A class of compressible non-Newtonian fluids with external force and vacuum under no compatibility conditions

Abstract

We are concerned with the Cauchy problem for a class of compressible non-Newtonian fluids on the whole one-dimensional space with external force and vacuum. It is proved that the Cauchy problem for a class of compressible non-Newtonian fluids with external force and vacuum admits a unique local strong solution under no compatibility conditions.

1 Introduction and main results

We consider the one-dimensional equations of compressible non-Newtonian fluids which read as follows:

$$ \textstyle\begin{cases} \rho_{t}+(\rho u)_{x}=0, \\ (\rho u)_{t}+(\rho u^{2})_{x}-[(u_{x}^{2}+\mu_{0})^{\frac{p-2}{2}}u_{x}]_{x}+\pi_{x}=f, \end{cases} $$

(1.1)

where $t\geq0$, $x\in\mathbb{R}$, $\mu_{0}>0$, the unknown functions $\rho =\rho(x,t)$, $u=u(x,t)$ and $\pi(\rho)=A\rho^{\gamma}$ ($A>0$, $\gamma>1$) denote the density, the velocity and the pressure, separately. Without loss of generality, we set $A=1$. We consider the Cauchy problem with $(\rho,u)$ vanishing at infinity. For given initial functions, we require that

$$ \rho(x,0)=\rho_{0}(x), \qquad u(x,0)=u_{0}(x), \quad x \in\mathbb{R}. $$

(1.2)

The motion of the fluid is driven by an external force $f(t,x,u)$. In this paper, for any $(t,x,u)\in(0,T_{0}]\times(-\infty,+\infty)\times (-\infty,+\infty)$, we assume $f(t,x,u)$ to satisfy

$$ \textstyle\begin{cases} fu\leq-u^{2}, \\ fu_{t}\leq-uu_{t}, \\ f_{t}u_{t}\leq-u_{t}^{2}, \end{cases} $$

(1.3)

and to satisfy the structural conditions

$$ \textstyle\begin{cases} f\in L^{\infty}(0,T,L^{2}(\mathbb{R})), \\ f_{x}\in L^{\infty}(0,T,L^{2}(\mathbb{R})), \\ f_{u}\leq-B, \end{cases} $$

(1.4)

where B is a positive constant.

In recent years, there has been an increasing recognition of the importance of non-Newtonian fluids. Non-Newtonian fluids arise in a large number of problems such as in the field of biomechanics, chemistry, hemorheology, glaciology, and geology. For the study of the non-Newtonian fluids, sparking the increasing interest, see [1–3].

When the initial vacuum is allowed, Lions [4] obtained the global weak solutions for the isentropic fluids for large initial data. Li and Xin [5] obtained that the Cauchy problem of the Navier-Stokes equations for viscous compressible barotropic flows in two or three spatial dimensions with vacuum as far field density admits global well-posedness and large time asymptotic behavior of strong and classical solutions. Huang and Li [6] establish the global existence and uniqueness of strong and classical solutions to the Cauchy problem for the barotropic compressible Navier-Stokes equations in two spatial dimensions with smooth initial data with vacuum. Liang and Lu [7] obtained the global-in-time existence of a unique classical solution with large initial data for the Cauchy problem for a compressible viscous fluid in one-dimensional space. Recently, Li and Liang [8] found that the two-dimensional Cauchy problem of the compressible Navier-Stokes equations admits a unique local classical solution provided the initial density decays not too slow at infinity.

Up to now, the results about non-Newtonian fluids are quite few. Recently, Yuan and Xu [9] obtained an existence result on local solutions. They obtained local existence and uniqueness of solution by using a classical energy method. For related results we refer the reader to [9–16] and the references therein.

For the Cauchy problem (1.1)-(1.2), it is still open even for the local existence of strong solutions under no compatibility conditions when the far field density is vacuum. Moreover, the system (1.1) is with strong nonlinearity, so we are facing another difficulty. In fact, this is the aim of this paper. Reference [8] motivated our study. Compared with [9, 15], the advantage of this paper is there is no need for compatibility conditions, and compared with [8], our problem is nonlinear. In this paper, we will obtain a unique strong solutions for (1.1) under $4< p<+\infty$.

The authors in [8] bounded the $L^{p} (\mathbb{R}^{2})$-norm of u just in the terms of $\|\rho^{1/2} u\|_{L^{2}(\mathbb{R}^{2})}$ and $\| u_{x}\|_{L^{2}(\mathbb {R}^{2})}$ by Hardy type and Poincaré type inequalities. In a similar way, we bounded the $L^{k} (\mathbb{R})\ (k>p)$-norm of u just in the terms of $\|\rho^{1/2} u\|_{L^{2}(\mathbb{R})}$ and $\| u_{x}\|_{L^{p}(\mathbb{R})}$. However, the application of a Sobolev embedding inequality in $\mathbb {R}$ is very different from $\mathbb{R}^{2}$. For this, we use truncation techniques which are needed to obtain the local existence of strong solutions.

The rest of the paper is organized as follows: Firstly, we shall give some elementary facts and inequalities which will be needed in later analysis in Section 2. Sections 3 is devoted to the a priori estimates which are needed to obtain the local existence and uniqueness of strong solution. Finally, Theorem 1.2 is proved in Section 4.

Definition 1.1

If all derivatives involved in (1.1) for $(\rho,u)$ are regular distributions, and equations. (1.1) hold almost everywhere in $\mathbb{R}\times(0,T)$, then $(\rho,u)$ is called a strong solution to (1.1).

Theorem 1.2

For constant $4< p<+\infty$, assume that the initial data $(\rho_{0},u_{0})$ satisfies

$$ \rho_{0}\geq0,\qquad u_{0}\in L^{2}( \mathbb{R}),\qquad u_{0x}\in L^{2}(\mathbb{R}),\qquad \rho_{0}^{\frac{1}{2}}u_{0}\in L^{2}( \mathbb{R}), $$

(1.5)

where $(t,x,u)\in(0,T_{0}]\times\mathbb{R}\times\mathbb{R}$. Further, for constant $p\leq q<+\infty$, assume that $\rho_{0}$ also satisfies

$$\Phi\rho_{0}\in L^{1}(\mathbb{R})\cap H^{1}( \mathbb{R})\cap W^{1,q}(\mathbb{R}), $$

where

$$ \Phi\triangleq\bigl(e+x^{2}\bigr)^{1+\zeta_{0}} $$

(1.6)

and $\zeta_{0}$ is a positive constant.

Then there exists a positive time $T_{0}$ such that the problem (1.1)-(1.2) has a unique strong solution $(\rho,u)$ on $\mathbb{R}\times(0,T_{0}]$ satisfying

$$ \textstyle\begin{cases} \rho\in C([0,T_{0}]; L^{1}(\mathbb{R})\cap H^{1}(\mathbb{R})\cap W^{1,q}(\mathbb{R})), \\ \Phi\rho\in L^{\infty}(0,T_{0};L^{1}(\mathbb{R})\cap H^{1}(\mathbb{R})\cap W^{1,q}(\mathbb{R})), \\ u,\sqrt{\rho}u,\sqrt{t}\sqrt{\rho}u_{t}\in L^{\infty}(0,T_{0};L^{2}(\mathbb {R})), \\ \sqrt{\rho}u_{t},\sqrt{t} u_{tx}\in L^{2}(\mathbb{R}\times(0,T_{0})), \\ u_{x}\in L^{\infty}(0,T_{0};L^{2}(\mathbb{R}))\cap L^{\infty}(0,T_{0};L^{p}(\mathbb {R})), \\ u_{x}\in L^{2}(0,T_{0};H^{1}(\mathbb{R}))\cap L^{(q+1)/q}(0,T_{0};W^{1,q}(\mathbb{R})), \\ \sqrt{t} u_{x}\in L^{2}(0,T_{0};W^{1,q}(\mathbb{R})), \\ \sqrt{t}u_{xx}\in L^{\infty}(0,T_{0};L^{2}(\mathbb{R})). \end{cases} $$

(1.7)

Moreover,

$$ {\inf_{0\leq t\leq T_{0}}} \int_{\Omega_{R}}\rho(x,t)\,\mathrm{d}x\geq\frac {1}{4} \int_{\mathbb{R}} \rho_{0}(x)\,\mathrm{d}x. $$

(1.8)

2 Preliminaries

We have the following results concerning local existence theory on bounded intervals whose existence can be found in [11, 17–19].

Lemma 2.1

For $R>0$ and $\Omega_{R}\triangleq \{x\in\mathbb{R} | |x|< R \}$, assume that $(\rho_{0},u_{0})$ satisfies

$$ \rho_{0}\in H^{1}(\Omega_{R}), \qquad \inf _{x\in\Omega_{R}}\rho_{0}(x)>0, \qquad u_{0}\in H_{0}^{1}(\Omega_{R})\cap H^{2}( \Omega_{R}). $$

(2.1)

Let f as in (1.3) and (1.4). Then there exist a small time $T_{R} > 0$ and a unique classical solution $(\rho,u)$ to the following initial-boundary value problem:

$$ \textstyle\begin{cases} \rho_{t}+(\rho u)_{x}=0, \\ (\rho u)_{t}+(\rho u^{2})_{x}-[(u_{x}^{2}+\mu_{0})^{\frac{p-2}{2}}u_{x}]_{x}+\pi_{x}=f, \\ \pi(\rho)=\rho^{\gamma},\quad \gamma>1, \\ (\rho,u)|_{t=0}=(\rho_{0},u_{0}), \quad x\in\Omega_{R}, \\ u(x)=0,\quad x\in\partial\Omega_{R}, t>0, \end{cases} $$

(2.2)

on $\Omega_{R}\times(0,T_{R}]$ such that

$$ \textstyle\begin{cases} \rho\in C([0,T_{R}]; H^{1}(\Omega_{R})),\qquad u\in C([0,T_{R}]; H_{0}^{1}(\Omega _{R}))\cap L^{\infty}(0,T_{R};H^{2}(\Omega_{R})), \\ \rho_{t}\in C([0,T_{R}]; L^{2}(\Omega_{R})) ,\qquad u_{t}\in L^{2}(0,T_{R}; H_{0}^{1}(\Omega _{R})), \\ \sqrt{\rho}u_{t}\in L^{\infty}(0,T_{R};L^{2}(\Omega_{R})), \\ [(u_{x}^{2}+\mu_{0})^{\frac{p-2}{2}}u_{x}]_{x}\in C([0,T_{R}]; L^{2}(\Omega_{R})). \end{cases} $$

(2.3)

Remark 2.2

Assume $v\in H_{0}^{1}(\Omega_{R})\cap H^{2}(\Omega_{R})$, then $v\in W_{0}^{1,p}(\Omega_{R})$ with $p>4$.

Lemma 2.3

For either $s=2$ or $s=p$, $m\in[s,\infty)$ and $\vartheta\in (m+1/2,\infty)$, there exists a positive constant C such that, for either $\Omega=\mathbb{R}$ or $\Omega=\Omega_{R}$ with $R\geq1$ and for any $v\in W_{\mathrm{loc}}^{1,s}(\Omega)$,

$$ \biggl( \int_{\Omega}|v|^{m}\bigl(e+x^{2} \bigr)^{-\vartheta}\,\mathrm{d}x \biggr)^{1/m}\leq C\|v \|_{L^{s}(\Omega_{1})}+ C\| v_{x}\|_{L^{s}(\Omega)}. $$

(2.4)

A consequence of Lemma 2.3 will play a crucial role in our analysis.

Proof

First, we begin with the case $v\in W_{\mathrm{loc}}^{1,2}(\Omega)$. For all $R\geq1$, since

$$\int_{-R}^{R}\bigl\vert v(x)\bigr\vert ^{2}\,\mathrm{d}x=R \int_{-1}^{1}\bigl\vert v(R\tau)\bigr\vert ^{2}\,\mathrm {d}\tau, $$

we observe there exists $R_{0}\in[\frac{1}{2},1]$ such that

$$ \biggl( \int_{-1}^{1}\bigl\vert v(R_{0}\tau) \bigr\vert ^{2}\,\mathrm{d}\tau \biggr)^{\frac {1}{2}}\leq2 \biggl( \int_{-1}^{1}\bigl\vert v(x)\bigr\vert ^{2}\,\mathrm{d}x \biggr)^{\frac{1}{2}}. $$

(2.5)

By calculation, we get

$$ \int_{R_{0}}^{R}\frac{\mathrm{d}}{\mathrm{d}s} \biggl( \int_{-1}^{1}\bigl\vert v(s\tau)\bigr\vert ^{2}\,\mathrm{d}\tau \biggr)^{\frac{1}{2}}\,\mathrm{d}s= \biggl( \int_{-1}^{1}\bigl\vert v(R\tau)\bigr\vert ^{2}\,\mathrm{d}\tau \biggr)^{\frac{1}{2}}- \biggl( \int_{-1}^{1}\bigl\vert v(R_{0}\tau) \bigr\vert ^{2}\,\mathrm{d}\tau \biggr)^{\frac{1}{2}}. $$

(2.6)

