Maximal $L^p$-regularity for perturbed evolution equations in Banach spaces

TL;DR

This paper studies conditions under which maximal $L^p$-regularity is maintained for perturbed evolution equations in Banach spaces, extending previous results to various classes of perturbations including boundary and integro-differential cases.

Contribution

It introduces new conditions ensuring preservation of maximal $L^p$-regularity under Miyadera-Voigt, Desch-Schappacher, and Staffans-Weiss perturbations, extending prior work.

Findings

Established preservation of maximal $L^p$-regularity for boundary perturbed heat equations.

Extended results to perturbed boundary integro-differential equations.

Provided examples demonstrating applicability to different perturbation classes.

Abstract

The main purpose of this paper is to investigate the concept of maximal $L^p$-regularity for perturbed evolution equations in Banach spaces. We mainly consider three classes of perturbations: Miyadera-Voigt perturbations, Desch-Schappacher perturbations, and more general Staffans-Weiss perturbations. We introduce conditions for which the maximal $L^p$-regularity can be preserved under these kind of perturbations. We give examples for a boundary perturbed heat equation in $L^r$-spaces and a perturbed boundary integro-differential equation. We mention that our results mainly extend those in the works: [P. C. Kunstmann and L. Weis, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30 (2001), 415-435] and [B.H. Haak, M. Haase, P.C. Kunstmann, Adv. Differential Equations 11 (2006), no. 2, 201-240].