Stability of an elastodynamic system with localized internal damping and acoustic boundary conditions

In this paper, we prove a stability result for an elastodynamic system with acoustic boundary conditions and localized internal damping, defined in a bounded domain $\Omega$ of $\mathbb{R}^3$. Here, the internal damping is only assumed to be locally distributed and satisfies suitable assumptions. The smooth boundary of $\Omega$ is $\Gamma=\Gamma_0\cup\Gamma_1$ such that $\overline{\Gamma_0}\cap\overline{\Gamma_1}=\emptyset$. On $\Gamma_0$, we consider the homogeneous Dirichlet boundary condition, and on $\Gamma_1$ , we consider the acoustic boundary condition without a damping term. More precisely, by making use of semigroup techniques, well-posedness results are discussed, as well as the asymptotic behavior of solutions. The difficulty in establishing the stability of the system arises from the presence of higher-order operators, normal derivatives, and some boundary terms. The key tools combine the multiplier approach, trace theorems, ideas from Frota and Vicent\'e \cite{FrotaVicente2018}, and new technical arguments.