Uniform stability of the damped wave equation with a confining potential in the Euclidean space

We investigate trend to equilibrium for the damped wave equation with a confining potential in the Euclidean space. We provide with necessary and sufficient geometric conditions for the energy to decay exponentially uniformly. The proofs rely on tools from semiclassical analysis together with the construction of quasimodes of the damped wave operator. In addition to the Geometric Control Condition, which is familiar in the context of compact Riemannian manifolds, our work involves a new geometric condition due to the presence of turning points in the underlying classical dynamics which rules the propagation of waves in the high-energy asymptotics.