Fixed Point Theorem Based Solvability of 2-Dimensional Dissipative Cubic Nonlinear Klein-Gordon Equation

Abstract

The purpose of this article is to establish the solvability of the 2-Dimensional dissipative cubic nonlinear Klein-Gordon equation (2DDCNLKGE) through periodic boundary value conditions (PBVCs). The analysis of this study is founded on the Galerkin’s method (GLK) and the Leray-Schauder’s fixed point theorem (LS). First, the GLK method is used to construct some uniform priori estimates of approximate solution to the corresponding equation of 2DDCNLKGE. Finally, the LS fixed point theorem is applied to obtain the efficient and straightforward existence and uniqueness criteria of time periodic solution to the 2DDCNLKGE.

1. Introduction

The nonlinear Klein-Gordon equation (NLKGE for short) has been obtained by a modification of nonlinear Schrödinger equation $i{\partial }_t\psi =-\frac{1}{2}{\partial }_x{}^2\psi +k{\left|\psi \right|}^2\psi$, where $\psi \left(x,t\right)$ is a complex field. This equation has extensively been used for modeling of various nonlinear physical and environmental phenomena; see for instance [1,2,3,4,5,6] and their cited references.

As vital nonlinear partial differential equations (NLPDEs), the NLKG types equations have received great consideration for developing solutions by applying various types of techniques; see for instance [3,4,7,8] and their cited references.

Certain nonlinear physical systems expressed with NLPDEs may be transformed into nonlinear ordinary differential equations by using traveling wave transformations, and the travelling wave solutions of these NLPDEs is analogous to the exact solutions of corresponding nonlinear ordinary differential equations. The 2-Dimension dissipative NLKGE is a practical instance of the above mentioned nonlinear physical system.

Throughout this paper, $R^{+}$ denotes a set of positive real numbers.

The general form of a 2-Dimension dissipative NLKGE is:

$$u_{tt}-\mathsf{\Delta }u+\alpha u_t+\beta u+\gamma {\left|u\right|}^{k}u=f\left(x,t\right),x\in \mathsf{\Omega },t\in R^{+}$$

(1)

where $u$ is a real valued, $\mathsf{\Omega }=\left[0,L\right]\times \left[0,L\right],\alpha ,\beta ,\gamma$ are real physical constants, $\mathsf{\Delta }=\frac{{\partial }^2}{\partial x_1^2}+\frac{{\partial }^2}{\partial x_2^2}$ and $k$ is a positive integer, which is used to measure the nonlinearity of 2D dissipative NLKGE. Here, it is also mentioned that $\alpha \mathrm{and}\beta$ may be considered as continuous functions.

For $k=2$, Equation (1) reduces to the following 2D dissipative quadratic NLKGE:

$$u_{tt}-\mathsf{\Delta }u+\alpha u_t+\beta u+\gamma {\left|u\right|}^{2}u=f\left(x,t\right),x\in \mathsf{\Omega },t\in R^{+}$$

(2)

For $k=3$ Equation (1), reduces to the following 2DDCNLKGE:

$$u_{tt}-\mathsf{\Delta }u+\alpha u_t+\beta u+\gamma {\left|u\right|}^{3}u=f\left(x,t\right),x\in \mathsf{\Omega },t\in R^{+}$$

(3)

Equation (3) is used to explain relativistic quantum mechanics; see for instance [9].

In 2004, Gao and Guo [10] established solvability of the time-periodic solution of a 2D dissipative quadratic NLKG equation given by (2) with time periodic boundary value conditions using the GLK method [11,12] and the LS fixed point theorem [13]. There exists a wide range of solvability for Equations (2) and (3), in case of $\alpha =0\mathrm{and}f=0$; see for instance [1,14,15,16] and their cited references. After Gao and Guo [10], in 2006, Fu and Guo [17] established the time-periodic solution of the following one-dimensional viscous Camassa-Holm equation:

$$u_t-u_{xxt}-\mu \left(u_{xx}-u_{xxxx}\right)+3u{u}_x=2u_x{u}_{xx}+u{u}_{xxx}+f\left(x,t\right),x\in \mathsf{\Omega },t\in R^{+},\mu >0$$

(4)

applying the GLK method and the LS fixed point theorem. Sequentially, in 2014, Gao et al. [16] proved the uniqueness of the time-periodic solution to 1D quadratic viscous modified Camassa-Holm equation:

$$u_t-u_{xxt}-\mu \left(u_{xx}-u_{xxxx}\right)+3u^2{u}_x-2u_x{u}_{xx}-u{u}_{xxx}=f\left(x,t\right),x\in \mathsf{\Omega },t\in R^{+},\mu >0$$

(5)

by means of the GLK method and the LS fixed point theorem.

In the last few decades, many researchers have devoted themselves to establishing the time-periodic solution for various nonlinear evolution equations; see for instance [10,11,17,18,19,20] and their cited references. Recently, Obinwanne and Collins [21] applied the LS fixed point theorem to obtain a solution of Duffing’s equation. Moreover, there is a certain focus on the uniqueness of the time-periodic solution of 2DDCNLKGE given by Equation (5), applying the GLK method and the LS fixed point theorem. Inspired by the above-mentioned works in this paper, we establish a solvability for the following 2DDCNLKGE with PBVCs applying the GLK method and the LS fixed point theorem:

$$\{\begin{array}{l}{u}_{tt}-\mathsf{\Delta }u+\alpha u_t+\beta u+\gamma {\left|u\right|}^{3}u=f\left(x,t\right),x\in \mathsf{\Omega },\\ u\left(x+L,t\right)=u\left(x,t\right),x\in \mathsf{\Omega },\\ f\left(x,t+\omega \right)=f\left(x,t\right),\end{array}$$

(6)

where $t\in R^{+},u=u\left(x,t\right)$ is a real value, $\mathsf{\Omega }=\left[0,L\right]\times \left[0,L\right],$ $\alpha ,\beta ,\gamma$ are real physical constants and $\mathsf{\Delta }=\frac{{\partial }^2}{\partial x_1^2}+\frac{{\partial }^2}{\partial x_2^2}$.

The outline of this article is as follows: The present section provides an introduction to this article. In Section 2, we provide some notations, the GLK method, and the LS fixed point theorem. Section 3 is used to formulate uniform priori estimates of the approximate solution of 2DDCNLKGE given in Equation (6), which will be applied in the next section. Section 4 is devoted to establishing a unique time-periodic solution criterion for 2DDCNLKGE given in Equation (6). Finally, we provide a conclusion.

