Book review

The brief book under consideration deals with global behaviour of solutions u(t) = u(t, x) of evolution equations dtu = A(u), u|(t = 0) = u0, in terms of the corresponding semigroup. The results are illustrated by typical examples: Navier-Stokes equations, equation of reaction-diffusion, semilinear wave equation with dissipation, other parabolic and hyperbolic equations. Non-autonomous dynamical systems with a quasi-periodic or almost periodic dependence on time are also studied in the Appendix. The book is based on a series of lectures delivered by the author at the universities of Rome and Pavia, and at the Scuola Normale Superiore in Pisa, and includes results obtained by the author in collaboration with A.V. Babin, M.Y. Skvortsov, V.Y. Skvortsov and V.V. Chepyzhov. The contents are as follows, Chapter I: Preliminaries (attractors of evolution equations and invariant manifolds); Chapter II: Local spectral asymptotics; Chapter III: Global spectral asymptotics; Chapter IV: Uniform approximation of trajectories of semigroups depending on a parameter; Chapter V: The asymptotics of solutions of reaction-diffusion equations with small parameter; Chapter VI: Asymptotics of elements lying on the attractor of solutions of the perturbed evolutionary equations; Chapter VII: Asymptotics of solutions of singular perturbed evolutionary equations; Appendix: Non-autonomous dynamical systems and their attractors (existence of attractors and Hausdorff dimension of these attractors).

There are other books on asymptotic behaviour of solutions of evolution equations, notably A. Haraux, Nonlinear evolution equations - Global behaviour of solutions, Springer, 1981; G.Morosanu, Nonlinear evolution equations and applications (in Romanian), Ed. Academiei, Bucuresti, 1986; A.V. Babin & M.I. Vishik, Attractors of evolution equations, Nauka, Moscow, 1989; A. Haraux, Systèmes dynamiques dissipatifs et applications, Masson, 1991. The present book is not intented to be exhaustive but to initiate the reader in the interesting problem of the asymptotic behaviour of solutions of evolution equations. This is achieved by making accessible some very recent results given by the author and his collaborators.

The lectures are carefully written with numerous instructive examples that provide concrete illustrations of the abstract notions and results. Thus, they are very readable for advanced graduate students.

The author is known as a real expert in the field and the style of these dense lectures provides that he is equally a good expositor. Indeed, the present book plenty satisfies the aim of the editor of the series Lezioni Lincee, to provide a "mise au point" for the subject it deals with and to adress to a broad audience of graduate students and faculty members.

We warmly recommend this book to all interested in dynamical systems and nonlinear partial differential equations.