Vol. 5, No. 4, 2012

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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 C 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(t,x) = 1 λ(N2)2(t)(Q + ε(t))( x λ(t)),with ε(t),tε(t)1×L2 1,

where Q is the extremizing profile of the Sobolev embedding 1 L2 , and a blow-up speed

λ(t) = (T t)e|log (Tt)|(1+o(1)) as t T.

Keywords
wave equation, blow-up
Mathematical Subject Classification 2010
Primary: 35Q51
Milestones
Received: 8 October 2010
Accepted: 13 May 2011
Published: 27 November 2012
Authors
Matthieu Hillairet
Ceremade
Université Paris Dauphine
75775 Paris Cedex 16
France
http://www.ceremade.dauphine.fr/~mhillair/
Pierre Raphaël
Université Paul Sabatier
Institut de Mathématiques de Toulouse
31062 Toulouse Cedex 9
France
http://www.math.univ-toulouse.fr/~raphael/