Dynamical behavior of the indirectly and locally memory-damped Timoshenko system.

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. 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 system, 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 compared with the ones in the literature, we still derive stronger conclusions. Finally, we present some results of numerical simulation to illustrate quantitatively the behavior of the solution energies of our system, which agree with our theoretical results well. • Timoshenko system in the paper is controlled only by an indirect local memory damping. • Ideal asymptotic decay results are established for the long-term dynamical behavior of the Timoshenko system. • The obtained theorems 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. • Although our conditions on the memory kernels are weaker compared with the ones in the literature, we still derive stronger conclusions. • Numerical simulation results agree with our theoretical results well. [ABSTRACT FROM AUTHOR]