Global Attractors for Autonomous Evolution Equations

Abstract

Chapter 2 is concerned with large time behaviour of solutions of evolution equations in terms of the global attractor, its existence and properties. Note that, good estimates on the dimension of attractors in terms of biological (medical, physical etc.) parameters are crucial for the finite-dimensional reduction and at present there exists a highly developed machinery for obtaining such estimates. The best known estimates are usually obtained by the so-called volume contraction method which is based on the differentiability of associated semigroup. We especially emphasize that, for a quite large class of degenerate parabolic systems arising in the modelling of life science problems the associate semigroup is not differentiable. In Chap. 2 we present theorem that play decisive role in the study of dimension of attractor, which in turn does not require differentiability of associated semigroup. Moreover, the Kolmogorov entropy and its asymptotics in functional spaces are presented in Chap. 2 as well.