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[double-struck] ^d with d ≥ 7. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one-dimensional semilinear heat equation ∂ _t =∂ _r ² u+(d-1)/r ∂ _r u-(d-1)/2r ² sin(2u) We construct for this equation a family of C ^∞ solutions which blow up in finite time via concentration of the universal profile u(r,t)Q(r/λ(t)), where Q is the stationary solution of the equation and the speed is given by the quantized rates λ(t) c _u (T-t) ^λ/γ , ℓ ∈ ℕ *, 2ℓ > γ=γ(d) ∈ (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, Raphaël and Rodnianski (Camb. J. Math. 3:4 (2015), 439-617) for the energy supercritical nonlinear Schrödinger equation and by Raphaël 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.