property (15.16), it is sufficient to show that v(t) = U (t)u0 for small t. Decreasing T1 if necessary we may thus assume that T1 ≤ T0 and v(s) q < M for all s ∈ [0, T1 ). Let T = T1 /2. For each τ ∈ (0, T ), since vτ := v(· + τ ) ∈ YT and vτ satisfies (15.6), property (15.16) implies v(t + τ ) = U (t)v(τ ) for all t ∈ (0, T ). Passing to the limit as τ → 0 and using (15.19), we obtain v(t) = U (t)u0 for all t ∈ (0, T ), hence the solution u is unique. Step 5. Smoothing estimate. Fix M = 2 u0 q and notice that T0 = T0 ( u0 q ) (provided we suppress the dependence of T0 on n, p, q). Choose r ≥ q. If r = q or r = pq, then (15.2) follows from (15.19) (with v0 = 0) or (15.15), respectively. Assume that u(t) m ≤ C u0 q t−αm (15.20) for some m ≥ max(p, q), where αm is defined in (15.2). We shall prove that we may increase the value of m in this estimate (by enlarging C if necessary) in such


83

a way that we can reach the value m = ∞ in a finite number of iterations. Then (15.2) follows for any r ∈ [q, ∞] from the interpolation inequality 1−q/r u(t) r ≤ u(t) q/r . q u(t) ∞

Similarly as above we obtain

u(t) r ≤ e−tA/2 u(t/2) r +

t

(t − s)−(n/2)(p/m−1/r) u(s) pm ds

t/2

≤ t−(n/2)(1/m−1/r) u(t/2) m

t p p + C u0 q (t − s)−(n/2)(p/m−1/r) s−p(n/2)(1/q−1/m) ds t/2 −αr

≤ C u0 q t

1−n(p−1)/(2q) × 1+t

1

1/2

(1 − s)−(n/2)(p/m−1/r) s−p(n/2)(1/q−1/m) ds

≤ C u0 q t−αr provided p/m − 1/r < 2/n. Since p/m − 1/m < 2/n due to m ≥ q, the conclusion follows. Step 6. Positivity. The positivity statement follows from the nonnegativity of the semigroup e−tA and the construction of the solution as a limit of nonnegative iterations uk+1 = Φu0 (uk ), u1 (t) ≡ 0. In view of the proof of Theorem 15.3, we prepare the following lemma from [535] (see also [529]). It implies in particular a (weighted) a priori estimate for any local nonnegative (integral) solution of (15.1) (see Corollary 15.8), which will be used in the proof of Theorem 18.3. Given a measurable function Φ : Ω → [0, ∞], we set

−tA (e Φ)(x) := G(x, y, t)Φ(y) dy, Ω

where G = GΩ is the Dirichlet heat kernel in Ω (see Appendix B). Lemma 15.6. Let u0 : Ω → [0, ∞] and u : Ω × [0, T ] → [0, ∞] be measurable and satisfy

t u(t) ≥ e−tA u0 + e−(t−s)A up (s) ds a.e. in QT . (15.21) 0

Assume that u(x, t) < ∞ for a.a. (x, t) ∈ QT . Then there holds t1/(p−1) e−tA u0 ∞ ≤ kp := (p − 1)−1/(p−1)

for all t ∈ (0, T ].

(15.22)

84


Proof. In this proof, operations such as interchange of integrals and moving of e−tA inside integrals are justified by Fubini’s theorem for nonnegative measurable functions. First notice that e−tA Φ = e−(t−s)A e−sA Φ for all 0 < s < t and any measurable Φ : Ω → [0, ∞]. Also, we deduce from Jensen’s inequality and

Ω

(15.23)

G(x, y, t) dy ≤ 1 that

e−tA Φp ≥ (e−tA Φ)p for all measurable Φ : Ω → [0, ∞].

(15.24)

Now, by redefining u on a null set, we may assume that (15.21) actually holds everywhere in Ω × (0, T ). By assumption, for a.a. τ ∈ (0, T ), we have u(·, τ ) < ∞ a.e. in Ω. Fix such τ and let Ωτ := {x ∈ Ω : u(x, τ ) < ∞}. For t ∈ [0, τ ], it follows from (15.21), (15.23) and (15.24) that e−(τ −t)A u(t) ≥ e−τ A u0 + ≥e

−τ A

t

0

u0 +

e−(τ −s)A up (s) ds

t

e

−(τ −s)A

p u(s) ds =: h(·, t).

(15.25)

0

By the second inequality in (15.25), we see that h(·, t) ≤ e−τ A u0 +

0

τ

e−(τ −s)A up (s) ds ≤ u(·, τ );

(15.26)

and so h(x, t) < ∞ for all (x, t) ∈ Ωτ × [0, τ ]. Fix x ∈ Ωτ . Then the function φ(t) := h(x, t) is absolutely continuous on [0, τ ] and (15.25) yields p φ (t) = e−(τ −t)A u(t) (x) ≥ φp (t)

for a.a. t ∈ [0, τ ].

(15.27)

Also φ(t) ≥ e−τ A u0 (x) > 0, and so (15.27) can be rewritten as [φ1−p ] ≤ −(p−1). Integrating this inequality over [0, τ ], we get −τ A 1−p e u0 (x) = φ1−p (0) ≥ φ1−p (τ ) + (p − 1)τ ≥ (p − 1)τ.

(15.28)

It follows that τ 1/(p−1) e−τ A u0 ∞ ≤ k. This guarantees in particular that e−tA u0 ∈ L∞ (Ω) for a.a. t ∈ (0, T ), Since t → e−tA v ∞ is continuous for v ∈ L∞ (Ω) and t > 0, we deduce from (15.23) that the function t → t1/(p−1) e−tA u0 ∞ is continuous in (0, T ), hence (15.22).


85

Remark 15.7. If 0 ≤ u0 ∈ L∞ (Ω), and u is a (sufficiently regular) supersolution of (14.1) on [0, T ], then estimate (15.22) can be alternatively obtained as follows (cf. [363]). Let

e−tA u0

u(x, t) :=

1−p

−1/(p−1)

(x) − (p − 1)t

+

,

which is finite in QT1 , where T1 := inf{t ∈ [0, T ] : t1/(p−1) e−tA u0 ∞ ≥ kp } ∈ (0, T ]. A direct computation reveals that ut − ∆u ≤ up in QT1 . In view of the comparison principle, since u = 0 on ST1 and u(·, 0) = u0 , we obtain the lower estimate u ≥ u in QT1 . (15.29) In particular, we have T1 = T , hence (15.22). On the other hand, let us observe that estimate (15.29) also follows from (15.26) and (15.28). Corollary 15.8. Assume that (15.21) is true with the inequality sign replaced by the equality sign. Then t1/(p−1) e−tA u(τ ) ∞ ≤ kp

for all t ∈ (0, T − τ ] and a.a. τ ∈ (0, T ).

Proof. Set v(t) := u(t + τ ). Then (15.23) and Fubini’s theorem guarantee, for a.a. τ ∈ (0, T ) and a.a. t ∈ (τ, T ), v(t) = e−(t+τ )A u0 +

t+τ

e−(t+τ −s)A up (s) ds

0

t+τ e−tA e−(τ −s)A up (s) ds + e−(t+τ −s)A up (s) ds 0 τ

τ t −tA −τ A −(τ −s)A p e u0 + e u (s) ds + e−(t−s)A v p (s) ds =e

=e

−tA −τ A

e

τ

u0 +

= e−tA u(τ ) +

0

t

0

e−(t−s)A v p (s) ds.

0

Hence, we may use Lemma 15.6 with u0 replaced by u(τ ) and T replaced by T − τ for a.a. τ ∈ (0, T ). Proof of Theorem 15.3. (i) Fix α ∈ (0, n/q), assume (without loss of generality) that B(0, 2ρ) ⊂ Ω, ρ > 0, and define u0 (y) = |y|−α χB(0,ρ) (y).

86


Clearly, we have 0 ≤ u0 ∈ Lq (Ω). Using the heat kernel estimate in Proposition 49.10, we obtain, for t > 0 small,

−tA e u0 (0) = G(0, y, t)|y|−α dy |y| 0. (ii) The assertion is a consequence of Proposition 28.1 below. For certain applications (see Section 26), it is useful to study well-posedness and regularization properties in different types of Lebesgue spaces. We first consider bounded domains and the spaces Lqδ (Ω), the Lebesgue spaces weighted by the function distance to the boundary. Based on the linear theory in these spaces (see Theorem 49.7 in Appendix C), we obtain the following results [200], in a completely similar manner as in Theorems 15.2 and 15.3(i). They show that the critical exponent for local well-posedness is now q = (n + 1)(p − 1)/2. Theorem 15.9. Assume Ω bounded and p > 1. Let u0 ∈ Lqδ (Ω), 1 ≤ q < ∞, q > (n + 1)(p − 1)/2. Then there exists T = T ( u0 q,δ ) > 0 such that problem (15.1) possesses a unique classical Lqδ -solution in [0, T ) and the following smoothing estimate is true: u(t) r,δ ≤ C u0 q,δ t−βr ,

βr :=

n + 1 1 1 − , 2 q r

(15.31)

for all t ∈ (0, T ] and r ∈ [q, ∞], with C = C(n, p, q, Ω) > 0. In addition, u ≥ 0 provided u0 ≥ 0. Theorem 15.10. Assume Ω bounded, p>1+

2 n+1

and

1≤q
0. In the case Ω = Rn , let us finally consider the uniformly local Lebesgue spaces Lqul (Rn ). Using the linear smoothing effect in these spaces (see Proposition 49.15 in Appendix C or [37]), we obtain the following smoothing estimate (cf. [253]) by similar arguments as in the proof of Theorem 15.2. Here e−tA denotes the heat semigroup in Rn .

16. Maximal existence time. Uniform bounds from Lq -estimates

87

Theorem 15.11. Let p > 1, q > qc and 1 ≤ q < ∞. Let u0 ∈ L∞ (Rn ), T > 0 and assume that u ∈ L∞ ((0, T ), L∞ (Rn )) is a solution of u(t) = e−tA u0 +

0

t

e−(t−s)A |u(s)|p−1 u(s) ds,

0 < t < T.

Then there exist C = C(n, p, q) > 0, T0 = T0 ( u0 q,ul ) > 0 such that u(t) r,ul ≤ C u0 q,ul t−αr ,

αr :=

n 1 1 − , 2 q r

for all t ∈ (0, min(T, T0 )] and r ∈ [q, ∞]. Remark 15.12. A local well-posedness result similar to Theorem 15.2 can also be proved in Lqul (Rn ) (see [253], and cf. also [28]).

16. Maximal existence time. Uniform bounds from Lq -estimates In this section we are interested in sufficient conditions guaranteeing global existence. More precisely, we want to show that any solution satisfying suitable bounds in the Lebesgue space Lq (Ω) is global. Let us start with a simple proposition which defines the maximal solution and existence time. We formulate the statement only for the model problem (14.1) but it will be clear from the proof that the same statement is true for a much more general class of equations and systems. Proposition 16.1. Let X be a Banach space of functions defined in Ω. Assume that problem (14.1) possesses for each u0 ∈ X a unique (classical X-) solution u on the interval [0, T ], where T = T (u0 ). Then there exists Tmax = Tmax (u0 ) ∈ (T, ∞] with the following properties. (i) The solution u can be continued (in a unique way) to a classical X-solution on the interval [0, Tmax). (ii) If Tmax < ∞, then u cannot be continued to a classical X-solution on [0, τ ) for any τ > Tmax . We call u the maximal (classical X-) solution starting from u0 and Tmax its maximal existence time. (iii) Assume further that T = T ( u0 X ). Then either Tmax = ∞ or limt→Tmax u(t) X = ∞.

(16.1)

Proof. Let u0 ∈ X be fixed. If u1 and u2 are solutions of (14.1) on [0, T1 ) and [0, T2 ), respectively, then u1 = u2 on [0, min(T1 , T2 )) due to the uniqueness. Let

88


{uα : [0, Tα ) → X} be the set of all solutions of (14.1) and T˜ := sup Tα . Define u : [0, T˜) → X by u(t) := uα (t), where α is any index such that Tα > t. Then u is obviously a solution of (14.1) on [0, T˜ ), and properties (i)(ii) are verified. Under the assumption in property (iii), suppose that T˜ < ∞

and

lim inf u(t) X < ∞. t→T˜

Choose C > 0 and tk → T˜ such that u(tk ) X < C for all k = 1, 2, . . . . Due to our assumptions there exists T > 0 independent of k such that the problem (14.1) with initial data u(tk ) possesses a unique solution uk : [0, T ] → X, k = 1, 2, . . . . By uniqueness, uk (t) = u(t + tk ) for t small. Fix k such that tk ∈ (T˜ − T, T˜ ) and set u(t), t ∈ [0, tk ], u ˜(t) := uk (t − tk ), t ∈ [tk , tk + T ]. Then u˜ is a solution of (14.1) on [0, tk + T ] and tk + T > T˜ which contradicts the definition of T˜. Remarks 16.2. (i) Maximal Lq -solution. Consider problem (15.1) and set X = Lq (Ω), where 1 ≤ q ≤ ∞ satisfies q > qc = n(p − 1)/2 or q = qc > 1. If u0 ∈ X, then Theorem 15.2 and Proposition 16.1 (or Remark 51.11 if q = ∞) guarantee the existence of a maximal (classical Lq -) solution u, up to a maximal existence time Tmax (u0 ). Moreover, property (16.1) is true if q > qc . Similarly as in Example 51.9, u in fact satisfies u ∈ BC 2,1 (Ω × [t1 , t2 ]),

0 < t1 < t2 < Tmax (u0 ).

(16.2)

If u0 ≥ 0, then u ≥ 0. If u0 is radial (resp. nonnegative and radial nonincreasing), then u enjoys the same property, as a consequence of Proposition 52.17. (ii) Independence of the maximal solution with respect to q. Let q, u0 and u be as in remark (i). We show that, if u0 belongs to several Lq -spaces, then u and Tmax (u0 ) do not depend on q. Thus assume that u0 ∈ Lq1 ∩ Lq2 (Ω) for some q1 , q2 as above, and denote by ui , i = 1, 2, the corresponding maximal, classical Lqi -solution, of existence time T i . We know that u1 = u2 for t > 0 small (cf. Remark 15.5). Using ui ∈ C([0, T i ), Lqi (Ω)), we deduce easily that u1 = u2 on [0, min(T 1 , T 2 )). Assume for contradiction that T 1 < T 2 (hence T 1 < ∞). Since, by the definition of a maximal 2 ∞ 1 p−1 1 classical Lq -solution, u2 ∈ L∞ u | ≤ C|u1 | loc ((0, T ), L (Ω)), it follows that ||u | 1 1 1 on (T /2, T ), which readily implies sup[T 1 /2,T 1 ] u (t) q1 < ∞. If q1 > qc , then this contradicts (16.1). If q = qc , then the variation-of-constants formula implies u1 ∈ C([0, T 1 ], Lq1 (Ω)). The local existence theorem can then be used to extend u1 after T 1 and we again reach a contradiction. (iii) Lower bounds on supercritical Lq -norms. By using the local theory of problem (14.1), developed in Section 15, it is actually possible to obtain lower

16. Maximal existence time. Uniform bounds from Lq -estimates

89

estimates of the supercritical Lq -norms, in case Tmax (u0 ) < ∞. Namely, let q ≥ 1 satisfy qc < q < ∞ and assume u0 ∈ Lq (Ω). Then the proof of Theorem 15.2 ≥ (see in particular formula (15.14)) shows that (Tmax (u0 ))1−n(p−1)/2q u0 p−1 q C(n, p, q) > 0. After a time shift, this yields u(t) q ≥ C(n, p, q)(Tmax (u0 ) − t)n/(2q)−1/(p−1) ,

0 ≤ t < T.

(16.3)

(iv) Critical Lq -space. If u0 ∈ Lq (Ω) with q = qc > 1, then it is not known in general whether Tmax (u0 ) < ∞ implies limt→Tmax (u0 ) u(t) q = ∞. A positive result in that direction (cf. [91], [535]) is given by the following proposition. The proof, which relies on simple energy arguments, is postponed to the next section. For further positive results, see [530], [535], [246], [357]. Proposition 16.3. Consider problem (15.1) with p = 1 + 4/n (so that qc = n(p − 1)/2 = 2). Let u0 ∈ L2 (Ω) and assume T := Tmax (u0 ) < ∞. Then u(t) 2 ≥ C(n, p)| log(T − t)|1/2 ,

t → T.

(16.4)

We say that (14.1) possesses a global solution if Tmax = ∞. Proposition 16.1 provides a simple criterion for global existence: If u(t) X remains bounded, then Tmax = ∞. Since the assumptions of Proposition 16.1 are satisfied with X = L∞ (Ω) if f ∈ C 1 (see Remark 51.11) we see that the boundedness of the solution in L∞ (Ω) is sufficient for its global existence. Note that the same statement is true for a much more general class of equations and systems. Unfortunately, it is not easy to obtain the L∞ -estimate for solutions of (14.1). As we shall see, standard methods usually yield only an Lq -estimate for some q < ∞. Therefore, it is important to find q as small as possible and such that the Lq -estimate guarantees the L∞ -estimate, hence global existence. We will call this property of Lq the continuation property. Theorem 15.2, Proposition 16.1 and Remarks 16.2 guarantee the global existence of a solution of the model problem (15.1) provided the solution is bounded in Lq (Ω) for some q > qc . As we shall see in Corollary 24.2, this condition is optimal (up to the equality sign): If 1 < q < qc , then there exists a radial positive solution of (15.1) in a ball such that Tmax < ∞ but the solution stays bounded in Lq (Ω). Therefore, the exponent qc for problem (15.1) is “critical” both for well-posedness and the global existence. This is due to the simple structure of the nonlinearity. We will see in Chapter III that for more complicated problems, the critical exponents for the wellposedness and the continuation property may differ. Therefore it is important to find methods guaranteeing the global existence of a solution under the assumption of its boundedness in Lq and not using any well-posedness result. In this section we present a method due to [11], [12] (cf. also [459]) which is based on Moser-type iterations and can be efficiently used for a very general class

90


of problems (including degenerate problems, problems on nonsmooth domains etc). In order to make it as clear as possible, we again restrict ourselves to the model problem (15.1). Another method for obtaining L∞ -bounds from Lq -bounds is presented in Appendix E (see Proposition 51.34). That method is based on the variation-ofconstants formula and interpolation-extrapolation spaces and is due to [14]. Hence, our aim is to prove the following theorem (which is a consequence of Theorem 15.2 and Proposition 16.1), without using the well-posedness results. Theorem 16.4. Let p > 1 and let u be a classical solution of (15.1) defined on [0, T ). Assume q > 1 and Uq := supt 0 such that e−ct

d d ct 2ρ(r) ˜r2rρ(r) . e w 22 = w 22 + c w 22 ≤ Cr2rσ(r) w 1 ≤ Cr2rσ(r) U dt dt

17. Blow-up

91

2 2r ˜ 2r Since w(0) 22 ≤ C u0 2r ∞ ≤ C Ur and w 2 = u 2r , integration of the above estimate implies the assertion.

Proof of Theorem 16.4. We shall use the notation from Lemma 16.5. Notice that γ := qσ(q) ≥ rσ(r) for any r ≥ q. Using repeatedly Lemma 16.5 with r := q, r := 2q, r := 4q etc, one can easily verify that, given ν ∈ {0, 1, 2, . . . }, ˜2ν+1 q ≤ (C1 q γ )k1 2k2 U ˜ k3 , U q where ρ(2ν q) ρ(2ν q) · · · · · ρ(2q) 1 + + · · · + , 2ν q 2ν−1 q q γ ν ν −1 1 ν ν 2 ρ(2 k2 = k2 (ν) = + ρ(2 q) + · · · + q) · · · · · ρ(2 q) , q 2ν 2ν−1 2 k3 = k3 (ν) = ρ(2ν q) · · · · · ρ(q). k1 = k1 (ν) =

Since σ(2r) ≤ σ(r)/2 we obtain ρ(2i q) ≤ 1 + (p − 1)σ(q)2−i . Now using the inequality log(1 + x) ≤ x for x ≥ 0 we get log k3 ≤

ν

(p − 1)

i=0

σ(q) ≤ 2(p − 1)σ(q) =: C3 < ∞. 2i

Finally, k1 ≤

ν k3 1 2 ≤ eC3 < ∞ i q i=0 2 q

k2 ≤

∞

i γ i γ k3 ≤ eC3 < ∞, i q i=1 2 q 2i i=1 ν

and

which concludes the proof.

17. Blow-up In this section we mainly consider the model problem ut − ∆u = λu + |u|p−1 u, u = 0, u(x, 0) = u0 (x),

x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0, x ∈ Ω,

⎫ ⎪ ⎬ ⎪ ⎭

(17.1)

with p > 1 and λ ∈ R, and we derive some criteria for u0 which guarantee blow-up of the solution of (17.1) in finite time. More general nonlinearities f (u) will be briefly considered. We will always assume that u0 belongs to a function space X where (17.1) is well-posed and we denote by Tmax (u0 ) the maximal existence time of the solution of (17.1). We start with a simple criterion. In the bounded domain case, it is based on the eigenfunction method due to [296].

92


Theorem 17.1. Consider problem (17.1) with p > 1 and λ ∈ R. (i) Assume Ω bounded. Let u0 ∈ L∞ (Ω) satisfy

1/(p−1) u0 ≥ 0, u0 ϕ1 dx > c := max(0, λ1 − λ) .

(17.2)

Ω

Then Tmax (u0 ) < ∞. (ii) Assume Ω = Rn . Then assertion (i) remains valid if we replace λ1 by 2n and 2 ϕ1 by the function ϕ(x) = π −n/2 e−|x| . Proof. (i) Recall that u ≥ 0 and denote y = y(t) := Ω u(t)ϕ1 dx. Multiplying the equation in (17.1) with ϕ1 , integrating by parts, and using ∆ϕ1 = −λ1 ϕ1 and Jensen’s inequality yields

y = ut ϕ1 dx = u∆ϕ1 dx + λ uϕ1 dx + up ϕ1 dx ≥ y p − cp−1 y. (17.3) Ω

Ω

Ω

Ω

Since y(0) > c, we infer from (17.3) that y ≥ εy p ,

0 < t < Tmax (u0 ),

with ε = 1 − (c/y(0))p−1 > 0. This differential inequality guarantees that u cannot exist globally. (ii) The proof is the same except that we now use ∆ϕ ≥ −2nϕ. The calculation in (17.3) can be easily justified by integrating by parts over BR and letting R → ∞, using property (16.2) and the exponential decay of ϕ and ∇ϕ. Remarks 17.2. (i) Estimation of the blow-up time. The proof of Theo rem 17.1 shows that if, for instance, λ = 0, Ω is bounded and Ω u0 ϕ1 dx ≥ (2λ1 )1/(p−1) , then

1−p 2 Tmax (u0 ) ≤ u0 ϕ1 dx . p−1 Ω We refer to [324], [262] for more precise results concerning upper estimates of the blow-up time. (ii) Neumann boundary conditions. If we replace the homogeneous Dirichlet boundary conditions in problem (17.1) with the homogeneous Neumann boundary conditions ∂ν u = 0, then all positive solutions blow up in finite time when λ ≥ 0. the over Ω, we see that the function y(t) := equation Indeed, by integrating p 1−p p u(t) dx satisfies y (t) ≥ u dx ≥ |Ω| y with y(0) > 0. Alternatively, for Ω Ω t0 > 0 small, the strong maximum principle guarantees that u(x, t0 ) ≥ ε > 0 in Ω, and it suffices to use the solution of the ODE z = z p , z(t0 ) = ε, as subsolution. The previous result can be easily extended to problem (14.1) under suitable convexity and superlinearity conditions.

17. Blow-up

93

Theorem 17.3. Consider problem (14.1) where f : R → R is a convex C 1 function and Ω is bounded. Assume that, for some a > 0, we have f (s) > 0 for all s ≥ a and

∞ ds < ∞. (17.4) f (s) Then Theorem 17.1(i) remains true provided the constant c in (17.2) is replaced by C = C(Ω, f ) > 0 large enough. Proof. Denote again y = y(t) := Ω u(t)ϕ1 dx. Arguing as in the previous proof, we obtain

y = ut ϕ1 dx = u∆ϕ1 dx + f (u)ϕ1 dx ≥ −λ1 y + f (y). (17.5) Ω

Ω

Ω

(a) Since f is convex, the function g(s) := f (s)−f is nondecreasing for s > a and s−a g(s) → ∞ as s → ∞, due to (17.4). Therefore, there exists C ≥ a such that f (s) ≥ 2λ1 s for all s ≥ C. If y(0) ≥ C, it follows from (17.5) that, as long as u exists, y(t) ≥ C and 1 y ≥ f (y) − λ1 y ≥ f (y), 2

hence

λ1 t/2 ≤

t 0

y (τ ) dτ = f (y(τ ))

y(t)

y(0)

ds ≤ f (s)

∞ y(0)

ds < ∞. f (s)

Therefore u cannot exist globally. Remark 17.4. It is well known that condition (17.4) is necessary and sufficient for the existence of blow-up solutions of the ODE u = f (u), t ≥ 0. The convexity condition in Theorem 17.3 can be replaced by the assumption that f ≥ f˜ for s large, where f˜ satisfies the assumptions of the theorem. As a typical “weakly superlinear” f satisfying (17.4), one may take a function f such that f (s) = (1 + s) logp (1 + s) for s ≥ 0, with p > 1. The next criterion is based on the fact that the energy functional E(u) =

1 2

Ω

|∇u|2 − λu2 dx −

1 p+1

Ω

|u|p+1 dx

(17.6)

is nonincreasing along any solution of (17.1). More precisely we have: Lemma 17.5. Consider problem (17.1) with p > 1, λ ∈ R, u0 ∈ L∞ ∩ H01 (Ω), and let T = Tmax (u0 ). Then E(u(·)) ∈ C([0, T )) ∩ C 1 ((0, T )) and d E u(t) = − dt

Ω

u2t (t) dx.

(17.7)

94


Proof. Example 51.28 guarantees u ∈ C [0, T ), H01 ∩ Lp+1 (Ω) , hence E(u(·)) ∈ C([0, T )), and u ∈ C (0, T ), H 2 (Ω) ∩ C 1 (0, T ), L2 ∩ Lp+1 (Ω) . (17.8) Denote E1 (t) = Ω |∇u(t)|2 dx. For t, s ∈ (0, T ), s = t, using integration by parts, we obtain

E1 (t) − E1 (s) 1 = ∇(u(t) − u(s)) · ∇(u(t) + u(s)) dx t−s t−s Ω

u(t) − u(s) ∆(u(t) + u(s)) dx → −2 =− ut (t)∆u(t) dx t−s Ω Ω as s → t, due to (17.8). Consequently, E(u(·)) ∈ C 1 ((0, T )) and

d E u(t) = (−∆u − λu − |u|p−1 u)ut dx = − u2t (t) dx. dt Ω Ω

The following result is due to [327]. The simpler proof in the case Ω bounded, λ = 0, is from [515], [51]. Theorem 17.6. Consider problem (17.1) with p > 1, λ ∈ R and u0 ∈ L∞ ∩H01 (Ω). Assume either Ω bounded or λ ≤ 0. If E(u0 ) < 0, then Tmax (u0 ) < ∞. Proof. (i) First assume that Ω is bounded. older’s Set ψ(t) := u(t) 22 . Multiplying the equation in (17.1) by u and using H¨ inequality we obtain

1 ψ (t) = uut (t) dx = − |∇u(t)|2 dx + λ u2 dx + |u(t)|p+1 dx 2 Ω Ω Ω Ω

(17.9) p−1 p+1 = −2E u(t) + |u(t)| dx ≥ −2E(u0 ) + cψ(t)(p+1)/2 , p+1 Ω where c := (p − 1)/[(p + 1)|Ω|(p−1)/2 ]. This inequality implies Tmax (u0 ) < ∞ provided E(u0 ) < 0 (or ψ(0)(p+1)/2 > 2E(u0 )/c). (ii) Next consider the case Ω unbounded, λ ≤ 0. (The following argument works also if λ ≤ λ1 and Ω is bounded). We will use the concavity method due to [327]. t Assume Tmax (u0 ) = ∞ and denote M (t) := 12 0 u(s) 22 ds. Then we have M (t) = 12 u(t) 22 and

2 2 M (t) = uut (t) dx = − |∇u(t)| dx + λ u (t) dx + |u(t)|p+1 dx Ω Ω Ω Ω

p−1 |∇u(t)|2 − λu2 (t) dx = −(p + 1)E u(t) + 2 Ω ≥ −(p + 1)E(u0 ) > 0,

17. Blow-up

95

which implies M (t) → ∞ and M (t) → ∞ as t → ∞. Moreover, this estimate and

ut (s) 22 ds = E(u0 ) − E u(t) < −E u(t)

t

0

(17.10)

(cf. (17.7)) imply M (t) ≥ −(p + 1)E u(t) ≥ (p + 1)

t

0

ut (s) 22 ds,

hence

t p + 1 t 2 M (t)M (t) ≥ ut (s) 2 ds u(s) 22 ds 2 0 0

2 p + 1 t u(x, s)ut (x, s) dx ds ≥ 2 Ω 0 2 p + 1 M (t) − M (0) . = 2

Since M (t) → ∞ as t → ∞, the last estimate implies existence of α, t0 > 0 such that 2 M (t)M (t) ≥ (1 + α) M (t) , t ≥ t0 . This inequality guarantees that the nonincreasing function t → M −α (t) is concave on [t0 , ∞) which contradicts the fact M −α (t) → 0 as t → ∞. Proof of Proposition 16.3. By Example 51.27 in Appendix E, after a timeshift, we may assume u0 ∈ H01 (Ω). Similarly as in (17.9) (but without assuming Ω bounded), we have 1 d 2 dt

Ω

u2 (t) dx ≥ −2E(u0 ) +

p−1 p+1

Ω

|u(t)|p+1 dx.

Integrating and using (16.3) with q = p + 1 > qc , it follows that

Ω

p−1 t |u(s)|p+1 dx ds p+1 0 Ω

t (T − s)(n/2)−(p+1)/(p−1) ds. ≥ −C + C(n, p)

u2 (t) dx ≥ −2E(u0 )t +

0

Since (n/2) − (p + 1)/(p − 1) = −1, (16.4) follows.

96


Remarks 17.7. (i) The proof of Theorem 17.6 does not imply blow-up of the L2 -norm of u. Indeed, as was observed in [51], the solution u might cease to exist before the time obtained by integrating the differential inequality in (17.9). Examples where the L2 -norm of u remains bounded will be given in Corollary 24.2. A similar remark holds concerning the quantity y(t) in the proof of Theorem 17.1. (ii) The first part of the proof of Theorem 17.6 shows that 1/(p+1) u(t) 2 ≤ 2E(u0 )/c for any global solution u of (17.1) provided Ω is bounded. Now the results of the preceding section guarantee that u(t) ∞ ≤ C u0 ∞ , E(u0 ) if p < 1 + 4/n. As we shall see later in Section 22, this assertion is true for any p < pS . (iii) If u is a global solution of (17.1) and Ω is bounded or λ ≤ 0, then Theorem 17.6 guarantees 0 ≤ E u(t) ≤ E(u0 ) for all t > 0. (iv) Inequality (17.9) also shows the following: Given δ > 0 there exists Cδ > 0 such that Tmax (u0 ) < δ whenever E(u0 ) < −Cδ . (v) Let ϕ ∈ L∞ ∩ H01 (Ω) be a fixed function, ϕ ≡ 0. Then Tmax (αϕ) < ∞ for α > 0 large enough. This follows from Theorem 17.6 and the fact that E(αϕ) = α2

Ω

|∇ϕ|2 − λϕ2 dx − αp+1 2

Ω

ϕp+1 dx. p+1

Note that if we assume 0 ≤ ϕ ∈ L∞ (Ω) instead, then the same conclusion follows from Theorem 17.1. Further blow-up conditions involving the energy will be given in Theorem 19.5. We now give a third criterion (cf. [342]), which guarantees blow-up if one starts above a positive equilibrium. Theorem 17.8. Assume Ω bounded, p > 1 and λ ∈ R. Assume that problem (17.1) has a (classical) equilibrium v, with v > 0 in Ω. If u0 ∈ L∞ (Ω) satisfies u0 ≥ v, u0 ≡ v, then Tmax (u0 ) < ∞. For the proof, we prepare the following separation lemma, which will be used again later. Lemma 17.9. Assume Ω bounded and consider problem (14.1) where f : R → R is a convex C 1 -function with f (0) = 0. Let u0 , u0 ∈ L∞ (Ω) be such that u0 ≥ u0 , u0 ≡ u0 . Let u, u be the corresponding solutions of (14.1), and fix τ ∈ (0, Tmax (u0 )). Then Tmax (u0 ) ≥ Tmax (u0 ) and there exists α > 1 such that u ≥ αu,

τ ≤ t < Tmax (u0 ).

(17.11)

17. Blow-up

97

Proof. Since u ≤ u by the comparison principle and f (s) ≥ f (0)s, s ∈ R, by the convexity of f and f (0) = 0, we have Tmax (u0 ) ≥ Tmax (u0 ). By the strong and the Hopf maximum principles, we have u(x, τ ) > u(x, τ ) in Ω

and

∂u ∂u (x, τ ) < (x, τ ) on ∂Ω. ∂ν ∂ν

Therefore, there exists α > 1 such that u(x, τ ) ≥ αu(x, τ ) in Ω. Since f (αu) ≥ αf (u), due to f convex and f (0) = 0, we infer that (αu)t − ∆(αu) − f (αu) ≤ α(ut − ∆u − f (u)) = 0, and the lemma follows from the comparison principle. Proof of Theorem 17.8. By Lemma 17.9, applied with u0 = v, there exist α > 1 and τ ∈ (0, Tmax (u0 )), such that u ≥ αv,

t ∈ [τ, Tmax (u0 )).

(17.12)

Denote z = z(t) := Ω u(t)v dx. Multiplying the equation in (15.1) with v, integrating by parts, and using (17.12) and H¨ older’s (or Jensen’s) inequality, we obtain

z = ut v dx = u∆v dx + (up + λu)v dx Ω Ω Ω

p p 1 − (v/u)p−1 up v dx = (u v − v u) dx = Ω Ω

1−p 1−p p ≥ (1 − α ) u v dx ≥ (1 − α1−p ) v dx z p, Ω

Ω

for t ∈ [τ, Tmax (u0 )). It follows that u cannot exist globally. By using an alternative linearization argument based on an idea from [311], one can extend Theorem 17.10 to more general convex nonlinearities. Theorem 17.10. Consider problem (14.1) with f and Ω as in Theorem 17.3. Assume in addition that f (0) = 0, f is nonconstant near 0, and that problem (14.1) has a (classical) equilibrium v, with v > 0 in Ω. If u0 ∈ L∞ (Ω) satisfies u0 ≥ v, u0 ≡ v, then Tmax (u0 ) < ∞. Proof. Let µ and ψ > 0 denote the first eigenvalue and the corresponding eigenfunction of the problem ∆ψ + f (v)ψ = µψ

in Ω,

ψ = 0 on ∂Ω,

ψ dx = 1. Multiplying the above equation by v, the equation ∆v + f (v) = 0 by Ω ψ, integrating and subtracting the resulting identities, we obtain

µ vψ dx = (vf (v) − f (v))ψ dx. Ω

Ω

98


Due to v, ψ > 0, f (0) = 0, f convex and f nonconstant near 0, the last integral is positive, hence µ > 0 (the solution v is linearly unstable). Since u ≥ v by the comparison principle, we have y(t) := Ω (u(t) − v)ψ dx ≥ 0. In addition,

y (t) =

(∆u + f (u))ψ dx =

Ω

= Ω

Ω

(u − v)∆ψ + (f (u) − f (v))ψ dx

µ(u − v)ψ + f (u) − f (v) − f (v)(u − v) ψ dx.

Since f (u) − f (v) − f (v)(u − v) ≥ 0 by convexity, we have y (t) ≥ µy(t). Assume for contradiction that Tmax (u0 ) = ∞. Then limt→∞ y(t) = ∞. Since ψ ≤ cϕ1 due to (1.4), it follows that limt→∞ Ω u(t)ϕ1 dx = ∞. But this contradicts Theorem 17.3. Remark 17.11. The proofs of Theorems 17.8 and 17.10 were based on the convexity of the nonlinearity. However, if v is a maximal, unstable equilibrium, then blow-up of solutions starting above v can be shown for general superlinear f with subcritical growth. Assume first f (cu) ≥ cf (u)

for c > 1

(17.13)

and let u0 ≥ v, u0 ≡ v. Fix τ > 0. Due to the maximum principle, there exists ε > 0 such that u(τ ) ≥ (1 + ε)v =: u ˜0 (cf. the proof of Lemma 17.9). Let u ˜ denote the solution with initial data u˜0 . Since u(t + τ ) ≥ u ˜(t) by the maximum principle, it suffices to prove Tmax (˜ u0 ) < ∞. Assume on the contrary that u˜ exists globally. Since ∆˜ u0 + f (˜ u0 ) ≥ 0, we have u ˜t ≥ 0. Lemma 53.10 and the maximality of v guarantee that u˜ cannot stay bounded, hence u(t) ∞ → ∞ as t → ∞. Since the growth of f is subcritical, we have also u(t) 1,2 → ∞ as t → ∞. Now a simple modification of the concavity method (cf. the proof of Theorem 17.6(ii) and see [186] for details) yields a contradiction. If f is a general function (not necessarily satisfying (17.13)), then [354] guarantees the existence of a time increasing solution w defined for t ∈ (−∞, 0] and satisfying w(t) → v in C 1 (Ω) as t → −∞. Fix τ > 0. Since u(τ ) ≥ w(t) for suitable t ≤ 0 we can proceed as above. This approach can be used for more general problems provided one can show boundedness of global increasing solutions (see [185], for example). Our last criterion [55], [324] concerns the Cauchy problem and asserts that finite-time blow-up occurs whenever the nonnegative initial data has a sufficiently slow decay at infinity. Theorem 17.12. Let p > 1 and consider problem (15.1) with Ω = Rn . Let −µ < 0 be the first eigenvalue of the Dirichlet Laplacian in the unit ball of Rn . If

17. Blow-up

99

0 ≤ u0 ∈ L∞ (Rn ) satisfies lim inf |x|2/(p−1) u0 (x) > µ1/(p−1) ,

(17.14)

|x|→∞

then Tmax (u0 ) < ∞. Remarks 17.13. (i) Slow decay in more general domains. A similar result holds (with a different constant on the RHS of (17.14)) if the inferior limit is taken on a cone Σ, instead of the whole space (see [497]). The proof of [497], is different, based on scaling and comparison arguments. Similar blow-up conditions still hold for more general domains Σ, typically a paraboloid of the form Σ = {x = (x , xn ) ∈ Rn : xn > 0, |xn | < xβn } for some 0 < β < 1, the power 2/(p − 1) in (17.14) being then replaced by the smaller number 2β/(p − 1) (see [386], [461]). Also, similar results can be proved when Ω itself is replaced by such a domain. (ii) Sign-changing initial data with slow decay. An extension of Theorem 17.12 to sign-changing solutions has been obtained in [386]. Proof of Theorem 17.12. Assume that Tmax (u0 ) = ∞. For R > 0, denote by λ1,R the first Dirichlet eigenvalue of −∆ in the ball BR . Let ϕ1,R be the corresponding eigenfunction satisfying BR ϕ1,R dx = 1. We know that Tmax (u0 ) = ∞ implies

1/(p−1)

BR

u0 ϕ1,R dx ≤ λ1,R

.

(17.15)

Indeed, this follows from the proof of Theorem 17.1, using the fact that

ϕ1,R ∆u dx = u ∆ϕ1,R dx − u ∂ν ϕ1,R dσ ≥ −λ1,R u ϕ1,R dx. BR

BR

∂BR

BR

Set ψ := ϕ1,1 . By standard scaling properties of eigenfunctions and eigenvalues, we have ϕ1,R (x) = R−n ψ(R−1 x), x ∈ BR , and λ1,R = R−2 µ. For each ε ∈ (0, 1), (17.15) implies

1 2 − p−1 p−1 µ R ≥ u0 (x)ϕ1,R (x) dx εR c˜5 + 1 or c˜1 v(s0 ) 2L2g > c˜5 + 1 + (2 + 2ε)E(u0 ). (18.25) Then (18.24), the inequality E(v(s0 )) ≤ E(u0 ) and the identity

s vs (t) 2L2g dt, s ≥ s0 , E(v(s)) = E(v(s0 )) −

(18.26)

s0

imply

s 1 d v(s) 2L2g ≥ (2 + 2ε) vs (t) 2L2g dt + 1, s ≥ s0 . 2 ds s0 s Set f (s) := 12 s0 v(t) 2L2 dt. Then the same arguments as in the proof of Theog

rem 17.6 show that the function s → f (s)−ε is concave for s large which contradicts the assumption T = ∞. Consequently, (18.25) fails and (18.17), (18.18) are true. Notice that (18.26) and (18.17) imply

∞ vs (s) 2L2g ds ≤ C1 + C0 (18.27) 0

and (18.24), (18.17), (18.18) and Cauchy’s inequality guarantee the existence of c6 , c7 > 0 such that vs (s) 2L2g ≥ c6 v(s) 4Hg1 − c7 . (18.28) Set Λt := {s ≥ t : c6 v(s) 4H 1 > c7 + 1} and let |Λt | denote the measure of g Λt . Then |Λt | → 0 as t → ∞ due to (18.27) and (18.28). The well-posedness of (18.14) in Hg1 (see Example 51.24) guarantees the existence of η, c8 > 0 such that / Λ0 . Fix t > 0 such that |Λt | < η. v(s + τ ) Hg1 ≤ c8 whenever τ ∈ [0, η] and s ∈ Then v(s) Hg1 ≤ c8 for all s ≥ t + η, which proves (18.19).

18. Fujita-type results

107

Remark 18.5. The constant C2 in (18.19) depends on u0 Hg1 only. In fact, let Λt be the set in the proof of Lemma 18.4(iii). Since |Λ0 | < C1 + C0 due to (18.27) and (18.28), in any interval of the form [s, s + C1 + C0 ], s > 0 we can find s0 such that v(s0 ) Hg1 ≤ C3 for some C3 = C3 ( u0 Hg1 ) (and the same is true for all s0 > 0 close to zero). Due to the smoothing estimates in Example 51.24 we may also assume v(s0 ) ∞ ≤ C3 . Now √ estimate (5.26) in [440, Theo−1/(p−1) v(x/ t + 1, log(t + 1) + s0 )) guaranrem 5.3] (used with u ˜(x, t) = (t + 1) tees v(s) ∞ ≤ C4 for some C4 = C4 ( u0 Hg1 ) and all s ∈ [s0 , s0 + 2(C0 + C1 )]. Consequently, ||v|p−1 v| ≤ C4p−1 |v| and an easy estimate based on the variation-ofconstants formula guarantees v(s) Hg1 ≤ C5 for some C5 = C5 ( u0 Hg1 ) and all s ∈ [s0 , s0 + 2(C0 + C1 )]. Another proof of Theorem 18.1(i) for classical solutions. Let p ≤ pF , 0 ≤ u0 ∈ L∞ (Rn ), u0 ≡ 0, and assume that the corresponding maximal classical solution u of (18.1) is global. Similarly as in the proof of Theorem 18.3, we have (18.11), hence u(·, 2) ≥ c0 φ1 for some c0 > 0. Due to the maximum principle, the solution v of (18.14) starting at v0 := c0 φ1 exists globally. First assume p < pF . Since the solution v is global, Lemma 18.4(iii) guarantees that it is bounded in Hg1 . On the other hand, L

v(t) ≥ e−t(L−β)(c0 φ1 ) = c0 et(β−λ1 ) φ1 and β − λL 1 > 0, which yields a contradiction. L Now assume p = pF . Using (Lφ1 , φ1 )g = λL 1 (φ1 , φ1 )g and β = λ1 we obtain E(c0 φ1 ) = −

cp+1 0 p+1

which contradicts Lemma 18.4(ii).

Rn

φp+1 (y)g(y) dy < 0, 1

Remarks 18.6. (i) Alternative proof. In [299], another contradiction argument starting at v0 := c0 φ1 . was used in the case p < pF : Let v be the global solution Set ψ := bε φ1+ε where ε > 0 and b > 0 is such that ψg dy = 1. Notice that ε 1 Rn ε−1 Lψ = (1 + ε)λL |∇φ1 |2 ≤ (1 + ε)λL 1 ψ − ε(1 + ε)bε φ1 1ψ

and set f (s) := (v(s), ψ)g . Then Jensen’s inequality implies d f (s) = ds

n

R

v(y, s)p ψ(y)g(y) dy + β v(s), ψ g − v(s), Lψ g

p v(s), ψ g v(y, s)ψ(y)g(y) dy + β − (1 + ε)λL 1 Rn = f (s)p + β − (1 + ε)λL 1 f (s). ≥

108


Due to p < pF there exists ε > 0 such that β = 1/(p − 1) ≥ (1 + ε)n/2 = (1 + ε)λL 1, hence f ≥ f p , f (0) > 0, which contradicts the global existence of f . (ii) Other domains. Consider problem (15.1) in the half-space Ω = Rn+ := {x ∈ Rn : xn > 0}. Repeating the last proof of Theorem 18.3 we obtain a self˜ ˜ with the first eigenvalue λL adjoint operator L 1 = (n + 1)/2 and the corresponding 2 −|y| /4 eigenfunction φ˜1 (y) = yn e . Consequently, the problem does not possess nontrivial nonnegative global solutions if p ≤ 1 + 2/(n + 1). of Of course, instead ˜ t (x, z) = Gt (x − z) 1 − e−xn zn /t . If Ω = Gt one has to work with the kernel G (0, ∞)n , then analogous arguments show nonexistence of global positive solutions 2 for p ≤ 1 + 1/n (the first eigenfunction is y1 y2 . . . yn e−|y| /4 ). (iii) A characterization of the critical exponent. It has been observed in [363] that, for any domain Ω, the critical Fujita exponent pF = pF (Ω) can be characterized in terms of maximal decay rate of the heat semigroup. Namely, denoting a∗ := sup a > 0 : sup ta e−tA u0 ∞ < ∞ for some 0 ≤ u0 ∈ L∞ (Ω), u0 ≡ 0 , t∈(0,∞)

(18.29) there holds

1 . a∗ Indeed, if 1 1 + (1/a∗ ), by taking u0 such that a in (18.29) satisfies 1/(p − 1) < a < a∗ , we deduce from the proof of Theorem 20.2 below (with Rn replaced by Ω) that Tmax (u0 ) = ∞. (iv) Sign-changing solutions. Consider problem (18.1) with n = 1 and set Λk = {u : u has exactly k sign changes}. Then there exists a global solution of (18.1) with u0 ∈ Λk if and only if p > 1 + 2/(k + 1). In addition, if p > 1 + 2/(k + 1), then there exists a global solution of (18.1) with u0 ∈ Λk ∩ Hg1 (see [384] and [385]). Notice that 1 + 2/(k + 1) = 1 + 1/λL k+1 . We close this section with an application of Theorem 18.1 to a model arising in population genetics [211], [35]. In that model, a biological species possesses a gene existing in two allelic forms A and a, leading to the three genotypes AA, Aa and aa. It is assumed that the death rate of the individuals is determined by this particular gene, and the death rates corresponding to the genotypes AA, Aa, aa are respectively denoted by k1 , k2 , k3 . Moreover, it is assumed that k2 = k3


109

and that the genotype AA is advantageous in the sense that k1 < k2 . Denote by u : Rn × [0, ∞) → [0, 1] the relative density of the gene A at point x and time t, and set k = k2 − k1 > 0. Under suitable physical assumptions, the equation for u is then given by ut − ∆u = ku2 (1 − u),

x ∈ Rn , t > 0.

(18.30)

This equation is supplemented with the initial condition u(x, 0) = u0 (x),

x ∈ Rn ,

(18.31)

where u0 ∈ X := {φ ∈ C(Rn ) : 0 ≤ φ(x) ≤ 1, x ∈ Rn }. It follows from Remark 51.11 and the comparison principle that problem (18.30)–(18.31) admits a unique, global, classical solution u and that 0 ≤ u(x, t) ≤ 1 in Rn × (0, ∞). We have the following result [35] concerning the asymptotic behavior of solutions (our proof is a simplification of arguments in [35]). Theorem 18.7. Consider problem (18.30)–(18.31). (i) If n = 1 or 2, then u = 1 is globally stable in the following sense: For any u0 ∈ X, u0 ≡ 0, there holds lim u(x, t) = 1, t→∞

uniformly on compact subsets. (ii) If n ≥ 3, then there exist positive u0 ∈ X such that lim u(t) ∞ = 0.

t→∞

Remarks 18.8. (i) The phenomenon displayed in Theorem 18.7(i) is called the “hair-trigger effect”: Any small perturbation from the rest-state u ≡ 0 drives the solution to the equilibrium u ≡ 1, leading to the eventual extinction of the gene a. (ii) Equation (18.30) is a special case of a more general class of equations of the form ut − ∆u = f (u), where the nonlinearity satisfies f (0) = f (1) = 0, which arise in various biological models and also in flame propagation models from combustion theory. An important case is the so-called Fisher-KPP equation, corresponding to f (u) = u(1 − u). Starting with the pioneering works [211], [308], a very large amount of literature has been devoted to these problems, in particular to the existence of traveling wave solutions and to their analysis. These are solutions of the form u(x, t) = w(x1 − ct), connecting the equilibria u ≡ 0 and u ≡ 1 (i.e. w(−∞) = 1, w(+∞) = 0). See [35], [179], and e.g. [266] and the references therein for more recent results. (iii) Simple modifications of the proof of Theorem 18.7 show the following. Assume that the nonlinearity in (18.1) is replaced by any C 1 function f : [0, ∞) → [0, ∞) such that f (u) ≥ kup ,

u ∈ [0, b],

for some k, b > 0 and 1 ≤ p ≤ pF .

(18.32)

110


Then any positive solution is either nonglobal or satisfies lim inf t→∞ u(x, t) ≥ b, uniformly on compact subsets. In that sense, the Fujita-type result can be seen as an instability property of u = 0 for small perturbations, which essentially depends on the behavior of f for small u. This explains the fact that the instable range corresponds to small p’s. On the other hand, if in addition to (18.32) we assume that f is convex and satisfies the blow-up condition (17.4), then it is easy to show that any positive solution is nonglobal. For the proof of assertion (i), we need the following lemma. Lemma 18.9. For all ε > 0, there exist Rε > 0 and a function φε ∈ C 2 (B Rε ) such that 0 ≤ φε (x) ≤ 1 − ε, x ∈ BRε and vε (0, t) ≥ 1 − ε,

t ≥ 0,

(18.33)

where vε is the solution of the problem vt − ∆v = v 2 (1 − v), v = 0,

x ∈ BRε , t > 0, x ∈ ∂BRε , t > 0, x ∈ BRε .

v(x, 0) = φε (x),

⎫ ⎪ ⎬ ⎪ ⎭

Proof. Assume ε ∈ (0, 1/2) without loss of generality. Fix a nontrivial nonnegative radial function h ∈ D(Rn ) such that h(x) = 0 for |x| ≥ 1/2. Let ϕ be the classical solution of −∆ϕ = h, |x| < 1, ϕ = 0,

|x| = 1,

and observe that ϕ is positive, radial nonincreasing. Let φ(x) = φε (x) := (1 − ε)

ϕ(x/R) ≤ 1 − ε, ϕ(0)

|x| ≤ R,

where R > 0 is to be fixed. For |x| ≤ R/2, we have φ ≥ c := (2ϕ(0))−1 ϕ(1/2) > 0, hence ∆φ + φ2 (1 − φ) ≥ ∆φ + εc2 ≥ −(ϕ(0))−1 R−2 ∆ϕ ∞ + εc2 > 0 provided we take R = Rε > 0 large enough. Since ∆φ = 0 for |x| ≥ R/2, we obtain ∆φ + φ2 (1 − φ) ≥ 0,

x ∈ BRε .


111

It follows from Proposition 52.19 that ∂t vε ≥ 0, hence in particular vε (0, t) ≥ φ(0) = 1 − ε,

t ≥ 0.

Proof of Theorem 18.7(i). We may assume k = 1 without loss of generality. Step 1. Let v0 ∈ X, v0 ≡ 0, be such that v0 (x0 + ·) is radial nonincreasing for some x0 ∈ Rn , and let v be the solution of (18.30) with initial data v0 . Then v(x + x0 , t) is also radial nonincreasing (cf. Proposition 52.17). We claim that lim sup v(x0 , t) = 1. t→∞

Assume the contrary. Then there exist ε ∈ (0, 1) and T > 0 such that v(x, t) ≤ 1 − ε in Rn × [T, ∞). Consequently, w := εv satisfies wt − ∆w ≥ w2 in Rn × [T, ∞). Since 2 ≤ pF due to n ≤ 2, it follows from Theorem 18.1(i) and Remark 18.2(i) that w is nonglobal: a contradiction. Step 2. Let u0 ∈ X, u0 ≡ 0. We claim that for all ε, R > 0, there exists t0 > 0 such that u(x, t0 ) ≥ 1 − ε, |x| ≤ R. (18.34) By a time shift, we may assume without loss of generality that u0 > 0 in Rn . Therefore, for any x0 ∈ Rn , u0 dominates some nontrivial v0 ∈ X such that v0 (x0 + ·) is radial nonincreasing. If follows from Step 1 and the comparison principle that for all x0 ∈ Rn , lim sup u(x0 , t) = 1. t→∞

If u0 is radial nonincreasing, then this readily implies (18.34). The general case follows from the fact that u0 dominates some nontrivial, radial nonincreasing v0 ∈ X. Step 3. Let u0 ∈ X, u0 ≡ 0. Fix ε ∈ (0, 1) and M > 0. Let Rε , φε be given by Lemma 18.9. By Step 2, applied with R = Rε + M , there exists t0 > 0 such that u(x0 + x, t0 ) ≥ 1 − ε ≥ φε (x),

|x| ≤ Rε , |x0 | ≤ M.

By the comparison principle and (18.33), we conclude that u(x0 , t) ≥ vε (0, t) ≥ 1 − ε,

|x0 | ≤ M, t ≥ t0 .

The assertion is proved. (ii) Since 2 > pF due to n ≥ 3, this is an immediate consequence of Theorem 18.1(ii) and of the comparison principle.

112


19. Global existence for the Dirichlet problem 19.1. Small data global solutions We start with a basic result of global existence for small initial data for the problem ⎫ x ∈ Ω, t > 0, ut − ∆u = f (u), ⎪ ⎬ u = 0, x ∈ ∂Ω, t > 0, (19.1) ⎪ ⎭ x ∈ Ω. u(x, 0) = u0 (x), Definition 19.1. Assume that f (0) = 0 (so that u ≡ 0 is a solution to (19.1)) and that (19.1) is locally well-posed in a space X. We say that the zero solution is asymptotically stable in X if there exists a constant η > 0 such that, for all u0 ∈ X with u0 X ≤ η, there holds Tmax (u0 ) = ∞ and lim u(t) X = 0.

t→∞

We say that the zero solution is exponentially asymptotically stable in X if there exist constants η, µ > 0 and K ≥ 1 such that, for all u0 ∈ X with u0 X ≤ η, there holds Tmax (u0 ) = ∞ and u(t) X ≤ K u0 X e−µt ,

t > 0.

Theorem 19.2. Consider problem (19.1), where Ω is bounded and f : R → R is a C 1 -function such that f (0) = 0 and f (0) < λ1 . Then the zero solution is exponentially asymptotically stable in L∞ (Ω). Theorem 19.2 can be given a simple proof based on the comparison principle (see [296] for similar arguments). Proof. By assumption, there exist η > 0, ε ∈ (0, λ1 /2) such that |f (s)| ≤ (λ1 − 2ε)|s|,

|s| ≤ η.

(19.2)

We claim that there exists a function ϕ ∈ C 2 (Ω) such that −∆ϕ = (λ1 − ε)ϕ

and

ϕ ≥ 1,

x ∈ Ω.

(19.3)

Indeed, it suffices to consider ϕ = 1 + ψ, where ψ is the solution of −∆ψ = (λ1 − ε)ψ + (λ1 − ε), ψ = 0,

x ∈ Ω, x ∈ ∂Ω,

19. Global existence for the Dirichlet problem

113

and to note that ψ ≥ 0 by the maximum principle. Next set u(x, t) = η e−εt ϕ(x),

−1 where η = max ϕ η. Ω

An obvious computation and (19.3), (19.2) yield ut − ∆u = (λ1 − 2ε)u ≥ f (u),

x ∈ Ω, t > 0.

Assume that u0 ∞ ≤ η, hence |u0 | ≤ u(·, 0). By the comparison principle, we deduce that u ≤ u in Ω × (0, Tmax (u0 )), and we get u ≥ −u by arguing on −u. The conclusion follows. Let us now consider in more detail the case of the model problem (15.1). We would like to extend Theorem 19.2 in two directions: • unbounded domains; • Lq - instead of L∞ -stability. Note that this is a legitimate question for q > qc := n(p − 1)/2 or q = qc > 1, since we know (cf. Theorems 15.2 and 15.3 and Remark 15.4) that problem (15.1) is locally well-posed in Lq (Ω) for (and only for) such q. Domains that admit such extension can be characterized geometrically through the notion of inradius. Recall (see Appendix D) that the inradius of Ω is defined by:

ρ(Ω) = sup r > 0 : Ω contains a ball of radius r = sup dist(x, ∂Ω) x∈Ω

and that, for any q ∈ [1, ∞], the condition ρ(Ω) < ∞ is equivalent to the Poincaré inequality φ q ≤ C(Ω, q) ∇φ q , φ ∈ W01,q (Ω) (19.4) (provided Ω is uniformly smooth). The following result of [481], [483] asserts in particular that for any qc < q ≤ ∞, the zero solution is asymptotically stable in Lq (Ω) if and only if Ω has finite inradius. Theorem 19.3. Consider problem (15.1) with p > 1 and let 1 ≤ q ≤ ∞. (i) Assume q > qc or q = qc > 1. If ρ(Ω) < ∞, then the zero solution is exponentially asymptotically stable in Lq (Ω). (ii) Assume q > qc . If ρ(Ω) = ∞, then the zero solution is not asymptotically stable in Lq (Ω). More precisely, there exist initial data u0 ∈ Lq (Ω) of arbitrarily small Lq -norm such that Tmax (u0 ) < ∞.

114


Remarks 19.4. (a) Critical case. The result of Theorem 19.3(ii) is no longer true in the critical case q = qc : We shall see later that for any domain Ω (including the whole space), the zero solution is asymptotically stable in Lqc (Ω) — see Corollary 20.20 and Remark 20.21. However, the stability is exponential only if ρ(Ω) < ∞ (see [481]). (b) Different methods of proof. Theorem 19.3(i) for 1 < q < ∞ can be proved by a multiplier argument, using multiplication by powers of u and the Poincaré inequality [481]. We shall employ this method here, but for simplicity we shall prove the result only in the range 2 ≤ q < ∞ (the idea for 1 < q < 2 is the same, but some additional technical difficulties arise). An alternative proof, covering the extremal cases q = 1 and q = ∞ as well, can be carried out by using the variation-of-constants formula and the exponential decay of the heat semigroup for ρ(Ω) < ∞ (see [483]). Such arguments can be used to prove more general results of linearized stability; see Theorems 51.17, 51.19 and 51.33 in Appendix E. As an advantage, the energy proof might also apply to certain quasilinear problems. Proof of Theorem 19.3(i) for 2 ≤ q < ∞. To simplify notation, if k is any positive number, we write uk for sign(u) |u|k . Since u0 ∈ Lq (Ω), it follows from Example 51.27 in Appendix E that u ∈ C([0, ∞), Lq (Ω))∩C((0, ∞), W 2,q ∩W01,q (Ω))∩ C 1 ((0, ∞), Lq (Ω)). Multiplying the equation by uq−1 and integrating by parts, we obtain 1 d 4(q − 1) u(t) qq = uq−1 , ∆u + u q+p−1 ∇(uq/2 ) 22 + u q+p−1 q+p−1 = − q+p−1 , q dt q2 (19.5) for all t ∈ (0, Tmax (u0 )). For the last term of inequality (19.5), we next establish the estimate 2(1−θ)/q u q+p−1 ≤ C u θq ∇(uq/2 ) 2 , (19.6) with θ =1−

n(p − 1) ∈ (0, 1). 2(q + p − 1)

(19.7)

To do so, let us consider separately the cases n ≥ 3 and n ≤ 2. If n ≥ 3, since q ≥ n(p − 1)/2, we have q + p − 1 ≤ nq/(n − 2) hence, by Hölder’s inequality, u q+p−1 ≤ u θq u 1−θ nq/(n−2) , with $ θ=

1 1 − q + p − 1 nq/(n − 2)

%$

1 1 − q nq/(n − 2)

%−1 =1−

n(p − 1) , 2(q + p − 1)


115

and Sobolev’s inequality then yields (19.6). If n ≤ 2, we use the Gagliardo-Nirenberg inequalities (a+2)/2a

v a ≤ Ca v 2 and

2/a

v a ≤ Ca v 2

(a−2)/2a

v 2

1−(2/a)

∇v 2

,

v ∈ H01 (Ω)

a ≥ 2,

,

a ≥ 2,

v ∈ H01 (Ω)

(n = 1)

(n = 2).

Applying this with v = uq/2 (t) and a = 2(q + p − 1)/q > 2 yields (19.6) with θ = q/(q + p − 1) if n = 2 and θ = (2q + p − 1)/2(q + p − 1) if n = 1, that is (19.7). The next step is to use the Poincaré inequality (50.2) in W01,q (Ω) (valid due to ρ(Ω) < ∞; see Proposition 50.1 in Appendix D) to obtain a lower estimate of the first term in the right-hand side of (19.5). It follows from (50.2) that, for all α ∈ [0, 1], 2(1−α)

q/2 ∇(uq/2 ) 22 ≥ C u qq + C ∇(uq/2 ) 2α ) 2 2 ∇(u 2(1−α)

q/2 ≥ C u qq + C u qα ) 2 q ∇(u

(19.8)

.

On the other hand, one has (1 − θ)(q + p − 1)/q = n(p − 1)/2q ≤ 1. Therefore, we may choose α = 1 − (1 − θ)(q + p − 1)/q, and by combining (19.5), (19.6) and (19.8), it follows that 1 d 2(1−α) p−1 u(t) qq ≤ −C u qq + C ∇(uq/2 ) 2 u qα − C . q u q q dt It follows from this differential inequality that if u0 q is sufficiently small, then for all t > 0, d u(t) qq ≤ −C u qq , dt hence

u(t) q ≤ e−C t u0 q ,

(19.9)

as long as the solution exists. If q > qc , we know from Theorem 15.2 that the Lq -norm must blow up if Tmax (u0 ) is finite. The estimate (19.9) thus ensures global existence. If q = qc , global existence when u0 q is small follows from Corollary 20.20 below. Proof of Theorem 19.3(ii). Fix a test-function ϕ ∈ D(Rn ), ϕ ≥ 0, ϕ ≡ 0 with supp(ϕ) ⊂ B := B(0, 1), and let w be the solution of problem (15.1) with Ω replaced by B and u0 replaced by ϕ. Due to e.g. Theorem 17.1, we can assume that w blows up in a finite time T (replacing ϕ by a sufficiently large multiple).

116


Now, since ρ(Ω) = ∞, Ω contains some ball Bk = B(xk , k) for any integer k ≥ 1. Let us set uk (x, t) := k −2/(p−1) w(k −1 (x − xk ), k −2 t),

u0,k (x) := k −2/(p−1) ϕ(k −1 (x − xk )).

Due to the invariance of the equation under this scaling, it is easily verified that uk solves the problem ⎫ x ∈ Bk , 0 < t < k 2 T, ∂t uk − ∆uk = |uk |p−1 uk , ⎪ ⎬ 2 uk = 0, x ∈ ∂Bk , 0 < t < k T, ⎪ ⎭ x ∈ Bk . uk (x, 0) = u0,k (x), Let u ˜k be the solution of problem (15.1) with u0 = u0,k . Since each Bk is included in Ω and u˜k ≥ 0 on ∂Bk , it follows from the comparison principle that u˜k ≥ uk , hence u ˜k blows up in finite time. Last, an easy calculation yields u0,k q = k −2/(p−1)+n/q ϕ q → 0,

k → ∞,

which concludes the proof. We shall now describe the potential well method. It will enable us to obtain alternative sufficient conditions for global existence (and nonexistence) for the model problem (15.1). In the rest of this subsection we assume Ω bounded and 1 < p ≤ pS . Recall that the energy functional E is given by

1 1 2 E(u) = |∇u| dx − |u|p+1 dx, u ∈ H01 (Ω). (19.10) 2 Ω p+1 Ω We define the Nehari functional I by

2 |∇u| dx − |u|p+1 dx, I(u) = Ω

Ω

u ∈ H01 (Ω).

The potential well associated with problem (15.1) is the set

W := u ∈ H01 (Ω) : E(u) < d, I(u) > 0 ∪ {0}, where d, the depth of the potential well, is defined by

d := inf E(u) : u ∈ H01 (Ω) \ {0}, I(u) = 0 .

(19.11)

We shall show in Lemma 19.7(i) below that d=

p − 1 2(p+1)/(p−1) Λ , 2(p + 1)

(19.12)


117

where Λ = Λp+1 (Ω) denotes the best constant in the Sobolev embedding H01 (Ω) → Lp+1 (Ω), i.e., ∇u 2 : u ∈ H01 (Ω), u = 0 . (19.13) Λ := inf u p+1 The exterior of the potential well is the set

Z := u ∈ H01 (Ω) : E(u) < d, I(u) < 0 . In what follows, for u0 ∈ H01 (Ω), u denotes the maximal Lp+1 -classical solution of problem (15.1) (recall from Section 15 that (15.1) is well-posed in Lp+1 (Ω), since p + 1 ≥ qc due to p ≤ pS ). Theorem 19.5. Consider problem (15.1) with Ω bounded. (i) Assume 1 0,

and u(t) ∞ → 0,

t → ∞.

(19.14)

(ii) Assume 1 < p ≤ pS . If u0 ∈ Z, then Tmax (u0 ) < ∞. The potential well method was introduced in [467] to obtain global existence results for nonlinear hyperbolic equations. As for parabolic problems, the global existence part in Theorem 19.5(i) is due to [515] and the decay property is essentially from [290]. Theorem 19.5(ii) is due to [409] (see also [290]), where the potential well method was extended to obtain nonexistence results for hyperbolic and parabolic problems. Remarks 19.6. (a) Theorem 19.5(i) provides in particular a sufficient smallness condition on u0 for global existence when √ p < pS . Indeed, we have u0 ∈ W whenever u0 ∈ H01 (Ω) satisfies ∇u0 < 2d (cf. Lemma 19.7(iii)). (b) The quantity d can be interpreted as a mountain-pass energy (cf. Section 7). Indeed, for p ≤ pS , it is easy to show that d=

inf

u∈H01 (Ω)\{0}

max E(su). s≥0

Note that for p < pS , there exist least-energy stationary solutions v, i.e.: such that E(v) = d (this follows from Theorem 7.2, applied with u0 = 0 and u1 such that E(u1 ) < 0, and from the easy fact that d = β, where β is defined in (7.1)). (c) The sets W and Z are invariant under the semiflow associated with problem (15.1) for p ≤ pS . This follows from the proof of Theorem 19.5. (d) Theorem 19.5 admits a converse (cf. [287]). Namely, if p ≤ pS and u is a global solution satisfying (19.14), then u(t) ∈ W for large t. If p < pS and u is a

118


blowing-up solution, then u(t) ∈ Z for t close to Tmax (u0 ). These facts respectively follow from smoothing effects and Theorem 19.5(ii), and from property (22.28) in Proposition 22.11 and I(u) ≤ 2E(u). (e) Theorems 17.6 and 19.5 give an essentially complete characterization of global existence/nonexistence in the subcritical range for initial data with energy less than d. See [236] and the references therein for additional information, including some partial results for higher energy data. It seems an open question whether or not I(u0 ) < 0 is a sufficient condition for blow-up. In view of the proof of Theorem 19.5, we need the following properties of the potential well. Lemma 19.7. Let Ω be bounded and let 1 0, there holds

dε := inf E(u) : u ∈ H01 (Ω), I(u) = −ε ≥ d −

ε . p+1

(19.15)

(iii) For all u ∈ H01 (Ω), we have ∇u 2
0, we may assume that I(uj ) = 0. Therefore

Ω

|∇uj |2 dx =

Ω

|uj |p+1 dx = (Λ + ηj )−(p+1)

Ω

(p+1)/2 |∇uj |2 dx ,


119

where ηj → 0+. Combining this with (19.17) for ε = 0, we obtain E(uj ) =

p−1 (Λ + ηj )2(p+1)/(p−1) → D, 2(p + 1)

hence d = D, i.e. (19.12). If p < pS , then the infimum in (19.13) is attained for some v ∈ H01 (Ω), due to the compactness of the embedding H01 (Ω) → Lp+1 (Ω). Arguing similarly as above, with uj replaced by v, we see that the infimum in (19.11) is also attained. Assertion (ii) follows from (19.18).

√ Finally, let us prove assertion (iii). 0 < ∇u 2 < 2d. Then E(u) < d. √ Assume Next, using (19.13) and ∇u 2 < 2d < Λ(p+1)/(p−1) , we obtain

|u|

p+1

Ω

dx ≤ Λ

−(p+1)

Ω

(p+1)/2

|∇u| dx < |∇u|2 dx. 2

Ω

Consequently I(u) > 0, hence u ∈ W . On the other hand, for any u ∈ W , the conditions E(u) < d and I(u) ≥ 0 imply p−1 2(p + 1)

Ω

|∇u|2 dx ≤ E(u) < d,

hence the last inequality in (19.16). Proof of Theorem 19.5. Set T := Tmax (u0 ). By (17.7), we have E(t) ≤ E(u0 ) < d,

t ∈ [0, T ).

(19.19)

(i) If u(t) = 0 for some t ≥ 0, then by uniqueness, u(s) = 0 for all s ≥ t and the conclusion is true. Hence we may assume that u(t) = 0 for all t ∈ [0, T ). Since I(u0 ) > 0, using (19.11) and (19.19), it follows by continuity that, for all t ∈ [0, T ), I(u(t)) > 0, hence u(t) ∈ W . By Lemma 19.7(iii), we deduce that u(t) is bounded in H01 (Ω), hence in Lp+1 (Ω). Remarks 16.2 then guarantee that T = ∞. On the other hand, by Example 53.7 (and in particular the existence of a strict Lyapunov functional given by (19.10)), the ω-limit set ω(u0 ) in the H01 (Ω)-topology is nonempty and consists of (classical) equilibria. But for any nontrivial equilibrium v, we have I(v) = 0, hence E(v) ≥ d by (19.11). Consequently v ∈ ω(u0 ) in view of (19.19). In other words, limt→∞ u(t) 1,2 = 0, hence limt→∞ u(t) p+1 = 0. By the smoothing estimate (15.2), this guarantees (19.14). (ii) Fix ε > 0 such that ε < min −I(u0 ), d − E(u0 ) .

120


By (19.15) and (19.19), we have E(t) ≤ E(u0 ) < dε for t ∈ [0, T ). Since I(u0 ) < −ε, using the definition of dε in (19.15), it follows by continuity that I(u(t)) < −ε for all t ∈ [0, T ), hence

1 d u2 dx = −I(u(t)) > ε (19.20) 2 dt Ω (cf. (17.9)). But on the other hand, we know from Remark 17.7(ii) that T = ∞ implies supt≥0 u(t) 2 < ∞. In view of (19.20), we conclude that T < ∞.

19.2. Structure of global solutions in bounded domains In this subsection we study some properties of the set of initial data giving rise to global solutions of problem (19.1). Throughout this subsection we assume that the domain Ω is bounded. We define the sets

G = u0 ∈ L∞ (Ω) : Tmax (u0 ) = ∞ ,

B = u0 ∈ L∞ (Ω) : Tmax (u0 ) = ∞ and sup u(t) ∞ < ∞ , t≥0

and

D = u0 ∈ L∞ (Ω) : Tmax (u0 ) = ∞ and u(t) ∞ → 0, t → ∞

(D is the domain of attraction of 0). When (19.1) admits both global and nonglobal solutions, these sets are natural and interesting objects. Clearly, D ⊂ B ⊂ G. In order to describe the properties of these sets, we first need some properties of the steady states of (19.1), i.e. (classical) solutions of −∆u = f (u),

x ∈ Ω,

u = 0,

x ∈ ∂Ω.

(19.21)

The following result implies in particular that two ordered positive steady states cannot exist when the nonlinearity is strictly convex. Note that the result fails in general if Ω = Rn (with u, v → 0 at infinity), as shown by Theorem 9.1 with p ≥ pJL . Proposition 19.8. Assume Ω bounded and let f : R → R be a strictly convex C 1 -function, with f (0) = 0. Assume that u, v ∈ C 2 (Ω) ∩ C 1 (Ω) are respectively sub- and supersolutions to (19.21), in the sense that −∆u ≤ f (u),

x ∈ Ω,

−∆v ≥ f (v), u = v = 0,

x ∈ Ω, x ∈ ∂Ω.

⎫ ⎪ ⎬ ⎪ ⎭

(19.22)


121

If v ≥ u > 0 in Ω, then u ≡ v. Proof. Multiplying the inequalities in (19.22) by v and u respectively, we obtain

Ω

f (u)v dx ≥

hence

(−∆u)v dx =

Ω

Ω

∇u · ∇v dx =

Ω

(−∆v)u dx ≥

f (v)u dx, Ω

f (u) f (v) uv dx ≥ 0. − u v Ω

But in view of the strict convexity of f , the integrand is nonpositive in Ω, and (strictly) negative at each x such that v(x) > u(x). The conclusion follows. The next result describes some basic geometrical and topological properties of the sets D, B, G. Here we refer to the L∞ -topology (but other choices are possible). Also, for a given convex subset K of a vector space, we recall that x ∈ K is called an extremal point if it cannot be written under the form x = θy + (1 − θ)z with y, z ∈ K and θ ∈ (0, 1). Theorem 19.9 is due to [342]. However the present proof of assertion (iv) is different and simpler than that in [342]. Theorem 19.9. Consider problem (19.1) where Ω is bounded and f : R → R is a C 1 -function, with f (0) = f (0) = 0. (i) Then D is an open neighborhood of 0. (ii) Assume that f is convex. Then the sets G, B and D are convex. Now assume that f is strictly convex. (iii) If u0 is not an extremal point of G (resp., of B), then u0 is an interior point. This is true in particular if 0 ≤ u0 ≤ v0 , with v0 ∈ G (resp., B), v0 ≡ u0 . (iv) There holds int(B) = D. Proof. (i) This is a consequence of Theorem 19.2 and of the continuous dependence of solutions on initial values. (ii) Let θ ∈ (0, 1), u0 , v0 ∈ L∞ (Ω), u0 ≡ v0 , w0 = θu0 + (1 − θ)v0 , and denote by u, v, w the solutions of (19.1) with initial data u0 , v0 , w0 , respectively. Set w = θu+ (1 − θ)v. By the convexity of f , for all x ∈ Ω and t ∈ (0, min(Tmax (u0 ), Tmax (v0 ))), we have w t − ∆w = θf (u) + (1 − θ)f (v) ≥ f (θu + (1 − θ)v) = f (w),

(19.23)

hence w ≤ θu + (1 − θ)v,

(19.24)

in view of the comparison principle. On the other hand, the assumptions on f imply f (s) ≥ 0, s ∈ R. Denoting by e−tA the heat semigroup in Ω with homogeneous

122


Dirichlet boundary conditions, the maximum principle and Proposition 48.5 in Appendix B imply w ≥ e−tA w0 ≥ −Ce−λ1 t . (19.25) It then follows from (19.24) and (19.25) that w0 ∈ G (resp., B, D) whenever u0 , v0 ∈ G (resp., B, D) and the convexity assertion is proved. Assume now that f is strictly convex. (iii) Let w0 be a nonextremal point of B, i.e. w0 = θu0 +(1−θ)v0 , with θ ∈ (0, 1), u0 , v0 ∈ B, u0 ≡ v0 . Then, by continuity and the strict convexity of f , we have θf (u(·, t)) + (1 − θ)f (v(·, t)) ≡ f (w(·, t)) for t ∈ [0, τ ], with τ > 0 small. By (19.23) and the strong maximum principle, we deduce that w(x, τ ) < w(x, τ ) in Ω

and

∂w ∂w (x, τ ) > (x, τ ) on ∂Ω. ∂ν ∂ν

Due to (51.28), we know that for small τ > 0, the map L∞ (Ω) u ˜0 → u(·, τ ; u˜0 ) ∈ C 1 (Ω) is well-defined and continuous on a neighborhood of u0 . Therefore, if w0 − u ˜0 ∞ is small enough, then Tmax (˜ u0 ) > τ and u ˜(τ ) ≤ w(τ ), where u ˜ := u(·; u ˜0 ). This, along with u ˜(t) ≥ e−tA u ˜0 guarantees that u ˜0 ∈ B and w0 is an interior point. The same argument applies for G. To justify the last part of assertion (iii), write u0 = θv0 + (1 − θ)˜ v0 , with v˜0 := (1 − θ)−1 (u0 − θv0 ) ≡ v0 . For θ > 0 small, we have v0 ≥ v˜0 ≥ −(1 − θ)−1 θv0 ∈ D due to Theorem 19.2, hence v˜0 ∈ B (resp., G) by comparison. (iv) Let u0 ∈ int(B). In particular, there exists v0 ∈ B, with u0 ≤ v0 , u0 ≡ v0 . Denote by u, v the solutions of (19.1) with initial data u0 , v0 , respectively. Due to Example 53.7, we know that uniformly bounded solutions are relatively compact in X := H 1 ∩ C0 (Ω) for t ≥ 1 and that the ω-limit set ω(u0 ) (in the X-topology) is nonempty and consists of equilibria. Let z ∈ ω(u0 ). By definition, there exists a sequence tk → ∞, such that u(tk ) → z in X. Since {v(t) : t ≥ 1} is precompact in X, there exist z˜ ∈ ω(v0 ) and a subsequence tkj such that v(tkj ) → z˜ in X. By Lemma 17.9, there exist τ > 0 and α > 1, such that v ≥ αu for t ≥ τ > 0, hence z˜ ≥ αz.

(19.26)

Due to f ≥ 0, we have z ≥ 0 by the maximum principle. Since z, z˜ are steady states of (19.1), we then deduce that z ≡ 0, because otherwise (19.26) would contradict Proposition 19.8. Consequently, u0 ∈ D. In particular, int(B) ⊂ D. Conversely, it is clear that D ⊂ int(B) since D is open. Remark 19.10. Instability of positive equilibria. Theorem 19.9 shows that the only way for the ω-limit set of a global bounded solution of (19.1) to contain positive equilibria is to have u(t) be an extremal (or boundary) point of B for all t ≥ 0. By the same token, if v is a positive steady state and u0 ≥ v, u0 ≡ v, then u0 ∈ B. In the special case f (u) = |u|p , we have the following stronger property (which generalizes Theorem 17.8).


123

Proposition 19.11. Consider problem (19.1) with Ω bounded and f (u) = |u|p , p > 1. Assume that v0 ∈ B \ D and that u0 ≥ v0 , u0 ≡ v0 . Then Tmax (u0 ) < ∞. Proof. Let u, v be the corresponding solutions and assume for contradiction that u is global. Then, by Lemma 17.9, we have u(t) ≥ αv(t), for all t ≥ 1 and some α > 1. On the other hand, by Example 53.7, ω(v0 ) contains a nonzero steady state z and there exists a sequence tk → ∞ such that v(tk ) → z in C 1 (Ω). Moreover, since v(tk ) ≥ e−tA v0 , we have z ≥ 0 in Ω, hence ∂z/∂ν < 0 on ∂Ω by the Hopf maximum principle. It follows that for large k, v(tk ) > α−1 z in Ω. Consequently, u(tk ) > z, contradicting Theorem 17.8. Remark 19.12. Further properties of D, B, G. Consider problem (19.1) with f (u) = |u|p and p > 1, and let us restrict ourselves to nonnegative initial data. Let us define G + := {u ∈ G : u ≥ 0} and B + , D+ similarly. We first note that the set D+ is unbounded, due to the existence of nonnegative global classical solutions such that u(t) ∞ → ∞ as t → 0+. Indeed, since 0 is an asymptotically stable solution of problem (15.1) in Lq for q ≥ qc by Theorem 19.3, this occurs for any 0 ≤ u0 ∈ Lq (Ω) \ L∞ (Ω) with u0 q small enough. On the other hand, for p < pS , it follows from Theorem 6.2 that B + = D+ . Further related results will be obtained later, in particular in Subsection 28.4, where we study the transition between global existence and blow-up along each ray of nonnegative initial data starting from 0. Among the consequences of these results, let us mention the following properties: (a) if p < pS , then G + = B + = D+ and G + is a closed subset of L∞ (Ω) (cf. Theorem 22.1); (b) if p ≥ pS and Ω is starshaped, then B + = D+ (cf. Corollary 5.2 and Theorem 28.7(iv)); (c) if p = pS and Ω is a ball, then G + = B + = D+ (cf. Theorem 28.7); (d) if p > pS , Ω is a ball and we consider radial solutions only, then G + = B = D+ (see Theorem 22.4). +

Remark 19.13. Stabilization towards an equilibrium. In the proofs of Theorem 19.9 and Proposition 19.11 we used the fact that the ω-limit set of any global bounded solution of (19.1) in a bounded domain Ω is a nonempty compact connected set consisting of equilibria. If all equilibria (at a given energy level) are isolated, then this fact guarantees that each global bounded solution converges to a single equilibrium. If n = 1 or if we consider radial solutions in a ball, then this convergence is true without any information on the set of equilibria, see [543], [353], [265], [102], [268], [127]. Similar stabilization result is true for general bounded domains in Rn , n ≥ 1, provided the nonlinearity is analytic, see [474], [291]. Other sufficient conditions can be found in the survey article [421]. Nonconvergent global bounded solutions were constructed in [426] and [427] for spatially inhomogeneous nonlinearities of the form f = f (x, u).

124


Remark 19.14. Global solutions and very weak stationary solutions. Consider problem (14.1), where f : [0, ∞) → (0, ∞) is a C 1 nondecreasing convex function satisfying the blow-up condition (17.4). It was shown in [94] that strong relations exist between the existence of global (classical) solutions of (14.1) and the existence of very weak solutions of the stationary problem (13.1) (throughout this remark, “solution” implicitly means “nonnegative solution”): (i) if Tmax (u0 ) = ∞ for some 0 ≤ u0 ∈ L∞ (Ω), then (13.1) admits a very weak solution; (ii) conversely, if (13.1) admits a very weak solution v, then for any u0 ∈ L∞ (Ω) with 0 ≤ u0 ≤ v, we have Tmax (u0 ) = ∞. Note that (i) provides a further blow-up criterion: if (13.1) has no very weak solution, then all solutions of (14.1) have to blow up in finite time. As for (ii), it gives a new sufficient condition for global existence (see Theorem 20.5 for a related result concerning the Cauchy problem). Assertion (i) is not immediate since no bound is assumed on u. As for assertion (ii), it would be a direct consequence of the comparison principle if we were assuming v ∈ L∞ (Ω), but it is far from obvious in general since the inequality u ≤ v in itself does not a priori prevent u(t) ∞ from blowing up in finite time. On the other hand, the existence of a global solution of (14.1) does not in general imply the existence of a classical steady state. In fact, there are situations where (13.1) has a singular (very weak) solution but no classical solution (see Remark 3.7) and where (14.1) admits global unbounded solutions which stabilize to a singular solution as t → ∞ (see Remark 22.6(b)). The idea of the proof of assertion (i) is as follows. Assume u0 = 0 without loss of generality (u is then also global by the comparison principle). Since ut ≥ 0 by Proposition 52.19, we may let v(x) := limt→∞ u(x, t) ≤ ∞. Theorem 17.3 implies u(t)ϕ dx ≤ C for t ≥ 0. Integrating (17.5) in time between t and t + 1 and 1 Ω using ut ≥ 0, it follows that

t+1

f (u(t))ϕ1 dx ≤ f (u)ϕ1 dx ds Ω

t

= λ1 t

Ω t+1

Ω

uϕ1 dx ds +

Ω

t+1 u(s)ϕ1 dx ≤ (1 + λ1 )C. t

Let now Θ ∈ C 2 (Ω), Θ ≥ 0, be the classical solution of the problem −∆Θ = 1 in Ω, Θ=0 on ∂Ω.

(19.27)

Multiplying the equation in (14.1) by the function Θ defined in (19.27), integrating over Ω × (t, t + 1), and using ut ≥ 0, we obtain

t+1

t+1

t+1 u(t) dx ≤ u dx ds = f (u)Θ dx ds − u(s)Θ dx ≤ C. Ω

t

Ω

t

Ω

Ω

t


125

In particular, we get f (v) ∈ L1δ (Ω) and v ∈ L1 (Ω). By arguing similarly as in the (alternative) proof of Lemma 53.10, using again ut ≥ 0, we then easily conclude that v is a very weak solution of (13.1). The proof of assertion (ii) is more delicate and will not be given here. It is based on a perturbation argument which relies on a variant of Lemma 27.4 and on Lemma 27.5 below (used in the study of complete blow-up).

19.3. Diffusion eliminating blow-up In Section 17, we used the convexity of the function f (u) = λu + up , u > 0, in order to prove blow-up of solutions of (17.1) for suitable initial data. On the other hand, it follows from Theorem 19.2 that any solution of (17.1) with Ω bounded, λ < λ1 and u0 small does exist globally and tends to zero as t → ∞. A similar assertion is true for Ω = Rn if, for example, λ = 0 and p > pF (see Theorem 18.1). Since all positive solutions of the ODE U = U p blow up in finite time, we see that diffusion and the Dirichlet boundary conditions (or just the diffusion if Ω = Rn ) can prevent blow-up for some initial data. Next we show that for some particular nonlinearities f , diffusion with the Dirichlet boundary condition can completely eliminate blow-up. This result (and its modification for unbounded domains) is due to [197]. Hence, let f : [0, ∞) → [0, ∞) be smooth, f (u) > 0 for u > 0 and consider the ODE Ut = f (U ), t > 0, (19.28) U (0) = U0 , where U0 > 0, and the related Cauchy-Dirichlet problem ut − d∆u = f (u), u = 0, u(x, 0) = u0 (x),

x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0, x ∈ Ω,

⎫ ⎪ ⎬ ⎪ ⎭

(19.29)

where Ω ⊂ Rn is a bounded domain, d > 0 and u0 ≥ 0 is a an L∞ -function. It is well known that condition (17.4) is sufficient and necessary for blow-up of the solution of (19.28), and we have seen in Theorem 17.3 that if f satisfies (17.4) and is convex, then the solution of (19.29) blows up for large initial data. We will prove that there exist (nonconvex) f satisfying (17.4) such that (19.29) possesses a global and bounded solution for any u0 and any d > 0. Theorem 19.15. There is a C ∞ -function f : [0, ∞) → [0, ∞), f (u) > 0 for u > 0, such that the following holds: (i) All solutions of (19.28) with U (0) > 0 blow up in finite time.

126


(ii) If Ω is bounded and d > 0, then all solutions of (19.29) with 0 ≤ u0 ∈ L∞ (Ω) exist and remain bounded for all t ≥ 0. Of course Theorem 19.15 cannot be true in the case of Neumann boundary conditions uν = 0, since solutions of (19.28) also solve the PDE. On the other hand, Theorem 19.15 remains true in the case of Robin boundary conditions κuν + (1 − κ)u = 0, κ ∈ (0, 1), see [196]. The idea of the construction of the function f for Theorem 19.15 is to start with a typical blow-up function satisfying (17.4), like f (u) = cup , with c > 0, p > 1, and then to modify it in an infinite number of intervals Ik = (ak , bk ) with ak < bk < ak+1 , ak → ∞. The modified function will be small enough in subintervals of Ik in order to provide us with suitable supersolutions of (19.29) but it will still satisfy condition (17.4). Lemma 19.16. Let {ak } be an increasing sequence, a1 ≥ 1, limk→∞ ak = ∞. Then there are a C ∞ function f : [0, ∞) → [0, ∞) with f (u) > 0 for u > 0, and a sequence {bk } such that ak < bk < ak+1 , (19.30)

∞ du < ∞, (19.31) f (u) 1

bk du ! ≥ k, (19.32) F (bk ) − F (u) ak for k = 1, 2, . . . , where F = f . Proof. Take any C 1 -function g : [0, ∞) → [0, ∞), with g(u) > 0 for u > 0, such that

∞ ds < ∞, g(s) ≥ 1 for s ≥ 1. g(s) 1 Choose also a positive sequence {βk } such that βk < ∞, βk < k 2 , 2βk2 g(ak )k −2 < ak+1 − ak , k

and define γk := 1 − βk k −2 > 0,

bk := ak + βk2 g(ak )k −2 < ak+1 .

We will also choose sequences {ck } and {dk } (ak < bk < ck < ak+1 , dk > 0) specified later. Then, we construct an auxiliary function g' by modifying the function g on the intervals on [ak , bk ] and [bk , ck ] in the following way (see Figure 8) ⎧ g(ak ) − dk ⎪ dk + (bk − u)γk for ak ≤ u ≤ bk , ⎪ ⎪ ⎪ (bk − ak )γk ⎨ g'(u) = ⎪ ⎪ ⎪ ⎪ ⎩ dk + g(ck ) − dk (u − bk ) for bk ≤ u ≤ ck . ck − b k


127

g˜(u)

0

ak

b k ck

ak+1

bk+1 ck+1 u

Figure 8: Graph of ' g. Set also ' G(u) = dk (u − bk ) −

g(ak ) − dk (bk − u)γk +1 (γk + 1)(bk − ak )γk

' =' on the interval [ak , bk ]. Then G g and

bk du & ak ' ' k ) − G(u) G(b %−1/2

bk $ g(ak ) − dk γk +1 dk (bk − u) + (bk − u) du = (γk + 1)(bk − ak )γk ak %−1/2

bk $ g(ak ) γk +1 (bk − u) du → (γk + 1)(bk − ak )γk ak as dk → 0. Thus we obtain (

bk du (γk + 1)(bk − ak ) 2 & . = lim dk →0 a 1 − γ g(ak ) k ' ' k G(bk ) − G(u) We choose dk ∈ (0, 1/2) small enough so that (

bk du b k − ak 1 & = k. ≥ 1 − γk g(ak ) ak ' ' k ) − G(u) G(b

(19.33)

128


Using g(s) ≥ 1 > 2dk > 0 for s ≥ 1 we obtain %−1

bk $

bk du g(ak ) − dk γk = dk + (bk − u) du '(u) (bk − ak )γk ak g ak %−1

bk $ g(ak ) − dk γk (bk − u) du ≤ (bk − ak )γk ak g(ak )βk b k − ak = ≤ 2βk , = (1 − γk )(g(ak ) − dk ) g(ak ) − dk and

ck

bk

$ %−1 g(ck ) − dk dk + (u − bk ) du ck − b k bk g(ck ) g(ck ) ck − b k log ≤ 2(ck − bk ) log ≤ βk = g(ck ) − dk dk dk

du = g'(u)

ck

provided ck ∈ (bk , ak+1 ) is sufficiently close to bk . The above estimates imply

ck du ≤ 3βk . g ' (u) ak This inequality and (19.33) guarantee that (19.31) and (19.32) are satisfied for g'. Take a C ∞ -function f such that 1 g(u) ≤ f (u) ≤ g'(u). ' 2

(19.34)

We can easily check (19.31). Since

bk f (s)ds = F (bk ) − F (u), u

we have

' k ) − G(u), ' F (bk ) − F (u) ≤ G(b

by integrating the second inequality in (19.34) over [bk , u]. This guarantees that f also satisfies (19.32). The existence of supersolutions immediately follows from the previous lemma. Lemma 19.17. Let f be as in Lemma 19.16 and d, L > 0. Then for sufficiently large k there is a solution uk of d(uk )xx + f (uk ) = 0 (uk )x (0) = 0,

for − L < x < L,

uk (x) ≥ ak

for − L < x < L.

(19.35) (19.36)

20. Global existence for the Cauchy problem

129

Proof. Since the solution of the initial value problem d(uk )xx + f (uk ) = 0, (uk )x (0) = 0, is given by

bk

uk (x)

uk (0) = bk ,

du ! = F (bk ) − F (u)

)

2 |x|, d

the assertion follows from (19.32). Proof of Theorem 19.15. Since Ω is bounded, we may choose L > 0 such that {x1 | x = (x1 , x ˆ) ∈ Ω} ⊂ [−L, L]. ∞

Let u0 ∈ L (Ω), u0 ≥ 0, and let u be the solution of (19.29). For large enough positive integer k, the function uk defined in Lemma 19.17 becomes a supersolution of (19.29) and we have u0 (x) < ak ≤ uk (x1 ),

x ∈ Ω.

Since there is no problem in comparing the data on the lateral boundary, the comparison principle thus implies u(x, t) ≤ uk (x1 ) for t ∈ (0, Tmax(u0 )), hence Tmax (u0 ) = ∞.

20. Global existence for the Cauchy problem 20.1. Small data global solutions As announced in Section 18 (cf. Theorem 18.1(ii)), we show that, when p > pF , small positive initial data yield global solutions. A simple example is provided by data dominated by a small multiple of a Gaussian, in which case the solution remains controlled by the heat kernel. In all this section we use the notation (Gt )t>0 set in (48.5). Theorem 20.1. Consider problem (18.1) with p > pF , u0 ∈ L∞ (Rn ), and let γ > 0. There exists ε = ε(γ) > 0 such that, if 0 ≤ u0 (x) ≤ εGγ (x),

x ∈ Rn ,

(20.1)

x ∈ Rn , t > 0

(20.2)

then Tmax (u0 ) = ∞ and u satisfies u(x, t) ≤ CGt+γ (x), for some C = C(γ) > 0. Theorem 20.1 is due to [220], where it was obtained by a contraction mapping argument. Here we shall derive Theorem 20.1 as a consequence of a more general criterion on u0 for global existence, due to [530].

130


Theorem 20.2. Consider problem (18.1) with p > pF . Assume that 0 ≤ u0 ∈ L∞ (Rn ) satisfies

∞ e−sA u0 p−1 (20.3) ∞ ds < 1/(p − 1). 0

Then Tmax (u0 ) = ∞ and u behaves like the solution of the linear part of the equation, up to multiplicative constants, i.e.:

e−tA u0 (x) ≤ u(x, t) ≤ C e−tA u0 (x),

x ∈ Rn , t > 0,

(20.4)

for some C > 1 (depending on u0 ). Remarks 20.3. (a) Inequality (20.2) corresponds in a sense to the minimal growth in time and space for positive solutions. Indeed, any positive solution u of (18.1) satisfies u(x, t + τ ) ≥ cGt+α (x), x ∈ Rn , t > 0 for some τ, α, c > 0 (this follows from the argument preceding formula (18.12)). (b) Since (20.5) e−tA u0 ∞ ≥ ct−n/2 , t → ∞ for any nontrivial u0 ≥ 0, it follows that condition (20.3) cannot be satisfied for p ≤ pF . (c) A different smallness condition on u0 ensuring global existence appears in Corollary 20.20 below. (d) The constant C in (20.4) can be explicitly computed from the proof below. In particular C converges to 1 as the LHS of (20.3) goes to 0. Proof of Theorem 20.2. We look for a supersolution of the form u(x, t) = h(t) e−tA u0 (x), where h(t) =

x ∈ Rn , t > 0,

%−1/(p−1) $

t e−sA u0 p−1 ds . 1 − (p − 1) ∞ 0

Since

h (t) = e

−tA

u0 p−1 ∞

$ %−1/(p−1)−1

t −sA p−1 1 − (p − 1) e u0 ∞ ds 0

p = e−tA u0 p−1 ∞ h (t),

it follows that p −tA ut = h(t) e−tA u0 t +h (t)e−tA u0 = ∆u + e−tA u0 p−1 u0 ≥ ∆u + up . ∞ h (t)e


131

Since u(x, 0) = u0 (x), we infer from the comparison principle that 0 ≤ u(x, t) ≤ u(x, t),

x ∈ Rn , t < Tmax (u0 ).

The conclusion follows. Proof of Theorem 20.1. By (48.6) we have e−tA Gγ = Gt ∗ Gγ = Gt+γ . Since −n(p−1)/2 and n(p − 1)/2 > 1, we deduce that (20.3) is Gt+γ p−1 ∞ = (4π(t + γ)) satisfied with strict inequality for ε > 0 small. The conclusion then follows from Theorem 20.2. Remarks 20.4. (i) Global existence under assumption (20.1) can be shown by a simpler comparison argument, by looking for a supersolution of the form v(x, t) = ηtα G(x, t), where α, η > 0. Using ∂t G − ∆G = 0, we obtain vt − ∆v − v p = ηαtα−1 G − η p tαp Gp 2 = ηtα−1 α − η p−1 t1+α(p−1)−n(p−1)/2 e−(p−1)|x| /(4t) G ≥ 0, provided we choose α = (n/2) − 1/(p − 1) > 0 and η = α1/(p−1) . It then suffices to compare u with v(x, t + γ). However, this argument does not yield estimate (20.2) nor the sharp decay rate in t−n/2 . (ii) Let Ω ⊂ Rn be an arbitrary domain and e−tA denote the Dirichlet heat semigroup in Ω. Then, for any p > 1 and u0 satisfying condition (20.3) (with 0 ≤ u0 ∈ L∞ (Ω), say), problem (15.1) has a unique global nonnegative (mild) solution and estimate (20.4) is true for x ∈ Ω and t > 0. Indeed, the local in time solution u is constructed by the Banach fixed point theorem as a limit of iterations uk+1 = Φu0 (uk ), u1 (t) ≡ 0 (cf. (15.12)). But one easily shows that the function u(t) in the proof of Theorem 20.2 satisfies u ≥ Φu0 (u). By induction, it follows that u ≥ uk . (iii) If p ≥ pS , then problem (18.1) possesses positive stationary solutions (see Theorem 9.1). If pF < p and p(n − 4) < n, then the existence of global positive solutions of (18.1) with exponentially decaying initial data also follows from Example 51.24 (the zero solution of (18.14) is exponentially stable), cf. Proposition 20.13 and Remark 20.14(ii) below. When n ≥ 3 and p > psg , we have a simple global existence criterion, for solutions starting below the singular steady state (cf. [232, Theorem 10.4], where a more general result is proved). Theorem 20.5. Consider problem (18.1) with n ≥ 3, p > psg , and u0 ∈ L∞ (Rn ). Assume that |u0 | ≤ u∗ in Rn \ {0}, where u∗ (x) := U∗ (|x|) is defined in (3.9). Then Tmax (u0 ) = ∞. Proof. The proof is based on the strong maximum principle, along with a spaceshift argument. Assume for contradiction that T := Tmax (u0 ) < ∞.

132


We first claim that x = 0, 0 < t < T.

u(x, t) ≤ u∗ (x),

(20.6)

Fix 0 < τ < T . Since u is bounded in Rn ×(0, τ ), there exists ε > 0 such that u ≤ u∗ in {0 < |x| ≤ ε} × (0, τ ). By applying the comparison principle in the domain {|x| > ε}, it follows that u ≤ u∗ in (Rn \ {0}) × (0, τ ), hence (20.6). In particular, by parabolic estimates, u extends to a continuous function in (Rn \ {0}) × (0, T ]. Now fix 0 < t0 < T . There exist a, ε > 0 such that u(x, t0 ) ≤ u∗ (x) − ε,

0 < |x| ≤ 3a.

(20.7)

As a consequence of (20.6), (20.7) and of the strong maximum principle, applied in the domain {a < |x| < 3a}, we deduce that u(x, t) ≤ u∗ (x) − η,

|x| = 2a, t0 ≤ t ≤ T

(20.8)

for some η > 0. By (20.7), (20.8) and continuity, one can find b ∈ Rn , 0 < |b| < a, such that v(x, t) := u(x + b, t) satisfies v(x, t0 ) < u∗ (x),

0 < |x| ≤ 3a

and v(x, t) < u∗ (x),

|x| = 2a, t0 ≤ t ≤ T.

Since v is a solution, arguing as for (20.6), we deduce that v(x, t) ≤ u∗ (x),

0 < |x| ≤ 2a, t0 ≤ t < T,

hence u(x, t) ≤ u∗ (x − b),

0 < |x| ≤ a, t0 ≤ t < T.

(20.9)

Finally, (20.6) together with (20.9) imply supRn ×(t0 ,T ) u < ∞. Applying the above argument to −u we obtain supRn ×(t0 ,T ) |u| < ∞ which contradicts T < ∞. We have seen in Theorem 17.12 that if the nonnegative initial data decays slower than |x|−2/(p−1) , then the solution of (18.1) blows up in finite time. We shall now show that if the initial data decays faster (and satisfies a global smallness condition), then the solution exists for all times. Moreover, we shall show that these global solutions exhibit a typical parabolic feature: they have a temporal decay whose exponent is precisely half that of the spatial decay of the initial data, with an upper limit of n/2. The following result is due to [324].


133

Theorem 20.6. Consider problem (18.1) with p > pF , and let k ≥ 2/(p − 1). There exists c = c(n, p, k) > 0 such that, if u0 ∈ L∞ (Rn ) satisfies 0 ≤ u0 (x) ≤ c(1 + |x|)−k ,

x ∈ Rn ,

(20.10)

then Tmax (u0 ) = ∞ and we have, for all t ≥ 1, ⎧ −n/2 if k > n, ⎪ ⎨t u(t) ∞ ≤ t−n/2 log t if k = n, ⎪ ⎩ −k/2 if 2/(p − 1) ≤ k < n. t If moreover k > 2/(p − 1), then (20.4) is satisfied. Remarks 20.7. (i) The decay rates in Theorem 20.6 are sharp for the choice u0 (x) = c(1 + |x|)−k . This follows from u(x, t) ≥ (e−tA u0 )(x) and the lower estimates in Lemma 20.8. (ii) For any p > 1, problem (18.1) admits some nontrivial global classical solutions. Of course, they have to change sign if p ≤ pF . For instance, for any √ p > 1, there exist self-similar solutions of the form u(x, t) = (t + 1)−1/(p−1) w(x/ t + 1), with w ∈ L∞ (Rn ) (see [269, Theorem 5]). (iii) All the solutions constructed in Theorem 20.6 decay at least like t−1/(p−1) . We shall see in Section 26 (see Theorem 26.9) that if p is less than a suitable exponent, then this is actually true for any nonnegative global classical solution of (18.1), cf. also Theorem 28.10. On the other hand, if p ≥ pJL and u∗ (x) = cp |x|−2/(p−1) denotes the singular steady state (see (3.9)), then there are bounded positive initial data u0 satisfying u0 < u∗ such that the corresponding solutions exist globally, decay to zero, but lim t1/(p−1) u(t) ∞ = ∞,

t→∞

see [261]. In addition, if p > pJL and k ∈ (0, 2/(p − 1)) , then one can find > 2/(p−1) such that for any bounded nonnegative continuous radial u0 satisfying u0 ≤ u∗ and u0 (x) − u∗ (x) ∼ |x|− for large |x|, there exist C1 , C2 > 0 such that C1 t−k/2 ≤ u(t) ∞ ≤ C2 t−k/2 ,

t ≥ 1,

(20.11)

see Section 29. Recent results (see [306]) indicate that similar behavior of suitable positive solutions can also be expected for p = pS provided n < 6 (if n = 4, then the decay rate t−k/2 should be replaced by t−k/2 log t). 2

(iv) If p < pS , u0 ≥ 0 has exponential decay (more precisely, u0 (x)e|x| /8 ∈ L (Rn )) and the corresponding solution u exists globally, then (20.11) is true with either k = n or k = 2/(p − 1) (and both possibilities occur). This follows from Theorem 28.9 below. 2

For the proof of Theorem 20.6 we need the following lemma concerning the linear heat equation. Here and in the rest of this subsection, f (t) ∼ g(t) means that C1 g(t) ≤ f (t) ≤ C2 g(t) for some constants C1 , C2 > 0.

134


Lemma 20.8. Let φ(x) = (1 + |x|)−k with k > 0. There holds ⎧ −n/2 if k > n, ⎪ ⎨t −tA −n/2 e φ ∞ ∼ t log t if k = n, ⎪ ⎩ −k/2 if k < n, t for t ≥ 1. Proof. Due (1 + |x|)−k ≤ (1 + |x|2 )−k/2 ≤ C(1 + |x|)−k , we may replace φ by φ(x) := (1 + |x|2 )−k/2 . For each t > 0, the function e−tA φ(x) is radially symmetric in x and nonincreasing in r = |x| (see Proposition 52.17). Consequently, we have

2 −tA −tA φ ∞ = (e φ)(0) = (4πt)−n/2 e−|y| /4t φ(y) dy e |y|≤1

2 (4πt)−n/2 e−|y| /4t φ(y) dy =: I1 (t) + I2 (t). + |y|>1

(t) ∼ t−n/2 for t ≥ 1. If k > n, the conclusion follows from First it is clear that I1−k −n/2 |y| dy = Ct−n/2 . Now assume k ≤ n and observe that I2 (t) ≤ t |y|>1

−|z|2 (1 + 4|z|2 t)−k/2 dz I2 (t) = π −n/2 √ e |z|>1/(2 t)

−|z|2 |z|−k dz, ∼ t−k/2 √ e |z|>1/(2 t)

2 for t ≥ 1. If k < n, we simply use Rn e−|z| |z|−k dz < ∞, hence I2 (t) ∼ t−k/2 ∞ 2 2 for t ≥ 1. If k = n, we use |z|>R e−|z| |z|−n dz = R e−r r−1 dr ∼ log(1/R) for

R ∈ (0, 1/2], hence I2 (t) ∼ t−n/2 log t for t ≥ 1. The lemma follows.

Proof of Theorem 20.6. In view of the comparison principle, it is sufficient to prove the theorem when u0 (x) = c(1 + |x|)−k for some c > 0. Let us first consider the case ∞k > 2/(p − 1). Since min(k, n)(p − 1)/2 > 1, it follows from Lemma 20.8 that 0 e−sA u0 p−1 ∞ ds < 1/(p−1) for c = c(n, p, k) > 0 small enough. The result is then a consequence of Theorem 20.2. Let us turn to the case k = 2/(p − 1). If n ≥ 3 and p > n/(n − 2), the result follows from the observation that the function u(x, t) = ε(1 + |x|2 + εt)−1/(p−1) is a (self-similar) supersolution for ε > 0 sufficiently small (which can be checked by a simple computation). If (n − 2)p ≤ n (or in the general case p > pF ), the result is a consequence of Theorem 20.19(i) and (ii) below (applied with u0 = c|x|−2/(p−1) and v0 = c min(1, |x|−2/(p−1) )) and of the comparison principle. Similar results as in Theorem 20.6 can also be obtained for sign-changing solutions. Let us first prove two auxiliary lemmas concerning the linear heat equation on a half-line and in a cone in R2 .


135

Lemma 20.9. Let k ∈ [1, 2) and φ(x) = (1 − e−x )(1 + x)−k for x ≥ 0. Let e−tA denote the Dirichlet heat semigroup in (0, ∞). Then e−tA φ ∞ ∼ t−k/2 ,

t ≥ 1.

Proof. We will use the formula e

−tA

1 φ(x, t) = √ 4πt

∞

0

2 1 − e−xy/t e−|x−y| /4t φ(y) dy

and estimates xy 1 (1 − e−xy/t φ(y) ≤ min 1, , t (1 + y)k 2 c (1 − e−xy/t e−|x−y| /4t φ(y) ≥ ≥ ct−k/2 , (1 + y)k

x, y, t > 0, for y ∈ [x, 2x], x2 = t ≥ 1,

where c > 0 denotes a generic constant which may change from step to step. If x2 = t ≥ 1, then the above estimates imply 1 e−tA φ(x, t) ≥ √ 4πt

2x

ct−k/2 dy = ct−k/2 .

x

On the other hand, if t ≥ x2 , t ≥ 1, then

∞ 1 1 1 x xy −|x−y|2 /4t x e−tA φ(x, t) ≤ √ dy + e dy t y k−1 4πt 0 t (1 + y)k x

x

∞ 2 x2−k c 1 1 √ e−|x−y| /4t dy ≤ ct−k/2 . dy + c ≤ k−1 t 0 (1 + y) t t x Finally, if 1 ≤ t < x2 , then

1 1 x/2 −x2 /16t x e dy e−tA φ(x, t) ≤ √ t (1 + y)k−1 4πt 0

∞ 2 c + e−|x−y| /4t dy (1 + x/2)k x/2 x2 (3−k)/2 2 c ≤c e−x /16t t−k/2 + k ≤ ct−k/2 , 16t x which concludes the proof. In the following lemma we will use polar coordinates (r, ϕ) in R2 .

136


Lemma 20.10. Let k ∈ N \ {0}, Ω = Ωk := {(r, ϕ) : r > 0, ϕ ∈ (−π/2k, π/2k)}. Then there exists u0 ∈ L∞ (Ω), u0 ≥ 0, such that e−tA u0 ∞ ∼ t−1−k/2 for t ≥ 1. Sketch of proof (see [363] for details). Let G(r, ρ; ϕ, ψ; t) denote the Dirichlet heat kernel in Ω and w(r, ϕ, t) := G(r, 1; ϕ, 0; t + t0 ), where t0 > 0 is fixed. Then w(r, ϕ, t) =

2k−1 (r cos ϕ − cos jπ/k)2 + (r sin ϕ − sin jπ/k)2 1 . (−1)j exp − 4π(t + t0 ) j=0 4(t + t0 )

Set s(r, t) := w(r, 0, t). Then one can show that w(·, ·, t) ∞ = supr>0 s(r, t) and s(r, t) = C0

r2 + 1 r k r 1 r 1+ , exp − R t + t0 4(t + t0 ) t + t0 t + t0 t + t0

where C0 is a positive constant and R is bounded on bounded sets. Obviously, √ s( t + t0 , t) ≥ ct−1−k/2 for t ≥ 1. On the other hand, one can √ also show that supr s(r, t) is attained at some rM (t) which satisfies rM (t) ≤ C t + t0 , hence s(rM (t), t) ≤ Ct−1−k/2 . Theorem 20.11. Let n ≥ 3, p > 1 and α > 1/(p − 1). Then there exists u0 ∈ L∞ (Rn ) such that the solution u of (18.1) is global and u(t) ∞ ∼ t−α for t ≥ 1. Proof. If α ≤ n/2, then p > pF and the assertion follows from Theorem 20.6. *3 Let n/2 < α < 2. Then n = 3. Set γ := α/3 and φ(x) := i=1 ψ(xi ), where ψ(r) := sign(r)(1 − e−|r| )(1 + |r|)−2γ .

(20.12)

*3 Let −Am denote the Laplacian in Rm . Then e−tA3 φ(x, t) = i=1 e−tA1 ψ(xi , t), hence e−tA3 φ ∞ ∼ t−α for t ≥ 1 due to Lemma 20.9 and the oddness of e−tA1 ψ(·, t). Now choosing u0 = εφ, ε > 0 small, we obtain the result from Remark 20.4(ii) used with Ω = (0, ∞)3 . Finally, let α ≥ 2. Fix k ∈ N such that γ := α − 1 − k/2 ∈ [1/2, 1) and consider the cone Ωk and the function w(t) := e−tA u0 from Lemma 20.10. Extend the function w to R2 × [0, ∞) by w(r, ϕ, t) = −w(r, π/k − ϕ, t) for ϕ ∈ (π/2k, 3π/2k) and w(r, ϕ + 2jπ/k, t) = w(r, ϕ, t), j = 1, 2, . . . , k − 1. Then w = w(x1 , x2 , t) is a solution of the heat equation in R2 and w(t) ∞ ∼ t−1−k/2 for t ≥ 1. Set φ(x) = w(x1 , x2 , 0)ψ(x3 ), where ψ is defined by (20.12). Then, similarly as above, e−tAn φ(x, t) = w(x1 , x2 , t)e−tA1 ψ(x3 , t), hence e−tA3 φ ∞ ∼ t−α for t ≥ 1. Now choosing u0 = εφ, ε > 0 small, we obtain the result from Remark 20.3(d) used with Ω = Ωk × (0, ∞) × Rn−3 .


137

Remark 20.12. (i) Solutions with exponential time decay. In addition to the solutions with polynomial time decay in Theorem 20.11 one can also easily construct sign-changing global √ solutions with exponential time decay. In fact, let 1 0, A = π/2 λ and let w be the positive solution of the problem w + wp = 0 in (−A, A), w(−A) = w(A) = 0. Choose α ∈ (0, 1) and set u0 (x) :=

αw(x − 4kA)

if x ∈ [(4k − 1)A, (4k + 1)A),

−αw(x − (4k + 2)A)

if x ∈ [(4k + 1)A, (4k + 3)A),

where k ∈ Z. Then the solution u of (18.1) with n = 1 satisfies u(t) ∞ ∼ e−λt . This follows from Theorem 51.19 and Theorem 19.9(iv). (ii) Decay of global solutions. Let 1 1. If the solution u of (18.1) is global, then u(t) ∞ → 0 as t → ∞ (see Theorem 26.9). This result is also true for all nonnegative data lying in the energy space E (see [478] and Example 51.27), but it fails for sign-changing radial solutions (consider the choice α = 1 in (i)). (iii) Notice that the radially symmetric initial data of the form c(1 + |x|)−k , k ≥ 2/(p − 1), appearing in Theorem 20.6 belong to the energy space E := {u ∈ Lp+1 (Rn ) : ∇u ∈ L2 (Rn )} if p < pS (see Example 51.27 for the wellposedness in this space).

20.2. Global solutions with exponential spatial decay We have seen in Theorem 20.11 and Remark 20.12 that there is a wide range of possibilities for the decay of global solutions of the Cauchy problem (18.1). In this subsection we show that the situation is much simpler if we restrict ourselves to the initial data with exponential spatial decay. More precisely, we will consider initial data in the space Hg1 (see (18.16)) and exponents p ∈ (1, pS ). We will use the rescaled solutions v (see (18.13)) and operator L (see (18.15)). As above, let L λL k = (n + k − 1)/2 denote the eigenvalues of L. In addition, we denote λ0 := 1/(p − 1). Proposition 20.13. (i) Let 1 1 + 1/λL k0 . If u is a global 1 solution of (18.1) with u0 ∈ Hg \ {0} and t0 > 0, then there exist C1 , C2 > 0 and k ∈ {0} ∪ {k0 , k0 + 1, k0 + 2, . . . } such that L

L

C1 t−λk ≤ u(t) ∞ ≤ C2 t−λk ,

t ≥ t0 .

(20.13)

138


Conversely, if k = 0 or k ≥ k0 , then there exists u0 ∈ Hg1 such that the corresponding solution of (18.1) is global and satisfies (20.13). (ii) Let pF n/2. Then there exists M = 0 such that u(t) − M (t + 1)−n/2 e−|x|

2

/4(t+1)

∞ ≤ C(t + 1)−q ,

t ≥ 0.

Remark 20.14. (i) If p = 1 + 1/λL k0 for some k0 ≥ 1, then the proof of Proposition 20.13(i) guarantees the following: Let u be a global solution of (18.1) with L u0 ∈ Hg1 \ {0}, t0 > 0. Then u(t) ∞ ≤ Ct−λ0 for t ≥ t0 . If there exist C > 0 −λ and λ > λL for t ≥ t0 , then there exist C1 , C2 > 0 0 such that u(t) ∞ ≤ Ct and k > k0 such that (20.13) is true. Conversely, if k = 0 or k > k0 , then there exists u0 ∈ Hg1 such that the corresponding solution of (18.1) is global and satisfies (20.13). (ii) Some of the results in Proposition 20.13(i) concerning the decay (20.13) with k > 0 can also be obtained for supercritical p, p(n − 4) < n (cf. Example 51.24). (iii) Sufficient conditions for the initial data u0 to satisfy the assumptions of Proposition 20.13(ii) can be found in Theorem 28.9. For related results see also the following subsection and [288], for example. Proof of Proposition 20.13. (i) Let u be a global solution of (18.1) with u0 ∈ Hg1 \ {0}. Then the rescaled solution v (see (18.13)) is a global solution of (18.14). If v(s) Hg1 → 0 as s → ∞, then (20.13) is true with some k ≥ k0 due to Example 51.24 (see (51.72)). If v(s) Hg1 → 0, then Lemma 18.4(iii) and Example 51.24 show that v(s) ∞ ≤ C2 for all s ≥ s0 and s0 > 0. Assume lim inf s→∞ v(s) ∞ = 0. Then the same estimates as at the end of Example 51.24 guarantee lim inf s→∞ v(s) Hg1 = 0. Consequently, choosing δ > 0 small, there exist sj → ∞ such that v(sj ) Hg1 = δ. Using the compactness of the semiflow we may assume v(sj ) → w in Hg1 , where w belongs to the ω-limit set of v, hence w is an equilibrium of (18.14), w Hg1 = δ. However, the zero equilibrium is isolated due to p ∈ / {1 + 1/λL k : k = 1, 2, . . . } which yields a contradiction. Hence C1 ≤ v(s) ∞ ≤ C2 for s ≥ s0 which implies (20.13) with k = 0. To prove the converse statement, assume first k = 0. By [174] there exist a sequence of nontrivial stationary solutions vj , j = 1, 2, . . . , of problem (18.14). The corresponding rescaled solutions uj satisfy (20.13) with k = 0. If k ≥ k0 , then the existence of u0 ∈ Hg1 such that the solution u satisfies (20.13) follows from Example 51.24. (ii) The proof is a direct consequence of assertion (ii) in Example 51.24.


139

20.3. Asymptotic profiles for small data solutions More information on the asymptotic behavior of positive solutions than in Subsection 20.1 can be gained if one considers suitably small initial data in L1 . Theorem 20.15. Consider problem (18.1) with p > pF . Assume that 0 ≤ u0 ∈ L∞ ∩ L1 (Rn ) satisfies (n(p−1)/2)−1 ≤ c(n, p), u0 1 u0 ∞

(20.14)

with c(n, p) > 0 sufficiently small. Then Tmax (u0 ) = ∞ and (20.4) is satisfied. Moreover, u(t) behaves like a multiple of the heat kernel. Namely, the limit I∞ := lim u(t) 1 exists and is finite, t→∞

(20.15)

and there holds u(t) − I∞ Gt 1 → 0,

t → ∞.

(20.16)

Theorem 20.15 is a variant of a result of [302] (see also [145], [322]). We prove (20.15), as a consequence of Theorem 20.2. This proof is simpler than those in [302] (based on energy estimates) or in [145], [322] (based on the variation-of-constants formula). As for property (20.16), it will be a consequence of the following lemma from [86] (see also [322]) concerning the inhomogeneous heat equation. Lemma 20.16. Let u0 ∈ L1 (Rn ), f ∈ L1 (Rn × (0, ∞)) with u0 , f ≥ 0, u0 ≡ 0 and let u be given by u(t) = e−tA u0 +

t

e−(t−s)A f (s) ds,

t > 0.

0

Then M := limt→∞ u(t) 1 exists in (0, ∞) and we have u(t) − M Gt 1 → 0, t → ∞. Proof. By the variation-of-constants formula, we have u(t) − e−(t−t0 )A u(t0 ) 1 =

t

f (s) 1 ds,

t ≥ t0 ≥ 0.

t0

Since e−(t−t0 )A u(t0 ) 1 = u(t0 ) 1 we see that limt→∞ u(t) 1 exists and is finite. Since u(t) ≥ e−tA u0 this limit is positive. Denoting M (t) = u(t) 1 , it follows that

u(t) − M Gt 1 ≤

∞

t0

" " f (s) 1 ds + "e−(t−t0 )A u(t0 ) − M (t0 ) Gt "1 +|M (t0 ) − M |.

140


Using

" −sA " "e φ − φ 1 Gs "1 → 0, s → ∞,

0 ≤ φ ∈ L1 (Rn )

(see Proposition 48.6 in Appendix B) and letting t → ∞, we obtain

lim sup u(t) − M Gt 1 ≤ t→∞

∞

f (s) 1 ds + |M (t0 ) − M |.

t0

The lemma follows by letting t0 → ∞. Proof of Theorem 20.15. Assume (20.14) with c = c(n, p) > 0 small. Using the Lp -Lq -estimate (cf. Proposition 48.4(d)) and choosing τ = ( u0 1 / u0 ∞ )2/n , we obtain

∞

τ

∞ −sA p−1 p−1 e u0 ∞ ds ≤ u0 ∞ ds + u0 p−1 s−n(p−1)/2 ds 1 0

0

τ

p−1 1−n(p−1)/2 = τ u0 p−1 τ ∞ + C(n, p) u0 1 2/n

≤ C(n, p) u0 1 u0 p−1−2/n ≤ 1/2(p − 1). ∞ By Theorem 20.2, we deduce that Tmax (u0 ) = ∞ and that u(t) ≤ Ce−tA u0 .

(20.17)

Applying the Lp -Lq -estimate again, we obtain u(t) pp ≤ C e−tA u0 pp ≤ C min(1, t−n(p−1)/2 ).

(20.18)

On the other hand, by Proposition 48.4(b) and the variation-of-constants formula, we have

t

u(t) dx = Rn

Rn

u0 dx +

0

u(s) pp ds.

(20.19)

We deduce from (20.18), (20.19) and n(p − 1)/2 > 1 that u(t) 1 is nondecreasing and bounded. The conclusion then follows from Lemma 20.16. Remark 20.17. Estimates similar to (20.16) are also true for other Lq -norms. In particular there holds tn/2 u(t) − I∞ Gt ∞ → 0,

t → ∞.

(20.20)

This is a consequence of [70, Theorem 4.1] and inequality (20.17). Estimates for all Lq -norms follow immediately by interpolating between (20.16) and (20.20). It follows from Theorems 17.12 and 20.6 that initial data which decay at the rate |x|−2/(p−1) constitute a borderline between blow-up and global existence. Our


141

next results concern some particular classes of initial data with this asymptotic behavior which are especially interesting. First it turns out that initial data which are homogeneous of degree −2/(p − 1) (and suitably small) give rise to (forward) self-similar solutions, cf. Remark 15.4(ii). Moreover, these solutions enjoy stability properties. For instance, if a (small) initial data coincides for large x with a homogeneous function of degree −2/(p − 1), then the solution is asymptotically self-similar. These results will be proved by semigroup techniques and suitable fixed point arguments, that are refinements of the methods introduced in Section 15. We will also describe the global properties of the equation in a space which naturally arises in this connection, namely the critical Lq -space. Its special role as an invariant space can be explained as follows (cf. e.g. [112], and see also [115] for earlier references and for similar considerations concerning the Navier-Stokes and nonlinear Schr¨ odinger equations). Consider the scaling transformations Sλ : u → uλ (x, t) := λ2/(p−1) u(λx, λ2 t) for λ > 0. Observe that the equation in (18.1) is invariant under these transformations. On the other hand, for spatial functions φ = φ(x), we have φλ q = λ2/(p−1)−(n/q) φ q ,

1 ≤ q ≤ ∞,

(20.21)

so that the only Lq -norm left invariant by the transformations Sλ is the critical norm, i.e. q = qc = n(p − 1)/2. Now assume that there exists q with the property that the solution of (18.1) is global whenever the initial data u0 is small in Lq . If q = qc , then, by (20.21) applied to φ = u0 , global existence will hold for any u0 ∈ Lq . But this is a contradiction to Theorem 17.1; hence q = qc is the only possible value with that property. In accordance with these observations, we will indeed prove global existence for small initial data in Lqc , provided that qc > 1. Note that the critical exponent p = pF corresponds to the case when qc = 1, and the requirement that qc > 1 is thus consistent with the Fujita-type result Theorem 18.1. Furthermore, still using the techniques mentioned in the previous paragraph, we will establish the asymptotic stability of the zero solution in the space Lqc . More generally, we will show that the above mentioned self-similar solutions are in a sense stable with respect to critical Lq -perturbations. Note, in turn, that the transformations Sλ also leave invariant the homogeneous functions of degree −2/(p − 1) (from which the self-similar solutions arise). We shall use the following definition of mild solution of problem (18.1). Definition 20.18. Let u0 ∈ L1 (Rn ) + L∞ (Rn ). We say that u is (global) mild r n solution of (18.1) if u ∈ L∞ loc ((0, ∞), L (R )) for some r ≥ p and satisfies

t u(t) = e−tA u0 + e−(t−s)A |u|p−1 u(s) ds, t > 0, 0

142


where for each t > 0 the integral is absolutely convergent in Lr (Rn ). In particular, there holds u(t) − e−tA u0 → 0 in Lr (Rn ) as t → 0. This definition is slightly different from that in Remark 15.4(iii). Note that the r n definition makes sense since e−(t−s)A |u|p−1 u(s) ∈ L∞ loc ((0, t), L (R )), due to r ≥ p p q and the L -L -estimates. The following result is due to [457], [115] for assertion (i), [115], [475] for assertion (ii). Assertion (iii) for u0 = 0 (i.e. Corollary 20.20) is from [481], improving on earlier results of [530], whereas the case u0 = 0 seems new. Theorem 20.19. Let p > pF , ω ∈ L∞ (S n−1 ) and set u0 (x) := |x|−2/(p−1) ω(x/|x|),

x ∈ Rn \ {0}.

(20.22)

There exists µ0 = µ0 (n, p) > 0 such that, if ω ∞ ≤ µ0 , then the following properties hold. (i) Problem (18.1) admits a global mild solution u (in the sense of Definition 20.18). Moreover, u is self-similar, i.e. is of the form √ u(x, t) = t−1/(p−1) f x/ t ,

x ∈ Rn , t > 0,

with f (y) = u(y, 1) ∈ L∞ (Rn ), and u is a classical solution for t > 0. Furthermore, the solution u is stable in the sense indicated in parts (ii) and (iii) hereafter. (ii) Let v0 = ϕu0 , where ϕ ∈ L∞ (Rn ) satisfies ϕ = 1 for |x| large. Assume that ω(·/| · |)ϕ ∞ ≤ µ0 . Then problem (18.1) with initial data v0 admits a global solution v with v(t) ∈ L∞ (Rn ) for each t > 0, and v is a classical solution for t > 0. Furthermore, v is asymptotically self-similar, with profile f , i.e.: √ t1/(p−1) u(t) − v(t) ∞ = sup t1/(p−1) v y t, t −f (y) → 0, y∈Rn

t → ∞.

(20.23)

(iii) Let q := qc . Assume that v0 ∈ L1 (Rn ) + L∞ (Rn ) satisfies u0 − v0 ∈ Lq (Rn ) and u0 − v0 q < µ0 . Then problem (18.1) with initial data v0 admits a global solution v which satisfies (20.23), together with sup u(t) − v(t) q ≤ 2 u0 − v0 q

(20.24)

t>0

and u(t) − v(t) q → 0,

t → ∞.

(20.25)


143

Corollary 20.20. Let p > pF and q := qc . Then the zero solution is asymptotically stable in Lq . More precisely, if v0 ∈ Lq (Rn ) satisfies v0 q ≤ µ0 with µ0 = µ0 (n, p) > 0 sufficiently small, then (18.1) admits a global mild solution v which satisfies sup v(t) q ≤ 2 v0 q t>0

and

v(t) q → 0,

t → ∞.

Furthermore v(t) ∈ L∞ (Rn ) for each t > 0, v is a classical solution for t > 0, and there holds t1/(p−1) v(t) ∞ → 0, t → ∞. Remarks 20.21. (i) Nonuniqueness. The solutions u and v constructed in Theorem 20.19 are unique in a suitable class of functions (see Lemma 20.22 and cf. also Remark 20.24(iii) below). When ω is a suitably small positive constant, nonuniqueness in a larger class of functions has been proved in [499]. Another nonuniqueness result can be found in [389]. (ii) Decay rates. The convergence statement in Theorem 20.19(ii) says that u(t) − v(t) decays in L∞ faster that u(t) or v(t) separately. More precise estimates on the decay of u(t) − v(t) when u is radial can be found in [205]. Observe that the asymptotic behaviors in Theorem 20.19 and in Corollary 20.20 are different (note that u0 in Theorem 20.19 just fails to be in Lq for q = qc if u0 ≡ 0). In particular, in Theorem 20.19 with u0 ≡ 0, v(t) ∞ decays like t−1/(p−1) as t → ∞, whereas in Corollary 20.20 it decays faster. (iii) Radial self-similar solutions. The self-similar solutions constructed in Theorem 20.19 are not radial unless u0 is radial. Radial self-similar solutions of (18.1) have been constructed by ODE or variational techniques (see Remarks 15.4 and the references there). In the radial case, the decay of the profile f (y) as y → ∞ has also been studied. The profile can decay either like |y|−2/(p−1) or exponentially. (iv) Other domains. Consider problem (15.1) in a (possibly unbounded) domain Ω. By the comparison principle and Corollary 20.20, it follows that the zero solution is asymptotically stable in Lq for q = qc . This is in contrast with the situation for q > qc (cf. Theorem 19.3). In view of the proof, we introduce the following notation. For p, q as above, we fix r such that 1 ≤ r/p < q < r. (20.26) Although r is not uniquely determined, we assume that it is fixed once and for all 1 n (see also Remark 20.24(ii) below). We let β = n2 ( 1q − 1r ) = p−1 − 2r and we define the following function spaces:

r n X = u ∈ L∞ where u X = sup tβ u(t) r , loc ((0, ∞), L (R )) : u X < ∞ , t>0

144


∞ n Y = u ∈ L∞ loc ((0, ∞), L (R )) : u Y < ∞ ,

1

where u Y = sup t p−1 u(t) ∞ , t>0

and Z = X ∩ Y , with norm u Z = u X + u Y . For δ ≥ 0, we also define

Eδ = u0 ∈ L1 (Rn ) + L∞ (Rn ) : Nδ (u0 ) < ∞ , where

(20.27)

Nδ (u0 ) = sup tβ+δ e−tA u0 r , t>0

and for 0 < T < ∞ we set u X,δ,T = sup tβ+δ u(t) r < ∞,

u ∈ X,

00

(iii) Let m ∈ (r/p, q] and set δ = n/2m−1/(p−1). There exists ε2 = ε2 (n, p, r, m) ∈ (0, ε0 ] such that if u0 , v0 ∈ E satisfy N (u0 ), N (v0 ) ≤ ε2 and u0 − v0 ∈ Lm (Rn ), then the corresponding solutions u, v of (18.1) given by part (i) satisfy sup tδ u(t) − v(t) q ≤ 2 u0 − v0 m . t>0

(20.30)


145

Proof. For u0 ∈ L1 (Rn ) + L∞ (Rn ) and u ∈ X, we define the mapping Tu0 u(t) = e−tA u0 +

t 0

e−(t−s)A |u|p−1 u(s) ds.

