Optimal decay rate of solutions to Timoshenko system with past history in unbounded domains (Q1624157)

Summary: 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 \((1 + t)^{-\frac{1}{8}}\) which is better than \((1 + t)^{-\frac{1}{12}}\) found in our work [Appl. Math. Optim. 75, No. 3, 403--428 (2017; Zbl 1378.35297)]. We also extend the recent results in [\textit{N. Mori} and \textit{S. Kawashima}, J. Hyperbolic Differ. Equ. 11, No. 1, 135--157 (2014; Zbl 1288.35084); Anal. Appl., Singap. 14, No. 3, 393--413 (2016; Zbl 1338.35052)] 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.