Local decay for the damped wave equation in the energy space

TL;DR

This paper enhances understanding of local energy decay for the damped wave equation on Euclidean space with long-range metric perturbations and absorption, using advanced resolvent estimates and Mourre theory.

Contribution

It introduces improved resolvent estimates and a limiting absorption principle for the damped wave equation with long-range perturbations.

Findings

Achieved decay rate of O(t^{--d+ε}) in weighted energy spaces.

Developed an improved Mourre theory for dissipative operators.

Proved the limiting absorption principle for resolvent powers.

Abstract

We improve a previous result about the local energy decay for the damped wave equation on R^d. The problem is governed by a Laplacian associated with a long range perturbation of the flat metric and a short range absorption index. Our purpose is to recover the decay O(t^{--d+$\epsilon$}) in the weighted energy spaces. The proof is based on uniform resolvent estimates, given by an improved version of the dissipative Mourre theory. In particular we have to prove the limiting absorption principle for the powers of the resolvent with inserted weights.