Stability Analysis of Coupled Structural Acoustics PDE Models under Thermal Effects and with no Additional Dissipation

Abstract

In this study we consider a coupled system of partial differential equations (PDE's) which describes a certain structural acoustics interaction. One component of this PDE system is a wave equation, which serves to model the interior acoustic wave medium within a given three dimensional chamber $% Ω$. This acoustic wave equation is coupled on a boundary interface ($% Γ_{0}$) to a two dimensional system of thermoelasticity: this thermoelastic PDE comprises a structural beam or plate equation, which governs the vibrations of flexible wall portion $Γ_{0}$ of the chamber $Ω$; the elastic dynamics is coupled to a heat equation which also evolves on $Γ_{0}$, and which imparts a thermal damping onto the entire structural acoustic system. As we said, the interaction between the wave and thermoelastic PDE components takes place on the boundary interface $% Γ_{0}$, and involves coupling boundary terms which are above the level of finite energy. We analyze the stability properties of this coupled structural acoustics PDE model, in the absence of \ any additive feedback dissipation on the hard walls $Γ_{1}$ of the boundary $\partial Ω$. Under a certain geometric assumption on $Γ_{1}$, an assumption which has appeared in the literature in conection with structural acoustic flow, and which allows for the invocation of a recently derived microlocal boundary trace estimate, we show that classical solutions of this thermally damped structural acoustics PDE decay uniformly to zero, with a rational rate of decay.