2. Preliminary Notes

Here, we provide some introductory truths that are needed to describe the main results of this article.

Let $B$ be a Banach space. For $1\le p\le \infty ,$ the space $L^p\left(B;\omega \right)$ is defined as the set of $\omega$-periodic $B$-measurable functions on $\Re$ (set of real numbers), such that:

$${\Vert u\Vert }_{L^p\left(B;\omega \right)}=\{\begin{array}{cc}{\left({\displaystyle {\int }_0^{\omega }{\Vert u\Vert }_B^{p}ds}\right)}^{1/p}<\infty ,& 1\le p<\infty \\ \underset{0\le t\le \omega }{\sup }{\Vert u\Vert }_B<\infty ,& p=\infty .\end{array}$$

The space $W^{k,p}\left(B;\omega \right)$ denote the set of functions that belong to $L^p\left(B;\omega \right)$ together with their derivatives up to order $k$; if $B$ is a Hilbert space, we write $W^{k,2}\left(B;\omega \right)=H^k\left(B;\omega \right)$.

During this study, we use these notations:

$$\begin{array}{l}{L}_{per}^2=\left\{u\in L^2\left(\mathsf{\Omega }\right):u\left(x+L,t\right)=u\left(x,t\right)\right\},\\ H_{per}^k=\left\{u\in H^k:u\left(x+L,t\right)=u\left(x,t\right)\right\},k=1,2,\end{array}$$

where $L^2\left(\mathsf{\Omega }\right)$ is obtained from $L^p\left(B;\omega \right)$ by putting $p=2$ and $H^k=H^k\left(B;\omega \right)$.

And

$$C^k\left(\omega ,X\right)=\left\{f:\left[0,\infty \right)\to X:f^{\left(i\right)}\mathrm{is}\mathrm{continuous},i=0,1,\cdots ,k,f\mathrm{is}\mathrm{an}\omega -\mathrm{periodic}\mathrm{function}\right\}.$$

where $X$ may be a real or complex space.

For $k=0,$ we replace $C^0\left(\omega ,X\right)\mathrm{with}C\left(\omega ,X\right)$. The inner product and norm of $L^2\left(\mathsf{\Omega }\right)$ are denoted by $\left(.,.\right)$ and $\Vert .\Vert$, respectively. We also denote that:

$$T=-\mathsf{\Delta },D\left(T\right)=H_{per}^2,N\left(u\right)=-\gamma {\left|u\right|}^{3}u,\mathrm{and}N$$

is a cubic nonlinear operator on $L^2\left(\mathsf{\Omega }\right)$.

Now, we state the LS fixed point theorem, which will be used as the main tools of this study.

Theorem 1 [13]. 

Let$B$be a Banach space and  $T:B\to B$ be a continuous and compact mapping with property “there exists  $R>0$ such that the statement ($u=rTuwithr\in \left[0,1\right)$) implies  ${\Vert u\Vert }_B<R$ ”. Then  $T$ has a fixed point  $u^{\ast }$ such that $\Vert u^{\ast }\Vert \le R$.

We now provide a brief discussion on the GLK method [11,12].

The GLK method is a strong and general method. Here, we introduce the GLK method with a nonconcrete problem modelled as a frail design on a Hilbert space $H,$ V {\displaystyle V} specifically searching for $x\in H$:

$$y\in H,\mathrm{and}b\left(x,y\right)=h\left(y\right)$$

where $b\left(.,.\right)$ is bilinear and $h\left(y\right)$ is a bounded linear functional on $H$.

Select a $n$ dimension subspace $H_n$ of the Hilbert space $H$ to solve the following problem: search $x_n\in H_n$ from:

$$b\left(x_n,y_n\right)=h\left(y_n\right),\mathrm{for}\mathrm{all}{y}_n\in H_n$$

(7)

Equation (7) is known as the GLK formula. The main theme of the GLK method is that the mistake is orthogonal to the preferred sub-spaces, since $H_n\subset H$ V n ⊂ V {\displaystyle V_{n}\subset V}, $y_n$ is used v n {\displaystyle v_{n}} as a trial vector in the main problem. If the mistake between the solution of the main problem $x$ u {\displaystyle u} and the solution of the GLK formula $x_n$ is $m_n=x-x_n$, ϵ n = u − u n {\displaystyle \epsilon _{n} = u − u_{n}}thenthethen we have:

$$b\left(m_n,y_n\right)=b\left(x,y_n\right)-b\left(x_n,y_n\right)=h\left(y_n\right)-h\left(y_n\right)=0$$

(8)

In the GLK method, we can represent the problem in matrix form and calculate the solution algorithmically. Regarding this matrix representation, if we consider $e_1,e_2,e_3,\cdots ,e_n$ as a basis of $H_n,$ then from Equation (7), we can obtain $x_n\in H_n$ from $b\left(x_n,e_i\right)=h\left(e_i\right),$ $i=1,2,3,\cdots ,n$.

Now, if we enlarge $x_n$ according to this basis, we get $x_n={\displaystyle \sum _{j=1}^n{x}_j{e}_j}$ and hence obtain

$$b\left({\displaystyle \sum _{j=1}^n{x}_j{e}_j},e_i\right)={\displaystyle \sum _{j=1}^n{x}_{j}b\left(e_j,e_i\right)}=h\left(e_i\right),i=1,2,3,\cdots ,n.$$

(9)

Equation (9) represents a system of equations given by $C_{ij}{x}_j=h_i,$ where the coefficient matrix $C_{ij}$ is given by ${\displaystyle \sum _{j=1}^{n}b\left(e_j,e_i\right)}$ and $h_i=h\left(e_i\right)$.

3. Existence of Uniform Priori Estimates for the Solution of 2DDCNLKGE

In this section, applying the GLK method and Theorem 1, we formulate uniform priori estimates for an approximate solution to the 2DDCNLKGE.

In space $C\left(\omega ,L_{per}^2\right)$, we write the problem given in Equation (6) as the following abstract problem:

$$u_{tt}+\alpha u_t+\beta u+Tu=N\left(u\right)+f,u\left(.,t\right)=u\left(.,t+\omega \right),\forall f\in C^1\left(\omega ,H_{per}^1\right).$$

(10)

Now, we obtain an approximate solution of 2DDCNLKGE given in Equation (6) using the GLK method. Let ${\left\{{\omega }_j\right\}}_{j=0}^{\infty }$ be a normal orthogonal basis of the space $L_{per}^2$ and satisfy $T{\omega }_j={\lambda }_j{\omega }_j,\left(j=1,2,\cdots \right),$ where ${\lambda }_j$ are eigenvalues for the map $T$ and the eigenvectors ${\omega }_j,\left(j=1,2,\cdots \right).$ We denote $H_n=\mathrm{span}\left\{{\omega }_1,{\omega }_2,\cdots ,{\omega }_n\right\},\forall n\in \aleph$ (set of natural numbers).