Let M > 0. We fix 0 ≤ δ < δ¯ (≤ 1) and u0 , v0 ∈ E, with u0 − v0 ∈ Eδ . Step 1. Estimates of the mapping T in X. For all u, v ∈ X and 0 < s < t < T < ∞, we have " −(t−s)A p−1 " "e |u| u(s) − |v|p−1 v(s) "r " " ≤ (t − s)−n(p−1)/2r " |u|p−1 u(s) − |v|p−1 v(s) "r/p u(s) − v(s) r ≤ (t − s)−q/r u(s) p−1 + v(s) p−1 r r p−1 p−1 −q/r −(βp+δ) u X + v X u − v X,δ,T ≤ (t − s) s On the other hand, using 1 − β(p − 1) − q/r = 0, we have

tβ+δ 0

t

(t − s)−q/r s−(βp+δ) ds = t1−β(p−1)−q/r

0

1

(1 − σ)−q/r σ −(βp+δ) dσ

= C(n, p, r, δ), where the integrals are finite, due to q/r < 1 and βp + δ < βp + δ¯ = 1. It follows that tβ+δ Tu0 u(t) − Tv0 v(t) r ≤ tβ+δ e−tA (u0 − v0 ) r

t " −(t−s)A p−1 " β+δ "e |u| u(s) − |v|p−1 v(s) "r ds +t 0 p−1 ≤ Nδ (u0 − v0 ) + C u p−1 u − v X,δ,T , X + v X hence Tu0 u − Tv0 v X,δ,T ≤ Nδ (u0 − v0 ) + C1 M p−1 u − v X,δ,T ,

u, v ∈ BX (M ), (20.31)

with C1 = C1 (n, p, r, δ) > 0. Step 2. Estimates of the mapping T in Z and fixed-point. For all u, v ∈ Z, 0 < t < T < ∞ and t/2 < s < t, we have " −(t−s)A p−1 " "e |u| u(s) − |v|p−1 v(s) "∞ " " ≤ "|u|p−1 u(s) − |v|p−1 v(s)"∞ p−1 u(s) − v(s) ∞ ≤ p u(s) p−1 ∞ + v(s) ∞ p p−1 − p−1 −δ u Y + v p−1 u − v Y,δ,T . ≤ p(t/2) Y

146


Using the fact that Tu0 u(t) = e−(t/2)A Tu0 u(t/2) +

t

e−(t−s)A |u|p−1 u(s) ds,

t/2

it follows that, for all u, v ∈ BZ (M ), 1

t p−1 +δ Tu0 u(t) − Tv0 v(t) ∞ " " 1 ≤ t p−1 +δ "e−(t/2)A Tu0 u( 2t ) − Tu0 v( 2t ) "∞

t " −(t−s)A p−1 " 1 "e |u| u(s) − |v|p−1 v(s) "∞ ds + t p−1 +δ t/2

1 p−1 +δ

n

1

n

(4π)− 2r ( 2t ) p−1 − 2r +δ Tu0 u( 2t ) − Tu0 v( 2t ) r 1 u − v Y,δ,T + p2 p−1 +δ u p−1 + v p−1 Y Y

≤2

≤ C(p) Tu0 u − Tu0 v X,δ,T + C(p)M p−1 u − v Y,δ,T . Taking supremum for t ∈ (0, T ) and combining this with (20.31), we obtain Tu0 u − Tu0 v Z,δ,T ≤ C2 Nδ (u0 − v0 ) + C3 M p−1 u − v Z,δ,T ,

u, v ∈ BZ (M ), (20.32) with C2 = C2 (p) ≥ 1 and C3 = C3 (n, p, r, δ) > 0. In particular, letting T → ∞ in (20.32) with δ = 0, we get Tu0 u − Tv0 v Z ≤ C2 N (u0 − v0 ) + C3 M p−1 u − v Z ,

u, v ∈ BZ (M ). (20.33)

Choose ε0 = ε0 (n, p, r) > 0 such that 2p C3 (n, p, r, 0)(C2 ε0 )p−1 ≤ 1 and assume that N (u0 ) ≤ ε0 . Taking M = 2C2 N (u0 ), we have C3 M p−1 ≤ 1/2 and C2 N (u0 ) + C3 M p ≤ M . It follows from (20.33) (with the choices v0 = 0, v = 0 and u0 = v0 ) that Tu0 is a strict contraction on the complete metric space BZ (M ), endowed with the distance induced by the norm · Z . Therefore it possesses a unique fixed point, that we denote by u(t) = Wt u0 . In particular u(t) ∈ L∞ (Rn ) for t > 0 and u is a classical solution of (18.1) for t > 0. This proves the existence-uniqueness statement of assertion (i). Next, assume in addition that N (v0 ) ≤ ε0 and put v(t) = Wt v0 . Replacing ε0 by ε1 > 0 possibly smaller and depending also on δ, we have C3 (n, p, r, δ)M p−1 ≤ 1/2. It then follows from (20.32) that u − v Z,δ,T ≤ 2C2 Nδ (u0 − v0 ). Assertion (ii) follows by letting T → ∞. n 1 − p−1 . Note Step 3. Lq -estimates. Fix m ∈ (r/p, q] and put δ = δ(m) = 2m ¯ Assume that u0 , v0 ∈ E satisfy N (u0 ), N (v0 ) ≤ ε2 (n, p, r, m) := that δ ∈ [0, δ).


147

ε1 (n, p, r, δ) and u0 − v0 ∈ Lm (Rn ). Let u, v be the corresponding solutions of (18.1) given by Steps 1 and 2. Similarly as in the beginning of Step 1, we obtain for 0 < s < t: " −(t−s)A p−1 " "e |u| u(s) − |v|p−1 v(s) "q n p 1 p−1 sup σ β+δ u(σ) − v(σ) r . ≤ (t − s)− 2 ( r − q ) s−(βp+δ) u p−1 X + v X σ>0

On the other hand, using 1 − n2 ( pr − 1q ) − βp = 0, we have

t 0

p 1 −n 2 ( r − q ) −(βp+δ)

(t − s)

s

ds = t

−δ

0

1

n

p

(20.34)

1

(1 − σ)− 2 ( r − q ) σ −(βp+δ) dσ

(20.35)

= C(n, p, r, δ)t−δ ,

where the integrals are finite, due to n2 ( pr − 1q ) < n(p − 1)/2q = 1 and βp + δ < βp + δ¯ = 1. Combining (20.34), (20.35) and (20.28) (and taking ε2 (n, p, r, m) possibly smaller), we obtain tδ u(t)−v(t) q

t"

" "e−(t−s)A |u|p−1 u(s) − |v|p−1 v(s) " ds q 0 p−1 p−1 β+δ ≤ u0 − v0 m + C(n, p, r, δ) u X + v X sup σ u(σ) − v(σ) r ≤ t e δ

−tA

(u0 − v0 ) q + t

δ

σ>0

≤ u0 − v0 m + C(n, p, r, δ)M

p−1

Nδ (u0 − v0 ) ≤ 2 u0 − v0 m .

This proves assertion (iii). The next lemma shows that the homogeneous initial data u0 belong to the class E used in Lemma 20.22. Lemma 20.23. Let 0 < k < n, L > 0, and let u0 ∈ L1 (Rn ) + L∞ (Rn ) satisfy |u0 (x)| ≤ L|x|−k . Then, for n/k < s ≤ ∞, there holds sup tk/2−n/(2s) e−tA u0 s ≤ cL t>0

where c = c(n, k, s) = e−A |x|−k s < ∞. Proof. Set φ(x) = |x|−k and decompose φ = φ1 + φ2 , where φ1 = χ{|x| n/k. Consequently, e−A φ = e−A φ1 + e−A φ2 ∈ Ls (Rn ),

s > n/k.

148


Now using φ(λx) = λ−k φ(x), we obtain

|(e−tA u0 )(x)| =

2 (4πt)−n/2 e−|y| /4t u0 (x − y) dy Rn

2 −n/2 ≤ (4π) L e−|z| /4 φ(x − zt1/2 ) dz n R

2 −k/2 −n/2 (4π) e−|z| /4 φ(xt−1/2 − z) dz = Lt n R = Lt−k/2 e−A φ (xt−1/2 ).

In particular, e−tA u0 s ≤ Lt(n/2s)−(k/2) e−A φ s ,

s > n/k,

and the lemma follows. Proof of Theorem 20.19. In this proof we shall take µ0 as small as necessary to apply Lemma 20.22, but µ0 will depend only on n, p, r. (i) Since N (u0 ) ≤ c(n, p, r) ω ∞ by Lemma 20.23, the existence of u follows from Lemma 20.22(i). Let us show that u is self-similar. This is equivalent to showing that, for each λ > 0, uλ (x, t) := λ2/(p−1) u(λx, λ2 t) satisfies uλ ≡ u (indeed, consider λ = t−1/2 ). Since uλ X = u X , it is thus sufficient, in view of the uniqueness part of Lemma 20.22(i), to check that uλ is also a mild solution of (18.1). To this end, we define the dilation operator (dλ f )(x) := f (λx) and note that uλ (t) = λ2/(p−1) dλ u(λ2 t). A direct computation involving the heat kernel yields 2 e−tA (dλ f ) = dλ eλ −tA f .

(20.36)

Applying (20.36) with f = up , we see that the function

t

e−(t−s)A up (s) ds

(Su)(t) := 0

satisfies S(uλ )(t) = λ2p/(p−1)

t

e−(t−s)A dλ up (λ2 s) ds

t 2 eλ −(t−s)A up (λ2 s) ds = λ2p/(p−1) dλ 0

0

= λ2p/(p−1) dλ

λ2 t

2

e−(λ

u (σ) λ−2 dσ

t−σ)A p

0

= λ2/(p−1) dλ (Su)(λ2 t) =: (Su)λ (t).

(20.37)


149

Now, since u0 satisfies (20.22), we have dλ u0 = λ−2/(p−1) u0 , hence 2

(e−tA u0 )λ := λ2/(p−1) dλ (eλ

−tA

u0 ) = λ2/(p−1) e−tA (dλ u0 ) = e−tA u0 .

(20.38)

Combining (20.37) and (20.38), it follows that e−tA u0 + S(uλ )(t) = (e−tA u0 )λ + (Su)λ (t) = uλ (t). We have thus shown that u is self-similar. (ii) Since N (v0 ) ≤ c(n, p, r) ω(·/| · |)ϕ ∞ by Lemma 20.23, the existence of v follows from Lemma 20.22(i). Next, since |v0 −u0 | ≤ C|x|−2/(p−1) χ{|x| 1, (n − 1)2 we have the following Liouville-type theorem in the general (nonradial) case. It is a consequence of [79, Theorem 0.1].

21. Parabolic Liouville-type results

151

Theorem 21.2. Let 1 < p < pB . Then equation (21.1) has no positive classical solution. Remark 21.3. Theorem 21.1 remains true for nontrivial nonnegative radial classical solutions, bounded or not, whereas Theorem 21.2 remains true for nontrivial nonnegative classical solutions (see Remark 26.10(i) and cf. [425]). The proofs of Theorems 21.1 and 21.2 are completely different, based on intersection-comparison and integral estimates, respectively. For the proof of Theorem 21.1, we need some simple preliminary observations concerning radial steady states. Let ψ1 be the solution of the equation ψ +

n−1 ψ + |ψ|p−1 ψ = 0, r

r > 0,

(21.2)

satisfying ψ(0) = 1, ψ (0) = 0. Obviously ψ1 (0) < 0. It is known that the solution is defined on some interval and it changes sign due to p < pS (this follows for instance from Theorem 8.1). We denote by r1 > 0 its first zero. By uniqueness for the initial-value problem, ψ1 (r1 ) < 0. We thus have ψ1 (r) > 0 in [0, r1 ) and ψ1 (r1 ) = 0 > ψ1 (r1 ). p−1

Clearly, ψα (r) := αψ1 (α 2 r) is the solution of (21.2) with ψ(0) = α, ψ (0) = 0, p−1 and with the first positive zero rα = α− 2 r1 . As an elementary consequence of the properties of ψ1 we obtain the following Lemma 21.4. Given any m > 0, we have lim (sup{ψα (r) : r ∈ [0, rα ] is such that ψα (r) ≤ m}) = −∞.

α→∞

Proof of Theorem 21.1. The proof is by contradiction. Assume that u is a positive, bounded classical solution of (21.1), u(x, t) = U (r, t), where r = |x|. By the boundedness assumption and parabolic estimates, U and Ur are bounded on [0, ∞) × R. It follows from Lemma 21.4 that if α is sufficiently large, then U (·, t) − ψα has exactly one zero in [0, rα ] for any t and the zero is simple. We next claim that z[0,rα ] (U (·, t) − ψα ) ≥ 1 t ≤ 0, α > 0,

(21.3)

where z[0,rα ] (w) denotes the zero number of the function w in the interval [0, rα ] (see Appendix F). Indeed, if not, then U (·, t0 ) > ψα in [0, rα ] for some t0 . By Theorem 17.8 we know that each solution of the Dirichlet problem ⎫ ut − ∆u = up , |x| < rα , t > 0, ⎪ ⎬ u = 0, |x| = rα , t > 0, ⎪ ⎭ u(x, t0 ) = U 0 (|x|), |x| < rα

152


blows up in finite time provided U 0 > ψα in [0, rα ). Choosing the initial function U 0 between ψα and U (·, t0 ) we conclude, by comparison, that u and u both blow up in finite time, in contradiction to the global existence assumption on u. This proves the claim. Set α0 := inf{β > 0 : z[0,rα ] (U (·, t) − ψα ) = 1 for all t ≤ 0 and α ≥ β}. In view of the above remark on large α, we have α0 < ∞. Also α0 > 0. Indeed, for small α > 0 we have ψα (0) 0 small. By the properties of the zero number (see Theorem 52.28), we can choose t < 0 small such that ψα (0) − U (·, t) has only simple zeros and then, by (21.3), z[0,rα ] (U (·, t) − ψα ) ≥ 2. By definition of α0 (and (21.3)), there are sequences αk α0 and tk ≤ 0 such that z[0,rαk ] (U (·, tk ) − ψαk ) ≥ 2, k = 1, 2, . . . . Using Theorem 52.28 again, we get z[0,rαk ] (U (·, tk + t) − ψαk ) ≥ 2,

t ≤ 0, k = 1, 2, . . . .

(21.4)

This in particular allows us to assume, choosing different tk if necessary, that tk → −∞. By the boundedness assumption and parabolic estimates, passing to a subsequence, we may further assume that u(x, tk + t) → v(x, t),

x ∈ Rn , t ∈ R,

with convergence in C 2,1 (Rn × R). Clearly then, there is δ > 0 such that for each fixed t, U (·, tk + t) − ψαk → V (·, t) − ψα0 in C 1 [0, rα0 + δ], where v(x, t) = V (|x|, t). This and (21.4) guarantee that for each t ≤ 0, V (·, t) − ψα0 has at least two zeros or a multiple zero in [0, rα0 ). By the properties of the zero number (see Theorem 52.28), we may choose t < 0 so that V (·, t) − ψα0 has only simple zeros (and, hence at least two of them). Since U (·, tk +t)−ψα0 is close to V (·, t)−ψα0 in C 1 [0, rα0 ], if k is large, it has at least two simple zeros in [0, rα0 ) as well. But then, for α > α0 , α close to α0 , the function u(·, tk + t) − ψα has at least two zeros in [0, rα ), contradicting the definition of α0 . We have thus shown that the assumption u ≡ 0 leads to a contradiction, which proves the theorem. We now turn to the proof of Theorem 21.2. It will be a direct consequence of the following space-time integral estimates [79] for (local) solutions of (21.1).


153

Proposition 21.5. Let 1 (n + 2)(p − 1)/2 such that if 0 0, v(x, t) := R2/(p−1) u(Rx, R2 t) solves (21.1) in B1 × (−1, 1). It follows from Proposition 21.5 that

R2 /2

ur (y, s) dy ds = Rn+2

−R2 /2

|y| 0.

(21.7)

The main ingredient in the proof of Lemma 21.6 is Lemma 8.9, proved in Section 8, which provides a family of integral estimates relating any C 2 -function with its gradient and its Laplacian. Proof. (i) We apply Lemma 8.9 with q = 0. Denoting

−1 2 J= ϕ u |∇u| ∆u, K = ϕ (∆u)2 , this gives us −

n+2 n−1 k + 1 kI + kJ − K n n

n

1 ≤ |∇u|2 ∆ϕ + ∆u − ku−1 |∇u|2 ∇u · ∇ϕ. 2

n − 1

(21.8)

Now, since ∆u = ut − up , integrating by parts in t and/or in x, we obtain

K= ϕ (ut )2 + ϕ u2p − 2 ϕ up ut

2 up+1 ϕt = ϕ (ut )2 + L + p+1 and

pJ = − ϕ ∇u · ∇(up ) + p ϕ ut u−1 |∇u|2

= ϕ (∆u)up + (∇ϕ · ∇u)up + p ϕ ut u−1 |∇u|2

1 p+1 p u ϕt + = −L − (∇ϕ · ∇u)u + p ϕ ut u−1 |∇u|2 . p+1 Substituting in (21.8), we obtain (21.5). (ii) For k < 0, the condition α, δ > 0 is equivalent to (n − 1)p/(n + 2) < −k < n/(n − 1). Such choice of k < 0 is clearly possible if p < pB . Proof of Proposition 21.5. Taking k as in Lemma 21.6(ii), we shall estimate the terms on the RHS of (21.5). Let us first prepare a suitable test-function. We


155

take ξ ∈ D(B1 × (−1, 1)), such that ξ = 1 in B1/2 × (−1/2, 1/2) and 0 ≤ ξ ≤ 1. By taking ϕ = ξ 4p/(p−1) , we have |∇ϕ| ≤ Cϕ(3p+1)/4p ,

|∆ϕ| ≤ Cϕ(p+1)/2p ,

|ϕt | ≤ Cϕ(3p+1)/4p ≤ Cϕ(p+1)/2p . (21.9)

We first observe that

|∇u|2 |∆ϕ| + ϕ−1 |∇ϕ|2 + |ϕt | ≤ η(I + L) + C(η),

η > 0.

(21.10)

Indeed, this follows from Young’s inequality and (21.9), by writing |∇u|2 |∆ϕ| + ϕ−1 |∇ϕ|2 + |ϕt |

2 ≤ ηϕ u−2 |∇u|4 + C(η)ϕ−1 u2 (|∆ϕ| + ϕ−1 |∇ϕ|2 + |ϕt |

≤ ηϕ u−2 |∇u|4 + C(η)ϕ1/p u2 ≤ ηϕ u−2 |∇u|4 + ηϕ u2p + C(η). Now fix ε > 0. Using Young’s inequality, (21.9) and (21.10), we estimate the RHS of (21.5) as follows:

ϕ (ut )2 + |ut | u−1 |∇u|2 + |∇u|2 |∆ϕ|

+ (up + |ut | + u−1 |∇u|2 )|∇u · ∇ϕ| + up+1 |ϕt |

≤ε ϕ u2p + u−2 |∇u|4

+ C(ε) ϕ(ut )2 + |∇u|2 (ϕ−1 |∇ϕ|2 + |∆ϕ|) + (ϕ−(p+1) |ϕt |2p )1/(p−1)

≤ 2ε(I + L) + C(ε) 1 + ϕ(ut )2 .

(21.11) Let us handle the last term in the above inequality. Multiplying equation (21.1) by ϕ ut , integrating by parts in x and t, and using Young’s inequality and (21.9), we get, for each η > 0,

up+1 |∇u|2 − − (∇ϕ · ∇u)ut ϕ (ut ) = ϕ ∂t p+1 2

up+1 |∇u|2 − ϕt − (∇ϕ · ∇u)ut = 2 p+1

1 1 1 |∇u|2 |ϕt | + |∇ϕ|2 ϕ−1 + ϕ (ut )2 + up+1 |ϕt |. ≤ 2 2 p+1 2

156


By (21.10) and (21.9), for all η > 0, it follows that

2 2 2 2 −1 up+1 |ϕt | ϕ (ut ) ≤ + |∇u| |ϕt | + |∇ϕ| ϕ p+1

ϕ−(p+1)/(p−1) |ϕt |2p/(p−1) ≤ η(I + L) + C(η) + η ϕ u2p + C(η) ≤ 2η(I + L) + C(η). (21.12) Combining (21.12), applied with η = ε(2C(ε))−1 , (21.11) and (21.5), we obtain αI + δL ≤ C(n, p)ε(I + L) + C(ε). Since α, δ > 0, by choosing ε = ε(n, p) sufficiently small, we conclude that I, L ≤ C. Remark 21.7. In the above proof, it is a priori possible to use Lemma 8.9 with values other than q = 0 (at the expense of additional complications in the estimate of the terms on the RHS of (21.5)). However, this does not seem to enable one to go beyond the condition p < pB . We now consider the case of a half-space. The following result was proved in [425]. Theorem 21.8. Let p > 1. Assume n ≤ 2, or p < (n − 1)(n + 1)/(n − 2)2 and n ≥ 3. Then the problem ut − ∆u = up , u = 0,

x ∈ Rn+ , t ∈ R, x ∈ ∂Rn+ , t ∈ R

(21.13)

has no positive bounded classical solution. Remarks 21.9. (a) We note that (n − 1)(n + 1)/(n − 2)2 is the exponent pB of Theorem 21.2 in dimension n − 1, and that this number is greater than pS . (b) Any nontrivial bounded classical solution u ≥ 0 of (21.13) is positive (this follows from the argument after (26.44) in the proof of Theorem 26.8). On the other hand, it can be shown [425] that if p < pB = n(n + 2)/(n − 1)2 , then problem (21.13) has no nontrivial nonnegative classical solution, bounded or not. Theorem 21.8 is a consequence of Theorem 21.2 and the following monotonicity result [425] concerning the more general problem ut − ∆u = f (u), x ∈ Rn+ , t ∈ R, (21.14) u = 0, x ∈ ∂Rn+ , t ∈ R, where f is a C 1 -function.


157

Theorem 21.10. Assume f : [0, ∞) → R is a C 1 -function satisfying f (0) = 0 and f (0) ≤ 0. Then the following statements hold true. (i) Each positive bounded solution u of (21.14) is increasing in x1 : x ∈ Rn+ , t ∈ R.

∂x1 u(x, t) > 0,

(ii) If there is a positive bounded solution of (21.14), then there exists a positive bounded solution of ut − ∆u = f (u),

x ∈ Rn−1 , t ∈ R.

(21.15)

For n = 1, equation (21.15) should be understood as the ordinary differential equation ut = f (u). The proofs of both statements (i) and (ii) use extensions of moving plane arguments of [150] to parabolic equations. A straightforward modification of the proof below shows that (i), (ii) hold for positive bounded solutions defined on (−∞, T ) for some T > 0. Proof. First we prove (i). We use the following notation. For λ > 0 let Tλ = {x ∈ Rn : 0 < x1 < λ}. For a function z defined on Rn+ let z λ and Vλ z be functions on Tλ defined by z λ (x) = z(2λ − x1 , x ), (21.16) Vλ z(x) = z λ (x) − z(x), where x = (x2 , x3 , . . . , xn ). Let u be a positive bounded solution of (21.14). Observe that for each λ > 0, v = Vλ u satisfies ⎫ vt − ∆v = cλ (x, t)v, x ∈ Tλ , t ∈ R, ⎪ ⎬ (21.17) v = 0, x1 = λ, x ∈ Rn−1 , t ∈ R, ⎪ ⎭ n−1 v > 0, x1 = 0, x ∈ R , t ∈ R, where

λ

c (x, t) = 0

1

f (u(x, t) + s(uλ (x, t) − u(x, t))) ds.

(21.18)

Our goal is to prove that the statement Vλ u(x, t) ≥ 0

(x ∈ Tλ , t ∈ R)

(S)λ

holds for each λ > 0. Once this is done, the maximum principle applied to the above linear problem guarantees that we have in fact the strict inequality in (S)λ and the Hopf boundary principle then gives ⏐ ⏐ 2∂x1 u(x, t)⏐ = −∂x1 Vλ u(x, t)⏐ >0 x1 =λ

for each λ > 0, proving (i). We shall use the following lemma [150].

x1 =λ

158


Lemma 21.11. Given any positive constants q, λ satisfying λ−2 π 2 > q, there ¯ λ such that exists a smooth function h on T

h(x) > 0,

x ∈ Tλ , ¯ λ, x∈T

h(x) → ∞,

¯ λ. |x| → ∞, x ∈ T

∆h + qh = 0,

⎫ ⎪ ⎬ ⎪ ⎭

(21.19)

Moreover, h satisfies h ≥ η for some constant η > 0. Proof. A straightforward computation shows that h = h(x1 , x2 , . . . , xn ) = cos

n π(2x − λ) , 1 cosh(εxi ) 2(λ + ε) i=2

satisfies the required properties for ε > 0 small. We first prove that (S)λ holds for λ small. Fix a positive constant γ and set q :=

sup

t∈R, x∈Rn +

f (u(x, t)) + γ.

(21.20)

If λ > 0 is sufficiently small, so that λ−2 π 2 > q, we can apply Lemma 21.11. With the resulting function h, we consider the problem satisfied by w := eγt v/h, where v = Vλ u. A simple computation using (21.17), (21.19) shows that wt − ∆w −

⎫ ⎪ x ∈ Tλ , t ∈ R, ⎪ ⎬ x ∈ ∂Tλ , t ∈ R, ⎪ ⎪ ¯ λ , t ∈ R. ⎭ |x| → ∞, x ∈ T

2∇h · ∇w − (γ + cλ (x, t) − q)w = 0, h w ≥ 0, w(x, t) → 0,

(21.21)

The choice of q implies γ + cλ − q ≤ 0 in Tλ × R. Applying the maximum principle on Tλ × (t0 , t), for each t0 < t, we obtain sup w− (x, t) ≤ sup w− (x, t0 ).

x∈Tλ

(21.22)

x∈Tλ

For v the above inequality means sup x∈Tλ

v− (x, t) v− (x, t0 ) ≤ e−γ(t−t0 ) sup . h(x) h(x) x∈Tλ

(21.23)

In view of boundedness of v = Vλ u, letting t0 → −∞ we obtain that v ≥ 0 everywhere. Using the maximum principle again we conclude that v is positive in Tλ × R, hence (S)λ holds.


159

In the next step we denote λ0 = sup{µ > 0 : (S)λ holds for all λ ∈ (0, µ)}.

(21.24)

As proved above, λ0 > 0. We now show by contradiction that λ0 = ∞. Assume λ0 < ∞. Then there is a sequence λk ≥ λ0 such that λk → λ0 and the set Zk := {(x, t) ∈ Tλk × R : Vλk u(x, t) < 0} is nonempty. Set mk := sup{u(y1 , x , t) : y1 ∈ (0, λk ), x ∈ Rn−1 , t ∈ R, and there exists x1 ∈ (0, λk ) such that (x1 , x , t) ∈ Zk }. We consider the following two possibilities. (a) mk → 0, (b) passing to a subsequence we have mk ≥ ε0 for some ε0 > 0. First assume that (b) holds. Then there are sequences xk1 , y1k ∈ (0, λk ), z k ∈ R , tk ∈ R such that Vλk u(xk1 , z k , tk ) < 0 and u(y1k , z k , tk ) ≥ ε0 . We may assume that xk1 → a and y1k → b for some a, b ∈ [0, λ0 ] . Consider the functions n−1

uk (x, t) := u(x1 , x + z k , t + tk ),

x = (x1 , x ) ∈ Rn , t ∈ R.

Each of them is a positive solution of (21.14) satisfying Vλk uk (xk1 , 0, 0) < 0, uk (y1k , 0, 0) ≥ ε0 and Vλ0 uk ≥ 0 in Tλ0 × R (the last inequality follows from the definition of λ0 and continuity). Moreover, the sequence uk is uniformly bounded. Using standard parabolic estimates, one shows that if uk is replaced by a subsequence, then it converges in C 2,1 (Rn+1 ) to a nonnegative solution u˜ of (21.14). The above properties of uk imply Vλ0 u ˜(a, 0, 0) ≤ 0, u ˜(b, 0, 0) ≥ ε0 , and Vλ0 u ˜≥0 in Tλ0 × R. Since u˜ is nontrivial and f (0) = 0 the maximum principle guarantees that u ˜ is positive everywhere. Consequently, v˜ := Vλ0 u ˜ solves the corresponding problem (21.17) with λ = λ0 and therefore v˜ > 0 in Tλ0 ×R. It follows in particular that necessarily a = λ0 . By the Hopf principle, ⏐ ˜(x1 , 0, 0)⏐x1 =λ0 > 0. 2˜ ux1 (λ0 , 0, 0) = −∂x1 Vλ0 u Consequently, u ˜x1 (x1 , 0, 0) is bounded below by a positive constant on an interval around λ0 and this remains valid if u ˜ is replaced by uk for k large. That is, there is δ > 0 such that ∂x1 u(x1 , z k , tk ) = ∂x1 uk (x1 , 0, 0) > 0,

x1 ∈ [λ0 − δ, λ0 + δ],

(21.25)

for all sufficiently large k. However, since 2λk −xk1 > xk1 both belong to [λ0 −δ, λ0 +δ] for large k, (21.25) contradicts the assumption that Vλk u(xk1 , z k , tk ) < 0.

160


We have shown that (b) leads to a contradiction. Assume now that (a) holds. Consider problem (21.17) with λ = λk and k sufficiently large. We are going to apply the maximum principle on the set Zk (assumed to be nonempty). The boundary conditions in (21.17) imply v = Vλk u = 0 on ∂Zk . Next observe that property (a), in conjunction with (21.18) and the definition of mk , guarantees that for q˜k := sup cλk (x, t) (x,t)∈Zk

we have lim sup q˜k ≤ 0. k→∞ 2 Fix k so large that q := q˜k + γ < λ−2 k π , where γ is some positive constant, and set λ = λk . Apply Lemma 21.11 and let h be the resulting function. As in our arguments above, w := eγt v/h satisfies problem (21.21). This time we know that γ + cλ − q ≤ 0 on Zk only. However, since v vanishes on ∂Zk , we can still apply the maximum principle on Zk to conclude that (21.22) holds and, consequently, that v ≥ 0 in Zk . This of course contradicts the definition of Zk . Thus possibility (a) leads to a contradiction, too, which proves that λ0 = ∞. We have completed the proof of assertion (i). To prove assertion (ii), let u be a positive bounded solution of (21.14). For k = 1, 2, . . . consider the functions

uk (x1 , x , t) := u(x1 + k, x , t),

(x1 , x , t) ∈ (−k, ∞) × Rn−1 × R.

Each of them solves the equation ut −∆u = f (u) on its domain. Since the sequence is uniformly bounded, using parabolic estimates one shows that a subsequence of uk converges uniformly on each compact to a bounded nonnegative solution u ˜ of ut − ∆u = f (u) on Rn × R. From the monotonicity of u proved in (c1), we further conclude that u˜ is positive and independent of x1 . This proves assertion (ii). Remark 21.12. Liouville-type result under a decay assumption at −∞. A different parabolic Liouville-type theorem was proved in [367] for 1 0,

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

x ∈ Ω,

⎫ ⎪ ⎬ ⎪ ⎭

(22.1)

where Ω is bounded and p > 1. We have seen that (22.1) admits both: • finite-time blow-up solutions — cf. Section 17; and • global bounded solutions (in particular small data solutions decaying to 0 as t → ∞, and stationary solutions if p < pS ) — cf. Sections 19 and 6. In order to understand the structure of solutions of problem (22.1), it is natural to investigate whether or not it admits other kinds of solutions (namely global unbounded classical solutions). In the case when all global solutions are bounded, one can further look for an a priori estimate of global solutions, that is, an estimate of the form sup u(t) ∞ ≤ C( u0 ∞ ),

with C bounded on bounded sets.

(22.2)

t≥0

This estimate means that, given K > 0, there exists C = C(K) > 0 such that all global solutions with u0 ∞ ≤ K satisfy u(t) ∞ ≤ C for all t ≥ 0. The existence of stronger universal bounds (independent of initial data) will be studied in Section 26. We shall see that the answers to these questions (boundedness of global solutions vs. existence of unbounded global solutions, existence vs. nonexistence of a priori estimates) strongly depend on the value of p. Besides the intrinsic interest of such questions, let us emphasize that the results and techniques of proofs have many applications (see Section 28 and cf. also, for instance, Theorem 22.13, the proof of Theorem 23.7, Remark 23.14, and the proof of Theorem 27.2).

22.1. A priori bounds in the subcritical case In this subsection we establish a priori estimates of global solutions in the subcritical case p < pS . As we shall see below, the assumption p < pS is necessary for the bound (22.2) (at least if Ω is starshaped). Theorem 22.1. Assume Ω bounded and 1 < p < pS . Then the bound (22.2) is true for all global solutions of (22.1). This result was proved in [243] for u0 ≥ 0 and in [437] in the general case. Earlier partial results in that direction can be found in [403], [396], [114], [186].

162


We shall first prove the above theorem under the additional assumption u0 ≥ 0. This proof is due to [243] and it is based on rescaling arguments (similar to those used in the proof of Theorem 12.1) and on the energy functional E. Proof of Theorem 22.1 for nonnegative solutions. Assume that the bound (22.2) does not hold for global nonnegative solutions. Then there exist tk > 0 and u0,k ≥ 0 such that u0,k ∞ ≤ C0 and the solutions uk := u(·; u0,k ) satisfy Mk := uk (xk , tk ) = sup{uk (x, t) : x ∈ Ω, t ∈ [0, tk ]} → ∞ as k → ∞.

(22.3)

Let ψ be the solution of ψ(0) = C0 , ψ (t) = ψ p (t) for t > 0, and let δ = δ(C0 , p) > 0 be such that ψ(δ) = 2C0 . Then the comparison principle shows uk (x, t) ≤ ψ(t) ≤ 2C0 for all x ∈ Ω and t ∈ [0, δ], hence tk ≥ δ for k large enough. Now the variationof-constants formula (15.5) and the estimate e−tA w 1,2 ≤ C1 t−1/2 w 2 ≤ C2 t−1/2 w ∞ easily imply uk (δ/2) 1,2 ≤ C, where by C we denote a positive constant which does not depend on k. This estimate and Theorem 17.6 guarantee 0 ≤ E uk (δ/2) < C. (22.4) −(p−1)/2

Denote νk := Mk

vk (y, s) :=

and set 1 uk (xk + νk y, tk + νk2 s), Mk

(y, s) ∈ Qk ,

where Qk := {(y, s) : (xk + νk y, tk + νk2 s) ∈ Ω × (0, tk )}. Then 0 ≤ vk (y, s) ≤ 1 = vk (0, 0) and vk solves the problem ∂s vk − ∆y vk = vkp vk = 0

in Qk , for (y, s) ∈ ∂Qk , −

tk < s < 0. νk2

Denote dk := dist (xk , ∂Ω). Passing to a subsequence we may assume that one of the following cases occurs: (i) dk /νk → ∞, (ii) dk /νk → c ≥ 0. Case (i). Set ˜ k := {(y, s) : |y| < dk , − tk < s < 0}. Q νk 2νk2 ˜ k ⊂ Qk and the parabolic Lp -estimates (see Appendix B) together with Then Q standard embedding theorems guarantee the boundedness of vk in the space C α,α/2 (Rn × (−∞, 0)) for some α > 0. Consequently, given β ∈ (0, α), we may assume vk → v in C β,β/2 (Rn × (−∞, 0)), where v is a classical solution of vs − ∆v = v p

in Rn × (−∞, 0)

(22.5)

22. A priori bounds

163

satisfying 0 ≤ v ≤ v(0, 0) = 1. Now setting σ := 4/(p − 1) − (n − 2) > 0 and using (22.4) we obtain

˜k Q

2

|∂s vk | dy ds =

tk

νkσ tk /2

2

|x−xk | −c} (see [243] for details and cf. also the proof of Theorem 12.1). As in Case (i) we obtain vs ≡ 0, hence (22.6) contradicts Theorem 8.2. Now we are going to prove Theorem 22.1 in the general case. The proof is based on energy estimates, interpolation, maximal regularity, and a bootstrap argument. The first two ingredients were first used in [114], where the authors had to assume p(3n − 4) < (3n + 8). The bootstrap argument (which enables one to get rid of this additional assumption on p) appeared for the first time in [437]. Proof of Theorem 22.1. Let M > 0 and let u be a global solution of (22.1) with u0 ∞ ≤ M . We shall denote by C, C1 , C2 various positive constants which depend on u0 through M only and which may vary from step to step. Also, by the word “bounded”, we mean that the bound depends on u0 through M only. As in the proof of Theorem 22.1 for nonnegative solutions, there exists δ = δ(M ) > 0 such that u(t) ∞ ≤ C for t ∈ [0, δ] and u(δ) 1,2 ≤ C. Hence we may assume u0 1,2 ≤ C. Since u is global, Theorem 17.6 and Remark 17.7 guarantee 0 ≤ E u(t) ≤ C,

t ≥ 0,

(22.7)

and u(t) 2 ≤ C, Consequently,

0

∞

Ω

t ≥ 0.

u2t dx dt = E(u0 ) − lim E u(t) ≤ C. t→∞

(22.8)

(22.9)

This estimate and (22.8) guarantee that u is bounded in W 1,2 [t, t + 1], L2 (Ω) uniformly for t ≥ 0.

(22.10)

164


Multiplying the equation in (22.1) by u we get

uut dx = −

Ω

Ω

|∇u(t)|2 dx +

Ω

|u(t)|p+1 dx = −2E u(t) +C

Ω

|u(t)|p+1 dx,

so that, for each r ≥ 1, (22.7) implies

t

t+1

Ω

t+1

r |u|p+1 dx ds ≤ C 1 +

Ω

t

r |uut | dx ds ,

t ≥ 0.

(22.11)

Notice that Cauchy’s inequality, (22.8) and (22.9) imply

t+1

Ω

t

2 |uut | dx ds ≤ t

t+1

Ω

u2 dx u2t dx ds ≤ C, Ω

hence we infer from (22.11) that u is bounded in L(p+1)r [t, t + 1], Lp+1 (Ω) uniformly for t ≥ 0,

(22.12)

if r = 2. Now (22.10), (22.12) and (51.6) guarantee u(t) q ≤ Cq

for all t ≥ 0 and q < qr := p + 1 −

p−1 , r+1

(22.13)

where r = 2. Theorem 15.2 or Remark 51.37(iii) (see also Theorem 16.4) imply our assertion provided supt≥0 u(t) q ≤ C for some q > n(p − 1)/2. This estimate follows from (22.13) if p−1 n (p − 1) < p + 1 − . (22.14) 2 r+1 If r = 2, then (22.14) is equivalent to p(3n − 4) < 3n + 8 (which is the condition of [114]). In what follows we shall use a bootstrap argument to show that (22.13) is true for any r ≥ 2. Since (22.14) reduces to p < pS if r → ∞ we shall be done. We already know (see the beginning of the proof) that there exists δ = δ(M ) > 0 such that u(t) ∞ ≤ C for t ∈ [0, δ]. We claim that for any interval I ⊂ [0, ∞) of length δ there exists τ ∈ I such that u(τ ) BC 2 ≤ C. In fact, let I = (t, t + δ) and set J := (t, t + δ/2). Then (22.7) and (22.12) with r = 2 imply

J

Ω

2 2 |∇u|2 dx ds ≤ C 1 + |u|p+1 dx ds ≤ C, J

Ω

hence there exist C1 > 0 and τJ ∈ J such that u(τJ ) 1,2 ≤ C1 . The well-posedness of (22.1) in W01,2 (Ω) (see Example 51.10 and Theorem 51.7) guarantees the existence of η = η(C1 ) > 0 and C2 = C2 (C1 ) > 0 such that η < δ/2 and u(s) 1,2 ≤ C2 for all s ∈ [τJ , τJ + η]. Now standard regularity results (see Example 51.27 and

22. A priori bounds

165

Appendix B) guarantee u(τJ + η) BC 2 ≤ C, where C = C(η, C2 ). Hence it is sufficient to put τ := τJ + η. Next assume that r ≥ 2 and

t+1

r |u|p+1 dx ds ≤ C

Ω

t

for all t ≥ δ.

(22.15)

We shall show that the same estimate is true with r replaced by r˜ for any r˜ ∈ (r, r + 2). Since (22.15) is true for r = 2, an obvious bootstrap argument will guarantee (22.15) for any r ≥ 2. Since (22.15) implies (22.13), the conclusion will follow. Hence let r˜ ∈ (r, r + 2), and consider q < qr (q close to qr ). Set pˆ := (p + 1)/p,

θ :=

p+1q−2 ∈ (0, 1), p−1 q

β := 2/(˜ r(1 − θ)) > 1.

Choose t ≥ δ and τ ∈ (t − δ, t) such that u(τ ) BC 2 ≤ C. Using successively (22.11), H¨ older’s inequality and (22.13), interpolation, H¨ older’s inequality, (22.9), the maximal regularity property (51.8), and u(τ ) BC 2 ≤ C, we obtain

τ

t+1

|u|

p+1

Ω

r˜ dx ds ≤ C 1 + t+1

≤C 1+

ut rq˜ ds

τ

τ

t+1

≤C 1+ τ

≤C 1+

τ

r˜(1−θ)

ut rp˜ˆθ ut 2

t+1

ut rp˜ˆθβ ds

≤ C 1 + u(τ ) BC 2 +

t+1

≤C 1+

τ

t+1

Ω

ds

1/β

t+1

τ

uut r1˜ ds

τ

t+1

ut 22 ds

|u|p−1 u rp˜ˆθβ ds

1/β

1/β

r˜θβ p/(p+1) 1/β . |u|p+1 dx ds

Since r˜ < r + 2 we can choose q close to qr so that r˜θβ p/(p+ 1) < r˜. Consequently, (22.15) and the last estimate guarantee (22.15) with r replaced by r˜. Remarks 22.2. Uniform bound in terms of the energy. Let Ω be bounded, p < pS , u be a solution of (22.1) on the time interval [0, T ), T < ∞ and M > 0. (i) If u0 ∞ ≤ M,

E(u(t)) ≥ −M

and u(t) 2 ≤ M,

t ∈ [0, T ),

166


then u(t) ∞ ≤ C(M ),

t ∈ [0, T ).

This follows from the above proof of Theorem 22.1 by replacing the interval [0, ∞) with [0, T ). (ii) If u0 ∞ ≤ M and E(u(t)) ≥ −M, t ∈ [0, T ), (22.16) then u(t) ∞ ≤ C(M, T ),

t ∈ [0, T ).

In fact, (22.16), (17.10) and Gronwall’s inequality guarantee u(t) 2 ≤ C(K, M, T ),

t < T,

where K stands for a bound on u0 2 . Therefore the assertion follows from (i). Remark 22.3. Cauchy problem. Let Ω = Rn and 1 < p < pS . Then (22.2) is still true for positive radial solutions (and for all positive solutions provided p < pB ), see Theorem 26.9 below. Using the same approach as in the proof of Theorem 22.1 for nonnegative solutions one can also show weaker estimate u(t) ∞ ≤ C( u0 ∞ , E(u0 ))

for all t ≥ 0

for nonnegative initial data u0 ∈ H 1 (Rn ). If we consider problem (17.1) with λ < 0, 1 < p < pS , Ω = Rn and initial data in X := L∞ ∩ L(p+1)/p ∩ H 1 (Rn ), then any global (not necessarily positive) solution satisfies the estimate u(t) X ≤ C( u0 X ) for all t ≥ 0, see [440]. The same result remains true for X := H 1 (Rn ) or X := L∞ ∩ H 1 (Rn ) due to a recent result in [537].

22.2. Boundedness of global solutions in the supercritical case Consider problem (22.1), where Ω is a ball and u0 ∈ L∞ (Ω) is a nonnegative radial function. If the solution u is global and p < pS , then Theorem 22.1 guarantees the boundedness of u, i.e.: sup u(t) ∞ < ∞. (22.17) t≥0

In this subsection we show that this property remains true for p > pS . Let us emphasize that the bound (22.17) does not imply the stronger a priori estimate (22.2). In fact, we will see in Theorem 28.7(iv) that estimate (22.2) fails whenever p ≥ pS .

22. A priori bounds

167

Theorem 22.4. Assume p > pS , Ω = BR , and let u0 ∈ L∞ (Ω), u0 ≥ 0, be radial. If the solution u of (22.1) is global, then property (22.17) is true. We will prove Theorem 22.4 only under the additional assumption p < pL , where ∞ if n ≤ 10, pL := (22.18) 6 1 + n−10 if n > 10. Notice that pL > pJL if n > 10, where pJL is defined in (9.3). If n > 10 and p > pJL , then the statement of Theorem 22.4 follows from [378]. See also Remark 23.13 for an alternative proof, due to [125], in the case p < pJL . In the proof of the above theorem we will need the following result. Proposition 22.5. Let pS 0,

satisfying limy→∞ ϕ(y)y 2/(p−1) = B ∈ (0, cp ). Given T ∈ R, set √ w(r, t) := (T − t)−1/(p−1) ϕ r/ T − t

for r ≥ 0, t < T,

w(r, T ) := lim w(r, t)

for r > 0.

t→T +

Then wt − wrr −

n−1 wr = wp , r w(r, T ) = Br−2/(p−1) ,

r > 0, t < T, r > 0.

The function w in the preceding proposition is a backward self-similar solution of problem (22.1). Proposition 22.5 follows from [105], [325] (if p < pJL ) and [326] (if p ≥ pJL ). Since the corresponding proofs are quite long, we will prove it just in the case p = 2 when one can find an explicit formula for ϕ (due to [225]). Let us note that in the case p < pJL there exist infinitely many functions ϕ with the required properties, and that both numerical and analytic results indicate that such solutions do not exist if p > pL , see [418], [376]. Proof of Proposition 22.5 for p = 2. Let p = 2 and 6 < n < 16 (this corresponds to pS < 2 < pL ). Set ϕ(y) :=

B A + , 2 2 (a + y ) a + y2

168


where A := 48(10D − (n + 14)),

B := 24(D − 2),

D :=

! 1 + n/2.

It is easy to see that ϕ possesses the required properties. In particular, B < c2 = 2(n − 4). The following proof is due to [232]. Proof of Theorem 22.4 for p < pL . Let U∗ (r) = cp r−2/(p−1) be the singular solution defined in (3.9). Assume on the contrary that u is a global unbounded classical solution. Since u is radial (see Remark 16.2(i)), we have u(x, t) = U (|x|, t) for some U : [0, R] × (0, ∞) → R.

U∗ (δ) U∗

Uδ

U (·, t1 ) 0

R

Figure 9: Graphs of U∗ , Uδ , U (·, t1 ) if z(U (·, t0 ) − U∗ ) = 0.

Assume z(U (·, t0 ) − U∗ ) ≤ 1 for some t0 > 0, where z(ψ) denotes the zero number of the function ψ in the interval (0, R) (see Appendix F). Since U (0, t0 ) < U∗ (0) = ∞ and 0 = U (R, t0 ) < U∗ (R) we have z(U (·, t0 ) − U∗ ) = 0. Consequently U (·, t0 ) ≤ U∗ . Fix t1 > t0 . Then by the maximum principle there exists ε > 0 such that U (·, t1 ) ≤ U∗ − ε and we may find δ > 0 such that the function Uδ (r) := U∗ (r + δ) lies above U (·, t1 ). Since −Uδ −

n−1 Uδ ≥ Uδp , r

0 < r < R,

22. A priori bounds

169

with −Uδ (0) > 0 and Uδ (R) > 0, it follows from the maximum principle that U (r, t) ≤ Uδ (r) ≤ U∗ (δ) for all r ∈ [0, R] and t ≥ t1 , see Figure 9. However, this contradicts our assumptions. Consequently, z(U (·, t) − U∗ ) ≥ 2 for all t > 0.

(22.19)

Fix τ > 0 small. Since U (r, τ ) > 0 for r ∈ [0, R) and Ur (R, τ ) < 0 by the maximum principle, we can find T large enough such that the backward self-similar solution w from Proposition 22.5 satisfies z(U (·, τ ) − w(·, τ )) = 1, see Figure 10.

U (·, τ ) w(·, τ ) 0

R Figure 10: Graphs of U (·, τ ), w(·, τ ).

Consequently, Theorem 52.28 implies z(U (·, t) − w(·, t)) ≤ 1 for all t ∈ [τ, T ).

(22.20)

However, w(·, T ) < U∗ so that (22.19) implies (see Figure 11) z(U (·, t) − w(·, t)) ≥ 2 for t < T, t close to T, which contradicts (22.20). Remarks 22.6. (a) Cauchy problem. Let Ω = Rn , u0 ∈ C(Rn ) be nonnegative, bounded and radially symmetric and let the solution u of (22.1) be global. Then the boundedness of u is known in each of the following (supercritical) cases: (i) pS < p < pL , u0 ∈ C 1 has compact support and its local minima are bounded away from zero (see [374]); (ii) pS pJL , u0 (x) ≤ U∗ (|x|) − c0 |x|−|α| , where c0 > 0, & 1 , α = −(n − 2) + β 2 − 4(p − 1)cp−1 p 2 β = n − 2 − 4/(p − 1) and cp , U∗ are defined in (3.9) (see [378]).

170


On the other hand, if p ≥ pJL it was shown in [428] that there exists a continuous radial function u0 satisfying 0 < u0 (x) ≤ U∗ (|x|) such that the corresponding solution u is global and unbounded. See also Section 29 for more precise information on the asymptotic behavior of such solutions.

U∗

U (·, t) w(·, T )

0

R

Figure 11: Graphs of U∗ , w(·, T ), U (·, t) if t is close to T .

(b) Inhomogeneous boundary conditions. The result in Theorem 22.4 is sensitive to the boundary conditions. Indeed consider problem (22.1) in Ω = B1 with the boundary conditions replaced by u = a > 0 on ∂Ω × (0, ∞). Note that this is equivalent to problem (14.1) with f (v) := (v + a)p

(resp., f (w) := λ(w + a)p , λ = a1−p )

via the transformation v = u − a (resp., w = a−1 u − 1). If pS pJL and a = cp , where cp is given by (3.9), then there exist unbounded global solutions [316]. More precisely, any initial data u0 ∈ L∞ (Ω) satisfying 0 ≤ u0 (x) ≤ u∗ (x) = U∗ (|x|) gives rise to an unbounded global classical solution, which stabilizes to u∗ as t → ∞. The rate of approach has been studied in [164]. Remark 22.7. Eventual radial monotonicity of global radial solutions. The following property was shown in [395] (actually for more general nonlinearities). Let p > 1, Ω = BR , and assume that u ≥ 0 is a radial, global classical

22. A priori bounds

171

solution of (22.1) (not necessarily bounded). Then there exists t0 > 0 such that u becomes radial nonincreasing for t ≥ t0 . Remark 22.8. Exponential nonlinearity. Consider the problem ⎫ x ∈ Ω, t > 0, ut − ∆u = λeu , ⎪ ⎬ u = 0, x ∈ ∂Ω, t > 0, ⎪ ⎭ x ∈ Ω, u(x, 0) = u (x),

(22.21)

0

∞

where Ω ⊂ R , λ > 0 and u0 ∈ L (Ω). Embedding theorems, intersection properties of stationary solutions of (22.21) for Ω = Rn (see [510]), bifurcation diagrams for stationary solutions of (22.21) for Ω being a ball (see Remark 6.10(ii)), and several results for time dependent solutions of (22.21) indicate that the cases n ≤ 2, 3 ≤ n ≤ 9 and n ≥ 10 correspond to the cases p < pS , pS pJL for problem (22.1), respectively. In fact, many (but not all) proofs in this chapter can be adapted to the case of exponential nonlinearity. Unfortunately, similarly as in the case of the power nonlinearity, a lot of basic questions for (22.21) remain open. For example, a priori bounds (22.2) are known if n = 1 (see [440]) but not for n = 2. On the other hand, the boundedness of global solutions of (22.21) with general bounded Ω is true if n ≤ 2 (see [186], [109]) and the boundedness of radial global solutions in a ball for 3 ≤ n ≤ 9 is known as well (see [198]). We refer to [188] for a survey on problem (22.21). n

22.3. Global unbounded solutions in the critical case The following result due to [232] shows that the situation in the Sobolev critical case is very different from both the subcritical and the supercritical cases. To formulate it, we introduce the notion of threshold solution. Let ϕ ∈ L∞ (Ω) be a fixed nonnegative function, ϕ ≡ 0, α > 0, and set u0 = αϕ. If α is small enough, then the solution u = u(t; αϕ) of (22.1) exists globally. (Moreover u(t) → 0 in L∞ (Ω), as t → ∞.) This follows from Theorem 19.2. We may thus define α∗ = α∗ (ϕ) := sup{α > 0 : Tmax (αϕ) = ∞}. ∗

∗

(22.22) ∗

Note that α ∈ (0, ∞) due to Remark 17.7(v). The function u = u(t; α ϕ) is called the threshold solution (associated with ϕ), due to the fact that u∗ lies on the borderline between blow-up and global existence. Further properties of threshold and non-threshold solutions will be studied in Sections 27 and 28. Theorem 22.9. Consider problem (22.1) with p = pS and Ω = BR . Let u0 = α∗ ϕ, where ϕ(x) = Φ(|x|), with 0 ≤ Φ ∈ L∞ (0, R), Φ nonincreasing, and α∗ defined by (22.22). Then the solution u∗ is global and unbounded. More precisely, lim u∗ (t) q = ∞

t→∞ ∗

lim inf u (t) pS +1 < ∞. t→∞

for any q > pS + 1, (22.23)

172


Proof. First assume that u∗ blows up in finite time T . Let αk α∗ , α1 > 0 and let vk , k = 1, 2, . . . , denote the (global) solution with the initial data αk ϕ. The solutions u∗ , vk are radial and radially decreasing, u∗ (x, t) = U ∗ (|x|, t), vk (x, t) = Vk (|x|, t). Let t1 ∈ (0, T ) be fixed. Since V1 is positive on Q1 := [0, R/2]×[t1, T +1], there exists c1 > 0 such that Vk ≥ V1 > c1 on Q1 for any k. In addition, U ∗ ≥ Vk on [0, R/2] × [t1 , T ). The functions U ∗ (·, t1 ) and Vk (·, t1 ), k = 1, 2, . . . , are uniformly bounded in C 1 ([0, R]). In particular, there exists c2 > 0 such that Vk (·, t1 ) ≤ U ∗ (·, t1 ) < c2 . Let UM be the unique positive solution of (9.2) satisfying UM (0) = M , see Theorem 9.1. Since UM (R/2) → 0 as M → ∞, there exists M1 > 0 such that UM (R/2) < c1 for all M ≥ M1 . Enlarging M1 if necessary we may also assume that the function M → UM (R/2) is decreasing for M ≥ M1 , and that M1 ≥ M0 (R/2), where the function M0 is defined in Remark 9.3. Finally, since UM (r) → −∞ as M → ∞ uniformly on {r : UM (r) ∈ [c1 , c2 ]}, we may assume that UM intersects any of the functions Vk (·, t1 ), k = 1, 2, . . . , exactly once in [0, R/2], for all M ≥ M1 , see Figure 12.

UM

c2 U ∗ (·, t1 )

Vk (·, t1 ) c1 0

R/2

Figure 12: Graphs of U ∗ (·, t1 ), Vk (·, t1 ), UM .

Consequently, denoting by z(ψ) the zero number of the function ψ in the interval [0, R/2] (see Appendix F), we have z(UM − Vk (·, t1 )) = 1,

k = 1, 2, . . . , M ≥ M1 .

(22.24)

22. A priori bounds

173

˜ be the solution of the problem Fix M2 > M1 (see Figure 13) and let U ˜rr − n − 1 U ˜ p, ˜t − U ˜r = U U r ˜ (R/2, t) = UM1 (R/2), ˜r (0, t) = 0, U U ˜ (r, 0) = max(UM2 (r), UM1 (r)), U

r ∈ (0, R/2), t > 0, t > 0, r ∈ (0, R/2).

UM2

UM1 Vk (·, t1 )

c1

0

R/2 Figure 13: Graphs of Vk (·, t1 ), UM1 , UM2 .

˜ (r, 0) ˜r ≤ 0 for t > 0 by Proposition 52.17. Moreover, the function U We have U is a subsolution for this problem, hence ˜t ≥ 0, U

for t > 0

(22.25)

(in fact this follows from a simple modification of the proof of Proposition 52.19). We claim that: ˜ blows up in a finite time T˜ . U (22.26) ˜ (r, t). ˜ exists globally and let V˜ (r) := limt→∞ U Assume for contradiction that U First we have V˜ (r) < ∞ for 0 < r ≤ R/2 (otherwise we would have ˜ (r, t) = ∞ lim U

t→∞

uniformly on [0, r0 ) for some r0 > 0,

174


which would imply finite-time blow-up by an eigenfunction argument — cf. the proof of Theorem 17.1). It follows from (22.25) and Lemma 53.10 that V˜ ∈ ˜p ˜ C 2 ((0, R/2]) is a solution of V˜rr + n−1 r Vr + V = 0 on 0 < r ≤ R/2. Moreover we have V˜ > max(UM2 , UM1 ) and V˜r ≤ 0 on (0, R/2), and V˜ (R/2) = UM1 (R/2). But since M1 ≥ M0 (R/2), Remark 9.3 implies V˜ (r0 ) = UM1 (r0 ) for some r0 ∈ (0, R/2): a contradiction. Consequently, (22.26) is true. Fix β ≥ 1 such that Tβ := T˜β 1−p < 1, set Rβ := β −(p−1)/2 R/2 and notice that ˜ (rβ (p−1)/2 , tβ p−1 ) is a solution of the problem W (r, t) := β U Wt − Wrr −

n−1 Wr = W p , r Wr (0, t) = 0, W (Rβ , t) = UβM1 (Rβ ), W (r, 0) = max(UβM2 (r), UβM1 (r)),

r ∈ (0, Rβ ), t > 0, t > 0, r ∈ (0, Rβ ),

which blows up at time Tβ < 1. Since U ∗ blows up at time T and is decreasing in r, and since Vk (0, t) → U ∗ (0, t) as k → ∞ for any t < T , there exist k and t0 ∈ (t1 , T ) such that Vk (0, t0 ) > UβM2 (0) = βM2 > UβM1 (0). Notice also that Vk (R/2, t) > c1 > UβM1 (R/2) > UβM2 (R/2) for all t ∈ [t1 , T + 1]. Now (22.24) and the monotonicity of the zero number (see Theorem 52.28) imply Vk (·, t) > UβMi on [0, R/2] for all t ∈ [t0 , T + 1] and i = 1, 2, hence Vk (·, t0 ) > W (·, 0) on [0, Rβ ]. Since Vk (Rβ , t) > UβM1 (Rβ ) = W (Rβ , t) for t ∈ [t0 , T + 1], we have Vk (·, t + t0 ) > W (·, t) whenever t > 0, t + t0 ≤ T + 1. However W blows up at Tβ < 1 which yields a contradiction. Consequently, u∗ is global. Next assume that lim inf t→∞ u∗ (t) q < ∞ for some q > pS + 1. Then there exist C > 0 and tk → ∞ such that u∗ (tk ) q < C. Fix γ ∈ (1/2, 1). Since q > n(pS − 1)/2, Theorem 51.25, Remark 51.26(vi) and Example 51.27 (with z = q and α = 1) show the existence of δ > 0 such that the sequence {u∗ (tk + δ)} is bounded in W 2γ,q ∩W01,q (Ω), hence relatively compact in X := H01 ∩Lq (Ω). Next Example 51.28 and Proposition 53.6 guarantee that a subsequence of {u∗ (tk + δ)} converges in X to an equilibrium v. The maximum principle implies v ≥ 0. Assume v = 0. Then α∗ ϕ belongs to the domain of attraction of the zero solution (which is an open set) hence the same is true for αϕ with some α > α∗ . But this contradicts the definition of α∗ . Consequently, v > 0. However, this contradicts Corollary 5.2. Finally assume limt→∞ u∗ (t) pS +1 = ∞. Then estimate (17.9) shows that the 2 L -norm of u∗ (t) has to blow up in finite time which is absurd. In fact, Theorem 17.6 also shows that the energy of u∗ (t) remains bounded and the proof of Theorem 22.1 guarantees that the norm of u∗ in L4 ((t, t + 1), H 1 (Ω)) is bounded uniformly with respect to t ≥ t0 > 0. Remarks 22.10. (i) Grow-up rates in a ball. The assertion in Theorem 22.9 remains true even if the function Φ is not monotone (cf. the proof of Theorem 28.7

22. A priori bounds

175

below). In addition, if R = 1, then all such global unbounded radial positive solutions exhibit the following asymptotic behavior as t → ∞ (see [223]): π2 t(1 + o(1)) 4√ = 2 t(1 + o(1))

log u∗ (·, t) ∞ =

if n = 3,

log u∗ (·, t) ∞

if n = 4,

u∗ (·, t) ∞ = γ0 t(n−2)/2(n−4) (1 + o(1))

if n ≥ 5,

where the constant γ0 > 0 depends only on the spatial dimension n. (ii) Grow-up rates for the Cauchy problem. Let Ω = Rn , p = pS , γ > 2/(p − 1), Φ : [0, ∞) → (0, ∞) satisfy Φ(r) ∼ Cr−γ for r large and consider initial data u0 (x) = α∗ Φ(|x|), where α∗ has the same meaning as in Theorem 22.9. If n = 3, then formal matched asymptotics expansions ([306]) suggest that for t large, u(t) ∞ behaves like t(γ−1)/2 or t1/2 provided γ ∈ (1/2, 2) or γ > 2, respectively. On the other hand, the same arguments indicate that this solution remains bounded if n > 3. (iii) Nonuniformity of the smoothing time in the critical Lq -space. Let ∗ u be the global unbounded solution from Theorem 22.9. Fix C1 > 0 and tk → ∞ such that u∗ (tk ) pS +1 < C1 . Since pS + 1 = qc = n(pS − 1)/2, Remark 15.4(i) guarantees that problem (22.1) is well-posed in LpS +1 (Ω) and, in particular, there exist C2 > 0 and Tk > 0 such that u∗ (tk + t) ∞ ≤ C2 u∗ (tk ) pS +1 t−α ≤ C1 C2 t−α ,

t ∈ (0, Tk ),

where α = (n/2)(pS + 1) = (n − 2)/4, cf. (15.2). Since u∗ (tk + t) ∞ → ∞ for any t ≥ 0, we see that Tk → 0 (in spite of the fact that Tmax (u∗ (tk )) = ∞ and u∗ (tk ) pS +1 < C1 ).

22.4. Estimates for nonglobal solutions The estimates in Theorem 22.1 can be extended to nonglobal solutions in the following way. Proposition 22.11. Assume Ω bounded, 1 0 and u0 ∞ ≤ K. If u is the solution of (22.1), then u(t) ∞ ≤ C(δ, K)

for all t ∈ [0, Tmax (u0 ) − δ),

(22.27)

(where Tmax (u0 ) − δ := ∞ if Tmax (u0 ) = ∞) and E u(t) → −∞

as t → Tmax (u0 ),

whenever Tmax (u0 ) < ∞.

(22.28)

176


Remark 22.12. Related blow-up rate estimates of the form u(t) ∞ ≤ M (T − t)−1/(p−1) ,

0 < t < T := Tmax (u0 ),

will be proved in Section 23. In some cases the constant M will be known to depend on u0 through a bound on u0 ∞ only (see e.g. Remark 23.9). However, up to now, such a priori estimates are not available under the general assumptions of Proposition 22.11 (one has to assume either Ω convex, or u ≥ 0 and a stronger restriction on p) Proof of Proposition 22.11. If Tmax (u0 ) = ∞, then estimate (22.27) follows from Theorem 22.1. Assume Tmax (u0 ) < ∞, and set T := Tmax (u0 ) − δ. As in the proof of Theorem 22.1 we may assume that u0 1,2 ≤ C, hence E(u(t)) ≤ C for t ≥ 0. Denoting ψ(t) = u(t) 22 we have (cf. (17.9)) 1 ψ (t) ≥ −2E(u(t)) + c1 ψ (p+1)/2 (t), 2 where c1 = c1 (p, Ω) > 0. Set M := ((p−1)c1 δ/2)−2/(p−1) /δ and assume E(u(t0 )) ≤ −M for some t0 ∈ [0, T ]. Then ψ (t) ≥ 4M for t ≥ t0 , hence ψ(t0 + δ/2) ≥ 2δM . Since ψ ≥ 2c1 ψ (p+1)/2 and the solution of the problem y(0) = 2δM,

y = 2c1 y (p+1)/2

blows up at t < δ/2, ψ cannot exist on the whole interval [t0 + δ/2, t0 + δ) which yields a contradiction. Consequently, E(u(t)) ≥ −M for all t ∈ [0, T ] and similar ! ˜ for all t ∈ [0, T ] and suitable M ˜ = arguments show u(t) 2 = ψ(t) ≤ M ˜ M (K, δ). Now (22.27) follows from Remark 22.2(i). Assertion (22.28) follows from Remark 22.2(ii). Estimates (22.27) and (22.28) can be proved for a fairly general class of superlinear subcritical parabolic problems in bounded domains, including problems with nonlocal nonlinearities (see [440]). Property (22.28) plays an important role in the proof of complete blow-up (see Remark 27.8(b) below). As an easy application of estimate (22.27) we obtain the following important theorem concerning the continuity of the existence time. Theorem 22.13. Assume Ω bounded, 1 < p < pS , and let Tmax (u0 ) denote the maximal existence time of the solution of (22.1). Then the function Tmax : L∞ (Ω) → (0, ∞] : u0 → Tmax (u0 ) is continuous.

23. Blow-up rate

177

Proof. If 0 < T < Tmax (u0 ), then the continuous dependence of solutions of (22.1) on initial data (see (51.28)) guarantees the existence of ε > 0 such that Tmax (v0 ) > T for any v0 satisfying u0 − v0 ∞ < ε. Hence Tmax is lower semicontinuous. Next assume u0,k → u0 in L∞ (Ω) and Tmax (u0,k ) > T + δ > 0 for some δ > 0 and all k. Then (22.27) guarantees that the corresponding solutions uk satisfy uk (t) ∞ ≤ C for all t ∈ [0, T ] and k = 1, 2, . . . . Passing to the limit we obtain Tmax (u0 ) ≥ T and u(t) ∞ ≤ C. Consequently, Tmax is upper semicontinuous. The function Tmax need not be continuous in the supercritical case even in the model case (22.1) (consider the threshold trajectory u∗ from Theorem 28.7 below: if u∗ blows up in finite time, then Tmax is not continuous at u∗ (0) = α∗ φ).

23. Blow-up rate In this section we consider the model problem (22.1) and assume that Tmax (u0 ) < ∞. The solution of the ODE y = yp,

t > 0,

y(0) = y0 > 0,

(23.1)

where k = (p − 1)−1/(p−1) ,

(23.2)

is given by y(t) = k(T − t)−1/(p−1) , 0 < t < T,

with T = (p−1)−1 y01−p . It is natural to ask whether the blow-up rate for (22.1) will be of the same order. More precisely, do there exist positive constants C1 , C2 > 0 such that C1 (T − t)−1/(p−1) ≤ u(t) ∞ ≤ C2 (T − t)−1/(p−1) , (23.3) where T := Tmax (u0 )? It is not difficult to show that the lower bound in (23.3) is always satisfied, in fact with the same constant as for the ODE. Proposition 23.1. Consider problem (22.1) with p > 1. Let u0 ∈ L∞ (Ω) and assume that T := Tmax (u0 ) < ∞. Then u(t) ∞ ≥ k(T − t)−1/(p−1) ,

0 < t < T.

Proof. Assume for contradiction that there exists t0 ∈ [0, T ) such that u(t0 ) ∞ < y(t0 ), where y is given by (23.2). Therefore u(t0 ) ∞ ≤ y(t0 − ε) for some ε > 0. Since y = y p , we deduce from the comparison principle that ±u(x, t) ≤ y(t−ε) for (x, t) ∈ Ω × (t0 , T ). If follows that u is bounded in Ω × (t0 , T ), a contradiction. We present an alternative proof from [219]. It is slightly less simple but the argument may be useful for other problems (see e.g. the proofs of Theorems 44.2(i), 44.17(ii) and 46.4(i)).

178


Alternative proof for Ω bounded and u0 ≥ 0. We may assume M (t) := max u(x, t) > 0 x∈Ω

for all t ∈ (0, T ) and pick x0 (t) ∈ Ω such that M (t) = u(x0 (t), t). For 0 < s < t < T , we have M (t) − M (s) ≤ u(x0 (t), t) − u(x0 (t), s) = (t − s)ut x0 (t), s + θ(t − s)

(23.4)

and ˜ − s) M (t) − M (s) ≥ u(x0 (s), t) − u(x0 (s), s) = (t − s)ut x0 (s), s + θ(t for some θ, θ˜ ∈ (0, 1). Since ut is locally bounded in Ω × (0, T ), it follows that the function M is locally Lipschitz. In particular, M is a.e. differentiable.4 Dividing (23.4) by t − s, passing to the limit s → t, and using ∆u(x0 (t), t) ≤ 0, we obtain M (t) ≤ ut (x0 (t), t) ≤ up (x0 (t), t) = M p (t),

a.e. in (0, T ).

Integrating between t and s ∈ (t, T ) we get M 1−p (t) ≤ M 1−p (s) + (p − 1)(s − t) and the conclusion follows by letting s → T and using lims→T M (s) = ∞. Remarks 23.2. (i) Radial case. In the case when Ω = BR and u ≥ 0 is radial decreasing in r, then the above proof is just reduced to the obvious observation that x0 (t) = 0 and M (t) = ut (0, t) ≤ up (0, t) = M p (t). (ii) Alternative proof. By simple arguments based on the variation-of-constants formula, one obtains still another proof (cf. [530]) of the lower bound in (23.3) (without the sharp constant). Indeed, by (15.5), we have

s

u(s) ∞ ≤ u(t) ∞ + t

u(τ ) p∞ dτ,

0 pJL : There exist radial nonincreasing solutions u ≥ 0 such that (23.5) fails (see [277], [278] and [377]). The proof of this important result is quite long and delicate and will not be given here. Formal arguments indicate that this upper estimate should also fail for some radial (sign-changing) solutions if Ω = Rn , 3 ≤ n ≤ 6 and p = pS (see [206]). If p ≥ pS , nothing seems to be known for solutions which are neither radial nor increasing in time. Remarks 23.3. (a) Extensions. The result of case (i) above remains true for Ω = Rn if we assume in addition that u0 is radial nonincreasing (see [358]). The result of case (ii) is true also for Ω bounded convex (see [245]) and without the assumption u ≥ 0 (see [247], [248]). If u ≥ 0 and p < pB , the convexity assumption can be removed (see Theorem 26.8 below). As for the result of case (iii), it remains true for all radial solutions if pS < p < pJL and for all positive radial solutions if p = pS (see [355]). In the case Ω = Rn , it is true under an additional assumption on u0 . (b) Different methods of proof. The three proofs corresponding to cases (i), (ii) and (iii) above are quite different. They are based respectively on the maximum principle (applied to a suitable auxiliary function), on similarity variables, rescaling and energy, and on rescaling and intersection-comparison. In particular cases, different rescaling (resp., intersection-comparison) arguments were used before in [534] (resp., [229]). (c) Neumann problem. For problem (22.1) with Neumann instead of Dirichlet boundary conditions, results on (type I) blow-up rate can be found in [219], [375], for example.

180


Remark 23.4. Refined blow-up rate estimates. Assume p < pS and u ≥ 0. When Ω = Rn or Ω is a bounded convex domain, the refined asymptotic behavior limt→T (T − t)1/(p−1) u(t) ∞ = k was proved in [367] (see also [368] for a higher order asymptotic expansion). On the other hand, estimates similar to (23.5), but with M independent of u, will be studied in Section 26 on universal bounds. Theorem 23.5. Consider problem (22.1) with p > 1, Ω bounded and 0 ≤ u0 ∈ L∞ (Ω). Assume that u is nondecreasing in time and nonstationary. Then T := Tmax (u0 ) < ∞ and blow-up is of type I, i.e. (23.5) is true. Remark 23.6. The assumption ut ≥ 0 is guaranteed if, for instance, 0 ≤ u0 ∈ C0 ∩ C 2 (Ω) and ∆u0 + up0 ≥ 0 (see Proposition 52.19, and also Proposition 52.20 for weaker regularity conditions on u0 ). Proof of Theorem 23.5. It is a modification of the corresponding proof in [219]. The idea is to apply the maximum principle to the auxiliary function J defined in (23.6) below. By Example 51.10, we have ut ∈ C 2,1 (Ω×(0, T )). Since v := ut ≥ 0 is a nontrivial solution of vt − ∆v = f (u)v in QT vanishing on ST , it follows from the Hopf maximum principle (cf. Proposition 52.7) that ut > 0 in QT and ∂ν ut < 0 on ST . Choosing η ∈ (0, T ) we can thus find δ > 0 such that ut (x, η) ≥ δup (x, η) for all x ∈ Ω. Set J := ut − δf,

where f = f (u) := up ,

(23.6)

and note that J ∈ C 2,1 (QT ) ∩ C(Ω × (0, T )) (due to u > 0 in QT ). We compute Jt − ∆J = f ut − δf f + δf |∇u|2 , hence

Jt − ∆J − f J = δf |∇u|2 ≥ 0

in Qη := Ω × (η, T ).

Since J ≥ 0 on the parabolic boundary of Qη , it follows from the maximum principle (cf. Proposition 52.4) that J ≥ 0 in Qη . Consequently, ut ≥ δup in Qη . For each x ∈ Ω, by integrating this inequality between t and s ∈ (t, T ), and then letting s → T , we obtain u1−p (x, t) ≥ (p − 1)δ(T − t),

η < t < T.

This gives T < ∞ and (23.5). Theorem 23.7. Consider problem (22.1) with Ω = Rn , 1 < p < pS , 0 ≤ u0 ∈ L∞ (Rn ), and assume that T := Tmax (u0 ) < ∞. Then blow-up is of type I, i.e. (23.5) is true. In view of the proof of Theorem 23.7 we introduce the notion of backward similarity variables (cf. [244], [228]). This is a fundamental tool in the study

23. Blow-up rate

181

of the asymptotic behavior of blow-up solutions to problem (22.1), and it will be used again in Section 25. Namely, let 0 < T < ∞ and let u be a solution of (22.1) with Ω = Rn , such that u exists on Rn × (0, T ). For each fixed a ∈ Rn , we set x−a y := √ , T −t

s := − log(T − t),

(23.7)

and we define the rescaled function w(y, s) = wa (y, s) := e−βs u(a + e−s/2 y, T − e−s ),

β :=

1 p−1

(23.8)

(in other words, w(y, s) = (T − t)β u(x, t)). Let s0 := − log T . Then w is a global solution of 1 ws − ∆w + y · ∇w = |w|p−1 w − βw, 2 with

y ∈ Rn , s ∈ (s0 , ∞),

√ w(y, s0 ) = T β u0 (a + y T ),

y ∈ Rn .

(23.9)

(23.10)

Observe that (23.5) is equivalent to the uniform estimate |wa (y, s)| ≤ M , where M does not depend on y, s. To prove (23.5) we shall use this fact and the rescaling arguments from the proof of Theorem 22.1 for nonnegative solutions. Note that equation (23.9) can be rewritten as ρws − ∇ · (ρ∇w) = ρ|w|p−1 w − βρw,

(23.11)

where the Gaussian weight ρ is defined by ρ(y) := e−|y|

2

/4

.

An important property of the rescaled equation (23.9) is the existence of a weighted energy functional, defined by

β 1 1 (23.12) E(w) := |∇w|2 + w2 − |w|p+1 ρ dy. 2 p+1 Rn 2 We shall first establish some auxiliary results involving this energy (these results will also be used in Section 25). Proposition 23.8. Let p > 1 and let w be a global solution of (23.11) with w(·, s0 ) ∈ BC 1 (Rn ). Then, for all s > s0 , we have 1 d 2 ds

p−1 w2 ρ dy = −2E w(s) + p+1 Rn

Rn

|w|p+1 ρ dy,

(23.13)

182


d E w(s) = − ws2 ρ dy, ds Rn E w(s) ≥ 0,

and

Rn

Moreover,

(23.15)

2/(p+1) w2 ρ dy ≤ C(n, p) E w(s0 ) .

∞

Rn

s0

and

(23.14)

ws2 ρ dy ds ≤ E w(s0 )

a → E wa (s0 )

(23.16)

(23.17)

is smooth and bounded.

(23.18)

Proof. Problems (18.1) and (23.9)-(23.10) are equivalent via the transformation (23.7)–(23.8). Let 0 < t1 < t2 < T . By Proposition 48.7 and a simple use of the variation-of-constants formula, we see that u, ∇u ∈ L∞ (Rn × (0, t2 )) (see also (51.29) in Remark 51.11). On the other hand, applying interior parabolic Lq - and Schauder estimates, we obtain that D2 u, ut ∈ L∞ (Rn × (t1 , t2 )). Next applying Remark 48.3(i) and, again, interior Schauder estimates, we get ∇u ∈ C 2,1 (Rn × (0, T )) and ∇ut ∈ L∞ (Rn × (t1 , t2 )). Consequently, given s2 ∈ (s0 , ∞), the rescaled function w = w(s) satisfies sup Rn ×(s0 ,s2 )

(|w| + |∇w|) < ∞

(23.19)

and

sup Rn ×(s1 ,s2 )

|D2 w| + (1 + |y|)−1 (|ws | + |∇ws |) < ∞,

We shall write shortly 1 d 2 ds and

w2 ρ =

f instead of

wws ρ =

1 d 2 ds

2

Rn

s0 < s 1 < s 2 .

(23.20)

f (y) dy. We compute

w ∇ · (ρ∇w) + ρ|w|p−1 w − βρw ,

|∇w| ρ =

ρ(∇ws · ∇w),

for s > s0 . Note that the differentiability of the integrals is guaranteed by (23.19), (23.20), and the exponential decay of ρ. By using integration by parts, we deduce that

1 d p−1 w2 ρ = |w|p+1 ρ −|∇w|2 − βw2 + |w|p+1 ρ = −2E(w) + 2 ds p+1

23. Blow-up rate

i.e., (23.13), and 1 d 2 ds

|∇w|2 ρ = −

183

ws (∇ · ρ∇w).

This procedure can be easily justified by using again (23.19), (23.20), and the exponential decay of ρ: It suffices to integrate by parts on BR and then let R → ∞. On the other hand, we have

d β 2 1 p+1 w − |w| ρ = (βw − |w|p−1 w)ws ρ. ds 2 p+1 Summing the last two identities and using equation (23.11), we obtain (23.14). Denote ψ(s) := w2 (s)ρ. Then (23.13) and Jensen’s inequality imply 1 dψ ≥ −2E w(s) +C(n, p)ψ (p+1)/2 (s). 2 ds Since E w(s) is nonincreasing due to (23.14), this guarantees (23.15) and (23.16) (otherwise ψ has to blow up in finite time). Next, (23.15) and (23.14) imply (23.17). Finally, to check (23.18), we note that

x − a dx 2 2β wa (s0 )ρ dy = T u20 (x)ρ √ ≤ T 2β (4π)n/2 sup u20 , T T n/2 Rn which shows the smoothness and boundedness of the second term appearing in the definition of E wa (s0 ) , see (23.12). The proof for the remaining terms is similar. Now we are ready to repeat the idea of the proof of Theorem 22.1 for nonnegative solutions. Proof of Theorem 23.7. By a time shift we may assume u0 ∈ BC 1 (Rn ), see (51.28). Assume, on the contrary, that there exist tk such that Mk :=

sup (T − t)β u(x, t) = sup(T − tk )β u(x, tk ) → ∞.

Rn ×[0,tk ]

Rn

We may assume tk ≥ t˜ for some t˜ > 0. Choose xk ∈ Rn such that (T − tk )β u(xk , tk ) ≥ Mk /2. Rewriting u in similarity variables around a = xk (cf. (23.7)–(23.8)), we denote wk := wxk , sk := − log(T − tk ). Then sk − s0 ≥ δ 2 for some δ > 0, 0 ≤ wk (y, s) ≤ Mk for s ≤ sk and wk (0, sk ) ∈ [Mk /2, Mk ]. Denote vk (z, τ ) :=

1 wk (νk z, νk2 τ + sk ), Mk

−(p−1)/2

νk := Mk

.

184


Then 0 ≤ vk (z, τ ) ≤ 1 for (z, τ ) ∈ Qk := Rn × (−(δ/νk )2 , 0], vk (0, 0) ∈ [1/2, 1] and ∂τ vk − ∆vk = vkp − νk2

1 2

z · ∇vk + βvk

in Qk .

Since Q(r) := {(z, τ ) : |z| < r, −r2 < τ ≤ 0} ⊂ Qk for k large enough, uniform parabolic Lp -estimates used for the operators Ak v := −∆v + 12 νk2 z · ∇v (see Appendix B) imply the boundedness of vk in C α . Consequently, we may pass to the limit to get a solution v of the problem vτ − ∆v = v p

in Rn × (−∞, 0),

satisfying 0 ≤ v ≤ 1 and v(0, 0) ≥ 1/2. Finally, setting σ := −n + 2 + 4/(p − 1) > 0 and using (23.17) and (23.18) we obtain

Q(δ/νk )

2

|∂τ vk | dz dτ =

νkσ

sk

sk −δ 2

≤ νkσ C(δ)

|y| 1 (see [395], [229]). Lemma 23.12. Consider problem (22.1) with p > 1 and Ω = BR . Let u0 ∈ L∞ (Ω), u0 ≥ 0, be radial nonincreasing, and set T := Tmax (u0 ). Assume that either T < ∞ or (23.23) holds. Denote by N (t) := z[0,R] (ut (·, t)) the zero number of the function ut (·, t) in the interval [0, R] (see Appendix F). Then there exists t0 ∈ (0, T ) such that ut (0, t) > 0

and

N (t) = Const.,

t0 < t < T.

Proof. We note that the function ut is a radial classical solution of x ∈ BR , 0 < t < T, Vt − ∆V = pup−1 V, V = 0, x ∈ ∂BR , 0 < t < T.

(23.25)

(23.26)

By Theorem 52.28, N (t) is finite and nonincreasing, hence constant on (t0 , T ) for some t0 ∈ (0, T ). Moreover, by the symmetry of the solution, if ut (0, t) = 0, then ut (·, t) has a degenerate zero at r = 0, so that the function N drops at time t. Consequently, ut (0, t) does not change sign on (t0 , T ). Since lim supt→T u(0, t) = ∞, the claim follows.

186


Proof of Proposition 23.11. By a time shift, we may assume that t0 = 0 in Lemma 23.12. Our assumptions imply the existence of a sequence tj → T such that ut (0, tj ) →0 (23.27) up (0, tj ) (otherwise we would have ut (0, t) ≥ cup (0, t) as t → T for some c > 0, due to (23.25); but this contradicts each of (23.22) and (23.23)). Set Mj = u(0, tj ). By comparing with the solution of the ODE ψ = ψ p , ψ(tj ) = Mj , we easily obtain the existence of s∗ = s∗ (p) > 0 such that t˜j := tj + s∗ Mj1−p < T −(p−1)/2

Let λj = Mj vj (y, s) =

and

u(0, t) ≤ 2Mj ,

tj ≤ t ≤ t˜j .

(23.28)

and define the rescaled solutions

1 u(λj y, tj + λ2j s), Mj

−2 . (y, s) ∈ Dj := BRλ−1 × −tj λ−2 j , (T − tj )λj j

Then ∂s vj − ∆vj = vjp ,

(y, s) ∈ Dj

∗ and, by (23.28) and (23.25), we have 0 ≤ vj ≤ 2 in BRλ−1 ×(−tj λ−2 j , s ). Moreover, j vj (0, 0) = 1 and ∂s vj (0, 0) → 0, due to (23.27). Let D := Rn × (−∞, s∗ ). By interior parabolic estimates, it follows that (some subsequence of) vj converges in C 2+α,1+α/2 (D) to a radial, nonnegative solution of

∂s v − ∆v = v p ,

(y, s) ∈ D,

such that v(0, 0) = 1 and ∂s v(0, 0) = 0. By using equation (23.26), we see also that ∂s vj → ∂s v in C 1,0 (D). (23.29) We shall now show that ∂s v(·, 0) ≡ 0. Suppose not. Then there exist A > 0 and ε ∈ (0, s∗ ) such that ∂s v(A, s) = 0, |s| ≤ ε. (23.30) Since ∂s v(·, 0) has a degenerate zero at r = 0, it follows from Theorem 52.28 that the zero number of ∂s v on [0, A] drops at s = 0. Namely, we can fix −ε < s1 < 0 < s2 < ε such that ∂s v(·, si ) has only simple zeroes on [0, A] and such that z[0,A] ∂s v(·, s1 ) ≥ z[0,A] ∂s v(·, s2 ) +1. Owing to (23.29), we deduce that for j large enough, z[0,A] ∂s vj (·, s1 ) ≥ z[0,A] ∂s vj (·, s2 ) +1,

23. Blow-up rate

hence

187

z[0,Aλj ] ut (·, tj + λ2j s1 ) ≥ z[0,Aλj ] ut (·, tj + λ2j s2 ) +1.

(23.31)

Since, on the other hand, (23.30) implies ut (Aλj , tj + λ2j s) = 0 for |s| ≤ ε, Remark 52.29(ii) implies z[Aλj ,R] ut (·, tj + λ2j s1 ) ≥ z[Aλj ,R] ut (·, tj + λ2j s2 ) . (23.32) By (23.31) and (23.32), we deduce that N (tj + λ2j s1 ) ≥ N (tj + λ2j s2 ) + 1, which contradicts (23.25). It follows that vs (·, 0) ≡ 0, hence v(·, 0) ≡ U1 due to v(0, 0) = 1, and the proposition follows. Proof of Theorem 23.10. Assume that (23.5) is false and let the sequence tj → T < ∞ be given by Proposition 23.11. We treat the supercritical and critical cases separately. Case 1: pS < p < pJL . By Theorem 52.28, there exists an integer K such that z[0,R] (u(·, tj ) − U∗ ) ≤ K,

j = 1, 2, . . . .

(23.33)

On the other hand, by Theorem 9.1, U1 and U∗ intersect infinitely many times. Moreover, these intersections are transversal by local uniqueness for the ODE p U + n−1 r U + U = 0. Pick A > 0 such that z[0,A] (U1 − U∗ ) ≥ K + 1. Also it is clear that z[0,R] u(r, tj ) − U∗ (r)

(23.34)

-

%. $ % $ 1 r 1 r p−1 u U∗ = z , tj − p−1 p−1 0,R m 2 (tj ) m(tj ) m(tj ) m 2 (tj ) m 2 (tj ) . $ % 1 r p−1 u = z , tj − U∗ (r) . p−1 0,R m 2 (tj ) m(tj ) m 2 (tj )

By (23.34) and (23.24), it follows that z[0,R] u(·, tj ) − U∗ ≥ K + 1 for j large: a contradiction with (23.33). Case 2: p = pS . Fix t0 ∈ (0, T ) and take c2 > c1 > 0 such that u(r, t) ≥ (e−tA u0 )(r) ≥ c1 ,

0 ≤ r ≤ R/2, t0 ≤ t < T

and u(r, t0 ) ≤ c2 ,

0 ≤ r ≤ R.

Let UM be the unique positive solution of (9.2) satisfying UM (0) = M , see Theorem 9.1. Since UM (R/2) → 0 as M → ∞ and UM (r) → −∞ as M → ∞

188


uniformly on {r : UM (r) ∈ [c1 , c2 ]}, there exists M0 > c2 such that, for all M ≥ M0 , u(R/2, t) > UM (R/2) for t ∈ [t0 , T ) and z[0,R/2] (u(·, t0 ) − UM ) = 1. By the nonincreasing property of the zero-number (see Theorem 52.28), we deduce that z[0,R/2] u(·, t) − UM ≤ 1, t0 ≤ t < T, M ≥ M0 . Since limt→T u(0, t) = ∞, for each M ≥ M0 , there exists a first τ (M ) ∈ (t0 , T ) such that u(0, τ (M )) = UM (0) = M . By symmetry of the solutions, u(·, τ (M )) has a double zero at the origin. Therefore, by Theorem 52.28(iii), z[0,R/2] (u(·, t) − UM ) must drop at t = τ (M ), hence u(r, t) > UM (r),

0 ≤ r ≤ R/2, τ (M ) < t < T.

(23.35)

Now, for large j, (23.24) implies u(0, tj ) >

m(tj ) U1 (0) = Um(tj )/2 (0). 2

Consequently, tj > τ (m(tj )/2). Using (23.35) and (9.4), it follows that u(r, tj ) > Um(tj )/2 (r) = m(tj )U1/2 (m

p−1 2

(tj )r),

0 ≤ r ≤ R/2, tj ≤ t < T.

Therefore,

$ % ρ 1 u , tj > U1/2 (ρ), p−1 m(tj ) m 2 (tj )

0 ≤ ρ ≤ (R/2) m

p−1 2

(tj ), tj ≤ t < T.

Using (23.24) again and letting j → ∞, we obtain U1 (ρ) ≥ U1/2 (ρ),

0 ≤ ρ < ∞,

contradicting Theorem 9.1. Remark 23.13. By combining Proposition 23.11 for T = ∞ and Case 1 of the proof of Theorem 23.10, we obtain an alternative proof [125] of Theorem 22.4 on boundedness of global solutions in the case pS < p < pJL and ur ≤ 0. Moreover, as a consequence of Remark 22.7, this proof can be used without assuming ur ≤ 0. Remarks 23.14. (i) Sign-changing solutions. The proof of Theorem 23.7 can be considered as an analogue to the proof of Theorem 22.1 for nonnegative solutions. In [247] the authors prove Theorem 23.7 without the positivity assumption on u0 and the proof is again an analogue of the (interpolation) proof of Theorem 22.1 in the general, sign-changing case. However, the localization of the arguments of this interpolation proof is nontrivial: The authors of [247] have to use two kinds of localized version of weighted energies,

2 2 1 1 2 2 Eϕ (w) := |∇(ϕw)| + βϕ − |∇ϕ| w ρ dy − ϕ2 |w|p+1 ρ dy, 2 Rn p + 1 Rn

1 1 Eϕ (w) := ϕ2 |∇w|2 + βw2 ρ dy − ϕ2 |w|p+1 ρ dy, 2 Rn p + 1 Rn

23. Blow-up rate

189

and the corresponding bounds Eϕ (w) ≥ 0, Eϕ (w) ≤ C and |Eϕ (w) − Eϕ (w)| ≤ C. (ii) Applications of blow-up rate estimates. The knowledge of the blowup rate has important consequences in the study of the blow-up behavior. In particular (23.5) is the first step in the description of asymptotically self-similar blow-up (see Section 25). On the other hand, it can be used for the proof of the Hölder continuity of the maximal existence time Tmax : L∞ (Ω) → (0, ∞] (see [254], [255] and cf. Theorem 22.13). We close this section with a simple result which shows that the upper blow-up estimate (23.5) implies a similar estimate for the gradient. This property will be useful in Section 25. Proposition 23.15. Consider problem (22.1) with p > 1 and u0 ∈ L∞ (Ω). Assume that T := Tmax (u0 ) < ∞ and that (23.5) is satisfied for some M > 0. Then ∇u(t) ∞ ≤ M1 (T − t)−1/(p−1)−1/2 ,

T /2 ≤ t < T

for some M1 = M1 (M, p, Ω, T ) > 0. Proof. Fix T /2 ≤ t < T and put s = 2t − T ∈ [0, t). By the variation-of-constants formula, the gradient estimate in Proposition 48.7 and (23.5), we have

t ∇u(t) ∞ ≤ ∇e−(t−s)A u(s) ∞ + ∇e−(t−τ )A|u|p−1 u(τ ) ∞ dτ s

t −1/2 u(s) ∞ + C (t − τ )−1/2 u(τ ) p∞ dτ ≤ C(t − s) s

t −1/2 −1/(p−1) (T − s) + CM p (t − τ )−1/2 (T − τ )−p/(p−1) dτ. ≤ CM (t − s) s

Since T − t = t − s = (T − s)/2, we have ∇u(t) ∞ ≤ 2−1/(p−1) CM (T − t)−1/(p−1)−1/2 + CM p (T − t)−p/(p−1)

s

≤ C[2−1/(p−1) M + 2M p ] (T − t)−1/(p−1)−1/2 and the proposition is proved.

t

(t − τ )−1/2 dτ

190


24. Blow-up set and space profile This and the subsequent section are devoted to the space and space-time description of singularities of blowing-up solutions of the model problem (22.1). Assume that the solution u blows up in finite time T := Tmax (u0 ) and denote by B(u0 ) its blow-up set: B(u0 ) := {x ∈ Ω : ∃(xk , tk ) ∈ Ω × (0, T ) such that tk → T and |u(xk , tk )| → ∞}. (24.1) The following theorem, due to [219], guarantees single-point blow-up for radial decreasing solutions and provides an upper estimate for the blow-up profile. Theorem 24.1. Consider problem (22.1) with p > 1 and Ω = BR . Let u0 ∈ L∞ (Ω), u0 ≥ 0, be radial nonincreasing, and assume T := Tmax (u0 ) < ∞. Then B(u0 ) = {0} and, for any α > 2/(p − 1), there exists Cα > 0 such that u(x, t) ≤ Cα |x|−α ,

0 < |x| < R, 0 < t < T.

Corollary 24.2. Under the assumptions of Theorem 24.1, we have lim sup u(t) q < ∞,

1 ≤ q < qc = n(p − 1)/2.

t→T

Proof of Theorem 24.1. As in the proof of Theorem 23.5, the idea is to apply the maximum principle to a (different) auxiliary function J, defined in (24.3) below. Note that our assumptions imply ur (r, t) < 0 for all r ∈ (0, R], t ∈ (0, T ),

(24.2)

due to Proposition 52.17. We split the proof in two steps. Step 1. Let γ ∈ (1, p), η ∈ (0, T ) and δ > 0. We will show that there exists ε > 0 such that J(r, t) ≤ 0 in Ω × (η, T ), where

J =J(u) := w + c(r)F (u), w(r) := rn−1 ur (r, t),

c(r) := εrn+δ ,

F (u) := uγ .

(24.3)

Denote f (u) := up , Ω1 := Ω ∩ {x : x1 > 0} and notice that v := ux1 satisfies vt − ∆v = f (u)v in Ω1 . Since v = 0 for x ∈ ∂Ω1 , x1 = 0, and v < 0 for x ∈ ∂Ω1 , x1 > 0, the maximum principle implies v < 0 for x ∈ Ω1 and vx1 (0, t) = ux1 x1 (0, t) < 0 for t > 0. Hence urr (0, t) < 0. Since J = rn−1 (ur + εr1+δ uγ ), this inequality and (24.2) imply J(r, η) ≤ 0 for all r provided ε is small enough. We have also J ∈ C 2,1 ((0, R) × (0, T )) ∩ C([0, R] × (0, T )) (due to u > 0), with

24. Blow-up set and space profile

191

J(R, t) ≤ 0 and J(0, t) = 0 for t > 0. Now the claim follows from the maximum principle in Proposition 52.4, provided we show Jt +

n−1 Jr − Jrr − bJ ≤ 0 r

in (0, R) × (η, T ),

(24.4)

where b is bounded on (0, R) × (η, T − τ ) for all τ > 0. Summing the identities Jt = wt + cF ut , n−1 n − 1 Jr = wr + cF ur + c F , r r −Jrr = − wrr + cF urr + cF u2r + 2c F ur + c F , and using the equations ut − urr −

n−1 ur = f (u), r

wt +

n−1 wr − wrr = f (u)w, r

and F ≥ 0 we get Jt +

n−1 2(n − 1) n−1 Jr − Jrr ≤ f w + cF f + cF ur + c F − 2c F ur − c F. r r r

Using w = −cF + J and ur = w/rn−1 = (−cF + J)/rn−1 we see that the RHS of the last inequality can be written in the form bJ − cH, where the function b = f + 2(n − 1)cF r−n − 2c F r1−n is bounded on (0, R) × (η, T − τ ) for all τ > 0 and 2 n − 1 F n − 1 H = F f − F f − n−1 F F c − c + c − c . r r c r Now H ≥ 0 is equivalent to (p − γ)up−1 + (n + δ)δr−2 ≥ 2εγ(1 + δ)uγ−1 rδ

(24.5)

which is obviously satisfied if ε is small enough. Consequently, (24.4) is true. Step 2. Let t ∈ (η, T ) and α := (2 + δ)/(γ − 1). Notice that J(u) ≤ 0 implies ur /uγ ≤ −εr1+δ . Integrating this inequality we arrive at u(r, t) ≤ Cr−α , where C := (α/ε)1/(γ−1) . This estimate guarantees the assertion. Under an additional assumption of monotonicity in time, the corresponding lower estimate on the blow-up profile can be established by relatively simple arguments (cf. [490]). For more precise information on the blow-up profiles, see Remark 25.8 below.

192


Theorem 24.3. Consider problem (22.1) with p > 1 and Ω = BR or Ω = Rn . Let u0 ∈ L∞ (Ω), u0 ≥ 0, be radial nonincreasing and such that T := Tmax (u0 ) < ∞. Assume in addition that ut ≥ 0 in QT . Then there holds u(x, T ) ≥ C|x|−2/(p−1) ,

0 < |x| < η,

(24.6)

for some C = C(p) > 0 and η = η(u0 ) > 0. Proof. We assume Ω = BR (the case Ω = Rn can be treated by straightforward modifications). Since ut ≥ 0 and ur ≤ 0, we have ∂ 1 2 1 n−1 ur + up+1 = (urr + up )ur = ut − ur ur ≤ 0, ∂r 2 p+1 r hence

1 2

Therefore, we get

u2r +

1 1 up+1 (r, t) ≤ up+1 (0, t). p+1 p+1

ur (t) ∞ ≤ C1 u(p+1)/2 (0, t).

For 0 < t < T , let r0 (t) be such that u(r0 (t), t) = 12 u(0, t). Note that, in view of (24.2), the implicit function theorem guarantees that r0 (t) is unique and is a continuous function of t. We may assume that 0 is the only blow-up point, since otherwise the result is trivial. Using u(0, t) = u(t) ∞ → ∞ as t → T , this implies r0 (t) → 0 as t → T . Now we have −ur ≤ C2 u(p+1)/2 ,

0 ≤ r ≤ r0 (t).

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. Remark 24.4. The lower bound (24.6) has been proved in [232] (see also [229]) for radial solutions without assuming ut ≥ 0 nor ur ≤ 0, but under the condition p ≤ pS . The method therein is different, based on intersection-comparison with stationary solutions. We get back to the question of the blow-up set. In the case of Rn , the following result, due to [246], gives a necessary condition, involving the weighted energy, for a given point to be a blow-up point and a sufficient condition for the blow-up set to be compact. The proof is postponed to the next section.

24. Blow-up set and space profile

193

Theorem 24.5. Consider problem (22.1) with 1 0 such that, if Ea (u0 ) < η, then a is not a blow-up point. (ii) Assume in addition that u0 (x), |∇u0 (x)| → 0 as |x| → ∞, then the blow-up set is compact. Remarks 24.6. (i) Single-point blow-up. The first example of a single-point blow-up for problem (22.1) was found in [531] with n = 1 and u0 = kψ, where ψ is a positive stationary solution of (22.1) and k, p 1. On the other hand, under the assumptions of Theorem 24.1 with Ω = Rn instead of a ball, we still have a single-point blow-up (see [388]; the proof is different from that of Theorem 24.1). (ii) Blow-up at infinity. By a careful reading of the proof of Theorem 24.5(ii) one obtains the stronger conclusion that sup{|u(x, t)| : |x| > R, 0 < t < T } < ∞ for some large R > 0. On the other hand, the result may fail if no decay is assumed on u0 , as shown by the simple example u = k(T − t)−1/(p−1) . Furthermore, it has been shown in [249] that, if lim|x|→∞ u0 (x) = L > 0 and 0 ≤ u0 ≤ L in Rn , u0 ≡ L, then u remains bounded on compact subsets of Rn up to t = Tmax (u0 ) < ∞ and blows up only at space infinity (see also [312] for related results). Under the same assumption, denoting by y the solution of the ODE (23.1) with y(0) = L, it was also proved in [249] that u and y share the same blow-up time T and that lim|x|→∞ u(x, t) = y(t), uniformly for t bounded away from T . (iii) One-dimensional case. Consider problem (22.1) with n = 1 and Ω bounded. Assume first u0 ≥ 0 and T = Tmax (u0 ) < ∞. Then the results of [126] guarantee that B(u0 ) is finite and its cardinality is bounded above by the number of local maxima of u0 . Moreover, given x ∈ / B(u0 ), there exists ϕ(x) := limt→T u(x, t) and ϕ ∈ C 2 Ω \ B(u0 ) . On the other hand, given x1 , x2 , . . . xk ∈ Ω, there exists u0 ≥ 0 such that B(u0 ) = {x1 , . . . , xk }, see [364]. The arguments in [209] and the universal bounds in Section 26 show that there exists T ∗ < ∞ with the following property: If u0 ≥ 0 and Tmax (u0 ) > T ∗ , then B(u0 ) consists of a single point. (iv) Blow-up in the interior. Consider problem (22.1) with Ω bounded and convex. If u0 ≥ 0 and T = Tmax (u0 ) < ∞, then B(u0 ) is always a compact subset of Ω (see [219]). The idea of the proof is the following: Choose y ∈ ∂Ω. Without loss of generality we may assume that y = 0 and that the hyperplane {x = (x1 , . . . , xn ) : x1 = 0} is tangential to ∂Ω at the origin, with x1 < 0 for all x ∈ Ω. The method of moving planes guarantees ux1 < 0 for all t ≥ η > 0 and x ∈ Σλ := {z ∈ Ω : z1 > −λ} provided λ > 0 is small enough. Now, similarly as in the proof of Theorem 24.1 one obtains J := ux1 + ε(x1 − λ)1+δ uγ ≤ 0 in

194


˜ −α for any λ ˜ ∈ (0, λ) and x ∈ Σ ˜ (where Σλ × (η, T ) which implies u(x) ≤ C(λ − λ) λ α, γ, δ, η, ε have the same meaning as in the proof of Theorem 24.1). This estimate guarantees B(u0 ) ∩ ∂Ω = ∅. (v) Global and regional blow-up. Consider problem (14.1) with b b > 1. f (u) = (1 + u) log(1 + u) , Assume first that Ω is bounded, u0 ≥ 0 and Tmax (u0 ) < ∞ (such functions do exist). If b < 2, then the blow-up is global, that is B(u0 ) = Ω. If b = 2, then the blow-up is either global or regional (that is B(u0 ) contains a nonempty open set, but B(u0 ) = Ω), depending on the size of Ω. These results were proved in [313]. Similarly, if Ω = R, b = 2 and u0 ≥ 0 is symmetric and radially nonincreasing, u0 ≡ 0, then Tmax (u0 ) < ∞, the measure of B(u0 ) is at least 2π (and B(u0 ) = [−π, π] under some additional assumptions on u0 ), see [230]. On the other hand, if Ω is a ball and b > 2, then there are positive initial data such that the corresponding solutions blow up at a single point (this follows from the proof of Theorem 24.1 b−1 with the choice F (u) = (1 + u) log(1 + u) ). Regional or global blow-up cannot happen for positive solutions of (18.1) if p < pS and u0 is continuous, bounded and nonconstant. In this case, the (n − 1)dimensional Hausdorff measure of B(u0 ) ∩ M is finite for any bounded measurable set M ⊂ Rn , see [520]. This is optimal, in view of examples from [246] of solutions blowing up on a sphere. Moreover, results on the regularity of B(u0 ) near a nonisolated blow-up point have been obtained in [541], [542]. (vi) Small and large diffusion limits. Consider positive solutions of problem (22.1) with a diffusion coefficient D > 0 in front of the Laplacian, and with either the Dirichlet or the Neumann boundary conditions. Then, under suitable additional assumptions, the blow-up set of u concentrates near the maxima of u0 as D → 0. In the limit D → ∞, for the Neumann case, it concentrates near M, where M is the set of maxima of the L2 -projection of u0 onto the second Neumann eigenspace (see [289] and the references therein). Remark 24.7. Limitations concerning comparison arguments. If Ω is a bounded domain, then two ordered sub-/supersolutions cannot share the same existence time unless both are global. For instance, if u is the solution of (22.1) with Tmax (u0 ) < ∞ and if v ≡ u is a supersolution of (22.1) on (0, T ) such that u0 ≤ v(·, 0) ∈ L∞ (Ω), then T < Tmax (u0 ). This follows from Proposition 27.3 below. In particular the knowledge of the blow-up rate or set of v does not provide direct information on that of u (but the situation can be different in unbounded domains, cf. the end of Remark 24.6(ii)). Nevertheless, in bounded domains, one can sometimes use indirect comparison arguments (see Proposition 23.1 for a simple case) or more sophisticated intersection-comparison arguments.

25. Self-similar blow-up behavior

195

25. Self-similar blow-up behavior In this section we apply the method of similarity variables, introduced in the proof of Theorem 23.7, to study the space-time behavior of solutions of the model problem (22.1) near blow-up points as t approaches the blow-up time. The following theorem is due to [244], [246]. A similar result for n = 1 was obtained independently in [229]. Theorem 25.1 can be extended to bounded convex domains (see [246]), but here we restrict ourselves to the case of the whole space for simplicity. Theorem 25.1. Consider problem (22.1) with Ω = Rn , 1 < p ≤ pS , u0 ∈ L∞ (Rn ), and let k = (p − 1)−1/(p−1) . Assume that T := Tmax (u0 ) < ∞ and that the upper blow-up rate estimate (23.5) is satisfied. If a is a blow-up point of u, then we have √ lim (T − t)1/(p−1) u(a + y T − t, t) = ±k, (25.1) t→T

uniformly on compact sets |y| ≤ C. Remark 25.2. Let 1 < p < pS , Ω = Rn , u0 ∈ L∞ (Rn ) and let a be a blowup point of u (in the sense of the definition in (24.1)). Theorem 23.7 and Remark 23.3(a) guarantee that the upper blow-up rate estimate (23.5) is satisfied. Consequently, it follows in particular from Theorem 25.1 that u does blow-up at x = a and the blow-up rate of u(a, t) is exactly that given by the ODE (cf. (23.1)– (23.2)). In the proof, among other things, we shall use another result from [246], which is of independent interest since it is valid for any p > 1. If a is a blow-up point, then this result provides a lower bound on the blow-up rate. Note that no boundary conditions are assumed and that this is a purely local result. The proof in [246] was based on a cut-off, the variation-of-constants formula, parabolic estimates and bootstrap. We here give a simpler proof, based on a comparison argument using a quadratic change of unknown and a cut-off (cf. (25.6)). Theorem 25.3. Let p > 1, T > 0, ρ > 0, a ∈ Rn and denote Q = B(a, ρ) × (T − ρ2 , T ). There exists ε = ε(n, p) > 0 such that if u is a classical solution of ut − ∆u = |u|p−1 u, and u satisfies

|u(x, t)| ≤ ε(T − t)−1/(p−1) ,

(x, t) ∈ Q, (x, t) ∈ Q,

(25.2)

then u is uniformly bounded in a neighborhood of (a, T ). Proof. By a space-time translation, we may assume a = 0 and T = ρ2 . By scaling, we may also assume ρ = 1. Indeed, u ˜(x, t) := ρ2/(p−1) u(ρx, ρ2 t) solves the same

196


equation in B1 × (0, 1), and (25.2) is equivalent to |˜ u(x, t)| ≤ ε(1 − t)−1/(p−1) for |x| < 1 and t ∈ (0, 1). Set α = min(1/2, (p − 1)/4).

(25.3)

For each R > 0, we may find φ ∈ C 2 (Rn ) such that √ φ(x) = 0 for |x| ≥ R/ 2, and

φ(x) ≥ 1 for |x| ≤ R/2,

|∇φ|2 + |∆φ2 | ≤ C(R, n)φ2(1−α)

(25.4)

(25.5)

(it suffices to consider φ(x) = (2 − 4R−2 |x|2 )m + for m > 2 large enough). Choose R = 1 and put v = u2 φ2 . (25.6) For (x, t) ∈ B1 × (0, T ), we have vt − ∆v = 2uut φ2 − 2φ2 (u∆u + |∇u|2 ) − 8uφ∇u · ∇φ − u2 ∆φ2 . Since 4|uφ∇u · ∇φ| ≤ φ2 |∇u|2 + 4u2 |∇φ|2 , we deduce that vt − ∆v ≤ 2φ2 |u|p+1 + u2 (8|∇φ|2 + |∆φ2 |).

(25.7)

Using (25.5) and assumption (25.2), it follows that vt − ∆v ≤ 2|u|p−1 v + u2α v 1−α φ−2(1−α) (8|∇φ|2 + |∆φ2 |) ≤ 2|u|p−1 v + Cu2α (1 + v) ≤ 2εp−1 (T − t)−1 v + Cε2α (T − t)−2α/(p−1) (1 + v). Here and below, C is a generic positive constant depending only on n, p, whereas K will denote a generic positive constant depending on n, p, u. Assuming 0 < ε < 1 and recalling (25.3), we obtain vt − ∆v ≤ Cε2α (T − t)−1 v + C(T − t)−2α/(p−1) . 2α

Let v = (T − t)−Cε

(25.8)

+ K(T − t)1−2α/(p−1) for 0 ≤ t < T . We have

v t = Cε2α (T − t)−1 v + K(1 − 2α/(p − 1) − Cε2α )(T − t)−2α/(p−1) . For ε = ε(n, p) > 0 small and K > 0 large, it follows that v is a supersolution to (25.8) and we also have v(1/2) ≥ v(·, 1/2) ∞. Since v = 0 on ∂B1 × (0, T ), we deduce from the comparison principle that v ≤ v in B1 × [1/2, T ), hence 2α

u ≤ K(T − t)−Cε ,

|x| < 1/2, 1/2 ≤ t < T.

(25.9)


197

Now considering v = u2 φ2 with R = 1/2 instead of R = 1 in (25.4), and taking a smaller ε(n, p) > 0, inequalities (25.7) and (25.9) imply vt − ∆v ≤ K(T − t)−1/2 in ˜ − K(T − t)1/2 , we conclude B1/2 × (1/2, T ). Using a supersolution of the form K that u is bounded in a neighborhood of (x = 0, t = T ). Before going into the proof of Theorem 25.1, let us first observe that, considering the rescaled solution in similarity variables (cf. (23.7)–(23.8)), the conclusion can be restated as: lims→∞ wa (y, s) = ±k, uniformly on compact sets |y| ≤ C. The basic idea of the proof is to apply dynamical systems arguments to show that the global bounded solution wa is attracted by the set of equilibria, i.e. solutions of 1 ∆z − y · ∇z + |z|p−1 z − βz = 0, z ∈ Rn (25.10) 2 (cf. Lemma 25.6(i)). On the other hand, under the assumption p ≤ pS , we shall show that the only (bounded) equilibria are the constant solutions z = k, −k and 0 (Proposition 25.4). The last task will then be to show the nondegeneracy, i.e. to exclude the possibility of wa approaching 0, which will be achieved by combining Theorem 25.3 with suitable energy arguments (cf. Lemma 25.6(ii)(iii)). Thus, let us define

S = z ∈ C 2 ∩ L∞ (Rn ) : z is a solution of (25.10) . For given a ∈ Rn , we denote

ω(wa ) = z ∈ S : ∃sn → ∞, wa (y, sn ) → z(y) in C 1 (Rn ). Proposition 25.4. If 1 pS (provided p < pL defined in (22.18)). Indeed, in that range, Proposition 22.5 shows the existence of backward self-similar solutions with (positive) bounded nonconstant profile. (ii) However, if Ω = BR and u ≥ 0 is radial and satisfies ut ≥ 0 and ur ≤ 0, then assertion (25.1) with a = 0 remains true for all p > pS (see [62], and also [358], [359] for further results). Proof of Proposition 25.4. Let w ∈ √ S. We first claim that |∇w| is bounded. Indeed, by setting u(x, t) = (1−t)−β w(x/ 1 − t), we define a (self-similar) solution √ of (18.1) in Rn × (0, 1), with u0 = w ∈ L∞ (Rn ). Since ∇u(x, 1/2) = 2β+1/2 w( 2x) and ∇u(·, 1/2) ∈ L∞ (Rn ) by smoothing effect, the claim follows.

198


Let us show that w satisfies the Pohozaev-type identity

n 1 1 n − 2 1 |∇w|2 ρ dy + |y|2 |∇w|2 ρ dy = 0. (25.11) − − p+1 2 2 2 p + 1 Rn Rn We shall obtain (25.11) as a linear combination of three other identities. The first one is

|∇w|2 ρ dy + β w2 ρ dy − |w|p+1 ρ dy = 0. (25.12) (Here and in what follows all integrals are taken over Rn .) Rewriting (25.10) as ∇ · (ρ∇w) − βρw + ρ|w|p−1 w = 0,

(25.13)

(25.12) is obtained by multiplying (25.13) by −w and using integration by parts. This procedure can be easily justified since w and |∇w| are bounded and ρ decays exponentially: It suffices to integrate by parts on BR and then let R → ∞. This argument will be used in the rest of the proof without further mention. The second identity is

|y|2 |∇w|2 ρ dy +

β+

1 2 |y| − n w2 ρ dy − 2

|y|2 |w|p+1 ρ dy = 0. (25.14)

It is obtained by multiplying (25.13) by −|y|2 w and using integration by parts, since

− |y|2 w∇ · (ρ∇w) dy = |y|2 |∇w|2 ρ dy + (y · ∇w2 )ρ dy

1 |y|2 w2 ρ dy. = |y|2 |∇w|2 ρ dy − n w2 ρ dy + 2 The third identity is

2

n − 2 nβ 2 |y| β|y|2 − |∇w|2 ρ dy + − w ρ dy 4 2 4 2

n p+1 |y|2 |w| ρ dy = 0. − − 2(p + 1) p + 1

(25.15)

To get (25.15), we multiply (25.13) by −(y · ∇w) and we use

βw2 |w|p+1 − dy ρy · ∇ 2 p+1

2 βw2 |y| |w|p+1 = −n ρ − dy 2 2 p+1

(y · ∇w)ρ(βw − |w|p−1 w) dy =


199

and

−

(y · ∇w)∇ · (ρ∇w) dy =

(ρ∇w) · ∇(y · ∇w) dy

1 2 (ρy) · ∇(|∇w|2 ) dy = ρ|∇w| dy + 2

1 |y|2 2 − n ρ|∇w|2 dy. = ρ|∇w| dy + 2 2

Now, to complete the proof of (25.11), we eliminate the terms involving w2 and n 1 |w|p+1 by taking the linear combination p+1 ·(25.12) − 2(p+1) ·(25.14) + (25.15). Finally, (25.11) and our assumption p ≤ pS imply ∇w ≡ 0, hence w ≡ 0, w ≡ k or w ≡ −k. Lemma 25.6. Consider problem (18.1) with p > 1 and u0 ∈ L∞ (Rn ). Assume that the upper blow-up rate estimate (23.5) is satisfied. Then we have: (i) For any sequence sj → ∞, there exists a subsequence, still denoted sj , and a function z ∈ S such that wa (·, sj ) → z in C 1 (Rn ). (ii) Assume that ω(wa ) 0 (resp., ±k). Then E wa (s) → 0 (resp., E wa (s) → η(n, p) > 0). (iii) If ω(wa ) 0, then a is not a blow-up point. (iv) If p ≤ pS , then ω(wa ) is one of the sets {0}, {k}, {−k}. Proof. (i) Assumption (23.5) implies |wa | ≤ M,

y ∈ R n , s ≥ s0 .

(25.16)

By Proposition 23.15, since ∇wa (y, s) = (T − t)β+1/2 ∇u(x, t), this implies |∇wa | ≤ M1 ,

y ∈ Rn , s ≥ s0 := s0 + log 2.

(25.17)

Let zj (y, s) = wa (y, s + sj ). By (23.9), (25.16), (25.17) and parabolic estimates, the sequence {zj } is precompact in C 2,1 (Rn × [0, 1]). Consequently, there exists a subsequence of sj (still denoted sj ) and a solution z of 1 zs − ∆z + z · ∇z = |z|p−1 z − βz, 2

y ∈ Rn , s ∈ [0, 1],

such that wa (·, · + sj ) → z in C 2,1 (Rn × [0, 1]). Moreover z, ∇z are bounded in Rn × [0, 1]. On the other hand, using (23.17), we have

0

1

Rn

2

(∂s zj ) ρ dy ≤

∞

sj

Rn

(∂s wa )2 ρ dy → 0

200


as j → ∞. By Fatou’s lemma we deduce that ∂s z = 0 and the assertion follows. for|y| (ii) Assume that wa (·, sj ) → 0 (resp., ±k) and ∇wa (·, sj ) → 0, uniformly bounded. Using (25.16), (25.17) and dominated convergence, we infer E wa (sj ) → 0, resp.

E wa (sj ) →

ρ dy

β 2

Rn

−

k p−1 2 (4π)n/2 k 2 k = =: η(n, p) > 0. p+1 2(p + 1)

The assertion then follows from the monotonicity of E wa (s) . (iii) Let b ∈ Rn . Similar to (25.16) and (25.17), we have |wb | ≤ M,

|∇wb | ≤ M1 ,

y ∈ Rn , s ≥ s 0 ,

(25.18)

with M, M1 independent of b. We shall use the interpolation inequality 2 v ∈ C 1 (B1 ), |v(0)| ≤ C(n, θ) v θL2 (B1 ) ∇v 1−θ L∞ (B1 ) + v L (B1 ) ,

(25.19)

where 0 < θ < 2/(n + 2) if n ≥ 2 and θ = 1/2 if n = 1. (Inequality (25.19) can be shown by applying the mean-value theorem to the difference |v(0)|1/(1−θ) − older’s inequality.) Fix θ as |v(x)|1/(1−θ) , integrating over x ∈ B1 and using H¨ above. By (23.16), since wb exists globally, we have, for any s1 ≥ s0 , wb (0, s) 2L2 (B1 )

≤ C(n, p)

Rn

wb2 ρ dy ≤ C(n, p)E 2/(p+1) (wb (s1 )),

s ≥ s1 . (25.20)

Using (25.18), (25.19) and (25.20), it follows that |wb (0, s)| ≤ C(n, p) M11−θ E θ/(p+1) (wb (s1 )) + E 1/(p+1) (wb (s1 )) ,

s ≥ s1 .

Consequently, there exists γ(M1 , ε) > 0 such that E wb (s1 ) < γ(M1 , ε) implies

|wb (0, s)| ≤ ε,

s ≥ s1 .

Assume that ω(wa ) 0.Assertion (ii) implies E wa (s1 ) < γ(M1 , ε) for s1 large. But since, for given s, E wb (s) depends continuously on b (cf. Proposition 23.8), we infer that E wb (s1 ) < γ(M1 , ε) for |b − a| small. It follows that |wb (y, s)| ≤ ε, s ≥ s1 , hence (T − t)1/(p−1) |u(b, t)| ≤ ε, for (b, t) close to (a, T ). By Theorem 25.3, we conclude that a is not a blow-up point. (iv) In view of Proposition 25.4, this follows from an obvious connectedness argument. Proof of Theorem 25.1. This is an immediate consequence of assertions (iii) and (iv) of Lemma 25.6.


201

As a consequence of the above arguments, we are also now in a position to prove the result stated in the previous section concerning the blow-up set. Proof of Theorem 24.5. (i) If E(u0 ) < η, where η is given by Lemma 25.6(ii), then lims→∞ E wa (s) < η. Consequently, wa cannot converge to ±k, due to Lemma 25.6(ii). So it converges to 0 and a is not a blow-up point in view of Theorem 25.1. (ii) By dominated convergence, under the current assumption on u0 , we have lima→∞ E wa (s0 ) = 0. The conclusion thus follows from assertion (i). Remark 25.7. Radial nonincreasing case. In Theorem 25.1, assume in addition that u0 ≥ 0 is radial nonincreasing. Since u(0, t) = u(t) ∞ ≥ k(T − t)−1/(p−1) by Proposition 23.1, the conclusion (with the + sign) follows directly from Lemma 25.6(i) and Proposition 25.4, and the nondegeneracy result is not needed. Remark 25.8. Blow-up profile. The results in Theorems 24.1 and 24.3 concerning the blow-up profile in the original variables can be strongly improved, at the expense of significant technical difficulty, though. In order to do this one has to linearize problem (23.9) around the nontrivial equilibrium w∞ ≡ k and study the convergence to w∞ in more detail. Assume Ω = R, u0 ∈ BC(R), u0 ≥ 0 and 0 ∈ B(u0 ). Then one obtains (see [275], [276] and cf. also [207], [208] for earlier results in that direction) that one of the following alternatives must hold: 1. (T − t)β u(x, t) ≡ k, 1/2 −β , t → k 1 + (p − 1)y 2 /(4p) as t → T , 2. (T − t)β u y (T − t)| log(T − t)| 3. there exists C > 0 and an even integer m ≥ 4 such that (T − t)β u y(T − t)1/m , t → k(1 + Cy m )−β

as t → T,

where the convergence is uniform for y lying in a bounded set. (Note also that the second alternative is always true if u0 is symmetric and has a unique local maximum at x = 0.) As a consequence of this assertion, the following general result was obtained in [519]. Assume that u(x, t) is a positive solution of ut − uxx = up for x ∈ (−R, R) and t ∈ (0, T ) which blows up at t = T . Assume also that its blow-up set B is contained in [−δ, δ] for some δ < R. Then B is isolated and for any x ¯ ∈ B one of the following holds |x − x 8p β ¯|2 β u(x, T ) = , x→¯ x | log |x − x ¯|| (p − 1)2 lim

lim |x − x ¯|mβ u(x, T ) = kC −β ,

x→¯ x

202


where β, k, C, m are as above. It is also known that any of the possibilities mentioned above may happen (see [100], [366]). For results in higher dimensions we refer to [61], [521], [367], [360].

26. Universal bounds and initial blow-up rates The a priori estimate (22.2) with a universal constant C cannot be true for all global solutions of (22.1) for the following reasons. First, such an estimate would imply an a priori bound for stationary solutions and we know from Theorem 7.8(ii) that such bound is not true for sign-changing solutions in the subcritical case. Second, we know from Remark 19.12 that there exist nonnegative global classical solutions such that u(t) ∞ → ∞ as t → 0+. Anyhow, in the subcritical case, we can still hope for a universal bound of global nonnegative solutions of (22.1) on the interval (τ, ∞), where τ > 0. In other words, we are interested in the estimate sup u(t) ∞ ≤ C(τ )

for all τ > 0.

(26.1)

t≥τ

(Note that (26.1) cannot be true in the critical or supercritical case — at least in starshaped domains — due to Theorem 28.7.) It will be natural at the same time to ask about the dependence of the constant C(τ ), as τ → 0. In fact, this question can be also considered from a different point of view, which gives rise to interesting connections and unifications with questions studied in Sections 23 and 21. Consider local nonnegative classical solutions of ut − ∆u = up , u = 0,

x ∈ Ω, 0 < t < T, x ∈ ∂Ω, 0 < t < T

(26.2)

(without any prescribed initial conditions). Do there exist estimates of the form u(t) ∞ ≤ Ct−α , and

0 < t ≤ T /2,

u(t) ∞ ≤ C(T − t)−β ,

T /2 ≤ t < T,

(26.3)

(26.4)

where C = C(p, Ω, T ) > 0 is a universal constant, independent of u ? If (26.4) were true with β = 1/(p − 1), one would in particular recover the (final) blow-up estimates of Section 23, now with a universal constant. Analogously, estimate (26.3) would provide (universal) initial blow-up rates. An interesting question is what should be the optimal value of α. Of course, (26.3) or (26.4) implies in particular the universal bound (26.1) for global nonnegative solutions. Furthermore, we will see that these estimates are strongly connected with parabolic Liouville-type theorems and decay of global solutions of the Cauchy problem (see Remark 26.10(i)).

26. Universal bounds and initial blow-up rates

203

The bound (26.1) for all global nonnegative solutions of (22.1) in bounded domains was first proved in [200] for p < pBT (note that this exponent has already appeared in an elliptic context in Section 10). As for the initial and final blow-up rate estimates (26.3) and (26.4), they have first been established in [28] for the Cauchy problem with p < pF . Those results have been improved and extended in a number of subsequent works, using various techniques. We shall present some of these results and techniques. Some of the proofs rely on rescaling arguments and apply essentially only to the model problem (26.2), while some others allow to treat nonlinearities f (u) without precise power behavior (see Remarks 26.5 and 26.12). We start with a result whose proof is relatively simple. Better results will be given later for the model problem (see Theorems 26.6 and 26.8), but the present approach, besides its simplicity, has the advantage to be applicable to more general nonlinearities (see Remark 26.5). It is based on integral bounds obtained by testfunction arguments (in particular using the first eigenfunction) and on smoothing properties in Lq - or Lqδ -spaces (see Theorem 26.14 below for further results obtained by using Lqδ -spaces). Theorem 26.1. Assume Ω bounded and 1 0, there exists C(Ω, p, τ ) > 0 such that any global nonnegative classical solution of (26.2) satisfies sup u(t) ∞ ≤ C(Ω, p, τ ).

(26.5)

t≥τ

Remarks 26.2. (i) Instantaneous attractors. In other words, Theorem 26.1 (and similar subsequent results) shows the existence of “instantaneous attractors” for global nonnegative trajectories of (26.2). Note that, by standard smoothing effects, (26.5) guarantees that for each τ > 0, there is a compact (absorbing) subset Kτ of C 2 (Ω) ∩ C0 (Ω), such that any global nonnegative solution of (26.2) remains in Kτ for t ≥ τ (otherwise u has to blow up in finite time). In terms of the set G + introduced in Remark 19.12, Theorem 26.1 says that, although G + itself is unbounded, for each τ > 0, S(τ )G + is a bounded subset of L∞ (Ω) (where S(t)u0 denotes the solution u(t) of problem (15.1)). (ii) Differences from equations with absorption. We emphasize that such localization results are of a quite different nature from what occurs in equations with absorption, such as ut − ∆u + |u|p−1 u = 0 with p > 1. Indeed, for this equation, it is straightforward that all solutions of the Dirichlet or Cauchy problem (with bounded initial data) satisfy the universal estimate u(t) ∞ ≤ C(p)t−1/(p−1) for all t > 0. This immediately follows by comparing with the solution y(t) ≡ C(p)t−1/(p−1) of the ODE y + y p = 0. In the case of problem (26.2), this is of course not true, due to the existence of blowing-up solutions. The universal bound (26.5) is verified by a solution u, under

204


the assumption that u exists globally (or on some time interval (0, T ) in the case of estimates (26.3) and (26.4)). We give a first proof of Theorem 26.1 based on Lqδ -spaces, due to [200]. We first derive some basic estimates for positive solutions of (26.2). Lemma 26.3. Assume Ω bounded, p > 1, and 0 < T < ∞. Let u be a nonnegative classical solution of (26.2) on (0, T ). Then for all t ∈ (0, T /2], there holds

Ω

and

t

Ω

0

u(t)ϕ1 dx ≤ C(p, Ω)(1 + T −1/(p−1) ),

up ϕ1 dx ds ≤ C(p, Ω) 1 + t 1 + T −1/(p−1) .

(26.6)

(26.7)

Proof. As in the proof of Theorem 17.1, denote y = y(t) := Ω u(t)ϕ1 dx, multiply the equation in (26.2) by ϕ1 and integrate by parts. We obtain

d dt

Ω

u(t)ϕ1 dx + λ1

Ω

u(t)ϕ1 dx =

Ω

up (t)ϕ1 dx.

(26.8)

By Jensen’s inequality, we infer that d dt

$

Ω

u(t)ϕ1 dx ≥

Ω

%p

u(t)ϕ1 dx − λ1 u(t)ϕ1 dx. Ω

Since u exists on (0, T ), we deduce easily that

Ω

u(t)ϕ1 dx ≤ C(p, Ω)(1 + (T − t)−1/(p−1) ),

0 < t < T,

hence (26.6). Integrating (26.8) in time over (τ, t) (0 < τ < t ≤ T /2) and using (26.6), we obtain

t

τ

Ω

t

uϕ1 dx ds + u(t)ϕ1 dx − u(τ )ϕ1 dx Ω Ω Ω τ ≤ C(p, Ω) 1 + t 1 + T −1/(p−1)

up ϕ1 dx ds = λ1

and (26.7) follows by letting τ → 0. Proof of Theorem 26.1. By Theorem 22.1, we know that global solutions of (22.1) satisfy the a priori estimate u(t) ∞ ≤ C(Ω, p, u(t0 ) ∞ ),

t ≥ t0 ≥ 0,


205

where C remains bounded for u(t0 ) ∞ bounded. Therefore, it is sufficient to show the existence of C(p, Ω, τ ) > 0 such that any global classical solution u of (26.2) satisfies inf u(t) ∞ ≤ C(p, Ω, τ ). (26.9) t∈(0,τ )

Moreover, by the Lqδ -smoothing estimate in Theorem 15.9, (26.9) will follow if we can show that, for some q > (n + 1)(p − 1)/2, inf t∈(0,τ /2)

u(t) q,δ ≤ C(p, Ω, q, τ ).

