A comparison of different DNA extraction methods and molecular techniques for the detection and identification of foodborne pathogens

Spyridon Andreas Papatheodorou; Panagiotis Halvatsiotis; Dimitra Houhoula; Spyridon Andreas Papatheodorou; Panagiotis Halvatsiotis; Dimitra Houhoula

doi:10.3934/microbiol.2021019

AIMS Microbiology

2021, Volume 7, Issue 3: 304-319. doi: 10.3934/microbiol.2021019

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Research article

A comparison of different DNA extraction methods and molecular techniques for the detection and identification of foodborne pathogens

1.
Department of Food Science & Technology, School of Food Sciences, University of West Attica
2.
2nd Propaedeutic Department of Internal Medicine, Medical School, National and Kapodistrian University of Athens, “ATTIKON” University Hospital, Chaidari Greece

Received: 02 June 2021 Accepted: 10 August 2021 Published: 09 September 2021

Foodborne infections continue to plague Europe. Food safety monitoring is in crisis as the existing techniques for detecting pathogens do not keep up with the global rising of food production and consumption. Thus, the development of innovative techniques for detecting and identifying pathogenic bacteria has become critical. The aim of the present study was firstly to develop an innovative simple and low cost method of extracting bacterial DNA from contaminated food and water samples with Salmonella enteric(a) subsp. enteric(a) serovar Typhimurium and Listeria monocytogenes and its comparison with two commercial DNA extraction kits (Qiagen, Macherey-Nagel). Finally, pathogens' detection using two molecular techniques (PCR-electrophoresis, LAMP), in order to evaluate the best combination of DNA extraction and identification based on their sensitivity, cost, rapidity and simplicity. Considering the above criteria, among them, best was proved an in-house bacterial DNA extraction method, based on the chloroform-isoamyl alcohol protocol, with certain modifications. This technique showed statistically similar results in terms of sensitivity, compared to the commercial kits, while at the same time maintained high rapidity and much lower cost. Lastly, between the molecular techniques, LAMP was found more promising considering its simplicity, high rapidity and sensitivity. Conclusively, the in-house DNA extraction method along with the LAMP technique, was proven to be the best among the presented combinations.

Keywords:

DNA extraction methods,
Salmonella enterica serovar Typhimurium,
Listeria monocytogenes,
PCR,
LAMP,
Limit of detection

Citation: Spyridon Andreas Papatheodorou, Panagiotis Halvatsiotis, Dimitra Houhoula. A comparison of different DNA extraction methods and molecular techniques for the detection and identification of foodborne pathogens[J]. AIMS Microbiology, 2021, 7(3): 304-319. doi: 10.3934/microbiol.2021019

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Abstract

1. Introduction

The equation:

$\begin{equation} \begin{cases} \partial_t u + \partial_x f(u)-\beta^2 \partial_{x}^2 u+\delta \partial_{x}^3 u+\kappa u +\gamma^2\vert u\vert u = 0, &\quad 0 < t < T, \quad x\in \mathbb{R},\\ u(0,x) = u_0(x), &\quad x\in \mathbb{R}, \end{cases} \end{equation}$

(1.1)

was originally derived in ^[14,17] with $f(u) = au^2$ focusing on microbubbles coated by viscoelastic shells. These structures are crucial in ultrasound diagnosis using contrast agents, and the dynamics of individual coated bubbles are explored, taking into account nonlinear competition and dissipation factors such as dispersion, thermal effects, and drag force.

The coefficients $\beta^2$ , $\delta$ , $\kappa$ , and $\gamma^2$ are related to the dissipation, the dispersion, the thermal conduction dissipation, and to the drag force, repsctively.

If $\kappa = \gamma = 0$ , we obtain the Kudryashov-Sinelshchikov ^[18] Korteweg-de Vries-Burgers ^[3,20] equation

$\begin{equation} \partial_t u +a \partial_x u^2-\beta^2 \partial_{x}^2 u+\delta \partial_{x}^3 u = 0, \end{equation}$

(1.2)

that models pressure waves in liquids with gas bubbles, taking into account heat transfer and viscosity. The mathematical results on Eq (1.2) are the following:

● analysis of exact solutions in ^[13],

● existence of the traveling waves in ^[2],

● well-posedness and asymptotic behavior in ^[7,11].

If $\beta = 0$ , we derive the Korteweg-de Vries equation:

$\begin{equation} \partial_t u +a \partial_x u^2+\delta \partial_{x}^3 u = 0, \end{equation}$

(1.3)

which describes surface waves of small amplitude and long wavelength in shallow water. Here, $u(t, x)$ represents the wave height above a flat bottom, $x$ corresponds to the distance in the propagation direction, and $t$ denotes the elapsed time. In ^{[4,6,10,12,15,16]}, the completele integrability of Eq (1.3) and the existence of solitary wave solutions are proved.

Through the manuscript, we will assume

● on the coefficients

$\begin{equation} \beta,\,\delta,\,\kappa,\,\gamma\in \mathbb{R}, \quad \beta,\,\delta,\,\gamma\ne0; \end{equation}$

(1.4)

● on the flux $f$ , one of the following conditions:

$\begin{align} &f(u) = a u^2+bu^3, \end{align}$

(1.5)

$\begin{align} &f\in C^1( \mathbb{R}), \quad \vert f'(u)\vert\le C_0\left(1+\vert u\vert\right), \quad u\in \mathbb{R}, \end{align}$

(1.6)

for some positive constant $C_0$ ;

● on the initial value

$\begin{equation} u_0\in H^1( \mathbb{R}). \end{equation}$

(1.7)

The main result of this paper is the following theorem.

Theorem 1.1. Assume Eqs (1.5)–(1.7). For fixed $T > 0$ , there exists a unique distributional solution $u$ of Eq (1.1), such that

$\begin{equation} \begin{split} u \in&\, L^{\infty}(0,T;H^1( \mathbb{R}))\cap L^4(0,\,T;W^{1,\,4}( \mathbb{R}))\cap L^6(0,\,T;W^{1,\,6}( \mathbb{R}))\\ \partial_{x}^2 u\in&\, L^2((0,\,T)\times \mathbb{R}). \end{split} \end{equation}$

(1.8)

Moreover, if $u_1$ and $u_2$ are solutions to Eq (1.1) corresponding to the initial conditions $u_{1, 0}$ and $u_{2, 0}$ , respectively, it holds that:

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

(1.9)

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

Observe that Theorem 1.1 gives the well-posedness of (1.1), without conditions on the constants. Moreover, the proof of Theorem 1.1 is based on the Aubin-Lions Lemma ^[5,21]. The analysis of Eq (1.1) is more delicate than the one of Eq (1.2) due to the presence of the nonlinear sources and the very general assumptions on the coefficients.

The structure of the paper is outlined as follows. Section 2 is dedicated to establishing several a priori estimates for a vanishing viscosity approximation of Eq (1.1). These estimates are crucial for proving our main result, which is presented in Section 3.

2. Vanishing viscosity approximation

To establish existence, we utilize a vanishing viscosity approximation of equation (1.1), as discussed in ^[19]. Let $0 < \varepsilon < 1$ be a small parameter, and denote by $u_\varepsilon\in C^\infty([0, T)\times \mathbb{R})$ the unique classical solution to the following problem ^[1,9]:

$\begin{equation} \begin{cases} \partial_t u_\varepsilon + \partial_x f( u_\varepsilon)-\beta^2 \partial_{x}^2 u_\varepsilon+\delta \partial_{x}^3 u_\varepsilon+\kappa u\\ \qquad\quad +\gamma^2\vert u\vert u = - \varepsilon \partial_{x}^4 u_\varepsilon, &\quad 0 < t < T, \quad x\in \mathbb{R},\\ u_{ \varepsilon}(0,x) = u_{ \varepsilon,\,0}(x), &\quad x\in \mathbb{R}, \end{cases} \end{equation}$

(2.1)

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

$\begin{equation} \left\| {u_{ \varepsilon,\,0}} \right\|_{H^1( \mathbb{R})}\le \left\| {u_0} \right\|_{H^1( \mathbb{R})}. \end{equation}$

(2.2)

Let us prove some a priori estimates on $u_\varepsilon$ , denoting with $C_0$ 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 lemma:

Lemma 2.1. Let $T > 0$ be fixed. There exists a constant $C(T) > 0$ , which does not depend on $\varepsilon$ , such that

