Geodesic
Well-posedness for a class of generalized Zakharov system
You, Shujun1 ; Ning, Xiaoqi1
1 School of Mathematical Sciences, Huaihua University, Huaihua 418008, China
Journal of nonlinear sciences and its applications, Tome 10 (2017) no. 4, p. 1289-1302.

Voir la notice de l'article provenant de la source International Scientific Research Publications

In this paper, we study the existence and uniqueness of the global smooth solution for the initial value problem of generalized Zakharov equations in dimension two. By means of a priori integral estimates and Galerkin method, we first construct the existence of global solution with some conditions. Furthermore, we prove that the global solution is unique.
DOI : 10.22436/jnsa.010.04.01
Classification : 35A01, 35A02
Keywords: Global solutions, Zakharov equations, well-posedness.

You, Shujun 1 ; Ning, Xiaoqi 1

1 School of Mathematical Sciences, Huaihua University, Huaihua 418008, China 
@article{JNSA_2017_10_4_a0,
     author = {You, Shujun and Ning, Xiaoqi},
     title = {Well-posedness for a class of generalized {Zakharov} system},
     journal = {Journal of nonlinear sciences and its applications},
     pages = {1289-1302},
     publisher = {mathdoc},
     volume = {10},
     number = {4},
     year = {2017},
     doi = {10.22436/jnsa.010.04.01},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.22436/jnsa.010.04.01/}
}
TY  - JOUR
AU  - You, Shujun
AU  - Ning, Xiaoqi
TI  - Well-posedness for a class of generalized Zakharov system
JO  - Journal of nonlinear sciences and its applications
PY  - 2017
SP  - 1289
EP  - 1302
VL  - 10
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.22436/jnsa.010.04.01/
DO  - 10.22436/jnsa.010.04.01
LA  - en
ID  - JNSA_2017_10_4_a0
ER  - 
%0 Journal Article
%A You, Shujun
%A Ning, Xiaoqi
%T Well-posedness for a class of generalized Zakharov system
%J Journal of nonlinear sciences and its applications
%D 2017
%P 1289-1302
%V 10
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.22436/jnsa.010.04.01/
%R 10.22436/jnsa.010.04.01
%G en
%F JNSA_2017_10_4_a0
You, Shujun; Ning, Xiaoqi. Well-posedness for a class of generalized Zakharov system. Journal of nonlinear sciences and its applications, Tome 10 (2017) no. 4, p. 1289-1302. doi : 10.22436/jnsa.010.04.01. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.010.04.01/

[1] Aslan, I. Generalized solitary and periodic wave solutions to a (2 + 1)-dimensional Zakharov-Kuznetsov equation, Appl. Math. Comput., Volume 217 (2010), pp. 1421-1429 | DOI | Zbl

[2] Aslan, I. The first integral method for constructing exact and explicit solutions to nonlinear evolution equations, Math. Methods Appl. Sci., Volume 35 (2012), pp. 716-722 | Zbl | DOI

[3] Bejenaru, I.; Herr, S.; Holmer, J.; Tataru, D. On the 2D Zakharov system with \(L^2\)-Schrödinger data, Nonlinearity, Volume 22 (2009), pp. 1063-1089 | DOI | Zbl

[4] Brézis, H.; Wainger, S. A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations, Volume 5 (1980), pp. 773-789 | DOI | Zbl

[5] Gana, Z.-H.; Guo, B.; Han, L.-J.; Zhang, J. Virial type blow-up solutions for the Zakharov system with magnetic field in a cold plasma, J. Funct. Anal., Volume 261 (2011), pp. 2508-2528 | Zbl | DOI

[6] Ginibre, J.; Tsutsumi, Y.; Velo, G. On the Cauchy problem for the Zakharov system, J. Funct. Anal., Volume 151 (1997), pp. 384-436 | DOI

[7] Glangetas, L.; Merle, F. Concentration properties of blow-up solutions and instability results for Zakharov equation in dimension two, II, Comm. Math. Phys., Volume 160 (1994), pp. 349-389 | Zbl | DOI

[8] Glassey, R. T. Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension, Math. Comp., Volume 58 (1992), pp. 83-102 | DOI | Zbl

[9] Guo, B.; Zhang, J.-J.; Pu, X.-K. On the existence and uniqueness of smooth solution for a generalized Zakharov equation, J. Math. Anal. Appl., Volume 365 (2010), pp. 238-253 | DOI | Zbl

[10] Lions, J.-L. Quelques méthodes de résolution des problémes aux limites non linéaires, (French) Dunod; Gauthier-Villars, Paris, 1969

[11] Merle, F. Blow-up results of virial type for Zakharov equations, Comm. Math. Phys., Volume 175 (1996), pp. 433-455 | DOI | Zbl

[12] Ozawa, T.; Tsutaya, K.; Tsutsumi, Y. Well-posedness in energy space for the Cauchy problem of the Klein-Gordon-Zakharov equations with different propagation speeds in three space dimensions, Math. Ann., Volume 313 (1999), pp. 127-140 | DOI | Zbl

[13] Ozawa, T.; Tsutsumi, Y. The nonlinear Schrödinger limit and the initial layer of the Zakharov equations, Differential Integral Equations, Volume 5 (1992), pp. 721-745 | Zbl

[14] Takaoka, H. Well-posedness for the Zakharov system with the periodic boundary condition, Differential Integral Equations, Volume 12 (1999), pp. 789-810 | Zbl

[15] You, S.-J. The posedness of the periodic initial value problem for generalized Zakharov equations, Nonlinear Anal., Volume 71 (2009), pp. 3571-3584 | Zbl | DOI

[16] You, S.-J.; Guo, B.-L.; Ning, X.-Q. Equations of Langmuir turbulence and Zakharov equations: smoothness and approximation, Appl. Math. Mech. (English Ed.), Volume 33 (2012), pp. 1079-1092 | Zbl | DOI

[17] Zakharov, V. E. Collapse of Langmuir waves, Sov. Phys. JETP, Volume 35 (1972), pp. 908-914

Cité par Sources :