Birkhäuser Advanced Texts Basler Lehrbücher Superlinear Parabolic Problems Blow-up, Global Existence and Steady States Bearbeitet von Pavol Quittner, Philippe Souplet 1. Auflage 2007. Buch. xii, 584 S. Hardcover ISBN 978 3 7643 8441 8 Format (B x L): 17 x 25 cm Gewicht: 1120 g Weitere Fachgebiete > Mathematik > Mathematische Analysis > Funktionalanalysis schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, ebooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte.
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 u + u p =0inR n... 50 10. A priori bounds via the method of Hardy-Sobolev inequalities... 55 11. A priori bounds via bootstrap in L p δ-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 L q -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 L p δ 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
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 q -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 vii APPENDICES 47. Appendix A: Linear elliptic equations... 429 1. Elliptic regularity... 429 2. L p -L q -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, L p -L q -estimates, decay, gradient estimates... 439 3. Weak and integral solutions... 443 49. Appendix C: Linear theory in L p δ-spaces and in uniformly local spaces.. 447 1. The Laplace equation in L p δ-spaces... 447 2. The heat semigroup in L p δ-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: Poincaré, Hardy-Sobolev, and other useful inequalities... 462 1. Basic inequalities... 462 2. The Poincaré inequality... 463 3. Hardy and Hardy-Sobolev inequalities... 465 51. Appendix E: Local existence, regularity and stability for semilinear parabolic 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 L q -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