Now, by the GLK method, for any $n\in \aleph$ and any sequence of functions ${\left\{a_{jn}\left(t\right)\right\}}_{j=1}^n$, where $a_{jn}\left(t\right)\in C^2\left(\omega ,\Re \right),$ $\left(j=1,2,3,\cdots ,n\right)$ and $\Re$ denotes the set of real numbers, we can say that the function $u_n={\displaystyle \sum _{j=1}^n{a}_{jn}\left(t\right){\omega }_j\in C^2}\left(\omega ,H_n\right)$ is an approximate solution of Equation (10), if the following system holds:

$$\left(u_{ntt}+\alpha u_{nt}+\beta u_n+T{u}_n,{\omega }_j\right)=\left(N\left(u_n\right)+f,{\omega }_j\right),j=1,2,3,\cdots ,n,$$

(11)

where,

$$N\left(u_n\right)=-\gamma {\left|u_n\right|}^3{u}_n\mathrm{and}{H}_n=span\left\{{\omega }_1,{\omega }_2,{\omega }_3,\cdots ,{\omega }_n\right\}.$$

In order to demonstrate that Equation (10) has an approximate solution, we use Theorem 1. A solution $u\left(x,t\right)$ of Equation (10) is said to be unique if it has a fixed value $u_1\left(x,t\right)$, which satisfies Equation (10) uniquely, that is the solution $u\left(x,t\right)$ has no any value that is not equal to $u_1\left(x,t\right)$ and this solution will be $\omega$-periodic if $u\left(x,t\right)=u\left(x,t+\omega \right)$.

Now, from the classical viewpoint of ordinary differential equations, it is clear that for any fixed $v_n\left(t\right)={\displaystyle \sum _{j=1}^n{b}_{jn}\left(t\right){\omega }_j\in C^2\left(\omega ,H_n\right)},$ the following linear ordinary equation system

$$\left(u_{ntt}+\alpha u_{nt}+\beta u_n+T{u}_n,{\omega }_j\right)=\left(\mu N\left(v_n\right)+f,{\omega }_j\right),0\le \mu \le 1,j=1,2,3,\cdots ,n,$$

(12)

offers a unique $\omega$-periodic solution $a_{jn}\left(t\right)$ and the map $F_{\mu }:v_n\left(t\right)\to u_n\left(t\right)$ is continuous and compact on $C^2\left(\omega ,H_n\right)$. Furthermore, the map $F_{\mu }$ is completely continuous and hence uniform for $0\le \mu \le 1$. Clearly for $\mu =0,$ the linear ordinary equation system given by Equation (12) has a unique solution. Therefore, to prove the existence of the time periodic solution of Equation (12) by applying Theorem 2, it is enough to show that the inequality

$$\underset{0\le t\le \omega }{\sup }\Vert u_{ntt}\left(t\right)\Vert \le c$$

(13)

holds for all possible solutions of Equation (12), and the nonlinear term $N\left(u_n\right)$ is replaced by $\mu N\left(u_n\right),\left(0\le \mu \le 1\right)$, and $c$ is a constant function of $\alpha ,\beta ,\gamma ,\omega ,f\mathrm{and}L$.

Now, we establish some lemmas that convey the required uniform priori estimators for the time periodic solution of Equation (11).

Lemma 1. 

If $f\in C^1\left(\omega ,H_{per}^1\right)$, then

$${\Vert u_{nt}\left(t\right)\Vert }^2+{\Vert \nabla u_n\left(t\right)\Vert }^2+{\Vert u_n\left(t\right)\Vert }^2+{\Vert u_n\left(t\right)\Vert }_5^5\le d_1$$

where $d_1$is a positive constant function of $\alpha ,\beta ,\gamma ,\omega ,f\mathrm{and}L$.

Proof. 

After multiplication by ${a^{\prime }}_{jn}\left(t\right)$ and taking sum over $j$ from $1$ to $n$ on both sides of Equation (12), we get:

$$\begin{array}{l}\frac{1}{2}\frac{d}{dt}\left({\Vert u_{nt}\Vert }^2+{\Vert \nabla u_n\Vert }^2+\beta {\Vert u_n\Vert }^2\right)+\alpha {\Vert u_{nt}\Vert }^2+\frac{\mu \gamma }{5}\frac{d}{dt}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_n\left(t\right)\right|}^{5}dx}\\ ={\displaystyle {\int }_{\mathsf{\Omega }}f{u}_{nt}dx\le }\frac{\alpha }{2}{\Vert u_{nt}\Vert }^2+\frac{2}{\alpha }\Vert f\Vert ,\end{array}$$

(14)

That is

$$\frac{d}{dt}\left({\Vert u_{nt}\Vert }^2+{\Vert \nabla u_n\Vert }^2+\beta {\Vert u_n\Vert }^2+\frac{2\mu \gamma }{5}{\Vert u_n\Vert }_5^5\right)+\alpha {\Vert u_{nt}\Vert }^2\le \frac{4}{\alpha }\Vert f\Vert .$$

(15)

Now, after multiplication by $a_{jn}\left(t\right)$ and taking sum over $j$ from $1$ to $n$ on both sides of Equation (12), we yield:

$$\left(u_{ntt}+\alpha u_{nt}+\beta u_n+T{u}_n,u_n\right)=\left(\mu N\left(u_n\right)+f,u_n\right),$$

This implies that

$$\frac{d}{dt}\alpha {\Vert u_n\Vert }^2+2{\Vert \nabla u_n\Vert }^2+\beta {\Vert u_n\Vert }^2+2{\displaystyle {\int }_{\mathsf{\Omega }}{u}_{ntt}{u}_{n}dx}+\frac{5\mu \gamma }{2}{\Vert u_n\Vert }_5^5\le \frac{4}{\beta }\Vert f\Vert .$$

(16)

Multiplying both sides of Equation (16) by $\delta$ and adding Equation (15), we have:

$$\frac{d}{dt}\left({\Vert u_{nt}\Vert }^2+{\Vert \nabla u_n\Vert }^2+\left(\beta +\delta \alpha \right){\Vert u_n\Vert }^2+\frac{2\mu \gamma }{5}{\Vert u_n\Vert }_5^5\right)+\alpha {\Vert u_{nt}\Vert }^2+2\delta {\Vert \nabla u_n\Vert }^2+\delta \beta {\Vert u_n\Vert }^2+\delta \mu \gamma {\Vert u_n\Vert }_5^5+2\delta {\displaystyle {\int }_{\mathsf{\Omega }}{u}_{ntt}\left(t\right)u_{n}dx}\le 4\left(\frac{1}{\alpha }+\frac{\delta }{\beta }\right)\Vert f\Vert =c_1$$