$\begin{equation} \begin{split} \left\| { u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}&+2\gamma^2e^{\vert\kappa\vert t}\displaystyle {\int}_{0}^{t}\!\!\displaystyle {\int}_{ \mathbb{R}}e^{-\vert\kappa\vert s} u_\varepsilon^2\vert u_\varepsilon\vert dsdx\\ &+2\beta^2e^{\vert\kappa\vert t}\displaystyle {\int}_{0}^{t}e^{-\vert\kappa\vert s}\left\| { \partial_x u_\varepsilon(s,\cdot)} \right\|^2_{L^2( \mathbb{R})} ds\\ &+2 \varepsilon e^{\vert\kappa\vert t}\displaystyle {\int}_{0}^{t}e^{-\vert\kappa\vert s}\left\| { \partial_{x}^2 u_\varepsilon(s,\cdot)} \right\|^2_{L^2(R)}\le C(T), \end{split} \end{equation}$

(2.3)

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

Proof. For $0\le t\le T$ . Multiplying equations (2.1) by $2 u_\varepsilon$ , and integrating over $\mathbb{R}$ yields

$\begin{align*} \frac{d}{dt}&\left\| { u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})} = 2\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon \partial_t u_\varepsilon dx\\ = &\underbrace{-2\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon f'( u_\varepsilon) \partial_x u_\varepsilon dx}_{ = 0}+2\beta^2\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon \partial_{x}^2 u_\varepsilon dx-2\delta\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon \partial_{x}^3 u_\varepsilon dx \\ &-\kappa\left\| { u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}-2\gamma^2\displaystyle {\int}_{ \mathbb{R}}\vert u_\varepsilon\vert u_\varepsilon^2 dx-2 \varepsilon\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon \partial_{x}^4 u_\varepsilon dx \\ = &-2\beta^2\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}+2\delta\displaystyle {\int}_{ \mathbb{R}} \partial_x u_\varepsilon \partial_{x}^2 u_\varepsilon dx\\ &-\kappa\left\| { u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}-2\gamma^2\displaystyle {\int}_{ \mathbb{R}}\vert u_\varepsilon\vert u_\varepsilon^2 dx+2 \varepsilon\displaystyle {\int}_{ \mathbb{R}} \partial_x u_\varepsilon \partial_{x}^3 u_\varepsilon dx\\ = &-2\beta^2\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}-\kappa\left\| { u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}\\ &-2\gamma^2\displaystyle {\int}_{ \mathbb{R}}\vert u_\varepsilon\vert u_\varepsilon^2 dx-2 \varepsilon\left\| { \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}. \end{align*}$

Thus, it follows that

$\begin{align*} \frac{d}{dt}\left\| { u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}+&2\beta^2\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}\\ &+2\gamma^2\displaystyle {\int}_{ \mathbb{R}}\vert u_\varepsilon\vert u_\varepsilon^2 dx+2 \varepsilon\left\| { \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}\\ = &\kappa\left\| { u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}\le \vert\kappa\vert \left\| { u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}. \end{align*}$

Therefore, applying the Gronwall's lemma and using Eq (2.2), we obtain

$\begin{align*} \left\| { u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}&+2\beta^2 e^{\vert\kappa\vert t}\displaystyle {\int}_{0}^{t}e^{-\vert\kappa\vert s}\left\| { \partial_x u_\varepsilon(s,\cdot)} \right\|^2_{L^2( \mathbb{R})}ds\\ &+ 2\gamma^2e^{\vert\kappa\vert t}\displaystyle {\int}_{0}^t\!\!\displaystyle {\int}_{ \mathbb{R}}e^{-\vert\kappa\vert t}\vert u_\varepsilon\vert u_\varepsilon^2 dsdx+2 \varepsilon\left\| { \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}\\ &+2 \varepsilon e^{\vert\kappa\vert t}\displaystyle {\int}_{0}^{t}e^{-\vert\kappa\vert s} \left\| { \partial_{x}^2 u_\varepsilon(s,\cdot)} \right\|^2_{L^2( \mathbb{R})}ds\\ \le& C_0e^{\vert\kappa\vert t}\le C(T), \end{align*}$

which gives Eq (2.3). □

Lemma 2.2. Fix $T > 0$ and assume (1.5). There exists a constant $C(T) > 0$ , independent of $\varepsilon$ , such that

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

(2.4)

$\begin{align} \left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}+\beta^2\displaystyle {\int}_{0}^{t}\left\| { \partial_{x}^2 u_\varepsilon(s,\cdot)} \right\|^2_{L^2( \mathbb{R})}ds& \end{align}$

(2.5)

$\begin{align} +2 \varepsilon\displaystyle {\int}_{0}^{t}\left\| { \partial_{x}^3 u_\varepsilon(s,\cdot)} \right\|^2_{L^2( \mathbb{R})}ds&\le C(T),\\ \displaystyle {\int}_{0}^{t}\left\| { \partial_x u_\varepsilon(s,\cdot)} \right\|^4_{L^4( \mathbb{R})} ds&\le C(T), \end{align}$

(2.6)

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

Proof. Let $0\le t\le T$ . Consider $A, \, B$ as two real constants, which will be specified later. Thanks to Eq (1.5), multiplying Eq (2.1) by

$\begin{equation*} -2 \partial_{x}^2 u_\varepsilon+A u_\varepsilon^2+B u_\varepsilon^3, \end{equation*}$

we have that

$\begin{align} &\left(-2 \partial_{x}^2 u_\varepsilon+A u_\varepsilon^2+B u_\varepsilon^3\right) \partial_t u_\varepsilon+ 2a\left(-2 \partial_{x}^2 u_\varepsilon+A u_\varepsilon^2+B u_\varepsilon^3\right) u_\varepsilon \partial_x u_\varepsilon\\ &\qquad\quad+3b\left(-2 \partial_{x}^2 u_\varepsilon+A u_\varepsilon^2+B u_\varepsilon^3\right) u_\varepsilon^2 \partial_x u_\varepsilon-\beta^2\left(-2 \partial_{x}^2 u_\varepsilon+A u_\varepsilon^2+B u_\varepsilon^3\right) \partial_{x}^2 u_\varepsilon \\ &\qquad\quad+\delta\left(-2 \partial_{x}^2 u_\varepsilon+A u_\varepsilon^2+B u_\varepsilon^3\right) \partial_{x}^3 u_\varepsilon+\kappa\left(-2 \partial_{x}^2 u_\varepsilon+A u_\varepsilon^2+B u_\varepsilon^3\right) u_\varepsilon\\ &\qquad\quad+\gamma^2\left(-2 \partial_{x}^2 u_\varepsilon+A u_\varepsilon^2+B u_\varepsilon^3\right)\vert u_\varepsilon\vert u_\varepsilon = - \varepsilon\left(-2 \partial_{x}^2 u_\varepsilon+A u_\varepsilon^2+B u_\varepsilon^3\right) \partial_{x}^4 u_\varepsilon. \end{align}$

(2.7)

