Dynamical Behavior of the Indirectly and Locally Memory-Damped Timoshenko System

Abstract

We are concerned with dynamical behavior of the indirectly and locally memory-damped Timoshenko system. The polynomial/exponential decay results for the long-term dynamical behavior of the Timoshenko system are established when the memory kernels decay polynomially/exponentially. As one can see, to obtain ideal asymptotic decay rates under basic conditions, we take some analysis processes specially designed for our issues. The obtained long-term dynamical behavior theorems, with the exact uniform decay rates for the solutions to the systems, indicate that for the Timoshenko system with indirect memory damping, a local memory effect is enough to produce an entire dissipation mechanism and to ensure the same decay rates as in the case of global memory effect. Moreover, although our conditions on the memory kernels are weaker than before, we still derive stronger conclusions than the former ones. Finally, we present some results of numerical simulation to illustrate quantitatively the behavior of the solution energies of the system under study, which verify our theoretical results well.