#### Vol. 12, No. 1, 2019

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

### Tej-eddine Ghoul, Slim Ibrahim and Van Tien Nguyen

Vol. 12 (2019), No. 1, 113–187
##### 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

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

We construct for this equation a family of ${\mathsc{C}}^{\infty }$ 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

$\lambda \left(t\right)\sim {c}_{u}{\left(T-t\right)}^{\frac{\ell }{\gamma }},\phantom{\rule{1em}{0ex}}\ell \in {ℕ}^{\ast },\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}2\ell >\gamma =\gamma \left(d\right)\in \left(1,2\right].$

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.

##### Keywords
harmonic heat flow, blowup, stability, differential geometry
##### Mathematical Subject Classification 2010
Primary: 35B40, 35K50
Secondary: 35K55, 35K57