Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States

738 Pages·2019·7.799 MB·English

by  Prof. Dr. Pavol Quittner, Prof. Dr. Philippe Souplet

Birkhäuser Advanced Texts
Basler Lehrbücher
Pavol Quittner
Philippe Souplet
Superlinear
Parabolic
Problems
Blow-up, Global Existence and
Steady States
Second Edition

BirkhäuserAdvancedTextsBaslerLehrbücher
Serieseditors
StevenG.Krantz,WashingtonUniversity,St.Louis,USA
ShrawanKumar,UniversityofNorthCarolinaatChapelHill,ChapelHill,USA
JanNekováˇr,Sorbonne Université,Paris,France
Moreinformationaboutthisseriesathttp://www.springer.com/series/4842
Pavol Quittner • Philippe Souplet
Superlinear Parabolic Problems
Blow-up, Global Existence and Steady States
Second Edition
Prof. Dr. Pavol Quittner Prof. Dr. Philippe Souplet
Department of Applied Mathematics Laboratoire Analyse Géométrie
and Statistics et Applications
Comenius University Université Paris 13 – Sorbonne Paris Cité
Mlynská Dolina CNRS UMR 7539
842 48 Bratislava 99, av. Jean Baptiste Clément
Slovakia 93430 Villetaneuse
France
ISSN 1019-6242 ISSN 2296-4894 (electronic)
Birkhäuser Advanced Texts Basler Lehrbücher
I SBN 978-3-030-18220-5 ISBN 978-3-030-18222-9 (eBook)
https://doi.org/10.1007/978-3-030-18222-9

