1 Introduction

In 2001, Dullin, Gottwald, and Holm studied the following \(1+1\) quadratically nonlinear equation,

$$\begin{aligned} m_t+c_0u_x+um_x+2mu_x+\gamma u_{xxx}=0, x\in \mathbb {R}, t>0, \end{aligned}$$
(1.1)

where \(m=u-\alpha ^2u_{xx}\) is a momentum variable. This equation was derived using asymptotic expansions directly in the Hamiltonian for Euler’s equation in the shallow water regime, and it is completely integrable with a bi-Hamiltonian as well as with a Lax pair [25].

Using the notating \(m=u-\alpha ^2u_{xx}\), Eq. (1.1) can be written as

$$\begin{aligned} u_t-\alpha ^2u_{txx}+2\omega u_x+3u_x+\gamma u_{xxx}=\alpha ^2(2u_xu_{xx}+uu_{xxx}), x\in \mathbb {R},t>0, \end{aligned}$$
(1.2)

where \(\omega \) and \(\alpha \) are two positive constants. Formally, when \(\alpha ^2=0\), Eq. (1.2) becomes the Korteweg–de-Vries(KdV) equation,

$$\begin{aligned} u_t+2\omega u_x+3uu_x+\gamma u_{xxx}=0, x\in \mathbb {R}, t>0. \end{aligned}$$
(1.3)

While when \(\gamma =0\) and \(\alpha =1\), Eq. (1.1) turns into the Camassa–Holm equation [8, 9],

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

The details concerning the hydrodynamical relevance of Camassa–Holm equation were mathematically rigorously described in [14], where, in addition, authors investigate in what sense model under consideration gives us insight into the wave breaking phenomenon. Alternative derivations of Camassa–Holm equation as a equation for geodesic flow on the diffeomorphism group of the circle were presented by Constantin and Kolev [13] and Ionescu-Kruse [33]. The equation has bi-Hamiltonian structure [27] and is completely integrable [2, 9, 15, 16, 24]. Note that local well-posedness for the initial datum \(u_0(x)\in H^s\) with \(s>\frac{3}{2}\) was proved by several authors (see, for example, [17, 37, 40]. For the initial data with lower regularity, we refer to papers [7] and [41]. Camassa–Holm equation possesses a solitary wave with discontinuous first derivatives [8], which is named peakon (travelling wave solutions with a corner at their peak). More importantly, the peakons are orbitally stable [19, 36], which means that the shape of the peakons is stable so that these wave patterns are physically recognizable. Wave breaking for a large class of initial data has been established in [17, 18, 37, 45, 46] and in the recent paper [35], where, in particular, new and direct proof for the result from [39] on the necessary and sufficient condition for wave breaking was presented.

The Camassa–Holm equation also admits many integrable multicomponent generalizations. The most popular one is

$$\begin{aligned} \left\{ \begin{array}{ll} m_t-Au_x+um_x+2mu_x+\rho \rho _x=0, m=u-u_{xx}\\ \rho _t+(u\rho )_x=0,\\ \end{array} \right. \end{aligned}$$
(1.5)

Notice that the Camassa–Holm equation can be obtained via the obvious reduction \(\rho \equiv 0\) and \(A = 0\). System (1.5) was derived in 1996 [43]. Recently, Constantin–Ivanov [20] and Ivanov [34] established a rigorous justification of the derivation of system (1.5). Mathematical properties of the system have also been studied further in many works [1, 11, 12, 22, 26, 28, 29, 31, 32, 38, 42, 44, 47,48,49]. The reciprocal transformation between the two-component Camassa–Holm system and the first negative flow of the Ablowitz–Kaup–Newell–Segur hierarchy is established [12]. Escher, Lechtenfeld, and Yin investigated local well-posedness for the two-component Camassa–Holm system with initial data \((u_0, \rho _0-1)\in H^{s}(\mathbb {R})\times H^{s-1}(\mathbb {R})\) with \(s\ge 2\) and provided some precise blow-up scenarios for strong solutions to the system (1.5) [26]. The local well-posedness is improved in the Besov spaces (especially in the Sobolev space \(H^{s}(\mathbb {R})\times H^{s-1}(\mathbb {R})\) with \(s>\frac{3}{2}\), and the finite time blow-up is determined by either the slope of the first component u or the slope of the second component \(\rho \) [29]. Chen and Liu have derived some conditions of blow-up solutions for the generalized two-component Camassa–Holm system, which was recently derived in [11], following Ivanovs modeling approach [34]. The blow-up criterion is made more precise in [38] where the authors showed that the wave breaking in finite time only depends on the slope u. In other words, the wave breaking in u must occur before that in \(\rho \). This blow-up criterion is further improved [28] to the lowest Sobolev space.

In this paper, we are concerned with the Cauchy problem of the generalized two-component Dullin–Gottwald–Holm (DGH) system [10, 30],

$$\begin{aligned} \left\{ \begin{array}{ll} m_t-Au_x+\sigma (2mu_x+um_x)+3(1-\sigma )uu_x+\gamma u_{xxx}+\rho \rho _x=0,&{}\quad t>0, x\in \mathbb {R},\\ \rho _t+(u\rho )_x=0,&{}\quad t>0, x\in \mathbb {R},\\ u(0,x)=u_0(x),&{}\quad x\in \mathbb {R},\\ \rho (0,x)=\rho _0(x),&{}\quad x\in \mathbb {R}, \end{array} \right. \nonumber \\ \end{aligned}$$
(1.6)

where \(m=u-u_{xx}\) and \(\sigma \) is a real parameter. It is a model from the shallow water theory with nonzero constant vorticity, where u(x, t) is the horizontal velocity and \(\rho (t,x)\) is related to the free surface elevation from equilibrium with the boundary assumptions, \(u\rightarrow 0\) and \(\rho \rightarrow 1\) as \(|x|\rightarrow \infty \). The scalar \(A>0\) characterizes a linear underlying shear flow and hence the system (1.6) models wave-current interactions. The real dimensionless constant \(\sigma \) is a parameter, which provides the competition, or balance, in fluid convection between nonlinear steeping and amplification due to stretching. System (1.6) can be written in terms of u and \(\rho \),

$$\begin{aligned} \left\{ \begin{array}{ll} u_t-u_{txx}-Au_x+3uu_x-\sigma (2u_xu_{xx}+uu_{xxx})+\gamma u_{xxx}+\rho \rho _x=0, &{}\quad t>0, x\in \mathbb {R},\\ \rho _t+(u\rho )_x=0,&{}\quad t>0, x\in \mathbb {R},\\ u(0,x)=u_0(x),&{}\quad x\in \mathbb {R},\\ \rho (0,x)=\rho _0(x), &{}\quad x\in \mathbb {R}. \end{array} \right. \nonumber \\ \end{aligned}$$
(1.7)

