Well-posedness of the classical solutions for a Kawahara–Korteweg–de Vries-type equation

Abstract

The Kawahara–Korteweg–de Vries-type equation occurs in the modelization of magneto-acoustic waves in plasmas and propagation of nonlinear water waves in the long-wavelength region as in the case of Korteweg–de Vries equation. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem associated with this equation.

Appendix A: \(u_0\in H^5({\mathbb {R}})\)

In this appendix, we consider the Cauchy problem (1.1), where, on the initial datum, we assume

$$\begin{aligned} u_0(x)\in H^5({\mathbb {R}}). \end{aligned}$$

(A.1)

The main result of this appendix is the following theorem.

Theorem A.1

Assume (1.2), (1.3) and (A.1). Fix $T>0$, there exists an unique solution u of (1.1) such that

$$\begin{aligned} u \in H^1((0,T)\times {\mathbb {R}})\cap L^{\infty }(0,T;H^5({\mathbb {R}})), \end{aligned}$$

(A.2)

Moreover, if $u_1$ and $u_2$ are two solutions of (1.1) satisfying (A.1), (1.12) holds.

To prove Theorem A.1, we consider the approximation (2.1), where $u_{\varepsilon ,0}$ is a $C^{\infty }$ approximation of $u_0$ such that

$$\begin{aligned} \left\| u_{\varepsilon ,0} \right\| _{H^5({\mathbb {R}})}\le \left\| u_0 \right\| _{H^5({\mathbb {R}})}, \quad \varepsilon \left\| \partial _{x}^3u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\le C_0, \end{aligned}$$

(A.3)

where $C_0$ is a positive constant, independent of $\varepsilon$.

Let us prove some a priori estimates on $u_\varepsilon$.

Since $H^4({\mathbb {R}}) \subset H^5({\mathbb {R}})$, then Lemmas 2.1, 2.2 and 2.4 hold also in this case.

We prove the following result.

Lemma A.1

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

$$\begin{aligned} \left\| \partial _{x}^5u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+2 \varepsilon e^{C(T)t}\int _{0}^{t}e^{-C(T)s}\left\| \partial _{x}^8u_\varepsilon (s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s\le C(T), \end{aligned}$$

(A.4)

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

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

(A.5)

Proof

Let $0\le t\le T$. Multiplying (2.1) by $-2\partial _{x}^{10}u_\varepsilon$, thanks to (1.2), we have that

$$\begin{aligned} \begin{aligned} -2\partial _{x}^{10}u_\varepsilon \partial _tu_\varepsilon&=2\kappa u_\varepsilon \partial _x u_\varepsilon \partial _{x}^{10}u_\varepsilon +2qu_\varepsilon ^2\partial _x u_\varepsilon \partial _{x}^{10}u_\varepsilon +2\alpha \partial _{x}^3u_\varepsilon \partial _{x}^{10}u_\varepsilon \\&\quad +2\gamma \partial _{x}^5u_\varepsilon \partial _{x}^{10}u_\varepsilon -2\varepsilon \partial _{x}^6u_\varepsilon \partial _{x}^{10}u_\varepsilon . \end{aligned} \end{aligned}$$

(A.6)

