Well-posedness of the classical solution for the Kuramto–Sivashinsky equation with anisotropy effects

Abstract

The Kuramto–Sivashinsky equation with anisotropy effects models the spinodal decomposition of phase separating systems in an external field, the spatiotemporal evolution of the morphology of steps on crystal surfaces and the growth of thermodynamically unstable crystal surfaces with strongly anisotropic surface tension. Written in terms of the step slope, it can be represented in a form similar to a convective Cahn–Hilliard equation. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem, associated with this equation.

1 Introduction

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

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu+\alpha \partial _{x}^2u+\beta ^2\partial _{x}^4u -\gamma ^2(\partial _x u)^2\partial _{x}^2u +\tau \partial _x u\partial _{x}^2u\\ \quad +\kappa (\partial _x u)^4 +q(\partial _x u)^2 +\delta \partial _x u\partial _{x}^3u=0, &{} \quad t>0, \quad x\in {\mathbb {R}},\\ u(0,x)=u_0(x), &{}\quad x\in {\mathbb {R}}, \end{array}\right. } \end{aligned}$$

(1.1)

with $\alpha ,\,\beta ,\,\gamma ,\,\tau ,\,\kappa ,\,q,\,\delta \in {\mathbb {R}}$, $\beta ,\,\gamma \ne 0$, such that

$$\begin{aligned} \delta ^2<4\beta ^2\gamma ^2. \end{aligned}$$

(1.2)

On the initial datum, we assume

$$\begin{aligned} u_0\in H^{\ell }({\mathbb {R}}), \quad \ell \in \{2,3,4\}. \end{aligned}$$

(1.3)

Observe that, using the variable (see [24, 57])

$$\begin{aligned} v=\partial _x u, \end{aligned}$$

(1.4)

Equation (1.1) is equivalent to the following one:

$$\begin{aligned} \partial _tv +\alpha \partial _{x}^2v +\beta ^2\partial _{x}^4u -\frac{\gamma ^2}{3}\partial _{x}^2\left( v^3\right) +\frac{\tau }{2}\partial _{x}^2\left( v^2\right) +\kappa \partial _x v^4 +q\partial _x v^2 +\delta \partial _x \left( v\partial _{x}^2v\right) =0, \end{aligned}$$

(1.5)

which is known as the convective Cahn–Hilliard equation (see [24, 33]).

From a physical point of view, (1.1) and (1.5) model the spinodal decomposition of phase separating systems in an external field [19, 42, 64], the spatiotemporal evolution of the morphology of steps on crystal surfaces [24, 33, 52], and the growth of thermodynamically unstable crystal surfaces with strongly anisotropic surface tension [25, 26, 28, 46].

In the case of a growing crystal surface with strongly anisotropic surface tension, the function u represents is the surface slope, while the constants $\kappa$ and q are the growth driving forces proportional to the difference between the bulk chemical potentials of the solid and fluid phases. They were also obtained by Watson [61] as a small-slope approximation of the crystal growth model obtained in [17].

Observe that, in [52], the authors deduce (1.1) in the case $\tau =\kappa =\delta =0$, while, in [24], (1.1) is done with $\delta =0$. The general case is considered in [33]. In particular, in [24, 33], the authors show the dependence of the coefficients on the anisotropy of the surface tension and on the velocity of the solidification front. It allows one to assess the effects of these parameters on the evolution of the instability.

Assuming $\kappa =q=\delta =0$, (1.5) reads

$$\begin{aligned} \partial _tv +\alpha \partial _{x}^2v +\beta ^2\partial _{x}^4u -\frac{\gamma ^2}{3}\partial _{x}^2\left( v^3\right) +\frac{\tau }{2}\partial _{x}^2\left( v^2\right) =0, \end{aligned}$$

(1.6)

known as the Cahn–Hilliard equation [8, 9, 50, 51]. It describes the process of spinodal decomposition. In this case, the function u is the concentration of one of the components of an alloy. [51] shows that (1.6) has an exact solution that describes the final stage of the spinodal decomposition, the formation of the interface between two stable state of an alloy with different concentrations.

It also describes the coarsening dynamics of the faceting of thermodynamically unstable surfaces [31, 56]. Moreover, [34] shows that Eq. (1.6) can be an effective tool in technological applications to design nanostructured materials.

From a mathematical point of view, in [2], the existence of some extremely slowly evolving solutions for (1.5) is proven, considering a bounded domain, while, in [6, 22], the problem of a global attractor is studied. Instead, in [27, 65], numerical schemes for (1.5) are analyzed, while, in [60], an approximate analytical solution is studied.

Observe that Eq. (1.5) is has been studied in the multidimensional case in the papers [7, 18, 66] and their references.

Taking $\tau =\kappa =\delta =0$ in (1.5), we have the following equation

$$\begin{aligned} \partial _tv +\alpha \partial _{x}^2v +\beta ^2\partial _{x}^4u -\frac{\gamma ^2}{3}\partial _{x}^2\left( v^3\right) +q\partial _x v^2=0. \end{aligned}$$

(1.7)

(1.7) describes a spinodal decomposition in the presence of an external (e.g., gravitational or electric) field, when the dependence of the mobility factor on the order parameter is important [19, 24, 42, 64].

From a mathematical point of view, the coarsening dynamics for (1.7) has been studied in the limit $0< q\ll 1$ in [19, 26] and analytically in [62].

In [1], a numerical scheme is studied for (1.7), while the existence of the periodic solution are analyzed in [20, 36]. In [42, 47], the existence of exact solutions for (1.7) and its viscous form have been investigated. Moreover, [26] shows that, when $q\rightarrow \infty$, (1.7) reduces to the Kuramoto–Sivashinsky equation (see Eq. (1.8)). Physically, it means that, with the growth of the driving force, there must be a transition from the coarsening dynamics to a chaotic spatiotemporal behavior.

Assuming $\gamma =\tau =\kappa =\delta =0$ in (1.5), we have the following equation:

$$\begin{aligned} \partial _tv+\alpha \partial _{x}^2v +\beta ^2\partial _{x}^4u +q\partial _x v^2 =0, \end{aligned}$$

(1.8)

(1.8) arises in interesting physical situations, for example as a model for long waves on a viscous fluid flowing down an inclined plane [59] and to derive drift waves in a plasma [16]. Equation (1.8) was derived also independently by Kuramoto [37,38,39] as a model for phase turbulence in reaction-diffusion systems and by Sivashinsky [55] as a model for plane flame propagation, describing the combined influence of diffusion and thermal conduction of the gas on the stability of a plane flame front.

Equation (1.8) also describes incipient instabilities in a variety of physical and chemical systems [11, 29, 40]. Moreover, (1.8), which is also known as the Benney–Lin equation [4, 43], was derived by Kuramoto in the study of phase turbulence in the Belousov–Zhabotinsky reaction [44].

The dynamical properties and the existence of exact solutions for (1.8) have been investigated in [21, 32, 35, 48, 49, 63]. In [3, 10, 23], the control problem for (1.8) with periodic boundary conditions, and on a bounded interval are studied, respectively. In [12], the problem of global exponential stabilization of (1.8) with periodic boundary conditions is analyzed. A generalization of optimal control theory for (1.8) was proposed in [30], while in [45] the problem of global boundary control of (1.8) is considered. In [53], the existence of solitonic solutions for (1.8) is proven. In [5, 13, 57], the well-posedness of the Cauchy problem for (1.8) is proven, using the energy space technique, a priori estimates together with an application of the Cauchy–Kovalevskaya and the fixed point methods, respectively. Finally, following [14, 41, 54], in [15], the convergence of the solution of (1.8) to the unique entropy one of the Burgers equation is proven when $\alpha ,\,\beta \rightarrow 0$.

Before stating our main result it is important to comment our assumption (1.2) on the coefficients. That condition guarantees the conservation of the $H^2$ norm of the solution in time, in other words thanks to (1.2) the map $t\mapsto u(t,\cdot )$ never leaves the energy space, that is $H^2$.

We use the following definition of solution.

Definition 1.1

A function $u:[0,\infty )\rightarrow {\mathbb {R}}$ is a solution of (1.1) if

$$\begin{aligned} u\in L^\infty (0,T;H^2({\mathbb {R}})),\quad T>0, \end{aligned}$$

and for every test function with compact support $\varphi \in C^\infty ({\mathbb {R}}^2)$

$$\begin{aligned} \int \limits _0^\infty \int \limits _{\mathbb {R}}&\Big (u\partial _t\varphi -\alpha u\partial _{x}^2\varphi -\beta ^2\varphi \partial _{x}^4\varphi -\frac{\gamma ^2}{3}(\partial _x u)^3\partial _x \varphi +\frac{\tau }{2}(\partial _x u)^2\partial _x \varphi \\&-\kappa (\partial _x u)^4\varphi -q(\partial _x u)^2\varphi +\delta (\partial _{x}^2u)^2+\delta u\partial _{x}^2u\partial _x \varphi \Big )\mathrm{d}t\mathrm{d}x+\int \limits _{\mathbb {R}}u_0(x)\varphi (0,x)\mathrm{d}x=0. \end{aligned}$$

The main result of this paper is the following theorem.

Theorem 1.1

Fix $T>0$. If (1.2) and

$$\begin{aligned} u_0\in H^4({\mathbb {R}}), \end{aligned}$$

(1.9)

hold there exists a unique solution u of (1.1), such that

$$\begin{aligned} u \in H^1((0,T)\times {\mathbb {R}})\cap L^{\infty }(0,T;H^4({\mathbb {R}})). \end{aligned}$$

(1.10)

Moreover, if $u_1$ and $u_2$ are two solutions of (1.1), we have that

$$\begin{aligned} \left\| u_1(t,\cdot )-u_2(t,\cdot ) \right\| _{H^1({\mathbb {R}})}\le e^{C(T)t}\left\| u_{1,0}-u_{2,0} \right\| _{H^1({\mathbb {R}})}, \end{aligned}$$

(1.11)

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

Assuming (1.2) and

$$\begin{aligned} u_0\in H^3({\mathbb {R}}), \quad \delta =0, \end{aligned}$$

(1.12)

there exists a unique solution u of (1.1), such that

$$\begin{aligned} u \in H^1((0,T)\times {\mathbb {R}})\cap L^{\infty }(0,T;H^3({\mathbb {R}})). \end{aligned}$$

(1.13)

Moreover, if $u_1$ and $u_2$ are two solutions of (1.1), we have that

$$\begin{aligned} \left\| u_1(t,\cdot )-u_2(t,\cdot ) \right\| _{L^2({\mathbb {R}})}\le e^{C(T)t}\left\| u_{1,0}-u_{2,0} \right\| _{L^2({\mathbb {R}})}, \end{aligned}$$

(1.14)

Under Assumptions (1.2) and

$$\begin{aligned} u_0\in H^2({\mathbb {R}}), \end{aligned}$$

(1.15)

there exists a solution u of (1.1), such that

$$\begin{aligned} u \in H^1((0,T)\times {\mathbb {R}})\cap L^{\infty }(0,T;H^2({\mathbb {R}})). \end{aligned}$$

(1.16)

The argument of Theorem 1.1 relies on deriving suitable a priori estimates together with an application of the Cauchy–Kovalevskaya Theorem [58]. Moreover, observe that the models studied in [24, 52] correspond to the case $\delta =0$ and satisfy (1.2).

The paper is organized as follows. In Sect. 2, we prove some a priori estimates of (1.1). Those play a key role in the proof of our main result, that is given in Sect. 3.

2 A priori estimates

In this section, we prove some a priori estimates on u. We denote with $C_0$ the constants which depend only on the initial data, and with C(T) the constants which depend also on T.

We begin by proving the following result

Lemma 2.1

Fix $T>0$. There exists a constant $C(T)>0$, such that

$$\begin{aligned} \left\| \partial _x u(t,\cdot ) \right\| _{L^2({\mathbb {R}})}&\le C(T), \end{aligned}$$

(2.1)

$$\begin{aligned} \int \limits _{0}^{t}\left\| \partial _{x}^3u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s&\le C(T), \end{aligned}$$

(2.2)

$$\begin{aligned} \int \limits _{0}^{t}\left\| \partial _x u(s,\cdot )\partial _{x}^2u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s&\le C(T), \end{aligned}$$

(2.3)

$$\begin{aligned} \int \limits _{0}^{t}\left\| \partial _{x}^2u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} \mathrm{d}s&\le C(T), \end{aligned}$$

(2.4)

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

Proof

Let $0\le t\le T$. Multiplying (1.1) by $-2\partial _{x}^2u$, an integration on ${\mathbb {R}}$ gives

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\left\| \partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}&=-2\int \limits _{{\mathbb {R}}}\partial _{x}^2u\partial _tu \mathrm{d}x\\&=2\alpha \left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+2\beta ^2\int \limits _{{\mathbb {R}}}\partial _{x}^2u\partial _{x}^4u \mathrm{d}x -2\gamma ^2\left\| \partial _x u(t,\cdot )\partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} \\&\quad +2\tau \int \limits _{{\mathbb {R}}}\partial _x u(\partial _{x}^2u)^2 \mathrm{d}x+2\kappa \int \limits _{{\mathbb {R}}}(\partial _x u)^4\partial _{x}^2u \mathrm{d}x +q\int \limits _{{\mathbb {R}}}(\partial _x u)^2\partial _{x}^2u \mathrm{d}x\\&\quad +2\delta \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^2u\partial _{x}^3u \mathrm{d}x \\&=2\alpha \left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} -2\beta ^2\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} -2\gamma ^2\left\| \partial _x u(t,\cdot )\partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\quad +2\tau \int \limits _{{\mathbb {R}}}\partial _x u(\partial _{x}^2u)^2 \mathrm{d}x+2\delta \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^2u\partial _{x}^3u \mathrm{d}x. \end{aligned}$$

Therefore, we have that

$$\begin{aligned} \begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left\| \partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+2\beta ^2\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+2\gamma ^2\left\| \partial _x u(t,\cdot )\partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad =2\alpha \left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+2\tau \int \limits _{{\mathbb {R}}}\partial _x u(\partial _{x}^2u)^2 \mathrm{d}x+2\delta \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^2u\partial _{x}^3u \mathrm{d}x. \end{aligned} \end{aligned}$$

(2.5)

Due to the Young inequality,

$$\begin{aligned}&2\vert \delta \vert \int \limits _{{\mathbb {R}}}\vert \partial _x u\partial _{x}^2u\vert \vert \partial _{x}^3u\vert \mathrm{d}x=2\int \limits _{{\mathbb {R}}}\left| \frac{\delta \partial _x u\partial _{x}^2u}{\sqrt{D_1}}\right| \left| \sqrt{D_1}\partial _{x}^3u\right| \mathrm{d}x\\&\qquad \le \frac{\delta ^2}{D_1}\left\| \partial _x u(t,\cdot )\partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +D_1\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}, \end{aligned}$$

where $D_1$ is a positive constant, which will be specified later. It follows from (2.5) that

$$\begin{aligned} \begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left\| \partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+\left( 2\beta ^2-D_1\right) \left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \quad +\left( 2\gamma ^2-\frac{\delta ^2}{D_1}\right) \left\| \partial _x u(t,\cdot )\partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le 2\vert \alpha \vert \left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+2\vert \tau \vert \int \limits _{{\mathbb {R}}}\vert \partial _x u\vert (\partial _{x}^2u)^2 \mathrm{d}x. \end{aligned} \end{aligned}$$

(2.6)

We search $D_1$ such that,

$$\begin{aligned} 2\beta ^2-D_1>0, \qquad 2\gamma ^2-\frac{\delta ^2}{D_1}>0, \end{aligned}$$

that is

$$\begin{aligned} D_1<2\beta ^2, \qquad D_1>\frac{\delta ^2}{2\gamma ^2}. \end{aligned}$$

(2.7)

By (2.7), we have that

$$\begin{aligned} \frac{\delta ^2}{2\gamma ^2}<D_1<2\beta ^2. \end{aligned}$$

(2.8)

Thanks to (1.2), $D_1$ does exist. Therefore, by (1.2), (2.6), (2.7) and (2.8), we have that

$$\begin{aligned} \begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left\| \partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+K_1^2\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\quad \qquad +K^2_2\left\| \partial _x u(t,\cdot )\partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le C_0\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+2\vert \tau \vert \int \limits _{{\mathbb {R}}}\vert \partial _x u\vert (\partial _{x}^2u)^2 \mathrm{d}x. \end{aligned} \end{aligned}$$

(2.9)

where $K_1^2,\,K_2^2$ are two appropriate positive constants. Due to the Young inequality,

$$\begin{aligned} 2\vert \tau \vert \int \limits _{{\mathbb {R}}}\vert \partial _x u\vert (\partial _{x}^2u)^2 \mathrm{d}x&=\int \limits _{{\mathbb {R}}}\left| K_2\partial _x u\partial _{x}^2u\right| \left| \frac{2\tau \partial _{x}^2u}{K_2}\right| \mathrm{d}x\\&\le \frac{K^2_2}{2}\left\| \partial _x u(t,\cdot )\partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} + \frac{2\tau ^2}{K_2^2}\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}. \end{aligned}$$

Consequently, by (2.9),

$$\begin{aligned} \begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left\| \partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+K_1^2\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\quad \qquad +\frac{K^2_2}{2}\left\| \partial _x u(t,\cdot )\partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\le C_0\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}. \end{aligned} \end{aligned}$$

(2.10)

Observe that

$$\begin{aligned} C_0\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}=C_0\int \limits _{{\mathbb {R}}}\partial _{x}^2u\partial _{x}^2u \mathrm{d}x=-C_0\int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^3u \mathrm{d}x. \end{aligned}$$

Therefore, by the Young inequality,

$$\begin{aligned} \begin{aligned} C_0\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}&\le \int \limits _{{\mathbb {R}}}\left| \frac{C_0\partial _x u}{K_1}\right| \left| K_1\partial _{x}^3u \right| \mathrm{d}x\\&\le C_0\left\| \partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} + \frac{K_1^2}{2}\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}. \end{aligned} \end{aligned}$$

(2.11)

Consequently, by (2.10),

$$\begin{aligned} \begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left\| \partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+\frac{K_1^2}{2}\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\quad \qquad +\frac{K^2_2}{2}\left\| \partial _x u(t,\cdot )\partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\le C_0\left\| \partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}. \end{aligned} \end{aligned}$$

(2.12)

Integrating on (0, t), by the Gronwall Lemma and (1.3), we have that

$$\begin{aligned}&\left\| \partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+\frac{K_1^2 e^{C_0t}}{2}\int \limits _{0}^{t}e^{-C_s}\left\| \partial _{x}^3u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s\\&\quad \qquad +\frac{K^2_2e^{C_0t}}{2}\int \limits _{0}^{t}e^{-C_0s}\left\| \partial _x u(s,\cdot )\partial _{x}^2u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s\le C_0e^{C_0t}\le C(T), \end{aligned}$$

which gives (2.1), (2.2), (2.3).

