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Zeitschrift für Analysis und ihre Anwendungen


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Volume 37, Issue 4, 2018, pp. 435–459
DOI: 10.4171/ZAA/1622

Published online: 2018-10-18

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

Maisa Khader[1] and Belkacem Said-Houari[2]

(1) Princess Sumaya University of Technology, Amman, Jordan
(2) University of Sharjah, Sharjah, United Arab Emirates

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.

Keywords: Timoshenko system, decay, regularity loss, heat conduction, Lyapunov functional

Khader Maisa, Said-Houari Belkacem: Optimal Decay Rate of Solutions to Timoshenko System with Past History in Unbounded Domains. Z. Anal. Anwend. 37 (2018), 435-459. doi: 10.4171/ZAA/1622