##
**Attractors of evolution equations. Transl. from the Russian by A. V. Babin.**
*(English)*
Zbl 0778.58002

Studies in Mathematics and its Applications. 25. Amsterdam etc.: North- Holland. x, 532 p. (1992).

This book is devoted to a chapter of the modern theory of infinite- dimensional dynamical systems. The authors study the asymptotic behaviour of solutions of evolution equations.

In chapter 1 (Quasilinear evolution equations and semigroups generated by them) semigroups related to the known equations of mathematical physics are constructed and investigated. The corresponding function spaces on which these semigroups act are introduced. Then notions as boundedness, compactness and continuity of a semigroup of operators are discussed in order to use them for the construction of the semigroup attractors.

The authors concentrate their attention in this book to the so called maximal (global, universal) attractors, i.e. to objects attracting any bounded set in the basic Banach space under the action of the considered semigroup.

In the second chapter (Maximal attractors of semigroups) basic theorems on existence of attractors are proved. Attractors of the semigroups corresponding to the two-dimensional Navier-Stokes system, to a reaction- diffusion equation, to a damped wave equation and to several parabolic equations are constructed and studied.

Chapter 3 (Attractors and unstable sets) is devoted to the investigation of the unstable sets of equilibrium points of the semigroup. The authors prove that if the semigroup has a global Lyapunov function, then the attractor of this semigroup coincides with the unstable set of all equilibrium points of the semigroup.

The small chapter 4 (Some information on semigroups of linear operators) contains some known facts used in what follows.

In chapter 5 (Invariant manifolds of semigroups and mappings at equilibrium points) invariant manifolds in a neighbourhood of an equilibrium point of a differentiable semigroup are constructed. Moreover, this chapter contains some consideration on the asymptotic behaviour of the trajectories generated by the semigroup acting on the initial point, which belongs to some bounded set.

Chapter 6 (Steady state solutions) is devoted to the investigation of equilibrium points of evolution equations and their semigroups. In particular, the authors discuss estimates of the index of instability of equilibrium points.

In chapter 7 (Differentiability of operators of semigroups generated by partial differential equations) the question of differentiability of the semigroup operators is first considered in some abstract situation. Then the authors consider concrete equations (reaction-diffusion systems, parabolic systems, damped wave and Navier-Stokes equations). If the differential operator and the initial data of an evolution equation depend on a parameter, then both the solution and the attractor of the corresponding semigroup also may depend on this parameter. The authors show that under very general conditions the attractor depends upper semicontinuously on the parameter. Furthermore, they give natural conditions for the stable dependence on the parameter of the solution to the evolution equation. This is the content of chapter 8 (Semigroups depending on a parameter).

Chapter 9 (Dependence on a parameter of attractors of differentiable semigroups and uniform asymptotics of trajectories) deals with the problem of the asymptotic behaviour of the solutions to the evolution equation as the parameter tends to zero for any index of the semigroup. The principal term of the asymptotics with respect to the parameter is the solution of the evolution equation with parameter zero. Under natural conditions an exponential estimate is shown.

The last chapter 10 (Hausdorff dimension of attractors) contains upper and lower estimates of the Hausdorff dimension of attractors for a series of evolution equations. In the preface to the English translation (which is more an appendix than a preface) the translator A. V. Babin writes: “Only minor changes were made during the translation. Some misprints were corrected and former Appendix was transformed into Section 6.6. After the original Russian manuscript was prepared for print, many new works have appeared, to some earlier papers our attention was drawn during recently intensified contacts with foreign mathematicians. Therefore I have added many new items to the bibliography…” Mathematicians working in the field of evolution equation will welcome this book. It contains lots of results known only by journal papers. The formal side of this work is not always sufficient. It is evident that the native tongue of the translator is Russian. There are misprints being obvious already for a non-mathematician.

In chapter 1 (Quasilinear evolution equations and semigroups generated by them) semigroups related to the known equations of mathematical physics are constructed and investigated. The corresponding function spaces on which these semigroups act are introduced. Then notions as boundedness, compactness and continuity of a semigroup of operators are discussed in order to use them for the construction of the semigroup attractors.

The authors concentrate their attention in this book to the so called maximal (global, universal) attractors, i.e. to objects attracting any bounded set in the basic Banach space under the action of the considered semigroup.

In the second chapter (Maximal attractors of semigroups) basic theorems on existence of attractors are proved. Attractors of the semigroups corresponding to the two-dimensional Navier-Stokes system, to a reaction- diffusion equation, to a damped wave equation and to several parabolic equations are constructed and studied.

Chapter 3 (Attractors and unstable sets) is devoted to the investigation of the unstable sets of equilibrium points of the semigroup. The authors prove that if the semigroup has a global Lyapunov function, then the attractor of this semigroup coincides with the unstable set of all equilibrium points of the semigroup.

The small chapter 4 (Some information on semigroups of linear operators) contains some known facts used in what follows.

In chapter 5 (Invariant manifolds of semigroups and mappings at equilibrium points) invariant manifolds in a neighbourhood of an equilibrium point of a differentiable semigroup are constructed. Moreover, this chapter contains some consideration on the asymptotic behaviour of the trajectories generated by the semigroup acting on the initial point, which belongs to some bounded set.

Chapter 6 (Steady state solutions) is devoted to the investigation of equilibrium points of evolution equations and their semigroups. In particular, the authors discuss estimates of the index of instability of equilibrium points.

In chapter 7 (Differentiability of operators of semigroups generated by partial differential equations) the question of differentiability of the semigroup operators is first considered in some abstract situation. Then the authors consider concrete equations (reaction-diffusion systems, parabolic systems, damped wave and Navier-Stokes equations). If the differential operator and the initial data of an evolution equation depend on a parameter, then both the solution and the attractor of the corresponding semigroup also may depend on this parameter. The authors show that under very general conditions the attractor depends upper semicontinuously on the parameter. Furthermore, they give natural conditions for the stable dependence on the parameter of the solution to the evolution equation. This is the content of chapter 8 (Semigroups depending on a parameter).

Chapter 9 (Dependence on a parameter of attractors of differentiable semigroups and uniform asymptotics of trajectories) deals with the problem of the asymptotic behaviour of the solutions to the evolution equation as the parameter tends to zero for any index of the semigroup. The principal term of the asymptotics with respect to the parameter is the solution of the evolution equation with parameter zero. Under natural conditions an exponential estimate is shown.

The last chapter 10 (Hausdorff dimension of attractors) contains upper and lower estimates of the Hausdorff dimension of attractors for a series of evolution equations. In the preface to the English translation (which is more an appendix than a preface) the translator A. V. Babin writes: “Only minor changes were made during the translation. Some misprints were corrected and former Appendix was transformed into Section 6.6. After the original Russian manuscript was prepared for print, many new works have appeared, to some earlier papers our attention was drawn during recently intensified contacts with foreign mathematicians. Therefore I have added many new items to the bibliography…” Mathematicians working in the field of evolution equation will welcome this book. It contains lots of results known only by journal papers. The formal side of this work is not always sufficient. It is evident that the native tongue of the translator is Russian. There are misprints being obvious already for a non-mathematician.

Reviewer: R.Manthey (Jena)