1 Introduction

In this paper, we study the Cauchy problem for the following generalized two-component Dullin–Gottwald–Holm (g2DGH) system [6, 27, 38]

$$\begin{aligned} {\left\{ \begin{array}{ll} u_t - u_{txx} - Au_x + 3uu_x - \sigma (2u_xu_{xx} + uu_{xxx}) + \gamma u_{xxx}\\ + \rho \rho _x = 0, &{}\quad t>0, \quad x \in {\mathbb {R}}, \\ \rho _t + (\rho u)_x = 0, &{}\quad t>0, \quad x \in {\mathbb {R}}, \\ u(0,x) = u_0(x), &{}\quad x \in {\mathbb {R}}, \\ \rho (0,x) = \rho _0(x), &{}\quad x \in {\mathbb {R}}, \end{array}\right. }\nonumber \\ \end{aligned}$$
(1.1)

which was derived in the shallow water regime using Ivanov’s modeling approach in [30]. Here u(t, x) stands for the horizontal velocity of the fluid, \(\rho (t,x)\) is in connection with the free surface elevation from equilibrium, the parameter A is related to a linear underlying shear flow, and \(\sigma \) is a dimensionless parameter providing the competition (or balance) in fluid convection between nonlinear steepening and amplification due to stretching. In the derivation of system (1.1), the boundary conditions \(u \rightarrow 0\) and \(\rho \rightarrow 1\) as \(|x| \rightarrow \infty \) are required.

System (1.1) includes several classical shallow water wave models. For example, if \(\sigma = 1\), system (1.1) becomes the two-component Dullin–Gottwald–Holm (2DGH) system [8, 24, 33, 49]

$$\begin{aligned} {\left\{ \begin{array}{ll} u_t - u_{txx} - Au_x + 3uu_x - 2u_xu_{xx} - uu_{xxx} + \gamma u_{xxx} + \rho \rho _x = 0, &{}\quad t>0, \quad x \in {\mathbb {R}}, \\ \rho _t + (\rho u)_x = 0, &{}\quad t>0, \quad x \in {\mathbb {R}}. \end{array}\right. }\nonumber \\ \end{aligned}$$
(1.2)

If we further take \(\gamma = 0\) in the 2DGH system (1.2), then it recovers the integrable two-component Camassa–Holm (2CH) system

$$\begin{aligned} {\left\{ \begin{array}{ll} u_t - u_{txx} - Au_x + 3uu_x - 2u_xu_{xx} - uu_{xxx} + \rho \rho _x = 0, &{}\quad t>0, \quad x \in {\mathbb {R}}, \\ \rho _t + (\rho u)_x = 0, &{}\quad t>0, \quad x \in {\mathbb {R}}, \end{array}\right. }\nonumber \\ \end{aligned}$$
(1.3)

which was originally introduced by Olver and Rosenau [36] and rigorously derived in the context of shallow water regime [11, 30]. The Cauchy problem of the 2CH system has been studied widely [5, 15, 17,18,19,20,21,22,23, 31, 40, 44]. Here we give a brief review. The local well-posedness of the 2CH system with initial data in the Sobolev spaces and Besov spaces was established in [15, 21, 22]. It was shown that the 2CH system admits global strong solutions [11, 19, 22] and also finite time blow-up solutions [11, 15, 17, 19, 22, 23, 40, 44]. Besides, it has global weak solutions [18, 20, 31]. Moreover, the Lipschitz continuous dependence of conservative solutions to the 2CH system was investigated in [5].

If \(\rho = 0\), \(\sigma = 1\), system (1.1) is reduced to the Dullin–Gottwald–Holm (DGH) equation [13], modeling the unidirectional propagation of shallow water waves over a flat bottom,

$$\begin{aligned} u_t - u_{txx} - Au_x + 3uu_x - 2u_xu_{xx} - uu_{xxx} + \gamma u_{xxx} = 0, \quad t>0, \quad x \in {\mathbb {R}}.\nonumber \\ \end{aligned}$$
(1.4)

Equation (1.4) was deduced by Dullin, Gottwald and Holm [13] using the asymptotic analysis and a near-identity normal form transformation from water wave theory. This equation is integrable in the sense that it admits the Lax pair and bi-Hamiltonian structure [13]. For the DGH equation, the local well-posedness of strong solutions was proved in [37, 41, 43]. The precise blow-up scenario and several blow-up results for strong solutions to the DGH equation were presented in [25, 28, 29, 34, 35, 39, 43, 45, 47, 48]. For the global existence of weak and strong solutions, we refer the readers to [34, 39, 41, 43, 46] and the references therein. Furthermore, the orbital stability of single peakon and the train of peakons for the DGH equation were proved in [26] and [32], respectively. It is worth pointing out that the study of the DGH equation was based on methods that were developed for the Camassa–Holm (CH) equation [4, 16], corresponding to the choice \(\gamma = 0\) in (1.4). In particular, the idea to associate diffeomorphisms to derive invariance properties was pioneered by Constantin–Escher [10] and Constantin [9], while Constantin and Strauss [12] verified that the peakons of the CH equation are orbitally stable.

Han et al. [27] investigated the blow-up mechanism to the g2DGH system (1.1) on the line and provided two sufficient conditions for wave breaking of strong solutions in finite time, they also classified traveling wave solutions for this system. Chen and Yan [6] studied the wave breaking phenomena and global existence of the g2DGH system (1.1) on the circle. Recently, inspired by the ideas of [1,2,3], Wang and Zhu [38] established a local-in-space blow-up criterion for the g2DGH system (1.1) with \(\sigma = 1\), \(\gamma = 0\) (corresponding to the 2CH system (1.3)), that is, a blow-up condition involving only the values of \(u_{0,x}(x_0)\) and \(u_0(x_0)\) in a single point \(x_0\) of the real line. One of our purposes in the present paper is to improve this result to \(1 \le \sigma \le 4\), \(\gamma \in {\mathbb {R}}\).