Observe that

$\begin{align*} &\displaystyle {\int}_{ \mathbb{R}}\left(-2 \partial_{x}^2 u_\varepsilon+A u_\varepsilon^2+B u_\varepsilon^3\right) \partial_t u_\varepsilon dx\\ &\qquad = \frac{d}{dt}\left(\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}+\frac{A}{3}\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^3dx +\frac{B}{4}\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^4 dx\right),\\ &2a\displaystyle {\int}_{ \mathbb{R}}\left(-2 \partial_{x}^2 u_\varepsilon+A u_\varepsilon^2+B u_\varepsilon^3\right) u_\varepsilon \partial_x u_\varepsilon dx = -4a\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon \partial_x u_\varepsilon \partial_{x}^2 u_\varepsilon dx,\\ &3b\displaystyle {\int}_{ \mathbb{R}}\left(-2 \partial_{x}^2 u_\varepsilon+A u_\varepsilon^2+B u_\varepsilon^3\right) u_\varepsilon^2 \partial_x u_\varepsilon dx = -6b\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^2 \partial_x u_\varepsilon \partial_{x}^2 u_\varepsilon dx,\\ &{-\beta^2\displaystyle {\int}_{ \mathbb{R}}\left(-2 \partial_{x}^2 u_\varepsilon+A u_\varepsilon^2+B u_\varepsilon^3\right) \partial_{x}^2 u_\varepsilon dx}\\ &\qquad = 2\beta^2\left\| { \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}+2A\beta^2\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon( \partial_x u_\varepsilon)^2 dx +3B\beta^2\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^2( \partial_x u_\varepsilon)^2dx, \\ &\delta\displaystyle {\int}_{ \mathbb{R}}\left(-2 \partial_{x}^2 u_\varepsilon+A u_\varepsilon^2+B u_\varepsilon^3\right) \partial_{x}^3 u_\varepsilon dx = -2A\delta\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon \partial_x u_\varepsilon \partial_{x}^2 u_\varepsilon dx -3B\delta\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^2 \partial_x u_\varepsilon \partial_{x}^2 u_\varepsilon dx, \\ &\kappa\displaystyle {\int}_{ \mathbb{R}}\left(-2 \partial_{x}^2 u_\varepsilon+A u_\varepsilon^2+B u_\varepsilon^3\right) u_\varepsilon dx\\ &\qquad = {2\kappa\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}+{A\kappa}\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^3 dx +{B\kappa}\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^4 dx,}\\ &\gamma^2\displaystyle {\int}_{ \mathbb{R}}\left(-2 \partial_{x}^2 u_\varepsilon+A u_\varepsilon^2+B u_\varepsilon^3\right)\vert u_\varepsilon\vert u_\varepsilon dx\\ &\qquad = -2\gamma^2\displaystyle {\int}_{ \mathbb{R}}\vert u_\varepsilon\vert u_\varepsilon \partial_{x}^2 u_\varepsilon dx +A\gamma^2\displaystyle {\int}_{ \mathbb{R}}\vert u\vert u_\varepsilon^3dx +B\gamma^2\displaystyle {\int}_{ \mathbb{R}}\vert u_\varepsilon\vert u^4 dx,\\ &- \varepsilon\displaystyle {\int}_{ \mathbb{R}}\left(-2 \partial_{x}^2 u_\varepsilon+A u_\varepsilon^2+B u_\varepsilon^3\right) \partial_{x}^4 u_\varepsilon dx\\ &\qquad = -2 \varepsilon\left\| { \partial_{x}^3 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}+2A \varepsilon\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon \partial_x u_\varepsilon{ \partial_{x}^3 u_\varepsilon} dx+3B \varepsilon\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^2 \partial_x u_\varepsilon \partial_{x}^3 u_\varepsilon dx\\ &\qquad = -2 \varepsilon\left\| { \partial_{x}^3 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}-A \varepsilon\displaystyle {\int}_{ \mathbb{R}}( \partial_x u_\varepsilon)^3 dx -6B \varepsilon\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon( \partial_x u_\varepsilon)^2 \partial_{x}^2 u_\varepsilon dx\\ &\qquad\quad -3B \varepsilon\left\| { u_\varepsilon(t,\cdot) \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}\\ &\qquad = -2 \varepsilon\left\| { \partial_{x}^3 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}-A \varepsilon\displaystyle {\int}_{ \mathbb{R}}( \partial_x u_\varepsilon)^3 dx +2B \varepsilon\displaystyle {\int}_{ \mathbb{R}}( \partial_x u_\varepsilon)^4dx\\ &\qquad\quad -3B \varepsilon\left\| { u_\varepsilon(t,\cdot) \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}. \end{align*}$

Therefore, an integration on $\mathbb{R}$ gives

$\begin{align*} &\frac{d}{dt}\left(\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}+\frac{A}{3}\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^3dx +\frac{B}{4}\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^4 dx\right)\\ &\qquad\quad +\beta^2\left\| { \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}+2 \varepsilon\left\| { \partial_{x}^3 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}\\ &\qquad = -\left(4a+A\delta\right)\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon \partial_x u_\varepsilon \partial_{x}^2 u_\varepsilon dx -3\left(2b+B\delta\right)\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^2 \partial_x u_\varepsilon \partial_{x}^2 u_\varepsilon dx\\ &\qquad\quad -2A\beta^2\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon( \partial_x u_\varepsilon)^2 dx -3B\beta^2\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^2( \partial_x u_\varepsilon)^2dx\\ &\qquad\quad -\kappa\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}-\frac{A\kappa}{3}\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^3 dx -\frac{B\kappa}{4}\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^4 dx\\ &\qquad\quad +2\gamma^2\displaystyle {\int}_{ \mathbb{R}}\vert u_\varepsilon\vert u_\varepsilon \partial_{x}^2 u_\varepsilon dx {-A\gamma^2\displaystyle {\int}_{ \mathbb{R}}\vert u_\varepsilon\vert u_\varepsilon^3dx-B\gamma^2\displaystyle {\int}_{ \mathbb{R}}\vert u_\varepsilon\vert u_\varepsilon^4 dx}\\ &\qquad\quad -A \varepsilon\displaystyle {\int}_{ \mathbb{R}}( \partial_x u_\varepsilon)^3 dx +2B \varepsilon\displaystyle {\int}_{ \mathbb{R}}( \partial_x u_\varepsilon)^4dx-3B \varepsilon\left\| { u_\varepsilon(t,\cdot) \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}. \end{align*}$

Taking

$\begin{equation*} \left(A,\,B\right) = \left(-\frac{4a}{\delta},\,-\frac{2b}{\delta}\right), \end{equation*}$

we get

$\begin{align} &\frac{d}{dt}\left(\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}-\frac{4a}{3\delta}\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^3dx -\frac{b}{\delta}\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^4 dx\right)\\ &\qquad\quad +2\beta^2\left\| { \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}+2 \varepsilon\left\| { \partial_{x}^3 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}\\ &\qquad = \frac{8a\beta^2}{\delta}\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon( \partial_x u_\varepsilon)^2 dx +\frac{6b\beta^2}{\delta}\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^2( \partial_x u_\varepsilon)^2dx\\ &\qquad\quad -\kappa\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}+\frac{4a\kappa}{3\delta}\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^3 dx +\frac{b\kappa}{2}\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^4 dx\\ &\qquad\quad +2\gamma^2\displaystyle {\int}_{ \mathbb{R}}\vert u_\varepsilon\vert u_\varepsilon \partial_{x}^2 u_\varepsilon dx {+\frac{4a\gamma^2}{\delta}\displaystyle {\int}_{ \mathbb{R}}\vert u_\varepsilon\vert u_\varepsilon^3dx +\frac{2b\gamma^2}{\delta}\displaystyle {\int}_{ \mathbb{R}}\vert u_\varepsilon\vert u_\varepsilon^4 dx} \\ &\qquad\quad +\frac{4a \varepsilon}{\delta}\displaystyle {\int}_{ \mathbb{R}}( \partial_x u_\varepsilon)^3 dx-\frac{4b \varepsilon}{\delta}\displaystyle {\int}_{ \mathbb{R}}( \partial_x u_\varepsilon)^4dx+\frac{6b \varepsilon}{\delta}\left\| { u_\varepsilon(t,\cdot) \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}. \end{align}$

(2.8)

Since $0 < \varepsilon < 1$ , due to the Young inequality and (2.3),

$\begin{align*} &\frac{8a\beta^2}{\delta}\displaystyle {\int}_{ \mathbb{R}}\vert u_\varepsilon\vert( \partial_x u_\varepsilon)^2 dx\\ &\qquad\le 4\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^2( \partial_x u_\varepsilon)^2 dx +\frac{4a^2\beta^4}{\delta^2}\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}\\ &\qquad\le 4\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{4a^2\beta^4}{\delta^2}\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}\\ &\qquad\le C_0\left(1+\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,T)\times \mathbb{R})}\right)\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})},\\ &\left\vert\frac{6b\beta^2}{\delta}\right\vert\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^2( \partial_x u_\varepsilon)^2dx\le \left\vert\frac{6b\beta^2}{\delta}\right\vert\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})},\\ &\vert\frac{4a\kappa}{3\delta}\vert\displaystyle {\int}_{ \mathbb{R}}\vert u_\varepsilon\vert^3 dx\le \left\vert\frac{4a\kappa}{3\delta}\right\vert\left\| { u_\varepsilon} \right\|_{L^{\infty}((0,T)\times \mathbb{R})}\left\| { u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}\\ &\qquad\le C(T)\left\| { u_\varepsilon} \right\|_{L^{\infty}((0,T)\times \mathbb{R})},\\ &\left\vert\frac{b\kappa}{2}\right\vert\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^4 dx\le \left\vert\frac{b\kappa}{2}\right\vert\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,T)\times \mathbb{R})}\left\| { u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}\\ &\qquad\le C(T)\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,T)\times \mathbb{R})}, \\ &2\gamma^2\displaystyle {\int}_{ \mathbb{R}}\vert u_\varepsilon\vert u_\varepsilon \partial_{x}^2 u_\varepsilon dx{\le}2\displaystyle {\int}_{ \mathbb{R}}\left\vert\frac{\gamma^2\vert u_\varepsilon\vert u_\varepsilon}{\beta}\right\vert\left\vert\beta \partial_{x}^2 u_\varepsilon\right\vert dx\\ &\qquad\le\frac{\gamma^4}{\beta^2}\displaystyle {\int}_{ \mathbb{R}}{ u_\varepsilon}^4dx +\beta^2\left\| { \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}\\ &\qquad \le\frac{\gamma^4}{\beta^2}\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0.T)\times \mathbb{R})}\left\| { u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})} +\beta^2\left\| { \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}\\ &\qquad\le C(T)\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,T)\times \mathbb{R})}+\beta^2\left\| { \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})},\\ &\left\vert\frac{4a\gamma^2}{\delta}\right\vert\displaystyle {\int}_{ \mathbb{R}}\vert { u_\varepsilon}\vert \vert u_\varepsilon\vert^3dx = \left\vert\frac{4a\gamma^2}{\delta}\right\vert\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^4 dx\\ &\qquad\le \left\vert\frac{4a\gamma^2}{\delta}\right\vert\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,T)\times \mathbb{R})}\left\| { u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})} \le C(T)\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,T)\times \mathbb{R})}, \\ &\left\vert\frac{2b\gamma^2}{\delta}\right\vert\displaystyle {\int}_{ \mathbb{R}}\vert u_\varepsilon\vert { u_\varepsilon}^4 dx\le \left\vert\frac{2b\gamma^2}{\delta}\right\vert\left\| { u_\varepsilon} \right\|^3_{L^{\infty}((0,\,T)\times \mathbb{R})}\left\| { u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}\\ &\qquad\le C(T)\left\| { u_\varepsilon} \right\|^3_{L^{\infty}((0,\,T)\times \mathbb{R})},\\ &\left\vert\frac{4a \varepsilon}{\delta}\right\vert\displaystyle {\int}_{ \mathbb{R}}\vert \partial_x u_\varepsilon\vert^3 dx\le\left\vert\frac{4a \varepsilon}{\delta}\right\vert\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})} +\left\vert\frac{4a \varepsilon}{\delta}\right\vert\displaystyle {\int}_{ \mathbb{R}}( \partial_x u_\varepsilon)^4 dx\\ &\qquad \le \left\vert\frac{4a}{\delta}\right\vert\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})} +\left\vert\frac{4a \varepsilon}{\delta}\right\vert\displaystyle {\int}_{ \mathbb{R}}( \partial_x u_\varepsilon)^4 dx. \end{align*}$

