International Mathematics Research Papers

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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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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This Article Abstract Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Alert me to new issues of the journal Add to My Personal Archive Download to citation manager Request Permissions How to cite this article Google Scholar Articles by Duyckaerts, T. Articles by Merle, F. Search for Related Content Social Bookmarking          What's this?