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

Hillairet, Matthieu; Raphaël, Pierre

doi:10.2140/apde.2012.5.777

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1948-206X (online)

ISSN 2157-5045 (print)

Author index

To appear

Other MSP journals

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

DOI: 10.2140/apde.2012.5.777

Abstract

We exhibit $C^{\infty}$ 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 (t, x) = \frac{1}{λ^{(N - 2) ∕ 2} (t)} (Q + ε (t)) (\frac{x}{λ (t)}), w i t h ∥ ε (t), \partial_{t} ε (t) ∥_{Ḣ^{1} \times L^{2}} ≪ 1,

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

λ (t) = (T - t) e^{- \sqrt{| log (T - t) |} (1 + o (1))} as t \to 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/

Vol. 5, No. 4, 2012