Slow blow-up solutions for the H1(R3) critical focusing semilinear wave equation

Abstract

Given ν>1/2 and δ>0 arbitrary, we prove the existence of energy solutions of ∂ttu−Δu−u5=0 (0.1) in R3+1 which blow up exactly at r=t=0 as t→0−. These solutions are radial and of the form u=λ(t)1/2W(λ(t)r)+η(r,t) inside the cone r≤t, where λ(t)=t−1−ν, W(r)=(1+r2/3)−1/2 is the stationary solution of (0.1), and η is a radiation term with ∫[r≤t](|∇η(x,t)|2+|ηt(x,t)|2+|η(x,t)|6)dx→0, t→0. Outside of the light-cone, there is the energy bound ∫[r>t](|∇u(x,t)|2+|ut(x,t)|2+|u(x,t)|6)dx<δ for all small t>0. The regularity of u increases with ν. As in our accompanying article on wave maps [10], the argument is based on a renormalization method for the “soliton profile” W(r)