Stability of the Abstract Thermoelastic System with Singularity

TL;DR

This paper investigates the stability and decay rates of an abstract thermoelastic system governed by the Cattaneo law, revealing conditions for spectral properties and optimal decay without inertial effects.

Contribution

It provides new spectral analysis and decay rate results for thermoelastic systems with Cattaneo law, including optimal decay without inertial terms.

Findings

Zero is a spectrum point under certain parameter conditions.

Decay rates are established for systems with and without inertial terms.

Decay rate without inertial terms is proven to be optimal.

Abstract

In this paper, we analyze an abstract thermoelastic system, where the heat conduction follows the Cattaneo law. Zero becomes a spectrum point of the system operator when the coupling and thermal damping parameters of system satisfy specific conditions. We obtain the decay rates of solutions to the system with or without the inertial term. Furthermore, the decay rate of the system without inertial terms is shown to be optimal.