Non-Well-Posed Problems, Unstable Manifolds, Lyapunov Functions, and Lower Bounds on Dimensions

Abstract

This corresponds to the case where the semigroup S(t) is not defined everywhere, or is not defined for all times t. In that case, a concept of functionalinvariant sets can be introduced by modifying slightly that given in Chapter I. Of course, we can no longer expect to prove the existence of absorbing sets and attractors, and the results of Chapter I do not apply anymore. However, the results of Chapter V concerning the dimensions of invariant sets have been proved at a level of generality which makes them applicable to such situations. Some examples are given without any attempt at generality.