Stability of Transmission Wave-Plate Equations with Local Indirect Damping

This paper is devoted to the energy decay estimates for the coupled wave-plate system with local frictional damping in a bounded domain. The frictional damping distributes in the plate or wave domain. When the frictional damping is only acting through the plate equation, the transmission system is showed not to be exponentially stable. By the frequency domain approach and multiplier technique, it is found that the system energy has the polynomial decay rate. On the other hand, when the frictional damping only act through the wave equation, it is showed that the system energy has the exponential decay. It can be seen that only the frictional damper acting on the wave equation can stabilize exponentially the transmission plate-wave equations.