Optimal Decay Rate of Solutions to Timoshenko System with Past History in Unbounded Domains

Abstract

In this paper, we investigate the Cauchy problem for the Timoshenko system in thermo-elasticity, where the heat conduction is given by the Gurtin–Pipkin thermal law in one-dimensional space. We show an optimal decay rate of the $L^2$-norm of the solution with the rate of $(1+t{)}^{-1/8}$ which is better than $(1+t{)}^{-1/12}$ found in [6]. We also extend the recent results in [7] and [8] and showed that those results are only particular cases of the one obtained here. Also, we prove that the decay rate is controlled by a crucial stability number ${\alpha }_g$ which depends on the parameters of the system.