Nonscattering solutions and blowup at infinity for the critical wave equation

We consider the critical focusing wave equation (−∂2t+Δ)u+u5=0 in R1+3 and prove the existence of energy class solutions which are of the form u(t,x)=tμ2W(tμx)+η(t,x) in the forward lightcone {(t,x)∈R×R3:|x|≤t,t≫1} where W(x)=(1+13|x|2)−12 is the ground state soliton, μ is an arbitrary prescribed real number (positive or negative) with |μ|≪1 , and the error η satisfies ∥∂tη(t,⋅)∥L2(Bt)+∥∇η(t,⋅)∥L2(Bt)≪1,Bt:={x∈R3:|x|