Spectral analysis of Volterra integrodifferential equations with the kernels, depending on parameter

TL;DR

This paper conducts spectral analysis of operator-functions related to Gurtin-Pipkin integrodifferential equations, examining their solvability in Sobolev spaces and the impact of kernels depending on parameters.

Contribution

It provides a spectral analysis of operator-functions for Gurtin-Pipkin equations with parameter-dependent kernels and studies their solvability in Sobolev spaces.

Findings

Spectral properties of operator-functions are characterized.

Correct solvability in Sobolev space $W_{2}^{2}((0, T), A^2)$ is established.

Results hold for arbitrary time interval T.

Abstract

Spectral analysis of operator-functions which are the symbols of the abstract integrodifferential equations of the Gurtin-Pipkin is provided. These equations represent abstract wave equations disturbed by terms involving Volterra operators. Correct solvability in the Sobolev space $W_{2}^{2}((0, T), A^2)$, for arbitrary $T>0$, of that abstract integrodifferential equations is also studied.