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

We consider the energy supercritical wave maps 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 wave equation ∂_t ²u=∂_r ²u+[Formula presented]∂_ru−[Formula presented]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([Formula presented]), where Q is the stationary solution of the equation and the speed is given by the quantized rates λ(t)∼c_u(T−t)^{[Formula presented]},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.