Future stability of large-data wave maps in energy-supercritical dimensions

TL;DR

This paper proves the nonlinear stability of a self-similar blowup solution for energy-supercritical wave maps, showing that certain large initial data lead to solutions with slow decay, challenging typical dispersive expectations.

Contribution

It establishes the nonlinear asymptotic stability of a specific self-similar blowup solution in energy-supercritical wave maps, a novel result in this context.

Findings

Existence of an open set of initial data leading to slow decay solutions.

Stability of the explicit self-similar blowup solution in forward light cones.

Identification of solutions with less than dispersive decay compared to free waves.

Abstract

We consider energy-supercritical co-rotational wave maps from Minkowski spacetime to the sphere in odd spatial dimensions. The equation admits an explicit co-rotational self-similar blowup solution, which also induces solutions that blow up in the past. In the region after the blowup the solution treated in this paper is remarkable, as it is smooth forward in time and exhibits less than dispersive decay. We prove nonlinear asymptotic stability of this large-data self-similar solution inside forward light cones. In particular, we identify an open set of initial data close to the explicit solution that give rise to forward-in-time wave maps whose decay is slower than that of generic free waves.