Perturbation of Analytic Semigroups and Applications to Partial Differential Equations

TL;DR

This paper improves perturbation results for analytic semigroups, simplifying conditions for PDE applications, and demonstrates their effectiveness on complex differential operators and boundary conditions.

Contribution

It replaces complex conditions with simpler assumptions for perturbation of analytic semigroups, enhancing applicability to PDEs.

Findings

Simplified perturbation conditions for analytic semigroups.

Application to degenerate differential operators with boundary conditions.

Analysis of reaction-diffusion equations with unbounded delays.

Abstract

In a recent paper we presented a general perturbation result for generators of $C_0$-semigroups. The aim of the present paper is to replace, in case the unperturbed semigroup is analytic, the various conditions appearing in this result by simpler assumptions on the domain and range of the operators involved. The power of our result to treat classes of PDE's systematically is illustrated by considering a generic example, a degenerate differential operator with generalized Wentzell boundary conditions and a reaction diffusion equation subject to Neumann boundary conditions with distributed unbounded delay.