Finally, we prove (2.4). Due to (2.2) and (2.11),

$$\begin{aligned} C_0\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\le C(T)+\frac{K_1^2}{2}\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} \end{aligned}$$

(2.13)

Integrating on (0, t), by (2.2), we have (2.4). $\square$

Lemma 2.2

Fix $T>0$. There exist a constant $C(T)>0$, such that

$$\begin{aligned} \left\| u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}&\le C(T), \end{aligned}$$

(2.14)

$$\begin{aligned} \left\| u(t,\cdot ) \right\| _{L^2({\mathbb {R}})}&\le C(T), \end{aligned}$$

(2.15)

$$\begin{aligned} \int \limits _{0}^{t}\left\| \partial _x u(s,\cdot ) \right\| ^4_{L^4({\mathbb {R}})}\mathrm{d}s&\le C(T), \end{aligned}$$

(2.16)

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

The proof of this lemma is based on the following result.

Lemma 2.3

We have that

$$\begin{aligned} \left\| \partial _x u(t,\cdot ) \right\| ^4_{L^4({\mathbb {R}})}\le 9\int \limits _{{\mathbb {R}}}u^2(\partial _{x}^2u)^2 \mathrm{d}x. \end{aligned}$$

(2.17)

Proof

We begin by observing that

$$\begin{aligned} \int \limits _{{\mathbb {R}}}(\partial _x u)^4 \mathrm{d}x=\int \limits _{{\mathbb {R}}}\partial _x u(\partial _x u)^3 \mathrm{d}x=-3\int \limits _{{\mathbb {R}}}u(\partial _x u)^2\partial _{x}^2u \mathrm{d}x. \end{aligned}$$

(2.18)

By the Young inequality,

$$\begin{aligned} 3\int \limits _{{\mathbb {R}}}\vert u\vert (\partial _x u)^2\vert \partial _{x}^2u\vert \mathrm{d}x\le \frac{1}{2}\int \limits _{{\mathbb {R}}}(\partial _x u)^4 \mathrm{d}x +\frac{9}{2}\int \limits _{{\mathbb {R}}}u^2(\partial _{x}^2u)^2 \mathrm{d}x. \end{aligned}$$

It follows from (2.18) that

$$\begin{aligned} \frac{1}{2}\int \limits _{{\mathbb {R}}}(\partial _x u)^4 \mathrm{d}x\le \frac{9}{2}\int \limits _{{\mathbb {R}}}u^2(\partial _{x}^2u)^2 \mathrm{d}x, \end{aligned}$$

which gives (2.17). $\square$

Proof of Lemma 2.2

Let $0\le t\le T$. Multiplying (1.1) by 2u, an integration on ${\mathbb {R}}$ gives

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\left\| u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}&=2\int \limits _{{\mathbb {R}}}u\partial _tu \mathrm{d}x\\&=-2\int \limits _{{\mathbb {R}}}u\partial _{x}^2u \mathrm{d}x -2\beta ^2\int \limits _{{\mathbb {R}}}u\partial _{x}^4u \mathrm{d}x +2\gamma ^2\int \limits _{{\mathbb {R}}}u(\partial _x u)\partial _{x}^2u \mathrm{d}x\\&\quad -2\tau \int \limits _{{\mathbb {R}}}u\partial _x u\partial _{x}^2u \mathrm{d}x-2\kappa \int \limits _{{\mathbb {R}}}u(\partial _x u)^4 \mathrm{d}x -2q\int \limits _{{\mathbb {R}}}u(\partial _x u)^2 \mathrm{d}x\\&\quad -2\delta \int \limits _{{\mathbb {R}}}u\partial _x u\partial _{x}^3u \mathrm{d}x\\&=2\alpha \left\| \partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +2\beta \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^3u \mathrm{d}x -\frac{2\gamma ^2}{3}\left\| \partial _x u(t,\cdot ) \right\| ^4_{L^4({\mathbb {R}})}\\&\quad -2\tau \int \limits _{{\mathbb {R}}}u\partial _x u\partial _{x}^2u \mathrm{d}x-2\kappa \int \limits _{{\mathbb {R}}}u(\partial _x u)^4 \mathrm{d}x -2q\int \limits _{{\mathbb {R}}}u(\partial _x u)^2 \mathrm{d}x \\&\quad -2\delta \int \limits _{{\mathbb {R}}}u\partial _x u\partial _{x}^3u \mathrm{d}x\\&=2\alpha \left\| \partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}-2\beta ^2\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} -\frac{2\gamma ^2}{3}\left\| \partial _x u(t,\cdot ) \right\| ^4_{L^4({\mathbb {R}})}\\&\quad -2\tau \int \limits _{{\mathbb {R}}}u\partial _x u\partial _{x}^2u \mathrm{d}x-2\kappa \int \limits _{{\mathbb {R}}}u(\partial _x u)^4 \mathrm{d}x -2q\int \limits _{{\mathbb {R}}}u(\partial _x u)^2 \mathrm{d}x \\&\quad -2\delta \int \limits _{{\mathbb {R}}}u\partial _x u\partial _{x}^3u \mathrm{d}x. \end{aligned}$$

Therefore, we have that

$$\begin{aligned} \begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left\| u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+2\beta ^2\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+\frac{2\gamma ^2}{3}\left\| \partial _x u(t,\cdot ) \right\| ^4_{L^4({\mathbb {R}})}\\&\qquad =2\alpha \left\| \partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}-2\tau \int \limits _{{\mathbb {R}}}u\partial _x u\partial _{x}^2u \mathrm{d}x-2\kappa \int \limits _{{\mathbb {R}}}u(\partial _x u)^4 \mathrm{d}x \\&\quad \qquad -2q\int \limits _{{\mathbb {R}}}u(\partial _x u)^2 \mathrm{d}x -2\delta \int \limits _{{\mathbb {R}}}u\partial _x u\partial _{x}^3u \mathrm{d}x. \end{aligned} \end{aligned}$$

(2.19)

Due to (2.2) and the Young inequality,

$$\begin{aligned}&2\tau \int \limits _{{\mathbb {R}}}\vert u\partial _x u\vert \vert \partial _{x}^2u\vert \mathrm{d}x\le \tau ^2\int \limits _{{\mathbb {R}}}u^2(\partial _x u)^2 \mathrm{d}x +\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le \tau ^2\left\| u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le C(T)\left\| u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}+\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert q\vert \int \limits _{{\mathbb {R}}}\vert u\vert (\partial _x u)^2 \mathrm{d}x\le 2\vert q\vert \left\| u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le C(T)\left\| u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\le C(T)\left\| u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}+C(T),\\&2\vert \delta \vert \int \limits _{{\mathbb {R}}}\vert u\partial _x u\vert \vert \partial _{x}^3u\vert \mathrm{d}x\\&\qquad \le \delta ^2\int \limits _{{\mathbb {R}}}u^2(\partial _x u)^2 \mathrm{d}x +\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le \delta ^2\left\| u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} + \left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le C(T)\left\| u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})} + \left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}. \end{aligned}$$

It follows from (2.2) and (2.19) that

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left\| u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+2\beta ^2\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+\frac{2\gamma ^2}{3}\left\| \partial _x u(t,\cdot ) \right\| ^4_{L^4({\mathbb {R}})}\nonumber \\&\qquad \le 2\vert \alpha \vert \left\| \partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +C(T)\left\| u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\nonumber \\&\qquad \quad +2\vert \kappa \vert \int \limits _{{\mathbb {R}}}\vert u\vert (\partial _x u)^4 \mathrm{d}x+C(T)\nonumber \\&\qquad \le \left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +C(T)\left( 1+\left\| u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\right) +2\vert \kappa \vert \int \limits _{{\mathbb {R}}}\vert u\vert (\partial _x u)^4 \mathrm{d}x\nonumber \\&\qquad \quad +C(T)\left\| u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}+\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}. \end{aligned}$$

(2.20)

Thanks to (2.17), we have that

$$\begin{aligned} 2\vert \kappa \vert \int \limits _{{\mathbb {R}}}\vert u\vert (\partial _x u)^4 \mathrm{d}x&\le 2\vert \kappa \vert \left\| u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _x u(t,\cdot ) \right\| ^4_{L^{4}({\mathbb {R}})}\\&\le 18\vert \kappa \vert \left\| u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}u^2(\partial _{x}^2u)^2 \mathrm{d}x\\&\le 18\vert \kappa \vert \left\| u \right\| ^3_{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}. \end{aligned}$$

Consequently, by (2.20),

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left\| u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+2\beta ^2\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+\frac{2\gamma ^2}{3}\left\| \partial _x u(t,\cdot ) \right\| ^4_{L^4({\mathbb {R}})}\nonumber \\&\qquad \le \left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +C(T)\left( 1+\left\| u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\right) \nonumber \\&\qquad \quad +18\vert \kappa \vert \left\| u \right\| ^3_{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\nonumber \\&\qquad \quad +C(T)\left\| u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}+\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}. \end{aligned}$$

(2.21)

Integrating on (0, t), by (1.3), (2.2) and (2.4), we have that

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left\| u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+2\beta ^2\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+\frac{2\gamma ^2}{3}\left\| \partial _x u(t,\cdot ) \right\| ^4_{L^4({\mathbb {R}})}\nonumber \\&\qquad \le C_0+ \int \limits _{0}^{t}\left\| \partial _{x}^3u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s +C(T)\left( 1+\left\| u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\right) t\nonumber \\&\qquad \quad +18\vert \kappa \vert \left\| u \right\| ^3_{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{0}^t\left\| \partial _{x}^2u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s\nonumber \\&\qquad \quad +C(T)\left\| u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}t+\int \limits _{0}^{t}\left\| \partial _{x}^2u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s\nonumber \\&\qquad \le C(T)\left( 1+ \left\| u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}+\left\| u \right\| ^3_{L^{\infty }((0,T)\times {\mathbb {R}})}\right) . \end{aligned}$$

(2.22)

Due to the Young inequality,

$$\begin{aligned} \left\| u \right\| ^3_{L^{\infty }((0,T)\times {\mathbb {R}})}&=\sqrt{D_2}\left\| u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\frac{1}{\sqrt{D_2}}\left\| u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\\&\le \frac{D_2}{2}\left\| u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})} +\frac{1}{2D_2}\left\| u \right\| ^4_{L^{\infty }((0,T)\times {\mathbb {R}})}, \end{aligned}$$

where $D_2$ is a positive constant, which will be specified later. Therefore, by (2.22),

$$\begin{aligned} \begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left\| u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+2\beta ^2\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+\frac{2\gamma ^2}{3}\left\| \partial _x u(t,\cdot ) \right\| ^4_{L^4({\mathbb {R}})}\\&\qquad \le C(T)\left( 1+\left( 1+\frac{D_2}{2}\right) \left\| u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}+\frac{2}{D_2}\left\| u \right\| ^4_{L^{\infty }((0,T)\times {\mathbb {R}})}\right) . \end{aligned} \end{aligned}$$

(2.23)

We prove (2.14). Thanks to (2.2), (2.23) and the Hölder inequality,

$$\begin{aligned} u(t,x)^2=2\int \limits _{-\infty }^{x}u\partial _x u \mathrm{d}y&\le 2\int \limits _{{\mathbb {R}}}\vert u\vert \vert \partial _x u\vert \mathrm{d}x\le 2\left\| u(t,\cdot ) \right\| _{L^2({\mathbb {R}})}\left\| \partial _x u(t,\cdot ) \right\| _{L^2({\mathbb {R}})}\\&\le C(T)\sqrt{\left( 1+\left( 1+\frac{D_2}{2}\right) \left\| u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}+\frac{2}{D_2}\left\| u \right\| ^4_{L^{\infty }((0,T)\times {\mathbb {R}})}\right) }. \end{aligned}$$

Therefore,

$$\begin{aligned} \left( 1-\frac{C(T)}{2D_2}\right) \left\| u \right\| ^4_{L^{\infty }((0,T)\times {\mathbb {R}})}-C(T)\left( 1+\frac{D_2}{2}\right) \left\| u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}-C(T)\le 0. \end{aligned}$$

Taking

$$\begin{aligned} D_2=C(T), \end{aligned}$$

(2.24)

we have that

$$\begin{aligned} \frac{1}{2}\left\| u \right\| ^4_{L^{\infty }((0,T)\times {\mathbb {R}})} -C(T)\left\| u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})} -C(T)\le 0, \end{aligned}$$

which gives (2.14).

Finally, (2.15) follows from (2.14), (2.23) and (2.24). $\square$

Lemma 2.4

Fix $T>0$. There exist a constant $C(T)>0$, such that

$$\begin{aligned} \left\| \partial _x u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}&\le C(T), \end{aligned}$$

(2.25)

$$\begin{aligned} \left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +\beta ^2\int \limits _{0}^{t}\left\| \partial _{x}^4u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s&\le C(T), \end{aligned}$$

(2.26)

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

Proof

Let $0\le t\le T$. Multiplying (1.1) by $2\partial _{x}^4u$, an integration on ${\mathbb {R}}$ gives

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}=2\int \limits _{{\mathbb {R}}}\partial _{x}^4u\partial _tu \mathrm{d}x\\&\qquad =-2\alpha \int \limits _{{\mathbb {R}}}\partial _{x}^2u\partial _{x}^4u \mathrm{d}x -2\beta ^2\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+2\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _x u)^2\partial _{x}^2u\partial _{x}^4u \mathrm{d}x \\&\quad \qquad -2\tau \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^2u\partial _{x}^4u \mathrm{d}x-2\kappa \int \limits _{{\mathbb {R}}}(\partial _x u)^4\partial _{x}^4u \mathrm{d}x -2q\int \limits _{{\mathbb {R}}}(\partial _x u)^2\partial _{x}^4u \mathrm{d}x \\&\quad \qquad -2\delta \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^3u\partial _{x}^4u \mathrm{d}x\\&\qquad =2\alpha \left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} -2\beta ^2\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+2\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _x u)^2\partial _{x}^2u\partial _{x}^4u \mathrm{d}x\\&\quad \qquad -2\tau \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^2u\partial _{x}^4u \mathrm{d}x+8\kappa \int \limits _{{\mathbb {R}}}(\partial _x u)^3\partial _{x}^2u\partial _{x}^3u \mathrm{d}x -2q\int \limits _{{\mathbb {R}}}(\partial _x u)^2\partial _{x}^4u \mathrm{d}x\\&\quad \qquad -2\delta \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^3u\partial _{x}^4u \mathrm{d}x. \end{aligned}$$

Therefore, we have that

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+2\beta ^2\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\nonumber \\&\qquad =2\alpha \left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+2\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _x u)^2\partial _{x}^2u\partial _{x}^4u \mathrm{d}x-2\tau \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^2u\partial _{x}^4u \mathrm{d}x\nonumber \\&\qquad \quad +8\kappa \int \limits _{{\mathbb {R}}}(\partial _x u)^3\partial _{x}^2u\partial _{x}^3u \mathrm{d}x -2q\int \limits _{{\mathbb {R}}}(\partial _x u)^2\partial _{x}^4u \mathrm{d}x -2\delta \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^3u\partial _{x}^4u \mathrm{d}x. \end{aligned}$$

(2.27)

