On the stability of Bresse and Timoshenko systems with hyperbolic heat conduction

TL;DR

This paper analyzes the stability and decay rates of various thermoelastic beam systems with hyperbolic heat conduction, providing necessary and sufficient conditions for exponential and polynomial stability, and recovering classical stability conditions.

Contribution

It offers a comprehensive stability analysis for Bresse and Timoshenko systems with hyperbolic heat conduction, including new stability conditions and decay rate characterizations.

Findings

Exponential stability condition for Bresse-Gurtin-Pipkin system.

Optimal polynomial decay rates when exponential stability fails.

Full stability characterization of Timoshenko-Maxwell-Cattaneo system.

Abstract

We investigate the stability of three thermoelastic beam systems with hyperbolic heat conduction. First, we study the Bresse-Gurtin-Pipkin system, providing a necessary and sufficient condition for the exponential stability and the optimal polynomial decay rate when the condition is violated. Second, we obtain analogous results for the Bresse-Maxwell-Cattaneo system, completing an analysis recently initiated in the literature. Finally, we consider the Timoshenko-Gurtin-Pipkin system and we find the optimal polynomial decay rate when the known exponential stability condition does not hold. As a byproduct, we fully recover the stability characterization of the Timoshenko-Maxwell-Cattaneo system. The classical "equal wave speeds" conditions are also recovered through singular limit procedures. Our conditions are compatible with some physical constraints on the coefficients as the positivity of the Poisson's ratio of the material. The analysis faces several challenges connected with the thermal damping, whose resolution rests on recently developed mathematical tools such as quantitative Riemann-Lebesgue lemmas.