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Functional evolution equations are used in the modeling of numerous physicalprocesses. In this work, our main tool is perturbation theory of stronglycontinuous semigroups. The advantage of this technique is that one can providefunctional evolution equations with the explicit representation formulas of thesolution. First, we introduce a closed form of the fundamental solution of theevolution equation with a discrete delay using the delayed Dyson-Phillipsseries. Then we set up the analytical representation formulas of the classicalsolutions of linear homogeneous/non-homogeneous evolution equations with aconstant delay in a Banach space. In the special case, when a stronglycontinuous group $\left\lbrace \mathcal{T}(t)\right\rbrace_{t\in \mathbb{R}}$commutes with a bounded linear operator $A_{1}$, we obtain an elegant formulafor the fundamental solution using the powers of the resolvent operator of$A_{0}$.  Furthermore, we consider delay evolution equations withpermutable/non-permutable linear bounded operators and derive crucial resultsin terms of non-commutative analysis. Finally, we present an example, in thecontext of a one-dimensional heat equation with a discrete delay to demonstratethe applicability of our theoretical results and give some comparisons withexisting results.

Perturbation theory and linear partial differential equations with delay