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, Thomas Duyckaerts Département de Mathématiques , Université de Cergy-Pontoise , Site de Saint Martin, 95302 Cergy-Pontoise Cedex , France 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@u-cergy.fr Search for other works by this author on: Oxford Academic Frank Merle Département de Mathématiques , Université de Cergy-Pontoise , Site de Saint Martin, 95302 Cergy-Pontoise Cedex , France Search for other works by this author on: Oxford Academic
International Mathematics Research Papers, Volume 2008, 2008, rpn002, https://doi.org/10.1093/imrp/rpn002
Published:
01 January 2008
Article history
Received:
13 September 2007
Revision received:
13 September 2007
Accepted:
13 December 2007
Published:
01 January 2008
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Thomas Duyckaerts, Frank Merle, Dynamics of Threshold Solutions for Energy-Critical Wave Equation, International Mathematics Research Papers, Volume 2008, 2008, rpn002, https://doi.org/10.1093/imrp/rpn002
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Abstract
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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© Oxford University Press 2008
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