Stabilization of the variable-coefficient structural acoustic model with curved middle surface and delay effects in the structural component

Abstract

This paper is concerned with the uniform energy decay rates of a structural acoustic model which describes the interactions between the acoustic medium and the elastic structural in an acoustic chamber where one “wall” is flexible and curved. The coupled system considered consists of a variable-coefficient wave equation and a variable-coefficient plate equation defined on the Riemannian manifold with the coupling on the interface between the acoustic medium and the flexible wall. Both the components of the dynamics are subject to nonlinear boundary damping. Furthermore, a nonlinear delay term acting in the boundary feedbacks of the structure component is considered. The Riemannian geometry method is applied to derive the energy estimates of the nonlinear coupled system with variable coefficients. The uniform energy decay rates of the nonlinear variable-coefficient structural acoustic system with delay effects on the Riemannian manifold are quantified by a solution to a constructed nonlinear ODE.