Nonlinear diffusion equations. (English) Zbl 0997.35001

Wu, Zhuoqun; Zhao, Junning; Yin, Jingxue; Li, Huilai

Singapore: World Scientific. xvii, 502 p. (2001).

This monograph is devoted to the study of quasilinear degenerate parabolic equations of second and fourth order. It gives a comprehensive presentation of methods that have been developed in order to overcome the mathematical difficulties caused by the degeneracy. The main issues are: existence of weak solutions, uniqueness, regularity, properties of the free boundary and propagation of disturbances. Chapter I deals with the porous medium equation, Chapter II with the parabolic \(p\)-Laplace equation and Chapter III with more general second-order equations which may have strong degeneracy and can be of mixed hyperbolic-parabolic type. The purpose of Chapter IV is to describe the similarities and differences between second- and fourth-order degenerate quasilinear equations with particular attention given to the Cahn-Hilliard and thin film equations. Although there are some shortcomings in the English, the book is readable and a particular reason why it becomes a valuable addition to the existing literature is that it discusses many contributions of Chinese mathematics to the study of degenerate parabolic equations.

Reviewer: M.Fila (Bratislava)

MSC:

35-02	Research exposition (monographs, survey articles) pertaining to partial differential equations
35Kxx	Parabolic equations and parabolic systems
35K65	Degenerate parabolic equations

Keywords:

quasilinear degenerate parabolic equations; existence of weak solutions; uniqueness; regularity; free boundary; propagation of disturbances; equations of second order; equations of fourth order