This book is based on the author’s lecture notes for graduate students. It serves well its purpose, which is to introduce the students to the methods for the study of nonlinear evolution equations and to present some of the basic results in this area.
The book consists of six chapters, a bibliography and an index. In the preface, the author explains the contents of the book. Chapter 1, Preliminaries, contains some examples of nonlinear PDE, some known facts from the theory of Sobolev spaces, elliptic boundary value problems, interpolation theory, and a formulation of several standard inequalities. Chapter 2, Semigroup Method, contains some results on linear semigroups, some applications to semilinear equations in operator form, and applications to nonlinear PDE of parabolic and hyperbolic types. Chapter 3, Compactness Method and Monotone Operator Method, presents the methods of solving problems with monotone (in the sense of Minty) operators and applications to nonlinear parabolic equations. Chapter 4, Monotone Iterative Method and Invariant Regions, presents the method of upper and lower solutions and the method of invariant regions. Applications of these methods are illustrated by several examples, which include, among others, some equations arising in mathematical biology. Chapter 5, Global Solutions With Small Initial Data, presents several results on the existence of local and global solutions to nonlinear evolution equations, a discussion of the blow-up phenomenon, and of the behavior of global solutions at large times. Chapter 6, Asymptotic Behavior of Solutions and Global Attractors, deals with the interesting question: when does a bounded solution have a limit as time grows? Convergence of the solutions of evolution equations as time grows to the solutions of the corresponding stationary problems is studied. The notion of global attractor is introduced and sufficient conditions for the existence of global attractors are given.
At the end of each chapter, there are some bibliographical remarks. The bibliography contains 171 entries. The book is a useful contribution to the large literature of the subject, which includes many books (by Amann, Bourgain, Hale, Henry, John, Kato, Krylov, Ladyzhenskaya, Uraltseva and Solonnikov, J. L. Lions, Pao, Pazy, Racke, Sell and You, Showalter, Smoller, Strauss, Tanabe, Temam, and others). Although of the names are misspelled (p. 2, Sine-Gorden means Sine-Gordon, p. 101, Fato means Fatou, etc.), it is always clear what the author means.