Stabilization of nonlinear Timoshenko system just with local damping and local coupling effects acting on arbitrarily chosen subintervals

We are concerned with the stabilization of nonlinear Timoshenko system just with local damping and local coupling effects, which come from the local memory/frictional terms and local coupling terms, acting on arbitrarily chosen subintervals. Especially, the subintervals do not necessarily include the ends and intersect each other, unlike what is usually required. For this challenging problem, we successfully establish a stability theorem with the exact uniform decay rates for the solutions to this system with the help of some specially designed analysis approaches for the problem. Also, polynomial/exponential decay rates are obtained, when the memory kernels decay polynomially/exponentially. Our results reveal an important phenomenon that for the nonlinear Timoshenko system, local coupling and damping effects are enough to produce an entire dissipation mechanism as those for the case of the global coupling and damping effects, and the same decay rates can be obtained. Moreover, under quite weak conditions on the memory kernels, our stability theorem gives stronger conclusions than the previous corresponding results, whether global or local memory effect is concerned.