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International Mathematics Research Papers (2008) Vol. 2008 : article ID rpn002, 67 pages, doi:10.1093/imrp/rpn002 published on February 11, 2008
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© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org.

Dynamics of Threshold Solutions for Energy-Critical Wave Equation

Thomas Duyckaerts and Frank Merle

Département de Mathématiques, Université de Cergy-Pontoise, Site de Saint Martin, 95302 Cergy-Pontoise Cedex, France

Correspondence: Correspondence to be sent to: Thomas Duyckaerts, Université de Cergy-Pontoise, Département de Mathématiques, Site de Saint Martin, 2 avenue Adolphe-Chauvin, 95302 Cergy-Pontoise cedex, France. e-mail: thomas.duyckaerts{at}u-cergy.fr

We consider the energy-critical nonlinear focusing wave equation in dimension N = 3, 4, 5. An explicit stationary solution, W, of this equation is known. In [8], the energy E(W, 0) has been shown to be a threshold for the dynamical behavior of solutions of the equation. In the present article we study the dynamics at the critical level E(u0, u1) = E(W, 0) and classify the corresponding solutions. We show in particular the existence of two special solutions, connecting different behaviors for negative and positive times. Our results are analogous to [3], which treats the energy-critical nonlinear focusing radial Schrödinger equation, but without any radial assumption on the data. We also refine the understanding of the dynamical behavior of the special solutions.


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