A note on exterior stability of isolated singularity formation for nonlinear wave equations

TL;DR

This paper investigates the stability of the exterior region near singularity formation in nonlinear wave equations, extending previous results to more general settings and employing scattering theory techniques.

Contribution

It establishes exterior stability results for wave maps and power nonlinear wave equations with minimal symmetry assumptions, using a coordinate change and scattering methods.

Findings

Existence of solutions up to the Cauchy horizon for certain initial data

Application of scattering results to scaling-critical potentials

Extension of stability analysis beyond symmetric cases

Abstract

We study the stability of the exterior of Type I and Type II singularity formation for the wave maps equation in $\mathbb{R}^{d+1}$ with $d\geq2$ and the power nonlinear wave equation in $\mathbb{R}^{d+1}$ with $d\geq3$:Given characteristic initial data on the backwards lightcone of the singularity $\mathcal{C}=\{t+r=0\}$ converging to the singular background solution along with suitable data on an outgoing cone, we establish existence in a region $\{t+r\in(0,v_1),t-r\in(-1,0)\}$ for some suitably small $v_1$, i.e. all the way to the Cauchy horizon. Our result hinges on a particular set of assumptions on the regularity properties of these initial data, which conjecturally can be recovered by a more detailed stability analysis of the behaviour inside the past light cone; indeed, in certain settings, this was achieved in [BDS21,KAD26], and we strongly expect they can be proved in many other settings as well. The proof goes via a suitable change of coordinates and an application of the scattering result of [KK25], which, in particular, also applies to scaling-critical potentials. While no symmetry assumption is made for the power nonlinear wave equation, we only provide the proof in the corotational symmetry class for the wave maps equation, but we also sketch how to lift this restriction.