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). \(\square \)