A decay estimate for a wave equation with trapping and a complex potential

TL;DR

This paper establishes uniform energy bounds and decay estimates for a wave equation with trapping and complex potential, modeling phenomena relevant to black hole physics.

Contribution

It provides the first decay estimates for such wave equations with trapping and complex potentials, relevant to black hole models.

Findings

Proved uniform energy bounds for the wave equation.

Established Morawetz (local energy decay) estimates.

Applicable to scalar equations in black hole spacetimes.

Abstract

In this brief note, we consider a wave equation that has both trapping and a complex potential. For this problem, we prove a uniform bound on the energy and a Morawetz (or integrated local energy decay) estimate. The equation is a model problem for certain scalar equations appearing in the Maxwell and linearised Einstein systems on the exterior of a rotating black hole.