Well-Posedness Results for the Continuum Spectrum Pulse Equation

Abstract

The continuum spectrum pulse equation is a third order nonlocal nonlinear evolutive equation related to the dynamics of the electrical field of linearly polarized continuum spectrum pulses in optical waveguides. In this paper, the well-posedness of the classical solutions to the Cauchy problem associated with this equation is proven.

1. Introduction

In this paper, we investigate the well-posedness of the classical solution of the following Cauchy problem:

$$\left\{\begin{array}{cc}{\displaystyle {\partial }_{t}u+3g{u}^2{\partial }_{x}u-a{\partial }_x^{3}u+q{\partial }_x\left(uv\right)=bP,}\hfill & t>0,x\in \mathbb{R},\hfill \\ {\displaystyle {\partial }_{x}P=u,}\hfill & t>0,x\in \mathbb{R},\hfill \\ {\displaystyle \alpha {\partial }_x^{2}v+\beta {\partial }_{x}v+\gamma v=\kappa u^2,}\hfill & t>0,x\in \mathbb{R},\hfill \\ {\displaystyle P(t,-\infty )=0,}\hfill & t>0,\hfill \\ {\displaystyle u(0,x)=u_0\left(x\right),}\hfill & x\in \mathbb{R},\hfill \end{array}\right.$$

(1)

where $g,a,q,b,\alpha ,\beta ,\gamma ,\kappa \in \mathbb{R}$.

On the initial datum, we assume that

$$u_0\in H^2\left(\mathbb{R}\right)\cap L^1\left(\mathbb{R}\right),{\int }_{\mathbb{R}}{u}_0\left(x\right)dx=0.$$

(2)

Following [1,2,3,4,5,6], on the function

$$P_0\left(x\right)={\int }_{-\infty }^x{u}_0\left(y\right)dy,$$

(3)

we assume that

$$\begin{array}{cc}\hfill {\int }_{\mathbb{R}}{P}_0\left(x\right)dx& ={\int }_{\mathbb{R}}\left({\int }_{-\infty }^x{u}_0\left(y\right)dy\right)=0,\hfill \\ \hfill {∥P_0∥}_{L^2\left(\mathbb{R}\right)}^2& ={\int }_{\mathbb{R}}{\left({\int }_{-\infty }^x{u}_0\left(y\right)dy\right)}^{2}dx<\infty .\hfill \end{array}$$

(4)

In addition, we assume that

$$\frac{q\beta }{\kappa }\ge 0,g\ne 0,a\ne 0,\alpha \ne 0.$$

(5)

Observe that, since $\alpha$ cannot vanish, we can factorize it and deal with only three constants.

In the physical literature (1) is termed the continuum spectrum pulse equation (see [7,8,9,10,11,12,13,14]). It is used to describe the dynamics of the electrical field u of linearly polarized continuum spectrum pulses in optical waveguides, including fused-silica telecommunication-type or photonic-crystal fibers, as well as hollow capillaries filled with transparent gases or liquids.

The constants $a,b,g,q,\alpha ,\kappa ,\beta ,\gamma$, in (1), take into account the frequency dispersion of the effective linear refractive index and the nonlinear polarization response, the excitation efficiency of the vibrations, the frequency and the decay time (see [7,8,14]).

Moreover, (1) generalizes the following system:

$$\left\{\begin{array}{cc}{\displaystyle {\partial }_{t}u+q{\partial }_x\left(uv\right)=bP,}\hfill & t>0,x\in \mathbb{R},\hfill \\ {\displaystyle {\partial }_{x}P=u,}\hfill & t>0,x\in \mathbb{R},\hfill \\ {\displaystyle \alpha {\partial }_x^{2}v+\beta {\partial }_{x}v+\gamma v=\kappa u^2,}\hfill & t>0,x\in \mathbb{R},\hfill \\ {\displaystyle P(t,-\infty )=0,}\hfill & t>0,\hfill \\ {\displaystyle u(0,x)=u_0\left(x\right),}\hfill & x\in \mathbb{R},\hfill \end{array}\right.$$

(6)

whose the well-posedness is studied in [15]. From a mathematical point of view, the presence of the term

$$3g{u}^2{\partial }_{x}u-a{\partial }_x^{3}u$$

in the first equation of (1) makes the analysis of such system more subtle than the one of (6).

Observe that, taking $b=\alpha =\beta =0$, (1) becomes the modified Korteweg-de Vries equation (see [16,17,18,19,20])

$${\partial }_{t}u+\left(g+\frac{q\kappa }{\gamma }\right){\partial }_x{u}^3-a{\partial }_x^{3}u=0.$$

(7)

In [8,9,21,22,23,24], it is proven that (7) is a non-slowly-varying envelope approximation model that describes the physics of few-cycle-pulse optical solitons. In [6,18], the Cauchy problem for (7) is studied, while, in [16,19], the convergence of the solution of (7) to the unique entropy solution of the following scalar conservation law

$${\partial }_{t}u+\left(g+\frac{q\kappa }{\gamma }\right){\partial }_x{u}^3=0$$

(8)

is proven.

On the other hand, taking $\alpha =\beta =0$ in (1), we have the following equations

$$\left\{\begin{array}{cc}{\displaystyle {\partial }_{t}u+\left(g+\frac{q\kappa }{\gamma }\right){\partial }_x{u}^3=bP,}\hfill & t>0,x\in \mathbb{R},\hfill \\ {\displaystyle {\partial }_{x}P=u,}\hfill & t>0,x\in \mathbb{R},\hfill \end{array}\right.$$

(9)

that were deduced by Kozlov and Sazonov [12] for the description of the nonlinear propagation of optical pulses of a few oscillations duration in dielectric media and by Schäfer and Wayne [25] for the description of the propagation of ultra-short light pulses in silica optical fibers. Moreover, (9) is a non-slowly-varying envelope approximation model that describes the physics of few-cycle-pulse optical solitons (see [22,23,24,26,27,28]), a particular Rabelo equation which describes pseudospherical surfaces (see [29,30,31,32]), and a model for the descriptions of the short pulse propagation in nonlinear metamaterials characterized by a weak Kerr-type nonlinearity in their dielectric response (see [33]).

Finally, (9) was deduced in [34] in the context of plasma physic and that similar equations describe the dynamics of radiating gases [35,36], in [37,38,39,40] in the context of ultrafast pulse propagation in a mode-locked laser cavity in the few-femtosecond pulse regime and in [41] in the context of Maxwell equations.

The Cauchy problem for (9) was studied in [42,43,44] in the context of energy spaces, [4,5,45,46] in the context of entropy solutions. The homogeneous initial boundary value problem was studied in [47,48,49,50]. Nonlocal formulations of (9) were analyzed in [15,51] and the convergence of a finite difference scheme proved in [52].

Observe that, taking $\alpha =\beta =0$, (1) reads

$$\left\{\begin{array}{cc}{\displaystyle {\partial }_{t}u+\left(g+\frac{q\kappa }{\gamma }\right){\partial }_x{u}^3-a{\partial }_x^{3}u=bP,}\hfill & t>0,x\in \mathbb{R},\hfill \\ {\displaystyle {\partial }_{x}P=u,}\hfill & t>0,x\in \mathbb{R}.\hfill \end{array}\right.$$

(10)

It was derived by Costanzino, Manukian and Jones [53] in the context of the nonlinear Maxwell equations with high-frequency dispersion. Kozlov and Sazonov [12] show that (10) is an more general equation than (9) to describe the nonlinear propagation of optical pulses of a few oscillations duration in dielectric media.

Mathematical properties of (10) are studied in many different contexts, including the local and global well-posedness in energy spaces [43,53] and stability of solitary waves [53,54], while, in [6], the well-posedness of the classical solutions is proven.

Observe that letting $a\to 0$ in (10), we obtain (9). Hence, following [19,55,56], in [5,57], the convergence of the solution of (10) to the unique entropy solution of (9).

The main result of this paper is the following theorem.

Theorem 1.

Assume (2), (3), (4) and (5). Fix $T>0$, there exists an unique solution $(u,v,P)$ of (1) such that

$$\begin{array}{c}u\in L^{\infty }(0,T;H^2\left(\mathbb{R}\right)),\hfill \\ v\in L^{\infty }(0,T;H^4\left(\mathbb{R}\right)),\hfill \\ P\in L^{\infty }(0,T;H^3\left(\mathbb{R}\right)).\hfill \end{array}$$

(11)

In particular, we have that

$${\int }_{\mathbb{R}}u(t,x)dx=0,t\ge 0.$$

(12)

Moreover, if $(u_1,v_1,P_1)$ and $(u_2,v_2,P_2)$ are two solutions of (1), we have that

$$\begin{array}{cc}\hfill {∥u_1(t,·)-u_2(t,·)∥}_{L^2\left(\mathbb{R}\right)}\le & e^{C\left(T\right)t}{∥u_{1,0}-u_{2,0}∥}_{L^2\left(\mathbb{R}\right)},\hfill \\ \hfill {∥v_1(t,·)-v_2(t,·)∥}_{H^2\left(\mathbb{R}\right)}\le & e^{C\left(T\right)t}{∥u_{1,0}-u_{2,0}∥}_{L^2\left(\mathbb{R}\right)},\hfill \\ \hfill {∥P_1(t,·)-P_2(t,·)∥}_{H^1\left(\mathbb{R}\right)}\le & e^{C\left(T\right)t}{∥P_{1,0}-P_{2,0}∥}_{H^1\left(\mathbb{R}\right)},\hfill \end{array}$$

(13)

where,

$$P_{1,0}\left(x\right)={\int }_{-\infty }^x{u}_{1,0}\left(y\right)dy,P_{2,0}\left(x\right)={\int }_{-\infty }^x{u}_{2,0}\left(y\right)dy,$$

(14)

for some suitable $C\left(T\right)>0$, and every $0\le t\le T$.

The proof of Theorem 1 is based on the Aubin–Lions Lemma (see [58,59,60]).

The paper is organized as follows. In Section 2, we prove several a priori estimates on a vanishing viscosity approximation of (1). Those play a key role in the proof of our main result, that is given in Section 3. Appendix A is an appendix, where we prove the posedness of the classical solutions of (1), under the assumption

$$u_0\in L^1\left(\mathbb{R}\right)\cap H^3\left(\mathbb{R}\right).$$

(15)

2. Vanishing Viscosity Approximation

Our existence argument is based on passing to the limit in a vanishing viscosity approximation of (1).

Fix a small number $0<\epsilon <1$ and let $u_{\epsilon }=u_{\epsilon }(t,x)$ be the unique classical solution of the following mixed problem [19,61,62]:

$$\left\{\begin{array}{cc}{\displaystyle {\partial }_t{u}_{\epsilon }+3g{u}_{\epsilon }^2{\partial }_x{u}_{\epsilon }-a{\partial }_x^3{u}_{\epsilon }+q{\partial }_x\left(v_{\epsilon }{u}_{\epsilon }\right)=b{P}_{\epsilon }-\epsilon {\partial }_x^4{u}_{\epsilon },}\hfill & t>0,x\in \mathbb{R},\hfill \\ {\displaystyle {\partial }_x{P}_{\epsilon }=u_{\epsilon },}\hfill & t>0,x\in \mathbb{R},\hfill \\ {\displaystyle \alpha {\partial }_x^2{v}_{\epsilon }+\beta {\partial }_x{v}_{\epsilon }+\gamma v_{\epsilon }=\kappa u_{\epsilon }^2,}\hfill & t>0,x\in \mathbb{R},\hfill \\ P_{\epsilon }(t,-\infty )=0,\hfill & t>0,\hfill \\ u_{\epsilon }(0,x)=u_{\epsilon ,0}\left(x\right),\hfill & x\in \mathbb{R},\hfill \end{array}\right.$$

(16)

where $u_{\epsilon ,0}$ is a $C^{\infty }$ approximation of $u_0$ such that

$$\begin{array}{c}{∥u_{\epsilon ,0}∥}_{H^2\left(\mathbb{R}\right)}\le {∥u_0∥}_{H^2\left(\mathbb{R}\right)},{\int }_{\mathbb{R}}{u}_{\epsilon ,0}dx=0,\hfill \\ {∥P_{\epsilon ,0}∥}_{L^2\left(\mathbb{R}\right)}\le {∥P_0∥}_{L^2\left(\mathbb{R}\right)},{\int }_{\mathbb{R}}{P}_{\epsilon ,0}dx=0.\hfill \end{array}$$

(17)

Let us prove some a priori estimates on $u_{\epsilon }$, $P_{\epsilon }$ and $v_{\epsilon }$. We denote with $C_0$ the constants which depend only on the initial data, and with $C\left(T\right)$, the constants which depend also on T.

Lemma 1.

For each $t\ge 0$,

$$P_{\epsilon }(t,\infty )=0.$$

(18)

In particular, we have that

$${\int }_{\mathbb{R}}{u}_{\epsilon }(t,x)dx=0.$$

(19)

Proof. 

We begin by proving (18). Thanks to the regularity of $u_{\epsilon }$ and the first equation of (16), we have that

$$\underset{x\to \infty }{\lim }\left({\partial }_t{u}_{\epsilon }+3g{u}_{\epsilon }^2{\partial }_x{u}_{\epsilon }-a{\partial }_x^3{u}_{\epsilon }+q{\partial }_x\left(v_{\epsilon }{u}_{\epsilon }\right)-\epsilon {\partial }_x^5{u}_{\epsilon }\right)=b{P}_{\epsilon }(t,\infty )=0,$$

which gives (18).

Finally, we prove (19). Integrating the second equation of (16) on $(-\infty ,x)$, by (16), we have that

$$P_{\epsilon }(t,x)={\int }_{-\infty }^x{u}_{\epsilon }(t,y)dy.$$

(20)

Equation (19) follows from (18) and (20).  □

Arguing as in ([15], Lemma 2.2), we have the following result.

Lemma 2.

Assume (5). We have that

$${\int }_{\mathbb{R}}{u}_{\epsilon }^2{\partial }_x{v}_{\epsilon }dx=\left\{\begin{array}{cc}{\displaystyle \frac{\beta }{\kappa }{∥{\partial }_x{v}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2,}\hfill & if \beta \ne 0,\hfill \\ 0,\hfill & if \beta =0.\hfill \end{array}\right.$$

(21)

Lemma 3.