By virtue of

$$\begin{aligned}& \frac{\mathrm{d}}{\mathrm{d}s} \biggl( \int_{-1}^{1}\bigl\vert v(s\tau)\bigr\vert ^{2}\,\mathrm{d}\tau \biggr) \\& \quad = \int_{-1}^{1}2\bigl\vert v(s\tau)\bigr\vert \biggl\vert \frac{\partial v(s\tau)}{\partial s}\biggr\vert \, \mathrm{d}\tau \\& \quad \leq 2 \biggl( \int_{-1}^{1}\bigl\vert v(s\tau)\bigr\vert ^{2}\,\mathrm{d}\tau \biggr)^{\frac {1}{2}} \biggl( \int_{-1}^{1}\biggl\vert \frac{\partial v(s\tau)}{\partial s}\biggr\vert ^{2}\, \mathrm{d}\tau \biggr)^{\frac{1}{2}} \\& \quad \leq 2 \biggl( \int_{-1}^{1}\bigl\vert v(s\tau)\bigr\vert ^{2}\,\mathrm{d}\tau \biggr)^{\frac {1}{2}} \biggl( \int_{-1}^{1}\bigl\vert v'(s\tau) \tau\bigr\vert ^{2}\,\mathrm{d}\tau \biggr)^{\frac {1}{2}} \\& \quad \leq 2 \biggl( \int_{-1}^{1}\bigl\vert v(s\tau)\bigr\vert ^{2}\,\mathrm{d}\tau \biggr)^{\frac {1}{2}} \biggl( \int_{-1}^{1}\bigl\vert v'(s\tau) \bigr\vert ^{2}\,\mathrm{d}\tau \biggr)^{\frac{1}{2}}, \end{aligned}$$

we have

$$ \frac{\mathrm{d}}{\mathrm{d}s} \biggl( \int_{-1}^{1}\bigl\vert v(s\tau)\bigr\vert ^{2}\,\mathrm{d}\tau \biggr)^{\frac {1}{2}}\leq \biggl( \int_{-1}^{1}\bigl\vert v'(s\tau) \bigr\vert ^{2}\,\mathrm{d}\tau \biggr)^{\frac{1}{2}}. $$

(2.7)

By some directly calculations, we get

$$\begin{aligned}& \int_{R_{0}}^{R} \biggl( \int_{-1}^{1}\bigl\vert v'(s\tau) \bigr\vert ^{2}\,\mathrm{d}\tau \biggr)^{\frac{1}{2}}\,\mathrm{d}s \\& \quad = \int_{R_{0}}^{R} \biggl(s \int_{-1}^{1}\bigl\vert v'(s\tau) \bigr\vert ^{2}\,\mathrm{d}\tau \biggr)^{\frac{1}{2}} \biggl( \frac{1}{s} \biggr)^{\frac{1}{2}}\,\mathrm{d}s \\& \quad \leq \biggl( \int_{R_{0}}^{R}s \int_{-1}^{1}\bigl\vert v'(s\tau) \bigr\vert ^{2}\,\mathrm{d}\tau\, \mathrm{d}s \biggr)^{\frac{1}{2}} \biggl( \int_{R_{0}}^{R}\frac{1}{s}\,\mathrm{d}s \biggr)^{\frac{1}{2}} \\& \quad \leq \biggl( \int_{R_{0}}^{R} \int_{-s}^{s}\bigl\vert v_{x}(x)\bigr\vert ^{2}\,\mathrm{d}x\,\mathrm {d}s \biggr)^{\frac{1}{2}} \biggl( \ln\frac{R}{R_{0}} \biggr)^{\frac{1}{2}} \end{aligned}$$

(2.8)

and

$$\begin{aligned}& \int_{R_{0}}^{R} \int_{-s}^{s}\bigl\vert v_{x}(x)\bigr\vert ^{2}\,\mathrm{d}x\,\mathrm{d}s \\& \quad \leq \int_{0}^{R} \int_{x}^{R}\bigl\vert v_{x}(x)\bigr\vert ^{2}\,\mathrm{d}s\,\mathrm{d}x+ \int _{-R}^{0} \int_{-x}^{R}\bigl\vert v_{x}(x)\bigr\vert ^{2}\,\mathrm{d}s\,\mathrm{d}x \\& \quad \leq \int_{0}^{R}\bigl\vert v_{x}(x)\bigr\vert ^{2}(R-x)\,\mathrm{d}x+ \int _{-R}^{0}\bigl\vert v_{x}(x) \bigr\vert ^{2}(R+x)\,\mathrm{d}x \\& \quad = R \int_{-R}^{R}\bigl\vert v_{x}(x)\bigr\vert ^{2}\,\mathrm{d}x. \end{aligned}$$

(2.9)

Combining (2.5)-(2.9), we get

$$\begin{aligned}& \biggl( \int_{-1}^{1}\bigl\vert v(R\tau)\bigr\vert ^{2}\,\mathrm{d}\tau \biggr)^{\frac{1}{2}} \\& \quad \leq CR^{\frac{1}{2}} \bigl(\ln(2R) \bigr)^{\frac{1}{2}} \biggl[ \biggl( \int _{-1}^{1}\bigl\vert v(x)\bigr\vert ^{2}\,\mathrm{d}x \biggr)^{\frac{1}{2}} + \biggl( \int_{-R}^{R}\bigl\vert v_{x}(x)\bigr\vert ^{2}\,\mathrm{d}x \biggr)^{\frac{1}{2}} \biggr]. \end{aligned}$$

(2.10)

We denote

$$ |\!|\!|v|\!|\!|_{2}\triangleq \biggl( \int_{-1}^{1}\bigl\vert v(x)\bigr\vert ^{2}\,\mathrm{d}x \biggr)^{\frac{1}{2}} + \biggl( \int_{-R}^{R}\bigl\vert v_{x}(x)\bigr\vert ^{2}\,\mathrm{d}x \biggr)^{\frac{1}{2}} $$

(2.11)

and rewrite (2.10) as

$$ \int_{-1}^{1}\bigl\vert v(R\tau)\bigr\vert ^{2}\,\mathrm{d}\tau\leq CR\ln(2R)|\!|\!|v|\!|\!|_{2}^{2}. $$

(2.12)

Multiplying (2.12) by R, we obtain

$$\begin{aligned} \int_{-R}^{R}\bigl\vert v(x)\bigr\vert ^{2}\,\mathrm{d}x \leq& CR^{2}\ln(2R)|\!|\!|v|\!|\!|_{2}^{2} \\ \leq& CR^{2}\ln\bigl(e+R^{2}\bigr)|\!|\!|v|\!|\!|_{2}^{2} \\ \leq& CR^{2}\bigl(e+R^{2}\bigr)|\!|\!|v|\!|\!|_{2}^{2}. \end{aligned}$$

(2.13)

The Gagliardo-Nirenberg inequality implies that, for all $m_{1}\in (2,+\infty)$,

$$ \biggl( \int_{-R}^{R}\bigl\vert v(x)\bigr\vert ^{m_{1}}\,\mathrm{d}x \biggr)^{\frac{1}{m_{1}}}\leq C\|v\|_{L^{2}(\Omega)}^{\frac{{m_{1}}+2}{2{m_{1}}}} \|v_{x}\|_{L^{2}(\Omega )}^{\frac{{m_{1}}-2}{2{m_{1}}}}. $$

(2.14)

Consequently, we obtain after using (2.11), (2.13), and (2.14)

$$\begin{aligned}& \int_{-R}^{R}\bigl\vert v(x)\bigr\vert ^{m_{1}}\,\mathrm{d}x \\& \quad \leq C\|v\|_{L^{2}(\Omega)}^{\frac{{m_{1}}+2}{2}}\|v_{x} \|_{L^{2}(\Omega )}^{\frac{{m_{1}}-2}{2}} \\& \quad \leq C \bigl[R^{2}\bigl(e+R^{2}\bigr)|\!|\!|v|\!|\!|_{2}^{2} \bigr]^{\frac{m_{1}+2}{4}} \bigl[R^{2} \bigl(e+R^{2}\bigr)|\!|\!|v|\!|\!|_{2}^{2} \bigr]^{\frac{{m_{1}}-2}{4}} \\& \quad \leq CR^{m_{1}}\bigl(e+R^{2}\bigr)^{\frac{m_{1}}{2}}|\!|\!|v |\!|\!|_{2}^{m_{1}}. \end{aligned}$$

(2.15)

Next, we discuss the case $v\in W_{\mathrm{loc}}^{1,p}(\Omega)$. Since

$$\int_{-R}^{R}\bigl\vert v(x)\bigr\vert ^{p}\,\mathrm{d}x=R \int_{-1}^{1}\bigl\vert v(R\tau)\bigr\vert ^{p}\,\mathrm {d}\tau, $$

we observe there exists $R_{0}\in[\frac{1}{2},1]$ such that

$$ \biggl( \int_{-1}^{1}\bigl\vert v(R_{0}\tau) \bigr\vert ^{p}\,\mathrm{d}\tau \biggr)^{\frac {1}{p}}\leq2 \biggl( \int_{-1}^{1}\bigl\vert v(x)\bigr\vert ^{p}\,\mathrm{d}x \biggr)^{\frac{1}{p}}. $$

(2.16)

By some direct calculations, we get

$$ \int_{R_{0}}^{R}\frac{\mathrm{d}}{\mathrm{d}s} \biggl( \int_{-1}^{1}\bigl\vert v(s\tau)\bigr\vert ^{p}\,\mathrm{d}\tau \biggr)^{\frac{1}{2}}\,\mathrm{d}s= \biggl( \int_{-1}^{1}\bigl\vert v(R\tau)\bigr\vert ^{2}\,\mathrm{d}\tau \biggr)^{\frac{1}{p}}- \biggl( \int_{-1}^{1}\bigl\vert v(R_{0}\tau) \bigr\vert ^{p}\,\mathrm{d}\tau \biggr)^{\frac{1}{p}}. $$

(2.17)

By virtue of

$$\begin{aligned}& \frac{\mathrm{d}}{\mathrm{d}s} \biggl( \int_{-1}^{1}\bigl\vert v(s\tau)\bigr\vert ^{p}\,\mathrm{d}\tau \biggr) \\& \quad = \int_{-1}^{1}p\bigl\vert v(s\tau)\bigr\vert ^{p-1}\biggl\vert \frac{\partial v(s\tau)}{\partial s}\biggr\vert \,\mathrm{d}\tau \\& \quad \leq p \biggl( \int_{-1}^{1}\bigl\vert v(s\tau)\bigr\vert ^{p}\,\mathrm{d}\tau \biggr)^{\frac {p-1}{p}} \biggl( \int_{-1}^{1}\biggl\vert \frac{\partial v(s\tau)}{\partial s}\biggr\vert ^{p}\, \mathrm{d}\tau \biggr)^{\frac{1}{p}} \\& \quad \leq p \biggl( \int_{-1}^{1}\bigl\vert v(s\tau)\bigr\vert ^{p}\,\mathrm{d}\tau \biggr)^{\frac {p-1}{p}} \biggl( \int_{-1}^{1}\bigl\vert v'(s\tau) \tau\bigr\vert ^{p}\,\mathrm{d}\tau \biggr)^{\frac{1}{p}} \\& \quad \leq p \biggl( \int_{-1}^{1}\bigl\vert v(s\tau)\bigr\vert ^{p}\,\mathrm{d}\tau \biggr)^{\frac {p-1}{p}} \biggl( \int_{-1}^{1}\bigl\vert v'(s\tau) \bigr\vert ^{p}\,\mathrm{d}\tau \biggr)^{\frac{1}{p}}, \end{aligned}$$

we have

$$ \frac{\mathrm{d}}{\mathrm{d}s} \biggl( \int_{-1}^{1}\bigl\vert v(s\tau)\bigr\vert ^{p}\,\mathrm{d}\tau \biggr)^{\frac {1}{p}}\leq \biggl( \int_{-1}^{1}\bigl\vert v'(s\tau) \bigr\vert ^{p}\,\mathrm{d}\tau \biggr)^{\frac{1}{p}}. $$

(2.18)

For all $R\geq1$, we get

$$\begin{aligned}& \int_{R_{0}}^{R} \biggl( \int_{-1}^{1}\bigl\vert v'(s\tau) \bigr\vert ^{p}\,\mathrm{d}\tau \biggr)^{\frac{1}{p}}\,\mathrm{d}s \\& \quad = \int_{R_{0}}^{R} \biggl(s \int_{-1}^{1}\bigl\vert v'(s\tau) \bigr\vert ^{p}\,\mathrm{d}\tau \biggr)^{\frac{1}{p}}\biggl( \frac{1}{s}\biggr)^{\frac{1}{p}}\,\mathrm{d}s \\& \quad \leq \biggl( \int_{R_{0}}^{R} \int_{-1}^{1}\bigl\vert v_{x}(x)\bigr\vert ^{p}\,\mathrm{d}x\,\mathrm {d}s \biggr)^{\frac{1}{p}} \biggl( \int_{R_{0}}^{R}\biggl(\frac{1}{s} \biggr)^{\frac{1}{p-1}}\,\mathrm{d}s \biggr)^{\frac{p-1}{p}} \\& \quad \leq \biggl( \int_{R_{0}}^{R} \int_{-1}^{1}\bigl\vert v_{x}(x)\bigr\vert ^{p}\,\mathrm{d}x\,\mathrm {d}s \biggr)^{\frac{1}{p}} \biggl[ \biggl( \int_{R_{0}}^{R}\biggl(\frac{1}{s} \biggr)^{\frac{3-p}{p-1}}\,\mathrm {d}s\biggr)^{{\frac{1}{2}}} \biggl( \int_{R_{0}}^{R}\frac{1}{s}\,\mathrm{d}s \biggr)^{{\frac{1}{2}}} \biggr]^{\frac{p-1}{p}} \\& \quad \leq \biggl( \int_{R_{0}}^{R} \int_{-1}^{1}\bigl\vert v_{x}(x)\bigr\vert ^{p}\,\mathrm{d}x\,\mathrm {d}s \biggr)^{\frac{1}{p}} \biggl( \ln\frac{R}{R_{0}} \biggr)^{\frac{p-1}{2p}}R^{\frac{p-2}{p}} \\& \quad \leq \biggl( \int_{R_{0}}^{R} \int_{-1}^{1}\bigl\vert v_{x}(x)\bigr\vert ^{p}\,\mathrm{d}x\,\mathrm {d}s \biggr)^{\frac{1}{p}} \bigl( \ln(2R) \bigr)^{\frac{p-1}{2p}}R^{\frac{p-2}{p}} \\& \quad \leq R^{\frac{p-1}{p}} \bigl(e+R^{2} \bigr)^{\frac{1}{2}} \biggl( \int _{R_{0}}^{R} \int_{-1}^{1}\bigl\vert v_{x}(x)\bigr\vert ^{p}\,\mathrm{d}x\,\mathrm{d}s \biggr)^{\frac{1}{p}}. \end{aligned}$$

