Optimal decay for a wave-heat system with Coleman-Gurtin thermal law

TL;DR

This paper analyzes the long-term energy decay of a coupled wave-heat system with Coleman-Gurtin thermal law, demonstrating polynomial stability with an optimal decay rate using semigroup theory.

Contribution

It introduces a novel feedback interconnection framework and proves polynomial stability with the optimal decay rate for the coupled system.

Findings

Semigroup representing the system is polynomially stable.

Energy decay rate is optimally estimated as t^{-2}.

System stability is characterized within the Dafermos history framework.

Abstract

We study the long-term behaviour of solutions to a one-dimensional coupled wave-heat system with Coleman-Gurtin thermal law. Our approach is based on the asymptotic theory of $C_0$-semigroups and recent results developed for coupled control systems. As our main results, we represent the system as a feedback interconnection between the wave part and the Coleman-Gurtin part and we show that the associated semigroup in the history framework of Dafermos is polynomially stable with optimal decay rate $t^{-2}$ as $t\to\infty$. In particular, we obtain a sharp estimate for the rate of energy decay of classical solutions to the problem.