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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 $(u, \partial_t u)$ which blows up at $t = 0$ in a neighborhood (in the energy norm) of the family of solitons $W_λ$, asymptotically decomposes in the energy space as a sum of a bubble $W_λ$ and an asymptotic profile $(u_0^*, u_1^*)$, where $\lim_{t\to 0}λ(t)/t = 0$ and $(u^*_0, u^*_1) \in \dot H^1\times L^2$. We construct a blow-up solution of this type such that $(u^*_0, u^*_1)$ is any pair of sufficiently regular functions with $u_0^*(0) > 0$. For these solutions the concentration rate is $λ(t) \sim t^4$. We also provide examples of solutions with concentration rate $λ(t) \sim t^{ν+ 1}$ for $ν> 8$, related to the behaviour of the asymptotic profile near the origin.