Another interesting issue investigated here is concerned with the traveling wave solutions of the g2DGH system. Following the ideas of [7, 14], we show that horizontally symmetric weak solutions of the g2DGH system must be traveling wave solutions. Denote \(G(x) = \frac{1}{2}e^{-|x|}\), the fundamental solution of the operator \((1 - \partial ^2_x)^{-1}\) on \({\mathbb {R}}\), which satisfies \((1 - \partial ^2_x)^{-1}f = G*f\) for all \(f \in L^2({\mathbb {R}})\). Therefore, the Cauchy problem (1.1) can be reformulated as

$$\begin{aligned} {\left\{ \begin{array}{ll} u_t + (\sigma u - \gamma )u_x = - \partial _xG*\left[ \frac{3 - \sigma }{2}u^2 + (\gamma - A)u + \frac{\sigma }{2}u^2_x + \frac{1}{2}\rho ^2\right] , &{}\quad t>0, \quad x \in {\mathbb {R}}, \\ \rho _t + u\rho _x = -u_x\rho , &{}\quad t>0, \quad x \in {\mathbb {R}}, \\ u(0,x) = u_0(x), &{}\quad x \in {\mathbb {R}}, \\ \rho (0,x) = \rho _0(x), &{}\quad x \in {\mathbb {R}}. \end{array}\right. }\nonumber \\ \end{aligned}$$
(1.5)

The entire paper is organized as follows. In the next section, we recall several useful results including the local well-posedness and the precise blow-up scenario for the g2DGH system, as well as the convolution estimates. Then we establish a new local-in-space blow-up criterion with suitable conditions on the initial data. Finally in Sect. 3, we prove that an x-symmetric weak solution of the g2DGH system is necessarily a traveling wave solution.

2 Local-in-space blow-up

This section is devoted to investigating the local-in-space blow-up criterion for the Cauchy problem (1.5). We first recall the local well-posedness result and the precise blow-up scenario of (1.5).

Lemma 2.1

([27]) If \((u_0,\rho _0 - 1) \in H^s({\mathbb {R}})\times H^{s - 1}({\mathbb {R}})\) with \(s \ge 2\), then there exists a maximal time \(T = T(\Vert (u_0,\rho _0 - 1)\Vert _{H^s\times H^{s - 1}}) > 0\) and a unique solution \((u,\rho - 1)\) to the Cauchy problem (1.5) such that \((u,\rho - 1) \in C([0,T); H^s({\mathbb {R}})\times H^{s - 1}({\mathbb {R}})) \cap C^1([0,T); H^{s - 1}({\mathbb {R}})\times H^{s - 2}({\mathbb {R}}))\). Moreover, the solution depends continuously on the initial data, and T can be chosen to be independent of s.

Lemma 2.2

([27]) Suppose that \(\sigma \ne 0\) and \((u_0,\rho _0 - 1) \in H^s({\mathbb {R}})\times H^{s - 1}({\mathbb {R}})\) with \(s \ge 2\). Then the corresponding solution \((u,\rho )\) to the initial value problem (1.5) blows up in finite time \(T > 0\) if and only if

$$\begin{aligned} \liminf _{t \rightarrow T}\left( \inf _{x \in {\mathbb {R}}}\sigma u_x(t,x)\right) = -\infty . \end{aligned}$$

We define two convolution operators \(G_{+}\) and \(G_{-}\) as

$$\begin{aligned} \begin{aligned}&G_{+}*f(x) = \frac{e^{-x}}{2}\int ^x_{-\infty } e^yf(y)\,dy, \\&G_{-}*f(x) = \frac{e^x}{2}\int ^\infty _{x} e^{-y}f(y)\,dy, \end{aligned} \end{aligned}$$
(2.1)

it is easy to see that

$$\begin{aligned} G = G_{+} + G_{-}, \quad G_x = G_{-} - G_{+}. \end{aligned}$$
(2.2)

The following convolution estimates are crucial to prove the local-in-space blow-up criterion.

Lemma 2.3

([1, 2, 42]) Let \(1 \le \sigma \le 4\).

  1. (i)

    If \(\gamma \ne A\) and \(\sigma \ne 3\), then

    $$\begin{aligned} G_{\pm }*\left[ \frac{3 - \sigma }{2}\left( u + \frac{\gamma - A}{3 - \sigma }\right) ^2 + \frac{\sigma }{2}u^2_x\right] \ge \frac{\sqrt{\sigma }}{8}(\sqrt{12 - 3\sigma } - \sqrt{\sigma })\left( u + \frac{\gamma - A}{3 - \sigma }\right) ^2. \end{aligned}$$
  2. (ii)

    If \(\gamma = A\), then

    $$\begin{aligned} G_{\pm }*\left( \frac{3 - \sigma }{2}u^2 + \frac{\sigma }{2}u^2_x\right) \ge \frac{\sqrt{\sigma }}{8}(\sqrt{12 - 3\sigma } - \sqrt{\sigma })u^2. \end{aligned}$$

We are now in a position to state a local-in-space type of blow-up mechanism for the Cauchy problem (1.5).

Theorem 2.1

Let \(1 \le \sigma \le 4\), \((u_0,\rho _0 - 1) \in H^s({\mathbb {R}})\times H^{s - 1}({\mathbb {R}})\), \(s \ge 2\) and \(T > 0\) be the maximal time of existence of the corresponding solution \((u,\rho )\) to (1.5). Assume that there exists a point \(x_0 \in {\mathbb {R}}\) such that

$$\begin{aligned} \rho _0(x_0) = 0, \end{aligned}$$

and

$$\begin{aligned} u_{0,x}(x_0) < {\left\{ \begin{array}{ll} \displaystyle -\beta _\sigma \left| u_0(x_0) + \frac{\gamma - A}{3 - \sigma }\right| , &{}\hbox {if} \quad \gamma \ne A, \; \sigma \ne 3, \\ -\beta _\sigma |u_0(x_0)|, &{}\hbox {if} \quad \gamma = A, \end{array}\right. } \end{aligned}$$
(2.3)

where

$$\begin{aligned} \beta _\sigma = \sqrt{-\frac{1}{2} + \frac{3}{\sigma } - \frac{\sqrt{12 - 3\sigma }}{2\sqrt{\sigma }}}. \end{aligned}$$
(2.4)

Then the solution \((u,\rho )\) blows up in finite time with an estimate of the blow-up time \(T^*\) as

$$\begin{aligned} T^* \le {\left\{ \begin{array}{ll} \dfrac{2}{\sigma \sqrt{u^2_{0,x}(x_0) - \beta ^2_\sigma \left( u_0(x_0) + \dfrac{\gamma - A}{3 - \sigma }\right) ^2}}, &{}\hbox {if} \quad \gamma \ne A, \; \sigma \ne 3, \\ \dfrac{2}{\sigma \sqrt{u^2_{0,x}(x_0) - \beta ^2_\sigma u^2_0(x_0)}}, &{}\hbox {if} \quad \gamma = A. \end{array}\right. } \end{aligned}$$

