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

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 $\ddot{I}{•}_{\mathrm{model}}=c_p{t}^{-\hat{I}{\pm }_p}$. Given a sufficiently regular spacelike hypersurface $\hat{I}{\pounds }_f$, together with auxiliary scattering data $\ddot{I}ˆ$, we construct the unique corresponding solution to the nonlinear wave equation that (locally) forms an ODE type singularity on $\hat{I}{\pounds }_f$ attaining $\ddot{I}ˆ$ 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.

45 pages