Mathematical Surveys and Monographs 1988; 198 pp; softcover Volume: 25 ISBN10: 0821849344 ISBN13: 9780821849347 List Price: US$82 Member Price: US$65.60 Order Code: SURV/25.S
 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 infinitedimensional 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, reactiondiffusion 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 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 MorseSmale 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 NavierStokes equation
 Neutral functional differential equations
 Some abstract evolutionary equations
 A onedimensional damped wave equation
 A threedimensional damped wave equation
 Remarks on other applications
 Dependence on parameters and approximation of the attractor
 Appendix. Stable and unstable manifolds
 References
 Index
