Asymptotic behavior of dissipative systems. (English) Zbl 0642.58013

Hale, Jack K.

Mathematical Surveys and Monographs, 25. Providence, RI: American Mathematical Society (AMS). ix, 198 p. $ 54.00 (1988).

Roughly speaking, a dissipative system in a Banach space X is one defined by a (nonlinear) evolution equation with the following property: there is a bounded set B of X into which every orbit eventually enters and remains.
The study of dissipative systems in an infinite dimensional space requires techniques which have been developed in the last twenty years. This monograph reports the advances that have been made in the area by the author and many other mathematicians; it is an important source of ideas for the researchers interested in the subject.
A partial list of topics treated in the monograph is: existence of global attractors and its dependence on parameters, existence of fixed points, regularity of the solutions on the attractor, finite dimensional attractors and inle groupoid.

Reviewer: A.Weinstein

MSC:

58D25	Equations in function spaces; evolution equations
34D10	Perturbations of ordinary differential equations
58-02	Research exposition (monographs, survey articles) pertaining to global analysis
34-02	Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34G20	Nonlinear differential equations in abstract spaces
37-XX	Dynamical systems and ergodic theory
34C25	Periodic solutions to ordinary differential equations
34E05	Asymptotic expansions of solutions to ordinary differential equations
34C05	Topological structure of integral curves, singular points, limit cycles of ordinary differential equations

Keywords:

dissipative system; Banach space; (nonlinear) evolution equation; global attractors; fixed points; regularity