Assume (5). If $\beta \ne 0$, then for each $t\ge 0$, there exists a constant $C_0>0$, independent on ε, such that

$$\begin{array}{cc}\hfill {∥u_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2& +\frac{q\beta }{\kappa }{\int }_0^t{∥{\partial }_x{v}_{\epsilon }(s,·)∥}_{L^2\left(\mathbb{R}\right)}^{2}ds\hfill \\ & +2\epsilon {\int }_0^t{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^{2}ds\le C_0.\hfill \end{array}$$

(22)

If $\beta =0$, then for each $t\ge 0$,

$${∥u_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+2\epsilon {\int }_0^t{∥{\partial }_x{u}_{\epsilon }(s,·)∥}_{L^2\left(\mathbb{R}\right)}^{2}ds\le {∥u_0∥}_{L^2\left(\mathbb{R}\right)}^2.$$

(23)

In particular, we have

$$\begin{array}{cc}\hfill {∥{\partial }_x{v}_{\epsilon }(t,·)∥}_{L^{\infty }\left(\mathbb{R}\right)},{∥{\partial }_x{v}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}\le & C_0,\hfill \\ \hfill {∥v_{\epsilon }(t,·)∥}_{L^{\infty }\left(\mathbb{R}\right)},{∥v_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}\le & C_0.\hfill \end{array}$$

(24)

Moreover, fixed $T>0$, there exists a constant $C\left(T\right)>0$, independent on ε, such that

$$\epsilon {\int }_0^t{∥{\partial }_x{u}_{\epsilon }(s,·)∥}_{L^2\left(\mathbb{R}\right)}^{2}ds\le C\left(T\right).$$

(25)

Proof. 

Multiplying by $2u_{\epsilon }$ the first equation of (16), an integration on $\mathbb{R}$ gives

$$\begin{array}{cc}\hfill \frac{d}{dt}{∥u_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2=& 2{\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_t{u}_{\epsilon }dx\hfill \\ \hfill =& -6g{\int }_{\mathbb{R}}{u}_{\epsilon }^3{\partial }_x{u}_{\epsilon }dx-2q{\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x\left(u_{\epsilon }{v}_{\epsilon }\right)dx+2b{\int }_{\mathbb{R}}{P}_{\epsilon }{u}_{\epsilon }dx\hfill \\ & +2a{\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x^3{u}_{\epsilon }dx-2\epsilon {\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x^4{u}_{\epsilon }dx\hfill \\ \hfill =& -2q{\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x\left(u_{\epsilon }{v}_{\epsilon }\right)dx-2a{\int }_{\mathbb{R}}{\partial }_x{u}_{\epsilon }{\partial }_x^2{u}_{\epsilon }dx\hfill \\ & +2b{\int }_{\mathbb{R}}{P}_{\epsilon }{u}_{\epsilon }dx+2\epsilon {\int }_{\mathbb{R}}{\partial }_x{u}_{\epsilon }{\partial }_x^3{u}_{\epsilon }dx\hfill \\ \hfill =& -2q{\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x\left(u_{\epsilon }{v}_{\epsilon }\right)dx-2a{\int }_{\mathbb{R}}{\partial }_x{u}_{\epsilon }{\partial }_x^2{u}_{\epsilon }dx\hfill \\ & +2b{\int }_{\mathbb{R}}{P}_{\epsilon }{u}_{\epsilon }dx-2\epsilon {∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2.\hfill \end{array}$$

Therefore,

$$\frac{d}{dt}{∥u_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+2\epsilon {∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2=2b{\int }_{\mathbb{R}}{P}_{\epsilon }{u}_{\epsilon }dx-2q{\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x\left(u_{\epsilon }{v}_{\epsilon }\right)dx.$$

Arguing as in ([15], Lemma 2.2), we have (22), (23) and (24).

Finally, arguing as in ([6], Lemma 2.3), we have (25).  □

Lemma 4.

Assume (5). Fix $T>0$. There exists a constant $C_0>0$, independent on ε, such that

$${∥{\partial }_x^2{v}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}\le C_0\left(1+{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right).$$

(26)

Proof. 

Let $0\le t\le T$. Thanks to the third equation of (16), we have that

$$\alpha {\partial }_x^2{v}_{\epsilon }=\kappa u_{\epsilon }^2-\beta {\partial }_x{v}_{\epsilon }-\gamma v_{\epsilon }.$$

(27)

Therefore, by (24),

$$\begin{array}{cc}\hfill \left|\alpha \right||{\partial }_x^2{v}_{\epsilon }|=& |\kappa u_{\epsilon }^2-\beta {\partial }_x{v}_{\epsilon }-\gamma v_{\epsilon }|\le |\kappa |u_{\epsilon }^2+\left|\beta \right||{\partial }_x{v}_{\epsilon }|+|\gamma \left|\right|v_{\epsilon }|\hfill \\ \hfill \le & \left|\kappa \right|{∥u_{\epsilon }∥}_{L^2((0,T)\times \mathbb{R})}^2+\left|\beta \right|{∥{\partial }_x{v}_{\epsilon }(t,·)∥}_{L^{\infty }\left(\mathbb{R}\right)}+\left|\gamma \right|{∥v_{\epsilon }(t,·)∥}_{L^{\infty }\left(\mathbb{R}\right)}\hfill \\ \hfill \le & \left|\kappa \right|{∥u_{\epsilon }∥}_{L^2((0,T)\times \mathbb{R})}^2+C_0\le C_0\left(1+{∥u_{\epsilon }∥}_{L^2((0,T)\times \mathbb{R})}^2\right),\hfill \end{array}$$

which gives (26).  □

Arguing as in ([6], Lemma 2.2), we have the following result.

Lemma 5.

For each $t\ge 0$, we have that

$$\begin{array}{cc}\hfill {\int }_0^{-\infty }{P}_{\epsilon }(t,x)dx& =A_{\epsilon }\left(t\right),\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill {\int }_0^{\infty }{P}_{\epsilon }(t,x)dx& =A_{\epsilon }\left(t\right),\hfill \end{array}$$

(29)

where

$$A_{\epsilon }\left(t\right)=-\frac{1}{b}{\partial }_t{P}_{\epsilon }(t,0)-\frac{g}{b}{u}_{\epsilon }^3(t,0)-\frac{a}{b}{\partial }_x^2{u}_{\epsilon }(t,0)-\frac{q}{b}{u}_{\epsilon }(t,0)v_{\epsilon }(t,0)+\frac{\epsilon }{b}{\partial }_x{u}_{\epsilon }(t,0).$$

In particular, we have

$${\int }_{\mathbb{R}}{P}_{\epsilon }(t,x)dx=0.$$

(30)

Lemma 6.

Assume (5). Fix $T>0$. There exists a constant $C\left(T\right)>0$, independent on ε, such that

$$\begin{array}{cc}\hfill {∥P_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+& 2\epsilon e^{C_{0}t}{\int }_0^t{e}^{-C_{0}s}{∥{\partial }_x{u}_{\epsilon }(s,·)∥}_{L^2\left(\mathbb{R}\right)}^{2}ds\hfill \\ \hfill \le & C\left(T\right)\left(1+{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right).\hfill \end{array}$$

(31)

for every $0\le t\le T$. In particular, we have that

$${∥P_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}\le \sqrt[4]{C\left(T\right)\left(1+{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right)}.$$

(32)

Proof. 

Let $0\le t\le T$. We begin by observing that, by (28), we can consider the following function:

$$F_{\epsilon }(t,x)={\int }_{-\infty }^x{P}_{\epsilon }(t,y)dy.$$

(33)

Integrating the second equation of (16) on $(-\infty ,x)$, we have

$$P_{\epsilon }(t,x)={\int }_{-\infty }^x{u}_{\epsilon }(t,y)dy.$$

(34)

Differentiating (34) with respect to t, we get

$${\partial }_t{P}_{\epsilon }(t,x)=\frac{d}{dt}{\int }_{-\infty }^x{u}_{\epsilon }(t,y)dy={\int }_{-\infty }^x{\partial }_t{u}_{\epsilon }(t,x)dx.$$

(35)

Equation (33), (35) and an integration on $(-\infty ,x)$ of the first equation of (16) give

$${\partial }_t{P}_{\epsilon }=b{F}_{\epsilon }-\epsilon {\partial }_x^3{u}_{\epsilon }-g{u}_{\epsilon }^3+a{\partial }_x^2{u}_{\epsilon }-q{v}_{\epsilon }{u}_{\epsilon }.$$

(36)

Multiplying (36) by $-2P_{\epsilon }$, an integration on $\mathbb{R}$ gives

$$\begin{array}{cc}\hfill \frac{d}{dt}{∥P_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2=& 2b{\int }_{\mathbb{R}}{F}_{\epsilon }{P}_{\epsilon }dx-2\epsilon {\int }_{\mathbb{R}}{P}_{\epsilon }{\partial }_x^3{u}_{\epsilon }dx-2g{\int }_{\mathbb{R}}{P}_{\epsilon }{u}_{\epsilon }^{3}dx\hfill \\ & +2a{\int }_{\mathbb{R}}{P}_{\epsilon }{\partial }_x^2{u}_{\epsilon }dx-2q{\int }_{\mathbb{R}}{P}_{\epsilon }{v}_{\epsilon }{u}_{\epsilon }dx.\hfill \end{array}$$

(37)

Observe that, by (18), (30), (33) and the second equation of (16),

$$\begin{array}{cc}\hfill 2b{\int }_{\mathbb{R}}{F}_{\epsilon }{P}_{\epsilon }dx=& 2b{\int }_{\mathbb{R}}{F}_{\epsilon }{\partial }_x{F}_{\epsilon }dx=b{F}_{\epsilon }^2(t,\infty )=b{\left({\int }_{\mathbb{R}}{P}_{\epsilon }(t,x)dx\right)}^{2}dx=0,\hfill \\ \hfill -2\epsilon {\int }_{\mathbb{R}}{P}_{\epsilon }{\partial }_x^3{u}_{\epsilon }dx=& 2\epsilon {\int }_{\mathbb{R}}{\partial }_x{P}_{\epsilon }{\partial }_x^2{u}_{\epsilon }dx=2\epsilon {\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x^2{u}_{\epsilon }dx=-2\epsilon {∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2,\hfill \\ \hfill 2a{\int }_{\mathbb{R}}{P}_{\epsilon }{\partial }_x^2{u}_{\epsilon }dx=& -2a{\int }_{\mathbb{R}}{\partial }_x{P}_{\epsilon }{\partial }_x{u}_{\epsilon }dx=-2a{\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x{u}_{\epsilon }=0,\hfill \\ \hfill -2q{\int }_{\mathbb{R}}{P}_{\epsilon }{v}_{\epsilon }{u}_{\epsilon }dx=& -2q{\int }_{\mathbb{R}}{P}_{\epsilon }{v}_{\epsilon }{\partial }_x{P}_{\epsilon }dx=2q{\int }_{\mathbb{R}}{\partial }_x{v}_{\epsilon }{P}_{\epsilon }^{2}dx.\hfill \end{array}$$

Consequently, by (24) and (37),

$$\begin{array}{cc}\hfill \frac{d}{dt}{∥P_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2& +2\epsilon {∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill =& 2q{\int }_{\mathbb{R}}{\partial }_x{v}_{\epsilon }{P}_{\epsilon }^{2}dx-2g{\int }_{\mathbb{R}}{P}_{\epsilon }{u}_{\epsilon }^{3}dx\hfill \\ \hfill \le & 2\left|q\right|{\int }_{\mathbb{R}}|{\partial }_x{v}_{\epsilon }|P_{\epsilon }^{2}dx+2\left|g\right|{\int }_{\mathbb{R}}|P_{\epsilon }\left|\right|u_{\epsilon }{|}^{3}dx\hfill \\ \hfill \le & 2\left|q\right|{∥{\partial }_x{v}_{\epsilon }(t,·)∥}_{L^{\infty }\left(\mathbb{R}\right)}{∥P_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+2\left|g\right|{\int }_{\mathbb{R}}|P_{\epsilon }\left|\right|u_{\epsilon }{|}^{3}dx\hfill \\ \hfill \le & C_0{∥P_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+2\left|g\right|{\int }_{\mathbb{R}}|P_{\epsilon }\left|\right|u_{\epsilon }{|}^{3}dx.\hfill \end{array}$$

(38)

Due to Lemma 3 and the Young inequality,

$$\begin{array}{cc}\hfill 2\left|g\right|{\int }_{\mathbb{R}}|P_{\epsilon }\left|\right|u_{\epsilon }{|}^{3}dx=& {\int }_{\mathbb{R}}|2P_{\epsilon }{u}_{\epsilon }\left|\right|g{u}_{\epsilon }^2|dx\le 2{\int }_{\mathbb{R}}{P}_{\epsilon }^2{u}_{\epsilon }^{2}dx+\frac{g^2}{2}{\int }_{\mathbb{R}}{u}_{\epsilon }^{4}dx\hfill \\ \hfill \le & 2{∥P_{\epsilon }(t,·)∥}_{L^{\infty }\left(\mathbb{R}\right)}^2{∥u_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ & +\frac{g^2}{2}{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2{∥u_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & C_0{∥P_{\epsilon }(t,·)∥}_{L^{\infty }\left(\mathbb{R}\right)}^4+C_0{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\hfill \\ \hfill \le & {∥P_{\epsilon }(t,·)∥}_{L^{\infty }\left(\mathbb{R}\right)}^4+C_0+C_0{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2.\hfill \end{array}$$

It follows from (38) that

$$\begin{array}{cc}\hfill \frac{d}{dt}{∥P_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2& +2\epsilon {∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & C_0{∥P_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+{∥P_{\epsilon }(t,·)∥}_{L^{\infty }\left(\mathbb{R}\right)}^4+C_0\left(1+{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right).\hfill \end{array}$$

(39)

Thanks to Lemma 3 and the Hölder inequality,

$$\begin{array}{cc}\hfill P_{\epsilon }^2(t,x)=2& {\int }_{-\infty }^x{P}_{\epsilon }{u}_{\epsilon }dy\le 2{\int }_{\mathbb{R}}|P_{\epsilon }\left|\right|u_{\epsilon }|dx\hfill \\ \hfill \le & {∥P_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}{∥u_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}\le C_0{∥P_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}.\hfill \end{array}$$

Hence,

$${∥P_{\epsilon }(t,·)∥}_{L^{\infty }\left(\mathbb{R}\right)}^4\le C_0{∥P_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2.$$

(40)

It follows from (39) and (40) that

$$\begin{array}{cc}\hfill \frac{d}{dt}{∥P_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2& +2\epsilon {∥u_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & C_0{∥P_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+C_0\left(1+{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right).\hfill \end{array}$$

The Gronwall Lemma and (17) give

$$\begin{array}{cc}\hfill {∥P_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2& +2\epsilon e^{C_{0}t}{\int }_0^t{e}^{-C_{0}s}{∥u_{\epsilon }(s,·)∥}_{L^2\left(\mathbb{R}\right)}^{2}ds\hfill \\ \hfill \le & C_0{e}^{C_{0}t}+C_0\left(1+{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right)e^{C_{0}t}{\int }_0^t{e}^{-C_{0}s}ds\hfill \\ \hfill \le & C\left(T\right)\left(1+{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right),\hfill \end{array}$$

which gives (31).

