Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations of Porous Medium Type

Contents......Page 10
Introduction......Page 16
Part I: Estimates for the PME/FDE......Page 22
1.1 Functional preliminaries......Page 24
1.2 Preliminaries on the PME and the FDE......Page 28
1.3 Main comparison results......Page 33
1.4 Comments and historical notes......Page 34
2.1 The model. Source-type solutions......Page 37
2.2 Smoothing effect and decay with L[sup(1)] functions or measures as data. Best constants......Page 40
2.3 Smoothing exponents and scaling properties......Page 44
2.4 Strong and weak smoothing effects......Page 45
2.5 Comparison for different diffusivities......Page 46
2.7 Estimating the smoothing effect into L[sup(p)]......Page 47
2.8 Asymptotic sharpness of the estimates......Page 48
2.9 The limit m → ∞. Mesa problem......Page 49
2.10 Comments and historical notes......Page 51
3.1 Strong smoothing effect......Page 57
3.2 Scaling and self-similarity......Page 59
3.3 New special solution. Marcinkiewicz spaces......Page 63
3.4 New smoothing effect......Page 65
3.5 General smoothing result......Page 66
3.6 The L[sup(p)]–L[sup(p)] problem. Estimates of weak type......Page 67
3.8 The question of local estimates for the FDE......Page 69
3.9 Comments, open problems, and notes......Page 71
4.1 Lower bounds and Harnack inequalities......Page 73
4.2 Contractivity and error estimates......Page 83
4.3 Comments and historical notes......Page 86
Part II: Study of the subcritical FDE......Page 88
5 Subcritical range of the FDE. Critical line. Extinction. Backward effect......Page 92
5.1 Preliminaries. Critical line......Page 93
5.2 Extinction and the critical line......Page 94
5.3 Some basic facts on extinction......Page 97
5.4 The fast-diffusion backward effect......Page 101
5.5 Explaining how mass is lost......Page 106
5.6 The end-point m = m[sub(c)]......Page 110
5.7 Extinction and blow-up......Page 113
5.8 Comments, extensions and historical notes......Page 114
6.1 The phenomenon of delayed regularity......Page 122
6.2 Immediate boundedness......Page 129
6.3 Comments and historical notes......Page 130
7 Extinction rates and asymptotics for 0 < m < m[sub(c)]......Page 131
7.1 Self-similarity of Type II and extinction......Page 132
7.2 Special solutions with anomalous exponents......Page 134
7.3 Admissible extinction rates......Page 144
7.4 Radial asymptotic convergence result......Page 147
7.5 FDE with Sobolev exponent m = (n – 2)/(n + 2)......Page 148
7.6 The Dirichlet problem in a ball......Page 150
7.7 Comments, extensions, and historical notes......Page 151
8.1 Intermediate range –1 < m ≤ 0 in n = 1......Page 155
8.2 Logarithmic diffusion in n = 2. Ricci flow......Page 157
8.3 Weak local effect in log-diffusion......Page 170
8.4 Comments and historical notes......Page 173
9.1 Preliminaries......Page 181
9.2 Instantaneous extinction......Page 182
9.3 The critical line. Local smoothing effects......Page 185
9.4 End-points of the critical line......Page 189
9.5 Comments and historical notes......Page 191
10.1 Supercritical range......Page 193
10.2 Subcritical ranges......Page 194
10.3 Evolution of Dirac masses. Existence of source solutions with a background......Page 195
10.4 Comments and historical notes......Page 199
Part III: Extensions and appendices......Page 202
11.1 The evolution p-Laplacian equation......Page 204
11.2 The doubly nonlinear diffusion equation......Page 205
11.4 Source-type solutions......Page 206
11.5 Smoothing estimates, best constants and decay rates for PLE and DNLE......Page 208
11.6 Comments and historical notes......Page 212
Appendices......Page 216
AI.1 Some integrals and constants......Page 218
AI.2 More on Marcinkiewicz spaces. Lorentz spaces......Page 219
AI.4 Maximal monotone graphs......Page 220
AII.1 Lagrangian approach in diffusion......Page 222
AII.2 The ZKB solutions and similar......Page 223
AII.3 Speed distributions......Page 224
AIII.1 The Yamabe problem......Page 226
AIII.3 Two-dimensional Ricci flow. Gauss–Bonnet formula......Page 227
AIII.4 Comments and historical notes......Page 228
Appendix IV. Some extensions and parallel topics......Page 229
Bibliography......Page 232
E......Page 247
S......Page 248
Z......Page 249