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

Kishimoto, Nobu; Maeda, Masaya

Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 5, pp. 791-824

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

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

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

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

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

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

Classification: 35Q55
Keywords: Zakharov system, Blow-up solution, Modified energy, Minimal mass blow-up solution

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