Finally, (32) follows from (31) and (40).  □

Following ([6], Lemma 2.5), we prove the following result.

Lemma 7.

Assume (5). Fix $T>0$. There exists a constant $C\left(T\right)>0$, independent on ε, such that

$${∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}\le C\left(T\right).$$

(41)

In particular, we have

$${∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+2\epsilon e^{C_{0}t}{\int }_0^t{e}^{-C_{0}s}{∥{\partial }_x^3{u}_{\epsilon }(s,·)∥}_{L^2\left(\mathbb{R}\right)}^{2}ds\le C\left(T\right),$$

(42)

for every $0\le t\le T$. Moreover,

$$\begin{array}{cc}\hfill {∥{\partial }_x^2{v}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}\le & C\left(T\right),\hfill \\ \hfill {∥P_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}\le & C\left(T\right),\hfill \\ \hfill {∥P_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}\le & C\left(T\right),\hfill \end{array}$$

(43)

for every $0\le t\le T$.

Proof. 

Let $0\le t\le T$. Multiplying the first equation of (1) by $-2{\partial }_x^2{u}_{\epsilon }+\frac{2g}{a}{u}_{\epsilon }^3$, we have that

$$\begin{array}{cc}\hfill \left(-2{\partial }_x^2{u}_{\epsilon }+\frac{2g}{a}{u}_{\epsilon }^3\right){\partial }_t{u}_{\epsilon }& +3g\left(-2{\partial }_x^2{u}_{\epsilon }+\frac{2g}{a}{u}_{\epsilon }^3\right)u_{\epsilon }^2{\partial }_x{u}_{\epsilon }\hfill \\ & -a\left(-2{\partial }_x^2{u}_{\epsilon }+\frac{2g}{a}{u}_{\epsilon }^3\right){\partial }_x^3{u}_{\epsilon }+q\left(-2{\partial }_x^2{u}_{\epsilon }+\frac{2g}{a}{u}_{\epsilon }^3\right){\partial }_x\left(v_{\epsilon }{u}_{\epsilon }\right)\hfill \\ \hfill =& b\left(-2{\partial }_x^2{u}_{\epsilon }+\frac{2g}{a}{u}_{\epsilon }^3\right)P_{\epsilon }-\epsilon \left(-2{\partial }_x^2{u}_{\epsilon }+\frac{2g}{a}{u}_{\epsilon }^3\right){\partial }_x^4{u}_{\epsilon }.\hfill \end{array}$$

(44)

Observe that, by (18) and the second equation of (16),

$$-2b{\int }_{\mathbb{R}}{P}_{\epsilon }{\partial }_x^2{u}_{\epsilon }dx=2b{\int }_{\mathbb{R}}{\partial }_x{P}_{\epsilon }{\partial }_x{u}_{\epsilon }dx=2b{\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x{u}_{\epsilon }dx=0.$$

(45)

Moreover,

$$\begin{array}{cc}\hfill {\int }_{\mathbb{R}}\left(-2{\partial }_x^2{u}_{\epsilon }+\frac{2g}{a}{u}_{\epsilon }^3\right){\partial }_t{u}_{\epsilon }dx=& \frac{d}{dt}\left({∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+\frac{g}{2a}{\int }_{\mathbb{R}}{u}_{\epsilon }^{4}dx\right),\hfill \\ \hfill 3g{\int }_{\mathbb{R}}\left(-2{\partial }_x^2{u}_{\epsilon }+\frac{2g}{a}{u}_{\epsilon }^3\right)u_{\epsilon }^2{\partial }_x{u}_{\epsilon }dx=& -6g{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\partial }_x{u}_{\epsilon }{\partial }_x^2{u}_{\epsilon }dx,\hfill \\ \hfill -a{\int }_{\mathbb{R}}\left(-2{\partial }_x^2{u}_{\epsilon }+\frac{2g}{a}{u}_{\epsilon }^3\right){\partial }_x^3{u}_{\epsilon }dx=& 6g{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\partial }_x{u}_{\epsilon }{\partial }_x^2{u}_{\epsilon }dx,\hfill \\ \hfill -\epsilon {\int }_{\mathbb{R}}\left(-2{\partial }_x^2{u}_{\epsilon }+\frac{2g}{a}{u}_{\epsilon }^3\right){\partial }_x^4{u}_{\epsilon }dx=& -2\epsilon {∥{\partial }_x^3{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ & +\frac{6g\epsilon }{a}{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\partial }_x{u}_{\epsilon }{\partial }_x^3{u}_{\epsilon }dx.\hfill \end{array}$$

(46)

Defined

$$G\left(t\right):={∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+\frac{g}{2a}{\int }_{\mathbb{R}}{u}_{\epsilon }^{4}dx,$$

(47)

it follows from (45), (46) and an integration on $\mathbb{R}$ of (44) that

$$\begin{array}{cc}\hfill \frac{dG\left(t\right)}{dt}+2\epsilon {∥{\partial }_x^3{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2=& \frac{2bg}{a}{\int }_{\mathbb{R}}{P}_{\epsilon }{u}_{\epsilon }^{3}dx+\frac{6g\epsilon }{a}{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\partial }_x{u}_{\epsilon }{\partial }_x^3{u}_{\epsilon }dx\hfill \\ & +2q{\int }_{\mathbb{R}}{\partial }_x\left(v_{\epsilon }{u}_{\epsilon }\right){\partial }_x^2{u}_{\epsilon }dx-2qga{\int }_{\mathbb{R}}{\partial }_x\left(v_{\epsilon }{u}_{\epsilon }\right)u_{\epsilon }^{3}dx.\hfill \end{array}$$

(48)

Observe that

$$\begin{array}{cc}\hfill 2q{\int }_{\mathbb{R}}{\partial }_x\left(v_{\epsilon }{u}_{\epsilon }\right){\partial }_x^2{u}_{\epsilon }dx=& 2q{\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x{v}_{\epsilon }{\partial }_x^2{u}_{\epsilon }dx+2q{\int }_{\mathbb{R}}{v}_{\epsilon }{\partial }_x{u}_{\epsilon }{\partial }_x^2{u}_{\epsilon }dx\hfill \\ \hfill =& -2q{\int }_{\mathbb{R}}{\partial }_x^2{v}_{\epsilon }{u}_{\epsilon }{\partial }_x{u}_{\epsilon }dx-3q{\int }_{\mathbb{R}}{\partial }_x{v}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^{2}dx,\hfill \\ \hfill -2qga{\int }_{\mathbb{R}}{\partial }_x\left(v_{\epsilon }{u}_{\epsilon }\right)u_{\epsilon }^{3}dx=& -2qga{\int }_{\mathbb{R}}{\partial }_x{v}_{\epsilon }{u}_{\epsilon }^{4}dx-2qga{\int }_{\mathbb{R}}{v}_{\epsilon }{\partial }_x{u}_{\epsilon }{u}_{\epsilon }^{3}dx\hfill \\ & -\frac{3qg}{2a}{\int }_{\mathbb{R}}{\partial }_x{v}_{\epsilon }{u}_{\epsilon }^{4}dx.\hfill \end{array}$$

Consequently, by (48),

$$\begin{array}{cc}\hfill \frac{dG\left(t\right)}{dt}+2\epsilon {∥{\partial }_x^3{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2=& \frac{2bg}{a}{\int }_{\mathbb{R}}{P}_{\epsilon }{u}_{\epsilon }^{3}dx+\frac{6g\epsilon }{a}{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\partial }_x{u}_{\epsilon }{\partial }_x^3{u}_{\epsilon }dx\hfill \\ & -3q{\int }_{\mathbb{R}}{\partial }_x{v}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^{2}dx-2q{\int }_{\mathbb{R}}{\partial }_x^2{v}_{\epsilon }{u}_{\epsilon }{\partial }_x{u}_{\epsilon }dx\hfill \\ & -\frac{3qg}{2a}{\int }_{\mathbb{R}}{\partial }_x{v}_{\epsilon }{u}_{\epsilon }^{4}dx.\hfill \end{array}$$

(49)

Due to (26), (32), Lemma 3 and the Young inequality,

$$\begin{array}{cc}\hfill \left|\frac{2bg}{a}\right|{\int }_{\mathbb{R}}{P}_{\epsilon }{u}_{\epsilon }^{3}dx=& {\int }_{\mathbb{R}}\left|2P_{\epsilon }{u}_{\epsilon }\right|\left|\frac{bg{u}_{\epsilon }^2}{a}\right|dx\hfill \\ \hfill \le & 2{\int }_{\mathbb{R}}{P}_{\epsilon }^2{u}_{\epsilon }^{2}dx+\frac{b^2{g}^2}{2a^2}{\int }_{\mathbb{R}}{u}_{\epsilon }^{4}dx\hfill \\ \hfill \le & 2{∥P_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2{∥u_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ & +\frac{b^2{g}^2}{2a^2}{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2{∥u_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & 2C_0{∥P_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2+C_0{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\hfill \\ \hfill \le & {∥P_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^4+C_0{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2+C_0\hfill \\ \hfill \le & C\left(T\right)\left(1+{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right),\hfill \\ \hfill 3\left|q\right|{\int }_{\mathbb{R}}\left|{\partial }_x{v}_{\epsilon }\right|{\left({\partial }_x{u}_{\epsilon }\right)}^{2}dx\le & 3\left|q\right|{∥{\partial }_x{v}_{\epsilon }(t,·)∥}_{L^{\infty }\left(\mathbb{R}\right)}{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & C_0{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2,\hfill \\ \hfill 2\left|q\right|{\int }_{\mathbb{R}}{\partial }_x^2{v}_{\epsilon }{u}_{\epsilon }{\partial }_x{u}_{\epsilon }dx=& 2{\int }_{\mathbb{R}}|q{\partial }_x^2{v}_{\epsilon }{u}_{\epsilon }\left|{\partial }_x{u}_{\epsilon }\right|dx\hfill \\ \hfill \le & q^2{\int }_{\mathbb{R}}{\left({\partial }_x^2{v}_{\epsilon }\right)}^2{u}_{\epsilon }^{2}dx+{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & q^2{∥{\partial }_x^2{v}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2{∥u_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & C_0{∥{\partial }_x^2{v}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2+{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & C\left(T\right)\left(1+{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right)+{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2,\hfill \\ \hfill \left|\frac{3qg}{2a}\right|{\int }_{\mathbb{R}}\left|{\partial }_x{v}_{\epsilon }\right|u_{\epsilon }^{4}dx\le & \left|\frac{3qg}{2a}\right|{∥{\partial }_x{v}_{\epsilon }(t,·)∥}_{L^{\infty }\left(\mathbb{R}\right)}{\int }_{\mathbb{R}}{u}_{\epsilon }^{4}dx\hfill \\ \hfill \le & C_0{\int }_{\mathbb{R}}{u}_{\epsilon }^{4}dx\le C_0{∥u_{\epsilon }^2∥}_{L^{\infty }((0,T)\times \mathbb{R})}{∥u_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & C_0{∥u_{\epsilon }^2∥}_{L^{\infty }((0,T)\times \mathbb{R})},\hfill \\ \hfill \left|\frac{6g\epsilon }{a}\right|{\int }_{\mathbb{R}}|u_{\epsilon }^2{\partial }_x{u}_{\epsilon }\left|\right|{\partial }_x^3{u}_{\epsilon }|dx=& 2\epsilon {\int }_{\mathbb{R}}\left|\frac{3g}{a}{u}_{\epsilon }^2{\partial }_x{u}_{\epsilon }\right|\left|{\partial }_x^3{u}_{\epsilon }\right|dx\hfill \\ \hfill \le & \frac{9g^2\epsilon }{a^2}{\int }_{\mathbb{R}}{u}_{\epsilon }^4{\left({\partial }_x{u}_{\epsilon }\right)}^{2}dx+\epsilon {∥{\partial }_x^3{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2.\hfill \end{array}$$

Consequently, by (49),

$$\begin{array}{cc}\hfill \frac{dG\left(t\right)}{dt}+\epsilon {∥{\partial }_x^3{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\le & C_0{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+\frac{9g^2\epsilon }{a^2}{\int }_{\mathbb{R}}{u}_{\epsilon }^4{\left({\partial }_x{u}_{\epsilon }\right)}^{2}dx\hfill \\ & +C\left(T\right)\left(1+{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right).\hfill \end{array}$$

(50)

Lemma 2.6 of [6] says that

$${\int }_{\mathbb{R}}{u}_{\epsilon }^4{\left({\partial }_x{u}_{\epsilon }\right)}^{2}dx\le 4{∥u_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^4{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2.$$

(51)

Hence, by Lemma 3, we have that

$$\begin{array}{cc}\hfill \frac{9g^2\epsilon }{a^2}{\int }_{\mathbb{R}}{u}_{\epsilon }^4{\left({\partial }_x{u}_{\epsilon }\right)}^{2}dx\le & \frac{36g^2\epsilon }{a^2}{∥u_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^4{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & C_0\epsilon {∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2.\hfill \end{array}$$

Therefore, by (50),

$$\begin{array}{cc}\hfill .\frac{dG\left(t\right)}{dt}+\epsilon {∥{\partial }_x^3{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\le & C_0{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+C_0\epsilon {∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill & +C\left(T\right)\left(1+{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right).\hfill \end{array}$$

(52)

Observe that, by (47) and Lemma 3,

$$\begin{array}{cc}\hfill C_0{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2=& C_{0}G\left(t\right)-\frac{g{C}_0}{2a}{\int }_{\mathbb{R}}{u}_{\epsilon }^{4}dx\hfill \\ \hfill \le & C_{0}G\left(t\right)+\left|\frac{g{C}_0}{2a}\right|{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2{∥u_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & C_{0}G\left(t\right)+C_0{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2.\hfill \end{array}$$

(53)

It follows from (52) and (53) that

$$\begin{array}{cc}\hfill \frac{dG\left(t\right)}{dt}+\epsilon {∥{\partial }_x^3{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\le & C_{0}G\left(t\right)+C_0\epsilon {∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ & +C\left(T\right)\left(1+{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right).\hfill \end{array}$$