Observe that

$$\begin{aligned} -2\int _{{\mathbb {R}}}\partial _{x}^{10}u_\varepsilon \partial _tu_\varepsilon \mathrm{d}x&=2\int _{{\mathbb {R}}}\partial _{x}^9u_\varepsilon \partial _t\partial _x u_\varepsilon \mathrm{d}x =-2\int _{{\mathbb {R}}}\partial _{x}^8u_\varepsilon \partial _t\partial _{x}^2u_\varepsilon \mathrm{d}x\\&=2\int _{{\mathbb {R}}}\partial _{x}^7u_\varepsilon \partial _t\partial _{x}^3u_\varepsilon \mathrm{d}x =-2\int _{{\mathbb {R}}}\partial _{x}^6u_\varepsilon \partial _t\partial _{x}^4u_\varepsilon \mathrm{d}x\\&=\frac{\mathrm{d}}{\mathrm{d}t}\left\| \partial _{x}^5u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\ 2\alpha \int _{{\mathbb {R}}}\partial _{x}^3u_\varepsilon \partial _{x}^{10}u_\varepsilon \mathrm{d}x&=-2\alpha \int _{{\mathbb {R}}}\partial _{x}^4u_\varepsilon \partial _{x}^9u_\varepsilon \mathrm{d}x =2\alpha \int _{{\mathbb {R}}}\partial _{x}^5u_\varepsilon \partial _{x}^8u_\varepsilon \mathrm{d}x\\&=-2\int _{{\mathbb {R}}}\partial _{x}^6u_\varepsilon \partial _{x}^7u_\varepsilon \mathrm{d}x =0,\\ 2\gamma \int _{{\mathbb {R}}}\partial _{x}^5u_\varepsilon \partial _{x}^{10}u_\varepsilon \mathrm{d}x&= -2\gamma \int _{{\mathbb {R}}}\partial _{x}^6u_\varepsilon \partial _{x}^9u_\varepsilon \mathrm{d}x =2\gamma \int _{{\mathbb {R}}}\partial _{x}^7u_\varepsilon \partial _{x}^8u_\varepsilon \mathrm{d}x =0,\\ -2\varepsilon \int _{{\mathbb {R}}}\partial _{x}^6u_\varepsilon \partial _{x}^{10}u_\varepsilon \mathrm{d}x&=2\varepsilon \int _{{\mathbb {R}}}\partial _{x}^7u_\varepsilon \partial _{x}^8u_\varepsilon \mathrm{d}x=-2\varepsilon \left\| \partial _{x}^8u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}. \end{aligned}$$

Consequently, an integration on ${\mathbb {R}}$ of (A.6) give

$$\begin{aligned} \begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left\| \partial _{x}^5u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+2\varepsilon \left\| \partial _{x}^8u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\quad =2\kappa \int _{{\mathbb {R}}}u_\varepsilon \partial _x u_\varepsilon \partial _{x}^{10}u_\varepsilon \mathrm{d}x +2q\int _{{\mathbb {R}}}u_\varepsilon ^2\partial _x u_\varepsilon \partial _{x}^{10}u_\varepsilon \mathrm{d}x. \end{aligned} \end{aligned}$$

(A.7)

