Automatic Well-Posedness with the Abstract Cauchy Problem on a Fréchet Space

Abstract

For an arbitrary closed linear operator, A, on a Fréchet space, we introduce what we shall call its solution space. This is a Fréchet space that contains all initial data for which the corresponding abstract Cauchy problem has a unique global mild solution. We show that A, restricted to this space, generates a locally equicontinuous strongly continuous semigroup. Corollaries include an almost immediate proof of the fundamental relationship between generating a strongly continuous semigroup and having a unique mild solution, for all initial data. More generally, we show how the solution space may be used to present a simplified and unified approach to different types of semigroups and their relationships to each other and the abstract Cauchy problem.