Indirect stabilization of Timoshenko systems with one local memory damping

Abstract

We investigate the indirect stabilization of Timoshenko systems with only one local memory effect. By the multiplier method and some specially designed analysis approaches for the problem, we obtain the exponential decay rates for the case in which the wave speeds are the same. Of particular notes are: This is a local memory damping problem for coupled systems without complementary frictional damping (or any other type of damping); It is also an indirect damping problem, which has memory damping only in the second equation and no damping in the first equation. The decay results here are established without the usual nonnegative/decreasing assumptions for the kernel, that is, the kernel may be oscillating or sign-varying. As can be seen, we weaken the hypotheses of the kernel and the memory-effect area, but we still establish the exponential decay rates of the energy. Therefore, we have improved and extended the related results to a considerable extent.