  We consider a model of an elastic body immersed between two layers of
incompressible viscous fluid. The elastic displacement $w$ is governed by the
damped wave equation $w_{tt} + \alpha w_t + \Delta w =0$ without any
stabilization terms, where $\alpha >0$, and the fluid is modeled by the
Navier-Stokes equations. We assume continuity of the displacement and the
stresses across the moving interfaces and homogeneous Dirichlet boundary
conditions on the outer fluid boundaries. We establish a~priori estimates that
provide the global-in-time well-posedness and exponential decay to a final
state of the system for small initial data. We prove that the final state must
be trivial, except for a possible small displacement of the elastic structure
in the horizontal direction.