System (1.7) has the following two Hamiltonians:

$$\begin{aligned}&H_1=\frac{1}{2}\int _{\mathbb {R}}\left( u^2+u_x^2+(\rho -1)\right) dx,\\&H_2=\frac{1}{2}\int _{\mathbb {R}}\left( u^3+\sigma uu_x^2-Au^2-\gamma u_x^2+2u(\rho -1)+u(\rho -1)\right) dx. \end{aligned}$$

The goal of the present paper is to study the persistence property of solutions and derive some conditions of blow-up solutions for the initial value problem (1.6). And the main tool to study the persistence property is the method of characteristics, which was initially used by Constantin and Escher [17, 21, 23] to investigate the question of global existence and the blow-up mechanism for the Camassa–Holm equation. In present paper, working with moderate weight functions that are commonly used in time-frequency analysis [3], we generalized the persistence result on the solution to Eq. (1.6) in the weighted \(L^p_{\phi }=L^p(\mathbb {R},\phi ^p(x)dx)\) space. The blow-up problem for the system (1.6) has been addressed in [10, 30]. The conditions on the initial datum \(u_0\) leading to the blow-up typically involves the computation of some global quantities (the Sobolev norm \(\Vert u_0\Vert _{H^1}\), or some other integral expressions of \(u_0\)). For \(\sigma =1, \gamma =0\), motivated by the recent paper [4,5,6], we establish a local-in-space blow-up criterion for the system (1.6), i.e., a blow-up condition involving only the properties of \(u_0\) in a neighborhood of a single point \(x_0\in \mathbb {R}\). Such criterion will be more general (and more natural)than the earlier blow-up results. The details can be found in Theorem 3.1.

The remainder of the paper is organized as follows. In Sect. 2, we established persistence properties and some unique continuous properties of the solution to Eq. (1.6) in weighted \(L^p_{\phi }:= L^p(\mathbb {R}, \phi ^p(x)dx)\) spaces. Finally, we construct initial data which leads to the local-in-space blow-up results.

Notation. In the sequel, we denote by \(*\) the convolution. For \(1\le p<\infty \), the norms in the Lebesgue space \(L^p(\mathbb {R})\) is \(\Vert f\Vert _{p}=\Big (\int _{\mathbb {R}}|f(x)|^pdx\Big )^{\frac{1}{p}}\), the space \(L^{\infty }(\mathbb {R})\) consists of all essentially bounded, Lebesgue measurable functions f equipped with the norm \(\Vert f\Vert _{\infty }=\inf _{\mu (e)=0}\sup _{x\in \mathbb {R}\setminus e}|f(x)|\). For a function f in the classical Sobolev spaces \(H^s(\mathbb {R})\;(s\ge 0)\) the norm is denoted by \( \Vert f\Vert _{H^s} \). We denote \(G(x) = {1\over 2} e^{-|x|}\) the fundamental solution of \( 1 - \partial ^2_x \) on \(\mathbb {R}\), and define the two convolution operators \(G_+, \;G_-\) as

$$\begin{aligned} \begin{aligned}&G_+ *f (x) = {e^{-x} \over 2} \int ^x_{-\infty } e^y f(y) dy,\\&G_- *f(x) = {e^{x}\over 2} \int ^\infty _{x} e^{-y} f(y) dy. \end{aligned} \end{aligned}$$
(1.8)

Then we have the relations \( \displaystyle G= G_+ + G_-, G_x = G_- - G_+. \mathbb {R}\)

2 Persistence property

In this section, we intend to find a large class of weight functions \(\phi \) such that

$$\begin{aligned} \sup _{t\in [0,T)}(\Vert u(t)\phi \Vert _{L^p}+\Vert u_x(t)\phi \Vert _{L^p}+\Vert \rho (t)\phi \Vert _{L^p})<\infty , \end{aligned}$$

this way we obtain a persistence results on solution \((u,\rho )\) to Eq. (1.6) in the weight \(L^p\) space \(L^p_{\phi }:=L^p(\mathbb {R},\phi ^p(x)dx)\). As a consequence and an application we determine the spatial asymptotic behavior of certain solutions to Eq. (1.6). We will work with moderate weight functions which appear with regularity in the theory of time-frequency analysis and have led to optimal results for the Camassa–Holm equation in [3]. Firstly, we list some knowledge in time frequency analysis for later use, for the details see [3].

Definition 2.1

([3]) An admissible weight function for system (1.6) is a local absolutely continuous function \(\phi :\mathbb {R}\rightarrow \mathbb {R}\) such that, for some \(A>0\) and almost all \(x\in \mathbb {R}, |\phi '(x)| \le A|\phi (x)|\), and that is \(v-\hbox {moderate}\) for some sub-multiplicative function v satisfying \(\inf _\mathbb {R}v>0\) and

$$\begin{aligned} \int _\mathbb {R}\frac{v(x)}{e^{|x|}}dx<\infty . \end{aligned}$$
(2.1)

We can now state our main result on admissible weights.

Theorem 2.1

Assume that \(u_0\phi , u_{0,x}\phi , \rho _0\phi \in L^p(\mathbb {R}), 1\le p\le \infty \) for an admissible weight function \(\phi \) of Eq. (1.6). Let \(({u_0},{\rho _{0 - 1}})\in {H^s}(\mathbb {R}) \times {H^{s - 1}}(\mathbb {R})\) with \(s\ge 2\), and \(T>0\) be the maximal existence time of the solution \(({u}, {\rho })\) to system (1.6) with the initial data \(({u_0}, {\rho _0})\). Then, for all \(t\in [0,T]\), there is a constant \(C>0\) depending only on weight \(\phi \) such that

$$\begin{aligned} \Vert u(t)\phi \Vert _{L^p}+\Vert u_x(t)\phi \Vert _{L^p}+\Vert \rho (t)\phi \Vert _{L^p}\le & {} (\Vert u_0\phi \Vert _{L^p}+\Vert u_{0,x}\phi \Vert _{L^p}\\&+\Vert \rho _0\phi \Vert _{L^p})\exp \{\{C(1+M)t\}, \end{aligned}$$

where

$$\begin{aligned} M\doteq \sup _{t\in [0,T]}(\Vert u(t)\Vert _{L^{\infty }}+\Vert u_x(t)\Vert _{L^{\infty }}+\Vert \rho (t)\Vert _{L^{\infty }})<\infty . \end{aligned}$$

First, we present some standard definitions. In general a weight function is simple a non-negative function. A weight function \(v: \mathbb {R}^n\rightarrow \mathbb {R}\) is sub-multiplicative if

$$\begin{aligned} v(x+y)\le v(x)v(y),\quad \forall x,y\in \mathbb {R}^n. \end{aligned}$$
(2.2)

Given a sub-multiplicative function \(v: \mathbb {R}^n\rightarrow \mathbb {R}\), by definition a positive function \(\phi \) is v-moderate if

$$\begin{aligned} \exists C_0>0: \phi (x+y)\le C_0v(x)\phi (y),\forall x,y\in \mathbb {R}^n. \end{aligned}$$
(2.3)

If \(\phi \) is v-moderate for some sub-multiplicative function v, then we say that \(\phi \) is moderate. Let us recall the most standard examples of such weights.

