On the time behavior of a porous thermoelastic system with only thermal dissipation given by Gurtin-Pipkin law

TL;DR

This paper investigates the stability and decay properties of a porous thermoelastic system with thermal dissipation modeled by the Gurtin-Pipkin law, extending previous results for classical heat conduction laws.

Contribution

It introduces a stability number and proves exponential stability when this number is zero, generalizing prior work to the Gurtin-Pipkin thermal law.

Findings

Exponential stability occurs if the stability number is zero.

Lack of exponential decay if the stability number is non-zero.

Generalizes previous results for Fourier and Cattaneo laws.

Abstract

In the present paper we consider a porous thermoelastic system with only one dissipative mechanism generated by the heat conductivity modelled by the Gurtin-Pipkin thermal law. By the use of a semigroup approach and the Lumer-Phillips theorem we prove the existence of a unique solution. We introduce a stability number $\chi_g$ depends on the coefficients of the system, and establish an exponential stability result provided that $\chi_g=0$. Otherwise, if $\chi_g\ne 0$, we prove the lack of exponential decay. Our result improves and generalizes the previous results in the literature obtained for Fourier's and Cattaneo's laws of thermal conductivity.