Advances in Mathematical Physics
Volume 2018 (2018), Article ID 4820601, 10 pages
https://doi.org/10.1155/2018/4820601

Research Article

The Global Existence and Uniqueness of the Classical Solution with the Periodic Initial Value Problem for One-Dimension Klein-Gordon-Zakharov Equations

Cong Sun¹ and Lixia Li²

¹School of Applied Mathematics, Jilin University of Finance and Economics, Changchun, Jilin 130117, China
²College of Mathematics and Physics, Bohai University, Liaoning 121013, China

Correspondence should be addressed to Cong Sun; 3211519@163.com

Received 2 January 2018; Accepted 15 March 2018; Published 28 June 2018

Academic Editor: Zhi-Yuan Sun

Copyright © 2018 Cong Sun and Lixia Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

The Klein-Gordon-Zakharov equations are used to show the interaction between Langmuir wave and ion wave in the plasma, where

with periodic initial condition

Here, (complex function) denotes the biggest moment scale component produced by electron in electric field. (real function) denotes the speed of deviations between the ion at any position and that at equilibrium position. ; .

Inspired by B. 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. In past decades, many scholars devoted their researches to Zakharov system and have obtained some important results [2–4].

In [5], H. Pecher has researched the following Klein-Gordon equation:

where , , . Given the existence of the Cauchy problem of (3), H. 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. Moreover, in [7], W. A. Strauss introduced a compactness lemma, which proved existence of the solitary wave of these equations. In [4], Ozawa, Tsutaya, and Tsutsumi studied the following Klein-Gordon-Zakharov system:

with initial condition

Through applying harmonic analysis method and contracting mapping principle, they obtained that the Cauchy problem (4)-(5) had local well-posedness about in energy space . Furthermore, based on conservation of energy, they also obtained global existence and uniqueness with small initial value of above Cauchy problem.

Furthermore, in [8], T. C. Wang et al considered the following Klein-Gordon-Zakharov equations:

By applying Leray-Schauder fixed-point theorem, they proved the existence of different solutions of system (6).

Recently, Z. Y. Zhang et al. studied [9] the following Klein-Gordon-Zakharov equations:

With different parameter conditions, they adopted the bifurcation method and dynamical systems approach to research bifurcation analysis and dynamic behaviour of travelling wave solutions of (7).

Besides, in [10], by combining the trigonometric function series method and the exp-function method, Y. Zhang et al. considered the travelling wave solutions of (7).

Now, let us describe our results. Inspired by [1], we first introduced the following equations:

with periodic initial data

Moreover, , , , , , , , and are periodic functions with about (). Firstly, applying Galerkin method, we structure the approximate solution of (8)–(11); the aim is to obtain the local classical solution. Next, we employ this method of integral estimates to prove the existence and uniqueness of the global solution of (1)-(2).

We divide this paper into four parts. In Section 2, we establish a priori estimate of (8)–(11). In Section 3, we give the existence of local and global smooth solutions of the above equations. Lastly, in Section 4, we prove existence and uniqueness of the global classical solution of (1)-(2).

Notation. In complex Hilbert function space, inner product and standard norm are defined as follows:

where is conjugate function of .

is complex function space with generalized differentiable function , and the norm is defined as follows:

denotes function space about complex function of . Moreover, for any , , and is bound. Particularly,

Furthermore, , are estimate constants, which relate to initial conditions.

2. A Priori Estimate of Integrating

Lemma 1. If , , and , then .

Proof. Equation (10) makes inner product with and takes real part; we have

By Gronwall inequality, we can obtain that

Lemma 2. If , then

where

Proof. Taking inner product (11) with and making real part, we have that

Combining (20)–(23) yields

Therefore,

Lemma 3. If , , and , then

where , , , and are constants, which only relate to , , , and .

