1 Introduction

In recent decades, the celebrated Camassa-Holm(CH)-type equations raised a lot of interest because of their specific properties, one of which being that they possess peakon solutions (peaked soliton solutions with discontinuous derivatives at the peaks). The most well-known member of them is the Camassa-Holm equation [3]

$$\begin{aligned} u_t-u_{txx}+3uu_x=2u_xu_{xx}+uu_{xxx} \end{aligned}$$
(1.1)

which describes the unidirectional propagation of waves at the free surface of a shallow water under the influence of gravity [3, 4], where u(x, t) stands for the fluid velocity at time t in the spatial x-direction, \(x\in \mathbb R\). This equation spontaneously exhibits emergence of singular solutions from smooth initial conditions. It has a bi-Hamilton structure [22] and is completely integrable [3]. Moreover, the Camassa-Holm equation has traveling wave solutions of the form \(ce^{-|x-ct|}\), called peakons, which describes an essential feature of the traveling waves of largest amplitude (see [9, 11, 12]). It is shown in [7, 10, 12]that the inverse spectral or scattering approach is a powerful tool to study the Camassa-Holm equation and analyze its dynamics. It is worthwhile to mention that Eq.(1.1) gives rise to geodesic flows of a certain invariant metric on the Bott-Virasoro group [29], and this geometric illustration leads to a proof that the Least Action Principle holds. It is shown in [10] that blow-up occurs in the form of breaking waves, namely the solution remains bounded but its slope becomes unbounded in finite time.

Another well-known integrable equation admitting peakons with quadratic nonlinearities is the Degasperis-Procesi (DP) equation [17], which takes the form

$$\begin{aligned} m_t+um_x+3u_xm=0, \quad m=u-u_{xx}. \end{aligned}$$

It is regarded as a model for nonlinear shallow water dynamics, which can also be obtained from the governing equations for water waves [14]. Wave breaking phenomena and global existence of solutions for the DP equation were investigated in [8, 18, 19, 25, 26].

Both the CH and DP equations are the third-order CH-type equations with quadratic nonlinearity. Recently, by the symmetry classification study of nonlocal partial differential equations with quadratic or cubic nonlinearity, Novikov discovered a new CH-type equation with cubic nonlinearity [30]

$$\begin{aligned} u_t-u_{txx}+4u^2u_x=3uu_xu_{xx}+u^2u_{xxx}. \end{aligned}$$
(1.2)

Different from the CH and DP equations with quadratic nonlinearity, the Novikov equation has cubic nonlinearity. It is locally well-posed in certain Sobolev spaces and Besov spaces [31, 32].

In recent year, two-component or multi-component peakon systems have been attracting significant attention [20, 21]. It was known that the CH equation allows some multi-component generalizations, the well-known two-component CH system takes the form [6, 13]

$$\begin{aligned} \left\{ \begin{array}{lll} m_t+um_x+2mu_x=-\rho \rho _x, \\ \rho _t+u\rho _x+u_x\rho =0,\\ m=u-u_{xx}, \end{array} \right. \end{aligned}$$
(1.3)

which describes the evolution of both the horizontal velocity of the fluid and the horizontal deviation of the surface from the equilibrium [13], and is a completely integrable system as it obeys the Lax-pair and bi-Hamiltonian structure [6]. Later Geng and Xue [23] introduced the following two-component Novikov system

$$\begin{aligned} \left\{ \begin{array}{ll} \begin{aligned} &{}m_t+3u_xvm+uvm_x=0, \\ &{}n_t+3v_xun+uvn_x=0,\\ &{}m=u-u_{xx},n=v-v_{xx}, \end{aligned} \end{array} \right. \end{aligned}$$
(1.4)

which was associated with a \(3\times 3\) matrix spectral problem, they also gave the N peakons, infinite sequence of conserved quantities and a Hamiltonian structure. The two-component Novikov equation may be reduced to the Novikov equation under the constraint \(u=v\). In [24], it is shown that the Novikov system (1.4) is well-posed in Sobolev spaces \(H^s\) for \(s>\frac{3}{2}\), in the sense of Hadamard. Also, peaked traveling wave solutions are used to prove that the solution map is not uniformly continuous in \(H^s\) for \(s<\frac{3}{2}\). The integrability [23], dynamics and structure of the peaked solitons of (1.4) [27, 28] were investigated recently.

In a recent paper[33], Xia et al. proposed the following generalised version of the two-component peakon system

$$\begin{aligned} \left\{ \begin{array}{ll} \begin{aligned} &{}m_t=(mF)_x+mF-\frac{1}{2}m(u-u_x)(v+v_x), \\ &{}n_t=(nF)_x-nF+\frac{1}{2}n(u-u_x)(v+v_x),\\ &{}m=u-u_{xx},n=v-v_{xx}, \end{aligned} \end{array} \right. \end{aligned}$$
(1.5)

where F is an arbitrary function of u, v, and their derivatives. System (1.5) is a multi-component extension of CH equation (1.1), since it can be reduced to CH equation (1.1) as \(v=2\) and \(F=u\). Because an arbitrary function F is involved in (1.5), we do not expect all those equations to have bi-Hamiltonian structures in general. Nevertheless, for some special choices of F we may find the corresponding bi-Hamiltonian structures. Such a system is interesting, because we may obtain quite a large number of integrable peakon equations by choosing different F. In this paper, by setting \(F=\frac{1}{3}(u-u_x)(v+v_x)\) in (1.5), we obtain the following generalized version of the two-component peakon system:

$$\begin{aligned} \left\{ \begin{array}{ll} \begin{aligned} &{}m_t=\frac{1}{3}\left[ m(u-u_x)(v+v_x)\right] _x-\frac{1}{6}m(u-u_x)(v+v_x), \\ &{}n_t=\frac{1}{3}\left[ n(u-u_x)(v+v_x)\right] _x+\frac{1}{6}n(u-u_x)(v+v_x),\\ &{}m=u-u_{xx},n=v-v_{xx}. \end{aligned} \end{array} \right. \end{aligned}$$
(1.6)

The system (1.6) has cubic nonlinearity. The integrability of this system (1.6), its bi-Hamiltonian structure, and infinitely many conservation laws were already presented in [33, 38]. Let us now set up the Cauchy problem for the above system as follows:

$$\begin{aligned} \left\{ \begin{array}{ll} \begin{aligned} &{}m_t=\frac{1}{3}\left[ m(u-u_x)(v+v_x)\right] _x-\frac{1}{6}m(u-u_x)(v+v_x), \\ &{}n_t=\frac{1}{3}\left[ n(u-u_x)(v+v_x)\right] _x+\frac{1}{6}n(u-u_x)(v+v_x),\\ &{}m=u-u_{xx},n=v-v_{xx},\\ &{} m(0,x)=m_0(x), n(0,x)=n_0(x). \end{aligned} \end{array} \right. \end{aligned}$$
(1.7)

The goal of the present paper is to study some analytical properties of this system, such as the breakdown mechanism of the corresponding solutions and what conditions can ensure the occurrence of the blow-up. Note that the system (1.7) involves a nonlocal structure and can be reformulated in a weak form of nonlinear nonlocal transport type. The main idea used in our analysis for blow-ups is to trace the dynamics of the blow-up quantity along the characteristics. Such an idea has been successfully used in the field, cf. [34,35,36,37].

One of the key ingredients in our analysis is to establish a new conservation law via an unusual technique, which turns out to be very important in studying the blow-up phenomena of the system (1.7). Standard transport theory asserts that singularities are often caused by focusing of the characteristics or the unboundedness of the characteristic speed. In this paper, we precisely confirm the former for system (1.7). On the other hand, we will consider the transport equation in terms of \(M=-(u_xn+v_xm-un+vm)\), which is the slope of the term \(-(uv-u_xv+uv_x-u_xv_x)\), to derive a new blow-up result with respect to the initial data. The approach we adopt here to investigate the breakdown mechanism of the system (1.7) starts by tracing the dynamics of the blow-up quantity along the characteristics.

A crucial step in our argument for the blow-up criterion and blow-up condition involves the control of \(\Vert u\Vert _{L^\infty }\) and \(\Vert v\Vert _{L^\infty }\). In most of the literatures on peakon type equations (like the CH equation), such a control can be inferred from the \(H^1\)-norm conservation. Unfortunately, the \(H^1\)-norm of the solution (u, v) to our system (1.7) is not conserved. More seriously, it is not clear whether the \(H^1\)-norm is bounded. Thus we can not guarantee the uniform boundedness of the solution. Instead, we appeal to the new conservation law (see Lemma 3.2)

$$\begin{aligned} \mathcal {H}_1=\frac{1}{6}\int _\mathbb {R} (u-u_x)ndx=\frac{1}{6}\int _\mathbb {R} (v+v_x)mdx \end{aligned}$$

and the special structure of system to prove bounds on \(\Vert m\Vert _{L^1}\) and \(\Vert n\Vert _{L^1}\) (see Lemma 3.3). These bounds, together with the sign-preservation of m and n allow us to obtain the necessary estimates on \(\Vert u\Vert _{L^\infty }\) and \(\Vert v\Vert _{L^\infty }\).

Another novelty in our method is that we do not prove that the quantity \(M=-(u_xn+v_xm-un+vm)\) blows up in the finite time. As a matter of fact, the complicated algebraic structure of the evolution equation for M prevents one from easily getting the blow-up condition. To get around this difficulty, we will focus instead on another quantity \(\sqrt{mn}\), which is much easier to analyze. More precisely, it turns out that the dynamics of m and n can be put in the rather clean forms, which provide the estimates for n/m and m/n. With the help of these estimates and the interaction between M and \(\sqrt{mn}\), we can show that \(\sqrt{mn}\) will blow up in the finite time, then either m or n blows up in finite time. Thus, with appropriate choice of initial data, the solution of the system (1.7) must cease to exist in finite time. Our main results on the blow-up can be formulated in the following theorem.

Theorem 1.1

Let \((m_0, n_0)\in (H^s(\mathbb {R} )\cap L^1(\mathbb {R} ))\times (H^s(\mathbb {R} )\cap L^1(\mathbb {R} ))\) with \( s>\frac{1}{2}\), and (m, n) be the corresponding solution to system (1.7). Assume that \(T>0\) is the maximum time of existence. Suppose that \(m_0(x), n_0(x)\ge 0\) for all \(x\in \mathbb {R}\), and \(m_0(x_0), n_0(x_0)> 0, \) for some \(x_0\in \mathbb {R}\). Define

$$\begin{aligned} h(t)=&\frac{1}{\sqrt{m_0(x_0)n_0(x_0)}} \\&\Bigg [ \frac{1}{12\mathcal {H}_1}m_0(x_0)\Vert m_0\Vert _{L^1}\Vert n_0\Vert ^2_{L^1}\left( \frac{e^{2\mathcal {H}_1t}-1}{2\mathcal {H}_1}-t\right) +1+\frac{1}{3}(u_0(x_0)-u_{0,x}(x_0))n_0(x_0)t\\&-\frac{1}{3}(v_0(x_0)+v_{0,x}(x_0))m_0(x_0)t+n_0(x_0)\Vert m_0\Vert _{L^1}\\&\times \left( \int ^t_0e^{\frac{1}{6\mathcal {H}_1} \Vert m_0\Vert _{L^1}\Vert n_0\Vert _{L^1}(e^{\mathcal {H}_1s}-1)}ds-t\right) \Bigg ], \end{aligned}$$

for \(0\le t\le T\), where \(\mathcal {H}_1\) in (3.7) is a conservation law.

If the initial data satisfy that

$$\begin{aligned} (u_0(x_0)-u_{0,x}(x_0))n_0(x_0)-(v_0(x_0)+v_{0,x}(x_0))m_0(x_0)<0 \quad \text { and } \quad \inf _{t\in [0, T)} h(t)<0, \end{aligned}$$
(1.8)

then the solution (m, n) to the system (1.7) blows up in finite time.

Remark 1.1

We want to point out that the second condition in (1.8) can indeed be met by some careful manipulation of the initial data. Indeed, for the function h(t) defined above, we can easily find that \(h(0)>0\) and if there exists a \(K>0\) such that \(\Vert m_0\Vert _{L^1}\thicksim O\left( \frac{1}{K^2}\right) , m_0(x_0)\thicksim O(K), \Vert n_0\Vert _{L^1}\thicksim O(1), n_0(x_0)\thicksim O\left( \frac{1}{K}\right) , v_0(x_0)+v_{0,x}(x_0)\thicksim O(1)\) as \( x\rightarrow x_0\), then

$$\begin{aligned} h(t)\thicksim&\frac{1}{12\mathcal {H}_1}O\left( \frac{1}{K}\right) \left( \frac{e^{2\mathcal {H}_1t}-1}{2\mathcal {H}_1}-t\right) +1+\frac{1}{3}O\left( \frac{1}{K^3}\right) t-\frac{1}{3}O(K)t\nonumber \\&+O\left( \frac{1}{K^3}\right) \left( \int ^{t}_0e^{\frac{1}{6\mathcal {H}_1} O\left( \frac{1}{K^2}\right) \left( e^{\mathcal {H}_1s}-1\right) }ds-t\right) , \end{aligned}$$
(1.9)

where \(t>0\). It follows from (1.9) that h(t) can be less than zero for sufficiently large K.