Proof

The two associated Lagrangian scales of the g2DGH system are established by

$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{\partial q_1}{\partial t} = \sigma u(t,q_1) - \gamma , &{}\quad 0< t < T, \\ q_1(0,x) = x, &{}\quad x \in {\mathbb {R}}, \end{array}\right. } \end{aligned}$$
(2.5)

and

$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{\partial q_2}{\partial t} = u(t,q_2), &{}\quad 0< t < T, \\ q_2(0,x) = x, &{}\quad x \in {\mathbb {R}}, \end{array}\right. } \end{aligned}$$
(2.6)

where \(u \in C^1([0,T), H^{s - 1}({\mathbb {R}}))\) is the first component of the solution \((u,\rho )\) to (1.5). A direct calculation yields

$$\begin{aligned} q_{1,tx} = \sigma u_x(t,q_1)q_{1,x}, \end{aligned}$$

and

$$\begin{aligned} q_{2,tx} = u_x(t,q_2)q_{2,x}. \end{aligned}$$

Hence for \(t > 0\), \(x \in {\mathbb {R}}\), we have

$$\begin{aligned} q_{1,x}(t,x) = e^{\sigma \int ^t_0 u_x(\tau ,q_1(\tau ,x))\,d\tau } > 0, \end{aligned}$$

and

$$\begin{aligned} q_{2,x}(t,x) = e^{\int ^t_0 u_x(\tau ,q_2(\tau ,x))\,d\tau } > 0, \end{aligned}$$

which implies that \(q_i(t,\cdot ): {\mathbb {R}}\rightarrow {\mathbb {R}}~(i=1,2)\) are two diffeomorphisms of the line for every \(t \in [0,T)\). It is inferred that there exists an \(x_1(t) \in {\mathbb {R}}\) such that

$$\begin{aligned} q_2(t,x_1(t)) = q_1(t,x_0). \end{aligned}$$

When \(t = 0\), we obtain

$$\begin{aligned} x_1(0) = q_2(0,x_1(0)) = q_1(0,x_0) = x_0. \end{aligned}$$

Along with the trajectory of \(q_2(t,x_1(t))\), we have

$$\begin{aligned} \dfrac{d}{dt}\rho (t,q_2(t,x_1(t))) = -u_x(t,q_2(t,x_1(t)))\rho (t,q_2(t,x_1(t))). \end{aligned}$$
(2.7)

Since \(\rho _0(x_0) = 0\), integrating the above equation gives

$$\begin{aligned} \begin{aligned} \rho (t,q_2(t,x_1(t)))&= \rho (0,q_2(0,x_1(0)))e^{-\int ^t_0 u_x(\tau ,q_2(\tau ,x_1(\tau )))\,d\tau } \\&= \rho _0(x_0)e^{-\int ^t_0 u_x(\tau ,q_2(\tau ,x_1(\tau )))\,d\tau } = 0, \end{aligned} \end{aligned}$$

that is

$$\begin{aligned} \rho (t,q_1(t,x_0)) = \rho (t,q_2(t,x_1(t))) = 0. \end{aligned}$$

Differentiating the first equation in system (1.5) with respect to x and using the identity \(-\partial ^2_xG*f = f - G*f\), one gets

$$\begin{aligned} \begin{aligned} u_{tx} + (\sigma u - \gamma )u_{xx}&= \frac{3 - \sigma }{2}u^2 + (\gamma - A)u - \frac{\sigma }{2}u^2_x + \frac{1}{2}\rho ^2 \\&\quad \; -\, G*\left[ \frac{3 - \sigma }{2}u^2 + (\gamma - A)u + \frac{\sigma }{2}u^2_x + \frac{1}{2}\rho ^2\right] . \end{aligned} \end{aligned}$$
(2.8)

From the convolution estimates in Lemma 2.3, we infer that

$$\begin{aligned}&\quad \dfrac{d}{dt}u_x(t,q_1(t,x_0)) \nonumber \\&= [u_{tx} + (\sigma u - \gamma )u_{xx}](t,q_1(t,x_0)) \nonumber \\&= \left\{ \frac{3 - \sigma }{2}u^2 + (\gamma - A)u - \frac{\sigma }{2}u^2_x - G*\left[ \frac{3 - \sigma }{2}u^2 + (\gamma - A)u + \frac{\sigma }{2}u^2_x \right. \right. \nonumber \\&\left. \left. +\, \frac{1}{2}\rho ^2\right] \right\} (t,q_1(t,x_0)) \nonumber \\&\le \left\{ \frac{3 - \sigma }{2}u^2 + (\gamma - A)u - \frac{\sigma }{2}u^2_x - (G_{+} + G_{-})*\left[ \frac{3 - \sigma }{2}u^2 + (\gamma - A)u\right. \right. \nonumber \\&\left. \left. +\, \frac{\sigma }{2}u^2_x\right] \right\} (t,q_1(t,x_0)) \nonumber \\&= \left\{ \frac{3 - \sigma }{2}\left( u + \frac{\gamma - A}{3 - \sigma }\right) ^2 - \frac{\sigma }{2}u^2_x - (G_{+} + G_{-})*\left[ \frac{3 - \sigma }{2}\left( u + \frac{\gamma - A}{3 - \sigma }\right) ^2\right. \right. \nonumber \\&\left. \left. +\, \frac{\sigma }{2}u^2_x\right] \right\} (t,q_1(t,x_0)) \nonumber \\&\le \left[ \frac{3 - \sigma }{2}\left( u + \frac{\gamma - A}{3 - \sigma }\right) ^2 - \frac{\sigma }{2}u^2_x - \frac{\sqrt{\sigma }}{4}(\sqrt{12 - 3\sigma } \right. \nonumber \\&\left. - \sqrt{\sigma })\left( u +\, \frac{\gamma - A}{3 - \sigma }\right) ^2 \right] (t,q_1(t,x_0)) \nonumber \\&= \frac{\sigma }{2}\left[ \beta ^2_\sigma \left( u + \frac{\gamma - A}{3 - \sigma }\right) ^2 - u^2_x\right] (t,q_1(t,x_0)), \end{aligned}$$
(2.9)

where \(\beta _\sigma \) is given in (2.4). Due to the right-hand side of (2.9), we set

$$\begin{aligned} \begin{aligned} M(t)&= \left[ \beta _\sigma \left( u + \frac{\gamma - A}{3 - \sigma }\right) - u_x\right] (t,q_1(t,x_0)), \\ N(t)&= \left[ \beta _\sigma \left( u + \frac{\gamma - A}{3 - \sigma }\right) + u_x\right] (t,q_1(t,x_0)). \end{aligned} \end{aligned}$$

Then (2.9) becomes

$$\begin{aligned} \dfrac{d}{dt}u_x(t,q_1(t,x_0)) \le \frac{\sigma }{2}M(t)N(t). \end{aligned}$$
(2.10)

