Evolution Semigroups for Nonautonomous Cauchy Problems

In this paper, we characterize wellposedness of nonautonomous, linear Cauchy problems (NCP ) { u̇(t) = A(t)u(t) u(s) = x ∈ X 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. 1. Basic definitions In this section, we introduce the basic definitions and notations in order to discuss nonautonomous Cauchy problems in terms of evolution families and evolution semigroups. In addition, we mention some of their fundamental properties. The solution of a nonautonomous Cauchy problem on some Banach space X can be given by a so called evolution family which can be defined as follows. Definition 1.1 (Evolution family). A family (U(t, s))t≥s of linear, bounded operators on a Banach space X is called an (exponentially bounded) evolution family if (i) U(t, r)U(r, s) = U(t, s), U(t, t) = Id for all t ≥ r ≥ s ∈ IR, (ii) the mapping (t, s) → U(t, s) is strongly continuous, (iii) ‖U(t, s)‖ ≤ Meω(t−s) for some M ≥ 1, ω ∈ IR and all t ≥ s ∈ IR. 1991 Mathematics Subject Classification. Primary 47D06; secondary 47A55.