Superlinear parabolic problems: blow-up global existence and steady states
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Basel [u.a.]
Birkhäuser
2007
|
| Schriftenreihe: | Birkhäuser advanced texts / Basler Lehrbücher
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Auch als Internetausgabe |
| Beschreibung: | XI, 584 S. |
| ISBN: | 9783764384418 |
Internformat
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| 100 | 1 | |a Quittner, Pavol |d ca. 20. Jh. |e Verfasser |0 (DE-588)124741282 |4 aut | |
| 245 | 1 | 0 | |a Superlinear parabolic problems |b blow-up global existence and steady states |c Pavol Quittner ; Philippe Souplet |
| 264 | 1 | |a Basel [u.a.] |b Birkhäuser |c 2007 | |
| 300 | |a XI, 584 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Birkhäuser advanced texts / Basler Lehrbücher | |
| 500 | |a Auch als Internetausgabe | ||
| 650 | 4 | |a Elliptisches System - Lösung <Mathematik> | |
| 650 | 4 | |a Parabolisches Differentialgleichungssystem - Lösung <Mathematik> | |
| 650 | 0 | 7 | |a Elliptisches System |0 (DE-588)4121184-4 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Parabolisches Differentialgleichungssystem |0 (DE-588)4833677-4 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Lösung |g Mathematik |0 (DE-588)4120678-2 |2 gnd |9 rswk-swf |
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| 689 | 0 | 1 | |a Lösung |g Mathematik |0 (DE-588)4120678-2 |D s |
| 689 | 0 | |5 DE-604 | |
| 689 | 1 | 0 | |a Elliptisches System |0 (DE-588)4121184-4 |D s |
| 689 | 1 | 1 | |a Lösung |g Mathematik |0 (DE-588)4120678-2 |D s |
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| 700 | 1 | |a Souplet, Philippe |e Verfasser |4 aut | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-015688434 | ||
Datensatz im Suchindex
| _version_ | 1804136572281421824 |
|---|---|
| adam_text | Contents
Introduction ix
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 20
7. Minimax methods 29
8. Liouville type results 36
9. Positive radial solutions of Au + up = 0 in K 50
10. A priori bounds via the method of Hardy Sobolev inequalities 55
11. A priori bounds via bootstrap in L^ spaces 61
12. A priori bounds via the rescaling method 65
13. A priori bounds via moving planes and Pohozaev s identity 68
II. MODEL PARABOLIC PROBLEMS
14. Introduction 75
15. Well posedness in Lebesgue spaces 75
16. Maximal existence time. Uniform bounds from L9 estimates 87
17. Blow up 91
18. Fujita type results 100
19. Global existence for the Dirichlet problem 112
1. Small data global solutions 112
2. Structure of global solutions in bounded domains 120
3. Diffusion eliminating blow up 125
20. Global existence for the Cauchy problem 129
1. Small data global solutions 129
2. Global solutions with exponential spatial decay 137
3. Asymptotic profiles for small data solutions 139
21. Parabolic Liouville type results 150
22. A priori bounds 161
1. A priori bounds in the subcritical case 161
2. Boundedness of global solutions in the supercritical case 166
3. Global unbounded solutions in the critical case 171
4. Estimates for nonglobal solutions 175
vi
23. Blow up rate 177
24. Blow up set and space profile 190
25. Self similar blow up behavior 195
26. Universal bounds and initial blow up rates 202
27. Complete blow up 218
28. Applications of a priori bounds 230
1. A nonuniqueness result 230
2. Existence of periodic solutions 234
3. Existence of optimal controls 236
4. Transition from global existence to blow up and stationary solutions. 237
5. Decay of the threshold solution of the Cauchy problem 239
29. Decay and grow up of threshold solutions in the super supercritical case 245
III. SYSTEMS
30. Introduction 251
31. Elliptic systems 251
1. A priori bounds by the method of moving planes and Pohozaev type
identities 253
2. Liouville type results for the Lane Emden system 260
3. A priori bounds by the rescaling method 263
4. A priori bounds by the Lps alternate bootstrap method 266
32. Parabolic systems coupled by power source terms 272
1. Well posedness and continuation in Lebesgue spaces 273