Due to the Young inequality,

$$\begin{aligned}&2\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _x u)^2\vert \partial _{x}^2u\vert \partial _{x}^4u \mathrm{d}x=2\int \limits _{{\mathbb {R}}}\left| \frac{\gamma ^2(\partial _x u)^2\partial _{x}^2u}{\beta \sqrt{D_3}}\right| \left| \beta \sqrt{D_3}\partial _{x}^4u \right| \mathrm{d}x\\&\qquad \le \frac{\gamma ^4}{\beta ^2 D_3}\int \limits _{{\mathbb {R}}}(\partial _x u)^4(\partial _{x}^2u)^2 \mathrm{d}x + \beta ^2 D_3\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le \frac{\gamma ^4}{\beta ^2 D_3}\left\| \partial _x u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _x u(t,\cdot )\partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} + \beta ^2 D_3\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert \tau \vert \int \limits _{{\mathbb {R}}}\vert \partial _x u\partial _{x}^2u\vert \vert \partial _{x}^4u\vert \mathrm{d}x=2\int \limits _{{\mathbb {R}}}\left| \frac{\tau \partial _x u\partial _{x}^2u }{\beta \sqrt{D_3}}\right| \left| \beta \partial _{x}^4u\right| \mathrm{d}x\\&\qquad \le \frac{\tau ^2}{\beta ^2 D_3}\left\| \partial _x u(t,\cdot )\partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +D_3\beta ^2\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&8\vert \kappa \vert \int \limits _{{\mathbb {R}}}\vert \partial _x u\vert ^3\vert \partial _{x}^2u\vert \vert \partial _{x}^3u\vert \mathrm{d}x\le 8\vert \kappa \vert \left\| \partial _x u \right\| ^3_{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u\vert \vert \partial _{x}^3u\vert \mathrm{d}x\\&\qquad \le 4\kappa ^2\left\| \partial _x u \right\| ^3_{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+4\left\| \partial _x u \right\| ^3_{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert q\vert \int \limits _{{\mathbb {R}}}(\partial _x u)^2\vert \partial _{x}^4u\vert \mathrm{d}x=2\int \limits _{{\mathbb {R}}}\left| \frac{q(\partial _x u)^2}{\beta \sqrt{D_3}}\right| \left| \beta \sqrt{D_3}\partial _{x}^4u\right| \mathrm{d}x\\&\qquad \le \frac{q^2}{\beta ^2D_3}\left\| \partial _x u(t,\cdot ) \right\| ^4_{L^4({\mathbb {R}})} + \beta ^2D_3\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert \delta \vert \int \limits _{{\mathbb {R}}}\vert \partial _x u\partial _{x}^3u\vert \vert \partial _{x}^4u\vert \mathrm{d}x=2\int \limits _{{\mathbb {R}}}\left| \frac{\delta \partial _x u\partial _{x}^3u}{\beta \sqrt{D_3}}\right| \left| \beta \sqrt{D_3}\partial _{x}^4u\right| \mathrm{d}x\\&\qquad \le \frac{\delta ^2}{\beta ^2D_3}\int \limits _{{\mathbb {R}}}(\partial _x u)^2(\partial _{x}^3u)^2 \mathrm{d}x +\beta ^2D_3\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le \frac{\delta ^2}{\beta ^2D_3}\left\| \partial _x u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} + \beta ^2D_3\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}, \end{aligned}$$

where $D_3$ is a positive constant, which will be specified later. It follows from (2.27) that

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+ 2\left( 1-2D_3\right) \beta ^2\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le C_0 \left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+ \left( \frac{\tau ^2}{\beta ^2 D_3}+\frac{\gamma ^4}{\beta ^2 D_3}\left\| \partial _x u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\right) \left\| \partial _x u(t,\cdot )\partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \quad +4\kappa ^2\left\| \partial _x u \right\| ^3_{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+\frac{q^2}{\beta ^2D_3}\left\| \partial _x u(t,\cdot ) \right\| ^4_{L^4({\mathbb {R}})}\\&\qquad \quad + \frac{\delta ^2}{\beta ^2D_3}\left\| \partial _x u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+4\left\| \partial _x u \right\| ^3_{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}. \end{aligned}$$

Taking $D_3=\frac{1}{4}$, we obtain that

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+ \beta ^2\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le C_0 \left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+ \left( \frac{4\tau ^2}{\beta ^2}+\frac{4\gamma ^4}{\beta ^2}\right) \left\| \partial _x u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _x u(t,\cdot )\partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \quad +4\kappa ^2\left\| \partial _x u \right\| ^3_{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+\frac{4q^2}{\beta ^2}\left\| \partial _x u(t,\cdot ) \right\| ^4_{L^4({\mathbb {R}})}\\&\qquad \quad + \frac{4\delta ^2}{\beta ^2}\left\| \partial _x u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+4\left\| \partial _x u \right\| ^3_{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}. \end{aligned}$$

Integrating on (0, t), by (1.3), (2.2), (2.4) and (2.15), we have that

$$\begin{aligned}&\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+ \beta ^2\int \limits _{0}^{t}\left\| \partial _{x}^4u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s\nonumber \\&\qquad \le C_0 + C_0 \int \limits _{0}^{t}\left\| \partial _{x}^3u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s+\frac{4q^2}{\beta ^2}\int \limits _{0}^{t}\left\| \partial _x u(s,\cdot ) \right\| ^4_{L^4({\mathbb {R}})}\mathrm{d}s \nonumber \\&\qquad \quad + \left( \frac{4\tau ^2}{\beta ^2}+\frac{4\gamma ^4}{\beta ^2}\left\| \partial _x u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\right) \int \limits _{0}^{t}\left\| \partial _x u(s,\cdot )\partial _{x}^2u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s\nonumber \\&\qquad \quad +4\kappa ^2\left\| \partial _x u \right\| ^3_{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{0}^{t}\left\| \partial _{x}^2u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s \nonumber \\&\qquad \quad + \frac{4\delta ^2}{\beta ^2}\left\| \partial _x u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{0}^{t}\left\| \partial _{x}^3u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s\nonumber \\&\qquad \quad +4\left\| \partial _x u \right\| ^3_{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{0}^{t}\left\| \partial _{x}^3u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s\nonumber \\&\qquad \le C(T)\left( 1+ \left\| \partial _x u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}+ \left\| \partial _x u \right\| ^3_{L^{\infty }((0,T)\times {\mathbb {R}})}\right) . \end{aligned}$$

(2.28)

Due to the Young inequality,

$$\begin{aligned} \left\| \partial _x u \right\| ^3_{L^{\infty }((0,T)\times {\mathbb {R}})}=&\sqrt{D_4}\left\| \partial _x u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\frac{1}{\sqrt{D_3}}\left\| \partial _x u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\\&\le \frac{D_4}{2}\left\| \partial _x u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}+\frac{1}{2D_2}\left\| \partial _x u \right\| ^4_{L^{\infty }((0,T)\times {\mathbb {R}})}, \end{aligned}$$

where $D_4$ is a positive constant, which will be specified later. Therefore, by (2.28),

$$\begin{aligned} \begin{aligned}&\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+ \beta ^2\int \limits _{0}^{t}\left\| \partial _{x}^4u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s\\&\qquad \le C(T)\left( 1+ \left( 1+\frac{D_4}{2}\right) \left\| \partial _x u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}+ \frac{1}{2D_4}\left\| \partial _x u \right\| ^4_{L^{\infty }((0,T)\times {\mathbb {R}})}\right) . \end{aligned} \end{aligned}$$

(2.29)

We prove (2.25). Thanks to (2.2), (2.29) and the Hölder inequality,

$$\begin{aligned} (\partial _x u(t,x))^2&=2\int \limits _{-\infty }^{x}\partial _x u\partial _{x}^2u \mathrm{d}x=2\int \limits _{{\mathbb {R}}}\vert \partial _x u\vert \vert \partial _{x}^2u\vert \mathrm{d}x\le 2\left\| \partial _x u(t,\cdot ) \right\| _{L^2({\mathbb {R}})}\left\| \partial _{x}^2u(t,\cdot ) \right\| _{L^2({\mathbb {R}})}\\&\le C(T)\sqrt{\left( 1+ \left( 1+\frac{D_4}{2}\right) \left\| \partial _x u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}+ \frac{1}{2D_4}\left\| \partial _x u \right\| ^4_{L^{\infty }((0,T)\times {\mathbb {R}})}\right) }. \end{aligned}$$

Therefore,

$$\begin{aligned} \left( 1-\frac{C(T)}{2D_4}\right) \left\| \partial _x u \right\| ^4_{L^{\infty }((0,T)\times {\mathbb {R}})} -C(T)\left( 1+\frac{D_4}{2}\right) \left\| \partial _x u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})} -C(T)\le 0. \end{aligned}$$

Taking

$$\begin{aligned} D_4=C(T), \end{aligned}$$

(2.30)

we have that

$$\begin{aligned} \frac{1}{2}\left\| \partial _x u \right\| ^4_{L^{\infty }((0,T)\times {\mathbb {R}})}-C(T)\left\| \partial _x u \right\| ^4_{L^{\infty }((0,T)\times {\mathbb {R}})}-C(T)\le 0, \end{aligned}$$

which gives (2.25).

Finally, (2.26) follows from (2.25), (2.29) and (2.30). $\square$

Lemma 2.5

Fix $T>0$. There exist a constant $C(T)>0$, such that

$$\begin{aligned} \beta ^2\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+\int \limits _{0}^{t}\left\| \partial _tu(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} \mathrm{d}s\le C(T), \end{aligned}$$

(2.31)

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

Proof

Let $0\le t\le T$. Multiplying (1.1) by $2\partial _tu$, an integration on ${\mathbb {R}}$ gives

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left( \beta ^2\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} -\alpha \left\| \partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\right) \\&\qquad =-2\beta ^2\int \limits _{{\mathbb {R}}}\partial _{x}^4u\partial _tu \mathrm{d}x +2\alpha \int \limits _{{\mathbb {R}}}\partial _{x}^2u\partial _tu \mathrm{d}x\\&\qquad =-2\left\| \partial _tu(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +2\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _x u)^2\partial _{x}^2u\partial _tu \mathrm{d}x-2\tau \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^2u \partial _tu \mathrm{d}x\\&\qquad \quad -2\kappa \int \limits _{{\mathbb {R}}}(\partial _x u)^4\partial _tu \mathrm{d}x-2q\int \limits _{{\mathbb {R}}}(\partial _x u)^2\partial _tu \mathrm{d}x -2\delta \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^3u \partial _tu \mathrm{d}x. \end{aligned}$$

Therefore, we have that

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left( \beta ^2\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} -\alpha \left\| \partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\right) +2\left\| \partial _tu(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\nonumber \\&\qquad = 2\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _x u)^2\partial _{x}^2u\partial _tu \mathrm{d}x-2\tau \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^2u \partial _tu \mathrm{d}x-2\kappa \int \limits _{{\mathbb {R}}}(\partial _x u)^4\partial _tu \mathrm{d}x\nonumber \\&\qquad \quad -2q\int \limits _{{\mathbb {R}}}(\partial _x u)^2\partial _tu \mathrm{d}x -2\delta \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^3u \partial _tu \mathrm{d}x. \end{aligned}$$

(2.32)

Due to (2.2), (2.25), (2.26) and the Young inequality,

$$\begin{aligned}&2\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _x u)^2\vert \partial _{x}^2u\vert \vert \partial _tu\vert \mathrm{d}x\le 2\gamma ^2\left\| \partial _x u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u\vert \vert \partial _tu\vert \mathrm{d}x\\&\qquad \le 2C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u\vert \vert \partial _tu\vert \mathrm{d}x=2\int \limits _{{\mathbb {R}}}\left| \frac{C(T)\partial _{x}^2u}{\sqrt{D_5}}\right| \left| \sqrt{D_5}\partial _tu \right| \mathrm{d}x\\&\qquad \le \frac{C(T)}{D_5}\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+D_5\left\| \partial _tu(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le \frac{C(T)}{D_5}+D_5\left\| \partial _tu(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert \tau \vert \int \limits _{{\mathbb {R}}}\vert \partial _x u\vert \vert \partial _{x}^2u\vert \vert \partial _tu\vert \mathrm{d}x\le 2\vert \tau \vert \left\| \partial _x u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u\vert \vert \partial _tu\vert \mathrm{d}x\\&\qquad \le 2C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u\vert \vert \partial _tu\vert \mathrm{d}x=2\int \limits _{{\mathbb {R}}}\left| \frac{C(T)\partial _{x}^2u}{\sqrt{D_5}}\right| \left| \sqrt{D_5}\partial _tu\right| \mathrm{d}x\\&\qquad \le \frac{C(T)}{D_5}\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +C(T)\left\| \partial _tu(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le \frac{C(T)}{D_5}+ C(T)\left\| \partial _tu(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}, \\&2\vert \kappa \vert \int \limits _{{\mathbb {R}}}(\partial _x u)^4\partial _tu \mathrm{d}x\le 2\vert \kappa \vert \left\| \partial _x u \right\| ^3_{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _x u\vert \vert \partial _tu\vert \mathrm{d}x\\&\qquad \le 2C(T)\int \limits _{{\mathbb {R}}}\vert \partial _x u\vert \vert \partial _tu\vert \mathrm{d}x=2\int \limits _{{\mathbb {R}}}\left| \frac{C(T)\partial _x u}{\sqrt{D_5}}\right| \left| \sqrt{D_5}\partial _tu \right| \mathrm{d}x\\&\qquad \le \frac{C(T)}{D_5}\left\| \partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+ D_5\left\| \partial _tu(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le \frac{C(T)}{D_5} + D_5\left\| \partial _tu(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert q\vert \int \limits _{{\mathbb {R}}}(\partial _x u)^2\vert \partial _tu\vert \mathrm{d}x=2\vert q\vert \left\| \partial _x u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _x u\vert \vert \partial _tu\vert \mathrm{d}x\\&\qquad \le 2C(T)\int \limits _{{\mathbb {R}}}\vert \partial _x u\vert \vert \partial _tu\vert \mathrm{d}x=2\int \limits _{{\mathbb {R}}}\left| \frac{C(T)\partial _x u}{\sqrt{D_5}}\right| \left| \sqrt{D_5}\partial _tu\right| \mathrm{d}x\\&\qquad \le \frac{C(T)}{D_5}\left\| \partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+ D_5\left\| \partial _tu(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le \frac{C(T)}{D_5}+D_5\left\| \partial _tu(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert \delta \vert \int \limits _{{\mathbb {R}}}\vert \partial _x u\vert \vert \partial _{x}^3u\vert \vert \partial _tu\vert \mathrm{d}x=2\vert \delta \vert \left\| \partial _x u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u\vert \vert \partial _tu\vert \mathrm{d}x\\&\qquad \le 2C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u\vert \vert \partial _tu\vert \mathrm{d}x=2\int \limits _{{\mathbb {R}}}\left| \frac{C(T)\partial _{x}^3u}{\sqrt{D_5}}\right| \left| \sqrt{D_5}\partial _tu\right| \mathrm{d}x\\&\qquad \le \frac{C(T)}{D_5}\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+ D_5\left\| \partial _tu(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}, \end{aligned}$$

where $D_5$ is a positive constant, which will be specified later. It follows from (2.32) that

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left( \beta ^2\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} -\alpha \left\| \partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\right) +\left( 2-5D_5\right) \left\| \partial _tu(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le \frac{C(T)}{D_5}+\frac{C(T)}{D_5}\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}. \end{aligned}$$

Taking $D_5=\frac{1}{5}$, we have that

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left( \beta ^2\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} -\alpha \left\| \partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\right) +\left\| \partial _tu(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le C(T)+C(T)\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}. \end{aligned}$$

(1.3), (2.2) and an integration on (0, t) give

$$\begin{aligned}&\beta ^2\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} -\alpha \left\| \partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+\int \limits _{0}^{t}\left\| \partial _tu(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s\\&\qquad \le C_0+C(T)t+C(T)\int \limits _{0}^{t}\left\| \partial _{x}^3u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s\le C(T). \end{aligned}$$

Therefore, by (2.2),

$$\begin{aligned}&\beta ^2\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+\int \limits _{0}^{t}\left\| \partial _tu(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s\\&\qquad \le C(T)+\alpha \left\| \partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\le C(T)+\vert \alpha \vert \left\| \partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\le C(T), \end{aligned}$$

which gives (2.31). $\square$

Lemma 2.6

Fix $T>0$ and assume (1.3), with $\ell \in \{3,4\}$. There exist a constant $C(T)>0$, such that

$$\begin{aligned} \left\| \partial _{x}^2u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\le C(T). \end{aligned}$$

(2.33)

In particular,

$$\begin{aligned} \beta ^2\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} + \int \limits _{0}^{t}\left\| \partial _t\partial _x u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s\le C(T), \end{aligned}$$

(2.34)

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

Proof

Let $0\le t\le T$. Multiplying (1.1) by $-2\partial _t\partial _{x}^2u$, an integration on ${\mathbb {R}}$ gives