(26.10)

But (26.7) guarantees that (26.10) is true for q = p and, since p < pBT , we have p > (n + 1)(p − 1)/2. We now give a second proof (see [200, Section 6]), which does not use Lqδ -spaces. Instead it requires the following estimate, whose proof uses the special test-function constructed in Lemma 10.4 by considering a singular elliptic problem. Lemma 26.4. Assume Ω bounded, p > 1, 0 < T < ∞, and ε ∈ (0, (p + 1)/2]. Let u be a nonnegative classical solution of (26.2) on (0, T ). Then for all t ∈ (0, T /2], there holds

t

p+1 u 2 −ε dx ds ≤ C(p, Ω, ε) 1 + t 1 + T −1/(p−1) . 0

Ω

Proof. For given 0 < α < 1, Lemma 10.4 ensures the existence of a function ξ ∈ C(Ω) ∩ C 2 (Ω) ∩ H01 (Ω) such that −∆ξ = ϕ−α in Ω. Moreover, ξ satisfies 1 ξ(x) ≤ C(Ω, α)δ(x),

x ∈ Ω.

(26.11)

4ε p−1+2ε .

Here we choose α = 1 − Taking ξ as a test-function in (26.2) (which is possible due to ξ ∈ H01 (Ω)) and integrating in time over (τ, t), we obtain

t

t

−α p uϕ1 dx ds = u ξ dx ds + u(τ )ξ dx − u(t)ξ dx. τ

Ω

τ

Ω

Ω

Ω

Due to (26.11), (26.6) and (26.7) readily imply

t

−1/(p−1) . uϕ−α 1 dx ds ≤ C(p, Ω, ε) 1 + t 1 + T 0

Ω

Using Hölder’s inequality, the last estimate and (26.7) imply the lemma.

Second proof of Theorem 26.1. As in the first proof, it is sufficient to show the existence of C(p, Ω, τ ) > 0 such that any global classical solution u of (26.2) satisfies (26.9). Moreover, by the smoothing estimate in Theorem 15.2, (26.9) will follow if we can show that, for some q > n(p − 1)/2, inf t∈(0,τ /2)

u(t) q ≤ C(p, Ω, q, τ ).

(26.12)

But Lemma 26.4 guarantees that (26.12) is true for all q ∈ [1, (p + 1)/2) and, since p < pBT , we have q > n(p − 1)/2 for q < (p + 1)/2 close to (p + 1)/2.

206


Remark 26.5. The assumption p < pBT in Theorem 26.1 is not optimal for the model problem (26.2), see Theorems 26.6 and 26.8 below. However, unlike the proofs of those theorems, the proof of Theorem 26.1 does not rely on rescaling and can be applied to more general nonlinearities f (x, u) satisfying C1 uq − C ≤ f (x, u) ≤ C2 up + C with p < pBT , under suitable assumption on q ∈ (1, p) (see [450]). Note that the proof uses a priori estimates of global solutions obtained in Theorem 22.1. However, the proof of Theorem 22.1 based on interpolation can be also extended to such nonlinearities. Now we give an optimal result [438] in dimensions n ≤ 3 concerning universal bounds of global nonnegative solutions of the Dirichlet problem. The method is completely different. It is based on energy, measure arguments, rescaling and elliptic Liouville-type theorems. Theorem 26.6. Let n ≤ 3 and 1 0 such that any global classical solution u of (26.2) satisfies (26.9). Moreover, since p + 1 > n(p − 1)/2 due to p < pS , by the smoothing property in Theorem 15.2, (26.9) will follow if we can show that inf t∈(0,τ /2)

u(t) p+1 ≤ C(p, Ω, τ ).

We argue by contradiction and assume that for each k = 1, 2, . . . , there exists a global solution uk ≥ 0 of (26.2) such that uk (t) p+1 p+1 > k

for all t ∈ (0, τ /2).

(26.13)

Denote

1 Ek (t) = E uk (t) = 2

1 |∇uk (t)| dx − p + 1 Ω 2

Ω

up+1 k (t) dx.

Recall that Ek (t) = − ∂t uk (t) 22 ≤ 0 and that uk satisfies the identity 1 d 2 dt

Ω

u2k (t) dx

= Ω

up+1 k (t) dx

= −2Ek (t) +

−

p−1 p+1

Ω

|∇uk (t)|2 dx

Ω

(26.14) up+1 k (t) dx.

We now proceed in several steps. From now on, C will denote a positive constant and k0 a positive integer, both depending only on p, Ω, τ (and also on q in Steps 4 and 5).


Step 1. We claim that

207

Ek (τ /4) ≥ k 1/2 ,

(26.15)

for all k ≥ k0 large enough. older’s inequality, we Assume (26.15) is false. Using (26.14), Ek ≤ 0 and H¨ obtain, for all t ≥ τ /4,

1 d p−1 u2k (t) dx ≥ −2k 1/2 + up+1 (t) dx, (26.16) 2 dt Ω p+1 Ω k hence

1 d 2 dt

Ω

u2k (t) dx ≥ −2k 1/2 + C

This implies

1

Ω

Ω

u2k (t) dx ≤ Ck p+1 ,

(p+1)/2 u2k (t) dx .

t ≥ τ /4,

(26.17)

since otherwise Ω u2k (t) dx has to blow up in finite time. Integrating (26.16) over (τ /4, τ /2) and using (26.13), (26.17), we obtain 1 kτ ≤ 4

τ /2

τ /4

1

Ω

up+1 dx dt ≤ C(k p+1 + k 1/2 τ ), k

a contradiction for k ≥ k0 large. Step 2. Let a > 0 to be fixed later and set Fk = {t ∈ (0, τ /4] : −Ek (t) ≥ 1+1/a Ek (t)}. We claim that |Fk | < τ /8 for all k ≥ k0 large enough. Note that Ek > 0 on (0, τ /4] for k ≥ k0 by (26.15), since Ek ≤ 0. By definition of Fk , it follows that −1/a

−1−1/a

) = −Ek Ek

(aEk

≥ χFk

on (0, τ /4].

−1/a

(τ /4) ≥ |Fk |. The claim then follows from By integration, we deduce that aEk (26.15). Step 3. Choose a ≥ (p + 1)/(p − 1). (26.18) We claim that for all k ≥ k0 large,

(a+1)/a up+1 ∂t uk (t) 22 ≤ C k (t) dx Ω

for all t ∈ (0, τ /4] \ Fk .

(26.19)

For all t ∈ (0, τ /4] \ Fk , we have 1+1/a

∂t uk (t) 22 = −Ek (t) ≤ Ek

2(1+1/a)

(t) ≤ ∇uk (t) 2

.

(26.20)

208


Hence, by (26.14) as well as Hölder’s and Young’s inequalities,

up+1 ∇uk (t) 22 ≤ k (t) dx + uk (t) 2 ∂t uk (t) 2 Ω

1+1/a ≤ up+1 k (t) dx + uk (t) 2 ∇uk (t) 2 Ω

1+1/a ≤ up+1 k (t) dx + C uk (t) p+1 ∇uk (t) 2

Ω 1 2a/(a−1) ≤ up+1 + ∇uk (t) 22 k (t) dx + C uk (t) p+1 2 Ω

1 2 ≤C up+1 k (t) dx + ∇uk (t) 2 , 2 Ω where we have used (26.18) and (26.13). Consequently,

up+1 ∇uk (t) 22 ≤ C k (t) dx. Ω

This along with (26.20) implies (26.19). Step 4. Let 0 < q < (p + 1)/2, b = (p + 1 − q)(a + 1)/a and

Gk = t ∈ (0, τ /4] : ∂t uk (t) 22 ≤ C uk (t) b∞ . We claim that |Gk | > 0. Due to Lemma 26.4, for A = A(p, q, Ω, τ ) > 0 large enough, the set

' uqk (t) dx ≥ A} Gk := {t ∈ (0, τ /4] : Ω

satisfies

'k | < τ /8. |G

(26.21)

'k , We deduce from (26.13) that, for all t ∈ (0, τ /4] \ G

p+1−q up+1 (t) dx ≤ C u (t) uqk (t) dx ≤ C uk (t) p+1−q . k ∞ ∞ k Ω

Ω

'k ) by Step 3. The claim then follows from Step 2 Therefore, Gk ⊃ (0, τ /4] \ (Fk ∪ G and (26.21). Step 5. We will now obtain a contradiction by using a rescaling argument. By Step 4, for each large k, we may pick tk ∈ Gk . By (26.13), we have Mk := uk (tk ) ∞ → ∞. Choose xk ∈ Ω such that uk (xk , tk ) = Mk , denote νk = −(p−1)/2 Mk and put wk (y) = Mk−1 uk (xk + νk y, tk ), w 'k (y) = Mk−p ∂t uk (xk + νk y, tk ).


Then the functions wk , w 'k satisfy ∆wk + wkp = w 'k wk = 0

in Ωk , on ∂Ωk ,

209

(26.22)

where Ωk = νk−1 (Ω − xk ). Moreover, 0 ≤ wk ≤ 1 = wk (0). Now passing to the limit we will obtain a contradiction in the same way as in [241]; we only have to show that the functions wk are (locally) uniformly H¨ older continuous and w 'k → 0 in an appropriate way. k Hence let R > 0, BR (x0 ) = {x ∈ Ω : |x−x0 | < R} and BR = {y ∈ Ωk : |y| < R}. Since tk ∈ Gk , we have

|w 'k (y)|2 dy = Mk−2p |∂t uk (xk + νk y, tk )|2 dy k BR

k BR

=

Mk−2p νk−n

|∂t uk (x, tk )|2 dx

BRνk (xk )

≤ CMk−2p Mk

n(p−1)/2

Mkb = CMkγ

for k ≥ k0 , where γ = −2p +

n(p − 1) a+1 (p + 1 − q) + . a 2

By taking q close to (p+1)/2 and a sufficiently large, γ will be negative provided (n − 3)p < n − 1. (In particular this is true due to p < pS if n ≤ 4.) Consequently,

k BR

|w 'k (y)|2 dy → 0

for any R > 0. Since 0 ≤ wk ≤ 1 and wk solves (26.22), standard regularity theory k guarantees that wk is uniformly bounded in W 2,2 (BR ). Since W 2,2 is embedded in the space of Hölder continuous functions due to n ≤ 3, we may pass to the limit in (26.22), similarly as in the proof of Theorem 12.1, in order to get a limiting solution w ≥ 0 satisfying the equation ∆w + wp = 0 either in Rn or in a half-space (and satisfying the homogeneous Dirichlet boundary conditions in the latter case). Moreover w ≤ 1 and w(0) = 1, which contradicts the Liouville-type Theorems 8.1 and 8.2. Remark 26.7. By a (nontrivial) modification of the proof of Theorem 26.6, one can show that the result remains true for n = 4, and for n ≥ 5 under the stronger restriction p < (n − 1)/(n − 3) < pS (see [450]). We now turn to universal initial and final blow-up rates. Recall that the exponent pB in (26.23) has appeared in Section 21.

210


Theorem 26.8. Let p > 1, T > 0 and u be a nonnegative classical solution of (26.2) on QT . Assume that either p < pB ,

or p < pS , Ω = Rn or Ω = BR , and u radial.

(26.23)

Then there holds u(x, t) ≤ C(n, p) t−1/(p−1) + (T − t)−1/(p−1) ,

x ∈ Rn ,

0 0.

(26.26)

Assume p < pB ,

or p < pS and u radial.

Then there holds 1

u(x, t) ≤ C(n, p) t− p−1 ,

x ∈ Rn ,

t > 0.

(26.27)

Remarks 26.10. (i) If the parabolic Liouville-type Theorem 21.2 were known for all p < pS , then this would imply Theorems 26.8 and 26.9 for all p < pS as well. Conversely, it is clear that estimate (26.24) or (26.27) implies nonexistence of positive solutions of (21.1). We see that Liouville-type theorems and these universal estimates are thus equivalent. On the other hand, Theorem 26.8 guarantees that Theorem 21.1 remains true for nontrivial nonnegative radial classical solutions, bounded or not, and that Theorem 21.2 remains true for nontrivial nonnegative classical solutions.


211

(ii) For all p < pS and without radial symmetry assumption, it is however known that the solution of the Cauchy problem (18.1) satisfies u(t) ∞ → 0 as t → ∞ (without a universal estimate), provided u is global and 0 ≤ u0 ∈ L∞ ∩ L2 (Rn ); see [478] and cf. also [299]. (iii) In Theorems 26.8 and 26.9, no conditions at space infinity are assumed on the solution u. (iv) Consider problem (26.2) with the nonlinearity up replaced by f (u). Assume that f : [0, ∞) → R is continuous and is such that lims→∞ s−p f (s) exists in (0, ∞). Then Theorem 26.8 remains valid (with C in (26.24)-(26.25) depending also on f and with an additive constant 1 in estimate (26.24) as well). If we assume in addition that f is C 1 and verifies |f (s)| ≤ C(1 + sp−1 ), s ≥ 0, then Theorem 26.6 remains valid. (v) When Ω is a convex bounded domain, estimate (26.4) with β = 1/(p−1) and C = C(p, Ω, T ) is known also for p < pS , n ≤ 4 [450]. This follows by combining Theorem 26.6 (cf. also Remark 26.7) with the a priori estimate of the blow-up rate (cf. Remark 23.9). Let us point out that the method of proof of Theorem 26.6 can be modified to establish initial blow-up rate estimates as well [450], but the values of α = α(n, p) obtained in this way are not optimal. We will use the following key doubling lemma [424]. Lemma 26.11. Let (X, d) be a complete metric space and let ∅ = D ⊂ Σ ⊂ X, with Σ closed. Set Γ = Σ \ D. Finally let M : D → (0, ∞) be bounded on compact subsets of D and fix a real k > 0. If there exists y ∈ D such that M (y) dist(y, Γ) > 2k,

(26.28)

then there exists x ∈ D such that M (x) dist(x, Γ) > 2k, and M (z) ≤ 2M (x)

M (x) ≥ M (y),

(26.29)

for all z ∈ D ∩ B X x, k M −1 (x) .

Proof. Assume that the lemma is not true. Then we claim that there exists a sequence (xj ) in D such that M (xj ) dist(xj , Γ) > 2k,

(26.30)

M (xj+1 ) > 2M (xj ),

(26.31)

d(xj , xj+1 ) ≤ kM −1 (xj )

(26.32)

and

212


for all j ∈ N. We choose x0 = y. By our contradiction assumption, there exists x1 ∈ D such that M (x1 ) > 2M (x0 ) and d(x0 , x1 ) ≤ k M −1 (x0 ). Fix some i ≥ 1 and assume that we have already constructed x0 , . . . , xi such that (26.30)–(26.32) hold for j = 0, . . . , i − 1. We have dist(xi , Γ) ≥ dist(xi−1 , Γ) − d(xi−1 , xi ) > (2k − k) M −1 (xi−1 ) > 2k M −1 (xi ), hence M (xi ) dist(xi , Γ) > 2k. By our contradiction assumption, it follows that there exists xi+1 ∈ D such that M (xi+1 ) > 2M (xi ) and d(xi , xi+1 ) ≤ k M −1 (xi ). We have thus proved the claim by induction. Now, we have M (xi ) ≥ 2i M (x0 )

and d(xi , xi+1 ) ≤ k 2−i M −1 (x0 ),

i ∈ N.

(26.33)

In particular, (xi ) is a Cauchy sequence, hence it converges to some a ∈ D ⊂ Σ. Moreover, d(x0 , xi ) ≤

i−1 j=0

d(xj , xj+1 ) ≤ k M −1 (x0 )

i−1

2−j ≤ 2k M −1 (x0 ),

j=0

hence dist(xi , Γ) ≥ dist(x0 , Γ) − 2k M −1 (x0 ) =: δ > 0. Therefore, K := {xi : i ∈ N} ∪ {a} is a compact subset of Σ \ Γ = D. Since M (xi ) → ∞ as i → ∞ by (26.33), this contradicts the assumption that M is bounded on compact subsets of D. The lemma is proved. Proof of Theorem 26.8. We first consider the nonradial case and assume p < pB . We will show (26.25). Note that if Ω√= Rn , by a simple scaling argument (replacing u(x, t) by u˜(y, s) := T 1/(p−1) u( T y, T s)), (26.25) with T = 1, implies (26.24) for any T > 0.


213

Assume that estimate (26.25) fails. Then, there exist sequences Tk ∈ (0, ∞), uk , yk ∈ Ω, sk ∈ (0, Tk ), such that uk solves (26.2) (with T replaced by Tk ) and the functions p−1 Mk := uk 2 , k = 1, 2, . . . , (26.34) satisfy

Mk (yk , sk ) > 2k (1 + d−1 k (sk )),

(26.35)

where dk (t) := (min(t, Tk − t))1/2 . We will use Lemma 26.11 with X = Rn+1 , equipped with the parabolic distance dP (x, t), (y, s) = |x − y| + |t − s|1/2 , Σ = Σk = Ω × [0, Tk ], D = Dk = Ω × (0, Tk ), and Γ = Γk = Ω × {0, Tk }. Notice that dk (t) = distP (x, t), Γk , (x, t) ∈ Σk . By Lemma 26.11, it follows that there exists xk ∈ Ω, tk ∈ (0, Tk ) such that Mk (xk , tk ) > 2k d−1 k (tk ),

(26.36)

Mk (xk , tk ) ≥ Mk (yk , sk ) > 2k, and Mk (x, t) ≤ 2Mk (xk , tk ), where

˜k , (x, t) ∈ Dk ∩ B

(26.37)

˜k := (x, t) ∈ Rn+1 : |x − xk | + |t − tk |1/2 ≤ k λk , B

and

λk := Mk−1 (xk , tk ) → 0.

(26.38)

˜k , we have |t− tk | ≤ k 2 λ2 < d2 (tk ) = min(tk , Tk − tk ) Observe that for all (x, t) ∈ B k k by (26.36), hence t ∈ (0, Tk ). It follows that

Ω ∩ {|x − xk |
0 in Rn+1 , or u = 0 in Rn × (−∞, t0 ]

and

u > 0 in Q := Rn × (t0 , ∞),

(26.44)

for some t0 < 0. But, in the latter case, since u ≤ C, we have ut − ∆u ≤ C p−1 u in Q and we infer from the maximum principle in Proposition 52.4 that u = 0 in Q, a contradiction. Therefore u > 0, which contradicts Theorem 21.1. In case (26.43) holds, denote Hc := {y ∈ Rn : y1 > −c}. By performing a suitable orthogonal change of coordinates, similarly as in the proof of Theorem 12.1, using (26.38), (26.40), (26.41), interior-boundary parabolic estimates and the embedding (1.2), we obtain a subsequence of vk which converges in C α (H c ), 0 < α < 1, to a bounded classical solution v ≥ 0 of ∂s v − ∆y v = v p , v = 0,

y ∈ Hc , s ∈ R, y ∈ ∂Hc , s ∈ R,

(26.45)

with v(0, 0) = 1 (hence c > 0). Similarly as in the previous case, we obtain v > 0, which contradicts Theorem 21.8. In the radial case, let us assume in addition that u(|x|, t) is nonincreasing as a function of |x|. Then we may take xk = 0 in the above proof and the rescaling procedure yields a positive, bounded, radial, classical solution of (21.1), contradicting the radial Liouville-type Theorem 21.2. For the general (nonmonotone) radial case, which is slightly more delicate, we refer to [425]. Remark 26.12. Lemma 26.11 and the method of proof of Theorem 26.8 are a generalization of an idea in [283] (see also, e.g., [130], [448], [361], [339]). In those works, blow-up estimates and a priori bounds of global solutions, with nonuniversal constants, were derived for various types of superlinear parabolic problems. By using a property similar to Lemma 26.11 (but concerning functions of the


215

time variable only), it was shown that if a solution u were violating a suitable estimate, then the function M (t) := u(t) ∞ would satisfy M (s) ≤ 2M (tk ) for all s ∈ [tk , tk + kM 1−p (tk )] and some sequence of times tk . Then, by a rescaling argument similar to that used in the proof of Theorem 26.8, one was led to a contradiction with a Fujita-type theorem. Note that these approaches do not use any variational structure of the problem, unlike the methods in the proofs of Theorems 22.1 and 23.7 for instance. This advantage will be exploited in Sections 38 and 44. A natural question is whether the exponent 1/(p−1) in Theorem 26.8 is optimal. As for the (final) blow-up rates, this is indeed the case, due to Proposition 23.1. Interestingly, the situation is different for the initial blow-up rate, as it appears from the following results, which show that for p close to 1, the optimal initial blow-up rate exponents are in fact less than 1/(p − 1). Moreover, they are different for the Cauchy and for the Dirichlet problems. Theorem 26.13. Let p > 1, T > 0, and Ω = Rn . (i) Assume p < pF . Then any nonnegative classical solution of (26.2) on Rn ×(0, T ) satisfies u(x, t) ≤ C(n, p, T ) t−n/2 , x ∈ Rn , 0 < t < T /2. (ii) Let

n 1 . α0 := min , 2 p−1

For all ε > 0, there exist T > 0, a positive classical solution u of (26.2), and C > 0 such that u(t) ∞ ≥ Ct−α0 +ε , for t > 0 small. Theorem 26.14. Let p > 1, T > 0, and assume Ω bounded. (i) Assume p < 1 + 2/(n + 1). Then any nonnegative classical solution of (26.2) on QT satisfies u(x, t) ≤ C(p, Ω, T )t−(n+1)/2 , (ii) Let α1 := min

n + 1 2

x ∈ Ω,

,

0 < t < T /2.

1 . p−1

For all ε > 0, there exist T > 0, a positive classical solution u of (26.2), and C > 0 such that u(t) ∞ ≥ Ct−α1 +ε , for t > 0 small.

216


Remarks 26.15. (i) The proof of Theorem 26.13 yields C(n, p, T ) = C(n, p)T n/2−1/(p−1) . (ii) As already mentioned in Remark 15.4(ii), it is known [269] that if pF < p < pS , then (26.26)√possesses global, positive self-similar solutions of the form u(x, t) = t−1/(p−1) w(|x|/ t), with w ∈ C 2 ([0, ∞)), radial and decreasing. In particular we have u(t) ∞ = w(0)t−1/(p−1) , t > 0 (compare with Theorem 26.9). (iii) By minor modifications of the proof, one can show that Theorem 26.14(i) remains valid for more general nonlinearities f (u) instead of up , see [450]. Namely one may assume that f , of class C 1 , satisfies C1 sq − C2 ≤ f (s) ≤ C2 (1 + sp ), s ≥ 0, for some 1 < q 0. A similar generalization is true for Theorem 26.13(i). Theorem 26.13(i) was proved in [79] by using Harnack inequality for the linear parabolic equation ut − ∆u = V (x, t)u, which holds under suitable integrability conditions on the potential V . An alternative proof relying on local regularity estimates from [29] (based on Moser’s iteration arguments) was also given in [79]. Here we provide a more elementary proof (based on a modification of ideas from [361]), which relies on smoothing in uniformly local Lebesgue spaces (cf. Section 15). The introduction of these spaces in our problem is natural. Indeed, a simple application of the eigenfunction method (cf. Section 17) provides the following uniformly local L1 a priori estimate. Lemma 26.16. Let u be a nonnegative classical solution of ut − ∆u = up in Rn × (0, T ). Then there holds

u(t) 1,ul = sup |u(y, t)| dy ≤ C(n, p)(1 + T −1/(p−1) ), 0 < t < T /2. a∈Rn

|y−a| 0 be such that t = t1 + t2 . Since p < pF , we have 1 > n(p − 1)/2. ∞ n Moreover, due to Theorem 26.8, we have u ∈ L∞ loc ((0, 1), L (R )). It then follows from Theorem 15.11 with q = 1 and (26.46) that −n/2

u(t) ∞ ≤ C(n, p) u(t1 ) 1,ul t2

−n/2

≤ C(n, p) t2

,

provided t2 ≤ τ = τ (n, p). If t ≤ τ , we take t1 = t2 = t/2. If τ < t ≤ 1/2, we take t2 = τ , t1 = t − τ . In both cases, we thus obtain u(t) ∞ ≤ C(n, p)t−n/2 . (ii) Let q ∈ (q0 , ∞) with q0 := max(1, n(p − 1)/2). By Theorem 15.2, we know that problem (22.1) is locally well-posed in Lq (Rn ). Let u0 (x) = |x|−2k χ{|x| q0 close to q0 and, by estimate (15.30), we have e−tA u0 ∞ ≥ e−tA u0 (0) = Ct−k for t > 0 small. It follows that the local solution u of (22.1) with initial data u0 satisfies u(t) ∞ ≥ Ct−k for small t > 0. Since k → α0 as k → n/2q0 , the conclusion follows. The proof of Theorem 26.14(i) is similar to that of Theorem 26.13(i), except that we now use smoothing in Lqδ -spaces (cf. Section 15). Proof of Theorem 26.14(i). Due to (1.4), estimate (26.6) can be restated as an L1δ -estimate: u(t) 1,δ ≤ M := C(p, Ω)(1 + T −1/(p−1) ),

0 < t ≤ T /2.

(26.47)

We can now apply Lqδ -smoothing results as follows. Fix t ∈ (0, T /2] and let t1 , t2 > 0 be such that t = t1 + t2 . Since 1 > (n + 1)(p − 1)/2, due to p < 1 + 2/(n + 1), we may apply Theorem 15.9 with q = 1, and we deduce from (26.47) that −(n+1)/2

u(t) ∞ ≤ C(p, Ω, M ) t2 provided t2 ≤ τM := τM (p, Ω, M ).

,

218


If t ≤ τM , we take t1 = t2 = t/2. If τM < t ≤ T /2, we take t2 = τM , t1 = t − τM , −(n+1)/2 and we note that t2 ≤ ( τTM )(n+1)/2 t−(n+1)/2 . In both cases we thus obtain u(t) ∞ ≤ C(p, Ω, T )t−(n+1)/2 ,

0 < t ≤ T /2.

(ii) Let q ∈ (q1 , ∞) with q1 := max(1, (n + 1)(p − 1)/2). By Theorem 15.9, we know that problem (22.1) is locally well-posed in Lqδ (Ω). By Theorem 49.7(ii) in Appendix C, for any k < (n + 1)/2q, there exists u0 ∈ Lqδ (Ω) such that e−tA u0 ∞ ≥ Ct−k for small t > 0. It follows that the local solution u of (22.1) with initial data u0 satisfies u(t) ∞ ≥ Ct−k for small t > 0. Since k → α1 as k → (n + 1)/2q1 , the conclusion follows.

27. Complete blow-up In this section we consider the question whether or not nonglobal classical solutions of problem (22.1) can be continued in some weak sense after the blow-up time Tmax (u0 ). A natural way to look at this question is via monotone approximation. Let u be the solution of problem (22.1) and assume u0 ≥ 0 and Tmax (u0 ) < ∞. Set v ≥ 0, k = 1, 2, . . .

fk (v) := min(v p , k), and let uk be the solution of the problem vt − ∆v = fk (v),

x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0,

v = 0,

x ∈ Ω.

v(x, 0) = u0 (x),

⎫ ⎪ ⎬ ⎪ ⎭

(27.1)

The function uk is globally defined and uk+1 ≥ uk . Define u ¯(x, t) := lim uk (x, t). k→∞

Notice that uk solves the integral equation

t

uk (x, t) =

Ω

G(x, y, t)u0 (y) dy +

0

Ω

G(x, y, t − s)fk (uk (y, s)) dy ds, x ∈ Ω, t > 0,

(27.2)

27. Complete blow-up

219

where G is the Dirichlet heat kernel in Ω. Since G > 0 and uk+1 ≥ uk , we may pass to the limit in (27.2) in order to get

u ¯(x, t) = Ω

t

G(x, y, t)u0 (y) dy+

G(x, y, t−s)¯ up (y, s) dy ds, 0

Ω

x ∈ Ω, t > 0,

(27.3) where the double integral may be infinite. Obviously u ¯(·, t) = u(·, t) for t < Tmax (u0 ). Set ¯(x, t) = ∞ for all x ∈ Ω} T c = T c (u0 ) := inf{t ≥ Tmax (u0 ) : u and notice that T c (u0 ) ≥ Tmax (u0 ). Moreover, due to (27.3) and

G(x, y, t − s)uk (y, s) dy, x ∈ Ω, t > s > 0, uk (x, t) ≥ Ω

¯(·, t) < ∞ a.e. in Ω for all t ∈ (0, T c), and we have u ¯ < ∞ a.e. in Ω × (0, T c), u c u¯ = ∞ in Ω × (T , ∞). Definition 27.1. We say that u blows up at t = Tmax (u0 ) completely if Tmax (u0 ) = T c (u0 ). As we shall see below the notion of complete blow-up is different from the notion of global blow-up in Remark 24.6(v) and Section 43. In fact, the following theorem guarantees that the solution u from Theorem 24.1 (satisfying u(x, T ) ≤ Cα |x|−α ) blows up completely. Theorem 27.2. Consider problem (22.1) with p > 1, Ω bounded, 0 ≤ u0 ∈ L∞ (Ω), and Tmax (u0 ) < ∞. Assume either (i)

p < pS

or (ii)

ut ≥ 0

in (0, Tmax (u0 )).

Then u blows up completely at t = Tmax (u0 ). See Remark 23.6 for conditions ensuring that ut ≥ 0. Theorem 27.2 is due to [54]. In Proposition 27.7 below, we shall see that the result may fail for p > pS . Before presenting the full proof of Theorem 27.2, we shall first give a proof of a special, one-dimensional case. This alternative approach is simpler than that in the general case and, as an advantage, it can be used for problems with nonconvex nonlinearities. However, although the argument can be extended to dimensions n > 1, the nonlinearity then has to satisfy severe growth restrictions and it requires the solution u to be increasing in time; see [54] for details.

220


Proof of Theorem 27.2 in a special case. We shall prove the assertion in case (ii), under the additional assumptions that n = 1, Ω = (−1, 1), and u0 is radial nonincreasing. These assumptions and Proposition 52.17 guarantee that ux ≤ 0 for x ∈ [0, 1) and t ∈ (0, T ). Denote T := Tmax (u0 ). Step 1. Denote f (u) = up . We shall prove f (u(t)) 1 → ∞ as t → T −. Since u ≥ 0 and ut ≥ 0 we see that the function ψ : t → f (u(t)) 1 is nondecreasing. Assume, by contrary, that ψ is bounded. Then the Lp -Lq -estimates and the variation-of-constants formula guarantee

u(t) q ≤ u0 q +

t

0

(t − s)−α f (u(s)) 1 ds,

α :=

1 n 1− . 2 q

Since n = 1, in the particular case q = ∞ we obtain

u(t) ∞ ≤ u0 ∞ + C

t 0

(t − s)−1/2 ds < C(T ),

which contradicts T < ∞. Step 2. Denote ϕ(x) := limt→T − u(x, t) and let ε ∈ (0, 1). Then

1−ε

1−ε

f (ϕ(x)) dx = lim

t→T −

−1+ε

f (u(x, t)) dx −1+ε

≥ lim inf (1 − ε) t→T −

1

−1

f (u(x, t)) dx = ∞,

where we have used successively the monotone convergence of u to ϕ, ux ≤ 0 for x ≥ 0 and Step 1. Step 3. Choose x ∈ (−1, 1), t > T . Then there exists ε > 0 such that t − T ≥ 2ε and |x| < 1 − ε. We have (e

−sA

1

G(x, y, s)w(y) dy ≥ C˜ε

w)(x) = −1

1−ε

w(y) dy, −1+ε

s ∈ (ε, 2ε),

and C˜ε := inf{G(x, y, s) : |x|, |y| < 1 − ε, s ∈ (ε, 2ε)} > 0. Since e−tA u0 ≥ 0 and fk (uk (y, s)) ≥ fk (uk (y, T )) for s ≥ T , we obtain

uk (x, t) ≥

t

0

e−(t−s)A fk (uk (s)) (x) ds

t−ε

≥ C˜ε t−2ε

1−ε

−1+ε

fk (uk (y, s)) dy ds ≥ Cε

(27.4)

1−ε

−1+ε

fk (uk (y, T )) dy.


221

Step 4. Using (27.4) and Step 2 we see that, as k → ∞,

uk (x, t) ≥ Cε

1−ε

−1+ε

fk (uk (y, T )) dy → Cε

1−ε

−1+ε

f (ϕ(y)) dy = ∞,

which proves the assertion. An essential ingredient in the proof of Theorem 27.2 in the general case is the following result (cf. [54, Lemma 2.1]), of independent interest, which is true for all p > 1 and without monotonicity assumption on u. Proposition 27.3. Consider problem (22.1) with p > 1 and Ω bounded. Assume that u0 , v0 ∈ L∞ (Ω) satisfy 0 ≤ v0 ≤ u0 and u0 ≡ v0 . Then either Tmax (v0 ) = T c (u0 ) = ∞ or Tmax (v0 ) > T c (u0 ). In [345], by employing arguments different from those in [54], complete blow-up was proved for problem (19.1) for a rather general class of (convex nondecreasing) nonlinearities f (u), but only for monotone solutions in time (cf. case (ii) of Theorem 27.2). Our proofs of Theorem 27.2 and Proposition 27.3 are based on a modification of ideas of [345] which enable one to cover case (i) as well. Let us also mention that the proof of Theorem 27.2(ii) in [54] can be used for more general functions f only if either f satisfies serious growth restrictions or f is convex and the function f (u)/uγ is nondecreasing for large u, where γ > 1, see [54, Theorem 1]. In the proof of Proposition 27.3, we shall use the following two lemmas from [94]. The first one is an approximation lemma which will enable us to construct a suitable perturbation of the equation in (22.1). Lemma 27.4. Let p > 1 and set ε0 := 1/(p + 1). For each ε ∈ (0, ε0 ), there exists a concave function φε ∈ C 2 ([0, ∞)) with the following properties: 0 < φε (s) ≤ s for all s > 0, 1 ≥ φε (s) ≥ s−p φpε (s) − (p + 1)ε + , s > 0,

(27.5)

φε (s)

(27.7)

φε (0) = 0,

lim

ε→0+

=1

uniformly on [0, M ], for every M > 0, sup φε (s) < ∞.

(27.6)

(27.8)

s≥0

Proof. Let z = zε be the solution of the ODE z (s) = s−p (z p (s) − ε), s ≥ 1,

with z(1) = 1 − ε.

(27.9)

We claim that z exists and satisfies 0 < z (s) < 1, z(s) < s

for all s ≥ 1.

(27.10)

222


First observe that ε1/p < z(1) < 1, due to (1 − ε)p > 1 − pε > ε. The claim thus easily follows from the fact that z(s)−s < 0 implies z (s)−1 ≤ (z(s)/s)p −εs−p −1 < 0, and that z(s) > ε1/p implies z (s) > 0. Differentiating (27.9) and using (27.10), we get z (s) = s−p pz p−1 z − ps−p−1 (z p − ε) = ps−p (z p−1 − sp−1 )z ≤ 0,

s ≥ 1.

Now extend z(s) to a concave function φε ∈ C 2 ([0, ∞)) verifying (27.5) and 0 ≤ φε ≤ 1. (This is clearly possible since z (1) < z(1) < 1.) We see that φε (s) ≥ φε (1) = (1 − ε)p − ε > 1 − (p + 1)ε ≥ s−p (φpε (s) − (p + 1)ε)+ ,

0 < s ≤ 1. (27.11) This along with (27.9), (27.10) and 0 ≤ φε ≤ 1 proves (27.6). Next, by (27.9), we have

z(s)

∞

∞

s dτ dτ dτ dτ < C := < = , s ≥ 1, p−ε p p p −ε τ τ τ τ z(1) 1 1 z(1) which yields (27.8). Finally, (27.7) is a consequence of the first inequality in (27.11), together with the continuous dependence of the solution zε of (27.9) on ε (observe that z(s) = s is solution of (27.9) for ε = 0). Lemma 27.5. Assume Ω bounded and let 0 < T0 < ∞. There exists K = K(T0 ) > 0 such that the solution of the problem ⎫ Zt − ∆Z = 1, x ∈ Ω, 0 < t ≤ T0 , ⎪ ⎬ Z = 0, x ∈ ∂Ω, 0 < t ≤ T0 , (27.12) ⎪ ⎭ Z(x, 0) = −Kδ(x), x ∈ Ω, satisfies Z ≤ 0 in Ω × [0, T0 ]. Proof. Decompose Z as Z = Z1 − KZ2 where Z1 solves (27.12) with K = 0 −λ1 t −λ1 t ϕ1 ≥ c1 c−1 δ. and Z2 = e−tA δ. In view of (1.4), we have Z2 (x, t) ≥ c−1 2 e 2 e 1,0 Combining this with Z1 ≤ c3 (T0 )δ (due to Z1 ∈ C (Ω × [0, T0 ])) we obtain Z ≤ −λ1 t λ1 T0 )δ, and the lemma follows by choosing K = c−1 . (c3 − Kc1 c−1 2 e 1 c2 c3 e Proof of Proposition 27.3. Assume for contradiction that Tmax (v0 ) ≤ T c (u0 ) < ∞ or Tmax (v0 ) < ∞ = T c(u0 ). Let v be the solution of (22.1) with initial data v0 . Notice that Tmax (u0 ) ≤ Tmax (v0 ) and fix τ ∈ (0, Tmax (u0 )). By the assumptions on v0 and Proposition 52.7, there exists η > 0 such that v(x, τ ) + 2ηδ(x) ≤ u(x, τ ),

x ∈ Ω.

(27.13)

Fix T ∈ (τ, Tmax (v0 )). Step 1. L1 - and Lpδ -estimates. We claim that u, up δ ∈ L1 (QT ).

(27.14)


223

Let uk be the solution of (27.1). Using Proposition 49.11 in Appendix C and T < T c (u0 ), we deduce that, for all small t > 0,

c(x, t) Ω

uk (y, T )δ(y) dy ≤

Ω

G(x, y, t)uk (y, T ) dy

≤ uk (x, T + t) ≤ u(x, T + t) < ∞

for a.e. x ∈ Ω and some constant c(x, t) > 0. It follows that

sup Ω

k

uk (y, T )δ(y) dy < ∞.

(27.15)

Now, for any 0 ≤ ϕ ∈ C 2,1 (Ω × [0, T ]) such that ϕ = 0 on ∂Ω × [0, T ], by testing (27.1) with ϕ we obtain

Ω

uk (y, T )ϕ(y, T ) dy =

u0 (y)ϕ(y, 0) dy+

Ω

0

T

Ω

uk (ϕt +∆ϕ)+fk (uk )ϕ dy ds.

First taking ϕ(x, t) = eλ1 t ϕ1 (x) and using (1.4) and (27.15), we obtain

T

sup k

0

Ω

fk (uk )δ dy ds ≤ C sup k

Ω

uk (y, T )δ(y) dy < ∞,

(27.16)

hence up δ ∈ L1 (QT ) by monotone convergence. Finally taking ϕ(x, t) = Θ(x), where Θ is defined by (19.27), and using (27.16), we similarly obtain u ∈ L1 (QT ). Step 2. Derivation of a penalized weak inequality for φε (u). Now fix ε ∈ (0, ε0 ) to be determined later and let φε be given by Lemma 27.4. For each k, a direct computation yields ∂t (φε (uk )) − ∆(φε (uk )) = φε (uk )(∂t uk − ∆uk ) − φε (uk )|∇uk |2 ≥ φε (uk )fk (uk ) in Ω × (τ, T ). For any 0 ≤ ϕ ∈ C 2,1 (Ω × [τ, T ]) such that ϕ = 0 on ∂Ω × [τ, T ] and ϕ(T ) = 0, multiplying by ϕ and integrating by parts, it follows that

Ω

φε (uk )ϕ (y, τ ) dy +

T

τ

Ω

φε (uk )(ϕt + ∆ϕ) + φε (uk )fk (uk )ϕ dy ds ≤ 0.

(27.17) Set w := φε (u) and observe that w ∈ L∞ (Ω × (0, ∞)) by (27.8). Using (27.14), (27.5), (27.6) and passing to the limit in (27.17) via dominated convergence, we obtain

Ω

(φε (u)ϕ)(y, τ ) dy + τ

T

Ω

w(ϕt + ∆ϕ) + φε (u)up ϕ dy ds ≤ 0.

224


For sufficiently small ε1 ∈ (0, ε0 ) and all ε ∈ (0, ε1 ], owing to (27.13) and (27.7), we have φε (u(·, τ )) ≥ v(·, τ ) + ηδ(·). Using (27.6), it follows that

(v+ηδ)ϕ (y, τ ) dy+

Ω

τ

T

Ω

w(ϕt +∆ϕ)+(wp −(p+1)ε)+ϕ dy ds ≤ 0. (27.18)

Step 3. Construction of a supersolution to the original problem and conclusion. Now let K and Z be given by Lemma 27.5 for T0 := Tmax (v0 ), select ε = min{ε1 , η(K(p + 1))−1 }, and set z(·, t) := w(·, t) + (p + 1)εZ(·, t − τ ) ≤ w(·, t),

τ < t < T0 .

By (27.8), we have sup z(t) ∞ ≤ M := sup φε (s) + (p + 1)ε sup Z(s) ∞ < ∞.

τ <s 0 in Ω, u0 = 0 and ∂u0 /∂n < 0 on ∂Ω, and v0 := αu0 for some α ∈ (0, 1) (otherwise, just replace u0 by u(τ ) for some small τ > 0 and observe that v(τ ) ≤ αu(τ ) for some α ∈ (0, 1)). Let T ∈ (0, ∞), T ≤ T c (u0 ), hence u¯(x, t) < ∞ a.e. in QT . We shall prove that there exists a constant Cα < ∞ such that u(x, t; αu0 ) ≤ Cα for all x ∈ Ω and t < T , which implies the conclusion (since then, Tmax (αu0 ) > T , hence Tmax (αu0 ) > T c (u0 ) if T c (u0 ) < ∞, Tmax (αu0 ) = ∞ otherwise). Let V := e−tA v0 , and let ukλ , λ ∈ {α, 1}, k = 1, 2, . . . , be given by p ∂t ukλ − ∆ukλ = (uk−1 λ )

ukλ ukλ (x, 0)

in QT ,

=0

on ST ,

= λu0 (x),

x ∈ Ω,

where u0λ :≡ 0. Notice that ukλ ∈ C 2,1 (Ω × (0, T )) and that the maximum principle implies 0 ≤ ukλ ≤ uk+1 ≤u ¯ λ in QT . (27.20) ukα ≤ αuk1 For m ∈ N, µ > 1, set Eµm := {(x, t) ∈ QT : um α (x, t) > µV (x, t)}, gkm (µ) :=

inf

m (x,t)∈Eµ

uk1 (x, t) , um α (x, t)

m p m w(x, t) := uk+1 (x, t) − gkm (µ)p um α (x, t) + µ gk (µ) − gk+1 (µ) V (x, t). 1

226


(Here and below, we write gkm (µ)p in place of (gkm (µ))p for simplicity.) Observe that Eµm ⊂ Eµm for µ > µ, hence the functions gkm are nondecreasing in µ. Set M := sup{µ > 1 : Eµm = ∅} = inf{µ > 1 : Eµm = ∅} and assume 1 < µ < M . Then w ∈ C 2,1 (Ω × (0, T )) and there exists δ = δ(m, µ) > 0 such that t > δ for all (x, t) ∈ Eµm . For k ≥ m > 1 we have p , wt − ∆w = (uk1 )p − gkm (µ)um−1 α

m p m m p m m w ≥ gk+1 (µ)um α − gk (µ) uα + µ gk (µ) − gk+1 (µ) V

in Eµm ,

and, by (27.20), gkm (µ) ≥ 1/α > 1, p m p ≥ gk (µ)um−1 (uk1 )p ≥ gkm (µ)um α α

(27.21) in Eµm ,

hence Since um α

in Eµm . wt − ∆w ≥ 0 = µV on ∂Eµm \ Ω × {T } , we also have w≥0

on ∂Eµm \ Ω × {T } ,

and we deduce from the maximum principle5 that w ≥ 0 in Eµm . Assume that M > µ > µ > 1. We claim that µ m m (µ ) ≥ gkm (µ)p − gkm (µ)p − gk+1 (µ) . gk+1 µ

(27.22)

m If gkm (µ)p − gk+1 (µ) ≥ 0, then (27.22) follows by combining w ≥ 0 on Eµm ,

V (x, t)
M , we have um α (x, t) ≤ cα V (x, t)

in QT .

Since the limit Uα (x, t) := limm→∞ um α (x, t) is a bounded integral solution of (22.1) with u0 replaced by αu0 , it coincides with u(x, t; αu0 ) for t < T . This concludes the proof. The following result shows that incomplete blow-up may occur when p > pS . Parts (i) and (ii) are respectively due to [396] and [232].

228


Proposition 27.7. Consider problem (22.1) with Ω bounded and p > 1. Let 0 ≤ ϕ ∈ L∞ (Ω), ϕ ≡ 0, let α∗ be defined by (22.22) and set u0 = α∗ ϕ. (i) Then T c (u0 ) = ∞. (ii) Assume in addition Ω = BR , ϕ radial and p > pS . Then Tmax (u0 ) < ∞. Consequently, u blows up incompletely as t = Tmax . Proof. (i) Let 0 ≤ α < α∗ . As a consequence of the definition of α∗ and of the comparison principle, we have Tmax (αϕ) = ∞. Let vα be the solution of (22.1) with initial data αϕ, and let vα,k and uk be the (global) solutions of (27.1), with initial data αϕ and u0 respectively. Since Tmax (αϕ) = ∞, Theorem 17.1 implies

vα (t)ϕ1 dx ≤ C = C(Ω, p), for all t > 0. Ω

Since vα,k ≤ vα , it follows that Ω vα,k (t)ϕ1 dx ≤ C, hence

uk (t)ϕ1 dx ≤ C, for all t > 0, Ω

by continuous dependence. Letting k → ∞ and using monotone convergence, we deduce that, for each t > 0, Ω u(t)ϕ1 dx ≤ C, hence u(x, t) < ∞ for a.e. x ∈ Ω. We conclude that T c (u0 ) = ∞. (ii) This assertion is a consequence of Theorem 28.7 below. Remarks 27.8. (a) If Ω is a ball and u0 is radial, then the assumption p < pS in Theorem 27.2 can be weakened to p ≤ pS (or can be removed if we know B(u0 ) = {0}), see [232, the proof of Theorem 5.1]. (b) For some class of problems (including (22.1)), property (22.28) is sufficient for complete blow-up, see [54, Corollary 3.1]. (c) Genericity of complete blow-up. Consider the situation in Proposition 27.7. If u0 = αϕ, α > α∗ , then T c(u0 ) < ∞. More precisely, Tmax (αϕ) ≤ T c (αϕ) < Tmax (βϕ) for any α > β ≥ α∗ . This follows from Proposition 27.3 (see also [317, Theorem 2] and [232, Theorem 14.1]). Since the function [α∗ , ∞) → (0, ∞) : α → Tmax (αϕ)

(27.27)

is decreasing, it has to be (left) continuous a.e., hence T c(αϕ) = Tmax (αϕ) for a.e. α > α∗ . Note that if we were able to prove the blow-up rate (23.5) for u0 = αϕ, α ≥ α∗ , with M = M (α) being locally bounded, then the proof of [254, Theorem 1.2] would guarantee the continuity of (27.27) everywhere. Recall also that (23.5) is true if pS ≤ p < pJL , cf. Theorem 23.10. (d) Peaking solutions. The facts mentioned in (c) show that solutions which blow up incompletely are rather exceptional. Many interesting results on the behavior of such solutions in the interval (Tmax (u0 ), T c (u0 )) can be found in [194], [379], [380], [382], [195].


229

In [232] the authors considered problem (18.1) with pS T, where f = f (ρ) and g = g(ρ) are suitable bounded positive solutions of the ODE’s f +

n−1 1 1 f − f ρ− f + f p = 0, ρ 2 p−1

ρ > 0,

f (0) = 0,

g +

n−1 1 1 g + g ρ+ g + g p = 0, ρ 2 p−1

ρ > 0,

g (0) = 0,

and

respectively. These solutions have singularity (peak) only at the point (0, T ) and their blow-up profile is limt→T u(r, t) = Cr−2/(p−1) for some C < cp . Similar solutions for problem (22.21) had been previously constructed in [318]. (e) Incomplete blow-up in the subcritical case. Another explicit example of incomplete blow-up, for the nonautonomous equation ut − ∆u = a(|x|, t)u2 ,

x ∈ Rn , t ∈ R,

(27.28)

(with a > 0 being bounded above and bounded away from zero), is due to [414]. Let ϕ ∈ BC 1 (R) be nonnegative. Set u(x, t) :=

1 , ϕ(t) + r2

where r = |x|.

Then a straightforward computation yields ut − ∆u = a(|x|, t)u2

if x = 0 or ϕ(t) = 0,

where a(r, t) := 2n − ϕ (t) − is bounded above and

8r2 ϕ(t) + r2

a ≥ 2(n − 4) − sup ϕ > 0

provided n > 4 and sup ϕ is small enough. In addition, it is easily verified that u is a weak solution of (27.28) if n > 4. Notice that p = 2 is subcritical if n = 5. In particular, if n > 4 and ϕ(t) = t2 for |t| < n − 4, then ∞ > C2 ≥ a(|x|, t) ≥ C1 > 0 in Rn × (−1/2, 1/2) and the solution u exhibits incomplete blow-up at t = 0. Similarly, the choices ϕ(t) = [t(1 − t)]2 or ϕ(t) = t3 (sin 1t )2 yield examples of functions u which blow up incompletely multiple or infinitely many times (cf. [380], [382] in the case a ≡ 1).

230


Using the example above one can easily construct explicit examples of incomplete blow-up for the problem ut − ∆u = a(x, t)u2 + b(x, t), u = 0, u(x, 0) = u0 (x),

x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0, x ∈ Ω,

where Ω ⊂ Rn is bounded, n > 4, a, b are bounded and a > 0 is bounded away from zero, see [442]. On the other hand, it was shown in [442] that incomplete blow-up cannot occur in such problems in the subcritical range p < pS if a, b ∈ BU C 1 . (f) Analytic continuation. A different possible way of continuing the solution after Tmax was studied in [351]. It is based on a suitable notion of analytic continuation where the time variable t is extended to a sector in the complex plane. The existence of such continuation was proved there for the equation ut − ∆u = u2 , under Neumann boundary conditions and suitable assumptions on the initial data. (g) Critical Lq -space. Like the classical existence time, the complete blow-up time T c (u0 ) is not uniformly positive for bounded sets of initial data in Lq (Ω) when q = qc = n(p − 1)/2 (cf. [36]). Indeed, assume for instance Ω bounded, Ω ⊃ B(0, 1), and let u˜0,j = 2u0,j , where u0,j is given by (15.3). Then Proposition 27.3 and Remark 15.4(i) imply that T c(˜ u0,j ) < Tmax (u0,j ) → 0 as j → ∞, while ˜ u0,j qc = Const.

28. Applications of a priori bounds We have seen in previous sections that a priori and universal estimates of solutions play a key role in the proofs of several important statements. In this section we provide further applications of such estimates. Other applications (concerning existence of nodal equilibria and connecting orbits) can be found in [128] and [3], for example. These articles are devoted to superlinear problems with nonlinear boundary conditions and indefinite nonlinearities, respectively.

28.1. A nonuniqueness result In this subsection we use universal bounds from Section 26 and arguments of [53] in order to prove Theorem 15.3(ii). More precisely, we prove the following proposition. Proposition 28.1. Let Ω = BR and pF qc = n(p − 1)/2 and assume that u0 ∈ Lr (Ω), u0 ≥ 0, is radial nonincreasing. Let Tm denote the maximal existence time of the corresponding classical Lr (Ω)-solution um of (15.1) (cf. Theorem 15.2 and Proposition 16.1) and let T ∈ (0, Tm ). Then there exists a

28. Applications of a priori bounds

231

function u ≥ um , u = um , such that u is a classical Lq (Ω)-solution of (15.1) for any q ∈ [1, qc ), u(·, t) is radial nonincreasing, lim u(·, t) q = ∞

for any q > qc ,

(28.1)

lim u(·, t) q = ∞

for any q > qc .

(28.2)

t→0

t→T

In the proof of Proposition 28.1 we will also need the following lemma. Lemma 28.2. Let Ω = BR , p > pF , 0 < T < ∞ and let u be a positive, radial nonincreasing classical solution of (15.1) in the time interval (0, T ). Let δ > 0 and T ∈ (0, T ). Then there exist constants c depending only on R, p, n and the indicated quantities, such that u(x, t) ≤ c(δ)|x|−2/(p−1) ,

|x| ≤ R, t ∈ (0, T − δ],

(28.3)

u(·, t) q ≤ c(q, δ), t ∈ (0, T − δ], 1 ≤ q < qc ,

T u(·, s) pp ds < c(T , T − T ).

(28.4) (28.5)

0

Proof. Let t ∈ (0, T − δ]. Denote β = 1/(p − 1) and v := u(·, t). Then (15.22) guarantees sβ e−sA v ∞ ≤ Cp , for all s ∈ (0, δ]. (28.6) Let us show the existence of constants C, k > 0 such that 2

v(x) ≤ Cekδ/R (R2 /δ)β |x|−2β ,

|x| ≤ R.

(28.7)

If k > 0 is sufficiently large, then there exists η ∈ D(B1 ) radial, radially decreasing, η ≡ 0, such that −∆η ≤ kη (one can take η(x) = exp −1/(1 − 2|x|2 )+ , for example). Set ηλ (x) := η(λx), λ ≥ 1/R. Then the support of ηλ is a subset of Ω and −∆ηλ ≤ kλ2 ηλ , hence

2

e−sA ηλ ≥ e−kλ s ηλ

by the maximum principle. Consequently, (28.6) guarantees

2 Cp ηλ dx ≥ sβ (e−sA v)ηλ dx = sβ v(e−sA ηλ ) dx ≥ sβ e−kλ s vηλ dx. Ω

Ω

Ω

Ω

Since v(x) ≥ v(1/λ) on the support of ηλ , we obtain v

1 λ

2

≤ Cp s−β ekλ s ,

λ≥

1 , s ∈ [0, δ]. R

232


Choosing λ = 1/|x| and s = δ|x|2 /R2 we obtain (28.7). Notice that (28.7) guarantees (28.3) and (28.4) is a consequence of (28.3). For τ ∈ (0, t), multiplying the variation-of-constants formula between τ and t by ηλ we obtain

Ω

t

(e−(t−τ )A u(τ ))ηλ dx +

τ

Ω

up (s)(e−(t−s)A ηλ ) dx ds =

Ω

u(t)ηλ dx.

It follows that

t

p

u (s)e 0

−kλ2 (t−s)

Ω

ηλ dx ds ≤

Ω

u(t)ηλ dx.

Fixing λ0 ≥ 1/R and r0 > 0 such that ηλ0 (r0 ) > 0 and using (28.3) we obtain 2

ηλ0 (r0 )e−kλ0 T

t

0

up (s) dx ds ≤

Br0

Ω

u(t)ηλ0 dx ≤ C(T − T ),

Since u(·, t) is radial decreasing, the last estimate guarantees (28.5).

t ≤ T < T.

Proof of Proposition 28.1. Fix T ∈ (0, Tm ). Let ηk ∈ D(B1/k ), k = 1, 2, . . . , be nonnegative, radial, radially decreasing and ηk ≡ 0. Fix k. Due to the continuous dependence on initial data (see Remark 51.8(iii)) we have Tmax (u0 + αηk ) > T for α > 0 small. On the other hand, as a consequence of Remark 17.2(i) (see also Remark 17.7(iv)), we have Tmax (u0 + αηk ) < T for α > 0 large. Since the mapping α → Tmax (u0 + αηk ) is continuous (see Theorem 22.13 and (51.92)) there exists αk > 0 such that Tmax (u0 + αk ηk ) = T . Let uk denote the Lr (Ω)-solution of (15.1) with initial data u0 + αk ηk . Due to Theorem 26.8 the sequence {uk } is uniformly bounded on Ω × [δ, T − δ] for any δ > 0. Now parabolic regularity estimates (see Theorems 48.1 and 48.2) imply a uniform bound in BU C 2+α,1+α/2 (Ω × [δ, T − δ]) for some α > 0, hence we may assume uk → u in C 2,1 (Ω× [δ, T − δ]) for all δ > 0, where u is a classical solution of (26.2). Passing to the limit in the variation-of-constants formula for uk we obtain u(t) = e−(t−s)A u(s) +

t

e−(t−σ)A up (σ) dσ.

0 < s < t < T.

(28.8)

s

Moreover, applying Lemma 28.2 to the uk ’s, and then Fatou’s lemma, we deduce that u satisfies (28.3)–(28.5). Next fix q ∈ [1, qc ) and let t ∈ (0, T ). Inequality (28.4) and the compactness of e−tA show that there exist sm → 0 and w ∈ Lq (Ω) such that u(sm ) → w weakly in Lq (Ω) and e−(t−sm )A u(sm ) → e−tA w in Lq (Ω).


233

Since (28.5) guarantees up ∈ L1 (QT ) for all T < T , using (28.8) with s = sm and passing to the limit we obtain u(t) = e

−tA

t

e−(t−σ)A up (σ) dσ,

w+

0 < t < T.

(28.9)

0

Next we show w = u0 a.e. Let φ ∈ D(Ω). Multiplying the equation for uk with φ and integrating we obtain

t

Ω

uk (t)φ dx−

Ω

(u0 +αk ηk )φ dx+

t

Ω

0

uk (s)(−∆φ) dx ds =

Ω

0

uk (s)p φ dx ds.

Assume φ ≡ 0 on Bε for some ε > 0. Since uk ≤ C(ε) on (Ω \ Bε ) × (0, T ), we may pass to the limit via dominated convergence in the above identity, and we arrive at

t

Ω

u(t)φ dx −

Ω

u0 φ dx +

t

up (s)φ dx ds. (28.10)

u(s)(−∆φ) dx ds = 0

Ω

0

Ω

On the other hand, (26.2) shows that

Ω

u(t)φ dx −

Ω

t

u(sm )φ dx +

t

up (s)φ dx ds.

u(s)(−∆φ) dx ds = sm

Ω

sm

Ω

(28.11) Passing to the limit in (28.11) and comparing the resulting identity with (28.10) yields Ω u0 φ dx = Ω wφ dx for all φ ∈ D(Ω) which vanish in a neighborhood of the origin, hence u0 = w a.e. Now (28.9) guarantees u0 − u(t) 1 → 0 as t → 0. This convergence, (28.4) and interpolation yield u0 − u(t) q → 0 as t → 0 for any q < qc , hence u is an Lq (Ω)-solution of (15.1) for any q < qc . It remains to prove (28.1) and (28.2). Fix q > qc . We know that uk (T /2) → u(T /2) in Lq (Ω). Due to the continuity of Tmax (cf. above) we have Tmax (u(T /2)) = lim Tmax (uk (T /2)) = T /2, k→∞

hence u(t) q → ∞ as t → T due to Remarks 16.2. Next assume that there exist C > 0 and tk → 0 such that u(tk ) q < C. Choose q˜ ∈ (qc , q). Then interpolation yields u(tk ) → u0 in Lq˜(Ω) and the continuity of Tmax in Lq˜(Ω) shows T = lim Tmax (u(tk )) = Tmax (u0 ) = Tm > T, k→∞

a contradiction. This shows (28.2) and concludes the proof.

234


Remark 28.3. Entire solutions. Let us mention another, simple application of the universal bounds in Section 26. Let Ω = BR , 1 < p < pS , and denote by φ the unique positive solution of (6.1) with λ = 0 (cf. Remark 6.9(ii)); we know that φ is radial. One can show that any entire, radial, positive classical solution u of (15.1) (i.e. defined for all t ∈ R) is either φ or a connection from φ to 0. Moreover, this remains true without the assumption that u be radial if we assume 1 < p < pB . Indeed, due to Theorem 26.8, any such solution satisfies supt∈R u(t) ∞ < ∞, hence supt∈R u(t) BUC 1+α (Ω) < ∞ by smoothing effects. Owing to the (strict) Lyapunov functional given by the energy E (cf. (17.6)), we know from Proposition 53.5 that the ω-limit set of u (in the BU C 1 -topology) is nonempty and consists of nonnegative equilibria. By the same token, this is also true for the α-limit set (obtained by taking tk → −∞ instead of +∞ in formula (53.1)). Now, using the fact that φ and 0 are the only nonnegative steady states, that E (t) ≤ 0, and that E(φ) > E(0) = 0, one easily obtains the conclusion.

28.2. Existence of periodic solutions Analogously as in Corollary 10.3, a priori estimates for positive periodic solutions of (suitable) parabolic problems with periodic superlinear nonlinearities guarantee their existence. For example, consider the problem ut − ∆u = f (x, t, u), x ∈ Ω, t > 0, (28.12) u = 0, x ∈ ∂Ω, t > 0. If Ω = BR , f = f (|x|, t, u) is continuous, T -periodic in t, 1 0, u ≥ 0,

and, for all (x, t) in the closure of Q := QT , lim

u→∞, Q (z,τ )→(x,t)

u−p f (z, τ, u) = m(x, t) ∈ (0, ∞),

then a straightforward generalization of Theorem 26.8 shows that any positive T periodic solution of (28.12) is bounded by a universal constant C = C(f, Ω) (see [425] for more general statements). Consequently, if f satisfies additional assumptions guaranteeing the well-posedness of (28.12) in a suitable function space, then a topological degree argument shows the existence of a positive T -periodic solution of (28.12) (see [176] for details concerning the use of the topological degree). Of course, instead of the radial symmetry assumption we could have assumed p < pB . Let us sketch another proof of universal estimates of positive T -periodic solutions of (28.12) in the general nonradial case and the full subcritical range 1 < p < pS . Unfortunately, this alternative proof requires quite restrictive assumptions concerning the nonlinearity f .


235

Proposition 28.4. Assume Ω bounded and f (x, t, u) = m(t)|u|p−1 u, where 1 0

m (t)− 2n − (n − 2)(p + 1) < , m(t) r2 (Ω)

(28.13)

where r(Ω) denotes the radius of the smallest ball containing Ω. Then there exists a constant C > 0 such that any positive T -periodic solution of (28.12) satisfies u(t) ∞ ≤ C

for all t > 0.

(28.14)

Consequently, there exists at least one positive T -periodic solution of (28.12). Sketch of proof. Let u be a positive T -periodic solution of (28.12). Multiplying by ϕ1 one easily gets

ψ ≥ −λ1 ψ + inf m(t) ψ p , ψ(t) := u(t)ϕ1 dx, t>0

Ω

hence Ω u(t)ϕ1 dx ≤ C. If Ω is convex, then the method of moving planes guarantees u(x, t), |∇u(x, t)| ≤ C for all x in a neighborhood of ∂Ω. If Ω is not convex, then the same estimate can be obtained by using the Kelvin transform (cf. the proof of Theorem 13.1). Now, the Pohozaev-type identity

T

T |x|2 n − 2 p+1 n m (t) p+1 − u m(t) dx dt = u dx dt + u2t 2 2 Ω p+1 Ω p+1 0 0

1 T + |∇u|2 x · ν(x) dσ dt, 2 0 ∂Ω the identity

0

T

Ω

u2t dx dt = −

1 p+1

0

T

m up+1 dx dt

Ω

and the assumption (28.13) guarantee an a priori bound for u in W 1,2 QT . Finally it is sufficient to use the bootstrap procedure from the proof of Theorem 22.1. Remarks 28.5. (i) The estimates in Proposition 28.4 were first proved in [176] and [177] under the additional assumptions p(3n − 4) < 3n + 8 and p(n − 2) < n, respectively. The general case was proved in [441], cf. also [286]. Analogous results for f (x, t, u) = |u|p−1 u + h(x, t), h “small”, can be found in [280]. (ii) If Ω, f are as in Proposition 28.4 and we consider problem (28.12) complemented with the initial condition u(x, 0) = u0 (x), then the a priori bound (22.27) is true for all solutions of (28.12) (not necessarily positive or periodic) and even without assuming (28.13), see [441, Theorem 5.1(i)].

236


28.3. Existence of optimal controls The a priori estimate (22.27) also plays an important role in the proof of existence of optimal controls for problems with final observation. Let Ω ⊂ Rn be bounded, T > 0, 1 < p < pS , q ≥ 2, ud ∈ Lq (Ω), u0 ∈ C 2 (Ω) ∩ C0 (Ω), and let us consider the model optimal control problem Minimize J(u(w), w) over w ∈ L2 (Ω),

where J(u, w) =

Ω

|u(x, T ) − ud (x)|q dx +

w2 dx,

Ω

and u(w) is the solution of the governing equation ut − ∆u = |u|p−1 u + w, u = 0, u(x, 0) = u0 (x),

x ∈ Ω, t ∈ (0, T ], x ∈ ∂Ω, t ∈ (0, T ], x ∈ Ω,

(28.15)

⎫ ⎪ ⎬ ⎪ ⎭

(28.16)

(we set J(u(w), w) := ∞ if (28.16) does not possess global solution up to time T ). Then we have the following Proposition 28.6. Under the above assumptions, let n 2n q∈ (p − 1), . 2 (n − 4)+ Assume that problem (28.16) possesses a global solution at least for one w ∈ L2 (Ω). Then the optimal control problem (28.15) has a solution. The statement of Proposition 28.6 remains true for more general time-dependent controls w ∈ Lr ([0, T ], L2 (Ω)) (where r is large enough) and more general cost functionals J, see [22]. In addition, one can also derive optimality conditions for optimal controls and show that the assumption p < pS is essentially optimal (see [22]). Sketch of proof of Proposition 28.6. Let {wk } ⊂ L2 (Ω) be a minimizing sequence for J and uk := u(wk ). Then {wk } is bounded in L2 (Ω) (and we may assume wk → w weakly in L2 (Ω)) and {uk (T )} is bounded in Lq (Ω), due to the boundedness of J(uk , wk ). Since the problem (28.16) is well-posed in Lq (Ω) we may find δ > 0 such that the solutions uk can be continued on the interval [T, T + δ]. A straightforward modification of the proof of estimate (22.27) shows that the solutions uk are uniformly bounded in L∞ ((0, T ), L2p (Ω)). The Sobolev maximal regularity (see Theorem 51.1(vi)) guarantees that uk are uniformly bounded in W 1,r ([0, T ], L2(Ω)) ∩ Lr ([0, T ], W 2,2 ∩ W01,2 (Ω)) for any r > 1. Since this space is compactly embedded in X := C([0, T ], W01,2 ∩ Lq (Ω)) for r sufficiently large (see Proposition 51.3), we may assume uk → u in X. Now it is easy to pass to the limit to show u = u(w) and J(u, w) ≤ limk→∞ J(uk , wk ).


237

28.4. Transition from global existence to blow-up and stationary solutions Let us consider problem (22.1) with either Ω bounded and p > 1, or Ω = Rn and p > pF , and let us go back to the situation introduced in Subsection 22.3. Namely, fix a function φ ≥ 0, φ ≡ 0, with φ ∈ L∞ (Ω) if Ω is bounded and, for instance, φ ∈ D(Ω) if Ω = Rn . Let α∗ be again defined by α∗ = α∗ (φ) := sup{α > 0 : Tmax (αφ) = ∞},

(28.17)

and note that α∗ ∈ (0, ∞) (cf. Subsection 22.3 for Ω bounded; when Ω = Rn this follows from similar arguments by using Theorem 20.1). By definition of α∗ , we have T (αφ) < ∞ for α > α∗ and, as a consequence of the comparison principle, T (αφ) = ∞ for 0 ≤ α < α∗ . Now if we consider the threshold solution u∗ := u(t; α∗ φ) of (22.1) starting at u0 = α∗ φ, we have the following three possibilities for u∗ : (a) u∗ is global and bounded in L∞ (Ω), (b) u∗ is global but unbounded, (c) u∗ blows up in finite time. It turns out that any of these three possibilities may occur. Theorem 28.7. Consider the situation described above. (i) Assume either 1 pS . Assume either Ω = BR and φ radial, or Ω = Rn and φ radial nonincreasing. Then case (c) occurs. (iv) If Ω is bounded and case (a) occurs, then the ω-limit set of the solution u∗ is a nonempty compact connected set consisting of positive equilibria. As a consequence, if Ω is a bounded starshaped domain and p ≥ pS , then (b) or (c) occurs, and the a priori bound (22.2) fails. Proof. First let us show that the bound (22.2) guarantees alternative (a). For any α ∈ (0, α∗ ), the solution uα (t) := u(t; αφ) exist globally. If (22.2) is true, then uα (t) ∞ ≤ C ∗ for some C ∗ independent of α and the continuous dependence of the solutions on the initial data shows u∗ (t) ∞ ≤ C ∗ , hence case (a) occurs. Since (22.2) is true if p < pS and either Ω is bounded or Ω = Rn and u0 is radial (see Theorem 22.1 or Theorem 26.9, respectively), we have (a) in these cases.

238


If (a) is true and Ω is bounded, then Example 53.7 guarantees that the ω-limit set ω(α∗ φ) consists of positive equilibria. Since (22.1) does not possess positive equilibria if Ω is starshaped and p ≥ pS (see Corollary 5.2), the alternative (a) and hence the estimate (22.2) cannot be true in this case. Now assume p = pS , Ω = BR and φ radial. If (c) occurred, then u would blow up completely at t = Tmax (u0 ), due to [232, the proof of Theorem 5.1]. But this would contradict Proposition 27.7(i). Consequently, (b) is the only remaining possibility and assertion (ii) is proved. In the case of radial nonincreasing functions the assertion follows from Theorem 22.9. Finally, let p > pS . If Ω is a ball and φ is radially symmetric, then again (a) cannot happen, and (b) is ruled out by Theorem 22.4. Consequently, (c) is true. If Ω = Rn and φ is radial nonincreasing, then the result follows from [374, Theorem 1.1]) provided p < pL . If p ≥ pL , then one can use [374, Lemma 3.2] and [378]. Remarks 28.8. (i) Non-threshold solutions. Assume that either p < pS and Ω is bounded, or p > pS , Ω = BR and φ is radial. Then limt→∞ u(t; αφ) ∞ = 0 for all 0 ≤ α < α∗ . This follows from Proposition 19.11 and the boundedness of global solutions (cf. Theorems 22.1 and 22.4). (ii) Dynamical proofs of existence of equilibria. Let Ω be bounded and p < pS . Then similarly as above, ω(α∗ φ) consists of nontrivial equilibria for any (possibly sign-changing) φ ∈ Lq (Ω) \ {0}, q > qc , and this fact (together with a topological degree argument) can be used for the proof of existence of positive and sign-changing stationary solutions of (22.1) and related problems (see [114], [436], [439], [441], [3]). (iii) Threshold solutions in the nonradial supercritical case. Assume that Ω is bounded and convex, p > pS , and φ ∈ L∞ (Ω) is nonnegative, φ ≡ 0. Let α∗ = α∗ (φ) and u∗ have the same meaning as above. Fix αk α∗ and denote u ¯(t) := lim u(t; αk φ). k→∞

Then estimates in [396] show that u ¯ is a global weak solution of (22.1) and u¯(t) = u∗ (t) for t ∈ [0, Tmax (α∗ φ)) (cf. also Section 27). Recent results in [132] guarantee that Tmax (α∗ φ) < ∞ and there exists a compact set S ⊂ Ω × [Tmax (α∗ φ), ∞) such ¯ is continuous in Ω×(0, ∞)\S, that the Hausdorff measure Hn−4/(p−1) (S) is zero, u limt→∞ ¯ u(t) ∞ = 0. In particular, Theorem 28.7(iii) remains true in the nonradial case if Ω is bounded and convex. (iv) Further results and references on threshold solutions can be found in the following subsection, Remarks 27.8 and Section 29.


239

28.5. Decay of the threshold solution of the Cauchy problem In this subsection we denote β :=

1 , p−1

and the notation f (t) ∼ g(t) for t ≥ t0 means that C1 g(t) ≤ f (t) ≤ C2 g(t) for all t ≥ t0 and some constants C1 , C2 > 0. Consider the Cauchy problem (18.1) with p > pF . We continue to study the situation described at the beginning of the previous subsection. In what follows, by non-threshold solutions we more specifically mean solutions corresponding to α ∈ (0, α∗ ). Let us first consider the case of initial data with exponential spatial decay, more precisely φ ∈ Hg1 , and assume also p < pS . Recall from Proposition 20.13 that if u is global and t0 > 0, then there exists k ≥ 0 such that L

u(t) ∞ ∼ t−λk ,

t ≥ t0 ,

(28.18)

L where λL k = (n + k − 1)/2 for k ≥ 1 and λ0 = β. The following theorem is due to [301].

Theorem 28.9. Let pF 0, denote by uα the solution of (18.1) with initial data u0 = αφ and let α∗ be defined by (28.17). Then α∗ ∈ (0, ∞). Moreover: (a) uα is global and uα (t) ∞ ∼ t−n/2 for t ≥ 1 if 0 < α < α∗ ; (b) uα∗ is global and uα∗ (t) ∞ ∼ t−β for t ≥ 1; (c) uα blows up in finite time if α > α∗ . Proof. Assertion (c) follows from the definition of α∗ . Let vα denote the rescaled solution (see (18.13)). The asymptotic stability of the zero equilibrium of (18.14) (see Example 51.24) shows that vα is global, vα (s) → 0 in Hg1 (and L∞ ) if α is small and vα∗ (s) → 0 in Hg1 as s → ∞. In particular, α∗ > 0. If φ is radial, then Theorem 26.9 guarantees that uα∗ (hence vα∗ ) are global. In the general case one can use the estimates in [301] or [478, Theorem 1] (see also [440, Theorem 1.2] in the case of sign-changing solutions). The arguments in the proof of Proposition 20.13 show C1 ≤ vα∗ (s) ∞ ≤ C2 , hence (28.18) is true with k = 0. In addition, the compactness of the semiflow for problem (18.14), the existence of the Lyapunov functional and the stability of the zero equilibrium guarantee that the ω-limit set ω(vα∗ ) of vα∗ in Hg1 is nonempty and consists of positive equilibria (cf. Theorem 28.7). For further reference fix w∗ ∈ ω(vα∗ ) and a sequence sj → ∞ such that vα∗ (sj ) → w∗ .

240


Fix α < α∗ and assume that vα (s) → 0. Then the arguments above show that there exists a subsequence of vα (sj ) which converges to a positive equilibrium w. Now the proof of Theorem 19.9(ii) guarantees that vα (s) ≤ (α/α∗ )vα∗ (s), hence w < w∗ . However, the proof of Proposition 19.8 shows that (18.14) does not possess ordered positive equilibria. Consequently, vα (s) → 0 as s → ∞. Now the upper bound in (28.18) with k = 1 follows from Example 51.24 and the lower bound from the comparison with the solution of the linear problem (cf. (20.5)). Theorem 28.9 shows that for positive φ ∈ Hg1 and p < pS , the threshold solution decays with the self-similar rate t−β while the non-threshold solutions decay with the same rate as the corresponding solutions of the linear heat equation. The next theorem [443] and subsequent remarks show that the same behavior of nonthreshold solutions can be expected in a more general case, while the behavior of the threshold solution strongly depends on the exponent p. Theorem 28.10. Assume p > pF . Let φ ∈ C(R+ ) be nonnegative, φ ≡ 0, and lim φ(r)r2β = 0.

r→∞

(28.19)

Denote by uα the solution of (18.1) with u0 (x) = αφ(|x|), α > 0, and let α∗ be defined by (28.17). Then α∗ ∈ (0, ∞) and the following assertions are true. (i) Let p < pS . Then uα∗ is global and uα∗ (t) ∞ ∼ t−β ,

t ≥ 1.

(28.20)

If α ∈ (0, α∗ ), then lim uα (t) ∞ tβ = 0.

t→∞

(28.21)

(ii) Let p ≥ pS . If uα∗ is global, then lim sup uα∗ (t) ∞ tβ = ∞.

(28.22)

t→∞

If α ∈ (0, α∗ ) and uα (t) ∞ ≤ ct−β for all t > 0, then (28.21) is true. Remarks 28.11. (i) If limr→∞ φ(r)r2β = ∞, then uα blows up in finite time for any α > 0 due to Theorem 17.12. If 0 < lim inf φ(r)r2β ≤ lim sup φ(r)r2β < ∞ r→∞

(28.23)

r→∞

and p < pS , then (28.20) remains true. In fact, the proof of Theorem 28.10 shows that the threshold solution uα∗ satisfies the upper bound in (28.20). The lower bound uα∗ (t) ≥ ct−β follows from the comparison with the solution of the linear


241

problem and Lemma 20.8. Similarly, if p ≥ pJL , then one can replace condition (28.19) in the proof of (28.22) with the condition lim sup α∗ φ(r)r2β < cp ,

(28.24)

r→∞

where cp is the constant from (3.9), see [443]. (ii) Assume that lim sup φ(r)r2η < ∞

for some η > β.

r→∞

Then estimate (28.21) guarantees that the solution behaves like the solution of the linear problem. In fact, set h(t) := uα (t) ∞ and notice that the function w(t) := t exp[− 0 h(s)p−1 ds]uα (t) is a subsolution of the linear heat equation (cf. [507, Proposition 2.6]). Assuming η ∈ (β, n) without loss of generality, Lemma 20.8 thus implies t t h(t) = exp h(s)p−1 ds w(t) ∞ ≤ Ct−η exp h(s)p−1 ds , 0

t > 1,

0

and (28.21) guarantees h(t)tβ → 0 as t → ∞. Choose ε > 0 such that κ := η − εp−1 > β and fix t0 > 1 such that h(t) ≤ εt−β for t ≥ t0 . Let t ≥ t0 . Then

0

t

h(s)p−1 ds ≤

0

t0

h(s)p−1 ds + εp−1 log

t t =: I0 + εp−1 log , t0 t0

∞ p−1 = C0 t−κ , thus H := 0 hp−1 (t) dt < ∞. Now we hence h(t) ≤ Ct−η eI0 (t/t0 )ε see that e−tA (αφ) ≤ uα (t) ≤ eH w(t) ≤ eH e−tA (αφ), t > 1. In particular, Lemma 20.8 implies uα (t) ∞ ≤ Ct−η for all t > 0 provided η < n/2. The proof of H < ∞ above is based on [191, Lemma 2.3]. (iii) Let p = pS and φ be as in Theorem 28.10. Then uα∗ exists globally (see [232]) so that either its time decay is slower than the self-similar one or the solution does not decay at all. Remark 22.10(ii) suggests that both possibilities can occur. Assume in addition that φ has only finitely many local minima and belongs to the energy space {u ∈ Lp+1 (Rn ) : |∇u| ∈ L2 (Rn )}. Then [423] guarantees that limt→∞ uα∗ (t) ∞ tβ = ∞ and all non-threshold global solutions satisfy (28.21). (iv) Let p > pS and let φ ∈ L∞ ∩ H 1 (Rn ) satisfy the assumptions in Theorem 28.10. If Tmax (αφ) = ∞, then [356] implies uα (t) ∞ ≤ Ct−n/4 for t > 1. Since n/4 > β, the threshold solution uα∗ has to blow up in finite time. In the proof of Theorem 28.10 we will need the following result on stationary solutions of the rescaled equation (see [269], [412], [538], [163] and [390]).

242


Proposition 28.12. Let p > 1, λ ≥ 0 and let wλ = wλ (ρ) be the solution of the problem w +

n−1 ρ w + w + βw + |w|p−1 w = 0 ρ 2

for ρ > 0,

w(0) = λ,

w (0) = 0.

Then wλ is defined for all ρ > 0 and there exists finite limρ→∞ wλ (ρ)ρ2β =: A(λ). Given λ > 0, set ρλ := sup{ρ > 0 : wλ > 0 on [0, ρ)}. Then the following is true: (i) If p ≤ pF and λ > 0, then ρλ < ∞. (ii) If pF λ0 . In addition, A(λ) > 0 for λ ∈ (0, λ0 ) and A(λ0 ) = 0. (iii) If p ≥ pS , then ρλ = ∞ and A(λ) > 0 for all λ > 0. (iv) If p ≥ pJL , then the mapping λ → wλ (ρ) is strictly increasing for each fixed ρ > 0 and supλ A(λ) = cp , where cp is the constant from (3.9). Proof of Theorem 28.10. We have α∗ > 0 due to Theorem 20.6 and α∗ < ∞ due to Theorem 17.1. Since the solutions uα are radial we will consider them as functions uα (t) = uα (r, t), where r = |x|. Set v(ρ, s) = eβs u(es/2 ρ, es − 1), ρ, s ≥ 0. Then v solves the equation n−1 ρ vs − vρρ − vρ = vρ + βv + v p , (28.25) ρ 2 cf. (18.13), (18.14). (i) Assume p < pS . Theorem 26.9 guarantees that any global positive radial solution u = u(r, t) satisfies u(t) ∞ ≤ C0 t−β ,

where C0 = C0 (n, p).

This estimate, continuous dependence on initial data and the definition of α∗ show that the solution uα∗ is global and satisfies the upper bound in (28.20). If u is a solution (18.1), then the rescaled solution v of (28.25) satisfies v(s) ∞ = (t + 1)β u(t) ∞ , hence

1 β v(s) ∞ ≤ C0 1 + t

t = es − 1,

for all s ≥ 0,

(28.26)

(28.27)

whenever u is global positive and radial. Since α∗ φ ∈ L∞ (Rn ), the solution uα∗ remains bounded in L∞ (Rn ) on a small time interval. Now using (28.26) and (28.27) we can find C1 > 0 such that vα∗ (s) ∞ < C1

for all s ≥ 0.

(28.28)


243

Let λ0 be from Proposition 28.12 and fix λ ∈ (0, λ0 ). Then A := A(λ) > 0. Fix a ∈ (0, A) and set Wa (ρ) := aρ−2β . Choose δ > 0 such that Wa (δ) > C1 + 1.

(28.29)

An easy computation shows that the function V (ρ) = Va (ρ) := Wa (ρ − R1 ) is a supersolution of (28.25) for ρ ≥ R1 + δ provided R1 > 0 is large enough. In fact, n−1 ρ ρ Vρ + Vρ + βV + V p ≤ Vρρ + Vρ + βV + V p ρ 2 2 −2β−2 p−1 2β(2β + 1) + a = a(ρ − R1 ) − βR1 (ρ − R1 ) < 0,

Vρρ +

provided ρ ≥ R1 + δ and R1 > (2β(2β + 1) + ap−1 )/βδ. Increasing R1 if necessary we may also assume V (ρ) > (α∗ + 1)φ(ρ)

for all ρ ≥ R1 ,

(28.30)

due to (28.19). Fix R2 > R1 + δ such that wλ (ρ) > V (ρ)

for ρ ≥ R2

where wλ is the solution from Proposition 28.12. We will show that vα∗ (·, s) and wλ intersect in [0, R2 ] for any s ≥ 0, cf. Figure 14. This intersection guarantees the lower estimate in (28.20).

C1 + 1 vα∗ (·, s) V

λ

0

wλ

R1 + δ

R2

Figure 14: Intersection of vα∗ (·, s) and wλ in [0, R2 ].

244


Assume on the contrary that vα∗ (ρ, s0 ) < wλ (ρ) for some s0 ≥ 0 and all ρ ∈ [0, R2 ] and set ε := inf wλ − vα∗ (·, s0 ) > 0. [0,R2 ]

We have vα (s) − vα∗ (s) ∞ < min(ε, 1) for all s ≤ s0 and α close to α∗ , due to the continuous dependence of solutions u of (18.1) on initial data. Fix such α ∈ (α∗ , α∗ + 1). Then vα (ρ, s0 ) < wλ (ρ),

ρ ∈ [0, R2 ],

(28.31)

ρ ∈ [0, R2 ], s ≤ s0 .

(28.32)

and vα (ρ, s) < C1 + 1,

Since vα (ρ, 0) = αφ(ρ) < V (ρ) for ρ ≥ R1 due to (28.30) and vα (R1 + δ, s) < C1 + 1 < V (R1 + δ) for s ≤ s0 due to (28.32) and (28.29), the comparison principle (see Proposition 52.6) implies vα (ρ, s) ≤ V (ρ)

for ρ ≥ R1 + δ, s ≤ s0 .

(28.33)

Since V (ρ) < wλ (ρ) for ρ ≥ R2 , estimates (28.33) and (28.31) imply vα (s0 ) < wλ , hence vα exists globally due to the comparison principle. But this contradicts the choice of α∗ and concludes the proof of (28.20). Next choose α ∈ (0, α∗ ]. Since vα is uniformly bounded due to vα ≤ vα∗ and (28.28), the ω-limit set of {vα (0, s)}s≥0 is a compact interval J ⊂ [0, C1 ]. Assume that J is not a singleton and fix λ ∈ (inf J, sup J) \ {λ0 }. Then there exist an infinite sequence s1 < s2 < s3 < . . . such that vα (0, sk ) = λ for k = 1, 2, . . . . If λ > λ0 , then wλ (ρλ ) = 0 and the zero number z[0,ρλ ] (vα (s) − wλ ) is finite for s > 0. However, this number has to drop at each sk , which yields a contradiction. Consequently, λ ∈ (0, λ0 ). Let A := limρ→∞ wλ (ρ)ρ2β , a ∈ (0, A), and let V = Va , δ and R1 be as above. Then vα (ρ, s) < V (ρ) for ρ ≥ R1 + δ and any s. Fix R2 > R1 + δ such that wλ (R2 ) > V (R2 ). Then we obtain the same contradiction as above by considering the zero number z[0,R2 ] (vα (s) − wλ ). Consequently, there exists λ = λ(α) ≥ 0 such that vα (0, s) → λ

as s → ∞.

(28.34)

Due to the parabolic estimates the trajectory {vα (s)}s≥0 is relatively compact in C(R+ ) (considered with the locally uniform convergence) and its ω-limit set ωα is a nonempty compact connected set, invariant under the semiflow generated by (28.25). In addition, (28.34) implies ψ(0) = λ for any ψ ∈ ωα . Assume that there exists ψ ∈ ωα \ {wλ } and consider the solution v = vψ of (28.25) with initial data ψ. Fix ρ0 > 0 and s0 > 0 such that vψ (ρ0 , s) = wλ (ρ0 ) for all s ∈ [0, s0 ]. Then the zero number z[0,ρ0 ] (vψ (s) − wλ ) is finite for s > 0 and has to drop at each s ∈ (0, s0 ) (due to vψ (0, s) = λ = wλ (0)) which yields a contradiction.

29. Decay and grow-up of threshold solutions in the super-supercritical case

245

Consequently, ωα = {wλ }. Since vα ≥ 0 we have λ ≤ λ0 . Similarly, estimates of the form vα (s) ≤ V for ρ ≥ R1 + δ show λ ∈ / (0, λ0 ). Hence, λ(α) ∈ {0, λ0 } for any α ∈ (0, α∗ ]. Given 0 < α1 < α2 ≤ α∗ , the function v˜ = (α2 /α1 )vα1 is a subsolution of (28.25) and v˜(·, 0) = vα2 (·, 0), hence vα2 ≥ v˜. Consequently, λ(α2 ) ≥ (α2 /α1 )λ(α1 ). This inequality guarantees λ(α) = 0 for all α < α∗ (and λ(α∗ ) = λ0 ). Hence, given α < α∗ , we have vα (s) → w0 = 0 locally uniformly in [0, ∞) as s → ∞ and the estimate vα (s) ≤ V on [R1 + δ, ∞) concludes the proof of vα (s) ∞ → 0. Consequently, (28.21) is true. (ii) Assume p ≥ pS . If α ∈ (0, α∗ ), then our assumptions imply the existence of C1 > 0 such that the rescaled solution vα satisfies vα (s) ∞ < C1 for all s ≥ 0. Now the same arguments as in the proof of (i) show the existence of λ ∈ [0, C1 ] such that vα (s) − wλ ∞ → 0 as s → ∞, where wλ is the solution from Proposition 28.12. However, for any a ∈ (0, 1) we have an estimate of the form vα (ρ, s) ≤ a(ρ − R1 )−2β ,

ρ > R1 + δ,

for some R1 = R1 (a) > 0 (cf. (28.33)). In particular, assuming λ > 0, the choice a < A(λ) leads to a contradiction. Hence λ = 0 and (28.21) is true. Finally consider the threshold solution uα∗ and assume on the contrary that vα∗ (s) ∞ ≤ C1 for all s ≥ 0. Then the arguments above guarantee vα∗ (s) ∞ → 0

as s → ∞.

(28.35)

Fix λ > 0, a ∈ (0, A(λ)) and choose δ, R1 and R2 as in the proof of (i). Then the same arguments as in that proof show that vα∗ (s) and wλ intersect in [0, R2 ] for all s ≥ 0, which contradicts (28.35).

29. Decay and grow-up of threshold solutions in the super-supercritical case In this section we consider positive solutions of the Cauchy problem ut − ∆u = up , x ∈ Rn , t > 0, u(x, 0) = u0 (x),

x ∈ Rn ,

(29.1)

where n ≥ 11 and p > pJL . Set m := 2/(p − 1) and let U∗ (r) = cp r−m be the singular stationary solution defined in (3.9). We will use matched asymptotics to study the asymptotic behavior of solutions of (29.1) with initial data u0 ∈ L∞ (Rn ) satisfying 0 ≤ u0 (x) ≤ U∗ (|x|)

for x = 0

(29.2)

246


and U∗ (|x|) − c1 |x|− ≤ u0 (x) ≤ U∗ (|x|) − c2 |x|−

for |x| > c3 ,

(29.3)

for some c1 , c2 , c3 > 0 and > m. Note that solutions u with such initial data are global (due to Theorem 20.5) and they are also threshold solutions in the sense of Subsections 22.3, 28.4 since Tmax (λu0 ) < ∞ for λ > 1 (due to [260]). Set λ± :=

! 1 n − 2 − 2m ± (n − 2 − 2m)2 − 8(n − 2 − m) . 2

Due to Remark 9.4 and (9.4) there exists a > 0 such that the positive radial steady state Uα = Uα (r) of (29.1) satisfying Uα (0) = α > 0 has the asymptotic expansion U (r) = U∗ (r) − aα r−m−λ− + o(r−m−λ− )

as r → ∞,

(29.4)

where aα := α−λ− /m a. We will sketch the proof of the following theorem due to [203], [189], [204]. Theorem 29.1. Let p > pJL , ∈ (m, m + λ+ + 2). Suppose that u0 ∈ L∞ (Rn ) satisfies (29.2) and (29.3). Then there exist positive constants C1 , C2 such that the solution of (29.1) satisfies C1 (t + 1)α ≤ u(·, t) ∞ ≤ C2 (t + 1)α

for all t ≥ 0,

(29.5)

where α := m( − m − λ− )/(2λ− ). Remarks 29.2. (i) The above theorem shows that threshold solutions can decay to zero with an arbitrarily slow decay rate (if ∈ (m, m + λ− )) and also can grow up with any rate of the form tα , α ∈ (0, m(2 + λ+ − λ− )/(2λ− )). The upper bound for α is known to be optimal. More precisely, if p > pJL and u0 ∈ L∞ (Rn ) satisfies ∗ (29.2) (but not necessarily (29.3)), then u is global and u(t) ∞ ≤ C(t + 1)α , where α∗ = m(2+λ+ −λ− )/(2λ− ). In addition, there exists u0 ∈ L∞ (Rn ) satisfying ∗ (29.2) such that u(t) ∞ ≥ c(t + 1)α (see [381]). (ii) Let p = pJL . Then λ− = λ+ =: λ. If u0 ∈ L∞ (Rn ) satisfies (29.2) and (29.3) with some ∈ (m + λ, m + λ + 2), then (29.5) remains true with u(·, t) ∞ −m/λ replaced by u(·, t) ∞ log(t + 2) , see [190]. (iii) Assume psg 0. This is a consequence of [232, Theorem 10.1(i)] (see also [499]). Therefore the condition p ≥ pJL , for grow-up or slow decay below the singular steady-state, is optimal. The idea of matched asymptotics is to find a suitable asymptotic expansion for the solution in an inner region (for “small” |x|) and an outer region (for “large” |x|).


247

Matching these expansions on the boundary of the inner and outer regions (that is, comparing the coefficients of the leading terms of the expansions) determines the quantity that we are looking for. This formal approach not only provides a guess for the behavior of solutions but often also suggests the form of suband supersolutions that enable one to prove the result rigorously. It should be mentioned that in many cases the approach is more complicated: For example, in addition to the inner and outer regions one also has to consider an intermediate region. We will only consider the case < m + λ− in Theorem 29.1 since the case > m + λ− can be treated by similar arguments and the proof in the case = m + λ− follows from the fact that the solution remains between two positive stationary solutions of (29.1) for t ≥ t0 > 0 due to the comparison principle and (29.4). In addition, we will only describe in detail the formal part of the proof; the rigorous part will be sketched. Although the detailed rigorous proof in [204] represents one of the simplest applications of matched asymptotics, it is still quite long and technical and lies beyond the scope of this book. Another relatively simple example of matched asymptotics is mentioned in Remark 40.9(c). Throughout the rest of this section we will write f ∼ g if C˜1 g ≤ f ≤ C˜2 g for some constants C˜1 , C˜2 > 0 and f ≈ g (or f = g + h.o.t.) if f − g = o(f ). Sketch of proof of Theorem 29.1 for < m + λ− . Part 1: Formal matched asymptotics. We will consider radial solutions u = u(r, t), r = |x| of (29.1). Such solutions satisfy ⎫ n−1 ⎬ p ur + u , r > 0, t > 0, ut = urr + r (29.6) ⎭ r > 0. u(r, 0) = u (r), 0

Assume that u0 is continuous and radial nonincreasing and that η(t) := u(0, t) behaves like (t + 1)α for some α ∈ (−m/2, 0) and t 1. (29.7) Notice that introducing a new variable ζ = ζ(t, r) := η 1/m (t)r and assuming that u can be written in the form u = η(t)ϕ(ζ), (29.6) is transformed to 1 n−1 ϕζ + ϕp , ηt η −p ϕ + ζϕζ = ϕζζ + m ζ where ηt η −p → 0 as t → ∞. Consequently, the solution u should asymptotically behave like η(t)ϕ(η(t)1/m r),

(29.8)

where ϕ is a solution of ϕζζ +

n−1 ϕζ + ϕp = 0, ζ

ζ > 0,

ϕ(0) = 1, ϕζ (0) = 0.

(29.9)

248


It turns out that before making the transformation mentioned above it is useful to apply the self-similar change of variables v(ρ, s) = (t + 1)m/2 u(r, t),

r , ρ= √ t+1

s = log(t + 1),

which transforms (29.6) into n−1 ρ m vρ + v p + vρ + v, ρ 2 2 v(ρ, 0) = v0 (ρ) := u0 (ρ), vs = vρρ +

ρ > 0, s > 0,

⎫ ⎬

ρ > 0.

⎭

(29.10)

Notice that v(0, s) = (t + 1)m/2 u(0, t) → ∞ as s → ∞ due to (29.7). Let us first consider the inner region (where ρ is small). The equation in (29.10) is “similar” to that in (29.6) for small ρ: The additional two terms at the end of the RHS are expected to be small in comparison to the remaining ones if v is large and ρ small. Therefore, taking into account (29.8) and (29.9), for small ρ we will look for solution v in the form v(ρ, s) = σ(s) ψ(ξ) − R(s, ξ) (29.11) where σ(s) := v(0, s), ξ := σ 1/m ρ, ψ is the solution of ψξξ +

n−1 ψξ + ψ p = 0, ξ

ξ > 0,

ψ(0) = 1, ψζ (0) = 0,

(29.12)

and R represents the higher order terms (remainder). Plugging the ansatz (29.11) into (29.10) we obtain R ≈ σs σ −p Ψ(ξ) for ρ small and s large, where Ψξξ +

mσ n−1 1 Ψξ + pψ p−1 Ψ = − 1 ψ + ξψξ , ξ 2σs m Ψ(0) = Ψξ (0) = 0.

ξ > 0,

⎫ ⎬ ⎭

(29.13)

Since we expect σ(s) to behave like e(m/2+α)s for some α ∈ (−m/2, 0) due to mσ (29.7), the coefficient ( 2σ − 1) in (29.13) behaves like a positive constant and s [204, Lemma 3.1], [203, Lemma 4.2] guarantee that there exists K > 0 such that Ψ(ξ) ≈ Kξ 2−m−λ− as ξ → ∞. Fixing ρ > 0, we have ξ = σ 1/m (s)ρ → ∞ as s → ∞, hence R(s, ξ) ≈

σs 1 Ψ(ξ) ≈ K1 p−1 Ψ(ξ) ≈ K2 ξ −2 Ψ(ξ) ≈ K3 ξ −m−λ− , p σ σ

where K1 , K2 , K3 are positive constants. Due to (29.4), the solution ψ of (29.12) satisfies ψ(ξ) = cp ξ −m − aξ −m−λ− + o(ξ −m−λ− ), as ξ → ∞,


249

where a > 0. Consequently, we obtain the two-term inner expansion v ≈ σ(cp ξ −m − a ˜ξ −m−λ− ) = cp ρ−m − a ˜σ −λ− /m ρ−m−λ− ,

(29.14)

where a ˜ = a + K3 > 0. Next we consider the formal expansion in the outer region (where ρ 1) as s → ∞. Setting v = cp ρ−m − w and assuming w ρ−m for ρ 1, we have ws = wρρ +

pcp−1 n−1 ρ m p wρ + w + wρ + w + h.o.t., 2 ρ ρ 2 2

ρ 1.

If we look for a solution w in the form ˜ (ρ) + h.o.t., w(ρ, s) = e−βs W ˜ has to solve the equation then W pcp−1 ˜ = 0. ˜ ρρ + n − 1 W ˜ + ρW ˜ρ + p W ˜ρ + β + m W W ρ ρ2 2 2

(29.15)

˜ is required to satisfy the condition In addition, due to our assumption (29.3), W ˜ (ρ) ≤ lim sup ρ W ˜ (ρ) < ∞. 0 < lim inf ρ W ρ→∞

(29.16)

ρ→∞

If ρ 1, then the last two terms in (29.15) are much greater than the remaining ˜ . Due to (29.16) we have ˜ρ ≈ − β + m W ones so that we have to guarantee ρ2 W 2 to set β := ( − m)/2. In order that the outer expansion matches with the inner ˜ should also satisfy expansion (29.14), W ˜ (ρ) ≤ lim sup ρm+λ− W ˜ (ρ) < ∞. 0 < lim inf ρm+λ− W ρ→0

(29.17)

ρ→0

It is known (see [1] or [204]) that the problem (29.15), (29.16), (29.17) with β = ˜ provided ∈ (m, m + λ+ + 2) (this ( − m)/2 possesses a positive solution W solution can be expressed explicitly in terms of Kummer’s functions). Hence, we obtain the two-term outer expansion ˜ (ρ). v ≈ cp ρ−m − e−( −m)s/2 W

(29.18)

If we now match the inner expansion (29.14) with the outer expansion (29.18) at ρ = ρ0 > 0, then we obtain σ(s) ∼ em( −m)s/(2λ− ) ,

(29.19)

250


hence

m( − m − λ− ) . 2λ− This gives a formal proof of Theorem 29.1 for < m + λ− . u(0, t) ∼ tα ,

where α =

Part 2: Sketch of the rigorous proof. We will find a subsolution v and a supersolution v for the solution v of (29.10) such that the estimates v ≤ v ≤ v will guarantee (29.5). It is relatively easy to check that the subsolution v can be chosen as ˜ (ρ) , v(ρ, s) := max 0, cp ρ−m − be−( −m)s/2 W ˜ is a fixed solution of (29.15), (29.16), (29.17) with β = ( − m)/2, and where W b > 0 is large enough. The supersolution v is defined by v 1 (ρ, s), s ≥ 0, ρ ≤ ρM (s), v(ρ, s) := v 2 (ρ, s), s ≥ 0, ρ > ρM (s), where ρM (s) := inf{ρ > 0 : v 2 (ρ, s) < v 1 (ρ, s)} and v 1 , v 2 are supersolutions in the corresponding domains. It is again relatively easy to check that the supersolution v 2 can be chosen in the form v 2 (ρ, s) := cp ρ−m − be−( −m)s/2 W (ρ), where W is the solution of n−1 ρ Wρ + Wρ + W = 0, Wρρ + ρ 2 2

ρ > 0,

W (0) = 1,

Wρ (0) = 0,

(which can be again expressed in terms of Kummer’s functions) and b is small enough. The most difficult part is the choice of the supersolution v 1 . Recall that in the inner region, we expect σs v(ρ, s) ≈ σ(s) ψ(ξ) − p Ψ(ξ) , σ where Ψ solves (29.13). Plugging (29.19) into (29.13) we see that Ψ solves the problem ⎫ 1 n−1 m + λ− − p−1 Ψξ + pψ ψ + ξψξ + RΨ , ξ > 0, ⎬ Ψξξ + Ψ= ξ −m m (29.20) ⎭ Ψ(0) = Ψξ (0) = 0, where RΨ represents higher order terms. Now it turns out that one can set σs v 1 (ρ, s) := σ(s) ψ(ξ) − p Ψ(ξ) , σ where Ψ is the solution of (29.20) with RΨ := A/(1 + ξ m+λ− ) and A is a suitable positive constant. (The term RΨ is purely technical.)

Chapter III

Systems 30. Introduction Chapter III is devoted to systems of elliptic and parabolic types. In Section 31, we study the questions of a priori estimates and existence for weakly coupled elliptic systems which are natural extensions of the scalar equations considered in Chapter I. In Section 32, we study a simple parabolic system which is the analogue of the scalar model problem (15.1) studied in Chapter II. For this system, we treat the questions of well-posedness, global existence and blow-up. In Section 33, we discuss the different possible effects of adding linear diffusion (and some boundary conditions) to a system of ODE’s. It will turn out that quite opposite effects can be observed. This will lead us to consider some systems arising in biological or physical contexts, such as mass-preserving and Gierer-Meinhardt systems.

31. Elliptic systems In Sections 10–13, we have studied several methods to derive a priori estimates of positive solutions of scalar elliptic equations. The aim of this section is to present analogous results and methods in the case of elliptic systems. The three methods that we shall describe are extensions of the methods of Sections 11–13 from the scalar case, but they require substantial additional work and several new ideas. As mentioned before, a priori estimates can be used for the proof of existence, and they do not require any variational structure of the problem. Therefore they are well-suited for elliptic systems, which do not possess such structure in general. We will devote our attention to the Dirichlet problem for superlinear systems, especially of cooperative type, of the form: ⎫ −∆u = f (x, u, v), x ∈ Ω, ⎪ ⎬ −∆v = g(x, u, v),

x ∈ Ω,

u = v = 0,

x ∈ ∂Ω.

⎪ ⎭

(31.1)

A simple model case of such systems, and the analogue of the scalar problem (3.10), is the Lane-Emden system: ⎫ −∆u = v p , x ∈ Ω, ⎪ ⎬ −∆v = uq , u = v = 0,

x ∈ Ω, x ∈ ∂Ω.

⎪ ⎭

(31.2)

252

III. Systems

Throughout this section we assume p, q > 1, and we denote 2(p + 1) 2(q + 1) α= , β= . (31.3) pq − 1 pq − 1 These numbers play a fundamental role in the analysis of (31.2). They represent scaling exponents, corresponding to the fact that, for each λ > 0, the differential equations in (31.2) are invariant under the transformation (u, v) → (uλ , vλ ), where uλ (x) = λα u(λx), vλ (x) = λβ v(λx), due to α + 2 = βp, β + 2 = αq. On the other hand we say that (u, v) is positive if u, v > 0 (a.e.) in Ω. Note that, of course, if (u, v) is a nontrivial nonnegative, say classical, solution of (31.2) in a domain Ω ⊂ Rn , then it is positive by the strong maximum principle. Remarks 31.1. (i) Other nonlinearities. Although we shall concentrate, for simplicity, on the model case (31.2) and on a few variants, the three methods that we describe below, or their modifications, can be applied to wide varieties of systems. Let us mention systems with products or sums of powers, respectively given by f = ur v p , g = v s uq (31.4) (see [371], [454], [136], [449]), and by f = ur + v p ,

g = v s + uq

(see [184], [547], [449]), with p, q, r, s > 0. Several systems arising in physical or biological applications are also tractable by these methods. Let us mention the cooperative logistic system given by f = auv + u(c − u),

g = buv + v(d − v)

with a, b, c, d > 0 constants (see e.g. [343] and the references therein), which arises in population dynamics, where u, v stand for the densities of two biological species. Another example is given by f = uv − au,

g = bu

(31.5)

with a, b > 0 constants (see [258], [122], [449]), which arises as a model of nuclear reactor, where u and v respectively represent the neutron flux and the temperature. Each of the three methods works under different (and generally non-comparable) sets of assumptions, and its applicability depends on the problem under consideration (see Theorem 31.17 for an example in the case of (31.5)). (ii) Noncooperative systems. Many interesting examples from the point of view of biological or chemical applications involve noncooperative systems or systems with balance law. Results and techniques concerning the questions of global existence and blow-up for the parabolic version of such systems are presented in Section 33 below. (iii) Singularities for elliptic systems. Some results on isolated singularities for systems (31.2) and (31.1), (31.4), extending those in Section 4, can be found in [82], [80], [424].

31. Elliptic systems

253

31.1. A priori bounds by the method of moving planes and Pohozaev-type identities We consider the Lane-Emden system (31.2). For this system, the method described in this subsection allows to obtain complete and optimal results in the case of convex domains. Theorem 31.2. Assume p, q > 1, Ω convex and bounded, and 1 1 n−2 + > , p+1 q+1 n

(31.6)

equivalently α + β > n − 2. (i) Then any positive classical solution of (31.2) satisfies the a priori estimate u ∞ , v ∞ ≤ C,

(31.7)

with C independent of (u, v). (ii) There exists a positive classical solution of (31.2). Theorem 31.3. Assume p, q > 1, n ≥ 3, Ω starshaped and bounded, and 1 1 n−2 + ≤ , p+1 q+1 n

(31.8)

equivalently α + β ≤ n − 2. Then (31.2) has no positive classical solution. Theorems 31.2 and 31.3 are respectively due to [134] (see also [413]) and to [370]. The critical curve in the (p, q) plane: 1 1 n−2 + = , p+1 q+1 n associated with condition (31.6), is called the Sobolev hyperbola. Note that in the scalar case, corresponding to p = q, condition (31.6) reduces to p < pS . The method of proof of Theorem 31.3 is a modification of that of Section 13 in the scalar case. A common ingredient to the proofs of Theorems 31.2 and 31.3 is the following variational identity of Pohozaev-type [370], which is the analogue of Theorem 5.1 in the scalar case.

254

III. Systems

Lemma 31.4. Assume Ω bounded. (i) For any functions u, v ∈ C 2 (Ω) such that u = v = 0 on ∂Ω, there holds

Ω

(x · ∇v) ∆u + (x · ∇u) ∆v − (n − 2)∇u · ∇v dx =

(x · ν) ∂Ω

∂u ∂v dσ. ∂ν ∂ν

(ii) For any nonnegative classical solution (u, v) of (31.2) and any θ ∈ [0, 1], there holds

n n − (n − 2)θ v p+1 + − (n − 2)(1 − θ) uq+1 dx p+1 q+1 Ω

(31.9) ∂u ∂v dσ. = (x · ν) ∂ν ∂ν ∂Ω Proof. (i) We compute div((x · ∇v) ∇u) = (x · ∇v) ∆u + (∇(x · ∇v) · ∇u) ∂ ∂v ∂u = (x · ∇v) ∆u + xj ∂xi ∂xj ∂xi i,j = (x · ∇v) ∆u +

i,j

xj

∂ 2 v ∂u ∂v ∂u + . ∂xi ∂xj ∂xi ∂xi ∂xi i

Therefore div (x · ∇v) ∇u + (x · ∇u) ∇v = (x · ∇v) ∆u + (x · ∇u) ∆v + x · ∇(∇u · ∇v) + 2∇u · ∇v. On the other hand, we have div x(∇u · ∇v) = (div x) (∇u · ∇v) + x · ∇(∇u · ∇v) = n(∇u · ∇v) + x · ∇(∇u · ∇v). By subtracting, we obtain div (x · ∇v) ∇u + (x · ∇u) ∇v − x(∇u · ∇v) = (x · ∇v) ∆u + (x · ∇u) ∆v − (n − 2)∇u · ∇v. Applying the divergence theorem, it follows that

Ω

(x · ∇v) ∆u + (x · ∇u) ∆v − (n − 2)∇u · ∇v dx


=

255

(x · ∇v) ∇u + (x · ∇u) ∇v − x(∇u · ∇v) · ν dσ.

∂Ω ∂v ∇v = ( ∂ν ) ν on ∂Ω, due to u = v = 0 on ∂Ω, assertion (i) Since ∇u = follows. (ii) For a solution (u, v) of (31.2), we have

( ∂u ∂ν ) ν,

(x · ∇v) ∆u + (x · ∇u) ∆v = −(x · ∇v) v p − (x · ∇u) uq v p+1 uq+1 + = −x · ∇ p+1 q+1 v p+1 v p+1 uq+1 uq+1 = −div x + +n + p+1 q+1 p+1 q+1 hence

Ω

p+1 uq+1 v + dx. (x · ∇v) ∆u + (x · ∇u) ∆v dx = n q+1 Ω p+1

On the other hand,

Ω

and

∇u · ∇v dx =

Ω

uq+1 dx

u ∆v dx = Ω

∇u · ∇v dx = −

Ω

Ω

Ω

yield

∇u · ∇v dx = −

(31.10)

v p+1 dx

v ∆u dx = Ω

Ω

(1 − θ) uq+1 + θ v p+1 dx.

(31.11)

In view of (i), assertion (ii) then follows by combining (31.10) and (31.11). We first prove Theorem 31.3, which follows easily from Lemma 31.4. n ∈ (0, 1), Proof of Theorem 31.3. In view of (31.8), by choosing θ = (n−2)(p+1) we get n n − (n − 2)θ = 0, − (n − 2)(1 − θ) ≤ 0. (31.12) p+1 q+1 ∂v Identity (31.9) in Lemma 31.4 then implies ∂Ω (x · ν) ∂u ∂ν ∂ν dσ ≤ 0. Now since Ω ∂v is starshaped around, say, x = 0, we have x · ν ≥ 0 on ∂Ω, along with ∂u ∂ν , ∂ν ≤ 0, hence

∂u ∂v dσ = 0. (31.13) (x · ν) ∂ν ∂ν ∂Ω

If the inequality in (31.8) is strict, then so is the inequality in (31.12) and we deduce from (31.9) that u ≡ 0, hence v ≡ 0. In the equality case, then since

256

III. Systems

∂v x · ν ≡ 0 on ∂Ω, (31.13) implies ∂u ∂ν = 0 or ∂ν = 0 at some point of ∂Ω. Since −∆u, −∆v ≥ 0, u, v ≥ 0 in Ω and u = v = 0 on ∂Ω, we infer from Hopf’s lemma that u ≡ 0 or v ≡ 0, hence u ≡ v ≡ 0. (Note that this last argument actually applies whenever (31.8) holds.)

Proof of Theorem 31.2. (i) It is more involved and requires several steps. Step 1. Basic L1loc estimates. We claim that

Ω

uϕ1 dx ≤ C,

Ω

vϕ1 dx ≤ C.

(31.14)

Multiplying by ϕ1 , integrating by parts, and using Jensen’s inequality, we obtain

p uϕ1 dx = v p ϕ1 dx ≥ vϕ1 dx λ1 Ω

Ω

and λ1

Ω

Ω

vϕ1 dx =

Ω

uq ϕ1 dx ≥

Ω

q uϕ1 dx .

Consequently, we have

Ω

pq uϕ1 dx ≤ λp+1 uϕ1 dx, 1 Ω

which yields the first inequality in (31.14). The second follows similarly. Step 2. Estimates near ∂Ω. We use the notation of Section 13 (see after Theorem 13.1). Since Ω is convex and smooth, we can find λ0 , c0 > 0 such that Σ (y, λ) ⊂ Ω,

λ ≤ λ0

and

(ν(x), ν(y)) > c0 ,

x ∈ ∂Σ(y, λ0 ) ∩ ∂Ω.

Similarly as in Theorem 13.1, we shall apply the moving planes method (cf. [514] in the case of systems) to show that u(R(y, λ)x) ≥ u(x),

v(R(y, λ)x) ≥ v(x),

y ∈ ∂Ω, x ∈ Σ(y, λ), λ ≤ λ0 . (31.15) Without loss of generality, we may assume that y = 0 and that ν(0) = −e1 . For each x = (x1 , x ), we denote xλ := R(0, λ)x = (2λ − x1 , x ), Σλ := Σ(0, λ) = Ω ∩ {x1 < λ}, and Σλ := Σ (0, λ) = R(0, λ)Σλ . Define wλ (x) = u(xλ ) − u(x),

z λ (x) = v(xλ ) − v(x),

for x ∈ Σλ , 0 < λ ≤ λ0 ,

and set

E := µ ∈ (0, λ0 ] : wλ (x) ≥ 0, z λ (x) ≥ 0 for all x ∈ Σλ and λ ∈ (0, µ) .


257

∂u ∂v Since ∂x (0) > 0, ∂x (0) > 0 by Hopf’s lemma, we have λ ∈ E for λ > 0 small. 1 1 ¯ := sup E < λ0 . We have Assume for contradiction that λ

wλ ≥ 0,

¯ for all x ∈ Σλ and λ ∈ (0, λ],

z λ ≥ 0,

(31.16)

¯ with λ ¯ < λi < λ0 , such that (for instance) and there exists a sequence λi → λ, λi λ min w < 0. Since w = 0 on {x1 = λ} ∩ Ω and Σλi

wλ > 0 on {x1 < λ} ∩ ∂Ω,

for all λ < λ0 ,

(31.17)

it follows that this minimum is attained at a point qi ∈ Σλi . Therefore ∇wλi (qi ) = ∂u = (e1 · ν) ∂u 0. On the other hand, since ∂x ∂ν ≥ c > 0 on {x1 ≤ λ0 } ∩ ∂Ω and 1 wλ (x) = u(2λ − x1 , x ) − u(x1 , x ) = 2(λ − x1 )

∂u (ξ(x)), ∂x1

with |ξ(x) − x| ≤ 2(λ − x1 ), we see that wλ (x) ≥ 0 for x in an ε-neighborhood of {x1 = λ} ∩ ∂Ω, with ε > 0 independent of λ ∈ (0, λ0 ]. Therefore, we may assume ¯ ∩ ∂Ω, and by continuity we get / {x1 = λ} that qi → q¯ ∈ Σλ¯ , q¯ ∈ ¯

wλ (¯ q) = 0

¯

∇wλ (¯ q ) = 0.

and

(31.18)

But (31.16) implies ¯ ¯ −∆wλ (x) = v p xλ − v p (x) ≥ 0

and

¯

wλ (x) ≥ 0,

x ∈ Σλ¯ .

¯

By Hopf’s lemma, this along with (31.18) implies wλ = 0 in Σλ¯ , contradicting ¯ = λ0 , which proves (31.15). This guarantees that (31.17). Consequently, λ (∇u(x), ν(y)) ≤ 0,

(∇v(x), ν(y)) ≤ 0,

y ∈ ∂Ω, x ∈ Σ(y, λ0 ).

(31.19)

By Lemma 13.2 and Step 1, we deduce that u, v ≤ C on Ωε = {z ∈ Ω : δ(z) < ε} for some ε, C > 0 depending only on Ω. Using interior-boundary elliptic Lp -estimates (see Appendix A) and the embedding W 2,k → BU C 1 for k > n, we deduce a uniform bound for ∇u, ∇v in Ωε/2 . In particular, we have shown that ∂u , ∂ν

∂v ≤ C, ∂ν

x ∈ ∂Ω.

Step 3. Energy estimates. We claim that

v p+1 dx ≤ C, uq+1 dx ≤ C. Ω

Ω

(31.20)

258

Since

III. Systems 1 p+1

+

1 q+1

>

n−2 n ,

we may choose θ ∈ (0, 1), such that

n − (n − 2)θ > 0, p+1

n − (n − 2)(1 − θ) > 0. q+1

Using assertion (i) of Lemma 31.4 and estimate (31.20), we deduce that

∂u ∂v p+1 q+1 dσ ≤ C. v dx + u dx ≤ C (x · ν) ∂ν ∂ν Ω Ω ∂Ω Step 4. Bootstrap. Pick ρ > 1 to be fixed later and consider the following induction hypothesis: u (q+1)ρi , v (p+1)ρi ≤ C. (Hi ) Step 3 guarantees that (H0 ) is verified. Assume that (Hi ) holds for some i ∈ N. Then, since (u, v) solves (31.2), the linear estimate in Proposition 47.5(i) implies (Hi+1 ) provided p 1 2 − < (p + 1)ρi (q + 1)ρi+1 n

and

q 1 2 − < . (q + 1)ρi (p + 1)ρi+1 n

and

q 1 2 − < , (q + 1) (p + 1)ρ n

It is thus sufficient that p 1 2 − < (p + 1) (q + 1)ρ n i.e.

n − 2 n − 2 1 1 1 > max (q + 1) − , (p + 1) − . ρ n p+1 n q+1

Since, by assumption,

1 p+1

+

1 q+1

=

n−2 n

+ ε for some ε > 0, it suffices to choose

1 > 1 − ε min(p + 1, q + 1). ρ After a finite number of steps, we obtain u qˆ ≤ C, v pˆ ≤ C for some qˆ > nq/2, pˆ > np/2, and a further application of Proposition 47.5(i) yields u ∞ ≤ C, v ∞ ≤ C. (ii) The proof is similar to that of Corollary 10.3 (see e.g. [180] or [449, Section 4] for details). Remarks 31.5. Limitations and extensions. (i) The above method does not extend to general systems of the form (31.1). Indeed (but for very special cases), f should not depend on u (nor g on v) because of the need of variational identities. Also, f, g cannot depend on x (at least in an arbitrary way) in order to apply the moving planes method. It can still be generalized to f = f (v), g = g(u), with f, g nondecreasing (in order for the system to admit a comparison principle


259

to apply the moving planes method), provided f, g also satisfy suitable growth conditions related with the Sobolev hyperbola. These conditions can be expressed as a relation between f, g and their primitives which enables one to control Ω vf (v) and Ω ug(u) from the variational identities. (ii) The method partially extends to nonconvex domains Ω (via the Kelvin transform). However, this requires additional growth restrictions if n ≥ 3, namely p, q ≤ pS in the case of (31.2). Remark 31.6. Variational methods. If the nonlinearities f, g in system (31.1) have the form f (x, u, v) = Hv (x, u, v), g(x, u, v) = Hu (x, u, v), then solutions of (31.1) can be found as critical points of the functional

Φ(u, v) := ∇u · ∇v dx − H(x, u, v) dx. Ω

Ω

Considering Φ as a strongly indefinite functional in W01,2 × W01,2 (Ω) (or, more generally, in spaces of the form Xα × X1−α , where Xα , α ∈ (0, 1), are suitable interpolation spaces between X0 := L2 (Ω) and X1 := W 2,2 ∩ W01,2 (Ω), see [285] or [182], for example, and cf. Section 51.1) often leads to unnecessary technical restrictions concerning the growth of the Hamiltonian H. To overcome these difficulties one can use a dual approach (see [137] in the case of systems or [24], [25] in the scalar case). In the particular case of the Lane-Emden system (31.2) we have H = |v|p+1 /(p + 1) + |u|q+1 /(q + 1) and the dual functional has the form

p1 |w| |z|q1 1 ˜ + − (K ∗ w)z dx, Φ(w, z) = p1 q1 2 Ω where p1 = 1 + 1/p, q1 = 1 + 1/q, w = |v|p−1 v, z = |u|q−1 u and K is the Green function for the negative Dirichlet Laplacian, that is u := K ∗ w is the solution of the problem −∆u = w in Ω, u = 0 on ∂Ω. (Notice that Ω (K ∗ w)z dx = Ω (K ∗ z)w dx.) The functional ˜ : Lp1 × Lq1 (Ω) → R Φ possesses a mountain-pass structure and, in particular, it is easy to show that the existence result in Theorem 31.2 remains true without the assumption Ω convex. However, this approach does not provide a priori estimates of solutions. For some particular nonlinearities f, g, system (31.1) can also be reduced to a single higher-order equation. This is for instance the case for the Lane-Emden system (31.2), which is equivalent to the problem −∆((−∆u)1/p ) = uq , x ∈ Ω, u = ∆u = 0, x ∈ ∂Ω, where u ≥ 0 ≥ ∆u. Again, this problem can solved by variational methods.

260

III. Systems

31.2. Liouville-type results for the Lane-Emden system In this subsection we state Liouville-type theorems for the Lane-Emden system (and prove some of them). These are statements about nonexistence of entire positive solutions in the whole space or in a half-space. As in the scalar case, they constitute essential pieces of information in view of the rescaling method (see next subsection). We thus consider the following problems:

or

−∆u = v p ,

x ∈ Rn ,

−∆v = uq ,

x ∈ Rn ,

−∆u = v p ,

x ∈ Rn+ ,

−∆v = uq ,

x ∈ Rn+ ,

u = v = 0,

x∈

∂Rn+ ,

(31.21) ⎫ ⎪ ⎬ ⎪ ⎭

(31.22)

where p, q > 1 and Rn+ := {x ∈ Rn : xn > 0}. Conjecture 31.7. Systems (31.21) and (31.22) do not admit any positive classical solutions if (p, q) lies below the Sobolev hyperbola, i.e. α + β > n − 2. Remarks 31.8. “Classical solutions” in Conjecture 31.7 means u, v ∈ C 2 (Rn ) and u, v ∈ C 2 (Rn+ )∩C(Rn+ ), respectively; no growth or decay conditions at infinity are imposed. However for the rescaling method, it is sufficient to know a Liouvilletype theorem for bounded positive solutions. Although the full Conjecture 31.7 has not been proved so far, it is strongly supported by the following results. Theorem 31.9. Let p, q > 1. (i) Assume α + β ≤ n − 2. Then system (31.21) admits some radial, bounded, positive classical solution. (ii) System (31.21) does not admit any positive classical solution in the following cases: (a) α + β > n − 2 and either u, v are radial or n = 3, (b) max(α, β) ≥ n − 2, (c) p, q ≤ pS , (p, q) = (pS , pS ). Assertion (i) is due to [472]. As for assertion (ii), part (a) is due to [371] in the radial case. In the nonradial case, part (a) settles Conjecture 31.7 for n = 3. This


261

is due to [471] for polynomially bounded solutions, and to [424] in the general case. Part (b) is actually valid for supersolutions (see Theorem 31.12 below). Part (c), which in particular recovers the (optimal) scalar case, is due to [181]. The Liouvilletype result is also known in some other parts of the region α+β > n−2 (see [106]). Since most of the proofs are long and technical, we shall only prove nonexistence under assumption (b), as a consequence of Theorem 31.12 below. As for the half-space case, we have the following reduction. The result is due to [87], generalizing the idea of Theorem 8.3 in the scalar case. Theorem 31.10. Let n ≥ 1. For given p, q > 1, if system (31.21) does not admit any bounded, positive classical solution in dimension n − 1, then system (31.22) does not admit any bounded, positive classical solution in dimension n. Remarks 31.11. (i) Here we have made the convention that functions of n = 0 variables are constants (and have null Laplacian). Consequently the conclusion of Theorem 31.10 is true for n = 1. (ii) If system (31.21) does not admit any bounded positive solution in dimension n, then this remains true in dimension n − 1, so that in particular, system (31.22) does not admit any bounded positive solution in dimension n. Indeed if (u, v) solves (31.21) in Rn−1 and if we let u˜(x) = u(x1 , . . . , xn−1 ), v˜(x) = v(x1 , . . . , xn−1 ), then (˜ u, v˜) solves (31.21) in Rn . (iii) On the other hand, it was shown in [424] that, for given p, q > 1 and n, if system (31.21) does not admit any bounded, positive classical solution, then it does not admit any positive classical solution at all, and system (31.22) does not admit any bounded, positive classical solution. Sketch of proof of Theorem 31.10. Assume that (31.22) admits a bounded positive solution (u, v). By using the moving planes method (see the proof of ∂u Theorem 21.10 for similar arguments in the scalar case), one can show that ∂x ≥0 n ∂v n n−1 and ∂xn ≥ 0 in R+ . Therefore, for each x ∈ R , U (x ) := lim u(x , xn ) xn →∞

and

V (x ) := lim v(x , xn ) xn →∞

are well defined and are bounded positive functions. Arguing exactly as in the proof of Theorem 8.3, we see that (U, V ) is a bounded, positive classical solution of system (31.21) in Rn−1 . The result follows. Case (b) of Theorem 31.9(ii) is actually true for the following system of inequalities (see [501], [372]): −∆u ≥ v p ,

x ∈ Rn ,

−∆v ≥ uq ,

x ∈ Rn .

(31.23)

262

III. Systems

Theorem 31.12. Let p, q > 1. System (31.23) does not admit any positive solution u, v ∈ C 2 (Rn ) if max(α, β) ≥ n − 2. Proof. It is based on the rescaled test-function method. Fix φ ∈ D(Rn ), 0 ≤ φ ≤ 1, such that φ(x) = 1 for |x| ≤ 1 and φ(x) = 0 for |x| ≥ 2. For each R > 0, put φR (x) = φ(x/R). Let m, k ≥ 2 to be fixed later. We note that ∆(φm R ) = 0 for |x| ≤ R and that m−1 ∇φR |2 ≤ CR−2 φm−2 . |∆(φm ∆φR + m(m − 1)φm−2 R )| = mφR R R Multiplying the first inequality in (31.23) by φm R and integrating by parts, we obtain

m m −2 v p φm ≤ − φ ∆u = − u ∆(φ ) ≤ CR u φm−2 R R R R R λ1 . (31.32) Here f, g : Ω × [0, ∞)2 → [0, ∞) are Carathéodory functions, p, q > 1, r, s ≥ 1, C1 > 0. In what follows, we refer to the notion of L1δ , or very weak, solution introduced in Definition 3.1. The following result is due to [449]. Theorem 31.14. Assume Ω bounded and (31.31), (31.32), with max(α, β) > n − 1 and r, s < pBT =

n+1 . n−1

(31.33)

(31.34)

Then any nonnegative very weak solution (u, v) of (31.1) belongs to L∞ × L∞ (Ω) and satisfies the a priori estimate (31.7). Similarly as in the scalar case (cf. Corollary 10.3), existence results can be deduced from Theorem 31.14 under suitable additional assumptions on f, g. Condition (31.32) can be weakened or replaced by other conditions of different form.


267

For instance, by applying the same method, we obtain regularity and a priori estimate for the following simple system: ⎫ −∆u = a(x)v p , x ∈ Ω, ⎪ ⎬ x ∈ Ω, −∆v = b(x)uq , (31.35) ⎪ ⎭ u = v = 0, x ∈ ∂Ω. Theorem 31.15. Assume Ω bounded, p, q > 1, a, b ∈ L∞ (Ω), a, b ≥ 0, a, b ≡ 0 and (31.33). Then any nonnegative very weak solution of (31.35) belongs to L∞ × L∞ (Ω) and satisfies the a priori estimate (31.7). Moreover, there exists a solution (u, v) of (31.35), with u, v ∈ C0 ∩ W 2,m (Ω) for all finite m, and u, v > 0. Theorem 31.15 is from [489] (see also [449]). The optimality of condition (31.33) in Theorems 31.14 and 31.15 is shown by the following result from [489], which will be proved at the end of this section (see Theorem 11.5 for the analogue in the scalar case). Theorem 31.16. Assume Ω bounded, p, q > 1 and max(α, β) < n − 1.

(31.36)

Then there exist functions a, b ∈ L∞ (Ω), a, b ≥ 0, a, b ≡ 0, such that system (31.35) admits a positive very weak solution (u, v) satisfying u ∈ L∞ (Ω),

v ∈ L∞ (Ω).

In the bootstrap procedure in the proof of Theorems 31.14 and 31.15, each equation is used alternatively. At each step, we make use of the Lpδ regularity theory (cf. Theorem 49.2 and Proposition 49.5 in Appendix C), and L∞ is reached after finitely many steps. The proof of Theorem 31.15 given below presents the simplest case of application of these ideas to systems. The proof of Theorem 31.14, although based on the same basic approach, is more involved and will not be given here. Proof of Theorem 1. Initialization. By testing with ϕ1 , we obtain 31.15. Step the basic estimate Ω u dxϕ1 , Ω v dxϕ1 ≤ C, i.e. u 1,δ + v 1,δ ≤ C

(31.37)

in view of (1.4). (In the case a(x), b(x) ≥ C > 0, this is Step 1 of the proof of Theorem 31.2. For general a, b, this can be done by a simple modification using the argument in the proof of Theorem 11.3.) We set f := a(x)v p and g := b(x)uq . Then (31.37) guarantees f 1,δ + g 1,δ ≤ C. Assume without loss of generality q ≥ p and β = 2(q+1) pq−1 > n − 1. In particular, there holds (p − 1)(q + 1) ≤ pq − 1
0 small will be chosen below. Step 2. Bootstrap on the first equation. Let k1 ∈ (k, ∞] satisfy 1 2 p . > − k1 k n+1

(31.42)

Using Theorem 49.2(i) and the first equation, we obtain u k1 ,δ ≤ C ∆u k/p,δ = C f k/p,δ ≤ C v p k/p,δ = C v pk,δ ≤ C.

(31.43)

For later use, we already note that if k>

(n + 1)pq , 2(q + 1)

(31.44)

then by taking ε = ε(n, p) > 0 in (31.41) sufficiently small, we may find k1 >

(n + 1)q 2

(31.45)

2 2 such that (31.43) is satisfied. Indeed, kp − n+1 < min (n+1)q , k1 and we may thus find k1 ∈ (k, ∞) satisfying (31.45) and (31.42), hence (31.43). Step 3. Bootstrap on the second equation. Now assume

and let k2 ∈ (k, ∞] satisfy

k1 > q

(31.46)

1 q 2 . > − k2 k1 n+1

(31.47)

Using Theorem 49.2(i), the second equation and (31.43), we obtain v k2 ,δ ≤ C ∆v k1 /q,δ = C g k1 /q,δ ≤ C uq k1 /q,δ = C u qk1 ,δ ≤ C.