Moreover, we prove that the solution maintains the corresponding properties at infinity within its lifespan provided the initial data decay exponentially.

The outline of the paper is as follows. In Section 2, we prove the local well-posedness of the Cauchy problem (1.7). The precise blow-up scenario and the conditions on initial data which could lead to finite time blow-up are established in Section 3. Finally in Section 4, the persistence properties are established in weighted spaces. Lastly in the Appendix we provide some technical details regarding the blow-up criteria.

Notation. In the sequel, we denote by \(*\) the convolution. \(\mathcal {S}\) stands for the Schwartz space of smooth functions over \(\mathbb {R} ^{d}\) whose derivatives of all order decay at infinity. The set \(\mathcal {S}'\) of temperate distributions is the dual set of \(\mathcal {S}\) for the usual pairing. \(1\le p<\infty \), the norms in the Lebesgue space \(L^p(\mathbb {R} )\) is \(\Vert f\Vert _{L^p}=\Big (\int _{\mathbb {R} }|f(x)|^pdx\Big )^{\frac{1}{p}}\), the space \(L^{\infty }(\mathbb {R} )\) consists of all essentially bounded, Lebesgue measurable functions f equipped with the norm \(\displaystyle \Vert f\Vert _{L^\infty }=\inf _{\mu (e)=0}\sup _{x\in \mathbb {R} \setminus e}|f(x)|\). Throughout this paper, we denote every position constant by same symbol C without confusion. For a function f in the classical Sobolev spaces \(H^s(\mathbb {R} )\;(s\ge 0)\) the norm is denoted by \( \Vert f\Vert _{H^s} \). Denote \(\Lambda (x) = {1\over 2} e^{-|x|}\) to be the fundamental solution of \((1 - \partial ^2_x)^{-1}\) on \(\mathbb {R} \), that is, \((1 - \partial ^2_x)^{-1} f = \Lambda *f\). Furthermore, we define two convolution operators \(\Lambda _+\) and \(\Lambda _-\) as

$$\begin{aligned} \begin{aligned}&\Lambda _+ *f (x) = {e^{-x} \over 2} \int ^x_{-\infty } e^y f(y) dy,\\&\Lambda _- *f(x) = {e^{x}\over 2} \int ^\infty _{x} e^{-y} f(y) dy. \end{aligned} \end{aligned}$$
(1.10)

It is easy to find

$$\begin{aligned} \Lambda = \Lambda _+ + \Lambda _-, \qquad \Lambda _x = \Lambda _- - \Lambda _+. \end{aligned}$$
(1.11)

2 Local Well-posedness

In this section, we shall establish local well-posedness of the initial-value problem (1.7) in the Besov spaces. First, for the convenience of the readers, we recall some facts on the Littlewood-Paley decomposition and some useful lemmas.

Proposition 2.1

[15] (Littlewood-Paley decomposition) Let \(\mathcal{B}:= \{\xi \in \mathbb {R}^d,\ |\xi |\le \frac{4}{3}\}\) and \(\mathcal{C}:= \{\xi \in \mathbb {R}^d,\ \frac{3}{4}\le |\xi |\le \frac{8}{3}\}.\) There exist two radial functions \(\chi \in C_c^\infty (\mathcal{B})\) and \( \varphi \in C_c^\infty (\mathcal{C})\) such that

$$\begin{aligned}&\chi (\xi )+ \sum _{q \ge 0}\varphi (2^{-q}\xi )=1,\quad \forall \ \ \ \xi \in \mathbb {R}^d ,\\&|q-q^{\prime }|\ge 2 \Rightarrow \mathrm{Supp} \varphi (2^{-q}\cdot )\cap \mathrm{Supp} \varphi (2^{-q^{\prime }}\cdot ) = \varnothing ,\\&q\ge 1 \Rightarrow \mathrm{Supp} {\chi }(\cdot )\cap \mathrm{Supp} \varphi (2^{-q}\cdot ) = \varnothing , \quad \text {and }\\&\frac{1}{3}\le \chi (\xi )^2+\sum _{q \ge 0}\varphi (2^{-q}\xi )^2 \le 1, \quad \forall \ \ \ \xi \in \mathbb {R}^d . \end{aligned}$$

Furthermore, let \(h := \mathcal{F}^{-1}\varphi \) and \(\tilde{h} := \mathcal{F}^{-1}\chi .\) Then the dyadic operators \(\Delta _q\) and \(S_q\) can be defined as follows

$$\begin{aligned}\begin{aligned}&\Delta _qf := \varphi (2^{-q}D)f=2^{qd}\int _{\mathbb {R} ^d}h(2^qy)f(x-y)dy \quad \text{ for }\quad q\ge 0, \nonumber \\&S_qf := \chi (2^{-q}D)f=\sum _{-1\le k\le q-1}\Delta _kf=2^{qd}\int _{\mathbb {R}^d}\tilde{h}(2^qy)f(x-y)dy, \nonumber \\&\Delta _{-1}f := S_{0}f \, \, \text{ and } \, \, \Delta _{q}f := 0 \quad \,\text{ for } \quad \, q \le -2. \end{aligned} \end{aligned}$$

We shall also use the notation \(S_q u:= \sum _{k\le q-1}\Delta _k u.\) The formal equality \(u=\sum _{q\ge -1}\Delta _q u\) holds in \(\mathcal {S}'(\mathbb {R}^d)\) and is called the Littlewood-Paley decomposition.

Let \(s\in \mathbb {R}, 1\le q,r\le \infty .\) We use \(B^s_{q,r}\) to denote the inhomogenous Besov space, which is defined by

$$\begin{aligned} B^s_{q,r}:= \{f\in {\mathcal S}^{\prime }(\mathbb {R}^{d}); \quad \Vert f\Vert _{B^s_{q,r}}<\infty \}, \end{aligned}$$
(2.1)

where

$$\begin{aligned} \Vert f\Vert _{B^s_{q,r}}:= \left\{ \begin{array}{l} \displaystyle \bigg (\sum _{k \in \mathbb {Z}} 2^{k s r}\Vert \Delta _k f\Vert _{L^q}^r\bigg )^{\frac{1}{r}},\quad \hbox {for}\quad r<\infty ,\\ \displaystyle \sup _{k \in \mathbb {Z}}2^{k s}\Vert \Delta _k f\Vert _{L^q}, \quad \quad \ \quad \hbox { for} \quad r=\infty . \end{array}\right. \end{aligned}$$

In the case of \(s=\infty ,\) \(B^{\infty }_{q, r} := \bigcap _{s \in \mathbb {R}} B^{s}_{q, r} .\)

Now we state some useful results in the transport equation theory, which are crucial to the proofs of our main theorems later.

Lemma 2.1

[15, 16] Suppose that \((q, r)\in [1, +\infty ]^2\) and \(s >-\frac{d}{q}.\) Let v be a vector field such that \(\nabla v\) belongs to \(L^1([0, T]; B^{s-1}_{q, r})\) if \(s >1+\frac{d}{q}\) or to \(L^1([0, T]; B^{\frac{d}{q}}_{q, r}\cap L^{\infty })\) otherwise. Suppose also that \(f_{0}\in B^{s}_{q, r}, \, F \in L^1([0, T]; B^{s}_{q, r})\) and that \(f \in L^{\infty }([0, T]; B^{s}_{q, r})\cap C([0, T]; \mathcal {S}^{\prime })\) solves the d-dimensional linear transport equations

$$\begin{aligned}(T)~~~~~~~~ {\left\{ \begin{array}{ll} \partial _{t}f + v \cdot \nabla f= F,\\ f|_{t=0}=f_0. \end{array}\right. } \end{aligned}$$

Then there exists a constant C depending only on \(s, \, q\) and d such that the following statements hold:

1) If \(r=1\) or \(s \ne 1+\frac{d}{q},\) then

$$\begin{aligned} \Vert f\Vert _{B^{s}_{q, r}}\le \Vert f_0\Vert _{B^{s}_{q, r}}+\int _{0}^{t} \Vert F(\tau )\Vert _{B^{s}_{q, r}}d\tau +C\int _{0}^{t} V'(\tau )\Vert f(\tau )\Vert _{B^{s}_{q, r}}d\tau , \end{aligned}$$

or

$$\begin{aligned} \Vert f\Vert _{B^{s}_{q, r}}\le e^{CV(t)}\left( \Vert f_0\Vert _{B^{s}_{q, r}}+\int _{0}^{t} e^{-CV(\tau )}\Vert F(\tau )\Vert _{B^{s}_{q, r}}d\tau \right) \end{aligned}$$
(2.2)

hold, where \(V(t)= \int _{0}^{t} \Vert \nabla v(\tau )\Vert _{B^{\frac{d}{q}}_{q, r} \cap L^{\infty }}d\tau \) if \(s < 1+ \frac{d}{q}\) and \(V(t)= \int _{0}^{t} \Vert \nabla v(\tau )\Vert _{B^{s-1}_{q, r} }d\tau \) else.

2) If \(s \le 1+\frac{d}{q}\) and, in addition, \(\nabla f_0 \in L^{\infty },\) \(\nabla f \in L^{\infty }([0, T] \times \mathbb {R}^{d})\) and \(\nabla F \in L^{1}([0, T]; L^{\infty }),\) then

$$\begin{aligned} \begin{aligned} \Vert f(t)\Vert _{B^{s}_{q, r}}&+\Vert \nabla f(t)\Vert _{L^{\infty }}\\ \le&e^{CV(t)}\left( \Vert f_0\Vert _{B^{s}_{q, r}}+\Vert \nabla f_0\Vert _{L^{\infty }} +\int _{0}^{t} e^{-CV(\tau )}(\Vert F(\tau )\Vert _{B^{s}_{q, r}}+\Vert \nabla F(\tau )\Vert _{L^{\infty }})d\tau \right) \end{aligned} \end{aligned}$$

with \(V(t)= \int _{0}^{t} \Vert \nabla v(\tau )\Vert _{B^{\frac{d}{q}}_{q, r} \cap L^{\infty }}d\tau .\)

3) If \(f=v,\) then for all \(s >0,\) the estimate (2.2) holds with \(V(t)=\int _{0}^{t}\Vert \partial _x v(\tau )\Vert _{L^{\infty }}d\tau .\)

4) If \(r <+\infty ,\) then \(f \in C([0, T];B^{s}_{q, r} ).\) If \(r =+\infty ,\) then \(f \in C([0, T]; B^{s^{\prime }}_{q, 1} )\) for all \(s^{\prime } <s.\)

Lemma 2.2

[16] Let \((q, q_1, r)\in [1, +\infty ]^3.\) Assume that \(s >-d\min \{\frac{1}{q_1}, \frac{1}{q^{\prime }}\}\) with \(q^{\prime } := (1-\frac{1}{q})^{-1}.\) Let \(f_0\in B^{s}_{q, r}\) and \(F \in L^1([0, T]; B^{s}_{q, r}).\) Let v be a time dependent vector field such that \(v \in L^{\rho }([0, T]; B^{-M}_{\infty , \infty })\) for some \(\rho >1,\) \(M>0\) and \(\nabla v \in L^{1}([0, T]; B^{\frac{d}{q_1}}_{q_1, \infty }\cap L^{\infty } )\) if \(s < 1+\frac{d}{q_1},\) and \(\nabla v \in L^{1}([0, T]; B^{s-1}_{q_1, r} )\) if \(s > 1+\frac{d}{q_1}\) or \(s = 1+\frac{d}{q_1}\) and \(r=1.\) Then the transport equations (T) has a unique solution \(f \in L^{\infty }([0, T]; B^s_{q, r}) \cap (\cap _{s^{\prime }<s}C([0, T]; B^{s^{\prime }}_{q, 1}))\) and the inequalities in Lemma 2.1 hold true. If, moreover, \(r < \infty ,\) then we have \(f \in C([0, T];B^s_{q, r}).\)

Lemma 2.3

[1, 5] (1-D Moser-type estimates) Assume that \(1 \le q, \, r \le +\infty ,\) the following estimates hold:

(i) for \(s>0,\)

$$\begin{aligned} \Vert fg\Vert _{B^{s}_{q, r}(\mathbb {R})}\le C (\Vert f\Vert _{B^{s}_{q, r}(\mathbb {R})}\Vert g\Vert _{L^{\infty }(\mathbb {R})}+ \Vert g\Vert _{B^{s}_{q, r}(\mathbb {R})}\Vert f\Vert _{L^{\infty }(\mathbb {R})}); \end{aligned}$$

(ii) for \(s_1 \le \frac{1}{q},\, s_2>\frac{1}{q}\) (\( s_2\ge \frac{1}{q}\) if \(r=1\)) and \(s_1+s_2>0,\)

$$\begin{aligned} \Vert fg\Vert _{B^{s_1}_{q, r}(\mathbb {R})}\le C \Vert f\Vert _{B^{s_1}_{q, r}(\mathbb {R})}\Vert g\Vert _{B^{s_2}_{q, r}(\mathbb {R})}, \end{aligned}$$

(iii) In Sobolev space \(H^s=B^s_{2,2}\), we have for \(s>0\),

$$\begin{aligned} \Vert f\partial _xg\Vert _{H^{s}} \le C \bigl (\,\Vert f\Vert _{H^{s+1}}\Vert g\Vert _{L^{\infty }}+\Vert f\Vert _{L^{\infty }}\Vert \partial _x g\Vert _{H^{s}}\,\bigr ) \end{aligned}$$

where C is a constant independent of f and g.