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left( \beta ^2\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} -\alpha \left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\right) \\&\qquad =-2\beta ^2\int \limits _{{\mathbb {R}}}\partial _{x}^4u\partial _t\partial _{x}^2u \mathrm{d}x +\alpha \int \limits _{{\mathbb {R}}}\partial _{x}^2u\partial _t\partial _{x}^2u \mathrm{d}x\\&\qquad =2\int \limits _{{\mathbb {R}}}\partial _t\partial _{x}^2u\partial _tu \mathrm{d}x -2\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _x u)^2\partial _{x}^2u\partial _t\partial _{x}^2u \mathrm{d}x +2\tau \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^2u\partial _t\partial _{x}^2u \mathrm{d}x\\&\qquad \quad +2\kappa \int \limits _{{\mathbb {R}}}(\partial _x u)^4\partial _t\partial _{x}^2u \mathrm{d}x +2q\int \limits _{{\mathbb {R}}}(\partial _x u)^2\partial _t\partial _{x}^2u \mathrm{d}x +2\delta \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^3u\partial _t\partial _{x}^2u \mathrm{d}x\\&\qquad =-2\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +4\gamma ^2\int \limits _{{\mathbb {R}}}\partial _x u(\partial _{x}^2u)^2\partial _t\partial _x u \mathrm{d}x +2\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _x u)^2\partial _{x}^3u\partial _t\partial _x u \mathrm{d}x\\&\qquad \quad -2\tau \int \limits _{{\mathbb {R}}}(\partial _{x}^2u)^2\partial _t\partial _x u \mathrm{d}x -2\tau \int \limits _{{\mathbb {R}}}\partial _x \partial _{x}^3u\partial _t\partial _x u \mathrm{d}x-8\kappa \int \limits _{{\mathbb {R}}}(\partial _x u)^3\partial _{x}^2u\partial _t\partial _x u \mathrm{d}x \\&\qquad \quad -4q\int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^2u\partial _t\partial _x u \mathrm{d}x-2\delta \int \limits _{{\mathbb {R}}}\partial _{x}^2u\partial _{x}^3u\partial _t\partial _x u \mathrm{d}x -2\delta \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^4u\partial _t\partial _x u \mathrm{d}x. \end{aligned}$$

Therefore, we have that

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left( \beta ^2\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} -\alpha \left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\right) +2\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\nonumber \\&\qquad =4\gamma ^2\int \limits _{{\mathbb {R}}}\partial _x u(\partial _{x}^2u)^2\partial _t\partial _x u \mathrm{d}x +2\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _x u)^2\partial _{x}^3u\partial _t\partial _x u \mathrm{d}x\\&\qquad \quad -2\tau \int \limits _{{\mathbb {R}}}(\partial _{x}^2u)^2\partial _t\partial _x u \mathrm{d}x -2\tau \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^3u\partial _t\partial _x u \mathrm{d}x\nonumber \\&\qquad \quad -8\kappa \int \limits _{{\mathbb {R}}}(\partial _x u)^3\partial _{x}^2u\partial _t\partial _x u \mathrm{d}x-4q\int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^2u\partial _t\partial _x u \mathrm{d}x\nonumber \\&\qquad \quad -2\delta \int \limits _{{\mathbb {R}}}\partial _{x}^2u\partial _{x}^3u\partial _t\partial _x u \mathrm{d}x-2\delta \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^4u\partial _t\partial _x u \mathrm{d}x.\nonumber \end{aligned}$$

(2.35)

Due to (2.2), (2.25), (2.26) and the Young inequality,

$$\begin{aligned}&4\gamma ^2\int \limits _{{\mathbb {R}}}\vert \partial _x u\vert (\partial _{x}^2u)^2\vert \partial _t\partial _x u\vert \mathrm{d}x\le 4\gamma ^2\left\| \partial _x u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}(\partial _{x}^2u)^2\vert \partial _t\partial _x u \vert \mathrm{d}x\\&\qquad \le 2C(T)\int \limits _{{\mathbb {R}}}(\partial _{x}^2u)^2\vert \partial _t\partial _x u \vert \mathrm{d}x=2\int \limits _{{\mathbb {R}}}\left| \frac{C(T)(\partial _{x}^2u)^2}{\sqrt{D_6}}\right| \left| \sqrt{D_6}\partial _t\partial _x u \mathrm{d}x\right| \mathrm{d}x\\&\qquad \le \frac{C(T)}{D_6}\int \limits _{{\mathbb {R}}}(\partial _{x}^2u)^4 \mathrm{d}x + D_6\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le \frac{C(T)}{D_6}\left\| \partial _{x}^2u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} + D_6\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le \frac{C(T)}{D_6}\left\| \partial _{x}^2u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}+ D_6\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _x u)^2\vert \partial _{x}^3u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\le 2\gamma ^2\left\| \partial _x u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u\vert \partial _t\partial _x u\vert \mathrm{d}x\\&\qquad \le 2C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x =2\int \limits _{{\mathbb {R}}}\left| \frac{C(T)\partial _{x}^3u}{\sqrt{D_6}}\right| \left| \sqrt{D_6}\partial _t\partial _x u \mathrm{d}x\right| \mathrm{d}x\\&\qquad \le \frac{C(T)}{D_6}\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} + D_6\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert \tau \vert \int \limits _{{\mathbb {R}}}(\partial _{x}^2u)^2\vert \partial _t\partial _x u\vert \mathrm{d}x =2\int \limits _{{\mathbb {R}}}\left| \frac{\tau (\partial _{x}^2u)^2 \mathrm{d}x}{\sqrt{D_6}}\right| \left| \sqrt{D_6}\partial _t\partial _x u \right| \mathrm{d}x\\&\qquad \le \frac{\tau ^2}{D_6}\int \limits _{{\mathbb {R}}}(\partial _{x}^2u)^4 \mathrm{d}x +D_6\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le \frac{\tau ^2}{D_6}\left\| \partial _{x}^2u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +D_6\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le \frac{C(T)}{D_6}\left\| \partial _{x}^2u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}+D_6\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert \tau \vert \int \limits _{{\mathbb {R}}}\vert \partial _x u\vert \vert \partial _{x}^3u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x=2\vert \tau \vert \left\| \partial _x u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\\&\qquad \le 2C(T) \int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x=\int \limits _{{\mathbb {R}}}\left| \frac{C(T)\partial _{x}^3u}{\sqrt{D_6}}\right| \left| \sqrt{D_6}\partial _t\partial _x u\right| \mathrm{d}x\\&\qquad \le \frac{C(T)}{D_6}\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +D_6\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&8\vert \kappa \vert \int \limits _{{\mathbb {R}}}\vert \partial _x u\vert ^3\vert \partial _{x}^2u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\le 8\vert \kappa \vert \left\| \partial _x u \right\| ^3_{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\\&\qquad \le 2C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\le 2\int \limits _{{\mathbb {R}}}\left| \frac{C(T)\partial _{x}^2u}{\sqrt{D_6}}\right| \left| \sqrt{D_6}\partial _t\partial _x u\right| \mathrm{d}x\\&\qquad \le \frac{C(T)}{D_6}\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +D_6\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le \frac{C(T)}{D_6}+D_6\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&4\vert q\vert \int \limits _{{\mathbb {R}}}\vert \partial _x u\vert \vert \partial _{x}^2u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\le 4\vert q\vert \left\| \partial _x u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\\&\qquad \le 2C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x=2\int \limits _{{\mathbb {R}}}\left| \frac{C(T)\partial _{x}^2u}{\sqrt{D_6}}\right| \left| \sqrt{D_6}\partial _t\partial _x u\right| \mathrm{d}x\\&\qquad \le \frac{C(T)}{D_6}\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} + D_6\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le \frac{C(T)}{D_6}+ D_6\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert \delta \vert \int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u\partial _{x}^3u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x=2\int \limits _{{\mathbb {R}}}\left| \frac{\delta \partial _{x}^2u\partial _{x}^3u}{\sqrt{D_6}}\right| \left| \sqrt{D_6}\partial _t\partial _x u \right| \mathrm{d}x\\&\qquad \le \frac{\delta ^2}{D_6}\int \limits _{{\mathbb {R}}}(\partial _{x}^2u)^2(\partial _{x}^3u)^2 \mathrm{d}x +D_6\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le \frac{\delta ^2}{D_6}\left\| \partial _{x}^2u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} + D_6\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert \delta \vert \int \limits _{{\mathbb {R}}}\vert \partial _x u\vert \vert \partial _{x}^4u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x=2\vert \delta \vert \left\| \partial _x u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^4u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\\&\qquad \le 2C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^4u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x=2\int \limits _{{\mathbb {R}}}\left| \frac{C(T)\partial _{x}^4u}{\sqrt{D_6}}\right| \left| \sqrt{D_6}\partial _t\partial _x u\right| \mathrm{d}x\\&\qquad \le \frac{C(T)}{D_6}\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +D_6\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}, \end{aligned}$$

where $D_6$ is a positive constant, which will be specified later. it follows from (2.35) that

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left( \beta ^2\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} -\alpha \left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\right) +2\left( 1-4D_6\right) \left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le \frac{C(T)}{D_6}\left( 1+ \left\| \partial _{x}^2u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\right) + \frac{C(T)}{D_6}\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\quad +\frac{C(T)}{D_6}\left( 1+ \left\| \partial _{x}^2u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\right) \left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}. \end{aligned}$$

Taking $D_6=\frac{1}{8}$, we have that

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left( \beta ^2\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} -\alpha \left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\right) +\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le C(T)\left( 1+ \left\| \partial _{x}^2u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\right) +C(T)\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \quad +C(T) \left( 1+ \left\| \partial _{x}^2u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\right) \left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}. \end{aligned}$$

Integrating on (0, t), by (1.3), (2.2) and (2.26), we obtain that

$$\begin{aligned}&\beta ^2\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} -\alpha \left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+\int \limits _{0}^{t}\left\| \partial _t\partial _x u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s\\&\qquad \le C_0 + C(T)\left( 1+ \left\| \partial _{x}^2u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\right) t +C(T)\int \limits _{0}^{t}\left\| \partial _{x}^4u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s\\&\qquad \quad +C(T) \left( 1+ \left\| \partial _{x}^2u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\right) \int \limits _{0}^{t}\left\| \partial _{x}^3u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s\\&\qquad \le C(T)\left( 1+ \left\| \partial _{x}^2u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\right) . \end{aligned}$$

Therefore, by (2.26),

$$\begin{aligned}&\beta ^2\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +\int \limits _{0}^{t}\left\| \partial _t\partial _x u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s\nonumber \\&\qquad \le C(T)\left( 1+ \left\| \partial _{x}^2u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\right) +\vert \alpha \vert \left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\nonumber \\&\qquad \le C(T)\left( 1+ \left\| \partial _{x}^2u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\right) . \end{aligned}$$

(2.36)

We prove (2.33). Thanks to (2.26), (2.36) and the Hölder inequality,

$$\begin{aligned} (\partial _{x}^2u(t,x))^2&=2\int \limits _{-\infty }^{x}\partial _{x}^2u\partial _{x}^3u \mathrm{d}y \le 2\int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u\vert \vert \partial _{x}^3u\vert \mathrm{d}x\le 2\left\| \partial _{x}^2u(t,\cdot ) \right\| _{L^2({\mathbb {R}})}\left\| \partial _{x}^3u(t,\cdot ) \right\| _{L^2({\mathbb {R}})}\\&\qquad \le C(T)\sqrt{\left( 1+ \left\| \partial _{x}^2u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\right) }. \end{aligned}$$

Hence,

$$\begin{aligned} \left\| \partial _{x}^2u \right\| ^4_{L^{\infty }((0,T)\times {\mathbb {R}})}-C(T)\left\| \partial _{x}^2u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}-C(T)\le 0, \end{aligned}$$

which gives (2.33).

Finally, (2.34) follows from (2.33) and (2.36). $\square$

Lemma 2.7

Fix $T>0$ and assume (1.3), with $\ell =4$. There exist a constant $C(T)>0$, such that

$$\begin{aligned} \left\| \partial _{x}^3u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}&\le C(T), \end{aligned}$$

(2.37)

$$\begin{aligned} \left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+\frac{\beta ^2}{2}\int \limits _{0}^{t}\left\| \partial _{x}^6u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s&\end{aligned}$$

(2.38)

$$\begin{aligned} +2\gamma ^2\int \limits _{0}^{t}\left\| \partial _x u(s,\cdot )\partial _{x}^5u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s&\le C(T),\nonumber \\ \int \limits _{0}^{t}\left\| \partial _{x}^5u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}&\le C(T), \end{aligned}$$

(2.39)

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

Proof

Let $0\le t\le T$. Multiplying (1.1) by $2\partial _{x}^8u$, we have that

$$\begin{aligned} \begin{aligned}&2\partial _{x}^8u\partial _tu +2\alpha \partial _{x}^2u\partial _{x}^8u +2\beta ^2\partial _{x}^4u\partial _{x}^8u -2\gamma ^2(\partial _x u)^2\partial _{x}^2u\partial _{x}^8u\\&\qquad +2\tau \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^2u\partial _{x}^8u \mathrm{d}x+2\kappa (\partial _x u)^4\partial _{x}^8u +2q(\partial _x u)^2\partial _{x}^8u \\&\qquad +2\delta \partial _x u\partial _{x}^3u\partial _{x}^8u=0. \end{aligned} \end{aligned}$$

(2.40)

Observe that

$$\begin{aligned} 2\int \limits _{{\mathbb {R}}}\partial _{x}^8u\partial _tu \mathrm{d}x&=-2\int \limits _{{\mathbb {R}}}\partial _{x}^7u\partial _t\partial _x u=2\int \limits _{{\mathbb {R}}}\partial _{x}^6u\partial _t\partial _{x}^2u \mathrm{d}x\nonumber \\&=-2\int \limits _{{\mathbb {R}}}\partial _{x}^5u\partial _t\partial _{x}^3u \mathrm{d}x=\frac{\mathrm{d}}{\mathrm{d}t}\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\nonumber \\ 2\alpha \int \limits _{{\mathbb {R}}}\partial _{x}^2u\partial _{x}^8u \mathrm{d}x=&-2\alpha \int \limits _{{\mathbb {R}}}\partial _{x}^3u\partial _{x}^7u \mathrm{d}x =2\alpha \int \limits _{{\mathbb {R}}}\partial _{x}^4u\partial _{x}^6u \mathrm{d}x\nonumber \\&=-2\alpha \left\| \partial _{x}^5u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\nonumber \\ 2\beta ^2\int \limits _{{\mathbb {R}}}\partial _{x}^4u\partial _{x}^8u \mathrm{d}x&=-2\beta ^2\int \limits _{{\mathbb {R}}}\partial _{x}^5u\partial _{x}^7udx=2\beta ^2\left\| \partial _{x}^6u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\nonumber \\ -2\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _x u)^2\partial _{x}^2u\partial _{x}^8u&=4\gamma ^2\int \limits _{{\mathbb {R}}}\partial _x u(\partial _{x}^2u)^2\partial _{x}^7u \mathrm{d}x +2\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _x u)^2\partial _{x}^3u\partial _{x}^7u \mathrm{d}x\nonumber \\&=-4\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _{x}^3u)^3\partial _{x}^6u \mathrm{d}x -12\gamma ^2\int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^2u\partial _{x}^3u\partial _{x}^6u \mathrm{d}x\nonumber \\&\quad -2\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _x u)^2\partial _{x}^4u\partial _{x}^6u \mathrm{d}x\nonumber \\&=-4\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _{x}^3u)^3\partial _{x}^6u \mathrm{d}x -12\gamma ^2\int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^2u\partial _{x}^3u\partial _{x}^6u \mathrm{d}x\nonumber \\&\quad +4\gamma ^2\int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^2u\partial _{x}^4u\partial _{x}^5u \mathrm{d}x +2\gamma ^2\left\| \partial _x u(t,\cdot )\partial _{x}^5u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\nonumber \\ 2\tau \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^2u\partial _{x}^8u \mathrm{d}x&=-2\tau \int \limits _{{\mathbb {R}}}(\partial _{x}^2u)^2\partial _{x}^7u \mathrm{d}x -2\tau \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^3u\partial _{x}^7u \mathrm{d}x\nonumber \\&=6\tau \int \limits _{{\mathbb {R}}}\partial _{x}^2u\partial _{x}^3u\partial _{x}^6u \mathrm{d}x+2\tau \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^4u\partial _{x}^6u \mathrm{d}x,\nonumber \\ 2\kappa \int \limits _{{\mathbb {R}}}(\partial _x u)^4\partial _{x}^8u \mathrm{d}x&=-8\kappa \int \limits _{{\mathbb {R}}}(\partial _x u)^3\partial _{x}^2u\partial _{x}^7u \mathrm{d}x \nonumber \\&=24\kappa \int \limits _{{\mathbb {R}}}(\partial _x u)^2(\partial _{x}^2u)^2\partial _{x}^6u \mathrm{d}x +8\kappa \int \limits _{{\mathbb {R}}}(\partial _x u)^3\partial _{x}^3u \partial _{x}^6u \mathrm{d}x\nonumber \\ 2q\int \limits _{{\mathbb {R}}}(\partial _x u)^2\partial _{x}^8u \mathrm{d}x&=-4q\int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^2u\partial _{x}^7u \mathrm{d}x\nonumber \\&=4q\int \limits _{{\mathbb {R}}}(\partial _{x}^2u)^2\partial _{x}^6u \mathrm{d}x +4q\int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^3u\partial _{x}^6u \mathrm{d}x,\nonumber \\ 2\delta \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^3u\partial _{x}^8u \mathrm{d}x&=-2\delta \int \limits _{{\mathbb {R}}}\partial _{x}^2u\partial _{x}^3u\partial _{x}^7u \mathrm{d}x-2\delta \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^4u \partial _{x}^7u \mathrm{d}x\nonumber \\&=2\delta \int \limits _{{\mathbb {R}}}(\partial _{x}^3u)^2\partial _{x}^6u \mathrm{d}x +4\delta \int \limits _{{\mathbb {R}}}\partial _{x}^2u\partial _{x}^4u\partial _{x}^6u \mathrm{d}x\nonumber \\&\quad +2\delta \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^5u\partial _{x}^6u \mathrm{d}x\nonumber \\&=-4\delta \int \limits _{{\mathbb {R}}}\partial _{x}^3u\partial _{x}^4u\partial _{x}^5u \mathrm{d}x +4\delta \int \limits _{{\mathbb {R}}}\partial _{x}^2u\partial _{x}^4u\partial _{x}^6u \mathrm{d}x\nonumber \\&\quad -\delta \int \limits _{{\mathbb {R}}}\partial _{x}^2u(\partial _{x}^5u)^2 \mathrm{d}x. \end{aligned}$$

