Threshold phenomenon for the quintic wave equation in three dimensions

TL;DR

This paper proves that for the critical radial focusing wave equation in three dimensions, the center stable manifold acts as a threshold separating solutions that blow up from those that scatter to zero, confirming a long-standing conjecture.

Contribution

It rigorously establishes the threshold role of the center stable manifold near the ground state for the critical wave equation in three dimensions.

Findings

The center stable manifold separates blowup and scattering solutions.

The topology used is stronger than the energy norm.

Confirms a conjecture from numerical studies.

Abstract

For the critical focusing wave equation $\Box u = u^5$ on $\R^{3+1}$ in the radial case, we establish the role of the "center stable" manifold $\Sigma$ constructed in \cite{KS} near the ground state $(W,0)$ as a threshold between type I blowup and scattering to zero, establishing a conjecture going back to numerical work by Bizo\'n, Chmaj, Tabor. The underlying topology is stronger than the energy norm.