Asymptotic behavior of dissipative systems.

*(English)*Zbl 0642.58013
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.

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 |