Boundary stabilizability of nonlinear structural acoustic models with thermal effects on the interface

Abstract

A three-dimensional structural acoustic model is considered. This model consists of a wave equation defined on a 3-dimensional bounded domain $\text{Ω}$ coupled with a thermoelastic plate equation defined on Γ₀ – a flat surface of the boundary $\text{∂Ω}$. The main issue studied here is that of uniform stabilizability of the overall interactive model. Since the original (uncontrolled) model is only strongly stable, but not uniformly stable, the question becomes: what is the `minimal amount' of dissipation necessary to obtain uniform decay rates for the energy of the overall system? Our main result states that boundary nonlinear dissipation placed only on a suitable portion of the part of the boundary which is complementary to Γ₀, suffices for the stabilization of the entire structure.