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 d into the d-sphere Sd with d 7. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one-dimensional semilinear heat equation

tu = r2u + (d 1) r ru (d 1) 2r2 sin(2u).

We construct for this equation a family of C solutions which blow up in finite time via concentration of the universal profile

u(r,t) Q( r λ(t)),

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

λ(t) cu(T t) γ , ,2 > γ = γ(d) (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 (1)-codimension stable under perturbations of the initial data. As a consequence, the case = 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