The optimal polynomial decay in the extensible Timoshenko system

Abstract

In this paper, we derive the equations that constitute the nonlinear mathematical model of an extensible thermoelastic Timoshenko system. The nonlinear governing equations are derived by applying the Hamilton principle to full von Kármán equations. The model takes account of the effects of extensibility, where the dissipations are entirely contributed by temperature. Based on the semigroups theory, we establish existence and uniqueness of weak and strong solutions to the derived problem. By using a resolvent criterion, developed by Borichev and Tomilov, we prove the optimality of the polynomial decay rate of the considered problem under the condition (65). Moreover, by an approach based on the Gearhart–Herbst–Prüss–Huang theorem, we show the non-exponential stability of the same problem; but strongly stable by following a result due to Arendt–Batty. In the absence of additional mechanical dissipations, the system is often not highly stable. By adding a damping frictional function to the first equation of the nonlinear derived model with extensibility and using the multiplier method, we show that the solutions decay exponentially if Equation (85) holds.