A note on local energy decay results for wave equations with a potential

TL;DR

This paper investigates local energy decay for wave equations with short-range potentials, emphasizing L^2 bounds and weighted energy estimates without relying on finite speed of propagation or resolvent estimates.

Contribution

It introduces a simple multiplier method to analyze variable coefficient cases, avoiding traditional reliance on resolvent estimates and finite speed of propagation.

Findings

Established local energy decay results without finite speed of propagation.

Developed a straightforward multiplier approach for variable coefficients.

Provided new insights into energy estimates for wave equations with potentials.

Abstract

In this paper we consider the local energy decay result for wave equations with a short-range potential. It is important to note that one never uses a finite speed of propagation property unlike the historical previous papers. The essential parts of analysis are in getting L^2-bound of the solution itself, and deriving the weighted energy estimates. In this paper we only use a simple multiplier method to treat the variable coefficient case, and do not rely on any resolvent estimates.