An Introduction to Semiflows
This book is devoted to a study of the asymptotic behaviour of semilinear dissipative evolution equations. Attracting sets of interest for such equations include attractors, in particular exponential ones, and inertial manifolds. Existence of these sets is studied in the book. The first two chapters contain a brief introduction to dynamical systems and their dissipativity properties. The core of the book consists of the next three chapters. Existence of global attractors is proved in chapter 3 for a semilinear reaction-diffusion equation and a dissipative wave equation. These results are refined in chapter 4, where geometric conditions for exponential attractivity are described. The regularity (Lipschitz continuity) of attractors is examined in chapter 5 in a general context. The notions of a squeezing property, cone invariance and a spectral gap condition are introduced. With their help, the construction of an inertial manifold is presented.
As concrete examples, semilinear parabolic and hyperbolic equations with one-dimensional space variables are investigated. As a counterpoint, the result (due to Mora and Sola-Morales) on non-existence of inertial manifolds for hyperbolic equations is proved in chapter 7. In chapter 6, various types of attractor are constructed for famous nonlinear equations (Cahn-Hilliard, extensible beam, two dimensional Navier-Stokes and Maxwell).
The book is carefully written and it can be strongly recommended to graduate students as a guide before reading papers oriented to specialists. The book also contains eight short appendices that include basic facts about differential equations and functional analysis. The text will also be of use to researchers in applied fields, since they will find here a rigorous treatment of basic notions and results on dissipative systems together with many references.