Regularity of the Gurtin-Pipkin equation

TL;DR

This paper investigates the regularity properties of solutions to the Gurtin-Pipkin equation, demonstrating their smoothness in Sobolev spaces and analyzing the smoothness of their perturbation relative to wave equation solutions.

Contribution

It establishes the Sobolev space regularity of solutions to the Gurtin-Pipkin equation and compares their smoothness to wave equation solutions.

Findings

Solutions are smooth in Sobolev spaces.

The difference between the solution and wave equation solution is smoother.

Regularity results are proved for the first-order Gurtin-Pipkin equation.

Abstract

We study regularity of the solution $\theta$ to the Gurtin-Pipkin integral-differential equation of the first order in time. The solution smoothness in Sobolev spaces is proved. Also it is proved that the 'perturbation' part, namely, the difference of $\theta$ and the solution to the corresponding wave equation is smoother than $\theta$.