Energy decay estimates of elastic transmission wave/beam systems with a local Kelvin-Voigt damping

We consider the beam equation coupled by a transmission condition with a wave equation in an elastic beam. The beam has clamped boundary conditions and the wave equation has Dirichlet boundary conditions. The damping which is locally distributed acts through one of the equations only; its effect is transmitted to the other equation through the coupling. First, we consider the case where the dissipation acts through the beam equation. We show that in this case the coupled system is polynomially stable by using a recent result of Borichev and Tomilov on polynomial decay characterisation of bounded semigroups, we provide precise decay estimates showing that the energy of this coupled system decays polynomially as the time variable goes to infinity. Second, we discuss the case where the damping acts through the wave equation. Proceeding as in the first case, we prove that this system is also polynomially stable and we provide precise polynomial decay estimates for its energy. Finally, we show the lack of uniform exponential decay of solutions for both models.