Hyperboloidal evolution and global dynamics for the focusing cubic wave equation

TL;DR

This paper investigates the stability of the explicit blowup solution for the focusing cubic wave equation in three dimensions, using hyperboloidal evolution and conformal invariance to analyze global dynamics without symmetry restrictions.

Contribution

It introduces a hyperboloidal initial value framework and identifies a codimension-1 manifold of initial data leading to solutions converging to Lorentz boosts of the blowup profile.

Findings

Existence of a codimension-1 manifold of initial data

Stability of blowup solutions under hyperboloidal evolution

Solutions exhibit slow nondispersive decay

Abstract

The focusing cubic wave equation in three spatial dimensions has the explicit solution $\sqrt{2}/t$. We study the stability of the blowup described by this solution as $t \to 0$ without symmetry restrictions on the data. Via the conformal invariance of the equation we obtain a companion result for the stability of slow decay in the framework of a hyperboloidal initial value formulation. More precisely, we identify a codimension-1 Lipschitz manifold of initial data leading to solutions which converge to Lorentz boosts of $\sqrt{2}/t$ as $t\to\infty$. These global solutions thus exhibit a slow nondispersive decay, in contrast to small data evolutions.