The Gronwall Lemma, (17), (47) and Lemma 3 that

$$\begin{array}{c}{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+\frac{g}{2a}{\int }_{\mathbb{R}}{u}_{\epsilon }^{4}dx+\epsilon e^{C_{0}t}{\int }_0^t{e}^{-C_{0}s}{∥{\partial }_x^3{u}_{\epsilon }(s,·)∥}_{L^2\left(\mathbb{R}\right)}^{2}ds\hfill \\ \le C_0{e}^{C_{0}t}+C\left(T\right)\left(1+{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right)e^{C_{0}t}{\int }_0^t{e}^{-C_{0}s}ds\hfill \\ +C_0\epsilon e^{C_{0}t}{\int }_0^t{e}^{-C_{0}s}{∥{\partial }_x^2{u}_{\epsilon }(s,·)∥}_{L^2\left(\mathbb{R}\right)}^{2}ds\hfill \\ \le C\left(T\right)\left(1+{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right)+C\left(T\right)\epsilon {\int }_0^t{∥{\partial }_x^2{u}_{\epsilon }(s,·)∥}_{L^2\left(\mathbb{R}\right)}^{2}ds\hfill \\ \le C\left(T\right)\left(1+{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right).\hfill \end{array}$$

Consequently, by Lemma 3,

$$\begin{array}{cc}\hfill {∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2& +2\epsilon e^{C_{0}t}{\int }_0^t{e}^{-C_{0}s}{∥{\partial }_x^3{u}_{\epsilon }(s,·)∥}_{L^2\left(\mathbb{R}\right)}^{2}ds\hfill \\ \hfill \le & C\left(T\right)\left(1+{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right)-\frac{g}{2a}{∥u_{\epsilon }(t,·)∥}_{L^4\left(\mathbb{R}\right)}^4\hfill \\ \hfill \le & C\left(T\right)\left(1+{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right)+\left|\frac{g}{2a}\right|{\int }_{\mathbb{R}}{u}_{\epsilon }^{4}dx\hfill \\ \hfill \le & C\left(T\right)\left(1+{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right)+\left|\frac{g}{2a}\right|{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2{∥u_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & C\left(T\right)\left(1+{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right)+C_0{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\hfill \\ \hfill \le & C\left(T\right)\left(1+{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right).\hfill \end{array}$$

(54)

We prove (41). Thanks to (54), Lemma 3 and the Hölder inequality,

$$\begin{array}{cc}\hfill u_{\epsilon }^2(t,x)=& 2{\int }_{-\infty }^x{u}_{\epsilon }{\partial }_x{u}_{\epsilon }dx\le {\int }_{\mathbb{R}}|u_{\epsilon }\left|\right|{\partial }_x{u}_{\epsilon }|dx\hfill \\ \hfill \le & {∥u_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\le C\left(T\right)\sqrt{\left(1+{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right)}.\hfill \end{array}$$

Hence,

$${∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^4-C\left(T\right){∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2-C\left(T\right)\le 0,$$

which gives (41).

Finally, (42) follows from (41) and (54), while (26), (31), (32) and (41) give (43).  □

Arguing as in ([15], Lemmas 2.8 and 2.9), we have the following result.

Lemma 8.

Assume (5). Fix $T>0$. There exists a constant $C\left(T\right)>0$, independent on ε, such that

$${∥{\partial }_x^2{v}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)},{∥{\partial }_x^3{v}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}\le C\left(T\right),$$

(55)

for every $0\le t\le T$.

Lemma 9.

Assume (5). Fix $T>0$. There exists a constant $C\left(T\right)>0$, independent on ε, such that

$${∥{\partial }_x{u}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}\le C\left(T\right),$$

(56)

In particular, we have that

$${∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+2\epsilon e^{C_{0}t}{\int }_0^t{e}^{-C_{0}s}{∥{\partial }_x^4{u}_{\epsilon }(s,·)∥}_{L^2\left(\mathbb{R}\right)}^{2}ds\le C\left(T\right),$$

(57)

for every $0\le t\le T$.

Proof. 

Let $0\le t\le T$. Consider two real constants $D,E$ which will be specified later Multiplying the first equation of (16) by

$$2a^2{\partial }_x^4{u}_{\epsilon }+Dag{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^2+Eag{u}_{\epsilon }^2{\partial }_x^2{u}_{\epsilon },$$

we have that

$$\begin{array}{c}\left(2a^2{\partial }_x^4{u}_{\epsilon }+Dag{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^2+Eag{u}_{\epsilon }^2{\partial }_x^2{u}_{\epsilon }\right){\partial }_t{u}_{\epsilon }\hfill \\ +3g\left(2a^2{\partial }_x^4{u}_{\epsilon }+Dag{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^2+Eag{u}_{\epsilon }^2{\partial }_x^2{u}_{\epsilon }\right)u_{\epsilon }^2{\partial }_x{u}_{\epsilon }\hfill \\ -a\left(2a^2{\partial }_x^4{u}_{\epsilon }+Dag{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^2+Eag{u}_{\epsilon }^2{\partial }_x^2{u}_{\epsilon }\right){\partial }_x^3{u}_{\epsilon }\hfill \\ +q\left(2a^2{\partial }_x^4{u}_{\epsilon }+Dag{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^2+Eag{u}_{\epsilon }^2{\partial }_x^2{u}_{\epsilon }\right)u_{\epsilon }{\partial }_x{v}_{\epsilon }\hfill \\ +q\left(2a^2{\partial }_x^4{u}_{\epsilon }+Dag{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^2+Eag{u}_{\epsilon }^2{\partial }_x^2{u}_{\epsilon }\right)v_{\epsilon }{\partial }_x{u}_{\epsilon }\hfill \\ =b\left(2a^2{\partial }_x^4{u}_{\epsilon }+Dag{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^2+Eag{u}_{\epsilon }^2{\partial }_x^2{u}_{\epsilon }\right)P_{\epsilon }\hfill \\ -\epsilon \left(2a^2{\partial }_x^4{u}_{\epsilon }+Dag{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^2+Eag{u}_{\epsilon }^2{\partial }_x^2{u}_{\epsilon }\right){\partial }_x^4{u}_{\epsilon }.\hfill \end{array}$$

(58)

Observe that

$$\begin{array}{c}{\int }_{\mathbb{R}}\left(2a^2{\partial }_x^4{u}_{\epsilon }+Dag{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^2+Eag{u}_{\epsilon }^2{\partial }_x^2{u}_{\epsilon }\right){\partial }_t{u}_{\epsilon }dx\hfill \\ =a^2\frac{d}{dt}{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+Dag{\int }_{\mathbb{R}}{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^2{\partial }_t{u}_{\epsilon }dx+Eag{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\partial }_x^2{u}_{\epsilon }{\partial }_t{u}_{\epsilon }dx,\hfill \\ 3g{\int }_{\mathbb{R}}\left(2a^2{\partial }_x^4{u}_{\epsilon }+Dag{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^2+Eag{u}_{\epsilon }^2{\partial }_x^2{u}_{\epsilon }\right)u_{\epsilon }^2{\partial }_x{u}_{\epsilon }dx\hfill \\ =-12a^{2}g{\int }_{\mathbb{R}}{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^2{\partial }_x^3{u}_{\epsilon }dx-6a^{2}g{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\partial }_x^2{u}_{\epsilon }{\partial }_x^3{u}_{\epsilon }dx\hfill \\ +\left(3D-6E\right)a{g}^2{\int }_{\mathbb{R}}{u}_{\epsilon }^3{\left({\partial }_x{u}_{\epsilon }\right)}^{3}dx\hfill \\ =30a^{2}g{\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x{u}_{\epsilon }{\left({\partial }_x^2{u}_{\epsilon }\right)}^{2}dx+\left(3D-6E\right)a{g}^2{\int }_{\mathbb{R}}{u}_{\epsilon }^3{\left({\partial }_x{u}_{\epsilon }\right)}^{3}dx,\hfill \\ -a{\int }_{\mathbb{R}}\left(2a^2{\partial }_x^4{u}_{\epsilon }+Dag{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^2+Eag{u}_{\epsilon }^2{\partial }_x^2{u}_{\epsilon }\right){\partial }_x^3{u}_{\epsilon }dx\hfill \\ =-\left(2D+E\right)a^{2}g{\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x{u}_{\epsilon }{\left({\partial }_x^2{u}_{\epsilon }\right)}^{2}dx,\hfill \\ q{\int }_{\mathbb{R}}\left(2a^2{\partial }_x^4{u}_{\epsilon }+Dag{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^2+Eag{u}_{\epsilon }^2{\partial }_x^2{u}_{\epsilon }\right)u_{\epsilon }{\partial }_x{v}_{\epsilon }dx\hfill \\ =-2a^{2}q{\int }_{\mathbb{R}}{\partial }_x{u}_{\epsilon }{\partial }_x{v}_{\epsilon }{\partial }_x^3{u}_{\epsilon }dx-2a^{2}q{\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x^2{v}_{\epsilon }{\partial }_x^3{u}_{\epsilon }dx\hfill \\ +\left(D-3E\right)agq{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\left({\partial }_x{u}_{\epsilon }\right)}^2{\partial }_x{v}_{\epsilon }dx-agqE{\int }_{\mathbb{R}}{u}_{\epsilon }^3{\partial }_x{u}_{\epsilon }{\partial }_x^2{v}_{\epsilon }dx\hfill \\ =2a^{2}q{\int }_{\mathbb{R}}{\partial }_x{v}_{\epsilon }{\left({\partial }_x^2{u}_{\epsilon }\right)}^{2}dx+4a^{2}q{\int }_{\mathbb{R}}{\partial }_x{u}_{\epsilon }{\partial }_x^2{v}_{\epsilon }{\partial }_x^2{u}_{\epsilon }dx\hfill \\ +2a^{2}q{\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x^3{v}_{\epsilon }{\partial }_x^2{u}_{\epsilon }dx+aq\left(D-3E\right){\int }_{\mathbb{R}}{u}_{\epsilon }^2{\left({\partial }_x{u}_{\epsilon }\right)}^2{\partial }_x{v}_{\epsilon }dx,\hfill \\ q{\int }_{\mathbb{R}}\left(2a^2{\partial }_x^4{u}_{\epsilon }+Dag{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^2+Eag{u}_{\epsilon }^2{\partial }_x^2{u}_{\epsilon }\right)v_{\epsilon }{\partial }_x{u}_{\epsilon }dx\hfill \\ =-2a^{2}q{\int }_{\mathbb{R}}{\partial }_x{v}_{\epsilon }{\partial }_x{u}_{\epsilon }{\partial }_x^3{u}_{\epsilon }dx-2a^{2}q{\int }_{\mathbb{R}}{v}_{\epsilon }{\partial }_x^2{u}_{\epsilon }{\partial }_x^3{u}_{\epsilon }dx\hfill \\ +\left(D-E\right)agq{\int }_{\mathbb{R}}{u}_{\epsilon }{v}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^{3}dx-\frac{Eagq}{2}{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\left({\partial }_x{u}_{\epsilon }\right)}^3{\partial }_x^2{v}_{\epsilon }dx\hfill \\ =2a^{2}q{\int }_{\mathbb{R}}{\partial }_x^2{v}_{\epsilon }{\partial }_x{u}_{\epsilon }{\partial }_x^2{u}_{\epsilon }dx+3a^{2}q{\int }_{\mathbb{R}}{\partial }_x{v}_{\epsilon }{\left({\partial }_x^2{u}_{\epsilon }\right)}^{2}dx\hfill \\ +\left(D-E\right)agq{\int }_{\mathbb{R}}{u}_{\epsilon }{v}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^{3}dx-\frac{Eagq}{2}{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\left({\partial }_x{u}_{\epsilon }\right)}^3{\partial }_x^2{v}_{\epsilon }dx,\hfill \\ -\epsilon {\int }_{\mathbb{R}}\left(2a^2{\partial }_x^4{u}_{\epsilon }+Dag{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^2+Eag{u}_{\epsilon }^2{\partial }_x^2{u}_{\epsilon }\right){\partial }_x^4{u}_{\epsilon }dx\hfill \\ =-2a^2\epsilon {∥{\partial }_x^4{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+Dag\epsilon {\int }_{\mathbb{R}}{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^2{\partial }_x^4{u}_{\epsilon }dx\hfill \\ +Eag\epsilon {\int }_{\mathbb{R}}{u}_{\epsilon }^2{\partial }_x^2{u}_{\epsilon }{\partial }_x^4{u}_{\epsilon }dx.\hfill \end{array}$$

Consequently, an integration on $\mathbb{R}$ of (58) gives

$$\begin{array}{c}{a}^2\frac{d}{dt}{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+Dag{\int }_{\mathbb{R}}{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^2{\partial }_t{u}_{\epsilon }dx+Eag{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\partial }_x^2{u}_{\epsilon }{\partial }_t{u}_{\epsilon }dx\hfill \\ +2a^2\epsilon {∥{\partial }_x^4{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ =-a^{2}g\left(30+2D+E\right){\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x{u}_{\epsilon }{\left({\partial }_x^2{u}_{\epsilon }\right)}^{2}dx-\left(3D-6E\right)a{g}^2{\int }_{\mathbb{R}}{u}_{\epsilon }^3{\left({\partial }_x{u}_{\epsilon }\right)}^{3}dx\hfill \\ -5a^{2}q{\int }_{\mathbb{R}}{\partial }_x{v}_{\epsilon }{\left({\partial }_x^2{u}_{\epsilon }\right)}^{2}dx-6a^{2}q{\int }_{\mathbb{R}}{\partial }_x^2{v}_{\epsilon }{\partial }_x{u}_{\epsilon }{\partial }_x^2{u}_{\epsilon }dx\hfill \\ -2a^{2}q{\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x^3{v}_{\epsilon }{\partial }_x^2{u}_{\epsilon }dx-aq\left(D-3E\right){\int }_{\mathbb{R}}{u}_{\epsilon }^2{\left({\partial }_x{u}_{\epsilon }\right)}^2{\partial }_x{v}_{\epsilon }dx\hfill \\ -\left(D-E\right)agq{\int }_{\mathbb{R}}{u}_{\epsilon }{v}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^{3}dx+\frac{Eagq}{2}{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\left({\partial }_x{u}_{\epsilon }\right)}^3{\partial }_x^2{v}_{\epsilon }dx\hfill \\ +2a^{2}b{\int }_{\mathbb{R}}{P}_{\epsilon }{\partial }_x^4{u}_{\epsilon }dx+Dagb{\int }_{\mathbb{R}}{P}_{\epsilon }{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^{2}dx+Eagb{\int }_{\mathbb{R}}{P}_{\epsilon }{u}_{\epsilon }^2{\partial }_x^2{u}_{\epsilon }dx\hfill \\ -Dag\epsilon {\int }_{\mathbb{R}}{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^2{\partial }_x^4{u}_{\epsilon }dx+Eag\epsilon {\int }_{\mathbb{R}}{u}_{\epsilon }^2{\partial }_x^2{u}_{\epsilon }{\partial }_x^4{u}_{\epsilon }dx.\hfill \end{array}$$

