On the maximal regularity for a class of Volterra integro-differential equations

TL;DR

This paper develops a perturbation theory-based approach to establish maximal L^p-regularity for a class of Volterra integro-differential equations, including applications to heat equations with Laplacian operators.

Contribution

It introduces a novel method combining perturbation theory and semigroup analysis to achieve maximal regularity results for integro-differential equations.

Findings

Maximal L^p-regularity is established for autonomous and non-autonomous equations.

The approach applies to equations involving the left shift semigroup.

Applications to heat equations with Dirichlet or Neumann Laplacian are demonstrated.

Abstract

We propose an approach based on perturbation theory to establish maximal $L^p$-regularity for a class of integro-differential equations. As the left shift semigroup is involved for such equations, we study maximal regularity on Bergman spaces for autonomous and non-autonomous integro-differential equations. Our method is based on the formulation of the integro-differential equations to a Cauchy problems, infinite dimensional systems theory and some recent results on the perturbation of maximal regularity (see \cite{AmBoDrHa}). Applications to heat equations driven by the Dirichlet (or Neumann)-Laplacian are considered.