On the stability of type II blowup for the 1-corotational energy supercritical harmonic heat flow

TL;DR

This paper constructs and analyzes finite-time blowup solutions for a supercritical harmonic heat flow in high dimensions, demonstrating stability properties and quantized blowup rates under symmetry assumptions.

Contribution

It introduces a new family of stable, finite-time blowup solutions with quantized rates for the supercritical harmonic heat flow, extending previous methods to this setting.

Findings

Constructed smooth solutions blowing up in finite time.

Identified quantized blowup rates depending on an integer parameter.

Proved stability of the blowup regime under perturbations.

Abstract

We consider the energy supercritical harmonic heat flow from $\mathbb{R}^d$ into the $d$-sphere $\mathbb{S}^d$ with $d \geq 7$. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear heat equation $$\partial_t u = \partial^2_r u + \frac{(d-1)}{r}\partial_r u - \frac{(d-1)}{2r^2}\sin(2u).$$ We construct for this equation a family of $\mathcal{C}^{\infty}$ solutions which blow up in finite time 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 quantized rates $$\lambda(t) \sim c_u(T-t)^\frac{\ell}{\gamma}, \quad \ell \in \mathbb{N}^*, \;\; 2\ell > \gamma = \gamma(d) \in (1,2].$$ The construction relies on two arguments: the reduction of the problem to a finite-dimensional one thanks to a robust universal energy method and modulation techniques developed by Merle, Rapha\"el and Rodnianski [Camb. Jour. Math, 3(4):439-617, 2015] for the energy supercritical nonlinear Schr\"odinger equation and by Rapha\"el and Schweyer [Anal. PDE, 7(8):1713-1805, 2014] for the energy critical harmonic heat flow, then we proceed by contradiction to solve the finite-dimensional problem and conclude using the Brouwer fixed point theorem. Moreover, our constructed solutions are in fact $(\ell - 1)$ codimension stable under perturbations of the initial data. As a consequence, the case $\ell = 1$ corresponds to a stable type II blowup regime.