Nonlinear Diffusion Equations

By: Huilai Li, Zhuoqun Wu, Jingxue Yin, Junning Zhao

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Hardcover | 1 December 2000

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520 Pages

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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.

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Preface	p. vii
Newtonian Filtration Equations	p. 1
Introduction	p. 1
Physical examples	p. 1
Definitions of generalized solutions	p. 4
Special solutions	p. 12
Existence and Uniqueness of Solutions: One Dimensional Case	p. 13
Uniqueness of solutions	p. 13
Existence of solutions	p. 16
Comparison theorems	p. 21
Some extensions	p. 22
Existence and Uniqueness of Solutions: Higher Dimensional Case	p. 26
Comparison theorem and uniqueness of solutions	p. 26
Existence of solutions	p. 31
Some extensions	p. 34
Regularity of Solutions: One Dimensional Case	p. 39
Lemma	p. 39
Regularity of solutions	p. 44
Some extensions	p. 47
Regularity of Solutions: Higher Dimensional Case	p. 48
Generalized class B[subscript 2]	p. 49
Some lemmas	p. 52
Properties of functions in the generalized class B[subscript 2]	p. 55
Holder continuity of solutions	p. 65
Properties of the Free Boundary: One Dimensional Case	p. 71
Finite propagation of disturbances	p. 71
Localization and extinction of disturbances	p. 76
Differential equation on the free boundary	p. 80
Continuously differentiability of the free boundary	p. 83
Some further results	p. 95
Properties of the Free Boundary: Higher Dimensional Case	p. 96
Monotonicity and Holder continuity of the free boundary	p. 96
Lipschitz continuity of the free boundary	p. 114
Differential equation on the free boundary	p. 122
Initial Trace of Solutions	p. 123
Harnack inequality	p. 124
Main result	p. 132
Extension of existence and uniqueness theorem	p. 138
Other Problems	p. 138
Equations with strongly nonlinear sources	p. 139
Asymptotic properties of solutions	p. 143
Non-Newtonian Filtration Equations	p. 147
Introduction Preliminary Knowledge	p. 147
Introduction Physical example	p. 147
Basic spaces and some lemmas	p. 149
Definitions of generalized solutions	p. 155
Special solutions	p. 159
Existence of Solutions	p. 160
The case u[subscript 0] [set membership] C[superscript [infinity] subscript 0](R[superscript N]) or u[subscript 0] [set membership] L[superscript 1](R[superscript N]) [intersection of] L[superscript [infinity](R[superscript N])	p. 161
The case u[subscript 0] [set membership] L[superscript 1 subscript loc](R[superscript N])	p. 175
Some remarks	p. 188
Harnack Inequality and the Initial Trace of Solutions	p. 189
Local Harnack inequality	p. 189
Global Harnack inequality	p. 196
Initial trace of solutions	p. 201
Regularity of Solutions	p. 204
Boundedness of solutions	p. 204
Boundedness of the gradient of solutions	p. 208
Holder continuity of solutions	p. 212
Holder continuity of the gradient of solutions	p. 215
Uniqueness of Solutions	p. 230
Auxiliary propositions	p. 230
Uniqueness theorem and its proof	p. 246
Properties of the Free Boundary	p. 253
Monotonicity and Holder continuity of the free boundary	p. 253
Lipschitz continuity of the free boundary	p. 262
Other Problems	p. 263
p-Laplacian equation with strongly nonlinear sources	p. 263
Asymptotic properties of solutions	p. 266
General Quasilinear Equations of Second Order	p. 269
Introduction	p. 269
Weakly Degenerate Equations in One Dimension	p. 271
Uniqueness of bounded and measurable solutions	p. 272
Existence of continuous solutions	p. 279
Holder continuity of solutions	p. 284
Some extensions	p. 286
Weakly Degenerate Equations in Higher Dimension	p. 287
Existence of continuous solutions for equations with two points of degeneracy	p. 289
Uniqueness of BV solutions	p. 293
Existence of BV solutions	p. 301
Some extensions	p. 305
Strongly Degenerate Equations in One Dimension	p. 306
Definitions of solutions with discontinuity	p. 306
Interior discontinuity condition	p. 308
Uniqueness of BV solutions of the Cauchy problem	p. 314
Formulation of the boundary value problem	p. 323
Boundary discontinuity condition	p. 324
Uniqueness of BV solutions of the first boundary value problem	p. 327
Existence of BV solutions of the first boundary value problem	p. 328
Some extensions	p. 332
Equations with degeneracy at infinity	p. 333
Properties of the curves of discontinuity	p. 334
Degenerate Equations in Higher Dimension without Terms of Lower Order	p. 335
Uniqueness of bounded and integrable solutions	p. 336
A lemma on weak convergence	p. 341
Existence of solutions	p. 344
Finite propagation of disturbances	p. 349
General Strongly Degenerate Equations in Higher Dimension	p. 353
Existence of BV solutions	p. 354
Some extensions	p. 362
Appendix Classes BV and BV[subscript x]	p. 363
Nonlinear Diffusion Equations of Higher Order	p. 369
Introduction	p. 369
Similarity Solutions of a Fourth Order Equation	p. 371
Definition of similarity solutions	p. 372
Existence and uniqueness of global solutions of the Cauchy problem	p. 373
Regularity of solutions	p. 376
Properties of solutions at zero points	p. 377
Properties of unbounded solutions	p. 378
Bounded solutions on the half line	p. 379
Bounded solutions on the whole line	p. 386
Properties of solutions in typical cases k = 1,2,3,4	p. 388
Behavior of similarity solutions as t [right arrow] 0[superscript +]	p. 397
Equations with Double-Degeneracy	p. 399
Existence of solutions	p. 399
Uniqueness of solutions	p. 408
Weighted energy equality of solutions	p. 416
Some auxiliary inequalities	p. 417
Finite propagation of disturbances	p. 418
Asymptotic behavior of solutions	p. 421
Extinction of solutions at finite time	p. 422
Nonexistence of nonnegative solutions	p. 423
Infinite propagation case	p. 424
Cahn-Hilliard Equation with Constant Mobility	p. 425
Existence of classical solutions	p. 425
Blowing-up of solutions	p. 430
Global existence of solutions for small initial value	p. 432
Cahn-Hilliard Equations with Positive Concentration Dependent Mobility	p. 437
A modified Campanato space	p. 438
Holder norm estimates for a linear problem	p. 440
Zero potential case	p. 448
General case	p. 453
Thin Film Equation	p. 454
Definition of generalized solutions	p. 455
Approximate solutions	p. 455
Existence of solutions	p. 459
Nonnegativity of solutions	p. 461
Zeros of nonnegative solutions	p. 465
Regularity of solutions	p. 466
Monotonicity of the support of solutions	p. 468
Cahn-Hilliard Equation with Degenerate Mobility	p. 470
Models with degenerate mobility	p. 470
Definition of physical solutions	p. 471
Existence of solutions	p. 473
Physical solutions	p. 474
Bibliography	p. 479
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Nonlinear Diffusion Equations

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