Stability and Optimal Decay Rates for Abstract Systems with Thermal Damping of Cattaneo's Type

TL;DR

This paper investigates the stability and decay rates of an abstract thermoelastic system with Cattaneo's law, identifying parameter regions with distinct polynomial decay behaviors and proving their optimality.

Contribution

It introduces a parameter region framework for analyzing stability and decay rates in thermoelastic systems with thermal damping of Cattaneo's type, including spectral analysis and optimality results.

Findings

Distinct polynomial decay rates are established for different parameter subregions.

Optimality of the decay rates is rigorously proved.

Applications to coupled PDE systems demonstrate the theoretical results.

Abstract

This paper studies the stability of an abstract thermoelastic system with Cattaneo's law, which describes finite heat propagation speed in a medium. We introduce a region of parameters containing coupling, thermal dissipation, and possible inertial characteristics. The region is partitioned into distinct subregions based on the spectral properties of the generator of the corresponding semigroup. By a careful estimation of the resolvent operator on the imaginary axis, we obtain distinct polynomial decay rates for systems with parameters located in different subregions. Furthermore, the optimality of these decay rates is proved. Finally, we apply our results to several coupled systems of partial differential equations.