For \(1 \le \sigma \le 4\), notice that \(0 \le \beta _\sigma \le 1\). Similarly using Lemma 2.3 it follows that

$$\begin{aligned}&\quad \dfrac{dM}{dt} = \left\{ \beta _\sigma [u_t + (\sigma u - \gamma )u_x] - [u_{tx} + (\sigma u - \gamma )u_{xx}]\right\} (t,q_1(t,x_0)) \nonumber \\&= \left\{ -\frac{3 - \sigma }{2}u^2 - (\gamma - A)u + \frac{\sigma }{2}u^2_x \right. \nonumber \\&\quad \left. + G*\left[ \frac{3 - \sigma }{2}u^2 + (\gamma - A)u + \frac{\sigma }{2}u^2_x + \frac{1}{2}\rho ^2\right] \right. \nonumber \\&\quad \, \left. -\, \beta _\sigma \partial _xG*\left[ \frac{3 - \sigma }{2}u^2 + (\gamma - A)u + \frac{\sigma }{2}u^2_x + \frac{1}{2}\rho ^2\right] \right\} (t,q_1(t,x_0)) \nonumber \\&= \left\{ -\frac{3 - \sigma }{2}u^2 - (\gamma - A)u + \frac{\sigma }{2}u^2_x + (1 - \beta _\sigma )G_{-}\right. \nonumber \\&\quad \left. *\left[ \frac{3 - \sigma }{2}u^2 + (\gamma - A)u + \frac{\sigma }{2}u^2_x\right] \right. \nonumber \\&\quad \, \left. +\, (1 + \beta _\sigma )G_{+}*\left[ \frac{3 - \sigma }{2}u^2 + (\gamma - A)u + \frac{\sigma }{2}u^2_x\right] + (1 - \beta _\sigma )G_{-}*\left( \frac{1}{2}\rho ^2\right) \right. \nonumber \\&\quad \, \left. +\, (1 + \beta _\sigma )G_{+}*\left( \frac{1}{2}\rho ^2\right) \right\} (t,q_1(t,x_0)) \nonumber \\&\ge \left\{ -\frac{3 - \sigma }{2}\left( u + \frac{\gamma - A}{3 - \sigma }\right) ^2 + \frac{\sigma }{2}u^2_x + (1 - \beta _\sigma )G_{-}\right. \nonumber \\&\quad \left. *\left[ \frac{3 - \sigma }{2}\left( u + \frac{\gamma - A}{3 - \sigma }\right) ^2 + \frac{\sigma }{2}u^2_x\right] \right. \nonumber \\&\quad \, \left. +\, (1 + \beta _\sigma )G_{+}*\left[ \frac{3 - \sigma }{2}\left( u + \frac{\gamma - A}{3 - \sigma }\right) ^2 + \frac{\sigma }{2}u^2_x\right] \right\} (t,q_1(t,x_0))\nonumber \\&\ge \left[ -\frac{3 - \sigma }{2}\left( u + \frac{\gamma - A}{3 - \sigma }\right) ^2 + \frac{\sigma }{2}u^2_x + \frac{\sqrt{\sigma }}{4}(\sqrt{12 - 3\sigma } \right. \nonumber \\&\quad \left. - \sqrt{\sigma })\left( u +\, \frac{\gamma - A}{3 - \sigma }\right) ^2 \right] (t,q_1(t,x_0)) \nonumber \\&= -\frac{\sigma }{2}\left[ \beta ^2_\sigma \left( u + \frac{\gamma - A}{3 - \sigma }\right) ^2 - u^2_x\right] (t,q_1(t,x_0))\nonumber \\&= -\frac{\sigma }{2}MN, \end{aligned}$$
(2.11)

and

$$\begin{aligned}&\quad \dfrac{dN}{dt} = \left\{ \beta _\sigma [u_t + (\sigma u - \gamma )u_x] + u_{tx} + (\sigma u - \gamma )u_{xx}\right\} (t,q_1(t,x_0)) \nonumber \\&= \left\{ \frac{3 - \sigma }{2}u^2 + (\gamma - A)u - \frac{\sigma }{2}u^2_x - G\right. \nonumber \\&\quad \left. *\left[ \frac{3 - \sigma }{2}u^2 + (\gamma - A)u + \frac{\sigma }{2}u^2_x + \frac{1}{2}\rho ^2\right] \right. \nonumber \\&\quad \, \left. -\, \beta _\sigma \partial _xG*\left[ \frac{3 - \sigma }{2}u^2 + (\gamma - A)u + \frac{\sigma }{2}u^2_x + \frac{1}{2}\rho ^2\right] \right\} (t,q_1(t,x_0)) \nonumber \\&= \left\{ \frac{3 - \sigma }{2}u^2 + (\gamma - A)u - \frac{\sigma }{2}u^2_x - (1 + \beta _\sigma )G_{-}\right. \nonumber \\&\quad \left. *\left[ \frac{3 - \sigma }{2}u^2 + (\gamma - A)u + \frac{\sigma }{2}u^2_x\right] \right. \nonumber \\&\quad \, \left. -\, (1 - \beta _\sigma )G_{+}*\left[ \frac{3 - \sigma }{2}u^2 + (\gamma - A)u + \frac{\sigma }{2}u^2_x\right] - (1 + \beta _\sigma )G_{-}*\left( \frac{1}{2}\rho ^2\right) \right. \nonumber \\&\quad \, \left. -\, (1 - \beta _\sigma )G_{+}*\left( \frac{1}{2}\rho ^2\right) \right\} (t,q_1(t,x_0))\nonumber \\&\le \left\{ \frac{3 - \sigma }{2}\left( u + \frac{\gamma - A}{3 - \sigma }\right) ^2 - \frac{\sigma }{2}u^2_x - (1 + \beta _\sigma )G_{-}\right. \nonumber \\&\quad \left. *\left[ \frac{3 - \sigma }{2}\left( u + \frac{\gamma - A}{3 - \sigma }\right) ^2 + \frac{\sigma }{2}u^2_x\right] \right. \nonumber \\&\quad \, \left. -\, (1 - \beta _\sigma )G_{+}*\left[ \frac{3 - \sigma }{2}\left( u + \frac{\gamma - A}{3 - \sigma }\right) ^2 + \frac{\sigma }{2}u^2_x\right] \right\} (t,q_1(t,x_0)) \nonumber \\&\le \left[ \frac{3 - \sigma }{2}\left( u + \frac{\gamma - A}{3 - \sigma }\right) ^2 - \frac{\sigma }{2}u^2_x - \frac{\sqrt{\sigma }}{4}(\sqrt{12 - 3\sigma } \right. \nonumber \\&\quad \left. - \sqrt{\sigma })\left( u +\, \frac{\gamma - A}{3 - \sigma }\right) ^2 \right] (t,q_1(t,x_0)) \nonumber \\&= \frac{\sigma }{2}\left[ \beta ^2_\sigma \left( u + \frac{\gamma - A}{3 - \sigma }\right) ^2 - u^2_x\right] (t,q_1(t,x_0)) \nonumber \\&= \frac{\sigma }{2}MN. \end{aligned}$$
(2.12)