Observe that

$$\begin{aligned} 2\kappa \int _{{\mathbb {R}}}u_\varepsilon \partial _x u_\varepsilon \partial _{x}^{10}u_\varepsilon \mathrm{d}x&=-2\kappa \int _{{\mathbb {R}}}(\partial _x u_\varepsilon )^2\partial _{x}^9u_\varepsilon \mathrm{d}x -2\kappa \int _{{\mathbb {R}}}u_\varepsilon \partial _{x}^2u_\varepsilon \partial _{x}^9u_\varepsilon \mathrm{d}x\\&=6\kappa \int _{{\mathbb {R}}}\partial _x u_\varepsilon \partial _{x}^2u_\varepsilon \partial _{x}^8u_\varepsilon \mathrm{d}x +2\kappa \int _{{\mathbb {R}}}u_\varepsilon \partial _{x}^3u_\varepsilon \partial _{x}^8u_\varepsilon \mathrm{d}x\\&=-6\kappa \int _{{\mathbb {R}}}(\partial _{x}^2u_\varepsilon )^2\partial _{x}^7u_\varepsilon \mathrm{d}x -8\kappa \int _{{\mathbb {R}}}\partial _x u_\varepsilon \partial _{x}^3u_\varepsilon \partial _{x}^7u_\varepsilon \mathrm{d}x\\&\quad -2\kappa \int _{{\mathbb {R}}}u_\varepsilon \partial _{x}^4u_\varepsilon \partial _{x}^7u_\varepsilon \mathrm{d}x\\&=20\kappa \int _{{\mathbb {R}}}\partial _{x}^2u_\varepsilon \partial _{x}^3u_\varepsilon \partial _{x}^6u_\varepsilon \mathrm{d}x +10\kappa \int _{{\mathbb {R}}}\partial _x u_\varepsilon \partial _{x}^4u_\varepsilon \partial _{x}^6u_\varepsilon \mathrm{d}x\\&\quad +2\kappa \int _{{\mathbb {R}}}u_\varepsilon \partial _{x}^5u_\varepsilon \partial _{x}^6u_\varepsilon \mathrm{d}x\\&=-20\kappa \int _{{\mathbb {R}}}(\partial _{x}^3u_\varepsilon )^2\partial _{x}^5u_\varepsilon \mathrm{d}x -30\kappa \int _{{\mathbb {R}}}\partial _{x}^2u_\varepsilon \partial _{x}^4u_\varepsilon \partial _{x}^5u_\varepsilon \mathrm{d}x\\&\quad -11\kappa \int _{{\mathbb {R}}}\partial _x u_\varepsilon (\partial _{x}^5u_\varepsilon )^2 \mathrm{d}x\\&=-20\kappa \int _{{\mathbb {R}}}(\partial _{x}^3u_\varepsilon )^2\partial _{x}^5u_\varepsilon \mathrm{d}x+15\kappa \int _{{\mathbb {R}}}\partial _{x}^3u_\varepsilon (\partial _{x}^4u_\varepsilon )^2 \mathrm{d}x\\&\quad -11\kappa \int _{{\mathbb {R}}}\partial _x u_\varepsilon (\partial _{x}^5u_\varepsilon )^2 \mathrm{d}x,\\ 2q\int _{{\mathbb {R}}}u_\varepsilon ^2\partial _x u_\varepsilon \partial _{x}^{10}u_\varepsilon \mathrm{d}x&=-4q\int _{{\mathbb {R}}}u_\varepsilon (\partial _x u_\varepsilon )^2\partial _{x}^9u_\varepsilon \mathrm{d}x -2q\int _{{\mathbb {R}}}u_\varepsilon ^2\partial _{x}^2u_\varepsilon \partial _{x}^9u_\varepsilon \mathrm{d}x\\&=4q\int _{{\mathbb {R}}}(\partial _x u_\varepsilon )^3\partial _{x}^8u_\varepsilon \mathrm{d}x +10q\int _{{\mathbb {R}}}u_\varepsilon \partial _x u_\varepsilon \partial _{x}^2u_\varepsilon \partial _{x}^8u_\varepsilon \mathrm{d}x\\&\quad +2q\int _{{\mathbb {R}}}u_\varepsilon ^2\partial _{x}^3u_\varepsilon \partial _{x}^8u_\varepsilon \mathrm{d}x \\&=-22q\int _{{\mathbb {R}}}(\partial _x u_\varepsilon )^2\partial _{x}^2u_\varepsilon \partial _{x}^7u_\varepsilon \mathrm{d}x -10q\int _{{\mathbb {R}}}u_\varepsilon (\partial _{x}^2u_\varepsilon )^2\partial _{x}^7u_\varepsilon \mathrm{d}x\\&\quad -14q\int _{{\mathbb {R}}}u_\varepsilon \partial _x u_\varepsilon \partial _{x}^3u_\varepsilon \partial _{x}^7u_\varepsilon \mathrm{d}x -2q\int _{{\mathbb {R}}}u_\varepsilon ^2\partial _{x}^4u_\varepsilon \partial _{x}^7u_\varepsilon \mathrm{d}x \\&=54q\int _{{\mathbb {R}}}\partial _x u_\varepsilon (\partial _{x}^2u_\varepsilon )^2\partial _{x}^6u_\varepsilon \mathrm{d}x +36q\int _{{\mathbb {R}}}(\partial _x u_\varepsilon )^2\partial _{x}^3u_\varepsilon \partial _{x}^6u_\varepsilon \mathrm{d}x \\&\quad +34q\int _{{\mathbb {R}}}u_\varepsilon \partial _{x}^2u_\varepsilon \partial _{x}^3u_\varepsilon \partial _{x}^6u_\varepsilon \mathrm{d}x+18q\int _{{\mathbb {R}}}u_\varepsilon \partial _x u_\varepsilon \partial _{x}^4u_\varepsilon \partial _{x}^6u_\varepsilon \mathrm{d}x \\&\quad +2q\int _{{\mathbb {R}}}u_\varepsilon ^2\partial _{x}^5u_\varepsilon \partial _{x}^6u_\varepsilon \mathrm{d}x\\&=-54q\int _{{\mathbb {R}}}(\partial _{x}^2u_\varepsilon )^3\partial _{x}^5u_\varepsilon \mathrm{d}x -214q\int _{{\mathbb {R}}}\partial _x u_\varepsilon \partial _{x}^2u_\varepsilon \partial _{x}^3u_\varepsilon \partial _{x}^5u_\varepsilon \mathrm{d}x\\&\quad -36q\int _{{\mathbb {R}}}(\partial _x u_\varepsilon )^2\partial _{x}^4u_\varepsilon \partial _{x}^5u_\varepsilon \mathrm{d}x-34q\int _{{\mathbb {R}}}u_\varepsilon (\partial _{x}^2u_\varepsilon )^2\partial _{x}^3u_\varepsilon \partial _{x}^5u_\varepsilon \mathrm{d}x\\&\quad -34q\int _{{\mathbb {R}}}u_\varepsilon \partial _{x}^2u_\varepsilon \partial _{x}^4u_\varepsilon \partial _{x}^5u_\varepsilon \mathrm{d}x -18q\int _{{\mathbb {R}}}(\partial _x u_\varepsilon )^2\partial _{x}^4u_\varepsilon \partial _{x}^5u_\varepsilon \mathrm{d}x\\&\quad -18q\int _{{\mathbb {R}}}u_\varepsilon \partial _{x}^2u_\varepsilon \partial _{x}^4u_\varepsilon \partial _{x}^5u_\varepsilon \mathrm{d}x -22q\int _{{\mathbb {R}}}u_\varepsilon \partial _x u_\varepsilon (\partial _{x}^5u_\varepsilon )^2 \mathrm{d}x\\&=-54q\int _{{\mathbb {R}}}(\partial _{x}^2u_\varepsilon )^3\partial _{x}^5u_\varepsilon \mathrm{d}x-214q\int _{{\mathbb {R}}}\partial _x u_\varepsilon \partial _{x}^2u_\varepsilon \partial _{x}^3u_\varepsilon \partial _{x}^5u_\varepsilon \mathrm{d}x\\&\quad +106q\int _{{\mathbb {R}}}\partial _x u_\varepsilon \partial _{x}^2u_\varepsilon (\partial _{x}^4u_\varepsilon )^2 \mathrm{d}x -34q\int _{{\mathbb {R}}}u_\varepsilon (\partial _{x}^2u_\varepsilon )^2\partial _{x}^3u_\varepsilon \partial _{x}^5u_\varepsilon \mathrm{d}x\\&\quad +52q\int _{{\mathbb {R}}}u_\varepsilon \partial _{x}^3u_\varepsilon (\partial _{x}^4u_\varepsilon )^2 \mathrm{d}x -22q\int _{{\mathbb {R}}}u_\varepsilon \partial _x u_\varepsilon (\partial _{x}^5u_\varepsilon )^2 \mathrm{d}x. \end{aligned}$$

