#### Vol. 7, No. 8, 2014

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Quantized slow blow-up dynamics for the corotational energy-critical harmonic heat flow

### Pierre Raphaël and Remi Schweyer

Vol. 7 (2014), No. 8, 1713–1805
##### Abstract

We consider the energy-critical harmonic heat flow from ${ℝ}^{2}$ into a smooth compact revolution surface of ${ℝ}^{3}$. For initial data with corotational symmetry, the evolution reduces to the semilinear radially symmetric parabolic problem

${\partial }_{t}u-{\partial }_{r}^{2}u-\frac{{\partial }_{r}u}{r}+\frac{f\left(u\right)}{{r}^{2}}=0$

for a suitable class of functions $f$. Given an integer $L\in {ℕ}^{\ast }$, we exhibit a set of initial data arbitrarily close to the least energy harmonic map $Q$ in the energy-critical topology such that the corresponding solution blows up in finite time by concentrating its energy

at a speed given by the quantized rates

$\left(t\right)=c\left({u}_{0}\right)\left(1+o\left(1\right)\right)\frac{{\left(T-t\right)}^{L}}{|\phantom{\rule{0.3em}{0ex}}log\left(T-t\right){|}^{2L∕\left(2L-1\right)}},$

in accordance with the formal predictions of van den Berg et al. (2003). The case $L=1$ corresponds to the stable regime exhibited in our previous work (CPAM, 2013), and the data for $L\ge 2$ leave on a manifold of codimension $L-1$ in some weak sense. Our analysis is a continuation of work by Merle, Rodnianski, and the authors (in various combinations) and it further exhibits the mechanism for the existence of the excited slow blow-up rates and the associated instability of these threshold dynamics.

##### Keywords
blow-up heat flow
Primary: 35K58
##### Milestones
Received: 10 January 2013
Accepted: 22 December 2013
Published: 5 February 2015
##### Authors
 Pierre Raphaël Laboratoire J. A. Dieudonné Université de Nice Sophia Antipolis Institut Universitaire de France 06000 Nice France Remi Schweyer Institut de Mathématiques de Toulouse Université Paul Sabatier 31000 Toulouse France