(59)

Thanks to the second equation of (16) and (18), we have that

$$\begin{array}{cc}\hfill 2a^{2}b{\int }_{\mathbb{R}}{P}_{\epsilon }{\partial }_x^4{u}_{\epsilon }dx=& -2a^{2}b{\int }_{\mathbb{R}}{\partial }_x{P}_{\epsilon }{\partial }_x^3{u}_{\epsilon }=-2a^{2}b{\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x^3{u}_{\epsilon }dx\hfill \\ \hfill =& 2a^{2}b{\int }_{\mathbb{R}}{\partial }_x{u}_{\epsilon }{\partial }_x^2{u}_{\epsilon }dx=0,\hfill \\ \hfill Eagb{\int }_{\mathbb{R}}{P}_{\epsilon }{u}_{\epsilon }^2{\partial }_x^2{u}_{\epsilon }dx=& -Eagb{\int }_{\mathbb{R}}{\partial }_x{P}_{\epsilon }{u}_{\epsilon }^2{\partial }_x{u}_{\epsilon }dx-2Eagb{\int }_{\mathbb{R}}{P}_{\epsilon }{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^{2}dx\hfill \\ \hfill =& -2Eagb{\int }_{\mathbb{R}}{P}_{\epsilon }{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^{2}dx-Eagb{\int }_{\mathbb{R}}{u}_{\epsilon }^3{\partial }_x{u}_{\epsilon }dx\hfill \\ \hfill =& -2Eagb{\int }_{\mathbb{R}}{P}_{\epsilon }{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^{2}dx.\hfill \end{array}$$

Therefore, by (59),

$$\begin{array}{c}{a}^2\frac{d}{dt}{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+Dag{\int }_{\mathbb{R}}{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^2{\partial }_t{u}_{\epsilon }dx+Eag{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\partial }_x^2{u}_{\epsilon }{\partial }_t{u}_{\epsilon }dx\hfill \\ +2a^2\epsilon {∥{\partial }_x^4{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ =-a^{2}g\left(30+2D+E\right){\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x{u}_{\epsilon }{\left({\partial }_x^2{u}_{\epsilon }\right)}^{2}dx-\left(3D-6E\right)a{g}^2{\int }_{\mathbb{R}}{u}_{\epsilon }^3{\left({\partial }_x{u}_{\epsilon }\right)}^{3}dx\hfill \\ -5a^{2}q{\int }_{\mathbb{R}}{\partial }_x{v}_{\epsilon }{\left({\partial }_x^2{u}_{\epsilon }\right)}^{2}dx-6a^{2}q{\int }_{\mathbb{R}}{\partial }_x^2{v}_{\epsilon }{\partial }_x{u}_{\epsilon }{\partial }_x^2{u}_{\epsilon }dx\hfill \\ -2a^{2}q{\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x^3{v}_{\epsilon }{\partial }_x^2{u}_{\epsilon }dx-aq\left(D-3E\right){\int }_{\mathbb{R}}{u}_{\epsilon }^2{\left({\partial }_x{u}_{\epsilon }\right)}^2{\partial }_x{v}_{\epsilon }dx\hfill \\ -\left(D-E\right)agq{\int }_{\mathbb{R}}{u}_{\epsilon }{v}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^{3}dx+\frac{Eagq}{2}{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\left({\partial }_x{u}_{\epsilon }\right)}^3{\partial }_x^2{v}_{\epsilon }dx\hfill \\ +\left(D-2E\right)agb{\int }_{\mathbb{R}}{P}_{\epsilon }{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^{2}dx+Dag\epsilon {\int }_{\mathbb{R}}{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^2{\partial }_x^4{u}_{\epsilon }dx\hfill \\ +Eag\epsilon {\int }_{\mathbb{R}}{u}_{\epsilon }^2{\partial }_x^2{u}_{\epsilon }{\partial }_x^4{u}_{\epsilon }dx.\hfill \end{array}$$

(60)

Observe that

$$\begin{array}{c}Dag{\int }_{\mathbb{R}}{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^2{\partial }_t{u}_{\epsilon }dx+Eag{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\partial }_x^2{u}_{\epsilon }{\partial }_t{u}_{\epsilon }dx\hfill \\ =\frac{Dag}{2}{\int }_{\mathbb{R}}{\partial }_t\left(u_{\epsilon }^2\right){\left({\partial }_x{u}_{\epsilon }\right)}^{2}dx-Eag{\int }_{\mathbb{R}}{\partial }_x{u}_{\epsilon }{\partial }_x\left(u_{\epsilon }^2{\partial }_t{u}_{\epsilon }\right)dx\hfill \\ =\frac{Dag}{2}{\int }_{\mathbb{R}}{\partial }_t\left(u_{\epsilon }^2\right){\left({\partial }_x{u}_{\epsilon }\right)}^{2}dx-2Eag{\int }_{\mathbb{R}}{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^2{\partial }_t{u}_{\epsilon }dx-Ea{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\partial }_x{u}_{\epsilon }{\partial }_{tx}^2{u}_{\epsilon }dx\hfill \\ ag\left(\frac{D}{2}-E\right){\int }_{\mathbb{R}}{\partial }_t\left(u_{\epsilon }^2\right){\left({\partial }_x{u}_{\epsilon }\right)}^{2}dx-\frac{Eag}{2}{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\partial }_t\left({\left({\partial }_x{u}_{\epsilon }\right)}^2\right)dx.\hfill \end{array}$$

Consequently, by (60),

$$\begin{array}{c}{a}^2\frac{d}{dt}{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+ag\left(\frac{D}{2}-E\right){\int }_{\mathbb{R}}{\partial }_t\left(u_{\epsilon }^2\right){\left({\partial }_x{u}_{\epsilon }\right)}^{2}dx\hfill \\ -\frac{Eag}{2}{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\partial }_t\left({\left({\partial }_x{u}_{\epsilon }\right)}^2\right)dx+2a^2\epsilon {∥{\partial }_x^4{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ =-a^{2}g\left(30+2D+E\right){\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x{u}_{\epsilon }{\left({\partial }_x^2{u}_{\epsilon }\right)}^{2}dx-\left(3D-6E\right)a{g}^2{\int }_{\mathbb{R}}{u}_{\epsilon }^3{\left({\partial }_x{u}_{\epsilon }\right)}^{3}dx\hfill \\ -5a^{2}q{\int }_{\mathbb{R}}{\partial }_x{v}_{\epsilon }{\left({\partial }_x^2{u}_{\epsilon }\right)}^{2}dx-6a^{2}q{\int }_{\mathbb{R}}{\partial }_x^2{v}_{\epsilon }{\partial }_x{u}_{\epsilon }{\partial }_x^2{u}_{\epsilon }dx\hfill \\ -2a^{2}q{\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x^3{v}_{\epsilon }{\partial }_x^2{u}_{\epsilon }dx-aq\left(D-3E\right){\int }_{\mathbb{R}}{u}_{\epsilon }^2{\left({\partial }_x{u}_{\epsilon }\right)}^2{\partial }_x{v}_{\epsilon }dx\hfill \\ -\left(D-E\right)agq{\int }_{\mathbb{R}}{u}_{\epsilon }{v}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^{3}dx+\frac{Eagq}{2}{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\left({\partial }_x{u}_{\epsilon }\right)}^3{\partial }_x^2{v}_{\epsilon }dx\hfill \\ +\left(D-2E\right)agb{\int }_{\mathbb{R}}{P}_{\epsilon }{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^{2}dx+Dag\epsilon {\int }_{\mathbb{R}}{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^2{\partial }_x^4{u}_{\epsilon }dx\hfill \\ +Eag\epsilon {\int }_{\mathbb{R}}{u}_{\epsilon }^2{\partial }_x^2{u}_{\epsilon }{\partial }_x^4{u}_{\epsilon }dx.\hfill \end{array}$$

(61)

We search $D,E$ such that

$$\frac{D}{2}-E=-\frac{E}{2},30+2D+E=0,$$

that is

$$D=E,30+2D+E=0.$$

(62)

Since $D=E-10$ is the unique solution of (62), it follows from (61) that

$$\begin{array}{c}{a}^2\frac{d}{dt}{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+5ag{\int }_{\mathbb{R}}{\partial }_t\left(u_{\epsilon }^2\right){\left({\partial }_x{u}_{\epsilon }\right)}^{2}dx+5ag{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\partial }_t\left({\left({\partial }_x{u}_{\epsilon }\right)}^2\right)dx\hfill \\ +2a^2\epsilon {∥{\partial }_x^4{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ =-30a{g}^2{\int }_{\mathbb{R}}{u}_{\epsilon }^3{\left({\partial }_x{u}_{\epsilon }\right)}^{3}dx-5a^{2}q{\int }_{\mathbb{R}}{\partial }_x{v}_{\epsilon }{\left({\partial }_x^2{u}_{\epsilon }\right)}^{2}dx-6a^{2}q{\int }_{\mathbb{R}}{\partial }_x^2{v}_{\epsilon }{\partial }_x{u}_{\epsilon }{\partial }_x^2{u}_{\epsilon }dx\hfill \\ -2a^{2}q{\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x^3{v}_{\epsilon }{\partial }_x^2{u}_{\epsilon }dx-20aq{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\left({\partial }_x{u}_{\epsilon }\right)}^2{\partial }_x{v}_{\epsilon }dx-5agq{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\left({\partial }_x{u}_{\epsilon }\right)}^3{\partial }_x^2{v}_{\epsilon }dx\hfill \\ +10agb{\int }_{\mathbb{R}}{P}_{\epsilon }{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^{2}dx-10ag\epsilon {\int }_{\mathbb{R}}{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^2{\partial }_x^4{u}_{\epsilon }dx\hfill \\ -10ag\epsilon {\int }_{\mathbb{R}}{u}_{\epsilon }^2{\partial }_x^2{u}_{\epsilon }{\partial }_x^4{u}_{\epsilon }dx,\hfill \end{array}$$

that is

$$\begin{array}{c}\frac{d}{dt}\left(a^2{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+5ag{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\left({\partial }_x{u}_{\epsilon }\right)}^{2}dx\right)+2a^2\epsilon {∥{\partial }_x^4{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ =-30a{g}^2{\int }_{\mathbb{R}}{u}_{\epsilon }^3{\left({\partial }_x{u}_{\epsilon }\right)}^{3}dx-5a^{2}q{\int }_{\mathbb{R}}{\partial }_x{v}_{\epsilon }{\left({\partial }_x^2{u}_{\epsilon }\right)}^{2}dx\hfill \\ -6a^{2}q{\int }_{\mathbb{R}}{\partial }_x^2{v}_{\epsilon }{\partial }_x{u}_{\epsilon }{\partial }_x^2{u}_{\epsilon }dx-2a^{2}q{\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x^3{v}_{\epsilon }{\partial }_x^2{u}_{\epsilon }dx\hfill \\ -20aq{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\left({\partial }_x{u}_{\epsilon }\right)}^2{\partial }_x{v}_{\epsilon }dx-5agq{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\left({\partial }_x{u}_{\epsilon }\right)}^3{\partial }_x^2{v}_{\epsilon }dx\hfill \\ +10agb{\int }_{\mathbb{R}}{P}_{\epsilon }{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^{2}dx-10ag\epsilon {\int }_{\mathbb{R}}{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^2{\partial }_x^4{u}_{\epsilon }dx\hfill \\ -10ag\epsilon {\int }_{\mathbb{R}}{u}_{\epsilon }^2{\partial }_x^2{u}_{\epsilon }{\partial }_x^4{u}_{\epsilon }dx.\hfill \end{array}$$

(63)