(2.41)

Therefore, thanks to (2.41), an integration of (2.40) on ${\mathbb {R}}$ gives

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+2\beta ^2\left\| \partial _{x}^6u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+2\gamma ^2\left\| \partial _x u(t,\cdot )\partial _{x}^5u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\nonumber \\&\qquad =2\alpha \left\| \partial _{x}^5u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+4\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _{x}^3u)^3\partial _{x}^6u \mathrm{d}x+12\gamma ^2\int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^2u\partial _{x}^3u\partial _{x}^6u \mathrm{d}x\nonumber \\&\qquad \quad -4\gamma ^2\int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^2u\partial _{x}^4u\partial _{x}^5u \mathrm{d}x-6\tau \int \limits _{{\mathbb {R}}}\partial _{x}^2u\partial _{x}^3u\partial _{x}^6u \mathrm{d}x-2\tau \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^4u\partial _{x}^6u \mathrm{d}x\nonumber \\&\qquad \quad -24\kappa \int \limits _{{\mathbb {R}}}(\partial _x u)^2(\partial _{x}^2u)^2\partial _{x}^6u \mathrm{d}x-8\kappa \int \limits _{{\mathbb {R}}}(\partial _x u)^3\partial _{x}^3u \partial _{x}^6u \mathrm{d}x\nonumber \\&\qquad \quad -4q\int \limits _{{\mathbb {R}}}(\partial _{x}^2u)^2\partial _{x}^6u \mathrm{d}x-4q\int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^3u\partial _{x}^6u \mathrm{d}x\nonumber \\&\qquad \quad +4\delta \int \limits _{{\mathbb {R}}}\partial _{x}^3u\partial _{x}^4u\partial _{x}^5u \mathrm{d}x-4\delta \int \limits _{{\mathbb {R}}}\partial _{x}^2u\partial _{x}^4u\partial _{x}^6u+\delta \int \limits _{{\mathbb {R}}}\partial _{x}^2u(\partial _{x}^5u)^2 \mathrm{d}x. \end{aligned}$$

(2.42)

Due to (2.2), (2.25), (2.26), (2.33), (2.34) and the Young inequality,

$$\begin{aligned}&4\gamma ^2\int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u\vert ^3\vert \partial _{x}^6u\vert \mathrm{d}x=4\gamma ^2\left\| \partial _x u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _x u\vert \partial _{x}^6u\vert \mathrm{d}x\\&\qquad \le 2C(T)\int \limits _{{\mathbb {R}}}\vert \partial _x u\vert \vert \partial _{x}^6u\vert \mathrm{d}x=2\int \limits _{{\mathbb {R}}}\left| \frac{C(T)\partial _x u}{\beta \sqrt{D_7}}\right| \left| \beta \sqrt{D_7}\partial _{x}^6u \right| \mathrm{d}x\\&\qquad \le \frac{C(T)}{D_7}\left\| \partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} + \beta ^2D_7\left\| \partial _{x}^6u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le \frac{C(T)}{D_7}+ \beta ^2D_7\left\| \partial _{x}^6u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&12\gamma ^2\int \limits _{{\mathbb {R}}}\vert \partial _x u\vert \vert \partial _{x}^2u\vert \vert \partial _{x}^3u\vert \vert \partial _{x}^6u \vert \mathrm{d}x=12\gamma ^2\left\| \partial _x u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u\vert \vert \partial _{x}^3u\vert \vert \partial _{x}^6u \vert \mathrm{d}x\\&\qquad \le 2C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u\vert \vert \partial _{x}^3u\vert \vert \partial _{x}^6u \vert \mathrm{d}x\le 2C(T)\left\| \partial _{x}^2u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u\vert \vert \partial _{x}^6u \vert \mathrm{d}x\\&\qquad \le 2C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u\vert \vert \partial _{x}^6u \vert \mathrm{d}x=2\int \limits _{{\mathbb {R}}}\left| \frac{C(T)\partial _{x}^3u}{\beta \sqrt{D_7}}\right| \left| \beta \sqrt{D_7}\partial _{x}^6u\right| \mathrm{d}x\\&\qquad \le \frac{C(T)}{D_7}\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +\beta ^2D_7\left\| \partial _{x}^6u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le \frac{C(T)}{D_7} +\beta ^2D_7\left\| \partial _{x}^6u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&4\gamma ^2\int \limits _{{\mathbb {R}}}\vert \partial _x u\vert \vert \partial _{x}^2u\vert \vert \partial _{x}^4u\vert \vert \partial _{x}^5u\vert \mathrm{d}x\le 4\gamma ^2\left\| \partial _x u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u\vert \vert \partial _{x}^4u\vert \vert \partial _{x}^5u\vert \mathrm{d}x\\&\qquad \le 2C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u\vert \vert \partial _{x}^4u\vert \vert \partial _{x}^5u\vert \mathrm{d}x\le 2C(T)\left\| \partial _{x}^2u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^4u\vert \vert \partial _{x}^5u\vert \mathrm{d}x\\&\qquad \le 2C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^4u\vert \vert \partial _{x}^5u\vert \mathrm{d}x\le C(T)\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +C(T)\left\| \partial _{x}^5u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&24\vert \kappa \vert \int \limits _{{\mathbb {R}}}(\partial _x u)^2(\partial _{x}^2u)^2\vert \partial _{x}^6u\vert \mathrm{d}x\le 24\vert \kappa \vert \left\| \partial _x u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}(\partial _{x}^2u)^2\vert \partial _{x}^6u\vert \mathrm{d}x\\&\qquad \le 2C(T)\int \limits _{{\mathbb {R}}}(\partial _{x}^2u)^2\vert \partial _{x}^6u\vert \mathrm{d}x \le 2C(T)\left\| \partial _{x}^2u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u\vert \vert \partial _{x}^6u\vert \mathrm{d}x\\&\qquad \le 2C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u\vert \vert \partial _{x}^6u\vert \mathrm{d}x=2\int \limits _{{\mathbb {R}}}\left| \frac{C(T)\partial _{x}^2u}{\beta \sqrt{D_7}}\right| \left| \beta \sqrt{D_7}\partial _{x}^6u\right| \mathrm{d}x\\&\qquad \le \frac{C(T)}{D_7}\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +\beta ^2D_7\left\| \partial _{x}^6u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le \frac{C(T)}{D_7}+ \beta ^2D_7\left\| \partial _{x}^6u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&6\vert \tau \vert \int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u\vert \vert \partial _{x}^3u\vert \vert \vert \partial _{x}^6u\vert \mathrm{d}x\le 6\vert \tau \vert \left\| \partial _{x}^2u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u\vert \vert \partial _{x}^6u\vert \mathrm{d}x\\&\qquad \le 2C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u\vert \vert \partial _{x}^6u\vert \mathrm{d}x=2\int \limits _{{\mathbb {R}}}\left| \frac{C(T)\partial _{x}^3u}{\beta \sqrt{D_7}}\right| \left| \beta \sqrt{D_7}\partial _{x}^6u \right| \mathrm{d}x\\&\qquad \le \frac{C(T)}{D_7}\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} + \beta ^2D_7\left\| \partial _{x}^6u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le \frac{C(T)}{D_7}+ \beta ^2D_7\left\| \partial _{x}^6u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert \tau \vert \int \limits _{{\mathbb {R}}}\vert \partial _x u\vert \vert \partial _{x}^4u\vert \vert \partial _{x}^6u\vert \mathrm{d}x\le 2\vert \tau \vert \left\| \partial _x u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^4u\vert \vert \partial _{x}^6u\vert \mathrm{d}x\\&\qquad \le 2C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^4u\vert \vert \partial _{x}^6u\vert \mathrm{d}x=2\int \limits _{{\mathbb {R}}}\left| \frac{C(T)\partial _{x}^4u}{\beta \sqrt{D_7}}\right| \left| \beta \sqrt{D_7}\partial _{x}^6u\right| \mathrm{d}x\\&\qquad \le \frac{C(T)}{D_7}\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} + \beta ^2D_7\left\| \partial _{x}^6u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&8\vert \kappa \vert \int \limits _{{\mathbb {R}}}\vert \partial _x u\vert ^3\vert \partial _{x}^3u\vert \vert \partial _{x}^6u\vert \mathrm{d}x\le 8\kappa \left\| \partial _x u \right\| ^3_{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u\vert \vert \partial _{x}^6u\vert \mathrm{d}x\\&\qquad \le 2C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u\vert \vert \partial _{x}^6u\vert \mathrm{d}x=2\int \limits _{{\mathbb {R}}}\left| \frac{C(T)\partial _{x}^3u}{\beta \sqrt{D_7}}\right| \left| \beta \sqrt{D_7}\partial _{x}^6u\right| \mathrm{d}x\\&\qquad \le \frac{C(T)}{D_7}\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} + \beta ^2D_7\left\| \partial _{x}^6u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le \frac{C(T)}{D_7}+ \beta ^2D_7\left\| \partial _{x}^6u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&4\vert q\vert \int \limits _{{\mathbb {R}}}(\partial _{x}^2u)^2\vert \partial _{x}^6u\vert \mathrm{d}x\le 4\vert q\vert \left\| \partial _{x}^2u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u\vert \vert \partial _{x}^6u \vert \mathrm{d}x\\&\qquad \le 2C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u\vert \vert \partial _{x}^6u \vert \mathrm{d}x\le 2\int \limits _{{\mathbb {R}}}\left| \frac{C(T)\partial _{x}^2u}{\beta \sqrt{D_7}}\right| \left| \beta \sqrt{D_7}\partial _{x}^6u\right| \mathrm{d}x\\&\qquad \le \frac{C(T)}{D_7}\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +\beta ^2D_7\left\| \partial _{x}^6u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le \frac{C(T)}{D_7}+\beta ^2D_7\left\| \partial _{x}^6u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&4\vert q\vert \int \limits _{{\mathbb {R}}}\vert \partial _x u\vert \vert \partial _{x}^3u\vert \vert \partial _{x}^6u\vert \mathrm{d}x \le 4\vert q\vert \left\| \partial _x u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u\vert \vert \partial _{x}^6u\vert \mathrm{d}x\\&\qquad \le 2C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u\vert \vert \partial _{x}^6u\vert \mathrm{d}x =2\int \limits _{{\mathbb {R}}}\left| \frac{C(T)\partial _{x}^3u}{\beta \sqrt{D_7}}\right| \left| \beta \sqrt{D_7}\partial _{x}^6u\right| \mathrm{d}x\\&\qquad \le \frac{C(T)}{D_7}\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +\beta ^2D_7\left\| \partial _{x}^6u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le \frac{C(T)}{D_7}+\beta ^2D_7\left\| \partial _{x}^6u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&4\vert \delta \vert \int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u\partial _{x}^4u\vert \vert \partial _{x}^5u\vert \mathrm{d}x\le 2\delta ^2\int \limits _{{\mathbb {R}}}(\partial _{x}^3u)^2 (\partial _{x}^4u)^2 \mathrm{d}x +2\left\| \partial _{x}^5u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le 2\delta ^2\left\| \partial _{x}^3u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} + 2\left\| \partial _{x}^5u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&4\vert \delta \vert \int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u\vert \vert \partial _{x}^4u\vert \vert \partial _{x}^6u\vert \mathrm{d}x\le 4\vert \delta \vert \left\| \partial _{x}^2u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^4u\vert \vert \partial _{x}^6u\vert \mathrm{d}x\\&\qquad \le 2C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^4u\vert \vert \partial _{x}^6u\vert \mathrm{d}x=2\int \limits _{{\mathbb {R}}}\left| \frac{C(T)\partial _{x}^4u}{\beta \sqrt{D_7}}\right| \left| \beta \sqrt{D_7}\partial _{x}^6u\right| \mathrm{d}x\\&\qquad \le \frac{C(T)}{D_7}\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +\beta ^2D_7\left\| \partial _{x}^6u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&\vert \delta \vert \int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u\vert (\partial _{x}^5u)^2 \mathrm{d}x\vert \delta \vert \left\| \partial _{x}^2u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^5u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le C(T)\left\| \partial _{x}^5u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}, \end{aligned}$$

where $D_7$ is a positive constant, which will be specified later. It follows from (2.42) that

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +\beta ^2\left( 2-7D_7\right) \left\| \partial _{x}^6u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+2\gamma ^2\left\| \partial _x u(t,\cdot )\partial _{x}^5u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le \frac{C(T)}{D_7} +C(T)\left\| \partial _{x}^5u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\quad \qquad +C(T)\left( 1+\frac{1}{D_7}+\left\| \partial _{x}^3u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\right) \left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}. \end{aligned}$$

Taking $D_7=\frac{1}{7}$, we have that

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+\beta ^2\left\| \partial _{x}^6u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+2\gamma ^2\left\| \partial _x u(t,\cdot )\partial _{x}^5u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\nonumber \\&\qquad \le C(T)+C(T)\left( 1+\left\| \partial _{x}^3u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\right) \left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\nonumber \\&\quad \qquad +C(T)\left\| \partial _{x}^5u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}. \end{aligned}$$

(2.43)

Observe that

$$\begin{aligned} C(T)\left\| \partial _{x}^5u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}=C(T)\int \limits _{{\mathbb {R}}}\partial _{x}^5u\partial _{x}^5u \mathrm{d}x=-C(T)\int \limits _{{\mathbb {R}}}\partial _{x}^4u\partial _{x}^6u \mathrm{d}x. \end{aligned}$$

Therefore, by the Young inequality,

$$\begin{aligned} \begin{aligned} C(T)\left\| \partial _{x}^5u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}&\le \int \limits _{{\mathbb {R}}}\left| \frac{C(T)\partial _{x}^4u}{\beta }\right| \left| \beta \partial _{x}^6u \right| \mathrm{d}x\\&\le C(T)\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} + \frac{\beta ^2}{2}\left\| \partial _{x}^6u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}. \end{aligned} \end{aligned}$$

(2.44)

Consequently, by (2.43),

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+\frac{\beta ^2}{2}\left\| \partial _{x}^6u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+2\gamma ^2\left\| \partial _x u(t,\cdot )\partial _{x}^5u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le C(T)+C(T)\left( 1+\left\| \partial _{x}^3u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\right) \left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}. \end{aligned}$$

Integrating on (0, t), by (2.26), we have that

$$\begin{aligned}&\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+\frac{\beta ^2}{2}\int \limits _{0}^{t}\left\| \partial _{x}^6u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s+2\gamma ^2\int \limits _{0}^{t}\left\| \partial _x u(s,\cdot )\partial _{x}^5u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s\nonumber \\&\qquad \le C_0+ C(T)t + C(T)\left( 1+\left\| \partial _{x}^3u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\right) \int \limits _{0}^{t}\left\| \partial _{x}^4u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s\nonumber \\&\qquad \le C(T)\left( 1+\left\| \partial _{x}^3u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\right) . \end{aligned}$$

(2.45)

We prove (2.37). Thanks to (2.34), (2.45) and the Hölder inequality,

$$\begin{aligned} (\partial _{x}^3u(t,x))^2&=2\int \limits _{-\infty }^{x}\partial _{x}^3u\partial _{x}^4u \mathrm{d}y\le 2\int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u\vert \vert \partial _{x}^4u\vert \mathrm{d}x\\&\le 2\left\| \partial _{x}^3u(t,\cdot ) \right\| _{L^2({\mathbb {R}})}\left\| \partial _{x}^4u(t,\cdot ) \right\| _{L^2({\mathbb {R}})} \le C(T)\sqrt{\left( 1+\left\| \partial _{x}^3u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\right) }. \end{aligned}$$

Hence,

$$\begin{aligned} \left\| \partial _{x}^3u \right\| ^4_{L^{\infty }((0,T)\times {\mathbb {R}})}-C(T)\left\| \partial _{x}^3u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}-C(T)\le 0, \end{aligned}$$

which gives (2.37).

