Construction of blow-up solutions for Zakharov system on $ {\mathbb{T}}^{2}$

Kishimoto, Nobu; Maeda, Masaya

Construction of blow-up solutions for Zakharov system on

𝕋^{2}

Kishimoto, Nobu ; Maeda, Masaya

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013), p. 791-824 / Harvested from Numdam

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Résumé
Abstract

Nous considérons le système de Zakharov dans lʼespace à deux dimensions avec la condition périodique au bord : $\begin{matrix} {\begin{matrix} i \partial_{t} u = - Δ u + n u, \\ \partial_{t t} n = Δ n + Δ {| u |}^{2}, (t, x) \in [0, T) \times 𝕋^{2} . \end{matrix} & (Z) \end{matrix}$ Nous prouvons lʼexistence de solutions de (Z) explosant au temps fini. En outre, nous prouvons quʼil nʼy a aucune solution explosive de masse minimale.

We consider the Zakharov system in two space dimension with periodic boundary condition: $\begin{matrix} {\begin{matrix} i \partial_{t} u = - Δ u + n u, \\ \partial_{t t} n = Δ n + Δ {| u |}^{2}, (t, x) \in [0, T) \times 𝕋^{2} . \end{matrix} & (Z) \end{matrix}$ We prove the existence of finite time blow-up solutions of (Z). Further, we show there exists no minimal mass blow-up solution.

Publié le : 2013-01-01

MR 3103171 | Zbl 06295442

DOI : https://doi.org/10.1016/j.anihpc.2012.09.003
Classification: 35Q55

@article{AIHPC_2013__30_5_791_0,
     author = {Kishimoto, Nobu and Maeda, Masaya},
     title = {Construction of blow-up solutions for Zakharov system on $ {\mathbb{T}}^{2}$},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {30},
     year = {2013},
     pages = {791-824},
     doi = {10.1016/j.anihpc.2012.09.003},
     mrnumber = {3103171},
     zbl = {06295442},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2013__30_5_791_0}
}

Kishimoto, Nobu; Maeda, Masaya. Construction of blow-up solutions for Zakharov system on $ {\mathbb{T}}^{2}$. Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) pp. 791-824. doi : 10.1016/j.anihpc.2012.09.003. http://gdmltest.u-ga.fr/item/AIHPC_2013__30_5_791_0/

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