Uniform stability of transmission of wave-plate equations with source on Riemannian manifold

This paper is concerned with the semilinear transmission of wave-plate system with source term on Riemannian manifold. We prove the existence of weak solutions by using Faedo-Galerkin's method. Furthermore, by introducing nonlinear boundary feedbacks acting only on plate, we establish the explicit and general decay rates of the system. Our proofs are based on the geometric multiplier method and the Riemannian geometry method.