Consequently, by (A.7),

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left\| \partial _{x}^5u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+2\varepsilon \left\| \partial _{x}^8u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\nonumber \\&\quad =-20\kappa \int _{{\mathbb {R}}}(\partial _{x}^3u_\varepsilon )^2\partial _{x}^5u_\varepsilon \mathrm{d}x+15\kappa \int _{{\mathbb {R}}}\partial _{x}^3u_\varepsilon (\partial _{x}^4u_\varepsilon )^2 \mathrm{d}x\nonumber \\&\qquad -11\kappa \int _{{\mathbb {R}}}\partial _x u_\varepsilon (\partial _{x}^5u_\varepsilon )^2 \mathrm{d}x-54q\int _{{\mathbb {R}}}(\partial _{x}^2u_\varepsilon )^3\partial _{x}^5u_\varepsilon \mathrm{d}x\nonumber \\&\qquad -214q\int _{{\mathbb {R}}}\partial _x u_\varepsilon \partial _{x}^2u_\varepsilon \partial _{x}^3u_\varepsilon \partial _{x}^5u_\varepsilon \mathrm{d}x+106q\int _{{\mathbb {R}}}\partial _x u_\varepsilon \partial _{x}^2u_\varepsilon (\partial _{x}^4u_\varepsilon )^2 \mathrm{d}x\nonumber \\&\qquad -34q\int _{{\mathbb {R}}}u_\varepsilon (\partial _{x}^2u_\varepsilon )^2\partial _{x}^3u_\varepsilon \partial _{x}^5u_\varepsilon \mathrm{d}x+52q\int _{{\mathbb {R}}}u_\varepsilon \partial _{x}^3u_\varepsilon (\partial _{x}^4u_\varepsilon )^2 \mathrm{d}x\nonumber \\&\qquad -22q\int _{{\mathbb {R}}}u_\varepsilon \partial _x u_\varepsilon (\partial _{x}^5u_\varepsilon )^2 \mathrm{d}x. \end{aligned}$$

(A.8)

Due to (2.5), (2.6), (2.24), (2.25) and the Young inequality,

$$\begin{aligned}&20\vert \kappa \vert \int _{{\mathbb {R}}}(\partial _{x}^3u_\varepsilon )^2\vert \partial _{x}^5u_\varepsilon \vert \mathrm{d}x\\&\quad \le 10\kappa ^2\int _{{\mathbb {R}}}(\partial _{x}^3u_\varepsilon )^4 \mathrm{d}x + 10\left\| \partial _{x}^5u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\quad \le 10\kappa ^2\left\| \partial _{x}^3u_\varepsilon \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^3u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+10\left\| \partial _{x}^5u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\quad \le C(T)+10\left\| \partial _{x}^5u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&15\vert \kappa \vert \int _{{\mathbb {R}}}\vert \partial _{x}^3u_\varepsilon \vert (\partial _{x}^4u_\varepsilon )^2 \mathrm{d}x\le 15\vert \kappa \vert \left\| \partial _{x}^3u_\varepsilon \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^4u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\le C(T),\\&11\vert \kappa \vert \int _{{\mathbb {R}}}\vert \partial _x u_\varepsilon \vert (\partial _{x}^5u_\varepsilon )^2 \mathrm{d}x\\&\quad \le 11\vert \kappa \vert \left\| \partial _x u_\varepsilon \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^5u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\le C(T)\left\| \partial _{x}^5u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&54\vert q\vert \int _{{\mathbb {R}}}\vert \partial _{x}^2u_\varepsilon \vert ^3\vert \partial _{x}^5u_\varepsilon \vert \mathrm{d}x\\&\quad \le 27q^2\int _{{\mathbb {R}}}(\partial _{x}^2u_\varepsilon )^6 \mathrm{d}x+27\left\| \partial _{x}^5u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\quad \le 27q^2\left\| \partial _{x}^2u_\varepsilon \right\| ^4_{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^2u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+ 27\left\| \partial _{x}^5u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\quad \le C(T)+27\left\| \partial _{x}^5u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&214\vert q\vert \int _{{\mathbb {R}}}\vert \partial _x u_\varepsilon \partial _{x}^2u_\varepsilon \vert \vert \partial _{x}^3u_\varepsilon \partial _{x}^5u_\varepsilon \vert \mathrm{d}x\\&\quad \le 72q^2\int _{{\mathbb {R}}}(\partial _x u_\varepsilon )^2(\partial _{x}^2u_\varepsilon )^2\mathrm{d}x+72\int _{{\mathbb {R}}}(\partial _{x}^3u_\varepsilon )^2(\partial _{x}^5u_\varepsilon )^2 \mathrm{d}x\\&\quad \le 72q^2\left\| \partial _x u_\varepsilon \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^2u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})} \\&\qquad + 72\left\| \partial _{x}^3u_\varepsilon \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^5u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\quad \le C(T) +C(T)\left\| \partial _{x}^5u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\ \end{aligned}$$

