Semigroups and linear partial differential equations with delay

TL;DR

This paper establishes a framework linking delayed partial differential equations to abstract Cauchy problems, providing conditions for well-posedness, stability, and continuity, with applications to reaction-diffusion systems.

Contribution

It introduces an equivalence between delayed PDEs and abstract Cauchy problems, enabling new analysis methods for stability and well-posedness.

Findings

Conditions for well-posedness derived

Exponential stability criteria established

Applications to reaction-diffusion equations included

Abstract

We prove the equivalence of the well-posedness of a partial differential equation with delay and an associated abstract Cauchy problem. This is used to derive sufficient conditions for well-posedness, exponential stability and norm continuity of the solutions. Applications to a reaction-diffusion equation with delay are given.