It follows from Eq (2.8) that

$\begin{align} &\frac{d}{dt}\left(\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}-\frac{4a}{3\delta}\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^3dx -\frac{b}{\delta}\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^4 dx\right)\\ &\qquad\quad +\beta^2\left\| { \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}+2 \varepsilon\left\| { \partial_{x}^3 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}\\ &\qquad\le C_0\left(1+\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,T)\times \mathbb{R})}\right)\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}+C(T)\left\| { u_\varepsilon} \right\|_{L^{\infty}((0,T)\times \mathbb{R})}\\ &\qquad\quad +C(T)\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,T)\times \mathbb{R})}+C(T)\left\| { u_\varepsilon} \right\|^3_{L^{\infty}((0,T)\times \mathbb{R})}\\ &\qquad\quad+C_0 \varepsilon\displaystyle {\int}_{ \mathbb{R}}( \partial_x u_\varepsilon)^4 dx +C_0 \varepsilon\left\| { u_\varepsilon(t,\cdot) \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}\\ &\qquad\quad {+C_0\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}}. \end{align}$

(2.9)

[8, Lemma 2.3] says that

$\begin{equation} \displaystyle {\int}_{ \mathbb{R}}( \partial_x u_\varepsilon)^4 dx\le 9\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^2( \partial_{x}^2 u_\varepsilon)^2 dx\le 9\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,\,T)\times \mathbb{R})}\left\| { \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}. \end{equation}$

(2.10)

Moreover, we have that

$\begin{equation} \left\| { u_\varepsilon(t,\cdot) \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})} = \displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^2( \partial_{x}^2 u_\varepsilon)^2 dx {\le}\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,\,T)\times \mathbb{R})}\left\| { \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}. \end{equation}$

(2.11)

Consequentially, by Eqs (2.9)–(2.11), we have that

$\begin{align*} &\frac{d}{dt}\left(\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}-\frac{4a}{3\delta}\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^3dx -\frac{b}{\delta}\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^4 dx\right)\\ &\qquad\quad +\beta^2\left\| { \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}+2 \varepsilon\left\| { \partial_{x}^3 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}\\ &\qquad\le C_0\left(1+\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,T)\times \mathbb{R})}\right)\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}+C(T)\left\| { u_\varepsilon} \right\|_{L^{\infty}((0,T)\times \mathbb{R})}\\ &\qquad\quad +C(T)\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,T)\times \mathbb{R})}+C(T)\left\| { u_\varepsilon} \right\|^3_{L^{\infty}((0,T)\times \mathbb{R})}\\ &\qquad\quad +C_0 \varepsilon\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,\,T)\times \mathbb{R})}\left\| { \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})} {+C_0\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}.}\notag \end{align*}$

An integration on $(0, t)$ and Eqs (2.2) and (2.3) give

$\begin{align*} &\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}-\frac{4a}{3\delta}\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^3dx -\frac{b}{\delta}\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^4 dx\\ &\qquad\quad +\beta^2\displaystyle {\int}_{0}^{t}\left\| { \partial_{x}^2 u_\varepsilon(s,\cdot)} \right\|^2_{L^2( \mathbb{R})}ds+2 \varepsilon\displaystyle {\int}_{0}^{t}\left\| { \partial_{x}^3 u_\varepsilon(s,\cdot)} \right\|^2_{L^2( \mathbb{R})}ds\\ &\qquad\le C_0\left(1+\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,T)\times \mathbb{R})}\right)\displaystyle {\int}_{0}^{t}\left\| { \partial_x u_\varepsilon(s,\cdot)} \right\|^2_{L^2( \mathbb{R})}ds\\ &\qquad\quad+C(T)\left\| { u_\varepsilon} \right\|_{L^{\infty}((0,T)\times \mathbb{R})}t +C(T)\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,T)\times \mathbb{R})}t+C(T)\left\| { u_\varepsilon} \right\|^3_{L^{\infty}((0,T)\times \mathbb{R})}t\\ &\qquad\quad +C_0 \varepsilon\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,\,T)\times \mathbb{R})}\displaystyle {\int}_{0}^{t}\left\| { \partial_{x}^2 u_\varepsilon(s,\cdot)} \right\|^2_{L^2( \mathbb{R})}ds {+C_0\displaystyle {\int}_{0}^{t}\left\| { \partial_x u_\varepsilon(s,\cdot)} \right\|^2_{L^2( \mathbb{R})}ds}\\ &\qquad\le C(T)\left(1+\left\| { u_\varepsilon} \right\|_{L^{\infty}((0,T)\times \mathbb{R})} +\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,T)\times \mathbb{R})}+\left\| { u_\varepsilon} \right\|^3_{L^{\infty}((0,T)\times \mathbb{R})}\right). \end{align*}$

Therefore, by Eq (2.3),

$\begin{align} &\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}+\beta^2\displaystyle {\int}_{0}^{t}\left\| { \partial_{x}^2 u_\varepsilon(s,\cdot)} \right\|^2_{L^2( \mathbb{R})}ds+2 \varepsilon\displaystyle {\int}_{0}^{t}\left\| { \partial_{x}^3 u_\varepsilon(s,\cdot)} \right\|^2_{L^2( \mathbb{R})}ds\\ &\qquad\le C(T)\left(1+\left\| { u_\varepsilon} \right\|_{L^{\infty}((0,T)\times \mathbb{R})} +\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,T)\times \mathbb{R})}+\left\| { u_\varepsilon} \right\|^3_{L^{\infty}((0,T)\times \mathbb{R})}\right)\\ &\qquad\quad +\frac{4a}{3\delta}\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^3dx +\frac{b}{\delta}\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^4 dx \\ &\qquad\le C(T)\left(1+\left\| { u_\varepsilon} \right\|_{L^{\infty}((0,T)\times \mathbb{R})} +\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,T)\times \mathbb{R})}+\left\| { u_\varepsilon} \right\|^3_{L^{\infty}((0,T)\times \mathbb{R})}\right)\\ &\qquad\quad +\left\vert\frac{4a}{3\delta}\right\vert\displaystyle {\int}_{ \mathbb{R}}\vert u_\varepsilon\vert^3dx +\left\vert\frac{b}{\delta}\right\vert\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^4 dx \\ &\qquad\le C(T)\left(1+\left\| { u_\varepsilon} \right\|_{L^{\infty}((0,T)\times \mathbb{R})} +\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,T)\times \mathbb{R})}+\left\| { u_\varepsilon} \right\|^3_{L^{\infty}((0,T)\times \mathbb{R})}\right)\\ &\qquad\quad +\left\vert\frac{4a}{3\delta}\right\vert \left\| { u_\varepsilon} \right\|_{L^{\infty}((0,T)\times \mathbb{R})}\left\| { u_\varepsilon({t},\cdot)} \right\|^2_{L^2( \mathbb{R})} +\left\vert\frac{b}{\delta}\right\vert\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,T)\times \mathbb{R})}\left\| { u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}\\ &\qquad\le C(T)\left(1+\left\| { u_\varepsilon} \right\|_{L^{\infty}((0,T)\times \mathbb{R})} +\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,T)\times \mathbb{R})}+\left\| { u_\varepsilon} \right\|^3_{L^{\infty}((0,T)\times \mathbb{R})}\right). \end{align}$

