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

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 C0-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→∞. In particular, we obtain a sharp estimate for the rate of energy decay of classical solutions to the problem.