We consider the energy-supercritical harmonic heat flow from
into the
-sphere
with
. Under
an additional assumption of 1-corotational symmetry, the problem reduces to the
one-dimensional semilinear heat equation
We construct for this equation a family of
solutions which blow up in finite time via concentration of the universal profile
where
is the stationary solution of the equation and the speed is given by the quantized rates
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
-codimension
stable under perturbations of the initial data. As a consequence, the case
corresponds to a stable type II blowup regime.
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