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Initial boundary value problem for generalized Zakharov equations with nonlinear function terms

Abstract

In this paper, we consider the initial boundary value problem for generalized Zakharov equations. Firstly, we prove the existence and uniqueness of the global smooth solution to the problem by a priori integral estimates, the Galerkin method, and compactness theory. Furthermore, we discuss the approximation limit of the global solution when the coefficient of the strong nonlinear term tends to zero.

Introduction

The Zakharov system, derived by Zakharov in 1972 [1], describes the interaction between Langmuir (dispersive) and ion acoustic (approximately nondispersive) waves in an unmagnetized plasma. The usual Zakharov system defined in the space \(\mathbb{R}^{d+1}\) is given by

$$\begin{gathered} i\varepsilon_{t}+\Delta\varepsilon=n\varepsilon, \\ n_{tt}-\Delta n=\Delta \vert \varepsilon \vert ^{2},\end{gathered} $$

where the wave fields \(\varepsilon(x,t)\) and \(n(x,t)\) are complex and real, respectively. It has become commonly accepted that the Zakharov system is a general model to govern interaction of dispersive and nondispersive waves.

The generalized Zakharov system has found a number of applications in various physical problems, such as interaction of intramolecular vibrations giving rise to Davydov solitons with acoustic disturbances [2], interaction of high- and low-frequency gravity disturbances in an atmosphere [3], and so on. In the past decades, the Zakharov system has been studied by many authors [413].

Gajewski and Zacharias [14] studied the following generalized Zakharov system and established the global existence for initial value problem:

$$\begin{aligned}& i\varepsilon_{t}+\varepsilon_{xx}+(\alpha-n) \varepsilon=0, \\& v_{t}+\biggl(\frac{1}{2}v^{2}-\beta v_{x}+n+ \vert \varepsilon \vert ^{2} \biggr)_{x}=0, \\& n_{t}+v_{x}=0,\quad t>0, \end{aligned}$$

where the parameters \(\beta>0\) and α are real numbers.

You and Ning [15] considered the existence and uniqueness of the global smooth solution for the initial value problem of the following generalized Zakharov equations in dimension two:

$$\begin{gathered} i\varepsilon_{t}+\triangle\varepsilon-n\varepsilon=0, \\ v_{t}+\sum^{2}_{i=1} \frac{\partial}{\partial x_{i}}\operatorname {grad}\varphi(v)-\triangle v+\nabla\bigl(n+ \vert \varepsilon \vert ^{2}\bigr)=0, \\ n_{t}+\nabla\cdot v=0, \quad t>0,\end{gathered} $$

with initial data

$$\varepsilon|_{t=0}=\varepsilon_{0}(x),\qquad v|_{t=0}=v_{0}(x), \qquad n|_{t=0}=n_{0}(x), $$

where \(\varepsilon(x,t)=(\varepsilon_{1}(x,t), \varepsilon _{2}(x,t),\ldots,\varepsilon_{N}(x,t))\) is an N-dimensional complex-valued unknown functional vector, \(v(x,t)=(v_{1}(x,t), v_{2}(x,t))\) is a two-dimensional real-valued unknown functional vector, \(n(x,t)\) is a real-valued unknown function, \(x\in\mathbb {R}^{2}\), and \(\varphi(s)\) is a real function.

In the present paper, we study the following initial boundary value problem for generalized Zakharov equations:

$$\begin{aligned}& i\varepsilon_{t}+\varepsilon_{xx}+(\alpha-n)\varepsilon +\delta \vert \varepsilon \vert ^{p}\varepsilon=0, \end{aligned}$$
(1.1)
$$\begin{aligned}& v_{t}+\bigl[\varphi(v)-\beta v_{x}+n+ \vert \varepsilon \vert ^{2}\bigr]_{x}=0, \end{aligned}$$
(1.2)
$$\begin{aligned}& n_{t}+v_{x}=0, \quad t>0, x\in[0,L], \end{aligned}$$
(1.3)

with initial data

$$ \varepsilon|_{t=0}=\varepsilon_{0}(x),\qquad v|_{t=0}=v_{0}(x),\qquad n|_{t=0}=n_{0}(x),\quad x\in[0,L], $$
(1.4)

and boundary conditions

$$ \varepsilon(0,t)=\varepsilon (L,t)=v(0,t)=v(L,t)=n(0,t)=n(L,t)=0, $$
(1.5)

where the parameters \(p>0\), \(\beta>0\), α, and δ are real numbers, and \(\varphi(s)\) is a real function. Taking \(\delta=0\), \(\beta =0\), and \(\varphi^{\prime}(s)=\) Constant in this system, it becomes the classical Zakharov equation system. From a physical point of view, this system has stronger nonlinear excitation and interaction. It also can be considered as a further generalization of the generalized Zakharov system discussed in [14]. From the perspective of both mathematical research and physical applications, the problem is of great significance.

For convenience of the following contexts, we set some notations. For \(1\leq p\leq\infty\), we denote by \(L^{p}[0,L]\) the space of all pth-power integrable functions in \([0,L]\) equipped with norm \(\|\cdot \|_{L^{p}}\), and by \(H^{s,p}\) the Sobolev space with norm \(\|\cdot\| _{H^{s,p}}\). For \(p=2\), we write \(H^{s}\) instead of \(H^{s,2}\). \(C^{k}(R)\) is the space of k times continuously differentiable functions on R. If \(k=0\), then we write \(C(R)\) instead of \(C^{0}(R)\). Let \((f,g)=\int_{0}^{L}f(x)\overline{g(x)}\,dx\), where \(\overline {g(x)}\) denotes the complex conjugate function of \(g(x)\). The real and imaginary parts of a complex number A are denoted, respectively, by ReA and ImA. Throughout the paper, C is a generic constant, which may have different meanings in different places.

This paper is organized as follows. In Sect. 2, we establish a priori estimations for problem (1.1)–(1.5). In Sect. 3, we study the existence and uniqueness of global generalized solutions for problem (1.1)–(1.5). In Sect. 4, we discuss the regularity of global generalized solution for problem (1.1)–(1.5). In Sect. 5, we give the approximation limit of the global solution when the coefficient of the strong nonlinear term tends to zero.

A priori estimations for problem (1.1)–(1.5)

Lemma 2.1

Let\(\varepsilon_{0}\in L^{2}[0,L]\). Then for the solution of problem (1.1)(1.5), we have

$$\Vert \varepsilon \Vert ^{2}_{L^{2}}= \Vert \varepsilon_{0} \Vert ^{2}_{L^{2}}. $$

Proof

Taking the inner product of (1.1) and ε, we have

$$ \bigl(i\varepsilon_{t}+\varepsilon_{xx}+(\alpha-n) \varepsilon +\delta \vert \varepsilon \vert ^{p}\varepsilon, \varepsilon\bigr)=0. $$
(2.1)

Since \(\operatorname{Im}(i\varepsilon_{t},\varepsilon)=\frac{1}{2}\frac {d}{dt}\|\varepsilon\|^{2}_{L^{2}}\), \(\operatorname{Im}(\varepsilon _{xx}+(\alpha-n)\varepsilon+\delta|\varepsilon|^{p}\varepsilon ,\varepsilon)=0\), and hence from (2.1) we get

$$\frac{d}{dt} \Vert \varepsilon \Vert ^{2}_{L^{2}}=0, $$

that is,

$$\Vert \varepsilon \Vert ^{2}_{L^{2}}= \Vert \varepsilon_{0} \Vert ^{2}_{L^{2}}. $$

 □

Lemma 2.2

Suppose that (1) \(\varepsilon_{0}\in H^{1}[0,L]\), \(v_{0}\in L^{2}[0,L]\), \(n_{0}\in L^{2}[0,L]\), and (2) \(\varphi(v)\in C(R)\). Then for the solution of problem (1.1)(1.5), we have

$$\begin{gathered} \Vert \varepsilon_{x} \Vert ^{2}_{L^{2}}+ \int_{0}^{L}n \vert \varepsilon \vert ^{2}\,dx -\frac{2\delta}{p+2} \int_{0}^{L} \vert \varepsilon \vert ^{p+2}\,dx+\frac{1}{2} \Vert v \Vert ^{2}_{L^{2}} +\frac{1}{2} \Vert n \Vert ^{2}_{L^{2}} \\ \quad{}+\beta \int^{t}_{0} \bigl\Vert v_{x}(x, \tau) \bigr\Vert ^{2}_{L^{2}}\,d\tau=M_{1}.\end{gathered} $$

Proof

Taking the inner product of (1.1) and \(-\varepsilon_{t}\), we get that

$$ \bigl(i\varepsilon_{t}+\varepsilon_{xx}+(\alpha-n) \varepsilon +\delta \vert \varepsilon \vert ^{p}\varepsilon,- \varepsilon_{t}\bigr)=0. $$
(2.2)

Since

$$\operatorname{Re}(i\varepsilon_{t},-\varepsilon_{t})=0, \operatorname {Re}(\varepsilon_{xx},-\varepsilon_{t}) = \frac{1}{2}\frac{d}{dt} \Vert \varepsilon_{x} \Vert ^{2}_{L^{2}}, $$

we have

$$\begin{aligned}& \begin{aligned}[b] \operatorname{Re}\bigl((\alpha-n)\varepsilon,- \varepsilon_{t}\bigr)&=-\frac{\alpha }{2} \int_{0}^{L} (\varepsilon\overline{ \varepsilon}_{t}+\overline{\varepsilon}\varepsilon _{t}) \,dx+\frac{1}{2}\frac{d}{dt} \int_{0}^{L}n \vert \varepsilon \vert ^{2}\,dx \\ &\quad-\frac{1}{2} \int_{0}^{L}n_{t} \vert \varepsilon \vert ^{2}\,dx \\ &=-\frac{\alpha}{2}\frac{d}{dt} \Vert \varepsilon \Vert ^{2}_{L^{2}}+ \frac {1}{2}\frac{d}{dt} \int_{0}^{L}n \vert \varepsilon \vert ^{2}\,dx -\frac{1}{2} \int_{0}^{L}n_{t} \vert \varepsilon \vert ^{2}\,dx \\ &=\frac{1}{2}\frac{d}{dt} \int_{0}^{L}n \vert \varepsilon \vert ^{2}\,dx -\frac{1}{2} \int_{0}^{L}n_{t} \vert \varepsilon \vert ^{2}\,dx, \end{aligned} \\& \operatorname{Re}\bigl(\delta \vert \varepsilon \vert ^{p} \varepsilon,-\varepsilon _{t}\bigr)=-\frac{\delta}{p+2} \frac{d}{dt} \int_{0}^{L} \vert \varepsilon \vert ^{p+2}\,dx, \end{aligned}$$

and hence from (2.2) we get

$$ \frac{d}{dt}\biggl( \Vert \varepsilon_{x} \Vert ^{2}_{L^{2}}+ \int _{0}^{L}n \vert \varepsilon \vert ^{2}\,dx -\frac{2\delta}{p+2} \int_{0}^{L} \vert \varepsilon \vert ^{p+2}\,dx\biggr)- \int _{0}^{L}n_{t} \vert \varepsilon \vert ^{2}\,dx=0. $$
(2.3)

Taking the inner product of (1.2) and v, we have

$$ \bigl(v_{t}+\bigl[\varphi(v)-\beta v_{x}+n+ \vert \varepsilon \vert ^{2}\bigr]_{x},v\bigr)=0. $$
(2.4)

Since

$$\begin{gathered} (v_{t},v)=\frac{1}{2}\frac{d}{dt} \Vert v \Vert ^{2}_{L^{2}},\qquad (-\beta v_{xx},v)=\beta \Vert v_{x} \Vert ^{2}_{L^{2}}, \\ \bigl(\bigl[\varphi(v)\bigr]_{x},v\bigr)= \int_{0}^{L}\bigl[\varphi(v) \bigr]_{x}v\,dx=-\frac{1}{2} \int _{0}^{L}\varphi(v)v_{x}\,dx=- \frac{1}{2}\varPhi\bigl(v(x,t)\bigr)\bigg|^{L}_{0}=0,\end{gathered} $$

where

$$\begin{gathered} \varPhi(x)= \int^{x}_{0}\varphi(s)\,ds, \\ (n_{x},v)=- \int_{0}^{L}nv_{x}\,dx= \int_{0}^{L}nn_{t}\,dx= \frac {1}{2}\frac{d}{dt} \Vert n \Vert ^{2}_{L^{2}}, \\ \bigl( \vert \varepsilon \vert ^{2}_{x},v\bigr)=- \int_{0}^{L} \vert \varepsilon \vert ^{2}v_{x}\,dx= \int _{0}^{L} \vert \varepsilon \vert ^{2}n_{t}\,dx,\end{gathered} $$

and hence from (2.4) we get

$$ \frac{d}{dt}\biggl(\frac{1}{2} \Vert v \Vert ^{2}_{L^{2}}+\frac{1}{2} \Vert n \Vert ^{2}_{L^{2}}\biggr)+ \int_{0}^{L} \vert \varepsilon \vert ^{2}n_{t}\,dx +\beta \Vert v_{x} \Vert ^{2}_{L^{2}}=0. $$
(2.5)

By (2.3) and (2.5) we get

$$\begin{gathered} \frac{d}{dt}\biggl( \Vert \varepsilon_{x} \Vert ^{2}_{L^{2}}+ \int_{0}^{L}n \vert \varepsilon \vert ^{2}\,dx -\frac{2\delta}{p+2} \int_{0}^{L} \vert \varepsilon \vert ^{p+2}\,dx+\frac{1}{2} \Vert v \Vert ^{2}_{L^{2}}\biggr) \\ \quad{}+\frac{d}{dt}\biggl(\frac{1}{2} \Vert n \Vert ^{2}_{L^{2}}\biggr)+\beta \Vert v_{x} \Vert ^{2}_{L^{2}}=0.\end{gathered} $$

Thus

$$ \begin{aligned}[b] & \Vert \varepsilon_{x} \Vert ^{2}_{L^{2}}+ \int_{0}^{L}n \vert \varepsilon \vert ^{2}\,dx -\frac{2\delta}{p+2} \int_{0}^{L} \vert \varepsilon \vert ^{p+2}\,dx \\ &\qquad+\frac{1}{2} \Vert v \Vert ^{2}_{L^{2}}+ \frac{1}{2} \Vert n \Vert ^{2}_{L^{2}}+\beta \int^{t}_{0} \bigl\Vert v_{x}(x, \tau) \bigr\Vert ^{2}_{L^{2}}\,d\tau \\ &\quad= \Vert \varepsilon_{0x} \Vert ^{2}_{L^{2}} + \int_{0}^{L}n_{0} \vert \varepsilon_{0} \vert ^{2}\,dx-\frac{2\delta}{p+2} \int _{0}^{L} \vert \varepsilon_{0} \vert ^{p+2}\,dx \\ &\qquad+\frac{1}{2} \Vert v_{0} \Vert ^{2}_{L^{2}}+\frac{1}{2} \Vert n_{0} \Vert ^{2}_{L^{2}} \\ &\quad=M_{1}. \end{aligned} $$

 □

Lemma 2.3

(Sobolev estimates)

  1. (1)

    Assuming that\(u\in L^{q}(R^{n})\), \(D^{m}u\in L^{r}(R^{n})\), \(1\leq q, r\leq\infty\), \(0\leq j< m\), we have the estimates

    $$\bigl\Vert D^{j}u \bigr\Vert _{L^{p}}\leq C \bigl\Vert D^{m}u \bigr\Vert ^{\theta}_{L^{r}} \Vert u \Vert ^{1-\theta}_{L^{q}}, $$

    where

    $$\frac{1}{p}=\frac{j}{n}+\theta\biggl(\frac{1}{r}- \frac{m}{n}\biggr)+(1-\theta)\frac {1}{q},\quad\frac{j}{m} \leq\theta< 1, $$

    andCis a positive constant depending only onn, m, j, q, r, andθ.

  2. (2)

    For\(\gamma>0\)and\(s\in Z^{+}\), we can get a constantC (it only depends onγands) such that

    $$\begin{gathered} \biggl\Vert \frac{\partial^{k}u}{\partial x^{k}} \biggr\Vert _{L^{\infty}}\leq C \Vert u \Vert _{L^{2}}+\gamma \biggl\Vert \frac{\partial^{s}u}{\partial x^{s}} \biggr\Vert _{L^{2}},\quad k< s, \\ \biggl\Vert \frac{\partial^{k}u}{\partial x^{k}} \biggr\Vert _{L^{2}}\leq C \Vert u \Vert _{L^{2}}+\gamma \biggl\Vert \frac{\partial^{s}u}{\partial x^{s}} \biggr\Vert _{L^{2}},\quad k\leq s.\end{gathered} $$

Lemma 2.4

Suppose that the conditions of Lemma 2.2are satisfied and\(0< p<4\). Then for the solution of problem (1.1)(1.5), we have

$$\sup_{t\in[0,T]}\bigl( \Vert \varepsilon \Vert _{H^{1}}+ \Vert v \Vert _{L^{2}}+ \Vert n \Vert _{L^{2}}\bigr) +\beta \int^{T}_{0} \bigl\Vert v_{x}(x,t) \bigr\Vert ^{2}_{L^{2}}\,dt\leq C. $$

Proof

From Lemmas 2.12.3 and Young’s inequality we get

$$\begin{aligned} & \Vert \varepsilon_{x} \Vert ^{2}_{L^{2}}+\frac{1}{2} \Vert v \Vert ^{2}_{L^{2}} +\frac{1}{2} \Vert n \Vert ^{2}_{L^{2}}+\beta \int^{t}_{0} \bigl\Vert v_{x}(x, \tau) \bigr\Vert ^{2}_{L^{2}}\,d\tau \\ &\quad\leq \vert M_{1} \vert + \biggl\vert \int^{L}_{0}n \vert \varepsilon \vert ^{2}\,dx \biggr\vert +\frac{2 \vert \delta \vert }{p+2} \int^{L}_{0} \vert \varepsilon \vert ^{p+2}\,dx \\ &\quad\leq \vert M_{1} \vert + \Vert n \Vert _{L^{2}} \Vert \varepsilon \Vert ^{2}_{L^{4}}+ \frac{2 \vert \delta \vert }{p+2} \Vert \varepsilon \Vert ^{p+2}_{L^{p+2}} \\ &\quad\leq \vert M_{1} \vert + \Vert n \Vert _{L^{2}} \bigl(C \Vert \varepsilon \Vert ^{\frac{3}{2}}_{L^{2}} \Vert \varepsilon_{x} \Vert ^{\frac{1}{2}}_{L^{2}}\bigr) + \frac{2 \vert \delta \vert }{p+2}\bigl(C \Vert \varepsilon_{x} \Vert _{L^{2}}^{\frac{p}{2}} \Vert \varepsilon \Vert ^{\frac{p+4}{2(p+2)}}_{L^{2}}\bigr) \\ &\quad\leq \vert M_{1} \vert + \Vert n \Vert _{L^{2}} \bigl(C \Vert \varepsilon \Vert ^{\frac{3}{2}}_{L^{2}} \Vert \varepsilon_{x} \Vert ^{\frac{1}{2}}_{L^{2}}\bigr) +C \Vert \varepsilon_{x} \Vert _{L^{2}}^{\frac{p}{2}} \\ &\quad\leq \vert M_{1} \vert +\frac{1}{4} \Vert n \Vert ^{2}_{L^{2}}+C \Vert \varepsilon_{x} \Vert _{L^{2}} +\frac{1}{2} \Vert \varepsilon_{x} \Vert ^{2}_{L^{2}}+C \\ &\quad\leq \vert M_{1} \vert +\frac{1}{4} \Vert n \Vert ^{2}_{L^{2}}+\frac{3}{4} \Vert \varepsilon _{x} \Vert ^{2}_{L^{2}}+C \\ &\quad\leq\frac{1}{4} \Vert n \Vert ^{2}_{L^{2}}+ \frac{3}{4} \Vert \varepsilon_{x} \Vert ^{2}_{L^{2}}+C, \end{aligned}$$

and hence

$$ \Vert \varepsilon_{x} \Vert ^{2}_{L^{2}}+ \Vert v \Vert ^{2}_{L^{2}} + \Vert n \Vert ^{2}_{L^{2}}+\beta \int^{t}_{0} \bigl\Vert v_{x}(x, \tau) \bigr\Vert ^{2}_{L^{2}}\,d\tau \leq C. $$
(2.6)