Proof. From (18), we can easily obtain that

So that,

By (9) and (18), we have

Hence,

Furthermore, (9) takes inner product with ; we get

By Gronwall inequality, we have

Hence, in combination with (18), we can deduce that

Lemma 4. If , , , and , then

Proof. The first equation of (1) makes inner product with and takes real part; we have that

where and .
So that, by Gronwall inequality, we obtain that

Corollary 5. , , , and .

Lemma 6. If , then one has

Proof. We differentiate with respect to the second equation of (1); we obtain that

Equation (38) takes inner product with ; we have that

Furthermore,

where .
So,

Differentiating (1) with respect to , we have that

Next, (42), respectively, makes inner product with and and takes real part; we can obtain that

Here,

Combining (39), (41), (43), and (44) yields

Therefore, applying Gronwall inequality, we can obtain

Lemma 7. If , then

Proof. Differentiating (8) with respect to thrice and differentiating (9) with respect to and twice and eliminating , we have

Furthermore, also differentiating (10) with respect to thrice and differentiating (11) with respect to twice and eliminating , we can obtain

Equation (49) makes inner product with ; then

Furthermore,

Due to Lemmas 4 and 6, Corollary 5, and (53) and (54), we have that

where

Equation (50) makes inner product with and takes real part; we have that

Combining (51), (56), and (58) and Corollary 5 and applying Gronwall inequality, Lemma 7 can easily be proven.

3. Existence of Local Classical Solution

Firstly, we apply Galerkin method to establish approximating solution for (8)–(11) as follows:

Moreover, , , , and satisfy the following equations and initial condition:

Next, by implementing integration estimates for the approximation solution of (60)–(63), we establish local classical solution of (60)–(63).

Lemma 8. If , , and , then we have

Proof. Equation (62) multiplies and makes summation for ; we obtain

Taking real part, we have that

Therefore, by Gronwall inequality, we have .

Lemma 9. If , , , , and , then

Proof. Equation (60) multiplies and makes summation for ; we can obtain

Furthermore, (60) differentiates with respect to and (61) differentiates with respect to twice; we have that

Next, (74) multiplies and makes summation for ; it is easy to obtain the following equation:

So, we obtain that

Furthermore,

Therefore,

where and .
Equation (61) multiplies (); we have that

Equation (82) multiplies ; then

Furthermore,

So,

where .
Combining (76), (77), (78), (81), (84), and (86), we can obtain that

where .
So, by Gronwall inequality, we have that

Lemma 10. , , , and .

Proof. The proof is similar to that of Lemma 9. For details, see Lemma 9.

Definition 11. If , , , , and . Moreover, they satisfy the following integral identity:

and

Then, , , , and are local generalized solutions of (60)–(63). Besides, , , , and are periodic functions with about .

Theorem 12. If , , , , , , , , and are period functions with about . Then, (60)–(63) have local generalized solution.

Proof. By Lemmas 8, 9, and 10, , , , , and have uniform boundedness. So, we can select a subsequence and satisfy , , , and . Moreover, there exists subsequences, , , and .
In (60)–(63), let ; we have

Moreover and it is dense in , approximates to . We complete the proof of this theorem. Similarly, we also can prove (64)–(67).

Lemma 13. If , , , and are bound, then

Proof. Equation (74) differentiates with respect to . Next, it multiplies and makes summation for . We can easily obtain the following equation:

Furthermore,

Meanwhile, by CoóojieB inequality,

Here,

Moreover, (62) differentiates with respect to twice, multiplies , and makes summation for ; we can obtain that

Next, (62) differentiates with respect to twice, multiplies (), and makes summation for ; we have that

and (60) differentiates with respect to and ; we can obtain that

where multiplies and makes summation for ; we can obtain that

Taking real part, we have that

By (97), (98), (99), (100), (103), (104), (107), (110), and (111) and applying Gronwall inequality, we can obtain the result of Lemma 13.

Corollary 14. If , , and are bound, then there exist , , , , and .

Theorem 15. Assume that the conditions of Lemmas 8, 9, 10, and 13 are satisfied, then initial value problem (1) has local classical solution.

