Large-Time Behavior of Momentum Density Support of a Family of Weakly Dissipative Peakon Equations with Higher-Order Nonlinearity

Abstract

In this paper, we mainly study the weakly dissipative peakon equations with higher-order nonlinearity. Under the effect of dissipation, we first derive the infinite propagation speed if the initial datum has a nonnegative compact support. Furthermore, we obtain the large-time behavior of the support of momentum density with the initial data compactly supported. The corresponding results are obtained by using some prior estimates and the energy method. It is worth noting that we need to overcome the difficulties caused by the high-order nonlinear structure and the dissipative effect of the equation. The obtained results generalize the previous results to a certain degree.

1. Introduction

A family of peakon equations with higher-order nonlinearity can be expressed in the following form

$$\left\{\begin{array}{cc}{\displaystyle u_t-u_{txx}+(b+c)u^a{u}_x-b{u}^{a-1}{u}_x{u}_{xx}-c{u}^a{u}_{xxx}=0,}\hfill & t>0,x\in \mathbb{R},\hfill \\ u(0,x)=u_0\left(x\right),\hfill & x\in \mathbb{R},\hfill \end{array}\right.$$

(1)

where $a>0,b\in \mathbb{R}$ and $c\ne 0$. The single peakon and multi-peakon solutions of (1) have been obtained in [1] if and only if $a⩾0$ and $c\ne 0$. The meaning of “Peakon” solution will be given below. In [2], Yan first gave the local well-posedness regime for the Cauchy problem by using the Littlewood–Paley decomposition and transport equation theory. Then, the precise blow-up scenario and global existence results for strong solutions to (1) were established. Finally, the wave-breaking phenomenon in the Besov space was studied.

In fact, Equation (1) is an evolution equation with $(N+1)$-order nonlinearities, which can be transformed into three famous integrable dispersive equations: the Camassa–Holm (CH) equation, the Degasperis–Procesi (DP) equation, and the Novikov equation (NE).

For $a=c=1$, $b=2$, Equation (1) turns into the classical Camassa–Holm (CH) equation. In the recent years, Fokas and Fuchssteiner deduced the CH equation in [3], and then, in [4], it was derived as a unidirectional propagation model in shallow water by Camassa and Holm. Moreover, the CH equation can be viewed as a model of axisymmetric wave propagation in hyperelastic rods [5]. The CH equation not only has a bi-Hamilton structure [3], but is also completely integrable [4,6]. Particularly, the CH equation has infinite conservation laws, and it can be solved by its corresponding inverse scattering transformation. After the birth of the CH equation, it has been extensively investigated due to its important physical significance in the past decades. For example, the CH equation possesses traveling wave solutions of the form $c{e}^{-|x-ct|}$ which are called peakons [4]. It describes an essential feature of the traveling waves of largest amplitude [7,8,9]. Several authors [10,11] have considered the local well-posedness for initial data $u_0\left(x\right)\in H^s$ with $s>\frac{3}{2}$. Wave-breaking phenomena for a large class of initial data have been studied in [9,10,11]. Constantin and Strauss proved that the peakon solutions of the CH equation are orbitally stable [12,13]. In fact, orbital stability refers to Poincare stability. Roughly speaking, if the particle is very close to the orbital r at a certain time, it will be very close to the orbital r at any later time. It is different from the Lyapunov stability. The orbital stability implies that the shape of the peakons is stable so that these wave patterns are physically identifiable. Furthermore, the CH equation has a global conservative solution [14] and a dissipative solution [15]. We also focus on the Liouville-type property. The Liouville-type property also refers to unique continuous property, which is an important problem in the research field of nonlinear partial differential equations. A lot of important work has been done, for example, in [16,17,18,19,20]. In [21], Linares and Pone studied the Liouville-type property of the CH equation in the periodic and nonperiodic conditions. For the related generalized CH equation, we refer to [22,23,24,25,26] and their references.

For $a=c=1$, $b=3$, Equation (1) turns into the Degasperis–Procesi (DP) equation, which can be regarded as a model for nonlinear shallow water dynamics and its asymptotic accuracy is the same as the CH Equation [27,28]. It is formally integrable and has a bi-Hamiltonian structure [29]. The Cauchy problem of the DP equation has also been extensively studied. Local well-posedness of this equation with initial data $u_0\in H^s\left(\mathbb{R}\right),s>\frac{3}{2}$ was established in [30]. The DP equation also has blow-up phenomena at finite time and global strong solutions [31], which are similar to the properties of the CH equation. However, the two equations are really different [32].

For $a=2$, $b=3$, $c=1$, Equation (1) becomes the Novikov equation (NE) with cubic nonlinearity which was proposed in [33]. Its integrability, Lax pair, bi-Hamiltonian structure, and peakon solutions have been shown in [33]. The main difference between the NE, the CH and DP equations is that the former one has cubic nonlinearity and the latter ones have quadratic nonlinearity. The local well-posedness, blow-up of strong solutions, and global weak solutions for the Cauchy problem of the NE have been studied in [33,34,35].

In fact, it is really difficult to avoid energy dissipation in the real world. Ghidaglia [36] studied the long-time behavior of solutions to the weakly dissipative KdV equation as a finite-dimensional dynamical system. Wu and Yin [37] discussed the blow-up, blow-up rate, and decay of the solution to the weakly dissipative periodic CH equation and blow-up phenomena of the weakly dissipative DP equation on the line [32] or circle [38]. Hu and Yin [39] focused on the blow-up and blow-up rate of solutions to a weakly dissipative periodic rod equation.