By the assumption (2.3) on \(u_0(x_0)\), we know that

$$\begin{aligned} \begin{aligned}&M(0)> 0, \quad N(0)< 0, \\&M'(0) \ge -\frac{\sigma }{2}M(0)N(0) > 0, \\&N'(0) \le \frac{\sigma }{2}M(0)N(0) < 0. \end{aligned} \end{aligned}$$
(2.13)

We now claim for \(t \in [0,T)\)

$$\begin{aligned} M'(t) > 0, \quad N'(t) < 0. \end{aligned}$$
(2.14)

If (2.14) does not hold, then there exists \(t_0 \in [0,T)\) such that

$$\begin{aligned} t_0 = \min \{t \in [0,T)| M'(t) = 0 \;\; \mathrm{or}\;\; N'(t) = 0\}. \end{aligned}$$
(2.15)

Then (2.13) implies that \(t_0 > 0\). The definition of \(t_0\) given in (2.15) together with (2.11) and (2.12) leads to

$$\begin{aligned} 0 = M'(t_0) \ge -\frac{\sigma }{2}M(t_0)N(t_0) \quad \mathrm{or} \quad 0 = N'(t_0) \le \frac{\sigma }{2}M(t_0)N(t_0). \end{aligned}$$
(2.16)

On the other hand, since M(t) is increasing and N(t) is decreasing on \([0,t_0]\), we have

$$\begin{aligned} M(t_0) \ge M(0) > 0, \quad N(t_0) \le N(0) < 0. \end{aligned}$$

Thus,

$$\begin{aligned} -\frac{\sigma }{2}M(t_0)N(t_0) > 0 \quad \mathrm {and} \quad \frac{\sigma }{2}M(t_0)N(t_0) < 0, \end{aligned}$$

which contradict with (2.16). Therefore (2.14) holds for all \(t \in [0,T)\).

Furthermore from (2.13) we see that

$$\begin{aligned} M(t) \ge M(0) > 0, \quad N(t) \le N(0) < 0. \end{aligned}$$
(2.17)

Let

$$\begin{aligned} H(t) = \sqrt{-M(t)N(t)}. \end{aligned}$$

In view of (2.11), (2.12) and (2.17), it follows that

$$\begin{aligned} \begin{aligned} H'(t)&= -\frac{M'N + MN'}{2\sqrt{-MN}} \\&\ge -\frac{-\frac{\sigma }{2}MN^2 + \frac{\sigma }{2}M^2N}{2\sqrt{-MN}} \\&= \frac{M - N}{2\sqrt{-MN}}\left( -\frac{\sigma }{2}MN\right) \\&\ge \frac{\sigma }{2}H^2(t). \end{aligned} \end{aligned}$$
(2.18)

Solving the above differential inequality, we get

$$\begin{aligned} H(t) \ge \frac{2H(0)}{2 - \sigma H(0)t}. \end{aligned}$$

It is thereby inferred that

$$\begin{aligned} -u_x(t,q_1(t,x_0)) = \frac{M - N}{2} \ge H(t) \ge \frac{2H(0)}{2 - \sigma H(0)t} \rightarrow + \infty , \quad \mathrm{as} \quad t \rightarrow \frac{2}{\sigma H(0)}, \end{aligned}$$

which, in view of Lemma 2.2, implies that the solution \((u,\rho )\) blows up at a finite time \(T^*\) with

$$\begin{aligned} T^* \le \frac{2}{\sigma H(0)}. \end{aligned}$$

The proof of Theorem 2.1 is completed. \(\square \)

3 Symmetry of traveling wave solutions

In this section, we will consider the unique x-symmetric weak solution of the g2DGH system and prove that such a solution must be a traveling wave solution. We first rewrite system (1.1) with \(\rho = 1 + \eta \) (\(\eta \rightarrow 0\) as \(|x| \rightarrow \infty \)) as

$$\begin{aligned} {\left\{ \begin{array}{ll} (1 - \partial ^2_x)u_t - Au_x + 3uu_x - \sigma (2u_xu_{xx} + uu_{xxx}) + \gamma u_{xxx} + (1 + \eta )\eta _x = 0, \\ \eta _t + ((1 + \eta )u)_x = 0. \end{array}\right. } \end{aligned}$$
(3.1)

A weak solution of system (3.1) is defined as follows.

Definition 3.1

If \((u,\eta ) \in C({\mathbb {R}}_{+}, H^1({\mathbb {R}})\times L^2({\mathbb {R}}))\) satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} \int \int _{{\mathbb {R}}_{+}\times {\mathbb {R}}}\left[ u(1 - \partial ^2_x)\phi _t + \left( \frac{3}{2}u^2 - Au + \frac{\sigma }{2}u^2_x + \frac{(1 + \eta )^2}{2}\right) \phi _x \right. \\ \quad \left. -\, \left( \frac{\sigma }{2}u^2 - \gamma u\right) \phi _{xxx}\right] dtdx = 0, \\ \int \int _{{\mathbb {R}}_{+}\times {\mathbb {R}}}\left[ \eta \phi _t + (1 + \eta )u\phi _x\right] dtdx = 0, \end{array}\right. } \end{aligned}$$
(3.2)

for all \(\phi \in C^\infty _0({\mathbb {R}}_{+}\times {\mathbb {R}})\). Then \((u,\eta )\) is a weak solution to system (3.1).

Using the notation \(\langle \cdot ,\cdot \rangle \) for distributions, (3.2) can be rewritten as

$$\begin{aligned} {\left\{ \begin{array}{ll} \langle u,(1 - \partial ^2_x)\phi _t\rangle + \langle \frac{3}{2}u^2 - Au + \frac{\sigma }{2}u^2_x + \frac{(1 + \eta )^2}{2},\phi _x\rangle - \langle \frac{\sigma }{2}u^2 - \gamma u,\phi _{xxx}\rangle = 0, \\ \langle \eta ,\phi _t\rangle + \langle (1 + \eta )u,\phi _x\rangle = 0. \end{array}\right. } \end{aligned}$$
(3.3)

Now we give the definition of x-symmetric solution.

