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

TL;DR

The paper demonstrates exterior stability for finite-time singularities in nonlinear wave dynamics by recasting the exterior problem as a semi-global scattering task via a coordinate map that sends the singularity to spacelike infinity. Under suitably regular, conormal initial data on the backwards lightcone and an outgoing cone, it proves local existence up to a small exterior time $v_1$ and demonstrates extendibility across the Cauchy horizon as a weak solution, with potential extension to future null infinity under extra data. The analysis builds on the global scattering framework of KK25, employing a robust weighted conormal function space formalism, peeling, and regularity estimates to handle scaling-critical potentials and nonlinearities. While interior regularity results for the lightcone remain conjectural, the exterior stability results apply to both the power nonlinear wave equation and the wave maps equation in corotational symmetry, covering Type I and Type II singularities across several dimensions via concrete admissibility conditions. This work advances understanding of how exterior regions can be controlled by scattering techniques, with implications for the maximal globally hyperbolic development and potential insights into CH and null infinity behavior.

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 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.

Figures (1)

• Figure 1: On the left: We assume the existence of a solution in $\mathcal{X}=(\{v\leq 0\}\cup \{t\leq -0.9\})\cap\{-1\leq u<0\}$ forming a singularity at $u=0=v$. We then show that under suitable assumptions, the solution can be extended to all of $\mathcal{D}_{v_1}$ (shaded in grey) and, in fact, can be extended beyond the Cauchy horizon. The crucial observation of this paper is that a coordinate transformation transforms this problem into a scattering problem from $\mathcal{I}^-$ to $\mathcal{I}^+$ on the Minkowski spacetime, depicted on the right.

Theorems & Definitions (47)

• Theorem 1.1: Type I blow-up for \ref{['in:eq:wave_map']} shatah_weak_1988biernat_generic_2015 and \ref{['in:eq:nonlin']} glogic_co-dimension_2021
• Theorem 1.2: Stability of Type I blow-up solutions donninger_stable_2011costin_proof_2016costin_mode_2017glogic_co-dimension_2021
• Conjecture 1.1: Type I: Stability with additional conjectured regularity
• Remark 1.1: The relevance of $r\partial_t$-regularity.
• Conjecture 1.2: Type II singularity formation, energy-critical
• Remark 1.2: Explanation of the rates along $\mathcal{C}$
• Remark 1.3: Regularity across $\mathcal{C}$: Conormality vs Smoothness
• Remark 1.4: Polynomial vs logarithmic perturbation
• Conjecture 1.3: Type II singularity formation, energy-supercritical
• Remark 1.5: Long-range potentials and self-similar singularities