In this paper, we study the following weakly dissipative peakon equations:

$$\left\{\begin{array}{c}{\displaystyle u_t-u_{txx}+(b+c)u^a{u}_x-b{u}^{a-1}{u}_x{u}_{xx}-c{u}^a{u}_{xxx}}\hfill \\ +\lambda (u-u_{xx})=0,\hfill & t>0,x\in \mathbb{R},\hfill \\ u(0,x)=u_0\left(x\right),\hfill & x\in \mathbb{R},\hfill \end{array}\right.$$

(2)

where $a⩾1,b>0,c>0$ and $\lambda$ is a nonnegative dissipative constant. In [40], Zhang et al. obtained the local well-posedness result and blow-up criteria for Equation (2) when $\lambda =0$. Our aim in the present paper is to investigate the large-time behavior of the support of momentum density of (2).

The large-time behavior for the support of momentum density of the CH equation was studied by Jiang, Zhou and Zhu in [41]. After having established global existence of solutions in Theorem 2 below, inspired by [41], we use a similar idea to study the large-time behavior for the support of momentum density of (2) under the dissipation effect.

It is also observed that those parameters $a,b,c$ and $\lambda$ have an important role in the effects of nonlocal nonlinearities on large-time behavior of momentum density support of (2), which makes our study on large-time behavior of momentum density support of a family of weakly dissipative peakon equations with higher-order nonlinearity more interesting and challenging. In this process, we obtain the results by overcoming the difficulties caused by the high-order nonlinear structure and the dissipative effect of the equation, which generalizes the corresponding results in [42] to a certain extent.

In this paper, the norm of Lebesegue space of $L^p\left(\mathbb{R}\right),p\in [1,\infty ]$, is denoted by $\left|\right|·{\left|\right|}_{L^p}$ and the Sobolev space $H^s\left(\mathbb{R}\right),s\in \mathbb{R}$, by $\left|\right|·{\left|\right|}_{H^s}$.

The rest of this paper is organized as follows. In Section 2, we recall some useful preliminary results which will be used later. In Section 3, we present the infinite propagation speed for a family of weakly dissipative peakon equations with higher-order nonlinearity. In Section 4, large-time behavior of the support of momentum density of a family of weakly dissipative peakon equations with higher-order nonlinearity is obtained.

2. Preliminaries

In this section, the aim is to present a detailed description of the solution u in its lifespan with the initial datum $u_0$ being compactly supported.

For convenience and future use, the Cauchy problem for the nonlocal form of Equation (2) can be rewritten as follows by using the Green function $G\left(x\right)=\frac{1}{2}{e}^{-\left|x\right|}$ and the identity ${(1-{\partial }_x^2)}^{-1}f=G\ast f$ for all $f\in L^2\left(\mathbb{R}\right)$

$$\left\{\begin{array}{cc}{\displaystyle u_t+\left(c{u}^a\right)u_x=ϝ(t,x)-\lambda u,}\hfill & t>0,x\in \mathbb{R},\hfill \\ u(0,x)=u_0\left(x\right),\hfill & x\in \mathbb{R},\hfill \end{array}\right.$$

(3)

where

$$ϝ(t,x)=-G_x\ast \left(\frac{3ac-b}{2}{u}^{a-1}{u}_x^2+\frac{b}{a+1}{u}^{a+1}\right)-G\ast \left(\frac{(a-1)(b-ac)}{2}{u}^{a-2}{u}_x^3\right),$$

∗ represents spatial convolution.

Applying a similar proof as presented in [2] (Corollary 2.1), we can obtain the following local well-posedness result for Equation (3).

Theorem 1.

Assume that $u_0\in H^s\left(\mathbb{R}\right)$ with $s>\frac{3}{2}$. There exist a maximal existence time $T=T\left(\right|\left|u_0\right|{|}_{H^s\left(\mathbb{R}\right)})>0$ and a unique solution u to (3) such that $u\in C([0,T);H^s\left(\mathbb{R}\right))\cap C^1([0,T);H^{s-1}\left(\mathbb{R}\right))$. Moreover, the solution depends continuously on the initial data, that is, the mapping

$$u_0\mapsto u:H^s\left(\mathbb{R}\right)\to C([0,T);H^s\left(\mathbb{R}\right))\cap C^1([0,T);H^{s-1}\left(\mathbb{R}\right))$$

is continuous.

Proof. 

The conclusion can be proved by using the classical Kato’s semigroup theory [43]. □

Lemma 1.

Suppose that $u_0\in H^s\left(\mathbb{R}\right)$ with $s>\frac{3}{2}$. $m=(1-{\partial }_x^2)u$ is the momentum density. Let T be the maximal existence time of the corresponding solution u to Equation (3). Then, for all $(t,x)\in [0,T)\times \mathbb{R}$, we have

$$m(t,q(t,x))q_x^{\frac{b}{ac}}(t,x)=m_0\left(x\right)e^{-\lambda t}.$$

(4)

Proof. 

