Remarks on Exponential Stability for a Coupled System of Elasticity and Thermoelasticity with Second Sound

TL;DR

This paper analyzes the exponential stability of a coupled thermoelastic and elastic system with second sound, proving solutions approach equilibrium exponentially without large wave speed assumptions, using semigroup theory.

Contribution

It removes previous large wave speed restrictions by establishing exponential decay for the coupled system using resolvent bounds and semigroup methods.

Findings

Solutions approach equilibrium exponentially over time.

Uniform resolvent bounds are established for the system generator.

Exponential stability is proven even without large wave speed assumptions.

Abstract

We study the large time behavior of solutions to a linear transmission problem in one space dimension. The problem at hand models a thermoelastic material with second sound confined by a purely elastic one. We shall characterize all equilibrium states of the considered system and prove that every solution approaches one designated equilibrium state with an exponential rate as time goes to infinity. Hereto, we apply methods from the theory of strongly continuous semigroups. In particular, we obtain uniform resolvent bounds for the underlying generator. This removes the largeness assumption of elastic wave speeds imposed in [Y.P. Meng and Y.G. Wang, Anal. Appl. (Singap.) 13 (2015)] for having an exponential energy decay rate when the problem only has the trivial equilibrium. In an appendix we provide a similar exponential stability result for the case where heat conduction is modeled using Fourier's law.