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

Nobu Kishimoto; Masaya Maeda

doi:10.1016/j.anihpc.2012.09.003

JournalsaihpcVol. 30, No. 5pp. 791–824

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

Nobu Kishimoto
Department of Mathematics, Kyoto University, Kitashirakawa-Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan
Masaya Maeda
Mathematical Institute, Tohoku University, Sendai 980-8578, Japan

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Abstract

We consider the Zakharov system in two space dimension with periodic boundary condition:

{i \partial_{t} u = - Δ u + n u, \partial_{tt} n = Δ n + Δ ∣ u ∣^{2}, (t, x) \in [0, T) \times T^{2} . (Z)

We prove the existence of finite time blow-up solutions of (Z). Further, we show there exists no minimal mass blow-up solution.

Résumé

Nous considérons le système de Zakharov dans lʼespace à deux dimensions avec la condition périodique au bord :

{i \partial_{t} u = - Δ u + n u, \partial_{tt} n = Δ n + Δ ∣ u ∣^{2}, (t, x) \in [0, T) \times T^{2} . (Z)

Nous prouvons lʼexistence de solutions de (Z) explosant au temps fini. En outre, nous prouvons quʼil nʼy a aucune solution explosive de masse minimale.

Cite this article

Nobu Kishimoto, Masaya Maeda, Construction of blow-up solutions for Zakharov system on $T^{2}$ . Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013), no. 5, pp. 791–824

DOI 10.1016/J.ANIHPC.2012.09.003