Blow-up of a structural acoustics model

TL;DR

This paper investigates conditions under which solutions to a coupled structural acoustics model blow up in finite time, especially when the source forces dominate damping, considering complex initial energy scenarios.

Contribution

It establishes blow-up results for a coupled wave-plate system with supercritical source terms, addressing the challenge of nonlinear coupling and non-Lipschitz source operators.

Findings

Blow-up occurs when source terms outweigh damping.

Negative initial energy leads to blow-up.

Small positive initial energy with large quadratic energy also causes blow-up.

Abstract

This article studies the finite time blow-up of weak solutions to a structural acoustics model consisting of a semilinear wave equation defined on a bounded domain $\Omega\subset\mathbb{R}^3$ which is strongly coupled with a Berger plate equation acting on the elastic wall, namely, a flat portion of the boundary. The system is influenced by several competing forces, including boundary and interior source and damping terms. We stress that the power-type source term acting on the wave equation is allowed to have a supercritical exponent, in the sense that its associated Nemytskii operators is not locally Lipschitz from $H^1$ into $L^2$. In this paper, we prove the blow-up results for weak solutions when the source terms are stronger than damping terms, by considering two scenarios of the initial data: (i) the initial total energy is negative; (ii) the initial total energy is positive but small, while the initial quadratic energy is sufficiently large. The most significant challenge in this work arises from the coupling of the wave and plate equations on the elastic wall.