Stable type II blowup for the 7 dimensional 1-corotational energy supercritical harmonic map heat flow

TL;DR

This paper constructs a stable finite-time blowup solution for the 7-dimensional energy-supercritical harmonic map heat flow with 1-corotational symmetry, revealing a precise blowup rate at the borderline between Type I and Type II regimes.

Contribution

It provides the first stable blowup solution in the 7D case with a detailed blowup rate, advancing understanding of supercritical harmonic map heat flows.

Findings

Stable finite-time blowup solution constructed.

Blowup rate matches theoretical predictions.

Identifies the 7D case as a critical borderline.

Abstract

We consider the energy-supercritical harmonic map heat flow from $\mathbb{R}^d$ into $\mathbb{S}^d$, under an additional assumption of 1-corotational symmetry. We are interested by the 7 dimensional case which is the borderline between the Type I blowup regime. We construct for this problem a stable finite time blowup solution under the condition of corotational symmetry that blows up via concentration of the universal profile $$u(r,t) \sim Q\left(\frac{r}{\lambda(t)}\right),$$ where $Q$ is the stationary solution of the equation and the speed is given by the rate $$\lambda(t) \sim \frac{\sqrt{(T-t)}}{|\log(T-t)|},$$ which corresponds to the speed predicted by Biernat.