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Bulletin de la SMF - Parutions - 145 (2017) 503-573

Parutions < 145

Blow-up of the critical Sobolev norm for nonscattering radial solutions of supercritical wave equations on R^3
Thomas Duyckaerts, Tristan Roy
Bulletin de la SMF 145, fascicule 3 (2017), 503-573

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Résumé :
Explosion d'une norme de Sobolev critique pour les solutions radiales non-dispersives de l'équation des ondes surcritique sur R^3
Considérons l'équation des ondes avec une non-linéarité surcritique pour l'énergie, focalisante ou défocalisante, en dimension 3 d'espace. On démontre que toute solution radiale de l'équation, avec un temps d'existence maximal T, vérifie une des deux propriétés suivantes: (i) la norme de la solution dans l'espace de Sobolev critique tend vers l'infini quand t tend vers T; (ii) T est infini, et la solution est asymptotiquement proche d'une solution linéaire pour des temps infiniment grands. La démonstration utilise une variante de la méthode des canaux d'énergie basée sur une énergie généralisée (définie dans un espace L^p à poids) qui est presque conservée par le flot de l'équation des ondes linéaires.

Mots-clefs : Équation des ondes non-linéaire, diffusion, explosion, comportement asymptotique.

Abstract:
We consider the wave equation in space dimension 3, with an energy-supercritical nonlinearity which can be either focusing or defocusing. For any radial solution of the equation, with positive maximal time of existence T, we prove that one of the following holds: (i) the norm of the solution in the critical Sobolev space goes to infinity as t goes to T, or (ii) T is infinite and the solution scatters to a linear solution forward in time. We use a variant of the channel of energy method, relying on a generalized L^p-energy which is almost conserved by the flow of the radial linear wave equation.

Keywords: Nonlinear wave equation, scattering, blow-up, asymptotic behavior.

Class. math. : 35L05, 35L71, 35B40, 35B44.


ISSN : 0037-9484
DOI : 10.24033/bsmf.2746
Publié avec le concours de : Centre National de la Recherche Scientifique

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