(2.19)

We denote

$$ |\!|\!|v|\!|\!|_{p}\triangleq \biggl( \int_{-1}^{1}\bigl\vert v(x)\bigr\vert ^{p}\,\mathrm{d}x \biggr)^{\frac{1}{p}} + \biggl( \int_{-R}^{R}\bigl\vert v_{x}(x)\bigr\vert ^{p}\,\mathrm{d}x \biggr)^{\frac{1}{p}}. $$

(2.20)

Combining (2.17)-(2.20), we get

$$ \int_{-1}^{1}\bigl\vert v(R\tau)\bigr\vert ^{p}\,\mathrm{d}\tau \leq CR^{p-1}\bigl(e+R^{2} \bigr)^{\frac{p}{2}}|\!|\!|v|\!|\!|_{p}^{p}. $$

(2.21)

Multiplying (2.21) by R, we obtain

$$ \int_{-R}^{R}\bigl\vert v(x)\bigr\vert ^{p}\,\mathrm{d}x \leq CR^{p}\bigl(e+R^{2} \bigr)^{\frac{p}{2}}|\!|\!|v|\!|\!|_{p}^{p}. $$

(2.22)

The Gagliardo-Nirenberg inequality implies that, for all $m_{2}\in (p,+\infty)$,

$$\begin{aligned}& \int_{-R}^{R}\bigl\vert v(x)\bigr\vert ^{m_{2}}\,\mathrm{d}x \\& \quad \leq C\|v\|_{L^{p}(\Omega)}^{\frac{p{m_{2}}-{m_{2}}+p}{p}}\|v_{x} \|_{L^{p}(\Omega )}^{\frac{m_{2}-p}{p}} \\& \quad \leq C\bigl(R^{p}\bigl(e+R^{2}\bigr)^{\frac{p}{2}} |\!|\!|v|\!|\!|_{p}^{p}\bigr)^{\frac{m_{2}}{p}} \\& \quad \leq CR^{m_{2}}\bigl(e+R^{2}\bigr)^{\frac{m_{2}}{2}}|\!|\!|v |\!|\!|_{p}^{m_{2}}. \end{aligned}$$

(2.23)

By a simple integration by parts, (2.13), (2.15), (2.22), and (2.23), we obtain for all $R\geq1$, either $s=2$ or $s=p$, $m\in[s,\infty)$, and, for any $v\in W_{\mathrm{loc}}^{1,s}(\Omega)$,

$$\begin{aligned}& \int_{-R}^{R}\bigl\vert v(x)\bigr\vert ^{m}\bigl(e+x^{2}\bigr)^{-\vartheta}\,\mathrm{d}x \\& \quad \leq \int_{-R}^{R}\bigl\vert v(x)\bigr\vert ^{m}\,\mathrm{d}x\cdot\bigl(e+R^{2}\bigr)^{-\vartheta}+ \int_{-R}^{R}\biggl( \int_{0}^{x}\bigl\vert v(\tau)\bigr\vert ^{m}\,\mathrm{d}\tau\biggr)\frac {2\vartheta x}{(e+x^{2})^{\vartheta+1}}\,\mathrm{d}x \\& \quad \leq |\!|\!|v|\!|\!|_{s}^{m}\frac{(R^{2})^{\frac{m}{2}}(e+R^{2})^{\frac {m}{2}}}{(e+R^{2})^{\vartheta}}+ C |\!|\!|v|\!|\!|_{s}^{m} \int_{-R}^{R} \frac{2x\cdot x^{m-1}(e+x^{2})^{\frac {m}{2}}}{(e+x^{2})^{\vartheta}} \,\mathrm{d}x \\& \quad \leq |\!|\!|v|\!|\!|_{s}^{m}\bigl(e+R^{2} \bigr)^{m-\vartheta}+ C|\!|\!|v|\!|\!|_{s}^{m} \int_{-R}^{R} \frac{2x(e+x^{2})^{m-\frac {1}{2}}}{(e+x^{2})^{\vartheta}} \,\mathrm{d}x \\& \quad \leq |\!|\!|v|\!|\!|_{s}^{m}+ C|\!|\!|v|\!|\!|_{s}^{m} \int_{-R}^{R} \bigl(e+x^{2} \bigr)^{m-\frac{1}{2}-\vartheta} \,\mathrm {d}\bigl(e+x^{2}\bigr) \\& \quad \leq C|\!|\!|v|\!|\!|_{s}^{m}\biggl(1+ \int_{e+R^{2}}^{e+R^{2}} y^{m-\frac{1}{2}-\vartheta} \, \mathrm{d}y\biggr) \\& \quad = C|\!|\!|v|\!|\!|_{s}^{m}, \end{aligned}$$

if $\vartheta>m+\frac{1}{2}$. □

Lemma 2.4

Let Φ and $\zeta_{0}$ be as in (1.5) and Ω as in Lemma  2.3. For $\gamma> 1$, assume that $\rho\in L^{1}(\Omega)\cap L^{\gamma}(\Omega)$ is a non-negative function such that

$$ \int_{\Omega_{N_{1}}}\rho\,\mathrm{d}x\geq Q_{1}, \qquad \int_{\Omega}\rho^{\gamma}\, \mathrm{d}x\leq Q_{2}, $$

(2.24)

for positive constants $Q_{1}$, $Q_{2}$, and $N_{1}\geq1$ with $\Omega_{N_{1}}\subset\Omega$. Then there is a positive constant C depending only on $Q_{1}$, $Q_{2}$, $N_{1}$, γ, and $\zeta_{0}$ such that

$$ \bigl\Vert v\Phi^{-1}\bigr\Vert _{L^{s}(\Omega)}\leq C\bigl\Vert \rho^{1/2} v\bigr\Vert _{L^{2}(\Omega)}+C\Vert v_{x} \Vert _{L^{s}(\Omega)}, $$

(2.25)

for every $v\in W_{\mathrm{loc}}^{1,2}(\Omega)\cap W_{\mathrm{loc}}^{1,p}(\Omega)$. Moreover, for $\varepsilon>0$ and $\zeta>0$ there is a positive constant C depending only on ε, ζ, $Q_{1}$, $Q_{2}$, $N_{1}$, γ, and $\zeta_{0}$ such that

$$ \bigl\Vert v\Phi^{-\zeta}\bigr\Vert _{L^{(s+\varepsilon)/\tilde{\zeta}}(\Omega)}\leq C\bigl\Vert \rho^{1/2} v\bigr\Vert _{L^{2}(\Omega)}+C\Vert v_{x} \Vert _{L^{s}(\Omega)}, $$

(2.26)

with $\tilde{\zeta}=\min\{1,\zeta\}$.

Proof

It follows from (2.24) and similar arguments to Lemma 3.2 of [20] that there exists a positive constant C depending only on $Q_{1}$, $Q_{2}$, $N_{1}$, and γ, such that

$$ \|v\|_{L^{s}(\Omega_{N_{1}})}^{2}\leq C \int_{\Omega_{N_{1}}}\rho v^{2}\,\mathrm {d}x+C\| v_{x}\|_{L^{s}(\Omega_{N_{1}})}^{2}. $$

(2.27)

There exists a positive constant $\zeta_{0}$ such that

$$\zeta\frac{s+\varepsilon}{\tilde{\zeta}}(1+\zeta_{0})>\frac{s+\varepsilon }{\tilde{\zeta}}+ \frac{1}{2}, $$

which together with (2.4) and (2.27) gives (2.25) and (2.26). The proof of Lemma 2.4 is finished. □

3 A priori estimates

For $R>4R_{0}\geq4$, assume that the smooth $(\rho_{0},u_{0})$ satisfies, in addition to (2.1),

$$ 1/2\leq \int_{\Omega_{R_{0}}}\rho_{0}(x)\,\mathrm{d}x\leq \int_{\Omega _{R}}\rho_{0}(x)\,\mathrm{d}x \leq3/2. $$

(3.1)

Lemma 2.1 thus shows that there exists some $T_{R}>0$ such that the initial-boundary value problem (2.2) has a unique classical solution $(\rho,u)$ on $\Omega_{R}\times[0,T_{R}]$ satisfying (2.3).

For Φ, $\zeta_{0}$, and q as in Theorem 1.2, the main aim of this section is to derive the key a priori estimate on ψ defined by

$$\begin{aligned} \psi(t) \triangleq& 1+\bigl\Vert \rho^{1/2} u\bigr\Vert _{L^{2}(\Omega_{R})}+\|u\|_{L^{2}(\Omega_{R})}+\| u_{x}\|_{L^{2}(\Omega_{R})}+ \|u_{x}\|_{L^{P}(\Omega_{R})} \\ &{} +\|\Phi\rho\|_{L^{1}(\Omega_{R})\cap H^{1}(\Omega_{R})\cap W^{1,q}(\Omega_{R})}. \end{aligned}$$

(3.2)

Proposition 3.1

Assume that $(\rho_{0},u_{0})$ satisfies (2.1) and (3.1). Let $(\rho,u)$ be the solution to the initial-boundary value problem (2.2) on $\Omega_{R}\times(0,T _{R} ]$ obtained by Lemma  2.1. Then there exist positive constants $T_{0}$ and M both depending only on γ, q, $\mu_{0}$, $\zeta _{0}$, $N_{0}$, and $K_{0}$ such that

$$\begin{aligned}& \sup_{0\leq t\leq T_{0}}\psi(t)+ \int_{0}^{T_{0}} \bigl(\| u_{xx}\| _{L^{q}(\Omega_{R})}^{(q+1)/q} +t\| u_{xx}\|_{L^{q}(\Omega_{R})}^{2}+ \|u_{xx}\|_{L^{2}(\Omega_{R})}^{2} \bigr)\, \mathrm{d}t \\& \quad {}+ \sup_{0\leq t\leq T_{0}}t\|u_{xx}\|_{L^{2}(\Omega_{R})}^{2} \leq M, \end{aligned}$$

(3.3)

where

$$\begin{aligned} K_{0} \triangleq& \bigl\Vert \rho_{0}^{1/2} u_{0}\bigr\Vert _{L^{2}(\Omega_{R})}+\| u_{0x}\| _{L^{2}(\Omega_{R})}+\| u_{0x}\|_{L^{p}(\Omega_{R})}+\|u_{0} \|_{L^{2}(\Omega_{R})} \\ &{}+\|\Phi\rho_{0}\|_{L^{1}(\Omega_{R})\cap H^{1}(\Omega_{R})\cap W^{1,q}(\Omega_{R})}. \end{aligned}$$

The proof of Proposition 3.1 will be postponed to the end of this section; we begin with the following standard energy estimate for $(\rho,u)$.

Lemma 3.2

Let $(\rho,u)$ be a smooth solution to the initial-boundary value problem (2.2). Then there exist a $T^{*}=T^{*}(R _{0},K _{0})>0$ and a positive constant $\beta=\beta(\gamma,q)>1$ such that, for all $t\in(0,T^{*}]$,

$$\begin{aligned}& {\sup_{0\leq s\leq t}} \biggl( \int_{-R}^{R}|u|^{2}\,\mathrm{d}x+ \int _{-R}^{R}|u_{x}|^{2}\, \mathrm{d}x+ \int_{-R}^{R}|u_{x}|^{p}\, \mathrm{d}x \biggr)+ \int_{0}^{t} \int_{-R}^{R} \rho|u_{t}|^{2}\, \mathrm{d}x\,\mathrm{d}s \\& \quad \leq C+C \int_{0}^{t} \psi^{\beta}\,\mathrm{d}s, \end{aligned}$$

(3.4)

where (as in the following) C denotes a generic positive constant depending only on γ, q, $\zeta_{0}$, $\mu_{0}$, $R_{0}$ and $K_{0}$, under the conditions of Proposition  3.1.

Proof

First, applying standard energy estimate to (2.2) gives

$$ {\sup_{0\leq s\leq t}} \bigl(\|\sqrt{\rho}u\|_{L^{2}(\Omega_{R})}^{2}+ \|\rho\|_{L^{\gamma}(\Omega _{R})}^{\gamma}\bigr)+ \int_{0}^{t} \biggl( \int_{-R}^{R}|u_{x}|^{p}\, \mathrm{d}x+ \int_{-R}^{R}|u|^{2}\, \mathrm{d}x \biggr)\,\mathrm{d}s \leq C. $$

(3.5)

Next, for $R> 1$ and $\tilde{\xi}_{R}\in C_{0}^{\infty}(\Omega_{R})$ such that

$$ \begin{aligned} &0\leq\tilde{\xi}_{R}\leq1,\qquad \tilde{ \xi}_{R}(x)=1,\qquad |x|\leq\frac{R}{2}, \\ &|\tilde{\xi}_{Rx}|\leq\frac{C}{R}, \qquad |\tilde{ \xi}_{Rxx}|\leq \frac{C}{R^{2}}, \end{aligned} $$

(3.6)

it follows from (3.5) and (3.1) that

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} \int_{-R}^{R} \rho\tilde{\xi}_{2R_{0}}\, \mathrm{d}x =& \int_{-R}^{R} \rho u \tilde{ \xi}_{2R_{0}x}\,\mathrm{d}x \\ \geq& -C (R_{0} )^{-1} \biggl( \int_{-R}^{R} \rho\,\mathrm {d}x \biggr)^{1/2}\biggl( \int_{-R}^{R} \rho|u|^{2} \,\mathrm{d}x \biggr)^{1/2} \\ \geq& -\tilde{B}, \end{aligned}$$

(3.7)

where in the last inequality we have used

$$\int_{-R}^{R} \rho\,\mathrm{d}x= \int_{-R}^{R} \rho_{0} \,\mathrm{d}x, $$

due to (2.2)₁ and (2.2)₅. Integrating (3.7) gives

$$\begin{aligned} {\inf_{0\leq t\leq T^{*}}} \int_{-R}^{R} \rho\,\mathrm{d}x \geq& { \inf _{0\leq t\leq T^{*}}} \int_{-R}^{R} \rho\tilde{\xi }_{2R_{0}} \,\mathrm{d}x \\ \geq& \int_{-R}^{R} \rho_{0}\tilde{ \xi}_{2R_{0}}\,\mathrm{d}x-\tilde {B}T^{*} \\ \geq& 1/4, \end{aligned}$$

(3.8)

where $T^{*}\triangleq\min\{1,(4\tilde{B})^{-1}\}$.