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Contents
Introduction to the first edition........................................ xi
Introduction to the second edition ..................................... xiv
1. Preliminaries.......................................................... 1
I. MODEL ELLIPTIC PROBLEMS
2. Introduction........................................................... 7
3. Classical and weak solutions........................................... 7
4. Isolated singularities................................................... 12
5. Pohozaev’s identity and nonexistence results........................... 18
6. Homogeneous nonlinearities............................................ 21
7. Minimax methods..................................................... 30
8. Liouville-type results................................................... 36
1. Statements of the Liouville-type results............................. 37
2. Proofs of Liouville-type theorems for elliptic inequalities............. 40
3. Proof of Theorem 8.1(i) based on integral bounds, and related singu-
larity estimates..................................................... 42
4. Proofs of Liouville-type theorems based on moving planes........... 49
9. Positive radial solutions of ∆u+up =0 in Rn.......................... 59
10. A priori bounds via the method of Hardy-Sobolev inequalities.......... 65
11. A priori bounds via bootstrap in Lp-spaces............................. 71
δ
12. A priori bounds via the rescaling method............................... 75
13. A priori bounds via moving planes and Pohozaev’s identity............. 78
II. MODEL PARABOLIC PROBLEMS
14. Introduction........................................................... 85
15. Well-posedness in Lebesgue spaces..................................... 85
16. Maximal existence time. Uniform bounds from Lq-estimates............ 98
17. Blow-up............................................................... 104
18. Fujita-type results..................................................... 113
19. Global existence for the Dirichlet problem.............................. 124
1. Small data global solutions......................................... 124
Asymptotic stability of the zero solution......................... 124
Potential well theory............................................ 129
2. Structure of global solutions in bounded domains................... 135
3. Diffusion eliminating blow-up....................................... 141
v
vi Contents
20. Global existence for the Cauchy problem............................... 146
1. Small data global solutions......................................... 146
2. Global solutions with exponential spatial decay..................... 154
3. Asymptotic profiles for small data solutions......................... 156
4. Small data in scale-invariant Morrey spaces......................... 168
5. Blow-up for large Morrey norm and the separation problem......... 170
21. Parabolic Liouville-type results........................................ 173
22. A priori bounds........................................................ 188
1. A priori bounds in the subcritical case.............................. 189
2. Boundedness of global solutions in the supercritical case............. 194
3. Global unbounded solutions in the critical case...................... 200
4. Estimates for nonglobal solutions................................... 205
5. Partial results in the supercritical case for nonconvex domains....... 207
23. Blow-up rate.......................................................... 210
1. The lower estimate................................................. 210
2. The upper estimate: summary ..................................... 212
3. The upper estimate for time-increasing solution..................... 215
4. The upper estimate in the subcritical case: the method of backward
similarity variables ................................................. 217
5. The upper estimate for pS ≤p<pJL: intersection-comparison ...... 222
6. Some other applications of backward similarity variables ............ 227
24. Blow-up set and space profile.......................................... 233
1. Single-pointblow-upforradialdecreasingsolutionsandfirstestimates
of the space profile................................................. 233
2. Properties of the blow-up set....................................... 239
3. Refined single-point blow-up space profiles.......................... 242
25. Self-similar blow-up behavior.......................................... 244
1. Space-time profile in similarity variables in the subcritical case...... 244
2. Refined space-time blow-up behavior for radially decreasing solutions. 252
3. Other blow-up profiles in the sub- and supercritical cases............ 265
26. Universal bounds and initial blow-up rates............................. 269
27. Complete blow-up..................................................... 286
28. Applications of a priori and universal bounds........................... 300
1. A nonuniqueness result............................................. 300
2. Existence of periodic solutions...................................... 304
3. Existence of optimal controls....................................... 305
4. Transition from global existence to blow-up and stationary solutions. 306
5. Decay of the threshold solution of the Cauchy problem.............. 311
6. Parabolic Liouville-type theorems for radial solutions................ 318
29. Decay and grow-up of threshold solutions in the super-supercritical case 320
Contents vii
III. SYSTEMS
30. Introduction........................................................... 327
31. Elliptic systems........................................................ 327
1. A priori bounds by the method of moving planes and Pohozaev-type
identities........................................................... 329
2. Liouville-type results for the Lane-Emden system................... 336
2a. Liouville-type results for other systems.............................. 341
3. A priori bounds by the rescaling method............................ 343
4. A priori bounds by the Lp alternate bootstrap method.............. 346
δ
32. Parabolic systems coupled by power source terms...................... 352
1. Well-posedness and continuation in Lebesgue spaces................. 353
2. Blow-up and global existence....................................... 358
3. Fujita-type results.................................................. 360
4. Blow-up asymptotics............................................... 364
33. The role of diffusion in blow-up........................................ 369
1. Diffusion preserving global existence................................ 370
Systems with dissipation of mass................................ 370
Systems of Gierer-Meinhardt type............................... 384
2. Diffusion inducing blow-up.......................................... 389
Systems with dissipation of mass and unequal diffusions.......... 389
Systems with dissipation of mass, equal diffusions and mixed
boundary conditions......................................... 394
Systems with equal diffusions and homogeneous Neumann bound-
ary conditions................................................ 397
Diffusion-induced blow-up for other systems..................... 400
3. Diffusion eliminating blow-up....................................... 402
IV. EQUATIONS WITH GRADIENT TERMS
34. Introduction........................................................... 405
35. Well-posedness and gradient bounds................................... 406
36. Perturbations of the model problem: blow-up and global existence...... 411
37. Fujita-type results..................................................... 422
38. A priori bounds and blow-up rates..................................... 430
39. Blow-up sets and profiles............................................... 441
40. Diffusive Hamilton-Jacobi equations and gradient blow-up on the bound-
ary.................................................................... 448
1. Gradient blow-up and global existence.............................. 448
2. Asymptotic behavior of global solutions............................. 451
3. Space profile of gradient blow-up.................................... 457
4. Time rate of gradient blow-up...................................... 463
viii Contents
41. An example of interior gradient blow-up................................ 472
V. NONLOCAL PROBLEMS
42. Introduction........................................................... 475
43. Problems involving space integrals (I).................................. 475
1. Blow-up and global existence....................................... 476
2. Blow-up rates, sets and profiles..................................... 479
3. Uniform bounds from Lq-estimates.................................. 492
4. Universal bounds for global solutions................................ 493
44. Problems involving space integrals (II)................................. 496
1. Transition from single-point to global blow-up....................... 496
2. A problem with control of mass..................................... 501
3. A problem with variational structure................................ 510
4. A problem arising in the modeling of Ohmic heating................ 511
45. Fujita-type results for problems involving space integrals............... 517
46. A problem with memory term.......................................... 521
1. Blow-up and global existence....................................... 521
2. Blow-up rate....................................................... 523
APPENDICES
47. Appendix A: Linear elliptic equations.................................. 527
1. Elliptic regularity................................................... 527
2. Lp-Lq-estimates.................................................... 529
3. Some elliptic operators in weighted Lebesgue spaces (I)............. 532
4. Some elliptic operators in weighted Lebesgue spaces (II)............. 536
48. Appendix B: Linear parabolic equations................................ 541
1. Parabolic regularity................................................ 541
2. Heat semigroup, Lp-Lq-estimates, decay, gradient estimates......... 542
3. Weak and integral solutions......................................... 547
49. Appendix C: Linear theory in Lp-spaces and in uniformly local spaces.. 551
δ
1. The Laplace equation in Lp-spaces.................................. 552
δ
2. The heat semigroup in Lp-spaces.................................... 554
δ
3. Some pointwise boundary estimates for the heat equation........... 556
4. Proof of Theorems 49.2, 49.3 and 49.7.............................. 560
5. The heat equation in uniformly local Lebesgue spaces............... 564
6. The heat equation in Morrey spaces................................. 566
50. Appendix D: Poincar´e, Hardy-Sobolev, and other useful inequalities.... 567
1. Basic inequalities................................................... 567
2. The Poincar´e inequality............................................ 568
3. Hardy and Hardy-Sobolev inequalities.............................. 570
Contents ix
51. AppendixE:Localexistence, regularityandstabilityforsemilinearpara-
bolic problems........................................................ 572
1. Analytic semigroups and interpolation spaces...................... 572
2. Local existence and regularity for regular data..................... 576
3. Stability of equilibria.............................................. 591
4. Self-adjoint generators with compact resolvent..................... 594
5. Singular initial data............................................... 601
6. Uniform bounds from Lq-estimates................................. 611
7. An elementary proof of local well-posedness for problem (14.1) in
L∞(Ω)............................................................ 613
52. Appendix F: Maximum and comparison principles. Zero number ...... 614
1. Maximum principles for the Laplace equation...................... 615
2. Comparison principles for classical and strong solutions............ 616
3. Comparison principles via the Stampacchia method................ 620
4. Comparison principles via duality arguments....................... 622
5. Monotonicity of radial solutions.................................... 626
6. Monotonicity of solutions in time.................................. 627
7. Systems and nonlocal problems.................................... 629
8. Zero number...................................................... 634
53. Appendix G: Dynamical systems...................................... 636
53a. Appendix Ga: Summary of positive radial steady states and self-similar
profiles of (18.1)...................................................... 640
54. Appendix H: Methodological notes.................................... 644
55. Appendix I: Selection of open problems............................... 657
Bibliography.......................................................... 661
List of Symbols....................................................... 717
Index................................................................. 719