Perturbation and an abstract characterization of evolution semigroups

Abstract

Evans and Howland used an abstract characterization of evolution semigroups and a perturbation result for script C sign0-semigroups to treat non-autonomous Schrödinger equations and problems in scattering theory. Here we proceed in a similar way. At first we characterize evolution semigroups on spaces of vector-valued functions E(X) which are induced by a strongly continuous evolution family (U(t, s))t ≥ s defined on the Banach space X. From this and a perturbation result of Voigt we obtain a sufficient condition for an operator on LP(ℝ, X) to be the generator of an evolution semigroup. This will be applied to non-autonomous heat equations with absorption. © 1996 Academic Press, Inc.