Dynamics at the threshold for blowup for supercritical wave equations outside a ball

TL;DR

This paper investigates the precise conditions under which solutions to supercritical wave equations outside a ball blow up, identifying a critical stable manifold that governs the threshold behavior.

Contribution

It combines analytical and numerical methods to characterize the blowup threshold as a stable manifold of a unique static solution with one unstable direction.

Findings

The blowup threshold is a codimension-one stable manifold.

Solutions near the threshold converge to a static solution.

The static solution has exactly one unstable mode.

Abstract

We consider spherically symmetric supercritical focusing wave equations outside a ball. Using mixed analytical and numerical methods, we show that the threshold for blowup is given by a codimension-one stable manifold of the unique static solution with exactly one unstable direction. We analyze in detail the convergence to this critical solution for initial data fine-tuned to the threshold.