Let $u(t,x)$ be a strong solution to Equation (3) guaranteed by Theorem 1. Let us now consider the following initial value problem

$$\left\{\begin{array}{cc}{\displaystyle \frac{dq(t,x)}{dt}=c{u}^a(t,q(t,x)),}\hfill & t\in (0,T),x\in \mathbb{R},\hfill \\ q(0,x)=x,\hfill & x\in \mathbb{R},\hfill \end{array}\right.$$

(5)

where T is the lifespan of the solution. Differentiating Equation (5) with respect to x, we obtain

$$\frac{d{q}_t}{dx}=ac{u}^{a-1}{u}_x{q}_x,t\in (0,T).$$

(6)

Therefore,

$$\left\{\begin{array}{cc}{\displaystyle q_x=e^{{\int }_0^{t}ac{u}^{a-1}{u}_x(s,q)ds},}\hfill & t\in (0,T),x\in \mathbb{R},\hfill \\ q_x(0,x)=1,\hfill & x\in \mathbb{R}.\hfill \end{array}\right.$$

(7)

So, $q(t,·):\mathbb{R}\to \mathbb{R}$ is an increasing diffeomorphism of the line before blow-up. In fact, according to (5) and (6), a direct calculation yields

$$\begin{array}{ccc}\hfill \frac{d}{dt}\left[m(t,q)q_x^{\frac{b}{ac}}\right]& =& (m_t+m_x{q}_t)q_x^{\frac{b}{ac}}+\frac{b}{ac}m(t,q)q_x^{\frac{b}{ac}-1}{q}_{xt}\hfill \\ \hfill & =& (m_t+m_x{q}_t)q_x^{\frac{b}{ac}}+\frac{b}{ac}m(t,q)q_x^{\frac{b}{ac}-1}{\left(c{u}^a\right)}_x{q}_x\hfill \\ & =& [m_t+c{u}^a{m}_x+bm{u}^{a-1}{u}_x]q_x^{\frac{b}{ac}}=-\lambda m{q}_x^{\frac{b}{ac}},\hfill \end{array}$$

(8)

which implies that

$$m(t,q(t,x))q_x^{\frac{b}{ac}}(t,x)=m_0\left(x\right)e^{-\lambda t}.$$

□

Theorem 2.

Suppose that $u_0\in H^s\left(\mathbb{R}\right)$ with $s>\frac{3}{2}$. Let T be the maximal existence time of the corresponding solution u to the Equation (3) with $b=(a+1)c$. If $m_0\left(x\right)=(1-{\partial }_x^2)u_0\left(x\right)$ does not change sign for all $x\in \mathbb{R}$, then the solution exists globally.

Proof. 

The proof of this theorem is similar to the one presented in [2] (Theorem 4.2). For completeness, we give it here. We suppose $s>3$ and $m_0\left(x\right)⩾0$ to prove this theorem. In fact, if $u(t,x)$ is a smooth solution to Equation (3), then it follows from (4) that

$$m(t,x)=(1-{\partial }_x^2)u(t,x)⩾0.$$

Moreover, $u(t,x)$ and $u_x(t,x)$ can be expressed as

$$u(t,x)=\frac{1}{2}{e}^{-x}{\int }_{-\infty }^x{e}^{\xi }m(t,\xi )d\xi +\frac{1}{2}{e}^x{\int }_x^{\infty }{e}^{-\xi }m(t,\xi )d\xi .$$

(9)

$$u_x(t,x)=-\frac{1}{2}{e}^{-x}{\int }_{-\infty }^x{e}^{\xi }m(t,\xi )d\xi +\frac{1}{2}{e}^x{\int }_x^{\infty }{e}^{-\xi }m(t,\xi )d\xi .$$

(10)

Then, we have

$$u+u_x=e^x{\int }_x^{\infty }{e}^{-\xi }m(t,\xi )d\xi ⩾0,u-u_x=e^{-x}{\int }_{-\infty }^x{e}^{\xi }m(t,\xi )d\xi ⩾0.$$

(11)

In addition,

$$|u_x(t,x)|⩽u(t,x).$$

(12)

On the other hand, by using (2) and integration by parts, we have

$$\frac{d}{dt}{\int }_{\mathbb{R}}(u^2+u_x^2)dx+2\lambda {\int }_{\mathbb{R}}(u^2+u_x^2)dx=2(b-c(a+1)){\int }_{\mathbb{R}}{u}^a{u}_x{u}_{xx}dx=0.$$

(13)

Solving Equation (13), we obtain

$${\int }_{\mathbb{R}}(u^2+u_x^2)dx=e^{-2\lambda t}{\int }_{\mathbb{R}}(u_0^2+u_{0x}^2)dx.$$

(14)

Applying the Sobolev embedding theorem, (12) and (14), we have

$$|ac{u}^{a-1}{u}_x{|⩽ac|\left|u\right||}_{L^{\infty }}^a⩽\frac{ac}{2^{a/2}}{\left|\right|u\left|\right|}_{H^1}^a=\frac{ac}{2^{a/2}}{e}^{-\lambda at}\left|\right|u_0{\left|\right|}_{H^1}^a⩽\frac{ac}{2^{a/2}}\left|\right|u_0{\left|\right|}_{H^1}^a.$$

□

3. Infinite Propagation Speed

In this section, we can now give the behavior of solutions at infinity when a non-positive compactly supported initial momentum density is given.

Theorem 3.