Example 2.1

([3]) let \(\phi (x)=\phi _{a,b,c,d}=e^{a|x|^b}(1+|x|)^c \log (e+|x|)^d\). Then

(1) For \(a,c,d>0\) and \(0\le b\le 1\), such weight is sub-multiplicative.

(2) For \(|a|\le \alpha ,b\le \beta , |c|\le \gamma \) and \(|d|\le \delta , \phi _{a,b,c,d}\) is \(\phi _{\alpha ,\beta ,\gamma , \delta }-\hbox {moderate}\).

If v and \(\phi \) are continuous, then they admit the following properties, which may shed some light on Definition 2.1.

(1) If \(v\ne 0\) is an even sub-multiplicative weight function, then \(inf_\mathbb {R}v\ge 1\).

(2) Every nontrivial sub-multiplicative or moderate weight grows and decays not faster than exponentially: there exists \(a\ge 0\) such that

$$\begin{aligned} e^{-a}e^{-a|x|}\le \phi (x)\le e^ae^{a|x|}. \end{aligned}$$
(2.4)

(3) Let \(\phi \) be a locally continuous \(v-\) moderate weight such that \(C_0v(0)=1\) (where \(C_0\) is the constant in (2.8). If v has both left and right derivatives at the origin, then for a.e. \(y\in \mathbb {R}\),

$$\begin{aligned} |\phi '(y)|\le \phi (y) \end{aligned}$$
(2.5)

where \(A=C_0\max \{|v'(0-)|, |v'(0+)|\}\).

The interest of imposing the sub-multiplicativity condition on a weight function is also made clear by the following proposition:

Proposition 2.1

([3]) Let \(v: \mathbb {R}^n\rightarrow \mathbb {R}^+\) and \(C_0>0\). Then the following conditions are equivalent:

(1) \(\forall x,y: v(x+y)\le C_0v(x)v(y).\)

(2) For all \(1\le p,q,r\le \infty \) and for any measurable function \(f_1,f_2:\mathbb {R}^n\rightarrow {\mathcal {C}}\) the weighted Young inequality hold:

$$\begin{aligned} \Vert (f_1*f_2)v\Vert _r\le C_0\Vert f_1 v\Vert _p\Vert f_2 v\Vert _{q}, 1+\frac{1}{r}=\frac{1}{p}+\frac{1}{q}. \end{aligned}$$
(2.6)

The moderateness of a weight function is the good condition for weighed Young inequalities with two different weights.

Proposition 2.2

([3]) Let \(1\le p\le \infty \) and v be a sub-multiplicative weight on \(\mathbb {R}^n\). Then the following two conditions are equivalent:

(1) \(\phi \) is \(v-\) moderate weight function (with constant \(C_0\)).

(2) For all measurable function \(f_1\) and \(f_2\) the weighted Young estimate holds

$$\begin{aligned} \Vert (f_1*f_2)\phi \Vert _p\le C_0\Vert f_1 v\Vert _1\Vert f_2 \phi \Vert _{p}. \end{aligned}$$
(2.7)

Definition 2.2

([3]) An admissible weight function for the problem (1.1) is a locally absolutely continuous function \(\phi : \mathbb {R}\rightarrow \mathbb {R}\) s.t. for some \(A>0\) and a.e. \(x\in \mathbb {R}, |\phi '(t)|\le A |\phi |\), and that is \(v-moderate\), for some sub-multiplicative weight function v satisfying \(\inf _{x\in \mathbb {R}} v>0\) and

$$\begin{aligned} ve^{-|\cdot |}\in L^p(\mathbb {R}). \end{aligned}$$
(2.8)

Now, we give the needed results to pursue our goal. The DGH system (1.6) can be written in the following “transport” type:

$$\begin{aligned} \left\{ \begin{array}{ll} u_t+(\sigma u-\gamma )u_{x}=-\partial _x G*\left[ \frac{3-\sigma }{2}u^2+\frac{\sigma }{2}u_x^2+(\gamma -A)u+\frac{1}{2}\rho ^2\right] ,t>0,x\in \mathbb {R},\\ \rho _t+(u\rho )_x=0,t>0,x\in \mathbb {R},\\ u(0,x)=u_0(x), x\in \mathbb {R},\\ \rho (0,x)=\rho _0(x), x\in \mathbb {R}. \end{array} \right. \end{aligned}$$
(2.9)

The local well-posedness results for the system (1.6) has been established in the Sobolev space \(H^s\times H^{s-1}\), let us recall it as

Theorem 2.2

[30] If \((u_0, \rho _0-1)\in H^s(\mathbb {R})\times H^{s-1}(\mathbb {R}), s\ge 2\), 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 \((u,\rho -1)\) of system (1.6) in \(C([0,T); H^{s}(\mathbb {R})\times H^{s-1}(\mathbb {R}))\cap C^1([0,T); H^{s-1}(\mathbb {R})\times H^{s-2}(\mathbb {R}))\) with \((u,\rho -1)|_{t=0}=(u_0, \rho _0)\). Moreover, the solution depends continuously on the initial data, and T is independent of s.

As introduced in the introduction, we consider the following two associated Lagrangian scales of the generalized two component system (1.6):

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{\partial q_1}{\partial t}=u(t,q_1), &{}\quad 0<t<T,\\ q_1(0,x)=x,&{}\quad x\in \mathbb {R}\end{array} \right. \end{aligned}$$
(2.10)

and

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{\partial q_2}{\partial t}=\sigma u(t,q_2)-\gamma ,&{}\quad 0<t<T,\\ q_2(0,x)=x,&{}\quad x\in \mathbb {R}, \end{array} \right. \end{aligned}$$
(2.11)

where \(u\in C^1([0, T), H^{s-1})\) is the first component of the solution \((u, \rho )\) to Eq. (1.6) with initial data \((u_0, \rho _0) \mathbb {R}\) and \(T>0\) is the maximal time of existence.