Our main result of the local well-posedness in Besov space is the following.

Theorem 2.1

Suppose that \(1 \le q, \, r \le +\infty \), \(s>\max \left\{ 2+\frac{1}{q}, \frac{5}{2} \right\} \) and \((u_0, v_0)\in B^{s}_{q, r}\times B^{s}_{q, r}\). Then there exists a time \(T> 0\) such that the initial-value problem (1.7) has a unique solution \((u,v) \in E^s_{q, r}(T)\times E^s_{q, r}(T) ,\) and the map \((u_0, v_0)\mapsto (u,v)\) is continuous from a neighborhood of \((u_0, v_0)\) in \(B^s_{q, r}\times B^s_{q, r}\) into

$$\begin{aligned} C([0, T]; B^{s^{\prime }}_{q, r}) \cap C^1([0, T]; B^{s^{\prime }-1}_{q, r})\times C([0, T]; B^{s^{\prime }}_{q, r}) \cap C^1([0, T]; B^{s^{\prime }-1}_{q, r}) \end{aligned}$$

for every \(s^{\prime } < s\) when \(r =+\infty \) and \(s^{\prime } =s\) whereas \(r <+\infty .\)

In the following, we denote \(C>0\) a generic constant only depending on p, q, r, s. The following proposition gives the uniqueness and continuity with respect to the initial data. The proof follows a standard Gronwall-type argument, and hence we omit it.

Proposition 2.2

Let \(1 \le q, \, r \le +\infty \) and \(s > \max \left\{ 2+\frac{1}{q}, 3-\frac{1}{q}, \, \frac{5}{2} \right\} .\) Let \((u^{(i)}; \,v^{(i)})\in L^\infty ([0,T]; B^s_{p,r})\cap C([0,T]; \mathcal {S}')\times L^\infty ([0,T]; B^s_{p,r})\cap C([0,T]; \mathcal {S}')(i=1,2)\) be two solutions of the initial-value problem (1.7) with the initial data \((u^{(i)}_{0}; \, v^{(i)}_{0}) \in B^{s}_{q, r}\times B^{s}_{q, r})(i=1,2)\). Then for every \(t \in [0, T]:\)

$$\begin{aligned} \begin{aligned}&\Vert (u^{(2)}-u^{(1)})(t)\Vert _{B^{s-1}_{q, r}}+\Vert (v^{(2)}-v^{(1)})(t)\Vert _{B^{s-1}_{q, r}}\\&\quad \le \left( \Vert u^{(2)}_0-u^{(1)}_0\Vert _{B^{s-1}_{q, r}}+\Vert v^{(2)}_0-v^{(1)}_0\Vert _{B^{s-1}_{q, r}}\right) \\&\qquad \times \exp \left\{ C\int _{0}^{t}\left( \Vert u^{(1)}(\tau )\Vert ^{2}_{B^{s}_{q, r}}+\Vert u^{(2)}(\tau )\Vert ^{2}_{B^{s}_{q, r}}+\Vert v^{(1)}(\tau )\Vert ^{2}_{B^{s}_{q, r}}+\Vert v^{(2)}(\tau )\Vert ^{2}_{B^{s}_{q, r}}\right) \;d\tau \right\} . \end{aligned}\nonumber \\ \end{aligned}$$
(2.3)

Motivated by the method in [15], the following lemma can be also derived, and we omit details of its proof.

Lemma 2.4

Suppose that \(1 \le q, \, r \le +\infty \), \(s>\max \left\{ 2+\frac{1}{q}, \frac{5}{2} \right\} \) and \((u_0, v_0)\in B^{s}_{q, r}\times B^{s}_{q, r}\). Assume that \(u^{(0)} := 0\) and \(v^{(0)} := 0\). Then there exists a sequence of smooth functions

$$\begin{aligned} (u^{(l)}, v^{(l)})_{l \in \mathbb {N}} \in C(\mathbb {R}^{+}; B^{\infty }_{q, r})\times C(\mathbb {R}^{+}; B^{\infty }_{q, r}) \end{aligned}$$

solving the following linear transport equation by induction:

$$\begin{aligned} (T_l)~~~~~~ {\left\{ \begin{array}{ll} \left\{ \partial _t -\frac{1}{3}\left[ (u^{(l)}-u^{(l)}_x)(v^{(l)}+v^{(l)}_x)\right] \partial _x\right\} m^{(l+1)}\\ \qquad \qquad =\frac{1}{3}\left[ (u^{(l)}-u^{(l)}_x)(v^{(l)}+v^{(l)}_x)\right] _xm^{(l)} -\frac{1}{6}(u^{(l)}-u^{(l)}_x)(v^{(l)}+v^{(l)}_x)m^{(l)},\,\, &{} t> 0,\;x\in \mathbb {R},\\ \left\{ \partial _t -\frac{1}{3}\left[ (u^{(l)}-u^{(l)}_x)(v^{(l)}+v^{(l)}_x)\right] \partial _x\right\} n^{(l+1)}\\ \qquad \qquad =\frac{1}{3}\left[ (u^{(l)}-u^{(l)}_x)(v^{(l)}+v^{(l)}_x)\right] _xn^{(l)} +\frac{1}{6}(u^{(l)}-u^{(l)}_x)(v^{(l)}+v^{(l)}_x)n^{(l)},\,\, &{} t> 0,\;x\in \mathbb {R},\\ u^{(l+1)}|_{t=0} = u_{0}^{(l+1)}(x)=S_{l+1}u_0,&{} x\in \mathbb {R}\\ v^{(l+1)}|_{t=0} = v_{0}^{(l+1)}(x)=S_{l+1}v_0,&{} x\in \mathbb {R} \end{array}\right. } \end{aligned}$$
(2.4)

where \(S_{l+1}\) is the low frequency cut-off operator. Moreover, there exists a \(T>0\) such that the solutions satisfying the following properties:

  1. (i)

    \(\;\;(u^{(l)}, v^{(l)})_{l \in \mathbb {N}}\) is uniformly bounded in \(E^{s}_{q, r}(T)\times E^{s}_{q, r}(T)\).

  2. (ii)

    \(\;\;(u^{(l)}, v^{(l)})_{l \in \mathbb {N}}\) is a Cauchy sequence in \(C([0, T]; B^{s-1}_{q, r})\times C([0, T]; B^{s-1}_{q, r})\).

Proof of Theorem 2.1

Due to Lemma 2.4, we obtain that \((u^{(l)}, v^{(l)})_{l \in \mathbb {N}}\) is a Cauchy sequence in \(C([0, T]; B^{s-1}_{q, r})\times C([0, T]; B^{s-1}_{q, r}),\) so it converges to some function \((u,v) \in C([0, T]; B^{s-1}_{q, r})\times C([0, T]; B^{s-1}_{q, r}).\) We now have to verify that (u, v) belongs to \(E^s_{q,r}(T)\times E^s_{q,r}(T)\) and solves the Cauchy problem (1.7). Since \((u^{(l)}, v^{(l)})_{l \in \mathbb {N}}\) is uniformly bounded in \(L^{\infty }([0, T];B^s_{ q,r})\times L^{\infty }([0, T];B^s_{ q,r})\) according to Lemma 2.4, the Fatou property for the Besov spaces implies that (u, v) also belongs to \(L^{\infty }([0, T];B^s_{ q,r})\times L^{\infty }([0, T];B^s_{ q,r}) .\)

On the other hand, as \((u^{(l)}, v^{(l)})_{l \in \mathbb {N}}\) converges to (u, v) in \(C([0, T]; B^{s-1}_{q, r})\times C([0, T]; B^{s-1}_{q, r}),\) an interpolation argument guarantees that the convergence holds in \(C([0, T]; B^{s^{\prime }}_{q, r})\times C([0, T]; B^{s^{\prime }}_{q, r}) ,\) for any \(s^{\prime } <s\). It is then easy to pass to the limit in the equation \((T_l)\) and to obtain that (u, v) is indeed a solution to the Cauchy problem (1.7). Due to the fact that (u, v) belongs to \(L^{\infty }([0, T];B^s_{ q,r})\times L^{\infty }([0, T];B^s_{ q,r}) ,\) the right-hand side of the equations

$$\begin{aligned} m_t-\frac{1}{3}[(u-u_x)(v+v_x)]m_x =\frac{1}{3}(u_xn+v_xm-un+vm)m-\frac{1}{6}m(u-u_x)(v+v_x) \end{aligned}$$

belongs to \(L^{\infty }([0, T];B^{s-2}_{ q,r}),\) and the right-hand side of the equation

$$\begin{aligned} n_t-\frac{1}{3}[(u-u_x)(v+v_x)]n_x =\frac{1}{3}(u_xn+v_xm-un+vm)n+\frac{1}{6}n(u-u_x)(v+v_x) \end{aligned}$$

belongs to \(L^{\infty }([0, T];B^{s-2}_{ q,r})\). In particular, for the case \(r < \infty ,\) Lemma 2.2 leads to the fact that \((u,v) \in C([0, T]; B^{s^{\prime }}_{q, r})\times C([0, T]; B^{s^{\prime }}_{q, r}) \) for any \(s^{\prime } < s.\) Applying the equation again, we see that \((\partial _t u, \partial _t v )\, \in C([0, T]; B^{s-1}_{q, r})\times C([0, T]; B^{s-1}_{q, r})\) if \(r < \infty ,\) and in \(L^{\infty }([0, T];B^{s-1}_{q,r})\times L^{\infty }([0, T];B^{s-1}_{q,r})\) otherwise. Moreover, a standard use of a sequence of viscosity approximate solutions \((u_{\epsilon }, v_{\epsilon })_{\epsilon >0}\) for the Cauchy problem (1.7) which converges uniformly in

$$C([0, T]; B^{s}_{q, r}) \cap C^{1}([0, T]; B^{s-1}_{q, r}) \times C([0, T]; B^{s}_{q, r}) \cap C^{1}([0, T]; B^{s-1}_{q, r})$$

implies the continuity of the solution (u, v) in \(E^s_{ q,r}(T)\times E^s_{ q,r}(T) .\) The proof of Theorem 2.1 is complete.

Note that for every \(s\in \mathbb R, B^{s}_{2,2}=H^s\) in Theorem (2.1) holds true in Sobolev spaces \(H^s\) with \(s>\frac{5}{2}\), we have the following result.

Theorem 2.2

Let \((m_0,n_0)\in H^s(\mathbb R)\times H^s(\mathbb R)\) with \(s>\frac{1}{2}\). Then there exists a time \(T> 0\) such that the initial-value problem (1.7) has a unique solution \((m,n) \in C([0,T]; H^s(\mathbb R)\times H^s(\mathbb R))\cap C^1([0,T];H^{s-1}(\mathbb R)\times H^{s-1}(\mathbb R)),\) and the map \((m_0, n_0)\mapsto (m,n)\) is continuous from a neighborhood of \((m_0, n_0)\) in \(H^s(\mathbb R)\times H^s(\mathbb R)\) into \(C([0,T]; H^s(\mathbb R)\times H^s(\mathbb R))\cap C^1([0,T]; H^{s-1}(\mathbb R)\times H^{s-1}(\mathbb R))\).

3 Blow-up Phenomena

3.1 Blow-up Criterion

We are now in a position to state a blow-up criterion for the system (1.7), the proof is deferred in Appendix.

Theorem 3.1

Let \((m_0, n_0)\in H^s\times H^s\) with \( s>\frac{1}{2}\), and (m, n) be the corresponding solution to system (1.7). Assume that \(T>0\) is the maximum time of existence. If \(T<\infty \), then

$$\begin{aligned} \int ^{T}_{0}(\Vert m(t)\Vert _{L^{\infty }}^2+\Vert n(t)\Vert _{L^{\infty }}^2)dt=+\infty . \end{aligned}$$
(3.1)

Let us now turn our attention to the precise blow-up scenario for sufficiently regular solutions to the system (1.7), which is required for discussion in the remaining parts of this section. To do so, let us first consider the following initial value problem:

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{dq(t,x)}{dt} = -\frac{1}{3}(u-u_x)(v+v_x)(t,q(t,x)), \;\; x \in \mathbb {R},\quad t\in [0,T).\\ q(0,x)=x,\;\; x \in \mathbb {R}. \end{array}\right. } \end{aligned}$$
(3.2)

for the flow q generated by \(-\frac{1}{3}(u-u_x)(v+v_x)\).

