15 March 2009 Slow blow-up solutions for the $H^1({\mathbb R}^3)$ critical focusing semilinear wave equation
Joachim Krieger, Wilhelm Schlag, Daniel Tataru
Duke Math. J. 147(1): 1-53 (15 March 2009). DOI: 10.1215/00127094-2009-005

## Abstract

Given $\nu>1/2$ and $\delta>0$ arbitrary, we prove the existence of energy solutions of $\begin{equation}\partial_{tt} u - \Delta u - u^5 =0~~~~(0.1)\end{equation}$ 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 $$\int_{[r\le t]} \big(|\nabla \eta(x,t)|^2 + |\eta_t(x,t)|^2+|\eta(x,t)|^6\big) dx \to 0, \quad t \to 0.$$ Outside of the light-cone, there is the energy bound $$\int_{[r>t]} \big( |\nabla u(x,t)|^2+|u_t(x,t)|^2+|u(x,t)|^6\big) dx \lt \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” $W(r)$

## Citation

Joachim Krieger. Wilhelm Schlag. Daniel Tataru. "Slow blow-up solutions for the $H^1({\mathbb R}^3)$ critical focusing semilinear wave equation." Duke Math. J. 147 (1) 1 - 53, 15 March 2009. https://doi.org/10.1215/00127094-2009-005

## Information

Published: 15 March 2009
First available in Project Euclid: 26 February 2009

zbMATH: 1170.35066
MathSciNet: MR2494455
Digital Object Identifier: 10.1215/00127094-2009-005

Subjects:
Primary: 35L75
Secondary: 58J50