Effects of Kelvin–Voigt Damping on the Stability of (Thermo)Elastic Timoshenko System with Second Sound

In this paper, we investigate the stabilization of Timoshenko systems with or without Kelvin–Voigt damping, in the presence of second sound. We have drawn conclusions that provide new insights into the effect of local Kelvin–Voigt damping. Our primary conclusion is that the exponential stability of the system achieved through local thermal damping will be destroyed by the local Kelvin–Voigt damping with a discontinuous coefficient at the interface. In particular, when local dissipation solely originates from the second sound, the system exhibits exponential stability. However, when Kelvin–Voigt damping is introduced into the local thermoelastic component, the system, surprisingly, loses its exponential stability. In this case, the corresponding solution decays polynomially with the rate t-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^{-2}$$\end{document}.