Definition 3.2

A solution \((u(t,x),\eta (t,x))\) is x-symmetric if there exists a function \(b(t) \in C^1({\mathbb {R}}_{+})\) such that for every \(t > 0\),

$$\begin{aligned} (u(t,x),\eta (t,x)) = (u(t,2b(t) - x),\eta (t,2b(t) - x)) \end{aligned}$$

for a.e. \(x \in {\mathbb {R}}\). We say that b(t) is the symmetric axis of \((u(t,x),\eta (t,x))\).

The next lemma gives the form of a weak solution of (3.1).

Lemma 3.1

Suppose that \((U(x),V(x)) \in H^1({\mathbb {R}})\times L^2({\mathbb {R}})\) and satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} \int _{{\mathbb {R}}}\left[ -cU(1 - \partial ^2_x)\varphi _x + \left( \frac{3}{2}U^2 - AU + \frac{\sigma }{2}U^2_x + \frac{(1 + V)^2}{2}\right) \varphi _x \right. \\ \quad \left. - \left( \frac{\sigma }{2}U^2 - \gamma U\right) \varphi _{xxx}\right] dx = 0, \\ \int _{{\mathbb {R}}}\left[ -cV\varphi _x + (1 + V)U\varphi _x\right] dx = 0, \end{array}\right. } \end{aligned}$$
(3.4)

for all \(\varphi \in C^\infty _0({\mathbb {R}})\). Then \((u,\eta )\) given by

$$\begin{aligned} (u(t,x),\eta (t,x)) = (U(x - c(t - t_0)),V(x - c(t - t_0))) \end{aligned}$$
(3.5)

is a weak solution of system (3.1) for any fixed \(t_0 \in {\mathbb {R}}\).

Proof

Without loss of generality, suppose \(t_0 = 0\). Following the arguments in [14], we obtain that \((u,\eta )\) belongs to \(C({\mathbb {R}}, H^1({\mathbb {R}})\times L^2({\mathbb {R}}))\). For any \(\phi \in C^\infty _0({\mathbb {R}}_{+}\times {\mathbb {R}})\), letting \(\phi _c(t,x) = \phi (t,x + ct)\), we find that

$$\begin{aligned} \partial _x(\phi _c) = (\phi _x)_c, \quad \partial _t(\phi _c) = (\phi _t)_c + c(\phi _x)_c. \end{aligned}$$
(3.6)

Assume \((u(t,x),\eta (t,x)) = (U(x - ct),V(x - ct))\), it is easy to see that

$$\begin{aligned} \begin{aligned}&\langle u,\phi \rangle = \langle U,\phi _c\rangle , \quad \langle u^2,\phi \rangle = \langle U^2,\phi _c\rangle , \quad \langle u^2_x,\phi \rangle = \langle U^2_x,\phi _c\rangle , \\&\langle \eta ,\phi \rangle = \langle V,\phi _c\rangle , \\&\langle (1 + \eta )^2,\phi \rangle = \langle (1 + V)^2,\phi _c\rangle , \quad \langle (1 + \eta )u,\phi \rangle = \langle (1 + V)U,\phi _c\rangle , \end{aligned} \end{aligned}$$
(3.7)

where \((U,V) = (U(x),V(x))\). It follows from (3.6) and (3.7) that

$$\begin{aligned}&\langle u,(1 - \partial ^2_x)\phi _t\rangle = \left\langle U,\left( (1 - \partial ^2_x)\phi _t\right) _c\right\rangle = \langle U,(1 - \partial ^2_x)(\partial _t\phi _c - c\partial _x\phi _c)\rangle , \nonumber \\&\left\langle \frac{3}{2}u^2 - Au + \frac{\sigma }{2}u^2_x + \frac{(1 + \eta )^2}{2},\phi _x\right\rangle = \left\langle \frac{3}{2}U^2 - AU + \frac{\sigma }{2}U^2_x + \frac{(1 + V)^2}{2},\partial _x\phi _c\right\rangle , \nonumber \\&\left\langle \frac{\sigma }{2}u^2 - \gamma u,\phi _{xxx}\right\rangle = \left\langle \frac{\sigma }{2}U^2 - \gamma U,\partial ^3_x\phi _c\right\rangle , \nonumber \\&\langle \eta ,\phi _t\rangle = \langle V,(\phi _t)_c\rangle = \langle V,\partial _t\phi _c - c\partial _x\phi _c\rangle , \nonumber \\&\langle (1 + \eta )u,\phi _x\rangle = \langle (1 + V)U,\partial _x\phi _c\rangle . \end{aligned}$$
(3.8)

Notice that (U, V) is independent of time, for T sufficiently large such that it does not belong to the support of \(\phi _c\), it is then deduced that

$$\begin{aligned} \begin{aligned} \langle U,(1 - \partial ^2_x)\partial _t\phi _c\rangle&= \int _{{\mathbb {R}}}U(x)\int _{{\mathbb {R}}_{+}}\partial _t(1 - \partial ^2_x)\phi _c\,dtdx \\&= \int _{{\mathbb {R}}}U(x)[(1 - \partial ^2_x)\phi _c(T,x) - (1 - \partial ^2_x)\phi _c(0,x)]\,dx = 0, \\ \langle U,\partial _t\phi _c\rangle&= \int _{{\mathbb {R}}}U(x)\int _{{\mathbb {R}}_{+}}\partial _t\phi _c\,dtdx = \int _{{\mathbb {R}}}U(x)[\phi _c(T,x) \\&\quad -\, \phi _c(0,x)]\,dx = 0, \\ \langle V,\partial _t\phi _c\rangle&= \int _{{\mathbb {R}}}V(x)\int _{{\mathbb {R}}_{+}}\partial _t\phi _c\,dtdx = \int _{{\mathbb {R}}}V(x)[\phi _c(T,x) \\&\quad -\, \phi _c(0,x)]\,dx = 0. \end{aligned} \end{aligned}$$
(3.9)