From now on, we will always assume that $t\leq T^{*}$. The combination of (3.8), (3.5), and (2.26) shows, for $\varepsilon >0$, $\zeta>0$, $\forall v \in W_{0}^{1,2}(\Omega_{R})\cap W_{0}^{1,p}(\Omega_{R})$ satisfies

$$ \bigl\Vert v\Phi^{-\zeta}\bigr\Vert _{L^{(s+\varepsilon)/\tilde{\zeta}}(\Omega_{R})}^{2} \leq C(\varepsilon,\zeta) \int_{-R}^{R} \rho|v|^{2} \,\mathrm{d}x+ C(\varepsilon,\zeta)\|v_{x}\|_{L^{s}(\Omega_{R})}^{2}, $$

(3.9)

with $\tilde{\zeta}=\min\{1,\zeta\}$. In particular, we have

$$ \bigl\Vert \rho^{\zeta}v\bigr\Vert _{L^{(s+\varepsilon)/\tilde{\zeta}}(\Omega_{R})}+\bigl\Vert v \Phi ^{-\zeta}\bigr\Vert _{L^{(s+\varepsilon)/\tilde{\zeta}}(\Omega_{R})} \leq C(\varepsilon,\zeta) \psi^{1+\zeta}. $$

(3.10)

Multiplying (2.2)₂ by $u _{t}$, and integrating it over $(-R,R)$ on x and integrating over $(0,t)$ on the time variable

$$\begin{aligned}& \int_{0}^{t} \int_{-R}^{R} \rho|u_{t}|^{2}\, \mathrm{d}x\,\mathrm{d}s+ \int _{0}^{t} \int_{-R}^{R} \bigl[\bigl((u_{x})^{2}+ \mu_{0}\bigr)^{\frac{p-2}{2}}u_{x}\bigr]u_{xt}\, \mathrm {d}x\,\mathrm{d}s - \int_{0}^{t} \int_{-R}^{R} fu_{t}\,\mathrm{d}x\, \mathrm{d}s \\& \quad = - \int_{-R}^{R} \pi u_{x}(0)\, \mathrm{d}x+ \int_{-R}^{R} \pi u_{x}(t)\, \mathrm{d}x+ \int_{0}^{t} \int_{-R}^{R}\bigl|(-\rho uu_{x})u_{t}- \pi_{t}u_{x}\bigr|\,\mathrm {d}x\,\mathrm{d}s. \end{aligned}$$

(3.11)

We first compute the second term of (3.11), and we get

$$\begin{aligned}& \int_{-R}^{R} \bigl(u_{x}^{2}+ \mu_{0}\bigr)^{\frac{p-2}{2}}u_{x}u_{xt}\, \mathrm{d}x \\& \quad = \frac{1}{2} \int_{-R}^{R} \bigl(u_{x}^{2}+ \mu_{0}\bigr)^{\frac{p-2}{2}}(u_{x})_{t}^{2} \, \mathrm{d}x \\& \quad = \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \int_{-R}^{R} \biggl( \int_{0}^{u_{x}^{2}}(s+\mu_{0})^{\frac{p-2}{2}} \,\mathrm{d}s\biggr)\,\mathrm{d}x \end{aligned}$$

(3.12)

and

$$\begin{aligned}& \int_{0}^{u_{x}^{2}}(s+\mu_{0})^{\frac{p-2}{2}} \,\mathrm{d}s \\& \quad \geq \int_{0}^{u_{x}^{2}}s^{\frac{p-2}{2}}\,\mathrm{d}s \\& \quad = \frac{2}{p}|u_{x}|^{p}. \end{aligned}$$

(3.13)

Substituting (3.12), and (3.13) into (3.11), and by using (1.3), we obtain

$$\begin{aligned}& \int_{0}^{t}\bigl\Vert \sqrt{ \rho}u_{t}(t)\bigr\Vert _{L^{2}(\Omega_{R})}^{2}\,\mathrm{d}s+ \frac {1}{p} \int_{-R}^{R}\bigl\vert u_{x}(t)\bigr\vert ^{p}\,\mathrm{d}x +\frac{1}{2} \int _{-R}^{R}\bigl\vert u(t)\bigr\vert ^{2}\,\mathrm{d}x \\& \quad \leq \frac{2}{p}\bigl\Vert u_{x}(0)\bigr\Vert _{L^{p}(\Omega_{R})}^{p}+\bigl\Vert \pi(\rho_{0})\bigr\Vert _{L^{2}(\Omega_{R})}^{2}+ \int_{-R}^{R} \bigl\vert \pi u_{x}(t) \bigr\vert \,\mathrm{d}x \\& \qquad {}+ \int_{0}^{t} \int_{-R}^{R}\bigl\vert (-\rho uu_{x})u_{t}-\pi_{t}u_{x}\bigr\vert \,\mathrm{d}x\, \mathrm{d}s \\& \quad \leq C+ \int_{-R}^{R}\bigl\vert \pi u_{x}(t) \bigr\vert \,\mathrm{d}x+ \int_{0}^{t} \int _{-R}^{R}\bigl\vert (-\rho uu_{x})u_{t}-\pi_{t}u_{x}\bigr\vert \,\mathrm{d}x\,\mathrm{d}s. \end{aligned}$$

Using Young’s inequality, we obtain

$$\begin{aligned} \begin{aligned}[b] &\int_{0}^{t}\bigl\Vert \sqrt{ \rho}u_{t}(t)\bigr\Vert _{L^{2}(\Omega_{R})}^{2}\,\mathrm{d}s+ \int _{-R}^{R}\bigl\vert u_{x}(t) \bigr\vert ^{p}\,\mathrm{d}x+ \int_{-R}^{R}\bigl\vert u(t)\bigr\vert ^{2}\,\mathrm{d}x \\ &\quad \leq C+C \int_{0}^{t}\Vert \sqrt{\rho}uu_{x} \Vert _{L^{2}(\Omega_{R})}^{2}\,\mathrm {d}s+C \int_{0}^{t} \int_{-R}^{R}\vert \pi_{x}uu_{x} \vert \,\mathrm{d}x\,\mathrm{d}s+ \int _{0}^{t} \int_{-R}^{R}\bigl\vert \pi u_{x}^{2} \bigr\vert \,\mathrm{d}x\,\mathrm{d}s \\ &\qquad {}+C\Vert \pi \Vert _{L^{\frac{p}{p-1}}(\Omega_{R})}^{{\frac{p}{p-1}}}. \end{aligned} \end{aligned}$$

(3.14)

First, the Gagliardo-Nirenberg inequality implies that, for all $k\in (p,+\infty)$,

$$ \|u_{x}\|_{L^{k}(\Omega_{R})}\leq C\|u_{x} \|_{L^{p}(\Omega_{R})}^{\theta}\|u_{xx}\| _{L^{2}(\Omega_{R})}^{1-\theta}, $$

(3.15)

where $\theta=\frac{2k-2p}{2k+kp}$.

Next, we estimate each term on the right-hand side of (3.14) as follows: We deduce from (3.10), (3.2), and Hölder’s inequality that

$$\begin{aligned} \Vert \sqrt{\rho} uu_{x}\Vert _{L^{2}(\Omega_{R})} \leq& \Vert u_{x}\Vert _{L^{p}(\Omega_{R})}\Vert \rho\Phi \Vert _{L^{\infty}(\Omega_{R})}^{\frac {1}{2}}\bigl\Vert \Phi^{-\frac{1}{4}}u\bigr\Vert _{L^{2p}(\Omega_{R})}\bigl\Vert \Phi^{-\frac {1}{4}}\bigr\Vert _{L^{\frac{2p}{p-3}}(\Omega_{R})} \\ \leq& C\psi^{\beta}, \end{aligned}$$

(3.16)

where (as in the following) we use $\beta= \beta(\gamma,q) > 1$ to denote a generic constant depending only on γ and q, which may be different from line to line.

By (3.9),we have

$$\begin{aligned} \int_{-R}^{R}|\pi_{x}uu_{x}| \,\mathrm{d}x \leq& C\Vert \rho \Vert _{L^{\infty}(\Omega_{R})}^{\gamma-1}\Vert \Phi\rho_{x}\Vert _{L^{2}(\Omega_{R})}\Vert u_{x}\Vert _{L^{p}(\Omega_{R})}\bigl\Vert \Phi^{-\frac{1}{2}}u\bigr\Vert _{L^{2p}(\Omega_{R})} \bigl\Vert \Phi^{-\frac{1}{2}}\bigr\Vert _{L^{\frac{2p}{p-3}}(\Omega _{R})} \\ \leq& C\psi^{\beta}. \end{aligned}$$

(3.17)

Using (2.2)₂, we get

$$\bigl[\bigl((u_{x})^{2}+\mu_{0} \bigr)^{\frac{p-2}{2}}u_{x}\bigr]_{x}=\rho u_{t}+ \rho uu_{x}+\pi_{x}+f. $$

By virtue of

$$\mu_{0}^{\frac{p-2}{2}}|u_{xx}|\leq\bigl[ \bigl((u_{x})^{2}+\mu_{0}\bigr)^{\frac{p-2}{2}}u_{x} \bigr]_{x}, $$

we have

$$|u_{xx}|\leq C|\rho u_{t}+\rho uu_{x}+ \pi_{x}+f|. $$

Taking the above inequality by $L _{2}$ norm and $L _{p}$ norm, we obtain

$$\begin{aligned}& \Vert u_{xx}\Vert _{L^{2}(\Omega_{R})} \\& \quad \leq C\bigl\Vert \bigl(\bigl(u_{x}^{2}+ \mu_{0}\bigr)^{\frac{p-2}{2}}u_{x}\bigr)_{x} \bigr\Vert _{L^{2}(\Omega_{R})} \\& \quad \leq C\Vert \rho u_{t}\Vert _{L^{2}(\Omega_{R})}+C\Vert \rho uu_{x}\Vert _{L^{2}(\Omega_{R})}+C\Vert \pi_{x}\Vert _{L^{2}(\Omega_{R})}+\Vert f\Vert _{L^{2}(\Omega_{R})} \\& \quad \leq C\psi^{\frac{1}{2}}\Vert \sqrt{\rho} u_{t}\Vert _{L^{2}(\Omega_{R})} \\& \qquad {}+C\Vert u_{x}\Vert _{L^{p}(\Omega_{R})}\Vert \rho \Phi \Vert _{L^{q}(\Omega_{R})} \bigl\Vert \Phi^{-{\frac{1}{2}}}u\bigr\Vert _{L^{\frac{2pq}{\varepsilon_{0}}}(\Omega_{R})} \bigl\Vert \Phi^{-{\frac{1}{2}}}\bigr\Vert _{L^{\frac{2pq}{pq-2p-2q-\varepsilon_{0}}}(\Omega _{R})} +C\psi^{\beta} \\& \quad \leq C\psi^{\frac{1}{2}}\Vert \sqrt{\rho} u_{t}\Vert _{L^{2}(\Omega_{R})}+C\psi ^{\beta}, \end{aligned}$$

(3.18)

where $0<\varepsilon_{0}<\min\{1,pq-2p-2q\}$, and

$$ \Vert u_{xx}\Vert _{L^{q}(\Omega_{R})}\leq C\bigl(\Vert \rho u_{t}\Vert _{L^{q}(\Omega_{R})}+\Vert \rho uu_{x}\Vert _{L^{q}(\Omega_{R})}+\Vert \pi_{x}\Vert _{L^{q}(\Omega_{R})}+\Vert f \Vert _{L^{q}(\Omega_{R})}\bigr). $$

(3.19)

By (3.15), we have

$$\begin{aligned} \int_{-R}^{R}\bigl\vert \pi u_{x}^{2} \bigr\vert \,\mathrm{d}x \leq& \Vert \rho \Vert _{L^{\infty}(\Omega_{R})}^{\gamma-1} \Vert \Phi\rho \Vert _{L^{2}(\Omega _{R})}\Vert u_{x}\Vert _{L^{p}(\Omega_{R})}\Vert u_{x}\Vert _{L^{\frac{p^{2}}{p-2}}(\Omega_{R})}\bigl\Vert \Phi ^{-1}\bigr\Vert _{L^{\frac{2p^{2}}{(p-2)^{2}}}(\Omega_{R})} \\ \leq& C\psi^{\beta} \Vert u_{xx}\Vert _{L^{2}(\Omega_{R})}^{\theta} \\ \leq& C(\varepsilon)\psi^{\beta}+C\varepsilon\psi^{-1}\Vert u_{xx}\Vert _{L^{2}(\Omega_{R})}^{2}, \end{aligned}$$