Assume that $u_0\in H^s\left(\mathbb{R}\right)$ with $s>\frac{3}{2}$ and let T be the maximal time of the solution u to (3) with the initial data $u_0$. Let $\lambda \in \mathbb{R}$, $a,b$ and c satisfy the following relation:

$$max\left\{0,\frac{c(a-4)(a+1)}{a-1}\right\}<b<\frac{ac(a+1)}{a-1}.$$

(15)

Assume that the initial datum $m_0=(1-{\partial }_x^2)u_0$ has the same sign in an interval $[\alpha ,\beta ]$, then the corresponding solution $u(t,x)$ to Equation (3) has the following property:

$$\begin{array}{c}\hfill u(t,x)=\left\{\begin{array}{cc}\frac{1}{2}{e}^{-x}E\left(t\right),\hfill & {\displaystyle x>q(t,\beta ),}\hfill \\ \frac{1}{2}{e}^{x}F\left(t\right),\hfill & {\displaystyle x<q(t,\alpha ),}\hfill \end{array}\right.\end{array}$$

where $E\left(t\right)={\int }_{\mathbb{R}}{e}^{\xi }m(t,\xi )d\xi$ and $F\left(t\right)={\int }_{\mathbb{R}}{e}^{-\xi }m(t,\xi )d\xi$ denote continuous non-vanishing functions. Moreover, if a is odd, $e^{\lambda t}E\left(t\right)$ is a strictly increasing function while $e^{\lambda t}F\left(t\right)$ is a strictly decreasing function for all $t\in [0,T]$. If a is even, $e^{\lambda t}E\left(t\right)$ is a strictly decreasing function while $e^{\lambda t}F\left(t\right)$ is a strictly increasing function for all $t\in [0,T]$.

Proof. 

Since $m_0\left(x\right)$ is compactly supported in the interval $[\alpha ,\beta ]$, it follows from (4) that $m(t,x)$ is also compactly supported with its support contained in the interval $\left[q\right(t,\alpha ),q(t,\beta \left)\right]$. Hence, the following functions are well-defined:

$$E\left(t\right)={\int }_{\mathbb{R}}{e}^{\xi }m(t,\xi )d\xi ,F\left(t\right)={\int }_{\mathbb{R}}{e}^{-\xi }m(t,\xi )d\xi .$$

(16)

Using (4), we have

$$m(t,q(t,x\left)\right)=0,x\in (-\infty ,\alpha )\cup (\beta ,+\infty ),$$

(17)

then, for $x>q(t,\beta )$, from (9), we have

$$u(t,x)=\frac{1}{2}{e}^{-\left|x\right|}\ast m(t,x)=\frac{1}{2}{e}^{-x}{\int }_{q(t,\alpha )}^{q(t,\beta )}{e}^{\xi }m(t,\xi )d\xi =\frac{1}{2}{e}^{-x}E\left(t\right),$$

(18)

and for $x<q(t,\alpha )$,

$$u(t,x)=\frac{1}{2}{e}^{-\left|x\right|}\ast m(t,x)=\frac{1}{2}{e}^x{\int }_{q(t,\alpha )}^{q(t,\beta )}{e}^{-\xi }m(t,\xi )d\xi =\frac{1}{2}{e}^{x}F\left(t\right).$$

(19)

From (3), a direct calculation yields

$$\begin{array}{ccc}\hfill m_t+c{u}^a{u}_x-{\partial }_x^2\left(c{u}^a{u}_x\right)& =& -{\partial }_x\left(\frac{3ac-b}{2}{u}^{a-1}{u}_x^2+\frac{b}{a+1}{u}^{a+1}\right)\hfill \\ & & -\frac{(a-1)(b-ac)}{2}{u}^{a-2}{u}_x^3-\lambda m.\hfill \end{array}$$

(20)

Now, we prove the monotonicity of $e^{\lambda t}E\left(t\right)$ and $e^{\lambda t}F\left(t\right)$. It follows from the (16) and (20) that

$$\begin{array}{ccc}\hfill \frac{dE\left(t\right)}{dt}& =& {\int }_{\mathbb{R}}{e}^{\xi }{m}_{t}d\xi \hfill \\ \hfill & =& {\int }_{\mathbb{R}}{e}^{\xi }(-c{u}^a{u}_{\xi }+{\partial }_{\xi }^2\left(c{u}^a{u}_{\xi }\right)-{\partial }_{\xi }\left(\frac{3ac-b}{2}{u}^{a-1}{u}_{\xi }^2+\frac{b}{a+1}{u}^{a+1}\right)\hfill \\ & & -\frac{(a-1)(b-ac)}{2}{u}^{a-2}{u}_{\xi }^3)d\xi -\lambda {\int }_{\mathbb{R}}{e}^{\xi }md\xi .\hfill \end{array}$$

(21)

Then,

$$\begin{array}{ccc}\hfill \frac{dE\left(t\right)}{dt}+\lambda E\left(t\right)& =& {\int }_{\mathbb{R}}{e}^{\xi }(-c{u}^a{u}_{\xi }+{\partial }_{\xi }^2\left(c{u}^a{u}_{\xi }\right)-{\partial }_{\xi }\left(\frac{3ac-b}{2}{u}^{a-1}{u}_{\xi }^2+\frac{b}{a+1}{u}^{a+1}\right)\hfill \\ & & -\frac{(a-1)(b-ac)}{2}{u}^{a-2}{u}_{\xi }^3)d\xi .\hfill \end{array}$$