(31.48)


269

Step 4. Fulfillment of the bootstrap conditions. Let ρ = ρ(p, q, n) ∈ (0, 1) to be determined. Conditions (31.42), (31.46), (31.47), together with the bootstrap condition k min(k1 , k2 ) > , ρ are equivalent to A := and

ρ 1 2 1 p − < < min , k n+1 k1 k q 1 q 2 ρ < − < . k1 n+1 k2 k

Assume k≤

(n + 1)pq 2(q + 1)

(31.49)

(31.50)

(31.51)

hence, in particular, A > 0. Then condition (31.49) can be solved in k1 ∈ [1, ∞), and 1/k1 can be taken arbitrarily close to A, provided

and

p−ρ 2 < k n+1

(31.52)

2 1 p − < . k n+1 q

(31.53)

Since k ≥ p, condition (31.52) is satisfied whenever n−1 p < ρ < 1, n+1

(31.54)

which is allowable in view of (31.38). Due to β > n − 1, we have (pq − 1)(n − 1) < 2 1 2q + 2 hence n−1 n+1 p − n+1 < q . Taking ε = ε(p, q, n) > 0 small in (31.41), we thus get (31.53). On the other hand, condition (31.50) can be solved in k2 ∈ [1, ∞) if q ρ 2 < . − k1 n+1 k

(31.55)

Taking 1/k1 in (31.49) close enough to its lower bound A (cf. after (31.51)), (31.55) becomes equivalent to ρ > 1 − η,

where η :=

2(q + 1) k − (pq − 1). n+1

(31.56)

Observe that η > 0 is equivalent to k > (n + 1)/β and, since β > n − 1, this is true for ε = ε(p, q, n) > 0 small in (31.41). We may thus choose ρ = ρ(p, q, n) ∈ (0, 1) satisfying (31.54) and (31.56).

270

III. Systems

Step 5. Conclusion. We deduce from Step 4 that if (31.40) holds for some k satisfying (31.41) and (31.51), then (31.40) is true with k replaced by k/ρ. Starting from (31.39), we see that some value k > (n + 1)pq/2(q + 1) of k is reached after a finite number of steps. It then follows from the second paragraph in Step 2 that u k1 ,δ ≤ C for some k 1 > (n + 1)q/2 ≥ (n + 1)p/2. By Step 3 with k1 := k1 and k2 := ∞, it follows that v ∞ ≤ C. We may then apply Step 2 with k := k1 and k1 := ∞ to conclude that u ∞ ≤ C. The proof is complete. As an application of the methods in this section, one obtains the following result [449] concerning the system ⎫ −∆u = uv − au, x ∈ Ω, ⎪ ⎬ −∆v = bu, x ∈ Ω, (31.57) ⎪ ⎭ u = v = 0, x ∈ ∂Ω, mentioned in Remark 31.1(i). Theorem 31.17. Assume Ω bounded, a, b > 0, and n ≤ 4. Then any nonnegative very weak solution of (31.57) belongs to L∞ × L∞ (Ω) and satisfies the a priori estimate (31.7). Moreover, there exists a classical solution of (31.57) with u, v > 0. Sketch of proof (see [449] for details). We use a variant of Theorem 31.14. In fact, without assuming (31.32), the growth conditions (31.31), (31.33), (31.34) alone ensure that any very weak solution satisfies u, v ∈ L∞ ∩ W 2,m (Ω) for all finite m. Moreover, if we know an a priori estimate of u and v in L1δ (Ω), then this implies an a priori estimate in L∞ (Ω) (the only role of assumption (31.32) in Theorem 31.14 is to guarantee the L1δ -estimate). Take 1 < r < pBT , p = r/(r − 1) and q = 1. Using uv ≤ v p + ur , and noting that max(α, β) = 2(p + 1)/(pq − 1) = 4r − 2 > n − 1 for r close to pBT due to n < 5, we see that f = uv − au, g = bu satisfy (31.31), (31.33), (31.34). On the other hand, the L1δ a priori estimate can be shown as follows. We have −∆u = −b−1 v∆v − au ≥ −

b−1 ∆(v 2 ) − au. 2

Testing this inequality and the second equation in (31.57) with ϕ1 , we obtain

2 b−1 λ1 b−1 λ1 (λ1 + a) uϕ1 dx ≥ v 2 ϕ1 dx ≥ vϕ1 dx 2 2 Ω Ω Ω

2 b λ−1 = 1 uϕ1 dx . 2 Ω This implies the desired estimate.


271

Remarks 31.18. Comparison with other methods. (i) The method of Section 10 based on Hardy-Sobolev inequalities has also been extended to certain systems, see [141], [258], [135], [144]. Like the Lpδ bootstrap method, it essentially requires only upper bounds on the growth of the nonlinearities f, g. However, the growth restrictions on the nonlinearities are much stronger, unlike in the scalar case (roughly, min(α, β) > n − 1 instead of max(α, β) > n − 1; cf. [135]). The reason for this is that the bootstrap procedure in that method is based on an H 1 × H 1 -estimate and is carried out simultaneously on the two components. Consequently, unlike in the above proof, the possible compensation effects between the two equations are not fully exploited. (ii) Condition (31.33) also appears in the work [85], where existence and a priori estimates are studied for system (31.2) with extra (measure) terms added in the RHS and in the boundary conditions. The method in [85] is different from that described in this section. In particular, it uses maximum principle arguments to derive comparison estimates of the form uq+1 ≤ C(1 + v p+1 ). In the case of system (31.1) (without measures in the RHS), it applies typically when 0 ≤ f ≤ C2 v p and C1 uq ≤ g ≤ C2 uq , with C2 ≥ C1 > 0 and p, q satisfying (31.33). We now turn to the proof of Theorem 31.16. Like that of Theorem 11.5, it is mainly a consequence of Lemma 49.13, where a singular solution of the linear Laplace equation with an appropriate right-hand side belonging to L1δ is constructed. Proof of Theorem 31.16. Set φ := |x|−(α+2) χΣ and ψ := |x|−(β+2) χΣ , with Σ as in Lemma 49.13 and let u, v > 0 be the very weak solutions of (47.8) with f = φ, ψ, respectively. By (49.29), we have u ∈ L∞ , v ∈ L∞ and v p ≥ C|x|−βp χΣ = C|x|−(α+2) χΣ = Cφ, uq ≥ C |x|−αq χΣ = C |x|−(β+2) χΣ = C ψ. Setting a(x) = φ/v p ≥ 0, b(x) = ψ/uq ≥ 0, we get −∆u = φ = a(x)v p , −∆v = ψ = b(x)uq and a(x) ≤ 1/C, b(x) ≤ 1/C hence a, b ∈ L∞ . The proof is complete. Remark 31.19. Localization of singularities. The observations in Remarks 11.6 extend to the case of systems. In particular, in spite of the imposed homogeneous Dirichlet boundary condition, the singularities of the solution in Theorem 31.15 occur at a (single) boundary point. In fact, when n − 2 < max(α, β) < n − 1, system (31.1) cannot have purely interior singularities. On the contrary, for max(α, β) < n − 2, examples of similar systems which possess unbounded weak solutions with purely interior singularities can be easily constructed. Namely the pair (u, v) = (r−α − 1, r−β − 1), r = |x|, is a weak solution of system (31.1) with f = c1 (v + 1)p and g = c2 (u + 1)q for Ω = B1 and suitable constants c1 , c2 > 0 (note that the right-hand sides are in L1 ).

272

III. Systems

32. Parabolic systems coupled by power source terms In this section, as a simple superlinear parabolic system and an analogue of the scalar model problem (15.1), we study the system: ut − ∆u = |v|p−1 v,

x ∈ Ω, t > 0,

vt − ∆v = |u|q−1 u, u = v = 0,

x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0,

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

x ∈ Ω,

v(x, 0) = v0 (x),

x ∈ Ω,

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(32.1)

where p, q > 0. We set X = L∞ × L∞ (Ω)

and

X+ = {(u0 , v0 ) ∈ X : u0 , v0 ≥ 0}.

(32.2)

In all this section, when pq > 1, the scaling exponents α, β are defined by (31.3). Assume p, q ≥ 1. Then problem (32.1) is locally well-posed in X (see Example 51.12). In particular, if Tmax < ∞, then

lim

t→Tmax

u(t) ∞ + v(t) ∞ = ∞.

(32.3)

Also the solution satisfies u, v ∈ BC 2,1 (Ω × [t1 , t2 ]),

0 < t1 < t2 < Tmax .

(32.4)

Furthermore, problem (32.1) admits a comparison principle (cf. Proposition 52.22). Next consider the case p, q > 0 and min(p, q) < 1. For (u0 , v0 ) ∈ X, local existence can be proved easily by approximation arguments (similar to those in the proof of Proposition 51.16 for instance). Turning to the question of (non-) uniqueness, which has been studied in [171], [172], let us assume (u0 , v0 ) ∈ X+ , and Ω bounded or Ω = Rn . Local uniqueness is true in the class of nonnegative classical solutions if either pq ≥ 1 or (u0 , v0 ) = (0, 0), but the proof is nontrivial. On the contrary, there exist infinitely many nonnegative classical solutions if pq < 1 and (u0 , v0 ) = (0, 0). On the other hand, if p, q > 0 and (u, v) is any maximal classical solution of (32.1) with existence time denoted by Tmax , then we still have (32.3) and (32.4).

32. Parabolic systems coupled by power source terms

273

32.1. Well-posedness and continuation in Lebesgue spaces We consider system (32.1) with initial values in the space Y = Lr1 × Lr2 (Ω). For (u0 , v0 ) ∈ Y , by a local solution of (32.1) (on [0, T ]), we understand a function (u, v) ∈ C([0, T ], Y ) which is a classical solution of (32.1) for 0 < t ≤ T and which fulfills the initial conditions. (Actually the nonexistence result below will still hold for a weaker notion of solution, see [446] for details.) The optimal condition for local existence/nonexistence for system (32.1) can be expressed in terms of the numbers p q 1 1 P=n , Q=n . − − r2 r1 r1 r2 Theorem 32.1. (i) (Well-posedness) Let p, q > 1, r1 , r2 > 1 and assume max(P, Q) ≤ 2. For all (u0 , v0 ) ∈ Lr1 × Lr2 (Ω), there exist T > 0 and a unique local solution of system (32.1) on [0, T ]. (ii) (Local nonexistence) Let p, q > 0, r1 , r2 ≥ 1 and assume max(P, Q) > 2. Then there exists (u0 , v0 ) ∈ Lr1 × Lr2 (Ω), u0 , v0 ≥ 0, such that system (32.1) admits no local solution (u, v) with u, v ≥ 0. As in Section 16, it is natural to look for sufficient conditions, in terms of Lr bounds, guaranteeing global existence. Theorem 32.2. (Continuation) Let p, q ≥ 1, pq > 1, n ≥ 2 and assume Ω bounded. Let (u, v) be a maximal classical solution of (32.1) and denote by T its existence time. Assume that either n(pq − 1) n r1 > = and sup u(t) r1 < ∞, α 2(p + 1) (0,T ) or r2 >

n(pq − 1) n = β 2(q + 1)

and

sup v(t) r2 < ∞.

(0,T )

Then T = ∞. Theorems 32.1 and 32.2 are from [446] and [447], respectively. Observe that the inequality max(P, Q) < 2 implies r1 > n/α and r2 > n/β, but that this can be true also when max(P, Q) > 2. Therefore, the continuation property is valid under weaker assumptions on r1 , r2 than well-posedness. This is in sharp contrast with the situation in the scalar case (cf. Theorems 15.2 and 15.3, and Corollary 24.2). Note also that an Lr -bound on a single component is enough to guarantee global existence.

274

III. Systems

Remark 32.3. The gap between conditions guaranteeing well-posedness and continuation can be heuristically explained as follows. The final profiles of a solution around a blow-up point x0 are expected to verify the lower estimates u(x, T ) ≥ c1 |x − x0 |−α ,

v(x, T ) ≥ c2 |x − x0 |−β

for |x−x0 | small (cf. Remark 32.12(ii) for a partial result), hence (u(·, T ), v(·, T )) ∈ / Lr1 × Lr2 whenever r1 > n/α or r2 > n/β. On the other hand, if a local solution exists, then u0 and v0 have to satisfy suitable integral estimates as a consequence of the variation-of-constants formula (see (32.8) below), and this leads to necessary conditions involving r1 and r2 if (32.1) is well-posed in Lr1 × Lr2 . Theorem 32.1(i) is proved in Example 51.32 of Appendix E. As for the proof of Theorem 32.1(ii), the main ingredient is the following lemma, which provides lower estimates for certain time-space averages of solutions of the linear heat equation with positive singular initial data. Lemma 32.4. Assume 0 2. − r2 r1

Then there exists v0 ∈ Lr2 (Ω), v0 ≥ 0, such that " t p " " " e−(t−s)A e−sA v0 ds" " 0

r1

→ ∞, as t → 0+ .

Proof. Assume B(0, 2ρ) ⊂ Ω, ρ > 0, let k ∈ (0, n/r2 ), and define v0 (y) = |y|−k χB(0,ρ) (y). Clearly, v0 ∈ Lr2 (Ω). Using the √heat kernel estimate in Proposition 49.10, we obtain, for s > 0 small and |x| ≤ s/2,

G(x, y, s)|y|−k dy ≥ c1 s−n/2 |y|−k dy √ |y| 0 small.

(32.5)


275

√ √ Next, let t/4 ≤ s ≤√ t/2, with t > 0 small, and |x| ≤ s/2. For |y| ≤ s/2 we have |x − y| ≤ t − s, hence G(x, y, t − s) ≥ c1 (t − s)−n/2 ≥ c1 s−n/2 by Proposition 49.10. It follows that

−(t−s)A √ e χB(0, s/2) (x) = G(x, y, t − s) dy ≥ c > 0. √ |y|< s/2

Combining this with (32.5), we deduce that, for t > 0 small, p e−(t−s)A e−sA v0 (x) ≥ cs−kp/2 ≥ ct−kp/2 , t/4 ≤ s ≤ t/2,

|x| ≤

√ s/2. (32.6)

√ Now if |x| ≤ t/4 and t is small, it follows from (32.6) that t t/2 p p kp e−(t−s)A e−sA v0 ds (x) ≥ e−(t−s)A e−sA v0 ds (x) ≥ Ct1− 2 0

t/4

√

√

(note that s ≥ t/4 implies s/2 ≥ t/4 ≥ |x|). Therefore, for t > 0 small, we obtain " t p " " "r1 e−(t−s)A e−sA v0 ds" " r1 0

t p r1 kp n −(t−s)A −sA e ≥ e v ds (x) dx ≥ Ct 2 +r1 (1− 2 ) . 0 √ {|x|≤ t/4}

0

Since the assumption of the lemma implies n2 + r1 1 − follows by choosing k sufficiently close to n/r2 .

np 2r2

< 0, the conclusion

Proof of Theorem 32.1(ii). Similarly as in Remark 15.4(iii), any nonnegative solution of (32.1) in the sense of Theorem 32.1 satisfies the variation-of-constants formula: ⎫

t ⎪ −tA −(t−s)A p−1 ⎪ ⎪ u(t) = e u0 + e |v(s)| v(s) ds, 0 ≤ t < T, ⎬ 0 (32.7)

t ⎪ ⎪ −tA −(t−s)A q−1 ⎪ v0 + e |u(s)| u(s) ds, 0 ≤ t < T. v(t) = e ⎭ 0

In particular, we have u(t) ≥ e−tA u0 ≥ 0, v(t) ≥ e It follows that

0≤

e

0≤

t

−tA

v0 ≥ 0,

−(t−s)A

e 0

−(t−s)A

v0 ) ds ≤ u(t),

⎫ ⎪ ⎪ ⎪ ⎬

−sA

u0 ) ds ≤ v(t).

⎪ ⎪ ⎪ ⎭

(e (e

0 ≤ t < T.

−sA

0 t

0 ≤ t < T,

p

q

(32.8)

276

III. Systems

Since (u, v) ∈ C([0, T ], Lr1 × Lr2 ), the right-hand sides in (32.8) remain bounded in Lr1 or Lr2 , respectively, hence " t p " " " e−(t−s)A e−sA v0 ds" "

r1

0

" t q " " " +" e−(t−s)A e−sA u0 ds"

r2

0

≤ C,

0 < t < T.

If either P > 2 or Q > 2, that is, n

p 1 >2 − r2 r1

or

n

q 1 > 2, − r1 r2

then, by choosing u0 ∈ Lr1 or v0 ∈ Lr2 as given by Lemma 32.4, we conclude that no solution of (32.1) can exist. Proof of Theorem 32.2. We shall prove the result only for n ≥ 4. The proof for n = 2, 3 is more involved and relies on suitable interpolation spaces. (However, the proof below applies also if n = 3 and p, q ≥ 2, or if n ≤ 3 and r1 > q − (1/p), r2 > p − (1/q).) By Propositions 48.4 and 48.5, there exists ω > 0 such that n

1

1

e−tA L(Lm1 ,Lm2 ) ≤ C1 t− 2 ( m1 − m2 ) e−ωt ,

1 ≤ m1 ≤ m2 ≤ ∞.

(32.9)

By a time shift, we may assume that (u, v) is smooth up to t = 0. In particular, it satisfies the variation-of-constants formula (32.7). We fix τ ∈ (0, T ) and we denote |u|m := sup u(t) m < ∞, t∈(0,τ )

1≤m≤∞

(and similarly for v). In the rest of the proof, C denotes a generic constant independent of τ . Assume that |v|r ≤ C (32.10) for some r>

n(pq − 1) . 2(q + 1)

(32.11)

Let k, l satisfy 1 ≤ k ≤ l < ∞,

k≥

r p

and

1 1 2 − < . k l n

By the first equation in (32.7) and the smoothing property (32.9) with m1 = k, m2 = l it follows that |u|l ≤ C(1 + |v|p k ) = C(1 + |v|pkp )


277

hence, by (32.10) and by the interpolation inequality, p−(r/k) ). |u|l ≤ C(1 + |v|∞

(32.12)

If in addition

nq , 2 then the second equation in (32.7) and (32.9) with m1 = l/q, m2 = ∞ imply |v|∞ ≤ C 1 + |u|q l/q = C(1 + |u|ql ); l>

hence, by (32.12), q(p−(r/k)) |v|∞ ≤ C(1 + |v|∞ ).

It follows that |v|∞ ≤ C if

1 pq − 1 < . qr k

The sufficient conditions are thus 1 2 1 2 1 max 0, − < < min , (32.13) k n l nq k and p 1 pq − 1 < < min 1, . (32.14) qr k r Condition (32.13) can be solved in l if 2 2 1 − < , k n nq i.e., 1 2(q + 1) < . k nq Since pq − 1 p < , qr r it then suffices to satisfy pq − 1 2(q + 1) pq − 1 < and < 1, qr nq qr that is, n(pq − 1) 1 r> and r > p − . 2(q + 1) q Finally, note that n(pq − 1) 1 ≥p− if (n − 2)q ≥ 2, 2(q + 1) q which is true for all q ≥ 1 if n ≥ 4, and for q ≥ 2 if n = 3. The hypothesis (32.11) thus implies the solvability of (32.13)–(32.14). Consequently, |v|∞ ≤ C, hence |u|∞ ≤ C by the first equation in (32.7). Since C is independent of τ , we deduce that u and v are uniformly bounded on QT , hence T = ∞.

278

III. Systems

32.2. Blow-up and global existence The following result provides the conditions on the exponents p, q which imply or prevent blow-up for system (32.1) in bounded domains. Theorem 32.5. Assume Ω bounded, p, q > 0, (u0 , v0 ) ∈ X+ , and set p˜ = min(p, 1), q˜ = min(q, 1). Let (u, v) be a maximal classical solution of (32.1) and denote by T its existence time. (i) 1, then there exists C(p, q, Ω) > 0 with the following property: If Ifq˜ pq > p˜ (u + v )ϕ 1 dx > C(p, q, Ω), then T < ∞. 0 0 Ω (ii) If pq ≤ 1, then T = ∞. Moreover, if pq < 1, then u(t), v(t) are uniformly bounded for t ≥ 0. Theorem 32.5 is a modification of a result from [173] (see also [224], [226] for p, q > 1). Proof. (i) Denote y = y(t) := Ω u(t)ϕ1 dx, z = z(t) := Ω v(t)ϕ1 dx. We may assume q = max(p, q) > 1 without loss of generality. Multiplying the second equation in (32.1) with ϕ1 , we have

vt ϕ1 dx = v∆ϕ1 dx + uq ϕ1 dx. z = Ω

Ω

Ω

Using ∆ϕ1 = −λ1 ϕ1 and Jensen’s inequality yields z ≥ −λ1 z + y q .

(32.15)

We first consider the easier case p > 1. Similarly as above, we have y ≥ −λ1 y + z p . Therefore φ := y + z satisfies φ = y + z ≥ −λ1 φ + z p + y q ≥ −λ1 φ + z p + y p − y ≥ −(1 + λ1 )φ + 21−p φp . It follows that T < ∞ whenever φ(0) > C(λ1 , p). Next consider the case p ≤ 1. In what follows, the constants ci > 0 will depend only on p, q, Ω. Recall that (u, v) satisfies the variation-of-constants formula (32.7). p By (15.24), for each 0 < σ < s < t, we have e−(s−σ)A upq (σ) ≤ e−(s−σ)A uq (σ) , hence

s

s −(s−σ)A q p e e−(s−σ)A upq (σ) dσ ≤ u (σ) dσ 0 0 s p ≤ s1−p e−(s−σ)A uq (σ) dσ . 0


279

Using (32.7), we deduce that s p e−(t−s)A e−(s−σ)A uq (σ) dσ ds 0 0

t s −tA p−1 −(t−s)A u0 + t e e−(s−σ)A upq (σ) dσ ds, ≥e

u(t) ≥ e−tA u0 +

t

0

0

hence u(t) ≥ e

−tA

t

u0 + t

s

p−1 0

e−(t−σ)A upq (σ) dσ ds

(32.16)

0

by Fubini’s theorem. Put γ = pq > 1. It follows from Jensen’s inequality that (e−(t−σ)A uγ (σ), ϕ1 ) = e−λ1 (t−σ) (uγ (σ), ϕ1 ) ≥ e−λ1 (t−σ) y γ (σ). Multiplying (32.16) with ϕ1 , we thus obtain y(t) ≥ e

−λ1 t

t

y(0) + t

s

p−1 0

e−λ1 (t−σ) y γ (σ) dσ ds.

0

Assume that T ≥ 1. We have

t

y(t) ≥ c1 y(0) + c1 hence

s

y γ (σ) dσ ds =: h(t), 0

0 < t < 1,

0

h (t) ≥ c1 hγ ,

0 < t < 1.

(32.17)

To conclude, it suffices to show that this inequality cannot be satisfied whenever

(u0 + v0p )ϕ1 dx ≥ M, where M = M (p, q, Ω) is large enough. Ω

Multiplying (32.17) by h ≥ 0 and integrating, we have 2

h (t) ≥ c2 hγ+1 (t) − c3 y γ+1 (0),

0 < t < 1.

(32.18)

On the other hand, using (32.7) and p ≤ 1 again, we get

p −tA p v (t)ϕ1 dx ≥ (e v0 ) ϕ1 dx ≥ (e−tA v0p )ϕ1 dx = e−λ1 t z˜(0) z˜(t) := Ω

Ω

Ω

and next, y(t) = e−λ1 t y(0) +

0

t

e−λ1 (t−s)

Ω

v p (s)ϕ1 dx ds ≥ c4 (y(0) + t z˜(0)),

0 < t < 1.

280

III. Systems

Therefore, since h (t) = c1 y γ , we have h(1/2) ≥ c5 (y(0) + z˜(0))γ .

(32.19)

Due to γ > 1, if y(0) + z˜(0) ≥ M (where M is large enough), we deduce from (32.18) that h ≥ c6 h(γ+1)/2 on (1/2, 1), which contradicts (32.19) for M large. (ii) Let us first assume pq < 1. Let Θ be the classical solution of (19.27), and put M = Θ ∞ . We observe that (u, v) = (a(1 + Θ), b(1 + Θ)) is a supersolution, whenever the constants a, b > 0 satisfy a ≥ [b(1 + M )]p and b ≥ [a(1 + M )]q . It is thus sufficient that (1 + M )p bp ≤ a ≤ (1 + M )−1 b1/q . Since p < 1/q, for a given (u0 , v0 ), one can take a, b as above and such that a ≥ u0 ∞ , b ≥ u0 ∞ . The assertion then follows from the comparison principle (note that since u ≥ a > 0 and v ≥ b > 0, it applies even though p, q may be < 1 — see Remark 52.11(c)). Finally assume pq = 1, and p ≥ 1 without loss of generality. We claim that for all a > 0, (u, v) = (ap ept , aet ) is a supersolution. Indeed, this is equivalent to pap ept ≥ ap ept and aet ≥ apq epqt , which is true due to pq = 1 and p ≥ 1. It then 1/p suffices to choose a ≥ max u0 ∞ , v0 ∞ ). Remark 32.6. Results on a priori estimates and universal bounds for global positive solutions of (32.1) can be found in [448] (see also [425, Section 6]).

32.3. Fujita-type results In this subsection we consider nonnegative solutions of the Cauchy problem associated with (32.1), i.e.: ⎫ ut − ∆u = v p , x ∈ Rn , t > 0, ⎪ ⎪ ⎪ ⎬ x ∈ Rn , t > 0, vt − ∆v = uq , (32.20) ⎪ x ∈ Rn , u(x, 0) = u0 (x), ⎪ ⎪ ⎭ x ∈ Rn . v(x, 0) = v0 (x), We give a Fujita-type result for problem (32.20), i.e. we find the (optimal) conditions on p, q depending on n, which guarantee that the solution blows up in finite time for all u0 , v0 ≥ 0, (u0 , v0 ) ≡ (0, 0). Theorem 32.7. Let p, q > 0 satisfy pq > 1, and let (u0 , v0 ) ∈ X+ , (u0 , v0 ) ≡ (0, 0). (i) If max(α, β) ≥ n, then (32.20) admits no nontrivial global solution. (ii) If max(α, β) < n, then (32.20) admits global, bounded solutions for suitably small initial data. This result is from [171]. We will prove it only under the additional assumption p, q ≥ 1, and we will not treat the critical value max(α, β) = n. However in


281

this special case, the present proof, based on arguments from [192], is considerably simpler than that in [171]. We shall use Gaussian test-functions of x and differential inequalities. See [372] for a different proof in the case p, q ≥ 1, based on rescaled test-functions of x and t. As a preliminary to the proof, we prepare the following lemma concerning the system of differential inequalities: y (t) ≥ z p − λy,

t ≥ 0,

z (t) ≥ y q − λz,

t ≥ 0.

(32.21)

Lemma 32.8. Let p, q > 0 satisfy pq > 1 and λ > 0. Then there exists K = K(p, q) > 0 such that (32.21) has no global nonnegative solution y, z ∈ C([0, ∞)) ∩ C 1 ((0, ∞)) with y(0) ≥ Kλα/2 . Proof. Put τ = λ−1 and assume that (y, z) exists on [0, τ ]. Then there exists C1 = C1 (q) > 0 such that y(τ ) ≥ C1 y(0) and z(τ ) ≥ C1 λ−1 y q (0).

(32.22)

Indeed, we have (yeλt ) ≥ 0, hence y(t) ≥ y(0)e−λt ≥ e−1 y(0) on [0, τ ]. This implies (zeλt ) ≥ eλt y q (t) ≥ e−q y q (0) on [0, τ ], hence z(τ ) ≥ e−(q+1) λ−1 y q (0), and (32.22) follows. Next, since pq > 1, we may choose A, B > 1 depending only on p, q, such that p(B − 1) > A and q(A − 1) > B. We claim that if, for some t0 , there exist a, b > 0 such that y(t0 ) > a, z(t0 ) > b, bp > Aλa, and aq > Bλb, (32.23) then (y(t), z(t)) cannot exist globally. To prove the claim, assume for contradiction that (y, z) exists for all t > 0. By a time shift, we may assume t0 = 0. Let (˜ y , z˜) be the unique, positive local solution of ⎫ y˜ (t) = z˜p − λ˜ y, t ≥ 0, ⎪ ⎬ q z, t ≥ 0, z˜ (t) = y˜ − λ˜ ⎪ ⎭ y˜(0) = a, z˜(0) = b. By an easy comparison argument (using the fact that z → z p and y → y q are increasing functions), it follows that (˜ y , z˜) exists for all t > 0 and we have y(t) ≥ y˜(t) > 0 and z(t) ≥ z˜(t) > 0. Set φ(t) = z˜p − Aλ˜ y and ψ(t) = y˜q − Bλ˜ z . We have φ(0) > 0 and ψ(0) > 0 by (32.23). Assume that φ, ψ > 0 on [0, T ] for some T > 0. Then y˜ ≥ (A − 1)λ˜ y and z˜ ≥ (B − 1)λ˜ z on (0, T ]. On the other hand, for all t ∈ (0, T ], we have φ (t) = p˜ z p−1 z˜ − Aλ˜ y ≥ (p(B − 1) − A)λ˜ zp > 0

282

III. Systems

and

ψ (t) = q y˜q−1 y˜ − Bλ˜ z ≥ (q(A − 1) − B)λ˜ y q > 0.

z p and z˜ (t) ≥ c˜ yq We deduce that φ, ψ > 0 on [0, ∞). It follows that y˜ (t) ≥ c˜ with c = 1 − max(A−1 , B −1 ) > 0. But, as a consequence of Lemma 32.10 below, this guarantees that (˜ y , z˜) cannot exist for all t > 0. This contradiction proves the claim. Let us now show that, for suitable K, ε, η > 0 (independent of λ), y(0) ≥ Kλα/2 guarantees that a := ελα/2 and b := ηλβ/2 satisfy (32.23) for t0 = τ . In view of the last claim, this will prove the lemma. The last two conditions in (32.23) are equivalent to ηp λ

p(q+1) pq−1

p+1

> Aλελ pq−1 = Aελ

p(q+1) pq−1

εq λ

,

q(p+1) pq−1

q+1

> Bληλ pq−1 = Bηλ

q(p+1) pq−1

,

that is η p > Aε and εq > Bη; such η, ε > 0 clearly exist since pq > 1. Due to (32.22), the first two conditions in (32.23) are satisfied if p+1

p+1

ελ pq−1 < C1 Kλ pq−1 ,

q+1

ηλ pq−1 < C1 λ−1 K q λ −1/q 1/q

It thus suffices to choose K > max(C1−1 ε, C1

η

q(p+1) pq−1

q+1

= C1 K q λ pq−1 .

).

Proof of Theorem 32.7. (i) Without loss of generality, we may assume p ≥ q. As mentioned before, we shall prove the assertion under the stronger assumptions p ≥ q ≥ 1 (p > 1) and max(α, β) = α > n. For each λ > 0, let φλ (x) = 2 (4π)−n/2 λn/2 e−λ|x| /4 . We have ∂xi φλ =

−λxi φλ , 2

∂x2i xi φλ =

λ2 x2 i

4

−

λ φλ , 2

hence

∆φλ ≥

−nλ φλ , 2

and Rn φλ = 1. Multiplying the differential equations in (32.1) by φλ , integrating by parts, and using Jensen’s inequality, we obtain

p d nλ uφλ dx = u∆φλ dx + v p φλ dx ≥ − uφλ dx + vφλ dx dt Rn 2 Rn Rn Rn Rn and similarly d dt

Rn

vφλ dx ≥ −

nλ 2

Rn

vφλ dx +

Rn

q uφλ dx .

(the calculations can be justified similarly as in the proof of Theorem 17.1). Therefore, the functions

yλ (t) := u(t)φλ dx and zλ (t) := v(t)φλ dx Rn

Rn


283

˜ := nλ/2. By shifting the time origin, we satisfy system (32.21) with λ replaced by λ 2 may assume u0 ≡ 0. Consequently, since Rn e−λ|x| /4 u0 dx → Rn u0 dx ∈ (0, ∞] as λ → 0, there exists c0 > 0 such that yλ (0) ≥ c0 λn/2 for λ > 0 small. Since ˜ α/2 for λ > 0 small, where K is given by Lemma 32.8. α > n, we have yλ (0) ≥ K λ We then deduce from that lemma that (yλ , zλ ), hence (u, v), cannot exist globally. (ii) We assume p ≥ q ≥ 1, p > 1 and α < n. We look for a supersolution under the form u(x, t) = ε(t + 1)a φ(x, t), v(x, t) = ε(t + 1)b φ(x, t), with a, b, ε > 0 and φ(x, t) = (t + 1)−n/2 ψ(x, t), where ψ(x, t) = e−|x| Using φt − ∆φ = 0 and ψ ≤ 1, we obtain

2

/4(t+1)

.

ut − ∆u − v p = aε(t + 1)a−1 φ − εp (t + 1)bp φp = [a(t + 1)a−bp−1+n(p−1)/2 − εp−1 ψ p−1 ]ε(t + 1)bp−pn/2 ψ ≥ 0 for t ≥ 0, whenever a − bp ≥ 1 − n(p − 1)/2

and

εp−1 ≤ a.

(32.24)

εq−1 ≤ b.

(32.25)

Symmetrically, we have v t − ∆v − uq ≥ 0 whenever b − aq ≥ 1 − n(q − 1)/2

and

p+1 q+1 Choosing a = n2 − pq−1 and b = n2 − pq−1 , the first conditions in (32.24) and (32.25) are satisfied (with equalities) and since a, b > 0 due to max(α, β) = α < n, it suffices to choose ε > 0 small. It then follows from the comparison principle that (u, v) is global if u0 ≤ u(·, 0) and v0 ≤ v(·, 0).

32.4. Blow-up asymptotics As compared with the scalar model problem (15.1), less is known concerning the asymptotic behavior of blowing-up solutions to system (32.1). We shall establish the following theorem concerning type I blow-up rate for monotone solutions in time. A few results concerning other aspects of the blow-up behavior will be mentioned in Remarks 32.12. Theorem 32.9. Consider problem (32.1) with Ω bounded, p, q ≥ 1, pq > 1, and 0 ≤ u0 , v0 ∈ L∞ (Ω). Assume that u, v are nondecreasing in time and (u, v) is nonstationary. Then T := Tmax (u0 , v0 ) < ∞ and we have C1 (T − t)−α/2 ≤ u(t) ∞ ≤ C2 (T − t)−α/2 , C3 (T − t)−β/2 ≤ v(t) ∞ ≤ C4 (T − t)−β/2 ,

(32.26)

284

III. Systems

in (0, T ) for some C1 , C2 , C3 , C4 > 0. This result was proved in [158] and its proof is based on a modification of the maximum principle arguments of [219]. The assumption ut , vt ≥ 0 is guaranteed if, for instance, 0 ≤ u0 , v0 ∈ C0 ∩ C 2 (Ω) and ∆u0 + v0p ≥ 0, ∆v0 + uq0 ≥ 0 (see Remark 52.23(ii)). For the proof, we need the following lemmas concerning the systems of differential inequalities: y (t) ≥ εz p , (32.27) z (t) ≥ εy q , and

y (t) ≤ z p ,

(32.28)

z (t) ≤ y q .

Lemma 32.10. Let p, q, ε > 0 satisfy pq > 1, and let 0 < T ≤ ∞. Assume that 0 ≤ y, z ∈ C 1 (0, T ), (y, z) ≡ (0, 0), and that (y, z) solves (32.27) on (0, T ). Then T < ∞ and there holds y(t) ≤ C1 (T − t)−α/2 ,

z(t) ≤ C1 (T − t)−β/2 ,

0 < t < T,

(32.29)

with C1 = C1 (p, q, ε) > 0. Lemma 32.11. Let p, q > 0 satisfy pq > 1, and let 0 < T < ∞. Assume that 0 ≤ y, z are locally absolutely continuous and nondecreasing on (0, T ), and that (y, z) solves (32.28) a.e. on (0, T ). Assume also that sup(0,T ) (y + z) = ∞ and that (32.29) is satisfied for some C1 > 0. Then there holds y(t) ≥ η(T − t)−α/2 ,

z(t) ≥ η(T − t)−β/2 ,

T − η < t < T,

with η = η(p, q, C1 ) > 0. Proof of Lemma 32.10. We have

t

t

−p−1 −p p y(t) ≥ ε z (s) ds ≥ ε 0

0

s

y q (σ) dσ

p ds =: h(t).

0

Therefore, t p ] = (p + 1) y q (s) ds y q (t) ≥ (p + 1)εq(p+1) h hq = C(hq+1 ) .

(p+1)/p

[(h )

0

Since h(0) = h (0) = 0, it follows that (h )(p+1)/p ≥ Chq+1 . Moreover, due to (y, z) ≡ (0, 0), we may assume h > 0 on (t0 , T ) for some t0 ∈ (0, T ). Putting q+1 > 1, we get [h1−γ ] = −(γ − 1)h h−γ ≤ −C < 0. Integrating over (t, s) γ = p p+1


285

with t0 < t < s < T , we obtain h1−γ (t) ≥ h1−γ (s) + C(s − t) ≥ C(s − t). It follows that T < ∞. By letting s → T , we obtain h(t) ≤ C(T − t)−1/(γ−1) = C(T − t)−α/2 ,

t0 < t < T.

(32.30)

Next, fix t0 < t < T and let τ = (T − t)/4. Since y ≥ 0, we have t+2τ s

h(t + 2τ ) = 0

≥τ

y q (σ) dσ

0 t+τ

y q (σ) dσ

0

p

p ds ≥ τ (τ y q (t))p = τ p+1 y pq (t).

In view of (32.30), we deduce y pq (t) ≤ τ −(p+1) h(t + 2τ ) ≤ Cτ −(p+1) (T − t − 2τ )−(p+1)/(pq−1) = C(T − t)−pq(p+1)/(pq−1) , hence the estimate of y on (t0 , T ). The estimate of z follows symmetrically. Since the constant C is independent of t0 and y = z = 0 in (0, t) if h(t) = 0, the estimates above (obtained in (t0 , T )) remain true in (0, T ). Proof of Lemma 32.11. We first observe that for suitable a, b > 0 (depending on p, q) the functions y(t) := a(T − t)−α/2 ,

z(t) := b(T − t)−β/2

satisfy y (t) = z p (t), z (t) = y q (t) on (0, T ). We deduce that, for each t ∈ (0, T ), either y(t) ≥ y(t) or z(t) ≥ z(t).

(32.31)

(Indeed, if this failed for some t ∈ (0, T ), then we would have y(t) < y(t − η) and z(t) < z(t − η) for some η > 0 so that, by a simple comparison argument, y(s) ≤ y(s − η) and z(s) ≤ z(s − η), t ≤ s < T , contradicting the fact that (y, z) is unbounded on (0, T ).) Assume for contradiction that there exist sequences ηi → 0+ and ti → T such that z(ti ) ≤ ηi (T − ti )−β/2 . Fix K > 1 and put ti := ti − K(T − ti ). Then (32.31), (32.29) and z ≥ 0 guarantee that, for large i, −α/2

a(T −ti )

≤ y(ti ) ≤

y(ti )+

ti

ti

z p (s) ds ≤ C1 (T −ti )−α/2 +Kηip (T −ti )1−p(β/2) .

286

III. Systems

Using 1 − p(β/2) = −α/2 and noting that T − ti = (1 + K)(T − ti ), we get a ≤ C1 (1 + K)−α/2 + Kηip . Letting i → ∞, we get a contradiction for K = K(p, q, a) large enough. Consequently, there exists η = η(p, q) > 0 such that z(t) ≥ η(T − t)−β/2 on [T − η, T ). The estimate for y follows symmetrically. Proof of Theorem 32.9. We first prove the upper estimates. Using the maximum principle in a similar way as in the proof of Theorem 23.5, we obtain ut , vt > 0 in QT and ∂ν ut , ∂ν vt < 0 on ST . Choosing τ ∈ (0, T ) we can find ε > 0 such that ut (x, τ ) ≥ εv p (x, τ ) and vt (x, τ ) ≥ εuq (x, τ ) for all x ∈ Ω. Set f = f (v) := v p , g = g(u) := uq and J := ut − εf , H := vt − εg. Then Jt − ∆J = f vt − εf g + εf |∇v|2 , hence

Jt − ∆J ≥ f H

in Qτ

(32.32)

in Qτ .

(32.33)

where Q := Ω × (τ, T ), and symmetrically τ

Ht − ∆H ≥ g J

Since f (v) and g (u) and nonnegative and locally bounded in Ω × [τ, T ), we may apply the maximum principle (Proposition 52.21) to system (32.32)–(32.33). As J, H ≥ 0 on the parabolic boundary of Qτ , we thus have J, H ≥ 0 in Qτ . Consequently, ut ≥ εv p , vt ≥ εuq in Qτ . Applying Lemma 32.10 with y(t) = u(x, t), z(t) = v(x, t) (for each fixed x ∈ Ω) yields T < ∞ and the upper estimates in (32.26). Let us turn to the lower estimates. We now set y(t) = max u(x, t),

z(t) = max v(x, t).

x∈Ω

x∈Ω

Arguing as in the (alternative) proof of Proposition 23.1, we obtain y ≤ z p and z ≤ y q a.e. in (0, T ). Consequently, the lower estimates in (32.26) are guaranteed by Lemma 32.11. Remarks 32.12. (i) Blow-up rate. The blow-up estimates in Theorem 32.9 were obtained before in [110] under stronger restrictions on p, q and the initial data. On the other hand, when Ω = Rn , (32.26) is valid for all nonglobal nonnegative solutions if p, q > 1 satisfy max(α, β) ≥ n [130]. This remains true for general domains if max(α, β) ≥ n + 1 [199]. The proofs rely on rescaling arguments and Fujita-type theorems (cf. Remark 26.12). In the case Ω = Rn and max(α, β) > n, the upper estimate was proved before in [29] by different arguments based on Moser-type iteration. (ii) Blow-up set. Concerning the blow-up set for problem (32.1), the following has been proved recently in [493]. Let Ω = BR and p, q > 1. Assume that u, v ≥ 0

33. The role of diffusion in blow-up

287

are radially symmetric and decreasing in |x|. If (u, v) blows up in finite time and satisfies the upper blow-up estimates in (32.26), then single-point blow-up occurs at x = 0. If, moreover, ut , vt ≥ 0, then the final blow-up profiles satisfy the lower estimates u(x, T ) ≥ c1 |x|−α , v(x, T ) ≥ c2 |x|−β for |x| small. In the special case p = q > 1 and n = 1, an earlier result on singlepoint blow-up appeared in [216]. On the other hand, fine asymptotic properties of blow-up solutions of (32.1), including a classification of blow-up profiles, have been obtained in [29], [540] when Ω = Rn under the assumption that |p − q| 1. (iii) Nonsimultaneous blow-up. For system (32.1), it is easy to see that blowup is always simultaneous: If T = Tmax < ∞, then both components blow up in the sense that lim sup u(t) ∞ = lim sup v(t) ∞ = ∞. t→T

t→T

Indeed if u, say, were uniformly bounded on QT , then the second equation would yield a uniform bound on v, hence contradicting (32.3). For different systems with a weaker coupling, nonsimultaneous blow-up may occur. For instance, if the nonlinearities in (32.1) are replaced with f (u, v) = ur v p , g(u, v) = v s uq , or with f (u, v) = ur + v p , g(u, v) = v s + uq , then for suitable p, q, r, s > 0 and suitable nonnegative initial data, one component may blow up while the other remains bounded (see [466, pp. 467-472], [432], [494], [458]).

33. The role of diffusion in blow-up In this section, we discuss the different possible effects of adding linear diffusion (and some boundary conditions) to a system of ODE’s. It will turn out that quite opposite effects can be observed: a. In the case of an ODE system whose solutions all exist globally, it can either happen that: • diffusion preserves global existence (for all initial data), or that: • diffusion induces blow-up (for some initial data). b. Consider the case of ODE systems for which (at least some) solutions blow up in finite time. We already know examples where the addition of diffusion (even with homogeneous Dirichlet conditions) will not prevent the blow-up of (some) solutions (cf. Theorem 32.5). Of course, we have encountered in Section 17 a similar situation in the scalar case. We will see that at the opposite, for certain such ODE systems, the addition of diffusion and homogeneous Dirichlet conditions can make all solutions global and bounded (again, a similar example in the scalar case was given in Section 19).

288

III. Systems

All the systems appearing in this section are locally well-posed under the assumptions that will be made on the data (this will be a consequence of Example 51.12). The existence time of the unique, maximal, classical solution is denoted by Tmax or Tmax (u0 , v0 ), and the continuation and regularity properties (32.3) and (32.4) are valid. Also, we only consider nonnegative initial data and solutions. On the other hand, the systems in this section do not satisfy the comparison principle in general, and we shall need to rely on other techniques.

33.1. Diffusion preserving global existence Let us consider the following system ut − a∆u = f (u, v),

x ∈ Ω, t > 0,

vt − b∆v = g(u, v), uν = vν = 0,

x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0,

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

x ∈ Ω,

v(x, 0) = v0 (x),

x ∈ Ω,

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(33.1)

where a, b are positive constants. Here Ω is either a bounded domain or the whole of Rn (in which case the boundary conditions are of course empty), and (u0 , v0 ) ∈ X+ , defined in (32.2). We assume that f, g : [0, ∞)2 → R are C 1 -functions and that they satisfy f (0, v), g(u, 0) ≥ 0, for u, v ≥ 0, (33.2) which ensures that system (33.1) preserves positivity. (Indeed, we may extend f to R2 by f (u, v) = f (|u|, |v|), and (33.2) then implies ut − a∆u ≥ b1 (x, t)u, where b1 = fu (θu, v), θ = θ(x, t) ∈ (0, 1), and similarly for v.) In this subsection, we consider two classes of systems of the form (33.1): systems with dissipation of mass and systems of Gierer-Meinhardt type. Systems with dissipation of mass This class corresponds to nonlinearities satisfying the structure condition f (u, v) + g(u, v) ≤ 0,

for all u, v ≥ 0.

(33.3)

In case Ω is bounded, condition (33.3) guarantees that system (33.1) possesses the so-called mass-dissipation property:

u(x, t) dx + v(x, t) dx. t → M (t) is nonincreasing, where M (t) := Ω

Ω

Indeed, this follows immediately by integrating the differential equations in (33.1) over Ω and using the boundary conditions. This property is natural in the context


289

of chemical or biological applications, where systems of this form arise. If one looks at the corresponding kinetic system without diffusion, i.e. the ODE counterpart of (33.1): ⎫ U = f (U, V ), t > 0, ⎪ ⎪ ⎪ ⎪ ⎬ t > 0, V = g(U, V ), (33.4) ⎪ U (0) = U0 ≥ 0, ⎪ ⎪ ⎪ ⎭ V (0) = V0 ≥ 0, then it is clear that solutions of (33.4) are global since 0 ≤ U (t) + V (t) ≤ U0 + V0 . A central issue is to determine whether or not the mass-dissipation structure condition still guarantees the global existence of solutions for the diffusive system (33.1). In the case of equal diffusions a = b, it is easy to see that the answer is yes. Indeed, the function w = u + v then satisfies wt − a∆w = f + g ≤ 0, so 0 ≤ u + v ≤ u0 ∞ + v0 ∞ by the maximum principle and global existence follows. In the case of different diffusions a = b, a case often encountered in applications, this has long remained open and has motivated a large amount of work, along with related questions (see e.g. the survey article [348] and references therein). It will turn out that the answer is no in general (see Theorem 33.12 and the preceding comments). However, we shall now see that global existence is ensured if we make some additional assumption. An important particular case is that when f ≤ 0, which means that the first substance is absorbed by the reaction (systems with so-called “triangular” structure). Then one immediately obtains a uniform bound for u, since u(x, t) ≤ u0 ∞ ,

x ∈ Ω, t ∈ (0, Tmax)

(33.5)

by the maximum principle. The problem is then reduced to obtaining a uniform estimate of v. A simple case when this can be done is when a > b. This means that the absorbed substance diffuses faster than the other one. The following result for Ω = Rn was proved in [348]. A similar result was obtained in [295] for Ω bounded, but the proof is more delicate. Theorem 33.1. Let Ω = Rn , a > b > 0 and assume f (u, v) ≤ 0 ≤ g(u, v),

for all u, v ≥ 0,

(33.6)

along with (33.2), (33.3). Then, for all (u0 , v0 ) ∈ X+ , the solution of problem (33.1) is global. Moreover, u, v are uniformly bounded in Rn × [0, ∞). The proof is based on a simple comparison property concerning the kernels associated with the operators ∂t − a∆ and ∂t − b∆.

290

III. Systems

Proof. Let us denote by Sλ (t) the semigroup (say, on L∞ (Rn )) corresponding to the operator ∂t − λ∆. We observe that for all 0 ≤ φ ∈ L∞ (Rn ), λ → λn/2 [Sλ (t)φ](x) is nondecreasing for all (x, t).

(33.7)

This follows from the fact that −n/2

[Sλ (t)φ](x) = (4πλt)

Rn

exp[−|y|2 /4λt]φ(x − y) dy.

Denoting

za (t) = −

t

0

Sa (t − s)f (u(s), v(s)) ds,

zb (t) =

0

t

Sb (t − s)g(u(s), v(s)) ds,

we have u(t) + za (t) = Sa (t)u0 hence za (t) ≤ Sa (t)u0 ≤ u0 ∞ . Due to f + g ≤ 0, f ≤ 0 and (33.7), it follows that

zb (t) ≤ −

t

0

Sb (t − s)f (u(s), v(s)) ds ≤ (a/b)n/2 za (t) ≤ (a/b)n/2 u0 ∞ .

Therefore v(t) = Sb (t)v0 + zb (t) ≤ v0 ∞ + (a/b)n/2 u0 ∞ . This along with (33.5) yields the conclusion. Still in the case f ≤ 0 but without assuming a > b, the answer is again positive under a polynomial growth assumption on g: g(u, v) ≤ C(1 + u + v)γ ,

for all u, v ≥ 0 and some γ ≥ 1.

(33.8)

Theorem 33.2. Assume Ω bounded and let a, b > 0, a = b, and γ ≥ 1. Assume (33.2), (33.3), (33.6) and (33.8). Then, for all (u0 , v0 ) ∈ X+ , the solution of problem (33.1) is global. This result was proved in [281]. It can be shown in addition that u, v are uniformly bounded in Ω × [0, ∞). The main ingredient of the proof is the following lemma, which guarantees that whenever f + g ≤ 0, v is controlled by u in Lp for any finite p. The proof is based on a duality argument.


291

Lemma 33.3. Assume Ω bounded, 1 0 and T > 0. There exists C = C(T, p, a, b, Ω) > 0 such that, if u, v ∈ C 2,1 (Ω × (0, T ]) ∩ C(QT ) satisfy (u + v)t − a∆u − b∆v ≤ 0, uν = vν = 0,

x ∈ Ω, 0 < t < T, x ∈ ∂Ω, 0 < t < T,

(33.9)

then there holds v+ Lp (QT ) ≤ C u(·, 0) + v(·, 0) Lp (Ω) + u Lp(QT ) .

(33.10)

Proof. Let q = p . Fix χ ∈ D(QT ), χ ≥ 0, and let ϕ be the unique solution of the problem ⎫ −ϕt − b∆ϕ = χ, x ∈ Ω, 0 < t < T, ⎪ ⎬ x ∈ ∂Ω, 0 < t < T, ϕν = 0, (33.11) ⎪ ⎭ ϕ(x, T ) = 0, x ∈ Ω. We have ϕ ≥ 0 by the maximum principle. Parabolic Lq -estimates (see Remark 48.3(ii)) guarantee ϕt Lq (QT ) + D2 ϕ Lq (QT ) ≤ C χ Lq (QT ) .

(33.12)

Since ϕ(·, T ) = 0, this implies in particular ϕ(·, 0) Lq (Ω) ≤ C χ Lq (QT ) .

(33.13)

Multiplying the inequality in (33.9) by ϕ, integrating by parts, and using the boundary conditions and ϕ(x, T ) = 0, we obtain

0≥

(ut + vt − a∆u − b∆v)ϕ dx dt QT

u(−ϕt − a∆ϕ) + v(−ϕt − b∆ϕ) dx dt QT

− (u(x, 0) + v(x, 0))ϕ(x, 0) dx.

=

Ω

Therefore, by (33.12) and (33.13), we get

v(−ϕt − b∆ϕ)

vχ dx = QT

QT

≤

u(ϕt + a∆ϕ) dx dt +

(u(x, 0) + v(x, 0))ϕ(x, 0) dx ≤ C u Lp(QT ) + u(·, 0) + v(·, 0) Lp (Ω) χ Lq (QT ) . QT

Ω

292

III. Systems

Since χ ≥ 0 is arbitrary in D(QT ), the lemma follows. Proof of Theorem 33.2. Fix r > (n + 2)/2. By (33.5) and Lemma 33.3, we have

T

r (1 + u + v)rγ dx dt ≤ C(T ), g(u, v) Lr (QT ) ≤ C 0

Ω

for all finite T ≤ Tmax . Using the variation-of-constants formula, it follows that

t (t − s)−n/2r g(u(s), v(s)) Lr (Ω) ds v(t) ∞ ≤ Ct−n/2 v0 1 + C 0 t (r−1)/r (t − s)−n/2(r−1) ds g(u, v) Lr (Qt ) ≤ Ct−n/2 v0 1 + C 0

≤ Ct−n/2 v0 1 + C(T )tθ , for all 0 < t < T , with θ = 1 − (n + 2)/2r > 0. This along with (33.5) yields Tmax = ∞. Remark 33.4. The duality argument in the proof of Lemma 33.3 has other applications. For instance, under the assumptions of Theorem 33.2, it yields global existence for the system with inhomogeneous Neumann boundary conditions: ⎫ ut − a∆u = f (u, v), x ∈ Ω, t > 0, ⎪ ⎪ ⎪ ⎪ ⎪ x ∈ Ω, t > 0, vt − b∆v = g(u, v), ⎪ ⎪ ⎪ ⎪ ⎬ x ∈ ∂Ω, t > 0, uν = α1 (t), (33.14) ⎪ x ∈ ∂Ω, t > 0, vν = α2 (t), ⎪ ⎪ ⎪ ⎪ ⎪ x ∈ Ω, u(x, 0) = u0 (x), ⎪ ⎪ ⎪ ⎭ x ∈ Ω, v(x, 0) = v0 (x), where αi are arbitrary smooth functions. This works also in the case of Dirichlet boundary conditions u = α1 (t), v = α2 (t) [349]. The argument can also be used to study the case of nonlinearities of the form g = −f = c(x)up v q , with signchanging c(x) (see [282], [417]). Another system of physical interest, the so-called Brusselator, corresponding to the choices f = −uv 2 + Bv, g = uv 2 − (B + 1)v + A, can also be handled by similar techniques [281]. If f, g do not have a sign, it is still possible to show global existence modulo the additional dissipation condition: λf (u, v) + g(u, v) ≤ 0,

for u, v ≥ 0,

(33.15)

with sufficiently large λ > 1, assuming also that f, g have at most polynomial (upper) growth: f (u, v), g(u, v) ≤ C(1 + u + v)γ ,

for all u, v ≥ 0 and some γ ≥ 1.

(33.16)


293

Theorem 33.5. Assume Ω bounded and let a, b > 0, a = b, and γ ≥ 1. Assume (33.2), (33.3), (33.15), (33.16), with /

(a + b)2 λ ≥ λ0 (a, b, n, γ) := 4ab

0m−1 ≥ 1,

(33.17)

where m in the smallest integer such that m > n(γ − 1)/2. Then, for all (u0 , v0 ) ∈ X+ , the solution of problem (33.1) is global. Moreover, u, v are uniformly bounded in Ω × [0, ∞). Theorem 33.5 is from [309]. A typical example (without sign condition) to which it applies is given by f = up v q − ur v s , g = ur v s − λup v q for any p, q, r, s ≥ 1 and λ > 1 large enough (depending on p, q, r, s, n, a, b). Interestingly, it will turn out that the largeness assumption on λ is in some sense necessary (see Theorem 33.12 and the preceding paragraphs). Remarks 33.6. (i) The proof of Theorem 33.5 is based on a suitable Lyapunov functional, cf. Lemma 33.7 below, whereas Theorem 33.2 was based on a duality argument. Note that (33.15) is satisfied in particular if f + g ≤ 0 and f ≤ 0. Therefore, Theorem 33.5 is more general than Theorem 33.2. However, the duality argument has other applications (cf. Remark 33.4) which do not seem to be tractable by the Lyapunov functional approach. Also, in the case of homogeneous Dirichlet boundary conditions, results similar to Theorem 33.5 have been obtained in [468] by duality techniques, but the largeness condition on λ is not explicit. (ii) Note that if γ < (n + 2)/n, then λ0 = 1 in (33.17) (with m = 1), so that condition (33.15) disappears. For earlier results related to Theorems 33.2 and 33.5, see [11], [12] (based on Moser’s iteration), [460] (based on bootstrap) and [350] (based on a Lyapunov functional). On the other hand, the global existence result of Theorem 33.2 has been extended to f, g satisfying some exponential growth conditions, for instance for g = −f = uev . For this we refer to [270], [52] (relying on suitable Lyapunov functionals) and to [274], [294] (based on a delicate analysis using parabolic BMO estimates). The key of the proof of Theorem 33.5 is the following Lyapunov functional. Lemma 33.7. Assume Ω bounded, a, b > 0, a+b K≥ √ 2 ab

(33.18)

and let m be any positive integer. Assume that f, g satisfy (33.2), (33.3), (33.15), (33.16), for some λ ≥ K 2(m−1) . Let (u, v) be a solution of (33.1) and let

L(t) = Ω

Hm (u(x, t), v(x, t)) dx,

294

III. Systems

where Hm (u, v) =

m

2

i Cm Ki

−i i m−i

uv

i Cm =

,

i=0

m! . i!(m − i)!

Then L (t) ≤ 0 on the interval (0, Tmax ). Proof. Set w = Kv and L1 (t) = K m L(t). We have K m Hm (u, v) =

m

2

i Cm K i ui wm−i

i=0

and w solves wt − b∆w = Kg(u, v). Differentiating L1 with respect to t yields L1 (t) =

m m−1 2 2 i i iCm (m − i)Cm = K i ui−1 wm−i ut + K i ui wm−i−1 wt dx Ω

m Ω i=1

+

i=1

i iCm K i ui−1 wm−i (a∆u + f (u, v)) dx

m Ω i=1

=

m Ω

+

i=0

2

i−1 (i−1)2 i−1 m−i (m − i + 1)Cm (b∆w + Kg(u, v)) dx K u w

2 i i−1 (i−1)2 i−1 m−i aiCm K i ui−1 wm−i ∆u + b(m − i + 1)Cm K u w ∆w dx

i=1

m Ω

2

i iCm K i ui−1 wm−i f (u, v)

i=1 i−1 (i−1) K + (m − i + 1)Cm

2

+1 i−1

u

wm−i g(u, v) dx

=: I + J. By using Green’s formula we obtain

2 2 A |∇u| + B∇u∇w + C |∇w| dx, I=− Ω

where A=

m

2

i ai(i − 1)Cm K i ui−2 wm−i ,

i=2

B=

m−1 i=1

2

i ai(m−i)Cm K i ui−1 wm−i−1 +

m i=2

2

i−1 (i−1) i−2 m−i b(i−1)(m−i+1)Cm K u w ,


295

and C=

m−1

2

i−1 (i−1) i−1 m−i−1 b(m − i)(m − i + 1)Cm K u w .

i=1

Using the fact that i−1 i = mCm−1 , iCm

(m −

i i)Cm

=

i = 1, . . . , m, i = 0, . . . , m − 1,

i mCm−1 ,

we get A = am(m − 1)

m

2

i−2 Cm−2 K i ui−2 wm−i ,

i=2

B = am(m − 1)

m−1

2

i−1 Cm−2 K i ui−1 wm−i−1

i=1

+ bm(m − 1)

m

2

i−2 Cm−2 K (i−1) ui−2 wm−i

i=2

=: B1 + B2 , and C = bm(m − 1)

m−1

2

i−1 Cm−2 K (i−1) ui−1 wm−i−1 .

i=1

Putting j = i − 2, we have A = am(m − 1)

m−2

2

j Cm−2 K (j+2) uj wm−j−2 ,

j=0

B2 = bm(m − 1)

m−2

2

j Cm−2 K (j+1) uj wm−j−2 ,

j=0

and putting j = i − 1, we get B1 = am(m − 1)

m−2

2

j Cm−2 K (j+1) uj wm−j−2 ,

j=0

C = bm(m − 1)

m−2 j=0

2

j Cm−2 K j uj wm−j−2 .

(33.19)

296

III. Systems

Therefore, I = − m(m − 1)

m−2

j Cm−2

j=0

× aK

(j+2)2

uj wm−j−2 Ω

2 2 |∇u|2 + (a + b) K (j+1) ∇u∇w + bK j |∇w|2 dx.

The quadratic forms (with respect to ∇u and ∇w) are positive since 2 2 2 2 2 (a + b)K (j+1) − 4abK j K (j+2) = K 2j +4j+2 (a + b)2 − 4abK 2 ≤ 0 for j = 0, 1, . . . , m − 2, due to (33.18). It follows that I ≤ 0. On the other hand, by (33.19), we have J =m

m

i−1 Cm−1

i=1

2

K i f (u, v) + K (i−1)

2

+1

g(u, v) ui−1 wm−i dx.

Ω

Since (33.3) and (33.15) imply µf + g ≤ 0 for all µ ∈ [1, λ], we obtain 2

K i f (u, v) + K (i−1)

2

+1

g(u, v) = K (i−1)

2

+1

(K 2(i−1) f (u, v) + g(u, v)) ≤ 0

for i = 1, . . . , m, hence J ≤ 0. Proof of Theorem 33.5. In Lemma 33.7, we take K = 2(m−1)

a+b √ 2 ab

and m as in the

and we deduce from Lemma 33.7 statement of the theorem. Then λ0 = K that u(t) and v(t) are bounded in Lm (Ω). Since m > n(γ − 1)/2, by similar arguments as in the proof of Theorem 16.4, one deduces that they are bounded in L∞ (Ω). (Alternatively one could use modifications of arguments in the proof of (15.2) in Theorem 15.2.) In particular, this implies Tmax = ∞ and the theorem is proved. Remarks 33.8. (i) Simple modifications of the proofs of Theorems 33.2 and 33.5, show that global existence (without boundedness) is still true if the conditions f +g and/or λf + g ≤ 0 are replaced by f + g and/or λf + g ≤ C(1 + u + v). (ii) Under the assumptions of Theorem 33.5, if u, v ≥ 0 and (u, v) solves (33.1) for t ∈ (0, T ), with the boundary conditions replaced by uν ≤ 0,

vν ≤ 0,

then u, v are uniformly bounded in Ω × [0, T ). This follows from a simple modification of the proof of Lemma 33.7 and Theorem 33.5.


297

Systems of Gierer-Meinhardt type We consider the system ut − a∆u = −µ1 u + up /v q + σ,

x ∈ Ω, t > 0,

vt − b∆v = −µ2 v + ur /v s , uν = vν = 0,

x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0,

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

x ∈ Ω,

v(x, 0) = v0 (x),

x ∈ Ω,

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(33.20)

where p > 1, q, r, s ≥ 0, a, b > 0, µ1 , µ2 , σ ≥ 0, and u0 , v0 ∈ C(Ω) with u0 , v0 > 0. By the maximum principle, we immediately obtain the lower bounds u(x, t) ≥ min u0 e−µ1 t , v(x, t) ≥ min v0 e−µ2 t , x ∈ Ω, 0 < t < Tmax . Ω

Ω

(33.21)

These bounds imply in particular that if Tmax < ∞, then lim sup u(t) ∞ = ∞. t→Tmax

System (33.20) (for instance with p = r = 2, q = s = 4, n = 1) arises in a biological model of pattern formation, due to [242]. Here u and v represent the concentrations of an activator and an inhibitor, respectively. The peaks of high concentration of activator give the positional information for the development of a structure, e.g. a tentacle in the polyp Hydra. An essentially complete answer to the global existence/nonexistence question for system (33.20) is provided by the following theorem from [333] (see also [292], [398] for related results). Earlier partial results of global existence had been proved in [460], [352]. We note also that a large amount of literature has been devoted to the singular perturbation problem associated with the study of “spike-layers” (stationary solutions developing a finite number of concentration peaks as a → 0). We refer for this to the surveys [391], [392]. Theorem 33.9. Assume Ω bounded. (i) Assume that q p−1 < min ,1 . r s+1

(33.22)

Then, for all u0 , v0 ∈ C(Ω) with u0 , v0 > 0, the solution (u, v) of problem (33.20) is global. If in addition µ1 , µ2 , σ > 0, then u, v are uniformly bounded in Ω×[0, ∞). (ii) Assume that q p−1 p−1 > min ,1 ,

= 1. (33.23) r s+1 r Then there exist space-independent initial data u0 , v0 > 0 such that the solution (u, v) = (u(t), v(t)) of problem (33.20) satisfies Tmax < ∞.

298

III. Systems

Remark 33.10. Diffusion preserving global existence. It follows from Theorem 33.9 that, for any p, q, r, s such that the solutions of the ODE system are all global, the addition of diffusion preserves global existence (except perhaps for p − 1 = r and for the equality case in (33.22), which seem to be open). No example of blow-up seems to be known for system (33.20) beside the space-independent solutions. The proof of assertion (i) relies on multiplier arguments and on the following consequence of Young’s inequality. Lemma 33.11. Assume that p, q, r, s satisfy (33.22). For all η, α, β > 0, there exist C = C(η, α, β) > 0 and θ = θ(α) ∈ (0, 1) such that α

xα θ xp−1+α xr+α ≤ β s+1+β + C β , q+β y y y

x ≥ 0,

y ≥ η.

(33.24)

Proof. Let x > 0 and y ≥ η. Inequality (33.24) is equivalent to α Write α

y β 1−θ xr xp−1 ≤ β + C . yq y s+1 xα

xr (p−1)/r xr γ xp−1 (p−1)(s+1)/r−q = y = C β s+1 y −m , yq y s+1 y

where γ = (p − 1)/r < 1 and m = q − (p − 1)(s + 1)/r > 0. For each 0 < ε < min(m/(s + 1), 1 − γ), using y ≥ η and Young’s inequality, we obtain α

xr γ+ε xr γ+ε y β rε/α xp−1 −m+(s+1)ε −rε = C β y x ≤ C β s+1 yq y s+1 y xα xr y β rε/(1−γ−ε)α ≤ β s+1 + C α , y x

and (33.24) follows by taking ε sufficiently small.

Proof of Theorem 33.9(i). We shall only prove global existence and uniform boundedness in the case µ1 , µ2 , σ > 0. Global existence in the general case can be shown by simple modifications (using the lower bound (33.21) on finite time intervals instead of (33.25) below). Step 1. Lower estimates. We claim that there exists c1 > 0 such that u, v ≥ c1 ,

x ∈ Ω,

0 < t < Tmax .

(33.25)

Since u satisfies ut − a∆u > 0 on {u < σ/µ1 } (along with homogeneous Neumann conditions), the maximum principle implies u ≥ δ := min(σ/µ1 , minΩ u0 ) > 0 in


299

Ω × [0, ∞). Then, v satisfies vt − b∆v > 0 on {v < (δ r /µ2 )1/(s+1) } and the lower bound for v follows similarly. Step 2. Bound for a quotient. We claim that, for all large α, β > 0, the function

α u φ = φα,β (t) := dx β Ω v satisfies sup t∈(0,Tmax )

φ(t) < ∞.

(33.26)

By (33.20), we have

α−1 ut u uα vt α dx − β φ (t) = vβ v β+1 Ω

α−1

u uα up ur =α a∆u − µ dx − β b∆v − µ dx. u + σ + v + 1 2 β β+1 vq vs Ω v Ω v Using Green’s formula, we deduce that

p−1+α u ur+α uα−1 α q+β − β s+1+β + ασ β dx φ (t) = (−αµ1 + βµ2 )φ + v v v Ω

uα−2 uα + −aα(α − 1) β |∇u|2 − bβ(β + 1) β+2 |∇v|2 v v Ω α−1 u + (a + b)αβ β+1 ∇u · ∇v dx. v

(33.27)

The last integrand can be rewritten as uα−2 Q := −aα(α − 1)v 2 |∇u|2 − bβ(β + 1)u2 |∇v|2 + (a + b)αβ(v∇u) · (u∇v) β+2 . v Consequently we have Q ≤ 0, provided we assume 4ab αβ ≤ , (α − 1)(β + 1) (a + b)2

(33.28)

which guarantees that the discriminant (a + b)2 (αβ)2 − 4abαβ(α − 1)(β + 1) of the quadratic form Q is nonpositive. Owing to (33.25), we also have uα (α−1)/α uα−1 uα (α−1)/α −β/α = v ≤C β . β β v v v Using (33.25), Lemma 33.11, (33.29) and H¨ older’s inequality, we obtain

α θ

α−1 u u dx + ασ dx φ (t) ≤ (−αµ1 + βµ2 )φ + C β β Ω v Ω v (α−1)/α

≤ (−αµ1 + βµ2 )φ + C(φ + φ θ

)

(33.29)

(33.30)

300

III. Systems

for some θ ∈ (0, 1). Now assume α ≥ 2 max(1, µ2 /µ1 ) and β ≤ 2ab/(a + b)2 ≤ 1. Then (33.28) is satisfied and, since −αµ1 + βµ2 < 0, the function f (Y ) := (−αµ1 + βµ2 )Y + C(Y θ + Y (α−1)/α ) has a largest positive zero, say Y = K. Since, by (33.30), φ (t) < 0 whenever φ(t) > K, we deduce easily that supt∈(0,Tmax ) φ(t) ≤ max(φ(0), K), hence (33.26). Since v is bounded below, it is clear that (33.26) remains true if we enlarge β. The claim is proved. Step 3. L∞ -bounds. By (33.26), we have up v −q and ur v −s ∈ L∞ ((0, Tmax ), L (Ω)) for all m < ∞. By a simple argument using the variation-of-constants formula and the Lp -Lq -estimate (Proposition 48.4), we deduce that u and v are uniformly bounded and that Tmax = ∞. m

Proof of Theorem 33.9(ii). We consider space-independent solutions of (33.20), i.e. solutions of the corresponding ODE system without diffusion. For spatially homogeneous initial data u0 , v0 ≥ 1 to be determined later, we assume for contradiction that Tmax (u0 , v0 ) > 1. In what follows, all the positive constants C, c, . . . are independent of u0 , v0 . For fixed α, β > 0, let λ = αµ1 − βµ2 and w(t) = uα /v β . By direct calculation using (33.20) (cf. the first line of (33.27)), we have w + λw = α

up−1+α ur+α uα−1 − β + ασ . v q+β v s+1+β vβ

(33.31)

We consider two cases separately. Case 1: p − 1 > r. We apply (33.31) with α = 1. Taking β large enough and using (33.21) and v0 ≥ 1, we have for all t ∈ [0, 1], α ur+1 p/(r+1) k ur+1 α up = v ≥ β s+1+β − C, q+β s+1+β 2v 2 v v p − q − β > 0, and where k = (s + 1 + β) r+1

u p α up α u p m = v ≥ c , 2 v q+β 2 vβ vβ where m = (p − 1)β − q > 0. It follows that w ≥ cwp − λw − C. Taking w(0) large enough, this implies blow-up of u before t = 1; a contradiction.


301

Case 2: p − 1 < r, (p − 1)(s + 1) > qr. We claim that that there exist constants C1 , C2 > 0 such that, if ≥ C1 v0s+1−q , (33.32) ur−p+1 0 then ur−p+1 ≥ C2 v s+1−q ,

0 < t ≤ 1.

(33.33)

To prove this, letting z = eλt w and applying (33.31) with α=r−p+1>0

and

β = s + 1 − q > 0,

we see that, for all t ∈ [0, 1], z (t) ≤ 0 =⇒

α ur+α up−1+α −1 −|λ| u ≥ αβ =⇒ z(t) ≥ e ≥ e−|λ| αβ −1 =: C1 . v s+1+β v q+β vβ

Consequently we have z(t) ≥ min(C1 , w(0)) on [0, 1], and the claim follows with C2 = e−|λ| C1 . Now assume (33.32). Using the first equation in (33.20) and (33.33), we deduce that up u + µ1 u ≥ q ≥ cup−q(r−p+1)/(s+1−q) = cuγ , 0 < t ≤ 1, v where γ = 1 + (p−1)(s+1)−qr > 1. But, taking u0 larger, this implies blow-up of u s+1−q before t = 1; a contradiction.

33.2. Diffusion inducing blow-up In this subsection we show that certain parabolic systems admit a blowing-up solution for some particular initial data, although the corresponding system of ODE’s has only global bounded solutions. We shall give three different examples, each of them involving a different method. We first consider systems of the form ut − a∆u = f (u, v),

x ∈ Ω, t > 0,

vt − b∆v = g(u, v),

x ∈ Ω, t > 0,

uν = α1 (t),

x ∈ ∂Ω, t > 0,

vν = α2 (t),

x ∈ ∂Ω, t > 0,

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

x ∈ Ω,

v(x, 0) = v0 (x),

x ∈ Ω,

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(33.34)

under the mass-dissipation structure condition f + g ≤ 0. Sufficient conditions ensuring global existence for such systems were studied in the previous subsection.

302

III. Systems

Recall that, under a polynomial growth assumption on the nonlinearities, global existence of nonnegative solutions is true if f + g ≤ 0 and if either: • λf + g ≤ 0 with λ > 1 large enough, under homogeneous boundary conditions (or more generally uν , vν ≤ 0); or • f ≤ 0, with arbitrary (smooth) functions αi (or also under Dirichlet-Dirichlet boundary conditions). The following result [417] shows that in case of unequal diffusions, the condition f + g ≤ 0 is not sufficient to ensure global existence, even if the additional dissipation property λf + g ≤ 0 is also satisfied (with some λ > 1 not too large) and f and g have polynomial growth. We point out that the example below involves nonnegative solutions and functions αi ≤ 0, so that the condition f + g ≤ 0 still guarantees the mass-dissipation property (d/dt) Ω (u(t) + v(t)) dx ≤ 0. The result has to be compared with Theorem 33.5, which is therefore in a sense optimal. Theorem 33.12. Let Ω = B1 ⊂ Rn . There exist constants a, b, T > 0, a = b, functions f, g ∈ C ∞ (R2 , R) and α1 , α2 ∈ C ∞ ([0, T ], R), satisfying α1 , α2 ≤ 0, f + g ≤ 0,

λf + g ≤ 0,

for all u, v ≥ 0 and some λ > 1,

f (u, v), g(u, v) ≤ C(1 + u + v)γ ,

for all u, v ≥ 0 and some γ ≥ 1,

(33.35) (33.36)

and such that for some C ∞ initial data u0 , v0 ≥ 0, system (33.34) admits a classical nonnegative solution (u, v) on (0, T ), with lim u(0, t) = lim v(0, t) = ∞.

t→T

t→T

Moreover, u and v blow up only at x = 0 as t → T . The proof is based on the construction of an explicit solution, of self-similar form, and involves some relatively heavy numerical computations (still doable by hand, but the construction was initially carried out with the help of the formal computation software Maple). Sketch of proof (for n = 10). An explicit solution is searched under the form u(x, t) =

A(T − t) + B|x|2 , (T − t + |x|2 )5/4

v(x, t) =

C(T − t) + D|x|2 , (T − t + |x|2 )5/4

with constants A, B, C, D > 0 to be determined. Note that this is actually a selfsimilar solution, since it can be rewritten under the form u = (T − t)−1/4 U (y), with U (y) =

v = (T − t)−1/4 V (y),

A + B|y|2 , (1 + |y|2 )5/4

V (y) =

y = x(T − t)−1/2 ,

C + D|y|2 . (1 + |y|2 )5/4


303

A direct calculation yields ut − a∆u = (T − t)−5/4

A1 + B1 |y|2 + C1 |y|4 (1 + |y|2 )(5/4)+2

vt − b∆v = (T − t)−5/4

A2 + B2 |y|2 + C2 |y|4 , (1 + |y|2 )(5/4)+2

and

where A1 , B1 , C1 and A2 , B2 , C2 are computed in terms of n, a, A, B and n, b, C, D, respectively. As for the functions f, g, one looks for polynomials, homogeneous and of total degree 5, of the form

f (u, v) =

5

λi u

5−i i

v,

i=0

g(u, v) =

5

µi u5−i v i .

i=0

The PDE’s in system (33.34) then become equivalent to 5

λi (A + B|y|2 )5−i (C + D|y|2 )i = (1 + |y|2 )3 (A1 + B1 |y|2 + C1 |y|4 )

(33.37)

µi (A + B|y|2 )5−i (C + D|y|2 )i = (1 + |y|2 )3 (A2 + B2 |y|2 + C2 |y|4 ).

(33.38)

i=0

and 5 i=0

Choosing n = 10 (other choices are possible) and a = 1, b = 1/10, A = 1/25, B = 1, C = 11/2, D = 1/10, it turns out that it is possible to adjust the constants λi , µi in such a way that (33.37), (33.38) be satisfied, with moreover λi + µi < 0, so that 5 λf + g = (λλi + µi )u5−i v i ≤ 0 i=0

for λ equal or close to 1. Finally, for r = |x| = 1, we compute α1 (t) = uν (x, t) = ur (1, t) =

(4B − 5A)(T − t) − B 2(T − t + 1)9/4

and an analogous expression for α2 (t) (with C, D in place of A, B). Taking T > 0 small enough, it follows that αi (t) ≤ 0 on [0, T ].

304

III. Systems

Remark 33.13. (i) Other examples of blow-up. An example similar to that of Theorem 33.12 is also constructed in [416] for nonlinearities f (x, t, u, v) = c1 (x, t)up v q , g(x, t, u, v) = c2 (x, t)up v q , with n = 1, p, q > 1 and sign-changing functions ci such that c1 +c2 ≤ 0. However, it remains an open problem to construct similar examples of blow-up in the case of homogeneous boundary conditions. On the other hand, beyond the special examples, there is a lack of general blow-up criteria for such systems, as well as of a description of possible singularities, in comparison with the scalar problems studied in Chapter II. (ii) Global weak solutions. Consider problem (33.1) under the assumptions (33.2), (u0 , v0 ) ∈ X+ , f + g ≤ 0 and λf + g ≤ 0 for some λ > 1. Theorem 33.12 suggests that this problem need not admit a global classical solution. However, it was shown in [415] that there exists a global weak solution in some appropriate L1 sense. Also, it is to be noted that in the example constructed in Theorem 33.12, it is possible to extend the solution across the blow-up time to a global weak solution. Still for mass-dissipative systems of the form (33.34), here with f ≤ 0 ≤ g and f +g = 0, the next result [64], [65] shows that mixed Dirichlet-Neumann conditions can lead to finite-time blow-up, even for equal diffusions. Namely we consider the one-dimensional problem ut − uxx = −uv p ,

x ∈ (−1, 1), t > 0,

vt − vxx = uv p ,

x ∈ (−1, 1), t > 0,

u(±1, t) = 1,

t > 0,

vx (±1, t) = 0,

t > 0,

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

x ∈ (−1, 1),

v(x, 0) = v0 (x),

x ∈ (−1, 1).

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(33.39)

Theorem 33.14. Assume p > 2. Let u0 , v0 ∈ C 2 ([−1, 1]) be even and satisfy u0 (±1) = 1, (v0 )x (±1) = 0, 0 < u0 ≤ 1,

v0 > 0

(u0 + v0 )x ,

(v0 )x ≥ 0

(u0 + v0 )xx ≥ 0,

in [−1, 1], in [0, 1],

(v0 )xx + u0 v0p ≥ 0

in [−1, 1].

(33.40)

Then the solution of (33.39) satisfies Tmax < ∞ and limt→Tmax v(1, t) = ∞. Remark 33.15. It has been shown in [349] that solutions of (33.39) exist globally if 0 0 (this follows from a


305

simple modification of the proof of Theorem 33.2 and Lemma 33.3). Analogues of Theorem 33.14 in higher dimension can be found in [65]. The proof of Theorem 33.14 is based on monotonicity and subsolution arguments — to obtain pointwise lower bounds for u, v — and on the use of a simple differential inequality. Proof of Theorem 33.14. Step 1. Absence of steady states. We easily verify that (33.39) has no nonnegative stationary solution except (U, V ) ≡ (1, 0). Indeed, if (U, V ) is a nonnegative stationary solution, then Vxx = −U V p ≤ 0 and Vx (±1) = 0, hence Vx ≡ 0, so that V ≡ 0 (since U ≡ 0). But then Uxx ≡ 0, hence U ≡ 1 due to U (±1) = 1. Step 2. Monotonicity properties. From the assumptions on u0 , v0 , the functions u, v are symmetric in x, and we have 0 ≤ u ≤ 1. We next observe that (w, v), with w := u + v, solves the equivalent system wt − wxx = 0,

x ∈ (−1, 1), t > 0,

vt − vxx = (w − v)v p ,

x ∈ (−1, 1), t > 0,

w(±1, t) = 1 + v(±1, t),

t > 0,

vx (±1, t) = 0,

t > 0,

w(x, 0) = (u0 + v0 )(x),

x ∈ (−1, 1),

v(x, 0) = v0 (x),

x ∈ (−1, 1).

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(33.41)

Now we claim that wt , vt ≥ 0 in [−1, 1] × [0, Tmax ) and wx , vx ≥ 0 in [0, 1] × [0, Tmax ).

(33.42)

Let (y, z) = (wt , vt ). By continuous dependence, it suffices to prove that y, z ≥ 0 when the second inequality in (33.40) is assumed to be strict. By continuity, we have z > 0 in [−1, 1] for t small. Assume for contradiction that there is a first t0 > 0 such that z(x0 , t0 ) = 0 for some x0 ∈ [−1, 1], and denote Q0 := [0, 1] × [0, t0 ] and S0 := {−1, 1} × [0, t0 ]. Since yt − yxx = 0

in Q0 ,

y(·, 0) ≥ 0 and y = z ≥ 0 in S0 , the maximum principle implies y ≥ 0 in Q0 . But we then have zt − zxx = v p y + b(x, t)z ≥ b(x, t)z in Q0 , with b = (pw − (p + 1)v)v p−1 , and z ≥ 0 in Q0 . Therefore, x0 = ±1 by the strong maximum principle (since z(·, 0) ≡ 0). But this is impossible in view of Hopf’s lemma, since zx = 0 on S0 . We have thus proved the first part of (33.42).

306

III. Systems

Next, we have wxx = wt ≥ 0 and wx (0, t) = 0. Therefore wx ≥ 0 in [0, 1] × [0, Tmax ). By differentiating the second equation of (33.41) in x, we see that h := vx satisfies ht − hxx = v p wx + b(x, t)h ≥ b(x, t)h

in [0, 1] × [0, Tmax ),

with h(0, t) = h(1, t) = 0. The second part of (33.42) then follows from the maximum principle. We now denote M (t) :=

max

[−1,1]×[0,t]

v = v(1, t)

and we assume for contradiction that Tmax = ∞. Step 3. Unboundedness of v. We claim that M (t) → ∞,

t → ∞.

Otherwise u, v are bounded (recall that u ≤ 1) and since w, v are nondecreasing in time by Step 2, there would exist bounded functions (W, V ) such that W (x) = lim w(x, t), t→∞

V (x) = lim v(x, t). t→∞

But the monotonicity of w, v guarantees that (W, V ) is a stationary solution of (33.41) (see Proposition 53.8). Letting U := W − V , (U, V ) is thus a stationary solution of (33.39), hence V ≡ 0 by Step 1. This is a contradiction, since V ≥ v0 > 0. Step 4. Pointwise lower bounds for u and v and differential inequality. For fixed p/2 T > 0, put M = M (T ), δ = min[−1,1] u0 ≤ 1 and u(x, t) = δeM (x−1) . Then u satisfies ⎫ ut − uxx + M p u = 0 ≤ ut − uxx + M p u, x ∈ (−1, 1), 0 < t < T, ⎪ ⎬ 0 < t < T, u(±1, t) ≤ 1, ⎪ ⎭ x ∈ (−1, 1). u(x, 0) ≤ u0 (x), It follows from the maximum principle that u ≤ u in [−1, 1] × [0, T ]. We deduce that, for all t large, u(x, t) ≥ η := δ/e > 0, with

x0 (t) ≤ x ≤ 1,

(33.43)

x0 (t) := 1 − M −p/2 (t) ∈ (−1, 1).

On the other hand, we have vxx = vt − uv p ≥ −v p ≥ −M p (t). Consequently, by Taylor expansion, for some ξ ∈ (x, 1), we have v(x, t) = v(1, t) + vx (1, t)(x − 1) + vxx (ξ, t)

(x − 1)2 (x − 1)2 ≥ M (t) − M p (t) . 2 2


307

Therefore, for t large, v(x, t) ≥ M (t)/2,

x0 (t) ≤ x ≤ 1.

(33.44)

Integrating the second equation in (33.39) over (−1, 1) and using (33.43), (33.44), 1 we see that φ(t) := −1 v(t) dx satisfies φ (t) =

1

−1

uv p (t) dx ≥ (1 − x0 (t))η(M/2)p (t) = CM p/2 (t).

Since on the other hand φ(t) ≤ 2M (t), we obtain φ (t) ≥ Cφp/2 (t) for t large. Since p > 2, this contradicts Tmax = ∞. In our last example, we consider a system without the structure f + g ≤ 0, but with homogeneous Neumann conditions (unlike in the previous two examples) and equal diffusions, and for which blowing-up solutions exist for some particular initial data, although the corresponding system of ODE’s has only global bounded solutions. Namely, we consider the system ut − duxx = h(u, v)(1 + u) − δu,

x ∈ (−1, 1), t > 0,

vt − dvxx = −h(u, v)(1 + v) − δv, ux = vx = 0,

x ∈ (−1, 1), t > 0, x = ±1, t > 0,

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

x ∈ (−1, 1),

v(x, 0) = v0 (x),

x ∈ (−1, 1),

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(33.45)

with d > 0 and δ ≥ 0. Here the function h : [0, ∞)2 → R, of class C 1 , is assumed to satisfy: h(u, v) = −h(v, u), (33.46) h(u, 0) = h(0, v) = 0, h(u, v) ≥ 0,

u ≥ v ≥ 0,

h(u, v) ≥ k(u − v)γ ,

u ≥ v ≥ 1,

(33.47) (33.48) (33.49)

for some k, γ > 0. These assumptions apply for instance to the function h(u, v) = (uv)m |u − v|p (u − v), for any m ≥ 1, p ≥ 0. As for the initial data (u0 , v0 ), we assume u0 , v0 ∈ C 1 ([−1, 1]),

u0 , v0 > 0 in [−1, 1],

(u0 )x , (v0 )x = 0, x = ±1, (33.50)

308

III. Systems

v0 (x) = u0 (−x), u0 ≥ v0

in [0, 1].

(33.51) (33.52)

Due to (33.47) and the maximum principle, we have u, v > 0 in [−1, 1] × [0, T ). Let us first observe that for the corresponding system of ODE’s ⎫ y = h(y, z)(1 + y) − δy, ⎪ ⎬ z = −h(y, z)(1 + z) − δz, ⎪ ⎭ y(0) = y0 ≥ 0, z(0) = z0 ≥ 0, all solutions are global, and that they decay exponentially to (0, 0) if moreover δ > 0. This follows immediately from the fact that (y + z + yz) = y (1 + z)+ (1 + y)z = −δ(y + z + 2yz). The following result essentially comes from [527], where it was given for h(u, v) = uv(u − v). We here present the simplified proof from [492] (with more general nonlinearities). Theorem 33.16. Assume (33.46)–(33.52). There exists C = C(d, δ, k, γ) > 0 such that, if

1

1 (u0 − v0 ) sin(πx/2) dx ≥ C and log(1 + u0 ) dx ≥ C, (33.53) 0

−1

then T := Tmax (u0 , v0 ) < ∞. The idea of the proof is to derive differential inequalities for two different functionals on some interval (0, T0 ). Integrating them yields upper estimates for the measures of two complementary subsets of (0, T0 ), whose sum is less than T0 , leading to a contradiction with existence up to t = T0 . Remarks 33.17. Unequal diffusions. (i) The fact that the diffusion coefficients are equal in the two equations is used crucially in the proof (via the sign and symmetry properties of the two components). An example of a system with blow-up induced by unequal diffusions and homogeneous Neumann conditions can be found in [383]. The proof therein is more delicate. On the other hand, it is unknown whether or not Theorem 33.16 remains true in the case of homogeneous Dirichlet boundary conditions. As for the asymptotic blow-up behavior of solutions of (33.45), this is an essentially open problem. (ii) In the fundamental article [516], it had been shown that unequal diffusions can destabilize an otherwise stable constant equilibrium (global existence being however preserved). Other results on diffusion-induced blow-up can be found in [133], [387], [259]. Proof of Theorem 33.16. First note that, since (˜ u, v˜) := (v(−x, t), u(−x, t)) solves the same system due to (33.46), (33.51), we have by uniqueness: v(x, t) = u(−x, t).

(33.54)


309

Next, we put m(t) = min u(x, t) = x∈[−1,1]

min v(x, t),

M (t) = max u(x, t) = max v(x, t),

x∈[−1,1]

x∈[−1,1]

and we claim that

x∈[−1,1]

0 < t < min(T, δ −1 ).

M (t) ≥ 1,

(33.55)

Indeed, by adding the equations for u and v, we get (u + v)t − d(u + v)xx = h(u, v)(u − v) − δ(u + v). Integrating and using the boundary conditions, we deduce that d dt

1 −1

(u + v) dx ≥ −δ

1

(u + v) dx, −1

hence M (t) ≥

1 1 max (u + v) ≥ 2 [−1,1] 4

1

−1

(u + v) dx ≥

1 −δt e 4

1 −1

(u0 + v0 ) dx,

and (33.55) follows by taking C ≥ 2e in (33.53). Now we derive two differential inequalities for the auxiliary functions φ and ψ, defined as follows:

1

1 (δ+λd)t wϕ dx, ψ(t) := z dx + 2δ(t − T1 ), 0 ≤ t < T, φ(t) := e −1

−1

where w = u − v,

z = log

1 + u 2

,

and λ := π 2 /4,

ϕ(x) = (π/4) sin(πx/2),

T1 = min δ −1 , (γ(δ + λd))−1 .

We also put T0 = min(T, T1 ) and we set E = {t ∈ (0, T0 ) : m(t) ≥ 1},

F = (0, T0 ) \ E.

Claim 1. We have φ(t) > 0

and

φ (t) ≥ ke−1 φ1+γ χE ,

0 < t < T0 .

(33.56)

To show this, we subtract the equations for u and v to obtain wt − dwxx = h(u, v)(2 + u + v) − δw.

(33.57)

310

III. Systems

Note in particular that since w(0, t) = wx (1, t) = 0, (33.48), (33.52) and the maximum principle imply w ≥ 0 on [0, 1] × (0, T ). Therefore, φ ≥ 0 on (0, T ) and h(u, v)ϕ ≥ 0 on [−1, 1] × (0, T ) by (33.54) and (33.46). Multiplying by ϕ and integrating by parts yields

1

1 1 d 1 wϕ dx = d wx ϕ − ϕx w −1 + h(u, v)(2 + u + v)ϕ dx − (δ + λd) wϕ dx. dt −1 −1 −1 Since h(u, v)(2 + u + v)ϕ ≥ h(u, v)(u − v)|ϕ|, by using (33.46), (33.48), (33.49) and Jensen’s inequality we get

1 |u − v|1+γ |ϕ| dx ≥ e−γ(δ+λd)t k φ1+γ ≥ e−1 k φ1+γ , t ∈ E, φ (t) ≥ e(δ+λd)t k −1

and φ (t) ≥ 0 if t ∈ E. This, along with φ(0) ≥ C > 0 (cf. (33.53)), proves the claim. Claim 2. We have d ψ(t) > 0 and ψ (t) ≥ ψ 2 χF , 0 < t < T0 . (33.58) 8 By a simple computation, we get u . zt − dzxx = h(u, v) + d(zx )2 − δ 1+u 1 Since h(u(x, t), v(x, t)) is odd due to (33.46) and (33.54), we have −1 h(u, v) dx = 0 hence,

1 d 1 z dx + 2δ ≥ d (zx )2 dx ≥ 0. (33.59) ψ (t) = dt −1 −1 1 Since ψ(0) = −1 log(1 + u0 ) dx − 2 log 2 − 2δT1 > 0 by taking C > 2 log 2 + 2δT1 in (33.53), it follows in particular that ψ > 0. Now, if t ∈ F , i.e. m(t) < 1, then (33.55) implies the existence of ξ(t) ∈ [−1, 1] such that u(ξ(t), t) = 1, hence z(ξ(t), t) = 0. Therefore

1 1 2 1 2 2 |z| dx ≤ 4(max |z(x, t)|) ≤ 4 |zx | dx ≤ 8 (zx )2 dx. (33.60) −1

1

−1

−1

Since −1 z dx ≥ ψ on [0, T0 ) by the definition of ψ, (33.58) follows from (33.59) and (33.60). To complete the proof of Theorem 1, we integrate (33.56) and (33.58), to obtain

T0

T0 d φ φ−1−γ ds ≥ γke−1 |E|, ψ −1 (0) ≥ ψ ψ −2 ds ≥ |F |. φ−γ (0) ≥ γ 8 0 0 We deduce that min(T, δ −1 , [γ(δ + λd)]−1 ) = T0 = |E| + |F | ≤ (γk)−1 eφ−γ (0) + 8d−1 ψ −1 (0). We conclude that if φ(0) and ψ(0) ≥ C(d, δ, k, γ) > 0 large enough, then T ≤ (γk)−1 eφ−γ (0) + 8d−1 ψ −1 (0) < ∞.


311

33.3. Diffusion eliminating blow-up In this subsection we consider the following system ut − d1 ∆u = f (u − v),

x ∈ Ω, t > 0,

vt − d2 ∆v = f (u − v) − v,

x ∈ Ω, t > 0,

u = v = 0,

x ∈ ∂Ω, t > 0,

⎫ ⎪ ⎬ (33.61)

⎪ ⎭

where Ω ⊂ Rn is bounded, d1 , d2 > 0, d1 −d2 > 1/λ1 , f (w) = |w|p−1 w, 1 < p < pS , and the initial data belong to Z := H01 ×H01 (Ω). We also consider the corresponding system of ODE’s Ut = f (U − V ), (33.62) Vt = f (U − V ) − V. The following theorem is due to [197]. Theorem 33.18. Let the assumptions above be satisfied. Then: (i) there exists a solution of (33.62) which blows up in finite time; (ii) for all (u0 , v0 ) ∈ Z, the solution of (33.61) is global and converges to the trivial solution (0, 0) in Z as t → ∞. Proof. (i) Denote W := U −V and assume V (0) > 1, W (0) > We will prove that the solution (U, V ) blows up in finite time.

p+1 2

V (0)2

1/(p+1)

.

Since V = f (W ) − V and W = V , the functions W, V remain positive. Multiplying the equation W + W = W p by W we see that the function E(t) := 1 1 2 p+1 is nonincreasing, hence E(t) ≤ E(0) < 0. In particular, 2 (W (t)) − p+1 W (t) W p+1 >

p+1 (W )2 ≥ (W )2 = V 2 , 2

hence V = W p −V > V 2p/(p+1) −V . Since V (0) > 1, the last differential inequality guarantees blow-up of V . (ii) Similarly as in Example 51.27 we get that problem (33.61) is well-posed in Y := Lp+1 ×Lp+1 (Ω). In addition, if the initial data (u0 , v0 ) ∈ Z and the solution is bounded in Y , then this solution is global and its trajectory is relatively compact in Z, see Example 51.38. Finally, ut , vt ∈ C 1 ((0, ∞), L2 (Ω)) ∩ C((0, ∞), H 2 ∩ H01 (Ω)). Fix (u0 , v0 ) ∈ Z and set w := u − v. Then (w, v) solves the problem wt − d1 ∆w = (d1 − d2 )∆v + v, vt − d2 ∆v = −v + |w|p−1 w, w = v = 0,

x ∈ Ω, t > 0,

⎫ ⎪ ⎬

x ∈ Ω, t > 0, ⎪ ⎭ x ∈ ∂Ω, t > 0.

(33.63)

312

III. Systems

Let −A denote the Dirichlet Laplacian in L2 (Ω). Due to d1 − d2 > 1/λ1 , (d1 − d2 )A − 1 is a positive self-adjoint operator and its inverse K := ((d1 − d2 )A − 1)−1 is compact, positive and commutes with both A and A1/2 . The first equation in (33.63) can be rewritten as v = K(d1 ∆w − wt ). Now the second equation in (33.63) guarantees K(d1 ∆wt − wtt ) = d2 K∆(d1 ∆w − wt ) − K(d1 ∆w − wt ) + |w|p−1 w.

(33.64).

Define the norm ϕ −1 := K 1/2 ϕ L2 (Ω)

for ϕ ∈ L2 (Ω).

Multiplying (33.64) by wt and integrating in x over Ω, we have 1 d −d1 A1/2 wt 2−1 − wt 2−1 2 dt d1 d2 d d1 d = Aw 2−1 + d2 A1/2 wt 2−1 + A1/2 w 2−1 2 dt 2 dt

1 d + wt 2−1 + |w|p+1 dx, p + 1 dt Ω which implies d L(t) = −(d1 + d2 ) A1/2 wt 2−1 − wt 2−1 ≤ 0, (33.65) dt where

1 d1 d2 d1 1 Aw 2−1 + A1/2 w 2−1 + L(t) := wt 2−1 + |w|p+1 dx. 2 2 2 p+1 Ω Consequently, L is a Lyapunov functional (see Appendix G) and the function w(t) stays bounded in Lp+1 (Ω). Now the second equation in (33.63) and a simple estimate based on the variation-of-constants formula shows that v(t) stays bounded in W 2−ε,(p+1)/p (Ω) for any ε > 0 and t ≥ t0 > 0. Since this space is embedded in Lp+1 (Ω) for ε small due to p < pS , we see that the solution (u(t), v(t)) remains bounded in Y . Consequently, it exists globally and is relatively compact in Z. Consequently, the ω-limit set ω(u0 , v0 ) of this solution is a compact nonempty connected and invariant set in Z (see Proposition 53.3). Fix (˜ u0 , v˜0 ) ∈ ω(u0 , v0 ) and let (˜ u, v˜) be the solution of problem (33.61) with initial data (˜ u0 , v˜0 ). Set w ˜ = u ˜ − v˜. Since the Lyapunov functional L is constant on ω(u0 , v0 ), (33.65) guarantees w ˜t = 0, hence w ˜tt = 0. Now multiplying (33.64) (with w replaced by w) ˜ with w ˜ and denoting by (·, ·) the scalar product in L2 (Ω) we obtain

1/2 1/2 ˜ Aw) ˜ + d1 (KA w, ˜ A w) ˜ + |w| ˜ p+1 dx = 0, d1 d2 (KAw, Ω

which implies w ˜ = 0. Now the first equation in (33.63) shows v˜ = 0, hence u ˜ = 0. This concludes the proof.

Chapter IV

Equations with Gradient Terms 34. Introduction In Chapter IV, we consider problems with nonlinearities depending on u and its space derivatives: ⎫ ut − ∆u = F (u, ∇u), x ∈ Ω, t > 0, ⎪ ⎬ u = 0, x ∈ ∂Ω, t > 0, (34.1) ⎪ ⎭ x ∈ Ω. u(x, 0) = u0 (x), Here F = F (u, ξ) : R × Rn → R is a C 1 -function (except for problem (34.4) with 1 < q < 2, see below). In Sections 35–39, we consider perturbations of the model problem (15.1) by terms involving first-order derivatives: ⎫ ut − ∆u = up + g(u, ∇u), x ∈ Ω, t > 0, ⎪ ⎬ u = 0, x ∈ ∂Ω, t > 0, (34.2) ⎪ ⎭ x ∈ Ω. u(x, 0) = u0 (x), We will only consider nonnegative solutions of (34.2) (but up can be interpreted as |u|p−1 u for definiteness). In many results, g might depend also on x, t, but we restrict to (34.2) for simplicity. Typical examples that we shall pay a particular attention to, are given by: ⎫ ut − ∆u = up − µ|∇u|q , x ∈ Ω, t > 0, ⎪ ⎬ u = 0, x ∈ ∂Ω, t > 0, (34.3) ⎪ ⎭ x ∈ Ω, u(x, 0) = u0 (x), with p, q > 1, µ > 0 (dissipative gradient term) and ut − ∆u = up − a · ∇(uq ), u = 0, u(x, 0) = u0 (x),

x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0, x ∈ Ω,

⎫ ⎪ ⎬ ⎪ ⎭

(34.4)

with p > 1, q ≥ 1, a ∈ Rn (convective gradient term). A motivation for studying (34.3), (34.4) is to investigate the effect of a dissipative or convective gradient term

314

IV. Equations with Gradient Terms

on global existence or nonexistence of solutions, and on their asymptotic behavior, in finite or infinite time. We refer to [484], [490] for surveys on equations of the form (34.2). It will turn out that problems of this form reveal a number of interesting, qualitatively new phenomena, in comparison with the unperturbed model problem, such as new critical exponents, or changes in the parameters involved in the asymptotic blow-up behavior. In Sections 40 and 41, we consider problems whose essential superlinear character comes from the gradient term. A simple model case is given by: ⎫ ut − ∆u = |∇u|p , x ∈ Ω, t > 0, ⎪ ⎬ u = 0, x ∈ ∂Ω, t > 0, (34.5) ⎪ ⎭ x ∈ Ω, u(x, 0) = u0 (x), with p > 1. Problem (34.5) is often referred to as a viscous Hamilton-Jacobi equation. Also, (34.5) is related with the Kardar-Parisi-Zhang equation in the physical theory of growth and roughening of surfaces (see [252], [72] for details and references). Note that it is one of the simplest examples of a parabolic PDE with a nonlinearity depending on the first-order spatial derivatives, and can thus be considered as an analogue of the model problem (15.1). The case where the nonlinearity is replaced by um |∇u|p will also be studied. We will see that these equations exhibit phenomena qualitatively different from (15.1), such as (boundary or interior) gradient blow-up.

35. Well-posedness and gradient bounds Throughout Chapter IV we denote X := {u ∈ BC 1 (Ω) : u = 0 on ∂Ω},

(35.1)

equipped with the norm w X := w ∞ + ∇w ∞ , and X+ := {w ∈ X : w ≥ 0}. Problem (34.1), with F of class C 1 , is locally wellposed in X (see Remark 51.11). As for problem (34.4), with Ω bounded or Ω = Rn for simplicity, it is also locally well-posed in X for all q ≥ 1 (see Example 51.15 and Proposition 51.16). In particular, if Tmax = Tmax (u0 ) < ∞, then

lim

t→Tmax

u(t) X = ∞.

(35.2)

Moreover the solution enjoys the regularity property u ∈ BC 2,1 (Ω × [t1 , t2 ]),

0 < t1 < t2 < Tmax (u0 )

(35.3)

35. Well-posedness and gradient bounds

315

(cf. the corresponding proof in Example 51.9). Furthermore, problem (34.1) admits a comparison principle, cf. Propositions 52.6, 52.10 and Remarks 52.11. In the case of problem (34.4), see Proposition 52.16. Those results will be frequently used throughout this chapter without explicit reference. In particular, if F (0, 0) ≥ 0 and u0 ∈ X+ , then we have u ≥ 0. On the other hand, in the case of problem (34.3) in a ball or in Rn , if u0 is radial (resp. radial nonincreasing), then u enjoys the same property, as a consequence of Proposition 52.17. In the case of problems (34.3)–(34.5) well-posedness may actually hold true in some larger spaces, but this question is not our main concern in this chapter. However, in view of the study of the large time behavior, it will be very useful to know weaker continuation properties than (35.2). In the case of the general problem (34.1) this requires some structure assumptions on F . A rather sharp result in that direction is given by the following theorem. Here, for k > 0, we write F ≤ O(|ξ|k ) if F (u, ξ) ≤ C(u)(|ξ|k + 1) and F ≤ o(|ξ|k ) if for all ε > 0, F (u, ξ) ≤ ε|ξ|k + Cε (u), where C(u) and Cε (u) remain bounded on bounded sets of u ≥ 0. Theorem 35.1. Consider problem (34.1) with F (0, 0) ≥ 0 and F = f + g, where f, g ∈ C 1 satisfy |f | ≤ O(|ξ|2 ),

|fξ | ≤ O(|ξ|),

|fu | ≤ o(|ξ|2 )

(35.4)

and g(0, ξ) ≤ 0,

gu ≤ 0,

ξ·

∂ g ≤ 0, ∂ξ |ξ|

for all u ≥ 0, ξ ∈ Rn \ {0}. (35.5)

Let u0 ∈ X+ . If Tmax (u0 ) < ∞, then lim

t→Tmax (u0 )

u(t) ∞ = ∞.

Theorem 35.1 is a consequence of the following Bernstein-type gradient estimate from [59], which provides a pointwise a priori estimate of ∇u assuming a bound on u. Proposition 35.2. Let T > 0 and assume that F = f + g, where f, g ∈ C 1 satisfy (35.4) and (35.5). Let u ∈ C 2,1 (QT ) ∩ C(QT ), with ∇u ∈ C(QT ) ∩ L∞ (QT ), be a solution of (34.1), such that 0 ≤ u ≤ M in QT

and

|∇u0 | ≤ M in Ω

for some 0 < M < ∞. Then there holds |∇u| ≤ C = C(M, T, F, Ω) in QT .

(35.6)

316


Remarks 35.3. (a) Theorem 35.1 reduces the proof of global existence to the derivation of a uniform estimate of u (on bounded time intervals). It also guarantees that finite-time blow-up, in case it occurs, takes place in the L∞ -norm. (b) Assumptions (35.4), (35.5) in Theorem 35.1 can be viewed as one-sided quadratic growth restrictions. Theorem 35.1 applies for instance with F (u, ∇u) = f (u) + a|∇u|m − λur |∇u|q with f of class C 1 , 1 < m ≤ 2, r ≥ 1 or r = 0, q > 1 and λ ≥ 0. This includes in particular problem (34.3) for any p, q > 1 and µ > 0. In the special case of problem (34.3), the result was proved before in [433], [434], [498] by different techniques. (c) As for problem (34.4), Theorem 35.1 applies when q ≥ 2, but not if 1 < q < 2, since the nonlinearity is then not Lipschitz. However, it is proved in Proposition 51.16 (by different arguments) that an L∞ -estimate is sufficient to prevent blow-up of solutions. (d) The growth and sign assumptions in (35.4), (35.5) are essentially optimal. Indeed, the conclusion of Theorem 35.1 fails for problem (34.5) if p > 2 (cf. Section 40; see also Section 41 and [485] for other examples). The other assumptions on F can be slightly weakened. For instance, it is enough to assume F to be C 1 for ξ large. (e) For earlier results under two-sided quadratic growth conditions on F , see e.g. [319], [469]. Note that when g = 0, the nonnegativity of u0 and the assumption F (0, 0) ≥ 0 are not needed. Like in [319], [469], the proof of Proposition 35.2 relies on the classical Bernstein technique, which consists in applying the maximum principle to the function ∂v/∂xi (or to |∇v|2 ), where u = φ(v). Gradient estimates can be obtained by various other techniques. Approaches based on elaborate test-function arguments are used in [320, Theorem V.4.1 and Lemma VI.3.1], where a two-sided quadratic growth assumption is made on F (but no assumption on the derivatives Fu , Fξ ), and in [88]. If |F | ≤ O(|ξ|m ) with m < 2, results of this kind can be obtained via the variation-of-constants formula, or derived from well-posedness results in L∞ (cf. Example 51.30, and see also [10] and [361, Lemma 5.1]). For related results in the radial case under (different) one-sided quadratic growth assumptions, see [512]. The technique used there is still different, based on Kruzhkov’s idea of adding a new space variable. Results concerning sign-changing solutions under one-sided quadratic growth assumptions can also be found in [59], [512]. In view of the proof of Proposition 35.2, we start with a preliminary result (under weaker assumptions) which provides control of the gradient on the boundary. The proof is based on a barrier argument (cf. [320, Lemma VI.3.1]). Lemma 35.4. Assume that F (u, ξ) ≤ O(|ξ|2 ). Let T, M > 0 and let u ∈ C 2,1 (QT ) ∩ C 1,0 (QT ) be a solution of (34.1) satisfying (35.6). Then there holds |∇u| ≤ C = C(M, F, Ω) on ST .

35. Well-posedness and gradient bounds

317

Proof. Let U be the solution of −∆U = 1,

1 < |x| < 2,

U = 0,

|x| = 1,

U = 1,

|x| = 2.

⎫ ⎪ ⎬ ⎪ ⎭

It is easily checked that 0 < U (x) < 1

and

c1 (|x| − 1) ≤ U (x) ≤ c2 (|x| − 1),

1 < |x| < 2. (35.7)

Let x0 ∈ ∂Ω. Since Ω is uniformly smooth, there exists ρ0 ∈ (0, 1) depending only on Ω (independent of x0 ) with the following property: For any ρ ∈ (0, ρ0 ], there exists y = y(ρ) ∈ Rn such that B(y, ρ) ∩ Ω = {x0 }. Next put , V (x) = β −1 log 1 + eβM U x−y ρ

ρ ≤ |x − y| ≤ 2ρ,

with β ≥ 1, ρ ∈ (0, ρ0 ]. (Observe that, up to affine changes of variables, this is just the usual Hopf-Cole exponential transformation U = eβV .) ' × (0, T ], where 'T = Ω We want to compare u and V in the set Q ' = {x ∈ Ω : |x − y| < 2ρ}, Ω for suitably small ρ and y = y(ρ). Due to 0 ≤ V ≤ M + 1 and F (u, ξ) ≤ O(|ξ|2 ), a simple calculation shows that −∆V ≥ β|∇V |2 + (2ρ2 β)−1 ≥ F (V, ∇V ),

ρ < |x − y| < 2ρ,

(35.8)

by taking β ≥ 1 large and then ρ ∈ (0, ρ0 ] small (depending only on F and M ). On the other hand, using (35.6), (35.7) and imposing in addition ρ ≤ c1 /2M β, we have β −1 eβM U x−y U x−y ρ ρ ' ≥ V (x) ≥ ≥ M (|x − y| − ρ) ≥ u0 (x), x ∈ Ω, 2β 1 + eβM U x−y ρ and V (x) > M ≥ u(x, t) for x ∈ Ω ∩ {|x − y| = 2ρ}. In view of (35.8), and since ' ⊂ (Ω ∩ {|x − y| = 2ρ}) ∪ ∂Ω, we may then apply the comparison principle in ∂Ω ' QT to deduce that u(x, t) ≤ V (x) ≤ β −1 eβM U

x−y ρ

≤ c2 (βρ)−1 eβM (|x − y| − ρ),

'T , (x, t) ∈ Q

where we also used (35.7). Due to u(x0 , t) = V (x0 ) = 0, it follows that |∇u(x0 , t)| = −

∂u ∂V (x0 , t) ≤ − (x0 , t) ≤ c2 (βρ)−1 eβM , ∂ν ∂ν

0 < t < T.

318


Proof of Proposition 35.2. Consider a function φ of class C 3 on some compact interval J, with φ > 0 and φ(J) ⊃ [0, M ], and a constant K > 0 (φ and K will be specified later on). Let h ∈ Rn , with |h| = 1. Given a solution u satisfying the assumptions of the proposition, we set v := φ−1 (u),

w := ∂h v = h · ∇v,

z(x, t) := e−Kt w.

We want to apply the maximum principle to the function z. Step 1. Derivation of the equation for z. We have F (u, ∇u) = ut − ∆u = φ (v)(vt − ∆v) − φ (v)|∇v|2 , hence vt − ∆v =

F (φ(v), φ (v)∇v) φ (v) + |∇v|2 . φ (v) φ (v)

Note that z ∈ C(QT ) ∩ L∞ (QT ). Since F is C 1 , by differentiating the equation for 2,1;q (QT ) for all finite v in the direction h and using Remark 48.3(i), we get z ∈ Wloc q. In addition, direct computation yields wt − ∆w = a ˜(x, t) w + b(x, t) · ∇w

a.e. in QT ,

with a ˜ = Fu +

1 φ 2 φ ξ · F − F + 2 |ξ| ξ φ φ 2 φ

and b = Fξ + 2

φ ξ, φ 2

where F and its derivatives are evaluated at u = u(x, t), ξ = ∇u(x, t), while φ and its derivatives are evaluated at v(x, t). Setting a = a ˜ − K, we obtain zt − ∆z = a(x, t) z + b(x, t) · ∇z

a.e. in QT .

(35.9)

Step 2. Construction of a function φ such that a ≤ 0. Since ξ · gξ − g ≤ 0, we look for a function φ such that φ ≥ 0. We take

s φ(s) = eM exp(−e−λσ ) dσ, s ∈ J := [0, 1], 0

where λ > 0 will be chosen below. For s ∈ J, we compute φ = eM exp(−e−λs ),

φ = λe−λs φ ,

φ φ

= −λ2 e−λs .

Note that M ≤ φ ≤ eM , s ∈ J. In particular, we have [0, M ] ⊂ φ(J). By (35.4), (35.5), there exist a0 , a1 > 0 and, for each η > 0, there exists Cη > 0, such that g ∂ 2 Fu ≤ η|ξ|2 + Cη , ξ · Fξ − F = ξ · fξ − f + |ξ|ξ · ∂ξ |ξ| ≤ a0 |ξ| + a1 ,

36. Perturbations of the model problem: blow-up and global existence

319

for 0 ≤ u ≤ M , ξ ∈ Rn . Take λ = 2a0 eM , η = 2a20 e−λ , K ≥ Cη + λa1 /M . Using gu ≤ 0, 0 ≤ u(x, t) ≤ M and 0 ≤ v(x, t) ≤ 1, it follows that, for all (x, t) ∈ QT , λe−λv λ ξ · Fξ − F − |ξ|2 − K φ (v) φ (v) −λv λe λ (a0 − )|ξ|2 + a1 − K ≤ η|ξ|2 + Cη + φ (v) eM λa1 − K ≤ 0. ≤ (η − 2a20 e−λ )|ξ|2 + Cη + M

a(x, t) = Fu +

Step 3. Conclusion. Due to (35.6), Lemma 35.4, and φ (v) ≥ M , we have z ≤ C = C(F, Ω, M ) on PT . Applying the maximum principle to equation (35.9) and using a ≤ 0, we deduce that z ≤ C in QT . Getting back to ∂h u = eKt φ (v)z, and since h was arbitrary, the proposition follows. Proof of Theorem 35.1. Assume for contradiction that T := Tmax < ∞ and lim inf t→T u(t) ∞ < M for some M ∈ (0, ∞). By (35.4), (35.5), there exists K > 0 such that F (u, ξ) ≤ K(|ξ|2 + 1),

0 ≤ u ≤ M + 1, ξ ∈ Rn .

1 , T ) such that u(t0 ) ∞ ≤ M and let u(x, t) := M + K(t − t0 ) for Pick t0 ∈ [T − K (x, t) ∈ Q, where Q := Ω × (t0 , T ). For (x, t) ∈ Q, we have 0 ≤ u(x, t) ≤ M + 1, hence ut − ∆u − F (u, ∇u) = K − F (M + K(t − t0 ), 0) ≥ 0.

By the comparison principle, we deduce that 0 ≤ u ≤ u ≤ M + 1 in Q. Due to Proposition 35.2 it follows that supt∈(0,T ) u(t) X < ∞: a contradiction.

36. Perturbations of the model problem: blow-up and global existence In this section, we discuss the conditions on the perturbation terms which imply or prevent blow-up. We start with a simple criterion for equation (34.4) in bounded domains, which is based on a modification of the eigenfunction method (see Theorem 17.1). The idea of the proof is from [215], [330]. Theorem 36.1. Consider problem (34.4) with Ω bounded, p > 1, q ≥ 1, and u0 ∈ X+ . (i) Assume p > q and set m = p/(p − q). If Ω u0 ϕm 1 dx > C1 = C1 (Ω, p, q, a) > 0, then Tmax (u0 ) < ∞.

320


(ii) Assume q ≥ p. Then Tmax (u0 ) = ∞ and supt≥0 u(t) ∞ < ∞. Proof. (i) Denote y = y(t) := Ω u(t)ϕm 1 dx. Multiplying the differential equation in (34.4) with ϕm yields, for 0 < t < T := Tmax (u0 ), 1

m m p m q ut ϕ1 dx = ϕ1 ∆u dx + u ϕ1 dx + (a · ∇(ϕm y = 1 ))u dx. Ω

Ω

We claim that

Ω

Ω

Ω

ϕm 1 ∆u dx ≥ −mλ1

Ω

uϕm 1 dx.

(36.1)

2 m Since ϕm and observe that 1 ∈ C (Ω) (when 1 < m < 2), we consider (ϕ1 + ε)

∆(ϕ1 + ε)m = m(ϕ1 + ε)m−1 ∆ϕ1 + m(m − 1)(ϕ1 + ε)m−2 |∇ϕ1 |2 ≥ −mλ1 (ϕ1 + ε)m−1 ϕ1 . Integrating by parts, we obtain

(ϕ1 + ε)m ∆u dx = u∆(ϕ1 + ε)m dx + (ϕ1 + ε)m ∂ν u dσ Ω Ω ∂Ω

m−1 m ≥ −mλ1 u(ϕ1 + ε) ϕ1 dx + ε ∂ν u dσ Ω

∂Ω

and (36.1) follows upon letting ε → 0. Now by H¨ older’s inequality we have

q/p m−1 q q m p ))u dx ≤ m|a| |∇ϕ |ϕ u dx ≤ C ϕ u dx , (a · ∇(ϕm 1 1 1 1 Ω

Ω

Ω

for some C = C(Ω, p, q, a) > 0. Combining this with Jensen’s inequality, we obtain

1/p

q/p 1 p m ˜ y ≥ up ϕm dx − C u ϕ dx − C up ϕm ≥ y p − C, 1 1 1 dx 2 Ω Ω Ω ˜ for some C˜ = C(Ω, p, q, a) > 0. We infer that u cannot exist globally whenever 1/p ˜ y(0) > (2C) . (ii) Without loss of generality, we may assume that a = |a|e1 and that Ω ⊂ {x ∈ Rn : 0 < x1 < L} for some L > 0. We seek for a (stationary) supersolution of (34.4) of the form v(x) = Keαx1 , for arbitrarily large K > 0 (to guarantee K ≥ u0 ∞ ) and some α > 0. The condition to ensure this is thus −α2 Keαx1 ≥ K p eαpx1 − |a|αqK q eαqx1 ,

0 < x1 < L,

which is satisfied if αq|a|K q−p eα(q−p)x1 ≥ 1 + α2 K 1−p eα(1−p)x1 ,

0 < x1 < L.


321

Since q ≥ p > 1, it is thus sufficient that αq|a|K q−p ≥ 1 + α2 K 1−p . This is true for α = 2/q|a| and all large K > 1. It then follows from the comparison principle in Proposition 52.16, that 0 ≤ u(x, t) ≤ v(x) ≤ KeαL in Ω, as long as u(t) exists. By Proposition 51.16, this implies global existence. We now turn to problem (34.3) (in bounded and unbounded domains). We begin with a result from [497] which shows that finite-time blow-up occurs for large initial data when p > q. Theorem 36.2. Consider problem (34.3) with p > q > 1, µ > 0. Let u0 = λφ, with φ ∈ X+ , φ ≡ 0, λ > 0. If λ is sufficiently large, then Tmax (u0 ) < ∞. Remarks 36.3. (i) For problem (34.3) the eigenfunction method does not seem to apply, and the proof of Theorem 36.2 relies on a different technique, based on self-similar blowing-up subsolutions. For earlier results in that direction (and other methods), see for instance [129], [304], [433]. (ii) When p > q, by Young’s inequality, we have |a · ∇(uq )| ≤ q|a|uq−1 |∇u| ≤ 12 up + µ|∇u|m ,

m = p/(p − q + 1) < p,

for some µ = µ(a, p, q) > 0, so that any solution of (34.4) is a supersolution of ut − ∆u = 12 up − µ|∇u|m . Consequently, Theorem 36.2 implies blow-up of the solution of (34.4) for large initial data. However, the criterion in Theorem 36.1(i) is more precise. (iii) Blow-up for slow decay initial data. For problems (34.3) with p > q ≥ 2p/(p + 1), and (34.4) with q = (p + 1)/2, there are blow-up results for slow decay initial data in Ω = Rn , similar to those known for the model problem (15.1). In fact, the conclusion of Theorem 17.12 remains valid in this case, with a different constant on the RHS of (17.14) [497]. The proof is based on Theorem 36.2, rescaling and comparison arguments. Such results extend to more general unbounded domains (containing a cone or a paraboloid); see [497], [461]. Proof of Theorem 36.2. We seek a (self-similar) subsolution of (34.3) of the form: % $ 1 |x| v(x, t) = , t0 ≤ t < 1/ε, V (1 − εt)k (1 − εt)m where V is defined by V (y) = 1 +

A y2 − , 2 2A

y ≥ 0.

1/2 , Here A, k, m, t0 , ε > 0 (with t0 < 1/ε) are to be determined. Set R = A(2+A) so that V (R) = 0. Note that v(x, t) > 0 if and only if (x, t) ∈ D, where

D := (x, t) : t0 ≤ t < 1/ε, |x| < R(1 − εt)m ,

322


and that v is smooth in D. We will verify that P v := vt − ∆v − v p + µ|∇v|q ≤ 0 in D. We compute, setting y = |x|/(1 − εt)m for convenience: Pv =

n−1 ε(kV (y) + myV (y)) V (y) + y V (y) V p (y) |V (y)|q − − +µ . k+1 k+2m kp (1 − εt) (1 − εt) (1 − εt) (1 − εt)(k+m)q

The function V obviously satisfies 1 ≤ V (y) ≤ 1 + A/2, −1 ≤ V (y) ≤ 0,

0 ≤ V (y) ≤ 1, −R/A ≤ V (y) ≤ −1,

V (y) + (n − 1)V (y)/y = −n/A,

for 0 ≤ y ≤ A, for A ≤ y ≤ R, for 0 < y < R.

We first choose k=

1 , p−1

0 < m < min

1 2

,

p−q , q(p − 1)

so that kp = k + 1 > k + 2m and k + 1 > (k + m)q, and next we choose: A > k/m,

ε
0. Therefore, for t0 close to 1/ε and λ > 0 large enough, we have u0 ≥ v(·, t0 ) in B(0, R(1 − εt0 )m ), hence in Ω. Moreover, we have


323

v ≤ 0 on ∂Ω × (t0 , 1/ε). If Tmax (u0 ) ≥ 1/ε − t0 , it follows from the comparison principle that u(x, t − t0 ) ≥ v(x, t) in D. Since v(0, t) → ∞ as t → 1/ε, we conclude that Tmax (u0 ) ≤ 1/ε − t0 < ∞. The next result from [498], [481] shows in particular that the blow-up condition p > q in Theorem 36.2 is optimal for bounded domains (see [185], [434] for earlier results in that direction). However, for general unbounded domains, the issue depends in a crucial way on the geometry of the domain, through the notion of inradius ρ(Ω) (cf. Section 19 and Appendix D). Theorem 36.4. Consider problem (34.3) with q ≥ p > 1, µ > 0. (i) Assume ρ(Ω) < ∞. Then for all u0 ∈ X+ , there holds Tmax (u0 ) = ∞ and sup u(t) ∞ < ∞. t≥0

Assume in addition that u0 ∈ W01,r (Ω) for some finite r > n max(1, q − 1). There exist µ0 , λ > 0 (depending only on Ω, p, q, r) such that, if µ ≥ µ0 , then u(t) s ≤ C(u0 ) e−λt ,

t ≥ 0,

r ≤ s ≤ ∞.

(36.2)

(ii) Assume ρ(Ω) = ∞. Then there exists u0 ∈ X+ , such that either Tmax (u0 ) < ∞,

or

Tmax (u0 ) = ∞ and

lim u(t) ∞ = ∞.

t→∞

Furthermore, u0 can be taken in W01,r (Ω) for r large. We start with assertion (i). The proof of global existence and boundedness is based on comparison arguments. The idea is to construct a stationary supersolution v in the exterior of a ball of small radius ε, which is radial and whose minimum is larger than u0 ∞ . The solution u is thus dominated by all the translates of v, centered at points y such that B(y, ε) ⊂ Ωc (these supersolutions play the role of a barrier). Since ρ(Ω) < ∞ and Ω is uniformly regular, any point x of Ω is at bounded distance of such a point y. This guarantees a uniform bound for u, hence global existence in view of the gradient estimates in Section 35. The decay will be proved by a multiplier argument, using multiplication by a power of u and the Poincaré inequalities (which are valid due to ρ(Ω) < ∞). Remark 36.5. Although the comparison function v below is unbounded, v and ∇v are bounded on the set {(x, t) ∈ Ω × [0, T ] : u > v} for each T < Tmax (u0 ), due to u ∈ L∞ (QT ). Consequently the comparison principle can be applied in view of Remark 52.11(i). Proof of Theorem 36.4(i). Applying the finiteness assumption on ρ(Ω) and the uniform regularity of Ω, we may choose ε ∈ (0, 1) such that for any ball B of

324


radius ρ(Ω) + 1, B ∩ Ωc contains a ball of radius ε. Let a be a fixed point in Ω, and let us pick xa such that B(xa , ε) ⊂ Ωc and |xa − a| ≤ ρ(Ω) + 1.

(36.3)

We seek for a supersolution of (34.3) of the form v(x, t) = Ke , r = |x − xa |, α ≥ 0. The inequality P v := vt − ∆v + µ|∇v|q − v p ≥ 0 needs to be checked only for r ≥ ε. The condition to ensure is thus αr

−α2 Keαr − α

n−1 Keαr + µαq K q eαqr − K p eαpr ≥ 0, r

r > ε,

which is satisfied if µαq K q−1 eα(q−1)r ≥ K p−1 eα(p−1)r + α2 + α

n−1 , ε

r > ε.

Since q ≥ p > 1, this is achieved whenever µαq K q−1 ≥ 2K p−1

µαq K q−1 ≥ 2α2 + 2α

and

n−1 . ε

It thus suffices to choose α = (2/µ)1/q and next 1/(q−1) . K = max u0 ∞ , 1, α2 + α(n − 1)/ε It then follows from the comparison principle that 0 ≤ u(x, t) ≤ v(x, t) in Ω, as long as u(t) exists. In particular, using (36.3), we have 0 ≤ u(a, t) ≤ K exp[(2/µ)1/q (ρ(Ω) + 1)]. Since a was an arbitrary point in Ω, we deduce that u(t) remains bounded in L∞ on its existence interval. By virtue of Theorem 35.1, this implies global existence. Let us next prove the exponential decay statement. Since we now assume that u0 ∈ W01,r (Ω), it follows from Example 51.29 that u ∈ C([0, ∞), W01,r (Ω)) ∩ C((0, ∞), W 2,r ∩ W01,r (Ω)) ∩ C 1 ((0, ∞), Lr (Ω)). We multiply the equation by ur−1 and integrate over Ω, which yields, for t > 0,

1 d ur dx = ur−1 ∆u dx + up+r−1 dx − µ ur−1 |∇u|q dx. r dt Ω Ω Ω Ω Integrating by parts, it follows that6

1 d ur dx = up+r−1 dx − (r − 1) ur−2 |∇u|2 dx − µ ur−1 |∇u|q dx, r dt Ω Ω Ω Ω

q+r−1 q p+r−1 r/2 2 ∇ u q dx. = u dx − C1 |∇u | dx − µC2 Ω

6 Note

Ω

Ω

that we have r > 2 if n ≥ 2, thus integration by parts can be carried out without difficulty. If n = 1 and 1 < r < 2, this can still be done easily.


325

Here and in what follows, C, C1 , C2 denote any constant depending only on p, q, r and Ω, but not on µ. Now, due to Proposition 50.1, we may apply the Poincaré inequality in H01 (Ω) and in W01,q (Ω) to get 1 d r dt

Ω

ur dx ≤

Ω

up+r−1 dx − C

Ω

ur dx − µC

uq+r−1 dx.

(36.4)

Ω

Using the inequality xp+r−1 ≤ εxr + C(p, q)ε−(q−p)/(p−1) xq+r−1 ,

x ≥ 0, ε > 0

in case q > p, it follows from (36.4) that d dt

Ω

ur dx ≤ −C

ur dx Ω

whenever q ≥ p and µ > µ0 (Ω, p, q, r) large enough. Consequently,

u (t) dx ≤ exp(−Ct) r

Ω

Ω

ur0 dx,

t > 0.

(36.5)

To prove exponential decay in L∞ , we use an argument of comparison with the model problem (15.1). Fix t0 > 0. By (36.5), we have u(t0 ) r ≤ M := u0 r . Therefore, since r > n(p − 1)/2, by Theorem 15.2, the solution v of (15.1) with initial data v(0) = u(t0 ) exists on a time interval [0, τ ] with τ = τ (M ) (independent of t0 ) and satisfies v(t) ∞ ≤ C v(0) r t−n/2r on (0, τ ]. Since u(t0 + t) ≤ v(t) on [0, τ ] by the comparison principle, it follows from (36.5) that u(t) ∞ ≤ C u(t − τ ) r τ −n/2r ≤ C(M ) exp(−C (t − τ )),

t ≥ τ,

hence (36.2) with s = ∞. The general case r ≤ s ≤ ∞ follows by interpolating between s = r and s = ∞. The main ingredient of the proof of Theorem 36.4(ii) (and of Theorem 36.7 below) is the following lemma. Lemma 36.6. Let p > 1, q > 2p/(p + 1) and µ ≥ 0. There exist η, ε, R > 0 and a (radial) function v ≥ 0, of class C 2 on Rn × R+ , satisfying: Pµ v := vt − ∆v − v p + µ|∇v|q ≤ 0, supp (v(t)) ⊂ B(0, R + ηt), v(t) ∞ = v(0, t) ≥ εt,

x ∈ Rn , t ≥ 0,

(36.6)

t ≥ 0,

(36.7)

t ≥ 0,

(36.8)

326


lim v(x, t) = ∞,

x ∈ Rn ,

t→∞

vt (x, t) ≥ 0,

(36.9)

x ∈ Rn , t ≥ 0,

(36.10)

and ∇v L∞ (Rn ×R+ ) ≤ 1.

