  This article concerns the long term behavior of solutions to a structural
acoustic model consisting of a semilinear wave equation defined on a smooth
bounded domain $\Omega\subset\mathbb{R}^3$ which is coupled with a Berger plate
equation acting on a flat portion of the boundary of $\Omega$. The system is
influenced by several competing forces, in particular a source term acting on
the wave equation which is allowed to have a supercritical exponent.
  Our results build upon those obtained by Becklin and Rammaha [8]. With some
restrictions on the parameters in the system and with careful analysis
involving the Nehari manifold we obtain global existence of potential well
solutions and establish either exponential or algebraic decay rates of energy,
dependent upon the behavior of the damping terms. The main novelty in this work
lies in our stabilization estimate, which notably does not generate lower-order
terms. Consequently, the proof of the main result is shorter and more concise.