By (2.6) it follows that

$$\sup_{t\in[0,T]}\bigl( \Vert \varepsilon \Vert _{H^{1}}+ \Vert v \Vert _{L^{2}}+ \Vert n \Vert _{L^{2}}\bigr) +\beta \int^{T}_{0} \bigl\Vert v_{x}(x,t) \bigr\Vert ^{2}_{L^{2}}\,dt\leq C. $$

 □

Corollary 2.1

Suppose that the conditions of Lemma 2.4are satisfied. Then we have

$$\sup_{t\in[0,T]} \Vert \varepsilon \Vert _{L^{\infty}} \leq C. $$

Proof

By Lemmas 2.3 and 2.4, the result of Corollary 2.1 is obvious. □

Lemma 2.5

Suppose that the conditions of Lemma 2.4are satisfied, and assume that (1) \(\varepsilon_{0}\in H^{2}[0,L]\), \(v_{0}\in H^{1}[0,L]\), \(n_{0}\in H^{1}[0,L]\), and (2) \(\varphi(v)\in C^{1}(R)\), \(|\varphi'(v)|\leq C(|v|^{q}+1)\), \(0\leq q\leq2\). Then for the solution of problem (1.1)(1.5), we have

$$\sup_{t\in[0,T]}\bigl( \Vert \varepsilon \Vert _{H^{2}} + \Vert v \Vert _{H^{1}}+ \Vert n \Vert _{H^{1}}+ \Vert \varepsilon_{t} \Vert _{L^{2}}+ \Vert n_{t} \Vert _{L^{2}}\bigr) + \beta \int^{T}_{0} \bigl\Vert v_{xx}(x,t) \bigr\Vert ^{2}_{L^{2}}\,dt\leq C. $$

Proof

Differentiating (1.1) with respect to t, we get

$$ i\varepsilon_{tt}+\varepsilon_{xxt}+\alpha\varepsilon _{t}-n_{t}\varepsilon-n\varepsilon_{t}+ \bigl(\delta \vert \varepsilon \vert ^{p}\varepsilon \bigr)_{t}=0. $$
(2.7)

Taking the inner product of (2.7) and \(\varepsilon_{t}\), it follows that

$$ \bigl(i\varepsilon_{tt}+\varepsilon_{xxt}+\alpha \varepsilon _{t}-n_{t}\varepsilon-n\varepsilon_{t}+ \bigl(\delta \vert \varepsilon \vert ^{p}\varepsilon \bigr)_{t},\varepsilon_{t}\bigr)=0. $$
(2.8)

Since

$$\begin{gathered} \operatorname{Im}(i\varepsilon_{tt},\varepsilon_{t})= \frac{1}{2}\frac {d}{dt} \Vert \varepsilon_{t} \Vert ^{2}_{L^{2}},\qquad \operatorname{Im}(\varepsilon _{xxt}+\alpha\varepsilon_{t}-n\varepsilon_{t}, \varepsilon_{t})=0, \\ \begin{aligned}[b] \operatorname{Im}\bigl(\bigl(\delta \vert \varepsilon \vert ^{p}\varepsilon\bigr)_{t}, \varepsilon_{t}\bigr) &=\operatorname{Im} \int^{L}_{0}\biggl[\biggl(1+ \frac{p}{2}\biggr)\delta \vert \varepsilon \vert ^{p}{ \varepsilon}_{t}+\frac{p\delta}{2} \vert \varepsilon \vert ^{p-2}\varepsilon ^{2}\overline{\varepsilon}_{t} \biggr]\overline{\varepsilon}_{t}\,dx \\ &=\frac{p\delta}{2}\operatorname{Im} \int^{L}_{0} \vert \varepsilon \vert ^{p-2}\varepsilon^{2}\overline{\varepsilon}_{t}^{2} \,dx, \end{aligned} \end{gathered}$$

and hence from (2.8), (1.3), and Corollary 2.1 we get

$$ \begin{aligned}[b] \frac{1}{2}\frac{d}{dt} \Vert \varepsilon_{t} \Vert ^{2}_{L^{2}}&= \operatorname {Im}(n_{t}\varepsilon,\varepsilon_{t}) - \frac{p\delta}{2}\operatorname{Im} \int^{L}_{0} \vert \varepsilon \vert ^{p-2}\varepsilon^{2}\overline{\varepsilon}_{t}^{2} \,dx \\ &\leq \Vert \varepsilon \Vert _{L^{\infty}} \Vert n_{t} \Vert _{L^{2}} \Vert \varepsilon_{t} \Vert _{L^{2}} +\frac{p \vert \delta \vert }{2} \Vert \varepsilon \Vert _{L^{\infty}}^{p} \Vert \varepsilon_{t} \Vert _{L^{2}}^{2} \\ &\leq C\bigl( \Vert n_{t} \Vert _{L^{2}}^{2}+ \Vert \varepsilon_{t} \Vert _{L^{2}}^{2} \bigr) \\ &\leq C\bigl( \Vert v_{x} \Vert _{L^{2}}^{2}+ \Vert \varepsilon_{t} \Vert _{L^{2}}^{2} \bigr). \end{aligned} $$
(2.9)

Taking the inner product of (1.2) and \(-v_{xx}\), it follows that

$$ \bigl(v_{t}+\bigl[\varphi(v)-\beta v_{x}+n+ \vert \varepsilon \vert ^{2}\bigr]_{x},-v_{xx} \bigr)=0. $$
(2.10)

Since

$$\begin{gathered} (v_{t},-v_{xx})=\frac{1}{2} \frac{d}{dt} \Vert v_{x} \Vert ^{2}_{L^{2}},\qquad (-\beta v_{xx},-v_{xx})=\beta \Vert v_{xx} \Vert ^{2}_{L^{2}}, \\ \begin{aligned}[b] \bigl\vert \bigl(\bigl[\varphi(v) \bigr]_{x},-v_{xx}\bigr) \bigr\vert &= \biggl\vert \int_{0}^{L}\varphi'(v)v_{x}v_{xx} \,dx \biggr\vert \\ &\leq C \int_{0}^{L}\bigl( \vert v \vert ^{q}+1\bigr) \vert v_{x} \vert \vert v_{xx} \vert \,dx \\ &\leq\bigl( \Vert v \Vert ^{q}_{L^{4q}} \Vert v_{x} \Vert _{L^{4}}+ \Vert v_{x} \Vert _{L^{2}}\bigr) \Vert v_{xx} \Vert _{L^{2}} \\ &\leq C\bigl[\bigl( \Vert v \Vert ^{\frac{3q}{4}+\frac{1}{8}}_{L^{2}} \Vert v_{xx} \Vert ^{\frac {q}{4}-\frac{1}{8}}_{L^{2}}\bigr) \bigl( \Vert v \Vert ^{\frac{3}{8}}_{L^{2}} \Vert v_{xx} \Vert ^{\frac{5}{8}}_{L^{2}}\bigr)\bigr] \Vert v_{xx} \Vert _{L^{2}} \\ &\quad+C \Vert v_{x} \Vert _{L^{2}} \Vert v_{xx} \Vert _{L^{2}} \\ &\leq C \Vert v_{xx} \Vert ^{\frac{q}{4}+\frac{3}{2}}_{L^{2}}+C \Vert v_{x} \Vert _{L^{2}} \Vert v_{xx} \Vert _{L^{2}} \\ &\leq\frac{\beta}{4} \Vert v_{xx} \Vert _{L^{2}}^{2}+C\bigl( \Vert v_{x} \Vert ^{2}_{L^{2}}+1\bigr), \end{aligned} \\ (n_{x},-v_{xx})=- \int_{0}^{L}n_{x}v_{xx} \,dx= \int_{0}^{L}n_{x}n_{xt} \, dx=\frac{1}{2}\frac{d}{dt} \Vert n_{x} \Vert ^{2}_{L^{2}}, \\ \begin{aligned}[b] \bigl\vert \bigl( \vert \varepsilon \vert ^{2}_{x},-v_{xx}\bigr) \bigr\vert &\leq2 \int_{0}^{L} \vert \varepsilon \vert \vert \varepsilon_{x} \vert \vert v_{xx} \vert \,dx \\ &\leq2 \Vert \varepsilon \Vert _{L^{\infty}} \Vert \varepsilon_{x} \Vert _{L^{2}} \Vert v_{xx} \Vert _{L^{2}} \\ &\leq\frac{\beta}{4} \Vert v_{xx} \Vert _{L^{2}}^{2}+C, \end{aligned}\end{gathered} $$

and hence from (2.10) we get

$$ \frac{d}{dt}\bigl( \Vert v_{x} \Vert ^{2}_{L^{2}}+ \Vert n_{x} \Vert ^{2}_{L^{2}}\bigr) +\beta \Vert v_{xx} \Vert ^{2}_{L^{2}}\leq C\bigl( \Vert v_{x} \Vert ^{2}_{L^{2}}+1\bigr). $$
(2.11)

By (2.9) and (2.11) we obtain

$$\frac{d}{dt}\bigl( \Vert \varepsilon_{t} \Vert ^{2}_{L^{2}} + \Vert v_{x} \Vert ^{2}_{L^{2}}+ \Vert n_{x} \Vert ^{2}_{L^{2}}\bigr) +\beta \Vert v_{xx} \Vert ^{2}_{L^{2}} \leq C\bigl( \Vert v_{x} \Vert _{L^{2}}^{2}+ \Vert \varepsilon_{t} \Vert _{L^{2}}^{2}+ \Vert n_{x} \Vert _{L^{2}}^{2}+1\bigr), $$

and thus by Gronwall’s inequality we obtain

$$ \sup_{t\in[0,T]}\bigl( \Vert \varepsilon_{t} \Vert ^{2}_{L^{2}} + \Vert v_{x} \Vert ^{2}_{L^{2}}+ \Vert n_{x} \Vert ^{2}_{L^{2}}\bigr) +\beta \int^{T}_{0} \bigl\Vert v_{xx}(x,t) \bigr\Vert ^{2}_{L^{2}}\,dt\leq C. $$
(2.12)

By (1.1), Lemmas 2.1, 2.3, and 2.4, Corollary 2.1, Young’s inequality, and (2.12) we obtain

$$ \begin{aligned}[b] \Vert \varepsilon_{xx} \Vert _{L^{2}} &\leq \vert \alpha \vert \Vert \varepsilon \Vert _{L^{2}}+ \Vert n\varepsilon \Vert _{L^{2}}+ \Vert \varepsilon_{t} \Vert _{L^{2}} + \vert \delta \vert \Vert \varepsilon \Vert _{L^{2p+2}}^{2p+2} \\ &\leq \vert \alpha \vert \Vert \varepsilon \Vert _{L^{2}}+ \Vert \varepsilon \Vert _{L^{\infty}} \Vert n \Vert _{L^{2}}+ \Vert \varepsilon_{t} \Vert _{L^{2}} \\ &\quad+ \vert \delta \vert \bigl(C \Vert \varepsilon_{x} \Vert _{L^{2}}^{\frac{p}{2}} \Vert \varepsilon \Vert ^{\frac{p+4}{2(p+2)}}_{L^{2}}\bigr) \\ &\leq \vert \alpha \vert \Vert \varepsilon \Vert _{L^{2}}+ \Vert \varepsilon \Vert _{L^{\infty}} \Vert n \Vert _{L^{2}}+ \Vert \varepsilon_{t} \Vert _{L^{2}} +C \Vert \varepsilon_{x} \Vert _{L^{2}}^{\frac{p}{2}} \\ &\leq C\bigl( \Vert \varepsilon_{t} \Vert _{L^{2}}+1 \bigr) \\ &\leq C. \end{aligned} $$
(2.13)

By (1.3), (2.12), and (2.13) we obtain

$$\sup_{t\in[0,T]}\bigl( \Vert \varepsilon_{xx} \Vert ^{2}_{L^{2}} + \Vert v_{x} \Vert ^{2}_{L^{2}}+ \Vert n_{x} \Vert ^{2}_{L^{2}}+ \Vert \varepsilon_{t} \Vert ^{2}_{L^{2}}+ \Vert n_{t} \Vert ^{2}_{L^{2}}\bigr) +\beta \int^{T}_{0} \bigl\Vert v_{xx}(x,t) \bigr\Vert ^{2}_{L^{2}}\,dt\leq C. $$

The Lemma 2.5 is proved. □

Corollary 2.2

Suppose that the conditions of Lemma 2.5are satisfied. Then we have

$$\sup_{t\in[0,T]}\bigl( \Vert \varepsilon_{x} \Vert _{L^{\infty}}+ \Vert v \Vert _{L^{\infty}}+ \Vert n \Vert _{L^{\infty}}\bigr)\leq C. $$

Proof

By Lemmas 2.3 and 2.5 the result of Corollary 2.2 is obvious. □

Lemma 2.6

Suppose that the conditions of Lemma 2.5are satisfied, and assume that (1) \(\varepsilon_{0}\in H^{3}[0,L]\), \(v_{0}\in H^{2}[0,L]\), \(n_{0}\in H^{2}[0,L]\), and (2) \(\varphi(v)\in C^{2}(R)\). Then for the solution of problem (1.1)(1.5), we have

$$\begin{gathered} \sup_{t\in[0,T]}\bigl( \Vert \varepsilon \Vert _{H^{3}} + \Vert v \Vert _{H^{2}}+ \Vert n \Vert _{H^{2}}+ \Vert \varepsilon_{t} \Vert _{H^{1}}+ \Vert v_{t} \Vert _{L^{2}}+ \Vert n_{t} \Vert _{H^{1}}\bigr) \\ \quad{}+\beta \int^{T}_{0} \bigl\Vert v_{xxx}(x,t) \bigr\Vert ^{2}_{L^{2}}\,dt\leq C.\end{gathered} $$

Proof

Taking the inner product of (2.7) and \(-\varepsilon_{txx}\), it follows that

$$ \bigl(i\varepsilon_{tt}+\varepsilon_{xxt}+\alpha \varepsilon _{t}-n_{t}\varepsilon-n\varepsilon_{t} +\bigl(\delta \vert \varepsilon \vert ^{p}\varepsilon \bigr)_{t},-\varepsilon_{txx}\bigr)=0. $$
(2.14)

Since

$$\begin{aligned}& \operatorname{Im}(i\varepsilon_{tt},-\varepsilon_{txx})= \frac{1}{2}\frac {d}{dt} \Vert \varepsilon_{tx} \Vert ^{2}_{L^{2}}, \qquad\operatorname{Im}(\varepsilon_{xxt}+ \alpha\varepsilon_{t},-\varepsilon_{txx})=0, \\& \begin{aligned}[b] \bigl\vert \operatorname{Im}(n_{t} \varepsilon,-\varepsilon_{txx}) \bigr\vert &= \biggl\vert \operatorname{Im} \int^{L}_{0}(n_{t} \varepsilon)_{x}\overline {\varepsilon}_{tx}\,dx \biggr\vert \\ &= \biggl\vert \operatorname{Im} \int^{L}_{0}(n_{tx} \varepsilon+n_{t}\varepsilon _{x})\overline{ \varepsilon}_{tx}\,dx \biggr\vert \\ &\leq C\bigl( \Vert \varepsilon \Vert _{L^{\infty}} \Vert n_{tx} \Vert _{L^{2}} + \Vert \varepsilon_{x} \Vert _{L^{\infty}} \Vert n_{t} \Vert _{L^{2}} \bigr) \Vert \varepsilon_{tx} \Vert _{L^{2}} \\ &\leq C\bigl( \Vert \varepsilon_{tx} \Vert ^{2}_{L^{2}}+ \Vert n_{tx} \Vert ^{2}_{L^{2}}\bigr) \\ &\leq C\bigl( \Vert \varepsilon_{tx} \Vert ^{2}_{L^{2}}+ \Vert v_{xx} \Vert ^{2}_{L^{2}}\bigr), \end{aligned} \\& \begin{aligned}[b] \bigl\vert \operatorname{Im}(-n \varepsilon_{t},-\varepsilon_{txx}) \bigr\vert &= \biggl\vert \operatorname{Im} \int^{L}_{0}(n\varepsilon_{t})_{x} \overline {\varepsilon}_{tx}\,dx \biggr\vert \\ &= \biggl\vert \operatorname{Im} \int^{L}_{0}(n_{x} \varepsilon_{t}+n\varepsilon _{tx})\overline{ \varepsilon}_{tx}\,dx \biggr\vert \\ &\leq C\bigl( \Vert \varepsilon_{t} \Vert _{L^{\infty}} \Vert n_{x} \Vert _{L^{2}} \Vert \varepsilon _{tx} \Vert _{L^{2}} + \Vert n \Vert _{L^{\infty}} \Vert \varepsilon_{tx} \Vert ^{2}_{L^{2}} \bigr) \\ &\leq C\bigl( \Vert \varepsilon_{tx} \Vert ^{2}_{L^{2}}+1 \bigr), \end{aligned} \\& \begin{aligned}[b] \operatorname{Im}\bigl(\bigl(\delta \vert \varepsilon \vert ^{p}\varepsilon\bigr)_{t},- \varepsilon_{txx}\bigr) &=-\operatorname{Im} \int^{L}_{0}\biggl[\biggl(1+ \frac{p}{2}\biggr)\delta \vert \varepsilon \vert ^{p}{ \varepsilon}_{t}+\frac{p\delta}{2} \vert \varepsilon \vert ^{p-2}\varepsilon ^{2}\overline{\varepsilon}_{t} \biggr]\overline{\varepsilon}_{txx}\,dx \\ &=\operatorname{Im} \int^{L}_{0}\biggl[\biggl(1+ \frac{p}{2}\biggr)\delta \vert \varepsilon \vert ^{p}{ \varepsilon}_{t}+\frac{p\delta}{2} \vert \varepsilon \vert ^{p-2}\varepsilon ^{2}\overline{\varepsilon}_{t} \biggr]_{x}\overline{\varepsilon}_{tx}\,dx \\ &\leq C\bigl( \Vert \varepsilon_{tx} \Vert ^{2}_{L^{2}}+1 \bigr), \end{aligned} \end{aligned}$$

from (2.14) we get

$$ \begin{aligned}[b] \frac{d}{dt} \Vert \varepsilon_{tx} \Vert ^{2}_{L^{2}} &\leq C \bigl( \Vert \varepsilon_{tx} \Vert ^{2}_{L^{2}}+ \Vert v_{xx} \Vert ^{2}_{L^{2}}\bigr). \end{aligned} $$
(2.15)

