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

Abstract

We consider the energy-supercritical harmonic heat flow from ${ℝ}^d$ into the $d$-sphere ${\mathbb{S}}^d$ with $d\ge 7$. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one-dimensional semilinear heat equation

$${\partial }_{t}u={\partial }_r^{2}u+\frac{\left(d-1\right)}{r}{\partial }_{r}u-\frac{\left(d-1\right)}{2r^2}sin\left(2u\right).$$

We construct for this equation a family of ${\mathcal{C}}^{\infty }$ solutions which blow up in finite time via concentration of the universal profile

$$u\left(r,t\right)\sim Q\left(\frac{r}{\lambda \left(t\right)}\right),$$

where $Q$ is the stationary solution of the equation and the speed is given by the quantized rates

$$\lambda \left(t\right)\sim c_u{\left(T-t\right)}^{\frac{\ell }{\gamma }},\ell \in {\mathbb{N}}^{\ast },2\ell >\gamma =\gamma \left(d\right)\in \left(1,2\right].$$

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, Raphaël and Rodnianski (Camb. J. Math. 3:4 (2015), 439–617) for the energy supercritical nonlinear Schrödinger equation and by Raphaël and Schweyer (Anal. PDE 7:8 (2014), 1713–1805) 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 $\left(\ell -1\right)$-codimension stable under perturbations of the initial data. As a consequence, the case $\ell =1$ corresponds to a stable type II blowup regime.