Slow blow-up solutions for the \(H^1(\mathbb R^3)\) critical focusing semilinear wave equation
Given \(\nu >1/2\) and \(\delta >0\) arbitrary, existence of an energy solution of \(u_{tt}-\Delta u-u^{5}=0\) on \(\mathbb R^{3+1},\) which blow-up exactly at \(r=t=0\) as \(t\rightarrow 0-,\) is proved. These solutions are of the form \[ t^{-(\nu+1)/2} \Bigg(1+ \frac{1}{3} \bigg(\frac{r}{t^{\nu+1}}\bigg)^2\Bigg)^{-1/2}+\eta (r,t) \] inside the cone \(\{r\leq t\},\) radial and their energy is bounded by \(\delta \) for small \(t>0,\) whereas the energy of radiation term \(\eta (r,t)\) tends to \(0\) as \(t\rightarrow 0.\) The proof is based on a renormalization method for stationary solutions (see also the authors' article [Invent. Math. 171, No. 3, 543--615 (2008; Zbl 1139.35021)]).
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