Perturbation theory and linear partial differential equations with delay

TL;DR

This paper develops explicit solution formulas for linear delay evolution equations using perturbation theory of semigroups, including cases with non-commuting operators, and demonstrates applications to heat equations with delay.

Contribution

It introduces a fundamental solution formula for delay evolution equations via the delayed Dyson-Phillips series, extending perturbation methods to non-commutative operator cases.

Findings

Explicit fundamental solution formulas for delay equations

Representation formulas for classical solutions with delays

Application to heat equation with delay demonstrating practical relevance

Abstract

Functional evolution equations are used in the modeling of numerous physical processes. In this work, our main tool is perturbation theory of strongly continuous semigroups. The advantage of this technique is that one can provide functional evolution equations with the explicit representation formulas of the solution. First, we introduce a closed form of the fundamental solution of the evolution equation with a discrete delay using the delayed Dyson-Phillips series. Then we set up the analytical representation formulas of the classical solutions of linear homogeneous/non-homogeneous evolution equations with a constant delay in a Banach space. In the special case, when a strongly continuous group $\left\lbrace \mathcal{T}(t)\right\rbrace_{t\in \mathbb{R}}$ commutes with a bounded linear operator $A_{1}$, we obtain an elegant formula for the fundamental solution using the powers of the resolvent operator of $A_{0}$. Furthermore, we consider delay evolution equations with permutable/non-permutable linear bounded operators and derive crucial results in terms of non-commutative analysis. Finally, we present an example, in the context of a one-dimensional heat equation with a discrete delay to demonstrate the applicability of our theoretical results and give some comparisons with existing results.