# Well-posedness for a class of generalized Zakharov system

Volume 10, Issue 4, pp 1289--1302
Publication Date: April 20, 2017 Submission Date: February 01, 2017
• 2382 Views

### Authors

Shujun You - School of Mathematical Sciences, Huaihua University, Huaihua 418008, China. Xiaoqi Ning - School of Mathematical Sciences, Huaihua University, Huaihua 418008, China.

### Abstract

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.

### Share and Cite

##### ISRP Style

Shujun You, Xiaoqi Ning, Well-posedness for a class of generalized Zakharov system, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 4, 1289--1302

##### AMA Style

You Shujun, Ning Xiaoqi, Well-posedness for a class of generalized Zakharov system. J. Nonlinear Sci. Appl. (2017); 10(4):1289--1302

##### Chicago/Turabian Style

You, Shujun, Ning, Xiaoqi. "Well-posedness for a class of generalized Zakharov system." Journal of Nonlinear Sciences and Applications, 10, no. 4 (2017): 1289--1302

### Keywords

• Global solutions
• Zakharov equations
• well-posedness.

•  35A01
•  35A02

### References

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

• [2] I. Aslan, The first integral method for constructing exact and explicit solutions to nonlinear evolution equations, Math. Methods Appl. Sci., 35 (2012), 716–722.

• [3] I. Bejenaru, S. Herr, J. Holmer, D. Tataru, On the 2D Zakharov system with $L^2$-Schrödinger data, Nonlinearity, 22 (2009), 1063–1089.

• [4] H. Brézis, S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773–789.

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

• [6] J. Ginibre, Y. Tsutsumi, G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384–436.

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

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

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

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

• [11] F. Merle, Blow-up results of virial type for Zakharov equations, Comm. Math. Phys., 175 (1996), 433–455.

• [12] T. Ozawa, K. Tsutaya, Y. Tsutsumi, 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., 313 (1999), 127–140.

• [13] T. Ozawa, Y. Tsutsumi, The nonlinear Schrödinger limit and the initial layer of the Zakharov equations, Differential Integral Equations, 5 (1992), 721–745.

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

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

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

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