Finally, (2.38) follows from (2.37) and (2.45), while (2.26), (2.38) and an integration on (0, t) gives (2.39). $\square$

Lemma 2.8

Fix $T>0$ and assume (1.3), with $\ell =4$. There exist a constant $C(T)>0$, such that

$$\begin{aligned} \beta ^2\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+\frac{1}{42}\int \limits _{0}^{t}\left\| \partial _t\partial _{x}^2u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s\le C(T), \end{aligned}$$

(2.46)

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

Proof

Let $0\le t\le T$. Multiplying (1.1) by $2\partial _t\partial _{x}^4u$, an integration on ${\mathbb {R}}$ gives

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left( \beta ^2\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}-\alpha \left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\right) \\&\qquad =2\beta ^2\int \limits _{{\mathbb {R}}}\partial _{x}^4u\partial _t\partial _{x}^4u \mathrm{d}x +2\alpha \int \limits _{{\mathbb {R}}}\partial _{x}^2u\partial _t\partial _{x}^4u \mathrm{d}x\\&\qquad =-2\int \limits _{{\mathbb {R}}}\partial _tu\partial _t\partial _{x}^4u \mathrm{d}x +2\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _x u)^2\partial _{x}^2u\partial _t\partial _{x}^4u \mathrm{d}x -2\tau \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^2u\partial _t\partial _{x}^4u \mathrm{d}x\\&\qquad \quad -2\kappa \int \limits _{{\mathbb {R}}}(\partial _x u)^4\partial _t\partial _{x}^4u \mathrm{d}x-2q\int \limits _{{\mathbb {R}}}(\partial _x u)^2\partial _t\partial _{x}^4u \mathrm{d}x -2\delta \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^3u \partial _t\partial _{x}^4u \mathrm{d}x\\&\qquad =2\int \limits _{{\mathbb {R}}}\partial _t\partial _x u\partial _t\partial _{x}^3u \mathrm{d}x -4\gamma ^2\int \limits _{{\mathbb {R}}}\partial _x u(\partial _{x}^2u)^2\partial _t\partial _{x}^3u \mathrm{d}x -2\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _x u)^2\partial _{x}^3u\partial _t\partial _{x}^3u \mathrm{d}x\\&\qquad \quad +2\tau \int \limits _{{\mathbb {R}}}(\partial _{x}^2u)^2\partial _t\partial _{x}^3u \mathrm{d}x +2\tau \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^3u\partial _t\partial _{x}^3u \mathrm{d}x+8\kappa \int \limits _{{\mathbb {R}}}(\partial _x u)^3\partial _{x}^2u\partial _t\partial _{x}^3u \mathrm{d}x\\&\qquad \quad +4q\int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^2u\partial _t\partial _{x}^3u \mathrm{d}x +2\delta \int \limits _{{\mathbb {R}}}\partial _{x}^2u\partial _{x}^3u \partial _t\partial _{x}^3u \mathrm{d}x+2\delta \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^4u\partial _t\partial _{x}^3u \mathrm{d}x\\&\qquad =-2\left\| \partial _t\partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} + 4\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _{x}^2u)^3\partial _t\partial _{x}^2u \mathrm{d}x+12\gamma ^2\int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^2u\partial _{x}^3u \partial _t\partial _{x}^2u \mathrm{d}x\\&\qquad \quad +2\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _x u)^2\partial _{x}^4u\partial _t\partial _{x}^2u \mathrm{d}x -4\tau \int \limits _{{\mathbb {R}}}\partial _{x}^2u\partial _{x}^3u\partial _t\partial _{x}^2u \mathrm{d}x -2\tau \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^4u\partial _t\partial _{x}^2u \mathrm{d}x \\&\qquad \quad -24\kappa \int \limits _{{\mathbb {R}}}(\partial _x u)^2(\partial _{x}^2u)^2\partial _t\partial _{x}^2u \mathrm{d}x-8\kappa \int \limits _{{\mathbb {R}}}(\partial _x u)^3\partial _{x}^3u\partial _t\partial _{x}^2u \mathrm{d}x-4q\int \limits _{{\mathbb {R}}}(\partial _{x}^2u)^2\partial _t\partial _{x}^2u \mathrm{d}x \\&\qquad \quad -4q\int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^3u\partial _t\partial _{x}^2u \mathrm{d}x-2\delta \int \limits _{{\mathbb {R}}}(\partial _{x}^3u)^2\partial _t\partial _{x}^2u \mathrm{d}x -4\delta \int \limits _{{\mathbb {R}}}\partial _{x}^2u\partial _{x}^4u\partial _t\partial _{x}^2u \mathrm{d}x\\&\qquad \quad -2\delta \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^5u\partial _t\partial _{x}^2u \mathrm{d}x\\&\qquad =-2\left\| \partial _t\partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +\gamma ^2\frac{\mathrm{d}}{\mathrm{d}t}\int \limits _{{\mathbb {R}}}(\partial _{x}^2u)^4 \mathrm{d}x -12\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _{x}^2u)^2\partial _{x}^3u \partial _t\partial _x u \mathrm{d}x\\&\qquad \quad -12\gamma ^2\int \limits _{{\mathbb {R}}}\partial _x u(\partial _{x}^3u)^2\partial _t\partial _x u \mathrm{d}x -16\gamma ^2\int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^3u\partial _{x}^4u\partial _t\partial _x u \mathrm{d}x \\&\qquad \quad -2\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _x u)^2\partial _{x}^5u\partial _t\partial _x u \mathrm{d}x-4\tau \int \limits _{{\mathbb {R}}}\partial _{x}^2u\partial _{x}^3u\partial _t\partial _{x}^2u \mathrm{d}x\\&\qquad \quad -2\tau \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^4u\partial _t\partial _{x}^2u \mathrm{d}x+24\kappa \int \limits _{{\mathbb {R}}}(\partial _x u)^2\partial _{x}^2u\partial _{x}^3u\partial _t\partial _x u \mathrm{d}x\\&\qquad \quad +8\kappa \int \limits _{{\mathbb {R}}}(\partial _x u)^2\partial _{x}^4u\partial _t\partial _x u \mathrm{d}x-\frac{4q}{3}\frac{\mathrm{d}}{\mathrm{d}t}\int \limits _{{\mathbb {R}}}(\partial _{x}^2u)^3 \mathrm{d}x +4q\int \limits _{{\mathbb {R}}}\partial _{x}^2u\partial _{x}^3u\partial _t\partial _x u \mathrm{d}x\\&\qquad \quad +4q\int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^4u\partial _t\partial _x u \mathrm{d}x -2\delta \int \limits _{{\mathbb {R}}}(\partial _{x}^3u)^2\partial _t\partial _{x}^2u \mathrm{d}x -4\delta \int \limits _{{\mathbb {R}}}\partial _{x}^2u\partial _{x}^4u\partial _t\partial _{x}^2u \mathrm{d}x \\&\qquad \quad -2\delta \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^5u\partial _t\partial _{x}^2u \mathrm{d}x. \end{aligned}$$

Therefore, we have that

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left( \beta ^2\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}-\alpha \left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\right) \nonumber \\&\qquad \quad -\frac{\mathrm{d}}{\mathrm{d}t}\left( \gamma ^2\int \limits _{{\mathbb {R}}}(\partial _{x}^2u)^4 \mathrm{d}x-\frac{4q}{3}\int \limits _{{\mathbb {R}}}(\partial _{x}^2u)^3 \mathrm{d}x\right) +2\left\| \partial _t\partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad =-12\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _{x}^2u)^2\partial _{x}^3u \partial _t\partial _x u \mathrm{d}x-12\gamma ^2\int \limits _{{\mathbb {R}}}\partial _x u(\partial _{x}^3u)^2\partial _t\partial _x u \mathrm{d}x\nonumber \\&\qquad \quad -16\gamma ^2\int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^3u\partial _{x}^4u\partial _t\partial _x u \mathrm{d}x-2\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _x u)^2\partial _{x}^5u\partial _t\partial _x u \mathrm{d}x\nonumber \\&\qquad \quad -4\tau \int \limits _{{\mathbb {R}}}\partial _{x}^2u\partial _{x}^3u\partial _t\partial _{x}^2u \mathrm{d}x-2\tau \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^4u\partial _t\partial _{x}^2u \mathrm{d}x\nonumber \\&\qquad \quad +24\kappa \int \limits _{{\mathbb {R}}}(\partial _x u)^2\partial _{x}^2u\partial _{x}^3u\partial _t\partial _x u \mathrm{d}x+8\kappa \int \limits _{{\mathbb {R}}}(\partial _x u)^2\partial _{x}^4u\partial _t\partial _x u \mathrm{d}x\nonumber \\&\qquad \quad +4q\int \limits _{{\mathbb {R}}}\partial _{x}^2u\partial _{x}^3u\partial _t\partial _x u \mathrm{d}x+4q\int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^4u\partial _t\partial _x u \mathrm{d}x \nonumber \\&\qquad \quad -2\delta \int \limits _{{\mathbb {R}}}(\partial _{x}^3u)^2\partial _t\partial _{x}^2u \mathrm{d}x -4\delta \int \limits _{{\mathbb {R}}}\partial _{x}^2u\partial _{x}^4u\partial _t\partial _{x}^2u \mathrm{d}x\nonumber \\&\qquad \quad -2\delta \int \limits _{{\mathbb {R}}}\partial _x u\partial _{x}^5u\partial _t\partial _{x}^2u \mathrm{d}x.\nonumber \end{aligned}$$

(2.47)

Due to (2.25), (2.33), (2.34), (2.37), (2.38) and the Young inequality,

$$\begin{aligned}&12\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _{x}^2u)^2\vert \partial _{x}^3u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\le 12\gamma ^2\left\| \partial _{x}^2u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}|\partial _{x}^3u\partial _t\partial _x u \vert \mathrm{d}x\\&\qquad \le C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u\partial _t\partial _x u \vert \mathrm{d}x\le C(T)\left\| \partial _{x}^3u(t, \cdot ) \right\| ^2_{L^2({\mathbb {R}})}+C(T)\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le C(T) + C(T)\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&12\gamma ^2\int \limits _{{\mathbb {R}}}\vert \partial _x u\vert (\partial _{x}^3u)^2\vert \partial _t\partial _x u\vert \mathrm{d}x\le 12\gamma ^2\left\| \partial _x u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}(\partial _{x}^3u)^2\vert \partial _t\partial _x u\vert \mathrm{d}x\\&\qquad \le C(T)\int \limits _{{\mathbb {R}}}(\partial _{x}^3u)^2\vert \partial _t\partial _x u\vert \mathrm{d}x\le C(T)\left\| \partial _{x}^3u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\\&\qquad \le C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\le C(T)\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +C(T)\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le C(T)+ C(T)\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&16\gamma ^2\int \limits _{{\mathbb {R}}}\vert \partial _x u\vert \vert \partial _{x}^3u\vert \vert \partial _{x}^4u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\le 16\gamma ^2\left\| \partial _x u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u\vert \vert \partial _{x}^4u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\\&\qquad \le C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u\vert \vert \partial _{x}^4u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\le C(T)\left\| \partial _{x}^3u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^4u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\\&\qquad \le C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^4u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\le C(T)\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +C(T)\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le C(T)+ C(T)\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _x u)^2\vert \partial _{x}^5u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\le 2\gamma ^2\left\| \partial _x u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^5u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\\&\qquad \le C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^5u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\le C(T)\left\| \partial _{x}^5u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +C(T)\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&4\vert \tau \vert \int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u\vert \vert \partial _{x}^3u\vert \vert \partial _t\partial _{x}^2u\vert \mathrm{d}x\le 4\vert \tau \vert \left\| \partial _{x}^2u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u\vert \vert \partial _t\partial _{x}^2u\vert \mathrm{d}x\\&\qquad \le 2C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u\vert \vert \partial _t\partial _{x}^2u\vert \mathrm{d}x=2\int \limits _{{\mathbb {R}}}\left| \sqrt{3}C(T)\partial _{x}^3u\right| \left| \frac{\partial _t\partial _{x}^2u}{\sqrt{3}}\right| \mathrm{d}x\\&\qquad \le C(T)\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+\frac{1}{3}\left\| \partial _t\partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le C(T) +\frac{1}{3}\left\| \partial _t\partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert \tau \vert \int \limits _{{\mathbb {R}}}\vert \partial _x u\vert \vert \partial _{x}^4u\vert \vert \partial _t\partial _{x}^2u\vert \mathrm{d}x\le 2\vert \tau \vert \left\| \partial _x u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^4u\vert \vert \partial _t\partial _{x}^2u\vert \mathrm{d}x\\&\qquad \le 2C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^4u\vert \vert \partial _t\partial _{x}^2u\vert \mathrm{d}x=2\int \limits _{{\mathbb {R}}}\left| \sqrt{7}C(T)\partial _{x}^4u\right| \left| \frac{\partial _t\partial _{x}^2u}{\sqrt{7}}\right| \mathrm{d}x\\&\qquad \le C(T)\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +\frac{1}{7}\left\| \partial _t\partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le C(T) +\frac{1}{7}\left\| \partial _t\partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&24\vert \kappa \vert \int \limits _{{\mathbb {R}}}(\partial _x u)^2\vert \partial _{x}^2u\vert \vert \partial _{x}^3u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\le 24\vert \kappa \vert \left\| \partial _x u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u\vert \vert \partial _{x}^3u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\\&\qquad \le C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u\vert \vert \partial _{x}^3u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\le C(T)\left\| \partial _{x}^2u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\\&\qquad \le C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\le C(T)\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +C(T)\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le C(T) +C(T)\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&8\vert \kappa \vert \int \limits _{{\mathbb {R}}}(\partial _x u)^2\vert \partial _{x}^4u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\le 8\vert \kappa \vert \left\| \partial _x u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^4u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\\&\qquad \le C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^4u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\le C(T)\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +C(T)\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le C(T)+C(T)\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&4\vert q\vert \int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u\vert \vert \partial _{x}^3u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\le 4\vert q\vert \left\| \partial _{x}^2u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\\&\qquad \le C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\le C(T)\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +C(T)\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le C(T) +C(T)\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}, \\&4\vert q\vert \int \limits _{{\mathbb {R}}}\vert \partial _x u\vert \vert \partial _{x}^4u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x \le 4\vert q\vert \left\| \partial _x u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^4u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\\&\qquad \le C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^4u\vert \vert \partial _t\partial _x u\vert \mathrm{d}x\le C(T)\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +C(T)\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le C(T) +C(T)\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert \delta \vert \int \limits _{{\mathbb {R}}}(\partial _{x}^3u)^2\partial _t\partial _{x}^2u \mathrm{d}x\le 2\vert \delta \vert \left\| \partial _{x}^3u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u\vert \vert \partial _t\partial _{x}^2u\vert \mathrm{d}x\\&\qquad \le C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u\vert \vert \partial _t\partial _{x}^2u\vert \mathrm{d}x\le C(T)\left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} + \frac{1}{2}\left\| \partial _t\partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le C(T) + \frac{1}{2}\left\| \partial _t\partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}, \\&4\vert \delta \vert \int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u\vert \vert \partial _{x}^4u\vert \vert \partial _t\partial _{x}^2u \vert \mathrm{d}x\le 4\vert \delta \vert \left\| \partial _{x}^2u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^4u\vert \vert \partial _t\partial _{x}^2u \vert \mathrm{d}x\\&\qquad \le C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^4u\vert \vert \partial _t\partial _{x}^2u \vert \mathrm{d}x\le C(T)\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +\frac{1}{2}\left\| \partial _t\partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le C(T) + \frac{1}{2}\left\| \partial _t\partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert \delta \vert \int \limits _{{\mathbb {R}}}\vert \partial _x u\vert \vert \partial _{x}^5u\vert \vert \partial _t\partial _{x}^2u\vert \mathrm{d}x\le 2\vert \delta \vert \left\| \partial _x u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\int \limits _{{\mathbb {R}}}\vert \partial _{x}^5u\vert \vert \partial _t\partial _{x}^2u\vert \mathrm{d}x\\&\qquad \le C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^5u\vert \vert \partial _t\partial _{x}^2u\vert \mathrm{d}x\le C(T)\left\| \partial _{x}^5u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +\frac{1}{2}\left\| \partial _t\partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}. \end{aligned}$$

It follows from (2.47) that

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left( \beta ^2\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}-\alpha \left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\right) \nonumber \\&\qquad \quad -\frac{\mathrm{d}}{\mathrm{d}t}\left( \gamma ^2\int \limits _{{\mathbb {R}}}(\partial _{x}^2u)^4 \mathrm{d}x-\frac{4q}{3}\int \limits _{{\mathbb {R}}}(\partial _{x}^2u)^3 \mathrm{d}x\right) +\frac{1}{42}\left\| \partial _t\partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le C(T)+C(T)\left\| \partial _{x}^5u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} + C(T)\left\| \partial _t\partial _x u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}. \end{aligned}$$

(1.3), (2.34), (2.39) and an integration on (0, t) give

$$\begin{aligned}&\beta ^2\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}-\alpha \left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \quad -\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _{x}^2u)^4 \mathrm{d}x+\frac{4q}{3}\int \limits _{{\mathbb {R}}}(\partial _{x}^2u)^3 \mathrm{d}x+\frac{1}{42}\int \limits _{0}^{t}\left\| \partial _t\partial _{x}^2u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s\\&\qquad \le C_0+ C(T)\int \limits _{0}^{t}\left\| \partial _{x}^5u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s + C(T)\int \limits _{0}^{t}\left\| \partial _t\partial _x u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s\le C(T). \end{aligned}$$

Therefore, by (2.26), (2.33) and (2.34),

$$\begin{aligned}&\beta ^2\left\| \partial _{x}^4u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+\frac{1}{42}\int \limits _{0}^{t}\left\| \partial _t\partial _{x}^2u(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s\\&\qquad \le C(T) +\alpha \left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+ \gamma ^2\int \limits _{{\mathbb {R}}}(\partial _{x}^2u)^4 \mathrm{d}x-\frac{4q}{3}\int \limits _{{\mathbb {R}}}(\partial _{x}^2u)^3 \mathrm{d}x\\&\qquad \le C(T)+\vert \alpha \vert \left\| \partial _{x}^3u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+\gamma ^2\left\| \partial _{x}^2u \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \quad +\left| \frac{4q}{3}\right| \left\| \partial _{x}^2u \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^2u(t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\le C(T), \end{aligned}$$

which gives (2.46). $\square$

3 Proof of Theorem 1.1

This section is devoted to the proof of Theorem 1.1.

