Nonlinear Diffusion Equations
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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Chapter 1 Newtonian Filtration Equations
Chapter 2 NonNewtonian Filtration Equations
Chapter 3 General Quasilinear Equations of Second Order
Chapter 4 Nonlinear Diffusion Equations of Higher Order
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Anal Assume boundary value condition boundary value problem bounded BV solution Cahn-Hilliard Cahn–Hilliard equation Cauchy problem Cauchy problem 7.1 choose classical solution compact support comparison theorem complete the proof conclusion constant depending dadt dardt defined Definition degeneracy degenerate parabolic equations Denote derive Differential Equations diffusion equations eaſists equation 2.3 estimate exists filtration equation finite follows free boundary hence Hölder continuity Hölder's inequality holds implies initial data initial value condition integral Lemma Lipschitz continuous Math maximum principle p e Cô(QT porous medium equation proof is complete proof of Theorem Proposition prove quasilinear Radon measure right hand side satisfies smooth solution of 1.1 t)da uniformly yields Yin Jingxue Young's inequality zero