(22)

Combining (22) with integration by parts, we have

$$\begin{array}{ccc}\hfill \frac{d\left[e^{\lambda t}E\left(t\right)\right]}{dt}& =& e^{\lambda t}\left[{\int }_{\mathbb{R}}{e}^{\xi }\right(-c{u}^a{u}_{\xi }+{\partial }_{\xi }^2\left(c{u}^a{u}_{\xi }\right)-{\partial }_{\xi }\left(\frac{3ac-b}{2}{u}^{a-1}{u}_{\xi }^2+\frac{b}{a+1}{u}^{a+1}\right)\hfill \\ \hfill & & -\frac{(a-1)(b-ac)}{2}{u}^{a-2}{u}_{\xi }^3\left)d\xi \right]\hfill \\ \hfill & =& e^{\lambda t}[{\int }_{\mathbb{R}}{e}^{\xi }(-c{u}^a{u}_{\xi }+{\partial }_{\xi }^2\left(c{u}^a{u}_{\xi }\right))d\xi -e^{\xi }\left(\frac{3ac-b}{2}{u}^{a-1}{u}_{\xi }^2+\frac{b}{a+1}{u}^{a+1}\right){\mid }_{\mathbb{R}}\hfill \\ \hfill & & +{\int }_{\mathbb{R}}{e}^{\xi }\left(\frac{3ac-b}{2}{u}^{a-1}{u}_{\xi }^2+\frac{b}{a+1}{u}^{a+1}\right)d\xi -\frac{(a-1)(b-ac)}{2}{\int }_{\mathbb{R}}{e}^{\xi }{u}^{a-2}{u}_{\xi }^{3}d\xi ]\hfill \\ \hfill & =& e^{\lambda t}[-\frac{c}{a+1}{e}^{\xi }{u}^{a+1}{|}_{\mathbb{R}}+\frac{c}{a+1}{\int }_{\mathbb{R}}{e}^{\xi }{u}^{a+1}d\xi +e^{\xi }\left({\left(c{u}^a{u}_{\xi }\right)}_{\xi }-c{u}^a{u}_{\xi }+\frac{c}{a+1}{u}^{a+1}\right){\mid }_{\mathbb{R}}\hfill \\ \hfill & & -\frac{c}{a+1}{\int }_{\mathbb{R}}{e}^{\xi }{u}^{a+1}d\xi -e^{\xi }\left(\frac{3ac-b}{2}{u}^{a-1}{u}_{\xi }^2+\frac{b}{a+1}{u}^{a+1}\right){\mid }_{\mathbb{R}}\hfill \\ \hfill & & +{\int }_{\mathbb{R}}{e}^{\xi }\left(\frac{3ac-b}{2}{u}^{a-1}{u}_{\xi }^2+\frac{b}{a+1}{u}^{a+1}\right)d\xi -\frac{(a-1)(b-ac)}{2}{\int }_{\mathbb{R}}{e}^{\xi }{u}^{a-2}{u}_{\xi }^{3}d\xi ]\hfill \\ \hfill & =& e^{\lambda t}[\frac{b}{a+1}{\int }_{\mathbb{R}}{e}^{\xi }{u}^{a+1}d\xi +\frac{3ac-b}{2}{\int }_{\mathbb{R}}{e}^{\xi }{u}^{a-1}{u}_{\xi }^{2}d\xi +\frac{(a-1)(ac-b)}{2}\hfill \\ & & \times {\int }_{\mathbb{R}}{e}^{\xi }{u}^{a-2}{u}_{\xi }^{3}d\xi ].\hfill \end{array}$$

(23)

If $m_0$ is non-positive with compact support in an interval $[\alpha ,\beta ]$, then in view of (4), (9) and (10), for all $t\in [0,T]$ and $x\in \mathbb{R}$, we have

$$u(t,x)⩽0,u(t,x)+u_x(t,x)⩽0,u(t,x)-u_x(t,x)⩽0.$$

(24)

It follows that

$$u^3+u_x^3=(u+u_x)\left[{\left(u-\frac{1}{2}{u}_x\right)}^2+\frac{3}{4}{u}_x^2\right]⩽0,$$

(25)

and

$$u^3-u_x^3=(u-u_x)\left[{\left(u+\frac{1}{2}{u}_x\right)}^2+\frac{3}{4}{u}_x^2\right]⩽0.$$

(26)

We first treat the case with a being an odd number.

If $b⩽ac$, we have

$$\frac{b}{a+1}{\int }_{\mathbb{R}}{e}^{\xi }{u}^{a+1}d\xi >0,\frac{3ac-b}{2}{\int }_{\mathbb{R}}{e}^{\xi }{u}^{a-1}{u}_{\xi }^{2}d\xi >0.$$

