Construction of type II blowup solutions for the 1-corotational energy supercritical wave maps

Abstract

We consider the energy supercritical wave maps from Rd into the d-sphere Sd with d≥7. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear wave equation∂t2u=∂r2u+(d−1)r∂ru−(d−1)2r2sin⁡(2u). We construct for this equation a family of C∞ solutions which blow up in finite time via concentration of the universal profileu(r,t)∼Q(rλ(t)), where Q is the stationary solution of the equation and the speed is given by the quantized ratesλ(t)∼cu(T−t)ℓγ,ℓ∈N⁎,ℓ>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 [49] for the energy supercritical nonlinear Schrödinger equation, then we proceed by contradiction to solve the finite-dimensional problem and conclude using the Brouwer fixed point theorem.