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

Raphaël, Pierre; Hillairet, Matthieu (2010), Smooth type II blow up solutions to the four dimensional energy critical wave equation, Analysis & PDE, 5, 4, p. 777-829. http://dx.doi.org/10.2140/apde.2012.5.777

Type
Article accepté pour publication ou publié
External document link
http://hal.archives-ouvertes.fr/hal-00524838/fr/
Date
2010-10
Journal name
Analysis & PDE
Volume
5
Number
4
Publisher
Mathematical Sciences Publisher
Pages
777-829
Publication identifier
Metadata
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Author(s)
Raphaël, Pierre
Hillairet, Matthieu
Abstract (EN)
We exhibit $\mathcal C^{\infty}$ type II blow up solutions to the focusing energy critical wave equation in dimension $N=4$. These solutions admit near blow up time a decomposiiton $$u(t,x)=\frac{1}{\lambda^{\frac{N-2}{2}}(t)}(Q+\e(t))(\frac{x}{\lambda(t)}) \ \ \mbox{with} \ \ \|\e(t),\pa_t\e(t)\|_{\dot{H}^1\times L^2}\ll1 $$ where $Q$ is the extremizing profile of the Sobolev embedding $\dot{H}^1\to L^{2^*}$, and a blow up speed $$\lambda(t)=(T-t)e^{-\sqrt{|\log (T-t)|}(1+o(1))} \ \ \mbox{as} \ \ t\to T.$$
Subjects / Keywords
Sobolev embedding; energy critical wave equation; Equations aux dérivées partielles