Proof. The idea and method are similar to Theorem 3.2 of [1]. For details, please see this paper.

4. Initial Value Function and Its Derivative Estimate in

In this section, we apply the idea and method of [1] to discuss boundedness of , , , and .

Assume that , , , and , and take , , , and ; they satisfy , , , and .

Due to (64), (65), (66), and (67) and above assumption, it is easy to deduce that , , , and have uniform boundedness about .

Next, we denote an orthogonal projection operator , which is spanned by . By (60), (61), (62), and (63), we have that

and by , we have

So, we obtain that , , , and have uniform boundedness about .

Similarly, it is not difficult to obtain some higher-order derivative estimation, including , , , and . For details, we do not discuss. We have the following Lemma.

Lemma 16. Assume that , , , , then , , , , , , , , , , , , , , , , , , , and have uniform boundedness about .

5. Existence and Uniqueness of the Global Classical Solution

Theorem 17. Assume that , , , , and they are periodic functions with about ; then initial value problem (1) has global classical solution.

Proof. The idea and method are similar to Theorem 4.1 of [1]. For details, please see this paper.

Theorem 18. Global classical solution of (1)-(2) is unique.

Proof. Assume that , and , are both classical solutions with the same periodic initial value of (1)-(2). Let and ; by (1)-(2), we obtain the following equations about and :

where is a complex function and is a real function.
Equation (114) makes inner product with and takes real part; we have that

Meanwhile, (115) makes inner product with and takes real part; we have that

Combining (117)–(128), we have that

where

Integrating (130) from , we obtain that

Hence, and .

Data Availability

No data were used to support this study. In our manuscript, any reader can access the data supporting the conclusions of the study and this clearly outlines the reasons why unavailable data cannot be released.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This paper is funded by the new entry doctoral research project of Jilin University of Finance and Economics (2017B2013).

References

B. L. Guo and L. J. Shen, “The existence and uniqueness of the classical solution to the periodic initial value problem for the Zakharov equation,” Acta Mathematicae Applicatae Sinica, vol. 5, no. 3, pp. 310–324, 1982 (Chinese).

F. Oliveira, “Stability of the solitons for the one-dimensional Zakharov-Rubenchik equation,” Physica D: Nonlinear Phenomena, vol. 175, no. 3-4, pp. 220–240, 2003.

T. Ozawa, K. Tsutaya, and Y. Tsutsumi, Normal Form and Global Solutions for the Klein-Gordon- Zakharov Equation, University of Tokyo, 1994.

T. Ozawa, K. Tsutaya, and 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,” Mathematische Annalen, vol. 313, no. 1, pp. 127–140, 1999.

H. Pecher, “Lp Abschatzungen and klassiche Losungen fur nichtlieare wellengenichungen. I,” Mathematische Zeitschrift, vol. 150, no. 2, pp. 159–183, 1976.

H. A. Levine, “Instability and nonexistence of global solution to nonlinear wave equation of the form Putt=-Au-F(u),” Transactions of the American Mathematical Society, vol. 192, pp. 1–21, 1974.

W. A. Strauss, “Existence of solitary waves in higher dimensions,” Communications in Mathematical Physics, vol. 55, no. 2, pp. 149–162, 1977.

T. Wang, J. Chen, and L. Zhang, “Conservative difference methods for the Klein-Gordon-Zakharov equations,” Journal of Computational and Applied Mathematics, vol. 205, no. 1, pp. 430–452, 2007.

Z. Zhang, F.-L. Xia, and X.-P. Li, “Bifurcation analysis and the travelling wave solutions of the Klein-Gordon-Zakharov equations,” Pramana—Journal of Physics, vol. 80, no. 1, pp. 41–59, 2013.

Z. Y. Zhang, Y. H. Zhang, X. Y. Gan, and D. M. Yu, “A note on exact travelling wave solutions for the klein -gordon-zakharov equations,” Zeitschriftfü Naturforschung, vol. A67, no. 167, 2012.