The assumption (15) implies that

$$\frac{b}{a+1}+\frac{3ac-b}{2}>\frac{(a-1)(ac-b)}{2}.$$

Therefore, there exist non-negative constants ${\gamma }_1$ and ${\gamma }_2$ such that

$${\gamma }_1+{\gamma }_2=\frac{(a-1)(ac-b)}{2},$$

and

$$\begin{array}{ccc}\hfill \frac{d\left[e^{\lambda t}E\left(t\right)\right]}{dt}& =& e^{\lambda t}[\frac{b}{a+1}{\int }_{\mathbb{R}}{e}^{\xi }{u}^{a+1}d\xi +\frac{3ac-b}{2}{\int }_{\mathbb{R}}{e}^{\xi }{u}^{a-1}{u}_{\xi }^{2}d\xi \hfill \\ & & +\frac{(a-1)(ac-b)}{2}{\int }_{\mathbb{R}}{e}^{\xi }{u}^{a-2}{u}_{\xi }^{3}d\xi ]\hfill \\ & ⩾& e^{\lambda t}\left[{\gamma }_1{\int }_{\mathbb{R}}{e}^{\xi }{u}^{a-2}(u^3+u_{\xi }^3)d\xi +{\gamma }_2{\int }_{\mathbb{R}}{e}^{\xi }{u}^{a-2}{u}_{\xi }^2(u+u_{\xi })d\xi \right].\hfill \end{array}$$

It follows from (24) and (25) that

$$\frac{d\left[e^{\lambda t}E\left(t\right)\right]}{dt}>0,t\in [0,T].$$

If $ac<b<\frac{ac(a+1)}{a-1}$, a direct calculation yields

$$\frac{b}{a+1}+\frac{3ac-b}{2}>\frac{(a-1)(b-ac)}{2}.$$

Then, similar as above, there exist positive constants ${\gamma }_3$ and ${\gamma }_4$ such that

$${\gamma }_3+{\gamma }_4=\frac{(a-1)(b-ac)}{2}.$$

$$\begin{array}{ccc}\hfill \frac{d\left[e^{\lambda t}E\left(t\right)\right]}{dt}& =& e^{\lambda t}[\frac{b}{a+1}{\int }_{\mathbb{R}}{e}^{\xi }{u}^{a+1}d\xi +\frac{3ac-b}{2}{\int }_{\mathbb{R}}{e}^{\xi }{u}^{a-1}{u}_{\xi }^{2}d\xi \hfill \\ & & +\frac{(a-1)(b-ac)}{2}{\int }_{\mathbb{R}}{e}^{\xi }{u}^{a-2}{(-u_{\xi })}^{3}d\xi ]\hfill \\ & ⩾& e^{\lambda t}\left[{\gamma }_3{\int }_{\mathbb{R}}{e}^{\xi }{u}^{a-2}(u^3-u_x^3)d\xi +{\gamma }_4{\int }_{\mathbb{R}}{e}^{\xi }{u}^{a-2}{u}_{\xi }^2(u-u_{\xi })d\xi \right].\hfill \end{array}$$

Then, from (24) and (26), we have

$$\frac{d\left[e^{\lambda t}E\left(t\right)\right]}{dt}>0,t\in [0,T].$$

Similarly to (23), we have the following equality for $F\left(t\right)$:

$$\begin{array}{ccc}\hfill \frac{d\left[e^{\lambda t}F\left(t\right)\right]}{dt}& =& e^{\lambda t}[-\frac{b}{a+1}{\int }_{\mathbb{R}}{e}^{\xi }{u}^{a+1}d\xi -\frac{3ac-b}{2}{\int }_{\mathbb{R}}{e}^{\xi }{u}^{a-1}{u}_{\xi }^{2}d\xi \hfill \\ & & +\frac{(a-1)(ac-b)}{2}{\int }_{\mathbb{R}}{e}^{\xi }{u}^{a-2}{u}_{\xi }^{3}d\xi ].\hfill \end{array}$$

(27)

Under the initial condition and (24)–(26), the above equality leads to

$$\frac{d\left[e^{\lambda t}F\left(t\right)\right]}{dt}<0,t\in [0,T].$$

We then consider the case of a being an even number.

If $b⩽ac$, we have

$$\begin{array}{ccc}\hfill \frac{d\left[e^{\lambda t}E\left(t\right)\right]}{dt}& =& e^{\lambda t}[\frac{b}{a+1}{\int }_{\mathbb{R}}{e}^{\xi }{u}^{a+1}d\xi +\frac{3ac-b}{2}{\int }_{\mathbb{R}}{e}^{\xi }{u}^{a-1}{u}_{\xi }^{2}d\xi \hfill \\ & & +\frac{(a-1)(ac-b)}{2}{\int }_{\mathbb{R}}{e}^{\xi }{u}^{a-2}{u}_{\xi }^{3}d\xi ]\hfill \\ & ⩽& e^{\lambda t}\left[\overline{{\gamma }_1}{\int }_{\mathbb{R}}{e}^{\xi }{u}^{a-2}(u^3+u_x^3)d\xi +\overline{{\gamma }_2}{\int }_{\mathbb{R}}{e}^{\xi }{u}^{a-2}{u}_{\xi }^2(u+u_{\xi })d\xi \right],\hfill \end{array}$$

