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 ${\mathbb{R}}^d$ 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 $${\partial }_{t}u={\partial }_r^{2}u+\frac{(d-1)}{r}{\partial }_{r}u-\frac{(d-1)}{2r^2}\sin (2u).$$ 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 $$\lambda (t)\sim c_u(T-t{)}^{\frac{\ell }{\gamma }},\ell \in {\mathbb{N}}^{\ast },2\ell >\gamma =\gamma (d)\in (1,2].$$ 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, Raphaël and Rodnianski [Camb. Jour. Math, 3(4):439-617, 2015] for the energy supercritical nonlinear Schrödinger equation and by Raphaël 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.

68 pages

References in corpus (6)

• Stable blow up dynamics for the 1-corotational energy critical harmonic heat flow
• Dynamics near the ground state for the energy critical nonlinear heat equation in large dimensions
• Non radial type II blow up for the energy supercritical semilinear heat equation
• Non-self-similar blow-up in the heat flow for harmonic maps in higher dimensions
• Type II blow up manifolds for the energy supercritical wave equation
• Stability of infinite time blow up for the Patlak Keller Segel system

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• Blowup solutions for the shadow limit model of singular Gierer-Meinhardt system with critical parameters
• Sharp equivalent for the blowup profile to the gradient of a solution to the semilinear heat equation