Taking the inner product of (1.2) and \(v_{xxxx}\), it follows that

$$ \bigl(v_{t}+\bigl[\varphi(v)-\beta v_{x}+n+ \vert \varepsilon \vert ^{2}\bigr]_{x},v_{x^{4}} \bigr)=0. $$
(2.16)

Since

$$\begin{aligned}& (v_{t},v_{x^{4}})=\frac{1}{2} \frac{d}{dt} \Vert v_{xx} \Vert ^{2}_{L^{2}},\qquad (-\beta v_{xx},v_{x^{4}})=\beta \Vert v_{xxx} \Vert ^{2}_{L^{2}}, \\& \begin{aligned}[b] \bigl\vert \bigl(\bigl[\varphi(v) \bigr]_{x},v_{x^{4}}\bigr) \bigr\vert &= \biggl\vert \int_{0}^{L}\bigl[\varphi(v) \bigr]_{x}v_{x^{4}}\,dx \biggr\vert \\ &= \biggl\vert \int_{0}^{L}\bigl[\varphi(v) \bigr]_{xx}v_{xxx}\,dx \biggr\vert \\ &= \biggl\vert \int_{0}^{L}\bigl[\varphi''(v)v^{2}_{x}+ \varphi'(v)v_{xx}\bigr]v_{xxx}\,dx \biggr\vert \\ &\leq C\bigl( \Vert v_{x} \Vert ^{2}_{L^{4}}+ \Vert v_{xx} \Vert _{L^{2}}\bigr) \Vert v_{xxx} \Vert _{L^{2}} \\ &\leq C\bigl( \Vert v_{x} \Vert ^{\frac{3}{2}}_{L^{2}} \Vert v_{xx} \Vert ^{\frac{1}{2}}_{L^{2}} + \Vert v_{xx} \Vert _{L^{2}}\bigr) \Vert v_{xxx} \Vert _{L^{2}} \\ &\leq\frac{\beta}{4} \Vert v_{xxx} \Vert _{L^{2}}^{2}+C\bigl( \Vert v_{xx} \Vert ^{2}_{L^{2}}+1\bigr), \end{aligned} \\& \begin{aligned}[b] (n_{x},v_{x^{4}})&=- \int_{0}^{L}n_{xx}v_{xxx} \,dx= \int _{0}^{L}n_{xx}n_{xxt} \,dx=\frac{1}{2}\frac{d}{dt} \Vert n_{xx} \Vert ^{2}_{L^{2}}, \end{aligned} \\& \begin{aligned}[b] \bigl\vert \bigl( \vert \varepsilon \vert ^{2}_{x},v_{x^{4}}\bigr) \bigr\vert &= \biggl\vert \int_{0}^{L} \vert \varepsilon \vert ^{2}_{xx}v_{xxx}\,dx \biggr\vert \\ &\leq2 \int_{0}^{L} \vert \varepsilon \vert \vert \varepsilon_{xx} \vert \vert v_{xxx} \vert \,dx+2 \int _{0}^{L} \vert \varepsilon_{x} \vert ^{2} \vert v_{xxx} \vert \,dx \\ &\leq2 \Vert \varepsilon \Vert _{L^{\infty}} \Vert \varepsilon_{xx} \Vert _{L^{2}} \Vert v_{xxx} \Vert _{L^{2}}+2 \Vert \varepsilon_{x} \Vert ^{2}_{L^{4}} \Vert v_{xxx} \Vert _{L^{2}} \\ &\leq2 \Vert \varepsilon \Vert _{L^{\infty}} \Vert \varepsilon_{xx} \Vert _{L^{2}} \Vert v_{xxx} \Vert _{L^{2}} +2 \Vert \varepsilon \Vert ^{\frac{3}{4}}_{L^{2}} \Vert \varepsilon_{xx} \Vert ^{\frac {5}{4}}_{L^{2}} \Vert v_{xxx} \Vert _{L^{2}} \\ &\leq\frac{\beta}{4} \Vert v_{xxx} \Vert _{L^{2}}^{2}+C, \end{aligned} \end{aligned}$$

from (2.16) we get

$$ \frac{d}{dt}\bigl( \Vert v_{xx} \Vert ^{2}_{L^{2}}+ \Vert n_{xx} \Vert ^{2}_{L^{2}}\bigr) +\beta \Vert v_{xxx} \Vert ^{2}_{L^{2}}\leq C\bigl( \Vert v_{xx} \Vert _{L^{2}}^{2}+ \Vert n_{xx} \Vert _{L^{2}}^{2}+1\bigr). $$
(2.17)

By (2.15) and (2.17) we obtain

$$\begin{gathered} \frac{d}{dt}\bigl( \Vert \varepsilon_{tx} \Vert ^{2}_{L^{2}} + \Vert v_{xx} \Vert ^{2}_{L^{2}}+ \Vert n_{xx} \Vert ^{2}_{L^{2}}\bigr) +\beta \Vert v_{xxx} \Vert ^{2}_{L^{2}} \\ \quad\leq C\bigl( \Vert v_{xx} \Vert _{L^{2}}^{2}+ \Vert \varepsilon_{tx} \Vert _{L^{2}}^{2}+ \Vert n_{xx} \Vert _{L^{2}}^{2}+1\bigr),\end{gathered} $$

and thus by Gronwall’s inequality we obtain

$$ \sup_{t\in[0,T]}\bigl( \Vert \varepsilon_{tx} \Vert ^{2}_{L^{2}} + \Vert v_{xx} \Vert ^{2}_{L^{2}}+ \Vert n_{xx} \Vert ^{2}_{L^{2}}\bigr) +\beta \int^{T}_{0} \bigl\Vert v_{xxx}(x,t) \bigr\Vert ^{2}_{L^{2}}\,dt\leq C. $$
(2.18)

By (1.1), Young’s inequality, and (2.18) we obtain

$$ \begin{aligned}[b] \Vert \varepsilon_{xxx} \Vert _{L^{2}} &\leq \vert \alpha \vert \Vert \varepsilon_{x} \Vert _{L^{2}}+ \Vert (n\varepsilon)_{x} \Vert _{L^{2}}+ \Vert \varepsilon_{tx} \Vert _{L^{2}} + \bigl\Vert \delta\bigl( \vert \varepsilon \vert ^{p} \varepsilon\bigr)_{x} \bigr\Vert _{L^{2}} \\ &\leq \vert \alpha \vert \Vert \varepsilon_{x} \Vert _{L^{2}}+\bigl( \Vert n_{x}\varepsilon \Vert _{L^{2}}+ \Vert n\varepsilon_{x} \Vert _{L^{2}}\bigr)+ \Vert \varepsilon_{tx} \Vert _{L^{2}} \\ &\quad+ \biggl\Vert \biggl(1+\frac{p}{2}\biggr)\delta \vert \varepsilon \vert ^{p}{\varepsilon}_{x}+ \frac {p\delta}{2} \vert \varepsilon \vert ^{p-2} \varepsilon^{2}\overline{\varepsilon }_{x} \biggr\Vert _{L^{2}} \\ &\leq \vert \alpha \vert \Vert \varepsilon_{x} \Vert _{L^{2}}+\bigl( \Vert n_{x}\varepsilon \Vert _{L^{2}}+ \Vert n\varepsilon_{x} \Vert _{L^{2}}\bigr) + \Vert \varepsilon_{tx} \Vert _{L^{2}} \\ &\quad+\biggl(1+\frac{p}{2}\biggr) \vert \delta| \bigl\Vert |\varepsilon \vert ^{p}\varepsilon_{x} \bigr\Vert _{L^{2}} +\frac{p \vert \delta \vert }{2} \bigl\Vert \vert \varepsilon \vert ^{p}\varepsilon_{x} \bigr\Vert _{L^{2}} \\ &\leq \Vert \varepsilon_{tx} \Vert _{L^{2}}+C \\ &\leq C. \end{aligned} $$
(2.19)

By (1.3), (2.18), and (2.19) we obtain

$$\begin{aligned}[b] &\sup_{t\in[0,T]}\bigl( \Vert \varepsilon_{xxx} \Vert ^{2}_{L^{2}} + \Vert v_{xx} \Vert ^{2}_{L^{2}}+ \Vert n_{xx} \Vert ^{2}_{L^{2}}+ \Vert \varepsilon_{tx} \Vert ^{2}_{L^{2}}+ \Vert n_{tx} \Vert ^{2}_{L^{2}}\bigr) \\ &\quad+\beta \int^{T}_{0} \bigl\Vert v_{xxx}(x,t) \bigr\Vert ^{2}_{L^{2}}\,dt\leq C.\end{aligned} $$
(2.20)

By (1.2) we obtain

$$ \begin{aligned}[b] \Vert v_{t} \Vert _{L^{2}} &\leq C \bigl\Vert \bigl[\varphi(v)\bigr]_{x} \bigr\Vert _{L^{2}}+\beta \Vert v_{xx} \Vert _{L^{2}}+ \Vert n_{x} \Vert _{L^{2}}+\bigl\| | \varepsilon|_{xx}\bigr\| _{L^{2}} \\ &\leq C \bigl\Vert \varphi'(v)v_{x} \bigr\Vert _{L^{2}}+\beta \Vert v_{xx} \Vert _{L^{2}}+ \Vert n_{x} \Vert _{L^{2}} \\ &\quad+ \Vert \varepsilon_{xx}\overline{\varepsilon}+2\varepsilon \overline {\varepsilon}_{x} +\varepsilon\overline{ \varepsilon}_{xx} \Vert _{L^{2}} \\ &\leq C \Vert v_{x} \Vert _{L^{2}}+\beta \Vert v_{xx} \Vert _{L^{2}}+ \Vert n_{x} \Vert _{L^{2}} \\ &\quad+2 \Vert \varepsilon \Vert _{L^{\infty}} \Vert \varepsilon_{xx} \Vert _{L^{2}} +2 \Vert \varepsilon \Vert _{L^{\infty}} \Vert \varepsilon_{x} \Vert _{L^{2}} \\ &\leq C. \end{aligned} $$
(2.21)

By (2.20) and (2.21) we obtain

$$\begin{gathered} \sup_{t\in[0,T]}\bigl( \Vert \varepsilon \Vert _{H^{3}} + \Vert v \Vert _{H^{2}}+ \Vert n \Vert _{H^{2}}+ \Vert \varepsilon_{t} \Vert _{H^{1}}+ \Vert v_{t} \Vert _{L^{2}}+ \Vert n_{t} \Vert _{H^{1}}\bigr) \\ \quad{}+\beta \int^{T}_{0} \bigl\Vert v_{xxx}(x,t) \bigr\Vert ^{2}_{L^{2}}\,dt\leq C.\end{gathered} $$

 □

Corollary 2.3

Suppose that the conditions of Lemma 2.6are satisfied. Then we have

$$\sup_{t\in[0,T]}\bigl( \Vert E_{xx} \Vert _{L^{\infty}}+ \Vert v_{x} \Vert _{L^{\infty}}+ \Vert n_{x} \Vert _{L^{\infty}}+ \Vert \varepsilon_{t} \Vert _{L^{\infty}}+ \Vert n_{t} \Vert _{L^{\infty}} \bigr)\leq C. $$

Proof

By Lemmas 2.3 and 2.6 the result of Corollary 2.3 is obvious. □

Lemma 2.7

Suppose that (1) \(\varepsilon_{0}\in H^{l+2}[0,L]\), \(v_{0}\in H^{l+1}[0,L]\), \(n_{0}\in H^{l+1}[0,L]\), \(l\in Z^{+}\), (2) \(\varphi(v)\in C^{l+1}(R)\), \(|\varphi'(v)|\leq C(|v|^{q}+1)\), \(0\leq q\leq2 \), and (3) \(0< p<4\). Then for the solution of problem (1.1)(1.5), we have

$$\begin{gathered} \sup_{t\in[0,T]}\bigl( \Vert \varepsilon \Vert _{H^{l+2}} + \Vert v \Vert _{H^{l+1}}+ \Vert n \Vert _{H^{l+1}}+ \Vert \varepsilon_{t} \Vert _{H^{l}}+ \Vert v_{t} \Vert _{H^{l-1}}+ \Vert n_{t} \Vert _{H^{l}}\bigr) \\ \quad{}+\beta \int^{T}_{0} \bigl\Vert v_{x^{l+2}}(x,t) \bigr\Vert ^{2}_{L^{2}}\,dt\leq C.\end{gathered} $$

Proof

We prove this lemma by mathematical induction. By Lemma 2.6 the lemma is true for \(l=1\). Suppose it is is true for \(l=k\) (\(k\geq 1\)), that is,

$$\begin{gathered} \sup_{t\in[0,T]}\bigl( \Vert \varepsilon \Vert _{H^{k+2}} + \Vert v \Vert _{H^{k+1}}+ \Vert n \Vert _{H^{k+1}}+ \Vert \varepsilon_{t} \Vert _{H^{k}}+ \Vert v_{t} \Vert _{H^{k-1}}+ \Vert n_{t} \Vert _{H^{k}}\bigr) \\ \quad{}+\beta \int^{T}_{0} \bigl\Vert v_{x^{k+2}}(x,t) \bigr\Vert ^{2}_{L^{2}}\,dt\leq C.\end{gathered} $$

Next, we will show that the lemma is true for \(l=k+1\).

Taking the inner product of (1.2) and \((-1)^{k+2}v_{x^{2k+4}}\), it follows that

$$ \bigl(v_{t}+\bigl[\varphi(v)-\beta v_{x}+n+ \vert \varepsilon \vert ^{2}\bigr]_{x}, (-1)^{k+2}v_{x^{2k+4}} \bigr)=0. $$
(2.22)

Since

$$\begin{aligned}& \bigl(v_{t},(-1)^{k+2}v_{x^{2k+4}}\bigr)= \frac{1}{2}\frac{d}{dt} \Vert v_{x^{k+2}} \Vert ^{2}_{L^{2}}, \\& \begin{aligned}[b] \bigl\vert \bigl(\bigl[\varphi(v) \bigr]_{x},(-1)^{k+2}v_{x^{2k+4}}\bigr) \bigr\vert &= \bigl\vert \bigl(\bigl[\varphi(v)\bigr]_{x^{k+2}},(-1)^{k+2}v_{x^{k+3}} \bigr) \bigr\vert \\ &= \biggl\vert \int_{0}^{L}\bigl[\varphi(v) \bigr]_{x^{k+2}}v_{x^{k+3}}\,dx \biggr\vert \\ &\leq\frac{\beta}{4} \Vert v_{x^{k+3}} \Vert ^{2}_{L^{2}}+C\bigl( \Vert v_{x^{k+2}} \Vert ^{2}_{L^{2}}+1\bigr), \end{aligned} \\& \bigl(-\beta v_{xx},(-1)^{k+2}v_{x^{2k+4}}\bigr)= \beta \Vert v_{x^{k+3}} \Vert ^{2}_{L^{2}}, \\& \begin{aligned}[b] \bigl(n_{x},(-1)^{k+2}v_{x^{2k+4}} \bigr)&=- \int_{0}^{2L}n_{x^{k+2}}v_{x^{k+3}} \, dx \\ &= \int_{0}^{L}n_{x^{k+2}}n_{tx^{k+2}} \,dx \\ &\leq\frac{1}{2}\frac{d}{dt} \Vert n_{x^{k+2}} \Vert ^{2}_{L^{2}}, \end{aligned} \\& \begin{aligned}[b] \bigl\vert \bigl( \vert \varepsilon \vert ^{2}_{x},(-1)^{k+2}v_{x^{2k+4}}\bigr) \bigr\vert &= \biggl\vert \int_{0}^{L}\bigl( \vert \varepsilon \vert ^{2}\bigr)_{x^{k+2}}v_{x^{k+3}}\,dx \biggr\vert \\ &\leq\frac{\beta}{4} \Vert v_{x^{k+3}} \Vert ^{2}_{L^{2}}+C, \end{aligned} \end{aligned}$$

from (2.22) we get

$$\frac{d}{dt}\bigl( \Vert v_{x^{k+2}} \Vert ^{2}_{L^{2}}+ \Vert n_{x^{k+2}} \Vert ^{2}_{L^{2}}\bigr) +\beta \Vert v_{x^{k+3}} \Vert ^{2}_{L^{2}}\leq C\bigl( \Vert v_{x^{k+2}} \Vert _{L^{2}}^{2}+ \Vert n_{x^{k+2}} \Vert _{L^{2}}^{2}+1\bigr). $$

By Gronwall’s inequality we obtain

$$ \sup_{t\in[0,T]}\bigl( \Vert v_{x^{k+2}} \Vert ^{2}_{L^{2}}+ \Vert n_{x^{k+2}} \Vert ^{2}_{L^{2}}\bigr) +\beta \int^{T}_{0} \bigl\Vert v_{x^{k+3}}(x,t) \bigr\Vert ^{2}_{L^{2}}\,dt\leq C, $$
(2.23)

and by Eqs. (1.2) and (1.3) we get

$$ \sup_{t\in[0,T]}\bigl( \Vert v_{tx^{k}} \Vert ^{2}_{L^{2}}+ \Vert n_{tx^{k+1}} \Vert ^{2}_{L^{2}}\bigr)\leq C. $$
(2.24)

Taking the inner product of (2.7) and \((-1)^{k+1}\varepsilon ^{2(k+1)}_{t}\), it follows that

$$ \bigl(i\varepsilon_{tt}+\varepsilon_{xxt}+\alpha \varepsilon _{t}-n_{t}\varepsilon-n\varepsilon_{t} +\bigl(\delta \vert \varepsilon \vert ^{p}\varepsilon \bigr)_{t},(-1)^{k+1}\varepsilon _{tx^{2k+2}} \bigr)=0. $$
(2.25)

Since

$$\begin{aligned}& \operatorname{Im}\bigl(i\varepsilon_{tt},(-1)^{k+1} \varepsilon _{tx^{2k+2}}\bigr)=\frac{1}{2}\frac{d}{dt} \Vert \varepsilon_{tx^{k+1}} \Vert ^{2}_{L^{2}}, \\& \operatorname{Im}\bigl(\varepsilon_{xxt}+\alpha\varepsilon _{t},(-1)^{k+1}\varepsilon_{tx^{2k+2}}\bigr)=0, \\& \begin{aligned}[b] \bigl\vert \operatorname{Im} \bigl(n_{t}\varepsilon,(-1)^{k+1}\varepsilon_{tx^{2k+2}} \bigr) \bigr\vert &= \biggl\vert \operatorname{Im} \int^{L}_{0}(n_{t} \varepsilon)_{x^{k+1}}\overline {\varepsilon}_{tx^{k+3}}\,dx \biggr\vert \\ &\leq C\bigl( \Vert \varepsilon_{tx^{k+1}} \Vert ^{2}_{L^{2}}+1 \bigr), \end{aligned} \\& \begin{aligned}[b] \bigl\vert \operatorname{Im}\bigl(-n \varepsilon_{t},(-1)^{k+1}\varepsilon_{tx^{2k+2}} \bigr) \bigr\vert &= \biggl\vert \operatorname{Im} \int^{L}_{0}(n\varepsilon_{t})_{x^{k+1}} \overline {\varepsilon}_{tx^{k+1}}\,dx \biggr\vert \\ &\leq C\bigl( \Vert \varepsilon_{x^{k+1}} \Vert ^{2}_{L^{2}}+1 \bigr), \end{aligned} \\& \begin{aligned}[b] \bigl\vert \operatorname{Im}\bigl(\bigl(\delta \vert \varepsilon \vert ^{p}\varepsilon \bigr)_{t},(-1)^{k+1} \varepsilon_{tx^{2k+2}}\bigr) \bigr\vert &=\operatorname{Im} \int^{L}_{0}\bigl[\bigl(\delta \vert \varepsilon \vert ^{p}\varepsilon \bigr)_{t} \bigr]_{x^{k+1}}\overline{\varepsilon}_{tx^{k+1}}\,dx \\ &\leq C\bigl( \Vert \varepsilon_{tx^{k+1}} \Vert ^{2}_{L^{2}}+1 \bigr), \end{aligned} \end{aligned}$$

from (2.25) we get

$$ \begin{aligned}[b] \frac{d}{dt} \Vert \varepsilon_{tx^{k+1}} \Vert ^{2}_{L^{2}} &\leq C \bigl( \Vert \varepsilon_{tx^{k+1}} \Vert ^{2}_{L^{2}}+1 \bigr). \end{aligned} $$