(17)

Integrating inequality (17) over the closed interval $\left[0,\omega \right],$ we get:

$${\displaystyle {\int }_0^{\omega }\left[\left(\alpha -2\delta \right){\Vert u_{nt}\Vert }^2+2\delta {\Vert \nabla u_n\Vert }^2+\delta \beta {\Vert u_n\Vert }^2+\frac{5\delta \mu \gamma }{2}{\Vert u_n\Vert }_5^5\right]dt}\le \omega c_1.$$

Now, if we take $\delta <\alpha /2,$ then for $t^{\ast }\in \left(0,\omega \right),b>0,$ we have:

$${\Vert u_{nt}\left(t^{\ast }\right)\Vert }^2+{\Vert \nabla u_n\left(t^{\ast }\right)\Vert }^2+{\Vert u_n\left(t^{\ast }\right)\Vert }^2+{\Vert u_n\left(t^{\ast }\right)\Vert }_5^5\le \frac{c_1}{b}.$$

Again, integrating inequality (17) from $t^{\ast }$ to $t\in \left(t^{\ast },t^{\ast }+\omega \right)$ and for any $L>0,$ we have:

$${\Vert u_{nt}\left(t\right)\Vert }^2+{\Vert \nabla u_n\left(t\right)\Vert }^2+{\Vert u_n\left(t\right)\Vert }^2+{\Vert u_n\left(t\right)\Vert }_5^5\le 2c_1\omega +\left(c_{1}L/b\right)=d_1$$

Therefore, we deduce that:

$$\underset{0\le t\le \omega }{\sup }\left({\Vert u_{nt}\left(t\right)\Vert }^2+{\Vert \nabla u_n\left(t\right)\Vert }^2+{\Vert u_n\left(t\right)\Vert }^2+{\Vert u_n\left(t\right)\Vert }_5^5\right)\le d_1.$$

(18)

Which finishes the proof. □

Remark 1. 

From inequality (18), we can obtain the estimate $\underset{0\le t\le \omega }{\sup }\Vert u_{ntt}\left(t\right)\Vert \le d$. Hence, LS fixed point Theorem 2 and Lemma 1 offers the following result:

“If $f\in C\left(\omega ,H_{per}^1\right),$then for any positive integer $n$Equation (10) has an approximate solution $\left(u_n\left(t\right),{u^{\prime }}_n\left(t\right)\right)\in C^2\left(\omega ,H_n\right)\times C^1\left(\omega ,H_n\right)$”.

From the above established results, it is clear that ${\left\{u_n\right\}}_{n=1}^{\infty }$ is the sequence of an approximate solution of Equation (10). Now we have to prove that the sequence ${\left\{u_n\right\}}_{n=1}^{\infty }$ is convergent and that the converging point is a solution of Equation (10) and to fulfill the requirement, we have to establish a priori estimates for $u_n$.

Lemma 2. 

If $f\in C\left(\omega ,H_{per}^1\right),$then there exists a positive constant $d_2=d_2\left(\alpha ,\beta ,\gamma ,\omega ,L,f\right)$ such that:

$$\underset{0\le t\le \omega }{\sup }\left({\Vert \nabla u_{nt}\left(t\right)\Vert }^2+{\Vert \mathsf{\Delta }{u}_n\left(t\right)\Vert }^2+{\Vert \nabla u_n\left(t\right)\Vert }^2\right)\le d_2$$

Proof. 

The following inequalities, which are obtained from Ladyzhenskaya’s inequality [22,23,24,25], will be needed to prove this lemma:

$$\{\begin{array}{cc}{\Vert u_n\Vert }_4\le c{\Vert \nabla u_n\Vert }^{1/2}{\Vert u_n\Vert }^{2/2},& u_n\in H^1\left(\mathsf{\Omega }\right),\\ {\Vert u_n\Vert }_8\le c{\Vert \mathsf{\Delta }{u}_n\Vert }^{1/6}{\Vert u_n\Vert }^{5/6},& u_n\in H^2\left(\mathsf{\Omega }\right).\end{array}$$

(19)

Multiplying both sides of Equation (12) by $-{\lambda }_j{a}_{jn}\left(t\right)$ and taking the sum over $j$ from 1 to $n$, we have $\left(u_{ntt}+T{u}_n+\alpha u_{nt}+\beta u_m,-\mathsf{\Delta }{u}_n\right)=\left(\mu N\left(u_n\right)+f,-\mathsf{\Delta }{u}_n\right),$ and hence applying inequalities of (19), Hölder’s inequality, and Young’s inequality, we yield the following:

$$\begin{array}{l}\frac{1}{2}\frac{d}{dt}\alpha {\Vert \nabla u_n\Vert }^2+{\Vert \mathsf{\Delta }{u}_n\Vert }^2+\beta {\Vert \nabla u_n\Vert }^2+{\displaystyle {\int }_{\mathsf{\Omega }}\nabla u_n\nabla u_{ntt}\left(t\right)dx}\\ \le \mu \gamma d_2\Vert \mathsf{\Delta }{u}_n\Vert {\Vert u_n\Vert }_4{\Vert u_n\Vert }_8^3+\Vert f\Vert \Vert \mathsf{\Delta }{u}_n\Vert \\ \le \mu \gamma d_2{c}^2\Vert \mathsf{\Delta }{u}_n\Vert {\Vert \nabla u_n\Vert }^{1/2}{\Vert u_n\Vert }^{1/2}{\Vert \mathsf{\Delta }{u}_n\Vert }^{1/2}{\Vert u_n\Vert }^{5/2}+\Vert f\Vert \Vert \mathsf{\Delta }{u}_n\Vert \\ \le \mu \gamma e{d}_3{\Vert \mathsf{\Delta }{u}_n\Vert }^{3/2}+\Vert f\Vert \Vert \mathsf{\Delta }{u}_n\Vert \\ \le 2\epsilon {\Vert \mathsf{\Delta }{u}_n\Vert }^2+e\left(\epsilon ,\gamma ,c,d_3\right)+\frac{{\Vert f\Vert }^2}{\epsilon }.\end{array}$$

