On the asymptotic behavior of solutions to a structure acoustics model

TL;DR

This paper studies the long-term behavior of solutions to a coupled structural acoustic model involving a semilinear wave and Berger plate equations, establishing conditions for global existence and decay rates of energy.

Contribution

It introduces a novel stabilization estimate that simplifies the analysis and extends understanding of energy decay in coupled acoustic-structural systems.

Findings

Global existence of potential well solutions

Exponential or algebraic energy decay rates

Stabilization estimate without lower-order terms

Abstract

This article concerns the long term behavior of solutions to a structural acoustic model consisting of a semilinear wave equation defined on a smooth bounded domain $\Omega\subset\mathbb{R}^3$ which is coupled with a Berger plate equation acting on a flat portion of the boundary of $\Omega$. The system is influenced by several competing forces, in particular a source term acting on the wave equation which is allowed to have a supercritical exponent. Our results build upon those obtained by Becklin and Rammaha [8]. With some restrictions on the parameters in the system and with careful analysis involving the Nehari manifold we obtain global existence of potential well solutions and establish either exponential or algebraic decay rates of energy, dependent upon the behavior of the damping terms. The main novelty in this work lies in our stabilization estimate, which notably does not generate lower-order terms. Consequently, the proof of the main result is shorter and more concise.