(2.12)

We prove Eq (2.4). Thanks to the Hölder inequality,

$\begin{equation*} u_\varepsilon^2(t,x) = 2\displaystyle {\int}_{-\infty}^{x} u_\varepsilon \partial_x u_\varepsilon dx\le 2\displaystyle {\int}_{ \mathbb{R}}\vert u_\varepsilon\vert\vert \partial_x u_\varepsilon\vert dx\le 2\left\| { u_\varepsilon(t,\cdot)} \right\|_{L^2( \mathbb{R})}\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|_{L^2( \mathbb{R})}. \end{equation*}$

Hence, we have that

$\begin{equation} \left\| { u_\varepsilon(t,\cdot)} \right\|^4_{L^{\infty}( \mathbb{R})}\le 4\left\| { u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}. \end{equation}$

(2.13)

Thanks to Eqs (2.3) and (2.12), we have that

$\begin{equation} \begin{split} &\left\| { u_\varepsilon} \right\|^4_{L^{\infty}((0,T)\times \mathbb{R})}\\ &\qquad\le C(T)\left(1+\left\| { u_\varepsilon} \right\|_{L^{\infty}((0,T)\times \mathbb{R})} +\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,T)\times \mathbb{R})}+\left\| { u_\varepsilon} \right\|^3_{L^{\infty}((0,T)\times \mathbb{R})}\right). \end{split} \end{equation}$

(2.14)

Due to the Young inequality,

$\begin{align*} C(T)\left\| { u_\varepsilon} \right\|^3_{L^{\infty}((0,T)\times \mathbb{R})}\le& \frac{1}{2}\left\| { u_\varepsilon} \right\|^4_{L^{\infty}((0,T)\times \mathbb{R})}+C(T)\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,T)\times \mathbb{R})},\\ C(T)\left\| { u_\varepsilon} \right\|_{L^{\infty}((0,T)\times \mathbb{R})}\le& C(T)\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,T)\times \mathbb{R})}+C(T). \end{align*}$

By Eq (2.14), we have that

$\begin{equation*} \frac{1}{2}\left\| { u_\varepsilon} \right\|^4_{L^{\infty}((0,T)\times \mathbb{R})}-C(T)\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,T)\times \mathbb{R})}-C(T){\le0}, \end{equation*}$

which gives Eq (2.4).

Equation (2.5) follows from Eqs (2.4) and (2.12).

Finally, we prove Eq (2.6). We begin by observing that, from Eqs (2.4) and (2.10), we have

$\begin{equation*} \left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^4_{L^4( \mathbb{R})}\le C(T)\left\| { \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}. \end{equation*}$

An integration on $(0, t)$ and Eqs (2.5) give Eq (2.6). □

Lemma 2.3. Fix $T > 0$ and assume (1.6). There exists a constant $C(T) > 0$ , independent of $\varepsilon$ , such that Eq (2.4) holds. Moreover, we have Eqs (2.5) and (2.6).

Proof. Let $0\le t\le T$ . Multiplying Eq (2.1) by $-2 \partial_{x}^2 u_\varepsilon$ , an integration on $\mathbb{R}$ gives

$\begin{align*} \frac{d}{dt}&\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})} = -2\displaystyle {\int}_{ \mathbb{R}} \partial_{x}^2 u_\varepsilon \partial_t u_\varepsilon dx\\ = &-2\displaystyle {\int}_{ \mathbb{R}}f'( u_\varepsilon) \partial_x u_\varepsilon \partial_{x}^2 u_\varepsilon dx-2\beta^2\left\| { \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})} -2\delta\displaystyle {\int}_{ \mathbb{R}} \partial_{x}^2 u_\varepsilon \partial_{x}^3 u_\varepsilon dx\\ &-2\kappa\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon \partial_{x}^2 u_\varepsilon dx-2\gamma^2\displaystyle {\int}_{ \mathbb{R}}\vert u_\varepsilon\vert u_\varepsilon \partial_{x}^2 u_\varepsilon dx+2 \varepsilon\displaystyle {\int}_{ \mathbb{R}} \partial_{x}^2 u_\varepsilon \partial_{x}^4 u_\varepsilon dx\\ = &-2\displaystyle {\int}_{ \mathbb{R}}f'( u_\varepsilon) \partial_x u_\varepsilon \partial_{x}^2 u_\varepsilon dx -2\beta^2\left\| { \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}\\ &+2\kappa\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}+2\gamma^2\displaystyle {\int}_{ \mathbb{R}}\vert u_\varepsilon\vert u_\varepsilon \partial_{x}^2 u_\varepsilon dx-2 \varepsilon\left\| { \partial_{x}^3 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}. \end{align*}$

Therefore, we have that

$\begin{equation} \begin{split} &\frac{d}{dt}\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}+2\beta^2\left\| { \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}+2 \varepsilon\left\| { \partial_{x}^3 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}\\ &\qquad = -2\displaystyle {\int}_{ \mathbb{R}}f'( u_\varepsilon) \partial_x u_\varepsilon \partial_{x}^2 u_\varepsilon dx+2\kappa\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}+2\gamma^2\displaystyle {\int}_{ \mathbb{R}}\vert u_\varepsilon\vert u_\varepsilon \partial_{x}^2 u_\varepsilon dx. \end{split} \end{equation}$

(2.15)

Due Eqs (1.6) and (2.3) and the Young inequality,

$\begin{align*} &2\displaystyle {\int}_{ \mathbb{R}}\vert f'( u_\varepsilon)\vert\vert \partial_x u_\varepsilon\vert\vert \partial_{x}^2 u_\varepsilon\vert dx\le C_0\displaystyle {\int}_{ \mathbb{R}}\vert \partial_x u_\varepsilon \partial_{x}^2 u_\varepsilon\vert dx+C_0\displaystyle {\int}_{ \mathbb{R}}\vert u_\varepsilon \partial_x u_\varepsilon\vert\vert \partial_{x}^2 u_\varepsilon\vert dx\\ &\qquad = 2\displaystyle {\int}_{ \mathbb{R}}\left\vert\frac{C_0\sqrt{3} \partial_x u_\varepsilon}{2\beta}\right\vert\left\vert\frac{\beta \partial_{x}^2 u_\varepsilon}{\sqrt{3}}\right\vert dx+2\displaystyle {\int}_{ \mathbb{R}}\left\vert\frac{C_0\sqrt{3} u_\varepsilon \partial_x u_\varepsilon}{2\beta}\right\vert\left\vert\sqrt{3}\beta \partial_{x}^2 u_\varepsilon\right\vert dx\\ &\qquad\le C_0\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}+C_0\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^2( \partial_x u_\varepsilon)^2 dx +\frac{2\beta^2}{3}\left\| { \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}\\ &\qquad\le C_0\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}+C_0\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\beta^2}{3}\left\| { \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}\\ &\qquad\le C_0\left(1+\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,T)\times \mathbb{R})}\right)\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})} +\frac{2\beta^2}{3}\left\| { \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})},\\ &2\gamma^2\displaystyle {\int}_{ \mathbb{R}}\vert u_\varepsilon\vert u_\varepsilon \partial_{x}^2 u_\varepsilon dx\le 2\gamma^2\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^2\vert \partial_{x}^2 u_\varepsilon\vert dx = 2\displaystyle {\int}_{ \mathbb{R}}\left\vert\frac{\sqrt{3}\gamma^2 u_\varepsilon^2}{\beta}\right\vert\left\vert\frac{\beta \partial_{x}^2 u_\varepsilon}{\sqrt{3}}\right\vert dx\\ &\qquad\le \frac{3\gamma^4}{\beta^2}\displaystyle {\int}_{ \mathbb{R}} u_\varepsilon^4 dx + \frac{\beta^2}{3}\left\| { \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}\\ &\qquad\le\frac{3\gamma^4}{\beta^2}\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,T)\times \mathbb{R})}\left\| { u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})} + \frac{\beta^2}{3}\left\| { \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}\\ &\qquad\le C(T)\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,T)\times \mathbb{R})}+\frac{\beta^2}{3}\left\| { \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}. \end{align*}$

It follows from Eq (2.15) that

$\begin{align*} &\frac{d}{dt}\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}+\beta^2\left\| { \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}+2 \varepsilon\left\| { \partial_{x}^3 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}\\ &\qquad\le C_0\left(1+\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,T)\times \mathbb{R})}\right)\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}+C(T)\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,T)\times \mathbb{R})}. \end{align*}$

Integrating on $(0, t)$ , by Eq (2.3), we have that

(2.16)

Thanks to Eqs (2.3), (2.13), and (2.16), we have that

$\begin{equation*} \left\| { u_\varepsilon} \right\|^4_{L^{\infty}((0,T)\times \mathbb{R})}\le C(T)\left(1+\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,T)\times \mathbb{R})}\right). \end{equation*}$

Therefore,

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

which gives (2.4).