Due to (41), (42), (43), (55), Lemma 3 and the Young inequality,

$$\begin{array}{cc}\hfill |30a{g}^2|{\int }_{\mathbb{R}}|u_{\epsilon }{|}^3{\left|{\partial }_x{u}_{\epsilon }\right|}^{3}dx\le & |30a{g}^2|{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^3{\int }_{\mathbb{R}}{\left|{\partial }_x{u}_{\epsilon }\right|}^{3}dx\hfill \\ \hfill \le & C\left(T\right){\int }_{\mathbb{R}}{\left|{\partial }_x{u}_{\epsilon }\right|}^{3}dx\hfill \\ \hfill \le & C\left(T\right){∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+C\left(T\right){\int }_{\mathbb{R}}{\left({\partial }_x{u}_{\epsilon }\right)}^{4}dx\hfill \\ \hfill \le & C\left(T\right)+C\left(T\right){∥{\partial }_x{u}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & C\left(T\right)\left(1+{∥{\partial }_x{u}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right),\hfill \\ \hfill |5a^{2}q|{\int }_{\mathbb{R}}\left|{\partial }_x{v}_{\epsilon }\right|{\left({\partial }_x^2{u}_{\epsilon }\right)}^{2}dx\le & |5a^{2}q|{∥{\partial }_x{v}_{\epsilon }(t,·)∥}_{L^{\infty }\left(\mathbb{R}\right)}{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & C_0{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2,\hfill \\ \hfill |6a^{2}q|{\int }_{\mathbb{R}}|{\partial }_x^2{v}_{\epsilon }{\partial }_x{u}_{\epsilon }\left|\right|{\partial }_x^2{u}_{\epsilon }|dx\le & 3a^4{q}^2{\int }_{\mathbb{R}}{\left({\partial }_x^2{v}_{\epsilon }\right)}^2{\left({\partial }_x{u}_{\epsilon }\right)}^{2}dx+3{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & 3a^4{q}^2{∥{\partial }_x^2{v}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+3{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & C\left(T\right)+3{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2,\hfill \\ \hfill |2a^{2}q|{\int }_{\mathbb{R}}|u_{\epsilon }{\partial }_x^3{v}_{\epsilon }\left|\right|{\partial }_x^2{u}_{\epsilon }|dx\le & a^4{q}^2{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\left({\partial }_x^3{v}_{\epsilon }\right)}^{2}dx+{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & a^4{q}^2{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2{∥{\partial }_x^3{v}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & C\left(T\right)+{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2,\hfill \\ \hfill \left|20aq\right|{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\left({\partial }_x{u}_{\epsilon }\right)}^2\left|{\partial }_x{v}_{\epsilon }\right|dx\le & \left|20aq\right|{∥{\partial }_x{v}_{\epsilon }(t,·)∥}_{L^{\infty }\left(\mathbb{R}\right)}{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & C\left(T\right){∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\le C\left(T\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \left|5agq\right|{\int }_{\mathbb{R}}{u}_{\epsilon }^2|{\partial }_x{u}_{\epsilon }{|}^3\left|{\partial }_x^2{v}_{\epsilon }\right|dx\le & \left|5agq\right|{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2{∥{\partial }_x^2{v}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}{\int }_{\mathbb{R}}{\left|{\partial }_x{u}_{\epsilon }\right|}^{3}dx\hfill \\ \hfill \le & C\left(T\right){\int }_{\mathbb{R}}{\left|{\partial }_x{u}_{\epsilon }\right|}^{3}dx\hfill \\ \hfill \le & C\left(T\right){∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+C\left(T\right){\int }_{\mathbb{R}}{\left({\partial }_x{u}_{\epsilon }\right)}^{4}dx\hfill \\ \hfill \le & C\left(T\right)+C\left(T\right){∥{\partial }_x{u}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & C\left(T\right)\left(1+{∥{\partial }_x{u}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right),\hfill \\ \hfill \left|10agb\right|{\int }_{\mathbb{R}}\left|P_{\epsilon }{u}_{\epsilon }\right|{\left({\partial }_x{u}_{\epsilon }\right)}^{2}dx\le & \left|10agb\right|{∥P_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & C\left(T\right){∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\le C\left(T\right),\hfill \\ \hfill \left|10ag\right|\epsilon {\int }_{\mathbb{R}}|u_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^2\left|\right|{\partial }_x^4{u}_{\epsilon }|dx=& \epsilon {\int }_{\mathbb{R}}|10g{u}_{\epsilon }{\left({\partial }_x{u}_{\epsilon }\right)}^2\left|\right|a{\partial }_x^4{u}_{\epsilon }|dx\hfill \\ \hfill \le & 50g^2\epsilon {\int }_{\mathbb{R}}{u}_{\epsilon }^2{\left({\partial }_x{u}_{\epsilon }\right)}^{4}dx+\frac{a^2\epsilon }{2}{∥{\partial }_x^4{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & 50g^2\epsilon {∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2{\int }_{\mathbb{R}}{\left({\partial }_x{u}_{\epsilon }\right)}^{4}dx+\frac{a^2\epsilon }{2}{∥{\partial }_x^4{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & C\left(T\right)\epsilon {∥{\partial }_x{u}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ & +\frac{a^2\epsilon }{2}{∥{\partial }_x^4{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2,\hfill \\ \hfill \left|10ag\right|\epsilon {\int }_{\mathbb{R}}|u_{\epsilon }^2{\partial }_x^2{u}_{\epsilon }\left|\right|{\partial }_x^4{u}_{\epsilon }|dx=& \epsilon {\int }_{\mathbb{R}}|10g{u}_{\epsilon }^2{\partial }_x^2{u}_{\epsilon }\left|\right|a{\partial }_x^4{u}_{\epsilon }|dx\hfill \\ \hfill \le & 50g^2\epsilon {\int }_{\mathbb{R}}{u}_{\epsilon }^4{\left({\partial }_x^2{u}_{\epsilon }\right)}^{2}dx+\frac{a^2\epsilon }{2}{∥{\partial }_x^4{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & 50g^2\epsilon {∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^4{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+\frac{a^2\epsilon }{2}{∥{\partial }_x^4{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & C\left(T\right)\epsilon {∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+\frac{a^2\epsilon }{2}{∥{\partial }_x^4{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2.\hfill \end{array}$$

Therefore, defining

$$G_1\left(t\right)=a^2{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+5ag{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\left({\partial }_x{u}_{\epsilon }\right)}^{2}dx,$$

(64)

by (63) and (64), we have

$$\begin{array}{cc}\hfill \frac{d{G}_1\left(t\right)}{dt}+\epsilon {∥{\partial }_x^4{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\le & C_0{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+C\left(T\right)\left(1+{∥{\partial }_x{u}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right)\hfill \\ & +C\left(T\right)\epsilon {∥{\partial }_x{u}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ & +C\left(T\right)\epsilon {∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2.\hfill \end{array}$$

(65)

Observe that by (41), (42) and (64),

$$\begin{array}{cc}\hfill C_0{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2=& \frac{C_0{a}^2}{a^2}{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill =& \frac{C_0}{a^2}{G}_1\left(t\right)-\frac{5C_{0}g}{a}{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\left({\partial }_x{u}_{\epsilon }\right)}^{2}dx\hfill \\ \hfill \le & C_0{G}_1\left(t\right)+\left|\frac{5C_{0}g}{a}\right|{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\left({\partial }_x{u}_{\epsilon }\right)}^{2}dx\hfill \\ \hfill \le & C_0{G}_1\left(t\right)+\left|\frac{5C_{0}g}{a}\right|{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & C_0{G}_1\left(t\right)+C\left(T\right).\hfill \end{array}$$

(66)

It follows from (65) and (66) that

$$\begin{array}{cc}\hfill \frac{d{G}_1\left(t\right)}{dt}+\epsilon {∥{\partial }_x^4{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\le & C_0{G}_1\left(t\right)+C\left(T\right)\left(1+{∥{\partial }_x{u}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right)\hfill \\ & +C\left(T\right)\epsilon {∥{\partial }_x{u}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ & +C\left(T\right)\epsilon {∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2.\hfill \end{array}$$

The Gronwall Lemma, (17) and Lemma 3 give

$$\begin{array}{cc}\hfill {∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2& +5ag{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\left({\partial }_x{u}_{\epsilon }\right)}^{2}dx+2\epsilon e^{C_{0}t}{\int }_0^t{e}^{-C_{0}s}{∥{\partial }_x^4{u}_{\epsilon }(s,·)∥}_{L^2\left(\mathbb{R}\right)}^{2}ds\hfill \\ \hfill \le & C_0{e}^{C_{0}t}+C\left(T\right)\left(1+{∥{\partial }_x{u}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right)e^{C_{0}t}{\int }_0^t{e}^{-C_{0}s}ds\hfill \\ & +C\left(T\right)\epsilon {∥{\partial }_x{u}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2{e}^{C_{0}t}{\int }_0^t{e}^{-C_{0}s}{∥{\partial }_x{u}_{\epsilon }(s,·)∥}_{L^2\left(\mathbb{R}\right)}^{2}ds\hfill \\ & +C\left(T\right))\epsilon e^{C_{0}t}{\int }_0^t{e}^{-C_{0}s}{∥{\partial }_x^2{u}_{\epsilon }(s,·)∥}_{L^2\left(\mathbb{R}\right)}^{2}ds\hfill \\ \hfill \le & C\left(T\right)\left(1+{∥{\partial }_x{u}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right)\hfill \\ & +C\left(T\right)\epsilon {∥{\partial }_x{u}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2{\int }_0^t{∥{\partial }_x{u}_{\epsilon }(s,·)∥}_{L^2\left(\mathbb{R}\right)}^{2}ds\hfill \\ & +C\left(T\right))\epsilon {\int }_0^t{∥{\partial }_x^2{u}_{\epsilon }(s,·)∥}_{L^2\left(\mathbb{R}\right)}^{2}ds\hfill \\ \hfill \le & C\left(T\right)\left(1+{∥{\partial }_x{u}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right).\hfill \end{array}$$

Therefore, thanks to (41) and (42),

$$\begin{array}{cc}\hfill {∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2& +2\epsilon e^{C_{0}t}{\int }_0^t{e}^{-C_{0}s}{∥{\partial }_x^4{u}_{\epsilon }(s,·)∥}_{L^2\left(\mathbb{R}\right)}^{2}ds\hfill \\ \hfill =& C\left(T\right)\left(1+{∥{\partial }_x{u}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right)-5ag{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\left({\partial }_x{u}_{\epsilon }\right)}^{2}dx\hfill \\ \hfill \le & C\left(T\right)\left(1+{∥{\partial }_x{u}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right)+\left|5ag\right|{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\left({\partial }_x{u}_{\epsilon }\right)}^{2}dx\hfill \\ \hfill \le & C\left(T\right)\left(1+{∥{\partial }_x{u}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right)+\left|5ag\right|{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & C\left(T\right)\left(1+{∥{\partial }_x{u}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right).\hfill \end{array}$$

(67)

We prove (56). Due to (42), (67) and the Hölder inequality,

$$\begin{array}{cc}\hfill {\left({\partial }_x{u}_{\epsilon }(t,x)\right)}^2=& 2{\int }_{-\infty }^x{\partial }_x{u}_{\epsilon }{\partial }_x^2{u}_{\epsilon }dx\le 2{\int }_{\mathbb{R}}|{\partial }_x{u}_{\epsilon }\left|\right|{\partial }_x^2{u}_{\epsilon }|dx\hfill \\ \hfill \le & {∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}\le C\left(T\right)\sqrt{\left(1+{∥{\partial }_x{u}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\right)}.\hfill \end{array}$$

Therefore,

$${∥{\partial }_x{u}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^4-C\left(T\right){∥{\partial }_x{u}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2-C\left(T\right)\le 0,$$

which gives (56).

Finally, (57) follows from (56) and (67).  □

Lemma 10.

Assume (5). Fix $T>0$. There exists a constant $C\left(T\right)>0$, independent on ε, such that

$${∥{\partial }_x^4{v}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\le C\left(T\right),$$

(68)

for every $0\le t\le T$. In particular, we have that

$${∥{\partial }_x^3{v}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}\le C\left(T\right),$$

(69)

Proof. 

Let $0\le t\le T$. Differentiating the third equation of (16) twice with respect to x, we have

$$\alpha {\partial }_x^4{v}_{\epsilon }=2\kappa {\left({\partial }_x{u}_{\epsilon }\right)}^2+2\kappa u_{\epsilon }{\partial }_x^2{u}_{\epsilon }-\beta {\partial }_x^3{v}_{\epsilon }-\gamma {\partial }_x^2{v}_{\epsilon }.$$

(70)

Since

$$u_{\epsilon }(t,\pm \infty )={\partial }_x{u}_{\epsilon }(t,\pm \infty )={\partial }_x^2{u}_{\epsilon }(t,\pm \infty )=0,$$

(71)

it follows from (24) and (55) that

$${\partial }_x^4{v}_{\epsilon }(t,\pm \infty )=0.$$

(72)

Multiplying (70) by $2\alpha {\partial }_x^4{v}_{\epsilon }$, an integration on $\mathbb{R}$ gives

$$\begin{array}{cc}\hfill 2{\alpha }^2{∥{\partial }_x^4{v}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2=& 2\kappa \alpha {\int }_{\mathbb{R}}{\left({\partial }_x{u}_{\epsilon }\right)}^2{\partial }_x^4{v}_{\epsilon }dx+2\kappa \alpha {\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x^2{u}_{\epsilon }{\partial }_x^4{v}_{\epsilon }dx\hfill \\ & -2\beta \alpha {\int }_{\mathbb{R}}{\partial }_x^3{v}_{\epsilon }{\partial }_x^4{v}_{\epsilon }dx-2\gamma \alpha {\int }_{\mathbb{R}}{\partial }_x^2{v}_{\epsilon }{\partial }_x^4{v}_{\epsilon }dx.\hfill \end{array}$$

(73)

Observe that, thanks to (24), (55) and (72),

$$\begin{array}{cc}\hfill -2\beta \alpha {\int }_{\mathbb{R}}{\partial }_x^3{v}_{\epsilon }{\partial }_x^4{v}_{\epsilon }dx=& 0,\hfill \\ \hfill -2\gamma \alpha {\int }_{\mathbb{R}}{\partial }_x^2{v}_{\epsilon }{\partial }_x^4{v}_{\epsilon }dx=& 2\gamma \alpha {∥{\partial }_x^3{v}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2.\hfill \end{array}$$

Therefore, by (55) and (73),

$$\begin{array}{cc}\hfill 2{\alpha }^2{∥{\partial }_x^4{v}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\le & 2\left|\kappa \alpha \right|{\int }_{\mathbb{R}}{\left({\partial }_x{u}_{\epsilon }\right)}^2|{\partial }_x^4{v}_{\epsilon }|dx+2|\kappa \alpha |{\int }_{\mathbb{R}}|u_{\epsilon }{\partial }_x^2{u}_{\epsilon }\left|\right|{\partial }_x^4{v}_{\epsilon }|dx\hfill \\ & +2\left|\gamma \alpha \right|{∥{\partial }_x^3{v}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & 2\left|\kappa \alpha \right|{\int }_{\mathbb{R}}{\left({\partial }_x{u}_{\epsilon }\right)}^2|{\partial }_x^4{v}_{\epsilon }|dx+2|\kappa \alpha |{\int }_{\mathbb{R}}|u_{\epsilon }{\partial }_x^2{u}_{\epsilon }\left|\right|{\partial }_x^4{v}_{\epsilon }|dx+C\left(T\right).\hfill \end{array}$$

(74)

Due to (41), (42), (56), (57) and the Young inequality,

$$\begin{array}{cc}\hfill 2\left|\kappa \alpha \right|{\int }_{\mathbb{R}}{\left({\partial }_x{u}_{\epsilon }\right)}^2\left|{\partial }_x^4{v}_{\epsilon }\right|dx\le & {\kappa }^2{\int }_{\mathbb{R}}{\left({\partial }_x{u}_{\epsilon }\right)}^{4}dx+{\alpha }^2{∥{\partial }_x^4{v}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & {\kappa }^2{∥{\partial }_x{u}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+{\alpha }^2{∥{\partial }_x^4{v}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & C\left(T\right)+{\alpha }^2{∥{\partial }_x^4{v}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2,\hfill \\ \hfill 2\left|\kappa \alpha \right|{\int }_{\mathbb{R}}|u_{\epsilon }{\partial }_x^2{u}_{\epsilon }\left|\right|{\partial }_x^4{v}_{\epsilon }|=& {\int }_{\mathbb{R}}|2\kappa u_{\epsilon }{\partial }_x^2{u}_{\epsilon }\left|\right|\alpha {\partial }_x^4{v}_{\epsilon }|dx\hfill \\ \hfill \le & 2{\kappa }^2{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\left({\partial }_x^2{u}_{\epsilon }\right)}^{2}dx+\frac{{\alpha }^2}{2}{∥{\partial }_x^4{v}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & 2{\kappa }^2{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+\frac{{\alpha }^2}{2}{∥{\partial }_x^4{v}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & C\left(T\right)+\frac{{\alpha }^2}{2}{∥{\partial }_x^4{v}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2.\hfill \end{array}$$

Consequently, by (74),

$$\frac{{\alpha }^2}{2}{∥{\partial }_x^4{v}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\le C\left(T\right),$$

which gives (68).

