In this paper, we consider an elastodynamic system with acoustic boundary conditions and localized internal damping, defined on a smooth bounded domain \(\Omega \) of \(\mathbb {R}^3\) , with smooth boundary \(\Gamma =\overline{\Gamma }_0\cup \overline{\Gamma }_1\) such that \({\Gamma _0}\cap {\Gamma _1}=\emptyset \) . On \(\Gamma _0\) , we consider the homogeneous Dirichlet boundary condition, while on \(\Gamma _1\) , we consider a non-locally reacting acoustic boundary condition. By assuming that the internal damping is only acting on a subregion of the domain and satisfying suitable assumptions, existence and uniqueness as well as exponential stability results are established. More precisely, semigroup techniques are used to establish well-posedness, while the asymptotic behavior of solutions is derived through the multiplier method. Notably, these stability results are established without imposing restrictive geometric constraints on the controlled portion of the boundary. This allows us to extend previous findings that were traditionally limited to star-shaped domains. 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 Vicenté[14], and new technical arguments.