Lemma 3.1

Suppose \((m_0, n_0) \in H^s(\mathbb {R})\times H^s(\mathbb {R})\) with \(s>\frac{1}{2}\), and let \(T>0\) be the maximal existence time of the strong solution (m, n) to the system (1.7). Then (3.2) has a unique \(C^1\) solution \(q : [0,T)\times \mathbb {R} \rightarrow \mathbb {R}\) such that \(q(t,\cdot )\) is an increasing diffeomorphism over \(\mathbb {R} \) with

$$\begin{aligned} q_x(t,x)=\exp \left( -\frac{1}{3}\int _0^t (u_xn+v_xm-un+vm)(\tau ,q(\tau ,x)) \,d\tau \right) >0, \end{aligned}$$
(3.3)

for all \((t,x)\in [0,T)\times \mathbb {R}\). Furthermore,

$$\begin{aligned}&m(t,q(t,x))q_x(t,x)= m_{0}(x)\exp \left( -\frac{1}{6}\int ^t_0(u-u_x)(v+v_x)(\tau ,q(\tau ,x))\right) d\tau , \end{aligned}$$
(3.4)
$$\begin{aligned}&n(t,q(t,x))q_x(t,x)= n_{0}(x)\exp \left( \frac{1}{6}\int ^t_0(u-u_x)(v+v_x)(\tau ,q(\tau ,x))\right) d\tau , \end{aligned}$$
(3.5)

for all \((t,x)\in [0,T)\times \mathbb {R}\).

Proof

Since \((u,v) \in C \left( [0, T), H^{s}({\mathbb R})\times H^{s}({\mathbb R})\right) \cap C^1 \left( [0, T), H^{s-1}({\mathbb R})\times H^{s-1}({\mathbb R})\right) \), it follows from the fact \(H^{s-1}\hookrightarrow Lip({\mathbb R})\) with \(s>\frac{5}{2}\) that \(u(t,x),v(t,x),u_x(t,x)\) and \(v_x(t,x)\) are bounded, Lipschitz in the space variable x, and of class \(C^1\) in time t. Then the classical ordinary differential equations theory ensures that the initial value problem (3.2) has a unique solution \(q(t, x) \in C^1 \left( [0, T) \times {\mathbb R}\right) .\)

Differentiating (3.2) with respect to x yields

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{d q_x(t,x)}{dt} = -\frac{1}{3}(u_xn+v_xm-un+vm)(t,q(t,x))q_x(t,x), \;\; x \in \mathbb {R},\quad t\in [0,T).\\ q_x(0,x)=1,\;\; x \in \mathbb {R}. \end{array}\right. } \end{aligned}$$
(3.6)

Solving the above ODE yields (3.3).

For every \(T^{'}<T\), it follows from the Sobolev embedding theorem that

$$\begin{aligned} \sup _{(s,x)\in [0,T^{'}]\times {\mathbb R}}\frac{1}{3}|(u_xn+v_xm-un+vm)(s,x)|<\infty . \end{aligned}$$

Therefore, we infer from (3.3) that there exists a constant \(C>0\) such that \(q_x(t,x)\ge e^{-Ct}\), for all \((t,x)\in [0,T)\times {\mathbb R}\).

Making use of (3.6) and (1.7), we have

$$\begin{aligned}&\frac{d}{dt}\big [m(t,q(t,x))q_x(t,x)\big ]\\&\quad =(m_t(t,q)+m_x(t,q)q_t(t,x))q_x(t,x)+m(t,q)q_{xt}(t,x)\\&\quad =(m_t(t,q)-\frac{1}{3}((u-u_x)(v+v_x))(t,q)m_x(t,q))q_x(t,x)\\&\qquad -\frac{1}{3}(u_xn+v_xm-un+vm)(t,q)m(t,q)q_x(t,x)\\&\quad =-\frac{1}{6}(uv-u_xv+uv_x-u_xv_x)(t,q)m(t,q)q_x(t,x). \end{aligned}$$

Using the same argument, we also have

$$\begin{aligned} \frac{d}{dt}\big [n(t,q(t,x))q_x(t,x)\big ] =\frac{1}{6}(uv-u_xv+uv_x-u_xv_x)(t,q)n(t,q)q_x(t,x). \end{aligned}$$

Therefore, solving the above two equations gives the desired identities (3.4) and (3.5).

Now, we will derive a new conservation law \(\mathcal {H}_1\) by sufficiently exploiting the structure of system (1.7).

Lemma 3.2

Let \((m_0, n_0)\in H^s\times H^s\) with \( s>\frac{1}{2}\), and (m, n) be the corresponding solution to system (1.7). Assume that \(T>0\) is the maximum time of existence. Then for all \(t\in [0, T)\),

$$\begin{aligned} \mathcal {H}_1=\frac{1}{6}\int _\mathbb {R} (u-u_x)ndx=\frac{1}{6}\int _\mathbb {R} (v+v_x)mdx \end{aligned}$$
(3.7)

is a conservation law.

Proof

First of all, a simple computation shows that

$$\begin{aligned} \int _\mathbb {R} ((u-u_x)n-(v+v_x)m)dx=&\int _\mathbb {R} ((u-u_x)(v+v_x))_xdx=0, \end{aligned}$$
(3.8)
$$\begin{aligned} \int _\mathbb {R} (u-u_x)_tndx=&\int _\mathbb {R} (u_t-u_{xt})(v-v_{xx})dx\nonumber \\ =&\int _\mathbb {R} (v+v_{x})m_tdx \end{aligned}$$
(3.9)

and

$$\begin{aligned} \int _\mathbb {R} (v+v_x)_tmdx=\int _\mathbb {R} (u-u_{x})n_tdx. \end{aligned}$$
(3.10)

Next, using the system (1.7) and the above identities (3.8)-(3.10), we deduce that

$$\begin{aligned} \frac{d(6\mathcal {H}_1)}{dt} =&\frac{d}{dt}\int _\mathbb {R} (u-u_x)ndx\\ =&\frac{1}{2}\frac{d}{dt}\int _\mathbb {R} \big ((u-u_x)n+(v+v_x)m\big )dx\\ =&\int _\mathbb {R} \big ((u-u_x)n_t+(v+v_{x})m_t\big )dx\\ =&\frac{1}{3}\int _\mathbb {R} (u-u_x)(v+v_{x})\big [(u-u_x)_xn+(v+v_x)_xm\big ]dx\\&+\frac{1}{6}\int _\mathbb {R} (u-u_x)(v+v_{x})\big [(v+v_x)m-(u-u_x)n\big ]dx\\ =&\frac{1}{2}\int _\mathbb {R} (u-u_x)(v+v_{x})\big [(u-u_x)(v+v_x)\big ]_xdx=0. \end{aligned}$$

This completes the proof of Lemma 3.2.

With the new conservation law \(\mathcal {H}_1\) in hand, we can control the the quantities \(\Vert u\Vert _{L^\infty }\) and \(\Vert v\Vert _{L^\infty }\), which is very important in studying the blow-up phenomena of the system (1.7).

Lemma 3.3

Let \((m_0, n_0)\in (H^s(\mathbb {R} )\cap L^1(\mathbb {R} ))\times (H^s(\mathbb {R} )\cap L^1(\mathbb {R} ))\) with \(s>\frac{1}{2}\). Suppose that \(m_0(x)\ge 0, n_0(x)\ge 0\) for all \(x\in \mathbb {R}\) and \(T>0\) is the maximum time of existence, then for all \(t\in [0,T]\), we have

$$\begin{aligned}&0\le u(t,x)+|u_x(t,x)|\le \int _\mathbb {R} m(t,x)dx\le \Vert m_0\Vert _{L^1}, \end{aligned}$$
(3.11)
$$\begin{aligned}&0\le v(t,x)+|v_x(t,x)|\le \int _\mathbb {R} n(t,x)dx\le e^{\mathcal {H}_1t}\Vert n_0\Vert _{L^1}, \end{aligned}$$
(3.12)

where \(\mathcal {H}_1\) is given in (3.7).

Proof

From the assumption \(m_0(x) \ge 0, n_0(x)\ge 0\) and (3.4)-(3.5), we know

$$\begin{aligned} m(t,x)\ge 0, n(t,x)\ge 0, \forall x\in \mathbb {R} . \end{aligned}$$
(3.13)

Note that

$$\begin{aligned} u(t,x)=\Lambda *m(t,x)=\frac{1}{2}\int _\mathbb {R} e^{-|x-y|}m(y)dy, \end{aligned}$$

then

$$\begin{aligned}&u(t,x)=\frac{e^{-x}}{2}\int _{-\infty }^{x}e^{y}m(y)dy +\frac{e^{x}}{2}\int _x^{\infty }e^{-y}m(y)dy, \end{aligned}$$

and

$$\begin{aligned}&u_x(t,x)=-\frac{e^{-x}}{2}\int _{-\infty }^{x}e^{y}m(y)dy +\frac{e^{x}}{2}\int _x^{\infty }e^{-y}m(y)dy, \end{aligned}$$

which together with (3.13) yields

$$\begin{aligned} 0\le u(t,x)+u_x(t,x)=\int _x^{\infty }e^{x-y}m(t,y)dy\le \int _{-\infty }^x m(t,x)dx \end{aligned}$$
(3.14)

and

$$\begin{aligned} 0\le u(t,x)-u_x(t,x)=\int ^x_{-\infty }e^{y-x}m(t,y)dy\le \int _x^{\infty } m(t,x)dx. \end{aligned}$$
(3.15)

It follows from the system (1.7) that

$$\begin{aligned} m_t=\frac{1}{3}[m(u-u_x)(v+v_x)]_x-\frac{1}{6}m(u-u_x)(v+v_x). \end{aligned}$$
(3.16)

Integrating (3.16) over \({\mathbb R}\) with respect to x and applying (3.13)-(3.14), we can obtain

$$\begin{aligned} \int _{\mathbb R}m_tdx&=\frac{1}{3}\int _{\mathbb R}\big [m(u-u_x)(v+v_x)\big ]_xdx-\frac{1}{6}\int _{\mathbb R}m(u-u_x)(v+v_x)dx\\&=-\frac{1}{6}\int _{\mathbb R}m(u-u_x)(v+v_x)dx \le 0. \end{aligned}$$

Then, we get

$$\begin{aligned} \int _{\mathbb R}mdx\le \int _{\mathbb R}m_0dx=\int _{\mathbb R}|m_0|dx, \end{aligned}$$

which along with (3.14) and (3.15) imply (3.11).

Similarly, in view of

$$\begin{aligned} v(t,x)=\Lambda *n(t,x)=\frac{1}{2}\int _\mathbb {R} e^{-|x-y|}n(y)dy, \end{aligned}$$

we obtain

$$\begin{aligned}&0\le v(t,x)+v_x(t,x)=\int _x^{\infty }e^{x-y}n(t,y)dy\le \int _{-\infty }^x n(t,x)dx, \end{aligned}$$
(3.17)
$$\begin{aligned}&0\le v(t,x)-v_x(t,x)=\int ^x_{-\infty }e^{y-x}n(t,y)dy\le \int _x^{\infty } n(t,x)dx \end{aligned}$$
(3.18)

Moreover, we deduce from the system (1.7) that

$$\begin{aligned} n_t=\frac{1}{3}\big [n(u-u_x)(v+v_x)\big ]_x+\frac{1}{6}n(u-u_x)(v+v_x). \end{aligned}$$
(3.19)

Integrating (3.19) over \({{\mathbb R}}\) with respect to x yields and using (3.12), (3.2), we can obtain

$$\begin{aligned} \int _{\mathbb R}n_tdx =\frac{1}{6}\int _{\mathbb R}n(u-u_x)(v+v_x)dx. \end{aligned}$$

In view of (3.17) and (3.7), we deduce that

$$\begin{aligned} \int _{\mathbb R}n_tdx&\le \frac{1}{6}\Vert v+v_x\Vert _{L^\infty }\int _{\mathbb R}n(u-u_x)dx\nonumber \\&\le \mathcal {H}_1\big [(u_0,v_0)\big ]\int _{\mathbb R}ndx. \end{aligned}$$
(3.20)

Taking account of (3.17)-(3.18) and (3.20), one gets

$$\begin{aligned} 0\le v(t,x)+|v_x(t,x)|\le \int _\mathbb {R} n(t,x)dx\le e^{\mathcal {H}_1[(u_0,v_0)]t}\int _\mathbb {R} n_0dx=e^{\mathcal {H}_1t}\int _\mathbb {R} |n_0|dx. \end{aligned}$$

This completes the proof of Lemma 3.3.

