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

Ghoul, Tej-eddine; Ibrahim, Slim; Nguyen, Van Tien

doi:10.2140/apde.2019.12.113

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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

DOI: 10.2140/apde.2019.12.113

Abstract

We consider the energy-supercritical harmonic heat flow from $ℝ^{d}$ into the $d$ -sphere $S^{d}$ with $d \geq 7$ . 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)}{2 r^{2}} sin (2 u) .

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

u (r, t) \sim Q (\frac{r}{λ (t)}),

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

λ (t) \sim c_{u} {(T - t)}^{\frac{ℓ}{γ}}, ℓ \in ℕ^{*}, 2 ℓ > γ = γ (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. 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 $(ℓ - 1)$ -codimension stable under perturbations of the initial data. As a consequence, the case $ℓ = 1$ corresponds to a stable type II blowup regime.

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Keywords

harmonic heat flow, blowup, stability, differential geometry

Mathematical Subject Classification 2010

Primary: 35B40, 35K50

Secondary: 35K55, 35K57

Milestones

Received: 23 September 2017

Accepted: 9 April 2018

Published: 2 August 2018

Authors

Tej-eddine Ghoul
Department of Mathematics New York University in Abu Dhabi Abu Dhabi United Arab Emirates

Slim Ibrahim
Department of Mathematics and Statistics University of Victoria Victoria, BC Canada	Department of Mathematics New York University in Abu Dhabi Abu Dhabi United Arab Emirates

Van Tien Nguyen
Department of Mathematics New York University Abu Dhabi Abu Dhabi United Arab Emirates

Vol. 12, No. 1, 2019