By Gronwall’s inequality we obtain

$$ \sup_{t\in[0,T]} \Vert \varepsilon_{tx^{k+1}} \Vert ^{2}_{L^{2}}\leq C, $$
(2.26)

and by Eq. (1.1) we get

$$ \sup_{t\in[0,T]} \Vert \varepsilon_{x^{k+3}} \Vert ^{2}_{L^{2}}\leq C. $$
(2.27)

By (2.23), (2.24), (2.26), and (2.27) we get

$$\begin{gathered} \sup_{t\in[0,T]}\bigl( \Vert \varepsilon \Vert _{H^{k+3}} + \Vert v \Vert _{H^{k+2}}+ \Vert n \Vert _{H^{k+2}}+ \Vert \varepsilon_{t} \Vert _{H^{k+1}}+ \Vert v_{t} \Vert _{H^{k}}+ \Vert n_{t} \Vert _{H^{k+1}}\bigr) \\ \quad{}+\beta \int^{T}_{0} \bigl\Vert v_{x^{k+3}}(x,t) \bigr\Vert ^{2}_{L^{2}}\,dt\leq C.\end{gathered} $$

 □

The existence and uniqueness of global generalized solutions for problem (1.1)–(1.5)

Definition 1

The set of functions \(\varepsilon(x,t)\in L^{\infty}(0,T; H^{3}[0,L])\cap W^{1,\infty}(0,T; H^{1}[0,L])\), \(v(x,t)\in L^{\infty }(0,T; H^{2}[0,L])\cap L^{2}(0,T; H^{3}[0,L])\cap W^{1,\infty}(0,T; L^{2}[0,L])\), and \(n(x,t)\in L^{\infty}(0,T; H^{2}[0,L])\cap W^{1,\infty}(0,T; H^{1}[0,L])\) is called the generalized solution of problem (1.1)–(1.5) if for any \(\omega\in L^{2}[0,L]\), the functions satisfy

$$\begin{aligned}& (i\varepsilon_{t}, \omega)+(\varepsilon_{xx}, \omega )+(\alpha\varepsilon, \omega)-(n\varepsilon, \omega)+\bigl(\delta \vert \varepsilon \vert ^{p}\varepsilon, \omega\bigr)=0, \end{aligned}$$
(3.1)
$$\begin{aligned}& (v_{t}, \omega)+\bigl(\bigl[\varphi(v)\bigr]_{x}, \omega\bigr)-(\beta v_{xx}, \omega)+(n_{x}, \omega)- \bigl( \vert \varepsilon \vert ^{2}_{x}, \omega \bigr)=0, \end{aligned}$$
(3.2)
$$\begin{aligned}& (n_{t}, \omega)+(v_{x}, \omega)=0, \end{aligned}$$
(3.3)
$$\begin{aligned}& \bigl(\varepsilon(x,0),\omega\bigr)=\bigl(\varepsilon_{0}(x), \omega\bigr),\qquad\bigl(v(x,0),\omega \bigr)=\bigl(v_{0}(x), \omega\bigr), \\& \bigl(n(x,0),\omega\bigr)=\bigl(n_{0}(x), \omega \bigr), \end{aligned}$$
(3.4)
$$\begin{aligned}& \varepsilon(0,t)=\varepsilon (L,t)=v(0,t)=v(L,t)=n(0,t)=n(L,t)=0. \end{aligned}$$
(3.5)

Theorem 3.1

Suppose that the conditions of Lemma 2.6are satisfied. Then there exists a global generalized solution of the initial boundary value problem (1.1)(1.5),

$$\begin{gathered} \varepsilon(x,t)\in L^{\infty}\bigl(0,T; H^{3}[0,L]\bigr),\qquad \varepsilon _{t}(x,t)\in L^{\infty}\bigl(0,T; H^{1}[0,L]\bigr), \\ v(x,t)\in L^{\infty}\bigl(0,T; H^{2}[0,L]\bigr)\cap L^{2}\bigl(0,T; H^{3}[0,L]\bigr),\qquad v_{t}(x,t) \in L^{\infty}\bigl(0,T; L^{2}[0,L]\bigr), \\ n(x,t)\in L^{\infty}\bigl(0,T; H^{2}[0,L]\bigr),\qquad n_{t}(x,t)\in L^{\infty}\bigl(0,T; H^{1}[0,L] \bigr).\end{gathered} $$

Proof

By using the Galerkin method we choose a basis \(\{{\omega_{j}(x)}\} \subseteq H^{2}[0,L]\cap H_{0}^{1}[0,L]\) consisting of the eigenfunctions of the problem

$$\begin{aligned}& -\triangle\omega_{j}(x)=\lambda_{j} \omega_{j}(x),\quad j=1, 2,\ldots, m, \end{aligned}$$
(3.6)
$$\begin{aligned}& \omega_{j}(x)\bigl|_{x=0}=\omega_{j}(x)\bigr|_{x=L}=0. \end{aligned}$$
(3.7)

Then the approximate solution of problem (1.1)–(1.4) can be written as

$$\begin{aligned}[b] &\varepsilon_{m}(x,t)=\sum_{j=1}^{m} \alpha_{jm}(t){\omega_{j}(x)},\qquad v_{m}(x,t)= \sum_{j=1}^{m}\beta_{jm}(t){ \omega_{j}(x)}, \\ &n_{m}(x,t)=\sum_{j=1}^{m} \gamma_{jm}(t){\omega_{j}(x)}.\end{aligned} $$
(3.8)

According to Galerkin’s method, the undetermined coefficients \(\alpha _{jm}(t)\), \(\beta_{jm}(t)\), and \(\gamma_{jm}(t)\) need to satisfy the following initial value problem of ordinary differential equations:

$$\begin{aligned}& \bigl(i\varepsilon_{mt}+\varepsilon_{mxx}+(\alpha -n_{m})\varepsilon_{m}+\delta \vert \varepsilon_{m} \vert ^{p}\varepsilon_{m}, \omega\bigr)=0, \end{aligned}$$
(3.9)
$$\begin{aligned}& \bigl(v_{mt}+\bigl[\varphi(v_{m})\bigr]_{x}- \beta v_{mxx}+n_{mx}- \vert \varepsilon \vert ^{2}_{mx}, \omega\bigr)=0, \end{aligned}$$
(3.10)
$$\begin{aligned}& (n_{mt}+v_{mx}, \omega)=0, \end{aligned}$$
(3.11)
$$\begin{aligned}& \varepsilon_{m}(x,0)=\varepsilon_{m0}(x),\qquad v_{m}(x,0)=v_{m0}(x),\quad n_{m}(x,0)=n_{m0}(x),\quad x\in[0, L], \end{aligned}$$
(3.12)

where

$$\begin{gathered} \varepsilon_{m0}(x)\rightarrow\varepsilon_{0}(x) \quad\text{in }H^{3}[0,L], \qquad v_{m0}(x)\rightarrow v_{0}(x)\quad\text{in }H^{2}[0,L], \\ n_{m0}(x)\rightarrow n_{0}(x) \quad\text{in }H^{2}[0,L], m\rightarrow \infty.\end{gathered} $$

Similarly to the proof of Lemmas 2.1, 2.4, 2.5, and 2.6, for the solution \(\varepsilon_{m}(x,t)\), \(v_{m}(x,t)\), \(n_{m}(x,t)\) of problem (3.9)–(3.12), we can establish the following estimate:

$$\begin{aligned}[b] &\sup_{t\in[0,T]}\bigl( \Vert \varepsilon_{m} \Vert _{H^{3}}+ \Vert v_{m} \Vert _{H^{2}}+ \Vert n_{m} \Vert _{H^{2}} + \Vert \varepsilon_{mt} \Vert _{H^{1}}+ \Vert v_{mt} \Vert _{L^{2}}+ \Vert n_{mt} \Vert _{H^{1}}\bigr) \\ &\quad+\beta \int^{T}_{0} \Vert v_{m} \Vert ^{2}_{H^{3}}\,dt \leq C,\end{aligned} $$
(3.13)

where the constant C is independent of m. By compact argument we can choose a subsequence \(\varepsilon_{\nu}(x,t)\), \(v_{\nu}(x,t)\), \(n_{\nu }(x,t)\) such that, as \(\nu\rightarrow\infty\),

$$\begin{aligned}& \varepsilon_{\nu}(x,t)\rightarrow\varepsilon(x,t) \quad\text{in }L^{\infty}\bigl(0,T;H^{3}[0,L] \bigr)\text{ weakly star}, \\& \varepsilon_{\nu}(x,t)\rightarrow\varepsilon(x,t) \quad\text{in strong topology of }L^{2}(Q_{T}), \\& \varepsilon_{\nu t}(x,t)\rightarrow\varepsilon_{t}(x,t) \quad\text{in }L^{\infty}\bigl(0,T;H^{1}[0,L] \bigr)\text{ weakly star}, \\& v_{\nu}(x,t)\rightarrow v(x,t)\quad\text{in }L^{\infty } \bigl(0,T;H^{2}[0,L]\bigr)\cap L^{2}\bigl(0,T; H^{3}[0,L]\bigr)\text{ weakly star}, \\& v_{\nu}(x,t)\rightarrow v(x,t) \quad\text{in strong topology of }L^{2}(Q_{T}), \\& v_{\nu t}(x,t)\rightarrow v_{t}(x,t)\quad\text{in }L^{\infty } \bigl(0,T;L^{2}[0,L]\bigr)\text{ weakly star}, \\& n_{\nu}(x,t)\rightarrow n(x,t)\quad\text{in }L^{\infty } \bigl(0,T;H^{2}[0,L]\bigr)\text{ weakly star}, \\& n_{\nu}(x,t)\rightarrow n(x,t)\quad\text{in strong topology of }L^{2}(Q_{T}), \\& n_{\nu t}(x,t)\rightarrow n_{t}(x,t)\quad\text{in }L^{\infty } \bigl(0,T;H^{1}[0,L]\bigr)\text{ weakly star}, \\& \vert \varepsilon_{\nu} \vert ^{p} \varepsilon_{\nu}\rightarrow \vert \varepsilon \vert ^{p}\varepsilon\quad\text{in }L^{\infty} \bigl(0,T;L^{2}[0,L]\bigr)\text{ weakly star}, \\& \vert \varepsilon_{\nu} \vert ^{2}_{x} \rightarrow \vert \varepsilon \vert ^{2}_{x} \quad\text{in }L^{\infty}\bigl(0,T;L^{2}[0,L] \bigr)\text{ weakly star}, \\& n_{\nu}\varepsilon_{\nu}\rightarrow n\varepsilon \quad\text{in }L^{\infty}\bigl(0,T;L^{2}[0,L] \bigr)\text{ weakly star}, \\& \varphi(v_{\nu})\rightarrow\varphi(v)\quad\text{in }L^{\infty } \bigl(0,T;L^{2}[0,L]\bigr)\text{ weakly star}, \end{aligned}$$

where \(Q_{T}=[0,L]\times[0,T]\). Hence, taking \(m=\nu\rightarrow\infty\) in (3.9)–(3.13), by using the density of \(\omega _{j}(x)\) in \(L^{2}[0,L]\) we get the existence of a local generalized solution for problem (1.1)–(1.5). From the conditions of the theorem and a priori estimates in Sect. 2 we can get the existence of a global generalized solution for problem (1.1)–(1.5) by the continuation extension principle. □

Theorem 3.2

Suppose that the conditions of Theorem 3.1are satisfied. Then the global generalized solution of the initial boundary value problem (1.1)(1.5) is unique, and

$$\begin{gathered} \varepsilon(x,t)\in L^{\infty}\bigl(0,T; H^{3}[0,L]\bigr),\qquad E_{t}(x,t)\in L^{\infty }\bigl(0,T; H^{1}[0,L] \bigr), \\ v(x,t)\in L^{\infty}\bigl(0,T; H^{2}[0,L]\bigr)\cap L^{2}\bigl(0,T; H^{3}[0,L]\bigr),\qquad v_{t}(x,t) \in L^{\infty}\bigl(0,T; L^{2}[0,L]\bigr), \\ n(x,t)\in L^{\infty}\bigl(0,T; H^{2}[0,L]\bigr),\qquad n_{t}(x,t)\in L^{\infty}\bigl(0,T; H^{1}[0,L] \bigr).\end{gathered} $$

Proof

Suppose that there are two solutions \(\varepsilon_{1}\), \(n_{1}\), \(\varphi _{1}\) and \(\varepsilon_{2}\), \(n_{2}\), \(\varphi_{2}\). Let

$$\varepsilon=\varepsilon_{1}-\varepsilon_{2},\qquad v=v_{1}-v_{2},\qquad n=n_{1}-n_{2}. $$

From (1.1)–(1.5) we get

$$\begin{aligned}& i\varepsilon_{t}+\varepsilon_{xx}+\alpha\varepsilon -n_{1}\varepsilon_{1}+n_{2} \varepsilon_{2} +\delta \vert \varepsilon_{1} \vert ^{p}\varepsilon_{1}-\delta \vert \varepsilon _{2} \vert ^{p}\varepsilon_{2}=0, \end{aligned}$$
(3.14)
$$\begin{aligned}& v_{t}+\bigl[\varphi(v_{1})\bigr]_{x}- \bigl[\varphi(v_{2})\bigr]_{x}-\beta v_{xx}+n_{x}+ \vert \varepsilon_{1} \vert ^{2}_{x}- \vert \varepsilon_{2} \vert ^{2}_{x}=0, \end{aligned}$$
(3.15)
$$\begin{aligned}& n_{t}+v_{x}=0, \end{aligned}$$
(3.16)

with initial data

$$ \varepsilon|_{t=0}=0,\qquad v|_{t=0}=0,\qquad n|_{t=0}=0 $$
(3.17)

and boundary conditions

$$ \varepsilon(0,t)=\varepsilon (L,t)=v(0,t)=v(L,t)=n(0,t)=n(L,t)=0. $$
(3.18)

Taking the inner product of (3.14) and ε, it follows that

$$ \bigl(i\varepsilon_{t}+\varepsilon_{xx}+\alpha \varepsilon -n_{1}\varepsilon_{1}+n_{2} \varepsilon_{2} +\delta \vert \varepsilon_{1} \vert ^{p}\varepsilon_{1}-\delta \vert \varepsilon _{2} \vert ^{p}\varepsilon_{2},\varepsilon \bigr)=0. $$
(3.19)

Since

$$\begin{gathered} \operatorname{Im}(i\varepsilon_{t},\varepsilon)= \frac{1}{2}\frac {d}{dt} \Vert \varepsilon \Vert ^{2}_{L^{2}},\qquad \operatorname{Im}(\varepsilon _{xx}+\alpha\varepsilon,\varepsilon)=0, \\ \begin{aligned}[b] \bigl\vert \operatorname{Im}(n_{1} \varepsilon_{1}-n_{2}\varepsilon_{2}, \varepsilon) \bigr\vert &= \biggl\vert \operatorname{Im} \int^{L}_{0}n\varepsilon_{1} \overline{\varepsilon}\, dx \biggr\vert \\ &\leq\frac{1}{2} \Vert \varepsilon_{1} \Vert _{L^{\infty}}\bigl( \Vert n \Vert ^{2}_{L^{2}}+ \Vert \varepsilon \Vert ^{2}_{L^{2}}\bigr) \\ &\leq C\bigl( \Vert n \Vert ^{2}_{L^{2}}+ \Vert \varepsilon \Vert ^{2}_{L^{2}}\bigr). \end{aligned} \end{gathered}$$

By the Lagrange mean value theorem we get

$$ \begin{aligned}[b] \bigl\vert \vert \varepsilon_{1} \vert ^{p}\varepsilon_{1}- \vert \varepsilon_{2} \vert ^{p}\varepsilon_{2} \bigr\vert &= \bigl\vert \vert \varepsilon_{1} \vert ^{p}\varepsilon_{1}- \vert \varepsilon_{1} \vert ^{p}\varepsilon _{2}+ \vert \varepsilon_{1} \vert ^{p}\varepsilon_{2}- \vert \varepsilon _{2} \vert ^{p}\varepsilon_{2} \bigr\vert \\ &\leq \vert \varepsilon_{1} \vert ^{p} \vert \varepsilon_{1}-\varepsilon_{2} \vert + \vert \varepsilon _{2} \vert \bigl( \vert \varepsilon_{1} \vert ^{p}- \vert \varepsilon_{2} \vert ^{p}\bigr) \\ &\leq \vert \varepsilon_{1} \vert ^{p} \vert \varepsilon \vert +p \vert \varepsilon_{2} \vert \sup _{t\in [0,T]}\bigl( \vert \varepsilon_{1} \vert ^{p-1}, \vert \varepsilon_{2} \vert ^{p-1} \bigr) \vert \varepsilon \vert \\ &\leq(p+1)\sup_{t\in[0,T]}\bigl( \vert \varepsilon_{1} \vert ^{p}, \vert \varepsilon _{2} \vert ^{p} \bigr) \vert \varepsilon \vert . \end{aligned} $$

Therefore

$$ \begin{aligned}[b] \bigl\vert \operatorname{Im}\bigl(\delta\bigl( \vert \varepsilon_{1} \vert ^{p}\varepsilon _{1}- \vert \varepsilon_{2} \vert ^{p} \varepsilon_{2}\bigr),\varepsilon\bigr) \bigr\vert &\leq \vert \delta \vert \int^{L}_{0} \vert |\varepsilon_{1} \vert ^{p}\varepsilon _{1}- \vert \varepsilon_{2} \vert ^{p}\varepsilon_{2} \bigl\vert \vert \varepsilon \vert \,dx \bigr\vert \\ &\leq \vert \delta \vert \int^{L}_{0}(p+1)\sup_{t\in[0,T]} \bigl( \vert \varepsilon _{1} \vert ^{p}, \vert \varepsilon_{1} \vert ^{p}\bigr) \vert \varepsilon \vert ^{2}\,dx \\ &\leq \vert \delta \vert (p+1)\sup_{t\in[0,T]}\bigl( \Vert \varepsilon_{1} \Vert ^{p}_{L^{\infty }}, \Vert \varepsilon_{1} \Vert ^{p}_{L^{\infty}}\bigr) \Vert \varepsilon \Vert ^{2}_{L^{2}} \\ &\leq C \Vert \varepsilon \Vert ^{2}_{L^{2}}. \end{aligned} $$