A direct calculation shows

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

and

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

Thus for \(t>0, x\in \mathbb {R}\)

$$\begin{aligned} q_{1,x}(t,x)=\exp \left( \int ^t_0u_x(\tau , q_1(\tau ,x))\right) d\tau >0, (t,x)\in [0,T)\times \mathbb {R}\end{aligned}$$

and

$$\begin{aligned} q_{2,x}(t,x)=\exp \left( \int ^t_0\sigma u_x(\tau , q_2(\tau ,x))\right) d\tau >0, (t,x)\in [0,T)\times \mathbb {R}, \end{aligned}$$

indicating that \(q_1(t,\cdot ): \mathbb {R}\rightarrow \mathbb {R}\) and \(q_2(t,\cdot ): \mathbb {R}\rightarrow \mathbb {R}\) are diffeomorphisms of the line for each \(t\in [0, T)\). Hence, the \(L^{\infty }\) norm of any function \(v(t,\cdot )\in L^{\infty }(\mathbb {R}), T\in [0, t)\) is preserved under the family of diffeomorphisms \(q_1(t,\cdot )\) and \(q_2(t,\cdot )\) with \(t\in [0, T)\), that is

$$\begin{aligned} \Vert v(t,\cdot )\Vert _{L^{\infty }(\mathbb {R})}=\Vert v(t,q_1(t,\cdot ))\Vert _{L^{\infty }(\mathbb {R})}=\Vert v(t,q_2(t,\cdot ))\Vert _{L^{\infty }(\mathbb {R})}, t\in [0, T).\nonumber \\ \end{aligned}$$
(2.12)

Similarly, we obtain

$$\begin{aligned} \inf _{x\in \mathbb {R}} v(t,x)=\inf _{x\in \mathbb {R}}v(t,q_1(t,x))=\inf _{x\in \mathbb {R}}v(t,q_2(t,x)),t\in [0,T) \end{aligned}$$
(2.13)

and

$$\begin{aligned} \sup _{x\in \mathbb {R}} v(t,x)=\sup _{x\in \mathbb {R}}v(t,q_1(t,x))=\sup _{x\in \mathbb {R}}v(t,q_2(t,x)),t\in [0,T). \end{aligned}$$
(2.14)

Lemma 2.1

([26]) Let \(({u_0},{\rho _{0 - 1}})\in {H^s}(\mathbb {R}) \times {H^{s - 1}}(\mathbb {R})\) with \(s\ge 2\), and \(T>0\) be the maximal existence time of the solution \(({u}, {\rho })\) to system (1.6) with the initial data \(({u_0}, {\rho _0})\).

$$\begin{aligned} \rho (t, q(t,x))q_x(t,x)=\rho _0(x), (t,x)\in [0, T)\times \mathbb {R}. \end{aligned}$$
(2.15)

Moreover, if there exists \(x_0\in \mathbb {R}\) such that \(\rho _0(x_0)=0\), then \(\rho (t, q(t,x_0))=0\) for all \(t\in [0, T)\).

Proof of Theorem 2.1

Let \((u,\rho )\) be the solution to Eq. (1.6) with the initial data \((u_0, \rho _0)\), and T be the maximal existence times of the solution \((u,\rho )\), which is guaranteed by Theorem 2.2.

From the solution \((u,\rho )\in C([0, T), H^s(\mathbb {R}))\times C([0,T), H^{s-1}), s\ge 2.\) The Sobolev’s embedding theorem yields

$$\begin{aligned} M\equiv \sup _{t\in [0,T]}(\Vert u(t,\cdot )\Vert _{L^{\infty }}+\Vert u_x(t,\cdot )\Vert _{\infty }+\Vert \rho (t,\cdot )\Vert _{L^{\infty }})<\infty . \end{aligned}$$
(2.16)

For any \(N\in Z^+\), let us consider the N-trancations of \(\phi (x): f(x)=f_{N}(x)=\min \{\phi (x), N\}\), then \(f:\mathbb {R}\rightarrow \mathbb {R}\) is a locally absolutely function such that \(\Vert f\Vert _{\infty }\le N, |f'(x)|\le A|f(x)|\) a.e on \(\mathbb {R}\).

In addition, if \(C_1=\max \{C_0, \alpha ^{-1}\}\), where \(\alpha =\inf _{x\in \mathbb {R}}v(x)>0\), then

$$\begin{aligned} f(x+y)\le C_1v(x)f(y), \quad \forall x,y\in \mathbb {R}. \end{aligned}$$

Moreover, as shown in [3], the \(N-\)trunctions f of a \(v-\)moderate weight \(\phi \) are uniformly \(v-\) moderate with respect to N.

We rewrite Eq. (1.6) as the following form

$$\begin{aligned} \left\{ \begin{array}{ll} u_t+(\sigma u-\gamma )u_{x}=-\partial _x G*P(u,\rho ),&{}\quad t>0,x\in \mathbb {R},\\ \rho _t+(u\rho )_x=0, &{}\quad t> 0,x\in \mathbb {R},\\ u(0,x)=u_0(x),&{}\quad x\in \mathbb {R},\\ \rho (0,x)=\rho _0(x),&{}\quad x\in \mathbb {R}, \end{array} \right. \end{aligned}$$
(2.17)

where \(P(u, \rho )\doteq [\frac{3-\sigma }{2}u^2+\frac{\sigma }{2}u_x^2+(\gamma -A)u+\frac{1}{2}\rho ^2]\) and \(G(x) = {1\over 2} e^{-|x|}\).

Let us start from the case \(1\le p <\infty \), multiplying the first equation in Eq. (2.17) by \(|uf|^{p-1}sgn(uf)f\) and integrating it lead to

$$\begin{aligned} \int _\mathbb {R}|uf|^{p-1}sgn(uf)(\partial _t uf)dx=&\int _\mathbb {R}|uf|^{p-1}sgn (uf)f (\sigma u-\gamma ) u_x dx\nonumber \\&-\int _\mathbb {R}|uf|^{p-1}sgn(uf)f \partial _x(G*P(u,\rho ))dx. \end{aligned}$$
(2.18)

The first term on the left hand of (2.18) reads

$$\begin{aligned} \int _\mathbb {R}|uf|^{p-1}sgn(uf)(\partial _t uf)dx=\frac{1}{p}\frac{d}{dt}\Vert uf\Vert ^p_{L^p}=\Vert uf\Vert ^{p-1}_{L^p} =\Vert uf\Vert ^{p-1}_{L^p}\frac{d}{dt}\Vert uf\Vert _{L^p}. \end{aligned}$$