Equation (2.5) follows from (2.4) and (2.16), while, arguing as in Lemma 2.2, we have Eq (2.6).

□

Lemma 2.4. Fix $T > 0$ . There exists a constant $C(T) > 0$ , independent of $\varepsilon$ , such that

$\begin{equation} \displaystyle {\int}_{0}^{t}\left\| { \partial_x u_\varepsilon(s,\cdot)} \right\|^6_{L^6( \mathbb{R})}ds\le C(T), \end{equation}$

(2.17)

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

Proof. Let $0\le t\le T$ . We begin by observing that,

$\begin{equation} \displaystyle {\int}_{ \mathbb{R}}( \partial_x u_\varepsilon)^6 dx\le \left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^4_{L^{\infty}( \mathbb{R})}\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}. \end{equation}$

(2.18)

Thanks to the Hölder inequality,

$\begin{align*} ( \partial_x u_\varepsilon(t,x))^2 = &2\displaystyle {\int}_{-\infty}^{x} \partial_x u_\varepsilon \partial_{x}^2 u_\varepsilon dy \le 2\displaystyle {\int}_{ \mathbb{R}}\vert \partial_x u_\varepsilon\vert\vert \partial_{x}^2 u_\varepsilon\vert dx\\ \le&2\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|_{L^2( \mathbb{R})}\left\| { \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}. \end{align*}$

Hence,

$\begin{equation*} \left\| {u(t,\cdot)} \right\|^4_{L^{\infty}( \mathbb{R})}\le 4\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}\left\| { \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}. \end{equation*}$

It follows from Eq (2.18) that

$\begin{equation*} \displaystyle {\int}_{ \mathbb{R}}( \partial_x u_\varepsilon)^6 dx\le 4\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^4_{L^2( \mathbb{R})}\left\| { \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}. \end{equation*}$

Therefore, by Eq (2.5),

$\begin{equation*} \displaystyle {\int}_{ \mathbb{R}}( \partial_x u_\varepsilon)^6 dx\le C(T)\left\| { \partial_{x}^2 u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}. \end{equation*}$

An integration on $(0, t)$ and Eq (2.5) gives (2.17). □

3. Proof of Theorem 1.1

This section is devoted to the proof of Theorem 1.1.

We begin by proving the following result.

Lemma 3.1. Fix $T > 0$ . Then,

$\begin{equation} {{the\ family\ }} \{ u_\varepsilon\}_{ \varepsilon > 0} {{ \ is\ compact\ in\ }} L^2_{loc}((0,T)\times \mathbb{R}) . \end{equation}$

(3.1)

Consequently, there exist a subsequence $\{ u_{\varepsilon_k} \}_{k\in \mathbb{N}}$ and $u\in L^2_{loc}((0, T)\times \mathbb{R})$ such that

$\begin{equation} u_{\varepsilon_k}\to u\ { in }\ L^2_{loc}((0,T)\times \mathbb{R}) {{\ and\ a.e.\ in\ }} (0,T)\times \mathbb{R} . \end{equation}$

(3.2)

Moreover, $u$ is a solution of Eq (1.1), satisfying Eq (1.8).

Proof. We begin by proving Eq (3.1). To prove Eq (3.1), we rely on the Aubin-Lions Lemma (see ^[5,21]). We recall that

$\begin{equation*} H^1_{loc}( \mathbb{R})\hookrightarrow\hookrightarrow L^2_{loc}( \mathbb{R})\hookrightarrow H^{-1}_{loc}( \mathbb{R}), \end{equation*}$

where the first inclusion is compact and the second one is continuous. Owing to the Aubin-Lions Lemma ^[21], to prove Eq (3.1), it suffices to show that

$\begin{align} & \{ u_\varepsilon\}_{ \varepsilon > 0} {\rm{\ is\ uniformly\ bounded\ in\ }} L^2(0,T;H^1_{loc}( \mathbb{R})) , \end{align}$

(3.3)

$\begin{align} & \{ \partial_t u_\varepsilon\}_{ \varepsilon > 0} {\rm{\ is\ uniformly\ bounded\ in\ }} L^2(0,T;H^{-1}_{loc}( \mathbb{R})) . \end{align}$

(3.4)

We prove Eq (3.3). Thanks to Lemmas 2.1–2.3,

$\begin{equation*} \left\| { u_\varepsilon(t,\cdot)} \right\|^2_{H^1( \mathbb{R})} = \left\| { u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}+\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}\le C(T). \end{equation*}$

Therefore,

$\begin{equation*} \{ u_\varepsilon\}_{ \varepsilon > 0} {\rm{\ is\ uniformly\ bounded\ in }}\ L^{\infty}(0,T;H^{1}( \mathbb{R})) , \end{equation*}$

which gives Eq (3.3).

We prove Eq (3.4). Observe that, by Eq (2.1),

$\begin{equation*} \partial_t u_\varepsilon = - \partial_x\left( G( u_\varepsilon)\right)-f'( u_\varepsilon) \partial_x u_\varepsilon-\kappa u_\varepsilon -\gamma^2\vert u_\varepsilon\vert u_\varepsilon, \end{equation*}$

where

$\begin{equation} G( u_\varepsilon) = \beta^2 \partial_x u_\varepsilon-\delta \partial_{x}^2 u_\varepsilon- \varepsilon \partial_{x}^3 u_\varepsilon. \end{equation}$

(3.5)

Since $0 < \varepsilon < 1$ , thanks to Eq (2.5), we have that

$\begin{equation} \begin{split} &{\beta^2\left\| { \partial_x u_\varepsilon} \right\|^2_{L^2((0,T)\times \mathbb{R})}},\, \delta^2\left\| { \partial_{x}^2 u_\varepsilon} \right\|^2_{L^2((0,T)\times \mathbb{R})}\le C(T),\\ & \varepsilon^2\left\| { \partial_{x}^3 u_\varepsilon} \right\|^2_{L^2((0,T)\times \mathbb{R})}\le C(T). \end{split} \end{equation}$

(3.6)

Therefore, by Eqs (3.5) and (3.6), we have that

$\begin{equation} \{ \partial_x\left(G( u_\varepsilon)\right)\}_{ \varepsilon > 0} \quad {\rm{\ is\ bounded\ in}}\ { L^2(0, T ; H^{-1}( \mathbb{R})) }. \end{equation}$

(3.7)

We claim that

$\begin{equation} \displaystyle {\int}_{0}^{T}\!\!\!\displaystyle {\int}_{ \mathbb{R}}(f'( u_\varepsilon))^2( \partial_x u_\varepsilon)^2 dtdx\le C(T). \end{equation}$

(3.8)

Thanks to Eqs (2.4) and (2.5),

$\begin{equation*} \displaystyle {\int}_{0}^{T}\!\!\!\displaystyle {\int}_{ \mathbb{R}}(f'( u_\varepsilon))^2( \partial_x u_\varepsilon)^2 dtdx\le \left\| {f'} \right\|^2_{L^{\infty}(-C(T),C(T))}\displaystyle {\int}_{0}^{T}\left\| { \partial_x u_\varepsilon(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}dt\le C(T). \end{equation*}$

Moreover, thanks to Eq (2.3),

$\begin{equation} \vert\kappa\vert\displaystyle {\int}_{0}^{T}\!\!\!\displaystyle {\int}_{ \mathbb{R}}( u_\varepsilon)^2 dx \le C(T). \end{equation}$

(3.9)

We have that

$\begin{equation} \gamma^2\displaystyle {\int}_{0}^T\!\!\!\displaystyle {\int}_{ \mathbb{R}}(\vert u_\varepsilon\vert u_\varepsilon)^2 dsdx\le C(T). \end{equation}$

(3.10)

In fact, thanks to Eqs (2.3) and (2.4),

$\begin{align*} \gamma^2\displaystyle {\int}_{0}^T\!\!\!\displaystyle {\int}_{ \mathbb{R}}(\vert u_\varepsilon\vert u_\varepsilon)^2 dsdx\le& \gamma^2\left\| { u_\varepsilon} \right\|^2_{L^{\infty}((0,T)\times \mathbb{R})}\displaystyle {\int}_{0}^T\!\!\!\displaystyle {\int}_{ \mathbb{R}}( u_\varepsilon)^2 dsdx\\ \le&C(T)\displaystyle {\int}_{0}^T\!\!\!\displaystyle {\int}_{ \mathbb{R}}( u_\varepsilon)^2 dsdx\le C(T). \end{align*}$

Therefore, Eq (3.4) follows from Eqs (3.7)–(3.10).