Remark 3.1

The proof of Lemma 3.3 implies the following further estimates:

$$\begin{aligned} u(t,x)\le \frac{1}{2}\Vert m_0\Vert _{L^1}, v(t,x)\le \frac{1}{2} e^{\mathcal {H}_1t}\Vert n_0\Vert _{L^1}. \end{aligned}$$

The following theorem shows the precise blow-up scenario for sufficiently regular solutions to the system (1.7)

Theorem 3.2

Let \((m_0, n_0)\in H^s\times H^s\) with \( s>\frac{1}{2}\) and \(T>0\) is the maximum time of existence. If \(m_0(x)\ge 0, n_0(x)\ge 0\) for all \(x\in {\mathbb R}\), then the corresponding solution (m, n) blows up in finite time \(T<\infty \) if and only if

$$\begin{aligned} \lim _{t\rightarrow T}\sup _{x\in \mathbb {R} }\big ((u_x-u)n+(v+v_x)m\big )=+\infty . \end{aligned}$$
(3.21)

Proof

Let \(T>0\) be the maximal time of existence of the corresponding solution (m, n) to the system (1.7). If (3.21) holds, it follows from the Sobolev embedding theorem that the corresponding solution (m, n) blows up in finite time. Conversely, assume that \(T<\infty \) and (3.21) is not valid. Then there is some positive number \(M_1>0\), such that

$$\begin{aligned} (u_x-u)n+(v+v_x)m\le M_1,\qquad \forall (t,x)\in [0,T)\times \mathbb {R} . \end{aligned}$$

By Lemma 3.3, in view of (3.4) and (3.5), we obtain that

$$\begin{aligned} \Vert m(t,\cdot )\Vert _{L^\infty }&=\Vert m(t,q(t,\cdot ))\Vert _{L^\infty }\\&=\left\| q_x^{-1}(t,\cdot )m_{0}(x)\exp \left( -\frac{1}{6}\int ^t_0(u-u_x)(v+v_x)(\tau ,q(\tau ,x))\right) d\tau \right\| _{L^\infty }\\&\le M_1e^{\frac{1}{3}M_1t}e^{\frac{1}{6\mathcal {H}_1}\Vert m_0\Vert _{L^1}\Vert n_0\Vert _{L^1}\left( e^{\mathcal {H}_1t}-1\right) }\Vert m_0\Vert _{H^s} \end{aligned}$$

and

$$\begin{aligned} \Vert n(t,\cdot )\Vert _{L^\infty }&=\Vert n(t,q(t,\cdot ))\Vert _{L^\infty }\\&=\left\| q_x^{-1}(t,\cdot )n_{0}(x)\exp \left( \frac{1}{6}\int ^t_0(u-u_x)(v+v_x)(\tau ,q(\tau ,x))\right) d\tau \right\| _{L^\infty }\\&\le M_1e^{\frac{1}{3}M_1t}e^{\frac{1}{6\mathcal {H}_1}\Vert m_0\Vert _{L^1}\Vert n_0\Vert _{L^1}\left( e^{\mathcal {H}_1t}-1\right) }\Vert n_0\Vert _{H^s}. \end{aligned}$$

Thus, we have

$$\begin{aligned}&\int ^T_0(\Vert m\Vert ^2_{L^\infty }+\Vert n\Vert ^2_{L^\infty })d\tau \\&\quad \le \int ^T_0 \left[ M_1^2e^{\frac{2}{3}M_1\tau }e^{\frac{1}{3\mathcal {H}_1}\Vert m_0\Vert _{L^1}\Vert n_0\Vert _{L^1}\left( e^{\mathcal {H}_1\tau }-1\right) }\Vert m_0\Vert ^2_{H^s}\right. \\&\qquad \left. +M_1^2e^{\frac{2}{3}M_1\tau }e^{\frac{1}{3\mathcal {H}_1}\Vert m_0\Vert _{L^1}\Vert n_0\Vert _{L^1}\left( e^{\mathcal {H}_1\tau }-1\right) }\Vert n_0\Vert ^2_{H^s}\right] d\tau \\&\le M_1^2Te^{\frac{2}{3}M_1T}e^{\frac{1}{3\mathcal {H}_1}\Vert m_0\Vert _{L^1}\Vert n_0\Vert _{L^1}\left( e^{\mathcal {H}_1T}-1\right) }(\Vert m_0\Vert ^2_{H^s}+\Vert n_0\Vert ^2_{H^s}) <\infty . \end{aligned}$$

which contradicts to Theorem 3.1.

On the other hand, we can see that if \(\lim _{t\rightarrow T}\sup _{x\in \mathbb {R} }\big ((u_x-u)n+(v+v_x)m\big )=+\infty \), then either m or n blows up in finite time. Now, the proof of the theorem is completed.

In the blow up analysis of the CH-type system, a fruitful attempt is to investigate the dynamics of certain blow-up quantities along the characteristics.

3.2 Dynamics of Blow-up Quantities

The following lemma gives evolution of the quantities \(M(t,x)= -(u_xn+v_xm-un+vm)\) along the characteristics.

Lemma 3.4

Let \((m_0,n_0)\in H^s({\mathbb R})\times H^s({\mathbb R}) \) with \(s>\frac{1}{2}\), and \(T>0\) be the maximal existence time of the solution (m, n) to system (1.7). Then for all \((t,x)\in [0, T)\times {\mathbb R}\),

$$\begin{aligned} M^{'} =&-\frac{1}{3}M^2+\frac{n}{6}(u+u_x)(u-u_x)(v+v_x)+\frac{m}{2}(u-u_x)(v+v_x)^2 +\frac{n}{3}(u-u_x)^2(v+v_x)\nonumber \\&-\frac{m}{2}(\Lambda +\Lambda _x)*\big [(u-u_x)(v+v_x)n\big ]-\frac{n}{2}(\Lambda -\Lambda _x)*\big [(u-u_x)(v+v_x)m\big ]. \end{aligned}$$
(3.22)

where \('\) denotes the material derivative \(f' = f_t - \frac{1}{3}(u-u_x)(v+v_x) f_x\).

Proof

Using the first equation of system (1.7), we can obtain

$$\begin{aligned} u_t&-\frac{1}{3}(u-u_x)(v+v_x)u_x\nonumber \\ =&(1-\partial _x^2)^{-1}\big [m_t-\frac{1}{3}(1-\partial _x^2)(u-u_x)(v+v_x)u_x)\big ]\nonumber \\ =&(1-\partial _x^2)^{-1}\big [m_t-\frac{1}{3}(u-u_x)(v+v_x)u_x+\frac{1}{3}\partial _x^2((u-u_x)(v+v_x)u_x)\big ]\nonumber \\ =&(1-\partial _x^2)^{-1}\big [-\frac{1}{3}(u-u_x)(v+v_x)u_{xxx}-\frac{1}{3}Mm-\frac{1}{6}(u-u_x)(v+v_x)m\nonumber \\&+\frac{1}{3}\partial _x((u-u_x)(v+v_x)u_{xx})-\frac{1}{3}\partial _x(Mu_x)\big ]\nonumber \\ =&-\frac{1}{6}(1-\partial _x^2)^{-1}(u-u_x)(v+v_x) m-\frac{1}{3}\partial _x(1-\partial _x^2)^{-1}(Mu_x)-\frac{1}{3}(1-\partial _x^2)^{-1}Mu. \end{aligned}$$
(3.23)

Similarly,

$$\begin{aligned} v_t&-\frac{1}{3}(u-u_x)(v+v_x)v_x\nonumber \\&=\frac{1}{6}(1-\partial _x^2)^{-1}(u-u_x)(v+v_x) n-\frac{1}{3}\partial _x(1-\partial _x^2)^{-1}(Mv_x)-\frac{1}{3}(1-\partial _x^2)^{-1}Mv. \end{aligned}$$
(3.24)

By a direct computation, it follows that M(t, x) satisfies

$$\begin{aligned} M_t&-\frac{1}{3}(u-u_x)(v+v_x)M_x\\ =&-(u_{xt}n+v_{xt}m+(v_x+v)m_t-(u-u_x)n_t+v_tm-u_tn)\\&+\frac{1}{3}(u-u_x)(v+v_x)(un-2mn+u_xn_x+vm+v_xm_x-u_xn-un_x+v_xm+vm_x). \end{aligned}$$

It is inferred from (3.23)-(3.24) and (1.7) that

$$\begin{aligned}&u_{xt}n+v_{xt}m=\frac{1}{3}(u-u_x)(v+v_x)(un-2mn+vm) -\frac{1}{3}(\Lambda _{-}+\Lambda _{+})*(nu_xM+mv_xM)\nonumber \\&\quad +\frac{1}{6}(\Lambda _{-}-\Lambda _{+})*\big [m((u-u_x)(v+v_x)n-2vM)-n((u-u_x)(v+v_x) m+2uM)\big ], \end{aligned}$$
(3.25)
$$\begin{aligned}&(v_x+v)m_t+(u_x-u)n_t\nonumber \\&\quad =\frac{1}{3}M^2 +\frac{1}{3}(u-u_x)(v+v_x)(m_x v+m_xv_x-un_x+u_xn_x-\frac{1}{2}vm-\frac{1}{2}v_xm-\frac{1}{2}un-\frac{1}{2}u_xn) \end{aligned}$$
(3.26)

and

$$\begin{aligned} v_tm-u_tn=\frac{1}{3}&(u-u_x)(v+v_x)(v_xm-u_xn)+\frac{1}{3}(\Lambda _{-}-\Lambda _{+})*(nu_xM-mv_xM)\nonumber \\&+\frac{1}{6}(\Lambda _{-}+\Lambda _{+})*[m((u-u_x)(v+v_x)n-2vM)+n((u-u_x)(v+v_x)m+2um)]. \end{aligned}$$
(3.27)

It follows from (3.25)-(3.27) that

$$\begin{aligned}&M_t-\frac{1}{3}(uv-u_xv+uv_x-u_xv_x)M_x\nonumber \\&\quad =-\frac{1}{3}M^2+\frac{1}{6}(uv-u_xv+uv_x-u_xv_x)(un+vm+u_xn+v_xm)\nonumber \\&\qquad -\frac{1}{3}m\Lambda _{-}*\big [(u-u_x)(v+v_x) n-2vM-2v_xM\big ]\nonumber \\&\qquad -\frac{1}{3}n\Lambda _{+}*\big [(u-u_x)(v+v_x) m-2uM-2u_xM\big ]\nonumber \\&\quad =-\frac{1}{3}M^2+\frac{1}{6}(u-u_x)(v+v_x)\big [(u+u_x)n+(v+v_x)m\big ]\nonumber \\&\qquad -\frac{m}{6}(\Lambda +\Lambda _x)*\Big \{(u-u_x)(v+v_x)m-2(u-u_x)[(u-u_x)(v+v_x)]_x\Big \}\nonumber \\&\qquad -\frac{n}{6}(\Lambda -\Lambda _x)*\Big \{(u-u_x)(v+v_x)n+2(v+v_x)[(u-u_x)(v+v_x)]_x\Big \}. \end{aligned}$$
(3.28)

It is easily seen that

$$\begin{aligned}&(\Lambda +\Lambda _x)*\left( (u-u_x)\big [(u-u_x)(v+v_x)\big ]_x\right) \nonumber \\&\quad =(\Lambda +\Lambda _x)*\big [(u-u_x)^2(v+v_x)-(u-u_x)_x(u-u_x)(v+v_x)\big ]-(u-u_x)^2(v+v_x) \end{aligned}$$

and

$$\begin{aligned}&(\Lambda -\Lambda _x)*\left( (v+v_x)\big [(u-u_x)(v+v_x)\big ]_x\right) \nonumber \\&\quad =-(\Lambda -\Lambda _x)*\big [(u-u_x)(v+v_x)^2+(v+v_x)_x(u-u_x)(v+v_x)\big ]+(u-u_x)(v+v_x)^2. \end{aligned}$$

Therefore plugging the above into (3.28) we find

$$\begin{aligned} M^{'} =&-\frac{1}{3}M^2+\frac{1}{6}(uv-u_xv+uv_x-u_xv_x)(un+vm+u_xn+v_xm)\nonumber \\&-\frac{m}{6}(\Lambda +\Lambda _x)*\big [(u-u_x)(v+v_x)n\big ]-\frac{n}{6}(\Lambda -\Lambda _x)*\big [(u-u_x)(v+v_x)m\big ]\nonumber \\&-\frac{m}{3}(\Lambda +\Lambda _x)*\big [(u-u_x)(v+v_x)^2-(v+v_x)_x(v+v_x)(u-u_x)\big ]+\frac{m}{3}(u-u_x)(v+v_x)^2\nonumber \\&-\frac{n}{3}(\Lambda -\Lambda _x)*\big [(u-u_x)^2(v+v_x)+(v+v_x)(v+v_x)(u-u_x)_x\big ]+\frac{n}{3}(u-u_x)^2(v+v_x), \end{aligned}$$
(3.29)

which gives (3.22), and then the proof is completed.

3.3 Blow-up Data

Now we are ready to prove the main theorem.