Then, the \(\hbox {H}\ddot{o}\hbox {lder}\) inequality is followed by the estimate

$$\begin{aligned}&|\int _\mathbb {R}|uf|^{p-1}sgn(uf)(\sigma u-\gamma ) u_x f dx|\le \Vert uf\Vert ^{p-1}_{L^p}\Vert (\sigma u-\gamma ) u_x f\Vert _{L^p}\\&\quad \le (|\sigma M|+|\gamma |)\Vert uf\Vert ^{p-1}_{L^p}\Vert u_x f\Vert _{L^p}. \end{aligned}$$

For the nonlocal term, we have

$$\begin{aligned}&|\int _\mathbb {R}|uf|^{p-1}sgn(uf)(f \partial _x(G*P(u,\rho )))dx|\\&\quad \le \Vert uf\Vert ^{p-1}_{L^p}\Vert f \partial _x(G*P(u,\rho ))\Vert _{L^p}\\&\quad \le C_{\alpha ,b, k}\Vert uf\Vert ^{p-1}_{L^p}\{\Vert (\partial _x G) v\Vert _{L^1}\Vert f\cdot (u+u^2+u_x^2+\rho ^2)\Vert _{L^p}\}\\&\quad \le C(1+M)\Vert uf\Vert ^{p-1}_{L^p}(\Vert uf\Vert _{L^p}+\Vert u_xf\Vert _{L^p}+\Vert \rho f\Vert _{L^p}). \end{aligned}$$

where the \(\hbox {H}\ddot{o}\hbox {lder}\)’s inequality, Propositions 3.1 and 3.2 in [3], and condition (2.1) are applied in the first inequality, the second one, and the last one, respectively, and the constant C only depends on v and \(\phi \). From Eq. (2.18) one may get

$$\begin{aligned} \frac{d}{dt}\Vert uf\Vert _{L^p}\le C(1+M)(\Vert uf\Vert _{L^p}+\Vert u_xf\Vert _{L^p}+\Vert \rho f\Vert _{L^p}). \end{aligned}$$
(2.19)

Let us now give the estimate on \(u_xf\). Differentiating the first equation in Eq. (2.17) with respect to the variable x and then multiplying by f, we may arrive at

$$\begin{aligned} \partial _t(u_{x}f)+\sigma u_x^2f+uu_{xx}f+f\partial _x^2(G*P(u,\rho ))=0. \end{aligned}$$

which yields

$$\begin{aligned}&\int _\mathbb {R}|u_xf|^{p-1}sgn(u_xf)\partial _t(u_xf)dx =\Vert u_xf\Vert ^{p-1}_{L^p}\frac{d}{dt}\Vert u_xf\Vert _{L^p},\\&|\int _\mathbb {R}|u_xf|^{p-1}sgn(u_x f)f u_x^2dx|\le \Vert u_xf\Vert ^{p-1}_{L^p}\Vert u_xf u_x\Vert _{L^p}\le M\Vert u_xf\Vert ^{p-1}_{L^p}\Vert u_xf\Vert _{L^p} \end{aligned}$$

and

$$\begin{aligned} |\int _{\mathbb {R}}&|u_xf|^{p-1}sgn(u_xf)[f\partial _x^2(G*P(u,\rho ))]dx|\\&\le \Vert u_xf\Vert ^{p-1}_{L^p}\Vert f\partial ^2_x(G*P(u,\rho ))\Vert _{L^p}\\&\le C(1+M)\Vert u_x f\Vert ^{p-1}_{L^p}(\Vert uf\Vert _{L^p}+\Vert u_xf\Vert _{L^p}+\Vert \rho f\Vert _{L^p}). \end{aligned}$$

For the second order derivative term, we have

$$\begin{aligned} |\int _{\mathbb {R}}|u_xf|^{p-1}sgn(u_xf)u u_{xx}f dx|&=|\int _\mathbb {R}|u_xf|^{p-1}sgn(u_xf)u[\partial _x(u_xf)-u_xf_x]dx|\\&=|\int _\mathbb {R}u\partial _x\left( \frac{|u_xf|^p}{p}\right) dx \\&\quad -\int _\mathbb {R}|u_xf|^{p-1}sgn(u_xf)uu_xf_xdx|\\&\le M(1+A)\Vert u_xf\Vert ^p_{L^p} \end{aligned}$$

where the inequality \(|f_x(x)|\le A f(x)\) for a.e. x is applied. Thus, it follows that

$$\begin{aligned} \frac{d}{dt}\Vert u_x f\Vert _{L^p}\le C_3(1+M)(\Vert uf\Vert _{L^p}+\Vert u_xf\Vert _{L^p}+\Vert \rho f\Vert _{L^p}). \end{aligned}$$
(2.20)

We now multiply the second equation in Eq. (2.17) with \(|\rho f|^{p-1}sgn(\rho f)f\) and integrate to obtain the identity

$$\begin{aligned} \frac{1}{p}\frac{d}{dt}\Vert \rho f\Vert ^p_{L^p}+\int _\mathbb {R}|\rho f|^{p-1}sgn(\rho f)f u \rho _xdx+\int _\mathbb {R}|\rho f|^{p-1}sgn(\rho f)fu_x\rho dx=0. \end{aligned}$$

As above, we get

$$\begin{aligned} \int _\mathbb {R}|\rho f|^{p-1}sgn(\rho f)f u_x\rho dx|\le \Vert \rho f\Vert ^{p-1}_{L^p}\Vert fu_x\rho \Vert _{L^p}\le M\Vert \rho f\Vert ^{p-1}_{L^p}\Vert \rho f\Vert _{L^p}, \end{aligned}$$

and

$$\begin{aligned} \int _\mathbb {R}|\rho f|^{p-1}sgn(\rho f)f u\rho _x dx|&=|\int _\mathbb {R}|\rho f|^{p-1}sgn(\rho f)u[\partial _x(\rho f)-\rho f_x]dx|\\&=|\int _\mathbb {R}u\partial _x\left( \frac{|\rho f|^p}{p}\right) dx-\int _\mathbb {R}|\rho f|^{p-1}sgn(\rho f)u \rho f_x dx|\\&\le M(1+A)\Vert \rho f\Vert ^p_{L^p}, \end{aligned}$$

this yields

$$\begin{aligned} \frac{d}{dt}\Vert \rho f\Vert _{L^p}\le C_4M \Vert \rho f\Vert _{L^p}. \end{aligned}$$
(2.21)