Hence from (3.19) we get

$$ \frac{d}{dt} \Vert \varepsilon \Vert ^{2}_{L^{2}} \leq C\bigl( \Vert n \Vert ^{2}_{L^{2}}+ \Vert \varepsilon \Vert ^{2}_{L^{2}}\bigr). $$
(3.20)

Taking the inner product of (3.15) and v, it follows that

$$ \bigl(v_{t}+\bigl[\varphi(v_{1})\bigr]_{x}- \bigl[\varphi(v_{2})\bigr]_{x}-\beta v_{xx}+n_{x}+ \vert \varepsilon_{1} \vert ^{2}_{x}- \vert \varepsilon_{2} \vert ^{2}_{x},v \bigr)=0. $$
(3.21)

Since

$$\begin{aligned}& (v_{t},v)=\frac{1}{2}\frac{d}{dt} \Vert v \Vert ^{2}_{L^{2}},\qquad (-\beta v_{xx},v)=\beta \Vert v_{x} \Vert ^{2}_{L^{2}}, \\& \begin{aligned}[b] \bigl\vert \bigl(\bigl[\varphi(v_{1}) \bigr]_{x}-\bigl[\varphi(v_{2})\bigr]_{x},v \bigr) \bigr\vert &= \bigl\vert \bigl(\varphi'(\xi )v,v_{x}\bigr) \bigr\vert \\ &\leq \bigl\Vert \varphi'(\xi) \bigr\Vert _{L^{\infty}} \Vert v \Vert _{L^{2}} \Vert v_{x} \Vert _{L^{2}} \\ &\leq C\bigl( \vert \xi \vert ^{q}+1\bigr) \Vert v \Vert _{L^{2}} \Vert v_{x} \Vert _{L^{2}} \\ &\leq C\bigl( \Vert v_{1} \Vert ^{q}_{L^{\infty}}+ \Vert v_{2} \Vert ^{q}_{L^{\infty}}+1\bigr) \Vert v \Vert _{L^{2}} \Vert v_{x} \Vert _{L^{2}} \\ &\leq C\bigl( \Vert v \Vert ^{2}_{L^{2}}+ \Vert n_{t} \Vert ^{2}_{L^{2}}\bigr), \end{aligned} \\& \bigl\vert (n_{x},v) \bigr\vert \leq\frac{1}{2}\bigl( \Vert n \Vert ^{2}_{L^{2}}+ \Vert n_{t} \Vert ^{2}_{L^{2}}\bigr), \\& \begin{aligned}[b] \bigl( \vert \varepsilon_{1} \vert ^{2}_{x}- \vert \varepsilon_{2} \vert ^{2}_{x},v\bigr) &= \biggl\vert \int^{L}_{0}\bigl( \vert \varepsilon_{1} \vert ^{2}- \vert \varepsilon_{2} \vert ^{2}\bigr)v_{x}\,dx \biggr\vert \\ &= \biggl\vert \int^{L}_{0}(\varepsilon\overline{ \varepsilon}_{1} +\varepsilon_{2}\overline{ \varepsilon})v_{x}\,dx \biggr\vert \\ &\leq\bigl( \Vert \varepsilon_{1} \Vert _{L^{\infty}}+ \Vert \varepsilon_{2} \Vert _{L^{\infty }}\bigr) \Vert \varepsilon \Vert _{L^{2}} \Vert n_{t} \Vert _{L^{2}} \\ &\leq C\bigl( \Vert \varepsilon \Vert ^{2}_{L^{2}}+ \Vert n_{t} \Vert ^{2}_{L^{2}}\bigr), \end{aligned} \end{aligned}$$

from (3.21) we get

$$ \frac{d}{dt} \Vert v \Vert ^{2}_{L^{2}}\leq C \bigl( \Vert \varepsilon \Vert ^{2}_{L^{2}}+ \Vert v \Vert ^{2}_{L^{2}}+ \Vert n \Vert ^{2}_{L^{2}}+ \Vert n_{t} \Vert ^{2}_{L^{2}}\bigr). $$
(3.22)

Since

$$ \frac{d}{dt} \Vert n \Vert ^{2}_{L^{2}}= \frac{d}{dt} \int^{L}_{0}n^{2}\,dx\leq \Vert n \Vert ^{2}_{L^{2}}+ \Vert n_{t} \Vert ^{2}_{L^{2}}, $$
(3.23)

by (3.20), (3.22), and (3.23) we get

$$ \frac{d}{dt}\bigl( \Vert \varepsilon \Vert ^{2}_{L^{2}}+ \Vert v \Vert ^{2}_{L^{2}}+ \Vert n \Vert ^{2}_{L^{2}}\bigr)\leq C\bigl( \Vert \varepsilon \Vert ^{2}_{L^{2}}+ \Vert n \Vert ^{2}_{L^{2}}+ \Vert v \Vert ^{2}_{L^{2}}+ \Vert n_{t} \Vert ^{2}_{L^{2}}\bigr). $$
(3.24)

Taking the inner product of (3.15) and \(-v_{xx}\), it follows that

$$ \bigl(v_{t}+\bigl[\varphi(v_{1})\bigr]_{x}- \bigl[\varphi(v_{2})\bigr]_{x}-\beta v_{xx}+n_{x}+ \vert \varepsilon_{1} \vert ^{2}_{x}- \vert \varepsilon _{2} \vert ^{2}_{x},-v_{xx} \bigr)=0. $$
(3.25)

Since

$$\begin{aligned}& (v_{t},-v_{xx})=\frac{1}{2} \frac{d}{dt} \Vert v_{x} \Vert ^{2}_{L^{2}},\qquad (-\beta v_{xx},-v_{xx})=\beta \Vert v_{xx} \Vert ^{2}_{L^{2}}, \\& \begin{aligned}[b] \bigl\vert \bigl(\bigl[\varphi(v_{1}) \bigr]_{x}-\bigl[\varphi(v_{2})\bigr]_{x},-v_{xx} \bigr) \bigr\vert &= \biggl\vert \int_{0}^{L}\bigl[\varphi(v_{1})- \varphi(v_{2})\bigr]v_{xxx}\,dx \biggr\vert \\ &= \bigl\vert \bigl(\varphi'(\xi)v,v_{xxx}\bigr) \bigr\vert \\ &= \bigl\Vert \varphi'(\xi) \bigr\Vert _{L^{\infty}} \bigl\vert (v_{x},v_{xx}) \bigr\vert \\ &\leq \bigl\vert \varphi'(\xi) \bigr\vert \Vert v_{x} \Vert _{L^{2}} \Vert v_{xx} \Vert _{L^{2}} \\ &\leq C\bigl( \vert \xi \vert ^{q}+1\bigr) \Vert v_{x} \Vert _{L^{2}} \Vert v_{xx} \Vert _{L^{2}} \\ &\leq C\bigl( \Vert v_{1} \Vert ^{q}_{L^{\infty}}+ \Vert v_{2} \Vert ^{q}_{L^{\infty}}+1\bigr) \Vert v_{x} \Vert _{L^{2}} \Vert v_{xx} \Vert _{L^{2}} \\ &\leq\frac{\beta}{2} \Vert v_{xx} \Vert _{L^{2}}^{2}+C \Vert v_{x} \Vert _{L^{2}}^{2}, \end{aligned} \\& (n_{x},-v_{xx})=- \int_{0}^{L}n_{x}v_{xx} \,dx= \int_{0}^{L}n_{x}n_{xt} \, dx=\frac{1}{2}\frac{d}{dt} \Vert n_{x} \Vert ^{2}_{L^{2}}, \\& \begin{aligned}[b] \bigl\vert \bigl( \vert \varepsilon_{1} \vert ^{2}_{x}- \vert \varepsilon_{2} \vert ^{2}_{x},-v_{xx}\bigr) \bigr\vert &= \biggl\vert \int_{0}^{L}\bigl( \vert \varepsilon_{1} \vert ^{2}_{x}- \vert \varepsilon _{2} \vert ^{2}_{x} \bigr)v_{xx}\,dx \biggr\vert \\ &= \biggl\vert \int_{0}^{L}(\varepsilon_{x} \overline{\varepsilon}_{1}+ \varepsilon_{2x}\overline{ \varepsilon}+\varepsilon\overline{\varepsilon}_{1x} + \varepsilon_{2}\overline{\varepsilon}_{x})v_{xx} \,dx \biggr\vert \\ &= \biggl\vert \int_{0}^{L}(\varepsilon_{x} \overline{\varepsilon}_{1}+ \varepsilon_{2}\overline{ \varepsilon}_{x})v_{xx}\,dx \biggr\vert \\ &\quad+ \biggl\vert \int_{0}^{L}(\varepsilon\overline{ \varepsilon}_{1x} +\varepsilon_{2x}\overline{ \varepsilon}_{x})v_{xx}\,dx \biggr\vert \\ &= \biggl\vert \int_{0}^{L}(\varepsilon_{x} \overline{\varepsilon}_{1}+\varepsilon _{2}\overline{ \varepsilon}_{x})_{x}v_{x}\,dx \biggr\vert \\ &\quad+ \biggl\vert \int_{0}^{L}(\varepsilon\overline{ \varepsilon}_{1x} +\varepsilon_{2x}\overline{ \varepsilon}_{x})_{x}v_{x}\,dx \biggr\vert \\ &\leq \int_{0}^{L}\bigl( \vert \varepsilon_{1} \vert + \vert \varepsilon_{2} \vert \bigr) \vert \varepsilon _{x} \vert \vert v_{xx} \vert \,dx \\ &\quad+ \int_{0}^{L}\bigl( \vert \varepsilon_{1x} \vert + \vert \varepsilon_{2x} \vert \bigr) \vert \varepsilon \vert \vert v_{xx} \vert \,dx \\ &\leq\bigl( \Vert \varepsilon_{1} \Vert _{L^{\infty}}+ \Vert \varepsilon_{2} \Vert _{L^{\infty }}\bigr) \Vert \varepsilon_{x} \Vert _{L^{2}} \Vert v_{xx} \Vert _{L^{2}} \\ &\quad+\bigl( \Vert \varepsilon_{1x} \Vert _{L^{\infty}}+ \Vert \varepsilon_{2x} \Vert _{L^{\infty}}\bigr) \Vert \varepsilon \Vert _{L^{2}} \Vert v_{xx} \Vert _{L^{2}} \\ &\leq\frac{\beta}{2} \Vert v_{xx} \Vert _{L^{2}}^{2}+C \Vert \varepsilon \Vert _{L^{2}}^{2}, \end{aligned} \end{aligned}$$

from (1.3) and (3.25) we get

$$ \frac{d}{dt}\bigl( \Vert v_{x} \Vert ^{2}_{L^{2}}+ \Vert n_{x} \Vert ^{2}_{L^{2}}\bigr)=\frac{d}{dt}\bigl( \Vert n_{t} \Vert ^{2}_{L^{2}}+ \Vert n_{x} \Vert ^{2}_{L^{2}}\bigr)\leq C\bigl( \Vert \varepsilon \Vert _{L^{2}}^{2}+ \Vert v_{x} \Vert _{L^{2}}^{2}\bigr). $$
(3.26)

By (3.24) and (3.26) we obtain

$$\begin{gathered} \frac{d}{dt}\bigl( \Vert \varepsilon \Vert ^{2}_{L^{2}}+ \Vert v \Vert ^{2}_{L^{2}}+ \Vert n \Vert ^{2}_{L^{2}} + \Vert n_{t} \Vert ^{2}_{L^{2}}+ \Vert n_{x} \Vert ^{2}_{L^{2}}\bigr) \\ \quad\leq C\bigl( \Vert \varepsilon \Vert ^{2}_{L^{2}}+ \Vert n \Vert ^{2}_{L^{2}}+ \Vert v \Vert ^{2}_{L^{2}}+ \Vert n_{t} \Vert ^{2}_{L^{2}} + \Vert n_{x} \Vert ^{2}_{L^{2}}\bigr).\end{gathered} $$

By using Gronwall’s inequality we obtain

$$\varepsilon\equiv0,\qquad v\equiv0,\qquad n\equiv0, $$

and hence

$$\varepsilon_{1}=\varepsilon_{2},\quad\quad v_{1}=v_{2},\qquad n_{1}=n_{2}. $$

Therefore the proof of Theorem 3.2 is completed. □

The regularity of global generalized solution for problem (1.1)–(1.5)

To get the regularity of the global generalized solution for problem (1.1)–(1.5), we need the following lemma and corollary.

Lemma 4.1

Suppose that the conditions of Lemma 2.6are satisfied, and assume that (1) \(\varepsilon_{0}\in H^{4}[0,L]\), \(v_{0}\in H^{3}[0,L]\), \(n_{0}\in H^{3}[0,L]\), and (2) \(\varphi(v)\in C^{3}(R)\). Then for the solution of problem (1.1)(1.5), we have

$$\begin{gathered} \sup_{t\in[0,T]}\bigl( \Vert \varepsilon \Vert _{H^{4}} + \Vert v \Vert _{H^{3}}+ \Vert n \Vert _{H^{3}}+ \Vert \varepsilon_{t} \Vert _{H^{2}}+ \Vert v_{t} \Vert _{H^{1}}+ \Vert n_{t} \Vert _{H^{2}}\bigr) \\ \quad{}+\beta \int^{T}_{0} \bigl\Vert v_{x^{4}}(x,t) \bigr\Vert ^{2}_{L^{2}}\,dt\leq C.\end{gathered} $$

Proof

Taking the inner product of (2.7) and \(-\varepsilon_{tx^{4}}\), it follows that

$$ \bigl(i\varepsilon_{tt}+\varepsilon_{xxt}+\alpha \varepsilon _{t}-n_{t}\varepsilon-n\varepsilon_{t} +\bigl(\delta \vert \varepsilon \vert ^{p}\varepsilon \bigr)_{t},-\varepsilon_{tx^{4}}\bigr)=0. $$
(4.1)

Since

$$\begin{gathered} \operatorname{Im}(i\varepsilon_{tt},-\varepsilon_{tx^{4}})= \frac {1}{2}\frac{d}{dt} \Vert \varepsilon_{txx} \Vert ^{2}_{L^{2}}, \qquad\operatorname{Im}(\varepsilon_{xxt}+ \alpha\varepsilon_{t},-\varepsilon _{tx^{4}})=0, \\ \begin{aligned}[b] \bigl\vert \operatorname{Im}(n_{t} \varepsilon,-\varepsilon_{tx^{4}}) \bigr\vert &= \biggl\vert \operatorname{Im} \int^{L}_{0}(n_{t} \varepsilon)_{xx}\overline {\varepsilon}_{txx}\,dx \biggr\vert \\ &= \biggl\vert \operatorname{Im} \int^{L}_{0}(n_{xt} \varepsilon+n_{t}\varepsilon _{x})_{x} \overline{\varepsilon}_{txx}\,dx \biggr\vert \\ &= \biggl\vert \operatorname{Im} \int^{L}_{0}(n_{xxt} \varepsilon+n_{xt}\varepsilon _{x}+n_{xt} \varepsilon_{x}+n_{t}\varepsilon_{xx}) \overline{\varepsilon }_{txx}\,dx \biggr\vert \\ &\leq C\bigl( \Vert \varepsilon_{txx} \Vert ^{2}_{L^{2}}+ \Vert v_{xxx} \Vert ^{2}_{L^{2}}\bigr), \end{aligned} \\ \begin{aligned}[b] \bigl\vert \operatorname{Im}(-n \varepsilon_{t},-\varepsilon _{tx^{4}}) \bigr\vert &= \biggl\vert \operatorname{Im} \int^{L}_{0}(n\varepsilon _{t})_{xx} \overline{\varepsilon}_{txx}\,dx \biggr\vert \\ &\leq C\bigl( \Vert \varepsilon_{txx} \Vert ^{2}_{L^{2}}+1 \bigr), \end{aligned} \\ \begin{aligned}[b] \operatorname{Im}\bigl(\bigl(\delta \vert \varepsilon \vert ^{p}\varepsilon\bigr)_{t},- \varepsilon _{tx^{4}}\bigr) &=\operatorname{Im} \int^{L}_{0}\biggl[\biggl(1+ \frac{p}{2}\biggr)\delta \vert \varepsilon \vert ^{p}{ \varepsilon}_{t}+\frac{p\delta}{2} \vert \varepsilon \vert ^{p-2}\varepsilon ^{2}\overline{\varepsilon}_{t} \biggr]\overline{\varepsilon}_{t}^{(4)}\,dx \\ &=\operatorname{Im} \int^{L}_{0}\biggl[\biggl(1+ \frac{p}{2}\biggr)\delta \vert \varepsilon \vert ^{p}{ \varepsilon}_{t}+\frac{p\delta}{2} \vert \varepsilon \vert ^{p-2}\varepsilon ^{2}\overline{\varepsilon}_{t} \biggr]_{xx}\overline{\varepsilon}_{txx}\,dx \\ &\leq C\bigl( \Vert \varepsilon_{txx} \Vert ^{2}_{L^{2}}+1 \bigr), \end{aligned} \end{gathered}$$

from (4.1) we get

$$ \begin{aligned}[b] \frac{d}{dt} \Vert \varepsilon_{txx} \Vert ^{2}_{L^{2}} &\leq C \bigl( \Vert \varepsilon_{txx} \Vert ^{2}_{L^{2}}+ \Vert v_{xxx} \Vert ^{2}_{L^{2}}\bigr). \end{aligned} $$
(4.2)

Taking the inner product of (1.2) and \(v_{x^{6}}\), it follows that

$$ \bigl(v_{t}+\bigl[\varphi(v)-\beta v_{x}+n+ \vert \varepsilon \vert ^{2}\bigr]_{x},v_{x^{6}} \bigr)=0. $$
(4.3)

Since

$$\begin{aligned}& (v_{t},v_{x^{6}})=\frac{1}{2} \frac{d}{dt} \Vert v_{xxx} \Vert ^{2}_{L^{2}},\qquad (-\beta v_{xx},v_{x^{6}})=\beta \Vert v_{x^{4}} \Vert ^{2}_{L^{2}}, \\& \begin{aligned}[b] \bigl\vert \bigl(\bigl[\varphi(v) \bigr]_{x},v_{x^{6}}\bigr) \bigr\vert &= \biggl\vert \int_{0}^{L}\bigl[\varphi(v) \bigr]_{x}v_{x^{6}}\,dx \biggr\vert \\ &= \biggl\vert \int_{0}^{L}\bigl[\varphi(v) \bigr]_{xxx}v_{x^{4}}\,dx \biggr\vert \\ &\leq\frac{\beta}{4} \Vert v_{x^{4}} \Vert _{L^{2}}^{2}+C\bigl( \Vert v_{xxx} \Vert ^{2}_{L^{2}}+1\bigr), \end{aligned} \\& \begin{aligned}[b] (n_{x},v_{x^{6}})&=- \int_{0}^{L}n_{xxx}v_{x^{4}} \,dx= \int _{0}^{L}n_{xxx}n_{tx^{3}} \,dx=\frac{1}{2}\frac{d}{dt} \Vert n_{xxx} \Vert ^{2}_{L^{2}}, \end{aligned} \\& \begin{aligned}[b] &\bigl|\bigl(\vert \varepsilon|^{2}_{x},v_{x^{6}}\bigr) \bigr\vert = \biggl\vert \int_{0}^{L}\vert \varepsilon |^{2}_{xxx}v_{x^{4}}\,dx\biggr\vert \leq \frac{\beta}{4} \Vert v_{x^{4}} \Vert _{L^{2}}^{2}+C, \end{aligned} \end{aligned}$$

from (4.3) we get

$$ \frac{d}{dt}\bigl( \Vert v_{xxx} \Vert ^{2}_{L^{2}}+ \Vert n_{xxx} \Vert ^{2}_{L^{2}}\bigr) +\beta \Vert v_{x^{4}} \Vert ^{2}_{L^{2}}\leq C\bigl( \Vert v_{xxx} \Vert _{L^{2}}^{2}+ \Vert n_{xxx} \Vert _{L^{2}}^{2}+1\bigr). $$
(4.4)