Proof of Theorem 1.1

As is explained before, we will trace the dynamics along the characteristics emanating from \(x_0\). Denote

$$\begin{aligned}&\widehat{u}(t)\doteq u(t,q(t,x_0)), \widehat{v}(t)\doteq v(t,q(t,x_0)),\\ {}&\widehat{u_x}(t)\doteq u_x(t,q(t,x_0)), \widehat{v_x}(t)\doteq v_x(t,q(t,x_0)),\\ {}&\widehat{m}(t)\doteq m(t,q(t,x_0)), \widehat{n}(t)\doteq n(t,q(t,x_0)),\\ {}&\widehat{M}(t)\doteq M(t,q(t,x_0))=-(\widehat{u_x}(t)\widehat{n}(t)+\widehat{v_x}(t)\widehat{m}(t) -\widehat{u}(t)\widehat{n}(t)+\widehat{v}(t)\widehat{m}(t)). \end{aligned}$$

It follows from (3.22) and (3.11)-(3.12) that

$$\begin{aligned} \widehat{M}^{'} =&-\frac{1}{3}\widehat{M}^2+\frac{\widehat{n}}{6}(\widehat{u} +\widehat{u_x})(\widehat{u}-\widehat{u_x})(\widehat{v}+\widehat{v_x}) +\frac{\widehat{m}}{2}(\widehat{u}-\widehat{u_x})(\widehat{v}+\widehat{v_x})^2 +\frac{\widehat{n}}{3}(\widehat{u}-\widehat{u_x})^2(\widehat{v}+\widehat{v_x})\nonumber \\&-\frac{\widehat{m}}{2}(\Lambda +\Lambda _x)*\big [(\widehat{u}-\widehat{u_x})(\widehat{v}+\widehat{v_x})\widehat{m}\big ] -\frac{\widehat{n}}{2}(\Lambda -\Lambda _x)*\big [(\widehat{u}-\widehat{u_x})(\widehat{v}+\widehat{v_x})\widehat{m}\big ]\nonumber \\&\le -\frac{1}{3}\widehat{M}^2+\frac{\widehat{n}}{6}(\widehat{u}+\widehat{u_x})(\widehat{u} -\widehat{u_x})(\widehat{v}+\widehat{v_x})+\frac{\widehat{m}}{2}(\widehat{u}-\widehat{u_x})(\widehat{v} +\widehat{v_x})^2+\frac{\widehat{n}}{3}(\widehat{u}-\widehat{u_x})^2(\widehat{v}+\widehat{v_x}). \end{aligned}$$
(3.30)

Using the system (1.7), we can get

$$\begin{aligned} \widehat{m}^{'}(t)=-\frac{1}{3}\widehat{M}(t)\widehat{m}(t) -\frac{1}{6}(\widehat{u}-\widehat{u_x})(\widehat{v}+\widehat{v_x})\widehat{m}(t) \end{aligned}$$
(3.31)

and

$$\begin{aligned} \widehat{n}^{'}(t)=-\frac{1}{3}\widehat{M}(t)\widehat{n}(t) +\frac{1}{6}(\widehat{u}-\widehat{u_x})(\widehat{v}+\widehat{v_x})\widehat{n}(t). \end{aligned}$$
(3.32)

Apparently, (3.4) and (3.5) and the assumption imply \(\widehat{m}(t)>0\) and \(\widehat{n}(t)>0\) for all \(t\in [0,T)\). In view of (3.31) and (3.32), we obtain for \(t\in (0, T)\) that

$$\begin{aligned} \left( \frac{\widehat{m}(t)}{\widehat{n}(t)}\right) ^{'}=\frac{\widehat{m}^{'}(t)\widehat{n}(t)-\widehat{m}(t)\widehat{n}^{'}(t)}{\widehat{n}(t)^2} =-\frac{1}{3}(\widehat{u}-\widehat{u_x})(\widehat{v}+\widehat{v_x})\left( \frac{\widehat{m}(t)}{\widehat{n}(t)}\right) < 0, \end{aligned}$$

which implies that

$$\begin{aligned} \frac{\widehat{m}(t)}{\widehat{n}(t)}<\frac{\widehat{m}(0)}{\widehat{n}(0)}. \end{aligned}$$
(3.33)

On the other hand, we can get

$$\begin{aligned} \left( \frac{\widehat{n}(t)}{\widehat{m}(t)}\right) ^{'}=\frac{\widehat{n}^{'}(t)\widehat{m}(t)-\widehat{n}(t)\widehat{m}^{'}(t)}{\widehat{m}(t)^2} =\frac{1}{3}(\widehat{u}-\widehat{u_x})(\widehat{v}+\widehat{v_x})\left( \frac{\widehat{n}(t)}{\widehat{m}(t)}\right) . \end{aligned}$$

It follows from (3.11) and (3.12) that

$$\begin{aligned} \frac{\widehat{n}(t)}{\widehat{m}(t)}&=\frac{\widehat{n}(0)}{\widehat{m}(0)}\exp \left\{ \int ^t_0\frac{1}{3}(\widehat{u}-\widehat{u_x})(\widehat{v} +\widehat{v_x})ds\right\} \nonumber \\&\le \frac{\widehat{n}(0)}{\widehat{m}(0)} \exp \left\{ \frac{1}{3}\int ^t_0\Vert u-u_x\Vert _{L^{\infty }}\Vert v+v_x\Vert _{L^{\infty }}ds\right\} \nonumber \\&\le \frac{\widehat{n}(0)}{\widehat{m}(0)}\exp \left\{ \frac{1}{3}\Vert m_0\Vert _{L^1}\Vert n_0\Vert _{L^1}\int ^t_0e^{\mathcal {H}_1s}ds\right\} \nonumber \\&=\frac{\widehat{n}(0)}{\widehat{m}(0)}\exp \left\{ \frac{1}{3\mathcal {H}_1}\Vert m_0\Vert _{L^1}\Vert n_0\Vert _{L^1}(e^{\mathcal {H}_1t}-1)\right\} . \end{aligned}$$
(3.34)

Therefore, we deduce from (3.30)-(3.34) that

$$\begin{aligned} \left( \frac{\widehat{M}(t)}{\sqrt{\widehat{m}(t)\widehat{n}(t)}}\right) ^{'} =&\frac{\widehat{M}^{'}(t)\sqrt{\widehat{m}(t)\widehat{n}(t)}-\widehat{M}(t)(\sqrt{\widehat{m}(t)\widehat{n}(t)})^{'}}{\widehat{m}(t)\widehat{n}(t)} =\frac{\widehat{M}^{'}(t) +\frac{1}{3}\widehat{M}(t)^2}{\sqrt{\widehat{m}(t)\widehat{n}(t)}}\\ \le&\frac{\frac{\widehat{n}(t)}{6}(\widehat{u}^2-\widehat{u_x}^2)(\widehat{v}+\widehat{v_x}) +\frac{\widehat{n}(t)}{3}(\widehat{u}-\widehat{u_x})^2(\widehat{v}+\widehat{v_x}) +\frac{\widehat{m}(t)}{2}(\widehat{u}-\widehat{u_x})(\widehat{v}+\widehat{v_x})^2}{\sqrt{\widehat{m}(t)\widehat{n}(t)}}\\ \le&\frac{\frac{1}{2}\Vert m_0\Vert _{L^1}\Vert n_0\Vert ^2_{L^1}e^{2\mathcal {H}_1t}\widehat{m}(t)+\frac{1}{2}\Vert m_0\Vert ^2_{L^1}\Vert n_0\Vert _{L^1}e^{\mathcal {H}_1t}n(t)}{\sqrt{\widehat{m}(t)\widehat{n}(t)}}\\ =&\frac{1}{2}\Vert m_0\Vert _{L^1}\Vert n_0\Vert ^2_{L^1}e^{2\mathcal {H}_1t}\frac{\sqrt{\widehat{m}(t)}}{\sqrt{\widehat{n}(t)}} +\frac{1}{2}\Vert m_0\Vert ^2_{L^1}\Vert n_0\Vert _{L^1}e^{\mathcal {H}_1t}\frac{\sqrt{\widehat{n}(t)}}{\sqrt{\widehat{m}(t)}}\\ \le&\frac{1}{2}\Vert m_0\Vert _{L^1}\Vert n_0\Vert ^2_{L^1}e^{2\mathcal {H}_1t}\sqrt{\frac{\widehat{m}(0)}{\widehat{n}(0)}}\\ {}&+\frac{1}{2}\Vert m_0\Vert ^2_{L^1}\Vert n_0\Vert _{L^1}e^{\mathcal {H}_1t}\sqrt{\frac{\widehat{n}(0)}{\widehat{m}(0)}} e^{\frac{1}{6\mathcal {H}_1}\Vert m_0\Vert _{L^1}\Vert n_0\Vert _{L^1}\left( e^{\mathcal {H}_1t}-1\right) }. \end{aligned}$$

Integrating the above inequality from 0 to t yields that

$$\begin{aligned}&\frac{\widehat{M}(t)}{\sqrt{\widehat{m}(t)\widehat{n}(t)}}-\frac{\widehat{M}(0)}{\sqrt{\widehat{m}(0)\widehat{n}(0)}}\nonumber \\ \le&\frac{1}{2}\Vert m_0\Vert _{L^1}\Vert n_0\Vert ^2_{L^1} \sqrt{\frac{\widehat{m}(0)}{\widehat{n}(0)}}\int ^t_0e^{2\mathcal {H}_1s}ds\nonumber \\ {}&+\frac{1}{2}\Vert m_0\Vert ^2_{L^1}\Vert n_0\Vert _{L^1}\sqrt{\frac{\widehat{n}(0)}{\widehat{m}(0)}} \int ^t_0e^{\mathcal {H}_1s}e^{\frac{1}{6\mathcal {H}_1}\Vert m_0\Vert _{L^1}\Vert n(0)\Vert _{L^1}\left( e^{\mathcal {H}_1s}-1\right) }ds\nonumber \\ =&\frac{1}{4\mathcal {H}_1}\Vert m_0\Vert _{L^1}\Vert n_0\Vert ^2_{L^1}\sqrt{\frac{\widehat{m}(0)}{\widehat{n}(0)}}\left( e^{2\mathcal {H}_1t}-1\right) \nonumber \\ {}&+3\Vert m_0\Vert _{L^1}\sqrt{\frac{\widehat{n}(0)}{\widehat{m}(0)}}(e^{\frac{1}{6\mathcal {H}_1}\Vert m_0\Vert _{L^1}\Vert n_0\Vert _{L^1}\left( e^{\mathcal {H}_1t}-1\right) }-1),\end{aligned}$$
(3.35)

which implies

$$\begin{aligned} \widehat{M}(t)\le&\Big [\frac{\widehat{M}(0)}{\sqrt{\widehat{m}(0)\widehat{n}(0)}} +\frac{1}{4\mathcal {H}_1}\Vert m_0\Vert _{L^1}\Vert n_0\Vert ^2_{L^1}\sqrt{\frac{\widehat{m}(0)}{\widehat{n}(0)}}\left( e^{2\mathcal {H}_1t}-1\right) \nonumber \\&+3\Vert m_0\Vert _{L^1}\sqrt{\frac{\widehat{n}(0)}{\widehat{m}(0)}}(e^{\frac{1}{6\mathcal {H}_1} \Vert m_0\Vert _{L^1}\Vert n_0\Vert _{L^1}(e^{\mathcal {H}_1t}-1)}-1)\Big ]\sqrt{\widehat{m}(t)\widehat{n}(t)}. \end{aligned}$$
(3.36)

According to (3.35), we have

$$\begin{aligned} \left( \frac{1}{\sqrt{\widehat{m}(t)\widehat{n}(t)}}\right) ^{'}=&\frac{1}{3}\frac{\widehat{M}(t)}{\sqrt{\widehat{m}(t)\widehat{n}(t)}}\\ \le&\frac{1}{12\mathcal {H}_1}\Vert m_0\Vert _{L^1}\Vert n_0\Vert ^2_{L^1}\sqrt{\frac{\widehat{m}(0)}{\widehat{n}(0)}}\left( e^{2\mathcal {H}_1t}-1\right) +\frac{1}{3}\frac{\widehat{M}(0)}{\sqrt{\widehat{m}(0)\widehat{n}(0)}}\nonumber \\&+\Vert m_0\Vert _{L^1}\sqrt{\frac{\widehat{n}(0)}{\widehat{m}(0)}} \left( e^{\frac{1}{6\mathcal {H}_1}\Vert m_0\Vert _{L^1}\Vert n_0\Vert _{L^1}(e^{\mathcal {H}_1t}-1)}-1\right) , \end{aligned}$$

which implies

$$\begin{aligned} 0<&\frac{1}{\sqrt{\widehat{m}(t)\widehat{n}(t)}}\le \frac{1}{24\mathcal {H}_1^2}\sqrt{\frac{\widehat{m}(0)}{\widehat{n}(0)}}\Vert m_0\Vert _{L^1}\Vert n_0\Vert ^2_{L^1}(e^{2\mathcal {H}_1t}-1) +\frac{1}{\sqrt{\widehat{m}(0)\widehat{n}(0)}}\nonumber \\&+\left( \frac{1}{3}\frac{\widehat{M}(0)}{\sqrt{\widehat{m}(0)\widehat{n}(0)}}-\frac{1}{12\mathcal {H}_1}\sqrt{\frac{\widehat{m}(0)}{\widehat{n}(0)}}\Vert m_0\Vert _{L^1}\Vert n_0\Vert ^2_{L^1} -\sqrt{\frac{\widehat{n}(0)}{\widehat{m}(0)}}\Vert m_0\Vert _{L^1}\right) t\nonumber \\&+\sqrt{\frac{\widehat{n}(0)}{\widehat{m}(0)}}\Vert m_0\Vert _{L^1} \int ^t_0e^{\frac{1}{6\mathcal {H}_1}\Vert m_0\Vert _{L^1}\Vert n_0\Vert _{L^1}(e^{\mathcal {H}_1s}-1)}ds\nonumber \\ =&\frac{1}{\sqrt{\widehat{m}(0)\widehat{n}(0)}}\Big [\frac{1}{12\mathcal {H}_1}\widehat{m}(0)\Vert m_0\Vert _{L^1}\Vert n_0\Vert ^2_{L^1}\left( \frac{e^{2\mathcal {H}_1t}-1}{2\mathcal {H}_1}-t\right) +1+\frac{1}{3}(\widehat{u}(0)-\widehat{u_x}(0))\widehat{n}(0)t\nonumber \\&-\frac{1}{3}(\widehat{v}(0)+\widehat{v_x}(0))\widehat{m}(0)t+\widehat{n}(0)\Vert m_0\Vert _{L^1}\left( \int ^t_0e^{\frac{1}{6\mathcal {H}_1} \Vert m_0\Vert _{L^1}\Vert n_0\Vert _{L^1}(e^{\mathcal {H}_1s}-1)}ds-t\right) \Big ]\doteq h(t) \end{aligned}$$
(3.37)

Notice that \(h(0)=\frac{1}{\sqrt{\widehat{m}(0)\widehat{n}(0)}}>0\) and \(\inf h(t)<0\) by the assumption (1.8) and Remark 1.1, so there exists a \(T^* \in (0, T)\) such that \(h(T^*)=0\), and this indicates that \(h(t)\rightarrow 0^{+}\) as \(t\rightarrow T^*\), which together with (3.37) yields

$$\begin{aligned} \sqrt{\widehat{m}(t)\widehat{n}(t)}\rightarrow +\infty , \quad \text { as }\quad t\rightarrow T^*. \end{aligned}$$
(3.38)

So, according to Theorem 3.1, the solution (m, n) blows up at the time \(T^*\in (0, T)\), which completes the proof of the theorem.

