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.

arXiv comment70 pages. arXiv admin note: substantial text overlap with arXiv:1611.08877

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Cited by in corpus (1)

• Nonlinear stability of homothetically shrinking Yang-Mills solitons in the equivariant case