Evolution semigroups for nonautonomous Cauchy problems

Abstract

In this paper, we characterize wellposedness of nonautonomous, linear Cauchy problems $$\left(NCP\right)\{\begin{array}{l}\dot{u}\left(t\right)=A\left(t\right)u\left(t\right)\\ u\left(s\right)=x\in X\end{array}$$ on a Banach space $X$ by the existence of certain evolution semigroups.

Then, we use these generation results for evolution semigroups to derive wellposedness for nonautonomous Cauchy problems under some “concrete” conditions. As a typical example, we discuss the so called “parabolic” case.