Profiles for bounded solutions of dispersive equations, with applications to energy-critical wave and Schrödinger equations

Thomas Duyckaerts; Carlos E. Kenig; Frank Merle

doi:10.3934/cpaa.2015.14.1275

Article Contents

2015, Volume 14, Issue 4: 1275-1326. Doi: 10.3934/cpaa.2015.14.1275

This issue Previous Article A remark on the well-posedness of a degenerated Zakharov system Next Article Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces

Profiles for bounded solutions of dispersive equations, with applications to energy-critical wave and Schrödinger equations

1.
LAGA (UMR 7539), Institut Galilée, Université Paris 13, 99, avenue Jean-Baptiste Clément, 93430 Villetaneuse
2.
Department of Mathematics, University of Chicago, Chicago, Illinois, 60637–1514
3.
Université de Cergy-Pontoise and IHES, Laboratoire de mathématiques, UMR CNRS 8088, 2, av. Adolphe Chauvin, 95302 Cergy-Pontoise cedex

Received: September 30, 2013

Revised: February 28, 2014

Published: June 2015

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Abstract

Consider a bounded solution of the focusing, energy-critical wave equation that does not scatter to a linear solution. We prove that this solution converges in some weak sense, along a sequence of times and up to scaling and space translation, to a sum of solitary waves. This result is a consequence of a new general compactness/rigidity argument based on profile decomposition. We also give an application of this method to the energy-critical Schrödinger equation.

Keywords:

Nonlinear dispersive equations,
nonlinear wave and Schrödinger equations,
bounded solutions,
asymptotic behaviour,
profile decomposition.

Mathematics Subject Classification: Primary: 35L05, 35B40; Secondary: 35L71, 35Q55, 35B44.

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