Semilinear wave equations on extremal Reissner-Nordstr\"om black holes revisited

TL;DR

This paper presents a simplified method for proving global existence and decay of small solutions to semilinear wave equations on extremal Reissner-Nordström black holes, avoiding complex near-horizon estimates and applicable to broader settings.

Contribution

The authors develop a new approach that simplifies the proof of decay for semilinear wave equations on extremal black holes, avoiding sharp near-horizon estimates and extending applicability.

Findings

Proof of global existence and decay for small-data solutions

Method compatible with other extremal black hole settings

Potential extension to spacetimes approaching extremal Reissner-Nordström

Abstract

We revisit global existence and decay for small-data solutions of semilinear wave equations on extremal Reissner-Nordstr\"om black hole backgrounds satisfying the classical null condition, a problem which was previously addressed by the first author in joint work with Aretakis and Gajic (Ann. of PDE, 2020). In this paper, we develop a new approach based on propagating a significantly weaker set of estimates, which allows for a simpler and more streamlined proof. Our proof does not require tracking sharp estimates for the solution in the near-horizon region, which means that it is compatible with, but does not imply, the non-decay and growth hierarchy of derivatives of the solution along the event horizon expected from the Aretakis instability. In particular, this approach is in principle compatible with other settings where stronger horizon instabilities are expected, such as nonlinear charged scalar fields on extremal Reissner-Nordstr\"om, or nonlinear waves on extremal Kerr. We also sketch how our proof applies to semilinear problems on spacetimes settling down to extremal Reissner-Nordstr\"om, such as those constructed in our joint work with Kehle (arXiv:2410.16234, 2024).