**Mathematical Surveys and Monographs**

Volume: 25;
1988;
198 pp;
Softcover

MSC: Primary 58;
Secondary 34; 35

**Print ISBN: 978-0-8218-4934-7
Product Code: SURV/25.S**

List Price: $91.00

AMS Member Price: $72.80

MAA Member Price: $81.90

**Electronic ISBN: 978-1-4704-1252-4
Product Code: SURV/25.S.E**

List Price: $85.00

AMS Member Price: $68.00

MAA Member Price: $76.50

# Asymptotic Behavior of Dissipative Systems

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*Jack K. Hale*

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

# Table of Contents

## Asymptotic Behavior of Dissipative Systems

- Contents v6 free
- Acknowledgment ix10 free
- Chapter 1.Introduction 112 free
- Chapter 2. Discrete Dynamical Systems 819 free
- 2.1. Limit sets 819
- 2.2. Stability of invariant sets and asymptotically smooth maps 1021
- 2.3. Examples of asymptotically smooth maps 1324
- 2.4. Dissipativeness and global attractors 1627
- 2.5. Dependence on parameters 2132
- 2.6. Fixed point theorems 2334
- 2.7. Stability relative to the global attractor and Morse-Smale maps 2536
- 2.8. Dimension of the global attractor 2637
- 2.9. Dissipativeness in two spaces 2839
- Notes and Remarks 3344

- Chapter 3. Continuous Dynamical Systems 3546
- 3.1. Limit sets 3546
- 3.2. Asymptotically smooth and α-contracting semigroups 3647
- 3.3. Stability of invariant sets 3849
- 3.4. Dissipativeness and global attractors 3849
- 3.5. Dependence on parameters 4051
- 3.6. Periodic processes 4152
- 3.7. Skew product flows 4354
- 3.8. Gradient flows 4960
- 3.9. Dissipativeness in two spaces 5465
- 3.10. Properties of the flow on the global attractor 5667
- Notes and Remarks 6071

- Chapter 4. Applications 6172
- 4.1. Retarded functional differential equations (RFDE's) 6172
- 4.2. Sectorial evolutionary equations 7182
- 4.3. A scalar parabolic equation 7586
- 4.3.1. Existence and gradient 7586
- 4.3.2. Qualitative properties of the flow on the attractor 7990
- 4.3.3. Stability properties of equilibria 8495
- 4.3.4. A bifurcation problem—Dirichlet conditions 8798
- 4.3.5. A bifurcation problem—other boundary conditions 92103
- 4.3.6. Equations whose flow is equivalent to an ODE 94105
- 4.3.7. A method for determining stability 97108
- 4.3.8. Stable solutions in a singularly perturbed equation 99110
- 4.3.9. Quenching as a dynamic problem 105116

- 4.4. The Navier-Stokes equation 107118
- 4.5. Neutral functional differential equations 113124
- 4.6. Some abstract evolutionary equations 120131
- 4.7. A one dimensional damped wave equation 125136
- 4.8. A three dimensional damped wave equation 134145
- 4.9. Remarks on other applications 145156
- 4.9.1. Retarded equations with infinite delays 145156
- 4.9.2. Strongly damped quasilinear evolutionary equations 146157
- 4.9.3. A Beam equation 148159
- 4.9.4. Other hyperbolic systems 151162
- 4.9.5. Kuramoto-Sivashinsky equation 153164
- 4.9.6. A Nonlinear diffusion problem 154165
- 4.9.7. Age-dependent populations 155166

- 4.10. Dependence on parameters and approximation of the attractor 160171

- Appendix. Stable and Unstable Manifolds 179190
- References 187198
- Index 197208