(3.20)

where $\theta=\frac{4}{p^{2}+2p}$. We use (3.10) to get

$$\begin{aligned}& \int_{-R}^{R} \vert \pi \vert ^{\frac{p}{p-1}}\, \mathrm{d}x \\& \quad = \int_{-R}^{R} \bigl\vert \pi(0)\bigr\vert ^{\frac{p}{p-1}}\,\mathrm{d}x+ \int_{0}^{t} \frac {\partial}{\partial s}\biggl( \int_{-R}^{R}\bigl(\pi(s)\bigr)^{\frac{p}{p-1}}\, \mathrm {d}x\biggr)\,\mathrm{d}s \\& \quad = \int_{-R}^{R} \bigl\vert \pi(0)\bigr\vert ^{\frac{p}{p-1}}\,\mathrm{d}x+\frac {p}{p-1} \int_{0}^{t} \int_{-R}^{R}\gamma(\rho)^{\gamma-1} \pi^{\frac {1}{p-1}}(-\rho_{x}u-\rho u_{x})\,\mathrm{d}x\, \mathrm{d}s \\& \quad \leq C+C \int_{0}^{t}\Vert \rho \Vert _{L^{\infty}(\Omega_{R})}^{\gamma-1}\Vert \pi \Vert _{L^{\infty}(\Omega_{R})}^{\frac{1}{p-1}} \int_{-R}^{R}\bigl(\bigl\vert \rho_{x}\Phi\Phi ^{-1}u\bigr\vert +\bigl\vert \rho\Phi \Phi^{-1}u_{x}\bigr\vert \bigr)\,\mathrm{d}x\,\mathrm{d}s \\& \quad \leq C+C \int_{0}^{t}\Vert \rho \Vert _{L^{\infty}(\Omega_{R})}^{\gamma-1}\Vert \pi \Vert _{L^{\infty}(\Omega_{R})}^{\frac{1}{p-1}} \bigl(\Vert \rho_{x}\Phi \Vert _{L^{2}(\Omega_{R})}\bigl\Vert \Phi^{-\frac{1}{2}}u\bigr\Vert _{L^{2p}(\Omega_{R})} \bigl\Vert \Phi^{-\frac{1}{2}} \bigr\Vert _{L^{\frac{2p}{p-1}}(\Omega_{R})} \\& \qquad {} +\Vert \rho\Phi \Vert _{L^{2}(\Omega_{R})}\bigl\Vert \Phi^{-1}\bigr\Vert _{L^{\frac{2p}{p-2}}(\Omega _{R})}\Vert u_{x}\Vert _{L^{p}(\Omega_{R})}\bigr)\,\mathrm{d}s \\& \quad \leq C+ \int_{0}^{t}\psi^{\beta}\,\mathrm{d}s. \end{aligned}$$

(3.21)

By virtue of

$$\begin{aligned}& \int_{0}^{u_{x}^{2}}(s+\mu_{0})^{\frac{p-2}{2}} \,\mathrm{d}s \\& \quad \geq \int_{0}^{u_{x}^{2}}\mu_{0}^{\frac{p-2}{2}}\, \mathrm{d}s \\& \quad \geq \frac{1}{2}\mu_{0}^{\frac{p-2}{2}}|u_{x}|^{2}, \end{aligned}$$

(3.22)

and the arguments as in [8], we have

$$ {\sup_{0\leq s\leq t}}\| u_{x}\|_{L^{2}(\Omega_{R})}^{2} \leq C+C \int_{0}^{t} \psi^{\beta}\,\mathrm{d}s. $$

(3.23)

Putting (3.16), (3.17), (3.20), (3.21), and (3.23) into (3.14) and choosing ε suitably small yield

$$ \int_{-R}^{R}|u|^{2}\,\mathrm{d}x+ \int_{-R}^{R}|u_{x}|^{2}\, \mathrm{d}x+ \int _{-R}^{R}|u_{x}|^{p}\, \mathrm{d}x+ \int_{0}^{t} \int_{-R}^{R} \rho|u_{t}|^{2}\, \mathrm {d}x\,\mathrm{d}s \leq C+ \int_{0}^{t} \psi^{\beta}\,\mathrm{d}s. $$

The proof of Lemma 3.2 is finished. □

Lemma 3.3

Let $(\rho,u)$ and $T^{*}$ be as in Lemma  3.2. Then, for all $t \in (0,T^{*} ]$,

$$ \sup_{0\leq s\leq t}s \int_{-R}^{R} \rho|u_{t}|^{2}\, \mathrm{d}x + \int_{0}^{t} s \int_{-R}^{R} \bigl(|u_{xt}|^{2}+|u_{t}|^{2} \bigr) \,\mathrm{d}x\, \mathrm{d}s \leq C\exp \biggl\{ C \int_{0}^{t} \psi^{\beta}\,\mathrm{d}s \biggr\} . $$

(3.24)

Proof

We differentiate equation (2.2)₂ with respect to t, and multiply it by $u_{t}$, by using (1.3), and integrating it over $(-R,R)$ with respect to x we obtain

$$\begin{aligned}& \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \int_{-R}^{R} \rho|u_{t}|^{2}\, \mathrm{d}x + \int_{-R}^{R} |u_{xt}|^{2}\, \mathrm{d}x+ \int_{-R}^{R} |u_{t}|^{2}\, \mathrm {d}x \\& \quad \leq C \int_{-R}^{R} |\pi_{x}||u||u_{xt}| \,\mathrm{d}x +C \int_{-R}^{R} |\pi||u_{x}||u_{xt}| \,\mathrm{d}x +C \int_{-R}^{R} \rho|u||u_{t}||u_{xt}| \,\mathrm{d}x \\& \qquad {}+C \int_{-R}^{R} \rho|u||u_{x}|^{2}|u_{t}| \,\mathrm{d}x +C \int_{-R}^{R} \rho|u|^{2}|u_{xx}||u_{t}| \,\mathrm{d}x +C \int_{-R}^{R} \rho|u|^{2}|u_{x}||u_{xt}| \,\mathrm{d}x \\& \qquad {}+C \int_{-R}^{R} \rho|u_{x}||u_{t}|^{2} \,\mathrm{d}x \\& \quad = C\sum_{i=1}^{7}I_{j}. \end{aligned}$$

(3.25)

We estimate each term $I_{j}$. We deduce from the Gagliardo-Nirenberg inequality and (3.10) that

$$\begin{aligned} I_{1} =& \int_{-R}^{R}|\pi_{x}||u||u_{xt}| \,\mathrm{d}x \\ \leq& C\Vert \rho \Vert _{L^{\infty}(\Omega_{R})}^{\gamma-1}\Vert \Phi \rho_{x}\Vert _{L^{q}(\Omega_{R})}\Vert u_{xt}\Vert _{L^{2}(\Omega_{R})}\bigl\Vert \Phi^{-\frac{1}{2}}u\bigr\Vert _{L^{2q}(\Omega_{R})} \bigl\Vert \Phi^{-\frac{1}{2}}\bigr\Vert _{L^{\frac{2q}{q-3}}(\Omega_{R})} \\ \leq& \varepsilon \Vert u_{xt}\Vert _{L^{2}(\Omega_{R})}^{2}+C( \varepsilon)\psi^{\beta} \end{aligned}$$

(3.26)

and that

$$\begin{aligned} I_{2} =& \int_{-R}^{R} |\pi||u_{x}||u_{xt}| \,\mathrm{d}x \\ \leq& C\Vert \rho \Vert _{L^{\infty}(\Omega_{R})}^{\gamma-1}\Vert \Phi\rho \Vert _{L^{q}(\Omega _{R})}\Vert u_{xt}\Vert _{L^{2}(\Omega_{R})}\Vert u_{x}\Vert _{L^{\frac{q^{2}}{q-2}}(\Omega_{R})}\bigl\Vert \Phi^{-1}\bigr\Vert _{L^{\frac{2q^{2}}{(q-2)^{2}}}(\Omega_{R})} \\ \leq& C\psi^{\beta}\psi^{1-\theta} \Vert u_{xx}\Vert _{L^{2}(\Omega_{R})}^{\theta} \Vert u_{xt}\Vert _{L^{2}(\Omega_{R})} \\ \leq& C\psi^{\beta}\bigl(\psi+ \Vert u_{xx}\Vert _{L^{2}(\Omega_{R})}\bigr)\Vert u_{xt}\Vert _{L^{2}(\Omega_{R})} \\ \leq& \varepsilon \Vert u_{xt}\Vert _{L^{2}(\Omega_{R})}^{2}+C( \varepsilon)\psi^{\beta} \Vert u_{xx}\Vert _{L^{2}(\Omega_{R})}^{2} +C\psi^{\beta}, \end{aligned}$$

(3.27)

where $\theta=\frac{2q^{2}-2qp+4p}{q^{2}(2+p)}$.

Using the Gagliardo-Nirenberg inequality and (3.9), we have

$$\begin{aligned} I_{3} =& \int_{-R}^{R} \rho|u||u_{t}||u_{xt}| \,\mathrm{d}x \\ \leq& C\bigl\Vert \rho^{\frac{1}{2}}u\bigr\Vert _{L^{2p}(\Omega_{R})}\bigl\Vert \rho^{\frac {1}{2}}u_{t}\bigr\Vert _{L^{2}(\Omega_{R})}^{\frac{1}{2}} \bigl\Vert \rho^{\frac{1}{2}}u_{t}\bigr\Vert _{L^{\frac{4p}{p-2}}(\Omega_{R})}^{\frac{1}{2}} \Vert u_{xt}\Vert _{L^{2}(\Omega_{R})} \\ \leq& C\bigl\Vert \rho^{\frac{1}{2}}u_{t}\bigr\Vert _{L^{2}(\Omega_{R})}^{\frac{1}{2}}\Vert u_{xt}\Vert _{L^{2}(\Omega_{R})} \psi^{\beta}\bigl(\bigl\Vert \rho^{\frac{1}{2}}u_{t}\bigr\Vert _{L^{2}(\Omega_{R})}+\Vert u_{xt}\Vert _{L^{2}(\Omega_{R})} \bigr)^{\frac{1}{2}} \\ \leq& C\bigl\Vert \rho^{\frac{1}{2}}u_{t}\bigr\Vert _{L^{2}(\Omega_{R})}^{\frac{1}{2}}\Vert u_{xt}\Vert _{L^{2}(\Omega_{R})} \psi^{\beta}\bigl\Vert \rho^{\frac{1}{2}}u_{t}\bigr\Vert _{L^{2}(\Omega_{R})}^{\frac{1}{2}} \\ &{}+C\bigl\Vert \rho^{\frac{1}{2}}u_{t} \bigr\Vert _{L^{2}(\Omega_{R})}^{\frac{1}{2}}\Vert u_{xt}\Vert _{L^{2}(\Omega_{R})} \psi^{\beta} \Vert u_{xt}\Vert _{L^{2}(\Omega_{R})}^{\frac{1}{2}} \\ \leq& \varepsilon \Vert u_{xt}\Vert _{L^{2}(\Omega_{R})}^{2}+C \psi^{\beta}\bigl\Vert \rho^{\frac {1}{2}}u_{t}\bigr\Vert _{L^{2}(\Omega_{R})}^{2} \end{aligned}$$

(3.28)

and

$$\begin{aligned} I_{4} =& \int_{-R}^{R} \rho|u||u_{x}|^{2}|u_{t}| \,\mathrm{d}x \\ \leq& \Vert \rho\Phi \Vert _{L^{\infty}(\Omega_{R})}^{\frac{1}{2}}\bigl\Vert \Phi^{-\frac {1}{4}}u\bigr\Vert _{L^{2p}(\Omega_{R})}\bigl\Vert \Phi^{-\frac{1}{4}} \bigr\Vert _{L^{\frac {2p}{p-3}}(\Omega_{R})} \\ &{}\times\bigl\Vert \rho^{\frac{1}{2}}u_{t}\bigr\Vert _{L^{2}(\Omega_{R})}^{\frac{1}{2}} \bigl\Vert \rho^{\frac{1}{2}}u_{t} \bigr\Vert _{L^{\frac{2p}{p-2}}(\Omega_{R})}^{\frac{1}{2}}\Vert u_{x}\Vert _{L^{p}(\Omega_{R})}\Vert u_{x}\Vert _{L^{2p}(\Omega_{R})} \\ \leq& \psi^{\beta}\psi^{1-\theta} \Vert u_{xx}\Vert _{L^{2}(\Omega_{R})}^{\theta}\bigl\Vert \rho^{\frac{1}{2}}u_{t} \bigr\Vert _{L^{2}(\Omega_{R})}^{\frac{1}{2}} \bigl(\bigl\Vert \rho^{\frac{1}{2}}u_{t}\bigr\Vert _{L^{2}(\Omega_{R})}+\Vert u_{xt}\Vert _{L^{2}(\Omega _{R})}\bigr)^{\frac{1}{2}} \\ \leq& \psi^{\beta}\bigl(\psi+\Vert u_{xx}\Vert _{L^{2}(\Omega_{R})}\bigr)\bigl\Vert \rho^{\frac {1}{2}}u_{t}\bigr\Vert _{L^{2}(\Omega_{R})}^{\frac{1}{2}} \bigl(\bigl\Vert \rho^{\frac{1}{2}}u_{t} \bigr\Vert _{L^{2}(\Omega_{R})}^{\frac{1}{2}}+\Vert u_{xt}\Vert _{L^{2}(\Omega_{R})}^{\frac{1}{2}}\bigr) \\ \leq& \psi^{\beta}\bigl\Vert \rho^{\frac{1}{2}}u_{t}\bigr\Vert _{L^{2}(\Omega_{R})}+\psi^{\beta}\bigl\Vert \rho^{\frac{1}{2}}u_{t} \bigr\Vert _{L^{2}(\Omega_{R})}^{\frac{1}{2}}\Vert u_{xt}\Vert _{L^{2}(\Omega_{R})}^{\frac{1}{2}} \\ &{}+ \psi^{\beta} \Vert u_{xx} \Vert _{L^{2}(\Omega_{R})}\bigl\Vert \rho^{\frac{1}{2}}u_{t}\bigr\Vert _{L^{2}(\Omega_{R})}^{\frac{1}{2}}\Vert u_{xt}\Vert _{L^{2}(\Omega_{R})}^{\frac{1}{2}} \\ &{}+\psi^{\beta} \Vert u_{xx}\Vert _{L^{2}(\Omega_{R})}\bigl\Vert \rho^{\frac{1}{2}}u_{t}\bigr\Vert _{L^{2}(\Omega_{R})} \\ \leq& \varepsilon \Vert u_{xt}\Vert _{L^{2}(\Omega_{R})}^{2}+C( \varepsilon)\psi^{\beta}\bigl(\Vert u_{xx}\Vert _{L^{2}(\Omega_{R})}^{2}+ \bigl\Vert \rho^{\frac{1}{2}}u_{t} \bigr\Vert _{L^{2}(\Omega_{R})}^{2}+1\bigr), \end{aligned}$$

