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

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 and is therefore conjectural on the behaviour inside the lightcone. 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. In the case of the wave maps equation, we only provide the proof in the corotational symmetry class, but we also sketch how to lift this restriction.

💡 Research Summary

The paper investigates the exterior stability of finite‑time singularity formation for two classes of nonlinear wave equations: the wave‑maps equation in (\mathbb{R}^{d+1}) with (d\ge 2) and the power‑type nonlinear wave equation in (\mathbb{R}^{d+1}) with (d\ge 3). The authors focus on both Type I (self‑similar) and Type II (faster‑than‑self‑similar) blow‑up scenarios. Their main setting is a characteristic initial value problem posed on the backward light‑cone (\mathcal C={t+r=0}) of the singularity together with data on an outgoing cone ({t-r=-1}). The data on (\mathcal C) are assumed to converge to a known singular background solution (\phi_{0}) with a prescribed conormal regularity (expressed through vector fields (u\partial_{u},,x_{i}\partial_{x_{j}}-x_{j}\partial_{x_{i}},,r\partial_{t}) and higher derivatives). The outgoing cone data are taken to be small perturbations.

The key methodological step is a change of variables that maps the region ({0<v<v_{1},-1<u<0}) (in double‑null coordinates (u=t-r,;v=t+r)) onto a scattering problem on the whole Minkowski space. In this new framework the authors can invoke the recent global scattering theorem of Krieger–Krieger (referred to as