where $\overline{{\gamma }_1},\overline{{\gamma }_2}>0$, and satisfy that

$$\overline{{\gamma }_1}+\overline{{\gamma }_2}=\frac{(a-1)(ac-b)}{2}.$$

It follows from (24) and (25) that

$$\frac{d\left[e^{\lambda t}E\left(t\right)\right]}{dt}<0,t\in [0,T].$$

If $ac<b<\frac{ac(a+1)}{a-1}$, one obtains

$$\begin{array}{ccc}\hfill \frac{d\left[e^{\lambda t}E\left(t\right)\right]}{dt}& =& e^{\lambda t}[\frac{b}{a+1}{\int }_{\mathbb{R}}{e}^{\xi }{u}^{a+1}d\xi +\frac{3ac-b}{2}{\int }_{\mathbb{R}}{e}^{\xi }{u}^{a-1}{u}_{\xi }^{2}d\xi \hfill \\ & & +\frac{(a-1)(ac-b)}{2}{\int }_{\mathbb{R}}{e}^{\xi }{u}^{a-2}{u}_{\xi }^{3}d\xi ]\hfill \\ & ⩽& e^{\lambda t}\left[\overline{{\gamma }_3}{\int }_{\mathbb{R}}{e}^{\xi }{u}^{a-2}(u^3-u_x^3)d\xi +\overline{{\gamma }_4}{\int }_{\mathbb{R}}{e}^{\xi }{u}^{a-2}{u}_{\xi }^2(u-u_{\xi })d\xi \right],\hfill \end{array}$$

where $\overline{{\gamma }_3},\overline{{\gamma }_4}$ satisfy

$$\overline{{\gamma }_3}+\overline{{\gamma }_4}=\frac{(a-1)(b-ac)}{2}.$$

Due to (24) and (26), we find

$$\frac{d\left[e^{\lambda t}E\left(t\right)\right]}{dt}<0,t\in [0,T].$$

For $F\left(t\right)$, a similar argument in (27) leads to

$$\frac{d\left[e^{\lambda t}F\left(t\right)\right]}{dt}>0,t\in [0,T].$$

□

4. Large-Time Behavior for the Support of Momentum Density

In this section, we present large-time behavior of the support of momentum density of (3).

Theorem 4.

Let $\lambda <0,b=(a+1)c$ and a is an integer. Assume that $m_0\left(x_0\right)⩽0$ is non-trivial and is compactly supported in $[\alpha ,\beta ]$.

(1)

If a is odd, we have

$$e^{-aq(t,\alpha )}-e^{-aq(t,\beta )}\to +\infty ,t\to +\infty .$$

(2)

If a is even, we have

$$e^{aq(t,\beta )}-e^{aq(t,\alpha )}\to +\infty ,t\to +\infty .$$

Proof. 

Note that under the conditions of this theorem, the existence of solutions for (3) are global in time as has been proven in Theorem 2. For $\lambda <0$ and $m_0\left(x\right)⩽0$, from the relation (4), we have $E\left(t\right)<0,F\left(t\right)<0$ for all $t⩾0$.

(1) If a is odd, then $e^{\lambda t}E\left(t\right)<0$ and is strictly increasing so that the limit of $e^{\lambda t}E\left(t\right)$ exists as t goes to ∞. Then, we claim that

$$\underset{t\to +\infty }{lim}{e}^{\lambda t}E\left(t\right)=0.$$

(28)

Otherwise, there exists $\eta >0$ such that

$$\underset{t\to +\infty }{lim}{e}^{\lambda t}E\left(t\right)=-\eta ,$$

then, from (5) and (9), we have

$$e^{a\lambda t}\frac{dq(t,\beta )}{dt}=c{e}^{a\lambda t}{u}^a(t,q(t,\beta ))=\frac{c}{2^a}{e}^{-aq(t,\beta )}{e}^{a\lambda t}{E}^a\left(t\right)⩽\frac{c}{2^a}{e}^{-aq(t,\beta )}{(-\eta )}^a.$$

It follows that

$$e^{aq(t,\beta )}{q}_t⩽-\frac{c{\eta }^a}{2^a}{e}^{-a\lambda t}.$$

Then, since $\lambda <0$, we have

$$e^{aq(t,\beta )}⩽e^a+\frac{c{\eta }^a}{2^a\lambda }(e^{-a\lambda t}-1)\to -\infty ,t\to +\infty .$$

This is a contradiction. So, (28) holds.

Next, by using (4) and Hölder’s inequality, we have

$$\begin{array}{ccc}\hfill e^{-\lambda t}{\left[{\int }_{\alpha }^{\beta }{(-m_0)}^{\frac{a}{a+1}}dx\right]}^{\frac{a+1}{a}}& =& {\left[{\int }_{\alpha }^{\beta }{(-m_0{e}^{-\lambda t})}^{\frac{a}{a+1}}dx\right]}^{\frac{a+1}{a}}\hfill \\ & =& {\left[{\int }_{\alpha }^{\beta }{\left(-m{q}_x^{\frac{a+1}{a}}\right)}^{\frac{a}{a+1}}dx\right]}^{\frac{a+1}{a}}\hfill \\ & =& {\left[{\int }_{\alpha }^{\beta }{(-m)}^{\frac{a}{a+1}}{q}_{x}dx\right]}^{\frac{a+1}{a}}\hfill \\ & =& {\left[{\int }_{q(t,\alpha )}^{q(t,\beta )}{\left(-m(t,\xi )\right)}^{\frac{a}{a+1}}d\xi \right]}^{\frac{a+1}{a}}\hfill \\ & =& {\left[{\int }_{q(t,\alpha )}^{q(t,\beta )}{\left(-m(t,\xi )e^{\xi }\right)}^{\frac{a}{a+1}}{e}^{-\frac{a}{a+1}\xi }d\xi \right]}^{\frac{a+1}{a}}\hfill \\ & ⩽& {\int }_{q(t,\alpha )}^{q(t,\beta )}(-m)e^{\xi }d\xi {\left[{\int }_{q(t,\alpha )}^{q(t,\beta )}{e}^{-a\xi }d\xi \right]}^{\frac{1}{a}}\hfill \\ & =& -E\left(t\right){\left[\frac{1}{a}\left(e^{-aq(t,\alpha )}-e^{-aq(t,\beta )}\right)\right]}^{\frac{1}{a}}.\hfill \end{array}$$