Based on the inequalities (2.19)–(2.21), by Gronwall’s inequality

$$\begin{aligned}&\Vert u(t)f\Vert _{L^p}+\Vert u_x(t)f\Vert _{L^p}+\Vert \rho (t)f\Vert _{L^p}\\&\quad \le (\Vert u_0f\Vert _{L^p}+\Vert u_{0,x}f\Vert _{L^p}+\Vert \rho _0f\Vert _{L^p})\exp (C(1+M)t), \end{aligned}$$

for all \(t\in [0, T)\). Since \(f(x)=f_N(x)\uparrow \phi (x)\) as \(N\rightarrow \infty \) for a.e. \(x\in \mathbb {R}\) and \(u_0\phi , u_{0,x}\phi , \rho _0\phi \in L^p(\mathbb {R})\) the assertion of the theorem follows for the case \(p\in [1, \infty )\). Since \(\Vert \cdot \Vert _{L^\infty }=\lim _{p\rightarrow \infty }\Vert \cdot \Vert _{L^p}\) it is clear that the theorem also applies for \(p=\infty \). \(\square \)

Corollary 2.1

Let \(1\le p\le \infty \) and \(\phi \) be a \(v-\hbox {moderate}\) weight function as in Definition 2.1 satisfying \(ve^{-|\cdot |}\in L^p(\mathbb {R})\). Assume that \(u_0\phi , u_{0,x}\phi , \rho _0\phi \in L^p(\mathbb {R})\) and \(u_0\phi ^{\frac{1}{2}}, u_{0,x}\phi ^{\frac{1}{2}}, \rho _0\phi ^{\frac{1}{2}}\in L^2(\mathbb {R})\). Let also \((u,\rho )\in C([0, T), H^s(\mathbb {R}))\times C([0, T), H^{s-1}(\mathbb {R})), s\ge 2\) be the strong solution of the Cauchy problem for Eq. (1.6) emanating from \((u_0, \rho _0)\). Then

$$\begin{aligned} \sup _{t\in [0, T)}(\Vert u(t)\phi \Vert _{L^p}+\Vert u_x(t)\phi \Vert _{L^p}+\Vert \rho (t)\phi \Vert _{L^{p}})<\infty \end{aligned}$$

and

$$\begin{aligned} \sup _{t\in [0, T)}(\Vert u(t)\phi ^{1/2}\Vert _{L^p}+\Vert u_x(t)\phi ^{1/2}\Vert _{L^p}+\Vert \rho (t)\phi ^{1/2}\Vert _{L^{p}})<\infty . \end{aligned}$$

Proof

As explained in [3], if the function \(\phi \) is a \(v-\)moderate weight function, then the function \(\phi ^{\frac{1}{2}}\) is also a \(v^{1/2}-\)moderate weight satisfying \(|(\phi ^{1/2})'(x)|\le \frac{A}{2}\phi ^{1/2}(x), \inf v^{1/2}>0\) and \(v^{1/2}e^{-|\cdot |}\in L^1(\mathbb {R})\). We use Theorem 2.1 with \(p=2\) to the weight \(\phi ^{1/2}\) and obtain

$$\begin{aligned} \Vert&u(t)\phi ^{1/2}\Vert _{L^2}+\Vert u_x(t)\phi ^{1/2}\Vert _{L^2}+\Vert \rho (t)\phi ^{1/2}\Vert _{L^2}\nonumber \\&\le (\Vert u_0(t)\phi ^{1/2}\Vert _{L^2}+\Vert u_{0,x}(t)\phi ^{1/2}\Vert _{L^2}+\Vert \rho _0(t)\phi ^{1/2}\Vert _{L^2})\exp (C(1+M)t). \end{aligned}$$
(2.22)

In view of Proposition 3.2 in [3], noticing \(f(x)=f_N(x)=\min \{\phi (x), N\}\) admits

$$\begin{aligned}&\Vert f \partial _x(G*P(u, u_x))\Vert _{L^p}\nonumber \\&\quad \le C_{\alpha ,\kappa , b}\Vert f\partial _x(G(x)*(u+u^2+u^2_x+\rho ^2))\Vert _{L^p}\nonumber \\&\quad \le C\Vert f\partial _xG(x)\Vert _{L^p}\Vert f(u+u^2+u^3+u^4+u_x^2)\Vert _{L^1}\nonumber \\&\quad \le C\Vert fe^{-|x|}\Vert _{L^p}(\Vert fu\Vert _{L^1}+\Vert f^{1/2}u\Vert ^2_{L^2}+\Vert f^{1/2}u_x\Vert ^2_{L^2}+\Vert f^{1/2}\rho \Vert ^2_{L^2})\nonumber \\&\quad \le C_1\exp (C_2(1+M)t), \end{aligned}$$
(2.23)

where we used (2.22) and Theorem 2.1 with \(p=1\).

Similarly, noticing \(\partial _x^2G=G-\delta \) reveals

$$\begin{aligned}&\Vert f\partial _x^2(G*P(u,u_x))\Vert _{L^p}\nonumber \\&\quad \le C_1\exp (C_2(1+M)t)+C_3(1+M)(\Vert uf\Vert _{L^p}+\Vert fu_x\Vert _{L^p}+\Vert f\rho \Vert _{L^p}), \end{aligned}$$
(2.24)

where the constants on the right-hand side of Eqs. (2.23) and (2.24) are independent of N.

By using the procedure as shown in the proof of Theorem 2.1, we can readily obtain

$$\begin{aligned} \frac{d}{dt}\Vert uf\Vert _{L^p}\le C(1+M)\Vert uf\Vert _{L^p}+\Vert f\partial _x(G*P(u, u_x))\Vert _{L^p} \end{aligned}$$
(2.25)

and

$$\begin{aligned} \frac{d}{dt}\Vert u_xf\Vert _{L^p}\le C(1+M)\Vert u_xf\Vert _{L^p}+\Vert f\partial _x^2(G*P(u, u_x))\Vert _{L^p} \end{aligned}$$
(2.26)

for \(1\le p<\infty \). Plugging Eqs. (2.23) and (2.24) into Eqs. (2.25) and (2.26), respectively, and summing up them, we obtain

$$\begin{aligned}&\frac{d}{dt}(\Vert u(t)f\Vert _{L^p}+\Vert u_x(t)f\Vert _{L^p}+\Vert \rho (t)f\Vert _{L^p})\\&\quad \le K_1(1+M)(\Vert u_0f\Vert _{L^p}+\Vert u_{0,x}f\Vert _{L^p}+\Vert \rho _0f\Vert _{L^p})+C_1\exp (C_2(1+M)t), \end{aligned}$$

which is taken integration and limit \(N\rightarrow \infty \) to get the conclusion in the case \(1\le p<\infty \). The constants throughout the proof are independent of p. Therefore, for \(p=\infty \) one can rely on the result established for the finite exponents q and then let \(q\rightarrow \infty \). The argument is fully similar to that of Theorem 2.1.\(\square \)

3 Blow-up

For \(\sigma =1, \gamma =0\), we investigate the precise blow-up scenario of strong solution to system (1.6). Firstly, we present the following convolution estimates, which is the key technical issue for the blow-up analysis.