By (4.2) and (4.4) we obtain

$$\begin{gathered} \frac{d}{dt}\bigl( \Vert \varepsilon_{txx} \Vert ^{2}_{L^{2}} + \Vert v_{xxx} \Vert ^{2}_{L^{2}}+ \Vert n_{xxx} \Vert ^{2}_{L^{2}}\bigr) +\beta \Vert v_{x^{4}} \Vert ^{2}_{L^{2}} \\ \quad\leq C\bigl( \Vert v_{xxx} \Vert _{L^{2}}^{2}+ \Vert \varepsilon_{txx} \Vert _{L^{2}}^{2}+ \Vert n_{xxx} \Vert _{L^{2}}^{2}+1\bigr),\end{gathered} $$

and thus by using Gronwall’s inequality we obtain

$$ \sup_{t\in[0,T]}\bigl( \Vert \varepsilon_{txx} \Vert ^{2}_{L^{2}} + \Vert v_{xxx} \Vert ^{2}_{L^{2}}+ \Vert n_{xxx} \Vert ^{2}_{L^{2}}\bigr) +\beta \int^{T}_{0} \bigl\Vert v_{x^{4}}(x,t) \bigr\Vert ^{2}_{L^{2}}\,dt\leq C. $$
(4.5)

By (1.1) and Young’s inequality we obtain

$$ \begin{aligned}[b] \Vert \varepsilon_{x^{4}} \Vert _{L^{2}} &\leq \vert \alpha \vert \Vert \varepsilon_{xx} \Vert _{L^{2}}+ \bigl\Vert (n\varepsilon)_{xx} \bigr\Vert _{L^{2}}+ \Vert \varepsilon_{txx} \Vert _{L^{2}} + \bigl\Vert \delta\bigl( \vert \varepsilon \vert ^{p}\varepsilon\bigr)_{xx} \bigr\Vert _{L^{2}} \\ &\leq C\bigl( \Vert \varepsilon_{txx} \Vert _{L^{2}}+1 \bigr). \end{aligned} $$
(4.6)

By (1.3), (4.5), and (4.6) we obtain

$$ \begin{aligned}[b] &\sup_{t\in[0,T]}\bigl( \Vert \varepsilon_{x^{4}} \Vert ^{2}_{L^{2}} + \Vert v_{xxx} \Vert ^{2}_{L^{2}}+ \Vert n_{xxx} \Vert ^{2}_{L^{2}}+ \Vert \varepsilon_{txx} \Vert ^{2}_{L^{2}}+ \Vert n_{txx} \Vert ^{2}_{L^{2}}\bigr) \\ &\quad+\beta \int^{T}_{0} \bigl\Vert v_{x^{4}}(x,t) \bigr\Vert ^{2}_{L^{2}}\, dt\leq C.\end{aligned} $$
(4.7)

By (1.2) and Lemma 2.6 we obtain

$$ \begin{aligned}[b] \Vert v_{tx} \Vert _{L^{2}} &\leq C \bigl\Vert \bigl[\varphi(v)\bigr]_{xx} \bigr\Vert _{L^{2}}+\beta \Vert v_{xxx} \Vert _{L^{2}}+ \Vert n_{xx} \Vert _{L^{2}}+\bigl\| | \varepsilon|_{xxx}\bigr\| _{L^{2}} \\ &\leq C \bigl\Vert \bigl[\varphi(v)\bigr]_{xx} \bigr\Vert _{L^{2}}+\beta \Vert v_{xxx} \Vert _{L^{2}}+ \Vert n_{xx} \Vert _{L^{2}} \\ &\quad+ \bigl\Vert (\varepsilon_{xx}\overline{\varepsilon}+2 \varepsilon\overline {\varepsilon}_{x} +\varepsilon\overline{ \varepsilon}_{xx})_{x} \bigr\Vert _{L^{2}} \\ &\leq C\bigl( \Vert v_{xxx} \Vert _{L^{2}}+1\bigr) \\ &\leq C. \end{aligned} $$
(4.8)

By (4.7) and (4.8) we obtain

$$\begin{gathered} \sup_{t\in[0,T]}\bigl( \Vert \varepsilon \Vert _{H^{4}} + \Vert v \Vert _{H^{3}}+ \Vert n \Vert _{H^{3}}+ \Vert \varepsilon_{t} \Vert _{H^{2}}+ \Vert v_{t} \Vert _{H^{1}}+ \Vert n_{t} \Vert _{H^{2}}\bigr) \\ \quad{}+\beta \int^{T}_{0} \bigl\Vert v_{x^{4}}(x,t) \bigr\Vert ^{2}_{L^{2}}\,dt\leq C.\end{gathered} $$

 □

Corollary 4.1

Suppose that the conditions of Lemma 4.1are satisfied. Then we have

$$\sup_{t\in[0,T]}\bigl( \Vert E_{xxx} \Vert _{L^{\infty}}+ \Vert v_{xx} \Vert _{L^{\infty}}+ \Vert n_{xx} \Vert _{L^{\infty}}+ \Vert \varepsilon_{tx} \Vert _{L^{\infty}} + \Vert n_{tx} \Vert _{L^{\infty}}+ \Vert v_{t} \Vert _{L^{\infty}}\bigr)\leq C. $$

Proof

By Lemmas 2.3 and 4.1 the result of Corollary 4.1 is obvious. □

Theorem 4.1

Suppose that the conditions of Lemma 4.1are satisfied. Then there exists a unique global generalized solution of the initial boundary value problem (1.1)(1.5), and

$$\begin{gathered} \varepsilon(x,t)\in L^{\infty}\bigl(0,T; H^{4}[0,L]\bigr),\qquad E_{t}(x,t)\in L^{\infty }\bigl(0,T; H^{2}[0,L] \bigr), \\ v(x,t)\in L^{\infty}\bigl(0,T; H^{3}[0,L]\bigr)\cap L^{2}\bigl(0,T; H^{4}[0,L]\bigr),\qquad v_{t}(x,t) \in L^{\infty}\bigl(0,T; H^{1}[0,L]\bigr), \\ n(x,t)\in L^{\infty}\bigl(0,T; H^{3}[0,L]\bigr),\qquad n_{t}(x,t)\in L^{\infty}\bigl(0,T; H^{2}[0,L] \bigr).\end{gathered} $$

Proof

By using Theorem 3.1, Lemma 4.1, and Corollary 4.1 we can easily get this theorem. □

Theorem 4.2

Suppose that the conditions of Lemma 4.1are satisfied. Then there exists a unique global classical solution of the boundary value problem (1.1)(1.5).

Proof

By using Theorem 4.1 and the embedding theorems of Sobolev spaces we can easily get this theorem. □

Theorem 4.3

Suppose that the conditions of Lemma 2.7are satisfied. Then there exists a unique global smooth solution of the initial boundary value problem (1.1)(1.5), and

$$\begin{gathered} \varepsilon(x,t)\in L^{\infty}\bigl(0,T; H^{l+2}[0,L]\bigr),\qquad \varepsilon _{t}(x,t)\in L^{\infty}\bigl(0,T; H^{l}[0,L]\bigr), \\ v(x,t)\in L^{\infty}\bigl(0,T; H^{l+1}[0,L]\bigr),\qquad v_{t}(x,t)\in L^{\infty}\bigl(0,T; H^{l-1}[0,L] \bigr)\cap L^{2}\bigl(0,T; H^{l+2}[0,L]\bigr), \\ n(x,t)\in L^{\infty}\bigl(0,T; H^{l+1}[0,L]\bigr),\qquad n_{t}(x,t)\in L^{\infty}\bigl(0,T; H^{l}[0,L] \bigr).\end{gathered} $$

Proof

By Lemma 2.7 and the embedding theorems of Sobolev spaces the result of Theorem 4.3 is obvious. □

Approximation of solution

We now suppose that the generalized solution of initial boundary value problem (1.1)–(1.5) is approximated by the generalized solution of the following problem:

$$\begin{aligned}& i\eta_{t}+\eta_{xx}+(\alpha-m)\eta=0, \end{aligned}$$
(5.1)
$$\begin{aligned}& u_{t}+\bigl[\varphi(u)-\beta u_{x}+m+ \vert \eta \vert ^{2}\bigr]_{x}=0, \end{aligned}$$
(5.2)
$$\begin{aligned}& m_{t}+u_{x}=0,\quad t>0, x\in[0,L], \end{aligned}$$
(5.3)

with initial data

$$ \eta|_{t=0}=\eta_{0}(x),\qquad u|_{t=0}=u_{0}(x),\qquad m|_{t=0}=m_{0}(x),\quad x\in[0,L], $$
(5.4)

and boundary conditions

$$ \eta(0,t)=\eta(L,t)=u(0,t)=u(L,t)=m(0,t)=m(L,t)=0, $$
(5.5)

where the parameters \(p>0\), \(\beta>0\), and α are real numbers, and \(\varphi(s)\) is a real function.

Letting \(F(t,x)=\varepsilon(t,x)-\eta(t,x)\), \(G(t,x)=v(t,x)-u(t,x)\), \(H(t,x)=n(t,x)-m(t,x)\), we obtain

$$\begin{aligned}& iF_{t}+F_{xx}+\alpha F-(H\varepsilon+mF)+\delta \vert \varepsilon \vert ^{p}\varepsilon=0, \end{aligned}$$
(5.6)
$$\begin{aligned}& G_{t}+\bigl[\varphi(v)-\varphi(u)\bigr]_{x}-\beta G_{xx}+H_{x}+\bigl( \vert \varepsilon \vert ^{2}- \vert \eta \vert ^{2}\bigr)_{x}=0, \end{aligned}$$
(5.7)
$$\begin{aligned}& H_{t}+G_{x}=0,\quad t>0, x\in[0,L], \end{aligned}$$
(5.8)

with initial data

$$ F|_{t=0}=0,\qquad G|_{t=0}=0,\qquad H|_{t=0}=0,\quad x \in[0,L], $$
(5.9)

and boundary conditions

$$ F(0,t)=F(L,t)=G(0,t)=G(L,t)=H(0,t)=H(L,t)=0, $$
(5.10)

where the parameters \(p>0\), \(\beta>0\), α, and δ are real numbers, and \(\varphi(s)\) is a real function.

Lemma 5.1

Suppose that the conditions of Theorem 3.1are satisfied. Then for the solution of problem (5.6)(5.10), we have

$$\Vert F \Vert ^{2}_{H^{2}}+ \Vert G \Vert ^{2}_{H^{1}}+ \Vert H \Vert ^{2}_{H^{1}}+ \Vert F_{t} \Vert ^{2}_{L^{2}} + \Vert H_{t} \Vert ^{2}_{L^{2}}\leq \frac{C \vert \delta \vert }{ \vert \delta \vert +1}e^{C( \vert \delta \vert +1)t}+C \vert \delta \vert ^{2}. $$

Proof

Taking the inner product of (5.6) and F, it follows that

$$ \bigl(iF_{t}+F_{xx}+\alpha F-H\varepsilon-mF+\delta \vert \varepsilon \vert ^{p}\varepsilon,F\bigr)=0. $$
(5.11)

Since

$$\begin{aligned}& \operatorname{Im}(iF_{t},F)=\frac{1}{2} \frac{d}{dt} \Vert F \Vert ^{2}_{L^{2}},\qquad \operatorname{Im}(F_{xx}+\alpha F,F)=0, \\& \bigl\vert \operatorname{Im}(H\varepsilon+mF,F) \bigr\vert \leq C \Vert \varepsilon \Vert _{L^{\infty }} \Vert H \Vert _{L^{2}} \Vert F \Vert _{L^{2}}\leq C\bigl( \Vert H \Vert ^{2}_{L^{2}}+ \Vert F \Vert ^{2}_{L^{2}}\bigr), \\& \begin{aligned}[b] \bigl\vert \operatorname{Im}\bigl(\delta \vert \varepsilon \vert ^{p}\varepsilon,F\bigr) \bigr\vert &\leq|\delta| \Vert \varepsilon \Vert ^{p+1}_{L^{2p+2}} \Vert F \Vert _{L^{2}} \\ &\leq|\delta| \Vert \varepsilon_{x} \Vert ^{\frac{p}{2}}_{L^{2}} \Vert \varepsilon \Vert ^{\frac{p+1}{2}}_{L^{2}} \Vert F \Vert _{L^{2}} \\ &\leq C \vert \delta \vert \bigl( \Vert F \Vert ^{2}_{L^{2}}+1 \bigr), \end{aligned} \end{aligned}$$

from (5.11) we get

$$ \frac{d}{dt} \Vert F \Vert ^{2}_{L^{2}}\leq C \bigl[\bigl( \vert \delta \vert +1\bigr) \bigl( \Vert F \Vert ^{2}_{L^{2}}+ \Vert H \Vert ^{2}_{L^{2}} \bigr)+ \vert \delta \vert \bigr]. $$
(5.12)

Taking the inner product of (5.7) and G, it follows that

$$ \bigl(G_{t}+\bigl[\varphi(v)-\varphi(u)\bigr]_{x}- \beta G_{xx}+H_{x}+\bigl( \vert \varepsilon \vert ^{2}- \vert \eta \vert ^{2}\bigr)_{x},G \bigr)=0. $$
(5.13)

Since

$$\begin{gathered} (G_{t},G)=\frac{1}{2}\frac{d}{dt} \Vert G \Vert ^{2}_{L^{2}},\qquad (-\beta G_{xx},G)=\beta \Vert G_{x} \Vert ^{2}_{L^{2}}, \\ \begin{aligned}[b] \bigl\vert \bigl(\bigl[\varphi(v) \bigr]_{x}-\bigl[\varphi(u)\bigr]_{x},G\bigr) \bigr\vert &= \bigl\vert \bigl(\varphi'(\xi) (v-u),G_{x} \bigr) \bigr\vert \\ &\leq \bigl\Vert \varphi'(\xi) \bigr\Vert _{L^{\infty}} \Vert G \Vert _{L^{2}} \Vert G_{x} \Vert _{L^{2}} \\ &\leq C\bigl( \vert \xi \vert ^{q}+1\bigr) \Vert G \Vert _{L^{2}} \Vert G_{x} \Vert _{L^{2}} \\ &\leq C\bigl( \Vert v \Vert ^{q}_{L^{\infty}}+ \Vert u \Vert ^{q}_{L^{\infty}}+1\bigr) \Vert G \Vert _{L^{2}} \Vert G_{x} \Vert _{L^{2}} \\ &\leq C\bigl( \Vert G \Vert ^{2}_{L^{2}}+ \Vert H_{t} \Vert ^{2}_{L^{2}}\bigr), \end{aligned} \\ \bigl\vert (H_{x},G) \bigr\vert \leq\frac{1}{2}\bigl( \Vert H \Vert ^{2}_{L^{2}}+ \Vert H_{t} \Vert ^{2}_{L^{2}}\bigr), \\ \begin{aligned}[b] \bigl( \vert \varepsilon \vert ^{2}_{x}- \vert \eta \vert ^{2}_{x},G \bigr) &= \biggl\vert \int^{L}_{0}\bigl( \vert \varepsilon \vert ^{2}- \vert \eta \vert ^{2}\bigr)G_{x} \,dx \biggr\vert \\ &= \biggl\vert \int^{L}_{0}(F\overline{\varepsilon} +\eta \overline{F})G_{x}\,dx \biggr\vert \\ &\leq\bigl( \Vert \varepsilon \Vert _{L^{\infty}}+ \Vert \eta \Vert _{L^{\infty}}\bigr) \Vert F \Vert _{L^{2}} \Vert H_{t} \Vert _{L^{2}} \\ &\leq C\bigl( \Vert F \Vert ^{2}_{L^{2}}+ \Vert H_{t} \Vert ^{2}_{L^{2}}\bigr), \end{aligned}\end{gathered} $$

from (5.13) we get

$$ \frac{d}{dt} \Vert G \Vert ^{2}_{L^{2}}\leq C \bigl( \Vert G \Vert ^{2}_{L^{2}}+ \Vert F \Vert ^{2}_{L^{2}}+ \Vert H \Vert ^{2}_{L^{2}}+ \Vert H_{t} \Vert ^{2}_{L^{2}} \bigr). $$
(5.14)

Since

$$ \frac{d}{dt} \Vert H \Vert ^{2}_{L^{2}}= \frac{d}{dt} \int ^{L}_{0}H^{2}\,dx\leq \Vert H \Vert ^{2}_{L^{2}}+ \Vert H_{t} \Vert ^{2}_{L^{2}}, $$
(5.15)

by (5.12), (5.14), and (5.15) we get

$$ \begin{aligned}[b] \frac{d}{dt}\bigl( \Vert F \Vert ^{2}_{L^{2}}+ \Vert G \Vert ^{2}_{L^{2}}+ \Vert H \Vert ^{2}_{L^{2}}\bigr) &\leq C\bigl( \vert \delta \vert +1\bigr) \bigl( \Vert F \Vert ^{2}_{L^{2}}+ \Vert G \Vert ^{2}_{L^{2}}\bigr) \\ &\quad+C\bigl( \vert \delta \vert +1\bigr) \bigl( \Vert H \Vert ^{2}_{L^{2}}+ \Vert H_{t} \Vert ^{2}_{L^{2}}\bigr) \\ &\quad+C \vert \delta \vert . \end{aligned} $$
(5.16)

Differentiating (5.6) with respect to t, we get

$$ iF_{tt}+F_{txx}+\alpha F_{t}-(H\varepsilon +mF)_{t}+\delta\bigl( \vert \varepsilon \vert ^{p} \varepsilon\bigr)_{t}=0. $$
(5.17)

Taking the inner product of (5.17) and \(G_{t}\), it follows that

$$ \bigl(iF_{tt}+F_{txx}+\alpha F_{t}-(H \varepsilon +mF)_{t}+\delta\bigl( \vert \varepsilon \vert ^{p}\varepsilon\bigr)_{t},F_{t} \bigr)=0. $$
(5.18)