(36.11)

Intuitively, the idea is to seek an unbounded global subsolution, whose gradient remains uniformly bounded, so that the damping effect of the gradient term can never become too important even for large q. This subsolution will take the form of a spherical “expanding wave”, which propagates radially away from the origin with an increasing maximum at 0. Proof of Lemma 36.6. We need two auxiliary functions. Let us first define a function f : R → R, of class C 2 , by ⎧ s ≤ 0, ⎨ 0, f (s) = 4s3 (1 − s), 0 ≤ s ≤ 1/2, ⎩ s − 1/4, s ≥ 1/2. It is easily seen that f satisfies, for some ε > 0, 0 ≤ f ≤ 1, f ≥ 0, f + f p ≥ 3εf , s ≤ 1/2

and

s ∈ R, f p ≥ 3εf , s ≥ 1/2.

Next, we define β : R+ → R, as ⎧ (M − s)3 ⎨ s+ , 0 ≤ s ≤ M, β(s) = 3M 2 ⎩ s, s > M, with M = 2n/ε. The function β is of class C 2 on R+ , with the following properties: 0 ≤ β(s) ≤ M,

0 ≤ s ≤ M,

s ≤ β(s), 0 ≤ β ≤ 1, 0 ≤ β ≤ ε/n,

s ∈ R+ ,

β(0) = M/3, β (0) = 0. Now we set 1 U (x, t) = f M + + εt − β(|x|) , 2

x ∈ Rn , t ≥ 0,

which is of class C 2 on Rn × R+ . We compute (omitting the argument in f , f , f for simplicity): ∇U = −

x β (|x|)f |x|

(0 if x = 0),

2 ∆U = β (|x|)f − f ∆ β(|x|) ,


327

n−1 β (|x|) ≤ n sup β ≤ ε. ∆ β(|x|) = β (|x|) + |x| First taking µ = ε in (34.3), we have 2

2

Pε U = εf − β (|x|)f + ∆β(|x|)f − f p + ε|β (|x|)f |q ≤ 3εf − β (|x|)f − f p . If s = 1/2 + M + εt − β(|x|) ≥ 1/2, then f p ≥ 3εf hence Pε U (x, t) ≤ 0. On the other hand, if s ≤ 1/2, then β(|x|) ≥ M + εt ≥ M . Hence β (|x|) = 1 and Pε U (x, t) ≤ 3εf − f − f p ≤ 0. Now, for a general µ > 0, replacing U by Uα (x, t) = α2/(p−1) U (αx, α2 t), we get Pµ Uα = α2p/(p−1) Ut − ∆U − U p + µα(q(p+1)−2p)/(p−1) |∇U |q (αx, α2 t) ≤ α2p/(p−1) [Pε U ](αx, α2 t) ≤ 0, for α > 0 sufficiently small since q > 2p/(p + 1), which proves (36.6) with v = Uα . Finally, (36.7)–(36.11) are straightforward consequences of the definition of f (take R = (M + 1/2)/α and η = εα and replace ε in (36.8) by εα2p/(p−1) ). Proof of Theorem 36.4(ii). Let Rj be a sequence of positive reals, Rj → ∞. From the hypotheses, there is a sequence of disjoint balls Bj = B(xj , Rj ) ⊂ Ω, with Rj > Rj . We are going to construct a suitable subsolution w = w(x, t) of (34.3) on Ω by taking advantage of the scaling properties of the equation. With v as in Lemma 36.6, we set: wj (x, t) =

1

j

v 2/(p−1)

x − x

j

j

,

γj (t) , j2

x ∈ Rn , t ≥ 0, j ∈ N∗ ,

with γj (t) = Mj t/(Mj + t), where the constants Mj > 0 will be adjusted later. By (ii)–(iii) in Lemma 36.6, we have: supp (wj (t)) ⊂ B xj , j(R + ηMj /j 2 ) , t ≥ 0, wj (t) ∞ ≥

εγj (t) εMj → 2p/(p−1) as t → ∞. j 2p/(p−1) j

For (x, t) ∈ Rn × R+ , it follows from (i) that x − x γ (t) j j , 2 P wj = j −2p/(p−1) γj (t)vt − ∆v − v p + µj (2p−q(p+1))/(p−1) |∇v|q j j x − x γ (t) j j , 2 ≤ j −2p/(p−1) vt − ∆v − v p + µ|∇v|q ≤ 0, j j

328


where we have used the fact that q ≥ p > 2p/(p + 1), vt ≥ 0 and γj (t) = Mj2 /(Mj + t)2 ≤ 1. We now choose Mj = j 1+2p/(p−1)

and

and define the function w as: w=

Rj = j(R + ηMj /j 2 )

wj .

j≥1

Note that each wj is supported on Bj and that the Bj are disjoint. By Lemma 36.6, it is clear that w is C 2 on Rn × R+ , and hence is a classical subsolution of (34.3). Moreover, by the choice of γj , w is bounded on Rn × [0, T ] for each T > 0. We note that w(0) ∈ X+ . (Also, since wj (0) ∞ ≤ j −2/(p−1) v(0) ∞ and ∇wj (0) ∞ ≤ j −(p+1)/(p−1) ∇v(0) ∞ , it follows from the choice of Rj that w(0) ∈ W01,r (Ω) for all large r.) By the comparison principle, the solution of (34.3) with initial data w(0) remains above w(t) as long as it exists, which implies the desired conclusion. It is not known whether blow-up may occur in finite time when q ≥ p and ρ(Ω) = ∞ (except, of course, for the trivial example when u solves the corresponding ODE, i.e. u0 (x) = C in Ω = Rn ). The next result from [498] shows that infinite-time blow-up can occur in the case Ω = Rn . Theorem 36.7. Consider problem (34.3) with q ≥ p > 1, µ > 0 and Ω = Rn . (i) Assume that u0 ∈ X+ has compact support. Then Tmax (u0 ) = ∞. (ii) There exists u0 ∈ X+ with compact support, such that Tmax (u0 ) = ∞ and u is unbounded. Actually, it even holds lim u(x, t) = ∞,

t→∞

for all x ∈ Rn .

Proof. (i) We shall actually prove that the following exponential decay condition (instead of compact support) is sufficient for global existence: 0 ≤ u0 (x) ≤ Ce−ε|x.a| ,

x ∈ Ω,

for some C > 0, a ∈ Rn , |a| = 1,

where ε is any positive number if q > p, or ε = µ−1/p if q = p. Without loss of generality, we may assume that a is the unit vector in the x1 -direction. We claim that, for a suitable choice of α, the functions v± (x, t) = C exp(αt ± εx1 )


329

are (traveling wave) supersolutions. If q > p and β > 0, or if q = p and β = 1, we have the elementary inequality xq ≥ β (q−p)/(q−1) xp − βx,

x ≥ 0.

Therefore, p ∂t v± − ∆v± + µ|∇v± |q − v± p ≥ ∂t v± − ∆v± + µβ (q−p)/(q−1) |∇v± |p − µβ|∇v± | − v± = C exp(αt ± εx1 ) α − ε2 − µβε + C p exp p(αt ± εx1 ) µβ (q−p)/(q−1) εp − 1 .

It thus suffices to choose β = (µεp )−(q−1)/(q−p) and α = ε2 + µβε, if q > p and ε > 0, or β = 1 and α = ε2 + µε, if q = p and ε = µ−1/p . Then we get, thanks to the comparison principle 0 ≤ u(x, t) ≤ v± (x, t),

x ∈ Rn , 0 ≤ t < T,

where T = Tmax (u0 ) (note that the comparison principle applies, for the same reason as in Remark 36.5). Consequently, 0 ≤ u(x, t) ≤ C exp(αt − ε|x1 |),

x ∈ Rn , 0 ≤ t < T,

hence in particular u(t) ∞ ≤ C exp(αt),

0 ≤ t < T.

By virtue of Theorem 35.1, this implies global existence. (ii) Taking u0 = v(0), with v as in Lemma 36.6, it is an immediate consequence of that lemma and part (i). Remarks 36.8. (i) Blow-up set. In Theorem 36.7(ii), we have global blow-up (in infinite time), i.e. the blow-up set is the whole of Rn . Infinite-time blow-up for q ≥ p is also known to occur when Ω is a cone (see [498]). But in this case, blow-up takes place only at infinity (the solution remaining bounded for t ≥ 0 in compact subsets). (ii) In Theorem 36.4, the largeness assumption on µ0 for decay is necessary in general. Indeed, when q > 2p/(p + 1), p < pS and Ω is a ball, there exist positive stationary solutions (see [129, Corollary 5.4]). (iii) For problem (34.3) where the gradient term is replaced with −µur |∇u|q , related results can be found in [497], [58].

330


37. Fujita-type results We consider the Cauchy problems associated with (34.3) and (34.4), i.e.: x ∈ Rn , t > 0, ut − ∆u = up − µ|∇u|q , x ∈ Rn ,

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

ut − ∆u = up − a · ∇(uq ),

x ∈ Rn , t > 0, x ∈ Rn .

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

(37.1)

(37.2)

In this section, we give Fujita-type results for problems (37.1) and (37.2), i.e. we find conditions which guarantee that the solution blows up in finite time for all u0 ≥ 0, u0 ≡ 0 (and not only for large initial data as in the previous section). For (37.1), the following result is due to [373] and is based on a method of rescaled test-functions. Theorem 37.1. Consider problem (37.1) with p > 1, q = 2p/(p + 1). There exists µ0 (n, p) > 0 such that if p 1, and assume that at least one of the following assumptions holds: (i) (ii) (iii)

p>1+

2 ; n

2p ; p+1 2p q= p+1 q
µ1 (n, p) > 0 large enough.

Then Tmax (u0 ) = ∞ and supt≥0 u(t) ∞ < ∞ for some nontrivial u0 ∈ X+ . Remarks 37.3. (i) Critical exponents. The value q = p is critical for the blowup and global existence properties of equation (34.3), as shown in the previous section. Another particular role is played by q = 2p/(p + 1). Indeed, for this value of q, the differential equation in (34.3) enjoys the same scaling properties as for µ = 0. Namely, for any solution u and any α > 0, the rescaled function uα (x, t) := α2/(p−1) u(αx, α2 t) is still a solution. This property is reflected in the existence of blowing-up self-similar solutions (cf. Remark 39.8(i)).


331

(ii) It seems to be unknown whether (37.1) admits any global solutions when 2p/(p + 1) < q 2p/(p + 1) and p ≤ n/(n − 2)+ [470]. (iii) A stronger result than Theorem 37.1 actually holds: Under the assumptions of that theorem there exist no nontrivial nonnegative distributional solutions of ut − ∆u = up − µ|∇u|q in Q = Rn × (0, ∞), with u ∈ Lploc (Q) and ∇u ∈ L2loc (Q). This follows from a small modification of the proof below and from similar arguments as in Step 1 of the proof of Theorem 18.1(i). Proof of Theorem 37.1. Assume that u ≥ 0 is a global solution of (37.1), n classical for t > 0, with u ∈ L∞ loc (R × [0, ∞)). Step 1. Let α ∈ (0, 1), a = (p − α)/(p − 1) > 1 and Ai > 0, i = 1, . . . , 4, with (37.3) C1 := α − A1 − µA3 ≥ 0. for the time-space For simplicity, we shall write for the space integral Rn and ∞ integral 0 Rn . We claim that for any compactly supported ϕ ∈ C 1 (Rn × [0, ∞)), ϕ ≥ 0, there holds

p−α 2a 1−2a C2 u |∇ϕ| ϕ |ϕt |a ϕ1−a , ϕ ≤ C3 + C4 (37.4) where 1−a + µA−p C2 = 1 − C(p, α) A−1 3 + A4 , 1 A2

C3 = A2 /4A1 ,

C4 = C(p, α)A1−a 4

(the function ϕ will be later chosen such that the integrals on the RHS will be finite). Fix τ, ε > 0 and put uε = u + ε. Multiplying the equation by u−α ε ϕ and integrating by parts, we get

∞

∞

1 2 −1−α u1−α up u−α |∇u| ϕ + α u ϕ + (·, τ )ϕ(·, τ ) ε ε ε 1 − α τ τ

∞

∞

∞

1 q −α = u−α |∇u| u1−α ∇u · ∇ϕ + µ u ϕ + ϕt ε ε ε α−1 τ τ τ =: I1 + µI2 + I3 . (37.5) Let us estimate I1 , I2 , I3 in terms of the double integrals appearing on the LHS. Repeatedly using Young’s inequality, we obtain

∞

∞

I1 ≤ A1 |∇u|2 u−1−α |∇ϕ|2 u1−α ϕ + B ϕ−1 1 ε ε τ

τ ∞

∞

∞

2 −1−α |∇u| uε |∇ϕ|2a ϕ1−2a + B1 B2 up−α ≤ A1 ϕ + B1 A2 ϕ, ε τ

τ

τ

332


∞

I2 ≤ A3

|∇u|2 u−1−α ϕ + B3 ε

τ

up−α ϕ ε

τ

and

∞

∞

I3 ≤ A4

up−α ϕ + C4 ε

τ

∞

|ϕt |a ϕ1−a ,

τ

C(p, α)A1−a 2

and B3 = C(p)A−p where B1 = (4A1 ) , B2 = 3 . Plugging the above estimates in (37.5), we find that

∞

∞

∞

p−α up u−α u |∇u|2 u−1−α ϕ − (B B + µB + A ) ϕ + C ϕ 1 2 3 4 1 ε ε ε τ τ τ

∞

∞

≤ C3 |∇ϕ|2a ϕ1−2a + C4 |ϕt |a ϕ1−a . −1

τ

τ

Due to assumption (37.3), the third term in the LHS can be left out. Since ϕ is compactly supported, with the help of the monotone convergence theorem, we may pass to the limit ε → 0, and then τ → 0, in the first two terms of the LHS. This yields (37.4). Step 2. Choose 0 < α < 1 − n(p − 1)/2, (37.6) A1 = α/2 and A3 = 1. By taking A2 large, A4 small (depending only on n, p), and then µ < µ0 (n, p) small, we have (37.3) and C2 > 0. t 1 Now consider ϕ of the form ϕ(x, t) = ψ |x| R ψ R2 . Here R > 0, ψ ∈ C ([0, ∞)) satisfies ψ ≤ 0 and ⎧ 0 ≤ s ≤ 1, ⎨ 1, m ψ(s) = (2 − s) , 3/2 ≤ s ≤ 2, ⎩ 0, s ≥ 2, with m > 2a > 2. Inequality (37.4) implies

C2 up−α ≤ C3 |∇ϕ|2a ϕ1−2a + C4 Σ

Σ

Σ

|ϕt |a ϕ1−a ,

(37.7)

where

Σ = (x, t) : |x| ≤ R, 0 ≤ t ≤ R2 ,

Σ = (x, t) : |x| ≤ 2R, 0 ≤ t ≤ 2R2 .

Observe that the integrals on the RHS are finite (the integrands are continuous, including at |x| = 2R, t = 2R2 due to m > 2a). The substitutions x = Ry, t = R2 s into the integrals on the right-hand side of (37.7) then yield

C2 up−α ≤ CRn+2−2a . Σ


333

Since n + 2 − 2a < 0 due to (37.6), by letting R → ∞, we conclude that u ≡ 0. Proof of Theorem 37.2. By virtue of Theorem 35.1, it suffices to obtain a uniform estimate of u. If p > 1 + (2/n), then u is a subsolution of the same problem with µ = 0 and the same initial data. Global existence for small initial data then follows from Theorem 20.1 in view of the comparison principle. If q = 2p/(p + 1) and µ > µ1 (p) large enough we shall show that there exists a (bounded stationary) supersolution of the form U (x) = ε(1 + |x|2 )−a , with a = 1/(p − 1), which will imply the desired conclusion. We have ∇U = −2εax(1 + |x|2 )−(a+1) ,

−∆U = 2εa n + (n − 2 − 2a)|x|2 (1 + |x|2 )−(a+2) .

By choosing 0 < ε, r0 < 1 small enough (depending only on n, p), we first guarantee that −∆U ≥ anε ≥ εp ≥ U p , |x| ≤ r0 . (37.8) Next, for |x| > r0 , there holds ∆U ≤ C1 (1 + |x|2 )−(a+1) = C1 (1 + |x|2 )−p/(p−1) , and

U p ≤ (1 + |x|2 )−p/(p−1)

|∇U |q ≥ C2 (1 + |x|2 )−q(a+(1/2)) = C2 (1 + |x|2 )−p/(p−1)

for some C1 , C2 > 0 depending only on n, p. Therefore, −∆U + µ1 |∇U |q ≥ U p ,

|x| > r0 ,

provided µ1 = µ1 (n, p) is chosen large enough. This along with (37.8) guarantees that U is a supersolution. Finally, if q < 2p/(p + 1) and µ > 0, let us put V (x) = α2/(p−1) U (αx). Since |∇U | is bounded, we have |∇U |q ≥ c|∇U |2p/(p+1) in Rn for some c > 0. For α > 0 sufficiently small, it follows that −∆V + µ|∇V |q − V p (x)

= α2p/(p−1) −∆U + µα(q(p+1)−2p)/(p−1) |∇U |q − U p (αx) ≥ α2p/(p−1) −∆U + µ1 |∇U |2p/(p+1) − U p (αx) ≥ 0,

so that V is a supersolution. We now turn to problem (37.2). The following result from [5] shows that the critical number (for p) may depend on both n and q.

334


Theorem 37.4. Consider problem (37.2) with p, q > 1, a = 0, and set 2q 2 . p1 = p1 (n, q) := min 1 + , 1 + n n+1 (i) If q ≤ p ≤ p1 , then Tmax (u0 ) < ∞ for any nontrivial u0 ∈ X+ . (ii) If p > p1 , then Tmax (u0 ) = ∞ for some nontrivial u0 ∈ X+ . Remarks 37.5. (a) Critical exponents. It was also shown in [5] that when q = 1, the critical exponent becomes p = 1 + 2/n. We thus observe that the critical exponent p1 (n, q) is a discontinuous function of q (since p1 (n, q) → 1 + 2/(n + 1), as q → 1+, in view of Theorem 37.4). (b) It is known (see [498, Proposition 3.6] and its proof) that blow-up in finite or infinite time can occur for (37.2) whenever q ≥ p > 1 and that this actually occurs for all nontrivial u0 ≥ 0 when q > p > 1 and p < 1 + 2/n (see [498, Remark 3.4] and [5]). However it is unknown whether the blow-up time is finite or infinite. (c) The following proof is a simplification of the original proof of [5] (especially for part (ii) in Case 1 below). Moreover, it yields uniform decay rates for suitably small data in assertion (ii). Proof of Theorem 37.4. (i) We shall prove the result only for p < p1 , the equality case being more involved. Set 1 φ(x) = C exp − (1 + |x|2 )1/2 , n where C > 0 is chosen so that Rn φ(x) dx = 1. For i = 1, . . . , n, we have |∂xi φ| ≤

φ , n

φ ∂x2i xi φ ≥ − . n

(37.9)

Without loss of generality, we may assume that a = |a|e1 . Let γ ≥ 0 to be fixed below. Let λ ∈ (0, 1], and put φλ (x) = λn+γ φ(λ1+γ x1 , λx ), where x = (x1 , x ). By (37.9), we have

φλ (x) dx = 1, ∆φλ ≥ −λ2 φλ , |(φλ )x1 | ≤ λ1+γ φλ . Rn

Multiplying equation (37.2) by φλ and integrating on Rn , we obtain, for t > 0,

d p uφλ = φλ ∆u + u φλ − |a| (uq )x1 φλ dt Rn Rn Rn Rn

p = u∆φλ + u φλ + |a| uq (φλ )x1 Rn Rn Rn

≥ −λ2 uφλ + up φλ − |a|λ1+γ uq φλ Rn

Rn

Rn


335

(this can be easily justified by using the exponential decay of φ and the fact that u(·, t) ∈ BC 2 (Rn )). Denote yλ (t) = Rn u(t)φλ . If q < p, by Young’s inequality, we observe that |a|λ1+γ uq = up(q−1)/(p−1) (|a|λ1+γ u(p−q)/(p−1) ) ≤ 12 up + Cλ(1+γ)(p−1)/(p−q) u

(37.10)

for some C = C(p, q, |a|) > 0. If q = p, then (37.10) is obviously true with C = 0 for all λ small. Using Rn up φλ ≥ yλp (owing to Jensen’s inequality) and (37.10), we deduce that yλ (t) ≥ 12 yλp − (λ2 + Cλ(1+γ)(p−1)/(p−q) )yλ . It follows that yλ , and hence u, cannot exist for all t > 0 whenever the RHS in the previous inequality is positive at t = 0. This is satisfied if

Rn

p−1 u0 (x)φ(λ1+γ x1 , λx ) dx > 2λ−(n+γ)(p−1) (λ2 + Cλ(1+γ)(p−1)/(p−q) ).

(37.11) Now, since p < p1 , by choosing 0 < γ < γ+ := 2/(p − 1) − n close to γ+ , we get (n + γ)(p − 1) < 2

and

n + γ < (1 + γ)/(p − q).

p−1 Since, by monotone convergence, the LHS in (37.11) converges to φ(0) Rn u0 ∈ (0, ∞] as λ → 0, (37.11) holds for λ > 0 sufficiently small and we conclude that Tmax (u0 ) < ∞. (ii) By Proposition 51.16, it suffices to obtain a uniform estimate of u on bounded time intervals. Case 1: q > 1 + (1/n). This case is simple, since one can directly build a (selfsimilar) supersolution of (37.2) under the form ˜ t) v(x, t) = tα G(x, ˜ = (4π)n/2 G and G is the Gaussian heat kernel. for some 0 < α < n/2, where G Indeed, setting k = n/2 − α, the function v satisfies ˜ t − ∆G) ˜ + αtα−1 G ˜q ) ˜ − tαp G ˜ p + tαq a · ∇(G vt − ∆v − v p + a · ∇(v q ) = tα (G = αt−k−1 e−|x|

2

/4t

− t−kp e−p|x|

2

/4t

− qt−kq−1/2

x · a 2 √ e−q|x| /4t 2 t

2 ≥ αt−k−1 − t−kp − Ct−kq−1/2 e−|x| /4t ,

336

IV. Equations with Gradient Terms 2

2

where we used se−qs ≤ Ce−s , s ≥ 0. Now, since p > p1 = 1+2/n and q > 1+1/n, by taking α > 0 sufficiently small, it follows that kp > k + 1 and kq + 1/2 > k + 1, so that vt − ∆v − v p + a·∇(v q ) ≥ 0 in Rn for t ≥ t0 , where t0 ≥ 1 is large enough. If −n/2 u0 (x) ≤ t0 exp(−|x|2 /4t0 ), the comparison principle in Proposition 52.16 then guarantees that u(t) ≤ v(t0 + t) on [0, Tmax(u0 )) and u exists globally. Case 2: q ≤ 1 + (1/n). This case is more involved and requires the consideration of the auxiliary problem: vt − ∆v = −(1 + t)r a · ∇(v q ), v(x, 0) = u0 (x),

t > 0,

x ∈ Rn ,

x ∈ Rn ,

(37.12)

with r > 0. By Proposition 51.16, for any u0 ∈ X+ , problem (37.12) has a unique classical solution v ≥ 0. By the maximum principle, we have v(t) ∞ ≤ u0 ∞ ,

(37.13)

which guarantees the global existence of v, in view of (51.39). Moreover, v satisfies 2 n v ∈ L∞ loc ((0, ∞), BC (R )).

(37.14)

If in addition u0 ∈ L1 (Rn ), then v ∈ C([0, ∞), L1 (Rn )).

(37.15)

We shall use the following lemma: Lemma 37.6. For 1 < q ≤ 2 and u0 ∈ L∞ ∩ L1 (Rn ), u0 ≥ 0, the solution of (37.12) satisfies the estimate v(t) ∞ ≤ C( u0 1 + u0 ∞ )(1 + t)−(n+1+2r)/(2q) .

(37.16)

Proof. Assume that a = |a|e1 without loss of generality. Step 1. Set z := v q−1 and w := zx1 = (q − 1)v q−2 vx1 . We claim that w(x, t) ≤

r + 1 −r−1 t , q|a|

x ∈ Rn , t > 0.

(37.17)

By the strong maximum principle (apply Proposition 52.7 in any bounded subdomain) we have v > 0 in Rn × (0, ∞) (unless v ≡ 0). By continuous dependence, it thus suffices to establish (37.17) when u0 also satisfies u0 ≥ ε > 0, hence v ≥ ε. The function z verifies zt − ∆z +

q − 2 |∇z|2 = −q|a|(1 + t)r zzx1 . q−1 z


337

By parabolic regularity results, w ∈ C 2,1 (Rn × (0, ∞)). Differentiating in x1 , we get wt − ∆w +

2(q − 2) ∇z · ∇w 2 − q |∇z|2 + w = −q|a|(1 + t)r (w2 + zwx1 ). (37.18) q−1 z q − 1 z2

−r−1 Since 1 < q ≤ 2, for each t0 ∈ (0, 1], the function w(t) ˜ = r+1 is a q|a| (t + t0 ) supersolution of (37.18). On the other hand, for fixed τ > 0, by taking t0 small enough, we can ensure that w(τ ) < w(0). ˜ Since, for t ≥ τ , z, ∇z are bounded and z is bounded away from 0 (due to (37.14) and v ≥ ε), it follows from a small modification of the comparison principle in Proposition 52.6 that

w(x, τ + t) ≤ w(x, ˜ t) ≤

r + 1 −r−1 t , q|a|

x ∈ Rn , t > 0.

Claim (37.17) follows by letting τ → 0. Step 2. Write x = (x1 , x ). We claim that −(n−1)/2

h(t) ∞ ≤ u0 1 (4πt)

,

∞

where h(x , t) = −∞

v(x1 , x , t) dx1 . (37.19)

Formally, by integrating (37.12) on R with respect to x1 , we see that h solves ht − ∆h = 0 in Rn−1 × (0, ∞), so that (37.19) would follow as a consequence of the L1 -L∞ -estimate. However, integration needs to be justified and we shall proceed R instead as follows. For fixed R > 0, letting hR (x , t) = −R v(x1 , x , t) dx1 and integrating (37.12) on (−R, R) with respect to x1 , we obtain ∂t hR − ∆x hR =

R vx1 − |a|(1 + t)r v q (x1 , x , t) x =−R , 1

x ∈ Rn−1 , t > 0.

Fix 0 < τ < T < ∞. It follows from (37.14) and (37.15) that v(x, t), vx1 (x, t) → 0, |x| → ∞,

(37.20)

uniformly for t ∈ [τ, T ]. Therefore, ∂t hR − ∆x hR ≤ ε(R),

x ∈ Rn−1 , τ ≤ t ≤ T,

where limR→∞ ε(R) = 0. For x ∈ Rn−1 and t ∈ [τ, T ], it follows from the maximum principle that hR (x , t) ≤ (Gt−τ ∗ hR (τ ))(x ) + ε(R)(t − τ ). By the L1 -L∞ -estimate, we deduce that hR (x , t) ≤ (4π(t − τ ))−(n−1)/2 hR (τ ) L1 (Rn−1 ) + ε(R)t ≤ (4π(t − τ ))−(n−1)/2 v(τ ) L1 (Rn ) + ε(R)T.

338


Letting R → ∞ and then τ → 0, using (37.15), we deduce (37.19). Step 3. By (37.20) and (37.17), we have v q (x1 , x , t) = q/(q − 1)

x1

−∞

(v q−1 )x1 v(y1 , x , t) dy1 ≤ Ct−r−1 h(x , t).

This, combined with (37.19), yields (37.16) for t ≥ 1, whereas (37.13) gives (37.16) for t ≤ 1. Completion of proof of Theorem 37.4. Let U (x, t) = (1 + t)m v, where v is a solution of (37.12) for r = m(q − 1) and m > 0 to be fixed later on. We shall prove that if u0 1 + u0 ∞ is small enough, then U is a supersolution of (37.2) (hence v ≤ U by the comparison principle in Proposition 52.16). We have Ut − ∆U = (1 + t)m (vt − ∆v) + m(1 + t)m−1 v = −a · ∇(U q ) + m(1 + t)m−1 v. Therefore, it will be enough to see that m(1+t)m−1v ≥ (1+t)mp v p or equivalently: v(t) ∞ ≤ m1/(p−1) (1 + t)−m−1/(p−1) .

(37.21)

But, since p > p1 = 1+2q/(n+1), we may choose m > 0 so small that m+1/(p−1) ≤ (n + 1 + 2m(q − 1))/2q, and (37.21) follows from the lemma. The proof of Theorem 37.4 is complete.

38. A priori bounds and blow-up rates The following result shows that universal bounds of the form (26.25), known for the model problem (15.1), remain valid for the perturbed problem (34.2) if the perturbation term is not too strong. In particular, this implies a (universal) a priori bound for global solutions and the usual blow-up rate estimate. Theorem 38.1. Let p > 1 and T > 0. Assume that either p < pB or p < pS ,

Ω = Rn or BR ,

u = u(|x|, t),

g = g(u, |ξ|).

Assume in addition that the function g : R+ × Rn → R satisfies the growth assumption |g(u, ξ)| ≤ C0 (1 + |u|p1 + |ξ|q ), for some 1 ≤ p1 < p and 1 < q < 2p/(p + 1).

(38.1)

38. A priori bounds and blow-up rates

Then, for any nonnegative classical solution of ut − ∆u = up + g(u, ∇u),

x ∈ Ω, 0 < t < T,

339

(38.2)

x ∈ ∂Ω, 0 < t < T,

u = 0, there holds

u(x, t)+|∇u(x, t)|2/(p+1) ≤ C 1+t−1/(p−1) +(T −t)−1/(p−1) ,

x ∈ Ω, 0 < t < T,

with C = C(p, p1 , q, C0 , Ω) > 0. Assumption (38.1) is satisfied for instance for problems (34.3) and (34.4) when q < 2p/(p + 1) or q < (p + 1)/2, respectively. The method of proof is based on rescaling and doubling arguments, already used in the proof of Theorem 26.8. Note that this method does not use any variational structure, and is thus well adapted to problem (38.2). Proof. Since the proof is very similar to that of Theorem 26.8, we only sketch the main changes. Instead of (26.34), we define the functions Mk by (p−1)/2

Mk := uk

+ |∇uk |(p−1)/(p+1) .

Rescaling similarly as in (26.39) with again λk := Mk−1 (xk , tk ) → 0, the function vk is now a solution of the equation

˜ k, ∂s vk − ∆y vk = vkp + gk , (y, s) ∈ D 2 y ∈ λ−1 k (∂Ω − xk ), |y| < k/2, |s| < k /4,

vk = 0, with 2p/(p−1)

gk (y, s) := λk

−2/(p−1) −(p+1)/(p−1) g λk vk (y, s), λk ∇vk (y, s) ,

(p−1)/2

vk and

(p−1)/2

vk

(0) + |∇vk |(p−1)/(p+1) (0) = 1,

+ |∇vk |(p−1)/(p+1) ≤ 2,

˜ k. (y, s) ∈ D

The growth assumption (38.1) then implies ˜ |gk | ≤ Cλm k in Dk ,

2(p − p ) 2p − q(p + 1) 1 , > 0. where m := min p−1 p−1

Now, as in the proof of Theorem 26.8, we distinguish the cases (26.42) and (26.43). By using parabolic Lp -estimates, we obtain a subsequence of {vk } converging to a nonnegative solution v of (21.1) or (26.45). The difference is that we now use convergence in C 1+σ,σ/2 (Rn ×R), which is satisfied due to the embedding (1.2). Therefore we get v (p−1)/2 (0) + |∇v|(p−1)/(p+1) (0) = 1, so that v is nontrivial (moreover v and ∇v are bounded). As before, we reach a contradiction with a Liouville-type theorem.

340


Remarks 38.2. Blow-up rate. (i) For problem (34.2), the lower blow-up estimate u(t) ∞ ≥ C(p)(T − t)1/(p−1) is true whenever g satisfies g(u, 0) ≤ 0 and u blows up in L∞ -norm (see Theorem 35.1 for a sufficient condition). This follows from the proof of Proposition 23.1. (ii) By using different arguments, based on a modification of the method of the auxiliary function J of [219], the upper blow-up estimate 1

u(t) ∞ ≤ C(T − t)− p−1 ,

0≤t 1, but the solution is assumed to satisfy ut ≥ 0. In the rest of this section, we shall see that the conclusions of Theorem 38.1 may become false for stronger perturbation terms in equation (34.2) (so that the growth restriction q < 2p/(p + 1) in (38.1) is not purely technical — although it is presently unknown whether it is optimal). First, concerning a priori estimates of global solutions, we just recall Theorems 36.4 and 36.7, which already provide us with examples of global solutions of (34.3), unbounded as t → ∞, whenever q ≥ p (in, e.g., Ω = Rn ). A further counter-example in that direction can be found in [156] for (34.3) with p > q = 2, n = 1, Ω = (−1, 1). In that example, the solution stabilizes (monotonically) in infinite time to a stationary solution singular at x = 0. Next, we shall show that for stronger absorbing gradient terms, the blow-up rate may become faster, or type II [264]. Let us consider the following problem ⎫ u2x ⎪ p ⎪ ⎪ ut − uxx = (u + 1) − λ , − 1 < x < 1, t > 0, ⎬ u+1 (38.3) u = 0, x = ±1, t > 0, ⎪ ⎪ ⎪ ⎭ u(x, 0) = u0 (x), − 1 < x < 1, with p > 1 and λ ≥ 0. Note that (38.3) is of the form (34.2) (with g(u, ux) = u2x ). (u + 1)p − up − λ u+1 Theorem 38.3. Consider problem (38.3) with λ > p > 1. Assume that u0 ∈ X+ is even and nonincreasing in |x|. If T := Tmax (u0 ) < ∞, then (T − t)1/(p−1) u(t) ∞ → ∞

as t → T.

(38.4)

Remarks 38.4. (i) Instability of the blow-up rate. It was moreover proved in [264] that the assumption on λ in Theorem 38.3 is optimal: If 0 < λ ≤ p (and ut ≥ 0), then the usual blow-up rate is verified: C1 ≤ (T − t)1/(p−1) u(t) ∞ ≤ C2 ,

0 < t < T,


341

for some constants C1 , C2 > 0. This shows a phenomenon of strong sensitivity to gradient perturbations (with λ = p being the threshold value for problem (38.3)). (ii) It is unknown whether or not the value q = 2p/(p + 1) in Theorem 38.1 is optimal. However, observe that the PDE in (38.3), rewritten in terms of v := u+1, has the same scale invariance properties as that in (34.3) for q = 2p/(p + 1) (cf. Remark 37.3(i)). Theorem 38.3 thus suggests that the dividing line for problem (34.3) could be given by the scaling. Namely, global unbounded solutions might exist for q > 2p/(p + 1), and type II blow-up for 2p/(p + 1) < q < p (or even for q = 2p/(p + 1) and µ large). This conjecture is also supported by Theorem 39.1 below. Theorem 38.3 will be deduced from a result of [264] on dead-cores for the absorption problem wt − wxx = −wr , w(±1, t) = k,

− 1 < x < 1, t > 0, t > 0,

w(x, 0) = w0 (x),

⎫ ⎪ ⎬ ⎪ ⎭

− 1 < x < 1,

(38.5)

where 0 < r < 1 and k > 0. Indeed, following [304], we notice that (38.3) is transformed into (38.5) by the change of unknown u + 1 = aw−m , with r=

m=

λ−p ∈ (0, 1), λ−1

1 , λ−1

a = m1/(p−1) ,

k = (λ − 1)(1−λ)/(p−1) .

(38.6)

(38.7)

Now w is nondecreasing in |x| and blow-up of u at t = T is equivalent to the appearance of a dead-core for w, i.e. w(T, 0) = 0. Note that w ≥ 0 exists for all times t > 0, with w ∈ C 2,1 ([−1, 1] × (0, ∞)). The fast blow-up estimate (38.4) becomes equivalent to lim (T − t)−α w(0, t) = 0,

t→T

α=

1 . 1−r

(38.8)

The proof of (38.8) relies on backward similarity variables, a tool that we have already used in Section 25. Namely, following [244] (see also [228]) and [217], 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 r 2

in D,

(38.9)

342


where D := {(y, s) : − log T < s < ∞, |y| < es/2 }. Under the assumptions of Theorem 38.3, we shall actually show the more precise convergence statement lim v(y, s) = V1 (y) := kr |y|2α ,

s→∞

kr =

(1 − r)2 α 2(1 + r)

,

(38.10)

uniformly on {|y| < R} for each R > 0, from which (38.8) (and hence (38.4)) readily follows. A quick check reveals that the right-hand side V1 (y) of (38.10) provides an (unbounded) stationary solution of (38.9), more precisely a solution of y Vyy − Vy + αV − V r = 0, 2

y ∈ R.

(38.11)

Note that each solution of (38.11) corresponds to a self-similar solution of wt = √ wxx − wr in R × (−∞, T ) given by w(x, t) = (T − t)1/(1−r) V (x/ T − t). On the other hand V1 (x), restricted to [−1, 1], is also a stationary solution of (38.5) with k = kr . The proof of Theorem 38.3 will then be carried out in three steps. (i) Identify the stationary solutions of (38.9) (in a suitable set); (ii) Prove that all the global solutions of (38.9) are attracted by the set of stationary solutions of (38.9) (in the locally compact topology); (iii) Discard all the possible limits other than the stationary solution V1 . We need three lemmas. In what follows, we shall use the fact that, by parabolic regularity results, wx ∈ C 2,1 ((−1, 1) × (0, T )) ∩ C([−1, 1] × (0, T ]). We start with a lower estimate which is the key ingredient in step (iii). Lemma 38.5. Let u satisfy the hypotheses of Theorem 38.3, and w be defined by (38.6)–(38.7). There exists c1 > 0 (depending on u) such that α w(x, t) ≥ w1−r (0, t) + c1 x2 , |x| ≤ 1, T /2 ≤ t ≤ T. (38.12) Proof. The basic idea of the proof is similar to that in Theorem 24.1 (cf. [219]), but some special care is required and an auxiliary nonlocal parabolic equation has to be considered — cf. (38.17) below. We set J = wx − εxwr . It will be sufficient to show that, for ε > 0 small enough, there holds J ≥0

in [0, 1] × (T /2, T ).

(38.13)

Indeed, we will then have (w1−r )x = (1 − r)w−r wx ≥ ε(1 − r)x in [0, 1] × (T /2, T ) and the estimate will immediately follow by integrating in space between 0 and x.


343

To prove (38.13), we first claim that for ε > 0 sufficiently small, we have Jx (0, t) > 0 and J(1, t) > 0 on (T /2, T ).

J(x, T /2) > 0 in (0, 1],

(38.14)

We have wx ≥ 0 in [0, 1]×(0, T ] and wx (0, t) = 0 in (0, T ]. Next, since w(x, t) ≤ k (due to w0 ≤ k), we have wx (1, t) > 0 in (0, T ] by Hopf’s lemma (cf. Proposition 52.7). Moreover, since z := wx satisfies zt − zxx = −rwr−1 z in [0, 1] × (0, T ), the strong maximum principle (Proposition 52.7) then implies wx (x, t) > 0

in (0, 1] × (0, T ].

(38.15)

As z achieves its minimum value z = 0 at x = 0 for each t ∈ (0, T ), we also have wxx (0, t) = zx (0, t) > 0

in (0, T ),

(38.16)

in view of Hopf’s lemma. The claim (38.14) follows from (38.15) and (38.16). Let us now compute (xwr )t = xrwr−1 wt , (xwr )x = wr + xrwr−1 wx , (xwr )xx = 2rwr−1 wx + xrwr−1 wxx + xr(r − 1)wr−2 (wx )2 . We get Jt − Jxx = (wt − wxx )x − ε(xwr )t + ε(xwr )xx = −rwr−1 wx + ε −xrwr−1 wt + 2rwr−1 wx + xrwr−1 wxx + xr(r − 1)wr−2 (wx )2

= −rwr−1 (wx + εx(wt − wxx )) + εrwr−2 wx (2w + x(r − 1)wx )) = −rwr−1 J + εrwr−2 wx 2w + x(r − 1)J + εx2 (r − 1)wr ) . Putting a(x, t) = rwr−1 + εxr(1 − r)wr−2 wx we obtain

and

b(x, t) = 2εr(1 − r)w2r−2 wx ,

Jt − Jxx + aJ = εrwr−2 wx (2w − εx2 (1 − r)wr ) w1−r x2 −ε . =b 1−r 2

On the other hand, we note that w1−r 1−r

−ε

x2 = wx w−r − εx = w−r J. 2 x

344


It follows that $ Jt − Jxx + aJ = b

w1−r (0, t) + 1−r

x

w

−r

% J(y, t) dy .

(38.17)

0

We can then apply a simple nonlocal version of the maximum principle to deduce that J ≥ 0. The key point which allows this is that the function b is positive. Let us give the details to make everything safe. By continuity, using (38.14), we have

E := τ ∈ (T /2, T ) : J > 0 on (0, 1] × [T /2, τ ) = ∅. Assume for contradiction that t0 := sup E < T . Then there holds J(x, t0 ) ≥ 0 in [0, 1] and there exists x0 ∈ (0, 1) such that J(x0 , t0 ) = 0, Jt (x0 , t0 ) ≤ 0 and Jxx (x0 , t0 ) ≥ 0. Substituting this into (38.17) and noting that b(x0 , t0 ) > 0, we obtain $ 1−r %

x0 w (0, t0 ) 0 ≥ Jt − Jxx + aJ (x0 , t0 ) = b(x0 , t0 ) + w−r J(y, t0 ) dy > 0. 1−r 0 This contradiction shows that t0 = T , which gives the desired conclusion. Lemma 38.6. Let u satisfy the hypotheses of Theorem 38.3, and w be defined by (38.6)–(38.7). There exists c2 > 0 (depending on u) such that 2α w(x, t) ≤ w(1−r)/2 (0, t) + c2 |x|

(38.18)

for all T /2 ≤ t ≤ T , |x| ≤ 1. Moreover, the corresponding global solution v of (38.9) satisfies v(y, s) ≤ C(1 + |y|)2α

and

|vy (y, s)| ≤ C(1 + |y|)2α−1

(38.19)

for all − log(T /2) =: s0 < s < ∞, |y| < es/2 . Proof. We consider the function J(x, t) := constant to be determined later. We compute

1 2 2 wx

− Cwr+1 , where C > 1 is a

2 Jt − Jxx = wx (wt − wxx )x − wxx − C(r + 1) wr (wt − wxx ) − rwr−1 wx2 2 = −rwr−1 wx2 − wxx + C(r + 1) w2r + rwr−1 wx2 2 = C(r + 1)w2r + r(C(r + 1) − 1)wr−1 wx2 − wxx

in (0, 1) × (T /2, T ). Using the relation wx2 = 2(J + Cwr+1 ), we get 2 , Jt − Jxx + b1 J = C[1 − r + 2r(r + 1)C]w2r − wxx


345

with b1 (x, t) = 2(1−C(r +1))rwr−1 . Then using wxx = Jx /wx +C(r +1)wr (recall that wx > 0 in (0, 1) × (T /2, T )) and setting b2 (x, t) = Jx /wx2 + 2C(r + 1)wr /wx , we end up with Jt − Jxx + b2 Jx + b1 J = C(1 − r)[1 − (r + 1)C]w2r < 0. Now, for C > 0 sufficiently large, we have J < 0 on the parabolic boundary of Q := (0, 1) × (T /2, T ). The maximum principle then yields J ≤ 0 in Q, hence 1−r wx w−(r+1)/2 ≤ C 2

(w(1−r)/2 )x =

and the estimate (38.18) follows. Note that we get in turn the estimate |wx | ≤ Cw(r+1)/2 ,

|x| ≤ 1,

T /2 ≤ t ≤ T.

(38.20)

Let us next prove (38.19). Since wxx (0, t) ≥ 0, we have wt (0, t) ≥ −wr (0, t). By integrating between t and T , we easily get w(0, t) ≤ C(T − t)α . By combining this with (38.18), we obtain 2α √ √ v(y, s) = (T − t)−α w(y T − t, t) ≤ (T − t)−α w(1−r)/2 (0, t) + c2 |y| T − t √ 2α √ ≤ C(T − t)−α T − t + |y| T − t = C(1 + |y|)2α . The estimate of vy then follows from (38.20). The proof of the lemma is complete. Next, for step (i), we have the following. Lemma 38.7. Let V ∈ C 2 (R) be a solution of (38.11) such that V = V (|y|),

with

V ≥ 0,

V >0

for all y > 0,

and such that V is polynomially bounded. Then V = V1 := kr |y|2/(1−r)

or

V = V2 := κ := (1 − r)1/(1−r) .

Proof. Let W := V 1−r and denote = d/dy. Since V > 0 for y > 0, W is smooth there. The equation for W is: y r W 2 W − W + + W = 1 − r. 2 1−r W

(38.21)

346


By differentiating, we note that 2 2 y 1 r W r W (2W W − W ) W − W + W = − =− . 2 2 1−r W 1−r W2

(38.22) 1

Set H := W − y2 W and let D = {y > 0 : H(y) = 0}. For all y ∈ D, Z := |H| 1−r is smooth and we compute Z = hence

2r−1 1 |H| 1−r HH , 1−r

Z =

2r−1 1 r 2 |H| 1−r HH + H , 1−r 1−r

y 2r−1 y r 1 2 Z − Z = |H| 1−r H H − H − H , 2 1−r 2 1−r

and H =

1 y W − W , 2 2

y H = − W . 2

Using (38.22), it follows that, for all y ∈ D, y Z − Z 2 y 2 2r−1 y r W y 1 y W |H| 1−r W − W W − W + − − W = 1−r 2 2 2 2 1−r 2 2 2 2r−1 r W (2W W − W ) 2 1−r y(2W − yW |H| ) + (W − yW ) =− 4(1 − r)2 W2 2 2 2 2r−1 r 2 2 WW − W WW − W 1−r =− W |H| + y + 2yW 4(1 − r)2 W W 2 2 2r−1 r WW − W =− |H| 1−r W + y ≤0 2 4(1 − r) W hence

(e−y

2

/4

Z ) ≥ 0

in D.

(38.23)

We next claim that the function Z ≥ 0 is nonincreasing in (0, ∞). Indeed, otherwise, there would exist y0 such that Z(y0 ) > 0 and Z (y0 ) > 0, hence Z ≥ 2 Cey /4 for y ≥ y0 by (38.23). Due to |(y −2 W ) | = 2y −3 Z 1−r , we would get W ≥ 2 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 (hence y1 ∈ D). Therefore, at y = y1 , we 2 get 0 ≤ (y/2)f − f ≤ ε((y 2 /2) − 1 − y 2 )ey /2 < 0, a contradiction.


347

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 equation (38.21). Now, the rest of the proof of Theorem 38.3 via the dead-core rate estimate (38.8), and in particular Step (ii), will be a consequence of energy arguments close to those from Section 25 (cf. [244]) for blow-up problems. A difference with [244], [217] 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. Proof of Theorem 38.3. Let ρ(y) = e−y

R(s) v 2 y

E(s) =

2

0

+

2

/4

and define R(s) = es/2 and

v r+1 αv 2 − (y, s)ρ(y) dy. r+1 2

For s ≥ s0 = − log(T /2), we have

R(s) v2 v r+1 αv 2 y + − (R(s), s) + E (s) = R (s)ρ vy vys + (v r − αv)vs ρ dy 2 r+1 2 0 0 / v2 r+1 2 v αv y + − + vy vs (R(s), s) = ρ R (s) 2 r+1 2

R(s) + −(ρvy )y + ρ(v r − αv) vs dy 0

0 /

R(s) v2 v r+1 αv 2 y + − + vy vs (R(s), s) − = ρ R (s) ρvs2 dy 2 r+1 2 0

R(s) ≡ A(s) − ρvs2 dy. 0

On the other hand, by (38.19), we have

|E(s)| ≤

∞

Ce−y

2

0

/4

(1 + |y|)4α dy = C,

s ≥ s0

and, using vs (R(s), s) = αv − y2 vy (R(s), s) and (38.19), we obtain 1 |A(s)| ≤ C exp − es es/2 (1 + es/2 )4α , 4 hence A(s) ∈ L1 (s0 , ∞). It follows that

∞

s0

R(s) 0

ρvs2 dy ds < ∞.

348


Then, by arguing similarly as in the proof of Lemma 25.6(i), we deduce that, for each sequence sn → ∞, there exists a subsequence sn such that v(·, sn ) converges to a solution V of (38.11), uniformly on {|y| < R} for each R > 0. But on the other hand, by the lower bound (38.12), for each y ∈ R and s > 2 log |y|, we have √ α √ 2α v(y, s) = (T − t)−α w y T − t, t ≥ (T − t)−α c1 |y T − t|2 = cα 1 |y| . In view of Lemma 38.7 and (38.19), this shows that necessarily V = V1 . The conclusion follows.

39. Blow-up sets and profiles The following results show that there is a threshold q = 2p/(p + 1) above which the absorbing gradient term has a strong influence on the final blow-up profile of solutions of (34.3), making it more and more singular as q increases to p (observe that q/(p − q) > 2/(p − 1) for 2p/(p + 1) < q < p). Theorems 39.1 and 39.2 are from [131] and [490], respectively. Theorem 39.1. Consider problem (34.3) with 1 < q < p, µ > 0, and Ω = BR . Let u0 ∈ X+ be radial nonincreasing and such that T := Tmax (u0 ) < ∞. Then 0 is the only blow-up point. Moreover, for all α > α0 , there holds u(r, t) ≤ Cα r−α ,

with α0 =

0 ≤ t < T,

0 < r ≤ R,

2/(p − 1)

if 1 < q ≤ 2p/(p + 1),

q/(p − q)

if 2p/(p + 1) < q < p.

The optimality of Theorem 39.1 is shown by the following: Theorem 39.2. Under the hypotheses of Theorem 39.1, assume in addition that ut ≥ 0 in QT . Then there exist C, η > 0 such that u(r, T ) := lim u(r, t) ≥ Cr−α0 , t→T

0 < r < η.

(39.1)

Remarks 39.3. (i) The assumption ut ≥ 0 is guaranteed if u0 is a subsolution of the stationary problem (see Proposition 52.19), and it is not difficult to construct such initial data. (ii) We have η = η(u0 ) > 0 in (39.1), but we may take C = C(p) > 0 if 1 < q ≤ 2p/(p + 1), C = C(p, q, µ) > 0 if 2p/(p + 1) < q < p. The proof of Theorem 39.1 consists of two steps. The first one (Lemma 39.4) is a modification of the argument of [219] (cf. the proof of Theorem 24.1), which consists in estimating −ur from below near r = 0, by applying the maximum principle to an auxiliary function of the form J = ur + cε (r)F (u). The second step (in the case q > 2p/(p + 1)) is an additional bootstrap argument (on the value of γ for F (u) = uγ ), which enables one to reach the optimal exponent α0 .

39. Blow-up sets and profiles

349

Lemma 39.4. Consider problem (34.3) with 1 < q < p, µ > 0, Ω = BR , and let u0 be as in Theorem 39.1. Denote f (u) = up and let F ∈ C 2 ((0, ∞)) ∩ C 1 ([0, ∞)) satisfy F, F , F ≥ 0, with F F bounded near 0. Let δ > 0 and set K = (q − 1)µ. Assume that

∞ ds < ∞, y > 0 G(y) := F (s) y and that, for all sufficiently small ε > 0, f F − f F + ε2 r2+2δ F F 2 + δ(n + δ)r−2 F ≥ 2ε(1 + δ)rδ F F + 2q−1 Kεq rq+qδ F q F ,

u > 0,

0 < r ≤ R.

(39.2)

Then 0 is the only blow-up point and there exists ε0 > 0 such that u(r, t) ≤ G−1 (ε0 r2+δ )

in [0, R] × [T /2, T ).

Remark 39.5. If u is a given solution satisfying the assumptions of Lemma 39.4, then the conclusion remains valid if (39.2) is assumed to hold for all r and all u = u(r, t) such that (r, t) ∈ (0, R) × [T /2, T ) (instead of for all u > 0 and 0 < r < R). This fact will be used in the proof of Theorem 39.1(ii). Proof of Lemma 39.4. Set J = w + cε (r)F (u) where cε (r) = εr1+δ and w = ur ≤ 0. By parabolic regularity results, we have w ∈ C 2,1 ((0, R) × (0, T )) ∩ C([0, R] × (0, T )). Also, u > 0 in [0, R) × (0, T ) by the strong maximum principle. Differentiating the equation ut − urr −

n−1 ur = f (u) − µ|ur |q r

with respect to r yields wt − wrr −

n−1 n−1 wr = − 2 w + f (u)w + qµ|w|q−1 wr . r r

(39.3)

Using (39.3) and writing f, F for f (u), F (u), we compute the equation for J: Jt − Jrr −

n−1 n−1 Jr = − 2 w + f (u)w + qµ|w|q−1 wr + cε [f − µ|w|q ]F r r n − 1 cε + cε F − cε w2 F . − 2wcε F − r

Using the relations w = J − cε F , w2 = c2ε F 2 + (J − 2cε F )J and wr = Jr − cε F − cε F w,

350


we obtain Jt − Jrr −

where

n − 1 r

+ qµ|w|q−1 Jr − b0 J c n − 1 cε n − 1 cε = cε −f − ε qµ|w|q−1 + F − − cε r2 cε r cε + (f + K|w|q )F + 2cε F F − c2ε F F 2

b0 = f − (n − 1)r−2 − 2cε F − cε (J − 2cε F )F .

The assumption (39.2) is equivalent to f F − f F − 2cε F F + c2ε F F 2 − 2q−1 Kcqε F q F +

c ε

cε

+

n − 1 cε n − 1 − 2 F ≥ 0. r cε r

Combining this with |w|q ≤ 2q−1 (|J|q + cqε F q ), we obtain Jt − Jrr −

n − 1 r

+ qµ|w|q−1 Jr − bJ ≤ 0

in (0, R) × (0, T ],

where b = b0 + 2q−1 Kcε F |J|q−2 J. On the other hand, arguing as in the proof of Theorem 24.1, we obtain ur < 0 in (0, R] × (0, T ) and urr (0, t) < 0 in (0, T ). It follows that J(·, T /2) ≤ 0 in [0, R] for ε small. Obviously J(0, t) ≤ 0 and J(R, t) < 0 for all t ∈ (0, T ). Since b is bounded above in ((0, R) × (0, τ )) ∩ {(x, t) : J > 0} for each τ ∈ (0, T ), it follows from the maximum principle (cf. Proposition 52.4 and Remark 52.11(a)) that J ≤ 0 in [0, R] × [T /2, T ). Integrating this inequality between 0 and r yields the conclusion. We shall show that condition (39.2) in Lemma 39.4 is satisfied for F (u) = uγ with suitable choices of γ > 1. The inequality (39.2) takes the form (p − γ)up+γ−1 + (εr1+δ )2 γ(γ − 1)u3γ−2 + δ(n + δ)r−2 uγ ≥ 2εγ(1 + δ)rδ u2γ−1 + 2q−1 Kγ(εr1+δ )q uγq+γ−1 . In the proof of this inequality, we shall need the following elementary lemma. Lemma 39.6. Let n be a positive integer, R, K, δ > 0 and p > 1. (i) If 1 < γ < p, then for ε > 0 small enough, there holds 1 (p − γ)up+γ−1 + δ(n + δ)r−2 uγ 2 ≥ 2εγ(1 + δ)rδ u2γ−1 ,

(39.4) 0 < r ≤ R,

u ≥ 0.


351

(ii) If 1 < q < 2p/(p + 1) and γ ∈ (p/q, p), then for ε > 0 small enough, there holds 1 (p − γ)up+γ−1 + (εr1+δ )2 γ(γ − 1)u3γ−2 2 (39.5) ≥ 2q−1 Kγ(εr1+δ )q uγq+γ−1 , 0 < r ≤ R, u ≥ 0. (iii) If 1 < q 0 small enough, there holds 1 (p − γ)up+γ−1 ≥ 2q−1 Kγ(εr1+δ )q uγq+γ−1 , 2

0 < r ≤ R,

u ≥ 0.

(39.6)

Proof. Inequalities (39.4) and (39.5) are consequences of Young’s inequality aα bβ + ≥ ab, α β

a, b ≥ 0, α, β > 1,

1 1 + = 1, α β

where we choose α = (p − 1)/(γ − 1) in the case of (39.4) and α = (2γ − 1 − p)/ (2γ − 1 − γq) in the case of (39.5). In inequality (39.6), it is sufficient to choose 1/q −1−δ ε ≤ ( 2p−γ R . q Kγ ) We now continue with the proof of Theorem 39.1. Proof of Theorem 39.1. First assume 1 < q < 2p/(p + 1). In this case, we choose F (u) = uγ with 1 < γ < p and Lemma 39.6(i) and (ii) yields that (39.2) holds. Lemma 39.4 then implies 2+δ

u(r, t) ≤ Cr− γ−1 and

2+δ γ−1

can be made arbitrarily close to 2/(p − 1).

Next consider the case 2p/(p + 1) ≤ q < p. Now we first choose F (u) = uγ with γ = p/q and Lemma 39.6(i) and (iii) yields that (39.2) holds. Lemma 39.4 implies u(r, t) ≤ Cr−α ,

α = α(δ, γ) =

2+δ . γ−1

(39.7)

Inequality (39.6) is equivalent to uγq−p ≤

p−γ (2εr1+δ )−q Kγ

and, due to the estimate (39.7) on u, it is also true (for u = u(r, t) — cf. Remark 39.5) if γ is replaced by γ ∈ (γ, p) such that (γq − p)α < (1 + δ)q, or, equivalently, p 1+δ (γ − 1). γ< + q 2+δ

352


If δ is chosen small enough, this reduces to γ< Clearly, γ
1. On the one hand, since ut ≥ 0 and ur ≤ 0, we have ∂ 1 2 1 n−1 ur + up+1 = (urr + up )ur = ut + µ|ur |q − ur ur ≤ 0, ∂r 2 p+1 r hence

1 2

u2r +

1 1 up+1 (r, t) ≤ up+1 (0, t). p+1 p+1

Therefore, we get (39.8) with γ = (p + 1)/2 (and C1 = C1 (p)). On the other hand, 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 ur ≤ up , r

due to ut ≥ 0, ur ≤ 0 and urr (r, t) ≤ 0. This yields (39.8) with γ = p/q (and C1 = µ−1/q ), hence the claim. Step 2. For 0 < t < T := Tmax (u0 ), let r0 (t) be such that u(r0 (t), t) = 12 u(0, t). Note that, since ur < 0 for 0 < t < T and 0 < r ≤ R, the implicit function theorem guarantees that r0 (t) is unique and is a continuous function of t. Since Tmax (u0 ) < ∞, we have u(0, t) → ∞ as t → T , due to Theorem 35.1. Also, by Theorem 39.1, we know that 0 is the only blow-up point, hence r0 (t) → 0 as t → T . Now we have −ur ≤ C2 uγ , 0 ≤ r ≤ r0 (t). Integrating, we get u1−γ (r0 (t), t) ≤ u1−γ (0, t) + C3 r0 (t) = 21−γ u1−γ (r0 (t), t) + C3 r0 (t),


353

hence u(r0 (t), t) ≥ C4 (r0 (t))−1/(γ−1) . Using ut ≥ 0, it follows that u(r0 (t), T ) ≥ C4 (r0 (t))−1/(γ−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. For equation (38.3), the arguments in the proof of Theorem 38.3 provide precise information on the blow-up profile, which turns out to be slightly less singular than for the model problem (15.1) (cf. Remark 25.8). Theorem 39.7. Consider problem (38.3) with λ > p. Assume that u0 ∈ X+ is even and nonincreasing in |x|. If T := Tmax (u0 ) < ∞, then for each x = 0, u(x, T ) := limt→T u(x, t) exists and it satisfies C1 ≤ |x|2/(p−1) u(x, T ) ≤ C2 ,

x small, x = 0,

for some constants C1 , C2 > 0. Proof. The (globally defined) solution w ≥ 0 of the transformed problem (38.5) (cf. formulas (38.6)–(38.7)) satisfies (38.12) and (38.18). In particular, by (38.12), parabolic estimates and standard embeddings, we have u ∈ BU C α ({ε < |x| < 1}× (T /2, T )) for each ε > 0 and some α ∈ (0, 1). It follows that the limit u(x, T ) exists for x ∈ [−1, 1] \ {0}. On the other hand, since w(0, T ) = 0, (38.12) and (38.18) imply (c1 |x|2 )α ≤ w(x, T ) ≤ (c2 |x|)2α , −1 < x < 1. The assertion concerning u(x, T ) follows immediately. Remarks 39.8. (i) Self-similar blowing-up solutions. As mentioned in Remark 37.3(i) when q = 2p/(p+ 1), the equation (34.3) is scale-invariant. This property has been exploited in [495] to construct backward self-similar (blow-up) solutions by ODE methods. More precisely, for each 0 < µ < 2 and 1 < p < p0 (n, µ), there exists a solution of (34.3) of the form u(x, t) = (T − t)−1/(p−1) W x/(T − t)1/2 ,

(39.9)

for (x, t) ∈ Rn × (−∞, T ). Here W is a positive, C 2 , radially symmetric decreasing function on Rn . Moreover, for all such solutions, W satisfies lim |x|2/(p−1) W (x) = C > 0.

|x|→∞

This guarantees that u blows up at the single point x = 0 and admits a limiting profile, similar to that obtained for equation (38.3) in Theorem 39.7, given by u(x, T ) = C|x|−2/(p−1) ,

for all x = 0.

354


In contrast, recall that no nontrivial, backward, self-similar solutions exist for µ = 0 and p ≤ pS (cf. Proposition 25.4). Comparison of this result with Theorems 39.1–39.2 yields the interesting and a bit surprising observation that the gradient term can have opposite effects on the blow-up profile: When the perturbation is mild (q = 2p/(p + 1)), the profile is slightly less singular; when the perturbation is strong (2p/(p + 1) < q < p), it is more singular. (ii) Single-point vs. regional blow-up. We have seen several examples of single-point blow-up for equations with dissipative gradient terms in the radial case (cf. Theorems 39.1 and 39.7 and Remark 39.8(i)). Also, examples of single-point blow-up for the convective problem (34.4) can be found in [218]. On the other hand, it was proved in [131] that if Ω is convex bounded, µ > 0 and q < 2p/(p + 1), then the blow-up set of any solution of (34.3) is a compact subset of Ω. The situation is quite different when µ < 0. Namely, for q = 2 one has single-point blow-up if p > 2, regional blow-up if p = 2, and global blow-up if 1 < p < 2 (see [313], [304], [231]). The proof relies on the transformation v = eu − 1, which converts (34.3) into the equation with mildly superlinear source vt − ∆v = (1 + v) logp (1 + v). The authors of [304] 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. (iii) L∞ boundary blow-up for a Dirichlet problem. For the convective problem (34.4), a surprising example was recently constructed in [202], of a solution blowing up (only) at the boundary, in spite of the imposed homogeneous Dirichlet boundary condition. More precisely, consider problem (34.4) with n = 1, Ω = (0, ∞), p > 1 and q = (p + 1)/2. Then, for −a > 0 sufficiently large, there exists a positive solution u such that lim sup u(x, t) = ∞

t→T x>0

and

u(x, t) ≤ C|x|−2/(p−1) ,

x > 0, 0 < t < T.

This solution is constructed in the backward self-similar form (39.9), now with W (y) > 0, y > 0, and W (0) = 0 (note that (34.4) is scale-invariant for q = (p + 1)/2, similarly as (34.3) for q = 2p/(p + 1) — cf. Remark 39.8(i)). (iv) More self-similar profiles. Concerning self-similar profiles, still in the case µ < 0, q = 2, with Ω = Rn , it is proved in [231] that radial blow-up solutions to equation (34.3) behave asymptotically like a self-similar solution w of the following Hamilton-Jacobi equation without diffusion: wt = |∇w|2 + wp .

40. Viscous Hamilton-Jacobi equations and gradient blow-up on the boundary

355

The function w is of the form w(x, t) = (T − t)−1/(p−1) W x/(T − t)m ,

m = (2 − p)/2(p − 1).

Note that this kind of self-similar behavior is quite different from that in (i) above (or from those known for µ = 0 and p supercritical); indeed, m describes the range (−∞, 1/2) for p ∈ (1, ∞).

40. Viscous Hamilton-Jacobi equations and gradient blow-up on the boundary In this section we study problem (34.5), which exhibits quite different phenomena from the model problem (15.1) or its perturbations (34.3), (34.4). For simplicity we shall again only consider nonnegative solutions (this assumption is essential in some, but not all, of the results).

40.1. Gradient blow-up and global existence A basic fact about (34.5) is that solutions are uniformly bounded. Indeed, as a direct consequence of the maximum principle, for any u0 ∈ X+ , there holds 0 ≤ u(x, t) ≤ max u0 (x), x∈Ω

x ∈ Ω, 0 ≤ t < Tmax (u0 ).

(40.1)

In view of (40.1), and since (34.5) is well-posed in the space X, a solution can cease to exist in finite time Tmax (u0 ) < ∞ only if lim

t→Tmax (u0 )

∇u(t) ∞ = ∞.

(40.2)

This is what we call gradient blow-up (GBU for short). Unlike the model problem (15.1), for which nonglobal solutions exist if and only if p > 1, the following two results show that the dividing line for the Dirichlet problem associated with (34.5) is given by p = 2. Theorem 40.1. Consider problem (34.5) with 1 2 and Ω bounded, and let 1 ≤ q < ∞. There exists C = C(p, q, Ω) > 0 such that, if u0 ∈ X+ and u0 q ≥ C, then Tmax (u0 ) < ∞. Theorem 40.1 is an immediate consequence of the boundary gradient estimate from Lemma 35.4 and of the following simple result, which asserts that for problem (34.5), |∇u| achieves its maximal values on the parabolic boundary. Proposition 40.3. Assume p > 1 and u0 ∈ X+ . Let u be the solution of (34.5) and let 0 < T < Tmax (u0 ). Then sup ∇u(t) ∞ = sup |∇u|. PT

t∈[0,T ]

Proof. Fix h ∈ Rn , with |h| = 1, and put w := ∂h v = h · ∇v. We have w ∈ C(QT ) ∩ L∞ (QT ), and parabolic regularity results imply w ∈ C 2,1 (QT ). Taking the space derivative of the equation in the direction h, we obtain wt − ∆w = b(x, t) · ∇w

in QT ,

where b(x, t) = p|∇u|p−2 ∇u. By the maximum principle, we deduce that supQT w ≤ supPT w. Since h is arbitrary, the conclusion follows. Proof of Theorem 40.2. Put q0 := 2(p − 1)/(p − 2). It is obviously sufficient to show the assertion for q0 ≤ q < ∞. Let thus set k := q − 1 ∈ [p/(p − 2), ∞). Multiplying (34.5) by uk , we get

1 d k+1 p k u (t) dx = |∇u| u dx − k |∇u|2 uk−1 dx. (40.3) k + 1 dt Ω Ω Ω On the other hand, by Poincaré’s inequality, we have

p |∇u|p uk dx = C ∇u(p+k)/p ≥ C up+k dx. Ω

Ω

(40.4)

Ω

Since k ≥ p/(p − 2), by using H¨ older’s inequality and (40.4), we get

2/p

(p−2)/p |∇u|2 uk−1 dx ≤ |∇u|p uk dx uk−p/(p−2) dx Ω Ω Ω

(k+1)/(k+p) ≤C |∇u|p uk dx . Ω

Combining this with (40.3), (40.4) and H¨ older’s inequality, we obtain

(k+p)/(k+1) d k+1 p k u dx ≥ |∇u| u dx − C ≥ C1 uk+1 dx − C2 . dt Ω Ω Ω The conclusion follows.


357

Remarks 40.4. (i) Different methods of proof. Theorem 40.2, whose proof relies on multiplication by powers of u, is due to [279] for q = 2(p − 1)/(p − 2) and [500] in the general case. By a different argument, using the first eigenfunction, GBU for problem (34.5) was shown in [485] under a stronger condition on u0 (see also [6]). The first example of GBU seems to be due to [210], where a one-dimensional problem with time-dependent Dirichlet boundary conditions was considered. The proof was based on subsolution arguments. (ii) Sharp condition for GBU. A more precise growth condition for preventing GBU is known to be |F (u, ∇u)| ≤ C(u)(1 + |∇u|2 )h(|∇u|) where h is positive nondecreasing and satisfies

∞ ds = ∞, sh(s)

(40.5)

(40.6)

and C(u) is locally bounded (compare with condition (17.4) in the case of L∞ blow-up); see [320], [337], [512]. There are known examples showing that condition (40.5)–(40.6) is sharp. A GBU result for general (including homogeneous) Dirichlet data can be found in [485]. The proof relies on eigenfunction and convex conjugate functions arguments. Earlier examples involving particular time-dependent boundary data, and relying on subsolution methods, were given in [337]. Unlike in the Dirichlet problem, global existence for the Cauchy problem holds for any p > 1 (cf. [402], [26]): Proposition 40.5. Consider problem (34.5) with Ω = Rn and p > 1. Then Tmax (u0 ) = ∞ for any u0 ∈ X+ . Moreover, we have sup ∇u(t) ∞ = ∇u0 ∞ .

(40.7)

t≥0

Proposition 40.5 is an immediate consequence of Proposition 40.3. Remark 40.6. Unbounded domains. Although Theorem 40.2 is stated for bounded domains, GBU for large data when p > 2 occurs in any (regular) unbounded domain Ω other than Rn . (Thus, for p > 2, Proposition 40.5 is true only in Rn .) Indeed, this follows from a simple comparison argument: Choose a ball B ⊂ Ω such that ∂B ∩ ∂Ω consists of a single point, say x0 . Without loss of generality, we may assume that B = B(0, ρ) and x0 = ρ e1 . Let 0 ≤ φ ∈ C 1 (B) satisfy φ = 0 on ∂B, φ radially symmetric, and let v be the solution of problem (34.5) with Ω replaced by B and initial data φ. If φ 1 is sufficiently large, then v has GBU in a finite time T , due to Theorem 40.2. Since v is radially symmetric, it follows from

358


∂v Proposition 40.3 that lim inf t→T ∂x (x0 , t) = −∞. Take any u0 ∈ X+ such that 1 u0 ≥ φ in B and let u be the solution of (34.5). By the comparison principle, we have u(x, t) ≥ v(x, t) in B as long as u exists. Since u(x0 , t) = v(x0 , t) = 0, this ∂u ∂v (x0 , t) ≤ ∂x (x0 , t). Consequently GBU must occur for u no later than implies ∂x 1 1 at time T .

40.2. Asymptotic behavior of global solutions We start with the case of bounded domains. We have the following result on exponential decay. Assertions (i), (ii) follow from [146], [69], and (iii) from [485]. Theorem 40.7. Consider problem (34.5) with p > 1, Ω bounded and u0 ∈ X+ . (i) Assume that Tmax (u0 ) = ∞

and

sup u(t) X < ∞.

(40.8)

t≥0

Then there exists C > 0 (depending on u), such that u(t) X ≤ Ce−λ1 t ,

t ≥ 0.

(40.9)

(ii) If 1 2, then assertions (40.8) and (40.9) are true whenever u0 X is sufficiently small. In the proof we shall use the following simple observation about steady states of (34.5) (cf. [341]): Proposition 40.8. Assume Ω bounded and let p > 1. Then the only solution v ∈ C 2 ∩ C0 (Ω) of ∆v + |∇v|p = 0 is the trivial solution v ≡ 0. Proof. For ε > 0 small, let us denote ωε = {x ∈ Ω : δ(x) > ε}. By the maximum principle applied to the equation ∆v + b(x) · ∇v = 0 where b(x) = |∇v|p−2 ∇v, we have maxωε |v| = M (ε) := max∂ωε |v|. But v ∈ C0 (Ω) implies M (ε) → 0 as ε → 0. Consequently v ≡ 0. Proof of Theorem 40.7. (i) Let u0 satisfy (40.8). We first claim that lim u(t) X = 0.

t→∞

(40.10)

To see this, let us first observe that φ : X → [0, ∞), defined by φ(v) = v ∞ is a Lyapunov functional for problem (34.5) (cf. Appendix G). Indeed, as a consequence of (40.1), the function h(t) = u(t) ∞ is nonincreasing for t ≥ 0. Moreover, if h is constant, then for each t > 0, u(·, t) achieves the value u0 ∞ at some interior


359

point. Applying the strong maximum principle (Proposition 52.7), we infer that u is constant, hence 0, in Ω×(0, ∞). Therefore, φ is in fact a strict Lyapunov functional. Moreover, by (40.8) and parabolic estimates, u(t) is bounded in W 2,q (Ω) for each finite q, so that {u(t) : t ≥ 1} is precompact in X. By Propositions 53.3 and 53.5, it follows that ω(u0 ) (in the X topology) is nonempty and consists of equilibria. Property (40.10) thus follows from Proposition 40.8. Now the exponential decay in (40.9) follows from Remark 51.20(ii). (ii) This follows from Theorem 40.1 and assertion (i). (iii) Let Θ be the classical solution of (19.27). By Hopf’s lemma (cf. Proposition 52.1), we have Θ(x) ≥ c1 δ(x) in Ω. Letting M = ∇Θ ∞

and

φ = M −p/(p−1) Θ,

we find that −∆φ ≥ |∇φ|p

in Ω.

(40.11)

Now, assume that u0 X < c1 M −p/(p−1) . Consequently, u0 (x) ≤ u0 X δ(x) ≤ φ(x) in Ω. It then follows from (40.11) and the comparison principle (see Proposition 52.16) that u ≤ φ in Ω × [0, T ), where T = Tmax (u0 ). Since u = φ = 0 on ∂Ω, we deduce that ∂u ∂φ |∇u| = − ≤− ≤ C on ST . ∂ν ∂ν The result then follows from Proposition 40.3 and assertion (i). Remarks 40.9. (a) Boundedness of global solutions. In the case p > 2 and Ω bounded, the question of boundedness of global solutions has been studied, too. For n = 1, it was shown in [38] that all global solutions are bounded in X. Consequently, they satisfy (40.9). For n ≥ 2, this is an open problem. On the other hand, if one looks at weaker norms, an L∞ a priori estimate of global solutions is provided by (40.1), and it was shown in [500] that all global solutions actually decay in L∞ . Moreover, Theorem 40.2 implies the universal Lq -bound supt≥0 u(t) q ≤ C(Ω, p, q) for all finite q. Furthermore, under the stronger condition p > max(2, n), one actually has a universal L∞ -bound of the form u(t) ∞ ≤ C(Ω, p, q)(1 + t−α ) for all t > 0 and some α = α(n, p) > 0 [500]. (b) A priori estimates. Similarly to the model problem (15.1) (cf. Section 22) one can consider the question, not only of boundedness but of a priori estimates of global solutions in X norm, and the related problem of continuity of the existence time. Consider problem (34.5) with p > 2 and Ω = (0, 1), fix a nontrivial φ ∈ X+ and define

E := λ > 0 : Tmax (λφ) < ∞ and u(t) X → 0, as t → ∞ . By Theorems 40.7(iii) and 40.2, we have λ ∈ E for λ > 0 small and λ ∈ E for λ > 0 large. Therefore λ∗ := sup E ∈ (0, ∞) and, due to Theorem 40.7(iii) and continuous

360


dependence, λ∗ ∈ E. Since all global solutions decay in X (cf. Remark 40.9(a)), it follows that Tmax (λ∗ φ) < ∞. Consequently Tmax is discontinuous. Moreover global solutions fail to satisfy an a priori estimate of the form supt≥0 u(t) X ≤ C( u0 X ) (since, by continuous dependence, this would imply a bound for u(·; λ∗ φ) X , hence Tmax (λ∗ φ) = ∞). This exhibits a similar phenomenon as for the model problem in dimensions n ≥ 3, which does not occur in dimensions n = 1 or 2 (cf. Theorem 22.1, Theorem 28.7 for radial solutions in a ball and p > pS , and see after Theorem 22.13). (c) Unbounded global solutions. If one considers the modification of problem (34.5) where a (smooth) inhomogeneous term h(x) ≥ 0 is added on RHS, then boundedness of global solutions is still true in L∞ -norm, but may fail in the X norm. Indeed, examples of global solutions with |∇u(x, t)| becoming unbounded on the boundary as t → ∞ have been constructed in [500] for all n ≥ 1. In [496], for a variant of problem (34.5) with n = 1, the grow-up rate of ux is determined by techniques of matched asymptotics. (d) Consider the situation of Theorem 40.7 in the limiting case p = 1. Then all solutions are still global and decay exponentially, but the decay exponent can be smaller than λ1 (see [69]). On the other hand, decay no longer occurs in general for 0 < p < 1. Indeed, if Ω = (0, 1) for instance, it is easy to construct positive stationary solutions. We refer to [321] for results on the asymptotic behavior in this case. We turn to the Cauchy problem. Recall that now all solutions are global by Proposition 40.5. The most complete results available concern the case of solutions with finite mass: Unless otherwise specified, we shall assume in the rest of this subsection that u0 ∈ X+ ∩ L1 (Rn ), u0 ≡ 0. (40.12) Under this assumption, the solution of (34.5) satisfies u ∈ C([0, ∞), L1 (Rn )) (this can be shown by arguments similar to those in the proof of (51.42) in Proposition 51.16). Moreover, u(t) 1 is nondecreasing in time. This follows from

Rn

t

u(t) dx = Rn

u0 dx +

0

Rn

|∇u(y, s)|p dy ds,

(40.13)

due to Proposition 48.4(b) and the variation-of-constants formula. We may thus define I∞ = lim u(t) 1 ∈ (0, ∞], t→∞

and a natural question is then to determine whether the growth of mass is limited or not, i.e., I∞ < ∞ or I∞ = ∞. It turns out that the problem involves two critical exponents p = 2 and p = pc := (n + 2)/(n + 1). Recall that Gt denotes the Gaussian heat kernel, defined in (48.5).


361

Theorem 40.10. Consider problem (34.5) with Ω = Rn and u0 satisfying (40.12). (i) Assume p ≥ 2. Then, for all u0 , there holds I∞ < ∞. Moreover, u(t) − I∞ Gt 1 → 0, t → ∞.

(40.14)

(ii) Assume 1 < p ≤ pc . Then, for all u0 , there holds I∞ = ∞. (iii) Assume pc < p < 2. Then we have I∞ < ∞ for small data (in a suitable sense), and there also exist u0 such that I∞ = ∞. Furthermore, (40.14) is satisfied whenever I∞ < ∞. Assertions (i) and (ii) are due to [322]. As for assertion (iii), the fact that I∞ < ∞ under suitable smallness assumptions was proved in [145], [322], and the existence of at least one solution such that I∞ = ∞ is due to [72]. This was next shown to occur under suitable largeness conditions on u0 in [70]. We shall prove (i) and (ii) only. The proof of (iii) is more delicate and we refer for this to the above mentioned articles. Proof of Theorem 40.10(i). First observe that in view of (40.7), u satisfies ut − ∆u ≤ a|∇u|2 ,

x ∈ Rn , t > 0

au − 1. The with a = ∇u0 p−2 ∞ > 0. We use the Hopf-Cole transformation v := e function v satisfies

vt − ∆v = a(ut − ∆u − a|∇u|2 )eau ≤ 0,

x ∈ Rn , t > 0.

Therefore, v(t) ≤ e−tA v0 by the maximum principle, where e−tA denotes the heat semigroup in Rn . Using the inequalities x ≤ ex − 1 ≤ xex for x ≥ 0, it follows that a u(t) 1 ≤ v(t) 1 ≤ v0 1 ≤ au0 eau0 1 ≤ aea u0 ∞ u0 1 ,

t≥0

hence I∞ < ∞. Property (40.14) is then a consequence of Lemma 20.16 (and so is the last statement of assertion (iii)). Proof of Theorem 40.10(ii). Case 1: n ≥ 2. Since p < n, by the Sobolev inequality and (40.13), there holds

t u(s) pp∗ ds, t ≥ 0, where p∗ = np/(n − p). (40.15) u(t) 1 ≥ C 0

√ Also, for |x| ≤ t and t ≥ t0 (u0 ) large enough, we have

2 e−|x−y| /4t u0 (y) dy u(x, t) ≥ (4πt)−n/2 Rn

−n/2 −1 −n/2 ≥ (4πt) e u0 1 . √ u0 (y) dy ≥ Ct |y|< t

(40.16)

362


It follows that for all s ≥ t0 (u0 ),

∗

u(s) pp∗ ≥

∗

∗

n

∗

up (x, s) dx ≥ C u0 p1 s− 2 (p

√ |x|< s

−1)

,

which combined with (40.15) yields

u(t) 1 ≥ Since p ≤ pc , we have

np 2 (1

−

C u0 p1

1 p∗ )

=

t

s−

np 1 2 (1− p∗

)

ds.

t0

(n+1)p−n 2

≤ 1, hence I∞ = ∞.

Case 2: n = 1. We use the interpolation inequality v 2p−1 ≤ C v p−1 vx pp , ∞ 1

for all p ≥ 1 and v ∈ L1 such that vx ∈ Lp , (40.17)

which is a consequence of |v(x)|(p−1)/p v(x) =

2p − 1 p

x

−∞

|v|(p−1)/p vx dy

and of H¨ older’s inequality. Since u(t) 1 is nondecreasing, it follows from (40.13) and (40.17) that

u(t) p1

≥

u(t) p−1 1

≥

t 0

ux(s) pp ds

t

0

u(s) p−1 ux(s) pp 1

ds ≥ C

t

0

u(s) 2p−1 ds. ∞

But (40.16) implies u(t) ∞ ≥ C u0 1 t−1/2 for t ≥ t0 (u0 ) large enough, hence u(t) p1 ≥ C u0 2p−1 1

t

1

s−p+ 2 ds.

t0

Since p ≤ pc = 3/2, we conclude that I∞ = ∞. Remarks 40.11. (a) Nonlinear asymptotic behaviors. Theorem 40.10 shows that when p ≥ 2, or pc < p < 2 and u0 is small, then I∞ < ∞ and the asymptotic behavior is dominated by the diffusion. When I∞ = ∞, other behaviors are known. To describe this briefly, first observe that, since u(t) ∞ is nonincreasing in time due to (40.1), we may set N∞ := lim u(t) ∞ ∈ [0, ∞). t→∞


363

It was proved in [70] that if (and only if) N∞ > 0,

(40.18)

then I∞ = ∞ and u behaves like the viscosity solution z of the pure HamiltonJacobi equation zt = |∇z|p , with initial data N∞ χ{0} . More precisely, |x| p/(p−1) . lim u(t) − z(t) ∞ = 0, where z(x, t) = N∞ − c(p) 1/p t→∞ t + In the range 1 < p ≤ pc , property (40.18) is true for all nontrivial (not necessarily integrable) u0 ∈ X+ , see [251]. The situation is different in the range pc < p < 2: property (40.18) holds under a suitable largeness condition on u0 [70], but an example of a solution such that I∞ = ∞

and N∞ = 0

(40.19)

has been constructed in [72]. This solution is self-similar, of the form x 2−p , u(x, t) = (t + 1)−k V √ , k= 2(p − 1) t+1 where the profile V ∈ L1 (Rn ) decays exponentially at infinity (cf. Remark 15.4(ii) for an analogue in the model problem (15.1)). This corresponds to an intermediate behavior involving a balance between the diffusion and the nonlinear term. It is unknown whether this self-similar solution is unique, nor if there exist solutions satisfying (40.19) other than self-similar. (b) For estimates on I∞ (if finite) or on the growth rate of u(t) 1 otherwise, see [322], [70], [251]. An alternative proof of Theorem 40.10(i) based on multiplier arguments (instead of the Hopf-Cole transformation) is also given in [322]. (c) Estimates similar to (40.14) are also true for other Lq -norms [70]. Namely, for every q ∈ [1, ∞], there holds t(n/2)(1−(1/q)) u(t) − I∞ Gt q → 0,

t → ∞.

(d) For general solutions of (34.5) (assuming only u0 ∈ X+ but not u0 ∈ L1 (Rn )), some results on the asymptotic behavior can be found in [71], [252], [251], [491]. In particular it was shown in [71], [252] by Bernstein-type techniques, that any solution satisfies the global gradient estimate −1/p |∇u(x, t)| ≤ C(p) u0 1/p , ∞ t

x ∈ Rn , t > 0

(hence ∇u(t) ∞ → 0 as t → ∞). (e) The exponents p = 2 and p = pc are also critical in the local existenceuniqueness theory of problem (34.5) with irregular initial data u0 ; see [27], [71], [72].

364


40.3. Space profile of gradient blow-up In this subsection we study the space profile of GBU of solutions to (34.5) for p > 2. We shall restrict ourselves to one space dimension. Definition 40.12. Let Ω ⊂ Rn and consider problem (34.5). We say that x0 ∈ Ω is a GBU point (in finite or infinite time) if there exist sequences tj → Tmax (u0 ) and xj → x0 such that |ux (xj , tj )| → ∞. In order to formulate our results, it is convenient to introduce the steady states of (34.5) for n = 1. They will be useful again in the study of the time rate of GBU; see the proof of Theorem 40.19 in the next subsection. To describe these steady states, let us denote U (x) := dp x(p−2)/(p−1) ,

U (x) = dp x−1/(p−1) ,

x > 0,

(40.20)

where dp = (p − 2)−1 (p − 1)(p−2)/(p−1) and dp = (p − 1)−1/(p−1) . The function U ∈ C([0, ∞)) ∩ C 1 ((0, ∞)) is a “singular” steady state. Namely, it is a solution of V + V = 0, p

x > 0,

V (0) = 0,

(40.21)

which satisfies Ux (0) = ∞. Next, for each λ > 0, we put Uλ (x) := U (x + λ) − U (λ).

(40.22)

Each Uλ ∈ C 1 ([0, ∞)) also solves (40.21). Moreover we have Uλ (x) → ∞, as x → 0+ and λ → 0+, and Uλ (x) → U (x), uniformly for x ∈ [0, 1], as λ → 0+. Our first result gives bounds on ux away from x = 0 and 1. This shows in particular that GBU may occur only on the boundary. Proposition 40.13. Consider problem (34.5) with p > 2 and Ω = (0, 1). Let u0 ∈ X+ and 0 < t0 < T := Tmax (u0 ). There exists C1 > 0 such that, for all t0 ≤ t < T , (40.23) ux (x, t) ≤ U (x) + C1 x, 0 < x ≤ 1 and ux (x, t) ≥ −U (1 − x) − C1 (1 − x),

0 ≤ x < 1,

(40.24)

where U is defined by (40.20). In particular x = 0 and x = 1 are the only possible GBU points. The next result gives a precise description of the spatial profile around a GBU point. It is essentially due to [138] (where a slightly different problem, with nonhomogeneous boundary value at x = 1, was actually studied).


365

Theorem 40.14. Consider problem (34.5) with p > 2 and Ω = (0, 1). Let u0 ∈ X+ and assume that T := Tmax (u0 ) < ∞. (i) For each x ∈ (0, 1), the limits u(x, T ) := lim u(x, t) and ux (x, T ) := lim ux (x, t) exist and are finite. t→T

t→T

Moreover, the first (resp., second) limit is uniform (resp., locally uniform) for x ∈ (0, 1). (ii) If 0 is a GBU point, then lim ux (0, t) = ∞

(40.25)

t→T

and there exist C1 , C2 > 0 such that U (x) − C1 x ≤ u(x, T ) ≤ U (x) + C2 x2 , and

U (x) − C1 ≤ ux (x, T ) ≤ U (x) + C2 x,

0 < x ≤ 1/2

0 < x ≤ 1/2,

(40.26) (40.27)

where U is defined by (40.20). Similar estimates hold if 1 is a GBU point. As a preliminary to the proofs, we need the following simple properties of the time-derivative ut . (They are valid without restriction on n and will be used also in the next subsection.) We first note that ut ∈ C 2,1 (QT ) by parabolic regularity results, and that ut ∈ BC(Ω×[t0 , t1 ]), 0 < t0 < t1 < T , due to (35.3). The function w := ut satisfies wt − ∆w = a(x, t) · ∇w, x ∈ Ω, 0 < t < T, (40.28) w = 0, x ∈ ∂Ω, 0 < t < T, where a(x, t) = p|∇u|p−2 ∇u.

(40.29)

As an immediate consequence of (40.28) and of the maximum principle, we have: Lemma 40.15. Consider problem (34.5) with p > 1 and u0 ∈ X+ , and let 0 < t0 < T := Tmax (u0 ). There exists C1 > 0 such that |ut | ≤ C1 ,

x ∈ Ω, t0 ≤ t < T.

(40.30)

Proof of Proposition 40.13. Fix 0 < t0 < t < T and let y(x) = (ux (x, t) − C1 x)+ , where C1 is given by Lemma 40.15. The function y satisfies y + y p = (uxx − C1 )χ{ux >C1 x} + (ux − C1 x)p+ ,

for a.e. x ∈ (0, 1).

For each x such that ux (x, t) > C1 x, we have (y +y )(x) ≤ (uxx −C1 +|ux |p )(x) ≤ 0 by (40.30). Therefore, we have y + y p ≤ 0 a.e. on (0, 1). By integration, it follows 1 that y(x) ≤ ((p − 1)x)− p−1 , hence (40.23). As for (40.24), it follows by applying (40.23) to the function u(1 − x, t), which satisfies the same equation. p

For the proof of Theorem 40.14 we need the following lemma, which will provide a lower bound on the blow-up profile of ux at x = 0.

366


Lemma 40.16. Consider problem (34.5) with p > 2 and Ω = (0, 1). Let u0 ∈ X+ and 0 < t0 < T := Tmax (u0 ). There exists C3 > 0 such that, for all t0 ≤ t < T , 1−p 1−p ≤ ux (0, t) + C3 + (p − 1)x, 0 ≤ x ≤ 1. (40.31) (ux (x, t))+ + C3 Proof. Fix t ∈ [t0 , T ) and let 1/p

z(x) = (ux (x, t))+ + C1 , where C1 is given by Lemma 40.15. The function z satisfies 1/p p ≥ (uxx + |ux |p )χ{ux >0} + C1 ≥ 0 z + z p = uxxχ{ux >0} + (ux (x, t))+ + C1 a.e. on (0, 1) by (40.30). By integration, it follows that z 1−p (x) ≤ z 1−p (0)+(p−1)x. 1/p Using ux (0, t) ≥ 0, we obtain (40.31) with C3 = C1 . Proof of Theorem 40.14. (i) It follows from Proposition 40.13 that ux (·, t) is bounded in L1 (0, 1). This along with (40.1) implies {u(·, t) : t ∈ (0, T )} is precompact in C([0, 1]). Using Lemma 40.15, we deduce that limt→T u(x, t) exists, uniformly for x ∈ [0, 1]. Now fix t0 ∈ (0, T ). By Proposition 40.13 and Lemma 40.15, we deduce that ux , uxx ∈ L∞ ((ε, 1−ε)×(t0 , T )) for each ε ∈ (0, 1). Therefore, {ux (·, t) : t ∈ (t0 , T )} is precompact in C((0, 1)). Since uxt − uxxx = p|ux |p−2 ux uxx , parabolic estimates imply uxt ∈ Lq ((ε, 1−ε)×(t0 , T )) for each ε ∈ (0, 1) and each finite q. Consequently, ux ∈ BU C α ([ε, 1 − ε] × [t0 , T )) for each ε ∈ (0, 1) and some α ∈ (0, 1). We deduce that limt→T ux (x, t) exists, locally uniformly for x ∈ (0, 1). (ii) The upper estimates in (40.26), (40.27) follow from Proposition 40.13. To show the lower estimates, let us first note that, by assumption, there exist sequences tj → T and xj → 0 such that |ux (xj , tj )| → ∞, hence ux (xj , tj ) → ∞ due to (40.24). Moreover, by Lemma 40.15 and (34.5), we have uxx ≤ C1 , hence ux (0, t) ≥ ux (x, t) − C1 x,

(x, t) ∈ [0, 1] × [t0 , T ).

(40.32)

1−p

It follows that ux (0, tj ) → ∞. Put εj := (ux (0, tj ) + C3 ) → 0, where C3 is from Lemma 40.16. By that lemma, there exists η ∈ (0, 1) such that, for j large, ux (x, tj ) ≥ (εj + (p − 1)x)−1/(p−1) − C3 ,

0 < x < η,

hence

1 (p−2)/(p−1) (εj + (p − 1)x)(p−2)/(p−1) − εj − C3 x, 0 < x < η. p−2 Letting j → ∞, and since we already know that the limits in assertion (i) exist, we obtain the lower estimates in (40.26), (40.27) (η can be replaced by 1/2 by enlarging the constant C1 ). Finally, using (40.32), we may write, for each x ∈ (0, 1), u(x, tj ) ≥

lim inf ux (0, t) ≥ lim inf ux (x, t) − C1 x ≥ U (x) − C1 (x + 1), t→T

t→T

and (40.25) follows by letting x → 0.


367

Remark 40.17. The analogue of the upper estimate (40.23) is still true in higher dimensions. Namely, by using Bernstein-type techniques, it is shown in [500] that any solution of (34.5) with p > 2 satisfies |∇u(x, t)| ≤ C1 δ −1/(p−1) (x) + C2

in Ω × [0, Tmax (u0 )),

with C1 = C1 (p, n) > 0 and C2 = C2 (u) > 0.

40.4. Time rate of gradient blow-up We now study the time rate of GBU of solutions to (34.5) for p > 2, i.e.: the speed of divergence of ∇u(t) ∞ . We begin with lower estimates. The following theorem is due to [263]. Theorem 40.18. Consider problem (34.5) with p > 2 and Ω = Rn . Let u0 ∈ X+ and assume that T := Tmax (u0 ) < ∞. Then there exists C > 0 such that sup ∇u(s) ∞ ≥ C(T − t)−1/(p−2) ,

t → T.

(40.33)

s∈[0,t]

In one space dimension, we have the following more precise result from [138] (see also Remark 40.23 below). Theorem 40.19. Consider problem (34.5) with p > 2, n = 1 and Ω = (0, 1). Let u0 ∈ X+ and assume that T := Tmax (u0 ) < ∞. Then there exists C > 0 such that ux (t) ∞ ≥ C(T − t)−1/(p−2) ,

t → T.

(40.34)

Remarks 40.20. (i) Non self-similar GBU rate. The lower estimate (40.33) implies in particular that the GBU rate does not correspond to the one suggested by the self-similar invariance of the problem. Indeed, letting k = (p − 2)/(2(p − 1)), the scaling transformations Sλ : u → uλ (x, t) := λ2k u(λ−1 x, λ−2 t),

λ > 0,

leave invariant the equation in (34.5). This might allow for the existence of backward self-similar (classical) solutions of the form x (40.35) w(x, t) = (T − t)k V √ T −t (note that forward self-similar solutions in Rn exist for some p < 2, cf. Remark 40.11(a)). Now if there exists a nontrivial solution w of the form (40.35), say, on Ω = (0, ∞) with zero boundary condition at x = 0, and if ∇V ∈ L∞ , then

368


w will exhibit the GBU self-similar rate ∇w(t) ∞ = C(T − t)−1/(2(p−1)) . However, since 1/(p − 2) > 1/(2(p − 1)), Theorem 40.18 shows that no such solutions w exist and that the exponent of the GBU rate is always greater than that of the self-similar rate. A similar situation has been encountered for the supercritical model problem (cf. Section 23) and also for problem (38.3). (ii) A rough argument involving the variation-of-constants formula would also give a lower estimate (T − t)−1/(2(p−1)) . The upper blow-up rate estimate for problem (34.5) is still an open question. However, some results are known for the closely related one-dimensional problem: ⎫ ut − uxx = |ux |p + λ, 0 < x < 1, t > 0, ⎪ ⎬ u = 0, x ∈ {0, 1}, t > 0, (40.36) ⎪ ⎭ 0 < x < 1, u(x, 0) = u0 (x), with p > 2, λ > 0 and u0 ∈ X+ . Note that the local solution of (40.36) is nonnegative and uniformly bounded on finite time intervals (since u(x, t) := u0 ∞ + λt is a supersolution). Moreover, as a consequence of the proof of Theorem 40.2, gradient blow-up occurs whenever λ > λ0 (p) or u0 q ≥ C(p, q) for some q ∈ [1, ∞), where λ0 (p) and C(p, q) are suitable positive constants. Theorem 40.21. Consider problem (40.36) with p > 2 and λ > 0. Let u0 ∈ X+ ∩ C 2 ([0, 1]) be symmetric with respect to x = 1/2 and satisfy u0,xx + |u0,x |p + λ ≥ 0

in [0, 1].

(40.37)

If T := Tmax (u0 ) < ∞, then there exists C > 0 such that ux (t) ∞ ≤ C(T − t)−1/(p−2) ,

t → T.

(40.38)

Theorem 40.21 is a variant of a result of [263], where the authors considered the equation in (34.5) under inhomogeneous boundary conditions for n = 1. For that problem the upper GBU rate estimate was first conjectured in [138] on the basis of numerical simulations. Remarks 40.22. (i) Assumption (40.37) guarantees that the solution is nondecreasing in time. However, analogous assumption cannot be satisfied for problem (34.5). Indeed if u0 ∈ X+ ∩ C 2 ([0, 1]) verifies (40.37) with λ = 0, then u0 ≡ 0 (this follows for instance from the maximum principle). On the other hand, it is unknown if Theorem 40.21 (and the corresponding result in [263]) remains true without assumption (40.37) or in higher space dimensions. (ii) Estimate (40.38) is sharp. In fact, under the assumptions of Theorem 40.21, the lower estimate (40.34) follows from simple modifications of the proof of Theorem 40.18, along with (40.51) below. We first give (a variant of) the proof of Theorem 40.18 from [263]. It relies on linear regularity estimates applied to the equation for ut .


369

Proof of Theorem 40.18. Denote m(t) :=

sup Ω×[T /2,t]

|∇u| = max ∇u(s) ∞ , s∈[T /2,t]

T /2 ≤ t < T

(40.39)

(note that m(t) and ∇u(t) ∞ are continuous due to (51.27)). In this proof, C will denote positive constants, independent of t ∈ (T /2, T ), which may change from line to line and may depend on u. Step 1. We claim that w := ut satisfies ∇w(t) ∞ ≤ Cmp−1 (t),

T /2 < t < T.

(40.40)

Let t ∈ (T /2, T ), s ∈ (T /4, t), and put K = supσ∈[0,t−s] σ 1/2 ∇w(s + σ) ∞ . For τ ∈ (0, t − s), in view of (40.28), (40.29) and the variation-of-constants formula, we have

τ w(s + τ ) = e−τ A w(s) + e−(τ −σ)A (a · ∇w)(s + σ) dσ. 0

Using Proposition 48.7, Lemma 40.15, and the fact that 1 (1 − z)−1/2 z −1/2 dz, it follows that 0 ∇w(s + τ ) ∞ ≤ Cτ −1/2 w(s) ∞ + C

τ

0

τ 0

(τ − σ)−1/2 σ −1/2 dσ =

(τ − σ)−1/2 a · ∇w(s + σ) ∞ dσ

≤ Cτ −1/2 + Cmp−1 (t)K. Multiplying by τ 1/2 and taking the supremum for τ ∈ [0, t − s], we obtain K ≤ C + C(t − s)1/2 mp−1 (t)K. Now choosing s = t − (1/4) min T, (Cmp−1 (t))−2 ∈ (T /4, t), we obtain K ≤ 2C, hence ∇w(t) ∞ ≤ 2C(t − s)−1/2 ≤ 4C max(T −1/2 , Cmp−1 (t)). Since m is positive nondecreasing, this implies Claim (40.40). Step 2. We next claim that m is locally Lipschitz on (T /2, T ) and that m (t) ≤ Cmp−1 (t),

for a.e. t ∈ (T /2, T ).

(40.41)

Let T /2 < t < s < T . For any τ ∈ [t, s] and x ∈ Ω, it follows from the mean-value inequality and (40.40) that |∇u(x, τ ) − ∇u(x, t)| ≤ (τ − t) sup |∂t ∇u| ≤ C(τ − t)mp−1 (τ ) Ω×[t,τ ]

370


hence |∇u(x, τ )| ≤ |∇u(x, t)| + C(τ − t)mp−1 (τ ) ≤ m(t) + C(s − t)mp−1 (s). Taking supremum for (x, τ ) over Ω × [t, s], we get 0 ≤ m(s) − m(t) ≤ C(s − t)mp−1 (s). Since m is continuous, the claim follows. Finally, integrating (40.41) over (t, s) with T /2 < t < s < T , and using m(s) → ∞ as s → T , we infer that m(t) ≥ C(T − t)−1/(p−2) ,

(40.42)

which implies estimate (40.33).

We next give the proof of Theorem 40.19, based on (a simplification of) the idea from [138]. It relies on a completely different argument, involving the intersections of the solution with the singular steady state. Proof of Theorem 40.19. Recall that the steady states U and Uλ are defined in (40.20) and (40.22). Due to Proposition 40.3, we may assume, without loss of generality, that lim sup ux (0, t) = ∞. (40.43) t→T

Fix t0 ∈ [T /2, T ) and let

x0 := sup x ∈ (0, 1] : ux (·, t0 ) 0 small, x0 is well defined and x0 > 0. On the other hand, by definition, we have ux (x, t0 ) 0 small.

(40.44)

We claim that x0 ∈ (0, 1), hence ux (x0 , t0 ) = U (x0 ).

(40.45)

Indeed, otherwise x0 = 1, so that (40.44) implies u ≤ Uλ in [0, 1] × [t0 , T ) for λ > 0 small, due to the comparison principle. Therefore ux (0, t) ≤ Uλ (0) in [t0 , T ), contradicting (40.43). We next claim that sup u(x0 , t) ≥ U (x0 ). (40.46) t∈[t0 ,T )


371

Suppose the contrary. Then, for all λ > 0 small, we have u(x0 , t) ≤ Uλ (x0 ) in [t0 , T ). By (40.44) and the comparison principle, we deduce that u ≤ Uλ in [0, x0 ] × [t0 , T ), leading again to a contradiction. Now, using Lemma 40.15 and (40.46), we get

T

C1 (T − t0 ) ≥

ut (x0 , t) ≥ U (x0 ) − u(x0 , t0 ) = t0

0

x0

(U (x) − ux (x, t0 )) dx.

On the other hand, by (40.45), there clearly exists x1 ∈ (0, x0 ] such that U (x1 ) = max[0,1] ux (·, t0 ). Since U (x) − ux (x, t0 ) > 0 on (0, x0 ) by the definition of x0 , we obtain

x1 C1 (T − t0 ) ≥ (U (x) − ux (x, t0 )) dx 0

≥ U (x1 ) − x1 U (x1 ) =

2−p

ux (t0 ) 2−p U (x1 ) ∞ ≥ , (p − 1)(p − 2) (p − 1)(p − 2)

and the conclusion follows. Remark 40.23. More precise lower estimate. In addition to the hypotheses of Theorem 40.19, assume that u0 ∈ C 2 ([0, 1]) and denote C1 = max(u0,xx + |u0,x |p ). [0,1]

−1/(p−2) Then the proof of Theorem 40.19 provides the value C = (p−1)(p−2)C1 for the constant in (40.34). Finally, we give the proof of Theorem 40.21, based on the ideas in [263], which relies on the application of the maximum principle to a suitable auxiliary function. Note that this function (cf. w below) is quite different from the function J used in the proof of Theorem 24.1. Proof of Theorem 40.21. We consider the parabolic operator Lφ := φt − φxx − p|ux |p−2 ux φx . For σ ∈ (0, 1) and t0 ∈ (0, T ) to be chosen later, we introduce the auxiliary function w(x, t) := 1 +

ux 1 1 − , mσ (t) m(t)

x ∈ [0, 1],

t ∈ [t0 , T ),

where m(t) :=

max

(x,t)∈[0,1]×[t0 ,t]

|ux (x, t)| → ∞, as t → T.

(40.47)

372


Step 1. We shall show that for suitable t0 ∈ (0, T ) and C > 0, there holds w + u ≤ Cut

in (0, 1) × (t0 , T ).

(40.48)

We may assume m(t) ≥ 1 without loss of generality. Also, by the proof of Theorem 40.18, m is locally Lipschitz on (T /2, T ) and (40.41) is satisfied. A direct computation shows that Lw = −

σm 1 u x m ux + 1 + 1 − . mσ+1 m mσ m2

Since m ≥ 0 a.e., we have, in case |ux (x, t)| < Lw =

(40.49)

σ 1−σ (t), σ+2 m

ux ux m |ux | m + −σ + (σ + 1) ≤ −σ + (σ + 2) ≤0 mσ+1 m m1−σ mσ+1 m1−σ

(a.e. in t). On the other hand, if |ux (x, t)| ≥ (40.41), we have

σ 1−σ (t), σ+2 m

then by (40.49) and

1 u x m mp−1 C σ + 2 (p−2)/(1−σ) Lw ≤ 1 + σ ≤ C|ux | ≤ |ux |(p−1−σ)/(1−σ) . 2 2 m m m m σ We now choose σ = 1/(p − 1), so that (p − 1 − σ)/(1 − σ) = p. Thus, taking (p−2)/(1−σ) C˜ := C( σ+2 and using (40.47), we obtain, for t0 close to T , σ ) L(w + u) ≤

C˜ p−1 |ux |p − (p − 1)|ux |p + λ ≤ − |ux |p + λ, m 2

a.e. in (0, 1) × [t0 , T ).

If |ux (x, t)|p ≥ 2λ/(p − 1), then L(w + u) ≤ 0, whereas w(x, t) ≥ 1/2 otherwise (for t0 close to T ). In all cases we thus have L(w + u) ≤ 2λ(w + u), hence L e−2λt (w + u) ≤ 0 = Lut , a.e. in (0, 1) × [t0 , T ). (40.50) Due to our assumptions on u0 , u is symmetric with respect to x = 1/2, and we have ut ≥ 0 (and ut ≡ 0) in (0, 1) × (0, T ) by Proposition 52.19. In particular t → ux (0, t) is nonnegative and nondecreasing, and it follows from the proof of Proposition 40.3 that m(t) = ux (t) ∞ = ux (0, t),

t0 ≤ t < T,

(40.51)

by taking t0 closer to T if necessary. Consequently, [w + u](x, t) = 0 = ut (x, t),

x ∈ {0, 1}, t0 ≤ t < T.

(40.52)

On the other hand, the strong and Hopf maximum principles guarantee that ut (x, t0 ) > 0,

0 < x < 1,

utx (0, t0 ) > 0,

utx (1, t0 ) < 0.


In particular there exists C > 0 such that −2λt0 e (w + u) − Cut (x, t0 ) ≤ 0,

0 < x < 1.

373

(40.53)

Using (40.50), (40.52), (40.53), and the maximum principle (under the form of Proposition 52.8), we deduce e−2λt (w + u) ≤ Cut in (0, 1) × [t0 , T ), hence (40.48). Step 2. As a consequence of (40.48) and (40.51), we have Cut (x, t) [w + u](x, t) ≥ lim x→0+ x x w(x, t) 1 uxx (0, t) = wx (0, t) = − 1 + σ ≥ lim x→0+ x m (t) m(t) |ux (0, t)|p = up−1 ≥ x (0, t). m(t)

Cuxt (0, t) = lim

x→0+

By integration, we obtain ux (0, t) ≤ C(T − t)−1/(p−2) , which proves the result.

t → T,

Remarks 40.24. (a) Boundary layer. Under the assumptions of Theorem 40.21, for any K > 0, there exists c = c(K) > 0 such that ux (x, t) ≥ c(T − t)−1/(p−2)

for 0 < x < K(T − t)(p−1)/(p−2)

and t close to T (cf. [263]). This boundary layer estimate follows from the proof of Lemma 40.16, estimate (40.34) (cf. Remark 40.22(ii)) and (40.51). (b) Nonsymmetric initial data. In Theorem 40.21, the symmetry assumption on u0 can be removed. To show this, assuming that x = 0 (resp. x = 1) is a GBU point, one has to replace the interval [0, 1] by [0, 1/2] (resp. [1/2, 1]) in the proof of Theorem 40.21 (and in particular in the definition (40.47) of m(t)). One then uses the fact that ux is bounded away from the boundary (cf. the proof of Theorem 40.14) and that estimate (40.41) remains true (this follows from simple modifications of the proof of Theorem 40.18, using wφ instead of w in Step 1, where φ = φ(x) is a cut-off function equal to 1 on [0, 1/2] and to 0 near x = 1). On the other hand, by similar arguments, one can show that (under the hypotheses of Theorem 40.21 without the symmetry assumption) there holds |ux (x, t)| ≥ C(T − t)−1/(p−2) as t → T , for each GBU point x ∈ {0, 1}. (c) Continuation after GBU. For some results on continuation after GBU, see [193], [57], [201]. These references contain examples where all solutions can be continued in some sense after GBU (and not only threshold solutions, like in L∞ -blow-up — cf. Proposition 27.7 and Remark 27.8(c)).

374


41. An example of interior gradient blow-up In the previous section we studied the phenomenon of gradient blow-up on the boundary. The aim of this section is to provide a simple example of a different behavior, namely: interior gradient blow-up. Consider the following problem: ut − uxx = |u|m−1 u|ux|p ,

− 1 < x < 1, t > 0,

u(±1, t) = A± ,

t > 0,

⎫ ⎪ ⎬ ⎪ ⎭

− 1 < x < 1,

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

(41.1)

where p > 2, m ≥ 1, A− < 0 < A+ and u0 ∈ C 1 ([−1, 1]),

with u0 (−1) = A− ≤ u0 ≤ A+ = u0 (1) in [−1, 1].

(41.2)

Unlike in problem (34.5), the nonlinearity here changes sign, and this is the key feature that will allow for interior GBU rather than boundary GBU (see Remark 41.4 below). The examples in Remark 51.11 guarantee that (41.1) is locally well-posed (observe for instance that (41.1) can be converted to a problem with homogeneous boundary conditions by the change of unknown v(x, t) = u(x, t) − φ(x), where φ is an affine function such that φ(±1) = A± ). By the maximum principle and (41.2), we immediately obtain A− ≤ u(x, t) ≤ A+ ,

−1 ≤ x ≤ 1, 0 ≤ t < Tmax (u0 ).

(41.3)

Therefore Tmax (u0 ) < ∞ guarantees that GBU occurs (i.e. (40.2)). Theorem 41.1. Consider problem (41.1) with p > 2, m ≥ 1. There exists L = L(m, p) > 0 such that, if max(A+ , |A− |) > L, then Tmax (u0 ) < ∞ for any u0 satisfying (41.2). Theorem 41.1 is (a variant of) a result from [31]. The original proof was based on the construction of appropriate traveling wave sub- and supersolutions. We here present a simpler proof based on a multiplier argument similar to that in the proof of Theorem 40.2. Proof. Let k = (p + 2m)/(p − 2). In what follows, C and C1 denote any positive constant depending only on p, m. For all t ∈ (0, Tmax(u0 )), multiplying (41.1) by |u|k−1 u and integrating by parts, we get d dt

1

−1

1 |u|k+1 dx = ux |u|k−1 u −1 − k k+1

1 −1

2

|ux | |u|

k−1

1

dx + −1

|ux |p |u|m+k dx. (41.4)

41. An example of interior gradient blow-up

375

Next, by H¨ older’s inequality, we have

1

−1

|ux |p |u|m+k dx = C =C

1

(|u|(m+k)/p u)x p ≥ C

−1 (p+m+k)/p A+

1

p |u|(m+k)/p u x

−1 (p+m+k)/p p

+ |A− |

≥ CL

p+m+k

(41.5)

.

Moreover, since p(k − 1)/2 = m + k, Young’s inequality yields

1

k −1

|ux |2 |u|k−1 dx ≤

1 2

1

−1

|ux |p |u|m+k dx + C.

(41.6)

On the other hand, (41.3) implies ux (±1, t) ≥ 0.

(41.7)

Combining (41.4)–(41.7), and taking L = L(p, m) large enough, we obtain d dt

1

−1

|u|k+1 dx ≥ CLp+m+k − C1 ≥ 1.

Integrating and using (41.3), it follows that

t≤

1

−1

k+1 |u|k+1 dx ≤ 2 max(A+ , |A− |) ,

hence Tmax (u0 ) < ∞. Remark 41.2. It can be shown that if A+ and |A− | are small enough, then there exist stationary, hence global solutions (see [31]). In this case the argument of the above proof still shows that GBU occurs for suitably large initial data. The next result asserts that, for m = 1 and a suitable class of initial data, GBU occurs at a single interior point, namely x0 = 0. Moreover, an upper estimate is given for the final profile. A much more general result of interior GBU was proved in [31] (see Remark 41.4 below). However the proof therein is more delicate. Theorem 41.3. Consider problem (41.1) with p > 2, m = 1, and A± = ±A with A > L, where L is defined in Theorem 41.1. Let u0 ∈ C 2 ([−1, 1]) be an odd function satisfying u0,x ≥ 0,

u0,xx ≤ 0

in [0, 1],

u0 (1) = A,

u0,xx (1) + A|u0,x (1)|p = 0.

Then, there holds T := Tmax (u0 ) < ∞, lim ux (0, t) = ∞

t→T

(41.8)

376

and


0 ≤ ux (x, t) ≤ A|x|−1 ,

0 < |x| < 1, 0 < t < T.

(41.9)

Proof. By local uniqueness, we have u(−x, t) = −u(x, t). Let v = ux and w = uxx . By parabolic regularity results, we have v, w ∈ C 2,1 (QT ) ∩ C(QT ). We compute vt − vxx = (u|v|p )x = |v|p v + pu|v|p−2 vw. Due to (41.7) and u0,x ≥ 0 in [0, 1], the maximum principle implies v ≥ 0. Differentiating again in x, we get wt − wxx = (p + 1)v p w + p(uv p−1 w)x = a(x, t)w + b(x, t)wx , where a = (2p + 1)v p + p(p − 1)uv p−2 w and b = puv p−1 . Moreover w(1, t) = −uv p (1, t) ≤ 0 and, since u(0, t) = 0, we have w(0, t) = −uv p (0, t) = 0. We thus infer from the maximum principle that uxx = w ≤ 0,

0 ≤ x ≤ 1, 0 ≤ t < T.

For all 0 < x ≤ 1 and 0 ≤ t < T , it follows that ux (x, t) ≤ ux (0, t). Therefore, ux (0, t) = ux (t) ∞ , hence (41.8). On the other hand, by concavity, we have A ≥ u(x, t) − u(0, t) ≥ xux (x, t),

0 < x ≤ 1, 0 ≤ t < T,

hence (41.9). Remark 41.4. In fact, it was proved in [31] that for any m ≥ 1 and any initial data as in Theorem 41.1, interior GBU occurs, in the sense that ux remains bounded close to the boundary. Moreover, GBU may occur only at points “where u changes sign”; more precisely, for x0 ∈ (−1, 1), if u remains bounded away from 0 in a neighborhood of x0 as t → T , then ux remains bounded near x0 . The proof is based on Bernstein-type arguments.

Chapter V

Nonlocal Problems 42. Introduction In this chapter, we study various problems with nonlocal nonlinearities. The equations that we consider involve nonlocal terms taking the form of an integral in space, or in time. These terms may also be combined with local ones, either in an additive or in a multiplicative way. In Sections 43–44, we consider several problems with space integrals from the point of view of global existence, blow-up and a priori estimates. In particular, we study in some detail the asymptotic behavior of blowing-up solutions. The phenomenon of global blow-up appears as a typical feature of nonlocal problems. As an example of applied interest, we discuss the thermistor problem, which arises in the modeling of Ohmic heating. Fujita-type results for problems with space integrals are next described in Section 45. Finally, Section 46 is devoted to nonlocal problems (in time) with memory terms. Throughout this chapter we will only consider nonnegative solutions, except in Subsection 44.4. Unless otherwise specified, each of the problems below is locally well-posed for (nonnegative) L∞ initial data, and the (nonnegative) solution enjoys the regularity property (16.2) (see Example 51.13 and cf. Example 51.9). Also, we have the usual blow-up alternative in L∞ (cf. Proposition 16.1). As for the comparison principle, more care is necessary when considering nonlocal problems, since it may be valid for certain problems and fail for some others. This will be made precise whenever necessary.

43. Problems involving space integrals (I) We consider the following problem

ut − ∆u =

Ω

up (y, t) dy − kuq ,

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

x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0, x ∈ Ω,

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

(43.1)

where Ω ⊂ Rn is a bounded domain, p > 1, q ≥ 1, k ≥ 0 and u0 ∈ L∞ (Ω), u0 ≥ 0. Note that problem (43.1) with k = 0 is maybe the simplest analogue of the model

378

V. Nonlocal Problems

problem (15.1) for which the nonlocal, superlinear source term is given by a space integral.

43.1. Blow-up and global existence When k > 0, (43.1) involves a competition between nonlocal source and local damping terms. A basic question is to determine the conditions for global existence or nonexistence of solutions. An answer is provided by the following theorem [525], which in particular shows that the value q = p represents a critical blowup exponent. It also contains some information concerning the global asymptotic behavior. Theorem 43.1. Consider problem (43.1) with Ω bounded, p > 1, q ≥ 1, k ≥ 0, and 0 ≤ u0 ∈ L∞ (Ω). (i) Assume p > q or p = q and k < |Ω|. Then: (i1) there exists u0 such that Tmax (u0 ) < ∞; (i2) the trivial solution is locally exponentially stable, i.e.: for u0 ∞ small enough, u is global and u(t) ∞ decays exponentially to 0 as t → ∞. (ii) Assume p = q and k ≥ |Ω|. Then the trivial solution is globally exponentially stable, i.e.: all solutions of (43.1) are global, bounded and u(t) ∞ decays exponentially to 0 as t → ∞. (iii) Assume p < q and k > 0. Then: (iii1) all solutions of (43.1) are global and bounded; (iii2) if k is sufficiently large, then the trivial solution is globally exponentially stable; (iii3) if k is sufficiently small, then there exist positive stationary solutions. Proof. (i1) We first prove the existence of blowing-up solutions in the case p = q and k < |Ω|. Fix a subdomain Ω ⊂⊂ Ω, such that δ := (|Ω | − k)/2 > 0. There exists ψ ∈ D(Ω) such that ψ = 1 in Ω and we have

and

0 ≤ ψ ≤ 1 in Ω

ψ dx ≥ |Ω | = k + 2δ and ∆ψ ≥ −K Ω for some K > 0. Let y(t) = Ω u(t)ψ dx. Multiplying (43.1) by ψ, integrating by parts and using H¨ older’s inequality, we obtain

p p u∆ψ dx + ψ dx u dx − k u ψ dx ≥ −K u dx + 2δ up dx y (t) = Ω Ω Ω Ω Ω Ω

p−1

≥δ up dx + δ|Ω|1−p u dx −K u dx. Ω

Ω

Ω

43. Problems involving space integrals (I)

Setting g(t) = δ|Ω|1−p

Ω

379

p−1 1−p u dx − K and c = δ ψ −p , we deduce that ∞ |Ω|

y (t) ≥ cy p (t) + g(t)

u dx,

0 < t < Tmax (u0 ).

Ω

(43.2)

1−p p−1 . On the one hand ψ dx Now let u0 = µψ with µp−1 ≥ K δ max 1, |Ω| Ω we have g(0) ≥ 0. On the other hand, we get

p p p u0 dx − ku0 = µ ψ p dx − kψ p ≥ µp (|Ω | − k) = 2δµp ≥ Kµ ≥ −∆u0 , Ω

Ω

so that u ≥ u0 on (0, Tmax (u0 )) by the comparison principle (Proposition 52.25). It follows that g(t) ≥ 0 on (0, Tmax (u0 )) and (43.2) then implies Tmax (u0 ) < ∞. To prove the existence of blowing-up solutions in the case p > q, we just note that u satisfies

˜ p − Cu ut − ∆u ≥ up dx − ku Ω

for some 0 < k˜ < |Ω| and C > 0. The result then follows by an obvious modification of the proof in the case q = p. (i2) Let us prove the local exponential stability of the trivial solution. It obviously suffices to treat the case k = 0. Let Θ be defined in (19.27), and put z(x, t) = ε(1 + Θ(x))e−αt . Then, for α, ε > 0 sufficiently small, we have zt − ∆z = ε(−α(1 + Θ) + 1)e−αt ≥ 2ε e−αt

≥ εp e−pαt (1 + Θ)p dx = z p dx, Ω

Ω

t ≥ 0.

Therefore, if u0 ∞ ≤ ε, then z is a supersolution to (43.1), so that u is global and satisfies u(x, t) ≤ z(x, t) ≤ Ce−αt . The local exponential stability of the trivial solution also follows from abstract results in Appendix E (see Remark 51.20(ii)). (ii) Multiplying the equation by um (m ≥ 1), integrating by parts and using Hölder’s inequality, we have d dt

Ω

um+1 dx + m m+1

u Ω

m−1

2

|∇u| dx =

m

u dx Ω

≤ (|Ω| − k)

u dx − k p

Ω

Ω

um+p dx Ω

um+p dx ≤ 0.

It follows from the Poincaré inequality that

d 4m um+1 dx ≤ − |∇u(m+1)/2 |2 dx ≤ −C um+1 dx, dt Ω m+1 Ω Ω

380

hence


Ω

um+1 (t) dx ≤ M0 e−Ct ,

0 ≤ t < Tmax (u0 ).

If we choose m + 1 ≥ p, then (for different constants M0 , C > 0) u satisfies ut − ∆u ≤ M0 e−Ct , u = 0, u(x, 0) = u0 (x),

x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0, x ∈ Ω.

⎫ ⎪ ⎬ ⎪ ⎭

(43.3)

−αt Let now w(x, t) = M (1 + Θ(x))e , with Θ defined in (19.27). If we choose −1 α ≤ min (2(1 + Θ ∞ )) , C and M ≥ max(2M0 , u0 ∞ ), we then have

wt − ∆w = (−α(1 + Θ) + 1)M e−αt ≥ M0 e−Ct , and w(0, .) ≥ u0 , so that w is a supersolution of (43.3). It follows that u is global and decays exponentially to 0 as t → ∞. (iii1) To show that all solutions of (43.1) are global and bounded, it suffices to note that for any constant M > max u0 ∞ , (|Ω|k −1 )1/(q−p) , u ≡ M is a supersolution of (43.1). (iii2) Next, let us show the global stability of 0 for k large. By Young’s inequality, we have |Ω|up ≤ kuq + ε(k)u, where ε(k) = C(|Ω|, p, q)k −(p−1)/(q−p) , so that u satisfies

ut − ∆u ≤ up dx − |Ω|up + ε(k)u. Ω

For k ≥ k0 (|Ω|, p, q) sufficiently large (hence ε(k) small), the conclusion then follows from an easy modification of the proof of assertion (ii). (iii3) Finally, let us prove the existence of stationary solutions for k small. Let U = µΘ, with Θ again defined in (19.27) and µ = 2( Ω Θp dx)−1/(p−1) . For k > 0 sufficiently small we have

−∆U + kU q = (1 + kµq−1 Θq )µ ≤ 2p−1 µ =

U p dx. Ω

By a modification of Proposition 52.20, it follows that the solution of (43.1) with u0 = U is nondecreasing in time, and we already know that it is global and bounded. Now Example 51.39 and Proposition 53.8 guarantee that u(t) → V in L∞ (Ω) where V is a (classical) stationary solution, V ≥ U .


381

43.2. Blow-up rates, sets and profiles In this subsection, we study the blow-up asymptotics for problem (43.1). The methods and results of this subsection are from [480], except for Theorem 43.4 in the case p ≥ 2 [487] and Theorem 43.11(ii) (which is an improvement of [487]). We refer to [60], [119], [525] for earlier results on blow-up asymptotics for problem (43.1) and its variants. Our first result shows that blowing-up solutions to (43.1) exhibit global blow-up and can be described by a uniform blow-up profile in the interior of the domain. Theorem 43.2. Assume Ω bounded, p > q ≥ 1, k ≥ 0, and 0 ≤ u0 ∈ L∞ (Ω). Let u be the solution of (43.1) and assume that T := Tmax (u0 ) < ∞. Then we have − 1 1 1 lim (T − t) p−1 u(x, t) = lim (T − t) p−1 u(t) ∞ = (p − 1)|Ω| p−1 ,

t→T

t→T

(43.4)

uniformly on compact subsets of Ω. Since the solution vanishes on the boundary and blows up globally inside Ω, it follows that a boundary layer appears as t → T . The following result describes the behavior of the solution u near the blow-up time in the boundary layer. Theorem 43.3. Under the assumptions of Theorem 43.2, for all K > 0, there exist some constants C2 ≥ C1 > 0 and some t0 ∈ (0, T ), such that u satisfies δ(x) δ(x) u(t) ∞ ≤ u(x, t) ≤ C2 √ u(t) ∞ , C1 √ T −t T −t

(43.5)

√ for all (x, t) in Ω × [t0 , T ) such that δ(x) ≤ K T − t, From the right-hand side √ of (43.5), one deduces that the size of the boundary layer is at least of order T − t near √ the blow-up time, in the sense that u(x, t) = o( u(t) ∞ ), as t → T and δ(x)/ T − t → 0. However, estimate (43.5)√is not enough to conclude that the size of the boundary layer is exactly √ of order T − t, in the sense that u(x, t)/ u(t) ∞ → 1, as t → T and δ(x)/ T − t → ∞. The following theorem, though not very sharp regarding the actual behavior of the solution in the boundary layer, enables one to conclude that this is indeed true. Theorem 43.4. Under the assumptions of Theorem 43.2, for all ε > 0, there exists C(ε) > 0 such that T − t u(x, t) ≥ u(t) ∞ 1 − ε − C(ε) 2 , δ (x)

(x, t) ∈ Ω × [0, T ).

382


Therefore, we have ⎧ ⎪ ⎨ 1,

δ(x) → ∞, as t → T and √ T −t δ(x) → 0. as t → T and √ T −t √ In other words, the size of the boundary layer decays like T − t. u(x, t) −→ ⎪ u(t) ∞ ⎩ 0,

(43.6)

Remarks 43.5. (a) Comparison with the local model problem. For problem (15.1), we have seen that single-point blow-up occurs if Ω = BR and u ≥ 0 is radial nonincreasing (cf. Theorem 24.1). If moreover p < pS , then u(·, t) behaves like its maximum in space-time parabolas √ based at (0, T ), that is: u(x, t)/ u(t) ∞ → 1 as t → T , uniformly for |x| ≤ C T − t. In some cases (see Remark 25.8), it is even known that ⎧ |x| ⎪ 1, as t → T and ! → 0, ⎪ ⎨ (T − t)| log(T − t)| u(x, t) (43.7) −→ ⎪ u(t) ∞ |x| ⎪ ! ⎩ 0, as t → T and → ∞. (T − t)| log(T − t)| At the opposite, blow-up for problem (43.1) is global and solutions behave like their maximum everywhere outside of a space-time parabolic neighborhood of (∂Ω, T ) (compare formulas (43.6) and (43.7)). Problems (15.1) and (43.1) thus exhibit in some sense dual blow-up behaviors. (b) Asymptotic influence of the local damping term. It appears from Theorem 43.2 that the local damping term has no significant effect on the asymptotic behavior of solutions near the blow-up time if q < p. In the blow-up critical case q = p, k < |Ω|, which was studied in [487], this is no longer so: Blow-up is still global and uniform on compact sets, but the constant in the RHS of (43.4) −1/(p−1) has to be replaced by (p − 1)(|Ω| − k) . The proof of the above results relies on the study of linear problems with spatially homogeneous blowing-up source, of the form ⎫ x ∈ Ω, 0 < t < T, ut − ∆u = g(t), ⎪ ⎬ u = 0, x ∈ ∂Ω, 0 < t < T, (43.8) ⎪ ⎭ x ∈ Ω. u(x, 0) = u0 (x), If g is a given function, locally H¨ older continuous on [0, T ), and if u0 ∈ L∞ (Ω), then we know that (43.8) has a unique classical solution u ∈ C 2,1 (Ω × (0, T )), with u − e−tA u0 ∈ C(Ω × [0, T )). In what follows we shall use the following notation. We set

t

t g(s) ds and H(t) = G(s) ds. (43.9) G(t) = 0

0


383

We write u ∼ v for limt→T u(t)/v(t) = 1. As usual, λ1 and ϕ1 denote respec tively the first Dirichlet eigenvalue and eigenfunction, normalized by Ω ϕ1 dx = 1. Moreover we set Kρ = {x ∈ Ω : δ(x) ≥ ρ}, ρ > 0. For problem (43.8), we shall prove the following theorem. older Theorem 43.6. Assume Ω bounded, 0 ≤ u0 ∈ L∞ (Ω), g ≥ 0 locally H¨ continuous on [0, T ). Let u ≥ 0 be the solution of (43.8). Then we have lim sup u(x, t) ∞ = ∞

(43.10)

t→T

if and only if

T

0

g(s) ds = ∞.

(43.11)

Furthermore, if (43.10) or (43.11) is fulfilled, then lim

t→T

u(x, t) u(t) ∞ = lim = 1, t→T G(t) G(t)

(43.12)

uniformly on compact subsets of Ω. The proof of Theorem 43.6 is based on eigenfunction arguments, one-sided estimates of ∆u (obtained via the maximum principle), and the mean-value inequality for subharmonic functions. We need the following two simple lemmas. Lemma 43.7. (i) Under the assumptions of Theorem 43.6, we have u ≤ G(t) + u0 ∞ ,

(x, t) ∈ QT .

(43.13)

(ii) If moreover u0 ≡ 0, then ∆u ≤ 0,

(x, t) ∈ QT .

