Long-time behavior for a nonlinear Timoshenko system: Thermal damping versus weak damping of variable-exponents type

In this work, we consider a nonlinear thermoelastic Timoshenko system with a time-dependent coefficient where the heat conduction is given by Coleman-Gurtin [1]. Consequently, the Fourier and Gurtin-Pipkin laws are special cases. We prove that the system is exponentially and polynomially stable. The equality of the wave speeds is not imposed unless the system is not fully damped by the thermoelasticity effect. In other words, the thermoelasticity is only coupled to the first equation in the system. By constructing a suitable Lyapunov functional, we establish exponential and polynomial decay rates for the system. We noticed that the decay sometimes depends on the behavior of the thermal kernel, the variable exponent, and the time-dependent coefficient. Our results extend and improve some earlier results in the literature especially the recent results by Fareh [2], Mustafa [3] and Al-Mahdi and Al-Gharabli [4].