Since

$$\begin{gathered} \operatorname{Im}(iF_{tt},F_{t})=\frac{1}{2} \frac{d}{dt} \Vert F_{t} \Vert ^{2}_{L^{2}},\qquad \operatorname{Im}(F_{xxt}+\alpha F_{t},F_{t})=0, \\ \begin{aligned}[b] \bigl\vert \operatorname{Im}\bigl((H \varepsilon+mF)_{t},F_{t}\bigr) \bigr\vert &= \biggl\vert \operatorname{Im} \int^{L}_{0}(H_{t}\varepsilon+H \varepsilon _{t}+m_{t}F+mF_{t}) \overline{F}_{t}\,dx \biggr\vert \\ &= \biggl\vert \operatorname{Im} \int^{L}_{0}(H_{t}\varepsilon+H \varepsilon _{t}+m_{t}F)\overline{F}_{t} \,dx \biggr\vert \\ &\leq \Vert \varepsilon \Vert _{L^{\infty}} \Vert H_{t} \Vert _{L^{2}} \Vert F_{t} \Vert _{L^{2}}+ \Vert \varepsilon_{t} \Vert _{L^{\infty}} \Vert H \Vert _{L^{2}} \Vert F_{t} \Vert _{L^{2}} \\ &\quad+ \Vert m_{t} \Vert _{L^{\infty}} \Vert F \Vert _{L^{2}} \Vert F_{t} \Vert _{L^{2}} \\ &\leq C\bigl( \Vert H \Vert ^{2}_{L^{2}}+ \Vert H_{t} \Vert _{L^{2}}^{2}+ \Vert F \Vert ^{2}_{L^{2}}+ \Vert F_{t} \Vert ^{2}_{L^{2}}\bigr), \end{aligned} \\ \begin{aligned}[b] \bigl\vert \operatorname{Im}\bigl(\delta\bigl( \vert \varepsilon \vert ^{p}\varepsilon\bigr)_{t},F_{t} \bigr) \bigr\vert &\leq \vert \delta \vert \biggl\vert \operatorname{Im} \int^{L}_{0}\bigl( \vert \varepsilon \vert ^{p}\varepsilon \bigr)_{t}\overline{F}_{t} \,dx \biggr\vert \\ &=\operatorname{Im} \int^{L}_{0}\biggl[\biggl(1+ \frac{p}{2}\biggr)\delta \vert \varepsilon \vert ^{p}{ \varepsilon}_{t}+\frac{p\delta}{2} \vert \varepsilon \vert ^{p-2}\varepsilon ^{2}\overline{\varepsilon}_{t} \biggr]\overline{F}_{t}\,dx \\ &\leq C \vert \delta \vert \bigl( \Vert F_{t} \Vert _{L^{2}}^{2}+1\bigr), \end{aligned}\end{gathered} $$

from (5.18) we get

$$ \frac{d}{dt} \Vert F_{t} \Vert ^{2}_{L^{2}} \leq C\bigl[\bigl( \vert \delta \vert +1\bigr) \bigl( \Vert F \Vert ^{2}_{L^{2}}+ \Vert F_{t} \Vert _{L^{2}}^{2}+ \Vert H \Vert ^{2}_{L^{2}}+ \Vert H_{t} \Vert ^{2}_{L^{2}}\bigr)+ \vert \delta \vert \bigr]. $$
(5.19)

Taking the inner product of (5.7) and \(-G_{xx}\), it follows that

$$ \bigl(G_{t}+\bigl[\varphi(v)\bigr]_{x}-\bigl[ \varphi(u)\bigr]_{x}-\beta G_{xx}+H_{x}+ \vert \varepsilon \vert ^{2}_{x}- \vert \eta \vert ^{2}_{x},-G_{xx}\bigr)=0. $$
(5.20)

Since

$$\begin{aligned}& (G_{t},-G_{xx})=\frac{1}{2} \frac{d}{dt} \Vert G_{x} \Vert ^{2}_{L^{2}},\qquad (-\beta G_{xx},-G_{xx})=\beta \Vert G_{xx} \Vert ^{2}_{L^{2}}, \\& \begin{aligned}[b] \bigl\vert \bigl(\bigl[\varphi(v) \bigr]_{x}-\bigl[\varphi(u)\bigr]_{x},-G_{xx} \bigr) \bigr\vert &= \biggl\vert \int_{0}^{L}\bigl[\varphi(v)-\varphi(u) \bigr]G_{xxx}\,dx \biggr\vert \\ &= \bigl\vert \bigl(\varphi'(\xi)G,G_{xxx}\bigr) \bigr\vert \\ &= \bigl\Vert \varphi'(\xi) \bigr\Vert _{L^{\infty}} \bigl\vert (G_{x},G_{xx}) \bigr\vert \\ &\leq \bigl\Vert \varphi'(\xi) \bigr\Vert _{L^{\infty}} \Vert G_{x} \Vert _{L^{2}} \Vert G_{xx} \Vert _{L^{2}} \\ &\leq C\bigl( \vert \xi \vert ^{q}+1\bigr) \Vert G_{x} \Vert _{L^{2}} \Vert G_{xx} \Vert _{L^{2}} \\ &\leq C\bigl( \Vert v \Vert ^{q}_{L^{\infty}}+ \Vert u \Vert ^{q}_{L^{\infty}}+1\bigr) \Vert G_{x} \Vert _{L^{2}} \Vert G_{xx} \Vert _{L^{2}} \\ &\leq\frac{\beta}{2} \Vert G_{xx} \Vert _{L^{2}}^{2}+C \Vert G_{x} \Vert _{L^{2}}^{2}, \end{aligned} \\& (H_{x},-G_{xx})=- \int_{0}^{L}H_{x}G_{xx} \,dx= \int_{0}^{L}H_{x}H_{xt} \, dx=\frac{1}{2}\frac{d}{dt} \Vert H_{x} \Vert ^{2}_{L^{2}}, \\& \begin{aligned}[b] \bigl\vert \bigl( \vert \varepsilon \vert ^{2}_{x}- \vert \eta \vert ^{2}_{x},-G_{xx} \bigr) \bigr\vert &= \biggl\vert \int_{0}^{L}(F_{x}\overline{ \varepsilon}+ \eta_{x}\overline{F}+F\overline{\varepsilon}_{x} +\eta\overline{F}_{x})G_{xx}\,dx \biggr\vert \\ &= \biggl\vert \int_{0}^{L}(F_{x}\overline{ \varepsilon}+ \eta\overline{F}_{x})G_{xx}\,dx \biggr\vert \\ &\quad+ \biggl\vert \int_{0}^{L}(F\overline{\varepsilon}_{x} +\eta_{x}\overline{F}_{x})G_{xx}\,dx \biggr\vert \\ &= \biggl\vert \int_{0}^{L}(F_{x}\overline{ \varepsilon}+ \eta\overline{F}_{x})_{x}G_{x} \,dx \biggr\vert \\ &\quad+ \biggl\vert \int_{0}^{L}(F\overline{\varepsilon}_{x} +\eta_{x}\overline{F}_{x})_{x}G_{x} \,dx \biggr\vert \\ &\leq \int_{0}^{L}\bigl( \vert \varepsilon \vert + \vert \eta \vert \bigr) \vert F_{x} \vert \vert G_{xx} \vert \,dx \\ &\quad+ \int_{0}^{L}\bigl( \vert \varepsilon_{x} \vert + \vert \eta_{x} \vert \bigr) \vert \varepsilon \vert \vert G_{xx} \vert \, dx \\ &\leq\bigl( \Vert \varepsilon \Vert _{L^{\infty}}+ \Vert \eta \Vert _{L^{\infty}}\bigr) \Vert F_{x} \Vert _{L^{2}} \Vert G_{xx} \Vert _{L^{2}} \\ &\quad+\bigl( \Vert \varepsilon_{x} \Vert _{L^{\infty}}+ \Vert \eta_{x} \Vert _{L^{\infty}}\bigr) \Vert F \Vert _{L^{2}} \Vert G_{xx} \Vert _{L^{2}} \\ &\leq\frac{\beta}{2} \Vert G_{xx} \Vert _{L^{2}}^{2}+C\bigl( \Vert F \Vert _{L^{2}}^{2}+ \Vert F_{x} \Vert _{L^{2}}^{2}\bigr), \end{aligned} \end{aligned}$$

from (5.8) and (5.20) we get

$$ \begin{aligned}[b] \frac{d}{dt}\bigl( \Vert H_{t} \Vert ^{2}_{L^{2}}+ \Vert H_{x} \Vert ^{2}_{L^{2}}\bigr)&= \frac {d}{dt}\bigl( \Vert G_{x} \Vert ^{2}_{L^{2}}+ \Vert H_{x} \Vert ^{2}_{L^{2}}\bigr) \\ &\leq C\bigl( \Vert F \Vert _{L^{2}}^{2}+ \Vert F_{x} \Vert _{L^{2}}^{2}+ \Vert G_{x} \Vert _{L^{2}}^{2}\bigr). \end{aligned} $$
(5.21)

Taking the inner product of (5.11) and \(F_{xx}\), it follows that

$$ \bigl(iF_{t}+F_{xx}+\alpha F-H\varepsilon-mF+\delta \vert \varepsilon \vert ^{p}\varepsilon,F_{xx} \bigr)=0. $$
(5.22)

Since

$$\begin{aligned}& \operatorname{Im}(iF_{t},F_{xx})=\frac{1}{2} \frac{d}{dt} \Vert F_{x} \Vert ^{2}_{L^{2}},\qquad \operatorname{Im}(F_{xx}+\alpha F,F_{xx})=0, \\& \begin{aligned}[b] \bigl\vert \operatorname{Im}(H \varepsilon+mF,F_{xx}) \bigr\vert &= \biggl\vert \operatorname{Im} \int^{L}_{0}(H\varepsilon+mF)_{x}F_{x} \,dx \biggr\vert \\ &\leq \biggl\vert \operatorname{Im} \int^{L}_{0}(H_{x}\varepsilon+H \varepsilon _{x}+m_{x}F+mF_{x})F_{x} \,dx \biggr\vert \\ &\leq C\bigl( \Vert \varepsilon \Vert _{L^{\infty}} \Vert H_{x} \Vert _{L^{2}}+ \Vert \varepsilon _{x} \Vert _{L^{\infty}} \Vert H \Vert _{L^{2}} \bigr) \Vert F_{x} \Vert _{L^{2}} \\ &\quad+C\bigl( \Vert m_{x} \Vert _{L^{\infty}} \Vert F \Vert _{L^{2}}+ \Vert m \Vert _{L^{\infty}} \Vert F_{x} \Vert _{L^{2}}\bigr) \Vert F_{x} \Vert _{L^{2}} \\ &\leq C\bigl( \Vert H \Vert ^{2}_{L^{2}}+ \Vert H_{x} \Vert ^{2}_{L^{2}}+ \Vert F \Vert ^{2}_{L^{2}}+ \Vert F_{x} \Vert ^{2}_{L^{2}}\bigr), \end{aligned} \\& \begin{aligned}[b] \bigl\vert \operatorname{Im}\bigl(\delta \vert \varepsilon \vert ^{p}\varepsilon,F_{xx}\bigr) \bigr\vert &=\biggl|\operatorname{Im} \int^{L}_{0}\delta\bigl( \vert \varepsilon \vert ^{p}\varepsilon \bigr)_{x}F_{x}\,dx\biggr| \\ &\leq \vert \delta \vert \int_{0}^{L} \bigl\vert \bigl( \vert \varepsilon \vert ^{p}_{x}\varepsilon+ \vert \varepsilon \vert ^{p}\varepsilon_{x}\bigr) \bigr\vert \vert F_{x} \vert \,dx \\ &\leq \vert \delta \vert \int_{0}^{L} \biggl\vert \biggl[ \frac{p}{2} \vert \varepsilon \vert ^{p-2} ( \varepsilon_{x}\overline{\varepsilon}+\varepsilon\overline{ \varepsilon }_{x})+ \vert \varepsilon \vert ^{p} \varepsilon_{x}\biggr] \biggr\vert \vert F_{x} \vert \,dx \\ &\leq C \vert \delta \vert \bigl( \Vert F_{x} \Vert ^{2}_{L^{2}}+1\bigr), \end{aligned} \end{aligned}$$

from (5.22), we get

$$ \frac{d}{dt} \Vert F_{x} \Vert ^{2}_{L^{2}} \leq C\bigl[\bigl( \vert \delta \vert +1\bigr) \bigl( \Vert H \Vert ^{2}_{L^{2}}+ \Vert H_{x} \Vert ^{2}_{L^{2}}+ \Vert F \Vert ^{2}_{L^{2}}+ \Vert F_{x} \Vert ^{2}_{L^{2}}\bigr)+ \vert \delta \vert \bigr]. $$
(5.23)

By (5.16), (5.19), (5.21), and (5.23) we obtain

$$\begin{aligned}[b] &\quad\frac{d}{dt}\bigl( \Vert F \Vert ^{2}_{L^{2}}+ \Vert F_{x} \Vert ^{2}_{L^{2}}+ \Vert F_{t} \Vert ^{2}_{L^{2}}+ \Vert G \Vert ^{2}_{L^{2}}+ \Vert H \Vert ^{2}_{L^{2}} + \Vert H_{t} \Vert ^{2}_{L^{2}}+ \Vert H_{x} \Vert ^{2}_{L^{2}}\bigr) \\ &\quad\leq C\bigl( \vert \delta \vert +1\bigr) \bigl( \Vert F \Vert ^{2}_{L^{2}}+ \Vert F_{x} \Vert ^{2}_{L^{2}}+ \Vert F_{t} \Vert ^{2}_{L^{2}}\bigr)+C \vert \delta \vert \\ &\qquad+C\bigl( \vert \delta \vert +1\bigr) \bigl( \Vert G \Vert ^{2}_{L^{2}}+ \Vert H \Vert ^{2}_{L^{2}}+ \Vert H_{t} \Vert ^{2}_{L^{2}} + \Vert H_{x} \Vert ^{2}_{L^{2}}\bigr). \end{aligned} $$

By using Gronwall’s inequality we obtain

$$\begin{gathered} \Vert F \Vert ^{2}_{L^{2}}+ \Vert F_{x} \Vert ^{2}_{L^{2}}+ \Vert F_{t} \Vert ^{2}_{L^{2}}+ \Vert G \Vert ^{2}_{L^{2}}+ \Vert H \Vert ^{2}_{L^{2}}+ \Vert H_{x} \Vert ^{2}_{L^{2}} + \Vert H_{t} \Vert ^{2}_{L^{2}} \\ \quad\leq\frac{C \vert \delta \vert }{ \vert \delta \vert +1}e^{C( \vert \delta \vert +1)t}.\end{gathered} $$

By (5.8) we get

$$\begin{aligned}[b] &\Vert F \Vert ^{2}_{L^{2}}+ \Vert F_{x} \Vert ^{2}_{L^{2}}+ \Vert F_{t} \Vert ^{2}_{L^{2}}+ \Vert G \Vert ^{2}_{L^{2}}+ \Vert G_{x} \Vert ^{2}_{L^{2}}+ \Vert H \Vert ^{2}_{L^{2}} \\ &\quad+ \Vert H_{t} \Vert ^{2}_{L^{2}}+ \Vert H_{x} \Vert ^{2}_{L^{2}}\leq \frac {C \vert \delta \vert }{ \vert \delta \vert +1}e^{C( \vert \delta \vert +1)t}.\end{aligned} $$
(5.24)

By (5.6) and (5.24) we obtain

$$ \begin{aligned}[b] \Vert F_{xx} \Vert _{L^{2}}^{2} &\leq\bigl[ \vert \alpha \vert \Vert F \Vert _{L^{2}}+ \Vert H\varepsilon+mF \Vert _{L^{2}}+ \Vert F_{t} \Vert _{L^{2}} + \bigl\Vert \delta \vert \varepsilon \vert ^{p}\varepsilon \bigr\Vert _{L^{2}} \bigr]^{2} \\ &\leq C\bigl[ \vert \alpha \vert ^{2} \Vert F \Vert _{L^{2}}^{2}+ \Vert H\varepsilon \Vert ^{2}_{L^{2}}+ \Vert mF \Vert ^{2}_{L^{2}} \\ &\quad+ \Vert F_{t} \Vert ^{2}_{L^{2}}+ \vert \delta \vert ^{2}\bigl( \Vert \varepsilon \Vert _{L^{2p+2}}^{2p+2}\bigr)^{2}\bigr] \\ &\leq C\bigl[ \vert \alpha \vert ^{2} \Vert F \Vert ^{2}_{L^{2}}+ \Vert \varepsilon \Vert _{L^{\infty}} \Vert H \Vert ^{2}_{L^{2}}+ \Vert m \Vert _{L^{\infty}} \Vert F \Vert ^{2}_{L^{2}} \\ &\quad+ \Vert F_{t} \Vert ^{2}_{L^{2}}+ \vert \delta \vert ^{2}\bigl(C \Vert \varepsilon_{x} \Vert _{L^{2}}^{\frac{p}{2}} \Vert \varepsilon \Vert ^{\frac {p+4}{2(p+2)}}_{L^{2}}\bigr)^{2}\bigr] \\ &\leq C\bigl[ \vert \alpha \vert ^{2} \Vert F \Vert ^{2}_{L^{2}}+ \Vert \varepsilon \Vert _{L^{\infty}} \Vert H \Vert ^{2}_{L^{2}}+ \Vert m \Vert _{L^{\infty}} \Vert F \Vert ^{2}_{L^{2}} \\ &\quad+ \Vert F_{t} \Vert ^{2}_{L^{2}}+C \vert \delta \vert ^{2}\bigr] \\ &\leq C\bigl( \Vert F \Vert ^{2}_{L^{2}}+ \Vert F_{t} \Vert ^{2}_{L^{2}}+ \Vert H \Vert ^{2}_{L^{2}}+ \vert \delta \vert ^{2}\bigr) \\ &\leq\frac{C \vert \delta \vert }{ \vert \delta \vert +1}e^{C( \vert \delta \vert +1)t}+C \vert \delta \vert ^{2}. \end{aligned} $$
(5.25)

By (5.24) and (5.25) we obtain

$$ \Vert F \Vert ^{2}_{H^{2}}+ \Vert G \Vert ^{2}_{H^{1}}+ \Vert H \Vert ^{2}_{H^{1}}+ \Vert F_{t} \Vert ^{2}_{L^{2}}+ \Vert H_{t} \Vert ^{2}_{L^{2}} \leq \frac{C \vert \delta \vert }{ \vert \delta \vert +1}e^{C( \vert \delta \vert +1)t}+C \vert \delta \vert ^{2}. $$
(5.26)

 □

Lemma 5.2

Suppose that the conditions of Theorem 3.1are satisfied. Then for the solution of problem (5.6)(5.10), we have

$$\begin{gathered} \Vert F \Vert ^{2}_{H^{3}}+ \Vert G \Vert ^{2}_{H^{2}}+ \Vert H \Vert ^{2}_{H^{2}}+ \Vert F_{t} \Vert ^{2}_{H^{1}}+ \Vert G_{t} \Vert ^{2}_{L^{2}} + \Vert H_{t} \Vert ^{2}_{H^{1}} \\ \quad\leq\frac{C \vert \delta \vert }{ \vert \delta \vert +1}e^{C( \vert \delta \vert +1)}+C \vert \delta \vert ^{2}e^{Ct}+C \vert \delta \vert ^{2}.\end{gathered} $$

Proof

Taking the inner product of (5.17) and \(-F_{txx}\), it follows that

$$ \bigl(iF_{tt}+F_{txx}+\alpha F_{t}-(H \varepsilon +mF)_{t}+\delta\bigl( \vert \varepsilon \vert ^{p}\varepsilon\bigr)_{t},-F_{txx} \bigr)=0. $$
(5.27)

