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Scattering and stability for ODE-type blow-up surfaces for focusing nonlinear wave equations

Istvan Kadar, Warren Li

TL;DR

This work analyzes the focusing nonlinear wave equation $\Box\phi+|\phi|^{p-1}\phi=0$ in Minkowski space for arbitrary $p>1$ and dimension, constructing solutions that blow up on a given spacelike hypersurface with an ODE-type singularity. The authors develop a two-gear framework: a scattering construction that encodes singularities via smooth hypersurface data $f$ and scattering data $\psi$, and a stability analysis that shows these singularities persist under high-regularity perturbations away from the singularity, with blow-up surfaces and data depending continuously on the perturbations. Central to the approach is the $(\tau,z)$-coordinate formulation which converts the singular surface into a zero-set of $\tau$ and introduces Jacobian quantities $W,V^i$ and lapse $\Omega^2$, enabling energy-methods and transport estimates to control the solution and the geometry of the blow-up. The paper provides precise asymptotics and a robust energy-transport framework that handles derivative loss and yields a local scattering theory in high regularity spaces, applicable in any dimension and for all $p>1$. These results extend prior low-regularity blow-up analyses by delivering a general, local, hypersurface-based stability theory with a clear geometric interpretation and potential implications for threshold phenomena in nonlinear wave dynamics.

Abstract

We study the focusing power nonlinear wave equation with any power, in Minkowski space of any spacetime dimension. We present a complete understanding of the local stability and scattering theory (both in high regularity spaces) for solutions exhibiting ODE type blow-up on spacelike hypersurfaces, with the blow-up at each point modelled by the explicit solution $φ_{\mathrm{model}} = c_p t^{-α_p}$. Given a sufficiently regular spacelike hypersurface $Σ_f$, together with auxiliary scattering data $ψ$, we construct the unique corresponding solution to the nonlinear wave equation that (locally) forms an ODE type singularity on $Σ_f$ attaining $ψ$ as scattering data. Conversely, we show that such ODE type singularities are (locally) stable to suitably regular perturbations away from the singularity, and that the blow-up surface and scattering data remain regular, in a continuously dependent manner, following such perturbations.

Scattering and stability for ODE-type blow-up surfaces for focusing nonlinear wave equations

TL;DR

This work analyzes the focusing nonlinear wave equation in Minkowski space for arbitrary and dimension, constructing solutions that blow up on a given spacelike hypersurface with an ODE-type singularity. The authors develop a two-gear framework: a scattering construction that encodes singularities via smooth hypersurface data and scattering data , and a stability analysis that shows these singularities persist under high-regularity perturbations away from the singularity, with blow-up surfaces and data depending continuously on the perturbations. Central to the approach is the -coordinate formulation which converts the singular surface into a zero-set of and introduces Jacobian quantities and lapse , enabling energy-methods and transport estimates to control the solution and the geometry of the blow-up. The paper provides precise asymptotics and a robust energy-transport framework that handles derivative loss and yields a local scattering theory in high regularity spaces, applicable in any dimension and for all . These results extend prior low-regularity blow-up analyses by delivering a general, local, hypersurface-based stability theory with a clear geometric interpretation and potential implications for threshold phenomena in nonlinear wave dynamics.

Abstract

We study the focusing power nonlinear wave equation with any power, in Minkowski space of any spacetime dimension. We present a complete understanding of the local stability and scattering theory (both in high regularity spaces) for solutions exhibiting ODE type blow-up on spacelike hypersurfaces, with the blow-up at each point modelled by the explicit solution . Given a sufficiently regular spacelike hypersurface , together with auxiliary scattering data , we construct the unique corresponding solution to the nonlinear wave equation that (locally) forms an ODE type singularity on attaining as scattering data. Conversely, we show that such ODE type singularities are (locally) stable to suitably regular perturbations away from the singularity, and that the blow-up surface and scattering data remain regular, in a continuously dependent manner, following such perturbations.
Paper Structure (31 sections, 35 theorems, 217 equations, 5 figures)

This paper contains 31 sections, 35 theorems, 217 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background on focusing semilinear waves
  3. Related equations
  4. Ideas of the proof
  5. Ansatz:
  6. Exact solution:
  7. Uniqueness:
  8. Aside, solutions without scattering data:
  9. Stability:
  10. Overcoming derivative loss:
  11. Precise statements
  12. Outlook and overview
  13. Maximal globally hyperbolic development of finite energy solutions:
  14. Threshold behaviour:
  15. Overview:
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $s\geq s_0\coloneqq30 \kappa_p +\frac{n+1}{2}+5$ and $f\in H^{s}(\mathbb{B}_3)$$\psi\in H^{s-\lfloor 2\kappa_p \rfloor}(\mathbb{B}_3)$ with $\left\lvert \partial f\right\rvert<1/10$ and $f(0)=0$. Write $\mathbf{t}=t-f(x)$. Then there exist $\mathbf{t}_1 \in (0, 1)$, depending on $\left\lVert f\r See fig:backward for a visual representation.

Figures (5)

  • Figure 1: An illustration of \ref{['in:thm:main']}. We show the existence of a solution to \ref{['in:eq:main']} in the shaded domain, with the left picture showing the region in $(t, x)$ coordinates and the right picture showing the region in transformed $\mathbf{t} = t - f(x), y^i = x^i$ coordinates. The past boundary of the region is such that the solution $\phi$ exhibits ODE blow-up on $\{ t = f(x) \}$, with suitable scattering data $\psi$.
  • Figure 2: An illustration of \ref{['in:thm:main2']}. We show that regular perturbations of the model ODE-blow up data at $\{ t = 1 \}$ only has a perturbative effect on the (past) singular boundary $\mathscr{S}$.
  • Figure 3: Possible a priori structure of a singular surface $\Sigma^+$, with the solution existing in the grey region. In red, we have indicated the spacelike part $\mathscr{S}$; in blue, the null segments. We have also depicted a characteristic point $p$ in $\Sigma^+$, while $q_1$ and $q_2$ are examples of non-characteristic and non-spacelike singularities.
  • Figure 4: The solution constructed in \ref{['in:thm:main1_precise']} exists in the region bounded by $\mathcal{C}_1$, $\Sigma_{\mathbf{t}_1}$ and $\Sigma_0$ (indicated in gray), while the perturbed solution described in \ref{['in:cor:general']} will be constructed between $\Sigma_{\mathbf{t}_1/2}$, $\mathcal{C}_2$ and a new ODE blow-up hypersurface (indicated by the hatching). We have also indicated the slopes of the cones in the diagram. The red boxed region will be detailed in \ref{['fig:zoomed']} below.
  • Figure 5: On the left, we provide a zoom into the red boxed region of \ref{['fig:cor1.6']} where \ref{['gen:lemma:local']} is proved. On the right, we illustrate the same region in Lorentz boosted coordinates, so that $\Sigma_0$ has vanishing gradient at $p \in \Sigma_0$. We apply Cauchy stability in the cross hatched region $\mathcal{M}_p^{1/4}$, while stability in the cone below the region follows from \ref{['in:thm:main2_precise']}, as described below.

Theorems & Definitions (84)

  • Theorem 1.1: Rough version of \ref{['in:thm:main1_precise']}
  • Theorem 1.2: Rough version of \ref{['in:thm:main2_precise']}
  • Theorem 1.3: merle_openness_2008nouaili_1_2008
  • Theorem 1.4
  • Remark 1.1: Existence interval
  • Remark 1.2: Regularity
  • Remark 1.3: Solutions without scattering data
  • Theorem 1.5
  • Remark 1.4: Spectral theory
  • Corollary 1.6
  • ...and 74 more