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

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.

70 pages. arXiv admin note: substantial text overlap with arXiv:1611.08877

References in corpus (4)

• Stable blow up dynamics for the 1-corotational energy critical harmonic heat flow
• Type II blow up manifolds for the energy supercritical wave equation
• Blow up for the critical gKdV equation III: exotic regimes
• Stability of infinite time blow up for the Patlak Keller Segel system