Threshold for blowup for equivariant wave maps in higher dimensions

TL;DR

This paper investigates the conditions under which solutions to higher-dimensional equivariant wave maps blow up, identifying a critical threshold related to a specific self-similar solution using combined numerical and analytical techniques.

Contribution

It establishes the blowup threshold in supercritical dimensions for equivariant wave maps, linking it to the stable manifold of a self-similar solution with one instability.

Findings

Blowup threshold characterized by a codimension-one stable manifold.

Identification of a self-similar solution with one instability.

Combined numerical and analytical methods used to determine blowup conditions.

Abstract

We consider equivariant wave maps from $\mathbb{R}^{d+1}$ to $\mathbb{S}^d$ in supercritical dimensions $3\leq d\leq 6$. Using mixed numerical and analytic methods, we show that the threshold of blowup is given by the codimension-one stable manifold of a self-similar solution with one instability.