Thanks to the Aubin-Lions Lemma, Eqs (3.1) and (3.2) hold.

Consequently, arguing as in [, Theorem 1.1], $u$ is solution of Eq (1.1) and, thanks to Lemmas 2.1–2.3 and Eqs (2.4), (1.8) holds. □

Proof of Theorem 1.1. Lemma 3.1 gives the existence of a solution of Eq (1.1).

We prove Eq (1.9). Let $u_1$ and $u_2$ be two solutions of Eq (1.1), which verify Eq (1.8), that is,

$\begin{align*} &\begin{cases} \partial_t u_i+ \partial_x f(u_i)-\beta^2 \partial_{x}^2 u_i+\delta \partial_{x}^3 u_i +\kappa u_i +\gamma^2\vert u_i\vert u_i = 0,&\quad 0 < t < T, \quad x\in \mathbb{R},\\ u_i(0,x) = u_{i,\,0}(x), &\quad x\in \mathbb{R}, \end{cases}\quad i = 1,2. \end{align*}$

Then, the function

$\begin{equation} \omega(t,x) = u_1(t,x)-u_2(t,x), \end{equation}$

(3.11)

is the solution of the following Cauchy problem:

$\begin{equation} \begin{cases} \partial_t\omega+ \partial_x\left( f(u_1)-f(u_2)\right)-\beta^2 \partial_{x}^2\omega+\delta \partial_{x}^2\omega \\ \qquad\qquad+\kappa\omega+\gamma^2\left(\vert u_1\vert u_1-\vert u_2\vert u_2\right) = 0, &\quad 0 < t < T, \quad x\in \mathbb{R},\\ \omega(0,x) = u_{1,\,0}(x)-u_{2,\,0}(x), &\quad x\in \mathbb{R}. \end{cases} \end{equation}$

(3.12)

Fixed $T > 0$ , since $u_1, \, u_2\in H^1(\mathbb{R})$ , for every $0\le t\le T$ , we have that

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

(3.13)

We define

$\begin{equation} g = \frac{f(u_1)-f(u_2)}{\omega} \end{equation}$

(3.14)

and observe that, by Eq (3.13), we have that

$\begin{equation} \vert g\vert\le \left\| {f'} \right\|_{L^{\infty}(-C(T),\,C(T))}\le C(T). \end{equation}$

(3.15)

Moreover, by Eq (3.11) we have that

$\begin{equation} \left\vert \vert u_1\vert-\vert u_2\vert\right\vert\le \vert u_1-u_2\vert = \vert\omega\vert. \end{equation}$

(3.16)

Observe that thanks to Eq (3.11),

$\begin{equation} \begin{split} \vert u_1\vert u_1-\vert u_2\vert u_2 = &\vert u_1\vert u_1-\vert u_1\vert u_2 +\vert u_1\vert u_2 - \vert u_2\vert u_2\\ = &\vert u_1\vert\omega +u_2\left(\vert u_1\vert- \vert u_2\vert\right). \end{split} \end{equation}$

(3.17)

Thanks to Eqs (3.14) and (3.17), Equation (3.12) is equivalent to the following one:

$\begin{equation} \partial_t\omega+ \partial_x ({g}\omega) -\beta^2 \partial_{x}^2\omega+\delta \partial_{x}^3\omega+\kappa\omega+\gamma^2\vert u_1\vert\omega+\gamma^2 u_2\left(\vert u_1\vert- \vert u_2\vert\right) = 0. \end{equation}$

(3.18)

Multiplying Eq (3.18) by $2\omega$ , an integration on $\mathbb{R}$ gives

$\begin{align*} \frac{dt}{dt}&\left\| {\omega(t,\cdot)} \right\|^2_{L^2( \mathbb{R})} = 2\displaystyle {\int}_{ \mathbb{R}}\omega \partial_t\omega\\ = &-2\displaystyle {\int}_{ \mathbb{R}}\omega \partial_x ({g}\omega)dx+2\beta^2\displaystyle {\int}_{ \mathbb{R}}\omega \partial_{x}^2\omega dx-2\delta\displaystyle {\int}_{ \mathbb{R}}\omega \partial_{x}^3\omega dx \\ &-2\kappa\left\| {\omega(t,\cdot)} \right\|^2_{L^2( \mathbb{R})} -2\gamma^2\displaystyle {\int}_{ \mathbb{R}}\vert u_1\vert\omega^2 dx -2\gamma^2\displaystyle {\int}_{ \mathbb{R}}u_2\left(\vert u_1\vert- \vert u_2\vert\right)\omega dx\\ & = 2\displaystyle {\int}_{ \mathbb{R}}{g}\omega \partial_x\omega dx -2\beta^2\left\| { \partial_x\omega(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}+2\delta\displaystyle {\int}_{ \mathbb{R}} \partial_x\omega \partial_{x}^2\omega dx\\ &-2\kappa\left\| {\omega(t,\cdot)} \right\|^2_{L^2( \mathbb{R})} -2\gamma^2\displaystyle {\int}_{ \mathbb{R}}\vert u_1\vert\omega^2 dx -2\gamma^2\displaystyle {\int}_{ \mathbb{R}}u_2\left(\vert u_1\vert- \vert u_2\vert\right)\omega dx\\ = & 2\displaystyle {\int}_{ \mathbb{R}}{g}\omega \partial_x\omega dx -2\beta^2\left\| { \partial_x\omega(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}\\ & -2\kappa\left\| {\omega(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}-2\gamma^2\displaystyle {\int}_{ \mathbb{R}}\vert u_1\vert\omega^2 dx -2\gamma^2\displaystyle {\int}_{ \mathbb{R}}u_2\left(\vert u_1\vert- \vert u_2\vert\right)\omega dx. \end{align*}$

Therefore, we have that

$\begin{equation} \begin{split} \left\| {\omega(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}&+2\beta^2\left\| { \partial_x\omega(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}+2\gamma^2\displaystyle {\int}_{ \mathbb{R}}\vert u_1\vert\omega^2 dx\\ = & 2\displaystyle {\int}_{ \mathbb{R}}{g}\omega \partial_x\omega dx -\kappa\left\| {\omega(t,\cdot)} \right\|^2_{L^2( \mathbb{R})} -2\gamma^2\displaystyle {\int}_{ \mathbb{R}}u_2\left(\vert u_1\vert- \vert u_2\vert\right)\omega dx. \end{split} \end{equation}$

(3.19)

Due to Eqs (3.13), (3.15) and (3.16) and the Young inequality,

$\begin{align*} &2\displaystyle {\int}_{ \mathbb{R}}\vert {g}\vert\vert\omega\vert\vert \partial_x\omega\vert dx\le 2C(T)\displaystyle {\int}_{ \mathbb{R}}\vert\omega\vert\vert \partial_x\omega\vert dx\\ &\qquad = 2\displaystyle {\int}_{ \mathbb{R}}\left\vert\frac{C(T)\omega}{\beta}\right\vert\left\vert\beta \partial_x\omega\right\vert dx\le C(T)\left\| {\omega(t,\cdot)} \right\|^2_{L^2( \mathbb{R})} +\beta^2\left\| { \partial_x\omega(t,\cdot)} \right\|^2_{L^2( \mathbb{R})},\\ &2\gamma^2\displaystyle {\int}_{ \mathbb{R}}\vert u_2\vert\left\vert\left(\vert u_1\vert- \vert u_2\vert\right)\right\vert\vert\omega\vert dx\le 2\gamma^2\left\| {u_2} \right\|_{L^{\infty}((0,T)\times \mathbb{R})}\displaystyle {\int}_{ \mathbb{R}}\left\vert\left(\vert u_1\vert- \vert u_2\vert\right)\right\vert\vert\omega\vert dx\\ &\qquad\le C(T)\left\| {\omega(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}. \end{align*}$

It follows from Eq (3.19) that

$\begin{equation*} \left\| {\omega(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}+\beta^2\left\| { \partial_x\omega(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}+2\gamma^2\displaystyle {\int}_{ \mathbb{R}}\vert u_1\vert\omega^2 dx\le C(T)\left\| {\omega(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}. \end{equation*}$

The Gronwall Lemma and Eq (3.12) give

$\begin{equation} \begin{split} \left\| {\omega(t,\cdot)} \right\|^2_{L^2( \mathbb{R})}&+\beta^2 e^{C(T)t}\displaystyle {\int}_{0}^{t}e^{-C(T)s}\left\| { \partial_x\omega(s,\cdot)} \right\|^2_{L^2( \mathbb{R})}ds\\ &+2\gamma^2 e^{C(T)t}\displaystyle {\int}_{0}^{t}\!\!\!\displaystyle {\int}_{ \mathbb{R}}e^{-C(T)s}\vert u_1\vert\omega^2 dsdx\le e^{C(T)t}\left\| {\omega_0} \right\|^2_{L^2( \mathbb{R})}. \end{split} \end{equation}$

(3.20)

Equation (1.9) follows from Eqs (3.11) and (3.20). □

Author contributions

Giuseppe Maria Coclite and Lorenzo Di Ruvo equally contributed to the methodologies, typesetting, and the development of the paper.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

Giuseppe Maria Coclite is an editorial boardmember for [Networks and Heterogeneous Media] and was not involved inthe editorial review or the decision to publish this article.

