Abstract
Through applying Galerkin method, we establish the approximating solution for one-dimension Klein-Gordon-Zakharov equations and obtain the local classical solution. By applying integral estimates, we also obtain the existence and uniqueness of the global classical solution of Klein-Gordon-Zakharov equations.
Highlights
The Klein-Gordon-Zakharov equations are used to show the interaction between Langmuir wave and ion wave in the plasma, where utt − uxx + u + nu = 0, (1)
Guo’s idea and method [1], our paper focuses on the existence and uniqueness of the local and global classical solution of the periodic initial value problem for the above one-dimension Klein-Gordon-Zakharov equations
Assume that u1, n1 and u2, n2 are both classical solutions with the same periodic initial value of (1)-(2)
Summary
L. Guo’s idea and method [1], our paper focuses on the existence and uniqueness of the local and global classical solution of the periodic initial value problem for the above one-dimension Klein-Gordon-Zakharov equations. A. Levine in [6] adopted potential well method and concave function method to study the blowing up of the solution of the Cauchy problem of this equation. Based on conservation of energy, they obtained global existence and uniqueness with small initial value of above Cauchy problem. We structure the approximate solution of (8)–(11); the aim is to obtain the local classical solution We employ this method of integral estimates to prove the existence and uniqueness of the global solution of (1)-(2). C0, C1, C2, . . . , D0, D1, D2, . . . , E0, E1, E2, . . . are estimate constants, which relate to initial conditions
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