ON THE STABILITY OF TYPE II BLOWUP FOR THE 1-COROTATIONAL ENERGY-SUPERCRITICAL HARMONIC HEAT FLOW

We consider the energy-supercritical harmonic heat flow from R-d into the d-sphere S-d with d >= 7. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one-dimensional semilinear heat equation partial derivative(t)u = partial derivative(2)(r)u + (d-1)/r partial derivative(r)u - (d-1)/2y(2) sin(2u). We construct for this equation a family of C-infinity solutions which blow up in finite time via concentration of the universal profile u(r,t) similar to Q(r/lambda(t)), where Q is the stationary solution of the equation and the speed is given by the quantized rates lambda(t) similar to c(u)(T-t)l/gamma, l is an element of N*, 2l > gamma = gamma(d) is an element of(1, 2]. The construction relies on two arguments: the reduction of the problem to a finite-dimensional one thanks to a robust universal energy method and modulation techniques developed by Merle, Raphael and Rodnianski (Camb. J. Math. 3: 4 (2015), 439-617) for the energy supercritical nonlinear Schrodinger equation and by Raphael and Schweyer (Anal. PDE 7: 8 (2014), 1713-1805) for the energy critical harmonic heat flow. Then we proceed by contradiction to solve the finite-dimensional problem and conclude using the Brouwer fixed-point theorem. Moreover, our constructed solutions are in fact (l-1)-codimension stable under perturbations of the initial data. As a consequence, the case l = 1 corresponds to a stable type II blowup regime.