Well-posedness for a class of generalized Zakharov system
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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.
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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.
MSC
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.
