Vol. 5, No. 4, 2012

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Smooth type II blow-up solutions to the four-dimensional energy-critical wave equation

Matthieu Hillairet and Pierre Raphaël

Vol. 5 (2012), No. 4, 777–829
Abstract

We exhibit ${\mathsc{C}}^{\infty }$ type II blow-up solutions to the focusing energy-critical wave equation in dimension $N=4$. These solutions admit near blow-up time a decomposition

$u\left(t,x\right)=\frac{1}{{\lambda }^{\left(N-2\right)∕2}\left(t\right)}\left(Q+\epsilon \left(t\right)\right)\left(\frac{x}{\lambda \left(t\right)}\right),\phantom{\rule{1em}{0ex}}with\parallel \epsilon \left(t\right),{\partial }_{t}\epsilon \left(t\right){\parallel }_{{Ḣ}^{1}×{L}^{2}}\ll 1,$

where $Q$ is the extremizing profile of the Sobolev embedding ${Ḣ}^{1}\to {L}^{{2}^{\ast }}$, and a blow-up speed

Keywords
wave equation, blow-up
Primary: 35Q51