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 Wλ, decomposes in the energy space as(u(t),∂tu(t))=(Wλ(t)+u0⁎,u1⁎)+o(1), where limt→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)∼tν+1 for ν>8, related to the behavior of the asymptotic profile near the origin.