Taking $\epsilon$ to be small enough, we get:

$$\frac{1}{2}\frac{d}{dt}\alpha {\Vert \nabla u_n\Vert }^2+\frac{3}{4}{\Vert \mathsf{\Delta }{u}_n\Vert }^2+\beta {\Vert \nabla u_n\Vert }^2+{\displaystyle {\int }_{\mathsf{\Omega }}\nabla u_n\nabla u_{ntt}\left(t\right)dx}\le d_4.$$

(20)

Multiplying both sides of Equation (12) by $-{\lambda }_j{a^{\prime }}_{jn}\left(t\right)$ and taking the sum over $j$ from 1 to $n$, we obtain $\left(u_{ntt}+T{u}_n+\alpha u_{nt}+\beta u_m,-\mathsf{\Delta }{u}_{nt}\right)=\left(\mu N\left(u_n\right)+f,-\mathsf{\Delta }{u}_{nt}\right),$ and hence applying inequalities of (19), Hölder’s inequality, and Young’s inequality, we get the following:

$$\begin{array}{l}\frac{1}{2}\frac{d}{dt}\left[\beta {\Vert \nabla u_n\Vert }^2+{\Vert \mathsf{\Delta }{u}_n\Vert }^2+{\Vert \nabla u_{nt}\Vert }^2\right]+\alpha {\Vert \nabla u_{nt}\Vert }^2\\ =-\mu \gamma \mathrm{Re}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_n\right|}^3\nabla u_n.\nabla u_{nt}dx}+{\displaystyle {\int }_{\mathsf{\Omega }}\nabla f.\nabla u_{nt}dx},\left(0\le \mu \le 1\right)\\ \le e\gamma \Vert \nabla u_{nt}\Vert {\Vert u_n\Vert }_8^3{\Vert \nabla u_n\Vert }_4+\Vert \nabla f\Vert \Vert \nabla u_{nt}\Vert \\ \le e{d}_5{c}^2\Vert \nabla u_{nt}\Vert {\Vert \mathsf{\Delta }{u}_n\Vert }^{1/2}{\Vert u_n\Vert }^{5/2}{\Vert \nabla u_n\Vert }^{1/2}{\Vert \mathsf{\Delta }{u}_n\Vert }^{1/2}+\Vert \nabla f\Vert \Vert \nabla u_{nt}\Vert \\ \le \epsilon {\Vert \nabla u_{nt}\Vert }^2+e\left(\epsilon ,\gamma ,c,d_5\right)\Vert \mathsf{\Delta }{u}_n\Vert +\epsilon {\Vert \nabla u_{nt}\Vert }^2+e{\Vert \nabla f\Vert }^2.\end{array}$$

Taking $\epsilon$ to be small enough, we get:

$$\frac{1}{2}\frac{d}{dt}\left[\beta {\Vert \nabla u_n\Vert }^2+{\Vert \mathsf{\Delta }{u}_n\Vert }^2+{\Vert \nabla u_{ntt}\Vert }^2\right]+\frac{3\alpha }{4}{\Vert \nabla u_{nt}\Vert }^2\le e\Vert \mathsf{\Delta }{u}_n\Vert +e{\Vert \nabla f\Vert }^2.$$

(21)

Multiplying both sides of (20) by $\delta$ and by adding with (21), we have:

$$\begin{array}{l}\frac{1}{2}\frac{d}{dt}\left[{\Vert \nabla u_{nt}\Vert }^2+{\Vert \mathsf{\Delta }{u}_n\Vert }^2+\left(\beta +\alpha \delta \right){\Vert \nabla u_n\Vert }^2\right]+\frac{3\alpha }{4}{\Vert \nabla u_{nt}\Vert }^2+\frac{3\delta }{4}{\Vert \mathsf{\Delta }{u}_n\Vert }^2\\ +\beta \delta {\Vert \nabla u_n\Vert }^2+\delta {\displaystyle {\int }_{\mathsf{\Omega }}\nabla u_n\nabla u_{ntt}\left(t\right)dx}\\ \le \delta d_4+e\Vert \mathsf{\Delta }{u}_n\Vert +e{\Vert \nabla f\Vert }^2\le d_5+\frac{1}{4}\delta {\Vert \mathsf{\Delta }{u}_n\Vert }^2+d_6,\end{array}$$

(22)

where, $d_5\mathrm{and}{d}_6$ are constants.

Integrating both sides of (22) with respect to $t$ from $0\mathrm{to}\omega ,$ we get:

$${\displaystyle {\int }_0^{\omega }\left[\left(3\alpha -4\delta \right){\Vert \nabla u_{nt}\Vert }^2+2\delta {\Vert \mathsf{\Delta }{u}_n\Vert }^2+4\delta \beta {\Vert \nabla u_n\Vert }^2\right]dt}\le {\displaystyle {\int }_0^{\omega }4\left(d_5+d_6\right)dt}=d_7\omega .$$

Now, for $0<\delta <\frac{\alpha }{4}$ and $t^{\ast \ast }\in \left(0,\omega \right),$ we obtain:

$${\Vert \nabla u_{nt}\left(t^{\ast \ast }\right)\Vert }^2+{\Vert \mathsf{\Delta }{u}_n\left(t^{\ast \ast }\right)\Vert }^2+{\Vert \nabla u_n\left(t^{\ast \ast }\right)\Vert }^2\le d_8.$$

Again, integrating both sides of (22) from $t^{\ast \ast }$ to $t\in \left(t^{\ast \ast },t^{\ast \ast }+\omega \right)$, we have:

$${\Vert \nabla u_{nt}\left(t\right)\Vert }^2+{\Vert \mathsf{\Delta }{u}_n\left(t\right)\Vert }^2+{\Vert \nabla u_n\left(t\right)\Vert }^2\le d_9.$$

Thus, for a constant $d_2=d_2\left(\alpha ,\beta ,\gamma ,\omega ,f,L\right)$, we have:

$$\underset{0\le t\le \omega }{\sup }\left({\Vert \nabla u_{nt}\left(t\right)\Vert }^2+{\Vert \mathsf{\Delta }{u}_n\left(t\right)\Vert }^2+{\Vert \nabla u_n\left(t\right)\Vert }^2\right)\le d_2$$

which finishes the proof. □

The next lemma will establish the priori estimates of a higher order for $u_n.$

Lemma 3. 

If $f\in C^1\left(\omega ,H_{per}^1\right)$ and $d_{10}=d_{10}\left(\alpha ,\beta ,\gamma ,\omega ,L,f\right)$ is positive constant, then

$$\underset{0\le t\le \omega }{\sup }\Vert u_{ntt}\left(t\right)\Vert \le d_{10}$$

Proof. 