(3.29)

where $\theta=\frac{1}{p+2}$. By (3.9) and (3.10), we get

$$\begin{aligned} I_{5} =& \int_{0}^{1} \rho|u|^{2}|u_{xx}||u_{t}| \,\mathrm{d}x \\ \leq& C\bigl\Vert \rho^{\frac{1}{4}}u\bigr\Vert _{L^{4p}(\Omega_{R})}^{2} \bigl\Vert \rho^{\frac {1}{2}}u_{t}\bigr\Vert _{L^{2}(\Omega_{R})}^{\frac{1}{2}} \bigl\Vert \rho^{\frac{1}{2}}u_{t}\bigr\Vert _{L^{\frac{2p}{p-2}}(\Omega_{R})}^{\frac{1}{2}} \Vert u_{xx}\Vert _{L^{2}(\Omega_{R})} \\ \leq& C\psi \Vert u_{xx}\Vert _{L^{2}(\Omega_{R})}\bigl\Vert \rho^{\frac{1}{2}}u_{t}\bigr\Vert _{L^{2}(\Omega_{R})}^{\frac{1}{2}} \bigl(\bigl\Vert \rho^{\frac{1}{2}}u_{t}\bigr\Vert _{L^{2}(\Omega_{R})}^{\frac{1}{2}}+\Vert u_{xt}\Vert _{L^{2}(\Omega_{R})}^{\frac{1}{2}}\bigr) \\ \leq& C\psi^{\beta} \Vert u_{xx}\Vert _{L^{2}(\Omega_{R})}^{2}+\psi^{2}\bigl\Vert \rho^{\frac {1}{2}}u_{t}\bigr\Vert _{L^{2}(\Omega_{R})}^{2} + \bigl\Vert \rho^{\frac{1}{2}}u_{t}\bigr\Vert _{L^{2}(\Omega_{R})} \Vert u_{xt}\Vert _{L^{2}(\Omega_{R})} \\ \leq& \varepsilon \Vert u_{xt}\Vert _{L^{2}(\Omega_{R})}^{2}+C( \varepsilon)\psi^{\beta}\bigl(\Vert u_{xx}\Vert _{L^{2}(\Omega_{R})}^{2}+ \bigl\Vert \rho^{\frac{1}{2}}u_{t} \bigr\Vert _{L^{2}(\Omega_{R})}^{2}+1\bigr). \end{aligned}$$

(3.30)

Young’s inequality together with (3.9), (3.10) yields

$$\begin{aligned} I_{6} =& \int_{-R}^{R} \rho|u|^{2}|u_{x}||u_{xt}| \,\mathrm{d}x \\ \leq& C\Vert \rho\Phi \Vert _{L^{q}(\Omega_{R})}\bigl\Vert \Phi^{-\frac{1}{4}}u\bigr\Vert _{L^{\frac {4pq}{\varepsilon_{0}}}(\Omega_{R})}^{2}\Vert u_{x}\Vert _{L^{p}(\Omega_{R})}\Vert u_{xt}\Vert _{L^{2}(\Omega_{R})} \bigl\Vert \Phi^{-{\frac{1}{2}}}\bigr\Vert _{L^{\frac{2pq}{pq-2p-2q-\varepsilon_{0}}}(\Omega _{R})} \\ \leq& \varepsilon \Vert u_{xt}\Vert _{L^{2}(\Omega_{R})}^{2}+C( \varepsilon)\psi^{\beta}, \end{aligned}$$

(3.31)

where $0<\varepsilon_{0}<\min\{1,pq-2p-2q\}$, and

$$\begin{aligned} I_{7} =& \int_{-R}^{R}\rho|u_{x}||u_{t}|^{2} \,\mathrm{d}x \\ \leq& \Vert u_{x}\Vert _{L^{p}(\Omega_{R})}\bigl\Vert \rho^{\frac{1}{2}}u_{t}\bigr\Vert _{L^{2}(\Omega _{R})}^{\frac{1}{2}} \bigl\Vert \rho^{\frac{1}{2}}u_{t}\bigr\Vert _{L^{\frac{6p}{3p-4}}(\Omega_{R})}^{\frac{3}{2}} \\ \leq& \psi^{\beta}\bigl\Vert \rho^{\frac{1}{2}}u_{t}\bigr\Vert _{L^{2}(\Omega_{R})}^{\frac{1}{2}} \bigl(\bigl\Vert \rho^{\frac{1}{2}}u_{t} \bigr\Vert _{L^{2}(\Omega_{R})}^{\frac{3}{2}}+\Vert u_{xt}\Vert _{L^{2}(\Omega_{R})}^{\frac{3}{2}}\bigr) \\ \leq& \varepsilon \Vert u_{xt}\Vert _{L^{2}(\Omega_{R})}^{2}+C( \varepsilon)\psi^{\beta}\bigl\Vert \rho^{\frac{1}{2}}u_{t} \bigr\Vert _{L^{2}(\Omega_{R})}^{2}. \end{aligned}$$

(3.32)

Substituting (3.26)-(3.32) into (3.25) and choosing ε suitably small lead to

$$ \frac{\mathrm{d}}{\mathrm{d}t} \int_{-R}^{R} \rho|u_{t}|^{2}\, \mathrm{d}x+ \int_{-R}^{R} \bigl(|u_{xt}|^{2}+|u_{t}|^{2} \bigr) \,\mathrm{d}x \leq C\psi^{\beta}\bigl\Vert \rho^{1/2} u_{t}\bigr\Vert _{L^{2}(\Omega_{R})}^{2}+C\psi^{\beta}, $$

(3.33)

where in the last inequality we have used (3.18). Multiplying (3.33) by t, we obtain (3.24) after using Gronwall’s inequality and (3.4). The proof of Lemma 3.3 is completed. □

Lemma 3.4

Let $(\rho,u)$ and $T^{*}$ be as in Lemma  3.2. Then, for all $t \in (0,T^{*} ]$,

$$ { \sup_{0\leq s\leq t}} \bigl(\|\rho\Phi\|_{ L^{1}(\Omega_{R})\cap H^{1}(\Omega _{R})\cap W^{1,q}(\Omega_{R})} \bigr)\leq \exp \biggl\{ C\exp \biggl\{ C \int_{0}^{t}\psi ^{\beta}\,\mathrm{d}x \biggr\} \biggr\} . $$

(3.34)

Proof

First, multiplying (2.2)₁ by Φ and integrating the resulting equality over $\Omega_{R}$, we obtain after integration by parts and using (3.10)

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} \int_{-R}^{R} \rho\Phi\,\mathrm{d}x \leq& \int_{-R}^{R}\rho|u|\Phi_{x}\, \mathrm{d}x \\ \leq& C \int_{-R}^{R}\rho|u|\bigl(e+x^{2} \bigr)^{\zeta_{0}}x\,\mathrm{d}x \\ \leq& C \int_{-R}^{R}\rho|u|\bigl(e+x^{2} \bigr)^{\zeta_{0}}\bigl(e+x^{2}\bigr)^{\frac{1}{2}}\, \mathrm{d}x \\ \leq& C\Vert \rho\Phi \Vert _{L^{2}(\Omega_{R})}\bigl\Vert u \bigl(e+x^{2}\bigr)^{-\frac {1}{2}}\bigr\Vert _{L^{2}(\Omega_{R})} \\ \leq& C\Vert \rho\Phi \Vert _{L^{2}(\Omega_{R})}\bigl\Vert u \bigl(e+x^{2}\bigr)^{-\frac {1}{4}}\bigl(e+x^{2} \bigr)^{-\frac{1}{4}}\bigr\Vert _{L^{2}(\Omega_{R})} \\ \leq& C\Vert \rho\Phi \Vert _{L^{2}(\Omega_{R})}\bigl\Vert u \Phi^{-\frac {1}{4(1+\zeta_{0})}}\bigr\Vert _{L^{2p}(\Omega_{R})}\bigl\Vert \bigl(e+x^{2}\bigr)^{-\frac {1}{4}}\bigr\Vert _{L^{\frac{2p}{p-1}}(\Omega_{R})} \leq C\psi^{\beta}, \end{aligned}$$

which gives

$$ {\sup_{0\leq s\leq t}} \int_{-R}^{R}\rho\Phi\,\mathrm{d}x\leq C\exp \biggl\{ C \int_{0}^{t}\psi^{\beta}\,\mathrm{d}x \biggr\} . $$

(3.35)

Next, it follows from the Gagliardo-Nirenberg inequality and (3.10) that, for $0<\delta<1$,

$$\begin{aligned} \bigl\Vert u\Phi^{-\delta}\bigr\Vert _{L^{\infty}(\Omega_{R})} \leq& C\bigl\Vert u\Phi^{-\delta}\bigr\Vert _{L^{k}(\Omega_{R})}^{1-\theta}\bigl\Vert \bigl(u\Phi^{-\delta}\bigr)_{x}\bigr\Vert _{L^{k}(\Omega_{R})}^{\theta} \\ \leq& C \bigl(\bigl\Vert u\Phi^{-\delta}\bigr\Vert _{L^{k}(\Omega_{R})}+ \bigl\Vert u_{x}\Phi ^{-\delta}\bigr\Vert _{L^{k}(\Omega_{R})}+ \bigl\Vert u\Phi_{x}^{-\delta}\bigr\Vert _{L^{k}(\Omega_{R})} \bigr) \\ \leq& C \bigl(\psi^{\beta}+\Vert u_{xx}\Vert _{L^{2}(\Omega_{R})}+\bigl\Vert u\Phi ^{-\delta}\bigr\Vert _{L^{2k}(\Omega_{R})} \bigl\Vert \Phi^{-1}\Phi_{x}\bigr\Vert _{L^{2k}(\Omega_{R})} \bigr) \\ \leq& C \bigl(\psi^{\beta}+\Vert u_{xx}\Vert _{L^{2}(\Omega_{R})} \bigr), \end{aligned}$$

(3.36)

where $\theta=\frac{1}{k}$.

One derives from (1.1)₁ that $g\triangleq\rho\Phi$ satisfies

$$g_{t}+ug_{x}-agu(\ln\Phi)_{x}+gu_{x}=0, $$

which together with (3.36) gives

$$\begin{aligned}& \bigl( \Vert g_{x} \Vert _{L^{2}(\Omega_{R})} \bigr)_{t} \\& \quad \leq C \bigl(1+ \Vert u_{x} \Vert _{L^{\infty}(\Omega_{R})}+ \bigl\Vert u (\ln\Phi )_{x} \bigr\Vert _{L^{\infty}(\Omega_{R})} \bigr) \Vert g_{x} \Vert _{L^{2}(\Omega_{R})} \\& \qquad {}+C \bigl( \bigl\Vert \vert u_{x}\vert \bigl|(\ln \Phi)_{x}\bigr| \bigr\Vert _{L^{2}(\Omega_{R})} + \bigl\Vert \vert u \vert \bigl|(\ln\Phi)_{xx}\bigr| \bigr\Vert _{L^{2}(\Omega_{R})} + \Vert u_{xx} \Vert _{L^{2}(\Omega_{R})} \bigr) \Vert g \Vert _{L^{\infty}(\Omega _{R})} \\& \quad \leq C \bigl(\psi^{\beta}+ \Vert u_{x} \Vert _{ W^{1,q}(\Omega_{R})}+ \bigl\Vert u\Phi^{-\frac{1}{2(1+\zeta_{0})}} \bigr\Vert _{L^{\infty}(\Omega_{R})} \bigr) \Vert g_{x}\Vert _{L^{2}(\Omega_{R})} \\& \qquad {}+C \bigl( \Vert u_{xx} \Vert _{L^{2}(\Omega_{R})}+ \Vert u_{x} \Vert _{L^{p}(\Omega_{R})} \bigl\Vert \Phi^{-\frac{1}{2(1+\zeta _{0})}} \bigr\Vert _{L^{\frac{2p}{p-2}}(\Omega_{R})} \\& \qquad {}+ \bigl\Vert u \Phi^{-\frac {1}{2(1+\zeta_{0})}} \bigr\Vert _{L^{2p}(\Omega_{R})} \bigl\Vert \Phi^{-\frac{1}{2(1+\zeta_{0})}} \bigr\Vert _{L^{\frac{2p}{p-1}}(\Omega _{R})} \bigr) \Vert g \Vert _{L^{\infty}(\Omega_{R})} \\& \quad \leq C \bigl(\psi^{\beta}+ \Vert u_{xx} \Vert _{ L^{q}(\Omega_{R})} \bigr) \Vert g_{x} \Vert _{L^{2}(\Omega_{R})} +C \bigl(\psi^{\beta}+ \Vert u_{xx} \Vert _{ L^{2}(\Omega_{R})} \bigr) \Vert g \Vert _{L^{\infty}(\Omega_{R})} \\& \quad \leq C \bigl(\psi^{\beta}+ \Vert u_{xx} \Vert _{L^{2}(\Omega_{R})\cap L^{q}(\Omega_{R})} \bigr) \bigl(1+ \Vert g_{x} \Vert _{L^{2}(\Omega_{R})} + \Vert g_{x} \Vert _{L^{q}(\Omega_{R})} \bigr) \end{aligned}$$