Combining (3.8) with (3.9) gives rise to

$$\begin{aligned} \begin{aligned}&\langle u,(1 - \partial ^2_x)\phi _t\rangle + \left\langle \frac{3}{2}u^2 - Au + \frac{\sigma }{2}u^2_x + \frac{(1 + \eta )^2}{2},\phi _x\right\rangle - \left\langle \frac{\sigma }{2}u^2 - \gamma u,\phi _{xxx}\right\rangle \\&\quad = \langle U,-c(1 - \partial ^2_x)\partial _x\phi _c\rangle + \left\langle \frac{3}{2}U^2 - AU + \frac{\sigma }{2}U^2_x + \frac{(1 + V)^2}{2},\partial _x\phi _c\right\rangle \\&\qquad -\, \left\langle \frac{\sigma }{2}U^2 - \gamma U,\partial ^3_x\phi _c\right\rangle \\&\quad = \int _{{\mathbb {R}}_{+}}\int _{{\mathbb {R}}}\left[ -cU(1 - \partial ^2_x)\partial _x\phi _c + \left( \frac{3}{2}U^2 - AU + \frac{\sigma }{2}U^2_x + \frac{(1 + V)^2}{2}\right) \partial _x\phi _c \right. \\&\qquad \left. -\, \left( \frac{\sigma }{2}U^2 - \gamma U\right) \partial ^3_x\phi _c\right] dxdt \\&\quad = 0, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \langle \eta ,\phi _t\rangle + \langle (1 + \eta )u,\phi _x\rangle&= \langle V,-c\partial _x\phi _c\rangle + \langle (1 + V)U,\partial _x\phi _c\rangle \\&= \int _{{\mathbb {R}}_{+}}\int _{{\mathbb {R}}}\left[ -cV\partial _x\phi _c + (1 + V)U\partial _x\phi _c\right] dxdt = 0, \end{aligned} \end{aligned}$$

where we used (3.4) with \(\varphi (x) = \phi _c(t,x)\), which belongs to \(C^\infty _0({\mathbb {R}})\), for every given \(t \ge 0\). This completes the proof of the lemma. \(\square \)

Finally, we state the main result of this section.

Theorem 3.1

If \((u(t,x),\eta (t,x))\) is a unique weak solution of system (3.1) and is x-symmetric, then \((u(t,x),\eta (t,x))\) is a traveling wave solution.

Proof

Recalling Definition 3.1 and noting that \(C^\infty _0({\mathbb {R}}_{+}\times {\mathbb {R}})\) is dense in \(C^1_0({\mathbb {R}}_{+}, C^3_0({\mathbb {R}}))\), thus we only consider the test function \(\phi \) belonging to \(C^1_0({\mathbb {R}}_{+}, C^3_0({\mathbb {R}}))\). Let us introduce the notation

$$\begin{aligned} \phi _b(t,x) = \phi (t,2b(t) - x), \quad b(t) \in C^1({\mathbb {R}}). \end{aligned}$$

Then we derive that \((\phi _b)_b = \phi \) and

$$\begin{aligned} \begin{aligned}&\partial _xu_b = -(\partial _xu)_b, \quad \partial _x\phi _b = -(\partial _x\phi )_b, \\&\partial _t\phi _b = (\partial _t\phi )_b + 2{\dot{b}}(\partial _x\phi )_b, \end{aligned} \end{aligned}$$
(3.10)

where \({\dot{b}}\) denotes the derivative of b with respect to t. Furthermore, we obtain that

$$\begin{aligned}&\langle u_b,\phi \rangle =\langle u,\phi _b\rangle , \langle u^2_b,\phi \rangle = \langle u^2,\phi _b\rangle , \quad \langle (\partial _xu_b)^2,\phi \rangle = \langle (\partial _xu)^2,\phi _b\rangle , \nonumber \\&\langle \eta _b,\phi \rangle = \langle \eta ,\phi _b\rangle , \langle (1 + \eta _b)^2,\phi \rangle = \langle (1 + \eta )^2,\phi _b\rangle , \nonumber \\&\langle (1 + \eta _b)u_b,\phi \rangle = \langle (1 + \eta )u,\phi _b\rangle . \end{aligned}$$
(3.11)

Since \((u,\eta )\) is x-symmetric, using (3.10) and (3.11) we have

$$\begin{aligned} \begin{aligned} \langle u,(1 - \partial ^2_x)\phi _t\rangle = \langle u,((1 - \partial ^2_x)\partial _t\phi )_b\rangle&= \langle u,(1 - \partial ^2_x)(\partial _t\phi _b + 2{\dot{b}}\partial _x\phi _b)\rangle , \\ \left\langle \frac{3}{2}u^2 - Au + \frac{\sigma }{2}u^2_x + \frac{(1 + \eta )^2}{2},\phi _x\right\rangle&= -\left\langle \frac{3}{2}u^2 - Au + \frac{\sigma }{2}u^2_x + \frac{(1 + \eta )^2}{2},\partial _x\phi _b\right\rangle , \\ \left\langle \frac{\sigma }{2}u^2 - \gamma u,\phi _{xxx}\right\rangle&= -\left\langle \frac{\sigma }{2}u^2 - \gamma u,\partial ^3_x\phi _b\right\rangle , \\ \langle \eta ,\phi _t\rangle&= \langle \eta ,\partial _t\phi _b + 2{\dot{b}}\partial _x\phi _b\rangle , \\ \langle (1 + \eta )u,\phi _x\rangle&= -\langle (1 + \eta )u,\partial _x\phi _b\rangle . \end{aligned} \end{aligned}$$
(3.12)

In view of (3.3), one deduces

$$\begin{aligned}&\langle u,(1 - \partial ^2_x)\phi _t\rangle + \left\langle \frac{3}{2}u^2 - Au + \frac{\sigma }{2}u^2_x + \frac{(1 + \eta )^2}{2},\phi _x\right\rangle - \left\langle \frac{\sigma }{2}u^2 - \gamma u,\phi _{xxx}\right\rangle \nonumber \\&\quad = \langle u,(1 - \partial ^2_x)(\partial _t\phi _b + 2{\dot{b}}\partial _x\phi _b)\rangle - \left\langle \frac{3}{2}u^2 - Au + \frac{\sigma }{2}u^2_x + \frac{(1 + \eta )^2}{2},\partial _x\phi _b\right\rangle \nonumber \\&\qquad +\, \left\langle \frac{\sigma }{2}u^2 - \gamma u,\partial ^3_x\phi _b\right\rangle = 0, \nonumber \\&\langle \eta ,\phi _t\rangle + \langle (1 + \eta )u,\phi _x\rangle \nonumber \\&\quad = \langle \eta ,\partial _t\phi _b + 2{\dot{b}}\partial _x\phi _b\rangle - \langle (1 + \eta )u,\partial _x\phi _b\rangle = 0. \end{aligned}$$
(3.13)

Thanks to \((\phi _b)_b = \phi \), by taking \(\phi = \phi _b\) in (3.13), we get

$$\begin{aligned} {\left\{ \begin{array}{ll} \langle u,(1 - \partial ^2_x)(\partial _t\phi + 2{\dot{b}}\partial _x\phi )\rangle - \left\langle \frac{3}{2}u^2 - Au + \frac{\sigma }{2}u^2_x + \frac{(1 + \eta )^2}{2},\partial _x\phi \right\rangle \\ \quad +\, \left\langle \frac{\sigma }{2}u^2 - \gamma u,\partial ^3_x\phi \right\rangle = 0, \\ \langle \eta ,\partial _t\phi + 2{\dot{b}}\partial _x\phi \rangle - \langle (1 + \eta )u,\partial _x\phi \rangle = 0. \end{array}\right. } \end{aligned}$$
(3.14)