Differentiating Equation (11), we get:

$$\left(u_{nttt}+\alpha u_{ntt}+\beta u_{nt}-\mathsf{\Delta }{u}_{nt},{\omega }_j\right)=\left(-4\gamma {\left|u_n\right|}^3{u}_{nt}+f^{\prime },{\omega }_j\right),j=1,2,3,\cdots ,n.$$

(23)

After multiplying both sides of (23) with $2{a^{″}}_{jn}\left(t\right)$ and taking the sum over $j$ from 1 to $n$, we get:

$$\begin{array}{l}\frac{d}{dt}\left({\Vert u_{ntt}\Vert }^2+\beta {\Vert u_{nt}\Vert }^2+{\Vert \nabla u_{nt}\Vert }^2+{\Vert \nabla u_{nt}\Vert }^2\right)+2\alpha {\Vert u_{ntt}\Vert }^2\\ \le 8\gamma \Vert u_{ntt}\Vert {\Vert u_{nt}\Vert }_4{\Vert u_n\Vert }_8^3+2\Vert u_{ntt}\Vert \Vert f^{\prime }\Vert \le \alpha {\Vert u_{ntt}\Vert }^2+\frac{1}{\alpha }\left[8{\Vert f^{\prime }\Vert }^2+8\gamma {\left({\Vert u_{nt}\Vert }_4{\Vert u_n\Vert }_8^3\right)}^2\right].\end{array}$$

Using inequality (19) and Lemma 2, we have:

$$\frac{d}{dt}\left({\Vert u_{ntt}\Vert }^2+\beta {\Vert u_{nt}\Vert }^2+{\Vert \nabla u_{nt}\Vert }^2+{\Vert \nabla u_{nt}\Vert }^2\right)+2\alpha {\Vert u_{ntt}\Vert }^2\le d_{11}.$$

(24)

Again multiplying both sides of (23) with $2{a^{\prime }}_{jn}\left(t\right)$ and taking the sum over $j$ from 1 to $n$ and by lemma 2, we get:

$$\begin{gathered} \frac{d}{dt}\alpha {\Vert u_{nt}\Vert }^2+2{\Vert \nabla u_{nt}\Vert }^2+2\beta {\Vert u_{nt}\Vert }^2+2{\displaystyle {\int }_{\mathsf{\Omega }}{u}_{nttt}{u}_{nt}dx} \\ \le 8\gamma {\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_n\right|}^3{u}_{nt}^{2}dx}+2{\displaystyle {\int }_{\mathsf{\Omega }}{f}^{\prime }{u}_{nt}dx}\le 8\gamma {\Vert u_n\Vert }_4^3{\Vert u_{nt}\Vert }_4^2+2\Vert u_{nt}\Vert \Vert f^{\prime }\Vert =d_{12} \end{gathered}$$

(25)

Multiplying both sides of (25) by $\delta$ and on adding with (24), we have:

$$\begin{array}{l}\frac{d}{dt}\left({\Vert u_{nttt}\Vert }^2+{\Vert \nabla u_{nt}\Vert }^2+\left(\beta +\alpha \delta \right){\Vert u_{nt}\Vert }^2\right)+2\alpha {\Vert u_{ntt}\Vert }^2+2\delta {\Vert \nabla u_{nt}\Vert }^2\\ +2\beta \delta {\Vert u_{nt}\Vert }^2+2\delta {\displaystyle {\int }_{\mathsf{\Omega }}{u}_{nttt}{u}_{nt}dx}\le d_{11}+\delta d_{12}\doteq d_{13}.\end{array}$$

(26)

Integrating both sides of (26) from $0\mathrm{to}\omega ,$ we get:

$${\displaystyle {\int }_0^{\omega }\left[2\left(\alpha -\delta \right){\Vert u_{ntt}\Vert }^2+2\delta {\Vert \nabla u_{nt}\Vert }^2+2\delta \beta {\Vert u_{nt}\Vert }^2\right]dt}\le d_{14}\omega .$$

Now, for $0<\delta <\frac{\alpha }{2}$ and $t^{\ast \ast \ast }\in \left(0,\omega \right),$ we have:

$${\Vert u_{ntt}\left(t^{\ast \ast \ast }\right)\Vert }^2+{\Vert \nabla u_{nt}\left(t^{\ast \ast \ast }\right)\Vert }^2+{\Vert \nabla u_{nt}\left(t^{\ast \ast \ast }\right)\Vert }^2\le \frac{d_{15}}{n}.$$

Again, integrating Inequality (26) from $t^{\ast \ast \ast }$ to $t\in \left[t^{\ast \ast \ast },t^{\ast \ast \ast }+\omega \right]$, we have:

$${\Vert u_{ntt}\left(t\right)\Vert }^2+{\Vert \nabla u_{nt}\left(t\right)\Vert }^2+{\Vert u_{nt}\left(t\right)\Vert }^2\le d_{16}.$$

(27)

Thus, for a constant $d_{10}=d_{10}\left(\alpha ,\beta ,\gamma ,\omega ,f,L\right)$, we obtain:

$$\underset{0\le t\le \omega }{\sup }{\Vert u_{ntt}\left(t\right)\Vert }^2\le d_{10}.$$

This completes the proof. □

4. Solvability of Periodic Solution to the 2DDCNLKGE

In this section, we establish the existence and uniqueness of time periodic solutions to 2DDCNLKGE given in Equation (6).

The next theorem leads the existence criteria of a time periodic solution for 2DDCNLKGE given in Equation (6).

Theorem 2. 

For any $f\in C^1\left(\omega ,H_{per}^1\right),$ the time periodic solution $\left(u\left(x,t\right),u^{\prime }\left(x,t\right)\right)$ of 2DDCNLKGE given in Equation (6), is expressed in the following way:

$$u\left(x,t\right)\in C^2\left(\omega ,H_{per}^2\right),u^{\prime }\left(x,t\right)\in L_{per}^2$$

Proof. 

