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

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

tu r2u ru r + f(u) r2 = 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 L2

at a speed given by the quantized rates

(t) = c(u0)(1 + o(1)) (T t)L |log(T t)|2L(2L1),

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 2 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.

blow-up heat flow
Mathematical Subject Classification 2010
Primary: 35K58
Received: 10 January 2013
Accepted: 22 December 2013
Published: 5 February 2015
Pierre Raphaël
Laboratoire J. A. Dieudonné
Université de Nice Sophia Antipolis
Institut Universitaire de France
06000 Nice
Remi Schweyer
Institut de Mathématiques de Toulouse
Université Paul Sabatier
31000 Toulouse