4 Persistence Property

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

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

this way we obtain a persistence results on solution (u, v) to system (1.7) in the weight \(L^p\) space \(L^p_{\phi }:=L^p(R,\phi ^p(x)dx)\). As a consequence and an application we determine the spatial asymptotic behavior of certain solutions to system (1.7). 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 [2]. Firstly, we list some knowledge in time frequency analysis for later use, for the details see [2].

Definition 4.1

An admissible weight function for system (1.7) is a local absolutely continuous function \(\phi :R\rightarrow R\) such that, for some \(A>0\) and almost all \(x\in R, |\phi '(x)| \le A|\phi (x)|\), and that is \(\theta -\)moderate for some sub-multiplicative function v satisfying \(\inf _R \theta >0\) and

$$\begin{aligned} \int _R\frac{\theta (x)}{e^{|x|}}dx<\infty . \end{aligned}$$
(4.1)

We can now state our main result on admissible weights.

Theorem 4.1

Assume that \(u_0\phi , u_{0,x}\phi , v_0\phi , v_{0,x}\phi \in L^p(R), 2\le p\le \infty \) for an admissible weight function \(\phi \) of the system (1.7). Let \((u_0, v_0)\in H^s(\mathbb {R})\times H^{s}(\mathbb {R})\) with \(s>\frac{5}{2}\), and \(T>0\) be the maximal time of the solution (u, v) to system (1.7) with the initial data \((u_0, v_0)\). Then, for all \(t\in [0,T]\), we have the estimate

$$\begin{aligned}&\Vert u(t)\phi \Vert _{L^p}+\Vert u_x(t)\phi \Vert _{L^p}+ \Vert v(t)\phi \Vert _{L^p}+\Vert v_x(t)\phi \Vert _{L^p}\\&\quad \le (\Vert u_0\phi \Vert _{L^p} +\Vert u_{0,x}\phi \Vert _{L^p}+\Vert v_0\phi \Vert _{L^p}+\Vert v_{0,x}\phi \Vert _{L^p})\exp \left\{ CM_2^2t\right\} , \end{aligned}$$

for some constant \(C>0\) depending only on \(\phi \) and \(\theta \), and where

$$\begin{aligned} M_2\equiv \sup _{t\in [0,T]}(\Vert u(t)\Vert _{H^s}+\Vert v(t)\Vert _{H^s})<\infty . \end{aligned}$$

Remark 4.1

If we take standard weights \(\phi =\phi _{a,b,c,d}(x)=e^{a|x|^b}(1+|x|)^c\log (e+|x|)^d\) with the following conditions:

$$\begin{aligned} a\ge 0, c,d\in \mathbb {R},0\le b\le 1,ab<1, \end{aligned}$$

then, the restriction \(ab<1\) guarantees the validity of condition (4.1) for a multiplicative function \(\theta (x)\le 1\). Thus, we have the following special persistence properties.

  1. (1)

    Take \(\phi =\phi _{0,0,c,0}\) with \(c>0\), and choose \(p=\infty \). In this case Theorem 4.1 states that the condition

    $$\begin{aligned} |u_0(x)|+|u_{0,x}(x)|+|v_0(x)|+|v_{0,x}(x)|\le C(1+|x|)^{-c} \end{aligned}$$

    implies the uniform algebraic decay

    $$\begin{aligned} |u(x,t)|+|u_{x}(x,t)|+|v(x,t)|+|v_{x}(x,t)|\le C_1(1+|x|)^{-c} \end{aligned}$$

    in the time interval [0, T).

  2. (2)

    Choose \(\phi =\phi _{a,1,0,0}\) if \(x\ge 0\), and \(\phi (x)=1\) if \(x\le 0\) with \(0\le a<1\). It is easy to see that such weight satisfies the Definition 4.1. Let \(p=\infty \) in Theorem 4.1, we deduce that the system (1.7) preserves the decay \(O(e^{-ax})\) as \(x\rightarrow +\infty \) for any \(t>0\). Similarly, we have persistence of the decay \(O(e^{-ax})\) as \(x\rightarrow -\infty \).

In the following, we want to establish a variant of Theorem 4.1 that can be applied to some v-moderate weights \(\phi \) for which condition (4.1) does not hold.

Corollary 4.1

Let \(2\,\le \,p\,\le \,\infty \) and \(\phi \) be a \(\theta -\)moderate weight function as in Definition 4.1 satisfying \(\theta e^{-|\cdot |}\in L^p(\mathbb {R})\). Assume that \(u_0\phi , u_{0,x}\phi , v_0\phi , v_{0,x}\phi \in L^p(\mathbb {R})\) and \(u_0\phi ^{\frac{1}{2}},\)\(u_{0,x}\phi ^{\frac{1}{2}}, v_0\phi ^{\frac{1}{2}}, v_{0,x}\phi ^{\frac{1}{2}}\in L^2(\mathbb {R})\). Let also \((u,v)\in C([0, T), H^s(\mathbb {R}))\times C([0, T), H^{s}(\mathbb {R})), s>\frac{5}{2}\) be the strong solution of the Cauchy problem for (1.7) emanating from \((u_0, v_0)\). Then

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

and

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

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

$$\begin{aligned} \theta (x+y)\le \theta (x)\theta (y),\forall x,y\in \mathbb {R}^n. \end{aligned}$$
(4.2)

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

$$\begin{aligned} \exists C_0>0: \phi (x+y)\le C_0\theta (x)\phi (y),\forall x,y\in \mathbb {R}^n. \end{aligned}$$
(4.3)

If \(\phi \) is \(\theta \)-moderate for some sub-multiplicative function \(\theta \), then we say that \(\phi \) is moderate. Let \(\theta : \mathbb {R}^n\rightarrow \mathbb {R}^+\) and \(C_0>0\). Then the following conditions are equivalent:

  1. (1)

    \(\forall x,y: \theta (x+y)\le C_0\theta (x)\theta (y).\)

  2. (2)

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

    $$\begin{aligned} \Vert (f_1*f_2)\theta \Vert _r\le C_0\Vert f_1 \theta \Vert _p\Vert f_2 \theta \Vert _{q}, 1+\frac{1}{r}=\frac{1}{p}+\frac{1}{q}. \end{aligned}$$
    (4.4)

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

  1. (1)

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

  2. (2)

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

    $$\begin{aligned} \Vert (f_1*f_2)\phi \Vert _p\le C_0\Vert f_1 \theta \Vert _1\Vert f_2 \phi \Vert _{p}. \end{aligned}$$
    (4.5)

The system (1.7) can be written as follows:

$$\begin{aligned} \left\{ \begin{array}{ll} \begin{aligned} &{}{}u_t-\frac{1}{3}(u-u_x)(v+v_x)u_x=\Lambda *F_1(u,v)+\Lambda _x*F_2(u,v), \\ &{}{}v_t-\frac{1}{3}(u-u_x)(v+v_x)v_x=\Lambda *L_1(u,v)+\Lambda _x*L_2(u,v),\\ &{}{}u(0,x)=u_0(x),v(0,x)=v_0(x), \end{aligned} \end{array} \right. \end{aligned}$$
(4.6)

where

$$\begin{aligned}&F_1(u,v)\doteq -\frac{1}{6}(u-u_x)(v+v_x)u+\frac{1}{6}\left[ (u-u_x)(v+v_x)\right] _x(2u-u_x),\\ {}&F_2(u,v)\doteq \frac{1}{6}(u-u_x)(v+v_x)u_x+\frac{1}{3}\left[ (u-u_x)(v+v_x)\right] _xu_x\\ {}&L_1(u,v)\doteq \frac{1}{6}(u-u_x)(v+v_x)v+\frac{1}{6}\left[ (u-u_x)(v+v_x)\right] _x(2v+v_x),\\ {}&L_2(u,v)\doteq -\frac{1}{6}(u-u_x)(v+v_x)v_x+\frac{1}{3}\left[ (u-u_x)(v+v_x)\right] _xv_x. \end{aligned}$$

Proof of Theorem 4.1

For any \(N\in \mathbb {N}/\{0\}\), consider the \(N-\)truncation

$$\begin{aligned} f(x)=f_{N}(x)=\min \{\phi (x), N\}. \end{aligned}$$

Then, \(f: \mathbb {R} \rightarrow \mathbb {R} \) is a locally absolutely continuous function and such that

$$\begin{aligned} \Vert f\Vert _{L^\infty }\le N, |f'(x)|\le A|f(x)|\quad a.e. \quad x\in R. \end{aligned}$$

Furthermore, if \(C_1=\max \{C_0, \eta _1^{-1}\}\), where \(\eta _1=\inf _{x\in R} \theta (x)>0\), then

$$\begin{aligned} (x+y)\le C_1\theta (x)f(y), \forall x,y\in R. \end{aligned}$$
(4.7)

This shows that \(N-\) trunctions of a \(\theta ^{\frac{1}{2}}-\) moderate weight are uniformly \(\theta -\)moderate with respect to N. Let us start from the case \(2\le p <\infty \), multiplying the first equation in (1.7) by \(|uf|^{p-1}sgn(uf)f\) and integrating it lead to

$$\begin{aligned} \int _\mathbb {R} |uf|^{p-1}sgn(uf)(uf)_tdx=&\frac{1}{3}\int _\mathbb {R}|uf|^{p-1}sgn (uf)(u-u_x)(v+v_x)u_x f dx\nonumber \\ {}&+\int _\mathbb {R}|uf|^{p-1}sgn(uf)f \Lambda *F_1(u,v)dx\nonumber \\ {}&+\int _\mathbb {R}|uf|^{p-1}sgn(uf)f \Lambda _x*F_2(u,v)dx. \end{aligned}$$
(4.8)

The first term on the left hand of (4.8) reads

$$\begin{aligned} \int _\mathbb {R} |uf|^{p-1}sgn(uf)(uf)_tdx=\frac{1}{p}\frac{d}{dt}\Vert uf\Vert ^p_{L^p} =\Vert uf\Vert ^{p-1}_{L^p}\frac{d}{dt}\Vert uf\Vert _{L^p}. \end{aligned}$$
(4.9)

Using the H\(\ddot{o}\)lder inequality, we obtain

$$\begin{aligned} \frac{1}{3}|\int _\mathbb {R}|uf|^{p-1}sgn(uf)(u-u_x)(v+v_x)u_xfdx|&\le \frac{1}{3}\Vert uf\Vert ^{p-1}_{L^p}\Vert (u-u_x)(v+v_x) u_x f\Vert _{L^p}\nonumber \\&\le CM_2^2\Vert uf\Vert ^{p-1}_{L^p}\Vert u_x f\Vert _{L^p}. \end{aligned}$$
(4.10)

Furthermore, we have

$$\begin{aligned}&|\int _\mathbb {R} |uf|^{p-1}sgn(uf)[f\cdot \Lambda *F_1(u,v)]dx|\nonumber \\ {}&\quad \le C \Vert uf\Vert ^{p-1}_{L^p}\Vert f\cdot \Lambda *F_1(u,v))\Vert _{L^p}\nonumber \\ {}&\quad \le C\Vert uf\Vert ^{p-1}_{L^p}\Vert \Lambda v\Vert _{L^1}\Vert f\cdot F_1(u,v)\Vert _{L^p}\nonumber \\ {}&\quad \le C\Vert uf\Vert ^{p-1}_{L^p}(\Vert (u-u_x)(v+v_x)uf\Vert _{L^p}+\Vert [(u-u_x)(v+v_x)]_x(2u-u_x)f\Vert _{L^p})\nonumber \\ {}&\quad \le CM_2^2\Vert uf\Vert ^{p-1}_{L^p}\Vert uf\Vert _{L^p}(\Vert u_xf\Vert _{L^p}+\Vert u f\Vert _{L^p}), \end{aligned}$$
(4.11)

where the H\(\ddot{o}\)lder’s inequality, Propositions 3.1 and 3.2 in [2], and condition (4.1) are applied in the first inequality, the second one, and the last one, respectively, and the constant C only depends on \(\theta \) and \(\phi \). Similarly, we obtain

$$\begin{aligned}&|\int _\mathbb {R} |uf|^{p-1}sgn(uf)[f\cdot \partial _x\Lambda *F_2(u,v)dx|\nonumber \\&\quad \le \Vert uf\Vert ^{p-1}_{L^p}\Vert f \cdot \Lambda _x*F_2(u,v))\Vert _{L^p}\nonumber \\&\quad \le C\Vert uf\Vert ^{p-1}_{L^p}\Vert \Lambda _x v\Vert _{L^1}\Vert f\cdot F_2(u,v)\Vert _{L^p}\nonumber \\&\quad \le C\Vert uf\Vert ^{p-1}_{L^p}(\Vert (u-u_x)(v+v_x)u_xf\Vert _{L^p}+\Vert [(u-u_x)(v+v_x)]_xu_xf\Vert _{L^p})\nonumber \\&\quad \le CM_2^2\Vert uf\Vert ^{p-1}_{L^p}\Vert u_xf\Vert _{L^p}, \end{aligned}$$
(4.12)

