Blow-up of the critical Sobolev norm for nonscattering radial solutions of supercritical wave equations on $\protect \mathbb{R}^{3}$

Duyckaerts, Thomas; Roy, Tristan

doi:10.24033/bsmf.2746

Blow-up of the critical Sobolev norm for nonscattering radial solutions of supercritical wave equations on

ℝ^{3}

[Explosion d’une norme de Sobolev critique pour les solutions radiales non-dispersives de l’équation des ondes surcritique sur

ℝ^{3}

]

Duyckaerts, Thomas ¹ ; Roy, Tristan ²

¹ LAGA, Université Paris 13 (UMR 7539), Université Sorbonne Paris Cité.
² Nagoya University.

Bulletin de la Société Mathématique de France, Tome 145 (2017) no. 3, pp. 503-573

Abstract (VO)
Résumé

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.

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.

Reçu le : 2015-09-07
Accepté le : 2016-05-22
Publié le : 2017-07-21

MR Zbl

DOI : 10.24033/bsmf.2746

Classification : 35L05, 35L71, 35B40, 35B44
Keywords: Nonlinear wave equation, scattering, blow-up, asymptotic behavior.
Mots-clés : Équation des ondes non-linéaire, diffusion, explosion, comportement asymptotique.

Affiliations des auteurs :

Duyckaerts, Thomas ¹ ; Roy, Tristan ²

¹ LAGA, Université Paris 13 (UMR 7539), Université Sorbonne Paris Cité.
² Nagoya University.

@article{BSMF_2017__145_3_503_0,
     author = {Duyckaerts, Thomas and Roy, Tristan},
     title = {Blow-up of the critical {Sobolev} norm for nonscattering radial solutions of supercritical wave equations on~$\protect \mathbb{R}^{3}$
            },
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {503--573},
     year = {2017},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {145},
     number = {3},
     doi = {10.24033/bsmf.2746},
     mrnumber = {3766119},
     zbl = {1395.35042},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/bsmf.2746/}
}

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Duyckaerts, Thomas; Roy, Tristan. Blow-up of the critical Sobolev norm for nonscattering radial solutions of supercritical wave equations on $\protect \mathbb{R}^{3}$. Bulletin de la Société Mathématique de France, Tome 145 (2017) no. 3, pp. 503-573. doi: 10.24033/bsmf.2746

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