For all positive integers, we have proven that Equation (6) has an approximate solution $u_n\left(t\right),$ i.e., the system given by Equation (11) holds and we have estimates of the norm of $u_n\left(t\right)$. It is possible to consider a subsequence $\left\{\left(u_{n_k}\left(t\right),{u^{\prime }}_{n_k}\left(t\right)\right)\right\}$ converging weakly to $\left(u\left(t\right),u^{\prime }\left(t\right)\right)\in H_{per}^1\times L_{per}^2$, for fixed $t$ and uniform boundedness norms ${\Vert u_n\left(t\right)\Vert }_{H_{per}^1}$ and ${\Vert {u^{\prime }}_n\left(t\right)\Vert }_{per}^2$. We have to prove that $\left(u\left(t\right),u^{\prime }\left(t\right)\right)$ is a solution of the 2DDCNLKGE given in Equation (6). In fact, by weak convergence of $\left\{u_{n_k}\left(t\right)\right\}\mathrm{and}\left\{{u^{\prime }}_{n_k}\left(t\right)\right\}$ to $u\left(t\right)\mathrm{and}{u}^{\prime }\left(t\right)$ in spaces $H_{per}^1\mathrm{and}{L}_{per}^2$, respectively, we mean that the following are true:

$${\mathrm{u}}_{{\mathrm{n}}_{\mathrm{k}}}\left(\mathrm{t}\right)\rightharpoonup \mathrm{u}\left(\mathrm{t}\right)\mathrm{as}k\to \infty ,\mathrm{weakly}\mathrm{in}{H}_{per}^1,$$

(28)

$${\mathrm{u}\text{’}}_{{\mathrm{n}}_{\mathrm{k}}}\left(\mathrm{t}\right)\rightharpoonup \mathrm{u}\text{’}\left(\mathrm{t}\right)\mathrm{as}k\to \infty ,\mathrm{weakly}\mathrm{in}{L}_{per}^2.$$

(29)

Since $H_{per}^1$ $\hookrightarrow$ $L_{per}^2$ is compact, hence for a subsequence of $\left\{u_{n_k}\left(t\right)\right\}$ which is again denoted by $\left\{u_{n_k}\left(t\right)\right\}$ for convenience and for any $t\in \left[0,\omega \right),$ we have:

$$u_{n_k}\left(t\right)\mathrm{converges}\mathrm{strongly}\mathrm{to}u\left(t\right)\mathrm{as}k\to \infty \mathrm{in}{L}_{per}^2\left(\mathsf{\Omega }\right),$$

(30)

and

$$u_{n_k}\left(t\right)\mathrm{converges}\mathrm{to}u\left(t\right)\mathrm{as}k\to \infty ,\mathrm{a}.\mathrm{e}.\left(\mathrm{almost}\mathrm{every}\mathrm{where}\right)\mathsf{\Omega }.$$

(31)

By inequality (13), lemmas 2 and 3 and for any $t\in \left[0,\omega \right),$ we have $\left\{u_{n_k}\left(t\right)\right\}$ is uniformly bounded in $H_{per}^1$. Consequently, for a subsequence of $\left\{u_{n_k}\left(t\right)\right\}$, which is again denoted by $\left\{u_{n_k}\left(t\right)\right\}$ for convenience and for any $t\in \left[0,\omega \right),$ we get:

$$u_{n_k}\left(t\right)\mathrm{converges}\mathrm{to}u\left(t\right)\mathrm{as}k\to \infty ,\mathrm{a}.\mathrm{e}.\left(\mathrm{almost}\mathrm{every}\mathrm{where}\right)\mathsf{\Omega }.$$

$$\left\{u_{n_k}\left(t\right)\right\}\mathrm{converges}\mathrm{weakly}\mathrm{to}u\left(t\right)\mathrm{as}k\to \infty \mathrm{in}{H}_{per}^2,$$

and

$$\mathsf{\Delta }{\mathrm{u}}_{{\mathrm{n}}_{\mathrm{k}}}\left(\mathrm{t}\right)\rightharpoonup \mathsf{\Delta }\mathrm{u}\left(\mathrm{t}\right)\mathrm{as}k\to \infty ,\mathrm{converges}\mathrm{weakly}\mathrm{in}{L}_{per}^2.$$

(32)

Similar to (30), for a subsequence of $\left\{u_{n_k}\left(t\right)\right\}$, which is still denoted by $\left\{u_{n_k}\left(t\right)\right\}$, for convenience and for any $t\in \left[0,\omega \right),$ we obtain:

$$\nabla u_{n_k}\left(t\right)\mathrm{converges}\mathrm{strongly}\mathrm{to}w\in H_{per}^1\mathrm{as}k\to \infty \mathrm{in}{L}_{per}^2\left(\mathsf{\Omega }\right),$$

(33)

and

$$\nabla u_{n_k}\left(t\right)\to w\mathrm{as}k\to \infty ,\mathrm{a}.\mathrm{e}.\mathsf{\Omega }.$$

(34)

Combining Equations (28), (29) and (34), we obtain:

$$\nabla u_{n_k}\left(t\right)\to \nabla u\left(t\right)\mathrm{as}k\to \infty ,\mathrm{a}.\mathrm{e}.\mathsf{\Omega }.$$

(35)

Since $\left\{{u^{\prime }}_{n_k}\left(t\right)\right\}$ is uniform bound in $H_{per}^1$, similar to the above procedure, we obtain:

$$\nabla {u^{\prime }}_{n_k}\left(t\right)\mathrm{converges}\mathrm{strongly}\mathrm{to}{u}^{\prime }\left(t\right)\mathrm{as}k\to \infty \mathrm{in}{L}_{per}^2\left(\mathsf{\Omega }\right),$$

(36)

and

$${u^{\prime }}_{n_k}\left(t\right)\mathrm{converges}\mathrm{to}{u}^{\prime }\left(t\right)\mathrm{as}k\to \infty ,\mathrm{a}.\mathrm{e}.\mathsf{\Omega }.$$

(37)

According as inequality (13) and lemma 2, we have:

$$\Vert N\left(u_{n_k}\left(t\right)\right)\Vert \le d_{17},$$

(38)

where $d_{17}=d_{17}\left(\alpha ,\beta ,\gamma ,\omega ,f,L\right)$ is a constant.

