An extension of the $r^p$ method for wave equations with scale-critical potentials and first-order terms

TL;DR

This paper extends the $r^p$ method to prove decay for wave equations with scale-critical, time-dependent potentials and first-order terms, using a novel Grönwall argument to handle error absorption and expand the applicable range of $p$.

Contribution

It introduces a new extension of the $r^p$ method that effectively manages error terms for wave equations with scale-critical potentials and first-order terms.

Findings

Extended the $r^p$ method to a broader class of wave equations.

Developed a novel Grönwall-based error absorption technique.

Achieved a larger range of $p$ for decay estimates.

Abstract

The $r^p$ method, first introduced in [DR10], has become a robust strategy to prove decay for wave equations in the context of black holes and beyond. In this note, we propose an extension of this method, which is particularly suitable for proving decay for a general class of wave equations featuring a scale-critical time-dependent potential and/or first-order terms of small amplitude. Our approach consists of absorbing error terms in the $r^p$-weighted energy using a novel Gr\"{o}nwall argument, which allows a larger range of $p$ than the standard method. A spherically symmetric version of our strategy first appeared in [VdM22] in the context of a weakly charged scalar field on a black hole whose equations also involve a scale-critical potential.