Perturbation and an abstract characterization of evolution semigroups (Q1916797)

Evans and Howland used an abstract characterization of evolution semigroups and a perturbation result for \({\mathcal C}_0\) 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 \((U(t,s))_{t\geq s}\) defined on the Banach space \(X\). From this and a perturbation result of Voigt, we obtain a sufficient condition for an operator on \(L^p(\mathbb{R}, X)\) to be the generator of an evolution semigroup. This will be applied to non-autonomous heat equations with absorption.