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

Abstract

Given $\nu >1/2$ and $\delta >0$ arbitrary, we prove the existence of energy solutions of ${\displaystyle {\partial }_{\mathrm{tt}}u-\Delta u-u^5=0(0.1)}$ in ${\mathbb{R}}^{3+1}$ which blow up exactly at $r=t=0$ as $t\to 0-$. These solutions are radial and of the form $u=\lambda (t{)}^{1/2}W(\lambda (t)r)+\eta (r,t)$ inside the cone $r\le t$, where $\lambda (t)=t^{-1-\nu }$, $W(r)=(1+r^2/3{)}^{-1/2}$ is the stationary solution of (0.1), and $\eta$ is a radiation term with ${\displaystyle {\int }_{[r\le t]}(|\nabla \eta (x,t){|}^2+|{\eta }_t(x,t){|}^2+|\eta (x,t){|}^6)\mathrm{dx}\to 0,t\to 0.}$ Outside of the light-cone, there is the energy bound ${\displaystyle {\int }_{[r>t]}(|\nabla u(x,t){|}^2+|u_t(x,t){|}^2+|u(x,t){|}^6)\mathrm{dx}<\delta }$ for all small $t>0$. The regularity of $u$ increases with $\nu$. As in our accompanying article on wave maps [10], the argument is based on a renormalization method for the “soliton profile” $W(r)$