2. Blow up and global existence 278
3. Fujita type results 280
4. Blow up asymptotics 283
33. The role of diffusion in blow up 287
1. Diffusion preserving global existence 288
2. Diffusion inducing blow up 301
3. Diffusion eliminating blow up 311
IV. EQUATIONS WITH GRADIENT TERMS
34. Introduction 313
35. Well posedness and gradient bounds 314
36. Perturbations of the model problem: blow up and global existence 319
37. Fujita type results 330
38. A priori bounds and blow up rates 338
39. Blow up sets and profiles 348
vii
40. Viscous Hamilton Jacobi equations and gradient blow up on the boundary 355
1. Gradient blow up and global existence 355
2. Asymptotic behavior of global solutions 358
3. Space profile of gradient blow up 364
4. Time rate of gradient blow up 367
41. An example of interior gradient blow up 374
V. NONLOCAL PROBLEMS
42. Introduction 377
43. Problems involving space integrals (I) 377
1. Blow up and global existence 378
2. Blow up rates, sets and profiles 381
3. Uniform bounds from L estimates 394
4. Universal bounds for global solutions 395
44. Problems involving space integrals (II) 398
1. Transition from single point to global blow up 398
2. A problem with control of mass 403
3. A problem with variational structure 411
4. A problem arising in the modeling of Ohmic heating 412
45. Fujita type results for problems involving space integrals 418
46. A problem with memory term 421
1. Blow up and global existence 422
2. Blow up rate 424
APPENDICES
47. Appendix A: Linear elliptic equations 429
1. Elliptic regularity 429
2. Lp Z,« estimates 431
3. An elliptic operator in a weighted Lebesgue space 434
48. Appendix B: Linear parabolic equations 438
1. Parabolic regularity 438
2. Heat semigroup, LP L estimates, decay, gradient estimates 439
3. Weak and integral solutions 443
49. Appendix C: Linear theory in L^ spaces and in uniformly local spaces .. 447
1. The Laplace equation in L^ spaces 447
2. The heat semigroup in L^ spaces 450
3. Some pointwise boundary estimates for the heat equation 452
4. Proof of Theorems 49.2, 49.3 and 49.7 456
5. The heat equation in uniformly local Lebesgue spaces 460
viii
50. Appendix D: Poincare, Hardy Sobolev, and other useful inequalities .... 462
1. Basic inequalities 462
2. The Poincare inequality 463
3. Hardy and Hardy Sobolev inequalities 465
51. Appendix E: Local existence, regularity and stability for semilinear para¬
bolic problems 466
1. Analytic semigroups and interpolation spaces 466
2. Local existence and regularity for regular data 470
3. Stability of equilibria 485
4. Self adjoint generators with compact resolvent 488
5. Singular initial data 495
6. Uniform bounds from /^ estimates 505
52. Appendix F: Maximum and comparison principles. Zero number 507
1. Maximum principles for the Laplace equation 507
2. Comparison principles for classical and strong solutions 509
3. Comparison principles via the Stampacchia method 512
4. Comparison principles via duality arguments 515
5. Monotonicity of radial solutions 518
6. Monotonicity of solutions in time 520
7. Systems and nonlocal problems 522
8. Zero number 526
53. Appendix G: Dynamical systems 528
54. Appendix H: Methodological notes 532
Bibliography 543
List of Symbols 577
Index 579
|
| adam_txt |
Contents
Introduction ix
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 20
7. Minimax methods 29
8. Liouville type results 36
9. Positive radial solutions of Au + up = 0 in K" 50
10. A priori bounds via the method of Hardy Sobolev inequalities 55
11. A priori bounds via bootstrap in L^ spaces 61
12. A priori bounds via the rescaling method 65
13. A priori bounds via moving planes and Pohozaev's identity 68
II. MODEL PARABOLIC PROBLEMS
14. Introduction 75
15. Well posedness in Lebesgue spaces 75
16. Maximal existence time. Uniform bounds from L9 estimates 87
17. Blow up 91
18. Fujita type results 100
19. Global existence for the Dirichlet problem 112
1. Small data global solutions 112
2. Structure of global solutions in bounded domains 120
3. Diffusion eliminating blow up 125
