We consider the energy-critical harmonic heat flow from into a smooth compact revolution surface of . For initial data with corotational symmetry, the evolution reduces to the semilinear radially symmetric parabolic problem
for a suitable class of functions . Given an integer , we exhibit a set of initial data arbitrarily close to the least energy harmonic map 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
in accordance with the formal predictions of van den Berg et al. (2003). The case corresponds to the stable regime exhibited in our previous work (CPAM, 2013), and the data for leave on a manifold of codimension 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.
"Quantized slow blow-up dynamics for the corotational energy-critical harmonic heat flow." Anal. PDE 7 (8) 1713 - 1805, 2014. https://doi.org/10.2140/apde.2014.7.1713