(3.37)

and

$$\begin{aligned}& \bigl(\Vert g_{x}\Vert _{L^{q}(\Omega_{R})}\bigr)_{t} \\& \quad \leq C\bigl(1+\Vert u_{x}\Vert _{L^{\infty}(\Omega_{R})}+\bigl\Vert u (\ln\Phi )_{x}\bigr\Vert _{L^{\infty}(\Omega_{R})}\bigr)\Vert g_{x}\Vert _{L^{q}(\Omega_{R})} \\& \qquad {}+C\bigl(\bigl\Vert \vert u_{x}\vert \bigl|(\ln \Phi)_{x}\bigr|\bigr\Vert _{L^{q}(\Omega_{R})} +\bigl\Vert \vert u\vert \bigl|(\ln\Phi)_{xx}\bigr|\bigr\Vert _{L^{q}(\Omega_{R})} +\Vert u_{xx}\Vert _{L^{q}(\Omega_{R})}\bigr)\Vert g\Vert _{L^{\infty}(\Omega _{R})} \\& \quad \leq C\bigl(\psi^{\beta}+\Vert u_{x}\Vert _{ W^{1,q}(\Omega_{R})}+\bigl\Vert u\Phi^{-\frac{1}{2(1+\zeta_{0})}}\bigr\Vert _{L^{\infty}(\Omega_{R})} \bigr)\Vert g_{x}\Vert _{L^{q}(\Omega_{R})} \\& \qquad {}+C\bigl(\Vert u_{x}\Vert _{L^{q}(\Omega_{R})}+\bigl\Vert u\Phi^{-\frac {1}{2(1+\zeta_{0})}}\bigr\Vert _{L^{2q}(\Omega_{R})} \bigl\Vert \Phi^{-\frac{1}{2(1+\zeta_{0})}}\bigr\Vert _{L^{2q}(\Omega_{R})}+\Vert u_{xx}\Vert _{L^{q}(\Omega_{R})}\bigr)\Vert g\Vert _{L^{\infty}(\Omega_{R})} \\& \quad \leq C\bigl(\psi^{\beta}+\Vert u_{xx}\Vert _{ L^{q}(\Omega_{R})}\bigr)\Vert g_{x}\Vert _{L^{q}(\Omega_{R})} +C\bigl( \psi^{\beta}+\Vert u_{xx}\Vert _{L^{2}(\Omega_{R})\cap L^{q}(\Omega _{R})}\bigr) \Vert g\Vert _{L^{\infty}(\Omega_{R})} \\& \quad \leq C\bigl(\psi^{\beta}+\Vert u_{xx}\Vert _{L^{2}(\Omega_{R})\cap L^{q}(\Omega_{R})}\bigr) \bigl(1+\Vert g_{x}\Vert _{L^{q}(\Omega_{R})} \bigr), \end{aligned}$$

(3.38)

where in the last inequality we have used (3.35).

Next, we claim that

$$ \int_{0}^{t} \bigl(\|u_{xx} \|_{L^{2}(\Omega_{R})\cap L^{q}(\Omega_{R})}^{(q+1)/q}+t\| u_{xx}\|_{L^{2}(\Omega_{R})\cap L^{q}(\Omega_{R})}^{2} \bigr)\,\mathrm{d}x \leq C\exp \biggl\{ C \int_{0}^{t}\psi^{\beta}\,\mathrm{d}t \biggr\} , $$

(3.39)

which together with (3.35), (3.37), (3.38), and the Gronwall inequality yields

$${ \sup_{0\leq s\leq t}}\|\rho\Phi\|_{L^{1}(\Omega_{R})\cap H^{1}(\Omega _{R})\cap W^{1,q}(\Omega_{R})}\leq \exp \biggl\{ C \exp \biggl\{ C \int_{0}^{t}\psi^{\beta}\,\mathrm{d}x \biggr\} \biggr\} . $$

Finally, it only remains to prove (3.39). In fact, on the one hand, it follows from (3.18), (3.4), and (3.24) that

$$\begin{aligned}& \int_{0}^{t}\bigl(\Vert u_{xx}\Vert _{L^{2}(\Omega_{R})}^{\frac{q+1}{q}}+\Vert u_{xx}\Vert _{L^{2}(\Omega_{R})}^{2}\bigr)\,\mathrm{d}s \\& \quad \leq \int_{0}^{t}\bigl[\bigl(C\psi^{\frac{1}{2}}\Vert \sqrt{\rho} u_{t}\Vert _{L^{2}(\Omega _{R})}+C\psi^{\beta}\bigr)^{\frac{q+1}{q}}+s\bigl(C\psi^{\frac{1}{2}}\Vert \sqrt{\rho} u_{t}\Vert _{L^{2}(\Omega_{R})}+C\psi^{\beta}\bigr)^{2} \bigr] \,\mathrm{d}s \\& \quad \leq \int_{0}^{t}\bigl[C\psi^{\frac{q+1}{2q}}\Vert \sqrt{\rho} u_{t}\Vert _{L^{2}(\Omega _{R})}^{\frac{q+1}{q}}+C\bigl( \psi^{\beta}\bigr)^{\frac{q+1}{q}}+C\psi s\Vert \sqrt{\rho} u_{t}\Vert _{L^{2}(\Omega_{R})}^{2}+Cs\bigl( \psi^{\beta}\bigr)^{2}\bigr] \,\mathrm{d}s \\& \quad \leq \int_{0}^{t}\biggl[C\psi^{\frac{q+1}{q-1}}+{ \frac{q+1}{2q}}\Vert \sqrt{\rho} u_{t}\Vert _{L^{2}(\Omega_{R})}^{2}+C \psi \exp \biggl\{ C \int_{0}^{t}\psi^{\beta}\,\mathrm {d}s \biggr\} +Cs\psi^{\beta}\biggr]\,\mathrm{d}s \\& \quad \leq \int_{0}^{t}\bigl(\Vert \sqrt{ \rho}u_{t}\Vert _{L^{2}(\Omega_{R})}^{2}+\psi^{\beta}\bigr)\, \mathrm{d}s+ C\exp \biggl\{ C \int_{0}^{t}\psi^{\beta}\,\mathrm{d}s \biggr\} \int_{0}^{t}\psi^{\beta}\,\mathrm{d}s \\& \quad \leq C\exp \biggl\{ C \int_{0}^{t}\psi^{\beta}\,\mathrm{d}s \biggr\} . \end{aligned}$$

(3.40)

On the other hand, we denote

$$\dot{u}\triangleq u_{t}+u\cdot u_{x}. $$

By (3.19) and the Gagliardo-Nirenberg inequality, we have

$$\begin{aligned} \Vert u_{xx}\Vert _{L^{q}(\Omega_{R})} \leq& C\bigl(\Vert \rho \dot{u}\Vert _{L^{q}(\Omega_{R})}+\Vert \pi_{x}\Vert _{L^{q}(\Omega_{R})}+\Vert f\Vert _{L^{q}(\Omega_{R})}\bigr) \\ \leq& C\psi^{\beta}+C\Vert \rho\dot{u}\Vert _{L^{q}(\Omega_{R})}+ \Vert f\Vert _{L^{2}(\Omega _{R})}^{1-\theta} \Vert f_{x}\Vert _{L^{2}(\Omega_{R})}^{\theta} \\ \leq& C\psi^{\beta}+C\Vert \rho\dot{u}\Vert _{L^{q}(\Omega_{R})}, \end{aligned}$$

(3.41)

where $\theta=\frac{q-2}{2q}$.

By (3.9), (3.10), and (3.15), the last term on the right-hand side of (3.41) can be estimated as follows:

$$\begin{aligned} \Vert \rho\dot{u}\Vert _{L^{q}(\Omega_{R})} \leq& \Vert \rho u_{t} \Vert _{L^{q}(\Omega_{R})}+\Vert \rho u u_{x}\Vert _{L^{q}(\Omega_{R})} \\ \leq& \Vert \rho u_{t}\Vert _{L^{2}(\Omega_{R})}^{2(q-1)/(q^{2}-2)} \Vert \rho u_{t}\Vert _{L^{q^{2}}(\Omega_{R})}^{(q^{2}-2q)/(q^{2}-2)} +\Vert \rho u\Vert _{L^{2q}(\Omega_{R})}\Vert u_{x}\Vert _{L^{2q}(\Omega_{R})} \\ \leq& C\psi^{\beta}\bigl(\bigl\Vert \rho^{1/2} u_{t}\bigr\Vert _{L^{2}(\Omega _{R})}^{2(q-1)/(q^{2}-2)}\Vert u_{xt}\Vert _{L^{2}(\Omega_{R})}^{(q^{2}-2q)/(q^{2}-2)} +\bigl\Vert \rho^{1/2} u_{t}\bigr\Vert _{L^{2}(\Omega_{R})}\bigr) \\ &{}+C\psi^{\beta} \Vert u_{xx}\Vert _{L^{2}(\Omega_{R})}^{\frac{2q-p}{q(p+2)}}. \end{aligned}$$

This combined with (3.40), (3.24), and (3.4) yields

$$\begin{aligned}& \int_{0}^{t} \Vert \rho\dot{u}\Vert _{L^{q}(\Omega_{R})}^{\frac{q+1}{q}} \leq C \int_{0}^{t}\psi^{\beta}t^{-\frac{q+1}{2q}} \bigl(t\bigl\Vert \rho^{1/2} u_{t}\bigr\Vert _{L^{2}(\Omega_{R})}^{2}\bigr)^{\frac{q^{2}-1}{q(q^{2}-2)}} \bigl(t\Vert u_{xt}\Vert _{L^{2}(\Omega_{R})}^{2}\bigr)^{\frac{(q-2)(q+1)}{2(q^{2}-2)}}\, \mathrm {d}t \\& \hphantom{\int_{0}^{t} \Vert \rho\dot{u}\Vert _{L^{q}(\Omega_{R})}^{\frac{q+1}{q}}\leq{}}{}+C \int_{0}^{t}\bigl\Vert \rho^{1/2} u_{t}\bigr\Vert _{L^{2}(\Omega_{R})}^{2}\,\mathrm{d}t+C\exp \biggl\{ C \int_{0}^{t}\psi^{\beta}\,\mathrm{d}s \biggr\} \\& \hphantom{\int_{0}^{t} \Vert \rho\dot{u}\Vert _{L^{q}(\Omega_{R})}^{\frac{q+1}{q}}}\leq C\exp \biggl\{ C \int_{0}^{t}\psi^{\beta}\,\mathrm{d}s \biggr\} \int_{0}^{t}\bigl(\psi ^{\beta}+t^{-\frac{q^{3}+q^{2}-2q-2}{q^{3}+q^{2}-2q}} +t\Vert u_{xt}\Vert _{L^{2}(\Omega_{R})}^{2}\bigr)\, \mathrm{d}t \\& \hphantom{\int_{0}^{t} \Vert \rho\dot{u}\Vert _{L^{q}(\Omega_{R})}^{\frac{q+1}{q}}\leq{}}{}+C\exp \biggl\{ C \int_{0}^{t}\psi^{\beta}\,\mathrm{d}s \biggr\} \\& \hphantom{\int_{0}^{t} \Vert \rho\dot{u}\Vert _{L^{q}(\Omega_{R})}^{\frac{q+1}{q}}}\leq C\exp \biggl\{ C \int_{0}^{t}\psi^{\beta}\,\mathrm{d}s \biggr\} , \end{aligned}$$

(3.42)

$$\begin{aligned}& t^{\frac{1}{2}}\Vert u_{xx}\Vert _{L^{2}(\Omega_{R})} \leq C\psi^{\beta}+Ct\Vert \sqrt{\rho} u_{t} \Vert _{L^{2}(\Omega_{R})}^{2} \\& \hphantom{t^{\frac{1}{2}}\Vert u_{xx}\Vert _{L^{2}(\Omega_{R})}}\leq C\psi^{\beta}+C\exp \biggl\{ C \int_{0}^{t}\psi^{\beta}\,\mathrm{d}s \biggr\} , \end{aligned}$$

(3.43)

and that

$$ \int_{0}^{t} t\|\rho\dot{u}\|_{L^{q}(\Omega_{R})}^{2} \leq C\exp \biggl\{ C \int_{0}^{t}\psi^{\beta}\,\mathrm{d}s \biggr\} . $$

(3.44)

One thus obtains (3.39) from (3.40)-(3.44) and finishes the proof of Lemma 3.4. □

Proof of Proposition 3.1

It follows from (3.5), (3.4), and (3.33) that, for all $\forall t\in(0,T^{*}]$,

$$\psi(t)\leq \exp \biggl\{ C\exp \biggl\{ C \int_{0}^{t} \psi^{\beta}\,\mathrm{d}x \biggr\} \biggr\} . $$

Standard arguments thus yield, for $\tilde{K}\triangleq e^{Ce}$ and $T_{0}\triangleq\min \{T^{*},(CM^{\beta})^{-1} \}$,

$${ \sup_{0\leq t\leq T_{0}}}\psi(t)\leq\tilde{K}, $$

which together with (3.18), (3.39), and (3.4) gives (3.3). The proof of Proposition 3.1 is thus completed. □