20. Global existence for the Cauchy problem 129
1. Small data global solutions 129
2. Global solutions with exponential spatial decay 137
3. Asymptotic profiles for small data solutions 139
21. Parabolic Liouville type results 150
22. A priori bounds 161
1. A priori bounds in the subcritical case 161
2. Boundedness of global solutions in the supercritical case 166
3. Global unbounded solutions in the critical case 171
4. Estimates for nonglobal solutions 175
vi
23. Blow up rate 177
24. Blow up set and space profile 190
25. Self similar blow up behavior 195
26. Universal bounds and initial blow up rates 202
27. Complete blow up 218
28. Applications of a priori bounds 230
1. A nonuniqueness result 230
2. Existence of periodic solutions 234
3. Existence of optimal controls 236
4. Transition from global existence to blow up and stationary solutions. 237
5. Decay of the threshold solution of the Cauchy problem 239
29. Decay and grow up of threshold solutions in the super supercritical case 245
III. SYSTEMS
30. Introduction 251
31. Elliptic systems 251
1. A priori bounds by the method of moving planes and Pohozaev type
identities 253
2. Liouville type results for the Lane Emden system 260
3. A priori bounds by the rescaling method 263
4. A priori bounds by the Lps alternate bootstrap method 266
32. Parabolic systems coupled by power source terms 272
1. Well posedness and continuation in Lebesgue spaces 273
2. Blow up and global existence 278
3. Fujita type results 280
4. Blow up asymptotics 283
33. The role of diffusion in blow up 287
1. Diffusion preserving global existence 288
2. Diffusion inducing blow up 301
3. Diffusion eliminating blow up 311
IV. EQUATIONS WITH GRADIENT TERMS
34. Introduction 313
35. Well posedness and gradient bounds 314
36. Perturbations of the model problem: blow up and global existence 319
37. Fujita type results 330
38. A priori bounds and blow up rates 338
39. Blow up sets and profiles 348
vii
40. Viscous Hamilton Jacobi equations and gradient blow up on the boundary 355
1. Gradient blow up and global existence 355
2. Asymptotic behavior of global solutions 358
3. Space profile of gradient blow up 364
4. Time rate of gradient blow up 367
41. An example of interior gradient blow up 374
V. NONLOCAL PROBLEMS
42. Introduction 377
43. Problems involving space integrals (I) 377
1. Blow up and global existence 378
2. Blow up rates, sets and profiles 381
3. Uniform bounds from L' estimates 394
4. Universal bounds for global solutions 395
44. Problems involving space integrals (II) 398
1. Transition from single point to global blow up 398
2. A problem with control of mass 403
3. A problem with variational structure 411
4. A problem arising in the modeling of Ohmic heating 412
45. Fujita type results for problems involving space integrals 418
46. A problem with memory term 421
1. Blow up and global existence 422
2. Blow up rate 424
APPENDICES
47. Appendix A: Linear elliptic equations 429
1. Elliptic regularity 429
2. Lp Z,« estimates 431
3. An elliptic operator in a weighted Lebesgue space 434
48. Appendix B: Linear parabolic equations 438
1. Parabolic regularity 438
2. Heat semigroup, LP L' estimates, decay, gradient estimates 439
3. Weak and integral solutions 443
49. Appendix C: Linear theory in L^ spaces and in uniformly local spaces . 447
1. The Laplace equation in L^ spaces 447
2. The heat semigroup in L^ spaces 450
3. Some pointwise boundary estimates for the heat equation 452
4. Proof of Theorems 49.2, 49.3 and 49.7 456
5. The heat equation in uniformly local Lebesgue spaces 460
viii
50. Appendix D: Poincare, Hardy Sobolev, and other useful inequalities . 462
1. Basic inequalities 462
2. The Poincare inequality 463
3. Hardy and Hardy Sobolev inequalities 465
51. Appendix E: Local existence, regularity and stability for semilinear para¬
bolic problems 466
1. Analytic semigroups and interpolation spaces 466
2. Local existence and regularity for regular data 470
3. Stability of equilibria 485
4. Self adjoint generators with compact resolvent 488
5. Singular initial data 495