Plugging the estimates (4.9)-(4.12) into (4.8), we obtain

$$\begin{aligned} \frac{d}{dt}\Vert uf\Vert _{L^p}\le CM_2^2(\Vert uf\Vert _{L^p}+\Vert u_xf\Vert _{L^p}). \end{aligned}$$
(4.13)

Let us now give the estimate on \(u_xf\). Differentiating the first equation in (4.6) with respect to the variable x, we may arrive at

$$\begin{aligned}&\partial _tu_{x}-\frac{1}{3}(u-u_x)(v+v_x)u_{xx}-\frac{1}{3}[(u-u_x)(v+v_x)]_xu_x\\&\quad =\Lambda _x*F_1(u,v)+\Lambda *F_2(u,v)-F_2(u,v). \end{aligned}$$

Multiplying above equation by \(|u_xf|^{p-1}sgn(u_xf)f\) and then integrating on \(\mathbb {R}\) with respect to x, we arrive at

$$\begin{aligned}&\Vert u_xf\Vert _{L^p}^{p-1}\frac{d}{dt}\Vert u_xf\Vert _{L^p}\\ {}&\quad =\frac{1}{3}\int _\mathbb {R}|u_xf|^{p-1}sgn(u_xf)[(u-u_x)(v+v_x)]_xu_xfdx\\ {}&\qquad +\frac{1}{3}\int _\mathbb {R}|u_xf|^{p-1}sgn(u_xf)(u-u_x)(v+v_x)u_{xx}fdx\\ {}&\qquad -\int _\mathbb {R}|u_xf|^{p-1}sgn(u_xf)f\cdot (\Lambda _x*F_1(u,v)+\Lambda *F_2(u,v)+F_2(u,v))dx, \end{aligned}$$

which yields

$$\begin{aligned}&\left| \frac{1}{3}\int _\mathbb {R}\Vert u_xf\Vert ^{p-1}sgn(u_xf)[(u-u_x)(v+v_x)]_xu_xfdx\right| \le CM_2^2\Vert u_xf\Vert ^{p-1}_{L^p}\Vert u_xf\Vert _{L^p},\\ {}&\int _\mathbb {R}\Vert u_xf\Vert ^{p-1}sgn(u_xf)f\cdot F_2(u,v)dx|\le CM_2^2\Vert u_xf\Vert ^{p-1}_{L^p}\Vert u_xf\Vert _{L^p},\\ {}&\int _\mathbb {R}\Vert u_xf\Vert _{L^{p-1}}sgn(u_xf)f\cdot \Lambda _x*F_1(u,v)dx|\\ {}&\quad \le C\Vert u_xf\Vert ^{p-1}_{L^p}\Vert v\Lambda _x\Vert _{L^1}\Vert fF_1(u,v)\Vert _{L^p}\nonumber \\ {}&\quad \le C\Vert u_xf\Vert _{L^p} ^{p-1}\Vert \frac{1}{6}[(u-u_x)(v+v_x)]_x(2uf-u_xf)-\frac{1}{6}(u-u_x)(v+v_x)uf\Vert _{L^p}\nonumber \\ {}&\quad \le C M_2^2 \Vert u_xf\Vert ^{p-1}_{L^p}(\Vert uf\Vert _{L^p}+\Vert u_xf\Vert _{L^p}), \end{aligned}$$

and

$$\begin{aligned}&\int _\mathbb {R}\Vert u_xf\Vert ^{p-1}sgn(u_xf)f\cdot \Lambda *F_2(u,v)dx|\\&\quad \le C\Vert u_xf\Vert ^{p-1}_{L^p}\Vert v\Lambda \Vert _{L^1}\Vert fF_2(u,v)\Vert _{L^p}\nonumber \\&\quad \le C\Vert u_xf\Vert ^{p-1}_{L^p}\Vert \frac{1}{3}[(u-u_x)(v+v_x)]_xu_xf+\frac{1}{6}(u-u_x)(v+v_x)u_xf\Vert _{L^p}\nonumber \\&\quad \le CM_2^2\Vert u_xf\Vert ^{p-1}_{L^p}\Vert uf\Vert _{L^p},\\&\left| \frac{1}{3}\int _\mathbb {R}|u_xf|^{p-1}sgn(u_xf)f(u-u_x)(v+v_x)u_{xx}dx\right| \\&\quad =\left| \frac{1}{3}\int _\mathbb {R}|u_xf|^{p-1}sgn(u_xf)(u-u_x)(v+v_x)(\partial _x(u_xf)-u_xf_x)dx\right| \nonumber \\&\quad =\left| \frac{1}{3}\int _\mathbb {R}(u-u_x)(v+v_x)\partial _x\left( \frac{|u_xf|^p}{p}\right) dx -\frac{1}{3}\int _\mathbb {R}|u_xf|^{p-1}sgn(u_xf)(u-u_x)(v+v_x)u_xf_xdx\right| \nonumber \\&\quad \le CM_2^2\Vert u_xf\Vert ^p_{L^p}, \end{aligned}$$

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

$$\begin{aligned} \frac{d}{dt}\Vert u_x f\Vert _{L^p}\le CM_2^2(\Vert uf\Vert _{L^p}+\Vert u_xf\Vert _{L^p}), \end{aligned}$$

which along with (4.13) yield that

$$\begin{aligned} \frac{d}{dt}(\Vert uf\Vert _{L^p}+\Vert u_xf\Vert _{L^p})\le CM_2^2(\Vert uf\Vert _{L^p}+\Vert u_xf\Vert _{L^p}). \end{aligned}$$
(4.14)

Similarly, we get

$$\begin{aligned} \frac{d}{dt}(\Vert vf\Vert _{L^p}+\Vert v_xf\Vert _{L^p})\le CM_2^2(\Vert vf\Vert _{L^p}+\Vert v_xf\Vert _{L^p}). \end{aligned}$$
(4.15)

Now, together the inequalities (4.14) with (4.15) and then integrating yield

$$\begin{aligned}&\Vert uf\Vert _{L^p}+\Vert u_xf\Vert _{L^p}+\Vert vf\Vert _{L^p}+\Vert v_xf\Vert _{L^p}\nonumber \\&\quad \le (\Vert u_0f\Vert _{L^p}+\Vert u_{0,x}f\Vert _{L^p}+\Vert v_0f\Vert _{L^p}+\Vert v_{0,x}f\Vert _{L^p})e^{CM_2^2t}, \end{aligned}$$
(4.16)

for all \(x\in \mathbb R\). Since \(f(x)=f_N(x)\uparrow \phi (x)\) as \(x\in \mathbb R\) for a.e. \(x\in \mathbb R\) and \(u_0\phi , u_{0,x}\phi , v_0\phi , v_{0,x}\phi \in L^p(\mathbb R)\) 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 \(p=\infty \).

Proof of Corollary 4.1

As explained in [2], if the function \(\theta -\) is a \(\theta -\)moderate weight function, then the function \(\phi ^{\frac{1}{2}}\) is also a \(\theta ^{1/2}-\) moderate weight satisfying \(|(\phi ^{1/2})'(x)|\le \frac{A}{2}\phi ^{1/2}(x), \inf \eta ^{1/2}>0\) and \(\theta ^{1/2}e^{-|\cdot |}\in L^1(R)\). We use Theorem 4.1 with \(p=2\) to the weight \(\phi ^{1/2}\) and obtain

$$\begin{aligned}&\Vert u(t)\phi ^{1/2}\Vert _{L^2}+\Vert u_x(t)\phi ^{1/2}\Vert _{L^2}+\Vert v(t)\phi ^{1/2}\Vert _{L^2}+\Vert v_x(t)\phi ^{1/2}\Vert _{L^2}\nonumber \\&\quad \le (\Vert u_0(t)\phi ^{1/2}\Vert _{L^2}+\Vert u_{0,x}(t)\phi ^{1/2}\Vert _{L^2}+\Vert v_0(t)\phi ^{1/2}\Vert _{L^2}+\Vert v_{0,x}(t)\phi ^{1/2}\Vert _{L^2})\exp (CM_2^2t). \end{aligned}$$
(4.17)

By using the procedure as shown in the proof of Theorem 4.1, for \(2\le p<\infty \), we have

$$\begin{aligned} \frac{d}{dt}\Vert uf\Vert _{L^p}\le CM_2^2\Vert u_xf\Vert _{L^p}+\Vert f\cdot \Lambda *F_1(u,v)\Vert _{L^p}+\Vert f\cdot \partial _x\Lambda *F_2(u,v)\Vert _{L^p}, \end{aligned}$$
(4.18)

and

$$\begin{aligned} \frac{d}{dt}\Vert u_xf\Vert _{L^p}\le CM_2^2\Vert u_xf\Vert _{L^p}+\Vert f\cdot \partial _x\Lambda *F_1(u,v)\Vert _{L^p}+\Vert f\cdot \Lambda *F_2(u,v)\Vert _{L^p}. \end{aligned}$$
(4.19)

In view of Proposition 3.2 in [2], noticing \(f(x)=f_N(x)=\min \{\phi (x), N\}\) admits

$$\begin{aligned}&\Vert f\cdot \Lambda *F_1(u,v)\Vert _{L^p}\nonumber \\&\quad \le C_{\alpha ,\kappa , b}\Vert we^{-|\cdot |}\Vert _{L^p}\Vert fF_1(u,v)\Vert _{L^1}\nonumber \\&\quad \le C\Vert -(u-u_x)(v+v_x)uf+((u-u_x)(v+v_x))_x(2uf-u_xf)\Vert _{L^1}\nonumber \\&\quad \le C(\Vert u^3f\Vert _{L^1}+\Vert u_x^3f\Vert _{L^1}+\Vert v^3f\Vert _{L^1}+\Vert v_x^3f\Vert _{L^1})\nonumber \\&\qquad +CM_2(\Vert u^2f\Vert _{L^1}+\Vert u_x^2f\Vert _{L^1}+\Vert v^2f\Vert _{L^1}+\Vert v_x^2f\Vert _{L^1})\nonumber \\&\quad \le C(1+M_2)\exp (CM_2^2t) \end{aligned}$$
(4.20)

and

$$\begin{aligned}&\Vert f\cdot \partial _x\Lambda *F_2(u,v)\Vert _{L^p}\nonumber \\ {}&\quad \le C_{\alpha ,\kappa , b}\Vert w\partial _xe^{-|\cdot |}\Vert _{L^p}\Vert fF_2(u,v)\Vert _{L^1}\nonumber \\ {}&\quad \le C\Vert (u-u_x)(v+v_x)u_xf+((u-u_x)(v+v_x))_xu_xf\Vert _{L^1}\nonumber \\ {}&\quad \le C(1+M_2)\exp (CM_2^2t). \end{aligned}$$
(4.21)

Using the similar way, we have

$$\begin{aligned} \Vert f\cdot \partial _x\Lambda *F_1(u,v)\Vert _{L^p}+\Vert f\cdot \Lambda *F_2(u,v)\Vert _{L^p}\le C(1+M_2)\exp (CM_2^2t). \end{aligned}$$
(4.22)

Plugging (4.20)-(4.21) into inequality (4.18) and (4.22) into inequality (4.19), we get

$$\begin{aligned} \frac{d}{dt}(\Vert uf\Vert _{L^p}+\Vert u_xf\Vert _{L^p})\le CM_2^2(\Vert uf\Vert _{L^p}+\Vert u_xf\Vert _{L^p})+C(1+M_2)\exp (CM_2^2t). \end{aligned}$$
(4.23)

It follows from the similar argument that

$$\begin{aligned} \frac{d}{dt}(\Vert vf\Vert _{L^p}+\Vert v_xf\Vert _{L^p})\le CM_2^2(\Vert vf\Vert _{L^p}+\Vert v_xf\Vert _{L^p})+C(1+M_2)\exp (CM_2^2t). \end{aligned}$$
(4.24)

Therefore, we obtain

$$\begin{aligned} \frac{d}{dt}&(\Vert u(t)f\Vert _{L^p}+\Vert u_x(t)f\Vert _{L^p}+\Vert v(t)f\Vert _{L^p}+\Vert v_x(t)f\Vert _{L^p})\\ {}&\le CM_2^2(\Vert uf\Vert _{L^p}+\Vert u_xf\Vert _{L^p}+\Vert vf\Vert _{L^p}+\Vert v_xf\Vert _{L^p})+C(1+M_2)\exp (C_2M_2^2t), \end{aligned}$$

which is taken integration and limit \(N\rightarrow \infty \) to get the conclusion in the case \(2\le p<\infty \). 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 \(p=\infty \). The rest of the argument is fully similar to that of Theorem 4.1.