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

TL;DR

This paper provides a comprehensive analysis of the stability and scattering behavior of solutions with ODE-type blow-up for focusing nonlinear wave equations across various dimensions and powers, including explicit construction and perturbation stability.

Contribution

It introduces a complete framework for understanding local stability and scattering of ODE-type blow-up solutions, including explicit models and continuous dependence on perturbations.

Findings

Constructed unique solutions with prescribed scattering data.

Proved local stability of ODE-type singularities under perturbations.

Established continuous dependence of blow-up surface and scattering data.

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 $\phi_{\mathrm{model}} = c_p t^{-\alpha_p}$. Given a sufficiently regular spacelike hypersurface $\Sigma_f$, together with auxiliary scattering data $\psi$, we construct the unique corresponding solution to the nonlinear wave equation that (locally) forms an ODE type singularity on $\Sigma_f$ attaining $\psi$ 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.