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Asymptotic Behavior of Dissipative Systems
Jack K. Hale
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Mathematical Surveys and Monographs
1988; 198 pp; softcover
Volume: 25
ISBN-10: 0-8218-4934-4
ISBN-13: 978-0-8218-4934-7
List Price: US$82
Member Price: US$65.60
Order Code: SURV/25.S
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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.

--Zentralblatt MATH

Although advanced, this book is a very good introduction to the subject, and the reading of the abstract part, which is elegant, is pleasant. ... this monograph will be of valuable interest for those who aim to learn in the very rapidly growing subject of infinite-dimensional dissipative dynamical systems.

--Mathematical Reviews

This book is directed at researchers in nonlinear ordinary and partial differential equations and at those who apply these topics to other fields of science. About one third of the book focuses on the existence and properties of the flow on the global attractor for a discrete or continuous dynamical system. The author presents a detailed discussion of abstract properties and examples of asymptotically smooth maps and semigroups. He also covers some of the continuity properties of the global attractor under perturbation, its capacity and Hausdorff dimension, and the stability of the flow on the global attractor under perturbation. The remainder of the book deals with particular equations occurring in applications and especially emphasizes delay equations, reaction-diffusion equations, and the damped wave equations. In each of the examples presented, the author shows how to verify the existence of a global attractor, and, for several examples, he discusses some properties of the flow on the global attractor.

Table of Contents

  • Introduction
Discrete dynamical systems
  • Limit sets
  • Stability of invariant sets and asymptotically smooth maps
  • Examples of asymptotically smooth maps
  • Dissipativeness and global attractors
  • Dependence on parameters
  • Fixed point theorems
  • Stability relative to the global attractor and Morse-Smale maps
  • Dimension of the global attractor
  • Dissipativeness in two spaces
  • Notes and remarks
Continuous dynamical systems
  • Limit sets
  • Asymptotically smooth and \(\alpha\)-contracting semigroups
  • Stability of invariant sets
  • Dissipativeness and global attractors
  • Dependence on parameters
  • Periodic processes
  • Skew product flows
  • Gradient flows
  • Dissipativeness in two spaces
  • Properties of the flow on the global attractor
  • Notes and remarks
Applications
  • Retarded functional differential equations
  • Sectorial evolutionary equations
  • A scalar parabolic equation
  • The Navier-Stokes equation
  • Neutral functional differential equations
  • Some abstract evolutionary equations
  • A one-dimensional damped wave equation
  • A three-dimensional damped wave equation
  • Remarks on other applications
  • Dependence on parameters and approximation of the attractor
  • Appendix. Stable and unstable manifolds
  • References
  • Index
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