Stability of Laminated Timoshenko Beams with Local Viscoelastic Versus Frictional Damping

In this paper, we consider a two-layered beam system with an interfacial slip, stabilized only by one viscoelastic vs. frictional damping acting on a small portion of the beam. We show that the local damping is enough to induce the whole dissipation mechanism, and give a general and explicit energy decay rate only under basic conditions on the damping. Meanwhile, we obtain optimal decay rates, when the frictional damping is near linear or polynomial, and the behavior of the memory kernel at infinity is either unquantified or quantified in a quite general way, by means of quantifying the effectiveness of each type of the damping. In order to handle the difficulty caused by the local feature of the damping, we manage to find fitting weighted functions to process region segmentation, as well as to construct appropriate auxiliary functionals. Our results improve and generalize the existing related results for the system to a large extent, and they are novel even for the classical Timoshenko beam system (without slip).