Finally, we prove (69). Due to (55), (68) and the Hölder inequality,

$$\begin{array}{cc}\hfill {\left({\partial }_x^3{v}_{\epsilon }(t,x)\right)}^2=& 2{\int }_{-\infty }^x{\partial }_x^3{v}_{\epsilon }{\partial }_x^4{v}_{\epsilon }dx\le 2{\int }_{\mathbb{R}}|{\partial }_x^3{v}_{\epsilon }\left|\right|{\partial }_x^4{v}_{\epsilon }|dx\hfill \\ \hfill \le & {∥{\partial }_x^3{v}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}{∥{\partial }_x^4{v}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}\le C\left(T\right).\hfill \end{array}$$

Hence,

$${∥{\partial }_x^3{v}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\le C\left(T\right),$$

which gives (69).  □

3. Proof of Theorem 1

This section is devoted to the proof of Theorem 1.

We begin by proving the following lemma.

Lemma 11.

Fix $T>0$. Then,

$$the sequence {\left\{u_{\epsilon }\right\}}_{\epsilon >0} is compact in L_{loc}^2((0,\infty )\times \mathbb{R}).$$

(75)

Consequently, there exists a subsequence ${\left\{u_{{\epsilon }_k}\right\}}_{k\in \mathbb{N}}$ of ${\left\{u_{\epsilon }\right\}}_{\epsilon >0}$ and $u\in L_{loc}^2((0,\infty )\times \mathbb{R})$ such that, for each compact subset K of $(0,\infty )\times \mathbb{R})$,

$$\begin{array}{c}{u}_{{\epsilon }_k}\to u in L^2\left(K\right) and a.e.,\hfill \end{array}$$

(76)

$$\begin{array}{c}{v}_{{\epsilon }_k}\rightharpoonup v in H^1((0,T)\times \mathbb{R}),\hfill \end{array}$$

(77)

$$\begin{array}{c}{P}_{{\epsilon }_k}\rightharpoonup P in L^2((0,T)\times \mathbb{R}).\hfill \end{array}$$

(78)

Moreover, $(u,v,P)$ is a solution of (1) satisfying (11) and (12).

Proof. 

We begin by proving (75). To prove (75), we rely on the Aubin–Lions Lemma (see [58,59,60]). We recall that

$$H_{loc}^1\left(\mathbb{R}\right)\hookrightarrow \hookrightarrow L_{loc}^2\left(\mathbb{R}\right)\hookrightarrow H_{loc}^{-1}\left(\mathbb{R}\right),$$

where the first inclusion is compact and the second is continuous. Owing to the Aubin–Lions Lemma [60], to prove (75), it suffices to show that

$$\begin{array}{c}{\left\{u_{\epsilon }\right\}}_{\epsilon >0}\mathrm{is} \mathrm{uniformly} \mathrm{bounded} \mathrm{in} L^2(0,T;H_{loc}^1\left(\mathbb{R}\right)),\hfill \end{array}$$

(79)

$$\begin{array}{c}{\left\{{\partial }_t{u}_{\epsilon }\right\}}_{\epsilon >0}\mathrm{is} \mathrm{uniformly} \mathrm{bounded} \mathrm{in} L^2(0,T;H_{loc}^{-1}\left(\mathbb{R}\right)).\hfill \end{array}$$

(80)

We prove (79). Thanks to (42), (57) and Lemma 3,

$${∥u_{\epsilon }(t,·)∥}_{H^2\left(\mathbb{R}\right)}^2={∥u_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\le C\left(T\right).$$

Therefore,

$${\left\{u_{\epsilon }\right\}}_{\epsilon >0}\mathrm{is} \mathrm{uniformly} \mathrm{bounded} \mathrm{in} L^{\infty }(0,T;H^2\left(\mathbb{R}\right)),$$

which gives (79).

We prove (80). By the first equation of (16),

$${\partial }_t{u}_{\epsilon }={\partial }_x\left(-g{u}_{\epsilon }^3+a{\partial }_x^2{u}_{\epsilon }-q{v}_{\epsilon }{u}_{\epsilon }-\epsilon {\partial }_x^3{u}_{\epsilon }\right)+b{P}_{\epsilon }.$$

(81)

We have that

$${∥u_{\epsilon }∥}_{L^6((0,T)\times \mathbb{R})}\le C\left(T\right).$$

(82)

Indeed, thanks to (41) and Lemma 3,

$$\begin{array}{cc}\hfill g^2{\int }_0^T{\int }_{\mathbb{R}}{u}_{\epsilon }^{6}dtdx\le & g^2{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^4{\int }_0^T{\int }_{\mathbb{R}}{u}_{\epsilon }^{2}dtdx\hfill \\ \hfill \le & C\left(T\right){\int }_0^T{\int }_{\mathbb{R}}{u}_{\epsilon }^{2}dtdx\le C\left(T\right).\hfill \end{array}$$

We prove that

$$q^2{∥v_{\epsilon }{u}_{\epsilon }∥}_{L^2((0,T)\times \mathbb{R})}^2\le C\left(T\right).$$

(83)

Due to Lemma 3,

$$\begin{array}{cc}\hfill q^2{\int }_0^T{\int }_{\mathbb{R}}{v}_{\epsilon }^2{u}_{\epsilon }^{2}dtdx\le & q^2{∥v_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2{\int }_0^T{\int }_{\mathbb{R}}{u}_{\epsilon }^{2}dtdx\hfill \\ \hfill \le & C\left(T\right){\int }_0^T{\int }_{\mathbb{R}}{u}_{\epsilon }^{2}dtdx\le C\left(T\right).\hfill \end{array}$$

Observe that, since $0<\epsilon <1$, thanks to (42) and (57),

$$\epsilon {∥{\partial }_x^3{u}_{\epsilon }∥}_{L^2((0,T)\times \mathbb{R})}^2,{\beta }^2{∥{\partial }_x^2{u}_{\epsilon }∥}_{L^2((0,T)\times \mathbb{R})}^2\le C\left(T\right).$$

(84)

Therefore, by (82), (83) and (84),

$${\left\{{\partial }_x\left(-g{u}_{\epsilon }^3+a{\partial }_x^2{u}_{\epsilon }-q{v}_{\epsilon }{u}_{\epsilon }-\epsilon {\partial }_x^3{u}_{\epsilon }\right)\right\}}_{\epsilon >0}\mathrm{is} \mathrm{bounded} \mathrm{in}{H}^1((0,T)\times \mathbb{R}).$$

(85)

Moreover, by (43), we have that

$$b^2{∥P_{\epsilon }∥}_{L^2((0,T)\times \mathbb{R})}^2\le C\left(T\right).$$

(86)

Equation (80) follows from (85) and (86).

Thanks to the Aubin–Lions Lemma, (75) and (76) hold.

Observe that, (77) follows from Lemma 3, while, by (43), we have (78). Consequently, $(u,v,P)$ solves (1).

Observe again that, thanks to Lemmas 3, 7, 8, 9, (10) and the second equation of (16), we obtain (11).

Finally, we prove (12). Thanks to Lemmas 3 and 7, we have

$$u_{{\epsilon }_k}\rightharpoonup u \mathrm{in}{H}^1((0,T)\times \mathbb{R}).$$

(87)

Therefore, (12) follows from (19) and (87).  □

We are ready for the proof of Theorem 1.

Proof of Theorem 1.

Lemma 11 gives the existence of a solution of (1) satisfying (11) and (12). Let $(u_1,P_1)$ and $(u_2,P_2)$ be two solutions of (1) satisfying (11) and (12), namely

$$\left\{\begin{array}{cc}{\displaystyle {\partial }_t{u}_1+3g{u}_1^2{\partial }_x{u}_1-a{\partial }_x^3{u}_1+q{\partial }_x\left(u_1{v}_1\right)=b{P}_1,}\hfill & t>0,x\in \mathbb{R},\hfill \\ {\displaystyle {\partial }_x{P}_1=u_1,}\hfill & t>0,x\in \mathbb{R},\hfill \\ {\displaystyle \alpha {\partial }_x^2{v}_1+\beta {\partial }_x{v}_1+\gamma v_1=\kappa u_1^2,}\hfill & t>0,x\in \mathbb{R},\hfill \\ {\displaystyle P_1(t,-\infty )=0,}\hfill & t>0,\hfill \\ {\displaystyle u_1(0,x)=u_{1,0}\left(x\right),}\hfill & x\in \mathbb{R},\hfill \end{array}\right.$$

$$\left\{\begin{array}{cc}{\displaystyle {\partial }_t{u}_2+3g{u}_2^2{\partial }_x{u}_2-a{\partial }_x^3{u}_2+q{\partial }_x\left(u_2{v}_2\right)=b{P}_2,}\hfill & t>0,x\in \mathbb{R},\hfill \\ {\displaystyle {\partial }_x{P}_2=u_2,}\hfill & t>0,x\in \mathbb{R},\hfill \\ {\displaystyle \alpha {\partial }_x^2{v}_2+\beta {\partial }_x{v}_2+\gamma v_2=\kappa u_2^2,}\hfill & t>0,x\in \mathbb{R},\hfill \\ {\displaystyle P_2(t,-\infty )=0,}\hfill & t>0,\hfill \\ {\displaystyle u_2(0,x)=u_{2,0}\left(x\right),}\hfill & x\in \mathbb{R}.\hfill \end{array}\right.$$

Then, the triad $(\omega ,V,\mathsf{\Omega })$ defined by

$$\begin{array}{c}\omega (t,x)=u_1(t,x)-u_2(t,x),V(t,x)=v_1(t,x)-v_2(t,x),\hfill \\ \mathsf{\Omega }(t,x)={\int }_{-\infty }^x\omega (t,y)dy={\int }_{-\infty }^x{u}_1(t,y)dy-{\int }_{-\infty }^x{u}_2(t,y)dy,\hfill \\ \mathsf{\Omega }(0,x)={\int }_{-\infty }^x\omega (0,y)dy={\int }_{-\infty }^x{u}_1(0,y)dy-{\int }_{-\infty }^x{u}_2(0,y)dy.\hfill \end{array}$$

(88)

is solution of the following Cauchy problem:

$$\left\{\begin{array}{c}{\displaystyle {\partial }_t\omega +3g\left(u_1^2{\partial }_x{u}_1-u_2^2{\partial }_x{u}_2\right)}\hfill \\ -a{\partial }_x^3\omega +q{\partial }_x(u_1{v}_1-u_2{v}_2)=b\mathsf{\Omega },\hfill & t>0,x\in \mathbb{R},\hfill \\ {\displaystyle {\partial }_x\mathsf{\Omega }=\omega ,}\hfill & t>0,x\in \mathbb{R},\hfill \\ {\displaystyle \alpha {\partial }_x^{2}V+\beta {\partial }_{x}V+\gamma V=\kappa (u_1^2-u_2^2),}\hfill & t>0,x\in \mathbb{R},\hfill \\ {\displaystyle \mathsf{\Omega }(t,-\infty )=0,}\hfill & t>0,\hfill \\ {\displaystyle \omega (0,x)=u_{1,0}-u_{2,0}\left(x\right),}\hfill & x\in \mathbb{R}.\hfill \end{array}\right.$$

(89)

Arguing as in ([15], Theorem 1.1), we have that

$$\begin{array}{cc}\hfill {∥V(t,·)∥}_{H^2\left(\mathbb{R}\right)}^2\le & C\left(T\right){∥\omega (t,·)∥}_{L^2\left(\mathbb{R}\right)}^2,\hfill \end{array}$$

(90)

$$\begin{array}{cc}\hfill {∥V(t,·)∥}_{L^{\infty }\left(\mathbb{R}\right)}^2\le & C\left(T\right){∥\omega (t,·)∥}_{L^2\left(\mathbb{R}\right)}^2,\hfill \end{array}$$

(91)

$$\begin{array}{cc}\hfill {∥{\partial }_{x}V(t,·)∥}_{L^{\infty }\left(\mathbb{R}\right)}^2\le & C\left(T\right){∥\omega (t,·)∥}_{L^2\left(\mathbb{R}\right)}^2.\hfill \end{array}$$

(92)

Moreover, by (12) and (88),

$$\mathsf{\Omega }(t,\infty )={\int }_{\mathbb{R}}\omega (t,x)dx={\int }_{\mathbb{R}}{u}_1(t,y)dy-{\int }_{\mathbb{R}}{u}_2(t,x)dx=0.$$

(93)

Observe that, by (88)

$$\begin{array}{cc}\hfill 3g\left(u_1^2{\partial }_x{u}_1-u_2^2{\partial }_x{u}_2\right)=& 3g\left(u_1^2{\partial }_x{u}_1-u_2^2{\partial }_x{u}_1+u_2^2{\partial }_x{u}_1-u_2^2{\partial }_x{u}_2\right)\hfill \\ \hfill =& 3g\left({\partial }_x{u}_1\left(u_1^2-u_2^2\right)+u_2^2{\partial }_x\omega \right)\hfill \\ \hfill =& 3g\left({\partial }_x{u}_1\left(u_1+u_2\right)\omega +u_2^2{\partial }_x\omega \right).\hfill \end{array}$$

(94)

Moreover, arguing as in ([15], Theorem 1.1),

$$q{\partial }_x(u_1{v}_1-u_2{v}_2)=q{\partial }_x(u_1{v}_1-u_2{v}_1+u_2{v}_1-u_2{v}_2)=q{\partial }_x\left(v_1\omega \right)+q{\partial }_x\left(u_{2}V\right).$$

(95)

Therefore, thanks to (94) and (95), the first equation of (89) is equivalent to the following one:

$${\partial }_t\omega =b\mathsf{\Omega }-3g{\partial }_x{u}_1\left(u_1+u_2\right)\omega -3g{u}_2^2{\partial }_x\omega +a{\partial }_x^3\omega -q{\partial }_x\left(v_1\omega \right)-q{\partial }_x\left(u_{2}V\right).$$

(96)

