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. Presents the basic problems, main results, and typical methods for nonlinear diffusion equations with degeneracy. The authors, who are affiliated with Jilin University, develop Newtonian filtration equations, non-Newtonian filtration equations, general quasilinear degenerated parabolic equations of second order, and some special nonlinear diffusion equations of fourth order. Topics include properties of the free boundary, the regularity of solutions, the uniqueness of solutions, the evolution p-Laplacian equation, and the Cahn-Hilliard equation with mobility in different cases. Originally published in Chinese in 1996. No index. Annotation c. Book News, Inc., Portland, OR (booknews.com) Preface; Contents; Chapter 1 Newtonian Filtration Equations; 1.1 Introduction; 1.1.1 Physical examples; 1.1.2 Definitions of generalized solutions; 1.1.3 Special solutions; 1.2 Existence and Uniqueness of Solutions: One Dimensional Case; 1.2.1 Uniqueness of solutions; 1.2.2 Existence of solutions; 1.2.3 Comparison theorems; 1.2.4 Some extensions; 1.3 Existence and Uniqueness of Solutions: Higher Dimensional Case; 1.3.1 Comparison theorem and uniqueness of solutions; 1.3.2 Existence of solutions; 1.3.3 Some extensions; 1.4 Regularity of Solutions: One Dimensional Case; 1.4.1 Lemma