Do equal speed condition and exponential stability relate for the truncated thermoelastic Timoshenko system under Green Naghdi law?

Abstract Over the years, the stabilization Timoshenko systems with dissipative features have piqued the interest of researchers. The study of Timoshenko systems under various damping effects has resulted in a significant number of studies. When nonphysical assumptions of equal wave velocities are used in stabilization, the expected exponential decay of the energy solution is attained in all recent research. In this study, we analyze a one-dimensional thermooelastic Timoshenko type system in the setting of the second frequency spectrum, where the assumption of equal wave speed is not required for exponential decay to occur. In fact, According to Elishakoff’s studies [Elishakoff, Advances in Mathematical Modeling and Experimental Methods for Materials and Structures: The Jacob Aboudi Volume, Dordrecht, The Netherlands: Springer, pp. 249–254, 2009.], we consider the so-called truncated version of the Timoshenko system, and we added a thermoelastic damping according to Green Naghdi law of heat conduction. We first use Faedo–Galerkin approximation to verify the system’s global well-posedness. Using a Lyapunov functional we establish an exponential stability without assuming the condition of equal wave speed. A numerical scheme is introduced and analyzed. Finally, assuming extra regularity on the solution, we get some a priori error estimates and we present some numerical results which demonstrate the exponential behavior of the solution. This result significantly improves the previous results in the literature in which equal wave velocities are used to obtain exponential stability.