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

TL;DR

This paper analyzes the stability of a coupled structural acoustics PDE system with thermal effects, showing that solutions decay uniformly over time without additional damping, under certain geometric conditions.

Contribution

It provides the first stability analysis of a coupled PDE model with boundary thermal damping and no extra dissipation, using microlocal boundary trace estimates.

Findings

Solutions decay uniformly to zero over time.

Decay rate is rational under geometric assumptions.

No additional damping needed for stability.

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 $% \Omega $. This acoustic wave equation is coupled on a boundary interface ($% \Gamma _{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 $\Gamma _{0}$ of the chamber $\Omega $; the elastic dynamics is coupled to a heat equation which also evolves on $\Gamma _{0}$, 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 $% \Gamma _{0}$, 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 $\Gamma _{1}$ of the boundary $\partial \Omega $. Under a certain geometric assumption on $\Gamma _{1}$, 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.