Construction of type II blow-up solutions for the energy-critical wave equation in dimension 5

We consider the semilinear wave equation with focusing energy-critical nonlinearity in space dimension 5 with radial data. It is known that a solution $(u,{\mathrm{\partial }}_{t}u)$ which blows up at $t=0$ in a neighborhood (in the energy norm) of the family of solitons $W_{\lambda }$, asymptotically decomposes in the energy space as a sum of a bubble $W_{\lambda }$ and an asymptotic profile $(u_0^{\ast },u_1^{\ast })$, where ${\lim }_{t\to 0}\lambda (t)\mathrm{/}t=0$ and $(u^_0, u^_1) \in \dot H^1\times L^2$\mathrm{.}Weconstructablow-upsolutionofthistypesuchthat$(u^_0, u^_1)$isanypairofsufficientlyregularfunctionswith$u_0^*(0) > 0$\mathrm{.}Forthesesolutionstheconcentrationrateis$\lambda(t) \sim t^4$. We also provide examples of solutions with concentration rate $\lambda(t) \sim t^{\nu + 1}$for$\nu > 8$, related to the behaviour of the asymptotic profile near the origin.