Now, combining (31) and (35), we get:

$$N\left(u_{n_k}\left(t\right)\right)\mathrm{converges}\mathrm{to}N\left(u\left(t\right)\right)\mathrm{as}k\to \infty ,\mathrm{a}.\mathrm{e}.\mathsf{\Omega }.$$

(39)

Applying lemma 1.3 of Lions [16], we get:

$$N\left(u_{n_k}\left(t\right)\right)\mathrm{converges}\mathrm{weakly}\mathrm{to}N\left(u\left(t\right)\right)\mathrm{as}k\to \infty \mathrm{in}{L}_{per}^4.$$

(40)

Since $\left(u_n\left(t\right),{u^{\prime }}_n\left(t\right)\right)\in C^2\left(\omega ,H_{per}^2\right)\times C^1\left(\omega ,L_{per}^2\right),$ then from lemma 3, Equations (28) and(29), we get $\left(u\left(t\right),u^{\prime }\left(t\right)\right)\in C^2\left(\omega ,H_{per}^2\right)\times C^1\left(\omega ,L_{per}^2\right),$ and hence for some $t\in \left[0,\omega \right),$ we obtain:

$$u_{ntt}\left(t\right)\mathrm{converges}\mathrm{weakly}\mathrm{to}{u}_{tt}\left(t\right)\mathrm{as}k\to \infty \mathrm{in}{L}_{per}^2\left(\mathsf{\Omega }\right).$$

(41)

Multiplying each equation in (11) by any $a_{jn}\left(t\right)\in C^2\left(\omega ,\Re \right),$ and summing up over $j$ from $1$ to $n$, we get:

$$\left(u_{ntt}+\alpha u_{nt}+\beta u_n+T{u}_n,\tau \right)=\left(N\left(u_n\right)+f,\tau \right),\mathrm{for}\mathrm{all}\tau \in C^2\left(\omega ,H_n\right).$$

For any fixed $k_0\le k$, by $H_{n_{k_0}}\subset H_{n_{k_0}+1}\subset \cdots ,$ we have:

$$\left(u_{n_{k}tt}+\alpha u_{n_{k}t}+\beta u_{n_k}+T{u}_{n_k},\tau \right)=\left(N\left(u_{n_k}\right)+f,\tau \right),\mathrm{for}\mathrm{all}\tau \in C^2\left(\omega ,H_{n_{k_0}}\right).$$

(42)

Combining (32), (40), (41), and (42), we deduce:

$$\left(u_{tt}+\alpha u_t+\beta u+Tu,\tau \right)=\left(N\left(u\right)+f,\tau \right),\mathrm{for}\mathrm{all}\tau \in C^2\left(\omega ,H_{n_{k_0}}\right).$$

(43)

Here $k_0$ is an arbitrarily chosen number such that (43) holds for all $\tau \in C^2\left(\omega ,{\displaystyle \underset{n=1}{\overset{\infty }{\cup }}{H}_n}\right)$. Since ${\displaystyle \underset{n=1}{\overset{\infty }{\cup }}{H}_n}$ is dense in $L_{per}^2$, then $\left(u\left(t\right),u^{\prime }\left(t\right)\right)$ is a solution of (43), where $\tau \in C^2\left(\omega ,L_{per}^2\right),$ i.e., $\left(u\left(x,t\right),u^{\prime }\left(x,t\right)\right)$ is a solution of 2DDCNLKGE given by (6).

This completes the proof. □

The next theorem will form a new uniqueness criteria of time periodic solution to 2DDCNLKGE given in Equation (6).

Theorem 3. 

If the hypothesis of Theorem 2 holds, then the 2DDCNLKGE given in Equation (6) has a unique time periodic solution.

Proof. 

Let $\left(u\left(x,t\right),u^{\prime }\left(x,t\right)\right)\mathrm{and}\left(u^{\ast }\left(x,t\right),u^{\ast }{}^{\prime }\left(x,t\right)\right)$ be distinct time periodic solutions of (6).

If we set $\left(v\left(x,t\right),v^{\prime }\left(x,t\right)\right)=\left(u\left(x,t\right),u^{\prime }\left(x,t\right)\right)-\left(u^{\ast }\left(x,t\right),u^{\ast }{}^{\prime }\left(x,t\right)\right)$ then from (11), we get:

$$v_{tt}+\alpha v_t+\beta v+Tv=Nu-N{u}^{\ast }.$$

(44)

From (44), we obtain:

$$\frac{1}{2}\frac{d}{dt}\left({\Vert v_t\Vert }^2+\alpha {\Vert v\Vert }^2\right)+\beta {\Vert v\Vert }^2+{\Vert \nabla v\Vert }^2=\left(Nu-N{u}^{\ast },v\right).$$

(45)

Using lemmas 1, 2, and 3 in (45), we get:

$$\frac{d}{dt}\left({\Vert v_t\Vert }^2+{\Vert v\Vert }^2\right)+\delta \left({\Vert v\Vert }^2+{\Vert \nabla v\Vert }^2\right)\le 0,$$

(46)

where, $\delta \ge 0$.

Now, using Gronwall’s Inequality [26] in (46), we obtain:

$$\left({\Vert v_t\left(t\right)\Vert }^2+{\Vert v\left(t\right)\Vert }^2\right)\le \left({\Vert v_t\left(0\right)\Vert }^2+{\Vert v\left(0\right)\Vert }^2\right)e^{-\delta t},\forall t\ge 0$$

(47)

.

From $\omega$-periodicity of $v$, we get:

$$\left({\Vert v_t\left(t\right)\Vert }^2+{\Vert v\left(t\right)\Vert }^2\right)=\left({\Vert v_t\left(t+\kappa \omega \right)\Vert }^2+{\Vert v\left(t+\kappa \omega \right)\Vert }^2\right).$$

(48)

where, $\kappa$ is any positive integer.

Using (47) and (48), we get:

$$\left({\Vert v_t\left(t\right)\Vert }^2+{\Vert v\left(t\right)\Vert }^2\right)\le \left({\Vert v_t\left(0\right)\Vert }^2+{\Vert v\left(0\right)\Vert }^2\right)e^{-\delta \left(t+\kappa \omega \right)}.$$

This gives us:

$$v_t\left(0\right)=v\left(0\right)=0$$

.

Hence:

$$\left(u\left(x,t\right),u^{\prime }\left(x,t\right)\right)=\left(u^{\ast }\left(x,t\right),u^{\ast }{}^{\prime }\left(x,t\right)\right),$$

i.e., the time periodic solution of 2DDCNLKGE given by (6) is unique. This completes the proof. □

5. Conclusions

This article has proven a new solvability criterion for a time periodic solution for 2DDCNLKGE given in Equation (6) with the help of the GLK method and the LS fixed point theorem. The LS fixed point theorem helps us to determine the existence of approximate solution points within uniform priori estimates, whereas uniform priori estimates of the approximate solution of 2DDCNLKGE is constructed by using the GLK method. Theorem 2 provided an easy procedure to check the presence of a time periodic solution of 2DDCNLKGE given in Equation (6) and Theorem 3 ensured the uniqueness of that time periodic solution. The results of this article provided an easy and straightforward technique to identify a unique time periodic solution of 2DDCNLKGE given by Equation (6). Furthermore, these results extend the corresponding results of Gao and Guo [10], Kosecki [14], Geoggiev [15], Ozawa et al. [16], and Gao et al. [18].