Nonlinear diffusion equations, an important class of parabolic equations, come from a variety of diffusion phenomena which appear widely in nature. They are suggested as mathematical models of physical problems in many fields, such as filtration, phase transition, biochemistry and dynamics of biological groups. In many cases, the equations possess degeneracy or singularity. The appearance of degeneracy or singularity makes the study more involved and challenging. Many new ideas and methods have been developed to overcome the special difficulties caused by the degeneracy and singularity, which enrich the theory of partial differential equations. This book provides a comprehensive presentation of the basic problems, main results and typical methods for nonlinear diffusion equations with degeneracy. Some results for equations with singularity are touched upon.
Author(s): Wu Zhuoqun, Jingxue Yin, Huilai Li, Junning Zhao, Yin Jingxue, Li Huilai
Edition: 1st
Year: 2002
Language: English
Pages: 500
Contents ......Page 14
Preface ......Page 8
1.1.1 Physical examples ......Page 19
1.1.2 Definitions of generalized solutions......Page 22
1.1.3 Special solutions ......Page 30
1.2.1 Uniqueness of solutions ......Page 31
1.2.2 Existence of solutions ......Page 34
1.2.3 Comparison theorems ......Page 39
1.2.4 Some extensions ......Page 40
1.3.1 Comparison theorem and uniqueness of solutions ......Page 44
1.3.2 Existence of solutions ......Page 49
1.3.3 Some extensions ......Page 52
1.4.1 Lemma ......Page 57
1.4.2 Regularity of solutions ......Page 62
1.4.3 Some extensions ......Page 65
1.5 Regularity of Solutions: Higher Dimensional Case ......Page 66
1.5.1 Generalized class B2......Page 67
1.5.2 Some lemmas ......Page 70
1.5.3 Properties of functions in the generalized class B2 ......Page 73
1.5.4 Holder continuity of solutions ......Page 83
1.6.1 Finite propagation of disturbances ......Page 89
1.6.2 Localization and extinction of disturbances ......Page 94
1.6.3 Differential equation on the free boundary ......Page 98
1.6.4 Continuously differentiability of the free boundary ......Page 101
1.6.5 Some further results ......Page 113
1.7.1 Monotonicity and Holder continuity of the free boundary ......Page 114
1.7.2 Lipschitz continuity of the free boundary ......Page 132
1.7.3 Differential equation on the free boundary ......Page 140
1.8 Initial Trace of Solutions ......Page 141
1.8.1 Harnack inequality ......Page 142
1.8.2 Main result ......Page 150
1.9 Other Problems ......Page 156
1.9.1 Equations with strongly nonlinear sources ......Page 157
1.9.2 Asymptotic properties of solutions ......Page 161
2.1.1 Introduction Physical example ......Page 165
2.1.2 Basic spaces and some lemmas ......Page 167
2.1.3 Definitions of generalized solutions ......Page 173
2.1.4 Special solutions ......Page 177
2.2 Existence of Solutions ......Page 178
2.2.1 The case u0 E C∞0 (RN) or u0 E L1(RN) n L∞(RN)......Page 179
2.2.2 The case u0 E L1loc(RN)......Page 193
2.2.3 Some remarks ......Page 206
2.3.1 Local Harnack inequality ......Page 207
2.3.2 Global Harnack inequality ......Page 214
2.3.3 Initial trace of solutions ......Page 219
2.4.1 Boundedness of solutions ......Page 222
2.4.2 Boundedness of the gradient of solutions ......Page 226
2.4.3 Holder continuity of solutions ......Page 230
2.4.4 Holder continuity of the gradient of solutions ......Page 233
2.5.1 Auxiliary propositions ......Page 248
2.5.2 Uniqueness theorem and its proof ......Page 264
2.6.1 Monotonicity and Holder continuity of the free boundary ......Page 271
2.6.2 Lipschitz continuity of the free boundary ......Page 280
2.7.1 p-Laplacian equation with strongly nonlinear sources ......Page 281
2.7.2 Asymptotic properties of solutions ......Page 284
3.1 Introduction ......Page 287