(43.14)

Proof. To prove (43.13) it suffices to notice that u(x, t) := G(t) + u0 ∞ is a supersolution to (43.8). To show (43.14) one could apply the maximum principle to the equation satisfied by ∆u, after showing that ∆u ∈ C([0, T ), L2 (Ω)). Alternatively, one can use the following simple argument: For each h ∈ (0, T ), the function v(x, t) := u(x, t + h)− u(x, t) satisfies ⎫ vt − ∆v = g(t + h) − g(t), x ∈ Ω, 0 < t < T − h, ⎪ ⎬ v = 0, x ∈ ∂Ω, 0 < t < T − h, (43.15) ⎪ ⎭ v(x, 0) = u(x, h), x ∈ Ω.

384


Since u(·, h) ≤ G(h) due to (43.13), we see that v(x, t) := G(t + h) − G(t) is a supersolution to (43.15), hence u(x, t + h) − u(x, t) ≤ G(t + h) − G(t). Dividing by h and letting h → 0, we get ut ≤ g(t) in QT , hence (43.14). As for the next lemma, a more accurate inequality will be given below to obtain precise boundary estimates, see (43.35). However this one is sufficient for the purpose of Theorem 43.6. Lemma 43.8. Assume Ω bounded and let z ∈ C 2 (Ω) satisfy z≥0 Then z(x) ≤

∆z ≥ 0,

and

C(Ω) n+1 δ (x)

x ∈ Ω.

(43.16)

Ω

x ∈ Ω.

z(y)ϕ1 (y) dy,

Proof. Fix x ∈ Kρ ( = ∅). By the mean-value inequality for subharmonic functions, we have

1 C(n) z(x) ≤ z(y) dy = n z(y) dy. |B(x, ρ/2)| B(x,ρ/2) ρ B(x,ρ/2) Since inf Kρ ϕ1 ≥ c1 (Ω)ρ and z ≥ 0, we deduce that

C(Ω) C(Ω) z(x) ≤ n+1 z(y)ϕ1 (y) dy ≤ n+1 z(y)ϕ1 (y) dy ρ ρ B(x,ρ/2) Ω and the lemma follows. Proof of Theorem 43.6. We first consider the case u0 ≡ 0. From (43.13), it is clear that (43.10) implies (43.11). Conversely, assume that (43.11) holds. Our aim is then to prove (43.12). Define

z(x, t) = G(t) − u(x, t) and β(t) = z(y, t) ϕ1 (y) dy. Ω

By Green’s formula, we have

g(t) − ut (y, t) ϕ1 (y) dy = − ∆u(y, t) ϕ1 (y) dy β (t) = Ω Ω

=− u(y, t) ∆ϕ1 (y) dy = λ1 u(y, t) ϕ1 (y) dy = −λ1 β(t) + λ1 G(t). Ω

Ω

Integrating this equation and using β(0) = 0, we obtain

t β(t) = λ1 eλ1 (s−t) G(s) ds ≤ λ1 H(t), 0

(43.17)


385

where H is defined by (43.9). Since z ≥ 0 and ∆z ≥ 0 by (43.13) and (43.14), Lemma 43.8 implies

C(Ω) λ1 C(Ω)H(t) z(x, t) ≤ n+1 z(y, t)ϕ1 (y) dy ≤ , x ∈ Kρ , t ∈ (0, T ). δ (x) Ω ρn+1 (43.18) For t close enough to T , we have G(t) > 0 by (43.11), and (43.13) and (43.18) give us C(Ω) H(t) u(x, t) ≤ n+1 , x ∈ Kρ . 0≤1− (43.19) G(t) ρ G(t) On the other hand, since G is nondecreasing, for all ε > 0 we have

0≤

H(t) ≤ G(t)

T −ε

G(s) ds, 0

G(t)

+ ε.

Using (43.11), we deduce that limt→T H(t)/G(t) = 0. In view of (43.19), this proves (43.12) for u0 ≡ 0. Finally, for general u0 ≥ 0, we write u = U + e−tAu0 , where U is the solution of (43.8) corresponding to u0 ≡ 0. By using e−tA u0 ∞ ≤ u0 ∞ , the general case easily follows from the case u0 ≡ 0. We are now in a position to prove Theorem 43.2. Proof of Theorem 43.2. Case 1: k = 0. We apply Theorem 43.6 with

up (y, t) dy,

g(t) :=

G(t) =

Ω

t

g(s) ds.

(43.20)

0

By (43.12) in Theorem 43.6, it follows that ∀x ∈ Ω,

lim up (x, t)/Gp (t) = 1.

t→T

Moreover, (43.13) implies 0 ≤ up (x, t)/Gp (t) ≤ 2 in Ω for t close enough to T . By Lebesgue’s dominated convergence theorem, we infer that

up (y, t) dy ∼ |Ω|Gp (t), t → T, Ω

hence

G (t) = g(t) ∼ |Ω|Gp (t),

(43.21)

or (G1−p ) ∼ −(p − 1) |Ω|. After integrating this equivalence between t and T , we obtain −1/(p−1) G(t) ∼ (p − 1)|Ω|(T − t) . (43.22)

386


The result finally follows by returning to (43.12). Case 2: k > 0. It requires some modifications of the arguments from the case k = 0 and from the proof of Theorem 43.6 (in particular we no longer consider the case u0 ≡ 0 separately). We only indicate the necessary changes. We first note that (43.13) and consequently (43.11) are still valid. As an analogue of (43.14) in Lemma 43.7, we next establish the inequality ∆u ≤ C1 := ∆u(·, T /2) ∞,

(x, t) ∈ Ω × [T /2, T ).

(43.23)

By the strong maximum principle, we have u > 0 in QT . Set v = ∆u and note that v ∈ C 2,1 (QT ) ∩ C(Ω × (0, T )) by parabolic regularity. Taking the Laplacian of equation (43.1) then yields vt − ∆v = −q(uq−1 v + (q − 1)uq−2 |∇u|2 ) ≤ −quq−1 v

in Ω × (0, T ),

with v(x, t) = −g(t) ≤ 0 on the boundary, where g is still defined by (43.20). Therefore, by the maximum principle, v cannot achieve an interior positive maximum, hence (43.23). Now set

|x|2 + u0 ∞ and β(t) = z(x, t) = G(t) − u(x, t) + C1 z(y, t) ϕ1 (y) dy. 2n Ω By (43.13) and (43.23), we have z≥0

and

∆z ≥ 0,

(x, t) ∈ Ω × (T /2, T ).

(43.24)

On the other hand, arguing as in the proof of Theorem 43.6, we obtain

u(y, t)ϕ1 (y) dy + k uq (y, t)ϕ1 (y) dy. β (t) = λ1 Ω

Ω

Integrating and using H¨ older’s inequality and (43.11), we get $ %

t

β(t) ≤ C 1 + uq (y, s) dy ds 0

Ω

1−(q/p)

≤ C + C(T |Ω|)

$ t

%q/p p

u (y, s) dy ds 0

(43.25) = o(G(t)),

Ω

as t → T . Owing to (43.24) we may apply Lemma 43.8, and using (43.25) we then conclude in a similar way as in the proof of Theorem 43.6 and Case 1. To prove the boundary estimates in Theorems 43.3 and 43.4, we return to problem (43.8) and introduce the following definition.


387

Definition 43.9. We say that g is sub-standard, resp. super-standard, if it satisfies the following power-like growth assumption

resp.

g(t)/G(t) ≤ k1 (T − t)−1 ,

as t → T ,

(43.26)

g(t)/G(t) ≥ k2 (T − t)−1 ,

as t → T ,

(43.27)

with constants k1 , k2 > 0. We say that g is standard if it satisfies (43.26) and (43.27). Note that if (43.26) holds, then g(t) ≤ C1 (T − t)−(k1 +1) as t → T . If (43.27) T holds, then g(t) ≥ C2 (T −t)−(k2 +1) as t → T , so that in particular 0 g(s) ds = ∞. Conversely, g is standard whenever, for instance, c1 (T − t)−α ≤ g(t) ≤ c2 (T − t)−α as t → T , for some α > 1 and c2 ≥ c1 > 0. Theorem 43.10. Assume Ω bounded, 0 ≤ u0 ∈ L∞ (Ω), g ≥ 0 locally H¨ older continuous on [0, T ), and (43.11). Let u ≥ 0 be the solution of (43.8). (i) Assume that g is super-standard. Then for all K > 0 there exist C1 > 0 and t1 ∈ (0, T ), such that δ(x) G(t), u(x, t) ≥ C1 √ T −t √ for all (x, t) in Ω × [t1 , T ) such that δ(x) ≤ K T − t. (ii) Assume that g is sub-standard. Then for all K > 0 there exist C2 > 0 and t2 ∈ (0, T ), such that δ(x) G(t), u(x, t) ≤ C2 √ T −t √ for all (x, t) in Ω × [t2 , T ) such that δ(x) ≤ K T − t. Theorem 43.11. Assume Ω bounded, 0 ≤ u0 ∈ L∞ (Ω), g ≥ 0 locally H¨ older continuous on [0, T ). Let u ≥ 0 be the solution of (43.8). (i) Assume that G is super-standard. Then T − t − u0 ∞ u(x, t) ≥ u(t) ∞ 1 − C 2 (43.28) δ (x) in Ω × [0, T ). (ii) Assume that g is super-standard and nondecreasing. Then for all ε > 0, there exists C(ε) > 0 such that T − t u(x, t) ≥ u(t) ∞ 1 − ε − C(ε) 2 δ (x)

(43.29)

in Ω × [0, T ). Note that estimate (43.28) is stronger than (43.29) when u(t) ∞ is large. However, the assumption that G is super-standard is too restrictive in practice. For

388


instance in case of problem (43.1) with k = 0, we have g(t) := Ω up (t) dx ∼ C(T − t)−p/(p−1) and G(t) ∼ C (T − t)−1/(p−1) by (43.21), (43.22), so that G is super-standard only for p < 2, whereas g is standard for all p > 1. For the proof of Theorem 43.10 we construct blowing-up sub-/supersolutions of barrier type (relative to interior or exterior tangent balls at boundary points). As for Theorem 43.11, it is based on refinements of the arguments leading to Theorem 43.6. Proof of Theorem 43.10. Step 1. We first claim that we need only consider the case u0 ≡ 0. Indeed, for general u0 , u may be decomposed as u = e−tA u0 + U , where U solves Ut −∆U = g(t) with 0 initial and boundary values. Using e−tA √ u0 ∈ C 1,0 (Ω×[T /2, T ]) and (43.11), we have 0 ≤ e−tA u0 ≤ C δ(x) ≤ ε δ(x) G(t)/ T − t, for all x ∈ Ω and t close enough to T . The claim follows. Step 2. We prove the lower estimate when u0 = 0. The basic idea is to seek a suitable subsolution. Since Ω is smooth, ∂Ω satisfies a uniform interior and exterior sphere condition i.e., for some R, R > 0 depending only on Ω, and for each point ξ ∈ ∂Ω, there exist some balls Bi (ξ) of radius R and Be (ξ) of radius R such that Bi (ξ) ∩ Ωc = {ξ} = Be (ξ) ∩ Ω. Now fix x0 ∈ Ω. Let ξ ∈ ∂Ω be such that δ(x0 ) = |x0 − ξ|, and let B be the ball containing Bi (ξ), tangent to both Bi (ξ) and Be (ξ), of radius R = max(R, δ(x0 )). By the definition of δ(x0 ), it is clear that B ⊂ Ω and that δ(x0 ) = dist(x0 , ∂B), with R ≤ R ≤ diam(Ω). Without loss of generality, we may also assume that B is centered at the origin. Define the space-time domain D = B × [0, T ), and divide D into two sub-regions as follows: R √ D1 : 0 ≤ d(x) < √ T − t, 2 T

R √ D2 : d(x) ≥ √ T − t, 2 T

where d(x) = dist(x, ∂B) = R − r, r = |x|. We next define $ % ⎧ d(x) R d(x) ⎪ ⎪ in D1 , ⎨ 4G(t) √T − t √ − √T − t T v(x, t) = ⎪ 2 ⎪ ⎩ G(t) R in D2 . T

(43.30)

(43.31)

It is clear that v ∈ C 1 (D), and v(·, t) ∈ H 2 (B), 0 < t < T . Moreover, v(x, 0) = 0 in B and v(x, t) = 0 for x ∈ ∂B. One then computes: $ / % $ %0 ⎧ 4d(x) G(t) d(x) d(x) R R ⎪ √ √ √ √ √ ⎪ + g(t) in D1 , − − ⎨ T −t T −t T −t T −t 2 T T vt (x, t) = ⎪ 2 ⎪ ⎩ g(t) R in D2 . T (43.32)


389

We have vr (x, t) = vrr (x, t) = 0

in D2 ,

(43.33)

while in D1 (where r ≥ R/2), we find that $ % 4G(t) −R 2(R − r) √ + √ vr (x, t) = √ T −t T −t T and vrr (x, t) =

−8G(t) T −t ,

so that

$ % n−1 R √ n−1 4G(t) 8nG(t) √ vr ≤ 2+ −∆v(x, t) = −vrr − T −t ≤ r T −t r T −t T

in D1 .

Therefore, we get ⎧ G(t) R2 R2 8nG(t) ⎪ ⎪ + g(t) + ⎨ T − t 4T T T −t vt − ∆v ≤ 2 ⎪ ⎪ ⎩ g(t) R in D2 . T

in D1 ,

Using the fact that g is super-standard, it follows that vt − ∆v ≤ C(R)g(t) in D, where C(R) = R2 /T + (8n + R2 /4T ) k2−1. Therefore, C(R)−1 v is a subsolution in D, and since u ≥ 0, the maximum principle implies u ≥ C(R)−1 v in D. On the other hand, for any K > 0, we have ⎧ 2R √ √ d(x) ⎪ if d(x)/ T − t ≤ R/2 T , ⎪ ⎨ √T G(t) √T − t , v(x, t) ≥ ⎪ √ √ R2 d(x) ⎪ R2 ⎩ G(t) ≥ G(t) √ , if R/2 T ≤ d(x)/ T − t ≤ K. T TK T −t √ Since δ(x0 ) = d(x0 ), we deduce that if δ(x0 ) ≤ K T − t, then δ(x0 ) , u(x0 , t) ≥ C1 G(t) √ T −t with

√ min(2R/ T , R2 /T K) min(R, R2 ) ≥ C(T, K, n, k ) 2 −1 1 + R2 R2 /T + (8n + R2 /4T ) k2 2 ≥ C(T, K, n, k2 ) min R , diam−1 (Ω) ,

C1 =

where we have used R ≤ R ≤ diam(Ω). Therefore, C1 can be chosen independent of x0 , and the desired lower estimate follows. Step 3. We prove the upper estimate when u0 = 0. To do so, we show that the function v of Step 2, suitably modified and multiplied by a large constant, becomes a supersolution.

390


Fixing x0 ∈ Ω, and keeping the notation of Step 2, we now set D = B × [0, T ), with B = Be (ξ) the exterior ball, of radius R, associated with ξ, where ξ ∈ ∂Ω is such that δ(x0 ) = |x0 − ξ|. It is clear that δ(x0 ) = dist(x0 , B ) and we may again assume that B is centered at the origin. Consider the function v defined by (43.31), where now d(x) = dist(x, B ) = r − R and R = R/n, and where D1 , D2 are still defined by (43.30). Formulae (43.32) and (43.33) are unchanged, whereas in D1 we now have c

$ % R 4G(t) 2(r − R) √ − √ vr (x, t) = √ T −t T −t T and vrr (x, t) =

−8G(t) T −t ,

so that

$ % n−1 R √ 4G(t) 4G(t) √ 2− −∆v(x, t) ≥ T −t ≥ T −t T −t R T Therefore, we get

in D1 .

⎧ 4G(t) ⎪ ⎪ in D1 , ⎨ T −t vt − ∆v ≥ 2 ⎪ ⎪ ⎩ R g(t) in D2 . T

Using the fact that g is sub-standard, we find that vt − ∆v ≥ C (R)g(t) in D, 2 where C (R) = min(4k1−1 , R n−2 T −1 ). It follows that C (R)−1 v is a supersolution in D, hence in Ω × [0, T ), and the maximum principle implies u ≤ C (R)−1 v, so that δ(x0 ) u(x0 , t) ≤ C2 G(t) √ in [0, T ), T −t with C2 = 4Rn−1 T −1/2 C (R)−1 , which proves the upper estimate. Proof of Theorem 43.11. Step 1. We shall show that u(x, t) ≥ G(t) − C(n)

H(t) , δ 2 (x)

(x, t) ∈ Ω × [0, T ).

(43.34)

In view of the maximum principle, it suffices to establish (43.34) for u0 ≡ 0, which we assume in the rest of this step. Estimate (43.34) is equivalent to the following inequality, which is an improved version of Lemma 43.8: sup z(x, t) ≤ x∈Kρ

C(n) H(t), ρ2

where z(x, t) := G(t) − u(x, t). Note that z ≥ 0 due to (43.13).

(43.35)


391

We first establish (43.34) when Ω is a ball BR (x0 ). We may assume x0 = 0 without loss of generality. Fix t ∈ (0, T ), x ∈ Ω and set ρ := R − |x|. Since ∆z ≥ 0 by (43.14), the mean-value inequality for subharmonic functions implies z(x, t) ≤ If ρ ≥ R/2, then z(x, t) ≤

C(n) ρn

z(y, t) dy.

(43.36)

B(x,ρ/2)

C(n) R1−n ρ

z(y, t) dy.

(43.37)

Kρ/2

Next suppose that ρ < R/2. Note that u(·, t) is radially symmetric due to u0 ≡ 0. Switching to polar coordinates, with z(y, t) = z(r, t), r = |y|, we may write

|x|+ρ/2

z(y, t) dy =

z(r, t)M (r) dr, |x|−ρ/2

B(x,ρ/2)

where M (r) = Surf(B(x, ρ/2) ∩ S(0, r)) and “Surf” denotes the surface measure. Observing that M (r) ≤ Surf(S(x, ρ/2)) ≤ C(n)ρn−1 , it follows from (43.36) that z(x, t) ≤

C(n) ρ

R−ρ/2

z(r, t) dr ≤ R/4

C(n) R1−n ρ

R−ρ/2

z(r, t)rn−1 dr, R/4

so that (43.37) is true in all cases. Still assuming Ω = BR , fix ρ ∈ (0, R) and t ∈ (0, T ). Since the RHS in (43.37) is a decreasing function of ρ and, for each x ∈ Kρ , ρ˜ := R − |x| ≥ ρ, we see that C(n) R1−n sup z(x, t) ≤ ρ x∈Kρ

z(y, t) dy.

(43.38)

Kρ/2

On the other hand, by (43.17) we have

z(y, t)ϕR (y) dy ≤ λR H(t),

(43.39)

BR

where λR is the first eigenvalue in BR and ϕR is the corresponding eigenfunction, normalized by BR ϕR = 1. By straightforward scaling arguments, we have λR = C(n) R−2

and

inf ϕR ≥ c(n) R−(n+1) ρ.

Kρ/2

(43.40)

392


Inequality (43.35) then follows by combining (43.38), (43.39) and (43.40). Therefore (43.34) is proved when Ω = BR (and we stress that the constant C(n) does not depend on R). To extend (43.34) to a general domain Ω, we fix x0 ∈ Ω and consider B = B(x0 , R) ⊂ Ω with R = δ(x0 ). Letting u be the solution of ut − ∆u = g(t) in B × (0, T ), with 0 initial and boundary conditions, the maximum principle implies u ≥ u. Since δ(x0 ) = dist(x0 , ∂B), (43.34) follows from the same inequality in B. Step 2. Let us show assertion (i) of the theorem. Since H(t) ≤ k2−1 (T − t)G(t) for t close to T by assumption, (43.34) and u ≥ 0 imply T − t u(x, t) ≥ G(t) 1 − C 2 , δ (x) +

(x, t) ∈ Ω × [T0 , T ),

(43.41)

for some T0 ∈ (0, T ). By taking a larger constant C ≥ (T − T0 )−1 diam2 (Ω), we see that (43.41) becomes in fact valid in Ω × [0, T ). Estimate (43.28) then follows by combining (43.41) and (43.13). Step 3. To show assertion (ii) we shall use Step 1 to derive an estimate on ut similar to (43.34), and then integrate over carefully chosen time intervals. Take u0 ≡ 0. Fix h > 0 and, for t ∈ [0, T − h), put v(·, t) = u(·, t + h) − u(·, t) and g˜(t) = g(t + h) − g(t). Note that g˜ ≥ 0 by assumption. The function v satisfies vt − ∆v = g˜(t), v = 0, v(x, 0) = u(x, h),

x ∈ Ω, 0 < t < T − h, x ∈ ∂Ω, 0 < t < T − h, x ∈ Ω.

⎫ ⎪ ⎬ ⎪ ⎭

(43.42)

Applying the result of Step 1 to problem (43.42), we obtain u(x, t + h) − u(x, t) ≥ G(t + h) − G(t) − C(n)

H(t + h) − H(t) δ 2 (x)

in Ω × [0, T − h). Dividing by h and letting h → 0, and next using the assumption that g is super-standard and u ≥ 0, we obtain ut (x, t) ≥ g(t)−C(n)

T − t G(t) ≥ g(t) 1−C 2 , 2 δ (x) δ (x) +

(x, t) ∈ Ω×[T0 , T ), (43.43)

for some T0 ∈ (0, T ). Fix γ > 1 and let Tγ := T − γ −1 (T − T0 ). Let (x, t) ∈ Ω × [Tγ , T ) and set tγ := T − γ(T − t) ∈ [T0 , T ). Integrating (43.43) over (tγ , t) yields T − tγ u(x, t) − u(x, tγ ) ≥ 1 − C 2 δ (x) +

t tγ

T − t G(t) − G(tγ ) . g(s) ds = 1 − γC 2 δ (x) +


393

Now, g being super-standard guarantees that s → (T − s)k2 G(s) is nondecreasing −t k2 for s close to T . Taking Tγ closer to T , it follows that G(tγ ) ≤ G(t) TT−t = γ −k2 for t ∈ [Tγ , T ), hence G(t)γ T − t u(x, t) ≥ (1 − γ −k2 )G(t) 1 − γC 2 , (x, t) ∈ Ω × [Tγ , T ). (43.44) δ (x) + By the maximum principle, (43.44) obviously remains true for u0 ≥ 0. Using (43.12), we get T − t u(x, t) ≥ u(t) ∞ 1 − 2γ −k2 − γC 2 , (x, t) ∈ Ω × [Tγ , T ). (43.45) δ (x) Moreover, replacing γC by a larger constant C(γ), we see that (43.45) becomes in fact valid in Ω × [0, T ). Estimate (43.29) finally follows by choosing γ = (2/ε)1/k2 . − p 1 Proof of Theorems 43.3 and 43.4. Let g(t) = |Ω|− p−1 (p − 1)(T − t) p−1 . It follows from Theorem 43.2 that, for all ε ∈ (0, 1), u satisfies (1−ε)g(t) ≤ ut −∆u ≤ (1 + ε)g(t) in Ω × [Tε , T ) for Tε sufficiently close to T . Taking, say, ε = 1/2, the maximum principle implies v ≤ u ≤ w in Ω× [T1/2 , T ), where v and w solve vt − ∆v = 12 g(t) and wt − ∆w = 32 g(t) in Ω × [T1/2 , T ) with 0 boundary values and initial conditions v(T1/2 ) = w(T1/2 ) = u(T1/2 ). Since g is standard, we deduce from Theorem 43.10 that v and w, hence u, satisfy the conclusion of Theorem 43.3. Since g is standard and nondecreasing, by using the same comparison argument (from below) for each ε ∈ (0, 1), along with Theorem 43.11(ii), we obtain Theorem 43.4. Remark 43.12. Other nonlocal problems. We refer to e.g. [160], [331] for results on systems of equations involving space integral terms. A different kind of nonlocal equations, of “localized” type, have also been studied by several authors. A typical example is: ut − ∆u = up (x0 (t), t), (43.46) with Dirichlet boundary conditions. Here x0 : [0, ∞) → Ω is a given (smooth) curve, which may be thought of as representing the location of a sensor driving the reaction in the whole domain. For equation (43.46), results on global (non-) existence can be found in [119], [479]. It is known that blow-up is global and the asymptotics of blow-up was studied in [524], [480], [487] (the last two references contain results similar to Theorems 43.2–43.4). (Un-)boundedness of global solutions was investigated in [462], [488]. For other equations involving localized terms, the blow-up set has been studied in [401], [221] (see Remark 44.4 below). Finally, results on systems of equations of localized type can be found in e.g. [411], [332].

394


43.3. Uniform bounds from Lq -estimates In this subsection we derive smoothing estimates for problem (43.1), obtained in [463], which are similar to those obtained in Sections 15 and 16 for the model problem (15.1). These estimates will be one of the main ingredients in the derivation of (universal) a priori bounds for global solutions in the next subsection. It turns out that the critical value of q for smoothing from Lq into L∞ is smaller than for problem (15.1) with the same p. Theorem 43.13. Consider problem (43.1) with Ω bounded, p > 1, q ≥ 1, k ≥ 0, and 0 ≤ u0 ∈ L∞ (Ω). Assume q > q˜c :=

np . n+2

There exists T = T ( u0 q ) > 0 and C = C(Ω, p, q) > 0 such that Tmax (u0 ) > T and u(t) ∞ ≤ C u0 q t−n/2q , 0 < t < T. Remarks 43.14. (a) The number q˜c in Theorem 43.13 is optimal (up to the equality case). Indeed, it was shown in [463] that for 1 ≤ q < q˜c (hence p > 1+2/n) there exists a sequence of nonnegative initial data {u0,j } ∈ L∞ (Ω), bounded in Lq , and such that Tmax (u0,j ) → 0. (b) On the other hand, it is not difficult to modify the arguments in the proof to show a local well-posedness result in Lq for q > q˜c , similar to Theorem 15.2. Proof of Theorem 43.13. By the comparison principle (Proposition 52.25), it is sufficient to establish the result for k = 0. We proceed in two steps. Step 1. We estimate the Lm -norm for m = max(p, q), by considering the quantity 1 n 1 H(t) := sup sα u(s) m , − . where α = 2 q m s∈[0,t] Using the variation-of-constants formula, e−tA χΩ m ≤ C e−tA χΩ ∞ ≤ C, m ≥ p and the Lq -Lm -estimate (cf. Proposition 48.4), we have t u(t) m ≤ t e α

α

−tA

u0 m + t

≤ C u0 q + Ct

α 0

t

α 0

u(s) pp e−(t−s)A χΩ m ds

t

u(s) pm ds

≤ C u0 q + Ct H (t) α

t

p

s−pα ds.

0

Since pα < 1 due to q > q˜c , by taking the supremum over (0, τ ) we obtain H(τ ) ≤ C u0 q + Cτ 1−(p−1)α H p (τ ),

0 < τ < Tmax (u0 ).


Let

T := min 1, ((2C)−p u0 1−p )1/(1−(p−1)α) . q

395

(43.47)

We claim that H(τ ) ≤ 2C u0 q ,

0 < τ < min(T, Tmax (u0 )).

(43.48)

Indeed otherwise, since H(t) is continuous and H(0) = 0 (due to the regularity of u), there exists a first τ < min(T, Tmax(u0 )) such that 2C u0 q = H(τ ) ≤ C u0 q + Cτ 1−(p−1)α (2C u0 q )p hence τ ≥ T : a contradiction. Step 2. For 0 < t < min(T, Tmax (u0 )), arguing as in Step 1 and using (43.47), (43.48) and α ≤ n/2q, we get

t tn/2q u(t) ∞ ≤ tn/2q e−tA u0 ∞ + tn/2q u(s) pp e−(t−s)A χΩ ∞ ds 0

t n/2q u(s) pm ds ≤ C u0 q + Ct 0

t n/2q p H (t) s−pα ds ≤ C u0 q + Ct 0

≤ C u0 q + CT 1−pα+n/2q u0 pq ≤ C1 u0 q . It follows in particular that Tmax (u0 ) > T and the theorem is proved.

43.4. Universal bounds for global solutions In this subsection we prove universal bounds for global solutions of problem (43.1). It turns out that such bounds are true for all p > 1, in sharp contrast with the model problem (15.1) (where even a priori estimates fail for p ≥ pS , cf. Theorem 28.7). The following result is due to [463]. Theorem 43.15. Consider problem (43.1) with Ω bounded, p > 1 and k = 0. For all τ > 0, there exists C(Ω, p, τ ) > 0 such that any global nonnegative solution satisfies u(t) ∞ ≤ C(Ω, p, τ ), t ≥ τ. (43.49) As an important ingredient of the proof, we first establish uniform a priori estimates for global solutions. Note that the problem does not seem to admit an energy functional and that the proof, based on maximum principle arguments, is completely different from that of Theorem 22.1.

396


Proposition 43.16. Consider problem (43.1) with Ω bounded, p > 1 and k = 0. For all M > 0, there exists K(Ω, p, M ) > 0 such that any global nonnegative solution with u0 ∞ ≤ M satisfies u(t) ∞ ≤ K,

t ≥ 0.

(43.50)

Proof. In this proof, we denote g(t) := Ω up (t) dx and assume that u0 ∞ ≤ M . Step 1. We first establish a (universal) integral bound on the source term:

t+1

g(s) ds ≤ C(Ω, p),

t ≥ 0.

(43.51)

t

We argue as in the proof of Theorem 17.1 and denote y = y(t) := Ω u(t)ϕ1 dx. Multiplying the equation with ϕ1 , integrating by parts and using Ω ϕ1 dx = 1, we obtain

y + λ1 y = up dx. (43.52) Ω

By Hölder’s inequality, we deduce that y ≥ −λ1 y + C1 y p 1−p with C1 = ϕ1 −p . It follows that y(t) ≤ C2 := (λ1 /C1 )1/(p−1) for all t ≥ 0, ∞ |Ω| since otherwise u cannot exist globally. Integrating (43.52) in time, we deduce (43.51) with C = (1 + λ1 )C2 . Step 2. This is the main step: We shall show that u becomes eventually monotone if g(t) reaches a suitably large value. Comparison with the solution of the ODE y = |Ω|y p , y(0) = M , shows that there exists t0 = t0 (M ) > 0 such that

u(t) ∞ ≤ 2M,

0 < t ≤ t0 .

(43.53)

Now Lp - and Schauder estimates guarantee that there exists K1 = K1 (M ) > 0 such that ∆u(t0 ) ∞ ≤ K1 . (43.54) We claim that: if g(t1 ) ≥ K1 for some t1 ≥ t0 , then ut ≥ 0 in Ω × [t1 , ∞).

(43.55)

Thus assume t1 ≥ t0 and g(t1 ) ≥ K1 , and pick t2 ∈ [t0 , t1 ] such that g(t2 ) = max g(t) ≥ K1 . [t0 ,t1 ]

(43.56)


397

Let v := ∆u and w =: ut . By parabolic regularity results, we have v, w ∈ C 2,1 (QT ) ∩ C(Ω × (0, T )). Since v satisfies ⎫ x ∈ Ω, t > t0 , vt − ∆v = 0, ⎪ ⎬ v = −g(t), x ∈ ∂Ω, t > t0 , ⎪ ⎭ x ∈ Ω, v(x, t ) = ∆u(x, t ), 0

0

we deduce from the maximum principle, (43.54) and (43.56) that ∆u ≥ min min ∆u(·, t0 ), −g(t2 ) = −g(t2 ), x ∈ Ω, t ∈ [t0 , t1 ]. Ω

Consequently, ut (·, t2 ) = ∆u(·, t2 ) + g(t2 ) ≥ 0. Since w satisfies

wt − ∆w = p

up−1 w dy, Ω

w = 0, w(x, t2 ) ≥ 0,

x ∈ Ω, t > t2 , x ∈ ∂Ω, t > t2 , x ∈ Ω,

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

where up−1 ≥ 0, we deduce from the maximum principle for nonlocal equations (see Proposition 52.24) that ut ≥ 0 in Ω × [t2 , ∞), which implies the claim. Step 3. We next deduce a uniform estimate on the source term: there exists K2 = K2 (M ) > 0 such that g(t) ≤ K2 ,

t ≥ 0.

Indeed, if g(t1 ) ≥ K1 for some t1 ≥ t0 , then

t+1 g(s) ds ≤ C(Ω, p), g(t) ≤

(43.57)

t ≥ t1

t

by (43.55) and (43.51). Consequently, taking also (43.53) into account, we get (43.57) with K2 := max(K1 , C(Ω, p), |Ω|(2M )p ). Step 4. Conclusion. Let Θ be defined in (19.27). Owing to (43.57), we see that u := u0 ∞ + K2 Θ is a supersolution to (43.1). Consequently, (43.50) with K = u0 ∞ + K2 Θ ∞ follows from the comparison principle (Proposition 52.25). Proof of Theorem 43.15. By (43.51), there exists t0 ∈ (0, τ /2) such that u(t0 ) p ≤ C(Ω, p)τ −1/p . Since p > q˜c = np/(n + 2), applying Theorem 43.13, we infer the existence of t1 ∈ (t0 , τ ) such that u(t1 ) ∞ ≤ C(Ω, p, τ ). (43.58) Estimate (43.49) finally follows by combining (43.58) and (43.50) (taking t1 as initial time).

398


44. Problems involving space integrals (II) In this section, we consider a different class of nonlocal equations, of the form

m g(u) dx f (u), (44.1) ut − ∆u = Ω

with Ω bounded and m ∈ R, m = 0.

44.1. Transition from single-point to global blow-up We have seen in the previous section that purely nonlocal power nonlinearities give rise to global blow-up with a uniform profile (for all nonglobal solutions), whereas purely local power nonlinearities produce single-point blow-up in the radial nonincreasing case (cf. Theorem 24.1). In order to understand the transition between these two complementary situations, it is natural to consider equation (44.1) with f (u) = uq , g(u) = up−q , p > 1, 0 < q 0, ⎪ ⎬ Ω (44.2) u = 0, x ∈ ∂Ω, t > 0, ⎪ ⎪ ⎪ ⎭ x ∈ Ω. u(x, 0) = u0 (x), In what follows we assume u0 ∈ L∞ (Ω), u0 ≥ 0, u0 ≡ 0, and we shall denote

t

up−q (y, t) dy, G(t) := g(s) ds. g(t) = Ω

0

Remark 44.1. Non-Lipschitz case. Problem (44.2) is well-posed for q ≥ 1 and p ≥ q + 1. In the non-Lipschitz cases 0 < q < 1 and/or 0 < p − q < 1, existence of a local classical solution can still be shown, either by using the Schauder fixed point theorem, or by an approximation procedure (replacing the initial and boundary conditions by uε (x, 0) = u0 (x) + ε and uε (x, t) = ε, respectively). However, local uniqueness seems to be unknown in this case, and assertion (i) of Theorem 44.2 applies to any maximal solution starting from u0 . By a simple modification of the proof of Theorem 17.1, any solution u starting from suitably large u0 will blow up in a finite time T = T (u), in the sense that lim supt→T u(t) ∞ = ∞. The following result shows that the occurrence of single-point vs. (uniform or nonuniform) global blow-up depends in a precise way on the values of q. Observe that the rate of the (uniform) global blow-up does not change when q varies in [0, 1) and that the bifurcation to single-point for q in (1, p], occurs through a nonuniform global blow-up at q = 1. Theorem 44.2 is a variant of results combined from [157] and [331], except for the blow-up rate estimates (44.6)–(44.7) which are consequences of Proposition 44.3 below.

44. Problems involving space integrals (II)

399

Theorem 44.2. Assume Ω bounded, p > 1, 0 < q < p, and 0 ≤ u0 ∈ L∞ (Ω). Let u be a nonglobal solution of problem (44.2) and denote by T its maximal existence time. (i) If 0 < q < 1, then blow-up is global and uniform. More precisely: − 1 1 1 lim (T − t) p−1 u(x, t) = lim (T − t) p−1 u(t) ∞ = (p − 1)|Ω| p−1 ,

t→T

t→T

(44.3)

uniformly on compact subsets of Ω. (ii) If q = 1, then blow-up is global and nonuniform. More precisely: u(x, t) = k(t) e−tA u0 (x),

where k(t) ∼ C(T − t)−1/(p−1) as t → T

(44.4)

for some constant C > 0 depending on u0 . (iii) Assume 1 < q < p, Ω = BR , u0 ∈ C 1 (Ω) radial nonincreasing, with u0 (x) = 0 for |x| = R. Then single-point blow-up occurs at x = 0. More precisely, for any α > 2/(q − 1) there exists Cα > 0 such that u(x, t) ≤ Cα |x|−α ,

0 < |x| < R, 0 < t < T.

(44.5)

Assume in addition that p − q < n(q − 1)/2. Then we have u(t) ∞ ≥ C1 (T − t)−1/(p−1) ,

0 0 in QT by the strong maximum principle, we have vt − ∆v = u−q (ut − ∆u) + qu−q−1 |∇u|2 ≥ g(t).

400


By using (44.9), Theorem 43.6 and the maximum principle, it follows that, uniformly on compact subsets, lim inf t→T v(x, t)/G(t) ≥ 1, hence lim

t→T

v(x, t) =1 G(t)

(44.10)

by (44.8). Arguing as in the proof of Theorem 43.2 for k = 0, we obtain after some calculations −(1−q)/(p−1) G(t) ∼ (1 − q)−1 (p − 1)|Ω|(T − t) . Returning to (44.10), (44.8) and using u = ((1 − q)v)1/(1−q) we obtain (44.3). (ii) For q = 1, by direct calculation one checks that the solution of (44.2) can be written as u(x, t) = eG(t) e−tA u0 , and we have G(t) → ∞ as t → T . Consequently, g(t) = e

(p−1)G(t)

Ω

e−tA u0

p−1

dx

hence d −(p−1)G(t) e = −(p − 1)g(t)e−(p−1)G(t) → −C, dt

C := (p − 1)

Ω

(e−T A u0 )p−1 dx,

as t → T . By integration, we obtain eG(t) ∼ C 1/(p−1) (T − t)−1/(p−1) and (44.4) follows. (iii) The proof of (44.5) is very similar to that of Theorem 24.1. The variables f , f now stand for f = f (t, u) = g(t)uq , f = g(t)quq−1 , and J is defined by (24.3) with 1 < γ < q. The main difference is that the condition H ≥ 0 becomes equivalent to g(t)(q − γ)uq−1 + (n + δ)δr−2 ≥ 2εγ(1 + δ)uγ−1 rδ ,

(44.11)

instead of (24.5). Since

g(t) ≥

Ω

e−tA u0

p−q

dx ≥ c > 0,

0 ≤ t < T,

(44.12)

(44.11) is satisfied if ε is small enough. To show the blow-up estimates (44.6)–(44.7) in Theorem 44.2(iii), we first establish the following more general result, where the upper bound will be proved by using arguments from [425] (cf. Theorem 26.8).


401

Proposition 44.3. Let T > 0, p > 1, and let a ∈ C([0, T )) be nonnegative and bounded. Let 0 ≤ u ∈ C 2,1 (BR × (0, T )) be a radial nonincreasing solution of the equation ut − ∆u = a(t) up , x ∈ BR , 0 < t < T, such that limt→T u(t) ∞ = ∞. (i) There exists C1 > 0 such that u(t) ∞ ≥ C1 (T − t)−1/(p−1) ,

0 < t < T.

(ii) Assume in addition that := limt→T a(t) exists in (0, ∞) and that p < pS . Then there exists C2 > 0 such that u(t) ∞ ≤ C2 (T − t)−1/(p−1) ,

0 < t < T.

(44.13)

Proof. (i) Since N (t) := sup|x| 2k(T − sk )−1 = 2kd−1 (sk ). (44.14) It follows from Lemma 26.11 that there exists tk ∈ (0, T ) such that M (tk )d(tk ) > 2k,

(44.15)

M (tk ) ≥ M (sk )

(44.16)

and M (t) ≤ 2M (tk ) for all t ∈ (0, T ) ∩ (tk − kM −1 (tk ), tk + kM −1 (tk )).

(44.17)

Note that, by (44.14) and (44.16) we have tk → T.

(44.18)

For k large, we deduce from (44.15) that kM −1 (tk ) < d(tk ) = T − tk , so that (44.17) rewrites as M (t) ≤ 2M (tk ) for all t ∈ (tk − kM −1 (tk ), tk + kM −1 (tk )).

(44.19)

Now we rescale uk by setting λk := M −1 (tk ) → 0

(44.20)

402


and 1/(p−1)

vk (y, s) := λk

1/2

uk (λk y, tk + λk s),

˜ k := {|y| < Rλ−1/2 } × (−k, k). (y, s) ∈ D k

The function vk solves ˜ k. (y, s) ∈ D

∂s vk − ∆y vk = a(tk + λk s)vkp ,

(44.21)

Moreover we have vk (0, 0) = 1 and (44.19) implies 0 ≤ vk ≤ C := 21/(p−1) ,

˜ k. (y, s) ∈ D

(44.22)

By using (44.21), (44.22), (44.20), (44.18), interior parabolic estimates and the embedding (1.2), we deduce that some subsequence of vk converges in C α (Rn+1 ), 0 < α < 1, to a (bounded classical) solution v ≥ 0 of vt − ∆v = v p ,

x ∈ Rn , s ∈ R.

Moreover, v is radial nonincreasing and satisfies v(0, 0) = 1. This contradicts Theorem 21.1. End of proof of Theorem 44.2. In view of Proposition 44.3, to show (44.6) and (44.7), it suffices to verify that g(t) → ∈ (0, ∞),

as t → T .

(44.23)

Due to (44.5) and p − q < n(q − 1)/2, the function g(t) is bounded on [0, T ). By (44.5), parabolic estimates and the embedding (1.2), it follows that, for some ν ∈ (0, 1), u ∈ BU C ν ({γ < |x| < 1 − γ} × (T /2, T )) for each γ > 0. Consequently, for all x ∈ B(0, 1) \ {0}, limt→T u(x, t) exists and is finite. Using (44.5) and p − q < n(q − 1)/2 again, along with the dominated convergence theorem and (44.12), we obtain (44.23). Remark 44.4. Problems involving localized nonlinearities. A different type of competition between local and nonlocal reaction terms has been studied in [401], [221] for the following variant of equation (43.46): ut − ∆u = uq (x0 , t) + up p > 1, q > 0, x0 ∈ Ω, with Dirichlet boundary conditions, when Ω is a ball BR and u0 is radial decreasing. Interestingly, the critical condition is different depending on the location of x0 . Namely, for x0 = 0, blow-up is always global if p ≤ q + 1, while single-point blow-up occurs for some u0 if p > q + 1. Next assume x0 = 0. If p < q, then both global and single-point blow-ups occur, and there are no other possibilities. On the contrary, if p > q (or p = q > 2), then only single-point blow-up occurs.


403

44.2. A problem with control of mass We now consider equation (44.1) with f (u) = g(u) = up , p > 1, and m = −1, under Neumann boundary conditions, that is: ut − ∆u =

p

u (y, t) dy

−1

x ∈ Ω, t > 0,

p

u ,

Ω

x ∈ ∂Ω, t > 0,

uν = 0,

x ∈ Ω.

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

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

(44.24)

In what follows we assume u0 ∈ L∞ (Ω), u0 ≥ 0, u0 ≡ 0, and we shall denote

up (y, t) dy

k(t) =

−1

.

Ω

As in Chapters I and II we shall use the notation psg =

∞

if n ≤ 2,

n/(n − 2) if n > 2.

Let us first observe that, by integrating the equation, we immediately obtain

u(t) dx = t + Ω

Ω

u0 dx.

(44.25)

This means that the total “mass” is controlled. We shall first investigate under what conditions the solutions of (44.24) blow up or exist globally. On a heuristic level one can expect that, when u becomes large in some sense, then the factor k(t) might become large, too, and have a stabilizing effect which could prevent blow-up. Interestingly, whether or not this possible stabilizing effect is effective depends in a sharp way on the relation between the exponent p and the space dimension n. The following result is due to [284]. Theorem 44.5. Consider problem (44.24) with Ω bounded, p > 1, and 0 ≤ u0 ∈ L∞ (Ω), u0 ≡ 0. (i) If p < psg , then Tmax (u0 ) = ∞ for all u0 . (ii) Assume n ≥ 3, p > psg and let Ω = B1 . Then there exists u0 such that Tmax (u0 ) < ∞. Note that since the solution stays bounded in L1 , it is clear that for radial nonincreasing solutions, blow-up can occur only at the origin. As a corollary to the proof of Theorem 44.5, one obtains the following blow-up profile estimate.

404


Theorem 44.6. Consider problem (44.24) with n ≥ 3, p > psg and Ω = B1 . Then there exists 0 ≤ u0 ∈ L∞ (Ω), u0 ≡ 0, radial nonincreasing, such that T := Tmax (u0 ) < ∞, u exhibits single-point blow-up at x = 0, and u satisfies 2

u(x, t) ≤ Cε |x|− p−1 −ε ,

x ∈ Ω, 0 < t < T,

for each ε > 0.

(44.26)

Moreover, (44.26) is optimal, in the sense that it cannot be satisfied for any ε < 0. As for the blow-up rate, we have the following result, which is a consequence of Proposition 44.3. Theorem 44.7. Consider problem (44.24) with p > 1 and Ω = B1 . Let 0 ≤ u0 ∈ L∞ (Ω), u0 ≡ 0, be radial nonincreasing and assume that T := Tmax (u0 ) < ∞. (i) There exists C1 > 0 such that u(t) ∞ ≥ C1 (T − t)−1/(p−1) ,

0 < t < T.

(ii) Assume in addition that p < pS and that u satisfies (44.26). Then there exists C2 > 0 such that u(t) ∞ ≤ C2 (T − t)−1/(p−1) ,

0 < t < T.

(44.27)

Remarks 44.8. (a) Global solutions. For any p > 1, (44.24) admits global solutions for arbitrarily large initial data. Namely it suffices to take homogeneous initial data u0 = M (with any M > 0) and u is then given by solving the ODE, i.e.: uM (x, t) = M + |Ω|−1 t. We thus observe that it is the “shape” of u0 , rather than its size, which causes blow-up. On the other hand, all global solutions of (44.24) are unbounded, due to (44.25). (b) Failure of the comparison principle. Problem (44.24) admits no comparison principle. For instance taking Ω = B1 and u a blow-up solution as in Theorem 44.6 (ii), we see that u(·, 0) < M for M large but u eventually intersects the solution uM (x, t) = M + |Ω|−1 t. (c) Interpretation of the critical exponent. Observe that k(t) is bounded due to (44.25) and H¨ older’s inequality and that k(t) vanishes if and only if u(t) p blows up. This allows for a heuristic interpretation of the value of the critical exponent in Theorem 44.5 if we put problem (44.24) in parallel with the model equation ut − ∆u = up . Indeed the supercriticality condition for the Lp -norm is given by p > n(p − 1)/2 that is, p < psg . More precisely, for the model problem (15.1), if p < psg , then u(t) p blows up whenever u is nonglobal (cf. Theorem 15.2), whereas if p > psg , then there exist solutions (in a ball) such that u(t) p remains bounded (cf. Theorem 24.1 and Corollary 24.2). The idea of the proof of Theorem 44.5(ii) below is precisely to use (a nontrivial modification of) the method in


405

Theorem 24.1 to construct initial data which yield a blow-up profile belonging to Lp and provide a control of k(t) from below. (d) Critical case. In the critical case p = psg , it is proved in [284] that the solution exists globally if Ω u0 dx is large enough. (e) Explicit examples of initial data in Theorem 44.5(ii) are constructed in Lemma 44.10 below. Namely blow-up occurs whenever u0 ∈ C 2 (B 1 ) is radial and satisfies (44.33)–(44.37) with β > 0 small (depending on n, p) and M > 0 large (depending on n, p, β). Remarks 44.9. Other nonlocal problems. Results on blow-up for other non 1 local problems with control of mass, of the form ut − ∆u = f (u) − |Ω| f (u) dy Ω p p−1 with Neumann boundary conditions and f (u) = |u| or |u| u, can be found in [103], [284]. For physical motivation concerning such problems, see [465]. Proof of Theorem 44.5(i). The proof here is for n ≥ 3. The cases n = 1, 2 can be obtained with obvious modifications. Fix m > 1. For any 0 < a < 1 < q, we have

1/q

1/q up+m dx ≤ u(p+m)aq dx u(p+m)(1−a)q dx . Ω

Ω

Ω

We claim that we can find 0 < a < 1 and q > n/(n − 2) such that (p + m)aq = (m + 1)n/(n − 2)

and

(p + m)(1 − a)q ≤ p.

(44.28)

Indeed, (44.28) is equivalent to n m+1 p 1 ≥1− 1− , (n − 2)q m + p m+p q

a= i.e.:

q≤

n 1 n + −p n−2 m n−2

(44.29)

and, since p < n/(n − 2), we can choose q > n/(n − 2) satisfying (44.29) and the corresponding a then belongs to (0, 1). Now using H¨ older’s inequality, we obtain

u

p+m

Ω

dx ≤ C

u

(m+1)n/(n−2)

1/q

(p+m)(1−a)/p dx up dx .

Ω

(44.30)

Ω

Multiplying (44.24) by um and integrating by parts over Ω, we obtain d dt

Ω

um+1 dx + m m+1

Ω

um−1 |∇u|2 dx =

Ω

−1

up dx up+m dx. Ω

(44.31)

406


Set v = u(m+1)/2 . Since

Ω

up dx ≥ C

Ω

p

p u dx ≥ u0 dx

(44.32)

Ω

by (44.25), formulas (44.30), (44.31) and (p + m)(1 − a) n/(n − 2), we obtain

d v 2 dx ≤ C 1 + v 2 dx . dt Ω Ω By integration, it follows that for all m > 1, τ > 0,

m+1 u (t) dx = v 2 (t) dx ≤ C(m, τ ), 0 < t < min(τ, T ). Ω

Ω

Therefore, using also (44.32), the right-hand side of (44.24) remains bounded in Lr on bounded time intervals for each r < ∞. The L∞ -boundedness of u on bounded time intervals then follows easily from the variation-of-constants formula and the Lp -Lq -estimates (cf. Proposition 48.4). We conclude that u exists globally. The proof of part (ii) is more delicate. It requires carefully constructed initial data. This is achieved in the following lemma. Lemma 44.10. Let Ω = B1 and p > psg . Then, for all M, β > 0, one can find a radial function u0 ∈ C 2 (Ω) satisfying the following properties: u0 (0) ≥ M,

u0 (1) = β,

u0,r (1) = 0, u0,r < 0 on (0, 1),

u0 dx ≤ Cβ, Ω

k(0) = ∆u0 +

Ω λup0

up0 dy ≥ 0,

u0,r + µrup0 ≤ 0,

−1

(44.33) (44.34)

≥ Aβ −p ,

(44.35)

|x| ≤ 1,

(44.36)

0 ≤ r ≤ 1/2,

(44.37)

where λ = Kβ 1−p , µ = Lβ 1−p , and C, A, K, L > 0 depend only on n, p. Proof. Let α = 2/(p − 1) and fix a function U ∈ C 2 ((0, 1]) such that U (r) = r−α on (0, 1/2], Ur < 0 on (0, 1), Ur (1) = 0

and

U (1) = 1. (44.38)


407

Fix δ ∈ (0, 1/4) and β > 0. We define φ(r) :=

U (r), δ < r ≤ 1, − α(α+3) ( rδ )2 + δ −α 1 + α(α+5) 6 2

α(α+2) r 3 (δ) , 3

0 ≤ r ≤ δ,

(44.39)

and we set u0 = βφ. One can check that u0 ∈ C 2 (Ω), that 0 ≤ u0 ≤ βU on (0, 1], and that u0 satisfies (44.33) whenever 0 < δ ≤ (M/β)−1/α . Since pα < n, we have Ω U p dx < ∞, hence (44.34) and (44.35). On the other hand, we have ⎧ ∆U + K, 1/2 ≤ |x| ≤ 1, ⎪ ⎪ ⎨ −α−2 p α(α + 2 − n) + K r , δ ≤ |x| ≤ 1/2, ∆φ + Kφ ≥ ⎪ ⎪ −α−2 ⎩ −nα(α + 3) + K δ , |x| ≤ δ. Since ∆u0 + Kβ 1−p up0 = β(∆φ + Kφp ) this implies (44.36) for K = K(n, p) > 0 large. Next we have φr + Lrφp ≤ −αr−α−1 + Lr−αp+1 = (L − α)r−α−1 ,

δ ≤ r ≤ 1/2,

and + φr + Lrφp ≤ δ −α − α(α+3)r δ2

α(α+2)r 2 δ3

+ LC(α)δ −pα r ≤ δ −αp (LC(α) − α)r

for 0 ≤ r ≤ δ. Since u0,r + Lβ 1−p rup0 = β(φr + Lrφp ), this implies (44.37) for L = L(p) > 0 small. Next, we consider the auxiliary problem wt − ∆w = 2λwp , uν = 0, w(x, 0) = u0 (x),

x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0, x ∈ Ω.

⎫ ⎪ ⎬ ⎪ ⎭

(44.40)

We shall need the following upper estimate on the existence time of its solution. Lemma 44.11. Let Ω = B1 and p > psg . For M, β > 0, let λ and u0 be as in Lemma 44.10. Then the existence time Tw of the solution of problem (44.40) M 1−p satisfies Tw ≤ λ(p−1) . Proof. We use a similar idea as in the proof of Theorem 23.5, applying the maximum principle to the auxiliary function J := wt − λwp . By the maximum principle we have w ≥ β > 0 in Q := Ω × (0, Tw ). On the other hand, Example 51.9 shows

408


that w ∈ C([0, Tw ), W 1,q (Ω)) for any q > n, hence wp ∈ C([0, Tw ), W 1,2 (Ω)) and Theorem 51.1(v) guarantees wt ∈ C([0, Tw ), L2 (Ω)). In addition, wt ∈ C 2,1 (Ω × (0, Tw )) (cf. Example 51.10). Therefore, J ∈ C([0, Tw ), L2 (Ω)) ∩ C 2,1 (Ω × (0, Tw )). Now, J satisfies Jt − ∆J = (wt − ∆w)t − λp wp−1 wt − wp−1 ∆w − (p − 1)wp−2 |∇w|2 ≥ 2λpwp−1 wt − λpwp−1 2λwp = 2λpwp−1 J in Q. At t = 0, we have J = ∆u0 + λup0 ≥ 0 by (44.36). On ∂Ω, we have ∂J ∂ ∂w p−1 ∂w ∂ν = ∂t ( ∂ν )−λpw ∂ν = 0. It thus follows from the maximum principle (cf. Remark 52.9) that J ≥ 0 in Ω × (0, Tw ). But this implies (w1−p )t ≤ −λ(p − 1), hence (0) ≤ M 1−p on (0, Tw ) by (44.33). The in particular w1−p (0, t) + λ(p − 1)t ≤ u1−p 0 lemma follows. Observe that, for β ≤ A/4K, we have k(0) ≥ Aβ −p ≥ 4Kβ 1−p = 4λ. The idea of the proof is now to show that k(t) cannot become smaller than 2λ before the time t = β, independently of M . This will be achieved via the next lemma, where u is estimated from above by employing a modification of an argument from [219] (cf. Theorem 24.1). This will guarantee that u dominates the solution w of the auxiliary problem (44.40) for t ≤ β. But the blow-up time of w goes to 0 as M increases, which will imply blow-up of u if M is large. Lemma 44.12. Let Ω = B1 and p > psg . For M > 0 and β ∈ (0, 1), let A, K, λ and u0 be as in Lemma 44.10. Set T0 = min(β, T ). Assume in addition that β ≤ A/4K, so that k(0) ≥ 4λ, and define

T1 = sup τ ∈ [0, T0 ) : k(t) ≥ 2λ on [0, τ ] ∈ (0, T0 ]. For each 1 < q < p, we have u(r, t) ≤ C(n, p, q)βr−2/(q−1) ,

0 < r ≤ 1, 0 < t < T1 .

(44.41)

Proof. Step 1. By Example 51.13, we have ur ∈ C 2,1 ((0, 1) × (0, T )) ∩ C([0, 1] × [0, T )). Using (44.33) and the maximum principle (in particular Proposition 52.17), one deduces that u ≥ β and ur ≤ 0, Since ur ≤ 0, we have u(r, t) ≤ nr

−n

0 ≤ r ≤ 1, 0 < t < T.

r

u(ρ, t)ρ 0

n−1

dρ ≤ C(n)r

−n

(44.42)

u(t) dx. Ω

Using (44.25), (44.34) and T0 ≤ β, we deduce that u(r, t) ≤ C(n, p)βr−n ,

0 < r ≤ 1, 0 < t < T0 .

(44.43)


409

We next claim that ur (1/2, t) ≤ −c(n, p)β,

0 ≤ t < T0 .

(44.44)

To show this, observe that the function v := β −1 ur satisfies n−1 n−1 vr = pk(t)up−1 v ≤ 0, 1/4 < r < 1, 0 < t < T, v− 2 r r v(1/4, t) ≤ 0, v(1, t) = 0, 0 < t < T,

vt − vrr +

v(0, r) = Ur (r) < 0,

1/4 < r < 1,

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

where U , defined in (44.38), depends only on p for r ∈ (1/4, 1). By the strong maximum principle, recalling that T0 ≤ β < 1, it follows in particular that v(1/2, t) ≤ −c(n, p) < 0 for 0 ≤ t < T0 , hence (44.44). Step 2. Set J = ur + ηruq . We claim that for η = C(n, p, q)β 1−q ,

(44.45)

with C(n, p, q) > 0 sufficiently small, there holds J ≤0

in Q := (0, 1/2) × (0, T1 ).

(44.46)

We compute Jt − Jrr = ut − urr r +η r(uq )t − (ruq )rr n−1 n−1 urr = − 2 + k(t)pup−1 ur + r r + η qruq−1 (ut − urr ) − 2quq−1 ur − q(q − 1)ruq−2 (ur )2 n−1 ≤ − 2 + k(t)pup−1 + (n − 3)qηuq−1 ur r n − 1 J − ηruq r +qηk(t)rup+q−1 + r n−1 = − 2 + k(t)pup−1 − 2qηuq−1 J − ηruq r n−1 n−1 q Jr − η u + qηk(t)rup+q−1 + r r n−1 Jr + b(r, t), = a(r, t)J + r where a(r, t) = − and

n−1 + k(t)pup−1 − 2qηuq−1 r2

b(r, t) = ηrup+q−1 (q − p)k(t) + 2qηuq−p .

410


Using the definition of T1 , (44.42) and (44.45), we obtain b(r, t) ≤ ηrup+q−1 2qηβ q−p − 2(p − q)Kβ 1−p ≤ 0

in Q.

On the other hand, for t ∈ [0, T1 ), we have J(0, t) = 0 and, by (44.43) and (44.44), the choice (44.45) implies J(1/2, t) ≤ 0. Also, for t = 0, using (44.37), u0 ≥ β and p > q, (44.45) implies J(r, 0) ≤ 0 in [0, 1/2]. Since a is bounded from above in (0, 1/2) × (0, τ ) for each τ < T1 , Claim (44.46) thus follows from the maximum principle (see Proposition 52.4). By integrating (44.46), we have (u1−q )r ≥ (q − 1)ηr in (0, 1/2] × (0, T1 ). This combined with (44.43) yields (44.41). Proof of Theorems 44.5(ii) and 44.6. For M > 0, let β and u0 be as in Lemma 44.12. Since p > psg , we may fix q such that 1 + 2p/n < q < p. We deduce from Lemma 44.12 that

1 up (t) ≤ C(n, p)β p rn−1−2p/(q−1) dr = C(n, p)β p , 0 < t < T1 . |x|≤1

0

Taking 0 < β ≤ β0 (n, p) sufficiently small, we infer that k(t) ≥ C(n, p)β −p ≥ 4K(n, p)β 1−p = 4λ,

0 < t < T1 .

Consequently T1 = T0 = min(T, β). In particular, by the comparison principle (use Proposition 52.7), it follows that u ≥ w for t < min(T, β, Tw ). But we have Tw < β for M large by Lemma 44.11, and we know that w blows up in L∞ -norm. It follows that T ≤ Tw < ∞, which proves Theorem 44.5(ii). Since T1 = T , the first part of Theorem 44.6 is now a direct consequence of Lemma 44.12. Finally, let us show that estimate (44.26) cannot be satisfied for any ε < 0. Suppose the contrary. This implies sup u(t) q < ∞

t∈(0,T )

for some q > n(p − 1)/2.

(44.47)

older’s inequality, On the other hand, u is bounded on ST and, by (44.25) and H¨ we have k(t) ≤ C, 0 ≤ t < T. (44.48) Owing to (44.47), by comparison argument with (a variant of) the model problem (14.1), it follows from Theorem 16.4 (or, alternatively, Theorem 15.2 or Example 51.27 in Appendix E) that u is uniformly bounded in QT : a contradiction. Proof of Theorem 44.7. (i) Due to (44.48), the lower estimate follows from Proposition 44.3(i).


411

(ii) In view of Proposition 44.3(ii), to prove the upper estimate, it suffices to show that k(t) → ∈ (0, ∞), as t → T . (44.49) Using (44.48), (44.26), parabolic estimates and the embedding (1.2), for some ν ∈ (0, 1) we have u ∈ BU C ν ({γ < |x| < 1 − γ} × (T /2, T )) for each γ > 0. Consequently, for all x ∈ B(0, 1) \ {0}, limt→T u(x, t) exists and is finite. Since 2p/(p − 1) < n, using (44.26), the dominated convergence theorem and (44.48), we deduce (44.49). Remark 44.13. By the methods of this subsection, problem (44.24) with f (u) = up and g(u) = uq can be studied for more general values of p, q > 1 and m ∈ R, under either Neumann or Dirichlet boundary conditions.

44.3. A problem with variational structure We next consider equation (44.1) with

f (u) = |u|p−1 u,

u

g(u) = λ +

f (s) ds = λ + 0

|u|p+1 , p+1

where p > 1, m = −q < 0 and λ > 0, under Dirichlet boundary conditions. Taking λ = |Ω|−1 for simplicity, this leads to the problem %−q $

|u(y, t)|p+1 dy ut − ∆u = 1 + |u|p−1 u, p+1 Ω u = 0, u(x, 0) = u0 (x),

x ∈ Ω, t > 0,

⎫ ⎪ ⎪ ⎪ ⎬

x ∈ ∂Ω, t > 0, ⎪ ⎪ ⎪ ⎭ x ∈ Ω,

(44.50)

where u0 ∈ L∞ (Ω). Problem (44.50) possesses a variational structure. Namely, the energy functional 1 E(u) = 2

$ %1−q

|u|p+1 1 1+ dx |∇u| dx − 1−q Ω Ω p+1

2

(for q = 1, with an obvious modification if q = 1) is nonincreasing along any solution of (44.50). More precisely d E u(t) = − dt

Ω

u2t (t) dx

(this follows in the same way as in (17.7) and Example 51.28).

412


Theorem 44.14. Consider problem (44.50) with Ω bounded, p > 1 and 0 < q < (p − 1)/(p + 1). There exists C = C(p, q) > 0 such that, if u0 ∈ L∞ ∩ H01 (Ω) satisfies E(u0 ) < −C, then Tmax (u0 ) < ∞. Proof. Set ψ(t) := u(t) 22 . Multiplying the equation in (44.50) by u we obtain %−q

$

1 |u|p+1 ψ (t) = dx uut (t) dx = − |∇u(t)|2 dx + 1 + |u|p+1 dx 2 p + 1 Ω Ω Ω Ω %−q $

|u|p+1 dx = −2E u(t) + 1 + Ω p+1 % $

(p − 1) − q(p + 1) 2 p+1 × |u| dx − (p + 1)(1 − q) 1−q Ω %1−q $

p+1 |u| ≥ −2E(u0 ) + c1 1 + dx − c2 , Ω p+1 older’s inequality, we obtain where c1 , c2 > 0 depend only on p, q. Applying H¨ ψ ≥ cψ γ − 2E(u0 ) − c2 with c = c(p, q, Ω) > 0 and γ := (p + 1)(1 − q)/2 > 1. If E(u0 ) < −c2 /2 (or ψ γ (0) > 2(E(u0 ) + c2 )/c), then this inequality implies Tmax (u0 ) < ∞. Remark 44.15. A priori bounds. Results on boundedness and a priori estimates of global solutions and universal bounds for global nonnegative solutions for problems of the form (44.50) have been proved in [187], [440], [464].

44.4. A problem arising in the modeling of Ohmic heating We finally consider equation (44.1) with f (u) = λe−u , g(u) = e−u , λ > 0, m = −2, n = 1 and Ω = (−1, 1), under Dirichlet boundary conditions. Namely: ⎫ 1 −2 ⎪ −u −u ut − uxx = λ e (y, t) dy e , x ∈ (−1, 1), t > 0, ⎪ ⎪ ⎬ −1 (44.51) u(±1, t) = 0, t > 0, ⎪ ⎪ ⎪ ⎭ x ∈ (−1, 1), u(x, 0) = u0 (x), where we assume u0 ∈ L∞ (Ω). Problem (44.51) arises from a special case of the following elliptic-parabolic coupled system: ⎫ x ∈ Ω, t > 0, ut − ∆u = σ(u)|∇φ|2 , ⎪ ⎪ ⎪ ⎪ ⎪ div(σ(u)∇φ) = 0, x ∈ Ω, t > 0, ⎪ ⎬ u = 0, x ∈ ∂Ω, t > 0, (44.52) ⎪ ⎪ ⎪ x ∈ ∂Ω, t > 0, φ = φ0 , ⎪ ⎪ ⎪ ⎭ x ∈ Ω. u(x, 0) = u0 (x),


413

Here u and φ respectively represent the temperature and the electric potential in a thermistor, i.e. a conductor whose electric conductivity σ = σ(u) may vary with the temperature, and the RHS of the first equation in (44.52) stands for the heat production due to the Joule effect. We refer to [33] and the references therein for results on blow-up and global existence concerning system (44.52) and its variants. Now assume that the (thin) conductor can be represented by the interval x ∈ (−1, 1) and that the potential φ is imposed to be 0 at x = −1 and a constant V at x = 1. The second equation in (44.52) becomes (σ(u)φx )x = 0 and can be integrated in σ(u)φx = I(t) (which represents the electric current per cross-sectional area unit). Denoting ρ(u) = 1/σ(u) (electric resistivity), hence 1 1 φx = ρ(u)I(t), we obtain V = −1 φx dx = I(t) −1 ρ(u) dx. The first equation in (44.52) then rewrites as 1 −2 ρ(u) dx . ut − uxx = σ(u)|φx |2 = ρ(u)I 2 (t) = ρ(u)V 2 −1

u

If the conductivity law is given by σ(u) = e and thermal cooling is applied on the ends of the conductor, we thus arrive at problem (44.51). We shall see that the global behavior of solutions to (44.51) is closely related to the properties of the stationary problem 1 −2 e−w dy e−w = 0, x ∈ (−1, 1), with w(±1) = 0. (44.53) wxx + λ −1

Proposition 44.16. Let λ > 0. Problem (44.53) has a (classical) solution if and only if λ < 8. Moreover the solution is unique and it is given by wλ = zα , where cos(αx) , (44.54) zα (x) = 2 log cos α and α ∈ (0, π/2) satisfies λ = 8 sin2 α. Furthermore, for |x| < 1, zα (x) is an increasing function of α, hence wλ (x) is an increasing function of λ. −2 1 , we see that w solves Proof. Setting µ = λ −1 e−w dy zxx + µe−z = 0,

x ∈ (−1, 1),

with z(±1) = 0.

(44.55)

By direct calculation, we see that a solution of (44.55) is given by (44.54) where α is the unique number in (0, π/2) such that µ = 2α2 / cos2 α. On the other hand, the solution of (44.55) is unique. Indeed, if y and z are two solutions, by subtracting the equations for y and z and multiplying by z − y, we get

1

1 0= (yxx − zxx ) + µ(e−y − e−z ) (z − y) dx ≥ (yx − zx )2 dx, −1

1

−1

−zα

2 α

hence y = z. Finally, since −1 e dy = sin α cos α, the function zα solves 2 (44.53) with λ = 8 sin α, hence the necessary and sufficient condition on λ. The ∂ zα (x) = 2(tan α − x tan(αx)). remaining assertion follows from ∂α The following result is from [314].

414


Theorem 44.17. Consider problem (44.51) with λ > 0 and u0 ∈ L∞ (Ω). (i) If λ < 8, then the equilibrium wλ is globally asymptotically stable. Namely, all solutions are global and converge to wλ in L∞ (−1, 1) as t → ∞. (ii) If λ > 8, then all solutions blow up in finite time. Moreover, the blow-up is global, i.e. lim u(x, t) = ∞, −1 < x < 1. (44.56) t→Tmax (u0 )

(iii) If λ = 8, then all solutions are global and unbounded. Moreover, lim u(x, t) = ∞,

−1 < x < 1.

t→∞

Problem (44.51) admits a comparison principle (cf. Proposition 52.25). The proof of Theorem 44.17 is based on suitable sub- and supersolutions, which will be constructed under a “quasi-stationary” form (cf. (44.59) below). Proof of Theorem 44.17. By a time shift we may assume without loss of generality that u0 ∈ C 1 ([−1, 1]) and u0 (±1) = 0. (44.57) Also, denoting by ϕ1 the first eigenfunction, we observe that: if Tmax (u0 ) = ∞, then u(x, t) ≥ εϕ1 (x) for some ε > 0 small and all t large. (44.58) Indeed, fixing C > 0 such that u ≥ −Cϕ , one easily checks that u(x, t) := 0 1 2ε − (C + 2ε)e−λ1 t ϕ1 (x) is a subsolution to problem (44.51) for ε > 0 small enough. In order to construct sub-/supersolutions, we put v(x, t) = zα(t) (x),

(44.59)

where α(t) is a function to be determined. Plugging (44.54), (44.59) into equation (44.51), we compute

Pv := vt − vxx − λ

1

e−v dy

−2

e−v

−1

= 2 tan α − x tan(αx) α +

λα2 2α2 − 2 cos2 (αx) 4 sin α cos2 (αx)

(8 sin2 α − λ)α2 = 2 tan α − x tan(αx) α + . 4 sin2 α cos2 (αx) If λ < 8, then define α ¯ ∈ (0, π/2) by λ = 8 sin2 α ¯ (hence wλ = zα¯ ); otherwise set α ¯ := π/2.


415

We first assume λ < 8 and look for a decreasing supersolution v = v. Thus taking α ≤ 0, we have Pv ≥ 2α tan α +

(8 sin2 α − λ)α2 4 sin2 α

(44.60)

provided 8 sin2 α(t) ≥ λ. Due to (44.57), we may choose some α0 ∈ (¯ α, π/2) close enough to π/2 so that zα0 (x) ≥ u0 (x). Let α(t) be the solution of the ODE α (t) =

(λ − 8 sin2 α)α2 cos α , 8 sin3 α

t≥0

(44.61)

with α(0) = α0 . Since 8 sin2 α0 > λ, it is clear that α exists globally and satisfies α < 0 and limt→∞ α(t) = α. ¯ It follows from (44.60) that v is a supersolution to problem (44.51). Consequently u ≤ v,

0 < t < Tmax (u0 ),

(44.62)

hence in particular Tmax (u0 ) = ∞. Moreover there holds lim v(x, t) = wλ (x)

t→∞

uniformly in [−1, 1].

(44.63)

Now consider general λ again. Looking for an increasing subsolution v = v, hence α (t) ≥ 0, we have Pv ≤ 2α tan α +

(8 sin2 α − λ)α2 4 sin2 α

(44.64)

provided 8 sin2 α ≤ λ. Assuming Tmax (u0 ) = ∞ and using (44.58), we may choose some α1 ∈ (0, α ¯ ) small enough so that zα1 (x) ≤ u0 (x). Take now α(t) to be the solution of (44.61) with α(0) = α1 . Since 8 sin2 α1 < λ, it is clear that α exists globally and satisfies α > 0. Moreover, we have lim α(t) = α. ¯

t→∞

It follows from (44.64) that v is a subsolution to problem (44.51). Consequently u ≥ v,

0 < t < Tmax (u0 ).

If λ < 8, there holds in addition limt→∞ v(x, t) = wλ (x), uniformly in [−1, 1]. This, along with (44.62), (44.63), proves assertion (i). If λ ≥ 8, we have shown that either Tmax (u0 ) < ∞, or Tmax (u0 ) = ∞

and

u(x, t) ≥ 2 log

cos(α(t)x) cos α(t)

→ ∞,

t → ∞.

(44.65)

416


Assume λ > 8. We shall show by a further subsolution argument that (44.65) leads to a contradiction. We look for a modified subsolution of the form v(x, t) = p log

cos(αx) cos α

where the function α : [0, T0 ) → [α2 , π/2)

(44.66)

and the numbers p > 1, T0 > 0, α2 ∈ (0, π/2) are to be determined. We shall use the following elementary lemma: Lemma 44.18. For each p > 1 and ε > 0, there holds

1

I(a) := −1

4+ε dy ≤ , cosp (ay) π(p − 1) cosp−1 a

as a → ( π2 )− .

(44.67)

Proof. We write

1 dy dy = p π p 0 cos (a(1 − y)) 0 sin ( 2 − a + ay)

η dy 1 + ≤ p π sinp (aη) sin ( − a + ay) 0 2

1 I(a) = 2

1

(44.68)

for 0 < a < π2 and 0 < η < 1. Fix η = η(ε) > 0 small. Taking 0 < π/2 − a < η, and using sin x ∼ x as x → 0, we obtain

0

η

η dy dy ≤ (1 + ε/8) π sinp ( π2 − a + ay) ( − a + ay)p 0 2 1−p η 1−p 1 + ε/8 π 1 + ε/8 π − a + ay −a = ≤ . a(1 − p) 2 a(p − 1) 2 0

Since cos a ∼ ( π2 − a) as a → π/2, this combined with (44.68) and cosp−1 a =0 p a→π/2 sin (aη) lim

yields (44.67). Proof of Theorem 44.17 (continued). Assuming α (t) ≥ 0, we have Pv ≤ pα tan α +

λ pα2 − . 2 p p cos (αx) cos α cos (αx)I 2 (α)


417

For p ∈ (1, 2), by using (44.66), (44.67) and taking α2 close to π/2, we have pα2 λ pα2 λπ 2 (p − 1)2 cosp−2 α − ≤ − cos2 (αx) cosp α cosp (αx)I 2 (α) cos2 (αx) (4 + ε)2 cosp (αx) 2 p−2 π cos α p λ(p − 1)2 − . ≤ cosp (αx) 4 (4 + ε)2 Since λ > 8, we can choose p ∈ (1, 2) close to 2 and ε small such that γ := π 2

λ(p − 1)2 (4 + ε)2

−

p > 0. 4

Taking α2 still closer to π/2 and using tan a ∼ ( π2 − a)−1 as a → (π/2)− , it follows that π −1 p−2 π cosp−2 α ≤ 2p −α −α Pv ≤ pα tan α − γ α − γ . (44.69) p cos (αx) 2 2 Take now α(t) to be the solution of p−1 γ π , α(0) = α2 . −α 2p 2 Since 1 0. On the other hand, owing to (44.65), we may assume that u0 ≥ v(·, 0) (after a time shift) which, along with(44.69), guarantees that v is a subsolution to problem (44.51). Since limt→T0 v(x, t) = ∞ in (−1, 1), this contradicts Tmax = ∞. Let us finally prove global blow-up, i.e. (44.56). Denoting α (t) =

1

h(t) =

e−u dy

−2

−1

and arguing as in the (alternative) proof of Proposition 23.1, we see that M (t) := maxx∈[−1,1] u(x, t) satisfies M (t) ≤ g(t) := λh(t)e−M(t) ,

for a.e. 0 < t < T := Tmax (u0 ).

(44.70)

Since u ≥ min[−1,1] u0 by the maximum principle, T < ∞ implies lim supt→T M (t) T = ∞. Integrating (44.70), we deduce that 0 g(t) dt = ∞. Since ut − uxx = λh(t)e−u ≥ g(t), (44.56) then follows from Theorem 43.6(i). This completes the proof of assertion (ii). As for the critical case λ = 8, global existence can be shown by a modified supersolution argument. We refer for this to [314].

418


Remarks 44.19. (a) Formal results concerning the blow-up rate (and the behavior in the boundary layer) for problem (44.51) are given in [315]. (b) Results on problem (44.51) with more general conductivity functions σ(u) can be found in [315], [66]. For the analogue of problem (44.51) in dimension n = 2, results similar to Theorem 44.17 are proved in [518], [298] for radial solutions in a disk. (c) For problem (44.51) with Neumann boundary conditions, it is easy to see that all solutions blow up in finite time: Indeed the solution of the ODE y = λ4 ey with y(0) = inf u0 is a subsolution.

45. Fujita-type results for problems involving space integrals We consider Cauchy problems with nonlocal source terms involving space integrals, of the form ⎫

(p−1)/q ut − ∆u = K(y)uq (y, t) dy u1+r , x ∈ Rn , t > 0, ⎬ (45.1) Rn ⎭ x ∈ Rn . u(x, 0) = u0 (x), In what follows, we assume that p > 1, q ≥ 1, r ≥ 0, u0 ∈ L∞ (Rn ), u0 ≥ 0,

(45.2)

K is a positive, bounded continuous function.

If K ∈ L1 (Rn ), then we assume in addition, that u0 ∈ L1 (Rn ). Under these assumptions, problem (45.1) is locally well-posed (see Example 51.13). The critical exponent for problem (45.1) was studied in [227]. It will depend in a crucial way on whether or not the function K is integrable. In the integrable case we have the following result. Theorem 45.1. Assume (45.2), K ∈ L1 (Rn ), and let pc = 1 +

2 n

− r.

(i) If p < pc , then (45.1) admits no nontrivial global solution. (ii) If p > pc , then (45.1) admits both global positive and blowing-up solutions. In the non-integrable case we need some additional assumptions on the asymptotic behavior of K. Theorem 45.2. Assume (45.2) and u0 ∈ L1 (Rn ). Assume in addition that K satisfies c1 (1 + |x|)−β ≤ K(x) ≤ c2 (1 + |x|)−β , x ∈ Rn , (45.3)

45. Fujita-type results for problems involving space integrals

for some β ∈ [0, n) and some c1 , c2 > 0. Let pc = 1 +

419

q(2−nr) n(q−1)+β .

(i) If p < pc , then (45.1) admits no nontrivial global solution. (ii) If p > pc , then (45.1) admits both global positive and blowing-up solutions. The proofs are exclusively based on comparison with suitable (self-similar) suband supersolutions. Note that (45.1) does admit a comparison principle (this follows from Proposition 52.27). For results in the critical case p = pc , see [227]. Proof of Theorems 45.1 and 45.2. 1. Blow-up. We look for a blowing-up subsolution under the form 2 x u(x, t) = A(T − t)−α f (ξ), , f (ξ) = e−|ξ| , ξ= √ T −t where α, T, A > 0 are parameters. We compute ut = Aα(T −t)−α−1 f (ξ)+

A (T −t)−α−1 ξ·∇ξ f (ξ), 2

Denoting

I(t) =

K(y)e−q|y|

2

/(T −t)

∆u = A(T −t)−α−1 ∆ξ f (ξ).

dy,

Rn

the condition for u being a subsolution is thus given by αf +

p−1 ξ · ∇ξ f − ∆ξ f ≤ Ap+r−1 (T − t)1−(r+p−1)α I q (t)f r+1 , 2

that is α + 2n ≤ 5|ξ|2 + Ap+r−1 (T − t)1−(r+p−1)α I

p−1 q

2

(t)e−r|ξ| ,

ξ ∈ Rn , 0 < t < T. (45.4) 1 In the case K ∈ L , assume without loss of generality that K ≥ c0 χ{|y| 0. We then have

√ n/2 −q|z|2 n/2 −q|z|2 K(z T − t)e dz ≥ c0 (T − t) dz, I(t) = (T − t) √ e Rn

hence

|z| 0 (independent of T ). In the case when K satisfies (45.3), we have

√ 2 (1 + |z| T − t)−β e−q|z| dz I(t) ≥ C(T − t)n/2 n R

∞ 2 n/2 −β/2 ≥ C(T − t) T (T −1/2 + ρ)−β e−qρ ρn−1 dρ, 0

420


hence

I(t) ≥ C(T − t)n/2 T −β/2

for all T ≥ 1 and some C > 0 (independent of T ). Let us now take α=

2q + n(p − 1) 2q(p + r − 1)

⎧ ⎨

with γ=

⎩

A = BT γ

and

n(p−1) 2q(p+r−1)

if K ∈ L1 ,

β(p−1) 2q(p+r−1)

if K satisfies (45.3).

A sufficient condition for (45.4) is then that 2

α + 2n ≤ 5|ξ|2 + c2 B p+r−1 e−r|ξ| ,

ξ ∈ Rn .

This is satisfied for some large B > 0 and guarantees that u is a subsolution for all T ≥ 1. Finally assume for contradiction that u exists for all time. Since u is a positive 2 supersolution of the linear heat equation, it follows that u(x, 1) ≥ εσ −n/2 e−|x| /4σ 2 for some ε, σ > 0, hence u(x, t + 1) ≥ ε(σ + t)−n/2 e−|x| /4(σ+t) for all t > 0 (cf. (18.12)). Now, the assumption p < pc means that α − γ > n/2 in both cases. Taking T = 4(σ + t) and t > 0 sufficiently large, we thus get u(x, t + 1) ≥ εT −n/2 e−|x|

2

/T

≥ u(x, 0) = BT γ−α e−|x|

2

/T

and the comparison principle would then imply finite-time blow-up of u. Statement (i) of Theorems 45.1 and 45.2 follows. 2. Global existence. We look for a blowing-up supersolution under the form u(x, t) = (T + t)−α g(ξ),

ξ=√

x , T +t

2

g(ξ) = e−σ|ξ| ,

where α, T, σ > 0 are parameters. We compute 1 ut = −α(T + t)−α−1 g(ξ) − (T + t)−α−1 ξ · ∇ξ g(ξ), 2 Denoting

J(t) =

K(y)e−qσ|y|

2

/(T +t)

∆u = (T + t)−α−1 ∆ξ g(ξ).

dy,

Rn

the condition for u being a supersolution is thus given by −αg −

p−1 ξ · ∇ξ g − ∆ξ g ≥ (T + t)1−(r+p−1)α J q (t)g r+1 . 2

46. A problem with memory term

421

Taking σ = 1/4, this amounts to p−1 2 n − α ≥ (T + t)1−(r+p−1)α J q (t)e−rσ|ξ| , 2

ξ ∈ Rn , t > 0.

(45.5)

In the case K ∈ L1 , there obviously holds J(t) ≤ K L1 . Taking n 1 pc , we may take n 2q + (n − β)(p − 1) 0, x ∈ ∂Ω, t > 0, x ∈ Ω,

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

(46.1)

where Ω is bounded, p > 1, q ≥ 1, k ≥ 0, and u0 ∈ L∞ (Ω), u0 ≥ 0. Notice that the problem is well-posed due to Example 51.14.

422


46.1. Blow-up and global existence The following result [479] shows that q = p constitutes a critical blow-up exponent for problem (46.1). Moreover, blow-up (in finite or infinite time) occurs for all positive solutions of (46.1), and not only for solutions with large initial data, unlike in problems (43.1) and (15.1) for instance. Theorem 46.1. Consider problem (46.1) with Ω bounded, p > 1, q ≥ 1, k ≥ 0, and 0 ≤ u0 ∈ L∞ (Ω), u0 ≡ 0. (i) If p > q or k = 0, then all solutions of (46.1) blow up in finite time. (ii) If p ≤ q and k > 0, then all solutions of (46.1) are global and unbounded, that is, lim supt→∞ u(t) ∞ = ∞. The proof of Theorem 46.1 relies on the eigenfunction method (cf. the proof of Theorem 17.1), combined with the following lemma concerning the system of differential inequalities z ≥ yp,

y + λy + kz ≥ z. r

(46.2)

Lemma 46.2. Assume 0 < r < 1 0 and (46.2) on (0, T ). Then T < ∞. Proof. By translating the origin of time, we may assume that actually y, z ∈ C 1 ([0, T )) and z(0) > 0. Fix γ such that max(r, 1/p) < γ < 1. It follows from the first inequality in (46.2) that, for all ε > 0, there exists a constant Cε > 0 such that γ γ r Cε z ≥ y pγ + (3λ + 1)y − ε and Cε z ≥ 3kz − ε, hence

γ r 2Cε z + 3y ≥ 3 y + λy + kz +y pγ + y − 2ε.

By the second inequality in (46.2), we deduce that 2Cε z + 3y ≥ 3z + y pγ + y − 2ε. γ

(46.3)

Next take m ∈ (0, γ). By Young’s inequality, we have 2Cε z = 2Cε γ

z m m/(1−γ) z z ≤ εz + C , ε zm z m/γ γ

hence Cε (z θ ) + εz m/(1−γ) ≥ 2Cε z , γ

where θ = 1 − (m/γ) ∈ (0, 1),

(46.4)


423

for some large constant Cε > 0. Now assume further that m < 1 − γ and define φ = Cε z θ + 3y. By combining (46.3) and (46.4), for ε < 1, we get φ ≥ 3z + y pγ + y − 2ε − εz m/(1−γ) ≥ 2z + y pγ + y − 3ε,

0 ≤ t < T.

Choosing ε < z(0)/3, setting ν = min(pγ, 1/θ) > 1 and using the fact that z is nondecreasing, we then obtain 1−θν θν φ ≥ z + y pγ + y ≥ z(0) z + y ν ≥ Cφν on (0, T ), for some C > 0. We conclude that T < ∞. Proof of Theorem 46.1. (i) Define the functions

t

y(t) = u(x, t)ϕ1 (x) dx and z(t) = up (x, s)ϕ1 (x) dx ds, Ω

0

Ω

0 ≤ t < T.

Multiplying (46.1) by ϕ1 and integrating by parts over Ω, we get:

t

y + λ1 y = up (x, s)ϕ1 (x) dx ds − k uq (x, t)ϕ1 (x) dx, 0 < t < T. 0

Ω

Ω

We may assume q < p also if k = 0. Letting r = q/p < 1 and applying Jensen’s inequality yields r y + λ1 y + kz ≥ z and z ≥ y p . The conclusion thus follows from Lemma 46.2. (ii) If p < q, a simple calculation shows that v(x, t) = C(1 + t)1/(q−p) is a supersolution for all large C > 0. If p = q, the same holds with v(x, t) = CeCt . Taking C > u0 ∞ , it follows from the comparison principle (Proposition 52.25) that u must exist globally. Last, assume for contradiction that u is globally bounded by a constant M > 0. Then u satisfies

t ut − ∆u ≥ up (x, s) ds − aM q−1 u, x ∈ Ω, t > 0. 0

By the comparison principle, in view of part (i), this immediately implies finitetime blow-up: a contradiction. Remarks 46.3. (i) The assumption r < 1 in Lemma 46.2 is essential, at least if k > 0. Indeed, if r = 1, then z(t) = Ceµt , y(t) = (Cµ)1/p eµt/p is a global positive solution of (46.2) for µ = 1/k and any C > 0. (ii) Fujita-type results. For problem (46.1) and related equations, Fujita-type results have been recently obtained in [112].

424


46.2. Blow-up rate The following result shows a type I blow-up rate for monotone-in-time solutions and provides a sufficient condition for monotonicity. Theorem 46.4. Consider problem (46.1) with Ω bounded, p > 1, k = 0. Let u0 ∈ C 1 (Ω), u0 ≥ 0, u0 ≡ 0, and T := Tmax (u0 ). (i) Assume that: there exists t0 ∈ [0, T ) such that ut (x, t0 ) ≥ 0 for all x ∈ Ω.

(46.5)

Then T < ∞, ut ≥ 0 in Ω × [t0 , T ) and u satisfies the blow-up estimate C1 (T − t)−2/(p−1) ≤ u(t) ∞ ≤ C2 (T − t)−2/(p−1) ,

as t → T .

(46.6)

(ii) Assume that Φ ∈ C 2 (Ω) satisfies Φ > 0 in Ω, Φ|∂Ω = 0 and that there exist ε, η > 0 such that ∆Φ(x) ≥ εδ(x)

for all x ∈ Ω such that δ(x) ≤ η.

(46.7)

Then, for all λ > 0 large enough, the solution of (46.1) with initial data u0 = λΦ satisfies (46.5). Part (i) was proved in [335] (under the additional assumption Ω = BR and u0 radially symmetric decreasing). Part (ii) was proved in [486]. Note that (46.5) cannot be satisfied for 0 ≤ t0 T , due to ut (., 0) = ∆u0 , Proof. (i) Let

J(x, t) = ut − ε

t

up ds, 0

(x, t) ∈ Ω × (t1 , T ).

By Example 51.14 we have J ∈ C 2,1 (QT ) ∩ C(Ω × (0, T )). Pick t1 ∈ (t0 , T ). Taking ε > 0 small enough, using (46.5) and arguing as in the proof of Theorem 23.5, this time using the nonlocal maximum principle in Proposition 52.24, we obtain that J(·, t1 ) ≥ 0 in Ω. We compute

t

t up−1 ∆u ds + εp(p − 1) up−2 |∇u|2 ds Jt − ∆J = utt − εup − ∆ut + εp 0 0

t

s up−1 ut − up dσ ds ≥ (1 − ε)up + εp 0 0

t

s

t up−1 ut − ε up dσ ds ≥ p up−1 J ds. = (1 − ε)up0 + p 0

0

0


425

Since J = 0 on ∂Ω × (t1 , T ), it follows from Proposition 52.24 that J ≥ 0 in Ω × (t1 , T ). t Now, for each fixed x ∈ Ω, multiplying the inequality ut ≥ ε t1 up ds by up and integrating over (t1 , t), we obtain u (x, t) ≥ c p

t

up ds

2p/(p+1)

,

t1 < t < T.

t1

It follows that T < ∞ and, by integrating over (t, T ), we obtain

t

up ds ≤ C(T − t)−(p+1)/(p−1) ,

t1 < t < T.

(46.8)

t1

Setting t = t + (T − t)/2 and using ut ≥ 0, we deduce that

T −t p u (x, t) ≤ 2

t

up ds ≤ C(T − t )−(p+1)/(p−1) = C

t

hence

u(x, t) ≤ C(T − t)−2/(p−1) ,

T − t −(p+1)/(p−1) , 2

t1 < t < T,

with C = C(p, ε). The upper estimate in (46.6) follows. On the other hand, letting M (t) = u(t) ∞ and arguing as in the (alternative) proof of Proposition 23.1, we get

M (t) ≤

t

M p (s) ds,

for a.e. 0 < t < T .

0

Proceeding similarly as for (46.8), we obtain

t

0

M p ds ≥ c1 (T − t)−(p+1)/(p−1) ,

0 < t < T.

(46.9)

For t1 ≤ τ < t < T , by using (46.9), the upper estimate in (46.6) and M being nondecreasing on [t1 , T ), we obtain −(p+1)/(p−1)

c1 (T − t)

≤

τ p

M ds + 0

t

M p ds

τ −(p+1)/(p−1)

≤ C(T − τ )

+ (t − τ )M p (t).

For t close enough to T , taking τ = T − γ(T − t) with γ = (2C/c1 )(p−1)/(p+1) > 1, we get, M (t) ≥ (c1 /2γ)1/p (T − t)−2/(p−1) ,

426


which proves the lower estimate. (ii) Let v = vλ := ut . By Example 51.14 we have v ∈ C 2,1 (QT )∩C([0, T ), L2 (Ω)). The function v satisfies ⎫ vt − ∆v = up , x ∈ Ω, 0 < t < T, ⎪ ⎬ v = 0, x ∈ ∂Ω, 0 < t < T, ⎪ ⎭ x ∈ Ω, v(x, 0) = ∆u (x), 0

hence v(t) = e−tA (∆u0 ) + Since u(t) ≥ e

t

e−(t−s)A up (s) ds.

0

−tA

u0 and p p p e−(t−s)A e−sA u0 ≥ e−(t−s)A (e−sA u0 ) = e−tA u0 ,

we have v(t) ≥ e−tA (∆u0 ) +

t 0

p p e−(t−s)A e−sA u0 ds ≥ e−tA (∆u0 ) + t e−tA u0 .

Therefore, for all λ > 0, p 1 vλ (t) ≥ e−tA (∆Φ) + λp−1 t e−tA Φ , λ

0 ≤ t < T (λΦ).

(46.10)

We claim that there exists η1 > 0 such that e−tA (∆Φ)(x) > 0

for all (x, t) such that δ(x) ≤ η1 and 0 ≤ t ≤ η1 .

(46.11)

To prove the claim, observe that, by the assumption (46.7), there exist γ > 0 and ρ ∈ D(Ω), ρ ≥ 0, such that ∆Φ ≥ γϕ1 − ρ in Ω. Therefore, e−tA (∆Φ) ≥ γe−λ1 t ϕ1 − e−tA ρ. Using ϕ1 ≥ c1 δ(x) in Ω, ρ ∈ D(Ω) and the continuity of e−tA ρ in C 1 (Ω) at t = 0, claim (46.11) follows. −2/(p−1) −(p−1)/2 − kt is A straightforward calculation shows that w(t) := u0 ∞ −1/2 . Since blow-up takes place a supersolution of (46.1) for k = (p − 1)(2(p + 1)) −(p−1)/2 in L∞ -norm if it occurs, this implies in particular that T (u0 ) ≥ k1 u0 ∞ . −(p−1)/2 1 in (46.10), we obtain Taking t = tλ := 2k λΦ ∞ p 1 1 vλ (tλ ) ≥ e−tλ A (∆Φ) + Φ −(p−1)/2 λ(p−1)/2 e−tλ A Φ . ∞ λ 2k On the one hand, since Φ ≥ 0, by (46.11) we have 1 vλ (x, tλ ) ≥ 0 if δ(x) ≤ η1 and λ ≥ λ0 (p, Φ) > 0 large enough. λ


427

On the other hand, since Φ > 0 in Ω, we have e−tA Φ > 0 in Ω × [0, ∞) by the strong maximum principle. Therefore, there exists α > 0 such that e−tA Φ (x) ≥ α for all (x, t) such that δ(x) ≥ η1 and t ∈ [0, 1]. It follows that if δ(x) ≥ η1 and λ ≥ λ0 (p, Φ) (possibly larger), then 1 vλ (x, tλ ) ≥ − ∆Φ ∞ + C(p, Φ)αp λ(p−1)/2 > 0. λ We have thus shown that ut (x, tλ ) ≥ 0 in Ω whenever λ ≥ λ0 (p, Φ) and the theorem is proved. Remark 46.5. Blow-up set and profiles. Results on single-point blow-up and on blow-up profiles for equations similar to (46.1) have been obtained in [67] by employing methods from [219].

Appendices 47. Appendix A: Linear elliptic equations In this appendix we collect some fundamental estimates for linear elliptic equations.

47.1. Elliptic regularity We assume that Ω is an arbitrary domain in Rn and we consider second-order elliptic differential operators of the form Au = −

n

aij

i,j=1

n ∂2 ∂ u+ bi u + cu, ∂xi ∂xj ∂x i i=1

(47.1)

with measurable coefficients aij , bi , c satisfying the ellipticity condition

aij (x)ξi ξj ≥ λ|ξ|2

for all x ∈ Ω, ξ ∈ Rn ,

(47.2)

i,j

with λ > 0 and a uniform bound |aij |, |bi |, |c| ≤ Λ.

(47.3)

Au = f

(47.4)

We consider the linear problem in Ω,

where f = f (x) is a given function. 2,1 A strong solution of (47.4) is a function u ∈ Wloc (Ω) which satisfies (47.4) k,p a.e. We denote by u k,p;D the norm in W (D); in particular u k,p;Ω = u k,p . The following result (cf. [250, Theorems 9.11 and 9.13]) contains the basic interior and interior-boundary elliptic Lp -estimates.

Theorem 47.1. Let Ω be an arbitrary domain in Rn and assume (47.2) and 2,p ∩ Lp (Ω), 1 < p < ∞, be a strong solution of (47.4), where aij (47.3). Let u ∈ Wloc are continuous and f ∈ Lp (Ω). (i) Consider a subdomain Ω ⊂⊂ Ω. Then u 2,p;Ω ≤ C( u p + f p ),

(47.5)

430

Appendices

where C depends only on n, p, Ω, Ω , λ, Λ, and the moduli of continuity of the aij on Ω . (ii) Let Σ be an open subset of ∂Ω of class C 2 , u ∈ W 2,p (Ω) and u = 0 on Σ in the sense of traces. Let aij ∈ C(Ω ∪ Σ) and Ω ⊂⊂ Ω ∪ Σ. Then (47.5) is true, where C depends also on Σ. As for interior and interior-boundary elliptic Schauder estimates, we have the following theorem (cf. [250, Corollary 6.3, Theorems 6.6, 6.19 and Lemma 6.16]). Theorem 47.2. Let Ω be an arbitrary bounded domain in Rn , assume (47.2), and let f and the coefficients of A belong to BU C α (Ω), where α ∈ (0, 1). (i) Consider a subdomain Ω ⊂⊂ Ω. If u ∈ C 2 (Ω) is a solution of (47.4), then u ∈ BU C 2+α (Ω ) and u BUC 2+α (Ω ) ≤ C u ∞ + f BUC α (Ω) , where C depends only on n, α, λ, Ω, Ω and the norms of the coefficients of A in BU C α (Ω). (ii) Assume Ω of class C 2+α and let ϕ ∈ BU C 2+α (Ω). If u ∈ C 2 (Ω) ∩ C(Ω) is a solution of (47.4) satisfying u = ϕ on ∂Ω, then u ∈ BU C 2+α (Ω) and u BUC 2+α(Ω) ≤ C u ∞ + f BUC α (Ω) + ϕ BUC 2+α (Ω) , where C depends only on n, α, λ, Ω and the norms of the coefficients of A in BU C α (Ω). If we deal only with weaker type of solutions (say, variational), then the regularity assumptions u ∈ W 2,p (Ω) or u ∈ C 2 (Ω) in the above theorems can often be verified by means of the following existence-uniqueness theorem (cf. [250, Theorems 9.15 and 6.13]). See Remark 47.4(ii) for an example. Theorem 47.3. Assume (47.2), (47.3), c ≤ 0 and let Ω be a bounded domain of class C 2 . (i) Let aij ∈ C(Ω), f ∈ Lp (Ω) and ϕ ∈ W 2,p (Ω), where 1 < p < ∞. Then equation (47.4) has a unique (strong) solution u ∈ W 2,p (Ω) satisfying u − ϕ ∈ W01,p (Ω). (ii) Let f and the coefficients of A belong to B(Ω) ∩ C α (Ω), α ∈ (0, 1), and ϕ ∈ C(∂Ω). Then equation (47.4) has a unique (classical) solution u ∈ C 2+α (Ω)∩C(Ω) satisfying u = ϕ on ∂Ω. Remarks 47.4. (i) Assume that A has constant coefficients. Then the following regularity result can be deduced from Theorem 47.1(i): if u, f ∈ Lploc (Ω) for some 2,p (Ω). To show this, it suffices to 1 < p < ∞ and Au = f in D (Ω), then u ∈ Wloc

47. Appendix A: Linear elliptic equations

431

apply Theorem 47.1(i) to the convolution products u ∗ ρj , where ρj is a sequence of mollifiers, i.e.

n n ρ(x) dx = 1 (47.6) ρj (x) = j ρ(jx), 0 ≤ ρ ∈ D(R ), Rn

(see the end of the proof of Proposition 47.6 below for a more detailed, similar argument). Similarly, using Theorem 47.2(i), we obtain that u ∈ C 2+α (Ω) whenever u, f ∈ C α (Ω) for some 0 < α < 1 and Au = f is satisfied in D (Ω). (ii) For A with leading coefficients of class C 1 , Theorem 47.1(i) remains true 1,p ∩ Lp (Ω), equation (47.4) being understood in if we only assume that u ∈ Wloc the variational sense. The idea of the proof is as follows. Taking ψ a smooth cutoff function, the regularity of u and aij allows to apply Theorem 47.3(i) to the equation satisfied by the function uψ in a smooth domain Ω ⊂⊂ Ω. We can then conclude by using the uniqueness of variational solutions. (iii) As a useful consequence of the above theorems, we can prove the following property. Assume that A has constant coefficients, let Ω ⊂ Rn be a (possibly unbounded) domain of class C 2 , Σ an open subset of ∂Ω, and f ∈ Lploc (Ω ∪ Σ), 2,p with p > n. Assume that u ∈ Wloc (Ω) ∩ C(Ω ∪ Σ) satisfies Au = f a.e. in Ω and 2,p u = 0 on Σ. Then u ∈ Wloc (Ω ∪ Σ). If we further assume that Ω is of class C 2+α and that f ∈ C α (Ω ∪ Σ) for some α ∈ (0, 1), then u ∈ C 2+α (Ω ∪ Σ). Let us prove this in the case A = −∆ for simplicity. Let x0 ∈ Σ. One can find r > 0 and a bounded domain ω, as smooth as Ω, such that Ω ∩ B(x0 , r) ⊂ ω ⊂ Ω and ∂Ω ∩ B(x0 , r) ⊂ Σ. Let ϕ ∈ D(Rn ) be such that supp(ϕ) ⊂ B(x0 , r) and ϕ = 1 near x = x0 . Then v := uϕ satisfies (47.7) −∆v = f˜ := f ϕ − 2∇u · ∇ϕ − u∆ϕ in D (ω). Since f˜ ∈ W −1,p (ω), there exists a unique w ∈ W01,p (ω) ⊂ C0 (ω), such that 2,p (ω) due to f˜ ∈ Lploc (ω) and part (i). By the −∆w = f˜. Also, we have w ∈ Wloc maximum principle in Proposition 52.1(i), we deduce that w = v. It follows that 1,p u ∈ Wloc (Ω ∪ Σ). Getting back to equation (47.7), we now have f˜ ∈ Lp (ω). By Theorem 47.3(i) and the uniqueness of w, we deduce that w ∈ W 2,p (ω), hence 2,p (Ω ∪ Σ). Now, if also Ω ∈ C 2+α and f ∈ C α (Ω ∪ Σ), then f˜ ∈ BU C β (ω) u ∈ Wloc with β = min(α, 1 − n/p). By Theorem 47.2(ii), we get v ∈ BU C 2+β (ω), hence u ∈ C 2+β (Ω ∪ Σ). Iterating, we finally obtain f˜ ∈ BU C α (ω) and u ∈ C 2+α (Ω ∪ Σ).

47.2. Lp -Lq -estimates The following regularity results for the Laplacian are often used in bootstrap arguments in nonlinear problems. The notion of L1 -solution of the Laplace equation −∆u = f in Ω, (47.8) u=0 on ∂Ω has been introduced in Definition 3.1.

432

Appendices

Proposition 47.5. Let Ω ⊂ Rn be a bounded domain of class C 2+α for some α ∈ (0, 1), and assume that 1 ≤ p ≤ q ≤ ∞ satisfy 1 1 2 − < . p q n

(47.9)

Let f ∈ L1 (Ω) and u be the L1 -solution of (47.8). (i) If f ∈ Lp (Ω), then u ∈ Lq (Ω) and u q ≤ C(Ω, p, q) f p . (ii) If f+ ∈ Lp (Ω), then u+ ∈ Lq (Ω) and u+ q ≤ C(Ω, p, q) f+ p . Proposition 47.6. Let Ω be an arbitrary bounded domain in Rn , Ω ⊂⊂ Ω, and assume that 1 ≤ p ≤ q ≤ ∞ satisfy (47.9). Let u ∈ L1 (Ω) be such that −∆u =: f ∈ L1 (Ω) (where ∆u is understood in the sense of distributions). (i) If f ∈ Lp (Ω), then u ∈ Lq (Ω ) and u Lq (Ω ) ≤ C(Ω, Ω , p, q) f Lp(Ω) + u L1(Ω) .

(47.10)

(ii) If f+ ∈ Lp (Ω), then u+ ∈ Lq (Ω ) and u+ Lq (Ω ) ≤ C(Ω, Ω , p, q) f+ Lp (Ω) + u+ L1 (Ω) .

(47.11)

Proposition 47.5 will be proved in Appendix C, along with the analogous result in Lpδ -spaces (Theorem 49.2). As for Proposition 47.6, for p > 1, inequality (47.10) with u L1(Ω) replaced by u Lp(Ω) would follow from Theorem 47.3(i) and the Sobolev inequality. However, since we need the case p = 1 (and also (47.11)) in the applications, we have to rely on different, classical arguments, using the fundamental solution and Green’s formula. Proof of Proposition 47.6. We give the proof for n ≥ 3 only. The cases n = 1, 2 can be treated similarly. (i) say C 2 . Fix x ∈ Ω and 0 < r < R := We first assume that u is smooth, 2−n min 1, dist(Ω , ∂Ω) . Let Γr (y) = cn (|y| −r2−n ), where cn = ((n−2)|S n−1 |)−1 , be the fundamental solution of the Laplacian vanishing for |y| = r. It is well known that

1−n u(x) = Γr (y)f (x + y) dy + (n − 2)cn r u(x + y) dσ, (47.12) |y| 4, z ∈ [2, ∞) is arbitrary if n ≤ 4. L Lemma 47.13. Let λL 1 < λ2 < · · · denote all distinct eigenvalues of L. Then L λk = (n + k − 1)/2, k = 1, 2, . . . , and the eigenspaces are β Ker (L − λL k ) = Span {D φ1 : |β| = k − 1},

where φ1 (y) = e−|y|

2

/4

, Dβ = ∂1β1 · · · ∂nβn , |β| = β1 + · · · + βn .

Proof. Let u ∈ L2g and let uˆ denote the Fourier transform of u. Since | · |m u ∈ 1 ˆ ∈ m≥0 H m (Rn ) ⊂ C ∞ (Rn ). Assume Lu = λu. L2 (Rn ) for any m ≥ 0, we have u Applying the Fourier transform we obtain ˆ(ξ) + |ξ|2 u

1 n u ˆ(ξ) + ξ · ∇ˆ u(ξ) = λˆ u(ξ). 2 2

2

Set v(ξ) = e|ξ| u ˆ(ξ). Then v ∈ C ∞ (Rn ) and ξ · ∇v(ξ) = (2λ − n)v(ξ), which guarantees that v is a homogeneous function of degree (2λ − n) (cf. the Euler identity for homogeneous functions). As v ∈ C ∞ (Rn ), the degree (2λ − n) has to be a nonnegative integer, hence v = Pk−1 , where Pk−1 is a homogeneous 2 polynomial of degree (k − 1) and k ∈ {1, 2, . . . }. Then u ˆ(ξ) = Pk−1 (ξ)e−|ξ| , hence u = cPk−1 (D)φ1 .

438

Appendices

48. Appendix B: Linear parabolic equations This appendix is devoted to the estimates and various notions of solutions of linear parabolic equations.

48.1. Parabolic regularity Let Ω be an arbitrary domain in Rn and T > 0. We consider the problem ut + Au = f

in QT ,

(48.1)

where the operator A is defined in (47.1) and its coefficients aij , bi , c depend on z := (x, t) ∈ QT , aij (z)ξi ξj ≥ λ|ξ|2 for all z ∈ QT , ξ ∈ Rn . (48.2) i,j 2,1;1 A strong solution of (48.1) is a function u ∈ Wloc (QT ) satisfying (48.1) a.e. The following result (cf. [338, Theorems 7.13, 7.15, 7.17 and Corollary 7.16]) contains the basic interior and interior-boundary parabolic Lp -estimates, and an existence-uniqueness statement. See also [320] for additional results concerning parabolic Lp -theory.

Theorem 48.1. Let Ω be an arbitrary bounded domain in Rn . Assume (48.2) and 2,1;p ∩ Lp (QT ), 1 0, then u 2,1;p;Q ≤ C( u p;QT + f p;QT ),

(48.3)

where C depends only on n, p, QT , Q , λ, Λ, and the moduli of continuity of the aij . (ii) Let Ω be of class C 2 and either Σ be an open subset of ST or Σ = PT . Assume u ∈ W 2,1;p (QT ) and u = 0 on Σ. Let Q ⊂ QT , dist(Q , PT \ Σ) > 0 if Σ = PT . Then (48.3) is true, where C depends also on Σ. (iii) Let Ω be of class C 2 , ϕ ∈ W 2,1;p (QT ), f ∈ Lp (QT ). Then there exists a unique (strong) solution u of (48.1) satisfying u = ϕ on PT . Moreover, u satisfies the estimate u 2,1;p;QT ≤ C f p;QT + ϕ 2,1;p;QT . The following result (cf. [338, Theorems 4.28 and 5.14]) contains the basic interior-boundary parabolic Schauder estimate and an existence-uniqueness statement. We restrict ourselves to global estimates; local estimates can be easily derived by applying this theorem to the function uψ where ψ is a smooth cut-off function. See also [214] for additional results concerning parabolic Schauder theory.

48. Appendix B: Linear parabolic equations

439

Theorem 48.2. Assume (48.2). Let α ∈ (0, 1) and let Ω be a bounded domain of class C 2+α . Assume aij , bi , c, f ∈ BU C α,α/2 (QT ), ϕ ∈ BU C 2+α,1+α/2 (QT ). (i) If u ∈ BU C 2+α,1+α/2 (QT ) is a solution of (48.1) satisfying u = ϕ on PT , then |u|2+α;QT ≤ C u ∞ + |f |α;QT + |ϕ|2+α;QT , where C depends only on n, α, λ, Ω and the norms of aij , bi , c in BU C α,α/2 (QT ). (ii) There exists a unique solution u ∈ C(QT )∩C 2,1 (QT ) of (48.1) satisfying u = ϕ on PT . If ϕt + Aϕ = f on ∂Ω × {0}, then u ∈ BU C 2+α,1+α/2 (QT ) and |u|2+α;QT ≤ C |f |α;QT + |ϕ|2+α;QT . Remark 48.3. (i) Assume that A has constant coefficients. Then, by similar arguments as in Remark 47.4(i), one can deduce the following regularity results from Theorems 48.1(i) and 48.2(i). If u, f ∈ Lploc (QT ) for some 1 < p < ∞ and 2,1;p (QT ). If u, f ∈ C α,α/2 (QT ) for some ut + Au = f in D (QT ), then u ∈ Wloc 0 < α < 1 and ut + Au = f in D (QT ), then u ∈ C 2+α,1+α/2 (QT ). (ii) (Neumann boundary conditions) Under the assumptions Ω bounded, 2,1;p ∩ Lp (QT ) is (48.2), (47.3), aij ∈ C(QT ), 1 < p < ∞ and f ∈ Lp (QT ), if u ∈ Wloc a strong solution of (48.1) and satisfies ∂ν u = 0 on ST and u = 0 on Ω × {0}, then we have the estimate u 2,1;p;QT ≤ C f p;QT . Similarly, Theorem 48.2(i) remains valid if the condition u = ϕ on PT is replaced by ∂ν u = ∂ν ϕ on ST and u = ϕ on Ω × {0}. These facts follow from [337, Theorem 7.20] (see also [161, Theorem 8.2]) and [337, Theorem 4.31], respectively. For existence-uniqueness results analogous to Theorem 48.2(ii), see [337, Theorem 4.31]. In the rest of Appendix B and in Appendix C we shall restrict ourselves to the Laplace operator for simplicity, but many results can be extended to more general uniformly elliptic divergence form operators with sufficiently smooth coefficients.

48.2. Heat semigroup, Lp -Lq -estimates, decay, gradient estimates In this subsection we collect some useful properties of the Dirichlet heat semigroup. Let Ω be an arbitrary domain in Rn and let −A2 denote the Dirichlet Laplacian in L2 (Ω), that is the Laplacian on L2 (Ω) subject to homogeneous Dirichlet boundary conditions (see [154] for its precise definition and for the proof of the following statements). Then −A2 is a nonnegative self-adjoint operator and it generates a C 0 -semigroup e−tA2 on L2 (Ω). The space L1 ∩ L∞ (Ω) is invariant under e−tA2 and e−tA2 may be extended from L1 ∩ L∞ (Ω) to a positive contraction semigroup Tp (t) on Lp (Ω) for each 1 ≤ p ≤ ∞. These semigroups are strongly continuous if 1 ≤ p < ∞ and T∞ (t)f → f as t → 0+ in the weak-star topology. In addition,

440

Appendices

Tp (t)f = Tq (t)f for f ∈ Lp ∩ Lq (Ω) and p, q ∈ [1, ∞]. If no confusion seems likely, we will denote all the semigroups Tp , 1 ≤ p ≤ ∞, by the same symbol e−tA and call them the heat semigroup in Ω (more precisely, the Dirichlet heat semigroup in Ω, or the heat semigroup in Ω with homogeneous Dirichlet boundary conditions). Note that u = e−tA f solves the heat equation ut − ∆u = 0 in Ω × (0, ∞). In addition, if Ω is smooth enough (for instance if it satisfies an exterior cone condition at each point of ∂Ω), then u ∈ C(Ω × (0, ∞)) and u = 0 on ∂Ω × (0, ∞). There exists a positive C ∞ -function GΩ : Ω × Ω × (0, ∞) → R (Dirichlet heat kernel) such that (e−tA f )(x) = Ω GΩ (x, y, t)f (y) dy for any f ∈ Lp (Ω), 1 ≤ p ≤ ∞ (the subscript Ω in GΩ will be often omitted if no confusion is likely). In addition, GΩ1 (x, y, t) ≤ GΩ2 (x, y, t) (48.4) whenever Ω1 ⊂ Ω2 and x, y ∈ Ω1 , and GΩ (x, y, t) = GΩ (y, x, t) for all x, y ∈ Ω and t > 0. If Ω = Rn , then GRn (x, y, t) = G(x − y, t), where G(x, t) = Gt (x) := (4πt)−n/2 e−x

2

/4t

(48.5)

is the Gaussian heat kernel, hence e−tA f = Gt ∗ f . Note that the functions Gt satisfy the semigroup property under convolution: Gt+s = Gt ∗ Gs ,

s, t > 0.

(48.6)

Let us also observe that if λ > σ(−A2 ) and Bλ := (λ + A2 )−1 , then Bλ = e−λt e−tA2 dt and 0

t

KΩ,λ (x, y) :=

0

t

e−λt GΩ (x, y, t) dt

(48.7)

is the kernel of the operator Bλ , that is Bλ f (x) = Ω KΩ,λ (x, y)f (y) dy. Notice that for each f ∈ L2 (Ω), Bλ f is the unique solution of the problem λu − ∆u = f

in H −1 (Ω),

u ∈ H01 (Ω),

and KΩ,λ is the Green function of this problem. If Ω is bounded and if there is no risk of confusion, we denote simply K(x, y) = KΩ (x, y) = KΩ,0 (x, y), which is the (elliptic) Green kernel of the Dirichlet Laplacian. Moreover, for n ≥ 3, we have KΩ (x, y) ≤ Cn |x − y|2−n ,

(48.8)

as a consequence of (48.4), (48.5) and (48.7). The following Lp -Lq -estimate for the heat semigroup is of fundamental importance in the study of semilinear problems.


441

Proposition 48.4. Let (e−tA )t≥0 be the heat semigroup in Rn and Gt (x) = G(x, t) the Gaussian heat kernel. We have the following properties. (a) Gt 1 = 1 for all t > 0. (b) If Φ ≥ 0, then e−tA Φ ≥ 0 and e−tA Φ 1 = Φ 1 . (c) If 1 ≤ q ≤ ∞, then e−tA Φ q ≤ Φ q for all t > 0. (d) If 1 ≤ p < q ≤ ∞ and 1/r = 1/p − 1/q, then e−tA Φ q ≤ (4πt)−n/(2r) Φ p for all t > 0. (e) For an arbitrary domain Ω ⊂ Rn , assertions (c) and (d) remain valid if e−tA is replaced with the Dirichlet heat semigroup in Ω. Proof. Statement (a) is well known, statement (b) follows from Fubini’s theorem and part (a). Statement (c) follows from the contractivity of the semigroup Tq (t) (see above); it also easily follows from the estimate Gt ∗ Φ q ≤ Gt 1 Φ q . Interpolating between (b) and the inequality e−tA Φ ∞ ≤ (4πt)−n/2 Φ 1 we obtain e−tA Φ q ≤ (4πt)−(n/2)(1−1/q) Φ 1 . (48.9) Interpolating between (48.9) and (c) yields (d). To prove assertion (e), denote by e−tAΩ the Dirichlet heat semigroup in Ω. Let ˜ ˜ Φ(x) = Φ(x) if x ∈ Ω, Φ(x) = 0 otherwise. By (48.4) we have ˜ |e−tAΩ Φ| ≤ e−tAΩ |Φ| ≤ e−tA |Φ|. The conclusion follows from assertions (c) and (d).

In the case of bounded domains, we have the following classical property of uniform exponential decay. Proposition 48.5. Let Ω be an arbitrary bounded domain and let (e−tA )t≥0 be the Dirichlet heat semigroup in Ω. For all 1 ≤ p ≤ ∞ and all Φ ∈ Lp (Ω), there holds e−tA Φ p ≤ C(Ω)e−λ1 t Φ p , t ≥ 0. (48.10) Proof. If 0 < t < 2, then (48.10) follows from Proposition 48.4(c). We may thus assume t ≥ 2. It is well known that e−tA Φ 2 ≤ e−λ1 t Φ 2 ,

t ≥ 0.

(48.11)

Using (48.11), Proposition 48.4(d) for p = 2, q = ∞, H¨ older’s inequality and |Ω|1/p ≤ max(1, |Ω|), we get e−tA Φ p ≤ |Ω|1/p e−tA Φ ∞ ≤ (4π)−n/4 |Ω|1/p e−(t−1)A Φ 2 ≤ C(Ω)e−λ1 (t−2) e−A Φ 2 .

442

Appendices

The assertion then follows from e−A Φ 2 ≤ Φ 2 ≤ C(Ω) Φ p if p ≥ 2, and from e−A Φ 2 ≤ Φ p (owing to Proposition 48.4(d)) if p < 2. In the case of the whole space and integrable initial data, the asymptotic behavior is described by a multiple of the Gaussian heat kernel (see [175], or [168] for further results). Proposition 48.6. Let Φ ∈ L1 (Rn ) and put M = Rn Φ dx. (i) There holds e−tA Φ − M Gt 1 → 0, t → ∞. (ii) If, in addition, x Φ(x) ∈ L1 (Rn ), then e−tA Φ − M Gt 1 ≤ Ct−1/2 x Φ(x) 1 ,

t > 0,

where C = C(n) > 0.

Proof. We first establish assertion (ii). Let Φ ∈ L1 Rn ; (1 + |x|) dx .

−tA −|x−y|2 /4t 2 −n/2 e e Φ − M Gt (x) = (4πt) − e−|x| /4t Φ(y) dy Rn

−n/2

(4πt) √ = 2 t

Rn

1

0

y · (x − θy) −|x−θy|2/4t √ e Φ(y) dy dθ. t

−s2 /8

< ∞ and Fubini’s theorem, we deduce that

1

2 e−|x−θy| /8t |y||Φ(y)| dx dy dθ e−tA Φ − M Gt 1 ≤ Ct−(n+1)/2

Using sups>0 se

= Ct−1/2

0

1

0

Rn

Rn

Rn

|y||Φ(y)| dy dθ = Ct−1/2 x Φ(x) 1 .

1 n Let us next prove pick a sequence {ϕj } ∈ assertion (i). Fix Φ ∈ L (R 1 ) and n D(R ) such that Rn ϕj dx = M and ϕj → Φ in L (Rn ). For each j we write

e−tA Φ − M Gt 1 ≤ e−tA ϕj − M Gt 1 + e−tA (Φ − ϕj ) 1 ≤ e−tA ϕj − M Gt 1 + Φ − ϕj 1 . By assertion (ii), it follows that lim sup e−tA Φ − M Gt 1 ≤ Φ − ϕj 1 t→∞

and the conclusion follows by letting j → ∞. We conclude with a smoothing estimate for the gradient, which we state without proof (this follows from [320, Theorem IV.16.3, p. 413]). Proposition 48.7. Let Ω be a domain of class C 2+α for some α ∈ (0, 1) and let (e−tA )t≥0 be the Dirichlet heat semigroup in Ω. For all Φ ∈ L∞ (Ω), there holds ∇e−tA Φ ∞ ≤ C(Ω)(1 + t−1/2 ) Φ ∞ ,

t > 0.


443

48.3. Weak and integral solutions In this subsection we compare various notions of solutions of the inhomogeneous linear heat equation. Related semigroup and smoothing properties will be described in Appendix C (Subsection 49.2). Assume that Ω is a bounded domain of class C 2+α for some α ∈ (0, 1). Similarly as in Remarks 15.4(iv) and (v), we may define integral and weak L1δ -solutions of the linear problem ⎫ ut − ∆u = f, x ∈ Ω, t ∈ (0, T ), ⎪ ⎬ u = 0, x ∈ ∂Ω, t ∈ (0, T ), (48.12) ⎪ ⎭ x ∈ Ω, u(x, 0) = u0 (x), as follows. Definition 48.8. (i) Let u0 ∈ L1δ (Ω) and f ∈ L1loc ((0, T ), L1δ (Ω)). A function u ∈ C([0, T ), L1δ (Ω)) ∩ L1loc ((0, T ), L1 (Ω)) is a weak L1δ -solution of (48.12) if u(·, 0) = u0 and, for any 0 < τ < t < T ,

t

τ

t

Ω 2

fϕ = − τ

Ω

u(ϕt + ∆ϕ) −

u(τ )ϕ(τ ) Ω

(48.13)

for all ϕ ∈ C (Ω × [τ, t]) such that ϕ = 0 on ∂Ω × [τ, t] and ϕ(t) = 0. (ii) Let u0 , f be nonnegative measurable functions and let G denote the Dirichlet heat kernel in Ω. Then

t

u(x, t) := G(x, y, t)u0 (y) dy + G(x, y, t − s)f (y, s) dy ds ≤ ∞ Ω

0

Ω

is called the integral solution of (48.12). Proposition 48.9. Let Ω be as above and let u0 ∈ L1δ (Ω). (i) If f ∈ L1loc ([0, T ), L1δ (Ω)), then problem (48.12) possesses a unique weak L1δ solution. Moreover u ∈ L1loc ([0, T ), L1 (Ω)) and (48.13) is also satisfied for τ = 0. (ii) If f ∈ L1loc ((0, T ), L1δ (Ω)) and problem (48.12) possesses a weak L1δ -solution, then f ∈ L1loc ([0, T ), L1δ (Ω)). Proof. (i) Let f ∈ L1loc ([0, T ), L1δ (Ω)). We first prove the uniqueness. Assume that u1 , u2 are two weak solutions of (48.12) and set w := u1 − u2 . Then w is a weak solution of the homogeneous problem (48.12) (with f = 0 and u0 = 0). In particular,

t

τ

Ω

w(ϕt + ∆ϕ) +

w(τ )ϕ(τ ) = 0 Ω

(48.14)

444

Appendices

whenever 0 < τ < t < T and ϕ ∈ C 2,1 (Ω × [τ, t]) satisfies ϕ = 0 on ∂Ω × [τ, t] and ϕ(t) = 0. Fix t ∈ (0, T ). Let ψ ∈ D(Qt ) and let ϕ = ϕψ be the solution of the problem −ϕt − ∆ϕ = ψ

in Qt ,

ϕ=0

on St ,

ϕ(t) = 0

in Ω.

Then passing to the limit in (48.14) as τ → 0 we obtain

t

wψ = 0, 0

Ω

hence w = 0 a.e. In order to prove the existence, we may assume that u0 ≥ 0 and f ≥ 0 (otherwise we decompose u0 and f into their positive and negative parts and use the linearity of the problem (48.12)). Set u0,k := min(u0 , k) and fk := min(f, k), k = 1, 2, . . . . Let uk be the (strong) solution of (48.12) with f and u0 replaced by fk and u0,k , respectively. Then

t

uk (x, t) =

Ω

G(x, y, t)u0,k (y) dy +

Ω

0

G(x, y, t − s)fk (y, s) dy ds,

(48.15)

where G denotes the Dirichlet heat kernel in Ω. Passing to the limit in (48.15) as k → ∞ we get uk (x, t) u(x, t), where u satisfies

t

u(x, t) = Ω

G(x, y, t)u0 (y) dy +

0

Ω

G(x, y, t − s)f (y, s) dy ds.

(48.16)

2,1;r (Ω × (0, T )) for all r ∈ (1, ∞) Notice also that uk ∈ C([0, T ), Lr (Ω)) ∩ Wloc (see Theorem 48.1 and Appendix E). Let 0 ≤ τ < t < T , q > 1 and ϕ ∈ 2,1;q C([τ, t], Lq (Ω)) ∩ Wloc (Ω × (τ, t)) satisfy ϕ = 0 on ∂Ω × (τ, t) and (ϕt + ∆ϕ) ∈ 1 L (Ω × (τ, t)). Multiplying the equation for uk by ϕ, integrating over Ω × (τ , t ) with τ < τ < t < t and letting τ → τ , t → t, we obtain

t

τ

Ω

t

fk ϕ = − τ

Ω

uk (ϕt + ∆ϕ) +

Ω

uk (t)ϕ(t) −

Set ϕ := ψ, where ψ is the solution of the problem −ψt − ∆ψ = 1 ψ=0 ψ(t) = 0

in Qt , on St , in Ω.

Ω

uk (τ )ϕ(τ ).

(48.17)


445

Then (48.17) with τ = 0 implies

t

0

t

Ω

uk =

0

Ω

fk ψ +

Ω

u0,k ψ(0) ≤ C(t) < ∞,

hence the sequence {uk } is bounded in L1 (Qt ), uk → u in L1 (Qt ) and u ∈ L1loc ([0, T ), L1(Ω)). Next set ϕ := χ, where χ(x, s) = eλ1 (s−t) ϕ1 (x), which satisfies χs + ∆χ = 0 in Qt . For k ≥ j, it follows from (48.17) with τ = 0 and (1.4) that

c1

Ω

t

(uk − uj )(t)ϕ1 = (fk − fj )χ + (u0,k − u0,j )χ(0) Ω Ω Ω 0

t

(fk − fj )δ + c2 (u0,k − u0,j )δ. ≤ c2

(uk − uj )(t)δ ≤

0

Ω

Ω

This estimate guarantees that {uk } is a Cauchy sequence in C([0, t], L1δ (Ω)), hence u ∈ C([0, T ), L1δ (Ω)). Finally, fix 0 ≤ τ < t < T and ϕ ∈ C 2 (Ω × [τ, t]) satisfying ϕ = 0 on ∂Ω × [τ, t] and ϕ(t) = 0. Then passing to the limit in (48.17) as k → ∞ we see that u is a weak solution of (48.12) and that (48.13) is also satisfied for τ = 0. For future reference, we note that the solution u that we have just constructed satisfies

t

|f |δ ≤ C |u(t)|δ, 0 < t < T, (48.18) Ω

0

Ω

where C remains bounded for T bounded. Indeed (still assuming u0 , f ≥ 0 without loss of generality), (48.18) follows by passing to the limit k → ∞ in (48.17) with τ = 0 and ϕ = χ. (ii) Now assume that problem (48.12) possesses a weak L1δ -solution u. Then, for each τ ∈ (0, T ), u coincides with the weak L1δ -solution of (48.12) on (τ, T ) with initial data u(τ ), given by part (i). For each t ∈ (0, T ), estimate (48.18) guarantees that

t

τ

Ω

|f |δ ≤ C

Ω

|u(t)|δ,

0 < τ < t,

and the assertion follows by letting τ → 0. Corollary 48.10. Let Ω be as above, u0 ∈ L1δ (Ω), u ∈ L1loc (QT ), and f : QT → R be measurable. Assume that u0 , u, f ≥ 0. (i) If f ∈ L1loc ((0, T ), L1δ (Ω)) and u is a weak L1δ -solution of (48.12), then it is an integral solution of (48.12). (ii) If u is an integral solution of (48.12), then f ∈ L1loc ((0, T ), L1δ (Ω)) and u is a weak L1δ -solution of (48.12).

446

Appendices

Proof. If u is a weak solution, then f ∈ L1loc ([0, T ), L1δ (Ω)) by Proposition 48.9(ii), and the proof of Proposition 48.9(i) (cf. formula (48.16)) shows that u is an integral solution. Let u be an integral solution of (48.12). Again, the proof of Proposition 48.9(i) guarantees that u is a weak solution provided we show f ∈ L1loc ([0, T ), L1δ (Ω)). Let fk , uk be as in the proof of Proposition 48.9(i). Let 0 < t < T < T and ψ ∈ D(QT ), ψ ≥ 0, ψ(·, t) ≡ 0. Let ϕ be the solution of the problem −ϕt − ∆ϕ = ψ ϕ=0 ϕ(T ) = 0

in QT , on ST , in Ω.

Then there exists ε > 0 such that ϕ(x, s) ≥ εδ(x)

for all (x, s) ∈ Qt .

Multiplying the equation ∂t uk − ∆uk = fk by ϕ we obtain

t

T

T

ε fk δ ≤ fk ϕ = uk ψ − u0,k ϕ(0) ≤ 0

hence

Ω

t 0

Ω

0

Ω

0

Ω

Ω

f δ < ∞, which guarantees f ∈ L1loc ([0, T ), L1δ (Ω)).

T

0

Ω

uψ < ∞,

Corollary 48.11. Let Ω be as above, q ≥ 1 and let u be a mild Lq -solution of (48.12) (that is u ∈ C([0, T ), Lq (Ω)), u(0) = u0 , f ∈ L1loc ((0, T ), L1 (Ω)) and (15.5) is true with f (u) replaced by f ). Then u is a weak L1δ -solution of (48.12). Proof. Fix τ0 ∈ (0, T ) and set u0,1 := u+ (τ0 ), u0,2 := u− (τ0 ), f1 := f+ , f2 := f− ,

t vi (t) = vi (t; τ0 ) := e−(t−τ0 )A u0,i + e−(t−s)A fi (s) ds, τ0 ≤ t < T, i = 1, 2. τ0

Then vi , i = 1, 2, are nonnegative integral solutions of problem (48.12) with [0, T ), u0 , f replaced by [τ0 , T ), u0,i , fi . Consequently, vi , i = 1, 2, are weak solutions of those problems and v := v1 − v2 is a weak solution of (48.12) on [τ0 , T ) with initial data u(τ0 ). On the other hand, v1 (t; τ0 ) − v2 (t; τ0 ) = u(t) for any t ∈ (τ0 , T ), hence u = v is a weak solution of (48.12) on [0, T ). Remark 48.12. In the case of Ω = Rn , for instance, and of nonnegative data u0 , f , one can also study the relations between local classical nonnegative solutions of (48.12) and integral solutions. Let Ω = Rn , u0 ∈ L1loc (Rn ), f be locally H¨ older continuous in QT , with u0 ≥ 0 a.e. and f ≥ 0. Assume that 0 ≤ u ∈ C 2,1 (QT ) ∩ C([0, T ); L1loc (Rn )) is a solution of ut − ∆u = f in QT , with u(·, 0) = u0 . Then u satisfies (48.16) in QT , where all the integrals are in particular finite (see [482] and cf. also [526]). Such property may be useful, e.g., when considering problems of Fujita-type.

