Nonlinear Wave Equations With Null Condition On Extremal Reissner-Nordstr\"om Spacetimes I: Spherical Symmetry

TL;DR

This paper investigates the behavior of spherically symmetric solutions to nonlinear wave equations with null conditions on extremal Reissner-Nordström black holes, demonstrating global well-posedness and horizon instabilities.

Contribution

It establishes global existence and stability results for small data solutions and extends instability findings to nonlinear cases on extremal black hole backgrounds.

Findings

Solutions are globally well-posed up to the horizon.

Non-decay and blow-up phenomena occur along the horizon.

Results apply to spherically symmetric wave maps for various target spaces.

Abstract

We study spherically symmetric solutions of semilinear wave equations in the case where the nonlinearity satisfies the null condition on extremal Reissner--Nordstrom black hole spacetimes. We show that solutions which arise from sufficiently small compactly supported smooth data prescribed on a Cauchy hypersfurace \widetilde{{\Sigma}}_0 crossing the future event horizon \mathcal{H}^{+} are globally well-posed in the domain of outer communications up to and including \mathcal{H}^{+}. Our method allows us to close all bootstrap estimates under very weak decay results (compatible with those known for the linear case). Moreover we establish a certain number of non-decay and blow-up results along the horizon \mathcal{H}^{+} which generalize known instability results for the linear case. Our results apply to spherically symmetric wave maps for a wide class of target spaces.