This implies that

$$e^{-aq(t,\alpha )}-e^{-aq(t,\beta )}⩾a{\left[\frac{{\left({\int }_{\alpha }^{\beta }{(-m_0)}^{\frac{a}{a+1}}dx\right)}^{\frac{a+1}{a}}}{-E\left(t\right)e^{\lambda t}}\right]}^a.$$

Then, we obtain

$$e^{-aq(t,\alpha )}-e^{-aq(t,\beta )}\to +\infty ,t\to +\infty .$$

(2) If a is even, then $e^{\lambda t}F\left(t\right)<0$ and is strictly increasing so that the limit of $e^{\lambda t}F\left(t\right)$ exists as t goes to ∞. Then, we claim that

$$\underset{t\to +\infty }{lim}{e}^{\lambda t}F\left(t\right)=0.$$

(29)

Otherwise, if there exists $\delta >0$ such that

$$\underset{t\to +\infty }{lim}\left[e^{\lambda t}F\left(t\right)\right]=-\delta ,$$

then we have $e^{\lambda t}F\left(t\right)<-\delta$. Moreover, in view of (5) and (9), we have

$$e^{a\lambda t}\frac{dq(t,\alpha )}{dt}=c{e}^{a\lambda t}{u}^a(t,q(t,\alpha ))=\frac{c}{2^a}{e}^{aq(t,\alpha )}{e}^{a\lambda t}{F}^a\left(t\right)⩾\frac{c}{2^a}{e}^{aq(t,\alpha )}{\delta }^a.$$

It follows that

$$e^{-aq(t,\alpha )}{q}_t⩽\frac{c{\delta }^a}{2^a}{e}^{-a\lambda t}.$$

Then, we have

$$e^{-aq(t,\alpha )}⩽e^{-a}+\frac{c{\delta }^a}{2^a\lambda }(e^{-a\lambda t}-1)\to -\infty ,t\to +\infty .$$

It is a contradiction. Next, by using relation (4), we have

$$\begin{array}{ccc}\hfill e^{-\lambda t}{\left[{\int }_{\alpha }^{\beta }{(-m_0)}^{\frac{a}{a+1}}dx\right]}^{\frac{a+1}{a}}& =& {\left[{\int }_{\alpha }^{\beta }{(-m_0{e}^{-\lambda t})}^{\frac{a}{a+1}}dx\right]}^{\frac{a+1}{a}}\hfill \\ & =& {\left[{\int }_{\alpha }^{\beta }{\left(-m{q}_x^{\frac{a+1}{a}}\right)}^{\frac{a}{a+1}}dx\right]}^{\frac{a+1}{a}}\hfill \\ & =& {\left[{\int }_{\alpha }^{\beta }{(-m)}^{\frac{a}{a+1}}{q}_{x}dx\right]}^{\frac{a+1}{a}}\hfill \\ & =& {\left[{\int }_{q(t,\alpha )}^{q(t,\beta )}{\left(-m(t,\xi )\right)}^{\frac{a}{a+1}}d\xi \right]}^{\frac{a+1}{a}}\hfill \\ & =& {\left[{\int }_{q(t,\alpha )}^{q(t,\beta )}{\left(-m(t,\xi )e^{-\xi }\right)}^{\frac{a}{a+1}}{e}^{\frac{a}{a+1}\xi }d\xi \right]}^{\frac{a+1}{a}}\hfill \\ & ⩽& {\int }_{q(t,\alpha )}^{q(t,\beta )}(-m)e^{-\xi }d\xi {\left[{\int }_{q(t,\alpha )}^{q(t,\beta )}{e}^{a\xi }d\xi \right]}^{\frac{1}{a}}\hfill \\ & =& -F\left(t\right){\left[\frac{1}{a}\left(e^{aq(t,\beta )}-e^{aq(t,\alpha )}\right)\right]}^{\frac{1}{a}}.\hfill \end{array}$$

This implies that

$$e^{aq(t,\beta )}-e^{aq(t,\alpha )}⩾a{\left[\frac{{\left({\int }_{\alpha }^{\beta }{(-m_0)}^{\frac{a}{a+1}}dx\right)}^{\frac{a+1}{a}}}{-F\left(t\right)e^{\lambda t}}\right]}^a.$$

Then, we obtain

$$e^{aq(t,\beta )}-e^{aq(t,\alpha )}\to +\infty ,t\to +\infty .$$

□

5. Conclusions

In this paper, we study the large-time behavior of the momentum density support of (2) with a dissipative constant $\lambda <0$. For the result of $\lambda =0$ in Theorem 4, we can refer the readers to [42]. For $\lambda >0$, the conclusion of Theorem 4 may also be true in some special cases; we hope that it can be studied in the future. At the same time, it is worth noting that a similar result can also be obtained if the initial datum is nonnegative and compactly supported.