We begin by proving the following lemma.

Lemma 3.1

Fix $T>0$. Under Assumptions (1.2) and (1.9), there exists a unique solution u of (1.1), such that (1.10) and (1.11) hold.

Proof

Fix $T>0$. Thanks to Lemmas 2.1, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8 and the Cauchy–Kovalevskaya Theorem [58], we have that u is solution of (1.1) and (1.10) holds.

We prove (1.11). Let $u_1$ and $u_2$ be two solutions of (1.1), which verify (1.10), that is

$$\begin{aligned}&{\left\{ \begin{array}{ll} \displaystyle \partial _tu_1+ \alpha \partial _{x}^2u_1+\beta ^2\partial _{x}^4u_1 -\gamma ^2(\partial _x u_1)^2\partial _{x}^2u_1 +\tau \partial _x u_1\partial _{x}^2u_1 \\ \quad +\kappa (\partial _x u_1)^4 +q(\partial _x u_1)^2 +\delta \partial _x u_1\partial _{x}^3u_1=0, \quad &{}t>0, \quad x\in {\mathbb {R}},\\ \displaystyle u_1(0,x)=u_{1,\,0}(x),\quad &{}x\in {\mathbb {R}}, \end{array}\right. } \\&{\left\{ \begin{array}{ll} \displaystyle \partial _tu_2 +\alpha \partial _{x}^2u_2+\beta ^2\partial _{x}^4u_2 -\gamma ^2(\partial _x u_2)^2\partial _{x}^2u_2+\tau \partial _x u_2\partial _{x}^2u_2 \\ \quad +\kappa (\partial _x u_2)^4 +q(\partial _x u_2)^2 +\delta \partial _x u_2\partial _{x}^3u_2 =0, \quad &{}t>0, \quad x\in {\mathbb {R}},\\ \displaystyle u_1(0,x)=u_{1,\,0}(x),\quad &{}x\in {\mathbb {R}}, \end{array}\right. } \end{aligned}$$

Then, the function

$$\begin{aligned} \omega =u_1-u_2 \end{aligned}$$

(3.1)

is the solution of the following Cauchy problem:

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \partial _t\omega +\alpha \partial _{x}^2\omega +\beta ^2\partial _{x}^4\omega -\gamma ^2\left[ (\partial _x u_1)^2\partial _{x}^2u_1-(\partial _x u_2)^2\partial _{x}^2u_2\right] \\ \quad +\tau \left[ \partial _x u_1\partial _{x}^2u_1 -\partial _x u_2\partial _{x}^2u_2\right] \\ \quad +\kappa \left[ (\partial _x u_1)^4-(\partial _x u_2)^4\right] +q\left[ (\partial _x u_1)^2-(\partial _x u_2)^2\right] \\ \quad +\delta \left( \partial _x u_1\partial _{x}^3u_1-\partial _x u_2\partial _{x}^3u_2\right) =0, \quad &{}t>0, \quad x\in {\mathbb {R}},\\ \displaystyle \omega (0,x)=u_{1,\,0}(x)-u_{2,\,0}(x),\quad &{}x\in {\mathbb {R}}. \end{array}\right. } \end{aligned}$$

(3.2)

Observe that

$$\begin{aligned}&(\partial _x u_1)^2\partial _{x}^2u_1-(\partial _x u_2)^2\partial _{x}^2u_2\\&\qquad =(\partial _x u_1)^2\partial _{x}^2u_1-(\partial _x u_1)^2\partial _{x}^2u_2+(\partial _x u_1)^2\partial _{x}^2u_2-(\partial _x u_2)^2\partial _{x}^2u_2\\&\qquad =(\partial _x u_1)^2\partial _{x}^2\omega +\partial _{x}^2u_2\left[ (\partial _x u_1)^2 -(\partial _x u_2)^2\right] \\&\qquad =(\partial _x u)^2\partial _{x}^2\omega +\partial _{x}^2u_2(\partial _x u_1+\partial _x u_2)\partial _x \omega ,\\&\partial _x u_1\partial _{x}^2u_1-\partial _x u_2\partial _{x}^2u_2=\partial _x u_1\partial _{x}^2u_1-\partial _x u_1\partial _{x}^2u_2 +\partial _x u_1\partial _{x}^2u_2 -\partial _x u_2\partial _{x}^2u_2\\&\qquad =\partial _x u_1\partial _{x}^2\omega +\partial _{x}^2u_2\partial _x \omega ,\\&(\partial _x u_1)^4-(\partial _x u_2)^4=\left[ (\partial _x u_1)^2+(\partial _x u_2)^2\right] (\partial _x u_1+\partial _x u_2)\omega ,\\&(\partial _x u_1)^2-(\partial _x u_2)^2=(\partial _x u_1+\partial _x u_2)\partial _x \omega ,\\&\partial _x u_1\partial _{x}^3u_1-\partial _x u_2\partial _{x}^3u_2\\&\qquad =\partial _x u_1\partial _{x}^3u_1-\partial _x u_1\partial _{x}^3u_2+\partial _x u_1\partial _{x}^3u_2-\partial _x u_2\partial _{x}^3u_2\\&\qquad =\partial _x u_1\partial _{x}^3\omega +\partial _{x}^3u_2\partial _x \omega . \end{aligned}$$

Therefore, (3.2) is equivalent the following one:

$$\begin{aligned}&\partial _t\omega +\alpha \partial _{x}^2\omega +\beta ^2\partial _{x}^4\omega -\gamma ^2(\partial _x u_1)^2\partial _{x}^2\omega -\gamma ^2\partial _{x}^2u_2(\partial _x u_1+\partial _x u_2)\partial _x \omega \nonumber \\&\qquad +\tau \partial _x u_1\partial _{x}^2\omega +\tau \partial _{x}^2u_2\partial _x \omega +\kappa \left[ (\partial _x u_1)^2+(\partial _x u_2)^2\right] (\partial _x u_1+\partial _x u_2)\partial _x \omega \nonumber \\&\qquad +q(\partial _x u_1+\partial _x u_2)\partial _x \omega +\delta \partial _x u_1\partial _{x}^3\omega +\delta \partial _{x}^3u_2\partial _x \omega =0 \end{aligned}$$

(3.3)

Observe that, since $u_1,\,u_2\in H^4({\mathbb {R}})$, for every $0\le t\le T$, we have that

$$\begin{aligned} \begin{aligned}&\left\| \partial _x u_1 \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})},\,\left\| \partial _x u_2 \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})} \le C(T),\\&\left\| \partial _{x}^2u_2 \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})},\,\left\| \partial _{x}^3u_2 \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})} \le C(T). \end{aligned} \end{aligned}$$

(3.4)

Thanks to (3.4), we obtain

$$\begin{aligned} (\partial _x u_1)^2&\le C(T),\nonumber \\ \vert \partial _{x}^2u_2\vert \vert \partial _x u_1+\partial _x u_2\vert&\le C(T),\nonumber \\ \left[ (\partial _x u_1)^2+(\partial _x u_2)^2\right] \vert \partial _x u_1+\partial _x u_2\vert&\le C(T), \nonumber \\ \vert \partial _x u_1+\partial _x u_2\vert&\le C(T). \end{aligned}$$

(3.5)

Since

$$\begin{aligned} 2\int \limits _{{\mathbb {R}}}(\omega -\partial _{x}^2\omega )\partial _t\omega \mathrm{d}x&=\frac{\mathrm{d}}{\mathrm{d}t}\left( \left\| \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +\left\| \partial _x \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\right) =\frac{\mathrm{d}}{\mathrm{d}t}\left\| \omega (t,\cdot ) \right\| ^2_{H^1({\mathbb {R}})},\\ 2\alpha \int \limits _{{\mathbb {R}}}(\omega -\partial _{x}^2\omega )\partial _{x}^2\omega&=-2\alpha \left\| \partial _x \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} -2\alpha \left\| \partial _{x}^2\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\ 2\beta ^2\int \limits _{{\mathbb {R}}}(\omega -\partial _{x}^2\omega )\partial _{x}^4\omega \mathrm{d}x&=2\beta ^2\left\| \partial _{x}^2\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +2\beta ^2\left\| \partial _{x}^3\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}, \end{aligned}$$

multiplying (3.3) by $2\omega -2\partial _{x}^2\omega$, an integration on ${\mathbb {R}}$ gives,

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left\| \omega (t,\cdot ) \right\| ^2_{H^1({\mathbb {R}})}+2\beta ^2\left\| \partial _{x}^2\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +2\beta ^2\left\| \partial _{x}^3\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\nonumber \\&\qquad =2\alpha \left\| \partial _x \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+2\alpha \left\| \partial _{x}^2\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+2\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _x u_1)^2\omega \partial _{x}^2\omega \mathrm{d}x\nonumber \\&\qquad \quad -2\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _x u_1)^2(\partial _{x}^2\omega )^2 \mathrm{d}x +2\gamma ^2\int \limits _{{\mathbb {R}}}\partial _{x}^2u_2(\partial _x u_1+\partial _x u_2)\omega \partial _x \omega \mathrm{d}x\nonumber \\&\qquad \quad -2\tau \int \limits _{{\mathbb {R}}}\partial _x u_1\omega \partial _{x}^2\omega \mathrm{d}x+2\tau \int \limits _{{\mathbb {R}}}\partial _x u_1(\partial _{x}^2\omega )^2 \mathrm{d}x\nonumber \\&\qquad \quad -2\tau \int \limits _{{\mathbb {R}}}\partial _{x}^2u_2\omega \partial _x \omega \mathrm{d}x+2\tau \int \limits _{{\mathbb {R}}}\partial _{x}^2u_2\partial _x \omega \partial _{x}^2\omega \mathrm{d}x\nonumber \\&\qquad \quad -2\kappa \int \limits _{{\mathbb {R}}}\left[ (\partial _x u_1)^2+(\partial _x u_2)^2\right] (\partial _x u_1+\partial _x u_2)\omega \partial _x \omega \mathrm{d}x\nonumber \\&\qquad \quad +2\kappa \int \limits _{{\mathbb {R}}}\left[ (\partial _x u_1)^2+(\partial _x u_2)^2\right] (\partial _x u_1+\partial _x u_2)\partial _x \omega \partial _{x}^2u \mathrm{d}x \nonumber \\&\qquad \quad -2q\int \limits _{{\mathbb {R}}}(\partial _x u_1+\partial _x u_2)\omega \partial _x \omega \mathrm{d}x +2q\int \limits _{{\mathbb {R}}}(\partial _x u_1+\partial _x u_2)\partial _x \omega \partial _{x}^2\omega \mathrm{d}x\nonumber \\&\qquad \quad -2\delta \int \limits _{{\mathbb {R}}}\partial _x u_1\omega \partial _{x}^3\omega \mathrm{d}x +2\delta \int \limits _{{\mathbb {R}}}\partial _x u_1\partial _{x}^2\omega \partial _{x}^3\omega \mathrm{d}x\nonumber \\&\qquad \quad -2\delta \int \limits _{{\mathbb {R}}}\partial _{x}^3u_2\omega \partial _x \omega \mathrm{d}x +2\delta \int \limits _{{\mathbb {R}}}\partial _{x}^3u_2\partial _x \omega \partial _{x}^2\omega \mathrm{d}x. \end{aligned}$$

(3.6)

Due to (3.4), (3.5) and the Young inequality,

$$\begin{aligned}&2\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _x u_2)^2\vert \omega \vert \vert \partial _{x}^2\omega \vert \mathrm{d}x\le C(T)\int \limits _{{\mathbb {R}}}\vert \omega \vert \vert \partial _{x}^2\omega \vert \mathrm{d}x\\&\qquad \le C(T)\left\| \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +C(T)\left\| \partial _{x}^2\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _x u_1)^2(\partial _{x}^2\omega )^2 \mathrm{d}x\le C(T)\left\| \partial _{x}^2\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\gamma ^2\int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u_2\vert \vert \partial _x u_1+\partial _x u_2)\vert \vert \omega \vert \vert \partial _x \omega \vert \mathrm{d}x\le C(T)\int \limits _{{\mathbb {R}}}\vert \omega \vert \vert \partial _x \omega \vert \mathrm{d}x\\&\qquad \le C(T)\left\| \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +C(T)\left\| \partial _x \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}, \\&2\vert \tau \vert \int \limits _{{\mathbb {R}}}\vert \partial _x u_1\vert \vert \omega \vert \vert \partial _{x}^2\omega \vert \mathrm{d}x\le C(T)\int \limits _{{\mathbb {R}}}\vert \omega \vert \vert \partial _{x}^2\omega \vert \mathrm{d}x\\&\qquad \le C(T)\left\| \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +C(T)\left\| \partial _{x}^2\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert \tau \vert \int \limits _{{\mathbb {R}}}\vert \partial _x u_1\vert (\partial _{x}^2\omega )^2 \mathrm{d}x\le C(T)\left\| \partial _{x}^2\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert \tau \vert \int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u_2\vert \vert \omega \vert \vert \partial _x \omega \vert \mathrm{d}x\le C(T)\int \limits _{{\mathbb {R}}}\vert \omega \vert \vert \partial _x \omega \vert \mathrm{d}x\\&\qquad \le C(T)\left\| \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+ C(T)\left\| \partial _x \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert \tau \vert \int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u_2\vert \vert \partial _x \omega \vert \vert \partial _{x}^2\omega \vert \mathrm{d}x\le C(T)\int \limits _{{\mathbb {R}}}\vert \partial _x \omega \vert \vert \partial _{x}^2\omega \vert \mathrm{d}x\\&\qquad \le C(T)\left\| \partial _x \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +C(T)\left\| \partial _{x}^2\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert \kappa \vert \int \limits _{{\mathbb {R}}}\left[ (\partial _x u_1)^2+(\partial _x u_2)^2\right] \vert \partial _x u_1+\partial _x u_2\vert \vert \omega \vert \vert \partial _x \omega \vert \mathrm{d}x\le C(T)\int \limits _{{\mathbb {R}}}\vert \omega \vert \vert \partial _x \omega \vert \mathrm{d}x\\&\qquad \le C(T)\left\| \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} + C(T)\left\| \partial _x \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert \kappa \vert \int \limits _{{\mathbb {R}}}\left[ (\partial _x u_1)^2+(\partial _x u_2)^2\right] \vert \partial _x u_1+\partial _x u_2\vert \vert \partial _x \omega \vert \vert \partial _{x}^2u\vert \mathrm{d}x\le C(T)\int \limits _{{\mathbb {R}}}\vert \partial _x \omega \vert \vert \partial _{x}^2\omega \vert \mathrm{d}x \\&\qquad \le C(T)\left\| \partial _x \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +C(T)\left\| \partial _{x}^2\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert q\vert \int \limits _{{\mathbb {R}}}\vert \partial _x u_1+\partial _x u_2\vert \vert \omega \vert \vert \partial _x \omega \vert \mathrm{d}x\le C(T)\int \limits _{{\mathbb {R}}}\vert \omega \vert \vert \partial _x \omega \vert \mathrm{d}x\\&\qquad \le C(T)\left\| \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +C(T)\left\| \partial _x \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert q\vert \int \limits _{{\mathbb {R}}}\vert \partial _x u_1+\partial _x u_2\vert \vert \partial _x \omega \vert \vert \partial _{x}^2\omega \vert \mathrm{d}x\le C(T)\int \limits _{{\mathbb {R}}}\vert \partial _x \omega \vert \vert \partial _{x}^2\omega \vert \mathrm{d}x\\&\qquad \le C(T)\left\| \partial _x \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +C(T)\left\| \partial _{x}^2\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert \delta \vert \int \limits _{{\mathbb {R}}}\vert \partial _x u_1\vert \vert \omega \vert \vert \partial _{x}^3\omega \vert \mathrm{d}x\le C(T)\int \limits _{{\mathbb {R}}}\vert \omega \vert \vert \partial _{x}^3\omega \vert \mathrm{d}x\\&\qquad =\int \limits _{{\mathbb {R}}}\left| \frac{C(T)\omega }{\beta }\right| \left| \beta \partial _{x}^3\omega \right| \mathrm{d}x \le C(T)\left\| \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +\frac{\beta ^2}{2}\left\| \partial _{x}^3\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert \delta \vert \int \limits _{{\mathbb {R}}}\vert \partial _x u_1\vert \vert \partial _{x}^2\omega \vert \vert \partial _{x}^3\omega \vert \mathrm{d}x\le C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^2\omega \vert \vert \partial _{x}^3\omega \vert \mathrm{d}x\\&\qquad =\int \limits _{{\mathbb {R}}}\left| \frac{C(T)\partial _{x}^2\omega }{\beta }\right| \left| \beta \partial _{x}^3\omega \right| \mathrm{d}x\le C(T)\left\| \partial _{x}^2\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +\frac{\beta ^2}{2}\left\| \partial _{x}^3\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert \delta \vert \int \limits _{{\mathbb {R}}}\vert \partial _x u_1\vert \vert \partial _{x}^2\omega \vert \vert \partial _{x}^3\omega \vert \mathrm{d}x\le C(T)\int \limits _{{\mathbb {R}}}\vert \partial _{x}^2\omega \vert \vert \partial _{x}^3\omega \vert \mathrm{d}x\\&\qquad =\int \limits _{{\mathbb {R}}}\left| \frac{C(T)\partial _{x}^2\omega }{\beta }\right| \left| \beta \partial _{x}^3\omega \right| \mathrm{d}x\le C(T)\left\| \partial _{x}^2\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +\frac{\beta ^2}{2}\left\| \partial _{x}^3\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert \delta \vert \int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u_2\vert \vert \omega \vert \vert \partial _x \omega \vert \mathrm{d}x\le C(T)\int \limits _{{\mathbb {R}}}\vert \omega \vert \vert \partial _x \omega \vert \mathrm{d}x\\&\qquad \le C(T)\left\| \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +C(T)\left\| \partial _x \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert \delta \vert \int \limits _{{\mathbb {R}}}\vert \partial _{x}^3u_2\vert \vert \partial _x \omega \vert \vert \partial _{x}^2\omega \vert \mathrm{d}x\le C(T)\int \limits _{{\mathbb {R}}}\vert \partial _x \omega \vert \vert \partial _{x}^2\omega \vert \mathrm{d}x\\&\qquad \le C(T)\left\| \partial _x \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +C(T)\left\| \partial _{x}^2\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}. \end{aligned}$$