Lemma 3.1

Let \(-1\le a\le 3, -1\le \beta \le 1\). Then

$$\begin{aligned} (G\pm \beta \partial _x G)*\left( \frac{a}{2}u^2+\frac{3-a}{2}u_x^2-\gamma u\right) \ge \delta _a\left( u-\frac{\gamma }{a}\right) ^2-\frac{\gamma ^2}{2a}, \end{aligned}$$
(3.1)

where \(\delta _a=\frac{\sqrt{3-a}}{4}(\sqrt{3(1+a)}-\sqrt{3-a})\).

Proof

We denote by \(+\) and − the characteristic function of \(R^+\) and \(R^-\) respectively. Then, we obtain

$$\begin{aligned} G_+*\left( \frac{a}{2}u^2+\frac{3-a}{2}u_x^2-\gamma u\right)&=G_+*\left( \frac{a}{2}(u^2-\frac{2\gamma }{a}u)+\frac{3-a}{2}u_x^2\right) \\&=G_+*\left( \frac{a}{2}(u-\frac{\gamma }{a})^2+\frac{3-a}{2}u_x^2\right) -\frac{\gamma ^2}{4a}\\&=\frac{3-a}{2}G_+*\left( \frac{a}{3-a}(u-\frac{\gamma }{a})^2+u_x^2\right) -\frac{\gamma ^2}{4a}. \end{aligned}$$

Let \(r\in \mathbb {R}\), using Cauchy inequality, we have

$$\begin{aligned} G_+*\left( r^2\left( u-\frac{\gamma }{a}\right) ^2+u_x^2\right)&=\frac{e^{-x}}{2}\int ^x_{-\infty }e^{\xi } \left( r^2\left( u-\frac{\gamma }{a}\right) ^2+u_x^2\right) d\xi \\&\ge r e^{-x}\int ^x_{-\infty }\left( u-\frac{\gamma }{a}\right) u_x e^{\xi }d\xi \\&=\frac{r}{2}\left( u-\frac{\gamma }{a}\right) ^2 -rG_+*\left( u-\frac{\gamma }{a}\right) ^2. \end{aligned}$$

This leads to

$$\begin{aligned} G_+*\left( (r^2+r)\left( u-\frac{\gamma }{a}\right) ^2+u_x^2\right) \ge \frac{r}{2}\left( u-\frac{\gamma }{a}\right) ^2. \end{aligned}$$

Similarly, we get

$$\begin{aligned} G_-*\left( (r^2+r)\left( u-\frac{\gamma }{a}\right) ^2+u_x^2\right) \ge \frac{r}{2}\left( u-\frac{\gamma }{a}\right) ^2. \end{aligned}$$

Choose r such that \(r^2+r=\frac{a}{3-a}\). This is indeed possible if \(-1\le a<3\)( if \(a=3\), the proposition is trivial and there is nothing to prove). So,

$$\begin{aligned} G_+*\left( \frac{a}{2}u^2+\frac{3-a}{2}u_x^2-\gamma u\right)&\ge \frac{\delta _a}{2}\left( u-\frac{\gamma }{a}\right) ^2-\frac{\gamma ^2}{4a}, \end{aligned}$$
(3.2)
$$\begin{aligned} G_-*\left( \frac{a}{2}u^2+\frac{3-a}{2}u_x^2-\gamma u\right)&\ge \frac{\delta _a}{2}\left( u-\frac{\gamma }{a}\right) ^2-\frac{\gamma ^2}{4a}, \end{aligned}$$
(3.3)

where \(\delta _a=\frac{\sqrt{3-a}}{4}(\sqrt{3(1+a)}-\sqrt{3-a})\).

If \(-1\le \beta \le 1\), then from (3.2) and (3.3), we deduce

$$\begin{aligned} (G\pm \beta \partial _x G)*\left( \frac{a}{2}u^2+\frac{3-a}{2}u_x^2-\gamma u\right) \ge \delta _a\left( u-\frac{\gamma }{a}\right) ^2-\frac{\gamma ^2}{2a}. \end{aligned}$$
(3.4)

This completes the proof of Lemma 3.1. \(\square \)

The blow-up of solution will rely on the following basic property:

Lemma 3.2

([5]) Let \(0<T^{*}\le \infty \) and \(f,g\in C^1([0, T^{*}], \mathbb {R})\) be such that, for some constant \(c>0\) and all \(t\in [0, T^{*}]\),

$$\begin{aligned}&\frac{df}{dt}(t)\ge cf(t)g(t),\\&\frac{dg}{dt}(t)\ge cf(t)g(t). \end{aligned}$$

If \(f(0)>0\) and \(g(0)>0\), then

$$\begin{aligned} T^{*}\le \frac{1}{c\sqrt{f(0)g(0)}}. \end{aligned}$$

Next, we give the wave-breaking criterion for \(\sigma \ne 0\).

Theorem 3.1

[30] (Wave-breaking criterion for \(\sigma \ne 0\)). Let \(({u_0},{\rho _{0 - 1}})\in {H^s}(\mathbb {R}) \times {H^{s - 1}}(\mathbb {R})\) with \(s\ge 2\), and \(T>0\) be the maximal existence time of the solution \(({u}, {\rho })\) to system (1.6) with the initial data \(({u_0}, {\rho _0})\). Then the solution blows up in finite time if and only if

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

We are now in a position to give the following local-in-space criterion for finite time blow-up mechanism to system (1.6).