$$\begin{aligned}&106\vert q\vert \int _{{\mathbb {R}}}\vert \partial _x u_\varepsilon \vert \vert \partial _{x}^2u_\varepsilon \vert (\partial _{x}^4u_\varepsilon )^2 \mathrm{d}x\\&\quad \le 106\vert q\vert \left\| \partial _x u_\varepsilon \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^2u_\varepsilon \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^4u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\le C(T),\\&34\vert q\vert \int _{{\mathbb {R}}}\vert u_\varepsilon \vert (\partial _{x}^2u_\varepsilon )^2\vert \partial _{x}^3u_\varepsilon \partial _{x}^5u_\varepsilon \vert \mathrm{d}x\\&\quad \le 17q^2\int _{{\mathbb {R}}}u_\varepsilon ^2(\partial _{x}^2u_\varepsilon )^4 \mathrm{d}x+ 17\int _{{\mathbb {R}}}(\partial _{x}^3u_\varepsilon )^2(\partial _{x}^5u_\varepsilon )^2 \mathrm{d}x \\&\quad \le 17q^2\left\| u_\varepsilon \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^2u_\varepsilon \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^2u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\qquad + 17\left\| \partial _{x}^3u_\varepsilon \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^5u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\quad \le C(T) +C(T)\left\| \partial _{x}^5u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&52\vert q\vert \int _{{\mathbb {R}}}\vert u_\varepsilon \vert \vert \partial _{x}^3u_\varepsilon \vert (\partial _{x}^4u_\varepsilon )^2 \mathrm{d}x\\&\quad \le 52\vert q\vert \left\| u_\varepsilon \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^3u_\varepsilon \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^4u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\le C(T),\\&22\vert q\vert \int _{{\mathbb {R}}}\vert u_\varepsilon \vert \vert \partial _x u_\varepsilon \vert (\partial _{x}^5u_\varepsilon )^2 \mathrm{d}x\\&\quad \le 22\vert q\vert \left\| u_\varepsilon \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _x u_\varepsilon \right\| _{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _{x}^5u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\quad \le C(T)\left\| \partial _{x}^5u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}. \end{aligned}$$

It follows from (A.8) that

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left\| \partial _{x}^5u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+2\varepsilon \left\| \partial _{x}^8u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\quad \le C(T)\left\| \partial _{x}^5u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+C(T). \end{aligned}$$

The Gronwall Lemma and (A.1) give

$$\begin{aligned}&\left\| \partial _{x}^5u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+2\varepsilon e^{C(T)t}\int _{0}^{t}\left\| \partial _{x}^8u_\varepsilon (s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s\\&\quad \le C_0+ C(T)e^{C(T)t}\int _{0}^{t}e^{-C(T)s} \mathrm{d}s \le C(T), \end{aligned}$$

that is (A.4).

Finally, we prove (A.5). Thanks to (2.24), (A.4) and the Hölder inequality,

$$\begin{aligned} (\partial _{x}^4u_\varepsilon (t,x))^2&=2\int _{-\infty }^{x}\partial _{x}^4u_\varepsilon \partial _{x}^5u_\varepsilon \mathrm{d}x\le 2\int _{{\mathbb {R}}}\vert \partial _{x}^4u_\varepsilon \vert \vert \partial _{x}^5u_\varepsilon \vert \mathrm{d}x\\&\le 2\left\| \partial _{x}^4u_\varepsilon (t,\cdot ) \right\| _{L^2({\mathbb {R}})}\left\| \partial _{x}^5u_\varepsilon (t,\cdot ) \right\| _{L^2({\mathbb {R}})}\le C(T). \end{aligned}$$