6. Uniform bounds from /^ estimates 505
52. Appendix F: Maximum and comparison principles. Zero number 507
1. Maximum principles for the Laplace equation 507
2. Comparison principles for classical and strong solutions 509
3. Comparison principles via the Stampacchia method 512
4. Comparison principles via duality arguments 515
5. Monotonicity of radial solutions 518
6. Monotonicity of solutions in time 520
7. Systems and nonlocal problems 522
8. Zero number 526
53. Appendix G: Dynamical systems 528
54. Appendix H: Methodological notes 532
Bibliography 543
List of Symbols 577
Index 579 |
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| id | DE-604.BV022481083 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T17:48:18Z |
| indexdate | 2024-07-09T20:58:32Z |
| institution | BVB |
| isbn | 9783764384418 |
| language | English |
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| physical | XI, 584 S. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
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| publisher | Birkhäuser |
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| spelling | Quittner, Pavol ca. 20. Jh. Verfasser (DE-588)124741282 aut Superlinear parabolic problems blow-up global existence and steady states Pavol Quittner ; Philippe Souplet Basel [u.a.] Birkhäuser 2007 XI, 584 S. txt rdacontent n rdamedia nc rdacarrier Birkhäuser advanced texts / Basler Lehrbücher Auch als Internetausgabe Elliptisches System - Lösung <Mathematik> Parabolisches Differentialgleichungssystem - Lösung <Mathematik> Elliptisches System (DE-588)4121184-4 gnd rswk-swf Parabolisches Differentialgleichungssystem (DE-588)4833677-4 gnd rswk-swf Lösung Mathematik (DE-588)4120678-2 gnd rswk-swf Parabolisches Differentialgleichungssystem (DE-588)4833677-4 s Lösung Mathematik (DE-588)4120678-2 s DE-604 Elliptisches System (DE-588)4121184-4 s Souplet, Philippe Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015688434&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Quittner, Pavol ca. 20. Jh Souplet, Philippe Superlinear parabolic problems blow-up global existence and steady states Elliptisches System - Lösung <Mathematik> Parabolisches Differentialgleichungssystem - Lösung <Mathematik> Elliptisches System (DE-588)4121184-4 gnd Parabolisches Differentialgleichungssystem (DE-588)4833677-4 gnd Lösung Mathematik (DE-588)4120678-2 gnd |
| subject_GND | (DE-588)4121184-4 (DE-588)4833677-4 (DE-588)4120678-2 |
| title | Superlinear parabolic problems blow-up global existence and steady states |
| title_auth | Superlinear parabolic problems blow-up global existence and steady states |
| title_exact_search | Superlinear parabolic problems blow-up global existence and steady states |
| title_exact_search_txtP | Superlinear parabolic problems blow-up global existence and steady states |
| title_full | Superlinear parabolic problems blow-up global existence and steady states Pavol Quittner ; Philippe Souplet |
| title_fullStr | Superlinear parabolic problems blow-up global existence and steady states Pavol Quittner ; Philippe Souplet |
| title_full_unstemmed | Superlinear parabolic problems blow-up global existence and steady states Pavol Quittner ; Philippe Souplet |
| title_short | Superlinear parabolic problems |
| title_sort | superlinear parabolic problems blow up global existence and steady states |
| title_sub | blow-up global existence and steady states |
| topic | Elliptisches System - Lösung <Mathematik> Parabolisches Differentialgleichungssystem - Lösung <Mathematik> Elliptisches System (DE-588)4121184-4 gnd Parabolisches Differentialgleichungssystem (DE-588)4833677-4 gnd Lösung Mathematik (DE-588)4120678-2 gnd |
| topic_facet | Elliptisches System - Lösung <Mathematik> Parabolisches Differentialgleichungssystem - Lösung <Mathematik> Elliptisches System Parabolisches Differentialgleichungssystem Lösung Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015688434&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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