49. Appendix C: Linear theory in Lpδ -spaces and in uniformly local spaces

447

49. Appendix C: Linear theory in Lpδ -spaces and in uniformly local spaces In this section, we state and prove some useful properties of the Laplace and heat equations in weighted Lebesgue spaces Lpδ (Ω) and in uniformly local Lebesgue spaces Lpul (Rn ). We refer to Section 1 for the definition of these spaces. p

49.1. The Laplace equation in Lδ -spaces Very weak, or L1δ , solutions of the Laplace equation (47.8) have been introduced in Definition 3.1. We have the following existence-uniqueness result (see [94]; estimate (49.1) is proved there for q = 1 and in the general case in [107]). Theorem 49.1. Let Ω ⊂ Rn be a bounded domain of class C 2+α for some α ∈ (0, 1), and let f ∈ L1δ (Ω). Then there exists a unique u ∈ L1 (Ω) such that u is an L1δ -solution of problem (47.8). Moreover, for all 1 ≤ q < n/(n − 1), we have u ∈ Lq (Ω) and u q ≤ C(n, q, Ω) f 1,δ . (49.1) Furthermore, the maximum principle is satisfied, i.e.: f ≥ 0 a.e. implies u ≥ 0 a.e. Proof. We start by proving the uniqueness. Thus assume that u ∈ L1 (Ω) is an L1δ -solution of problem (47.8) with f = 0. Take any h ∈ D(Ω) and let ϕ ∈ C 2 (Ω) be the classical solution of −∆ϕ = h in Ω, ϕ=0 on ∂Ω. Then Ω uh dx = 0 by (3.3). It follows that u = 0, hence the uniqueness assertion. Let us show the existence. We may assume f ≥ 0 without loss of generality (writing f = f+ − f− ). Let fi = min(f, i) and denote by ui the strong solution of (47.8) with f replaced by fi . Let Θ be the classical solution of (19.27). For j ≥ i, we have fj ≥ fi ≥ 0, hence uj ≥ ui ≥ 0 by the maximum principle. Testing the equation for uj − ui with Θ, we have

(uj − ui ) dx = (fj − fi )Θ dx. uj − ui 1 = Ω

Ω

Since fi → f in L1δ (Ω), we deduce that {ui } is a Cauchy sequence in L1 (Ω), and we denote by u ∈ L1 (Ω) its limit. Observe that u ≥ 0. For any ϕ ∈ C 2 (Ω) with ϕ = 0 on ∂Ω, we then have

u(−∆ϕ) dx = lim ui (−∆ϕ) dx = lim fi ϕ dx = f ϕ dx, (49.2) Ω

i→∞

Ω

i→∞

Ω

Ω

448

Appendices

hence u is an L1δ -solution of (47.8). Next, the choice ϕ = Θ in (49.2) yields Ω u dx = Ω f Θ dx. Assuming again f ≥ 0, this implies estimate (49.1) for q = 1. The case 1 < q < n/(n − 1) will be proved along with Theorem 49.2. The following results describe the optimal regularity of the Dirichlet Laplacian in the scale of Lpδ -spaces (see [84], [200] for Theorem 49.2 and [489] for Theorem 49.3). The proofs will be given in Subsection 49.4 below. Theorem 49.2. Let Ω ⊂ Rn be a bounded domain of class C 2+α for some α ∈ (0, 1). Assume that 1 ≤ p ≤ q ≤ ∞ satisfy 1 1 2 − < . p q n+1

(49.3)

Let f ∈ L1δ (Ω) and let u be the L1δ -solution of (47.8). (i) If f ∈ Lpδ (Ω), then u ∈ Lqδ (Ω) and u q,δ ≤ C(p, q, Ω) f p,δ .

(49.4)

(ii) If f+ ∈ Lpδ (Ω), then u+ ∈ Lqδ (Ω) and u+ q,δ ≤ C(p, q, Ω) f+ p,δ .

(49.5)

Theorem 49.3. Let Ω ⊂ Rn be a bounded domain of class C 2+α for some α ∈ (0, 1). Assume that 1 ≤ p < q ≤ ∞ satisfy 2 1 1 − > . p q n+1 Then there exists f ∈ Lpδ (Ω) such that the L1δ -solution u of (47.8) satisfies u ∈ Lqδ (Ω). Remarks 49.4. (a) In Theorem 49.2 one may take in particular q = ∞ for p > (n + 1)/2 and any q < (n + 1)/(n − 1) for p = 1. 1 (b) By a density argument, it is easy to see that the Lδ -solution u of (47.8) is given by u(x) = Ω K(x, y)f (y) dy, where K(x, y) is the Dirichlet Green kernel in Ω. (c) By similar arguments as in the proofs of Theorems 49.2 and 49.3 (see [489] for details), one can obtain further optimal regularity properties of the solution u of (47.8). Namely, assuming 1 ≤ p ≤ q ≤ ∞:


• if

n+1 p

−

n q

< 2, then u ∈ Lq (Ω) and u q ≤ C f p,δ ;

• if

n+1 p

−

n q

> 2, then there exists f ∈ Lpδ such that u ∈ Lq (Ω);

• if

n+1 p

−

n q

< 1, then u ∈ W01,q (Ω) and u 1,q ≤ C f p,δ ;

• if

n+1 p

−

n q

> 1, then there exists f ∈ Lpδ such that u ∈ W01,q (Ω).

449

In particular, it follows that if f ∈ Lpδ for some p > 1, then u ∈ W01,q (Ω) for q > 1 close to 1, so that the boundary conditions in (47.8) are also satisfied in the sense of traces. We note that in the example constructed in the proof of Theorem 49.3, the solution u possesses a singularity at a (single) boundary point a ∈ ∂Ω and 2,m that u ∈ Wloc (Ω \ {a}) for all finite m. (d) Theorem 49.2 remains true in case of equality in (49.3) provided p > 1, q < ∞ and n = 2 (see [362], where equality cases in Remark (c) are also treated). (e) If f ∈ L1δ , then, for each α ∈ (0, 1), we have u/δ α ∈ L1 (Ω) and u/δ α 1 ≤ C(α) f 1,δ . This can be shown by using the singular test-function ξ from Lemma 10.4 for smooth f and the general case follows by density. We close this subsection by proving a useful, simple consequence of Theorem 49.2 (cf. [449, Proposition 2.3]). Proposition 49.5. Let Ω ⊂ Rn be a bounded domain of class C 2+α for some α ∈ (0, 1). Let f ∈ L1δ (Ω) and let u be the L1δ -solution of (47.8). Then, for any 1 ≤ k < (n + 1)/(n − 1), we have u k,δ ≤ C(Ω, k) u+ 1,δ + f− 1,δ .

Proof. Using (3.3) with ϕ = ϕ1 and (1.4), we obtain

Ω

|f |ϕ1 =

f ϕ1 + 2 (f− )ϕ1 Ω

= λ1 uϕ1 + 2 (f− )ϕ1 ≤ C(Ω) u+ 1,δ + f− 1,δ . Ω

Ω

Ω

Applying Theorem 49.2(i) with p = 1 and using ϕ1 ≥ c1 δ, we deduce that u k,δ ≤ C(Ω, k) f 1,δ ≤ C(Ω, k) u+ 1,δ + f− 1,δ .

450

Appendices

p

49.2. The heat semigroup in Lδ -spaces We start by introducing a natural extension of the Dirichlet heat semigroup. Here we also use the spaces Lpϕ1 (Ω), which are defined similarly as Lpδ (Ω). Note that if . Ω is C 2 -smooth, then Lpϕ1 (Ω) = Lpδ (Ω), due to (1.4). Proposition and Definition 49.6. Let Ω be an arbitrary bounded domain in Rn . The Dirichlet heat semigroup admits a unique extension to L1ϕ1 (Ω), still denoted by (e−tA )t≥0 . It is a contraction semigroup on L1ϕ1 (Ω), which satisfies e−tA φ 1,ϕ1 = e−λ1 t φ 1,ϕ1 ,

t ≥ 0,

φ ∈ L1ϕ1 (Ω).

(49.6)

Moreover the maximum principle is satisfied, i.e.: φ ∈ L1ϕ1 (Ω) and φ ≥ 0 a.e. imply e−tA φ ≥ 0 a.e.

(49.7)

Furthermore, for each 1 < p < ∞, (e−tA )t≥0 restricts to a contraction semigroup on Lpϕ1 (Ω), which satisfies e−tA φ p,ϕ1 ≤ e−(λ1 /p)t φ p,ϕ1 ,

t ≥ 0,

φ ∈ Lpϕ1 (Ω).

(49.8)

In addition, if Ω is of class C 2 , then we have e−tA φ p,δ ≤ C(Ω) e−(λ1 /p)t φ p,δ ,

t ≥ 0,

φ ∈ Lpδ (Ω),

1 ≤ p < ∞.

(49.9)

Proof. Let φ ∈ L2 (Ω) with φ ≥ 0. Since e−tA is self-adjoint on L2 (Ω) and e−tA ϕ1 = e−λ1 t ϕ1 , we have, for all t ≥ 0, e−tA φ 1,ϕ1 = (e−tA φ, ϕ1 ) = (φ, e−tA ϕ1 ) = e−λ1 t (φ, ϕ1 ) = e−λ1 t φ 1,ϕ1 . (49.10) Writing φ = φ+ −φ− and using the linearity and the positivity preserving property of e−tA , it follows that (49.10) is true for all φ ∈ L2 (Ω). Now fix φ ∈ L1ϕ1 (Ω) and pick a sequence {φi } in L2 (Ω), such that φi → φ in 1 Lϕ1 (Ω). For each fixed t > 0, (49.10) implies e−tA φi − e−tA φj 1,ϕ1 = e−λ1 t φi − φj 1,ϕ1 , thus {e−tA φi } is a Cauchy sequence in L1ϕ1 (Ω). Consequently, we may define e−tA φ := limi→∞ e−tA φi , and it follows from (49.10) that the limit is independent of the choice of the sequence {φi }, hence the uniqueness assertion. Moreover, (49.6) is satisfied. On the other hand, if φ ≥ 0, by choosing φi = min(φ, i) ≥ 0, we obtain (49.7).


451

p Finally let 1 ≤ p < ∞ and φ ∈ Lϕ1 (Ω). By using Jensen’s inequality and G(x, y, t) dy ≤ 1, we have Ω

|e−tA φ|p ≤ e−tA (|φ|p ),

φ ∈ Lpϕ1 (Ω)

(49.11)

(first assume φ ∈ Lp (Ω) and then argue by density). Therefore, using (49.6), we get e−tA φ pp,ϕ1 = |e−tA φ|p 1,ϕ1 ≤ e−tA (|φ|p ) 1,ϕ1 = e−λ1 t |φ|p 1,ϕ1 = e−λ1 t φ pp,ϕ1 , hence (49.8). If Ω is smooth, then (49.9) follows from (49.8) and (1.4). The following result provides optimal smoothing estimates for the Dirichlet heat semigroup in the scale of Lpδ -spaces (see [200] for assertion (i) and [200], [489] for assertion (ii)). Its proof is postponed to Subsection 49.4 below. Theorem 49.7. Let Ω ⊂ Rn be a bounded domain of class C 2 , let 1 ≤ p ≤ q ≤ ∞ 1 1 and set β = n+1 2 ( p − q ). (i) For all φ ∈ Lpδ (Ω), we have e−tA φ q,δ ≤ C(p, q, Ω) φ p,δ t−β ,

t > 0.

(49.12)

(ii) For all ε > 0, there exist a function φ ∈ Lpδ (Ω) and a constant C > 0, such that e−tA φ q,δ ≥ Ct−β+ε , for t > 0 small. Remarks 49.8. (a) The elliptic and parabolic estimates in Theorems 49.2 and 49.7 exhibit a remarkable dimension shift phenomenon: they are similar to those in standard Lp -spaces in n+1 dimensions (cf. Proposition 47.5 and Proposition 48.4). ∞ and interpolating between (b) Assume Ω smooth. Recalling that L∞ δ = L (49.9) with p = 1 and (48.10) with p = ∞, we see that there exists C = C(Ω) > 0 such that

e−tA φ p,δ ≤ C e−λ1 t φ p,δ ,

t ≥ 0,

φ ∈ Lpδ (Ω),

1 ≤ p ≤ ∞,

(49.13)

which is an alternative to (49.8). (c) Assume Ω smooth and let u0 ∈ L1δ (Ω). By a density argument, it is easy to see that u(t) := e−tA u0 satisfies u(x, t) = Ω G(x, y, t)u0 (y) dy in Ω× (0, ∞). Moreover u is a weak L1δ -solution of (48.12) with f = 0 in the sense of Remark 15.4(v).

452

Appendices

49.3. Some pointwise boundary estimates for the heat equation We here state and prove some pointwise estimates for the heat (and the Laplace) equation, involving the distance to the boundary, which are essential to establish the Lpδ -properties stated above. Some of them are also used at other places. Proposition 49.9. Let Ω be a bounded domain of class C 2 . There exists C = C(Ω) > 0 such that, for all φ ∈ L∞ (Ω), −tA δ(x) e φ (x) ≤ C φ ∞ √ , t

x ∈ Ω, t > 0.

(49.14)

Proposition 49.9 can be derived as a consequence of Gaussian estimates for the gradient of the heat kernel [477] (or of the reverse of estimate (49.17) below). However, we shall give a maximum principle based, self-contained proof relying on arguments from [346], [347]. Proof. Step 1. We consider the auxiliary problem Vt − ∆V = 1, V = 0, V (x, 0) = 0,

⎫ ⎪ ⎬

x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0,

⎪ ⎭

x ∈ Ω,

(49.15)

and we claim that, for some T = T (Ω) > 0, there holds √ V (x, t) ≤ 2 t δ(x),

x ∈ Ω, 0 < t ≤ T.

(49.16)

To show (49.16) we use a barrier argument based on the construction of a suitable supersolution. Fix x1 ∈ Ω and pick x2 ∈ ∂Ω such that δ(x1 ) =|x1 − x2 |. Since the domain Ω is C 2 -smooth and bounded, one can find 0 < ρ < R independent of x1 , and a ∈ Rn , such that Ω ⊂ D := {x ∈ Rn : ρ < |x − a| < R} and Ω ∩ B(a, ρ) = {x2 }. For (x, t) ∈ Q := D × (0, ∞), we consider the function V (x, t) = t ϕ(y), where y=

|x − a| − ρ √ t

and ϕ(y) =

y(2 − y)

if 0 ≤ y < 1,

1

if y ≥ 1.

˜ := {(x, t) ∈ Q : y = 1}. We The function V is C 1 in t on Q and C 2 in x on Q compute y 1 V t = ϕ(y) − ϕ (y) ≥ χ{y≥1} − χ{0≤y 0 and c2 ≥ 2 depending only on n, such that the Dirichlet heat kernel G(x, y, t) in Ω satisfies G(x, y, t) ≥ c1 t−n/2 , for all t > 0 and x, y ∈ Ω such that √ δ(x) ≥ c2 t

and

|x − y| ≤

√

t.

Proposition 49.10 is a consequence of the sharp estimate [545] −n/2 −C |x−y|2 /t t e 2 , G(x, y, t) ≥ C1 min 1, δ(x)δ(y) t

for t > 0 small,

(49.17)

454

Appendices

but the proof of (49.17) is much more delicate. (Estimate (49.17) is in fact proved in [545] for C 2 bounded domains and n ≥ 3; the reverse inequality, with different constants C1 , C2 , is also true [153].) Here we give an elementary and self-contained proof of Proposition 49.10 based only on the maximum principle. Proof. Fix y ∈ Ω, let ρ = δ(y), B = B(y, ρ), and denote u(x, t) = (4πt)−n/2 e−|x−y|

2

/4t

,

x ∈ B, t > 0.

For x ∈ ∂B, we have u(x, t) = ρ−n g(tρ−2 ), where g(s) = (4πs)−n/2 e−1/4s . Let a(n) := sups>0 g(s) (which is finite) and put M := a(n)ρ−n .

u(x, t) = u(x, t) − M, Then u satisfies ut − ∆u = 0, u ≤ 0,

x ∈ B, t > 0,

x ∈ ∂B, t > 0,

(49.18)

and moreover u(·, t) → δy − M in the sense of measures, as t → 0, where δy is the Dirac measure at point y. It follows from the maximum principle that G(x, y, t) ≥ u(x, t) in B × (0, ∞). (More precisely, one can easily show that inequality (52.16), with u(x, t) replaced by G(x, y, t) − u(x, t), is satisfied for f = 0 and u0 = √ 0; so t and the assertion follows from Proposition 52.13(ii).) In particular, if δ(x) ≥ c 2 √ √ |x − y| ≤ t, hence ρ = δ(y) ≥ (c2 − 1) t, we obtain G(x, y, t) ≥ (4π)−n/2 e−1/4 − a(n)(c2 − 1)−n t−n/2 ≥ c1 (n)t−n/2 provided we choose c2 = c2 (n) > 1 large enough. Proposition 49.11. Let Ω be a bounded domain of class C 2 . (i) The Dirichlet heat kernel G(x, y, t) in Ω satisfies G(x, y, t) ≥ c(t, Ω) δ(x)δ(y),

x, y ∈ Ω, t > 0

where the constant c(t, Ω) is uniformly positive for t bounded and bounded away from 0. (ii) There exists a constant c = c(Ω) > 0 such that the Dirichlet Green kernel K(x, y) in Ω satisfies K(x, y) ≥ c δ(x)δ(y),

x, y ∈ Ω.


455

Remarks 49.12. (i) Proposition 49.11 provides quantitative versions of Hopf’s lemma. Namely, let f ∈ L1δ (Ω) satisfy f ≥ 0 a.e. Then the L1δ (Ω) solution u of the Laplace equation (47.8) satisfies

u(x) = K(x, y)f (y) dy ≥ c(Ω) f 1,δ δ(x), x ∈ Ω. Ω

Likewise, for the heat equation, we have

−tA e f (x) = G(x, y, t)f (y) dy ≥ c(t, Ω) f 1,δ δ(x), Ω

x ∈ Ω, t > 0. (49.19)

(ii) Estimate (49.19) is sharp in the sense that, for all f ∈ L1δ (Ω), |(e−tA f )(x)| ≤ c(Ω) t−(n+2)/2 f 1,δ δ(x),

x ∈ Ω, t > 0.

This follows by writing e−tA f = e−(t/2)A (e−(t/2)A f ) and combining (49.14) with (49.12) for p = 1, q = ∞. Again, Proposition 49.11 is a consequence of estimate (49.17). We give a simple proof essentially based on [345] (see also [92]). Proof of Proposition 49.11. (i) We may assume, without loss of generality, that B(0, 4ρ) ⊂ Ω for some ρ > 0. In what follows, c(t) will denote any constant depending only on t and Ω (or ρ) and such that c(t) is uniformly positive for t bounded and bounded away from 0. For each y ∈ Rn , let us denote by (e−tAy )t≥0 the Dirichlet heat semigroup in B(y, 3ρ). Fix y ∈ B(0, ρ) and t > 0. Since B(y, 3ρ) ⊂ Ω, the maximum principle implies e−tA δy ≥ e−tAy δy

in B(y, 3ρ),

(49.20)

where δy is the Dirac measure at point y. Also, by the strong maximum principle, we have e−tA0 δ0 ≥ c(t) χB(0,2ρ) . (49.21) −tA −tA y 0 Since e δy (x) = e δ0 (x − y), it follows from (49.20) and (49.21) that e−tA δy ≥ c(t) χB(y,2ρ) ≥ c(t) χB(0,ρ) .

(49.22)

On the other hand, by Hopf’s lemma (see Proposition 52.7), we have e−tA χB(0,ρ) ≥ c(t) δ. Combining (49.22) and (49.23) (with t replaced by t/2), we obtain e−tA δy = e−(t/2)A e−(t/2)A δy ≥ c(t) e−(t/2)A χB(0,ρ) ≥ c(t) δ.

(49.23)

456

Appendices

In other words, we have shown that G(x, y, t) = (e−tA δy (x) ≥ c(t) δ(x)χB(0,ρ) (y), x, y ∈ Ω, t > 0. (49.24) Using G(x, y, t) = G(y, x, t) = e−tA δy (x) = e−tA δx (y), and (49.24), (49.23) (with t replaced by t/2), we then obtain G(x, y, t) = e−(t/2)A (e−(t/2)A δx ) (y) ≥ c(t) δ(x) e−(t/2)A χB(0,ρ) (y) ≥ c(t) δ(x)δ(y), hence assertion (i). ∞ (ii) Since K(x, y) = 0 G(x, y, t) dt, this is an immediate consequence of assertion (i).

49.4. Proof of Theorems 49.2, 49.3 and 49.7 We begin with the Lpδ -Lqδ -estimates. We first treat the parabolic case (Theorem 49.7(i)). The elliptic case (Theorem 49.2) will next be deduced as a consequence. Proof of Theorem 49.7(i). In this proof, C denotes any positive constant depending only on Ω (not on p, q). Let φ ∈ L2 (Ω) with φ ≥ 0. Since e−tA is selfadjoint on L2 (Ω) we deduce from Proposition 49.9 that, for all t > 0, e−tA φ 1 = (e−tA φ, χΩ ) = (φ, e−tA χΩ ) ≤ Ct−1/2 (φ, δ), hence

e−tA φ 1 ≤ Ct−1/2 φ 1,δ ,

t > 0.

(49.25)

Writing φ = φ+ −φ− and using the linearity and the positivity preserving property of e−tA , it follows that (49.25) is true for all φ ∈ L2 (Ω). Let now φ ∈ L1δ (Ω) and take φi ∈ L2 (Ω) such that φi → φ in L1δ (Ω). We have e−tA φi → e−tA φ in L1δ (Ω) by (49.6), hence a.e. (up to a subsequence). By Fatou’s lemma, we infer that (49.25) is true for all φ ∈ L1δ (Ω). Next using the L1 -L∞ -estimate (see Proposition 48.4), we deduce that e−tA φ ∞ = e−(t/2)A (e−(t/2)A φ) ∞ ≤ (2πt)−n/2 e−(t/2)A φ 1 ≤ Ct−(n+1)/2 φ 1,δ .

(49.26)

Now take φ ∈ Lpδ (Ω). Using (49.11) and applying (49.26) with φ replaced by |φ|p , we get e−tA φ p∞ = |e−tA φ|p ∞ ≤ e−tA (|φ|p ) ∞ ≤ Ct−(n+1)/2 |φ|p 1,δ = Ct−(n+1)/2 φ pp,δ ,


457

hence (49.12) for q = ∞. For p ≤ q < ∞, combining this with (49.9) yields −tA φ pp,δ ≤ C (q−p)/p C p t−(n+1)(q−p)/2p φ qp,δ . e−tA φ qq,δ ≤ e−tA φ q−p ∞ e

Raising to the power 1/q, we obtain (49.12). Proof of Theorem 49.2. (i) Let us first assume f ∈ D(Ω). Observe that u is a solution to the inhomogeneous heat equation with initial data u and right-hand side f . Therefore,

t

u = e−tA u +

e−sA f ds,

0

for all t > 0, by the variation-of-constants formula. Next, by (49.12) and (49.13), we have e−sA f q,δ ≤ Cs−β e−(s/2)A f p,δ ≤ Cs−β e−λ1 s/2 f p,δ . Consequently, −β −λ1 t/2

u q,δ ≤ Ct

e

u p,δ + C

t 0

s−β e−λ1 s/2 ds f p,δ ,

where the integral over (0, t) is convergent due to β < 1. Estimate (49.4) for f ∈ D(Ω) follows upon letting t → ∞. (Note that if q < ∞ one can also use (49.9) instead of (49.13).) Now, in the general case f ∈ Lpδ (Ω), the conclusion follows by a density argument: Take fi ∈ D(Ω) such that fi → f in Lpδ (Ω) and let ui be the solution of (47.8) with f replaced by fi . By (49.1) for q = 1 (which we already proved), we have ui → u in L1 (Ω), hence a.e. (up to a subsequence). Passing to the limit in ui q,δ ≤ C fi p,δ by Fatou’s lemma, the conclusion follows. (ii) Let v ≥ 0 be the L1δ -solution of (47.8) with f replaced by f+ . We have u ≤ v by the maximum principle (cf. Theorem 49.1), hence u+ ≤ v. Estimate (49.5) then follows from (49.4). Proof of Proposition 47.5. It is completely similar to that of Theorem 49.2, except that we use Propositions 48.5 and 48.4, instead of formulas (49.12) and (49.13). Proof of (49.1) in Theorem 49.1. For any φ ∈ L1δ (Ω), using inequality (49.25), the L1 -Lq -estimate and (49.9), we get n

1

e−tA φ q ≤ Ct− 2 (1− q ) e−(t/2)A φ 1 ≤ Ct− ≤ Ce−λ1 t/4 t−

n+1 n 2 + 2q

n+1 n 2 + 2q

e−(t/4)A φ 1,δ

φ 1,δ .

Arguing as in the proof of Theorem 49.2(i), we then obtain (49.1).

458

Appendices

We now proceed to prove the optimality results, namely Theorem 49.3 and Theorem 49.7(ii). The proofs are based on the construction of an appropriate right-hand side of the Laplace equation (or initial data of the heat equation), with suitable boundary singularities. It is supported in a conical subdomain of Ω with vertex at a boundary point. The following lemma provides key lower estimates of the corresponding solutions in the same cone. This construction is used also in Sections 11 and 31 to show the existence of unbounded solutions of nonlinear elliptic equations and systems. Lemma 49.13. Let n ≥ 2 and let Ω ⊂ Rn be a bounded domain of class C 2+γ for some γ ∈ (0, 1). Assume that 0 ∈ ∂Ω. Let α < n − 1. There exist R > 0 and a revolution cone Σ1 of vertex 0, with Σ := Σ1 ∩ B2R ⊂ Ω, such that the function φ := |x|−(α+2) χΣ

(49.27)

belongs to L1δ (Ω) and enjoys the following properties. (i) Denote V (t) = e−tA φ. Then (49.28) V (x, t) ≥ Ct−(α+2)/2 √ for all x, t such that x ∈ Σ, |x| ≤ R and σ|x| ≤ t ≤ 2σ|x|, where σ > 0 is a constant. (ii) The L1δ -solution U > 0 of (47.8) with f = φ satisfies U ≥ C|x|−α χΣ .

(49.29)

Proof. Write x = (x1 , x ), x = (x2 , . . . , xn ). Since Ω is a C 2 -domain, we may assume without loss of generality that Ω contains the (truncated) revolution cone

Σ0 := x : |x | ≤ 2θx1 , |x| ≤ 3R , for some θ, R > 0. Next define Σ1 := x : |x | ≤ θx1 },

Σ := Σ1 ∩ B2R ,

and let φ be defined by (49.27). The fact that φ ∈ L1δ will follow from Lemma 49.14 below. Let the constant c2 ≥ 2 be given by Proposition 49.10. We observe that there exists σ = σ(θ) ∈ (0, 1/c2 ) such that δ(x) ≥ dist(x, Σc0 ) ≥ 2c2 σ|x|,

for all x ∈ Σ.

(49.30)

(Indeed, dist(x, {z : |z | = 2θz1 }) ≥ |x| sin(β − β ), where β = arctan(2θ), β = arctan θ, and dist(x, {z : |z| = 3R}) ≥ R ≥ |x|/2.)


459

√ Let now x, t satisfy x ∈ Σ, |x| ≤ R and σ|x| ≤ √t ≤ 2σ|x|. In particular, we have t ≤ (2σ|x|)2 ≤ R2 and, by (49.30), δ(x) ≥ c2 t. By Proposition 49.10, it follows that

−(α+2) V (x, t) = G(x, y, t)φ(y) dy ≥ c1 t−n/2 χΣ (y) dy. √ |y| Ω

|x−y|≤ t

√ 2 Observe that, due , we have Σ ∩ B(x, t) ⊃ √ t ≤ Rn/2 √ √ to x ∈ Σ, |x| ≤ R and ≥ Ct . Since σ|x| ≤ t ≤ 2σ|x| (x + Σ) ∩ B(x, t), hence meas(Σ√∩ B(x, t)) √ √ (with 0 < σ < 1/2) and |x − y| ≤ t imply c t ≤ |y| ≤ C t, we obtain √ V (x, t) ≥ Ct−n/2 t−(α+2)/2 meas(Σ ∩ B(x, t)) ≥ Ct−(α+2)/2 . (49.31) This proves (i). Let x ∈ Σ. If |x| ≤ R, by (49.28), we have

4σ2 |x|2

∞ V (x, t) dt ≥ Ct−(α+2)/2 dt ≥ C|x|−α . U (x) = σ2 |x|2

0

If |x| ≥ R, then δ(x) ≥ 2c2 σR due to (49.30). By Remark 49.12(i), it follows that U (x) ≥ C with C > 0 independent of x. Thus (ii) is proved. As for the integrability properties of the functions φ, U , V , we have the following simple lemma. Lemma 49.14. Let Ω, α, φ, U be as in Lemma 49.13. (i) Assume α > −2. The function φ ∈ Lpδ (Ω) if and only if p < (n + 1)/(α + 2). (ii) Assume α > 0. If q ≥ (n + 1)/α, then U ∈ Lqδ (Ω). n+1

α+2

(iii) For 1 ≤ q ≤ ∞, there holds V (t) q,δ ≥ Ct 2q − 2 for t > 0 small. Proof. (i) We have φ pp,δ = C Σ |x|−(α+2)p δ(x) dx. By (49.30) and δ(x) ≤ |x|, the last integral has the same nature (finite or infinite) as

2R

|x|1−(α+2)p dx = rn−(α+2)p dr dω, Σ

Σ

0

where Σ = {x/|x| ∈ S : x ∈ Σ \ {0}}. Therefore, φ ∈ Lpδ if and only if p < (n + 1)/(α + 2). (ii) In view of (49.29), this follows from assertion (i). √ (iii) Due to (49.28) we may assume q < ∞. Let A(t) = {x ∈ Σ : σ|x| ≤ t ≤ 2σ|x|}. For t < σ 2 R2 , we have A(t) ⊂ BR . By (49.28) and (49.30), it follows that

1 q − α+2 q − α+2 q 2 2 2 V (x, t)δ(x) dx ≥ Ct δ(x) dx ≥ Ct dx n−1

Ω

1

≥ Ct 2 −

α+2 2 q

A(t)

√ √

t/σ

rn−1 dr t/2σ

A(t)

dω = Ct Σ

n+1 α+2 2 − 2 q

.

460

Appendices

After these preparations, we can now easily conclude. Proof of Theorem 49.3. By assumption, one can choose α ∈ (0, n − 1) such n+1 that n+1 q < α −2 such that n+1 p − 2 − 2ε < α < n+1 n+1 α+2 − 2. Then − < −β + ε and the result follows from Lemmas 49.13 and p 2q 2 49.14(i) and (iii).

49.5. The heat equation in uniformly local Lebesgue spaces We have the following smoothing property for the linear heat equation in uniformly local spaces. Proposition 49.15. Let 1 ≤ p < ∞. (i) The heat semigroup on Rn , given by e−tA φ = Gt ∗ φ, is well defined on Lpul and e−tA (Lpul ) ⊂ L∞ for all t > 0. (ii) Let 0 < T < ∞, p ≤ q ≤ ∞ and φ ∈ Lpul . Then n

1

1

e−tA φ q,ul ≤ C(n, p, q, T )t− 2 ( p − q ) φ p,ul ,

0 < t ≤ T.

We use the following simple lemma. Lemma 49.16. Let 1 ≤ p < ∞. The norms · p,ul and · p,∗ , where $

φ p,∗ := sup

a∈Rn

%1/p

= G1 ∗ |φ|p 1/p ∞ ,

|φ(a − y)| G1 (y) dy p

Rn

are equivalent on Lpul . Proof. On the one hand, we have

p −n/2 −1/4 |φ(y)| G1 (a − y) dy ≥ (4π) e Rn

|y−a| 0.

In what follows, Ω is an arbitrary domain in Rn . H¨ older’s inequality. Let 1 ≤ p ≤ ∞ and q = p = p/(p − 1). Then uv 1 ≤ u p v q ,

u ∈ Lp (Ω), v ∈ Lq (Ω).

A useful consequence is the following interpolation inequality. Let 1 ≤ p < r < q ≤ ∞. If u ∈ Lp ∩ Lq (Ω), then u ∈ Lr (Ω) and , u r ≤ u θp u 1−θ q

where θ =

1 r

−

1 1 1 −1 − ∈ (0, 1). q p q

Jensen’s inequality. Assume that F : R →[0, ∞) is a convex function, and that w : Ω → [0, ∞] is measurable and satisfies Ω w(x) dx = 1. If u is a measurable function on Ω such that uw, F (u)w ∈ L1 (Ω), then

F Ω

u(x)w(x) dx ≤ F (u(x))w(x) dx. Ω

Sobolev’s inequality. Let 1 ≤ p < n and denote p∗ = np/(n − p). Then u p∗ ≤ C(n, p) ∇u p ,

u ∈ W01,p (Ω).

(50.1)

50. Appendix D: Poincaré, Hardy-Sobolev, and other useful inequalities

463

50.2. The Poincar´ e inequality Let Ω be an arbitrary domain in Rn and let 1 ≤ q < ∞. The Poincaré inequality in W01,q (Ω) is the statement that v q ≤ Cq (Ω) ∇v q ,

for all v ∈ W01,q (Ω).

(50.2)

It is well known (see e.g. [90]) that (50.2) holds in any bounded domain, or more generally in any domain which is bounded in one direction. However, since this is a basic inequality in the study of elliptic and parabolic problems, it is important to have a characterization of those domains Ω such that (50.2) is true. It turns out that there is a simple geometric necessary condition, which is also almost sufficient. Moreover the equivalence is true for uniformly regular domains. To this end, let us introduce the notion of inradius ρ(Ω) of a domain Ω:

ρ(Ω) = sup r > 0 : Ω contains a ball of radius r = sup dist(x, ∂Ω). x∈Ω

We also define the strict inradius ρ (Ω) ≥ ρ(Ω), given by: ρ (Ω) = inf R > 0 : ∃ε > 0 such that for any ball B of radius R,

B ∩ Ωc contains a ball of radius ε . The relation between Poincaré inequalities and the inradius and strict inradius is given by the following result. Proposition 50.1. Let Ω be an arbitrary domain in Rn . (i) If (50.2) holds for some q ∈ [1, ∞), then ρ(Ω) < ∞. (ii) If ρ (Ω) < ∞, then (50.2) holds for all 1 ≤ q < ∞. (iii) Assume that Ω is uniformly regular or, more generally, that Ω satisfies a uniform exterior cone condition. Then for all 1 ≤ q < ∞, (50.2) holds if and only if ρ(Ω) < ∞. Examples 50.2. Let us give some simple examples concerning inradius and strict inradius in the case of unbounded domains. (a) If Ω is contained in a strip, then ρ and ρ are both finite, while if Ω contains an infinite cone, then they are both infinite. (b) If Ω is the complement of a periodic net of points, Ω = Rn \ {Rz}, for some R > 0, then ρ(Ω) = n1/2 R/2, ρ (Ω) = ∞.

z∈Zn

(c) If Ω is the complement of a periodic net of balls of constant radius, Ω = Rn \ B(Rz, ε), for some 0 < ε < R/2, then ρ(Ω) = ρ (Ω) = n1/2 R/2 − ε. z∈Zn

Part (i) of Proposition 50.1 is easy. The idea of the proof of part (ii) is due to [4, Lemma 7.4 p. 75], where this is done for q = 2 (see [481] for the general case).

464

Appendices

On the other hand, Proposition 50.1 (for all q) can be proved as a consequence of more general and difficult results, where the domain need not be uniformly regular (see [336, Corollary 2]). See also [154, Section 1.5] and the references therein for related results. Proof of Proposition 50.1. (i) Assume ρ(Ω) = ∞. This means that Ω contains some ball Bj = B(xj , j) for all j ≥ 1. Fixing a test-function w ∈ D(Rn ), w ≥ 0, w ≡ 0 with supp(w) ⊂ B(0, 1), and setting wj (x) = w((x − xj )/j), we get that wj ∈ D(Ω) and that wj q = j n/q w q

∇wj q = j (n/q)−1 ∇w q .

and

Consequently, (50.2) is false. (ii) By density, it obviously suffices to consider the case v ∈ D(Ω). Applying the definition of ρ := ρ (Ω), we may choose ε ∈ (0, 1) such that for any ball B of radius ρ + 1, B ∩ Ωc contains a ball of radius ε. Let Q be any cube of edge 2(ρ + 1), such that Q ∩ Ω = ∅. By translation we may assume that Q is centered at the origin. By the definition of ρ , there exists a point a such that B(a, ε) ⊂ B(0, ρ + 1) ∩ Ωc . In particular, B(a, ε) ∩ Ω = ∅ and d(0, a) < ρ + 1 (hence a ∈ Q). Using polar coordinates about a, denoted by (r, ω), we may represent the cube ˜ = {(r, ω) : ω ∈ S n−1 , 0 ≤ r < R(ω)}, where R(ω) is some Q by the set Q (continuous nonnegative) function. Using supp(v) ⊂ Ω and B(a, ε) ∩ Ω = ∅, we get

|v(x)|q dx = Q

S n−1

R(ω)

|v(r, ω)|q rn−1 dr dω. ε

Now, for all x ∈ Q, there holds d(a, x) ≤ d(a, 0) + d(0, x) ≤ R := (1 + n1/2 )(ρ + 1), ˜ hence R(ω) ≤ R. Using Hölder’s inequality, we have, for all (r, ω) ∈ Q,

|v(r, ω)|q =

r

q vr (σ, ω) dσ ≤ Rq−1

ε

≤ ε1−n Rq−1

R(ω)

|vr (σ, ω)|q dσ ε

R(ω)

|vr (σ, ω)|q σ n−1 dσ. ε

It follows that

S n−1

R(ω)

|v(r, ω)|q rn−1 dr dω ≤ ε

Rn+q−1 nεn−1

S n−1

R(ω)

|vr (r, ω)|q rn−1 dr dω, ε

50. Appendix D: Poincaré, Hardy-Sobolev, and other useful inequalities

hence

|v(x)|q dx ≤ Q

Rn+q−1 nεn−1

465

|∇v(x)|q dx. Q

Dividing Rn into a periodic net of cubes of edge 2(ρ + 1), and summing this inequality over all cubes yields the same inequality with Rn instead of Q, that is (50.2), with Cq (Ω) = (1 + n

1/2 1+(n−1)/q −1/q

)

n

%(n−1)/q $ 2 + ρ (Ω) (ρ (Ω) + 1) . ε

(iii) This follows immediately from (i) and (ii).

50.3. Hardy and Hardy-Sobolev inequalities The following lemma is a simple version of the Hardy inequality. Lemma 50.3. Let Ω ⊂ Rn be a bounded domain of class C 1 . Then there exists a positive constant C = C(Ω) such that u/δ 2 ≤ C ∇u 2 for all u ∈ W01,2 (Ω). Proof. First consider the case n = 1, Ω = (0, 1) and assume u(x) = 0 for x ∈ (0, ε]. Then integration by parts and the Cauchy inequality imply

1

0

1 1 u2 1/2 1 1/2 u2 1 1 2 1 2 u uu dx = − (x) + 2 dx ≤ 2 dx (u ) dx , 2 x2 x ε 0 x 0 x 0 "u" " " " " ≤ 2 u 2 . x 2

hence

(50.3)

If, in general, u ∈ W01,2 (0, 1), then there exist uk ∈ D(0, 1) such that uk → u a.e. and in W 1,2 (0, 1). Fatou’s lemma and (50.3) imply

0

1

u2 dx ≤ lim inf k→∞ x2

0

1

u2k dx ≤ lim inf 4 uk 22 = 4 u 22 . k→∞ x2

This inequality and the symmetric estimate u/(1 − x) 2 ≤ 2 u 2 imply the assertion in the case Ω = (0, 1). Let n > 1, Ω = (0, 1)n and u ∈ D(Ω). Writing x = (x1 , x ), x = (x2 , x3 , . . . , xn ), integrating the inequality

0

1

u2 (x1 , x ) dx1 ≤ 4 x21

1 2 ∂u (x1 , x ) dx1 ∂x1 0

466

Appendices

over x ∈ (0, 1)n−1 and using Fubini’s theorem we obtain

Ω

u2 dx ≤ 4 x21

∂u 2 dx ≤ 4 |∇u|2 dx. Ω ∂x1 Ω

Similarly as above, this implies the assertion in the case Ω = (0, 1)n . If Ω ⊂ Rn is a C 1 bounded domain, then one can use standard localization arguments (partition of unity and flattening the boundary ∂Ω) in order to prove the assertion. A combination of Lemma 50.3 and the Sobolev inequality (50.1) with p = 2 yields the following Hardy-Sobolev inequality. Lemma 50.4. Let Ω ⊂ Rn be a bounded domain of class C 1 , n ≥ 3, τ ∈ [0, 1], with 1/q = 1/2∗ + τ /n. Then there exists a positive constant C = C(Ω, τ ) such that u/δ τ q ≤ C ∇u 2 for all u ∈ W01,2 (Ω). Proof. Due to Lemma 50.3 and the Sobolev inequality we may assume τ ∈ (0, 1). Setting m := 2/τ and s := 2∗ /(1 − τ ) we have 1/q = 1/m + 1/s and H¨ older’s inequality implies "u" " uτ " " u "τ " " " " " " 1−τ τ = C ∇u 2 , " τ " ≤ " τ " u1−τ s = " " u 1−τ s(1−τ ) ≤ C ∇u 2 ∇u 2 δ q δ m δ mτ where we used Lemma 50.3 and the Sobolev inequality again. Remark 50.5. One can easily see that if n = 2 or n = 1, then the assertion of Lemma 50.4 remains true for any q ≥ 1 satisfying 1/q > τ /2 or 1/q > τ − 1/2, respectively.

51. Appendix E: Local existence, regularity and stability for semilinear parabolic problems 51.1. Analytic semigroups and interpolation spaces In this subsection we recall some basic facts on strongly continuous analytic semigroups and interpolation spaces. We refer to [77], [273], [410], [13], [14], [16], [344] and [513] for details. Let X0 be a Banach space endowed with the norm | · |0 and let A be a closed linear densely-defined operator in X0 . Denote by D(A) the domain of definition of A endowed with the graph norm x A := |x|0 + |Ax|0 and let X1 be a Banach . space endowed with the norm | · |1 and satisfying X1 = D(A). Then −A generates

51. Appendix E: Local existence, regularity and stability

467

a C 0 analytic semigroup e−tA in X0 if and only if there exist C > 0 and ω ∈ R such that ω + A : X1 → X0 is an isomorphism and |λ||x|0 + |x|1 ≤ C|(λ + A)x|0

for all x ∈ X1 , Re λ ≥ ω.

(51.1)

Set ω(−A) := sup{Re λ : λ ∈ σ(−A)}, where σ(−A) denotes the spectrum of −A. If −A generates a C 0 analytic semigroup in X0 and ω > ω(−A), then there exists C > 0 such that (51.1) is true. Unless explicitly stated otherwise, throughout the rest of Appendix E we shall assume that ⎫ X0 is a reflexive Banach space, ⎪ ⎬ −A generates a C 0 analytic semigroup in X0 , (51.2) ⎪ ⎭ ω > ω(−A). We will also consider the scale of spaces Xα and operators Aα , α ∈ [−1, 1], defined as follows. Let X−1 be the completion of X0 endowed with the norm |x|−1 := |(ω+A)−1 x|0 . Given θ ∈ (0, 1), set Xθ := (X0 , X1 )θ and X−1+θ := (X−1 , X0 )θ , where (·, ·)θ is either the complex interpolation functor [·, ·]θ or any of the real interpolation functors (·, ·)θ,p , 1 < p < ∞. Given θ ∈ [0, 1], let Aθ be the Xθ -realization of A (i.e. Aθ x = Ax for x ∈ D(Aθ ) := {x ∈ Xθ : Ax ∈ Xθ }) and let A−1+θ be the closure of A in X−1+θ (A is closable in X−1+θ ). The following theorem is a consequence of [14, Theorems 8.1, 8.3 and Corollary 8.2], [16, Theorems II.1.2.2, III.2.5.6, III.3.4.1, III.4.10.7 and Chapter V] and the proof of [344, Proposition 4.2.1]. Theorem 51.1. Let −1 ≤ β ≤ α ≤ 1. Then the following assertions are true. (i) The space Xα is densely embedded in Xβ ; the embedding Xα → Xβ is compact provided A has compact resolvent and α > β. (ii) We have (Xβ , Xα )η+ → X(1−η)β+ηα → (Xβ , Xα )η− for any 0 < η− < η < η+ < 1 and the embeddings are dense (almost reiteration property). (iii) Aα is the Xα -realization of Aβ and σ(Aα ) = σ(Aβ ). (iv) −Aα generates a C 0 analytic semigroup e−tAα in Xα . In addition, e−tAα = e−tAβ |Xα , and there exists C = C(ω, A) > 0 such that e−tAβ L(Xβ ,Xα ) ≤ Ctβ−α eωt

for all t > 0.

(51.3)

468

Appendices

(v) Let u0 ∈ X0 , η, ε > 0, η + ε < 1, f ∈ C ε ([0, T ], Xη ) + C([0, T ], Xη+ε ). Then there exists a unique u ∈ C([0, T ], X0 ) ∩ C 1 ((0, T ], X0 ) ∩ C((0, T ], X1 ) which solves the linear Cauchy problem u˙ + Au = f

in (0, T ],

u(0) = u0 .

(51.4)

In addition, u satisfies the variation-of-constants formula u(t) = e−tA u0 +

t

e−(t−s)A f (s) ds.

0

If u0 ∈ Xη , then u ∈ C([0, T ], Xη ). If u0 ∈ X1 , then u ∈ C 1 ([0, T ], X0 ). If ρ ∈ (0, 1), θ ∈ [0, 1] and f ∈ C ρ ((0, T ], Xθ ), then u ∈ C 1+ρ ((0, T ], Xθ ). (vi) Let X0 be a UMD space, ω(−A) < 0, Ait ≤ M eθ|t|

for some M > 0, θ ∈ [0, π/2) and all t ∈ R,

(51.5)

u0 ∈ X0 := (X0 , X1 )1−1/p,p and f ∈ Xf := Lp ([0, T ], X0 ), where 1 < p < ∞. Then the Cauchy problem (51.4) possesses a unique solution u ∈ X := Lp ([0, T ], X1 ) ∩ W 1,p ([0, T ], X0 ) and u X ≤ C( u0 X0 + f Xf ), where C does not depend on u0 , f and T . (vii) Let α ≥ 0, α − 1 < γ < α, f ∈ L∞ ((0, T ), Xα−1 ), and

v(t) :=

t

e−(t−s)A f (s) ds.

0

Then v ∈ C α−γ ([0, T ], Xγ ). The definition and properties of UMD spaces can be found in [16, Sections III.4.4–III.4.5]. For example, the Lebesgue spaces Lq (Ω), 1 < q < ∞, and Hilbert spaces are UMD spaces. For sufficient conditions for the boundedness of imaginary powers of A see Remark 51.5 and also [18], [161], [166], [169], [430], [476]. If α ∈ [−1, 1] and no confusion seems likely, then we will shortly write A and e−tA instead of Aα and e−tAα , respectively. We will also denote by | · |α the norm in Xα . Remarks 51.2. (i) In [273] the author uses the fractional power spaces X α , α ≥ 0, instead of the interpolation spaces Xα . However, if the operator A has bounded imaginary powers (that is if the estimate in (51.5) is true for some M > 0, θ ≥ 0 and all t ∈ R), then the fractional power spaces are equivalent to the interpolation spaces obtained by using the complex interpolation functor [·, ·]θ , see [16, Theorem V.1.5.4]. In the general case, we still have X α → Xβ and Xα → X β whenever 1 ≥ α > β ≥ 0.


469

(ii) The advantage of interpolation and extrapolation spaces becomes evident in Subsection 51.5 where we deal with singular initial data. Extrapolation spaces also naturally appear if one uses semigroup approach to problems with nonlinear boundary conditions (see [14], [15], [17], [444]). We will also need the following interpolation estimate (see [440, Proposition 2.1] and the references therein for a more general statement). We say that (E0 , E1 ) is an interpolation couple of Banach spaces if E0 , E1 are Banach spaces and there exists a locally convex space E such that E0 , E1 → E. Proposition 51.3. Let (E0 , E1 ) be an interpolation couple of Banach spaces. Let 1 ≤ p0 , p1 < ∞, θ ∈ (0, 1), 1/pθ = (1 − θ)/p0 + θ/p1 , s := 1 − θ, Eθ := (E0 , E1 )θ,pθ . Then W 1,p0 ([0, T ], E0 ) ∩ Lp1 ([0, T ], E1 ) → W s,pθ ([0, T ], Eθ ) and the norm of this embedding can be estimated by a constant C(T0 ) for all T ∈ (0, T0 ]. If E1 is compactly embedded in E0 and s < 1 − θ, then the above embedding is compact. If p > 1, r ≥ 1 and Ω ⊂ Rn is open, then Proposition 51.3 implies W 1,2 ([0, T ], L2 (Ω)) ∩ L(p+1)r ([0, T ], Lp+1 (Ω)) → L∞ ([0, T ], Lq (Ω))

(51.6)

for any q ∈ [2, p + 1 − (p − 1)/(r + 1)) (see [440] for details and see [114] for a direct proof). Examples 51.4. (See [13] and [14].) (i) Let Ω ⊂ Rn be uniformly regular of class C 2 , 1 < q < ∞, X0 = Lq (Ω), X1 = 2,q W ∩ W01,q (Ω) (this choice of X1 corresponds to Dirichlet boundary conditions). Let A be the unbounded linear operator in X0 with domain of definition X1 defined by n n ∂2 ∂ Au = − aij u+ bi u + cu, ∂x ∂x ∂x i j i i,j=1 i=1 where aij , bi , c ∈ BU C(Ω) and aij = aji are uniformly elliptic. Then −A generates a C 0 analytic semigroup in X0 . Let (·, ·)θ be the complex interpolation functor if θ = 1/2 and the real interpolation functor (·, ·)θ,q otherwise. Then . Xθ = Xθ (q) =

{u ∈ W 2θ,q (Ω) : u = 0 on ∂Ω}

if 2θ > 1/q,

W 2θ,q (Ω)

if 1/q > 2θ ≥ 0,

X1/2q (q) → W 1/q,q (Ω), and . Xθ (q) = X−θ (q )

if θ < 0.

(51.7)

470

Appendices

(ii) If we set X1 = {u ∈ W 2,q (Ω) : ∂u/∂n = 0 on ∂Ω} (Neumann boundary conditions), then the assertions in (i) remain true with {u ∈ W 2θ,q (Ω) : ∂u/∂n = 0 on ∂Ω} if 2θ > 1 + 1/q, . Xθ = Xθ (q) = if 1 + 1/q > 2θ ≥ 0, W 2θ,q (Ω) X1/2+1/2q (q) → W 1+1/q,q (Ω), and (51.7). Remark 51.5. Assume that Ω, A and Xα , α ∈ [−1, 1], are as in Examples 51.4. Then A satisfies (51.5) (see [17] and cf. also [161] and the references therein). If u solves (51.4), 1 0 does not depend on f, u0 and T . In what follows we will also need the following singular Gronwall inequality (see [16, Theorem 3.3.1]). Proposition 51.6. Let α, β ∈ [0, 1) and ε > 0. Then there exists a positive constant c := c(α, β, ε) such that the following is true: If A, B > 0 and u : [0, T ) → R+ satisfies [t → tβ u(t)] ∈ L∞ loc ([0, T )) and u(t) ≤ At−β + B then

0

t

(t − τ )−α u(τ ) dτ,

u(t) ≤ At−β 1 + cBt1−α e(1+ε)µt

for a.a. t ∈ (0, T ),

for a.a. t ∈ (0, T ),

where µ := (Γ(1 − α)B)1/(1−α) .

51.2. Local existence and regularity for regular data Recall that we assume (51.2) and that Xα , α ∈ [−1, 1] denote the corresponding interpolation-extrapolation scale of spaces. The proofs of the following theorem and Theorems 51.17, 51.19, 51.21, 51.25, 51.33 below are based on well-known and frequently used ideas (see [273], [344], for example). Theorem 51.7. Fix 1 ≥ α > β ≥ 0 and assume that F : Xβ → Xα−1 is locally Lipschitz continuous, uniformly on bounded subsets of Xβ . Let u0 ∈ Xβ . Then there exists T = T (|u0 |β ) > 0 such that the integral equation u(t) = e−tA u0 +

0

t

e−(t−s)A F (u(s)) ds

(51.9)


471

has a unique solution u ∈ C([0, T ], Xβ ). In addition, there exists C = C(A) > 0 such that |u(t)|β ≤ C|u0 |β + 1 for all t ∈ [0, T ]. If γ ∈ [β, α), γ > α − 1, then u ∈ C α−γ ((0, T ], Xγ ). Moreover we have the following continuous dependence property: if γ ∈ [β, α) and u and u ˜ are two solutions with initial data u0 and u ˜0 , respectively, then there exist T = T (|u0 |β , |˜ u0 |β ) > 0 and C > 0 independent of the initial data such that |u(t) − u ˜(t)|γ ≤ Ctβ−γ |u0 − u ˜0 |β

for all t ∈ (0, T ].

(51.10)

Finally, the solution can be continued on the maximal existence interval [0, Tmax), where either Tmax = ∞ or limt→Tmax |u(t)|β = ∞. Proof. Due to (51.3), there exists CA > 0 such that e−tA L(Xα1 ,Xα2 ) ≤ CA tα1 −α2

for all t ∈ (0, 1] and − 1 ≤ α1 ≤ α2 ≤ 1. (51.11) Let M > 2CA |u0 |β . The assumptions on F guarantee the existence of CF = CF (M ) > 0 such that |F (u)|α−1 ≤ CF

and |F (u) − F (v)|α−1 ≤ CF |u − v|β

(51.12)

for all u, v ∈ Xβ satisfying |u|β , |v|β ≤ M . Assume T ∈ (0, 1] and let BM = BM,T denote the closed ball in the Banach space YT := C([0, T ], Xβ ) with center 0 and radius M . We will use the Banach fixed point theorem for the mapping Φu0 : BM → BM , where

t Φu0 (u)(t) := e−tA u0 + e−(t−s)A F (u(s)) ds. (51.13) 0

Let u ∈ BM . Then

t |Φu0 (u)(t)|β ≤ e−tA L(Xβ ,Xβ ) |u0 |β + e−(t−s)A L(Xα−1 ,Xβ ) |F (u(s))|α−1 ds 0

t (t − s)α−1−β ds ≤ CA |u0 |β + CA CF 0

CA CF α−β 1 T ≤ M+ ≤ M, 2 α−β provided T ≤ τ0 , where τ0 = τ0 (M ) > 0 is small enough. Hence Φu0 maps BM into BM for T ≤ τ0 . Given u, v ∈ BM , we have

t | Φu0 (u) − Φu0 (v) (t)|β ≤ e−(t−s)A L(Xα−1 ,Xβ ) |F (u(s)) − F (v(s))|α−1 ds 0

t (t − s)α−1−β |u(s) − v(s)|β ds ≤ CA CF ≤ CA CF

0 α−β

T 1 u − v YT ≤ u − v YT , α−β 2

472

Appendices

provided T ≤ τ1 , where τ1 = τ1 (M ) > 0 is small enough. Consequently, Φu0 is a contraction in BM,T for T ≤ τ2 := min(τ0 , τ1 ) and possesses a unique fixed point u in BM,T . It is easily seen that this solution of (51.9) is unique in YT . Notice also that τ2 = τ2 (|u0 |β ) if we fix M = 2CA |u0 |β + 1, for example. Let γ ∈ [β, α). If γ > α−1, then u ∈ C α−γ ((0, T ], Xγ ) due to Theorem 51.1(vii). Next assume u0 , u ˜0 ∈ Xβ and fix M > 2CA max(|u0 |β , |˜ u0 |β ). Set u0 (t) := e−tA u0 ,

u ˜0 (t) := e−tA u ˜0 ,

uk+1 := Φu0 (uk ),

u ˜k+1 := Φu˜0 (˜ uk ),

k = 0, 1, 2, . . . .

˜k converge to the solutions u and u ˜ in Due to the above existence proof, uk and u BM for T small enough. Now (51.11) implies the following inequality for k = 0 and all t ∈ [0, T ]: |uk (t) − u ˜k (t)|β ≤ 2CA |u0 − u ˜0 |β . (51.14) Assume that (51.14) is true for some k ≥ 0. Then |uk+1 (t) − u˜k+1 (t)|β ≤ e−tA L(Xβ ,Xβ ) |u0 − u ˜0 |β

t + e−(t−s)A L(Xα−1 ,Xβ ) |F (uk (s)) − F (˜ uk (s))|α−1 ds 0

t ˜0 |β + CA CF (t − s)α−1−β |uk (s) − u ˜k (s)|β ds ≤ CA |u0 − u 0

T α−β 2 |u0 − u CF ˜0 |β ≤ 2CA |u0 − u ˜0 |β , ≤ CA + 2CA α−β provided T is small enough. Consequently, (51.14) is true for all k. Passing to the limit we obtain |u(t) − u ˜(t)|β ≤ 2CA |u0 − u ˜0 |β . Using this estimate we finally obtain |u(t) − u ˜(t)|γ = |Φu0 (u)(t) − Φu˜0 (˜ u)(t)|γ ≤ e−tA L(Xβ ,Xγ ) |u0 − u ˜0 |β

t + e−(t−s)A L(Xα−1 ,Xγ ) |F (u(s)) − F (˜ u(s))|α−1 ds 0

≤ CA t

β−γ

|u0 − u ˜0 |β + CA CF

0

t

(t − s)α−1−γ |u(s) − u ˜(s)|β ds

T α−β β−γ 2 t CF |u0 − u ˜0 |β ≤ 2CA tβ−γ |u0 − u ˜0 |β , ≤ CA + 2CA α−γ provided T is small enough. The existence of a maximal solution follows in the same way as in the proof of Proposition 16.1.


473

Remarks 51.8. (i) The solution u in Theorem 51.7 satisfies u ∈ C 1 ((0, Tmax ), Xγ−1 ) and u˙ + Aγ−1 u = F (u), t ∈ (0, Tmax ), ˜ 0 := Xγ−1 , X ˜ 1 := Xγ and let X ˜ η , η ∈ [−1, 1], for all γ ∈ [β, α). In fact, set X ˜ η for any be the corresponding interpolation-extrapolation scale. Then Xα−1 → X ˜ η ). Now the η ∈ (0, α − γ) due to Theorem 51.1(ii), hence F (u) ∈ C([0, Tmax ), X assertion follows from Theorem 51.1(v). (ii) It is straightforward to check that all statements in Theorem 51.7 remain true for nonautonomous nonlinearities of the form F = F (t, u) provided F : [0, ∞) × Xβ → Xα−1 is measurable in t, locally Lipschitz continuous in u (uniformly on bounded subsets of [0, ∞) × Xβ ) and F (·, 0) is bounded in Xα−1 on bounded subsets of [0, ∞). Similarly, if we assume that D ⊂ Xβ is open, F : D → Xα−1 is locally Lipschitz continuous (uniformly on bounded sets M ⊂ D satisfying distXβ (M, ∂D) > 0) and u0 ∈ D, then there exists a unique maximal solution u ∈ C([0, Tmax ), D) and (at least) one of the following possibilities occurs: (a) Tmax = ∞; (b) limt→Tmax |u(t)|β = ∞; (c) lim inf t→Tmax distXβ (u(t), ∂D) = 0. Finally, if ∞ > r > 1/(α − β), F : C([0, T ], Xβ ) → Lr ([0, T ], Xα−1 ) is uniformly Lipschitz continuous on bounded sets and has the Volterra property (that is F (u)|[0,t] depends on u|[0,t] only), and u0 ∈ Xα−1/r , then the problem ut + Au = F (u),

t > 0,

u(0) = u0 ,

(51.15)

1,r has a unique maximal strong solution u ∈ C([0, Tmax ), Xβ )∩Wloc ([0, Tmax ), Xβ−1 ) due to [21, Theorem 2.3]. Strong solution means that the equation ut + Au = F (u) is satisfied for a.e. t. Notice also that F (u) ∈ Lrloc ([0, Tmax ), Xα−1 ) is well defined for u ∈ C([0, Tmax ), Xβ ) due to the Volterra property of F . Additional regularity and stability results for solutions of (51.15) can be found in [21]. In particular, 1,r u ∈ C ρ ([0, Tmax ), Xβ ) ∩ Wloc ([0, Tmax ), Xγ−1 ) for all ρ < α − β − 1/r and γ ∈ (β, α) and the solution u is global (Tmax = T and u ∈ C([0, T ], Xβ )) whenever F (u) ∈ Lr ([0, Tmax ), Xα−1 ). (iii) Let α, β, γ, F and u0 be as in Theorem 51.7, and Tmax = Tmax (u0 ) be the maximal existence time of the solution u of (51.9). Fix t ∈ (0, Tmax ). Using (51.10) one can easily prove the existence of positive constants C, ε (depending on t and max0≤s≤t |u(s)|β ) such that Tmax (˜ u0 ) > t and

|˜ u(t) − u(t)|γ < C|˜ u0 − u0 |β for any u ˜0 ∈ Xβ satisfying |˜ u0 − u0 |β < ε. (iv) Let X0 be a (reflexive) ordered Banach space with a total positive cone P0 and let the semigroup e−tA0 be positive (note that P0 is total if P0 − P0 is dense in

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X0 ). Define positive cones Pθ in Xθ , θ ∈ [−1, 1] as follows: Pθ = P ∩Xθ if θ > 0, Pθ is the closure of P in Xθ if θ < 0. Then Xθ become ordered Banach spaces and the semigroups e−tAθ are positive. If, in addition, F maps Pβ into Pα−1 and u0 ∈ Pβ , then the corresponding solution u is obviously nonnegative. In fact, u = lim uk , where u0 = e−tA u0 ≥ 0 and uk+1 = Φu0 uk ≥ 0 whenever uk ≥ 0. In particular, if e−tA is positive, u0 ∈ Pβ and F : Pβ → Pα−1 , then F need not be defined for u ∈ / Pβ (any regular extension of F to Xβ leads to the same positive solution u). (v) A simple modification of the proof of Theorem 51.7 shows that the assumption β ≥ 0 can be replaced with β ≥ −1. Example 51.9. Let Ω, A and Xα , α ∈ [−1, 1], be as in Examples 51.4 and q > n. Let f ∈ C 1 (R) and let F be the Nemytskii mapping associated with f , that is F (u)(x) = f (u(x)). Assume also that either f (0) = 0 or Ω is bounded. Fix β = 1/2, α = 1 and let u0 ∈ Xβ . Recall from Examples 51.4 that Xβ = W01,q (Ω) or Xβ = W 1,q (Ω) if we consider Dirichlet or Neumann boundary conditions, respectively. Since W 1,q (Ω) → L∞ ∩ Lq (Ω), we see that the assumptions of Theorem 51.7 are satisfied and we obtain a unique maximal solution u ∈ C([0, Tmax ), Xβ ). In addition, u ∈ C 1−γ ((0, Tmax ), Xγ ) for γ ∈ [1/2, 1). Choose γ such that ρ := 1−γ = (1 − n/q)/3. Then Xγ → BU C 1+ρ (Ω), hence u ∈ C ρ ((0, Tmax ), BU C 1+ρ (Ω)). Remark 51.8(i) implies u˙ + Aγ−1 u = F (u) in (0, Tmax ). Fix 0 < δ < T < Tmax , choose ψ ∈ C ∞ (R) such that ψ(t) = 0 for t ≤ δ/2, ψ(t) = 1 for t ≥ δ, and set v(t) := ψ(t)u(t). Then v˙ + Aγ−1 v = f˜ in (0, Tmax ),

v(0) = 0,

(51.16)

where f˜(t) := ψ(t)F (u(t)) + ψt (t)u(t). Assume that the coefficients of the operator A belong to BU C ρ (Ω) and Ω is a bounded domain of class C 2+ρ . Since f˜ is also Hölder continuous, Theorem 48.2(ii) shows that there exists a classical solution w of problem (51.16). The uniqueness of solutions of (51.16) guarantees w = v, hence u is a classical solution for t > 0. Theorem 48.2(ii) also implies u ∈ BC 2,1 (Ω × [t1 , t2 ]) whenever 0 < t1 < t2 < Tmax .

(51.17)

If Ω is unbounded, then (51.17) can be shown by using a smooth cut-off function in the x-variable. This example can be straightforwardly modified for more general nonlinearities and systems (cf. also Remark 51.8(ii)). If F (t, u)(x) = f (x, t, u(x, t), ∇u(x, t)), for example, then one obtains the existence of a maximal solution u ∈ C([0, Tmax ), Xβ ) provided Ω is bounded, the function f = f (x, t, u, ξ) is C 1 , its derivatives satisfy the growth condition |∂t f | + |∂u f | + (1 + |ξ|)|∂ξ f | ≤ C(|u|)(1 + |ξ|p ) and q > n max(1, p − 1) (see [14] for details). Note that the regularity of f with respect to x and t can be considerably relaxed.


475

Example 51.10. Let Ω, A and Xα , α ∈ [−1, 1], be as in Example 51.4(i), and let p > 1 and n(p − 1) np q > max 1, , . p+1 n+p Fix z ∈ (max(1, q/p, nq/(n+q)), min(q, nq/p(n−q)+ )] and assume that F (u)(x) = f (x, u(x)) where f ∈ C 1 satisfies f (·, 0) ∈ Lz (Ω) and |∂u f (x, u)| ≤ a(x) + C|u|p−1 with a ∈ Lp z (Ω) (the regularity of f with respect to x can be relaxed). Set β=

1 2

and α =

1 n n 2+ − . 2 q z

Then α ∈ (β, 1], Xβ = W01,q (Ω) → Lpz (Ω) and Lz (Ω) → Xα−1 (due to X1−α (q )

→ W 2−2α,q (Ω) → Lz (Ω) and (51.7)). Since F considered as a map from Lpz (Ω) to Lz (Ω) is locally Lipschitz continuous (uniformly on bounded sets), it has the same properties as a map F : Xβ → Xα−1 . Consequently, given u0 ∈ W01,q (Ω), Theorem 51.7 guarantees the existence of a maximal solution u ∈ C([0, Tmax ), W01,q (Ω)) satisfying u ∈ C((0, Tmax ), Xγ ) for all γ < α. Next assume f = f (u) and notice that this restriction and our assumptions on f imply f (0) = f (0) = 0 if Ω is unbounded. In fact, if Ω is unbounded, then its measure has to be infinite (since Ω is uniformly regular of class C 2 ), hence the spaces Lz (Ω) and Lp z (Ω) do not contain nonzero constants. If q ≥ n or p ≤ n/(n − q), then we may set z = q, hence α = 1, and we obtain u ∈ C((0, Tmax ), Xγ )

for all γ ∈ [1/2, 1).

(51.18)

Assume q < n, p > n/(n − q), and consider t0 > 0 small and β˜ ∈ (β, α). Since u(t0 ) ∈ Xβ˜ we may repeat the considerations above with z˜ := min q,

nq , ˜ + p(n − 2βq)

α ˜ :=

n n 1 2+ − , 2 q z˜

˜ or p ≤ n/(n − 2βq), ˜ to obtain u ∈ C((0, Tmax ), Xγ ) for all γ < α ˜ . If q ≥ n/(2β) then z˜ = q, α ˜ = 1 and we obtain (51.18) again. Otherwise we notice that z˜ > z, ˜ z˜ and α) α ˜ > α, and use a bootstrap argument (enlarging β, ˜ to see that (51.18) is always true. Next choose γ ∈ (β, 1) . Since Xγ = W 2γ,q ∩ W01,q (Ω) → W01,˜q (Ω)

for some q˜ > q,

we can repeat the arguments above with q replaced by q˜. An obvious bootstrap w.r.t. q˜ shows u ∈ C((0, Tmax ), W 2γ,˜q ∩ W01,˜q (Ω))

for all γ < 1 and q˜ ∈ [q, ∞).

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Appendices

Notice also that the considerations in Example 51.9 guarantee now u ∈ C ρ ((0, Tmax ), BU C 1+ρ ∩ W01,˜q (Ω))

(51.19)

for some ρ ∈ (0, 1) and all q˜ ∈ [q, ∞). Fix t ∈ (0, Tmax ). Then the bootstrap argument used above, Remark 51.8(iii) and the embedding W 2γ,˜q (Ω) → BU C 1 (Ω) for suitable γ, q˜ guarantee the existence of ε, C > 0 (depending on t and max0≤s≤t u(s) W 1,q (Ω) ) such that given u ˜0 ∈ W01,q (Ω) satisfying ˜ u0 − u0 W 1,q (Ω) < ε, we have Tmax (˜ u0 ) > t and u0 − u0 W 1,q (Ω) . ˜ u(t) − u(t) BC 1 < C ˜

(51.20)

In fact, Remark 51.26(iii) and Example 51.27 below guarantee that the RHS in (51.20) can be replaced with C ˜ u0 −u0 r for any r > n(p−1)/2, r > 1. In addition, estimate (15.18) shows that the same is true if r = 1 > n(p − 1)/2. Next assume that f is locally H¨ older continuous. Then (51.19) implies the existence of ρ > 0 such that F (u) ∈ C ρ ((0, Tmax ), X1/2 ). Now Theorem 51.1(v) guarantees u ∈ C 1+ρ ((0, Tmax ), X1/2 ) ∩ C((0, Tmax ), X1 ), hence u ∈ Wq := C 1+ρ ((0, Tmax ), W01,q (Ω)) ∩ C((0, Tmax ), W 2,q (Ω)). Since u(t) ∈ W01,˜q (Ω) for all t > 0 and q˜ ≥ q the arguments above imply u ∈ Wq˜ for all q˜ ∈ [q, ∞). Next assume that f is locally Hölder continuous. We will show that ut ∈ Wq˜ for all q˜ ∈ [q, ∞). Fix 0 < δ < Tmax /2 and choose a cut-off function ψ ∈ C ∞ (R) such that ψ(t) = 0 for t ≤ δ and ψ(t) = 1 for t ≥ 2δ. Notice that the function ut ψ formally solves the linear problem wt + Aw = f (u)ut ψ + ut ψt in Ω × (0, Tmax ), w=0 w(·, 0) = 0

on ∂Ω × (0, Tmax ),

(51.21)

in Ω.

Theorem 51.1(v) guarantees that the solution w of (51.21) belongs to Wq˜. Set t W (t) := 0 (w(s) + u(s)ψt (s)) ds. Then is is easy to see that both W and uψ satisfy the same linear problem Wt + AW = f (u)ψ + uψt in Ω × (0, Tmax ), W =0 W (·, 0) = 0

on ∂Ω × (0, Tmax ), in Ω,

hence W ≡ uψ and Wt (s) = ut (s) for s > 2δ. Since Wt = w + uψt ∈ Wq˜ and δ > 0 was arbitrary, we see that ut ∈ Wq˜. If we only assumed that f is locally Hölder continuous (instead of f locally H¨ older continuous) and if the coefficients of A belong to BU C ρ (Ω) and Ω is of class C 2+ρ , then applying Lp - and subsequently Schauder estimates to (51.21) (along with a cut-off argument if Ω is unbounded) would yield ut ∈ C 2+ρ,1+ρ/2 (Ω × (0, T )) for suitable ρ > 0. Similar regularity properties can be derived in the same way in the case of Neumann boundary conditions.


477

Remark 51.11. Let Ω ⊂ Rn be uniformly regular of class C 2 , let X0 be any of the spaces L∞ (Ω), BC(Ω), BU C(Ω), C∗ (Ω) := {u ∈ BU C(Ω) : lim|x|→∞ u(x) = 0}, and n n ∂2 ∂ Au = − aij u+ bi u + cu, ∂x ∂x ∂x i j i i,j=1 i=1 where aij , bi , c ∈ BU C(Ω) and aij = aji are uniformly elliptic. Let A be the unbounded operator in X0 defined by Au = Au for u ∈ D(A), where 2 2,q D(A) = u ∈ Wloc (Ω) : u, A(u) ∈ X0 , u = 0 on ∂Ω . q≥1

Note that X0 is not reflexive and A is not densely defined, in general, since X0

D(A)

= {u ∈ BU C(Ω) ∩ X0 : u = 0 on ∂Ω}.

Nevertheless, [344, Corollary 3.1.21] guarantees that −A is sectorial, hence it generates an analytic semigroup e−tA in X0 (see [344] for the definition and properties of sectorial operators). Notice that this semigroup is not strongly continuous if D(A) X0

is not dense in X0 . However, its restriction to D(A) is strongly continuous, cf. [344, Remark 2.1.5]. Let X1 := D(A) be endowed with the graph norm and let (Xγ , | · |γ ), γ ∈ (0, 1), be Banach spaces satisfying (X0 , X1 )γ,1 → Xγ → (X0 , X1 )γ,∞ .

(51.22)

We will also assume that the spaces Xγ have the following property: if Aγ denotes the Xγ -realization of A, then −Aγ is sectorial in Xγ

and σ(Aγ ) ⊂ σ(A),

γ ∈ (0, 1).

(51.23)

For example, if Xγ = (X0 , X1 )γ , where (·, ·)γ is any of the real interpolation functors (·, ·)γ,p , 1 ≤ p ≤ ∞, or the complex interpolation functor [·, ·]γ , then both (51.22) and (51.23) are true, see [344]. Similarly, if X0 = BC(Ω), then the space X1/2 := {u ∈ BC 1 (Ω) : u = 0 on ∂Ω}

(51.24)

satisfies (51.22), (51.23) with γ = 1/2 due to [344, Propositions 3.1.27, 3.1.28, Theorem 3.1.25] and standard elliptic regularity theory. In general, [344, Proposition 3.1.28] and (51.22) imply Xγ → BU C 2γ−ε (Ω),

γ ∈ (0, 1], 0 < ε < 2γ.

(51.25)

The proofs of [344, Propositions 2.3.1, 2.2.9] show that the semigroup e−tA satisfies estimates (51.3) for 0 ≤ β ≤ α < 1. These estimates can be used for

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Appendices

the proof of similar existence results as above. To be more precise, assume that 1 = α > β ≥ 0, u ˜0 ∈ Xβ and F : Xβ → X0 is locally Lipschitz continuous, uniformly on bounded subsets of Xβ . Then [344, Theorem 7.1.2] guarantees the existence of r, T > 0 such that, given u0 ∈ Xβ , |u0 − u ˜0 |β < r, the integral equation (51.9) has a unique solution u ∈ L∞ ((0, T ), Xβ ). In addition, u ∈ C([0, T ], Xδ ) for δ ∈ [0, β), (u − e−tA u0 ) ∈ C([0, T ], Xβ ), (51.26) and

u ∈ C 1−γ ((0, T ], Xγ ),

γ ∈ (0, 1).

(51.27)

These results imply the existence of a maximal solution and one can also prove similar assertions to those in Remarks 51.8(ii)–(iv). Notice also that if β = 0 and u0 ∈ D(A)

X0

, then (51.26) and the strong continuity of the restriction of e−tA to

X0

D(A) guarantee u ∈ C([0, T ], X0 ). In particular, if F (u)(x) = f (u(x)) with f ∈ C 1 , then F : X0 → X0 is locally Lipschitz continuous, uniformly on bounded sets. Therefore, setting α = 1 and β = 0 we get a solution u of (51.9) on the maximal time interval [0, Tmax (u0 )) for any u0 ∈ X0 . In addition, the analogue of Remark 51.8(iii) and (51.25) guarantee the following: if u0 ∈ L∞ (Ω) and t ∈ (0, Tmax (u0 )) are fixed, then there exist C, ε > 0 such that Tmax (˜ u0 ) > t and u0 − u0 ∞ ˜ u(t) − u(t) BC 1 ≤ C ˜

(51.28)

u0 − u0 ∞ < ε. for any u ˜0 ∈ L∞ (Ω) satisfying ˜ If F (u)(x) = f (u(x), ∇u(x)) with f ∈ C 1 and X0 = BC(Ω), then F has obviously the required continuity properties as a map F : X1/2 → X0 , where X1/2 is defined in (51.24). Hence, setting α = 1 and β = 1/2 we get a maximal solution u ∈ C([0, Tmax ), X1/2 )

(51.29)

provided u0 ∈ X1/2 . Of course, analogous statements are true for nonlinearities of the form F (u)(x) = f (x, u(x), ∇u(x)) or F (t, u)(x) = f (x, t, u(x), ∇u(x)), cf. Remark 51.8(ii) and [344]. Finally, similar results are true if we consider Neumann boundary conditions instead of Dirichlet boundary conditions (that is, if we replace the condition u = 0 on ∂Ω in the definition of D(A) by ∂u/∂ν = 0 on ∂Ω) see [344, Corollary 3.1.24 and Theorem 3.1.26], for example. Example 51.12. Let Ω ⊂ Rn be uniformly regular of class C 2 , f, g ∈ C 1 , d1 , d2 > 0 and consider the system ut − d1 ∆u = f (u, v), x ∈ Ω, t > 0, (51.30) vt − d2 ∆v = g(u, v),


479

complemented with homogeneous Dirichlet boundary conditions if Ω = Rn . Consider also initial data u0 , v0 ∈ L∞ (Ω). Set X0 = L∞ × L∞ (Ω), 2 2,q ∞ Wloc (Ω), u, v, ∆u, ∆v ∈ L (Ω), u|∂Ω = v|∂Ω = 0 , X1 = (u, v) : u, v ∈ q≥1

Xγ = (X0 , X1 )γ , γ ∈ (0, 1), A(u, v) = (−d1 ∆u, −d2 ∆v) for (u, v) ∈ X1 and F (u, v) = (f (u, v), g(u, v)). Then F : X0 → X0 is locally Lipschitz continuous (uniformly on bounded sets) and a straightforward modification of Remark 51.11 shows that the problem has a unique maximal solution satisfying (u, v) − e−tA (u0 , v0 ) ∈ C([0, Tmax ), X0 ) and (u, v) ∈ C 1−γ ((0, Tmax ), Xγ ) for any γ < 1. Using the analogue of (51.25) we see that both u and v solve linear scalar problems with H¨ older continuous right-hand sides so that one can use Schauder estimates to prove higher regularity of these solutions. Analogous assertions as above are also true in the case of homogeneous Neumann conditions (or Dirichlet conditions for u and Neumann conditions for v). In addition, if we prescribe inhomogeneous Neumann boundary conditions of the form ∂ν u = h1 (t), ∂ν v = h2 (t), where h1 , h2 are smooth, then we can find smooth functions uh , vh satisfying these boundary conditions and we obtain the existence results by solving the system ∂t u ˜ − d1 ∆˜ u = f (˜ u + uh , v˜ + vh ) + d1 ∆uh − ∂t uh , v = g(˜ u + uh , v˜ + vh ) + d2 ∆vh − ∂t vh , ∂t v˜ − d2 ∆˜ with homogeneous Neumann boundary conditions (by using the analogue of Remark 51.8(ii)). Finally, using (the analogues of) Remarks 51.8(ii),(iv) one can also solve the problem if the functions f, g are defined for nonnegative (or positive) arguments only, provided the initial data are nonnegative (or positive) and either f, g ≥ 0 or we can guarantee the positivity of the solution by other means. Example 51.13. Let Ω ⊂ Rn be a bounded domain of class C 2+ρ for some ν ∈ (0, 1), let f, g be C 1 -functions and consider the equation

g(u) dx , ut − ∆u = f u, Ω

x ∈ Ω, t > 0,

(51.31)

complemented with homogeneous Dirichlet boundary conditions. Assume also that the initial data u0 ∈ L∞ (Ω). Since the nonlinearity

g(u) dx F : L∞ (Ω) → L∞ (Ω) : u → f u, Ω

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Appendices

is locally Lipschitz (uniformly on bounded sets), we can use Remark 51.11 in order to solve the problem. Similarly as in Example 51.12, we can also consider Neumann boundary conditions and nonlinearities defined for nonnegative or positive arguments only, for example

−m F (u) = up uq dx , (51.32) Ω

where p, q ≥ 1 and m > 0. The same arguments apply to the equation r

Kup uq , ut − ∆u = Rn

x ∈ Rn , t > 0,

(51.33)

where p, q ≥ 1, r > 0, K ∈ L1 (Rn ) is positive and continuous, the initial data u0 ∈ L∞ (Rn ) are nonnegative and not identically zero. If in addition u0 ∈ L1 (Rn ), then the assumption K ∈ L1 (Rn ) can be replaced by K ∈ L∞ (Rn ). In fact, the existence of a unique mild solution u ∈ L∞ ((0, T ), X), X := L1 ∩ L∞ (Rn ), follows by a direct application of the Banach fixed point theorem to the mapping defined in (51.13) in a ball of the space L∞ ((0, T ), X). Further regularity of solutions of the above problems can be obtained by considering those problems as linear problems with bounded (or H¨ older continuous) RHS, cf. Examples 51.9, 51.10. In particular, the solutions of (51.31) are classical for t > 0. Finally, let us consider the homogeneous Neumann problem for the nonlinearity (51.32) with p = q > 1, m = 1 (see (44.24)). Assume that Ω is the unit ball and u0 ∈ C 2 (Ω) is radial and positive, u0 (x) = U0 (|x|) where U0 (1) = 0. Then u(x, t) = U (|x|, t) for some U = U (r, t). As mentioned above, u is a classical solution for t > 0. Set T := Tmax . Theorem 51.7 (with the choice α = 1, β = 1 − ε, ε > 0 small, and X0 = Lr (Ω), r > n/(1 − 2ε)) also guarantees u ∈ C([0, τ ], W 2−2ε,r (Ω)) → C([0, τ ], C 1 (Ω)),

0 < τ < T.

The function v(x, t) := Ur (|x|, t) ∈ C(Ω × [0, T ))is a weak (and, consequently strong) solution of the linear equation vt − ∆v = ( Ω up )−1 pup−1 v complemented by homogeneous Dirichlet boundary conditions on ST . Now Schauder estimates imply v ∈ C 2,1 (QT ). Example 51.14. Let Ω ⊂ Rn be a bounded domain of class C 2+ρ for some ν ∈ (0, 1), let p > 1, q ≥ 1, k ≥ 0, and consider the problem ⎫

t ⎪ p−1 q ⎪ ut − ∆u = |u| u(x, s) ds − k|u| , x ∈ Ω, t > 0, ⎪ ⎬ 0 (51.34) u = 0, x ∈ ∂Ω t > 0, ⎪ ⎪ ⎪ ⎭ x ∈ Ω. u(x, 0) = u0 ,


481

t First notice that if F (u) = 0 F1 (u(s)) ds + F2 (u) where F1 , F2 satisfy the assumptions of Theorem 51.7, then a straightforward modification of the proof shows that the first part of that theorem remains true. More precisely, given u0 ∈ Xβ there exists a unique local solution u ∈ C([0, T ], Xβ ) and u ∈ C((0, T ], Xγ ) for all γ ∈ [β, α), |u(t)|γ ≤ Ctβ−γ for t > 0. Combining these arguments with Remark 51.11 we see that problem (51.34) is well-posed in X0 := L∞ (Ω) and the solution satisfies u ∈ C((0, T ], Xγ ) and |u(t)|γ ≤ Ct−γ for all γ ∈ [0, 1), where Xγ , γ ∈ (0, 1], are the spaces from Remark 51.11. Fix t0 ∈ (0, T ) and ε > 0 small. Since the nonlinearity F1 (u) = |u|p−1 u satisfies F1 (u) BC 1 ≤ C( u ∞ ) ∇u ∞ , {u ∈ BC 1 (Ω) : u|∂Ω = 0} → X1/2−ε and (51.25) is true, we see that |F1 (u(s))|1/2−ε ≤ F1 (u) BC 1 ≤ C|u(s)|1/2+ε ≤ Cs−1/2−ε and, in particular, F 0 :=

t0 0

F1 (u(s)) ds ∈ X1/2−ε → BU C 1−3ε (Ω).

(51.35)

Parabolic Lp -estimates and embedding (1.2) guarantee that u is Hölder continuous in both x and t for t ≥ t0 , hence F (u) − F 0 is Hölder continuous. Now (51.35) and Schauder estimates guarantee that u is a classical solution of (51.34) for t > 0. Obviously this remains true for the maximal solution on (0, Tmax). (Notice that the Hölder continuity of u for t > 0 also follows from Remark 51.8(ii) with the choice 1 > α > β > 0, r > max(1/(α − β), n/2β) and X0 = Lr (Ω).) Finally assume k = 0. Set T := Tmax . Similar arguments as at the end of Example 51.10 show that ut solves the linear problem vt − ∆v = |u|p−1 u, x ∈ Ω, 0 < t < Tmax , (51.36) v = 0, x ∈ ∂Ω, 0 < t < Tmax , hence Schauder estimates guarantee v = ut ∈ C 2,1 (QT ) ∩ C(Ω × (0, T )). Let ϕ ∈ C 2 (Ω × [0, T )), ϕ = 0 on ∂Ω × [0, T ). Multiplying the equation in (51.34) with ϕ, integrating by parts and passing to the limit as t → 0+ we obtain

lim ut (x, t)ϕ(x, t) dx = u0 (x)∆ϕ(x, 0) dx. t→0+

Ω 2

Ω

H01 (Ω),

In particular, if u0 ∈ H ∩ then

v(x, t)ϕ(x, t) dx = ∆u0 (x)ϕ(x, 0) dx. lim t→0+

Ω

Ω

Now we infer from the uniqueness proof in Proposition 48.9 that v is (a strong) solution of (51.36) with initial data ∆u0 . In particular v = ut ∈ C([0, T ), L2 (Ω)).

482

Appendices

Example 51.15. Consider the problem ut − ∆u = c|u|p−1 u − a · ∇(|u|q−1 u), u = 0,

x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0, x ∈ Ω,

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

⎫ ⎪ ⎬ ⎪ ⎭

(51.37)

where Ω ⊂ Rn is uniformly regular of class C 2+ρ , ρ ∈ (0, 1), p > 1, q ≥ 1, c ≥ 0 and a ∈ Rn , a = 0. Assume first that u0 ∈ W01,r (Ω) with r ∈ (n, ∞). Set X0 := Lr (Ω), X1 := W 2,r ∩ W01,r (Ω), Au := −∆u for u ∈ X1 , and F (u) := |u|p−1 u − a · ∇(|u|q−1 u). Let Xθ = Xθ (r) be defined as in Example 51.4(i), in particular X1/2 = W01,r (Ω). Choose ε > 0 such that (1−2ε)r > n and set α := 1/2 and β := 1/2−ε. Notice that F satisfies the assumptions of Theorem 51.7 since F can be viewed as F = F1 +F2 , where |u|p−1 u

F1 : Xβ → Lpr (Ω) −−−−−→ Lr (Ω) → Xα−1 , |u|q−1 u

a·∇r

F2 : Xβ → Lqr (Ω) −−−−−→ Lr (Ω) −−−→ (W01,r (Ω)) = Xα−1 , and ∇r is defined by

∇r w, ϕ := −

w∇ϕ dx Ω

ϕ ∈ W01,r (Ω), w ∈ Lr (Ω).

Consequently, (51.37) possesses a unique solution u ∈ C([0, T ], Xβ ). Next consider the case u0 ∈ X := {u ∈ BC 1 (Ω) : u|∂Ω = 0}.

(51.38)

Set X0 := BC(Ω), Au := −∆u for u ∈ D(A), where D(A) (and Xθ , θ ∈ (0, 1]) are as in Remark 51.11, X1/2 = X. If q ≥ 2 or q = 1, then the function f (u, ξ) := |u|p−1 u − q|u|q−1 (a · ξ) is C 1 , hence F : X1/2 → X0 has the required continuity properties and (51.37) possesses a unique solution u ∈ L∞ ((0, T ), X1/2 ) satisfying u ∈ C([0, T ], Xδ ) for δ < 1/2, (u−e−tA u0 ) ∈ C([0, T ], X1/2 ) and (51.27), see Remark 51.11. In addition, this solution can be continued on the maximal existence interval [0, Tmax(u0 )) and Remark 51.35 with the choice α = 1, γ = 0 and β = 1/2 (or Proposition 51.34 with α = 1, γ = 0 and β ∈ (1/2, 1)) guarantee that if Tmax (u0 ) < ∞, then

lim sup u(t) ∞ = ∞.

(51.39)

t→Tmax (u0 )

Finally, Schauder estimates show that u is a classical solution for t > 0 and the maximum principle guarantees that u ≥ 0 if u0 ≥ 0.


483

Obviously, all the assertions above concerning the case (51.38) remain true for all q ≥ 1 if we replace the nonlinearity |u|q−1 in the definition of f with (|u| + ε)q−1 , ε ∈ (0, 1]. Let uε denote the corresponding solution. Then the arguments in Remark 51.35 (with α = 1, γ = 0 and β = 1/2) guarantee that given T < ∞, C0 > 0, ∇uε (t) ∞ ≤ C1 , t ∈ [0, T ],

provided uε (t) ∞ ≤ C0 , t ∈ [0, T ],

(51.40)

where the constant C1 > 0 does not depend on ε. In the following proposition we use the approximation solutions uε in order to show the solvability of (51.37) in X1/2 for any q ≥ 1. For simplicity we restrict ourselves to nonnegative solutions and to the case Ω bounded or Ω = Rn . Proposition 51.16. Let Ω ⊂ Rn be a bounded domain of class C 2+γ for some γ ∈ (0, 1) or Ω = Rn . Consider the problem (51.37), where u0 ∈ X+ = {u ∈ BC 1 (Ω) : u|∂Ω = 0, u ≥ 0}. (i) There exists a unique, maximal classical solution u ∈ C 2,1 (Ω×(0, T )) of (51.37), such that u ∈ C([0, T ), C(Ω)) (u ∈ C([0, T ), BC(Rn )) if Ω = Rn ) and ∇u ∈ ∞ L∞ loc ([0, T ), L (Ω)). Moreover, (51.39) is true (with Tmax (u0 ) = T ). (ii) Let Ω = Rn . Then u also satisfies 2 n u ∈ L∞ loc ((0, T ), BC (R )).

(51.41)

If in addition u0 ∈ L1 (Rn ), then u ∈ C([0, T ), L1 (Rn )).

(51.42)

Proof. The uniqueness of the solution is guaranteed by the comparison principle in Proposition 52.16. To establish existence, we consider the approximating problem ⎫ ∂t uε − ∆uε = cupε − q(uε + ε)q−1 (a · ∇uε ), x ∈ Ω, t > 0, ⎪ ⎬ x ∈ ∂Ω, t > 0, uε = 0, (51.43) ⎪ ⎭ x ∈ Ω. uε (x, 0) = u0 (x), By Example 51.15, for each ε ∈ (0, 1], there exists Tε ∈ (0, 1] and a unique classical solution uε ∈ L∞ ([0, Tε ], X+ ) of (51.43) satisfying (uε − e−tA u0 ) ∈ C([0, Tε ], X+ ). Moreover, uε can be continued as long as uε (t) ∞ remains bounded and (51.40) is true. By comparing with the solution of the ODE y (t) = cy p , y(0) = M := u0 ∞ , we see that 0 ≤ uε (t) ≤ 2M, 0 < t ≤ T0 := C(p)M 1−p . (51.44)

484

Appendices

In particular, Tε ≥ T0 and ∇uε (t) ∞ ≤ C1 ,

0 < t ≤ T0 .

(51.45)

Now, (51.45) guarantees that the RHS of (51.43) is bounded in L∞ (QT0 ) independently of ε. In the case Ω bounded, by parabolic Lr -estimates and the embedding (1.2), it follows that uε is bounded in C 1+σ,σ/2 (Ω×(0, T0 ]) for some σ ∈ (0, 1). Applying this estimate to the RHS we deduce from Schauder estimates that uε is bounded in C 2+α,1+α/2 (Ω × (0, T0 ]) for some α ∈ (0, 1). Therefore (some subsequence of) uε converges to a classical solution u ∈ C 2,1 (Ω×(0, T0 ]) of (34.4). In the case Ω = Rn , we may apply the same argument in B(x0 , 1) for each x0 ∈ Rn and we obtain a bound of uε in C 2+α,1+α/2 (B(x0 , 1) × (0, T0 ]), with constant independent of x0 . This yields a solution u ∈ C 2,1 (Rn × (0, T0 ]) of (34.4), with ut , D2 u bounded in Rn × (τ, T0 ] for each τ > 0. Note that this implies u ∈ C((0, T0 ], BC(Rn )) and (51.41). Moreover, in both cases, (51.45) and the variation-of-constants formula imply uε (t) − e−tA u0 ∞ ≤ Ct M p + (M + 1)q−1 C1 . (51.46) Passing to the limit ε → 0, we get (51.46) with u(t) instead of uε (t), hence the continuity of u(t) in C(Ω) (or in BC(Rn )) at t = 0. Since the solution u satisfies the variation-of-constants formula for t > 0, assertion (51.39) follows from Proposition 51.34 or Remark 51.35 (cf. the same argument in Example 51.15). Finally, let us prove (51.42). Observe that for Ω = Rn and u ∈ C 2,1 (Rn × (0, T )) such that u ∈ C([0, T ), BC(Rn )) ∩ L∞ ((0, T ), W 1,∞ (Rn )), (51.37) is equivalent to the integral equation

u(t) = Gt ∗ u0 + c

0

t

Gt−s ∗ up (s) ds +

0

t

(a · ∇Gt−s ) ∗ uq (s) ds.

(51.47)

When u0 ∈ X+ ∩ L1 (Rn ), one can obtain a (unique) solution of (51.47) as a fixed point in a suitable ball of the metric space C([0, T ], L1 ∩ BC(Rn )) ∩ L∞ ((0, T ), W 1,∞ (Rn )) endowed with its natural norm. This can be done by using similar (and in fact simpler) arguments as in the proof of Theorem 15.3, using in particular the fact that ∇Gt ∗ f 1 ≤ Ct−1/2 f 1 , f ∈ L1 (Rn ). Moreover, it is easy to see that the solution of (51.47) can be continued in this space as long as u(t) ∞ remains bounded. By the already proved uniqueness statement, we deduce that the two solutions coincide, with same existence time, which completes the proof.


485

51.3. Stability of equilibria Theorem 51.17. Let α, β, F be as in Theorem 51.7. Let, in addition, ω(−A) < 0 and |F (u)|α−1 = o(|u|β ) as |u|β → 0. Then the zero solution of (51.9) is (locally) exponentially asymptotically stable. More precisely, given ω ˜ ∈ (ω(−A), 0) there exist δ ∗ > 0 and C > 0 such that the solution u with initial data u0 satisfying |u0 |β < δ ∗ exists globally and |u(t)|β ≤ Ceω˜ t |u0 |β

for all t ≥ 0.

(51.48)

Proof. Let ω ˜ ∈ (ω(−A), 0). Choose ω ∈ (ω(−A), ω ˜ ). Then (51.3) guarantees e−tA L(Xα−1 ,Xβ ) ≤ C(ω, A)tα−1−β eωt , for all t > 0, (51.49) e−tA L(Xβ ,Xβ ) ≤ C(ω, A)eωt , where C(ω, A) ≥ 1. Set

∗

∞

C = C(ω, A)

τ α−1−β e(ω−˜ω)τ dτ

0

and choose ε > 0 such that |F (u)|α−1 ≤

1 |u|β 2C ∗

whenever |u|β ≤ ε.

(51.50)

Choose δ ∗ = ε/2C(ω, A) and let |u0 |β < δ ∗ . We may assume u0 = 0. Set T = sup{t ∈ (0, Tmax(u0 )) : |u(s)|β ≤ 2C(ω, A)eω˜ s |u0 |β for all s ∈ [0, t]} and notice that T > 0 and |u(s)|β ≤ ε for all s ∈ [0, T ). If T = ∞, then (51.48) is true. Hence, assume T < ∞. Then T < Tmax (u0 ) due to the uniform bound of |u(s)|β for s ∈ [0, T ), hence |u(T )|β = 2C(ω, A)eω˜ T |u0 |β .

(51.51)

On the other hand, using (51.49), (51.50), the inequality in the definition of T and the definition of C ∗ we obtain

T |u(T )|β ≤ C(ω, A)eωT |u0 |β + C(ω, A) (T − s)α−1−β eω(T −s) |F (u(s))|α−1 ds 0

C(ω, A)2 ω˜ T e |u0 |β ≤ C(ω, A)eωT |u0 |β + C∗

T 0

(T − s)α−1−β e(ω−˜ω)(T −s) ds

< C(ω, A)eωT |u0 |β + C(ω, A)eω˜ T |u0 |β ≤ 2C(ω, A)eω˜ T |u0 |β , which contradicts (51.51) and concludes the proof.

486

Appendices

Remarks 51.18. (i) A combination of Theorem 51.17 with estimates of the form (51.10) or (51.20) shows that the solution u in Theorem 51.17 also decays exponentially to zero in stronger norms than | · |β . (ii) Theorem 51.17 can also be used in order to prove the stability of a non-zero equilibria. In fact, let w ∈ Xα , Aα−1 w = F (w). Assume that F : Xβ → Xα−1 is Fréchet differentiable at w and set Fw (v) := F (w + v) − F (w) − F (w)v, hence |Fw (v)|α−1 = o(|v|β )

as |v|β → 0.

˜ 0 = X0 , A˜ = ˜ 1 = X1 , X Let us first consider the special case α = 1. Set X A − F (w) (with domain X1 ) and assume that ˜ 0 and ω(−A) ˜ < 0. A˜ generates a C 0 analytic semigroup in X

(51.52)

Notice that if A has compact resolvent, then F (w) ∈ L(Xβ , X0 ) is a compact perturbation of A, hence the first part of (51.52) is automatically satisfied. Set v(t) = u(t) − w and v0 = u0 − w. Then (51.9) can be written as ˜

v(t) = e−tA v0 +

0

t

˜

e−(t−s)A Fw (v(s)) ds

and one can use Theorem 51.17 with A and F replaced by A˜ and Fw , respectively. If α < 1 and F (w)|X1 ∈ L(X1 , X0 ), then one can still use the same arguments as ˜ 1 = Xα , ˜ 0 = Xα−1 , X above (provided (51.52) is true). In the general case we set X ˜ ˜ A = Aα−1 − F (w) (with domain X1 ) and assume that (51.52) is true. Set also ˜ ˜ → Xβ and one can use Theorem 51.17 α ˜ = 1 and choose β˜ ∈ (β+1−α, 1). Then X β ˜ respectively. ˜ Fw , α ˜ , β, with A, F, α, β replaced by A, (iii) The conclusions of Theorem 51.17 remain true in the situation of Remark 51.11. Theorem 51.19. Let α, β, F be as in Theorem 51.7, p > 1 and |F (u)|α−1 = O(|u|pβ )

as |u|β → 0.

(51.53)

Assume that σ(−A) = {ω1 } ∪ σ2 , where ω1 < 0 is a simple eigenvalue of −A with eigenspace E1 and σ2 ⊂ {λ : Re λ ≤ ω2 } for some ω2 < ω1 . Fix ω ∈ (max(ω2 , ω1 p), ω1 ). Then there exist δ, C > 0 and a continuous map K : Xβ → E1 such that the solution u with initial data u0 satisfying |u0 |β < δ exists globally and |u(t) − K(u0 )eω1 t |β ≤ Ceωt |u0 |β

for all t ≥ 0.

(51.54)


487

Proof. Let P1 and P2 denote the spectral projections in Xβ corresponding to the spectral sets {ω1 } and σ2 , respectively, and E2 := P2 (Xβ ). Then Xβ = E1 ⊕ E2 , E2 is A (and e−tA ) invariant, σ(−A|E2 ) = σ2 and −A|E2 generates the analytic semigroup e−tA |E2 (see [273, Section 1.5] and the references therein), hence (51.3) implies |e−tA P2 u|β ≤ Ceωt min(|u|β , tα−1−β |u|α−1 ) (51.55) for all u ∈ Xβ . Choose ω ˜ ∈ (ω1 , ω/p). Then Theorem 51.17 guarantees the existence of δ ∈ (0, 1) and C > 0 such that |u(t)|β ≤ Ceω˜ t |u0 |β

(51.56)

whenever |u0 |β < δ. Assume |u0 |β < δ and denote ui (t) = Pi u(t), i = 1, 2. Since Pi e−tA = e−tA Pi we have u2 (t) = e−tA P2 u0 +

0

t

e−(t−s)A P2 F (u(s)) ds,

hence (51.53), (51.55), (51.56) and ω ˜ p < ω imply

|u2 (t)|β ≤ Ceωt |u0 |β + C

0

t

eω(t−s) (t − s)α−1−β |u(s)|pβ ds ≤ Ceωt |u0 |β . (51.57)

Set K(u0 ) := lim u1 (t)e−ω1 t = P1 u0 + t→∞

0

∞

e−ω1 s P1 F (u(s)) ds

(the integral is convergent since P1 F (u(s)) E1 ≤ Ceω˜ ps |u0 |pβ due to (51.53), (51.56) and ω ˜ p < ω < ω1 ). Now the assertion follows from (51.57) and the estimate " ∞ " " ω1 t ω1 t " u1 (t) − K(u0 )e E1 = e " e−ω1 s P1 F (u(s)) ds" ≤ Ceω˜ pt |u0 |pβ . t

E1

Remarks 51.20. (i) The proof of Theorem 51.19 implies K(u0 ) = P1 u0 + O(|u0 |pβ ). (ii) Let us consider the situation from Remark 51.11 with Ω bounded, Au = −∆u, X0 = BC(Ω) and X1/2 defined by (51.24). Set α = 1 and β = 1/2. Then the statement in Theorem 51.19 remains true for this choice of spaces since A has the required properties, ω1 is a simple eigenvalue of A1/2 and σ(A1/2 ) ⊂ σ(A).

488

Appendices

51.4. Self-adjoint generators with compact resolvent The proof of Theorem 51.21 below is based on an idea used in the construction of stable manifolds for general semilinear parabolic problems (see [273, Theorem 5.2.1] or [101, Lemma 4.1], for example). We will use this idea in a specific situation in order to obtain more precise information than that in [273] or [101]. In addition to (51.2) we will also assume that X0 is a Hilbert space, A : X0 → X0 is self-adjoint and has compact resolvent, ω1 > ω2 > · · · are all distinct eigenvalues of −A, (·, ·)θ is the complex interpolation functor for all θ ∈ [0, 1].

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

(51.58)

Then Xα , α ∈ [−1, 1], are Hilbert spaces and the operators Aα are self-adjoint (see [16, Theorem V.1.5.15]). Let Pi , Qi and Ri , i = 1, 2, . . . , denote the spectral projections in X0 corresponding to the spectral sets {ωi , ωi+1 , . . . }, {ω1 , . . . , ωi−1 } and {ωi }, respectively. Let Pi,α denote the restriction Pi |Xα if α > 0 and the closure of Pi in Xα if α < 0. Then Pi,α is the spectral projection in Xα corresponding to the spectral set {ωi , ωi+1 , . . . } and analogous assertions are true for Qi and Ri . Without fear of confusion we will write Pi , Qi , Ri instead of Pi,α , Qi,α , Ri,α . Since −A =

∞

ωj Rj

and

j=1

e−tA =

∞

eωj t Rj ,

j=1

it is easy to see that there exist Ci > 0, i = 1, 2, . . . , such that e−tA Pi L(Xα ,Xα ) ≤ eωi t , α ∈ [−1, 1], t ≥ 0, Ci ωi t e , α ∈ [0, 1], t > 0, e−tA Pi L(Xα−1 ,Xα ) ≤ t and, by interpolation, e−tA Pi L(Xβ ,Xα ) ≤ Ci tβ−α eωi t ,

−1 ≤ β ≤ α ≤ 1, t > 0.

(51.59)

Similarly, e−tA Qi L(Xβ ,Xα ) ≤ Ci eω1 t , e Qi L(Xβ ,Xα ) ≤ Ci e tA

where etA Qi :=

#i−1 j=1

−ωi−1 t

e−ωj t Rj if t ≥ 0.

β, α ∈ [−1, 1], t ≥ 0, ,

β, α ∈ [−1, 1], t ≥ 0,

(51.60)


489

Theorem 51.21. Assume (51.58). Let α, β, F be as in Theorem 51.7, p > 1, F (0) = 0 and |F (u) − F (v)|α−1 ≤ CF |u − v|β |u|p−1 for |u|β , |v|β small. (51.61) + |v|p−1 β β Fix i ≥ 1 with ωi < 0 and choose λ ∈ [ωi , 0], λ < ωi−1 if i > 1. Then there exist ρ = ρi > 0 small and C˜i > 0 with the following properties: given v0 ∈ Pi Xβ , |v0 |β ≤ ρ, there exists a unique z0 ∈ Qi Xβ such that the solution of (51.9) with u0 := v0 + z0 is global and satisfies |u(t)|β ≤ 2ρeλt for all t ≥ 0. In addition, |u(t)|β ≤ 2|v0 |β eωi t

for all t ≥ 0

(51.62)

and |z0 |β ≤ C˜i |v0 |pβ .

(51.63)

Finally, if |Ri v0 |β > C˜i |v0 |pβ , then there exists c = c(v0 ) > 0 such that |u(t)|β ≥ c|Ri v0 |β eωi t

for all t ≥ 0.

(51.64)

Proof. Let u be a global solution of (51.9). Then u can be written in the form u = v + z, where

t v(t) = e−tA v0 + e−(t−s)A Pi F (u(s)) ds, 0 (51.65)

t −tA −(t−s)A z0 + e Qi F (u(s)) ds, z(t) = e 0

v0 = Pi u0 , z0 = Qi u0 (z0 = 0 and z = 0 if i = 1). Assume first that |u(t)|β ≤ ceλt ,

t ≥ 0,

(51.66)

where c > 0 is small. If i > 1, then |etA z(t)|β = |etA Qi z(t)|β ≤ e−ωi−1 t |z(t)|β ≤ ce(λ−ωi−1 )t → 0 as t → ∞, |esA Qi F (u(s))|β ≤ Ci e−ωi−1 s |F (u(s))|α−1 ≤ Ci CF cp e(pλ−ωi−1 )s , (51.67) hence (51.65) guarantees

z0 = −

0

∞

esA Qi F (u(s)) ds

(51.68)

and u = Φv0 (u), where Φv0 (u)(t) := e−tA v0 +

0

t

e−(t−s)A Pi F (u(s)) ds −

∞

e−(t−s)A Qi F (u(s)) ds.

t

(51.69)

490

Appendices

On the other hand, if u is any function in C([0, ∞), Xβ ) satisfying (51.66) and u = Φv0 (u) for some v0 ∈ Pi Xβ , then obviously u solves (51.9), where u0 = v0 + z0 and z0 is given by (51.68). Denote u = sup |u(t)|β e−λt t≥0

and Bρ = {u ∈ C([0, ∞), Xβ ) : u ≤ 2ρ}. We will show that, given v0 ∈ Pi Xβ , |v0 |β ≤ ρ, the mapping Φv0 possesses a unique fixed point in Bρ provided ρ > 0 is small enough. Given u ∈ Bρ , we have e−λt |Φv0 (u)(t)|β ≤ e−λt e−tA Pi L(Xβ ,Xβ ) |v0 |β

t + e−λt e−(t−s)A Pi L(Xα−1 ,Xβ ) |F (u(s))|α−1 ds 0

∞ + e−λt e−(t−s)A Qi L(Xα−1 ,Xβ ) |F (u(s))|α−1 ds t

t

(t − s)α−1−β e−(λ−ωi )(t−s)+λ(p−1)s ds ≤ |v0 |β + Ci CF (2ρ) 0

∞ + e(ωi−1 −λ)(t−s)+λ(p−1)s ds p

t

< 2ρ, provided ρ is small enough and i > 1 (analogous estimates are true for i = 1). Notice that the above estimates also imply e−λt |Φv0 (u)(t) − e−tA v0 |β ≤ C˜i ρp for some C˜i > 0 and that similar estimates guarantee 1 ˜ for u, u ˜ ∈ Bρ . Φv0 (u) − Φv0 (˜ u) ≤ u − u 2 Consequently, Φv0 : Bρ → Bρ is a contraction and possesses a unique fixed point in Bρ . In addition, (51.68) and (51.67) imply |z0 |β ≤ C˜i ρp . Repeating the above arguments with ρ := |v0 |β and λ := ωi we obtain estimates (51.62), (51.63) and e−ωi t |u(t) − e−tA v0 |β ≤ C˜i |v0 |pβ .

(51.70)

Set w0 := Ri v0 , y0 := Pi+1 v0 = v0 − w0 and assume |w0 |β > C˜i |v0 |pβ . Then e−ωi t |e−tA y0 |β ≤ e−(ωi −ωi+1 )t |y0 |β , e−ωi t |e−tA w0 |β = |w0 |β , hence (51.70) yields e−ωi t |u(t)|β ≥ |w0 |β − e−(ωi −ωi+1 )t |y0 |β − C˜i |v0 |pβ > c|w0 |β , provided c < 1 − C˜i |v0 |pβ /|w0 |β and t is large enough. This concludes the proof.


491

Corollary 51.22. Assume (51.58). Let α, β, F, p be as in Theorem 51.21, and let u be a global solution of (51.9) satisfying |u(t)|β → 0 as t → ∞. Set Λ := inf{λ ≤ 0 : lim |u(t)|β e−λt = 0} t→∞

and assume Λ ∈ (−∞, 0). Then there exist C1 , C2 > 0 and i ≥ 1 such that Λ = ωi and C1 eωi t ≤ |u(t)|β ≤ C2 eωi t , t ≥ 0. Proof. The same arguments as in the proof of [2, Corollary A.11] guarantee the 1/t existence of i such that Λ = ωi , |u(t)|β → eωi and distXβ u(t)/|u(t)|β , Sβ → 0 as t → ∞, where Sβ := {v ∈ Ri Xβ : |v|β = 1}. Choose λ ∈ (ωi , 0), λ < ωi−1 if i > 1, 1/t and let ρ = ρi > 0 be the constant from Theorem 51.21. Then |u(t)|β → eωi implies |u(t + t0 )|β ≤ ρeλt for t0 ≥ 0 large enough and all t ≥ 0. Enlarging t0 if necessary we may assume |Ri u(t0 )|β > C˜i |v0 |pβ , where v0 := Pi u(t0 ). Now the assertion follows from Theorem 51.21. Remark 51.23. If 0 ∈ / σ(A), then the assumption Λ < 0 in Corollary 51.22 is automatically satisfied. In fact, using Theorem 51.21 with λ = 0 we obtain Λ ≤ ωi , where ωi is the largest negative eigenvalue of −A. The assumption Λ > −∞ can be verified by an argument guaranteeing backward uniqueness (see [2, Lemma A.16 and Lemma B.4], for example, and cf. Example 51.24 below). Example 51.24. Let L be the positive self-adjoint operator in the weighted space L2g defined by (47.16). Recall that the domain of definition of L equals Hg2 , and L has compact inverse. Consider the problem vt + Av = |v|p−1 v, v(y, 0) = v0 (y),

y ∈ Rn , t > 0, y ∈ Rn ,

(51.71)

where Av = Lv − λv and p > 1. Since L is self-adjoint and positive, it has bounded imaginary powers and −A generates a strongly continuous analytic semigroup in X0 := L2g (see [16]). In addition, the domain of definition of A equals X1 := Hg2 . Set Xθ = [X0 , X1 ]θ for θ ∈ (0, 1) and X−1+θ = [X−1 , X0 ]θ , where X−1 is the completion of X0 endowed with the norm |v|−1 = |L−1 v|0 . Then the abstract . results in [16] imply X−θ = Xθ for θ ∈ (0, 1]. One can also easily verify X1/2 = D(L1/2 ) = Hg1 (cf. Remark 51.2(i)). Let p < pS . For simplicity assume n ≥ 3 (the case n ≤ 2 is similar). Then Hg1 → ∩ L2g due to Lemma 47.11 and, by interpolation, Xθ → Lrg ∩ L2g for θ ∈ [0, 1/2] and 1/r = 2θ/2∗ +(1−2θ)/2. Using these embeddings, setting F (v) = |v|p−1 v, z = min(2, 2∗ /p), β = 1/2 and α = 1 + (n/z − n/2)/2 (cf. Example 51.10) one obtains

∗ L2g

492

Appendices

that F : Xβ → Xα−1 and A satisfy the assumptions of Theorem 51.21. (Notice that we could also choose α = 1 and β close to 1, β < 1, due to Remark 47.12(ii): in this case the assumption p < pS could be replaced by p(n − 4) < n.) Now assume λ ∈ / σ(L) and let v be a global solution of (51.71) satisfying |v(t)|β → 0 as t → ∞, v0 ≡ 0 (such solutions exist due to Theorem 51.21). We will show the following: (i) There exist C˜1 , C˜2 > 0 and ωi ∈ σ(−A), ωi < 0, such that C˜1 eωi t ≤ v(t) ∞ ≤ C˜2 eωi t ,

t ≥ 1.

(51.72)

2

(ii) Assume ω1 < 0 and let φ1 (y) = e−|y| /4 be the corresponding eigenfunction of A (see Lemma 47.13). Set ω ˆ := max(ω2 , pω1 ) < ω1 . Then there exists M = M (v0 ) ∈ R such that v(t) − M eω1 t φ1 ∞ ≤ Ceωˆ t ,

t ≥ 0.

(51.73)

Proof of (i). Set Λ := inf{κ ≤ 0 : lim |v(t)|β e−κt = 0}. t→∞

Then Λ < 0 due to Remark 51.23. Let us show that Λ > −∞. Choose κ < 0 and assume |v(t)|β ≤ Ceκt ,

t ≥ 0.

(51.74)

Let us prove that, given t0 > 0, estimate (51.74) and p < pS guarantee v(t) ∞ ≤ C(t0 )eκt ,

t ≥ t0 .

(51.75)

˜0 = X ˜ 0 (r) = Lr (Rn ), X ˜1 = X ˜ 1 (r) = W 2,r (Rn ) and let us rewrite Let r > 1, X (51.71) in the form ˜ = F˜ (v), vt + Av t > 0, (51.76) v(0) = v0 , ˜ = −∆v − λv is considered as an unbounded operator in X ˜ 0 with domain where Av ˜ α , α ∈ [−1, 1] be the scale ˜ 1 and F˜ (v)(y) = |v(y)|p−1 v(y) + (y · ∇v(y))/2. Let X X ˜ α → W 2α,r (Rn ) for α ≥ 0), of spaces constructed as in Example 51.4(i) (hence X ∼ and let | · |α denote the norm in this space. Since Xβ = X1/2 = Hg1 and y · ∇v(t) 2 ≤ C|v(t)|β ,

y · ∇v(t) 1 ≤ |y| · |∇v(t)| dy ≤ C Rn

|v(t)|p−1 v(t) r = v(t) ppr ≤ C|v(t)|pβ ,

Rn

(|∇v(t)g 1/2 )g −1/4 dy ≤ C|v(t)|β ,

r ∈ [max(1, 2/p), 2∗ /p],


493

(51.74) guarantees F˜ (v(t)) r ≤ Ceκt ,

t ≥ τ,

(51.77)

where r = min(2, 2∗ /p) and τ = 0. Similar estimates as above and Lemma 47.11 ∗ ˜ 0 (r) for any r ∈ (1, 2∗ ]. imply Hg1 → L1 ∩ L2 (Rn ), hence Hg1 → X Fix ε, T > 0 small and assume that (51.77) is true for some r ∈ [min(2, 2∗ /p), 2∗ ] ˜ we infer for all and τ ≥ 0. Then using estimates (51.3) (with A replaced by A) t ≥ τ, ˜

−T A |v(t + T )|∼ L(X˜ 0 ,X˜ 1−ε ) |v(t)|∼ 1−ε ≤ C e 0

t+T ˜ + e−(t+T −s)A L(X˜ 0 ,X˜ 1−ε ) |F˜ (v(s))|∼ 0 ds t

t+T

≤ C|v(t)|β + C

(t + T − s)ε−1 eκs ds ≤ Ceκ(t+T ) ,

t

where C depends on T, ε and the constants C in (51.74) and (51.77). If r > n/2, ˜ 1−ε → L∞ (Rn ) for ε small enough and (51.75) follows. If r ≤ n/2, then then X ˜ X1−ε → Lq1 ∩ W 1,q (Rn ), where 1/q1 = 1/r − (2 − 2ε)/n, 1/q = 1/r − (1 − 2ε)/n, hence v(t) q1 + ∇v(t) q ≤ Ceκt for t ≥ τ + T. Notice that choosing ε small enough we have q1 /p > r (due to p < pS and r ≥ min(2, 2∗ /p)), q˜ := ε + (2 − ε)q/2 > r and |v(t)|p−1 v(t) q1 /p = v(t) pq1 ≤ Cepκt ,

q˜ y · ∇v(t) q˜ ≤ C |∇v(t)|q˜−ε |∇v(t)|ε g ε/2 dy Rn

≤ C ∇v(t) qq˜−ε v(t) εHg1 ≤ Ceq˜κt ,

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

t ≥ τ + T,

hence F˜ (v(t)) r˜ ≤ Ceκt ,

t ≥ τ˜,

where τ˜ = τ + T and r˜ = min(q1 /p, q˜) > r. An obvious bootstrap argument concludes the proof of (51.75). Notice that similar estimates as above (or the choice α = 1 mentioned above) imply F (v) = |v|p−1 v ∈ C((0, ∞), X0 ). Consequently, Theorem 51.1(v) (used with ˆ 0 := X−ε , ε > 0 small) guarantees u ∈ C 1 ((0, ∞), X−ε ) ∩ C((0, ∞), X1−ε ). We X also have |F (v)(t)|−ε ≤ C|F (v)(t)|0 ≤ Ch(t)|v(t)|0 ≤ Ch(t)|v(t)|1/2−ε , where

h(t) := |v|p−1 (t) ∞ ≤ C(t0 )p−1 e(p−1)κt

t ≥ t0 ,

494

Appendices

belongs to L2 (t0 , ∞), hence [2, Lemma A.16] yields |v(t)|β ≥ c|v(t)|−ε ≥ cC1 |v(t0 )|−ε e−C2 (t−t0 ) for suitable c, C1 , C2 > 0. Consequently, Λ > −∞. Now we infer from Corollary 51.22 the existence of C1 , C2 > 0 and ωi ∈ σ(−A), ωi < 0, such that C1 eωi t ≤ |v(t)|β ≤ C2 eωi t , t ≥ 0. (51.78) Since (51.74) implies (51.75), we have v(t) ∞ ≤ C˜2 eωi t ,

t ≥ 1,

(51.79)

and simple estimates based on the variation-of-constants formula also yield |v(t)|β+ε ≤ Cˆ2 eωi t ,

t ≥ 1,

(51.80)

where ε > 0 is small. Since Xβ+ε is compactly embedded into Xβ , given δ > 0 we can find Cδ > 0 such that |v(t)|β ≤ δ|v(t)|β+ε + Cδ v(t) ∞ .

(51.81)

Choosing δ < C1 /Cˆ2 , estimates (51.78), (51.80) and (51.81) imply v(t) ∞ ≥ C˜1 eωi t ,

t ≥ 1,

for suitable C˜1 > 0. Consequently, (51.72) is true. Proof of (ii). Similarly as in (i), it suffices to prove (51.73) with · ∞ replaced by |·|β . The proof of this modified estimate is almost the same as the proof of Theorem 51.19; one just needs to replace estimate (51.55) by the more precise estimate (51.59) and estimate (51.56) by |v(t)|β ≤ Ceω1 t (which follows from the proof of (i) or from Theorem 51.19). The only difference appears in the case ω ˆ = pω1 = ω2 , −(t−s)A P2 F (v(s)). where one has to use a more precise estimate on the term e In fact, in this case Theorem 51.19 guarantees v(s) = M eω1 s φ1 + w(s), where |w(s)|β ≤ Ce(ω2 +ε)s and ε ∈ (0, ω1 − ω2 ) is such that δ := (p − 1)ω1 + ε < 0. Consequently, F (v(s)) = |M |p−1 M epω1 s φp1 + z(s), where |z(s)|α−1 ≤ Ce(ω2 +δ)s . Let P3 , R2 be the spectral projections introduced in the beginning of this subsection. Then R2 φp1 = 0 due to Lemma 47.13, hence P2 F (v(s)) = P3 (|M |p−1 M epω1 s φp1 ) + P2 z(s) and |e−(t−s)A P2 F (v(s))|β ≤ C(t − s)α−1−β eω3 (t−s) epω1 s + eω2 (t−s) e(ω2 +δ)s = C(t − s)α−1−β eω2 t e(ω3 −ω2 )(t−s) + eδs . This estimate is sufficient for the proof of (51.73) in the case ω ˆ = pω1 = ω2 .


495

51.5. Singular initial data In what follows we consider nonlinearities F : Xβ → Xα−1 with (at most) polynomial growth and we will show that under suitable assumptions one can obtain existence for initial data u0 ∈ Xδ with δ < β. The following theorem is an abstract analogue of Theorem 15.2. In addition, it also covers the critical case (cf. Remark 15.4(i)). We assume that β > δ, M, T > 0, and we define the Banach space β−δ YT := {u ∈ L∞ |u(t)|β < ∞}. loc ((0, T ], Xβ ) : u YT := sup t t∈(0,T )

We also denote by BM = BM,T the closed ball in YT with center 0 and radius M . Theorem 51.25. Assume that p > 1, 1 ≥ α > β > δ ≥ −1, δ > β − 1/p, β ≥ α − 1 and F : Xβ → Xα−1 satisfies . + |v|p−1 |F (u) − F (v)|α−1 ≤ CF |u − v|β 1 + |u|p−1 β β

(51.82)

Let u0 ∈ Xδ and let Φu0 be defined by (51.13). (i) If α > (β − δ)p + δ, then there exist M = M (|u0 |δ ) ≥ 1 and T = T (|u0 |δ ) > 0 such that Φu0 possesses a unique fixed point in BM,T . (ii) If α = (β − δ)p + δ, then there exist M = M (u0 ) > 0 and T = T (u0 ) > 0 such that Φu0 possesses a unique fixed point in BM,T . In both cases, u ∈ C([0, T ], Xδ ) ∩ C((0, T ], Xγ ) for any γ ∈ [δ, α). Proof. We will use the Banach fixed point theorem for the mapping Φu0 : BM → BM . Increasing CF if necessary we may assume |F (u)|α−1 ≤ CF (1 + |u|pβ ).

(51.83)

Let γ ∈ [δ, α), 0 < t ≤ T ≤ 1, M > 0, u ∈ BM , let CA be the constant from (51.11) and denote ξ+ := max(ξ, 0). Then t

γ−δ

|Φu0 (u)(t)|γ ≤ t

γ−δ

|e

−tA

≤ CA |u0 |δ + CA CF tγ−δ

0

≤ CA |u0 |δ + CA CF tγ−δ hence Φu0 (u)(t) ∈ Xγ .

t

0

t

u0 |γ + t

t e−(t−s)A F (u(s)) ds

γ−δ

0

γ

(t − s)−(γ+1−α)+ 1 + |u(s)|pβ ds (t − s)−(γ+1−α)+ 1 + M p s(δ−β)p ds < ∞

(51.84)

496

Appendices

(i) Let α > (β − δ)p + δ. Fix M ≥ max(1, 2CA |u0 |δ ). Since 1 + M p s(δ−β)p ≤ 2M p s(δ−β)p for s ∈ (0, 1], estimate (51.84) implies 1 M + 2CA CF M p B(α − β, 1 − (β − δ)p)T α−δ−(β−δ)p < M 2

tβ−δ |Φu0 (u)(t)|β ≤

provided T is small enough. Hence Φu0 maps BM into BM for such T . Now let u, v ∈ BM . Then, similarly as above, t β−δ β−δ t |Φu0 (u)(t) − Φu0 (v)(t)|β ≤ t e−(t−s)A F (u(s)) − F (v(s)) ds β 0

t ds ≤ CA CF tβ−δ (t − s)α−1−β |u(s) − v(s)|β 1 + |u(s)|p−1 + |v(s)|p−1 β β 0

≤ 3CA CF M p−1 B(α − β, 1 − (β − δ)p)T α−δ−(β−δ)p u − v YT , hence

1 u − v YT , 2 provided T is small enough. Consequently, Φu0 is a contraction in BM and it possesses a unique fixed point u. Assume 0 < t1 < t2 ≤ T and let either t1 → t2 − or t2 → t1 +. Then Φu0 (u) − Φu0 (v) YT ≤

|u(t2 ) − u(t1 )|γ ≤ e−t1 A L(Xδ ,Xγ ) |(e−(t2 −t1 )A − 1)u0 |δ

t2 + e−(t2 −s)A L(Xα−1 ,Xγ ) |F (u(s))|α−1 ds

t1 t1

+ → 0,

0

e−(t1 −s)A L(Xα−1 ,Xγ ) |(e−(t2 −t1 )A − 1)F (u(s))|α−1 ds

due to (51.11), (51.83), u ∈ BM , e−(t2 −t1 )A u0 → u0 e

−(t2 −t1 )A

F (u(s)) → F (u(s))

in Xδ , in Xα−1

and the Lebesgue theorem. Consequently, u ∈ C((0, T ], Xγ ). The continuity of u : [0, T ] → Xδ at t = 0 follows from the strong continuity of the semigroup e−tA in Xδ and the estimate t −tA |Φu0 (u)(t) − e u0 |δ = e−(t−s)A F (u(s)) ds δ 0

t ≤ CA CF (t − s)−(δ+1−α)+ 1 + |u(s)|pβ ds 0

≤ 2CA CF M p B(min(1, α − δ), 1 − (β − δ)p)tk ,


497

where k = min(1, α − δ) − (β − δ)p. (ii) Let α = (β − δ)p + δ. First let us prove that tβ−δ |e−tA u0 |β → 0

as t → 0.

(51.85)

e−tk A . Then Sk ∈ L(Xδ , Xβ ) are uniIn fact, choose tk → 0 and set Sk := tβ−δ k formly bounded due to (51.11) and, given w ∈ Xβ , |e−tk A w|β ≤ tβ−δ CA |w|β → 0 |Sk w|β ≤ tβ−δ k k

as k → ∞.

Since Xβ is dense in Xδ , we obtain |Sk w|β → 0 for any w ∈ Xδ . Consequently, (51.85) is true. Choose M > 0 such that 2CA CF M p−1 B(α−β, 1−(β −δ)p) ≤ 1/4 and T ∈ (0, 1] such that tβ−δ |e−tA u0 |β ≤ M/2 for all t ≤ T (this choice is possible due to (51.85)). Let u ∈ BM = BM,T . Then tβ−δ |Φu0 (u)(t)|β ≤ tβ−δ |e−tA u0 |β + CA CF tβ−δ

0

t

(t − s)α−1−β 1 + |u(s)|pβ ds

T α−δ + CA CF M p B(α − β, 1 − (β − δ)p) ≤ M/2 + CA CF α−β T α−δ ≤ M, ≤ 3M/4 + CA CF α−β (51.86) provided T is small enough. Consequently, Φu0 maps BM into BM . Similar estimates (cf. (i)) show that Φu0 is a contraction and the corresponding fixed point possesses the required continuity properties. Remarks 51.26. (i) It is easily seen from the proof that the existence time T in Theorem 51.25(ii) can be chosen uniform for initial data belonging to a compact subset of Xδ . (ii) Theorem 51.25 guarantees that the solution u is unique in the ball BM . However, similarly as in the proof of Theorem 15.2 one can prove the uniqueness of this solution in the space C := C([0, T ], Xδ ) ∩ C((0, T ], Xβ ). In fact, let v ∈ C be any solution of (51.9) on a (small) interval [0, τ ]. Then K :={v(t) : 0 ≤ t ≤ τ } is compact in Xδ hence (i) and the proof of Theorem 51.25 guarantee the solvability of (51.9) in BM,TK for some M, TK > 0 and for all initial data in K. Let U (t)v(s) denote the corresponding solution starting at v(s), s ∈ [0, τ ], t ∈ [0, TK ]. Then tβ−δ |U (t)v(s)|β ≤ M . Fix s ∈ (0, min(TK , τ )) and denote u1 (t) := U (t)v(s) and u2 (t) := v(t + s). Then u1 ∈ C([0, TK ], Xβ ) due to the existence part of Theorem 51.7 and the uniqueness in Theorem 51.25, and u2 ∈ C([0, τ − s], Xβ ). In addition, both u1 and u2 solve (51.9) with initial data v(s) ∈ Xβ . Hence

498

Appendices

u1 = u2 on [0, min(TK , τ − s)], due to the uniqueness in Theorem 51.7 (see also Remark 51.8(v)). Consequently, tβ−δ |v(t + s)|β = tβ−δ |U (t)v(s)|β ≤ M. Fix t > 0 small and let s → 0+ in the previous estimate. Then we obtain tβ−δ |v(t)|β ≤ M , hence v = u due to the uniqueness in Theorem 51.25. The example in Remark 15.4(iii) and Example 51.27 below show that the restriction v ∈ C((0, T ], Xβ ) in the uniqueness statement above is necessary, in general. (iii) Let the assumptions of Theorem 51.25(i) be fulfilled. Similarly as in the case of initial data in Xβ one can prove the existence of the maximal existence time Tmax = Tmax (u0 ), continuous dependence on initial data, positivity of the solution u if Xδ is ordered and e−tA0 is positive, etc. For example, given u0 , u ˜0 ∈ Xδ , there exists T = T (|u0 |δ , |˜ u0 |δ ) > 0 and C > 0 such that ˜0 |δ , |u(t) − u ˜(t)|γ ≤ Ctδ−γ |u0 − u

t ≤ T,

(51.87)

provided γ ∈ [δ, α). (iv) A simple modification of the proof of Theorem 51.25 shows that the assumption β ≥ α − 1 is superfluous (cf. also Remark 51.8(v)). (v) Assumption (51.82) in Theorem 51.25 can be replaced with |F (u) − F (v)|α−1 ≤ CF

k

|u − v|βi 1 + |u|pβii−1 + |v|pβii−1 ,

(51.88)

i=1

where (for all i = 1, 2, . . . , k) pi > 1, 1 ≥ α > βi > δ ≥ −1, δ > βi − 1/pi , 1 βi ≥ α − 1, α ≥ (βi − δ)pi + δ and F : ki=1 Xβi → Xα−1 . In this case, it is sufficient to use the fixed point argument in the space k 2 βi −δ YT := u ∈ L∞ (0, T ], : u X := max sup t |u(t)| < ∞ . β Y β i T i loc i=1

i

t∈(0,T )

For more complex generalizations of this type (and applications of such generalizations) see [446] and the references therein. (vi) Estimate (51.84) implies |u(t)|γ ≤ C(1 + t−(γ−δ) |u0 |δ ) where T is from Theorem 51.25.

for all γ ∈ [δ, α) and t ∈ (0, T ],

(51.89)


499

Example 51.27. Let Ω, A and Xα , α ∈ [−1, 1], be as in Example 51.4(i), p > 1 and q ≥ n(p − 1)/2, q > 1. Let F (u)(x) = f (x, u(x)), where f = f (x, u) is a C 1 -function satisfying f (·, 0) ∈ Lz (Ω) and the growth condition |∂u f (x, u)| ≤ a(x) + C|u|p−1 with a ∈ Lp z (Ω) and z ∈ (max(1, q/p), q], z < q if q = n(p − 1)/2 (cf. Example 51.10). Assume u0 ∈ X0 = Lq (Ω) and set β=

n 1 n − , 2 q pz

α=

1 n n 2+ − , 2 q z

δ = 0.

