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

We consider the energy supercritical wave maps from ${\mathbb{R}}^d$ into the $d$-sphere ${\mathbb{S}}^d$ with $d\ge 7$. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear wave equation $${\mathrm{\partial }}_t^{2}u={\mathrm{\partial }}_r^{2}u+\frac{(d-1)}{r}{\mathrm{\partial }}_{r}u-\frac{(d-1)}{2r^2}\sin (2u)\mathrm{.}$$ We construct for this equation a family of ${\mathcal{C}}^{\mathrm{\infty }}$ solutions which blow up in finite time via concentration of the universal profile $$u(r,t)\sim Q\left(\frac{r}{\lambda (t)}\right),$$ where $Q$ is the stationary solution of the equation and the speed is given by the quantized rates $$\lambda (t)\sim c_u(T-t{)}^{\frac{\mathrm{\ell }}{\gamma }},\mathrm{\ell }\in {\mathbb{N}}^{\ast },\mathrm{\ell }>\gamma =\gamma (d)\in (1,2]\mathrm{.}$$ 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, Rapha"el and Rodnianski for the energy supercritical nonlinear Schr"odinger equation, then we proceed by contradiction to solve the finite-dimensional problem and conclude using the Brouwer fixed point theorem.