Strong and weak solutions to a structural acoustic model with a $C^1$ source term on the plate

Abstract

In this manuscript, we consider a structural acoustic model consisting of a wave equation defined in a bounded domain $\Omega \subset \mathbb{R}^3$, strongly coupled with a Berger plate equation acting on the flat portion of the boundary of $\Omega$. The system is influenced by an arbitrary $C^1$ nonlinear source term in the plate equation. Using nonlinear semigroup theory and monotone operator theory, we establish the well-posedness of both local strong and weak solutions, along with conditions for global existence. With additional assumptions on the source term, we examine the Nehari manifold and establish the global existence of potential well solutions. Our primary objective is to characterize regimes in which the system remains globally well-posed despite arbitrary growth of the source term and the absence of damping mechanisms to stabilize the dynamics.