Then 1 ≥ α ≥ βp > 0 and α > βp if q > n(p − 1)/2, α < 1 if q = n(p − 1)/2. In addition, the choice of α and β guarantees Xβ → W 2β,q (Ω) → Lpz (Ω) and Lz (Ω) → Xα−1 (since X1−α (q ) → W 2−2α,q (Ω) → Lz (Ω)). Consequently, F satisfies (51.82) and Theorem 51.25 guarantees the existence of a unique solution u ∈ C([0, T ], Lq (Ω)) in the corresponding ball BM . In addition, u ∈ C((0, Tmax ), Xγ ) for any γ < α. Let f = f (u) where f is locally Hölder continuous. If q > n(p − 1)/2, then we may set z = q, hence α = 1 and u ∈ C((0, Tmax ), W 2γ,q ∩ W01,q (Ω))

for any γ < 1.

(51.90)

If q = n(p − 1)/2, then we may choose z < q arbitrarily close to q, hence α arbitrarily close to 1, so that (51.90) remains true as well. Now Example 51.10 guarantees u ∈ C 1 ((0, Tmax ), W01,˜q (Ω)) ∩ C((0, Tmax ), W 2,˜q (Ω))

for any q˜ ∈ [q, ∞). (51.91)

In addition, Remark 51.26(iii) and (51.20) show u(t; u0,k ) → u(t; u0 )

in BU C 1 (Ω)

(51.92)

provided u0,k → u0 in Lq (Ω) and t ∈ (0, Tmax (u0 )) is fixed. Example 51.28. Let us consider the situation in Example 51.27, where Au = −∆u − λu, f (u) = |u|p−1 , u0 ∈ Lq ∩ L2 (Ω) with q ≥ qc = n(p − 1)/2, q > 1. We will show that the corresponding energy function

1 λ 1 E(t) := |∇u(t)|2 − u2 (t) − |u(t)|p+1 dx 2 p+1 Ω 2 is differentiable for t > 0. In addition, we will also prove that the problem generates a dynamical system in H01 ∩ Lq (Ω) provided q ≥ max(qc , p + 1). Example 51.27 and Theorem 51.25 guarantee the existence of a unique maximal solution u ∈ C([0, Tmax )), Lq (Ω)). In what follows we set T := Tmax . Let us first show that u ∈ C([0, τ ], L2 (Ω)) for some τ ∈ (0, T ). (51.93)

500

Appendices

If 2 ≥ n(p − 1)/2, then this assertion follows from the well-posedness in L2 (Ω) (see Example 51.27). Hence assume 2 < n(p − 1)/2. Then q > 2. Let i ≥ 0 be the integer such that 2pi < q ≤ 2pi+1 . First assume i = 0. Observe that estimate (15.2) in Theorem 15.2 and Remark 15.4(i) remain obviously true for λ = 0. Applying (15.2) with r = 2p, we obtain up (s) 2 = u(s) p2p ≤ C u(s) pq s−θ ≤ Cs−θ ,

0 < s < τ,

for some τ > 0, where up denotes |u|p−1 u and 1 n p − (q/2) np 1 − = (p − (q/2)) ≤ < 1. 0 ≤ θ := 2 q 2p 2q p−1 t Therefore up ∈ L1 ((0, τ ), L2 (Ω)), hence g(t) := 0 e−(t−s)A up (s) ds ∈ C([0, τ ], L2 (Ω)), and (51.93) is satisfied. Next assume i ≥ 1. For any r ∈ [2p, qp], we have u ∈ C([0, τ ], Lr (Ω)) ⇒ up ∈ C([0, τ ], Lr/p (Ω)) ⇒ g ∈ C([0, τ ], Lr/p (Ω)) ⇒ u ∈ C([0, τ ], Lr/p (Ω)),

(51.94)

due to e−tA u0 ∈ C([0, τ ], L2 ∩ Lq (Ω)). First applying (51.94) with r = q, we obtain i u ∈ C([0, τ ], Lq/p (Ω)), hence u ∈ C([0, τ ], L2p (Ω)), due to q/p ≤ 2pi ≤ q. Then applying (51.94) iteratively with r = 2pi , 2pi−1 , . . . , 2p, we end up with (51.93). Due to (51.91) we know that there exists a positive constant C∞ such that |u| ≤ C∞ on Ω × [τ /2, T ]. Fix f˜ ∈ BC 1 (R) such that f˜(u) = f (u) for |u| ≤ C∞ . Then u is a solution of the equation ut − ∆u − λu = f˜(u) for t ≥ τ /2, hence estimate (51.91) (obtained with q = 2 and initial data u(τ /2)) implies u ∈ C 1 ([τ, T ), H01 (Ω)) ∩ C([τ, T ), H 2 (Ω)). Consequently, u ∈ C([0, T ), L2 (Ω)) ∩ C 1 ((0, T ), H01 (Ω)) ∩ C((0, T ), H 2 (Ω)). In particular, u ∈ C 1 ((0, T ), L2 (Ω)). Since also u ∈ C 1 ((0, T ), Lq˜(Ω)) for any q˜ ≥ q due to (51.91), we have u ∈ C 1 ((0, T ), Lp+1 (Ω)), hence E ∈ C 1 ((0, T )). Next assume u0 ∈ H01 ∩ Lq∗ (Ω), q ∗ := max(qc , p + 1). We already know that the solution satisfies u ∈ C([0, T ), L2 ∩ Lq (Ω)) for any q ∈ [qc , ∞), q > 1, in ∗ particular u ∈ C([0, T ), Lq (Ω)). Let us prove u ∈ C([0, T ), H01 (Ω)). Since u ∈ C((0, T ), H01 (Ω)) and e−tA u0 ∈ C([0, T ), H01 (Ω)) it is sufficient to show " t " " " e−(t−s)A |u(s)|p−1 u(s) ds" → 0 as t → 0. (51.95) " 0

1,2

Let Xθ (2), θ ∈ [0, 1], be the scale of spaces from Example 51.4(i) (in particular, X0 (2) = L2 (Ω) and X1/2 (2) = H01 (Ω)). If qc = p + 1, then there exists ε > 0 such ∗ that Lq /p (Ω) → X−1/2+ε (2), hence (51.95) follows from e−(t−s)A L(X−1/2+ε (2),X1/2 (2)) ≤ (t − s)−1+ε


501

∗

and |u(s)|p−1 u(s) q∗ /p = u pq∗ ≤ C. Let qc = p + 1. Then Lq /p (Ω) → X−1/2 (2). Since the estimate (51.85) is uniform for u0 lying in a compact set of Xδ and the ∗ set {|u(s)|p−1 u(s) : s ∈ [0, T ]} is compact in Lq /p (Ω), we have e−(t−s)A |u(s)|p−1 u(s) 1,2 = o (t − s)−1

as t → 0.

(51.96)

Now the smoothing estimate (15.2) with q = p + 1 = qc , r = 2p guarantees |u(s)|p−1 u(s) 2 = u(s) p2p ≤ C u(s) pp+1 s−1/2 ≤ Cs−1/2 , hence

e−(t−s)A |u(s)|p−1 u(s) 1,2 ≤ C(t − s)−1/2 s−1/2 .

(51.97)

Interpolation between (51.96) and (51.97) yields e−(t−s)A |u(s)|p−1 u(s) 1,2 = o (t − s)−3/4 s−1/4 ∗

which guarantees (51.95). Consequently, u ∈ C([0, T ), H01 ∩ Lq (Ω)) and E ∈ C([0, T )). Similar estimates as above show the continuous dependence of solutions ∗ on initial data in H01 ∩ Lq (Ω), hence the problem generates a dynamical system in this space. Obviously, the same remains true for the space H01 ∩ Lq (Ω) with q ∈ (q ∗ , ∞). If λ = 0, then the continuity properties of E can in some cases be proved without the assumption u0 ∈ L2 (Ω). For example, let 1 < p ≤ pS , Ω = Rn , u0 ∈ Lp+1 (Rn ) and ∇u0 ∈ L2 (Rn ). Set q := p + 1 ≥ n(p − 1)/2. Then (51.91) shows u ∈ C([0, T ), Lp+1 (Rn )) ∩ C 1 ((0, T ), Lp+1 (Rn )) ∩ C((0, T ), W 2,˜q (Rn )) for any q˜ ≥ p + 1. In addition, estimate (51.84) implies |u(t)|γ ≤ C( u0 p+1 )t−γ for any γ < 1. If p = pS set γ = 1/(2p). Otherwise fix γ < 1/(2p) such that Xγ = W 2γ,p+1 (Rn ) → L2p (Rn ) and set

v(t) :=

t

e−(t−s)A F (u(s)) ds.

0

Then

t (t − s)−1/2 F (u(s)) 2 ds ≤ (t − s)−1/2 u(s) p2p ds 0 0

t

t −1/2 p (t − s) |u(s)|γ ds ≤ M (t − s)−1/2 s−γp ds < ∞ ≤ t

v(t) 1,2 ≤

0

0

and v(t) 1,2 → 0

as t → 0,

(51.98)

502

Appendices

due to M = M (t) → 0 as t → 0 if p = pS (cf. estimates in (51.86)). Since also ∇e−tA u0 2 = e−tA ∇u0 2 ≤ ∇u0 2 , we obtain ∇u(t) ∈ L2 (Rn ). Similar estimates show the local Hölder continuity of v : (0, T ) → H 1 (Rn ) and ∇(e−tA u0 ) ∈ C 1 ((0, T ), L2 (Rn )). Since u : (0, T ) → BU C∩L2p (Rn ) is locally H¨ older continuous due to interpolation and ∇u : (0, T ) → L2 (Rn ) is also locally Hölder continuous, we have F (u) ∈ C ρ ((0, T ), H 1 (Rn )) for some ρ > 0. Finally, (51.98) and Theorem 51.1(v) imply v ∈ C 1 ((0, T ), H 1 (Rn )) ∩ C([0, T ), H 1 (Rn )). Since also ∇(e−tA u0 ) = e−tA (∇u0 ) ∈ C([0, T ), L2 (Rn )) we see that the energy function E belongs to C 1 ((0, T )) ∩ C([0, T )). Example 51.29. Let Ω, A and Xα , α ∈ [−1, 1], be as in Example 51.4(i), F (u) = |u|r−1 u − µ|∇u|p , where µ ∈ R, p, r > 1, q > n(p − 1), 1/r > 1/p − q/n. Assume u0 ∈ X1/2 = W01,q (Ω), choose z ∈ (max(1, q/p), q) such that 1/r ≥ 1/p − z/n and set 1 n n 1 n n 1 β= 1+ − , α= 2+ − , δ= . 2 q pz 2 q z 2 Then 1 > α > (β − δ)p + δ > 0 and δ ∈ (β − 1/p, β). Since F2 (u) := |∇u|p and F1 (u) := |u|r−1 u can be viewed as ∇

|·|p

F2 : Xβ → W 2β,q (Ω) → W 1,pz (Ω) → (Lpz (Ω))n −−→ Lz (Ω) → Xα−1 , |u|r−1 u

F1 : Xβ → W 2β,q (Ω) → Lrz (Ω) −−−−−→ Lz (Ω) → Xα−1 , we see that F satisfies (51.82). Now Theorem 51.26 and the same bootstrap argument as in Example 51.27 guarantee the existence of T := Tmax and of a solution u ∈ C([0, T ), W01,q (Ω)) ∩ C((0, T ), W 2γ,˜q (Ω)), where γ < 1 and q˜ ∈ [q, ∞) are arbitrary. Choose γ, q˜ such that (2γ − 1)˜ q > n. Then Wq˜ := W 2γ−1,˜q (Ω) → BU C(Ω) and |∇u| ∈ C((0, T ), Wq˜ ∩ Wq ). If w ∈ Wq˜ ∩ Wq , w ≥ 0, then w ≤ C in Ω, hence |wp (x) − wp (y)| ≤ pC p−1 |w(x) − w(y)|

(51.99)

and using the intrinsic norm in Wq (see [13], for example) we obtain wp ∈ Wq , wp Wq ≤ pC p−1 w Wq . Since |∇u| ∈ C((0, T ), Wq˜ ∩ Wq ) and Wq˜ → BU C(Ω), Wq → Lq (Ω), using (51.99) we obtain |∇u|p ∈ C((0, T ), Lq (Ω)). This fact, the local boundedness of |∇u|p : (0, T ) → Wq and interpolation yield |∇u|p ∈ C((0, T ), W s,q (Ω)) for s ∈ (0, 2γ − 1), hence F2 (u) ∈ C((0, T ), Xη ) for η small enough. Similar estimates show F1 (u) ∈ C((0, T ), Xη ) for η small. Now Theorem 51.1(v) guarantees u ∈ C 1 ((0, T ), Lq (Ω)) ∩ C((0, T ), W 2,q ∩ W01,q (Ω)).

(51.100)


503

Example 51.30. Let Ω, A and Xα , α ∈ [0, 1], be as in Remark 51.11, u0 ∈ X0 and F (u) = f (u, ∇u), where f ∈ C 1 , |fξ (u, ξ)| ≤ M (|u|)(1 + |ξ|p−1 ),

1 u0 ∞ . Since Xβ → BC 1 (Ω), (51.82) is true with CF = C(F, C∞ ) for all u, v ∈ Xβ satisfying u ∞ , v ∞ ≤ C∞ . Now an obvious modification of Theorem 51.25 shows the well-posedness of problem (51.4) in X0 ∈ {L∞ (Ω), BC(Ω)} (see also [344, Theorem 7.1.6]). Example 51.31. Let Ω, A and Xα , α ∈ [−1, 1], be as in Example 51.4(i), Ω bounded, p ∈ (1, 1 + 2/n), F (u) = ±|u|p−1 u and u0 be a bounded Radon measure in Ω. Fix q ∈ (1, p) and choose δ such that n−

n+2 n n < −2δ < − . q p q

Notice that δ ∈ (−1, 0). Set α = 1 + δ and choose β such that 1 n n 1 − < β < + δ. 2 q p p

Then u0 ∈ Xδ (since Xδ = (W0−2δ,q (Ω)) and W0−2δ,q (Ω) → C0 (Ω)) and F : Xβ → Xα−1 (since Xβ → Lp (Ω) and L1 (Ω) → Xδ = Xα−1 ). In addition, α > (β − δ)p + δ > 0 and δ ∈ (β − 1/p, β). Consequently, we can use Theorem 51.25 in order to get a solution u ∈ C([0, T ], Xδ )∩C((0, T ], Xγ ) for any γ < 1+δ. Choosing γ = 0 we obtain u ∈ C((0, T ], Lq (Ω)). Since q > 1 > n(p − 1)/2, Examples 51.27 and 51.9 guarantee that u is a classical solution for t > 0. Let us mention that the assumption p < 1 + 2/n is also necessary for the solvability of (51.9) if u0 is the Dirac distribution, see [95]. Example 51.32. Consider the system ∂t u1 − ∆u1 = |u2 |p1 −1 u2 ,

x ∈ Ω, t > 0,

∂t u2 − ∆u2 = |u1 |p2 −1 u1 ,

x ∈ Ω, t > 0,

u1 = u2 = 0, u1 (x, 0) = u0,1 (x),

u2 (x, 0) = u0,2 (x),

⎫ ⎪ ⎪ ⎪ ⎪ ⎬

x ∈ ∂Ω, t > 0, ⎪ ⎪ ⎪ ⎪ ⎭ x ∈ Ω,

(51.101)

where (u0,1 , u0,2 ) ∈ X0 := Lr1 × Lr2 (Ω), p1 , p2 , r1 , r2 > 1 and Ω ⊂ Rn is uniformly regular of class C 2 . Assume n n n n ≤ 2. p1 − , p2 − max r2 r1 r1 r2

(51.102)

504

Appendices

Set X1 := W 2,r1 ∩ W01,r1 (Ω) × W 2,r2 ∩ W01,r2 (Ω), Au := (−∆u1 , −∆u2 ) for u = (u1 , u2 ) ∈ X1 , and let Xα be defined similarly as in Example 51.4(i). We will use Remark 51.26(v) in order to prove the well-posedness of (51.101) in X0 . Choose α ∈ (0, 1) with α > maxi (1 − n/(2ri )), set δ = 0, βi = α/pi , i = 1, 2, and define zi ∈ (1, ri ) by n/zi = n/ri + 2 − 2α. Then Lz1 × Lz2 (Ω) → Xα−1 and (51.102) guarantees W 2β1 ,r2 (Ω) → Lp1 z1 (Ω), W 2β2 ,r1 (Ω) → Lp2 z2 (Ω). Now it is easy to verify (51.88), hence (51.101) is well-posed in X0 . Theorem 32.1(ii) shows that condition (51.102) is optimal. Theorem 51.33. Let α, β, δ, p, F be as in Theorem 51.25. Assume, in addition, that ω(−A) < 0 and |F (u)|α−1 = o(|u|β ) if α > (β − δ)p + δ,

as |u|β → 0

|F (u)|α−1 ≤ CF |u|pβ

if α = (β − δ)p + δ. Then, given ω ˜ ∈ (ω(−A), 0), there exists η > 0 and C > 0 such that the solution u with initial data u0 satisfying |u0 |δ < η exists globally and |u(t)|β ≤ Ctδ−β eω˜ t |u0 |δ

for all t ≥ 0.

(51.103)

Proof. Fix η1 > 0 and assume |u0 |δ ≤ η1 . If α > (β − δ)p + δ, then estimate (51.87) with u ˜0 = 0 shows that |u(t)|β ≤ C1 tδ−β |u0 |δ

for all t ∈ (0, T1 ],

(51.104)

∗

where T1 = T1 (η1 ) ∈ (0, 1]. Let δ > 0 be the constant from Theorem 51.17. Choose η > 0 such that C1 T1δ−β η < δ ∗ . Then the conclusion follows from (51.104) and Theorem 51.17 applied to the initial data u(T1 ). If α = (β−δ)p+δ choose η > 0 such that CF (C ∗ )p B(α−β, 1−(β−δ)p)η p−1 < 1, where C ∗ := 2CA . Assume |u0 |δ < η and set T = sup{t ∈ (0, Tmax(u0 )) : |u(s)|β ≤ C ∗ sδ−β |u0 |δ for all s ∈ (0, t]}. Notice that T > 0, since u ∈ BM,T (M) and the constant M can be chosen arbitrarily small in the proof of Theorem 51.25(ii). If T = ∞, then (51.104) is true for t ≤ T1 := 1 and we can proceed as in the case α > (β − δ)p + δ. Assume T < ∞. Then T < Tmax (u0 ), hence |u(T )|β = C ∗ T δ−β |u0 |δ . On the other hand, |u(T )|β ≤ CA T

δ−β

|u0 |δ + CA CF

0

(51.105)

T

(T − s)α−1−β |u(s)|pβ ds

≤ CA T δ−β |u0 |δ + CA CF (C ∗ )p B(α − β, 1 − (β − δ)p)|u0 |pδ T δ−β < C ∗ T δ−β |u0 |δ , which yields a contradiction and concludes the proof.


505

51.6. Uniform bounds from Lq -estimates In this part we present an abstract approach for obtaining L∞ -bounds of solutions from Lq -bounds. We will assume that (51.2) is true with ω < 0 and use the scale (Xα , Aα ) introduced above. The idea of the proof of the next proposition is contained in the proof of [14, Theorem 12.8]. Proposition 51.34. Let 0 ≤ β < α ≤ 1, −1 ≤ γ < β, T ∈ (0, ∞] and Cγ > 0. Let F : Xβ → Xα−1 be continuous and |F (u)|α−1 ≤ CF (|u|γ )(1 + |u|1−ε β ),

u ∈ Xβ ,

(51.106)

where ε ∈ (0, 1). Let u0 ∈ Xβ and let u ∈ C([0, T ), Xβ ) solve (51.9). If |u(t)|γ ≤ Cγ for all t ∈ [0, T ), then |u(t)|β ≤ Cβ for all t ∈ [0, T ), where Cβ depends on Cγ and |u0 |β but not on T . Proof. Let T˜ ∈ (0, T ) and t ≤ T˜. Using (51.3) we obtain

t

u0 |β + e−(t−s)A L(Xα−1 ,Xβ ) |F (u(s))|α−1 ds 0

t eω(t−s) (t − s)α−1−β CF (Cγ )(1 + |u(s)|1−ε ≤ c|u0 |β + c β ) ds 0

∞ ≤ c|u0 |β + cCF (Cγ ) eωτ τ α−1−β dτ 1 + sup |u(s)|1−ε β

|u(t)|β ≤ |e

−tA

0

0≤s≤T˜

and the assertion follows by choosing t such that |u(t)|β > sup0≤s≤T˜ |u(s)|β − 1 and letting T˜ → T . Remark 51.35. Let the hypothesis of Proposition 51.34 be satisfied with ε = 0. Then the proof and the singular Gronwall inequality in Proposition 51.6 guarantee that |u(t)|β ≤ C1 eC2 t . Lemma 51.36. Let p > 1, −1 ≤ δ < (1 − 1/p)γ + β/p and |F (u)|α−1 ≤ C 1 + |u|pδ ,

u ∈ Xβ .

(51.107)

Then the estimate (51.106) is true. Proof. We can find θ ∈ (0, 1/p) such that (1 − θ)γ + θβ > δ, hence (Xγ , Xβ )θ → |u|θβ . Now the assertion is obvious. Xδ and |u|δ ≤ |u|1−θ γ As an application we first give an alternative proof of Theorem 16.4. This proof will not require Ω to be bounded.

506

Appendices

Proof of Theorem 16.4. Let Ω, A and Xα = Xα (q), α ∈ [−1, 1], be as in Example 51.4(i). Notice that we can choose ω < 0 if Au = −∆u + au, a > 0 (or a = 0 if Ω is bounded). Set F (u) = |u|p−1 u + au, γ = 0 and α = 1. Using the assumption q > n(p − 1)/2 it is easy to find β < 1 close to 1 and δ < β/p close to β/p such that Xδ → Lpq ∩ Lq (Ω). Consequently, |F (u)|0 ≤ u ppq + a u q ≤ C(1 + |u|pδ ), hence (51.107) is true. Now assuming u(t) q ≤ C0 , Lemma 51.36 and Proposition 51.34 guarantee |u(t)|β < Cβ = Cβ (C0 , |u0 |β ). Since Xβ → Lq˜(Ω) for some q˜ > q, an obvious bootstrap argument shows u(t) ∞ < C∞ and concludes the proof. Remarks 51.37. (i) If the assumptions of Theorem 16.4 are satisfied, then the above proof guarantees the estimate U∞ ≤ C(u0 )Uqρ for suitable ρ ≥ 1. (ii) If we consider the more general problem ut − ∆u = f (x, t, u, ∇u), u = 0,

x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0, x ∈ Ω,

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

⎫ ⎪ ⎬ ⎪ ⎭

(51.108)

where f = f (x, t, u, ξ) is a C 1 -function satisfying the growth condition |f | ≤ C(1 + |u|p + |ξ|r ) with p < 1 + 2q/n and r < 1 + q/(n + q), then the Lq -bound for the solution of (51.108) guarantees the L∞ -bound. The same is true if one considers (51.108) with the nonlinear Neumann boundary condition ∂ν u = g(x, t, u) instead of the homogeneous Dirichlet condition, provided g ∈ C 1 satisfies the growth condition |g| ≤ C(1 + |u|z ), z < 1 + q/n (see [14] and [435]). (iii) Let 0 ≤ β < γ < α ≤ 1 and F : Xβ → Xα−1 be bounded on bounded sets. Let u0 ∈ Xβ and let u ∈ C([0, ∞), Xβ ) be a global solution of (51.9). If u(t) is uniformly bounded in Xβ for all t ≥ 0 and δ > 0, then the estimate |u(t)|γ ≤ |e−tA u0 |γ +

t

e−(t−s)A L(Xα−1 ,Xγ ) |F (u(s))|α−1 ds

t eω(t−s) (t − s)α−1−γ ds ≤ C(1 + tβ−γ ), ≤ Ctβ−γ |u0 |β + C 0

0

implies the boundedness of u(t) in Xγ for t ≥ δ. If A has compact resolvent, then the embedding Xγ → Xβ is compact (see Theorem 51.1(i)), hence the trajectory of u is relatively compact in Xβ . Example 51.38. Let Ω ⊂ Rn be bounded with C 2 -boundary, −A be the Dirichlet Laplacian and F (u) = f (u), where f ∈ C 1 , |f (u)| ≤ C(1 + |u|p−1 ), 1 < p < pS . Assume that u0 ∈ W01,2 (Ω) and u is a global solution of (51.9). If u(t) is uniformly bounded in Lp+1 (Ω), then the trajectory of u is relatively compact in W01,2 (Ω).

52. Appendix F: Maximum and comparison principles. Zero number

507

In fact, Example 51.10 shows the existence of T > 0 such that u ∈ C([0, T ], W01,2 (Ω)) ∩ C((0, T ], W 2β,q (Ω)) for any β < 1 and q ≥ 1. Fixing q := p + 1 > n(p − 1)/2, η ∈ (0, T ), and considering the solution u on the interval [η, ∞), the proof of Theorem 16.4 above shows that u(t) remains bounded in W01,q ∩ W 2β,q (Ω) for some β close to 1. In particular, the solution remains bounded in W01,2 (Ω) and the assertion follows from Remark 51.37(iii). Example 51.39. Let Ω ⊂ Rn be bounded with C 2 -boundary, −A be the Dirichlet Laplacian and F : L∞ (Ω) → L∞ (Ω) be uniformly Lipschitz continuous on bounded subsets of L∞ (Ω). Assume that u0 ∈ L∞ (Ω) and u is a global solution of (51.9) which is uniformly bounded in L∞ (Ω). Then the trajectory {u(t) : t ≥ 1} is relatively compact in L∞ (Ω). This follows from Remark 51.11 and (the corresponding analogue of) Remark 51.37(iii).

52. Appendix F: Maximum and comparison principles. Zero number Maximum and comparison principles represent a very useful tool in the study of scalar equations (and of some particular systems). Unfortunately, it is not easy to provide (or find in the literature) a general statement which would be applicable in all situations. We therefore prove — or at least formulate — various versions of these principles which we frequently use. For simplicity we have stated all the results for the case when the elliptic part of the equation is the Laplacian, but they remain true for more general operators (under suitable assumptions).

52.1. Maximum principles for the Laplace equation We first recall the weak and strong maximum principles and the Hopf boundary lemma for strong subsolutions (cf. [250, Theorems 9.1, 9.6, and the proof of Lemma 3.4]). Proposition 52.1. Let Ω be an arbitrary domain in Rn , b ∈ L∞ (Ω, Rn ), and let 2,n (Ω) satisfy u ∈ Wloc −∆u + b · ∇u ≤ 0 a.e. in Ω. (52.1) (i) If u ∈ C(Ω), u ≤ 0 on ∂Ω, and Ω is bounded, then u ≤ 0 in Ω. (ii) If u ≤ 0 in Ω, then either u ≡ 0 or u < 0 in Ω. (iii) Let x0 ∈ ∂Ω. Assume that Ω satisfies an interior sphere condition at x0 and that u is continuous at x0 . If u ≤ 0 in Ω and u(x0 ) = 0, then lim inf t−1 u(x0 − tν) < 0. t→0+

508

Appendices

(Here the outer normal ν is defined in the natural way via the interior sphere at x0 ). In particular, we have ∂ν u(x0 ) > 0 whenever this derivative exists. Remark 52.2. Assertions (ii) and (iii) of Proposition 52.1 remain valid if the inequality in (52.1) is replaced with −∆u + b · ∇u + cu ≤ 0 for some constant c > 0 (cf. e.g. [239]). This follows easily by applying Proposition 52.1(ii) and (iii) to the function v(x) = eαx1 u(x) with α > 0 large enough. We next give a useful maximum principle under weaker regularity assumptions, namely for variational or, even, distributional subsolutions. Proposition 52.3. Let Ω be an arbitrary domain in Rn and let u ∈ L1loc (Ω) satisfy −∆u ≤ 0 in D (Ω). Assume that either: (i) u ∈ H 1 (Ω) and u ≤ 0 on ∂Ω in the sense that u+ ∈ H01 (Ω); or (ii) Ω is bounded, u is continuous in a neighborhood of ∂Ω and u ≤ 0 on ∂Ω. Then u ≤ 0 a.e. in Ω. Proof. We first assume (i). We shall use the Stampacchia truncation argument. By assumption we have

∇u · ∇ϕ dx ≤ 0, for all 0 ≤ ϕ ∈ D(Ω). (52.2) Ω

Fix a C ∞ -function G : R → R+ such that G(s) = 0 for s ≤ 0 and 0 < G (s) ≤ 1 for s > 0. By our assumption that u+ ∈ H01 (Ω), there exists a sequence ψj ∈ D(Ω) such that ψj → u+ in H 1 (Ω) and a.e. Let ϕj = G ◦ ψj . We have 0 ≤ ϕj ∈ D(Ω). Writing |∇(G ◦ ψj ) − ∇(G ◦ u+ )| ≤ G (ψj )|∇ψj − ∇u+ | + |G (ψj ) − G (u+ )||∇u+ |, we obtain ∇ϕj → ∇(G ◦ u+ ) in L2 (Ω) by dominated convergence. Since ∇u+ = χ{u>0} ∇u, it follows from (52.2) that

2 G (u+ )|∇u+ | dx = ∇u · ∇(G ◦ u+ ) dx = lim ∇u · ∇ϕj dx ≤ 0. Ω

Ω

j→∞

Ω

Consequently, ∇(u+ )2 = 2u+ ∇u+ = 0 a.e. in Ω. Since u+ ∈ H01 (Ω), we conclude that u+ = 0 a.e. in Ω. Let us next consider case (ii). For ε > 0, denote ωε = {x ∈ Ω : δ(x) > ε}. By assumption, there exists ε0 > 0 small, such that u is continuous on Ω \ ωε0 . Now


509

set uj := ρj ∗ u, where ρj is a sequence of mollifiers defined by (47.6), and fix ε ∈ (0, ε0 ). For j ≥ j0 (ε) large, we have uj ∈ C 2 (ω ε ) and ∆uj = ∆u ∗ ρj ≥ 0 in ωε . Therefore, the assertion in case (i) implies supωε uj ≤ sup∂ωε uj . Since uj → u in L1 (ωε ) and in C(∂ωε ), it follows that ess supωε u ≤ sup∂ωε u. The conclusion follows by letting ε → 0 and using the fact that limε→0 (sup∂ωε u) ≤ 0. In the rest of Appendix F we shall only consider parabolic problems.

52.2. Comparison principles for classical and strong solutions We start with a basic maximum principle for classical solutions. Note that unbounded and singular first-order coefficients are allowed (this will be used in Lemma 52.18 below). Proposition 52.4. Let Ω be an arbitrary domain in Rn , T > 0, b : QT → Rn , c : QT → R, with supQT c < ∞. Assume that w = w(x, t) ∈ C 2,1 (QT ) ∩ C(QT ) satisfies w ≤ 0 on PT , supQT w < ∞, and wt − ∆w ≤ b · ∇w + cw

in QT .

(52.3)

If Ω is unbounded, assume in addition that either lim sup |x|→∞, (x,t)∈QT

or

w(x, t) ≤ 0,

|b(x, t)| ≤ C1 (1 + |x − a|−1 ),

x ∈ QT ,

(52.4)

(52.5)

for some a ∈ Rn and C1 > 0. Then w ≤ 0 in QT . Proof. We may assume c < 0 (if this is not true, then it is sufficient to consider the function w(x, ˜ t) = e−λt w(x, t), where λ > supQT c). Also, we may obviously assume w ∈ C 2,1 (Ω × (0, T ]). Case 1: Ω bounded. Assume on the contrary that w achieves a positive interior maximum at some point (x0 , t0 ) ∈ Ω×(0, T ]. At this point we have w > 0, ∇w = 0, ∆w ≤ 0, wt ≥ 0. Using c < 0 we obtain 0 ≤ wt − ∆w − b · ∇w ≤ cw < 0, which yields a contradiction. Case 2: Ω unbounded. If the conclusion is not true, then we have w(x0 , t0 ) > 0 for some (x0 , t0 ) ∈ QT . In case (52.4) is satisfied, then w achieves its positive maximum and we conclude as in case 1. In case (52.5) holds, arguing similarly as in [296], we set v(x, t) = w(x, t) − δt − ε(1 + |x − a|2 )1/2 ,

510

Appendices

where δ, ε > 0 are such that v(x0 , t0 ) > 0 and δ > ε(n + 2C1 ). We compute ∇(1 + |x − a|2 )1/2 = (x − a)(1 + |x − a|2 )−1/2 , ∆(1 + |x − a|2 )1/2 = (n + (n − 1)|x − a|2 )(1 + |x − a|2 )−3/2 ≤ n.

(52.6)

Since v ≤ 0 on St0 , v attains its (positive) maximum in Qt0 at some (x1 , t1 ) ∈ Ω × (0, t0 ]. At this point we have w > v > 0, ∇v = 0, ∆v ≤ 0, vt ≥ 0. Using c ≤ 0, it follows that 0 ≤ vt = wt − δ ≤ ∆w + b · ∇w + cw − δ ≤ ∆v + b · ∇v + nε + ε|b||x − a|(1 + |x − a|2 )−1/2 − δ ≤ ε(n + 2C1 ) − δ < 0, which yields a contradiction and concludes the proof. Remark 52.5. The assumption supQT c < ∞ in Proposition 52.4 is necessary (although it can be sometimes weakened). Consider for instance the simple examples u(x, t) = tϕ1 (x), c(x, t) = λ1 + t−1 (Ω bounded), or u(x, t) = t, c(x, t) = t−1 (Ω = Rn ), which satisfy ut − ∆u = cu and u > 0 in QT , with u ≡ 0 on PT . We next give a version of the comparison principle for classical (sub-/super-) solutions. Proposition 52.6. Let Ω be an arbitrary domain in Rn , T > 0, u, v ∈ C 2,1 (QT )∩ C(QT ). Assume that u ≤ v on PT and ∂t u − ∆u − f (x, u, ∇u) ≤ ∂t v − ∆v − f (x, v, ∇v)

in QT ,

(52.7)

where f = f (x, s, ξ) : Ω × R × Rn → R is continuous in x and C 1 in s and ξ. Assume also that u, v, ∇v ∈ L∞ (QT ),

|u|, |v| ≤ C1 , |∇v| ≤ C2

(52.8)

and |fs (x, s, ξ)| + (1 + |x|)−1 |fξ (x, s, ξ)| ≤ Cf

for all |s| ≤ C1 , |ξ| ≤ C2 + 1. (52.9)

Then u ≤ v in QT . Proof. Fix τ ∈ (0, T ) such that τ eCf τ < 1/8Cf . It is sufficient to prove u ≤ v in Qτ . Assume on the contrary δ := supQτ (u − v) > 0 and choose (x0 , t0 ) ∈ Qτ such that (u − v)(x0 , t0 ) > δ/2. Consider ε ∈ (0, 1) such that ε < min(Cf δ/(n + Cf ), e−Cf τ , e−Cf t0 δ/4ψ(x0 ))


511

and set z(x, t) = e−Cf t (u − v)(x, t) − 2Cf δt − εψ(x), where ψ(x) = (1 + |x|2 )1/2 . Then z(x0 , t0 ) > 0 and z attains its maximum in Qτ at x, t˜) > 0 some (˜ x, t˜) ∈ Qτ , since z(x, t) → −∞ as |x| → ∞, uniformly in t. Now z(˜ implies (˜ x, t˜) ∈ Qτ \Pτ , hence zt −∆z ≥ 0 and ∇z = 0 at this point. Consequently, |∇u(˜ x, t˜) − ∇v(˜ x, t˜)| ≤ eCf τ ε|∇ψ(˜ x)| < 1, since |∇ψ| ≤ 1. In addition, z(˜ x, t˜) > 0 implies ˜

ε|˜ x| ≤ εψ(˜ x) < e−Cf t δ ≤ δ. Now the mean value theorem guarantees the existence of s between u(˜ x, t˜), v(˜ x, t˜) and ξ between ∇u(˜ x, t˜), ∇v(˜ x, t˜) such that 0 ≤ (zt − ∆z)(˜ x, t˜) ˜ x, u(˜ x, t˜), ∇u(˜ x, t˜)) − f (˜ x, v(˜ x, t˜), ∇v(˜ x, t˜)) ≤ e−Cf t f (˜ − Cf (z(˜ x, t˜) + 2Cf δ t˜ + εψ(˜ x)) − 2Cf δ + εn ˜ ˜ x, s, ∇v(˜ x, t˜))eCf t (z(˜ x, t˜) + 2Cf δ t˜ + εψ(˜ x)) = e−Cf t fs (˜ ˜ t C f x, u(˜ x, t˜), ξ)e (∇z(˜ x, t˜) + ε∇ψ(˜ x)) + fξ (˜ − Cf (z(˜ x, t˜) + 2Cf δ t˜ + εψ(˜ x)) − 2Cf δ + εn x|)ε − 2Cf δ + εn ≤ Cf (1 + |˜ < −Cf δ + ε(n + Cf ) < 0, which yields a contradiction and concludes the proof. The following proposition is a version of the strong comparison principle and of the Hopf boundary lemma (for strong solutions, in bounded domains). A more general version can be derived by using the maximum principles in [152] (cf. the proof). Proposition 52.7. Let Ω be a bounded domain in Rn of class C 2 , p > n + 2, and 2,1;p (Ω × (0, T ]) ∩ C([0, T ], L2 (Ω)) ∩ L∞ (QT ). Assume T > 0. Let u, v ∈ Wloc ∂t u − ∆u − f (x, t, u, ∇u) ≤ ∂t v − ∆v − f (x, t, v, ∇v)

in QT ,

where f = f (x, t, s, ξ) : Ω × [0, T ] × R × Rn → R is continuous in x, t and C 1 in s and ξ. Assume also that u(·, 0) ≤ v(·, 0), u(·, 0) ≡ v(·, 0), and either u≤v

on ST

(52.10)

512

Appendices

or ∂ν u + bu ≤ ∂ν v + bv

on ST ,

(52.11)

where b ∈ C 1 (∂Ω). Finally, if f depends on ξ, we also assume that ∇u, ∇v ∈ L∞ (QT ). Then u < v in QT . In addition, if u(x0 , t0 ) = v(x0 , t0 ) for some x0 ∈ ∂Ω and t0 ∈ (0, T ), then ∂ν u(x0 , t0 ) > ∂ν v(x0 , t0 ). If (52.11) is true, then u < v in Ω × (0, T ). Proof. Setting w := v − u, the mean value theorem implies ∂wt − ∆w ≥ g1 (x, t)w + g2 (x, t) · ∇w, 1 1 where g1 (x, t) = 0 fu (x, t, u + θ(v − u), ∇v) dθ and g2 (x, t) = 0 fξ (x, t, u, ∇u + θ(∇v − ∇u)) dθ. Let us first consider the case u, v ∈ W 2,1;p (QT ) (hence in particular u, v ∈ 1,0 C (QT )). Then the assertion follows from [152, Propositions 13.1, 13.2 and Theorem 13.5]. Note that the proofs in [152] use a result from [165] and the strong maximum principle for classical solutions (cf. [429] and [214]). In the general case, since g1 , g2 ∈ L∞ (QT ) due to our assumptions, we may first apply Proposition 52.8 and Remark 52.9 below to deduce that u ≤ v in QT . Since u(·, t) ≡ v(·, t) for all sufficiently small t > 0 due to u, v ∈ C([0, T ], L2 (Ω)), the conclusion follows from the previous case.

52.3. Comparison principles via the Stampacchia method We now give versions of the weak maximum and comparison principles which apply 2,1;2 sub-/supersolutions and discontinuous initial data (as well as possibly to Wloc unbounded domains). Proposition 52.8. Let 0 < T < ∞. Let Ω be an arbitrary domain in Rn , c be measurable and a.e. finite on QT with supQT c < ∞, and K ≥ 0. Assume that w ∈ C(Ω × (0, T )) ∩ C([0, T ), L2loc (Ω)) satisfies sup w < ∞, QT

wt , ∇w, D2 w ∈ L2loc (QT ).

If w ≤ 0 on PT and wt − ∆w ≤ K|∇w| + cw

a.e. in QT ,


513

then w≤0

in QT .

Proof. Let ε > 0, λ = supQT c, and set z = we−λt − εψ, where ψ(x, t) = M t + (1 + |x|2 )1/2 ,

(52.12)

with M = n + K. We see that, a.e. in QT , there holds ∂t z − ∆z − K|∇z| ≤ e−λt (c − λ)w + ε(−ψt + ∆ψ + K|∇ψ|) ≤ ε(−M + n + K) ≤ 0.

(52.13)

We next apply the Stampacchia truncation method. Note that, for R > 0 large enough and for each τ > 0, there exists η = η(τ ) > 0 such that z≤0

in x ∈ Ω : δ(x) ≤ η or |x| ≥ R × (τ, T − τ ).

1 Our assumptions thus imply z+ ∈ C([0, T ), L2 (Ω))∩Hloc ((0, T ), L2 (Ω)), z+ (0) = 0 1 and, for a.e. t ∈ (0, T ), z+ (t) ∈ H0 (Ω ∩ BR ). For a.e. t ∈ (0, T ), since ∆z(·, t) ∈ L2 (Ω ∩ BR ), ∇(z+ )(·, t) = χ{z>0} ∇z(·, t), it follows from (52.13) that

1 d 2 dt

Ω

|∇(z+ )|2 dx + K |∇z|z+ dx Ω Ω

K2 |∇(z+ )|2 dx + |∇(z+ )|2 dx + (z+ )2 dx ≤− 4 Ω Ω Ω

K2 = (z+ )2 dx. 4 Ω

(z+ )2 (t) dx ≤ −

By integration, we conclude that z+ = 0 in QT and the conclusion follows by letting ε → 0. Remark 52.9. Proposition 52.8 can be extended to the case of Neumann boundary conditions. For instance, assume that Ω is smooth and bounded, and that w satisfies the assumptions of Proposition 52.8 with w ≤ 0 on PT replaced by w(·, 0) ≤ 0 and ∂ν w + bw ≤ 0 on ST , where ∇w ∈ C(Ω × (0, T )) and b ∈ L∞ (∂Ω). Then we conclude that w ≤ 0 in QT . This follows from simple modifications of the above proof, with ε = 0, using the trace inequality v L2 (∂Ω) ≤ η ∇v 2 +C(η) v 2 , v ∈ H 1 (Ω), applied with η > 0 small and v = z+ (t) for a.a. t.

514

Appendices

Proposition 52.10. Let 0 < T < ∞, Ω be an arbitrary domain in Rn , and let f = f (s, ξ) : R × Rn → R, be a C 1 -function. Let u ∈ C(Ω × (0, T )) satisfy u ∈ C([0, T ), L2loc (Ω)),

u ∈ L∞ (QT ),

ut , ∇u, D2 u ∈ L2loc (QT ),

and similarly for v. If f depends on ξ, we also assume that ∇u, ∇v ∈ L∞ (QT ). If u ≤ v on PT and ut − ∆u − f (u, ∇u) ≤ vt − ∆v − f (v, ∇v)

a.e. in QT ,

then u≤v

in QT .

Proof. Let w = u − v and set M := max ess sup(|u| + |∇u|), ess sup(|v| + |∇v|) < ∞ QT

(52.14)

QT

and K := sup{|fs (s, ξ)| + |fξ (s, ξ)| : |s|, |ξ| ≤ M } < ∞. Letting c(x, t) = (f (u, ∇u) − f (v, ∇u))/(u − v) (defined to be 0 whenever the denominator vanishes), we have |c| ≤ K and wt − ∆w ≤ f (u, ∇u) − f (v, ∇v) = c(u − v) + (f (v, ∇u) − f (v, ∇v)) ≤ cw + K|∇w| a.e. in QT . The result then follows from Proposition 52.8 applied to w.

Remarks 52.11. (a) In Proposition 52.4 (resp., Proposition 52.6) it is sufficient to ' T := {(x, t) ∈ QT : w(x, t) > 0} assume that (52.3) (resp., (52.7)) holds in the set Q ' (resp., QT := {(x, t) ∈ QT : u(x, t) > v(x, t)}). A similar remark holds for Propositions 52.8 and 52.10. Moreover any boundedness assumption on the functions 'T . u, v, ∇u, ∇v needs to be verified only on the set Q (b) The proof of Proposition 52.6 shows that we can assume ∇u ∈ L∞ (Ω) instead of ∇v ∈ L∞ (Ω). In addition, we do not need to assume the boundedness of ∇v (or ∇u) at all if f is independent of ξ. Similarly, the assumption u, v ∈ L∞ (Ω) can be replaced by supQT (u − v) < ∞ if f is independent of u. (c) In Proposition 52.10, assume f (s, ξ) to be only continuous (instead of C 1 ) at s = 0, and suppose in addition that inf QT |u| > 0 or inf QT |v| > 0. Then the conclusion remains valid. Indeed, assume for instance σ := inf QT |v| > 0 and let K0 := sup{|f (s, ξ)| : |s| ≤ M, |ξ| ≤ M } and K1 := sup{|fs (s, ξ)| + |fξ (s, ξ)| : σ/2 ≤ |s| ≤ M, |ξ| ≤ M },


515

with M defined by (52.14). Then the function c in the proof verifies |c(x, t)| ≤ K1 if |u(x, t)| ≥ σ/2, and |c(x, t)| ≤ 4K0 /σ if |u(x, t)| < σ/2. A similar remark holds concerning Proposition 52.22 (systems). (d) The proof of Proposition 52.10 shows that it is sufficient to assume that u or v ∈ L∞ (QT ), and that supQT (u − v) < ∞. (e) In Proposition 52.6, if f is of the form f = f (u) + g(x, ∇u), then the assumptions (52.8)–(52.9) can be replaced by lim sup |x|→∞, (x,t)∈QT

(u − v)(x, t) ≤ 0

and u or v ∈ L∞ (QT ). This can be proved easily by using Proposition 52.4 and Remark (a) above. (f) In Propositions 52.4 and 52.8, if c ≤ 0 and if, instead of w ≤ 0 on PT , we assume w ≤ M on PT for some M > 0, then the conclusion is w ≤ M in QT (just apply the result to the function w − M ). (g) When comparing a solution with a sub-/supersolution, the above (and similar) results are usually applied on the time interval (0, T ) for each T < Tmax (u0 ), hence guaranteeing the boundedness of the solution (and possibly of its derivatives).

52.4. Comparison principles via duality arguments We now provide “very weak” versions of the maximum and comparison principles, which are useful in particular in the study of complete blow-up (see Section 27). They can be also applied to show monotonicity of solutions in time (cf. Proposition 52.20 below). Assume that Ω is a bounded domain of class C 2+α for some α ∈ (0, 1). Let T > 0, u0 ∈ L1δ (Ω) and f ∈ L1loc ([0, T ), L1δ (Ω)). We say that u ∈ L1loc (Ω × [0, T )) is a very weak supersolution of ⎫ ut − ∆u = f, x ∈ Ω, t ∈ (0, T ), ⎪ ⎬ u = 0, x ∈ ∂Ω, t ∈ (0, T ), (52.15) ⎪ ⎭ x ∈ Ω, u(x, 0) = u0 (x), if

0

τ

Ω

u(ϕt + ∆ϕ) + f ϕ dx ds +

Ω

u0 ϕ(0) dx ≤ 0

(52.16)

for any 0 < τ < T and any 0 ≤ ϕ ∈ C 2,1 (Ω × [0, τ ]) such that ϕ = 0 on ∂Ω × [0, τ ] and ϕ(τ ) = 0. Subsolutions are defined similarly (namely, u is a subsolution if −u is a supersolution). Of course, the definition immediately carries over to the nonlinear case f = f (u).

516

Appendices

Remarks 52.12. (i) If u ∈ C 2,1 (Ω × (0, T )) ∩ C([0, T ), L1 (Ω)) is a classical supersolution of (52.15) (i.e., satisfies (52.15) with = signs replaced by ≥), then it is easy to show that it is a very weak supersolution. (ii) Alternatively, one could replace the integrability assumption on f near t = 0 by a continuity assumption on u (namely, u ∈ C([0, T ), L1δ (Ω)) and just f ∈ L1loc ((0, T ), L1δ (Ω)) and adopt a definition more similar to that of weak L1δ solution (cf. Definition 48.8). However, the present formulation seems better suited to certain applications, such as complete blow-up. Proposition 52.13. Let Ω be a bounded domain of class C 2+α for some α ∈ (0, 1). Let 0 < T < ∞ and c ∈ L∞ (QT ). (i) Assume that z ∈ Lqloc (Ω × [0, T )) If z is a very weak supersolution of zt − ∆z = cz, z = 0, z(x, 0) = 0,

for some 1 < q < ∞. x ∈ Ω, 0 < t < T, x ∈ ∂Ω, 0 < t < T, x ∈ Ω,

(52.17) ⎫ ⎪ ⎬ ⎪ ⎭

(52.18)

then z ≥ 0 a.e. in QT . (ii) If c = 0, then assertion (i) remains true for q = 1. Proof. (i) Fix m > max(n/2, q ) and a sequence of functions cj ∈ D(QT ) such that cj → c in Lm (QT ). For given 0 < τ < T and 0 ≤ h ∈ D(Qτ ), let ϕj ∈ C 2,1 (Qτ ) be the solution of ⎫ −∂t ϕj − ∆ϕj = cj ϕj + h, x ∈ Ω, 0 < t < τ, ⎪ ⎬ x ∈ ∂Ω, 0 < t < τ, ϕj = 0, (52.19) ⎪ ⎭ x ∈ Ω. ϕ (x, τ ) = 0, j

By Proposition 52.8, we have ϕj ≥ 0. Moreover, by using the variation-of-constants formula, the Lm -L∞ -estimate (Proposition 48.4), and m > n/2, one easily gets ϕj L∞ (Qτ ) ≤ C,

j = 1, 2, . . . .

(52.20)

Applying the definition of z being a (very weak) supersolution of (52.18), with ϕ = ϕj as a test-function, we obtain

τ

τ

hz + (cj − c)zϕj dx ds. (52.21) z(∂t ϕj + ∆ϕj + cϕj ) dx ds = 0≤− 0

Ω

0

Ω

Since(cj − c)zϕj → 0 in L1 (Qτ ) due to (52.17), (52.20) and m > q , we deduce τ that 0 Ω hz dx ds ≥ 0, and the conclusion follows. (ii) The argument is much simpler than in the previous case: It suffices to use (52.21) with c = cj = 0 and ϕ instead of ϕj , where ϕ is the solution of (52.19) with cj = 0. We have the following (very weak) comparison principle for the semilinear problem (14.1).


517

Proposition 52.14. Let Ω be a bounded domain of class C 2+α for some α ∈ (0, 1), 0 < T < ∞, and u0 ∈ L∞ (Ω). Assume that f : R → R is of class C 1 . Let u, v ∈ L∞ (QT ) be, respectively, very weak sub- and supersolutions to problem (14.1) on (0, T ). Then u ≤ v on (0, T ). Proof. This is an immediate consequence of Proposition 52.13 applied to z := v − u. The comparison results in the previous subsections do not apply in the case of convective equations like ut − ∆u = |u|p−1 u + a · ∇(|u|q−1 u) with 1 < q < 2, due to the fact that the nonlinearity is not C 1 at u = 0. For such problems, we shall rely instead on the following result, the proof of which involves a duality argument. For simplicity we restrict ourselves to the case Ω bounded or Ω = Rn . Proposition 52.15. Let Ω be a bounded domain of class C 2 or Ω = Rn . Let T > 0, b ∈ L∞ (QT , Rn ) and c ∈ L∞ (QT ). Assume that w = w(x, t) ∈ C 2,1 (Ω × (0, T )) ∩ L∞ (QT ) and that bw ∈ C 1,0 (Ω × (0, T )). If Ω = Rn assume in ∞ n addition that ∇w ∈ L∞ loc ((0, T ), L (R )). If w ≤ 0 on ST , lim supt→0 w(x, t) ≤ 0 for all x ∈ Ω, and wt − ∆w ≤ div(bw) + cw in QT , (52.22) then w ≤ 0 in QT . Proof. Fix h ∈ D(Ω), h ≥ 0, 0 < t2 < T . First consider the case Ω bounded and let ϕ be the solution of the adjoint problem ⎫ −ϕt − ∆ϕ = −b · ∇ϕ + cϕ, x ∈ Ω, 0 < t < t2 , ⎪ ⎬ ϕ = 0, x ∈ ∂Ω, 0 < t < t2 , (52.23) ⎪ ⎭ x ∈ Ω. ϕ(x, t ) = h(x), 2

By parabolic Lr -regularity, we have ϕ ∈ W 2,1:r (QT ), 1 < r < ∞, and ϕ ≥ 0 by Proposition 52.8. For each 0 < t1 < t2 , multiplying (52.22) by ϕ, integrating by parts and using w ≤ 0 on PT , ∂ϕ/∂ν ≤ 0 = ϕ on ∂Ω, we obtain 0t2 t2

/

wϕt + wt ϕ dx ds wϕ dx = Ω

t1

≤ ≤

t1 t2

t1

t2 t1

Ω

wϕt + (∆w + div(bw) + cw)ϕ dx ds

Ω

Ω

ϕt + ∆ϕ − b · ∇ϕ + cϕ w dx ds = 0.

(52.24)

518

Appendices

Letting t1 → 0, we obtain

Ω

w(t2 )h dx ≤ 0, hence w ≤ 0.

Next consider the case Ω = Rn . Observe that problem (52.23) still admits a solution ϕ ∈ C([0, T ], W 1,1 (Rn )), ϕ ≥ 0. This follows from a straightforward fixedpoint argument, using the variation-of-constants formula and simple estimates involving the Gaussian heat kernel G. Moreover, given 1 < r < ∞, we have 2,1:r ϕ ∈ C([0, T ], W 1,r (Rn )) due to Appendix E and ϕ ∈ Wloc (QT ) by Theorem 48.1, and a simple cut-off argument. For R > 0, arguing as in (52.24) with Ω replaced by BR , we get /

0t2

wϕ dx ≤

BR

t1

∂w ∂ϕ −w + (b · ν)wϕ dσds ϕ ∂ν ∂ν t1 ∂BR

t2

≤ C(t1 ) ϕ + |∇ϕ| dσds. t2

t1

(52.25)

∂BR

Since ϕ ∈ C([0, T ], W 1,1 (Rn )) there exists a sequence Rj → ∞ such that the RHS of (52.25) with R = Rj decays to 0. Then letting t1 → 0, we obtain Rn w(t2 )h dx = 0, hence w ≤ 0. As a direct consequence of Proposition 52.15, we obtain in particular: Proposition 52.16. Let Ω be a bounded domain of class C 2 or Ω = Rn . Let T > 0 and f, g : (t, u) [0, T ] × R → R be such that f, fu , g, gu are continuous. Let u, v ∈ C 2,1 (Ω × (0, T )) ∩ L∞ (QT ). If Ω = Rn assume in addition that ∇u, ∇v ∈ ∞ n L∞ loc ((0, T ), L (R )). If u ≤ v on ST , lim supt→0 (u − v)(x, t) ≤ 0 for all x ∈ Ω, and ∂t u − ∆u − f (t, u) − div(g(t, u)) ≤ ∂t v − ∆v − f (t, v) − div(g(t, v))

in QT ,

then u ≤ v in QT .

52.5. Monotonicity of radial solutions Assume that Ω is a symmetric domain and that problem (34.1) is well-posed in a space of functions X on Ω. If the C 1 -function F = F (s, ξ) depends on ξ through |ξ| only and if u0 ∈ X is radial, then the solution u of (34.1) is also radial. This follows immediately from the local uniqueness and the invariance of problem (34.1) by rotation. The same remains true in the case of Neumann boundary conditions. The following result provides sufficient conditions for the preservation of radial monotonicity. Proposition 52.17. Let Ω = BR or Ω = Rn . In what follows we use the notation T = Tmax (u0 ).


519

(i) Consider problem (34.1) with a C 1 -function F = F (s, ξ) : R+ × Rn → R such that F (s, ξ) = F˜ (s, |ξ|) and F (0, 0) ≥ 0. Assume that u0 ∈ BC 1 (Ω), u0 = 0 on ∂Ω, u0 ≥ 0, is radial nonincreasing. Then u ≥ 0 and u is radial nonincreasing in QT .

(52.26)

(ii) Consider problem (14.1) with f ∈ C 1 ([0, ∞)) such that f (0) ≥ 0. If u0 ∈ L∞ (Ω), u0 ≥ 0, is radial nonincreasing, then (52.26) is true. (iii) Consider problem (15.1) with p > 1, and let 1 ≤ q < ∞ satisfy q > qc = n(p − 1)/2 or q = qc > 1. If u0 ∈ Lq (Ω), u0 ≥ 0, is radial nonincreasing, then (52.26) is true. Moreover, in each case above, if u0 ≡ 0, then ur < 0

in (0, R] × (0, T )

(52.27)

(with (0, R] replaced by (0, ∞) if Ω = Rn ). We need the following lemma: Lemma 52.18. Let Ω = (0, R), 0 < R ≤ ∞, T > 0, f = f (s, ξ) ∈ C 1 (R+ × R). 2,1;2 (QT ) ∩ BC(QT ), Assume that u ∈ C 2,1 (Ω × (0, T )) ∩ BC(QT ) satisfies ur ∈ Wloc ur ≤ 0 on PT and ut − urr −

n−1 ur = f (u, ur ) r

in QT .

(52.28)

Then ur ≤ 0 in QT . If ur (·, 0) ≡ 0, then ur < 0 in QT and, assuming R < ∞, ur (R, t) < 0 for t ∈ (0, T ). Proof. For λ > 0 large enough, the function w := ur e−λt solves the equation wt − wrr = bwr + cw

a.e. in QT ,

n−1 with b = fξ (u, ur )+ n−1 r and c = fu (u, ur )−λ− r 2 ≤ 0. Assume for contradiction that supQT w > 0. Set m = supQT w/2, z = w − m, and choose r0 ∈ (0, R) such that w ≤ m in [0, r0 ] × [0, T ]. Then, for some K > 0, we have

zt − zrr = bzr + c(z + m) ≤ K|zr | + cz

a.e. in (r0 , R) × (0, T ).

Using Proposition 52.8 we infer that z ≤ 0 in (r0 , R) × [0, T ], contradicting the definition of m. Consequently, ur ≤ 0. ˜ = Finally, to show that ur < 0 it is sufficient to use Proposition 52.7 with Ω ˜ where 0 < ε < R ˜ < ∞, R ˜ ≤ R. (ε, R), Proof of Proposition 52.17. Let us first verify assertion (i). We know that u is radial and u satisfies (52.28) with f (u, ur ) = F (u, (ur , 0, . . . , 0)).

520

Appendices

We have ur ∈ BC([0, R] × [0, τ ]), 0 < τ < T (with [0, R] replaced by [0, ∞) 2,1;2 (QT ). Since u0,r ≤ if Ω = Rn ). Also, by Remark 48.3(i), we have ur ∈ Wloc 0, ur (0, t) = 0, and ur (R, t) ≤ 0 due to u ≥ 0, the assertion is then a direct consequence of Lemma 52.18. For the proof of assertions (ii) and (iii), we proceed in several steps. ' Step 1. Define D(Ω) = {v ∈ D(Ω) : v ≥ 0, v is radial nonincreasing} and q q ' L (Ω) = {v ∈ L (Ω) : v ≥ 0, v is radial nonincreasing} for 1 ≤ q ≤ ∞. It is ' q (Ω) and v ∈ D(Ω), ' then the convolution not too difficult to show that if u0 ∈ L ' q (Ω). The same is true product u0 ∗ v is radial nonincreasing, hence belongs to L if v is replaced by Gt , t > 0. Step 2. By standard arguments of truncation and convolution with a mollifier, ' ' q (Ω). using Step 1, one shows that D(Ω) is dense in L Step 3. We claim that for u0 satisfying the assumption of (ii) or (iii), the function z(x, t) := e−tA u0 is radial nonincreasing. If Ω = Rn , then this follows from Step 1, ' by assertion (i), since z(t) = Gt ∗ u0 . If Ω = BR , then this is true for u0 ∈ D(Ω) and the general case follows from the density property of Step 2. Step 4. First consider case (ii) with f ≥ 0, or case (iii). The solution u is constructed as the fixed point of a suitable contraction mapping (cf. the proof of Theorem 15.2 and Remark 51.11). Consequently, u is the limit of a sequence t uk+1 = Φu0 (uk ) := e−tA u0 + 0 e−(t−s)A f (u(s)) ds. The conclusion then follows from the fact that the operator Φu0 preserves the radial nonincreasing property, due to f ≥ 0. Step 5. In case (ii) for general f of class C 1 , let M := supQT u and set f˜(s) := f (s) + λs, where λ > 0 is chosen so large that f˜ ≥ 0 on [0, M ]. Noting that u solves ut − (∆ − λ)u = f˜(u), the conclusion follows from the argument in Step 4, where e−tA is replaced with e−λt e−tA . Finally, let u0 ≡ 0. Since, given δ > 0, u(δ) ∈ BC 1 (Ω) and u = 0 on ∂Ω, inequality (52.27) follows from (i).

52.6. Monotonicity of solutions in time We give two results which are useful to guarantee the monotonicity of solutions in time. Proposition 52.19. Let Ω ⊂ Rn be a uniformly regular domain of class C 2 , let F = F (s, ξ) : R × Rn → R be a C 1 -function, and consider problem (34.1). Assume 2 (Ω) satisfies u0 = 0 on ∂Ω and that u0 ∈ BC(Ω) ∩ Hloc ∆u0 + F (u0 , ∇u0 ) ≥ 0

a.e. in Ω.


521

If F depends on ξ, assume in addition that u0 ∈ BC 1 (Ω). Then ut ≥ 0 in QT , where T := Tmax (u0 ). Proof. In the case when F depends on ξ, first recall that problem (34.1) is wellposed in X = {u ∈ BC 1 (Ω) : u = 0 on ∂Ω}. By comparing u with the subsolution u(x, t) := u0 (x) via Proposition 52.10, we obtain u ≥ u0 in QT . Now fix h ∈ (0, T ) and put v(t) := u(t + h). Since v(0) = u(h) ≥ u0 , we infer from Proposition 52.6 that v ≥ u on (0, T − h). The result then follows by dividing by h and letting h → 0. In case Ω is bounded and the nonlinearity depends only on u, the following alternative approach guarantees monotonicity of solutions in time under much weaker regularity on the initial data. We say that u0 ∈ L∞ (Ω) satisfies ∆u0 + f (u0 ) ≥ 0, x ∈ Ω, (52.29) x ∈ ∂Ω u0 ≤ 0, in the very weak sense if, for all 0 ≤ ψ ∈ C 2 (Ω) such that ψ = 0 on ∂Ω, there holds

u0 ∆ψ + f (u0 )ψ dy ≥ 0. (52.30) Ω

(Of course, this is satisfied in particular if u0 belongs to H 2 ∩ H01 (Ω) and verifies ∆u0 + f (u0 ) ≥ 0 a.e. in Ω.) Proposition 52.20. Assume that Ω is a bounded domain of class C 2+α for some α ∈ (0, 1) and consider problem (14.1) with f ∈ C 1 (R). If u0 ∈ L∞ (Ω) satisfies (52.29) in the very weak sense, then ut ≥ 0 in QT , where T := Tmax (u0 ). Proof. Step 1. We claim that u ≥ u0 in QT . For 0 ≤ t < T , set v(t) := u(t) − u0 , c(x, t) = (f (u) − f (u0 ))/(u − u0 ) (defined to be 0 whenever the denominator vanishes), and notice that c ∈ L∞ (QT ). For given 0 < τ < T , let 0 ≤ ϕ ∈ C 2 (Ω × [0, τ ]) be such that ϕ = 0 on ∂Ω× [0, τ ] and ϕ(τ τ ) = 0. For each 0 < t < τ , by integrating by parts and using (52.30) with ψ = t ϕ as a test-function, we obtain

τ

τ

− vϕ (t) dx = vϕt + ut ϕ dx ds = vϕt + (∆u + f (u))ϕ dx ds Ω Ω t

t τ Ω

= v(ϕt + ∆ϕ) + f (u)ϕ + u0 ∆ϕ dx ds

t τ Ω

≥ v(ϕt + ∆ϕ) + (f (u) − f (u0 ))ϕ dx ds

t τ Ω

= v(ϕt + ∆ϕ + cϕ) dx ds. t

Ω

522

Appendices

Letting t → 0, hence u(t) − u0 1 → 0 (due to u(t) − e−tA u0 ∞ → 0), it follows that

τ

v(ϕt + ∆ϕ + cϕ) dx ds ≤ 0. t

Ω

By Proposition 52.13, we deduce that v ≥ 0, hence the claim. Step 2. As before, we fix h ∈ (0, T ) and let v(t) := u(t + h). Since 1 u, v ∈ C 2,1 (Ω × (0, T − h)) ∩ L∞ loc (Ω × [0, T − h)) ∩ C([0, T − h), L (Ω))

are classical solutions of the first two equations in (14.1) on (0, T − h), we deduce from Proposition 52.13(i) and Remark 52.12(i) that v ≥ u on (0, T − h). The result follows by dividing by h and letting h → 0.

52.7. Systems and nonlocal problems We first give extensions of some of the preceding results to systems of cooperative type. Proposition 52.21. Let 0 < T < ∞, Ω be an arbitrary domain in Rn , d1 , d2 > 0, and aij ∈ L∞ (QT ), i, j ∈ {1, 2}, with a12 , a21 ≥ 0. Assume that for i = 1, 2, the function wi satisfies wi ∈ C(Ω × (0, T )) ∩ C([0, T ), L2loc (Ω)), supQT wi < ∞, ∂t wi , ∇wi , D2 wi ∈ L2loc (QT ). If w1 , w2 ≤ 0 on PT and ∂t w1 − d1 ∆w1 ≤ a11 w1 + a12 w2 ∂t w2 − d2 ∆w2 ≤ a21 w1 + a22 w2

a.e. in QT , a.e. in QT ,

then w1 , w2 ≤ 0 Proof. Let ε > 0, λ = 2 max

in QT .

sup aij , and set

0≤i,j≤2 QT

zi = wi e−λt − εψ, where ψ defined in (52.12) with M = n. Since ∆ψ − ψt ≤ 0 by (52.6), it follows that a.e. in QT , there holds ∂t z1 − d1 ∆z1 = e−λt (∂t w1 − d1 ∆w1 − λw1 ) + ε(∆ψ − ψt ) ≤ e−λt (a11 − λ)w1 + a12 w2 ≤ (a11 − λ)z1 + a12 z2 + ε(a11 + a12 − λ)ψ ≤ (a11 − λ)z1 + a12 z2 and similarly, ∂t z2 − d2 ∆z2 ≤ a21 z1 + (a22 − λ)z2 .


523

Arguing as in the proof of Proposition 52.8, it follows that

1 d (z1,+ )2 (t) dx 2 dt Ω

|∇(z1,+ )|2 dx + (a11 − λ)(z1,+ )2 dx + a12 z2 z1,+ dx (52.31) ≤ −d1 Ω Ω

Ω (z1,+ )2 + (z1,+ )2 dx, ≤ a12 z2,+ z1,+ dx ≤ C Ω

Ω

where we used a11 − λ ≤ 0 and a12 ≥ 0. Similarly, we get

1 d (z1,+ )2 + (z1,+ )2 dx. (z2,+ )2 (t) dx ≤ C 2 dt Ω Ω

(52.32)

Adding up (52.31) and (52.32), integrating, and using z1,+ (0) = z2,+ (0) = 0, we infer that z1,+ = z2,+ = 0 in QT and the conclusion follows by letting ε → 0. By arguing similarly as in the proof of Proposition 52.10, one obtains a comparison principle for cooperative systems of the form ∂t ui − di ∆ui − fi (u1 , u2 ) = 0,

i = 1, 2.

(52.33)

Proposition 52.22. Let 0 < T < ∞, Ω be an arbitrary domain in Rn , and let fi = fi (u1 , u2 ) : R2 → R, i = 1, 2, be C 1 -functions such that ∂u2 f1 ≥ 0,

∂u1 f2 ≥ 0.

(52.34)

Let u = (u1 , u2 ), where ui ∈ C(Ω × (0, T )) satisfy ui ∈ L∞ (QT ), ui ∈ C([0, T ), L2loc (Ω)), and ∂t ui , ∇ui , D2 ui ∈ L2loc (QT ). Finally, let v satisfy the same hypotheses as u. If, for i = 1, 2, we have ui ≤ vi on PT and ∂t ui − di ∆ui − fi (u1 , u2 ) ≤ ∂t vi − di ∆vi − fi (v1 , v2 )

a.e. in QT ,

then ui ≤ vi

in QT , i = 1, 2.

Remarks 52.23. (i) The cooperativity assumption (52.34) (or a12 , a21 ≥ 0 in Proposition 52.21) is essential to ensure the order-preserving character of system (52.33), as shown by the following simple example. Consider system (52.33) under homogeneous Dirichlet boundary conditions, with f1 (u, v) = −v, f2 (u, v) = 0. If we take u0 = 0 and v0 ≥ 0, v0 ≡ 0, then, by the strong maximum principle, we have v > 0, hence u < 0, in Ω × (0, ∞). Therefore the order with the solution (0, 0) at t = 0 is not preserved. (ii) For system (52.33) with homogeneous Dirichlet boundary condition, under assumption (52.34), the analogues of Propositions 52.19 and 52.20 guaranteeing time-monotonicity of solutions can be established by simple modifications of the proofs. We next turn to nonlocal problems (with space or time integral nonlinearities).

524

Appendices

Proposition 52.24. Let 0 < T < ∞, Ω be an arbitrary bounded domain in Rn , and a, b, k ∈ L∞ (QT ), with b, k ≥ 0. Assume that the function w ∈ C(Ω × (0, T )) ∩ C([0, T ), L2 (Ω)) satisfies supQT w < ∞, ∂t w, ∇w, D2 w ∈ L2loc (Ω × (0, T )). If w ≤ 0 on PT and either

(52.35)

∂t w − ∆w ≤ aw + b or

Ω

∂t w − ∆w ≤ aw + b

k(y, ·)w(y, ·) dy

a.e. in QT ,

(52.36)

k(·, s)w(·, s) ds

a.e. in QT ,

(52.37)

t

0

then w≤0

in QT .

Proof. Our assumptions imply w+ ∈ C([0, T ), L2 (Ω)) ∩ C 1 ((0, T ), L2 (Ω)), w+ (0) = 0 and, for a.e. t ∈ (0, T ), w+ (t) ∈ H01 (Ω). Moreover, for a.e. t ∈ (0, T ), we have ∆w(·, t) ∈ L2 (Ω), and ∇(w+ )(·, t) = χ{w>0} ∇w(·, t). In the case of (52.36), by using b, k ≥ 0 and the Cauchy-Schwarz inequality, we obtain

1 d (w+ )2 (t) dx ≤ − |∇(w+ )|2 dx + a(w+ )2 dx + bw+ dx kw+ dy 2 dt Ω Ω Ω Ω Ω

≤ C (w+ )2 dx. Ω

In the case of (52.37), we obtain

1 d (w+ )2 (t) dx ≤ − |∇(w+ )|2 dx + a(w+ )2 dx 2 dt Ω Ω Ω

t bw+ kw+ ds dx + 0 Ω

t

a(w+ )2 dx + b2 (w+ )2 dx + T k 2 (w+ )2 dx ds. ≤ t

Ω

Ω

0

Ω

The function φ(t) := 0 Ω (w+ )2 dx ds thus satisfies φ ≤ C(φ + φ ) and φ, φ ≥ 0, hence [φ2 + (φ )2 ] = 2(φ + φ )φ ≤ C[φ2 + (φ )2 ], 0 < t < T, with φ(0) = φ (0) = 0. In both cases, by integration, we conclude that w+ = 0 in QT . As a consequence of Proposition 52.24 we obtain in particular the following comparison principle. The proof is similar to that of Proposition 52.10.


525

Proposition 52.25. Let 0 < T < ∞, Ω be an arbitrary bounded domain in Rn , and let f : R → R and g = g(s, z) : R2 → R be C 1 -functions, with either f , ∂z g ≥ 0 or f , ∂z g ≤ 0. Let u ∈ C(Ω×(0, T )) satisfy u ∈ L∞ (QT ), u ∈ C([0, T ), L2loc (Ω)), and ut , ∇u, D2 u ∈ L2loc (Ω × (0, T )), and let v satisfy the same hypotheses as u. Finally, denote

I(u, t) :=

f (u(y, t)) dy

t

(resp., I(u, t) :=

f (u(y, s)) ds). 0

Ω

If u ≤ v on PT and ut − ∆u − g u, I(u, ·) ≤ vt − ∆v − g v, I(v, ·)

a.e. in QT ,

then u≤v

in QT .

Remarks 52.26. (i) The positivity assumption on b, k is essential for the validity of the nonlocal maximum principle in Proposition 52.24, as shown by the following example from [525]: The function w(x, t) = x2 − t satisfies ⎫ 1 wt − wxx = −3 ≥ −18 −1 w(y, t) dy, 1 < x < 1, 0 < t < 1/4, ⎪ ⎪ ⎬ (52.38) w(±1, t) = 1 − t ≥ 0, 0 < t < 1/4, ⎪ ⎪ ⎭ 2 w(x, 0) = x ≥ 0, 1 < x < 1, but w(0, t) = −t < 0. (ii) Assumption (52.35) in Proposition 52.24 can be weakened to ∂t w, ∇w, D2 w ∈ L2loc (QT ) (and similarly in Proposition 52.25). To see this it suffices to replace w in the proof by z := w − εeλt with λ > 0 large (using the fact that z+ = 0 near the boundary similarly as in the proof of Proposition 52.8), and then let ε → 0. In the case of nonlocal problems in unbounded domains, we need a different statement. Proposition 52.27. Let 0 < T < ∞, Ω be an arbitrary domain in Rn , a, b ∈ L∞ (QT ), and k ∈ L∞ (Ω), with b, k ≥ 0. Assume that the function w ∈ C 2,1 (QT )∩ C(QT ) satisfies w ≤ 0 on PT ,

k(y)w(y, ·) dy a.e. in QT , (52.39) ∂t w − ∆w ≤ aw + b Ω

and w ∈ C([0, T ), L1 (Ω)),

k(y)w(y, 0) dy < 0. Ω

(52.40)

526

Appendices

Then w 0. Let

T0 := sup τ ∈ (0, T ) : (52.41) (hence (52.42)) is true and assume for contradiction that T0 < T . Then (52.41) and (52.42) hold for τ = T0 , hence in particular w(·, T0 ) < 0. Consequently, I(T0 ) < 0, so that (52.41) holds for some τ > T0 , contradicting the definition of T0 . This proves the result.

52.8. Zero number Zero number arguments can be viewed as a sophisticated form of the maximum principle. Although they are restricted to one-dimensional or radially symmetric problems, they represent a very powerful tool. The zero number of a function ψ ∈ C((0, R)) is defined as the number of sign changes of ψ in (0, R); z(ψ) = z[0,R] (ψ) = sup{k ∈ N : there are 0 < x0 < x1 < · · · < xk < R such that ψ(xi )ψ(xi+1 ) < 0 for 0 ≤ i < k}. Let BR = {x ∈ Rn : |x| < R}, t1 < t2 , q ∈ L∞ (BR , (t1 , t2 )), u ∈ C(BR × [t1 , t2 ]) ∩ W 2,1;∞ (BR × (t1 , t2 )) and ut − ∆u = qu

a.e. in BR × (t1 , t2 ).

(52.43)

Assume that q(·, t) and u(·, t) are radially symmetric for all t, hence q(x, t) = Q(|x|, t) and u(x, t) = U (|x|, t). Then Ut − Urr −

n−1 Ur = QU, r

and Ur (0, t) = 0 for all t ∈ (t1 , t2 ).

r ∈ (0, R), t ∈ (t1 , t2 ),

(52.44)


527

Theorem 52.28. Let q, u be as above, u ≡ 0, and either U (R, t) = 0 for all t ∈ [t1 , t2 ] or U (R, t) = 0 for all t ∈ [t1 , t2 ]. Let z = z[0,R] denote the zero number in (0, R). Then (i) z(U (·, t)) < ∞ for all t ∈ (t1 , t2 ), (ii) the function t → z(U (·, t)) is nonincreasing, (iii) if U (r0 , t0 ) = Ur (r0 , t0 ) = 0 for some r0 ∈ [0, R] and t0 ∈ (t1 , t2 ), then z(U (·, t)) > z(U (·, s)) for all t1 < t < t0 < s < t2 . Proof. If U (R, t) = 0 for all t, then the assertion follows from [127, Theorem 2.1]. If U (R, t) = 0 for all t, then we may assume U (R, t) > 0 for all t. Fix ε ∈ (0, R) such that U (r, t) > ε for all r ∈ [R − ε, R] and t ∈ [t1 , t2 ]. Let V = V (r) be the solution of Vrr +

n−1 Vr = 0 r

in [R − ε, R + ε],

V (R + ε) = 0, Vr (R + ε) = −1,

and notice that V (r) ≥ ε for r ≤ R. Choose ϕ ∈ C ∞ ([0, R+ε]) such that 0 ≤ ϕ ≤ 1, ϕ ≡ 1 on [0, R − ε], ϕ ≡ 0 on [R, R + ε], and set ˜ (r, t) = ϕ(r)U (r, t) + (1 − ϕ(r))V (r), U ⎧ t) ⎪ ⎨ Q(r, 1 ˜ n−1 ˜ ˜ ˜ Q(r, t) = ˜ Ut − Urr − r Ur U ⎪ ⎩ 0

r ∈ [0, R + ε], t ∈ [t1 , t2 ], if r ∈ [0, R − ε], if r ∈ (R − ε, R), if r ∈ [R, R + ε].

˜ solves (52.44) with Q replaced by Q ˜ and R by R + ε, U ˜ (R + ε, t) = 0 and Then U ˜ the assertion follows from z[0,R] (U (·, t)) = z[0,R+ε] (U (·, t)) Remarks 52.29. (i) The assertion of Theorem 52.28 remains true for more general problems of the form ut − ∆u = qu + bx · ∇u, where b ∈ W 1,∞ (BR × (t1 , t2 )), b(x, t) = B(|x|, t). This follows from the fact that |x| 1 the function v(x, t) := e 2 0 B(ξ,t)ξ dξ u(x, t) solves a problem of the form (52.43). (ii) If n = 1, then a more general statement (allowing Dirichlet, Neumann or periodic boundary conditions and more general coefficients of the differential operators) can be found in [30]. In particular, the arguments in [30] guarantee that if x1 < x2 , t1 < t2 and u = U ∈ C([x1 , x2 ] × [t1 , t2 ]) is a solution of ut − a(x, t)uxx = b(x, t)ux + c(x, t)u where

in (x1 , x2 ) × (t1 , t2 ),

a > 0, a, a−1 , at , ax , axx , b, bt , bx , c ∈ L∞ ,

528

Appendices

and, for any i ∈ {1, 2}, either

u(xi , t) = 0 for t ∈ [t1 , t2 ],

or u(xi , t) = 0 for t ∈ [t1 , t2 ], then statements (i)–(iii) in Theorem 52.28 hold with [0, R] replaced by [x1 , x2 ]. Let us mention that in the case of Neumann boundary conditions one has to assume a ≡ 1 and b ≡ 0.

53. Appendix G: Dynamical systems In this section we collect some basic definitions and properties of dynamical systems. Since the statements are usually proved only in the dissipative case in the literature (see [267], for example), we also provide detailed proofs. Definition 53.1. Let (X, d) be a complete metric space and τ : X → (0, ∞] be lower semicontinuous. A mapping ϕ : X × [0, ∞) → X defined for all (u, t) with u ∈ X and t ∈ [0, τ (u)) is called a (local) dynamical system on X if (i) ϕ(u, ·) : [0, τ (u)) → X is continuous, (ii) ϕ(·, t) : X → X is continuous at u for all u ∈ X and t < τ (u), (iii) ϕ(u, 0) = u for all u ∈ X, (iv) τ (ϕ(u, s)) = τ (u) − s and ϕ(u, t + s) = ϕ(ϕ(u, s), t) for all u ∈ X, s ∈ [0, τ (u)) and t ∈ [0, τ (u) − s). In our applications, X is typically a Banach space in which the studied problem is well-posed (or just the positive cone of such space), τ (u) is the maximal time of existence of the solution with initial data u and ϕ(u, t) is this solution at time t. Note that for most of our assertions, ϕ need not be continuous with respect to t at t = 0 so that we can also choose X = L∞ (Ω), for example. Given u ∈ X, the mapping ϕu : [0, τ (u)) : t → ϕ(u, t) is called the trajectory emanating from u. It is global if τ (u) = ∞. A point u ∈ X is called an equilibrium if τ (u) = ∞ and ϕ(u, t) = u for all t ≥ 0. We denote by S the set of all equilibria. If τ (u) = ∞, then we define the ω-limit set of ϕu by ω(ϕu ) = ω(u) := {v ∈ X : there exist tk → +∞ such that ϕ(u, tk ) → v as k → ∞}. It is easy to see that ω(u) =

2 3

{ϕ(u, t)}

s>0 t≥s

and ω(ϕ(u, t)) = ω(u) for all t > 0.

(53.1)

(53.2)

53. Appendix G: Dynamical systems

529

Proposition 53.2. Let τ (u) = ∞ and v ∈ ω(u). Then ϕ(v, t) ∈ ω(u) for all t ∈ [0, τ (v)). Proof. There exist tk → ∞ such that ϕ(u, tk ) → v. Given t ∈ [0, τ (v)), set τk := tk + t. Then ϕ(u, τk ) = ϕ(ϕ(u, tk ), t) → ϕ(v, t), hence ϕ(v, t) ∈ ω(u). Proposition 53.3. Assume τ (u) = ∞

and

3

{ϕ(u, t)} is relatively compact in X.

(53.3)

t≥0

Then τ (v) = ∞ for all v ∈ ω(u), ω(u) is compact, connected, nonempty, invariant (that is ϕ(ω(u), t) = ω(u) for all t > 0) and d(ϕ(u, t), ω(u)) → 0 as t → ∞. Proof. The set K := {ϕ(u, t) : t ≥ 0} is compact and ω(u) ⊂ K. The set ω(u) is closed due to (53.2), hence compact. Choose tk → ∞. Then {ϕ(u, tk )} is relatively compact, hence passing to a subsequence we may assume ϕ(u, tk ) → v. Consequently, v ∈ ω(u) and ω(u) is nonempty. Fix v ∈ ω(u). Proposition 53.2 guarantees ϕ(v, t) ∈ ω(u) for all t ∈ [0, τ (v)). Assume on the contrary that τ (v) < ∞. Choose tk ∈ [0, τ (v)), tk → τ (v). Then {ϕ(v, tk )} is relatively compact and passing to a subsequence we may assume ϕ(v, tk ) → v∞ . Then τ (ϕ(v, tk )) = τ (v) − tk → 0 and τ (v∞ ) > 0 which contradicts the lower semicontinuity of τ . Consequently, τ (v) = ∞. Due to Proposition 53.2, in order to show the invariance of ω(u) it is sufficient to prove ω(u) ⊂ ϕ(ω(u), t) for all t > 0. (53.4) Fix v ∈ ω(u), t > 0 and tk → ∞ such that ϕ(u, tk ) → v. Set τk := tk − t. Passing to a subsequence we may assume ϕ(u, τk ) → w ∈ ω(u). Then ϕ(w, t) = ϕ lim ϕ(u, τk ), t = lim ϕ(u, tk ) = v, k→∞

k→∞

which proves (53.4). Next we show that d(ϕ(u, t), ω(u)) → 0 as t → ∞. Assume on the contrary that there exist tk → ∞ and ε > 0 such that d(ϕ(u, tk ), ω(u)) ≥ ε. Passing to a subsequence we may assume ϕ(u, tk ) → v ∈ ω(u) which yields a contradiction. For any s > 0, the set {ϕ(u, t) : t ≥ s} is connected and relatively compact, hence its closure is connected and compact. Due to (53.2), ω(u) is a decreasing intersection of connected compact sets, hence ω(u) is connected. A continuous function E : X → R is called a Lyapunov function for ϕ if E(ϕ(u, t)) ≤ E(u) for all u ∈ X and t ∈ [0, τ (u)).

530

Appendices

Proposition 53.4. Let E be a Lyapunov function and (53.3) be true. Then the limit e := limt→∞ E(ϕ(u, t)) exists and E(v) = e for all v ∈ ω(u). Proof. The function t → E(ϕ(u, t)) is nonincreasing and bounded since the set {ϕ(u, t) : t ≥ 0} is relatively compact. Hence e := limt→∞ E(ϕ(u, t)) exists. If v ∈ ω(u), then there exist tk → ∞ such that ϕ(u, tk ) → v. Consequently, E(ϕ(u, tk )) → E(v) = e. A Lyapunov function E is called a strict Lyapunov function if the following condition is satisfied: If E(ϕ(u, t)) = E(u) for all t ∈ [0, τ (u)), then u is an equilibrium. The following two useful results are versions of Lasalle’s invariance principle. Proposition 53.5. Let E be a strict Lyapunov function and (53.3) be true. Then S is a closed nonempty set and d(ϕ(u, t), S) → 0 as t → ∞. In particular, ω(u) consists of equilibria. Proof. The continuity of ϕ guarantees that S is closed and Proposition 53.3 shows that ω(u) = ∅. Fix v ∈ ω(u) and t ≥ 0. Then τ (v) = ∞ and ϕ(v, t) ∈ ω(u) due to Proposition 53.3, hence Proposition 53.4 implies E(ϕ(v, t)) = E(v). Consequently, v ∈ S. Proposition 53.6. Assume τ (u) = ∞, tk → ∞ and ϕ(u, tk ) → v. Let there exist a strict Lyapunov function E. Then v ∈ S. Proof. The proof of Proposition 53.4 shows that e := limk→∞ E(ϕ(u, tk )) exists and E(v) = e. Fixing t ∈ [0, τ (v)), the continuity of ϕ implies ϕ(u, tk +t) → ϕ(v, t). As above, e˜ := limk→∞ E(ϕ(u, tk + t)) exists and E(ϕ(v, t)) = e˜. Fixing k ∈ N there exists j ∈ N such that tk+j ≥ tk + t ≥ tk , hence E(ϕ(u, tk+j )) ≤ E(ϕ(u, tk + t)) ≤ E(ϕ(u, tk )), thus e˜ = e and E(ϕ(v, t)) = E(v). Consequently, v ∈ S. Example 53.7. Consider problem (17.1) with p > 1 and λ ∈ R, and let q ∈ [max(qc , p + 1), ∞). By Example 51.28, we know that this problem generates a dynamical system on the space X := H01 ∩Lq (Ω). Moreover, by (17.7) in Lemma 17.5, the energy functional E(u) defined in (17.6) is a strict Lyapunov functional. Now assume Ω bounded and let u0 ∈ L∞ (Ω) be such that Tmax (u0 ) = ∞ and supt>0 u(t) ∞ < ∞. Then, as a consequence of parabolic estimates and Proposition 53.5, for each τ > 0, the set {u(t) : t ≥ τ } is relatively compact in X and the ω-limit set ω(u0 ) (in the X-topology) is nonempty and consists of (classical) equilibria. Moreover, by smoothing effects, the convergence in the definition of ω(u0 ) actually takes place (for instance) in C 1+β (Ω) for each β ∈ (0, 1). By similar arguments, the above facts remain true for the more general problem (14.1) with f ∈ C 1 if we replace the last integral in (17.6) by Ω F (u) dx, where

53. Appendix G: Dynamical systems

531

u F (u) = 0 f (s) ds, and X by H 1 ∩ C0 (Ω). (Note that, as far as Propositions 53.3– 53.6 are concerned, one could alternatively work with X := H 1 ∩L∞ (Ω): Although the continuity property at t = 0 in Definition 53.1(i) is not true, this is of no importance in those results.) Under an additional time monotonicity assumption, one obtains the convergence of the trajectory to a single equilibrium. See Remark 19.13 for other conditions guaranteeing convergence in the case of dynamical systems generated by parabolic differential equations. Proposition 53.8. Let (X, ≤) be an ordered Banach space with a closed positive cone X + := {u ∈ X : u ≥ 0}. Assume that (53.3) is true and the trajectory ϕu is nondecreasing, that is ϕ(u, t1 ) ≤ ϕ(u, t2 ) whenever t1 ≤ t2 . Then ω(u) is a singleton contained in S. Proof. Proposition 53.3 guarantees that ω(u) is nonempty and invariant. Let v 1 , v 2 ∈ ω(u). Then there exist t1k → ∞ and t2k → ∞ such that ϕ(u, tik ) → v i , i = 1, 2. Without loss of generality we may assume t1k < t2k < t1k+1 for all k. Then ϕ(u, t1k ) ≤ ϕ(u, t2k ) ≤ ϕ(u, t1k+1 ) and passing to the limit we obtain v 1 ≤ v 2 ≤ v 1 , hence v 1 = v 2 . Consequently, ω(u) is a singleton. Since it is an invariant set, we have ω(u) ⊂ S. Remark 53.9. For many dynamical systems generated by parabolic differential equations, the compactness assumption (53.3) in Proposition 53.8 can be replaced by a weaker boundedness assumption. In fact, the monotonicity of the solution in time usually enables one to pass to the limit and conclude that the limit is a stationary solution. For example one can often use the following lemma. In the case of monotonicity in space, similar arguments are used in the proofs of Theorems 8.3 and 21.10(ii). Lemma 53.10. Let f : R → R be locally H¨ older continuous, let Ω be an arbitrary domain in Rn , and set Q := Ω × (0, ∞). Let u ∈ C 2,1 (Q) satisfy ut − ∆u = f (u),

(x, t) ∈ Q,

(53.5)

and sup |u| < ∞. Q

Assume that ut ≥ 0 in Q, and let v(x) := limt→∞ u(x, t). Then v is a bounded classical solution of −∆v = f (v), x ∈ Ω. (53.6) Proof. Let wj (x, t) = u(x, t + j). The Lp and Schauder interior parabolic estimates guarantee that the sequence {wj } is relatively compact in C 2,1 (K) for each compact subset K of Q. It follows that some subsequence of wj converges to a

532

Appendices

classical solution w = w(x, t) of (53.5). But it is clear that v(x) = w(x, t) for each (x, t) ∈ Q. The conclusion follows. Alternative proof. Let ϕ ∈ D(Ω). For each t > 0, multiplying by ϕ and integrating by parts yields

Ω

t+1

uϕ dx + t

t

t+1

u∆ϕ + f (u)ϕ dx ds = 0.

Ω

Passing to the limit by dominated convergence as t → ∞, we obtain

v∆ϕ + f (v)ϕ dx = 0. Ω

It follows that v is a distributional solution of (53.6), hence a classical solution by standard elliptic regularity results (cf. Remarks 47.4). Let us ∈ S. The domain of attraction of us is the set D = D(us ) := {u : τ (u) = ∞ and ϕ(u, t) → us as t → ∞}. We say that us is asymptotically stable if D(us ) contains a neighborhood of us . If us is asymptotically stable, then D(us ) is obviously open. If u ∈ ∂D(us ), then the continuity of ϕ implies ϕ(u, t) ∈ ∂D(us ) for all t ∈ (0, τ (u)). Let u, v ∈ S, u = v. A function ψ : R → X is called a connecting (or heteroclinic) orbit between u and v if limt→−∞ ψ(t) = u, limt→+∞ ψ(t) = v and ψ(t) = ϕ(ψ(s), t − s) for all −∞ < s < t < ∞.

54. Appendix H: Methodological notes In this orientation section, we shall summarize the different methods employed in this monograph for each of the main questions that we address. Of course, there exist other important methods which are frequently exploited for related questions and which are not represented in this book for various reasons. For example, in the elliptic part we only use the simplest variational methods to prove those existence results which are needed in the parabolic part and we do not pay attention to linking or concentration compactness. Notice also that our summary does not even contain all important methods used in this monograph. For example, matched asymptotics is used in Section 29 in the study of decay and grow-up rates which is not a central theme for us. On the other hand, matched asymptotics plays a crucial role in several fundamental papers devoted to our main questions (like blow-up rates or blow-up profiles, see [277], [522], [275], [276], for example) but the corresponding rigorous proofs are too long and technical and lie beyond the scope of this book.

54. Appendix H: Methodological notes

533

Before providing the summary, let us briefly classify the main tools and techniques that are used throughout the book. We point out that, of course, many of these techniques are not specific to the field of superlinear elliptic and parabolic problems but are classical in various areas of PDE’s. A. Background tools A1. Tools from Functional Analysis (functional spaces and inequalities, interpolation) A2. Tools from ODE’s (differential and integral inequalities, phase plane analysis) A3. Linear elliptic and parabolic estimates (often used to obtain compactness properties) A4. Tools from Dynamical Systems (ω-limits, Lyapunov functionals) B. Main classes of techniques B1. Comparison techniques (based on maximum principles, including moving planes and zero-number) B2. Test-function and multiplier techniques7 (in particular including variational and energy methods). In the parabolic case this often leads to a differential inequality for a functional of the solution B3. Semigroup techniques (relying on the variation-of-constants formula) B4. Rescaling procedures (often leading to the use of a nonlinear Liouville-type theorem via a contradiction argument)8 B5. Bootstrap and iteration procedures C. Some other techniques C1. Changes of dependent and/or independent variables (cf., e.g., similarity variables, Hopf-Cole transformation, conversion to a problem with absorption, . . . ) C2. Differentiation of the PDE (cf., e.g., Bernstein-type techniques, auxiliary functions J, . . . ) C3. Monotonicity techniques: use of monotonicity properties of solutions (in time or in space — usually obtained via a maximum principle), monotone approximation (cf. complete blow-up, threshold trajectories) C4. Doubling arguments 7 By a test-function technique, we usually understand the space or space-time integration of the PDE after multiplication by a function independent of the solution itself, such as the first eigenfunction for instance. In a multiplier technique, the function may depend on the solution, e.g. a power of the solution. 8 Another aspect of the concept of scaling is the existence of self-similar solutions — cf. also similarity variables in C1.

534

Appendices

C5. Duality arguments We now turn to the more detailed summary of methods. For a given question, the choice of the applicable (and more appropriate) methods will depend on the particular properties of the problem: scale invariance, variational structure, monotonicity, convexity or boundedness of the domain, regularity assumed on solutions, . . . For some partial comparative discussion, see e.g. the beginning of sections 10– 13; Remark 26.5 and the end of Remark 26.12; the paragraph after Theorem 27.2; Remarks 31.5, 31.18(i) and the beginning of Subsection 31.4. We stress that the following list, which is mainly intended as a help and a guideline for readers, is necessarily schematic. Certain proofs may sometimes involve combinations of several methods, or some ad hoc arguments which do not appear in the list. On the other hand, some items below may partially overlap. The places where each method is used are mentioned in italic between brackets. I. METHODS FOR ELLIPTIC PROBLEMS M1. Methods to prove existence of solutions M1.1. Variational methods (a) Minimisation under constraint [Section 6] (b) Minimax methods [Section 7] M1.2. ODE methods [Section 9] M1.3. A priori estimates and topological degree argument [Corollary 10.3 and cf. M4] M1.4. Dynamical methods (a) Method based on a priori estimates of global solutions and threshold trajectories [Remark 28.8(ii)] (b) Stabilization of monotone bounded solutions [Theorem 43.1(iii)] M2. Methods to prove nonexistence of solutions Note: the methods which are mainly motivated by Liouville-type results appear only in M8.1–M8.3 below. M2.1. Variational identities of Pohozaev-type [Corollary 5.2, Proposition 25.4, Theorem 31.3] M2.2. Multiplication by the first eigenfunction [Remark 6.3] M2.3. ODE techniques [Section 9] Note: the methods in M2.1 and M2.3 may be combined with symmetry results [Remark 6.9(i)] M2.4. Maximum principle [Proposition 40.8]


535

M3. Methods to study regularity and singularities of solutions M3.1. To prove regularity: bootstrap procedures using linear elliptic estimates (W 2,p , Lp -Lq , Lpδ -Lqδ , . . . ). This can be combined with test-function, cut-off or truncation arguments [Propositions 3.3, 3.5, and see M4.2 below] M3.2. To establish pointwise singularity estimates (a) Integral estimates from M8.3 below, combined with a bootstrap procedure [Theorems 4.1 (psg < p < pS ), 8.7] (b) Rescaling, Liouville-type results and doubling arguments [Remark 8.8(i)] (c) Combination of the following three ingredients: the characterization of nonnegative distributions with point support (the nonnegativity being obtained by truncation and test-function techniques); a comparison argument involving the Newton potential; a bootstrap procedure [Theorems 4.2, 4.1 (1 < p < psg )] M3.3. To produce singular solutions (a) Method based on the construction and pointwise estimates of a singular solution of the linear Laplace equation [Theorems 11.5, 31.16] (b) Explicit singular solutions [Remarks 3.6(ii), 31.19, Formula (40.20)] (c) ODE methods [Remark 3.6(ii)] M4. Methods to prove a priori estimates M4.1. Method of Hardy-Sobolev inequalities. Also: variant based on the use of a singular test-function [Section 10, Remark 31.18(i)] M4.2. Bootstrap in Lpδ -spaces. Alternate bootstrap in the case of systems [Section 11, Subsection 31.4] M4.3. Method based on rescaling and Liouville-type theorems [Section 12, Subsection 31.3] M4.4. Method of moving planes and Pohozaev-type identities [Section 10, Theorem 31.2] II. METHODS FOR PARABOLIC PROBLEMS M5. Methods for local well-posedness M5.1. Local existence-uniqueness Note: we are mainly concerned with irregular initial data. The case of smooth data is standard. (a) Fixed-point in a metric space of functions of t, with a weight vanishing at t = 0, using Lp -Lq -estimates for the heat semigroup (variants: Lpδ spaces or uniformly local spaces, instead of Lp ) [Theorems 15.2, 15.9, 32.1(i), Remark 43.14(b)]

536

M5.2.

M5.3.

M5.4. M5.5.

Appendices

(b) Similar to M5.1(a), with Lp -spaces replaced by a scale of interpolationextrapolation spaces [Theorem 51.25] (c) Improvement of the uniqueness class (without temporal weight): one shows that any solution actually belongs to the fixed-point space; this is achieved by a time-shift and continuous dependence argument [same as M5.1(a)-M5.1(b)] Local nonexistence of positive solutions: for suitable singular initial data, contradiction between two pointwise estimates for the “free” part e−tA u0 (namely: a lower estimate for the linear heat equation as t → 0, and an upper a priori estimate depending on the nonlinear equation; see M8.5 for a closely related argument and more details) [Theorems 15.3, 15.10, 32.1(ii)] Local nonuniqueness (a) Nonuniqueness for zero initial data: construction of a forward selfsimilar solution with exponential decay in space by ODE (shooting) methods [Remarks 15.4(ii), 40.11(a)] (b) Construction of a singular stationary solution (which coexists with a classical solution for t > 0) [Remark 15.4(iii)] (c) Nonuniqueness for general initial data: method based on concentrated perturbations of an initial data, continuity of the existence time and universal bounds [Proposition 28.1] Regularity and smoothing: bootstrap procedure using (e.g.) Lp -Lq -estimates for the heat semigroup [Theorems 15.2, 15.9, 15.11, 43.13]. Continuation properties (in particular: uniform bounds from Lq -bounds) (a) Consequence of well-posedness in M5.1(a)–M5.1(b) and smoothing property in M5.4. Also, a lower estimate of the norm of u(·, t) near the blow-up time can be directly deduced from the fixed-point argument [Remark 16.2(iii), Theorem 33.5] (b) Moser-type iteration [Theorems 16.4, 33.5, Remark 33.6] (c) Variation-of-constants formula combined with interpolation inequality and interpolation-extrapolation spaces [Proposition 51.34] (or just Lp spaces [Theorem 32.2] ) Note: in the case of systems, better results can be obtained by alternate use of each equation (d) Consequence of lower estimates on the blow-up profile (cf. M12.3(a)) Note: non-continuation can be shown as a consequence of upper estimates on the blow-up profile [Corollary 24.2, Theorem 44.6, Remark 44.8(c)] (e) Energy arguments [Proposition 16.3] (f) Gradient bounds (in particular via Bernstein techniques): see details in [Section 35]


537

M6. Methods to prove global existence (and also boundedness, decay, stability) M6.1. Multiplier, energy and Lyapunov functional methods Note: the following three items partially overlap (a) Use of powers of the solution as multiplier, in combination with various functional inequalities [Theorems 19.3(i), 33.9(i), 36.4(i), Remark 40.11(b), Theorems 43.1, 44.5(i)] (b) Potential well method [Theorem 19.5(i)] (c) Use of a Lyapunov functional [Theorems 33.5, 33.18(ii), 40.7(i)] M6.2. Comparison methods (a) Supersolutions with separated variables; spatially homogeneous supersolutions [Theorems 19.2, 32.5(ii), 43.1, 46.1(ii)] (b) Stationary supersolutions (and families thereof); singular steadystates and their perturbations; quasi-stationary supersolutions [Theorem 19.15(ii), Remark 19.14, Theorems 20.5, 29.1 32.5(iii), 36.1(ii), 36.4(i), 37.2, 40.7(iii), 44.17(i)] (c) Supersolutions involving the heat semigroup (possibly self-similar) [Theorem 20.2, Remark 20.4(i), Theorems 20.6, 20.11, 32.5(ii) and 37.4(ii)] (d) Self-similar supersolutions [Theorem 20.6, Section 45] (e) Traveling wave supersolutions [Theorem 36.7] (f) Intersection-comparison with radial steady-states (to show global existence of a threshold solution) [Theorem 22.9] M6.3. Variation-of-constants formula and semigroup estimates (e.g., Lp -Lq or exponential decay) [Remark 19.4(b), Theorems 20.15, 33.1, 40.7(i), 40.10(i), Subsection 51.3] Note: sometimes combined with fixed-point arguments in spaces with temporal weight [Theorem 20.19, Corollary 20.20] M6.4. Forward similarity variables (combined with construction of stable manifolds) [Proposition 20.13] M6.5. Duality method [Theorem 33.2, Remarks 33.4, 33.6, 33.15] M6.6. Gradient estimates [Proposition 40.5, Theorem 40.7(iii), Remark 40.11(d)]

M7. Methods to prove blow-up (in finite -or sometimes infinite- time) Note 1: the methods in M7.1–M7.2 lead to a differential inequality for some functional of u(·, t). Note 2: the methods which are mainly motivated by (blow-up) results of Fujitatype do not appear here (see M8.4(a), M8.5, M8.6).

538

Appendices

M7.1. Eigenfunction method [Theorems 17.1, 17.3, 32.5(i), 33.16, 36.1(i), Remarks 40.4(i) and (ii), Theorem 46.1] Note: other test-functions independent of u can sometimes be used [cf. Theorem 43.1(i)] M7.2. Energy and multiplier methods (a) Energy and H¨ older’s inequality (in bounded domains) [Theorems 17.6, 44.14] (b) Energy and concavity argument (in general domains) [Theorem 17.6] (c) Potential well method [Theorem 19.5(ii)] (d) Use of a power of the solution as test-function, in combination with various functional inequalities [Theorems 40.2, 41.1] M7.3. Comparison methods (a) Blowing-up self-similar subsolutions [Theorems 36.2, Section 45] (b) Other forms of subsolutions (perturbation of singular or regular steady states, expanding waves, traveling waves, quasi-stationary, . . . ) [Theorems 29.1, 36.4(ii), Lemma 36.6, Theorem 41.1, Remark 40.4(i), Theorem 44.17(ii)] (c) Blow-up above a positive equilibrium [Theorem 17.8, Proposition 19.11] (d) Comparison between domains [Remark 17.14] (e) Cf. M11.2(a) [Theorem 23.5] Note: in spatially nonlocal problems, this is sometimes combined with the method in M12.1(a) to obtain preliminary estimates on the nonlocal term [Theorem 44.5(ii)] M7.4. Use of scaling properties of the equation (e.g. to prove blow-up for initial data with slow decay at infinity) (a) Rescaled eigenfunctions [Theorem 17.12] (b) Rescaled subsolutions [Remarks 17.13(i), 36.3(iii), Theorems 19.3(ii), 36.4(ii)] M7.5. Construction of explicit blowing-up solutions (often under self-similar form, or by solving an ODE) [Theorems 33.9(ii), 33.12, 33.18(i)] M7.6. Use of dynamical systems arguments (ω-limits via a strict Lyapunov functional, or via monotonicity) combined with the absence of steady-states (may lead to blow-up in finite or infinite time) [Remark 19.14, Theorems 28.7(iv), 33.14] M7.7. Direct estimation via integration in space-time parabolas and Sobolev inequality (to prove growth of mass for a Cauchy problem) [Theorem 40.10(ii)] M8. Methods to prove nonexistence in Liouville and Fujita-type results Note: the methods in M8.1–M8.3 concern both elliptic and parabolic problems


539

M8.1. Rescaled test-functions (a) Spatial test-functions [Theorems 8.4, 31.12] in the elliptic case; [Remark 18.2(i), Theorems 32.7, 37.4] in the parabolic case (where this leads to differential inequalities) (b) Space-time test-functions [Theorems 18.1(i), 37.1] M8.2. Moving plane methods (a) Via symmetry, using the Kelvin transform (elliptic; case of the whole space) [Theorem 8.1] (b) Via symmetry and reduction to a one-dimensional problem on a halfline (elliptic; case of a half-space) [Theorem 8.2] (c) Via monotonicity and reduction to an (n − 1)-dimensional problem in the whole space (elliptic and parabolic; case of a half-space) [Theorems 8.3, 21.8, 31.10] M8.3. Integral estimates, obtained by using Bochner’s identity, power change of dependent variable, and multipliers involving powers of u and cut-offs [Propositions 8.6, 21.5] M8.4. Comparison methods (a) Families of blowing-up self-similar subsolutions [Section 45] (b) Intersection-comparison with radial steady-states [Theorem 21.1] M8.5. Method based on the variation-of-constants formula (for Fujita-type results) [Theorem 18.3] More precisely, a contradiction is obtained by comparing two pointwise estimates for the “free” part e−tA u0 of the solution: the lower estimate from the linear heat equation as t → ∞, and an upper a priori estimate depending on the nonlinear equation [Lemma 15.6]. The latter is proved by taking the action of the heat semigroup on the variation-of-constants formula.9 In the critical case, the necessary additional information is provided by an L1 lower bound based on convolution properties of Gaussians. M8.6. Forward similarity variables [Lemma 18.4] M9. Methods to prove boundedness of global solutions and parabolic a priori estimates Note: the methods in M9.1(b), M9.2(b), M9.3(a) here yield only boundedness of global solutions 9 For Fujita-type problems, the methods in M8.1(a) and M8.5 are essentially equivalent. In fact, in M8.1(a), one also compares a lower asymptotic estimate with an upper a priori bound. The latter follows from differential inequalities obtained by multiplying with rescaled Gaussian test-functions, and these Gaussians are nothing but the heat kernel with time as a parameter. However the argument in M8.1(a) requires more regularity on the solution. Alternatively, the upper a priori bound can be obtained by a subsolution argument (see Remark 15.7).

540

Appendices

M9.1. Rescaling methods (a) Method based on rescaling, elliptic Liouville-type theorems and energy [Theorem 22.1] (b) Method based on rescaling and intersection-comparison, using the infinite intersection property of the singular and regular steady-states in Rn [Remark 23.13] (c) Cf. M10.2(a) [Theorem 38.1] M9.2. Energy methods (a) Method based on energy estimates and on a bootstrap argument using interpolation and maximal regularity [Theorem 22.1, Proposition 22.11, Remark 44.15] (b) Method based on energy estimates in forward similarity variables, and on a measure argument [Lemma 18.4] M9.3. Methods based on the maximum principle (a) Intersection-comparison with a backward self-similar solution and with the singular steady-state [Theorem 22.4] (b) Use of a monotonicity property: the solution becomes increasing in time if it reaches a sufficiently high level (for a nonlocal problem) [Proposition 43.16] M10. Methods to prove universal bounds of positive solutions and initial blow-up rates M10.1. Methods based on smoothing estimates (a) Smoothing in Lpδ -spaces (using integral bounds obtained by the eigenfunction method, and possibly combined with a priori estimates) [Theorems 26.1, 26.14] (b) Smoothing in Lp -spaces (using integral bounds obtained by using a singular test-function, or by the eigenfunction method, and combined with a priori estimates) [Theorems 26.1, 43.15] (c) Smoothing in uniformly local Lebesgue spaces (using integral bounds obtained by the eigenfunction method) [Theorem 26.13] M10.2. Rescaling methods (a) Method based on a doubling lemma and parabolic Liouville-type theorems [Theorems 26.8, 26.9, 38.1] (b) Method based on energy, measure arguments, and elliptic Liouvilletype theorems. [Theorem 26.6, Remark 26.7] M10.3. Methods based on space-time integral estimates (a) Via Moser-type iteration or Harnack inequality [Theorem 26.13(i)] (b) Via the method in M8.3 combined with Harnack inequality [Theorem 26.8]


541

M11. Methods to establish blow-up rates M11.1. Lower estimates (a) Comparison with solutions of the ODE [Proposition 23.1, Remark 38.2(i)] (b) Differential inequality obtained by considering points of maxima of u(·, t) [Proposition 23.1, Theorems 32.9, 44.2(i), 44.17(ii), 46.4(i), 44.2(i), Proposition 44.3(i), Theorem 44.17(ii)] (c) Variation-of-constants formula and use of the doubling time of u(t) ∞ [Remark 23.2(ii)] (d) Regularity estimates applied to the equation for ut (for the GBU problem) [Theorem 40.18] (e) Method using the intersections of the solution with a steady-state, and the boundedness of ut (for the GBU problem) [Theorem 40.19] M11.2. Upper estimates10 (a) Maximum principle applied to an auxiliary function J (for timeincreasing solutions) [Theorems 23.5, 32.9, Remark 38.2(ii), Theorem 46.4]. See also [Theorem 40.21] for a different type of auxiliary function in the GBU problem. (b) Methods based on backward similarity variables (b)-1 Via rescaling, elliptic Liouville-type theorems and energy [Theorem 23.7] (b)-2 Via localized energy estimates and bootstrap [Remark 23.14(i)]

(c)

(d) (e) (f)

Note: the last two methods are similar to M9.1(a) and M9.2(a), respectively, the question being equivalent to the boundedness of global solutions for the equation in similarity variables Method based on rescaling and intersection-comparison, using the infinite intersection property of the singular and regular steady-states in Rn [Theorem 23.10] Cf. M10.2(a) [Theorems 26.8, 38.1; see also Remarks 26.12, 32.12(i), Proposition 44.3(ii)] Cf. M10.3(a) [Remark 32.12(i)] Cf. M12.4 [Subsection 43.2, Theorem 44.2(i)]

M12. Methods to study blow-up sets and profiles M12.1. Methods based on the maximum principle (a) Maximum principle applied to an auxiliary function J (to obtain single-point blow-up and upper profile estimates for radial nonin10 One could alternatively classify the methods for upper blow-up estimates between: those using scaling and energy (M11.2(b)), those using scaling without energy (M11.2(c)– M11.2(d)), and those using neither energy nor scaling (M10.3(a), M11.2(a), M12.4).

542

Appendices

creasing solutions) [Theorem 24.1, Remark 32.12(ii), Theorems 39.7, 44.2(iii), 44.6 (ii), Remark 46.5] Note: sometimes combined with a bootstrap argument [cf. Theorem 39.1] (b) Moving plane method (to prove compactness of the blow-up set) [Remark 24.6(iv)] (c) Sub-/supersolutions of blowing-up barrier type, using the notion of sub-/super-standard functions (to obtain the blow-up behavior in the boundary layer for spatially nonlocal problems) [Theorem 43.10] (d) Bernstein-type techniques (for blow-up profile estimates in the GBU problem) [Remark 40.17, 41.4] M12.2. Method of backward similarity variables (a) Combined with weighted energy and dynamical systems arguments (to show asymptotically self-similar blow-up behavior; to exclude blow-up at a given point and prove compactness of the blow-up set) [Theorems 25.1, 24.5] Note: sometimes combined with comparison and cut-off arguments [Theorem 25.3] (b) Combined with linearization and spectral techniques (to obtain refined blow-up estimates and classification of profiles) [Remark 25.8] (c) Construction of exact backward self-similar solutions by ODE (phase plane) methods [Proposition 22.5, Remarks 39.8(i), (iii) and (iv)] M12.3. Methods based on ODE’s in space (a) ODE energy estimate and use of the point of half-maximum of u(·, t) (to obtain lower profile estimates for radial nonincreasing solutions) [Theorems 24.3, 39.2, Remark 32.12(ii)] (b) Differential inequalities in space, relying on the boundedness of the time derivative (for blow-up profile estimates in the GBU problem) [Theorem 40.14] M12.4. Method based on eigenfunction arguments, one-sided estimates of ∆u (via the maximum principle), and the mean-value inequality for subharmonic functions (to obtain the blow-up rate, set and profile for spatially nonlocal problems) [Subsection 43.2, Theorem 44.2(i)]

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List of Symbols Standard function spaces are defined in Preliminaries.

BR , BR (x), B(x, R) . . . . . . . . . . . 1 n−1

δy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

......................... 1

pS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

χM . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

D ⊂⊂ D . . . . . . . . . . . . . . . . . . . . . 1

2∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

s + , s− . . . . . . . . . . . . . . . . . . . . . . . . 1

psg . . . . . . . . . . . . . . . . . . . . . . . . . . 11

R+ . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

U∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

δ(x) . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Rn+ . . . . . . . . . . . . . . . . . . . . . . . . . . 36

S

ν(x) . . . . . . . . . . . . . . . . . . . . . . . . . . 1

pJL . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Q T , ST , P T . . . . . . . . . . . . . . . . . . . 1

pBT . . . . . . . . . . . . . . . . . . . . . . . . . 56

X . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

qc . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

p . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2

pF . . . . . . . . . . . . . . . . . . . . . . . . . .101

X → Y . . . . . . . . . . . . . . . . . . . . . . . 2

pB . . . . . . . . . . . . . . . . . . . . . . . . . 137

X → → Y . . . . . . . . . . . . . . . . . . . . 2 . X = Y . . . . . . . . . . . . . . . . . . . . . . . .2

pL . . . . . . . . . . . . . . . . . . . . . . . . . .167

L(X, Y ) . . . . . . . . . . . . . . . . . . . . . . 2

X, X+ (Chapter III). . . . . . . .272

· k,p . . . . . . . . . . . . . . . . . . . . . . . . 2

X, X+ (Chapter IV) . . . . . . . . 314

· p . . . . . . . . . . . . . . . . . . . . . . . . . 2

Lqg . . . . . . . . . . . . . . . . . . . . . . . . . .434

· p,δ . . . . . . . . . . . . . . . . . . . . . . . . 3

Hgk . . . . . . . . . . . . . . . . . . . . . . . . . 434

Lpul . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

GΩ (x, y, t), G(x, y, t) . . . . . . . . 440

· p,ul . . . . . . . . . . . . . . . . . . . . . . . 3

G(x, t), Gt (x). . . . . . . . . . . . . . .440

·α . . . . . . . . . . . . . . . . . . . . . . . . . . 4

KΩ (x, y), K(x, y) . . . . . . . . . . . 440

· 2,1;p . . . . . . . . . . . . . . . . . . . . . . .4

· A . . . . . . . . . . . . . . . . . . . . . . . 466

[·]α;Q . . . . . . . . . . . . . . . . . . . . . . . . . 4

ω(−A), σ(−A). . . . . . . . . . . . . .467

| · |a;Q . . . . . . . . . . . . . . . . . . . . . . . . 4

Xθ . . . . . . . . . . . . . . . . . . . . . . . . . 467

λk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

(·, ·)θ , [·, ·]θ , (·, ·)θ,p . . . . . . . . . 467

ϕk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

| · |α . . . . . . . . . . . . . . . . . . . . . . . . 468

e

−tA

..........................5

B(u0 ) . . . . . . . . . . . . . . . . . . . . . . 190

z[0,R] (ψ) . . . . . . . . . . . . . . . . . . . . 526

Index Lp -Lq -estimates – elliptic, 431 – parabolic, 440 Lp -estimates – elliptic, 429 – parabolic, 438 Lpδ -Lqδ -estimates, 451 Lpδ -spaces, 2, 61, 86, 203, 266, 447 ω-limit set, 122, 123, 138, 197, 199, 234, 237, 244, 312, 528

a – – –

priori estimates, a priori bounds applications of, 230 elliptic, 55, 61, 65, 68, 251 of global solutions (parabolic), 161, 166, 202, 280, 338, 359, 395, 412 – parabolic, 83, 150 absorption, 203 analytic semigroup, 466 annulus, 21, 25, 26 auxiliary function J – for blow-up rate, 180, 286, 340, 407, 424 – for space profile, 190, 342, 344, 348, 400, 409

barrier, 316, 323, 388, 452 Bernstein technique, 316, 363, 367, 376 bifurcation diagram, 26, 28, 55 blow-up – above a positive equilibrium, 96, 151 – at infinity, 193 – complete, 176, 219

– – – – – – –

diffusion-induced, 301 for slow decay initial data, 98, 321 global, 194, 219, 381, 398, 413 incomplete, 227, 229 nonsimultaneous, 287 regional, 194, 354 single-point, 190, 193, 194, 286, 302, 348, 354, 375, 398, 403, 427 blow-up criterion, 91, 94, 96, 98, 319, 321 blow-up profile, 190, 201, 274, 286, 348, 364, 398, 403, 427 – uniform, 381, 398 blow-up rate, 177, 195, 283, 286, 338, 367, 381, 398, 404, 418, 424 – applications, 189 – initial, 202 – refined estimate, 180 – type I/type II, 179, 184, 340 – universal estimate, 209, 211, 338 blow-up set, 190, 286, 302, 329, 348, 375, 381, 398, 403, 413, 427 – regularity of, 194 blow-up time, see existence time bootstrap – alternate (elliptic systems), 266 – elliptic, 9, 11, 14, 17, 39, 61, 72, 258, 293, 431 – parabolic, 81, 83, 163, 195, 235, 348, 352, 475, 493, 502, 506 boundary layer, 373, 381, 418 bounded imaginary powers, 468, 491

cap, 45 Carathéodory function, 7

580

Index

comparison principle, 315, 507, 509 – nonlocal problems, 377, 404, 414, 419, 523 – systems, 272, 288, 523 concavity method, 94 continuation after blow-up, 227, 373 continuation property, 89, 273 continuous dependence, 471, 473, 476, 478, 498 convection term, 313 convex – domain, 25, 68, 179, 180, 193, 195, 211, 354 – nonlinearity, 92, 120, 121, 125, 221 critical exponents, 314, 348, 360 – for blow-up, 319, 321, 323, 330, 403, 404, 422 – for scaling properties, 330 – for well-posedness, 76, 86, 113, 190, 230, 394 – Fujita, 101, 334, 418 – Sobolev, 7 critical value, 29, 31

dead-core, 341 decay – of initial data, 98, 131, 132, 137, 239 – time rate, 113, 132, 136, 142, 210, 239, 240, 246, 323, 358, 378, 441, 485, 489 deformation lemma, 30 degree – Brouwer, 69 – Leray-Schauder, 58 diffusion – eliminating blow-up, 125, 311 – large/small diffusion limit, 194 – preserving global existence, 288, 298 Dirac delta distribution, 13, 79

dissipative gradient term, 313 distance – parabolic, 213 – to the boundary (see also Lpδ -spaces), 1, 452 domain, 1 domain of attraction, 120, 174, 532 doubling lemma, 40, 211, 339, 401 duality argument, 290, 292, 293, 515 dynamical systems (see also ω-limit set), 197, 528 eigenfunction method, 91, 174, 216, 319, 321, 357, 422 eigenfunction, eigenvalue, 5, 26, 52, 99, 104, 137, 194, 437, 486 – first, 5 eigenvalue problem (nonlinear), 12, 28 energy, 29, 51, 89, 163, 165, 192, 206, 234, 347, 499 – functional, 9, 20, 93, 162, 395, 411 – space, 137 – weighted, 181, 199 existence time, 87, 471, 497, 498 – continuity, 176, 189, 224, 228, 360 – estimate, 92, 178, 179, 407 expanding wave, 326 exponential nonlinearity, 171 extremal point, 121 Fisher-KPP equation, 109 fixed-point theorem – Banach, 50, 79, 471, 480, 495 – Schauder, 398 Fujita-type theorem, 100, 215, 280, 330, 418 fundamental solution, 14, 432

Index

GBU, see gradient blow-up global existence – below the singular steady-state, 131 – for small data, 112, 129, 141 global solutions, 89, 528 – asymptotic profiles, 139, 362 – boundedness of, 161, 166, 169, 171, 188, 359, 360, 393, 412 – decay of, 137, 202, 210 – structure of, 120 – unbounded, 171, 184, 246, 328, 340, 360, 404, 414, 422 – weak, 227, 304 gradient blow-up (GBU), 100, 314, 355, 374 gradient estimate, gradient bound – elliptic, 43 – parabolic, 153, 189, 314, 323, 356, 363, 442, 452 gradient nonlinearities, 313 grow-up rates, 174, 246

Hölder continuous, 3 – locally, 3 – locally α-, 3 hair-trigger effect, 109 half-space, 36, 37, 67, 108, 156, 209, 260, 261, 265 homogeneous – initial data, 141, 147 – nonlinearities, 20 Hopf’s lemma – elliptic, 454, 507 – parabolic, 511 Hopf-Cole transformation, 317, 361

identity – Bochner, 40

581

– Pohozaev, 18, 22, 25, 26, 35, 68, 123, 174, 198, 235, 238, 253 – Rellich-Pohozaev type, 20 indefinite coefficients, 67, 230 inequality – Gagliardo-Nirenberg, 115 – H¨ older, 462 – Hardy, 465 – Hardy-Sobolev, 55, 271, 466 – Harnack, 216 – interpolation, 200, 362, 462 – Jensen, 462 – Poincaré, 113, 323, 325, 463 – singular Gronwall, 470, 505 – Sobolev, 462 – Sobolev, best constant, 23 – Young, 44, 462 – Young (for convolutions), 433 initial blow-up rate, 202 initial trace, 79 inradius, 113, 323, 463 instability – of equilibria, 113, 122 – of the blow-up rate, 340 instantaneous attractors, 203 interpolation, 163 – couple, 469 – embedding, 5 – functor, 467, 468, 488 – inequality, 200, 362, 462 interpolation-extrapolation spaces, 76, 90, 466 intersection-comparison, see zero number invariance principle, 530 invariant space, 141 isolated singularity, 12, 252

582

Index

Kelvin transform, 45, 47, 235, 259 kernel – (elliptic) Dirichlet Green, 5, 440, 454 – Dirichlet heat, 5, 78, 83, 86, 136, 219, 274, 440, 444, 453, 454, 459 – Gaussian heat, 5, 101, 102, 129, 131, 139, 289, 335, 360, 440, 460 Kranoselskii genus, 32

Lagrange multiplier, 20 Liouville-type theorem – elliptic equation, 36, 40, 65, 206 – elliptic inequality, 37 – elliptic system, 260, 266 – parabolic equation, 150, 210, 339 Ljusternik-Schnirelman, 26 localization of singularities, 64, 271 localization of trajectories, 210 localized nonlinearity, 393, 402 Lyapunov functional, 293, 529 – strict, 312, 359, 530

mass – dissipation of, 288 – growth of, 360 matched asymptotics, 247, 360 maximal regularity, 163, 165, 236, 470 maximum principle – elliptic, 507 – nonlocal, 344, 523 – parabolic, 509, 512 – strong (elliptic), 507 – strong (parabolic), 511 – systems, 522 – very weak (elliptic), 447 – very weak (parabolic), 450, 515 memory term, 421

minimax methods, 29 minimization, 236 model problem – elliptic, ix – parabolic, x mollifier, 431, 433, 509 monotonicity – of solutions in time, 174, 180, 191, 219, 283, 340, 424, 520, 531 – via moving planes, 38, 49, 157, 261 Moser-type iteration, 90, 216 mountain pass – energy, 117 – theorem, 29 moving planes – elliptic, 38, 45, 47, 49, 67, 68, 70, 253, 261 – parabolic, 157, 193, 235 multiplier argument (or technique), 533

Nehari functional, 116 Nemytskii mapping, 33, 474 Newton potential, see fundamental solution nonlinear boundary conditions, x, 225, 230, 469 nonuniqueness, 76, 77, 143, 230, 272

Ohmic heating, 412 operator – realization of, 2 optimal controls, 236

Index

Palais-Smale sequence, 29, 33 parabolic boundary, 1 peaking solution, 228 perturbation – of the model problem (elliptic), 35 – of the model problem (parabolic), 313, 319, 330, 338, 348 – singular, 297 population genetics, 108 potential well, 116 quenching, 100 radial – function, 2 – monotonicity, 518 – nonincreasing function, 2 realization of an operator, 2 reflection, 45 removable singularity, 13 rescaling method – elliptic, 36, 40, 65, 260, 263 – in similarity variables, 179, 183 – parabolic, 162, 206, 212, 215, 286, 339 scaling, 99, 327, 330, 391 – invariance, 116 scaling exponents, 252, 272 Schauder estimates – elliptic, 430 – parabolic, 438 Schwarz symmetrization, 22 self-adjoint operator, 104, 108, 312, 434, 439, 450, 456, 488, 491 self-similar – asymptotically, 142 – blow-up behavior, 189, 195, 347

583

– solution (backward), 167, 169, 197, 302, 342, 353, 355 – solution (forward), 77, 133, 141–143, 216, 330, 363 – subsolution, 321, 419 – supersolution, 131, 335, 420 semigroup – analytic, 466 – Dirichlet heat, 439 separation lemma, 96 similarity variables – backward, 180, 195, 341 – forward, 104 smoothing estimate, smoothing effect, 76 Sobolev hyperbola, 253, 260 solution – L1 , L1δ , very weak (elliptic), 8 – classical (elliptic), 7 – classical (parabolic), 75 – distributional (elliptic), 8 – distributional (parabolic), 101 – integral (parabolic), 77, 78, 79, 86, 227, 443 – periodic, 234 – singular (elliptic), 11, 50, 62, 131, 168, 185, 187, 267, 271, 364, 458 – strong (elliptic), 58, 65, 429 – strong (parabolic), 438, 473, 511 – variational (elliptic), 8, 56 – weak, weak-L1δ (parabolic), 78, 443 stability – of equilibria, 112, 113, 123, 131, 142, 308, 378, 485, 532 – of self-similar solutions, 142 stabilization, 123 Stampacchia method, 508, 512 standard function, 387 – sub-, 387

584

Index

– super-, 387 starshaped domain, 18, 19, 22, 26, 123, 161, 237, 253 subsolution, supersolution, 507 system – cooperative, 251, 522 – Gierer-Meinhardt, 297 – Lane-Emden, 251 – with mass-dissipation, 288

test-function, 40, 203 – Gaussian, 101, 281 – rescaled, 37, 102, 262, 330, 464 – singular, 59, 205 – torsion, 17, 124, 223, 280, 359, 379, 397, 447 test-function argument (or technique), 533 thermistor, 413 threshold solution, threshold trajectory, 171, 177, 228, 237, 239, 359 topology of the domain, 26 trajectory, 528 transition – from global existence to blow-up, 237 – from single-point to global blow-up, 398

uniformly local spaces, 3, 86, 217, 460 uniqueness – elliptic, 22, 25 – local (parabolic), 76, 87, 272, 398, 471, 495 universal bound, 193, 202, 230, 234, 280, 338, 359, 395

variation-of-constants formula, 77, 443, 446, 468 variational – identity, see identity (Pohozaev, Rellich-Pohozaev type) – methods, 20 – solution, see solution – structure, 55, 251, 258, 339, 411 viscous Hamilton-Jacobi equation, 314, 355 weighted spaces, see also Lpδ -spaces – Lebesgue, 104, 434, 491 – Sobolev, 104, 137, 239, 434, 491 well-posedness, 75, 89, 190, 230, 273, 314, 355, 394, 470, 495

zero number, 151, 168, 172, 185, 244, 526

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