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 N=5 ∂_ttu=Δu+|u|^4/3u with radial data. It is known [7] that a solution (u,∂_tu) which blows up at t=0 in a neighborhood (in the energy norm) of the family of solitons W_λ, decomposes in the energy space as (u(t),∂_tu(t))=(W_λ(t)+u₀ ^⁎,u₁ ^⁎)+o(1) where lim_t→0⁡λ(t)/t=0 and (u₀ ^⁎,u₁ ^⁎)∈H˙¹×L². We construct a blow-up solution of this type such that the asymptotic profile (u₀ ^⁎,u₁ ^⁎) is any pair of sufficiently regular functions with u₀ ^⁎(0)>0. For these solutions the concentration rate is λ(t)∼t⁴. We also provide examples of solutions with concentration rate λ(t)∼t^ν+1 for ν>8, related to the behavior of the asymptotic profile near the origin.