Logarithmic wave decay for short range wavespeed perturbations with radial regularity

Abstract

We establish logarithmic local energy decay for wave equations with a varying wavespeed in dimensions two and higher, where the wavespeed is assumed to be a short range perturbation of unity with mild radial regularity. The key ingredient is Hölder continuity of the weighted resolvent for real frequencies $\hat{I}»$, modulo a logarithmic remainder in dimension two as $\hat{I}»\to 0$. Our approach relies on a study of the resolvent in two distinct frequency regimes. In the low frequency regime, we derive an expansion for the resolvent using a Neumann series and properties of the free resolvent. For frequencies away from zero, we establish a uniform resolvent estimate by way of a Carleman estimate.

33 pages