Hence,

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

which gives (A.5). $\square$

Lemma A.2

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

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

(A.9)

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

Proof

Let $0\le t\le T$. Multiplying (2.1) by $2\partial _tu_\varepsilon$, thanks to (1.2), we get

$$\begin{aligned} \begin{aligned} -2\varepsilon \partial _tu_\varepsilon \partial _{x}^6u_\varepsilon +2(\partial _tu_\varepsilon )^2&= -2\kappa u_\varepsilon \partial _x u_\varepsilon \partial _tu_\varepsilon -3qu_\varepsilon ^2\partial _x u_\varepsilon \partial _tu_\varepsilon \\&\quad - 2\alpha \partial _{x}^3u_\varepsilon \partial _tu_\varepsilon -2\gamma \partial _{x}^5u_\varepsilon \partial _tu_\varepsilon . \end{aligned} \end{aligned}$$

(A.10)

Since

$$\begin{aligned} -2\varepsilon \int _{{\mathbb {R}}}\partial _tu_\varepsilon \partial _{x}^6u_\varepsilon \mathrm{d}x=\varepsilon \frac{\mathrm{d}}{\mathrm{d}t}\left\| \partial _{x}^3u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}, \end{aligned}$$

an integration on ${\mathbb {R}}$ of (A.10) gives

$$\begin{aligned} \begin{aligned}&\varepsilon \frac{\mathrm{d}}{\mathrm{d}t}\left\| \partial _{x}^3u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+2\left\| \partial _tu_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\quad =-4\kappa \int _{{\mathbb {R}}}u_\varepsilon \partial _x u_\varepsilon \partial _tu_\varepsilon \mathrm{d}x-6q\int _{{\mathbb {R}}}u_\varepsilon ^2\partial _x u_\varepsilon \partial _tu_\varepsilon \mathrm{d}x \\&\qquad - 2\alpha \int _{{\mathbb {R}}}\partial _{x}^3u_\varepsilon \partial _tu_\varepsilon \mathrm{d}x -2\gamma \int _{{\mathbb {R}}}\partial _{x}^5u_\varepsilon \partial _tu_\varepsilon \mathrm{d}x. \end{aligned} \end{aligned}$$

(A.11)

Due (2.6), (2.25), (A.4) and the Young inequality,

$$\begin{aligned}&2\vert \kappa \vert \int _{{\mathbb {R}}}\vert u_\varepsilon \partial _x u_\varepsilon \vert \vert \partial _tu_\varepsilon \vert \mathrm{d}x=2\int _{{\mathbb {R}}}\left| \kappa \sqrt{D_1}u_\varepsilon \partial _x u_\varepsilon \right| \left| \frac{\partial _tu_\varepsilon }{\sqrt{D_1}}\right| \mathrm{d}x\\&\quad \le 2\kappa ^2D_1\int _{{\mathbb {R}}}u_\varepsilon ^2(\partial _x u_\varepsilon )^2 \mathrm{d}x +\frac{2}{D_1}\left\| \partial _tu_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\quad \le 2\kappa ^2D_1\left\| u_\varepsilon \right\| ^2_{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _x u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+ \frac{2}{D_1}\left\| \partial _tu_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\quad \le D_1C(T)+\frac{2}{D_1}\left\| \partial _tu_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&6\vert q\int _{{\mathbb {R}}}\vert u_\varepsilon ^2\partial _x u_\varepsilon \vert \vert \partial _tu_\varepsilon \vert \mathrm{d}x=6\int _{{\mathbb {R}}}\left| q\sqrt{D_1}u_\varepsilon ^2\partial _x u_\varepsilon \right| \left| \frac{\partial _tu_\varepsilon }{\sqrt{D_1}}\right| \mathrm{d}x\\&\quad \le 3q^2D_1\int _{{\mathbb {R}}}u_\varepsilon ^4(\partial _x u_\varepsilon )^2 \mathrm{d}x + \frac{3}{D_1}\left\| \partial _tu_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\quad \le 3q^2D_1\left\| u_\varepsilon \right\| ^4_{L^{\infty }((0,T)\times {\mathbb {R}})}\left\| \partial _x u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+\frac{3}{D_1}\left\| \partial _tu_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\quad \le D_1C(T)+\frac{3}{D_1}\left\| \partial _tu_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert \alpha \vert \int _{{\mathbb {R}}}\vert \partial _{x}^3u_\varepsilon \vert \vert \partial _tu_\varepsilon \vert \mathrm{d}x=2\int _{{\mathbb {R}}}\left| \alpha \sqrt{D_1}\partial _{x}^3u_\varepsilon \right| \left| \frac{\partial _tu_\varepsilon }{\sqrt{D_1}}\right| \mathrm{d}x\\&\quad \le \alpha ^2D_1\left\| \partial _{x}^3u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+\frac{1}{D_1}\left\| \partial _tu_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\quad \le D_1C(T)+\frac{1}{D_1}\left\| \partial _tu_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})},\\&2\vert \gamma \vert \int _{{\mathbb {R}}}\vert \partial _{x}^5u_\varepsilon \vert \vert \partial _tu_\varepsilon \vert \mathrm{d}x=2\int _{{\mathbb {R}}}\left| \gamma \sqrt{D_1}\partial _{x}^5u_\varepsilon \right| \left| \frac{\partial _tu_\varepsilon }{\sqrt{D_1}}\right| \mathrm{d}x\\&\quad \le \gamma ^2D_1\left\| \partial _{x}^5u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+ \frac{1}{D_1}\left\| \partial _tu_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\\&\quad \le D_1C(T)+ \frac{1}{D_1}\left\| \partial _tu_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}, \end{aligned}$$