Since

$$\begin{aligned}& \operatorname{Im}(iF_{tt},-F_{txx})=\frac{1}{2} \frac{d}{dt} \Vert F_{tx} \Vert ^{2}_{L^{2}},\quad\quad \operatorname{Im}(F_{xxt}+\alpha F_{t},-F_{txx})=0, \\& \begin{aligned}[b] \bigl\vert \operatorname{Im}\bigl((H \varepsilon+mF)_{t},-F_{txx}\bigr) \bigr\vert &= \biggl\vert \operatorname{Im} \int^{L}_{0}(H_{t}\varepsilon+H \varepsilon _{t}+m_{t}F+mF_{t})_{x} \overline{F}_{tx}\,dx \biggr\vert \\ &= \biggl\vert \operatorname{Im} \int^{L}_{0}(H_{xt} \varepsilon+H_{t}\varepsilon _{x})\overline{F}_{tx} \,dx \biggr\vert + \biggl\vert \operatorname{Im} \int^{L}_{0}(H_{x} \varepsilon_{t}+H\varepsilon _{tx})\overline{F}_{tx} \,dx \biggr\vert \\ &\quad+ \biggl\vert \operatorname{Im} \int^{L}_{0}(m_{xt}F+m_{t}F_{x}) \overline {F}_{tx}\,dx \biggr\vert \\ &\quad+ \biggl\vert \operatorname{Im} \int^{L}_{0}(m_{x}F_{t}+mF_{xt}) \overline {F}_{tx}\,dx \biggr\vert \\ &\leq C\bigl( \Vert F \Vert ^{2}_{L^{2}}+ \Vert F_{x} \Vert ^{2}_{L^{2}}+ \Vert F_{t} \Vert ^{2}_{L^{2}} \bigr) \\ &\quad+C\bigl( \Vert H \Vert ^{2}_{L^{2}}+ \Vert H_{x} \Vert ^{2}_{L^{2}}+ \Vert H_{t} \Vert ^{2}_{L^{2}}\bigr) +C\bigl( \Vert F_{tx} \Vert ^{2}_{L^{2}}+ \Vert H_{tx} \Vert ^{2}_{L^{2}}\bigr) \\ &\leq\frac{C \vert \delta \vert }{ \vert \delta \vert +1}e^{C( \vert \delta \vert +1)t}+C \vert \delta \vert ^{2} +C\bigl( \Vert F_{tx} \Vert ^{2}_{L^{2}}+ \Vert H_{tx} \Vert ^{2}_{L^{2}}\bigr), \end{aligned} \\& \begin{aligned}[b] \operatorname{Im}\bigl(\bigl(\delta \vert \varepsilon \vert ^{p}\varepsilon\bigr)_{t},-F_{txx} \bigr) &=-\operatorname{Im} \int^{L}_{0}\biggl[\biggl(1+ \frac{p}{2}\biggr)\delta \vert \varepsilon \vert ^{p}{ \varepsilon}_{t}+\frac{p\delta}{2} \vert \varepsilon \vert ^{p-2}\varepsilon ^{2}\overline{\varepsilon}_{t} \biggr]\overline{F}_{txx}\,dx \\ &=\operatorname{Im} \int^{L}_{0}\biggl[\biggl(1+ \frac{p}{2}\biggr)\delta \vert \varepsilon \vert ^{p}{ \varepsilon}_{t}+\frac{p\delta}{2} \vert \varepsilon \vert ^{p-2}\varepsilon ^{2}\overline{\varepsilon}_{t} \biggr]_{x}\overline{F}_{tx}\,dx \\ &=\operatorname{Im} \int^{L}_{0}\biggl[\biggl(1+ \frac{p}{2}\biggr)\delta \vert \varepsilon \vert ^{p}_{x}{\varepsilon}_{t} +\biggl(1+ \frac{p}{2}\biggr)\delta \vert \varepsilon \vert ^{p}{ \varepsilon}_{tx}\biggr]\overline {F}_{tx}\,dx \\ &\quad+\operatorname{Im} \int^{L}_{0}\biggl(\frac{p\delta}{2} \vert \varepsilon \vert ^{p-2}_{x} \varepsilon^{2}\overline{\varepsilon}_{t} + \frac{p\delta}{2} \vert \varepsilon \vert ^{p-2} \varepsilon^{2}_{x}\overline {\varepsilon}_{t} \overline{\varepsilon}_{tx}\biggr)\overline{F}_{tx}\,dx \\ &\quad+\operatorname{Im} \int^{L}_{0}\frac{p\delta}{2} \vert \varepsilon \vert ^{p-2}\varepsilon^{2}\overline{ \varepsilon}_{tx}\overline{F}_{tx}\,dx \\ &\leq C \vert \delta \vert \bigl( \Vert F_{tx} \Vert ^{2}_{L^{2}}+1\bigr), \end{aligned} \end{aligned}$$

from (5.27) we get

$$ \frac{d}{dt} \Vert F_{tx} \Vert ^{2}_{L^{2}} \leq C\bigl( \vert \delta \vert +1\bigr) \bigl( \Vert F_{tx} \Vert ^{2}_{L^{2}}+ \Vert H_{tx} \Vert ^{2}_{L^{2}}\bigr) +\frac{C \vert \delta \vert }{ \vert \delta \vert +1}e^{C( \vert \delta \vert +1)t}+C \vert \delta \vert ^{2}. $$
(5.28)

Taking the inner product of (5.7) and \(G_{x^{4}}\), it follows that

$$ \bigl(G_{t}+\bigl[\varphi(v)-\varphi(u)\bigr]_{x}- \beta G_{xx}+H_{x}+\bigl( \vert \varepsilon \vert ^{2}- \vert \eta \vert ^{2}\bigr)_{x},G_{x^{4}} \bigr)=0. $$
(5.29)

Since

$$\begin{aligned}& (G_{t},G_{x^{4}})=\frac{1}{2} \frac{d}{dt} \Vert G_{xx} \Vert ^{2}_{L^{2}},\qquad (-\beta G_{xx},G_{x^{4}})=\beta \Vert G_{xxx} \Vert ^{2}_{L^{2}}, \\& \begin{aligned}[b] \bigl\vert \bigl(\bigl[\varphi(v)-\varphi(u) \bigr]_{x},G_{x^{4}}\bigr) \bigr\vert &= \biggl\vert \int_{0}^{L}\bigl[\varphi(v)-\varphi(u) \bigr]_{x}G_{x^{4}}\,dx \biggr\vert \\ &= \biggl\vert \int_{0}^{L}\bigl[\varphi(v)-\varphi(u) \bigr]G_{x^{5}}\,dx \biggr\vert \\ &= \biggl\vert \int_{0}^{L}\varphi'( \xi)GG_{x^{5}}\,dx \biggr\vert \\ &\leq \bigl\Vert \varphi'(\xi) \bigr\Vert _{L^{\infty}} \Vert G \Vert _{L^{2}} \Vert G_{x^{5}} \Vert _{L^{2}} \\ &\leq \bigl\Vert \varphi'(\xi) \bigr\Vert _{L^{\infty}} \Vert G_{xx} \Vert _{L^{2}} \Vert G_{xxx} \Vert _{L^{2}} \\ &\leq C\bigl( \vert \xi \vert ^{q}+1\bigr) \Vert G_{xx} \Vert _{L^{2}} \Vert G_{xxx} \Vert _{L^{2}} \\ &\leq C\bigl( \Vert v \Vert ^{q}_{L^{\infty}}+ \Vert u \Vert ^{q}_{L^{\infty}}+1\bigr) \Vert G_{xx} \Vert _{L^{2}} \Vert G_{xxx} \Vert _{L^{2}} \\ &\leq\frac{\beta}{4} \Vert G_{xxx} \Vert _{L^{2}}^{2}+C \Vert G_{xx} \Vert ^{2}_{L^{2}}, \end{aligned} \\& \begin{aligned}[b] (H_{x},G_{x^{4}})&=- \int_{0}^{2L}H_{xx}G_{xxx} \,dx= \int _{0}^{2L}H_{xx}H_{xxt} \,dx=\frac{1}{2}\frac{d}{dt} \Vert H_{xx} \Vert ^{2}_{L^{2}}, \end{aligned} \\& \begin{aligned}[b] \bigl\vert \bigl(\bigl( \vert \varepsilon \vert ^{2}- \vert \eta \vert ^{2} \bigr)_{x},G_{x^{4}}\bigr) \bigr\vert &= \biggl\vert \int_{0}^{L}\bigl( \vert \varepsilon \vert ^{2}- \vert \eta \vert ^{2}\bigr)_{x}G_{x^{4}} \,dx \biggr\vert \\ &= \biggl\vert \int_{0}^{L}(F_{x}\overline{ \varepsilon}+ \eta_{x}\overline{F}+F\overline{\varepsilon}_{x} +\eta\overline{F}_{x})G_{xxx}\,dx \biggr\vert \\ &\leq\frac{\beta}{4} \Vert G_{xxx} \Vert _{L^{2}}^{2}+C\bigl( \Vert F \Vert _{L^{2}}^{2}+ \Vert F_{x} \Vert _{L^{2}}^{2}\bigr) \\ &\leq\frac{\beta}{4} \Vert G_{xxx} \Vert _{L^{2}}^{2}+\frac{C \vert \delta \vert }{ \vert \delta \vert +1}e^{C( \vert \delta \vert +1)t}+C \vert \delta \vert ^{2}, \end{aligned} \end{aligned}$$

from (5.29) we get

$$ \begin{aligned}[b] \frac{d}{dt}\bigl( \Vert G_{xx} \Vert ^{2}_{L^{2}}+ \Vert H_{xx} \Vert ^{2}_{L^{2}}\bigr) &\leq C\bigl( \Vert G_{xx} \Vert _{L^{2}}^{2}+ \Vert H_{xx} \Vert _{L^{2}}^{2}\bigr) \\ &\quad+\frac{C \vert \delta \vert }{ \vert \delta \vert +1}e^{C( \vert \delta \vert +1)t}+C \vert \delta \vert ^{2}. \end{aligned} $$
(5.30)

By (5.28) and (5.30) we obtain

$$ \begin{aligned}[b] \frac{d}{dt}\bigl( \Vert F_{tx} \Vert ^{2}_{L^{2}} + \Vert G_{xx} \Vert ^{2}_{L^{2}}+ \Vert H_{xx} \Vert ^{2}_{L^{2}}\bigr) &\leq C\bigl( \Vert F_{tx} \Vert ^{2}_{L^{2}} + \Vert G_{xx} \Vert ^{2}_{L^{2}}+ \Vert H_{xx} \Vert ^{2}_{L^{2}}\bigr) \\ &\quad+\frac{C \vert \delta \vert }{ \vert \delta \vert +1}e^{C( \vert \delta \vert +1)t}+C \vert \delta \vert ^{2}, \end{aligned} $$

and thus by Gronwall’s inequality we obtain

$$ \begin{aligned}[b] \Vert F_{tx} \Vert ^{2}_{L^{2}} + \Vert G_{xx} \Vert ^{2}_{L^{2}}+ \Vert H_{xx} \Vert ^{2}_{L^{2}} &\leq\frac{C \vert \delta \vert }{( \vert \delta \vert +1)^{2}}e^{C( \vert \delta \vert +1)t}e^{Ct}+C \vert \delta \vert ^{2}e^{Ct} \\ &\leq\frac{C \vert \delta \vert }{ \vert \delta \vert +1}e^{C( \vert \delta \vert +1)t}+C \vert \delta \vert ^{2}e^{Ct}. \end{aligned} $$
(5.31)

By (5.8) we obtain

$$ \Vert F_{tx} \Vert ^{2}_{L^{2}} + \Vert G_{xx} \Vert ^{2}_{L^{2}}+ \Vert H_{xx} \Vert ^{2}_{L^{2}}+ \Vert H_{tx} \Vert ^{2}_{L^{2}}\leq \frac{C \vert \delta \vert }{ \vert \delta \vert +1}e^{C( \vert \delta \vert +1)t}+C \vert \delta \vert ^{2}e^{Ct}. $$
(5.32)

By (5.6), Young’s inequality, and (5.32) we obtain

$$ \begin{aligned}[b] \Vert F_{xxx} \Vert _{L^{2}}^{2} &\leq C\bigl[ \vert \alpha \vert \Vert F_{x} \Vert _{L^{2}}^{2}+ \bigl\Vert (H \varepsilon+mF)_{x} \bigr\Vert _{L^{2}}^{2} \bigr] \\ &\quad+C\bigl[ \Vert F_{tx} \Vert _{L^{2}}^{2} + \bigl\Vert \delta\bigl( \vert \varepsilon \vert ^{p} \varepsilon\bigr)_{x} \bigr\Vert _{L^{2}}^{2} \bigr] \\ &\leq C \Vert F_{x} \Vert _{L^{2}}^{2}+C \Vert F_{tx} \Vert _{L^{2}}^{2} \\ &\quad+C\bigl( \Vert H_{x}\varepsilon \Vert _{L^{2}}+ \Vert H\varepsilon_{x} \Vert _{L^{2}}+ \Vert m_{x}F \Vert _{L^{2}}+ \Vert mF_{x} \Vert _{L^{2}}\bigr)^{2} \\ &\quad+C \biggl\Vert \biggl(1+\frac{p}{2}\biggr)\delta \vert \varepsilon \vert ^{p}{\varepsilon}_{x}+ \frac {p\delta}{2} \vert \varepsilon \vert ^{p-2} \varepsilon^{2}\overline{\varepsilon }_{x} \biggr\Vert _{L^{2}}^{2} \\ &\leq C \Vert F_{x} \Vert _{L^{2}}^{2}+C \Vert F_{tx} \Vert _{L^{2}}^{2} \\ &\quad+C\bigl( \Vert H_{x}\varepsilon \Vert _{L^{2}}^{2}+ \Vert H\varepsilon_{x} \Vert _{L^{2}}^{2}+ \Vert m_{x}F \Vert _{L^{2}}^{2}+ \Vert mF_{x} \Vert _{L^{2}}^{2}\bigr) \\ &\quad+C\biggl(1+\frac{p^{2}}{4}\biggr) \vert \delta \vert ^{2} \bigl\Vert |\varepsilon \vert ^{p}\varepsilon _{x} \bigr\Vert _{L^{2}}^{2} +C \frac{p^{2} \vert \delta \vert ^{2}}{4} \bigl\Vert | \varepsilon \vert ^{p} \varepsilon_{x} \bigr\Vert _{L^{2}}^{2} \\ &\leq C\bigl( \Vert F \Vert ^{2}_{L^{2}}+ \Vert F_{x} \Vert ^{2}_{L^{2}}+ \Vert F_{tx} \Vert ^{2}_{L^{2}}\bigr) \\ &\quad+C\bigl( \Vert H \Vert ^{2}_{L^{2}}+ \|H_{x}\|^{2}_{L^{2}}\|_{L^{2}}\bigr)+C \vert \delta \vert ^{2} \\ &\leq\frac{C \vert \delta \vert }{ \vert \delta \vert +1}e^{C( \vert \delta \vert +1)t}+C \vert \delta \vert ^{2}e^{Ct}+C \vert \delta \vert ^{2}. \end{aligned} $$
(5.33)

By (1.3), (5.32), and (5.33) we obtain

$$ \begin{aligned}[b] & \Vert F_{xxx} \Vert ^{2}_{L^{2}}+ \Vert F_{tx} \Vert ^{2}_{L^{2}} + \Vert G_{xx} \Vert ^{2}_{L^{2}}+ \Vert H_{xx} \Vert ^{2}_{L^{2}}+ \Vert H_{tx} \Vert ^{2}_{L^{2}} \\ &\quad\leq\frac{C \vert \delta \vert }{ \vert \delta \vert +1}e^{C( \vert \delta \vert +1)t}+C \vert \delta \vert ^{2}e^{Ct}+C \vert \delta \vert ^{2}. \end{aligned} $$
(5.34)

By (5.7) we obtain

$$ \begin{aligned}[b] \Vert G_{t} \Vert ^{2}_{L^{2}} &\leq C\bigl[ \bigl\Vert \bigl[\varphi(v)- \varphi(u)\bigr]_{x} \bigr\Vert _{L^{2}}^{2}+ \beta \Vert G_{xx} \Vert ^{2}_{L^{2}}+ \Vert H_{x} \Vert ^{2}_{L^{2}} \\ &\quad+ \bigl\Vert \bigl( \vert \varepsilon \vert ^{2}- \vert \eta \vert ^{2}\bigr)_{x} \bigr\Vert ^{2}_{L^{2}}\bigr] \\ &\leq C\bigl[ \bigl\Vert \varphi'(\xi) \bigr\Vert _{L^{\infty}} \Vert G_{x} \Vert ^{2}_{L^{2}}+ \beta \Vert G_{xx} \Vert ^{2}_{L^{2}}+ \Vert H_{x} \Vert ^{2}_{L^{2}} \\ &\quad+ \bigl\Vert (F_{x}\overline{\varepsilon}+ \eta_{x}\overline{F}+F\overline {\varepsilon}_{x}+\eta \overline{F}_{x}) \bigr\Vert ^{2}_{L^{2}} \bigr] \\ &\leq\frac{C \vert \delta \vert }{ \vert \delta \vert +1}e^{C( \vert \delta \vert +1)t}+C \vert \delta \vert ^{2}e^{Ct}+C \vert \delta \vert ^{2}. \end{aligned} $$
(5.35)

By (5.34) and (5.35) we obtain

$$ \begin{aligned}[b] & \Vert F \Vert ^{2}_{H^{3}}+ \Vert G \Vert ^{2}_{H^{2}}+ \Vert H \Vert ^{2}_{H^{2}}+ \Vert F_{t} \Vert ^{2}_{H^{1}}+ \Vert G_{t} \Vert ^{2}_{L^{2}} + \Vert H_{t} \Vert ^{2}_{H^{1}} \\ &\quad\leq\frac{C \vert \delta \vert }{ \vert \delta \vert +1}e^{C( \vert \delta \vert +1)t}+C \vert \delta \vert ^{2}e^{Ct}+C \vert \delta \vert ^{2}. \end{aligned} $$

 □

By Lemmas 5.1 and 5.2 we obtain the following:

Theorem 5.1

Suppose that the conditions of Theorem 3.1are satisfied. If\((\varepsilon, v, n)\)is the solution to problem (1.1)(1.5) and\((\eta, u, m)\)is the solution to problem (5.1)(5.5), then

$$\Vert \varepsilon-\eta \Vert ^{2}_{H^{3}}+ \Vert v-u \Vert ^{2}_{H^{2}}+ \Vert n-m \Vert ^{2}_{H^{2}} \leq\frac{C \vert \delta \vert }{ \vert \delta \vert +1}e^{C( \vert \delta \vert +1)t}+C \vert \delta \vert ^{2}e^{Ct}+C \vert \delta \vert ^{2}, $$

whereCis a positive constant.

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Acknowledgements

The authors are grateful to the referees for useful remarks and suggestions that greatly improved the presentation of this manuscript.

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This work is supported by the National Natural Science Foundation of China (Grant 11401223, 11871172), the National Natural Science Foundation of Guangdong (Grant 2015A030313424), and the Science and Technology Program of Guangzhou (Grant 201607010005).

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XW carried out the studies and drafted the manuscript. All authors read and approved the final version of the manuscript.

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Correspondence to Yadong Shang.

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Wang, X., Shang, Y. & Lei, C. Initial boundary value problem for generalized Zakharov equations with nonlinear function terms. Bound Value Probl 2020, 85 (2020). https://doi.org/10.1186/s13661-020-01383-8

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Keywords

  • Generalized Zakharov equations
  • Initial boundary value
  • Existence and uniqueness
  • Global smooth solution
  • Approximation of solution