Dynamics of Threshold Solutions for Energy-Critical Wave Equation (2024)

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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

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Frank Merle

Département de Mathématiques

,

Université de Cergy-Pontoise

,

Site de Saint Martin, 95302 Cergy-Pontoise Cedex

,

France

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International Mathematics Research Papers, Volume 2008, 2008, rpn002, https://doi.org/10.1093/imrp/rpn002

Published:

01 January 2008

Article history

Revision received:

13 September 2007

Accepted:

13 December 2007

Published:

01 January 2008

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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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