Late-time asymptotics for geometric wave equations with inverse-square potentials

TL;DR

This paper develops a new physical-space method to determine the precise late-time decay behavior of solutions to geometric wave equations with inverse-square potentials on black hole backgrounds, aiding analysis of more complex models.

Contribution

It introduces a novel physical-space approach for deriving late-time asymptotics applicable to a broad class of geometric wave equations, including challenging nonlinear and higher-dimensional cases.

Findings

Established a method for precise late-time tail analysis in inverse-square potential settings.

Derived sharp, uniform decay estimates for solutions on Schwarzschild backgrounds.

Demonstrated the method's potential for more general geometric wave equations.

Abstract

We introduce a new, physical-space-based method for deriving the precise leading-order late-time behaviour of solutions to geometric wave equations on asymptotically flat spacetime backgrounds and apply it to the setting of wave equations with asymptotically inverse-square potentials on Schwarzschild black holes. This provides a useful toy model setting for introducing methods that are applicable to more general linear and nonlinear geometric wave equations, such as wave equations for electromagnetically charged scalar fields, wave equations on extremal Kerr black holes and geometric wave equations in even space dimensions, where existing proofs for deriving precise late-time asymptotics might not apply. The method we introduce relies on exploiting the spatial decay properties of time integrals of solutions to derive the existence and precise genericity properties of asymptotic late-time tails and obtain sharp, uniform decay estimates in time.