where $D_1$ is a positive constant which will be specified later. Therefore, by (A.11),

$$\begin{aligned} \varepsilon \frac{\mathrm{d}}{\mathrm{d}t}\left\| \partial _{x}^3u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+\left( 2-\frac{7}{D_1}\right) \left\| \partial _tu_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\le C(T). \end{aligned}$$

Choosing $D_1=7$, we have that

$$\begin{aligned} \varepsilon \frac{\mathrm{d}}{\mathrm{d}t}\left\| \partial _{x}^3u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+\left\| \partial _tu_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\le C(T). \end{aligned}$$

(A.3) and an integration on (0, t) give

$$\begin{aligned} \varepsilon \left\| \partial _{x}^3u_\varepsilon (t,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}+\int _{0}^t\left\| \partial _tu_\varepsilon (s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s\le C_0+C(T)t\le C(T), \end{aligned}$$

that is (A.9). $\square$

Using the Sobolev immersion theorem, we have the following result.

Lemma A.3

Fix $T>0$. There exist a subsequence $\{u_{\varepsilon _k}\}_{k\in {\mathbb {N}}}$ of $\{u_\varepsilon \}_{\varepsilon >0}$ and a limit function u which satisfies (A.2) such that

$$\begin{aligned} u_{\varepsilon _k}\rightarrow u \text { a.e. and in } L^{p}_{loc}((0,T)\times {\mathbb {R}}), 1\le p<\infty . \end{aligned}$$

(A.12)

Moreover, u is solution of (1.1).

Proof

Thanks to Lemmas 2.1, 2.2, 2.4, A.1 and A.2,

$$\begin{aligned} \{u_\varepsilon \}_{\varepsilon >0} \text { is uniformly bounded in } H^1((0,T)\times {\mathbb {R}}). \end{aligned}$$

(A.13)

Consequently, (A.13) gives (A.12).

Thanks to Lemmas 2.1, 2.2, 2.4 and A.1, we get

$$\begin{aligned} u\in L^{\infty }(0,T;H^5({\mathbb {R}})). \end{aligned}$$

Therefore, (A.2) holds and u is solution of (1.1). $\square$

Now, we prove Theorem A.1.

Proof of Theorem A.1

Lemma A.3 gives the existence of a solution of (1.1) satisfying (A.2). Arguing as in Theorem 1.1, we have (1.12). $\square$