Combining (3.3) and (3.14), we find

$$\begin{aligned} {\left\{ \begin{array}{ll} \langle u,{\dot{b}}(1 - \partial ^2_x)\partial _x\phi \rangle - \left\langle \frac{3}{2}u^2 - Au + \frac{\sigma }{2}u^2_x + \frac{(1 + \eta )^2}{2},\partial _x\phi \right\rangle + \left\langle \frac{\sigma }{2}u^2 - \gamma u,\partial ^3_x\phi \right\rangle = 0, \\ \langle \eta ,{\dot{b}}\partial _x\phi \rangle - \langle (1 + \eta )u,\partial _x\phi \rangle = 0. \end{array}\right. } \end{aligned}$$
(3.15)

For a fixed \(t_0 > 0\) and any \(\varphi \in C^\infty _0({\mathbb {R}})\), let \(\phi _{\epsilon }(t,x) = \varphi (x)\rho _\epsilon (t)\), where \(\rho _\epsilon \in C^\infty _0({\mathbb {R}}_{+})\) is a mollifier with the property that \(\rho _\epsilon \rightarrow \delta (t - t_0)\), the Dirac mass at \(t_0\), as \(\epsilon \rightarrow 0\). From (3.15), using the test function \(\phi _{\epsilon }(t,x)\), we get

$$\begin{aligned} \left\{ \begin{aligned}&\int _{{\mathbb {R}}}\left[ (1 - \partial ^2_x)\partial _x\varphi \int _{{\mathbb {R}}_{+}}{\dot{b}}u\rho _\epsilon (t)\,dt\right] dx \\&\quad -\, \int _{{\mathbb {R}}}\left[ \partial _x\varphi \int _{{\mathbb {R}}_{+}}\left( \frac{3}{2}u^2 - Au + \frac{\sigma }{2}u^2_x + \frac{(1 + \eta )^2}{2}\right) \rho _\epsilon (t)\,dt\right] dx \\&\quad +\, \int _{{\mathbb {R}}}\left[ \partial ^3_x\varphi \int _{{\mathbb {R}}_{+}}\left( \frac{\sigma }{2}u^2 - \gamma u\right) \rho _\epsilon (t)\,dt\right] dx = 0, \\&\int _{{\mathbb {R}}}\left( \partial _x\varphi \int _{{\mathbb {R}}_{+}}{\dot{b}}\eta \rho _\epsilon (t)\,dt\right) dx - \int _{{\mathbb {R}}}\left[ \partial _x\varphi \int _{{\mathbb {R}}_{+}}(1 + \eta )u\rho _\epsilon (t)\,dt\right] dx = 0. \end{aligned} \right. \end{aligned}$$
(3.16)

Notice that

$$\begin{aligned} \begin{aligned}&\lim _{\epsilon \rightarrow 0}\int _{{\mathbb {R}}_{+}}{\dot{b}}u\rho _\epsilon (t)\,dt = {\dot{b}}(t_0)u(t_0,x), \\&\lim _{\epsilon \rightarrow 0}\int _{{\mathbb {R}}_{+}}{\dot{b}}\eta \rho _\epsilon (t)\,dt = {\dot{b}}(t_0)\eta (t_0,x), \end{aligned} \end{aligned}$$

in \(L^2({\mathbb {R}})\), and

$$\begin{aligned} \begin{aligned}&\lim _{\epsilon \rightarrow 0}\int _{{\mathbb {R}}_{+}}\left( \frac{3}{2}u^2 - Au + \frac{\sigma }{2}u^2_x + \frac{(1 + \eta )^2}{2}\right) \rho _\epsilon (t)\,dt \\&\quad = \frac{3}{2}u^2(t_0,x) - Au(t_0,x) + \frac{\sigma }{2}u^2_x(t_0,x) + \frac{(1 + \eta (t_0,x))^2}{2}, \\&\lim _{\epsilon \rightarrow 0}\int _{{\mathbb {R}}_{+}}\left( \frac{\sigma }{2}u^2 - \gamma u\right) \rho _\epsilon (t)\,dt = \frac{\sigma }{2}u^2(t_0,x) - \gamma u(t_0,x), \\&\lim _{\epsilon \rightarrow 0}\int _{{\mathbb {R}}_{+}}(1 + \eta )u\rho _\epsilon (t)\,dt = (1 + \eta (t_0,x))u(t_0,x), \end{aligned} \end{aligned}$$

in \(L^1({\mathbb {R}})\). Hence, by taking the limit of (3.16) as \(\epsilon \rightarrow 0\), one obtains

$$\begin{aligned} \left\{ \begin{aligned}&\int _{{\mathbb {R}}}{\dot{b}}(t_0)u(t_0,x)(1 - \partial ^2_x)\partial _x\varphi \,dx - \int _{{\mathbb {R}}}\left( \frac{3}{2}u^2(t_0,x) - Au(t_0,x) + \frac{\sigma }{2}u^2_x(t_0,x) \right. \\&\quad \left. +\, \frac{(1 + \eta (t_0,x))^2}{2}\right) \partial _x\varphi \,dx + \int _{{\mathbb {R}}}\left( \frac{\sigma }{2}u^2(t_0,x) - \gamma u(t_0,x)\right) \partial ^3_x\varphi \,dx = 0, \\&\int _{{\mathbb {R}}}{\dot{b}}(t_0)\eta (t_0,x)\partial _x\varphi \,dx - \int _{{\mathbb {R}}}(1 + \eta (t_0,x))u(t_0,x)\partial _x\varphi \,dx = 0. \end{aligned} \right. \qquad \end{aligned}$$
(3.17)

Therefore \((u(t_0,x),\eta (t_0,x))\) satisfies (3.4) for \(c = {\dot{b}}(t_0)\). By means of Lemma 3.1, we see that \(({\tilde{u}}(t,x),{\tilde{\eta }}(t,x)) = (u(t_0,x - {\dot{b}}(t_0)(t - t_0)),\eta (t_0,x - {\dot{b}}(t_0)(t - t_0)))\) is a traveling wave solution of system (3.1). Due to \(({\tilde{u}}(t_0,x),{\tilde{\eta }}(t_0,x)) = (u(t_0,x),\eta (t_0,x))\) and the uniqueness of the solution of system (3.1), we deduce that \(({\tilde{u}}(t,x),{\tilde{\eta }}(t,x)) = (u(t,x),\eta (t,x))\) for any \(t > 0\). Thus, we complete the proof of Theorem 3.1. \(\square \)