Construction of blow-up solutions for Zakharov system on ${𝕋}^{2}$
Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 5, pp. 791-824.

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

 $\begin{array}{cc}\left\{\begin{array}{c}i{\partial }_{t}u=-\Delta u+nu,\hfill \\ {\partial }_{tt}n=\Delta n+\Delta {|u|}^{2},\phantom{\rule{1em}{0ex}}\left(t,x\right)\in \left[0,T\right)×{𝕋}^{2}.\hfill \end{array}& \text{(Z)}\end{array}$
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:

 $\begin{array}{cc}\left\{\begin{array}{c}i{\partial }_{t}u=-\Delta u+nu,\hfill \\ {\partial }_{tt}n=\Delta n+\Delta {|u|}^{2},\phantom{\rule{1em}{0ex}}\left(t,x\right)\in \left[0,T\right)×{𝕋}^{2}.\hfill \end{array}& \text{(Z)}\end{array}$
We prove the existence of finite time blow-up solutions of (Z). Further, we show there exists no minimal mass blow-up solution.

DOI : https://doi.org/10.1016/j.anihpc.2012.09.003
Classification : 35Q55
Mots clés : Zakharov system, Blow-up solution, Modified energy, Minimal mass blow-up solution
@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},
pages = {791--824},
publisher = {Elsevier},
volume = {30},
number = {5},
year = {2013},
doi = {10.1016/j.anihpc.2012.09.003},
zbl = {06295442},
mrnumber = {3103171},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2012.09.003/}
}
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) no. 5, pp. 791-824. doi : 10.1016/j.anihpc.2012.09.003. http://archive.numdam.org/articles/10.1016/j.anihpc.2012.09.003/

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