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

TL;DR

This paper establishes conditions under which the energy of the damped wave equation with a confining potential in Euclidean space decays exponentially, using semiclassical analysis and geometric conditions.

Contribution

It introduces a new geometric condition accounting for turning points, extending the geometric control condition to non-compact Euclidean settings.

Findings

Exponential energy decay under specific geometric conditions

Identification of a new geometric condition involving turning points

Application of semiclassical analysis to wave propagation

Abstract

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.