3.2 Weakly Degenerate Equations in One Dimension ......Page 289
3.2.1 Uniqueness of bounded and measurable solutions ......Page 290
3.2.2 Existence of continuous solutions ......Page 297
3.2.3 Holder continuity of solutions ......Page 302
3.2.4 Some extensions ......Page 304
3.3 Weakly Degenerate Equations in Higher Dimension ......Page 305
3.3.1 Existence of continuous solutions for equations with two points of degeneracy ......Page 307
3.3.2 Uniqueness of BV solutions ......Page 311
3.3.3 Existence of BV solutions ......Page 319
3.3.4 Some extensions ......Page 323
3.4.1 Definitions of solutions with discontinuity ......Page 324
3.4.2 Interior discontinuity condition ......Page 326
3.4.3 Uniqueness of BV solutions of the Cauchy problem ......Page 332
3.4.4 Formulation of the boundary value problem ......Page 341
3.4.5 Boundary discontinuity condition ......Page 342
3.4.6 Uniqueness of BV solutions of the first boundary value problem ......Page 345
3.4.7 Existence of BV solutions of the first boundary value problem ......Page 346
3.4.8 Some extensions ......Page 350
3.4.9 Equations with degeneracy at infinity ......Page 351
3.4.10 Properties of the curves of discontinuity ......Page 352
3.5 Degenerate Equations in Higher Dimension without Terms of Lower Order ......Page 353
3.5.1 Uniqueness of bounded and integrable solutions ......Page 354
3.5.2 A lemma on weak convergence ......Page 359
3.5.3 Existence of solutions ......Page 362
3.5.4 Finite propagation of disturbances ......Page 367
3.6 General Strongly Degenerate Equations in Higher Dimension ......Page 371
3.6.1 Existence of BV solutions......Page 372
3.6.2 Some extensions ......Page 380
3.7 Appendix Classes BV and BVx......Page 381
4.1 Introduction ......Page 387
4.2 Similarity Solutions of a Fourth Order Equation ......Page 389
4.2.1 Definition of similarity solutions ......Page 390
4.2.2 Existence and uniqueness of global solutions of the Cauchy problem ......Page 391
4.2.3 Regularity of solutions ......Page 394
4.2.4 Properties of solutions at zero points ......Page 395
4.2.5 Properties of unbounded solutions ......Page 396
4.2.6 Bounded solutions on the half line ......Page 397
4.2.7 Bounded solutions on the whole line ......Page 404
4.2.8 Properties of solutions in typical cases k = 1,2,3,4......Page 406
4.2.9 Behavior of similarity solutions as t -> 0+ ......Page 415
4.3.1 Existence of solutions ......Page 417
4.3.2 Uniqueness of solutions ......Page 426
4.3.3 Weighted energy equality of solutions ......Page 434
4.3.4 Some auxiliary inequalities ......Page 435
4.3.5 Finite propagation of disturbances ......Page 436
4.3.6 Asymptotic behavior of solutions ......Page 439
4.3.7 Extinction of solutions at finite time ......Page 440
4.3.8 Nonexistence of nonnegative solutions ......Page 441
4.3.9 Infinite propagation case ......Page 442
4.4.1 Existence of classical solutions ......Page 443
4.4.2 Blowing-up of solutions ......Page 448
4.4.3 Global existence of solutions for small initial value ......Page 450
4.5 Cahn-Hilliard Equations with Positive Concentration Dependent Mobility ......Page 455
4.5.1 A modified Campanato space ......Page 456
4.5.2 Holder norm estimates for a linear problem ......Page 458
4.5.3 Zero potential case ......Page 466
4.5.4 General case ......Page 471
4.6 Thin Film Equation ......Page 472
4.6.2 Approximate solutions ......Page 473
4.6.3 Existence of solutions ......Page 477
4.6.4 Nonnegativity of solutions ......Page 479
4.6.5 Zeros of nonnegative solutions ......Page 483
4.6.6 Regularity of solutions ......Page 484
4.6.7 Monotonicity of the support of solutions ......Page 486
4.7.1 Models with degenerate mobility ......Page 488
4.7.2 Definition of physical solutions ......Page 489
4.7.3 Existence of solutions ......Page 491
4.7.4 Physical solutions ......Page 492
Bibliography ......Page 497