Theorem 3.2

Let \(({u_0},{\rho _{0 - 1}})\in {H^s}(\mathbb {R}) \times {H^{s - 1}}(\mathbb {R})\) with \(s\ge 2\), and \(T>0\) be the maximal existence time of the solution \(({u}, {\rho })\) to system (1.6) with the initial data \(({u_0}, {\rho _0})\). Assume that there exists \(x_0\in \mathbb {R}\), such that

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

and

$$\begin{aligned} u_{0,x}(x_0)<-|u_0(x_0)-\frac{A}{2}|, \end{aligned}$$
(3.5)

then the corresponding solution \((u,\rho )\) of system (1.6) arising from \((u_0, \rho _0)\) blows up in finite time. More precisely, the following upper bound estimate for \(T^*\) holds:

$$\begin{aligned} T^*\le \frac{2}{\sqrt{(u_{0,x}(x_0))^2-(u_0(x_0)-\frac{A}{2})^2}}. \end{aligned}$$
(3.6)

Proof

First, differentiating the equation

$$\begin{aligned} u_t+uu_{x}=-\partial _x G*\left[ u^2+\frac{1}{2}u_x^2-Au+\frac{1}{2}\rho ^2\right] , \end{aligned}$$
(3.7)

with respect to x variable and applying the identity \(G*f-\partial ^2_x G*f=f\), we have

$$\begin{aligned}&u_{tx}+uu_{xx}+u_x^2\nonumber \\&\quad =\frac{1}{2}\rho ^2+u^2\frac{1}{2}u_x^2-Au-G* \left( u^2+\frac{1}{2}u_x^2-Au+\frac{1}{2}\rho ^2\right) . \end{aligned}$$
(3.8)

When \(\sigma =1\) and \(\gamma =0\) the two characteristics \(q_1(t,x)\) defined in (2.10) and \(q_2(t,x)\) defined in (2.11) coincide, so we can carry out the analysis along the trajectory.

Denote

$$\begin{aligned} \omega (t)=u(t, q(t;x_0)), n(t)=u_x(t, q(t;x_0)), \xi (t)=\rho (t,q(t;x_0)). \end{aligned}$$

Then, we use (3.7), (3.8) and the second equation of (3.1) to obtain the following time derivatives along the flow \(q(t,x_0)\),

$$\begin{aligned} \frac{d\omega (t)}{dt}=&-\partial _xG*\left( u^2+\frac{1}{2}u_x^2-Au+\frac{1}{2}\rho ^2\right) , \end{aligned}$$
(3.9)
$$\begin{aligned} \frac{d n(t)}{dt}=&-\frac{1}{2}n(t)^2+\omega (t)^2+\frac{1}{2}\xi (t)-A\omega (t) -G*\left( u^2+\frac{1}{2}u_x^2-Au+\frac{1}{2}\rho ^2\right) , \end{aligned}$$
(3.10)
$$\begin{aligned} \frac{d\xi (t)}{dt}=&-n(t)\xi (t). \end{aligned}$$
(3.11)

From the last equation above and the initial conditions on \(\rho _0\), we get

$$\begin{aligned} \xi (t)=\xi (0)e^{-\int ^t_0 n(\tau )d\tau }=\rho _0(x_0)e^{-\int ^t_0 n(\tau )d\tau }=0. \end{aligned}$$
(3.12)

Let us introduce the two \(C^1\) functions of the time variable depending on \(\beta \),

$$\begin{aligned} A(t)=\left( \beta \left( \omega (t)-\frac{A}{2}\right) -n(t)\right) (t,q_1(t,x_0)) \end{aligned}$$
(3.13)

and

$$\begin{aligned} B(t)=\left( -\beta \left( \omega (t)-\frac{A}{2}\right) -n(t)\right) (t,q_1(t,x_0)). \end{aligned}$$
(3.14)

Differentiating with respect to t and using (3.9), (3.10), we obtain

$$\begin{aligned} A_t(t) =&\,\beta \omega (t)_t-n(t)_t\nonumber \\ =&\,\beta \left[ -\partial _xG*\left( u^2+\frac{1}{2}u_x^2-Au+\frac{1}{2}\rho ^2\right) \right] \nonumber \\&-\left[ -\frac{1}{2}n(t)^2+\omega (t)^2-A\omega (t) -G*\left( u^2+\frac{1}{2}u_x^2-Au+\frac{1}{2}\rho ^2\right) (t,q_1(t;x_0))\right] \nonumber \\ =&\,\frac{1}{2}n(t)^2-\omega (t)^2+A\omega (t) +\left( G-\beta \partial _xG\right) *\left( u^2+\frac{1}{2}u_x^2-Au+\frac{1}{2}\rho ^2\right) (t,q_1(t;x_0)) \end{aligned}$$
(3.15)

and

$$\begin{aligned} B_t(t) =&-\beta \omega (t)_t-n(t)_t\nonumber \\ =&-\beta \left[ -\partial _xG*\left( u^2+\frac{1}{2}u_x^2-Au+\frac{1}{2}\rho ^2\right] \right. \nonumber \\&-\left[ -\frac{1}{2}n(t)^2+\omega (t)^2-A\omega (t) -G*\left( u^2+\frac{1}{2}u_x^2-Au+\frac{1}{2}\rho ^2\right) (t,q_1(t;x_0))\right] \nonumber \\ =&\frac{1}{2}n(t)^2-\omega (t)^2+A\omega (t) +(G+\beta \partial _xG)*\left( u^2+\frac{1}{2}u_x^2-Au+\frac{1}{2}\rho ^2\right) (t,q_1(t;x_0)). \end{aligned}$$
(3.16)

Choose \(\beta =1\), it follows from Lemma 3.1 and the fact \(G_{\pm }*\frac{1}{2}\rho ^2\ge 0\) that

$$\begin{aligned} A_t(t)&\ge \frac{1}{2}\left( n(t)^2-\left( \omega (t)-\frac{A}{2}\right) ^2\right) +(G-\partial _xG)*\frac{1}{2}\rho ^2\nonumber \\&\ge \frac{1}{2}\left[ n(t)^2-\left( \omega (t)-\frac{A}{2}\right) ^2\right] \nonumber \\&=\frac{1}{2}(AB)(t,x_0), \end{aligned}$$
(3.17)

and

$$\begin{aligned} B_t(t)&\ge \frac{1}{2}\left( n(t)^2-\left( \omega (t)-\frac{A}{2}\right) ^2\right) +(G+\partial _xG)*\frac{1}{2}\rho ^2\nonumber \\&\ge \frac{1}{2}\left[ n(t)^2-\left( \omega (t)-\frac{A}{2}\right) ^2\right] \nonumber \\&=\frac{1}{2}(AB)(t,x_0). \end{aligned}$$
(3.18)

By our assumption on the initial conditions made in Theorem 3.2,

$$\begin{aligned} u_{0,x}(x_0)<-|u_0(x_0)-\frac{A}{2}|, \end{aligned}$$

or, equivalently, as

$$\begin{aligned} A(0,x_0)>0 \quad and \quad B(0,x_0)>0. \end{aligned}$$

It follows from Lemma 3.2 that the corresponding solution \((u,\rho )\) of system (1.6) arising from \((u_0, \rho _0)\) blows up in finite time and the following upper bound estimate for \(T^*\) holds:

$$\begin{aligned} T^*\le \frac{2}{\sqrt{(u_{0,x}(x_0))^2-(u_0(x_0)-\frac{A}{2})^2}}. \end{aligned}$$

\(\square \)