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2014 Quantized slow blow-up dynamics for the corotational energy-critical harmonic heat flow
Pierre Raphaël, Remi Schweyer
Anal. PDE 7(8): 1713-1805 (2014). DOI: 10.2140/apde.2014.7.1713


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

t u r 2 u r u r + f ( u ) r 2 = 0

for a suitable class of functions f. Given an integer L, 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

u ( t , r ) Q ( r ( t ) ) u  in  L 2

at a speed given by the quantized rates

( t ) = c ( u 0 ) ( 1 + o ( 1 ) ) ( T t ) L | log ( T t ) | 2 L ( 2 L 1 ) ,

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 L2 leave on a manifold of codimension L1 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.


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Pierre Raphaël. Remi Schweyer. "Quantized slow blow-up dynamics for the corotational energy-critical harmonic heat flow." Anal. PDE 7 (8) 1713 - 1805, 2014.


Received: 10 January 2013; Accepted: 22 December 2013; Published: 2014
First available in Project Euclid: 20 December 2017

zbMATH: 1327.35196
MathSciNet: MR3318739
Digital Object Identifier: 10.2140/apde.2014.7.1713

Primary: 35K58

Keywords: blow-up heat flow

Rights: Copyright © 2014 Mathematical Sciences Publishers


Vol.7 • No. 8 • 2014
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