It follows from (3.6) that

$$\begin{aligned} \begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left\| \omega (t,\cdot ) \right\| ^2_{H^1({\mathbb {R}})}+2\beta ^2\left\| \partial _{x}^2\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +\beta ^2\left\| \partial _{x}^3\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le C(T)\left\| \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +C(T)\left\| \partial _x \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +C(T)\left\| \partial _{x}^2\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}. \end{aligned} \end{aligned}$$

(3.7)

Observe that

$$\begin{aligned} C(T)\left\| \partial _{x}^2\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}=C(T)\int \limits _{{\mathbb {R}}}\partial _{x}^2\omega \partial _{x}^2\omega \mathrm{d}x=-C(T)\int \limits _{{\mathbb {R}}}\partial _x \omega \partial _{x}^3\omega \mathrm{d}x. \end{aligned}$$

Therefore, by the Young inequality,

$$\begin{aligned} C(T)\left\| \partial _{x}^2\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}&\le \int \limits _{{\mathbb {R}}}\left| \frac{C(T)\partial _x \omega }{\beta }\right| \left| \beta \partial _{x}^3\omega \right| \mathrm{d}x\\&\le C(T)\left\| \partial _x \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +\frac{\beta ^2}{2}\left\| \partial _{x}^3\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}. \end{aligned}$$

Consequently, by (3.7),

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left\| \omega (t,\cdot ) \right\| ^2_{H^1({\mathbb {R}})}+2\beta ^2\left\| \partial _{x}^2\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +\frac{\beta ^2}{2}\left\| \partial _{x}^3\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le C(T)\left\| \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +C(T)\left\| \partial _x \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\le C(T)\left\| \omega (t,\cdot ) \right\| ^2_{H^1({\mathbb {R}})}. \end{aligned}$$

The Gronwall Lemma and (3.2) gives

$$\begin{aligned} \begin{aligned}&\left\| \omega (t,\cdot ) \right\| ^2_{H^1({\mathbb {R}})}+\beta ^2e^{C(T)t}\int \limits _{0}^{t}e^{-C(T)s}\left\| \partial _{x}^2\omega (s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} \mathrm{d}s\\&\qquad +\frac{\beta ^2 e^{C(T)t}}{2}\int \limits _{0}^{t}e^{-C(T)s}\left\| \partial _{x}^3\omega (s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s\le e^{C(T)t}\left\| \omega _0 \right\| _{H^1({\mathbb {R}})}. \end{aligned} \end{aligned}$$

(3.8)

(1.11) follows from (3.1) and (3.8). $\square$

Lemma 3.2

Fix $T>0$. Under Assumptions (1.2) and (1.12), there exists a unique solution u of (1.1), such that (1.13) and (1.14) hold.

Proof

We begin by observing that, since $\delta =0$, (1.1) reads

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu+\alpha \partial _{x}^2u+\beta ^2\partial _{x}^4u -\gamma ^2(\partial _x u)^2\partial _{x}^2u +\kappa (\partial _x u)^4 +q(\partial _x u)^2=0, &{}{}\quad t>0, \quad x\in {\mathbb {R}},\\ u(0,x)=u_0(x), &{}{}\quad x\in {\mathbb {R}}, \end{array}\right. } \end{aligned}$$

(3.9)

Thanks to Lemmas 2.1, (2.2), (2.4), (2.5), (2.6) and the Cauchy–Kovalevskaya Theorem [58], we have that u is solution of (3.9) and (1.13) holds.

We prove (1.14). Let $u_1$ and $u_2$ be two solutions of (3.9), which satisfy (1.13), that is

$$\begin{aligned}&{\left\{ \begin{array}{ll} \displaystyle \partial _tu_1+ \alpha \partial _{x}^2u_1+\beta ^2\partial _{x}^4u_1 -\gamma ^2(\partial _x u_1)^2\partial _{x}^2u_1 +\kappa (\partial _x u_1)^4 +q(\partial _x u_1)^2 =0, \quad &{}t>0, \quad x\in {\mathbb {R}},\\ \displaystyle u_1(0,x)=u_{1,\,0}(x),\quad &{}x\in {\mathbb {R}}, \end{array}\right. } \\&{\left\{ \begin{array}{ll} \displaystyle \partial _tu_2 +\alpha \partial _{x}^2u_2+\beta ^2\partial _{x}^4u_2 -\gamma ^2(\partial _x u_2)^2\partial _{x}^2u_2 +\kappa (\partial _x u_2)^4 +q(\partial _x u_2)^2 =0, \quad &{}t>0, \quad x\in {\mathbb {R}},\\ \displaystyle u_1(0,x)=u_{1,\,0}(x),\quad &{}x\in {\mathbb {R}}, \end{array}\right. } \end{aligned}$$

Then, the function $\omega$, defined in (3.1), is the solution of the following Cauchy problem:

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \partial _t\omega +\alpha \partial _{x}^2\omega +\beta ^2\partial _{x}^4\omega -\gamma ^2\left[ (\partial _x u_1)^2\partial _{x}^2u_1-(\partial _x u_2)^2\partial _{x}^2u_2\right] \\ \quad +\tau \left[ \partial _x u_1\partial _{x}^2u_1 -\partial _x u_2\partial _{x}^2u_2\right] \\ \quad +\kappa \left[ (\partial _x u_1)^4-(\partial _x u_2)^4\right] \\ \quad +q\left[ (\partial _x u_1)^2-(\partial _x u_2)^2\right] =0, \quad &{}t>0, \quad x\in {\mathbb {R}},\\ \displaystyle \omega (0,x)=u_{1,\,0}(x)-u_{2,\,0}(x),\quad &{}x\in {\mathbb {R}}. \end{array}\right. } \end{aligned}$$

(3.10)

Arguing as in Lemma 3.1, (3.2) is equivalent the following one:

$$\begin{aligned} \begin{aligned}&\partial _t\omega +\alpha \partial _{x}^2\omega +\beta ^2\partial _{x}^4\omega -\gamma ^2(\partial _x u_1)^2\partial _{x}^2\omega -\gamma ^2\partial _{x}^2u_2(\partial _x u_1+\partial _x u_2)\partial _x \omega \\&\qquad +\tau \partial _x u_1\partial _{x}^2\omega +\tau \partial _{x}^2u_2\partial _x \omega +\kappa \left[ (\partial _x u_1)^2+(\partial _x u_2)^2\right] (\partial _x u_1+\partial _x u_2)\partial _x \omega \\&\qquad +q(\partial _x u_1+\partial _x u_2)\partial _x \omega =0 \end{aligned} \end{aligned}$$

(3.11)

Observe that, since $u_1,\,u_2\in H^3({\mathbb {R}})$, for every $0\le t\le T$, we have that

$$\begin{aligned} \left\| \partial _x u_1 \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})},\,\left\| \partial _x u_2 \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})},\, \left\| \partial _{x}^2u_2 \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\le C(T). \end{aligned}$$

(3.12)

Therefore, by (3.12),

$$\begin{aligned} \vert \partial _{x}^2u_2\vert \vert \partial _x u_1+\partial _x u_2\vert&\le C(T),\nonumber \\ \left[ (\partial _x u_1)^2+(\partial _x u_2)^2\right] \vert \partial _x u_1+\partial _x u_2\vert&\le C(T),\nonumber \\ \vert \partial _x u_1+\partial _x u_2\vert&\le C(T). \end{aligned}$$

(3.13)

Since

$$\begin{aligned} 2\alpha \int \limits _{{\mathbb {R}}}\omega \partial _{x}^2\omega \mathrm{d}x&=-2\alpha \left\| \partial _x \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\ \beta ^2\int \limits _{{\mathbb {R}}}\omega \partial _{x}^4\omega \mathrm{d}x&=2\beta ^2\left\| \partial _{x}^4\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}, \end{aligned}$$

multiplying (3.11) by $2\omega$, an integration on ${\mathbb {R}}$ gives

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left\| \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +2\beta ^2\left\| \partial _{x}^2\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\nonumber \\&\qquad =2\alpha \left\| \partial _x \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+2\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _x u_1)^2\omega \partial _{x}^2\omega \mathrm{d}x\nonumber \\&\qquad \quad + 2\gamma ^2\int \limits _{{\mathbb {R}}}\partial _{x}^2u_2(\partial _x u_1+\partial _x u_2)\omega \partial _x \omega \mathrm{d}x\nonumber \\&\qquad \quad -2\tau \int \limits _{{\mathbb {R}}}\partial _x u_1\omega \partial _{x}^2\omega \mathrm{d}x -2\tau \int \limits _{{\mathbb {R}}}\partial _{x}^2u_2\omega \partial _x \omega \mathrm{d}x\nonumber \\&\qquad \quad -2\kappa \int \limits _{{\mathbb {R}}}\left[ (\partial _x u_1)^2+(\partial _x u_2)^2\right] (\partial _x u_1+\partial _x u_2)\omega \partial _x \omega \mathrm{d}x\nonumber \\&\qquad \quad -2q\int \limits _{{\mathbb {R}}}(\partial _x u_1+\partial _x u_2)\omega \partial _x \omega \mathrm{d}x. \end{aligned}$$

(3.14)

Due to (3.12), (3.13) and the Young inequality,

$$\begin{aligned}&2\gamma ^2\int \limits _{{\mathbb {R}}}(\partial _x u_1)^2\vert \omega \vert \vert \partial _{x}^2\omega \vert \mathrm{d}x\le C(T)\int \limits _{{\mathbb {R}}}\vert \omega \vert \vert \partial _{x}^2\omega \vert \mathrm{d}x \\&\qquad =\int \limits _{{\mathbb {R}}}\left| \frac{C(T)\omega }{\beta }\right| \left| \beta \partial _{x}^2\omega \right| \mathrm{d}x \le C(T)\left\| \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} + \frac{\beta ^2}{2}\left\| \partial _{x}^2\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\gamma ^2\int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u_2\vert \vert \partial _x u_1+\partial _x u_2\vert \vert \omega \vert \vert \partial _x \omega \vert \mathrm{d}x\le C(T)\int \limits _{{\mathbb {R}}}\vert \omega \vert \vert \partial _x \omega \vert \mathrm{d}x\\&\qquad \le C(T)\left\| \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +C(T)\left\| \partial _x \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert \tau \vert \int \limits _{{\mathbb {R}}}\vert \partial _x u_1\vert \vert \omega \vert \vert \partial _{x}^2\omega \vert \mathrm{d}x\le C(T)\int \limits _{{\mathbb {R}}}\vert \omega \vert \vert \partial _{x}^2\omega \vert \mathrm{d}x\\&\qquad =\int \limits _{{\mathbb {R}}}\left| \frac{C(T)\omega }{\beta }\right| \left| \beta \partial _{x}^2\omega \right| \mathrm{d}x \le C(T)\left\| \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +\frac{\beta ^2}{2}\left\| \partial _{x}^2\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert \tau \vert \int \limits _{{\mathbb {R}}}\vert \partial _{x}^2u_2\vert \vert \omega \vert \vert \partial _x \omega \vert \mathrm{d}x\le C(T)\int \limits _{{\mathbb {R}}}\vert \omega \vert \vert \partial _x \omega \vert \mathrm{d}x\le C(T)\left\| \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +C(T)\left\| \partial _x \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert \kappa \vert \int \limits _{{\mathbb {R}}}\left[ (\partial _x u_1)^2+(\partial _x u_2)^2\right] \vert \partial _x u_1+\partial _x u_2\vert \omega \partial _x \omega \mathrm{d}x\le C(T)\int \limits _{{\mathbb {R}}}\vert \omega \vert \vert \partial _x \omega \vert \mathrm{d}x\\&\qquad \le C(T)\left\| \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +C(T)\left\| \partial _x \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&\vert 2q\vert \int \limits _{{\mathbb {R}}}\vert \partial _x u_1+\partial _x u_2\vert \vert \omega \vert \vert \partial _x \omega \vert \mathrm{d}x\le C(T)\int \limits _{{\mathbb {R}}}\vert \omega \partial _x \omega \vert \mathrm{d}x\\&\qquad \le C(T)\left\| \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +C(T)\left\| \partial _x \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}. \end{aligned}$$

It follows from (3.14) that

$$\begin{aligned} \begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left\| \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +\beta ^2\left\| \partial _{x}^2\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad \le C(T)\left\| \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +C(T)\left\| \partial _x \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}. \end{aligned} \end{aligned}$$

(3.15)

Observe that

$$\begin{aligned} C(T)\left\| \partial _x \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}=C(T)\int \limits _{{\mathbb {R}}}\partial _x \omega \partial _x \omega \mathrm{d}x=-C(T)\int \limits _{{\mathbb {R}}}\omega \partial _{x}^2\omega \mathrm{d}x. \end{aligned}$$

Therefore, by the Young inequality,

$$\begin{aligned} C(T)\left\| \partial _x \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}&\le \int \limits _{{\mathbb {R}}}\left| \frac{C(T)\omega }{\beta }\right| \left| \beta \partial _{x}^2\omega \right| \mathrm{d}x\\&\le C(T)\left\| \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +\frac{\beta ^2}{2}\left\| \partial _{x}^2\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}. \end{aligned}$$

Consequently, by (3.15),

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\left\| \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +\frac{\beta ^2}{2}\left\| \partial _{x}^2\omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\le C(T)\left\| \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} \end{aligned}$$

The Gronwall Lemma and (3.10) gives

$$\begin{aligned} \left\| \omega (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} +\frac{\beta ^2 e^{C(T)t}}{2}\int \limits _{0}^{t}e^{-C(T)s}\left\| \partial _{x}^2\omega (s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s\le e^{C(T)t}\left\| \omega _0 \right\| _{L^2({\mathbb {R}})} \end{aligned}$$

(3.16)

(1.14) follows from (3.1) and (3.16). $\square$

Lemma 3.3

Fix $T>0$. Under Assumptions (1.2) and (1.15), there exists a solution u of (1.1), such that (1.16) holds.

Proof

Let $T>0$. Thanks to Lemmas 2.1, (2.2), (2.4), (2.5) and the Cauchy–Kovalevskaya Theorem [58], we have that u is solution of (3.9) and (1.13) holds. $\square$

Proof of Theorem 1.1

Theorem 1.1 follows from Lemmas 3.1, 3.2 and 3.3. $\square$