GMC is member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). GMC has been partially supported by the Project funded under the National Recovery and Resilience Plan (NRRP), Mission 4 Component 2 Investment 1.4 -Call for tender No. 3138 of 16/12/2021 of Italian Ministry of University and Research funded by the European Union -NextGenerationEUoAward Number: CN000023, Concession Decree No. 1033 of 17/06/2022 adopted by the Italian Ministry of University and Research, CUP: D93C22000410001, Centro Nazionale per la Mobilità Sostenibile, the Italian Ministry of Education, University and Research under the Programme Department of Excellence Legge 232/2016 (Grant No. CUP - D93C23000100001), and the Research Project of National Relevance "Evolution problems involving interacting scales" granted by the Italian Ministry of Education, University and Research (MIUR Prin 2022, project code 2022M9BKBC, Grant No. CUP D53D23005880006). GMC expresses its gratitude to the HIAS - Hamburg Institute for Advanced Study for their warm hospitality.

Conflict of interest

The authors declare there is no conflict of interest.

Acknowledgments

We would like to thank the Faculty of Food Science & Technology, University of West Attica, Greece and the molecular diagnosis (polydinamo) laboratory for the meritorious collaboration.

Conflicts of interest

The authors declare no conflicts of interest regarding the present manuscript.

References

[1]	WHO The Burden of Foodborne Diseases in the WHO European Region (2017) .
[2]	Baraketi A, Salmieri S, Lacroix M (2018) Foodborne pathogens detection: persevering worldwide challenge amina. Biosensing Technologies for the Detection of Pathogens-A Prospective Way For Rapid Analysis Croatia: InTech publishing, 53-72.
[3]	Sayad A, Ibrahim F, Uddin MS, et al. (2018) A microdevice for rapid, monoplex and colorimetric detection of foodborne pathogens using a centrifugal microfluidic platform. Biosens Bioel 100: 96-104. doi: 10.1016/j.bios.2017.08.060
[4]	EFSA and ECDC (2016) The European Union summary report on trends and sources of zoonoses, zoonotic agents and food-borne outbreaks in 2015. EFSA Journal 14: 4634.
[5]	Marquis H, Drevets DA, Bronze M S, et al. (2016) Pathogenesis of Listeria monocytogenes in Humans. Human Emerging and Re-emerging Infections: Viral and Parasitic Infections USA: Willey-Blackwell, 749-772.
[6]	Radoshevich L, Cossart P (2017) Listeria monocytogenes: towards a complete picture of its physiology and pathogenesis. Nat Rev Microbiol 16: 32-46. doi: 10.1038/nrmicro.2017.126
[7]	Santos RL, Tsolis RM, Adams LG, et al. (2003) Pathogenesis of Salmonella-induced enteritis. Braz J Med Biol Res 36: 3-12. doi: 10.1590/S0100-879X2003000100002
[8]	Zhang S, Kingsley RA, Santos RL, et al. (2003) Molecular pathogenesis of Salmonella enterica serotype typhimurium-induced diarrhea. Inf Immun 71: 1-12. doi: 10.1128/IAI.71.1.1-12.2003
[9]	Rajapaksha P, Elbourne A (2018) A review of methods for the detection of pathogenic microorganisms. Analyst 144: 396-411. doi: 10.1039/C8AN01488D
[10]	Liu Y, Cao Y, Wang T, et al. (2019) Detection of 12 common food-borne bacterial pathogens by TaqMan Real-Time PCR using a single set of reaction conditions. Front Microbiol 10: 1-9. doi: 10.3389/fmicb.2019.00001
[11]	Schirone M, Visciano P, Tofalo R, et al. (2019) Editorial: foodborne pathogens: hygiene and safety. Front Microbiol 10: 1-4. doi: 10.3389/fmicb.2019.01974
[12]	Charlermroj R, Makornwattana M, Phuengwas S, et al. (2019) DNA-based bead array technology for simultaneous identification of eleven foodborne pathogens in chicken meat. Food Cont 101: 81-88. doi: 10.1016/j.foodcont.2019.02.014
[13]	Rodrigues C, Desai N, Fernandes H (2016) Clinical Diagnosis in Resource-Limited Settings. Clinl Microbiol News 38: 51-56. doi: 10.1016/j.clinmicnews.2016.03.004
[14]	Klančnik A, Kovač M (2012) PCR in Food Analysis. Polymerase Chain Reaction China: Tech Publishing, 195-220.
[15]	Kaneko H, Kawana T, Fukushima E, et al. (2007) Tolerance of loop-mediated isothermal amplification to a culture medium and biological substances. J Biochem Biophys Meth 70: 499-501. doi: 10.1016/j.jbbm.2006.08.008
[16]	Lee JE, Kang LH, Kim JO, et al. (2019) Rapid detection of carbapenemase-producing Enterobacteriaceae (CPE) using a simple DNA extraction method and Loop-mediated isothermal amplification (LAMP). J Microbiol Meth 167: 105746. doi: 10.1016/j.mimet.2019.105746
[17]	Law JW, Mutalib NA, Chan K, et al. (2015) Rapid methods for the detection of foodborne bacterial pathogens: principles, applications, advantages and limitations. Front Microbiol 5: 1-20.
[18]	Mateus-Barros E, Meneghine AK, Bagatini IL, et al. (2019) Comparison of two DNA extraction methods widely used in aquatic microbial ecology. J Microbiol Meth 159: 12-17. doi: 10.1016/j.mimet.2019.02.005
[19]	Tanner NA, Zhang Y, Evans TC (2015) Visual detection of isothermal nucleic acidamplification using pH-sensitive dyes. Biotechn 58: 59-68. doi: 10.2144/000114253
[20]	Anju P, Latha C, Sunil B, et al. (2014) Detection of Salmonella and Yersinia spp . in uncooked retail chicken meat in Kerala by multiplex PCR. Int J Cur Microbiol Ap Sci 3: 1028-1034.
[21]	Khalil ZK (2017) A study of isolation and identification of Campylobacter species, Listeria monocytogenes, E. coli O157:H7 and Salmonella species from raw chicken carcasses and beef meat by multiplex polymer chain reaction in Baghdad city. Inte J Adv Res 5: 900-907. doi: 10.21474/IJAR01/5134
[22]	Srisawat M, Panbangred W (2015) Efficient and specific detection of Salmonella in food samples using a stn-Based Loop-Mediated isothermal amplification method. BioMed Res Int 176: 1-7. doi: 10.1155/2015/356401
[23]	Tang MJ, Zhou S, Zhang XY, et al. (2011) Rapid and sensitive detection of Listeria monocytogenes by Loop-Mediated isothermal amplification. Cur Microbiol 63: 511-516. doi: 10.1007/s00284-011-0013-3
[24]	Psi A, Dovas CI, Banos G (2010) A comparison of six methods for genomic DNA extraction suitable for PCR-based genotyping applications using bovine milk samples. Mol Cel Prob 24: 93-98. doi: 10.1016/j.mcp.2009.11.001
[25]	Khan M, Wang R, Li B, et al. (2018) Comparative Evaluation of the LAMP assay and PCR-Based assays for the rpid dtection of Alternaria solani. Front Microbiol 9: 2089. doi: 10.3389/fmicb.2018.02089
[26]	Wang X, Seo DJ, Lee MH, et al. (2014) Comparison of cnventional PCR, mltiplex PCR, and Loop-Mediated iothermal aplification asays for rpid dtection of Arcobacter secies. J Clin Microbiol 52: 557. doi: 10.1128/JCM.02883-13

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