Energy decay for the damped wave equation under a pressure condition

TL;DR

This paper proves exponential energy decay for the damped wave equation on negatively curved manifolds, under a pressure condition, by establishing a spectral gap near the real axis.

Contribution

It introduces a new dynamical condition involving topological pressure that guarantees a spectral gap and energy decay for the damped wave equation.

Findings

Spectral gap near the real axis established

Exponential energy decay proven for regular initial data

Decay rate linked to eigenvalues close to the real axis

Abstract

We establish the presence of a spectral gap near the real axis for the damped wave equation on a manifold with negative curvature. This results holds under a dynamical condition expressed by the negativity of a topological pressure with respect to the geodesic flow. As an application, we show an exponential decay of the energy for all initial data sufficiently regular. This decay is governed by the imaginary part of a finite number of eigenvalues close to the real axis.