Non-exponential stability to a Timoshenko system with heat conduction and Kelvin–Voigt damping

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We consider a Timoshenko system with partially Kelvin–Voigt damping, and with Cattaneo/Fourier type heat conduction. When the Timoshenko system with only heat conduction or only partially Kelvin–Voigt damping, some authors showed the non-exponential stability. In this paper, we add the Kelvin–Voigt damping in the thermoelastic Timoshenko system, and prove that the system is also not exponentially stable in cases of Fourier and Cattaneo type heat conduction, whether the wave speeds are equal or not. Additionally, we prove the semigroup decays with the optimal rate t−12.