Theoretical and Computational Advances in Nonlinear Dynamical Systems 2018View this Special Issue
The Global Existence and Uniqueness of the Classical Solution with the Periodic Initial Value Problem for One-Dimension Klein-Gordon-Zakharov Equations
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.
The Klein-Gordon-Zakharov equations are used to show the interaction between Langmuir wave and ion wave in the plasma, wherewith periodic initial conditionHere, (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 , 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 , H. Pecher has researched the following Klein-Gordon equation:where , , . Given the existence of the Cauchy problem of (3), H. A. Levine in  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 , W. A. Strauss introduced a compactness lemma, which proved existence of the solitary wave of these equations. In , Ozawa, Tsutaya, and Tsutsumi studied the following Klein-Gordon-Zakharov system:with initial conditionThrough 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 , 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  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).
Now, let us describe our results. Inspired by , we first introduced the following equations:with periodic initial dataMoreover, , , , , , , , 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 , thenwhere
Lemma 3. If , , and , then where , , , and are constants, which only relate to , , , and .
Proof. From (18), we can easily obtain thatSo that,By (9) and (18), we haveHence, 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 thatwhere 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 thatEquation (38) takes inner product with ; we have thatFurthermore, where .
So,Differentiating (1) with respect to , we have thatNext, (42), respectively, makes inner product with and and takes real part; we can obtain thatHere, 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 haveFurthermore, also differentiating (10) with respect to thrice and differentiating (11) with respect to twice and eliminating , we can obtainEquation (49) makes inner product with ; thenFurthermore,Due to Lemmas 4 and 6, Corollary 5, and (53) and (54), we have thatwhere Equation (50) makes inner product with and takes real part; we have thatCombining (51), (56), and (58) and Corollary 5 and applying Gronwall inequality, Lemma 7 can easily be proven.
3. Existence of Local Classical Solution
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 obtainFurthermore, (60) differentiates with respect to and (61) differentiates with respect to twice; we have thatNext, (74) multiplies and makes summation for ; it is easy to obtain the following equation: So, we obtain thatFurthermore,Therefore,where and .
Equation (61) multiplies (); we have thatEquation (82) multiplies ; thenFurthermore,So,where .
Combining (76), (77), (78), (81), (84), and (86), we can obtain thatwhere .
So, by Gronwall inequality, we have that
Lemma 10. , , , and .
Definition 11. If , , , , and . Moreover, they satisfy the following integral identity:andThen, , , , and are local generalized solutions of (60)–(63). Besides, , , , and are periodic functions with about .
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 haveMoreover 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 thatNext, (62) differentiates with respect to twice, multiplies (), and makes summation for ; we have that