Stable Blowup Dynamics for the 1-Corotational Energy Critical Harmonic Heat Flow

We exhibit a stable finite time blowup regime for the 1-corotational energy critical harmonic heat flow from R-2 into a smooth compact revolution surface of R-3 that reduces to the semilinear 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. The corresponding initial data can be chosen smooth, well localized, and arbitrarily close to the ground state harmonic map in the energy-critical topology. We give sharp asymptotics on the corresponding singularity formation that occurs through the concentration of a universal bubble of energy at the speed predicted by van den Berg, Hulshof, and King. Our approach lies in the continuation of the study of the 1-equivariant energy critical wave map and Schrodinger map with S-2 target by Merle, Raphael, and Rodnianski. (C) 2012 Wiley Periodicals, Inc.