Multiplying (96) by $2\omega$, an integration on $\mathbb{R}$ gives

$$\begin{array}{cc}\hfill \frac{d}{dt}{∥\omega (t,·)∥}_{L^2\left(\mathbb{R}\right)}^2=& 2b{\int }_{\mathbb{R}}\mathsf{\Omega }{\partial }_x\omega dx-6g{\int }_{\mathbb{R}}{\partial }_x{u}_1\left(u_1+u_2\right){\omega }^{2}dx-2a{\int }_{\mathbb{R}}\omega {\partial }_x^3\omega dx\hfill \\ & -6g{\int }_{\mathbb{R}}{u}_2^2\omega {\partial }_x\omega dx-2q{\int }_{\mathbb{R}}{\partial }_x\left(v_1\omega \right)\omega dx-2q{\int }_{\mathbb{R}}{\partial }_x\left(u_{2}V\right)\omega dx.\hfill \end{array}$$

(97)

Observe that, by (88) and (93),

$$\begin{array}{cc}\hfill 2b{\int }_{\mathbb{R}}\mathsf{\Omega }{\partial }_x\omega dx=& 2b{\int }_{\mathbb{R}}\mathsf{\Omega }{\partial }_x\mathsf{\Omega }dx=b{\mathsf{\Omega }}^2(t,\infty )=0,\hfill \\ \hfill -6g{\int }_{\mathbb{R}}{u}_2^2\omega {\partial }_x\omega dx=& 6g{\int }_{\mathbb{R}}{u}_2{\partial }_x{u}_2{\omega }^{2}dx,\hfill \\ \hfill -2a{\int }_{\mathbb{R}}\omega {\partial }_x^3\omega dx=& 2a{\int }_{\mathbb{R}}{\partial }_x\omega {\partial }_x^2\omega =0,\hfill \\ \hfill -2q{\int }_{\mathbb{R}}{\partial }_x\left(v_1\omega \right)\omega dx=& 2q{\int }_{\mathbb{R}}{v}_1\omega {\partial }_x\omega dx=-q{\int }_{\mathbb{R}}{\partial }_x{v}_1{\omega }^{2}dx,\hfill \\ \hfill -2q{\int }_{\mathbb{R}}{\partial }_x\left(u_{2}V\right)\omega dx=& -2q{\int }_{\mathbb{R}}{\partial }_x{u}_{2}V\omega dx-2q{\int }_{\mathbb{R}}{u}_2{\partial }_{x}V\omega dx.\hfill \end{array}$$

(98)

It follows from (97) and (98) that

$$\begin{array}{cc}\hfill \frac{d}{dt}{∥\omega (t,·)∥}_{L^2\left(\mathbb{R}\right)}^2=& -6g{\int }_{\mathbb{R}}{\partial }_x{u}_1\left(u_1+u_2\right){\omega }^{2}dx+6g{\int }_{\mathbb{R}}{u}_2{\partial }_x{u}_2{\omega }^{2}dx\hfill \\ & -q{\int }_{\mathbb{R}}{\partial }_x{v}_1{\omega }^{2}dx-2q{\int }_{\mathbb{R}}{\partial }_x{u}_{2}V\omega dx-2q{\int }_{\mathbb{R}}{u}_2{\partial }_{x}V\omega dx.\hfill \end{array}$$

(99)

Since (11) holds, we have that

$$\begin{array}{cc}\hfill {∥u_1∥}_{L^{\infty }((0,T)\times \mathbb{R})},{∥u_2∥}_{L^{\infty }((0,T)\times \mathbb{R})}\le & C\left(T\right),\hfill \\ \hfill {∥{\partial }_x{u}_1∥}_{L^{\infty }((0,T)\times \mathbb{R})},{∥{\partial }_x{u}_2∥}_{L^{\infty }((0,T)\times \mathbb{R})}\le & C\left(T\right),\hfill \\ \hfill {∥{\partial }_x{v}_1∥}_{L^{\infty }((0,T)\times \mathbb{R})},{∥u_2(t,·)∥}_{L^2\left(\mathbb{R}\right)},{∥{\partial }_x{u}_2(t,·)∥}_{L^2\left(\mathbb{R}\right)}\le & C\left(T\right),\hfill \end{array}$$

(100)

for evert $0\le t\le T$. Consequently, by (91), (100) and the Hölder inequality,

$$\begin{array}{cc}\hfill \left|6g\right|{\int }_{\mathbb{R}}|{\partial }_x{u}_1\left|\right|u_1+u_2|{\omega }^{2}dx\le & \left|6g\right|{∥{\partial }_x{u}_1∥}_{L^{\infty }((0,T)\times \mathbb{R})}{\int }_{\mathbb{R}}|u_1+u_2|{\omega }^{2}dx\hfill \\ \hfill \le & C\left(T\right)\left({∥u_1∥}_{L^{\infty }((0,T)\times \mathbb{R})}+{∥u_2∥}_{L^{\infty }((0,T)\times \mathbb{R})}\right){∥\omega (t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & C\left(T\right){∥\omega (t,·)∥}_{L^2\left(\mathbb{R}\right)}^2,\hfill \\ \hfill \left|6g\right|{\int }_{\mathbb{R}}|u_2\left|\right|{\partial }_x{u}_2|{\omega }^{2}dx\le & \left|6g\right|{∥u_1∥}_{L^{\infty }((0,T)\times \mathbb{R})}{\int }_{\mathbb{R}}\left|{\partial }_x{u}_2\right|{\omega }^{2}dx\hfill \\ \hfill \le & C\left(T\right){∥{\partial }_x{u}_2∥}_{L^{\infty }((0,T)\times \mathbb{R})}{∥\omega (t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & C\left(T\right){∥\omega (t,·)∥}_{L^2\left(\mathbb{R}\right)}^2,\hfill \\ \hfill \left|q\right|{\int }_{\mathbb{R}}\left|{\partial }_x{v}_1\right|{\omega }^{2}dx\le & \left|q\right|{∥{\partial }_x{v}_1∥}_{L^{\infty }((0,T)\times \mathbb{R})}{∥\omega (t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & C\left(T\right){∥\omega (t,·)∥}_{L^2\left(\mathbb{R}\right)}^2,\hfill \\ \hfill \left|2q\right|{\int }_{\mathbb{R}}|{\partial }_x{u}_2\left|\right|V\left|\right|\omega |dx\le & \left|2q\right|{∥V(t,·)∥}_{L^{\infty }\left(\mathbb{R}\right)}{\int }_{\mathbb{R}}|{\partial }_x{u}_2\left|\right|\omega |dx\hfill \\ \hfill \le & C\left(T\right){∥{\partial }_x{u}_2(t,·)∥}_{L^2\left(\mathbb{R}\right)}{∥\omega (t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & C\left(T\right){∥\omega (t,·)∥}_{L^2\left(\mathbb{R}\right)}^2,\hfill \\ \hfill \left|2q\right|{\int }_{\mathbb{R}}|u_2\left|\right|{\partial }_{x}V\left|\right|\omega |dx\le & \left|2q\right|{∥{\partial }_{x}V(t,·)∥}_{L^{\infty }\left(\mathbb{R}\right)}{\int }_{\mathbb{R}}|u_2\left|\right|\left|\omega \right|dx\hfill \\ \hfill \le & C\left(T\right){∥u_2(t,·)∥}_{L^2\left(\mathbb{R}\right)}{∥\omega (t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill \le & C\left(T\right){∥\omega (t,·)∥}_{L^2\left(\mathbb{R}\right)}^2.\hfill \end{array}$$

It follows from (99) that

$$\frac{d}{dt}{∥\omega (t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\le C\left(T\right){∥\omega (t,·)∥}_{L^2\left(\mathbb{R}\right)}^2.$$

(101)

The Gronwall Lemma and (89) give

$${∥\omega (t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\le e^{C\left(T\right)t}{∥\omega (0,x)∥}_{L^2\left(\mathbb{R}\right)}^2.$$

(102)

Since (11) holds, by (88), arguing as in Lemma 5, $\mathsf{\Omega }(t,·)$ is integrable at $\pm \infty$. Moreover, thanks to (93) and Lemma 5, we have that

$${\int }_{\mathbb{R}}\mathsf{\Omega }(t,x)dx=0.$$

(103)

Consider the following function:

$${\mathsf{\Omega }}_1(t,x)={\int }_{-\infty }^x\mathsf{\Omega }(t,y)dy,$$

(104)

since, by the second equation of (89),

$${\partial }_t\mathsf{\Omega }=\frac{d}{dt}{\int }_{-\infty }^x\omega (t,y)dy={\int }_{-\infty }^x{\partial }_t\omega (t,y)dy,$$

(105)

integrating the first equation of (89) on $(-\infty ,x)$, by (104) and (105), we have that

$${\partial }_t\mathsf{\Omega }=b{\mathsf{\Omega }}_1-g\left(u_1^3-u_2^3\right)+a{\partial }_x^2\omega -q\left(u_1{v}_1-u_2{v}_2\right).$$

(106)

Observe that, by (88),

$$\begin{array}{cc}\hfill u_1^3-u_2^3=& \left(u_1^2+u_2^2+u_1{u}_2\right)\omega ,\hfill \\ \hfill u_1{v}_1-u_2{v}_2=& v_1\omega +u_{2}V.\hfill \end{array}$$

Consequently, by (106),

$${\partial }_t\mathsf{\Omega }=b{\mathsf{\Omega }}_1-g\left(u_1^2+u_2^2+u_1{u}_2\right)\omega +a{\partial }_x^2\omega -q{v}_1\omega -q{u}_{2}V.$$

(107)

It follows from (88), (93), (103) and (104) that

$$\begin{array}{cc}\hfill 2b{\int }_{\mathbb{R}}{\mathsf{\Omega }}_1\mathsf{\Omega }dx=& 2b{\int }_{\mathbb{R}}{\mathsf{\Omega }}_1{\partial }_x{\mathsf{\Omega }}_{1}dx=b{\mathsf{\Omega }}_1^2(t,\infty )=b{\left({\int }_{\mathbb{R}}\mathsf{\Omega }(t,x)dx\right)}^2=0,\hfill \\ \hfill 2a{\int }_{\mathbb{R}}\mathsf{\Omega }{\partial }_x^2\omega dx=& -2a{\int }_{\mathbb{R}}{\partial }_x\mathsf{\Omega }{\partial }_x\omega dx=-2a{\int }_{\mathbb{R}}\omega {\partial }_x\omega dx=0.\hfill \end{array}$$

Therefore, multiplying (107) by $2\mathsf{\Omega }$, an integration on $\mathbb{R}$ gives

$$\begin{array}{cc}\hfill \frac{d}{dt}{∥\mathsf{\Omega }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2=& -2g{\int }_{\mathbb{R}}\left(u_1^2+u_2^2+u_1{u}_2\right)\omega \mathsf{\Omega }dx\hfill \\ & -2q{\int }_{\mathbb{R}}{v}_1\omega \mathsf{\Omega }dx-2q{\int }_{\mathbb{R}}{u}_{2}V\mathsf{\Omega }dx.\hfill \end{array}$$

(108)

Due to (91), (100) and the Young inequality,

$$\begin{array}{c}\left|2g\right|{\int }_{\mathbb{R}}\left|u_1^2+u_2^2+u_1{u}_2\right|\left|\omega \right|\left|\right|\mathsf{\Omega }|dx\hfill \\ \le g^2{\int }_{\mathbb{R}}{\left(u_1^2+u_2^2+u_1{u}_2\right)}^2{\omega }^{2}dx+{∥\mathsf{\Omega }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \le C\left(T\right){∥\omega (t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+{∥\mathsf{\Omega }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2,\hfill \\ \left|2q\right|{\int }_{\mathbb{R}}|v_1\omega \left|\right|\mathsf{\Omega }|dx\hfill \\ \le q^2{\int }_{\mathbb{R}}{v}_1^2{\omega }^{2}dx+{∥\mathsf{\Omega }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \le q^2{∥v_1∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2{∥\omega (t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+{∥\mathsf{\Omega }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \le C\left(T\right){∥\omega (t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+{∥\mathsf{\Omega }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2,\hfill \\ \left|2q\right|{\int }_{\mathbb{R}}\left|u_{2}V\right|\mathsf{\Omega }dx\hfill \\ \le q^2{\int }_{\mathbb{R}}{V}^2{u}_2^{2}dx+{∥\mathsf{\Omega }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \le q^2{∥V(t,·)∥}_{L^{\infty }\left(\mathbb{R}\right)}^2{∥u_2(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+{∥\mathsf{\Omega }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \le C\left(T\right){∥V(t,·)∥}_{L^{\infty }\left(\mathbb{R}\right)}^2+{∥\mathsf{\Omega }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \le C\left(T\right){∥\omega (t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+{∥\mathsf{\Omega }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2.\hfill \end{array}$$

Therefore, by (108),

$$\frac{d}{dt}{∥\mathsf{\Omega }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\le C\left(T\right){∥\omega (t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+3{∥\mathsf{\Omega }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2.$$

(109)

Adding (101) and (109), by (88) and the second equation of (89), we have that

$$\frac{d}{dt}{∥\mathsf{\Omega }(t,·)∥}_{H^1\left(\mathbb{R}\right)}^2\le C\left(T\right){∥\omega (t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+3{∥\mathsf{\Omega }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\le C\left(T\right){∥\mathsf{\Omega }(t,·)∥}_{H^1\left(\mathbb{R}\right)}^2$$

and

$${∥\mathsf{\Omega }(t,·)∥}_{H^1\left(\mathbb{R}\right)}^2\le e^{C\left(T\right)t}{∥\mathsf{\Omega }(0,·)∥}_{H^1\left(\mathbb{R}\right)}^2.$$

(110)

Therefore, (13) follows (14), (88), (89), (90), (102) and (110).  □

4. Conclusions

In this paper we studied the Cauchy problem for the Spectrum Pulse equation. It is a third order nonlocal nonlinear evolutive equation related to the dynamics of the electrical field of linearly polarized continuum spectrum pulses in optical waveguides. Our existence analysis is based on on passing to the limit in a fourth order perturbation of the equation. If the initial datum belongs to $H^2\left(\mathbb{R}\right)\cap L^1\left(\mathbb{R}\right)$ and has zero mean we use the Aubin–Lions Lemma while if it belongs to $H^3\left(\mathbb{R}\right)\cap L^1\left(\mathbb{R}\right)$ and has zero mean we use the Sobolev Immersion Theorem. Finally, we directly prove a stability estimate that implies the uniqueness of the solution.