QUANTIZED SLOW BLOW-UP DYNAMICS FOR THE COROTATIONAL ENERGY-CRITICAL HARMONIC HEAT FLOW

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 derivative(t)u - partial derivative(2)(r)u - partial derivative(r)u/r + f(u)/r(2) = 0 for a suitable class of functions f. Given an integer L is an element of N*, 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 del u(t, r) - del Q (r/lambda(t)) --> u* in L-2 at a speed given by the quantized rates lambda(t) = c(u(0)) (1 + o(1)) (T - t)(L)/\log(T - t)\(2L/(2L-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 L >= 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.