4 Proof of Theorem 1.2

Let $(\rho_{0},u_{0})$ be as in Theorem 1.2. Without loss of generality, assume that

$$\int_{-\infty}^{+\infty}\rho_{0}\,\mathrm{d}x=1, $$

which implies that there exists a positive constant $N_{0}$ such that

$$ \int_{-N_{0}}^{N_{0}}\rho_{0}\,\mathrm{d}x\geq \frac{3}{4} \int_{-\infty }^{+\infty}\rho_{0}\,\mathrm{d}x= \frac{3}{4}. $$

(4.1)

We construct $\rho_{0}^{R}=j_{\frac{1}{R}}*\rho_{0}+R^{-1}e^{-x^{2}}$ where $j_{\frac{1}{R}}*\rho_{0}$ satisfies

$$ \int_{-N_{0}}^{N_{0}}j_{\frac{1}{R}}* \rho_{0}\,\mathrm{d}x\geq1/2, $$

(4.2)

and

$$ \Phi(j_{\frac{1}{R}}*\rho_{0})\rightarrow\Phi\rho_{0},\quad \mbox{in }L^{1}(\mathbb {R})\cap H^{1}(\mathbb{R})\cap W^{1,q}(\mathbb{R}), $$

(4.3)

as $R\rightarrow\infty$, since $u_{0}\in L^{2}(\mathbb{R})$ and $u_{0x}\in L^{2}(\mathbb{R})$, choosing $v^{R}\in C_{0}^{\infty}(\Omega_{R})$ such that

$$ \textstyle\begin{cases} \lim_{R\rightarrow\infty} \|v^{R}- u_{0} \|_{L^{2}(\mathbb{R})}=0, \\ \lim_{R\rightarrow\infty} \|v_{x}^{R}- u_{0x} \|_{L^{2}(\mathbb {R})}=0, \\ \|v_{xx}^{R} \|_{L^{2}(\mathbb{R})}\leq C, \end{cases} $$

(4.4)

we consider a smooth solution $u_{0}^{R}$ of the following elliptic problem (see [21]):

$$\begin{aligned}& -u_{0xx}^{R}+u_{0}^{R}=- \rho_{0}^{R}u_{0}^{R}+\sqrt {\rho_{0}^{R}}h^{R} -v_{xx}^{R}+v^{R}, \end{aligned}$$

(4.5)

$$\begin{aligned}& u_{0}^{R}(-R)= u_{0}^{R}(R)=0, \end{aligned}$$

(4.6)

where $d^{R}=(\sqrt{\rho_{0}}u_{0})\ast j_{1/R}$, with $j_{\delta}$ being the standard mollifying kernel of width δ. Extending $u_{0}^{R}$ to $\mathbb{R}$ by defining 0 outside $\Omega_{R}$ and denoting $w_{0}^{R}\triangleq u_{0}^{R}\tilde{\xi}_{R}$ with $\tilde{\xi}_{R}$ as in (3.6), we claim that

$$ \lim_{R\rightarrow\infty} \Bigl(\bigl\Vert \bigl(w_{0}^{R}-u_{0} \bigr)\bigr\Vert _{L^{2}(\mathbb {R})}+\bigl\Vert \bigl(w_{0x}^{R}-u_{0x} \bigr)\bigr\Vert _{L^{2}(\mathbb{R})} +\Bigl\Vert \sqrt{ \rho_{0}^{R}}w_{0}^{R}-\sqrt{ \rho_{0}}u_{0}\Bigr\Vert _{L^{2}(\mathbb{R})} \Bigr)=0. $$

(4.7)

In fact, multiplying (4.5) by $u_{0}^{R}$ and integrating the resulting equation over $\Omega_{R}$ lead to

$$\begin{aligned}& \int_{\Omega_{R}}\bigl(\rho_{0}^{R}+1\bigr) \bigl\vert u_{0}^{R}\bigr\vert ^{2}\, \mathrm{d}x+ \int_{\Omega _{R}}\bigl\vert u_{0x}^{R}\bigr\vert ^{2}\,\mathrm{d}x \\& \quad \leq C\bigl\Vert v_{x}^{R}\bigr\Vert _{L^{2}(\Omega_{R})}\bigl\Vert u_{0x}^{R}\bigr\Vert _{L^{2}(\Omega _{R})} \\& \qquad {}+\Bigl\Vert \sqrt{\rho_{0}^{R}}u_{0}^{R} \Bigr\Vert _{L^{2}(\Omega_{R})}\bigl\Vert d^{R}\bigr\Vert _{L^{2}(\Omega_{R})} +\bigl\Vert v^{R}\bigr\Vert _{L^{2}(\Omega_{R})}\bigl\Vert u_{0}^{R}\bigr\Vert _{L^{2}(\Omega_{R})} \\& \quad \leq \varepsilon\bigl\Vert u_{0}^{R}\bigr\Vert _{L^{2}(\Omega_{R})}^{2}+\varepsilon \bigl\Vert u_{0x}^{R} \bigr\Vert _{L^{2}(\Omega_{R})}^{2}+ \varepsilon \int_{\Omega_{R}}\rho_{0}^{R}\bigl\vert u_{0}^{R}\bigr\vert ^{2}\,\mathrm{d}x+C( \varepsilon), \end{aligned}$$

which implies

$$ \int_{\Omega_{R}}\bigl\vert u_{0}^{R}\bigr\vert ^{2}\,\mathrm{d}x+ \int_{\Omega_{R}}\rho_{0}^{R}\bigl\vert u_{0}^{R}\bigr\vert ^{2}\, \mathrm{d}x+ \int_{\Omega_{R}}\bigl\vert u_{0x}^{R}\bigr\vert ^{2}\,\mathrm{d}x\leq C, $$

(4.8)

for some C independent of R.

By the Gagliardo-Nirenberg inequality and (4.3), we obtain

$$\begin{aligned} \bigl\Vert \rho_{0}^{R}\bigr\Vert _{L^{\infty}(\Omega_{R})} =& \bigl\Vert j_{\frac{1}{R}}*\rho_{0}^{R}+R^{-1}e^{-x^{2}} \bigr\Vert _{L^{\infty}(\Omega _{R})} \\ \leq& 1+ \Vert j_{\frac{1}{R}}*\rho_{0}\Vert _{L^{\infty}(\Omega_{R})} \\ \leq& 1+ \Vert j_{\frac{1}{R}}*\rho_{0}\Vert _{L^{\infty}(\mathbb{R})} \\ \leq& 1+ \Vert j_{\frac{1}{R}}*\rho_{0}\Vert _{L^{\infty}(\mathbb{R})} \\ \leq& 1+C\Vert \Phi j_{\frac{1}{R}}*\rho_{0}\Vert _{L^{2}(\mathbb {R})}+C\bigl\Vert (j_{\frac{1}{R}}*\rho_{0})_{x} \bigr\Vert _{L^{2}(\mathbb{R})} \\ \leq& C. \end{aligned}$$

(4.9)

Combining (4.4), (4.8), and (4.9), we get

$$\begin{aligned}& \bigl\Vert u_{0xx}^{R}\bigr\Vert _{L^{2}(\Omega_{R})} \\& \quad \leq \bigl\Vert u_{0}^{R}\bigr\Vert _{L^{2}(\Omega_{R})}+\bigl\Vert \rho_{0}^{R}\bigr\Vert _{L^{\infty}(\Omega_{R})}^{\frac{1}{2}}\Bigl\Vert \sqrt{\rho_{0}^{R}}u_{0}^{R} \Bigr\Vert _{L^{2}(\Omega _{R})} \\& \qquad {}+\bigl\Vert \rho_{0}^{R}\bigr\Vert _{L^{\infty}(\Omega_{R})}^{\frac{1}{2}}\bigl\Vert d^{R} \bigr\Vert _{L^{2}(\Omega_{R})} +\bigl\Vert v_{xx}^{R}\bigr\Vert _{L^{2}(\Omega_{R})}+\bigl\Vert v^{R}\bigr\Vert _{L^{2}(\Omega_{R})} \\& \quad \leq C. \end{aligned}$$

(4.10)

We deduce from (4.8) and (4.13) that there exist a subsequence $R_{j}\rightarrow\infty$ and a function $w_{0}\in \{w_{0}\in W_{\mathrm{loc}}^{1,2}(\mathbb{R}) | \sqrt{\rho_{0}}w_{0}\in L^{2}(\mathbb{R}) \}$ such that

$$ \textstyle\begin{cases} \sqrt{\rho_{0}^{R_{j}}}w_{0}^{R_{j}}\rightharpoonup\sqrt{\rho_{0}}w_{0},\quad \mbox{weakly in }L^{2}(\mathbb{R}), \\ w_{0}^{R_{j}}\rightarrow w_{0},\quad \mbox{weakly in }L^{2}(\mathbb{R}), \\ w_{0x}^{R_{j}}\rightarrow w_{0x},\quad \mbox{weakly in }L^{2}(\mathbb{R}). \end{cases} $$

(4.11)

It follows from (4.5), (4.6), and (4.8) that $w_{0}^{R}$ satisfies

$$ -w_{0xx}^{R}+w_{0}^{R} =- \rho_{0}^{R}w_{0}^{R}+\sqrt {\rho_{0}^{R}}h^{R}\tilde{ \xi}_{R}- v_{xx}^{R}\tilde{ \xi}_{R}+v^{R}\tilde{\xi}_{R}+R^{-1}F_{R}, $$

(4.12)

with $\int_{\mathbb{R}}(F_{R})^{2}\,\mathrm{d}x\leq C$.

Thus, one can deduce from (4.21)-(4.24), for any $\psi \in C_{0}^{\infty}(\mathbb{R})$,

$$\int_{\mathbb{R}}(w_{0}-u_{0})\cdot\psi\, \mathrm{d}x + \int_{\mathbb{R}} (w_{0x}-u_{0x} )\cdot \psi_{x}\,\mathrm{d}x+ \int_{\mathbb{R}}\rho_{0}(w_{0}-u_{0}) \cdot\psi\,\mathrm{d}x=0, $$

which yields

$$ w_{0}=u_{0}. $$

(4.13)

We get from (4.12)

$$\limsup_{R_{j}\rightarrow\infty} \int_{\mathbb{R}}\bigl(\bigl\vert w_{0}^{R_{j}} \bigr\vert ^{2}+\bigl\vert w_{0x}^{R_{j}}\bigr\vert ^{2}+\rho_{0}^{R_{j}}\bigl\vert w_{0}^{R_{j}}\bigr\vert ^{2}\bigr)\,\mathrm{d}x \leq \int_{\mathbb{R}}\bigl(\vert u_{0}\vert ^{2}+\vert u_{0x}\vert ^{2}+ \rho_{0}\vert u_{0}\vert ^{2}\bigr)\, \mathrm{d}x, $$

which combined with (4.11) implies

$$\begin{aligned}& \lim_{R_{j}\rightarrow\infty} \int_{\mathbb{R}}\bigl\vert w_{0}^{R_{j}}\bigr\vert ^{2}\,\mathrm {d}x= \int_{\mathbb{R}}\vert u_{0}\vert ^{2}\, \mathrm{d}x, \\& \lim_{R_{j}\rightarrow\infty} \int_{\mathbb{R}}\bigl\vert w_{0x}^{R_{j}}\bigr\vert ^{2}\,\mathrm {d}x= \int_{\mathbb{R}}\vert u_{0x}\vert ^{2}\, \mathrm{d}x, \end{aligned}$$

and

$$\lim_{R_{j}\rightarrow\infty} \int_{\mathbb{R}}\rho_{0}^{R_{j}}\bigl\vert w_{0}^{R_{j}}\bigr\vert ^{2}\, \mathrm{d}x= \int_{\mathbb{R}}\rho_{0}\vert u_{0}\vert ^{2}\,\mathrm{d}x. $$

This, along with (4.13) and (4.8), gives (4.7).

Then, in terms of Lemma 2.1, the initial-boundary value problem (2.2) with the initial data $(\rho_{0}^{R},w_{0}^{R})$ has a classical solution $(\rho^{R},u^{R})$ on $\Omega_{R}\times[0,T_{R}]$. Moreover, Proposition 3.1 shows that there exists a $T_{0}$ independent of R such that (3.2) holds for $(\rho^{R},u^{R})$. We first deduce from (3.3) that

$$\begin{aligned}& \int_{0}^{T_{0}}\bigl\Vert \Phi^{\frac{1}{2}}{ \rho}_{t}^{R}\bigr\Vert _{L^{2}(\Omega _{R})}^{2}\, \mathrm{d}t \\& \quad \leq C \int_{0}^{T_{0}}\bigl(\bigl\Vert \Phi^{\frac{1}{2}}u^{R} \rho_{x}^{R}\bigr\Vert _{L^{2}(\Omega_{R})}^{2}+\bigl\Vert \Phi^{\frac{1}{2}} \rho^{R} u_{x}^{R}\bigr\Vert _{L^{2}(\Omega_{R})}^{2}\bigr)\,\mathrm{d}t \\& \quad \leq C \int_{0}^{T_{0}}\bigl\Vert \Phi^{-\frac{1}{2}}u^{R} \bigr\Vert _{L^{\infty}(\Omega _{R})}^{2} \bigl\Vert \Phi\rho_{x}^{R} \bigr\Vert _{L^{2}(\Omega_{R})}^{2}\,\mathrm{d}t+C \\& \quad \leq C \end{aligned}$$

(4.14)

and

$$\begin{aligned}& \int_{0}^{T_{0}}\bigl\Vert \Phi^{\frac{1}{2}}{ \rho}_{t}^{R}\bigr\Vert _{L^{q}(\Omega _{R})}^{2}\, \mathrm{d}t \\& \quad \leq C \int_{0}^{T_{0}}\bigl(\bigl\Vert \Phi^{\frac{1}{2}}u^{R} \rho_{x}^{R}\bigr\Vert _{L^{q}(\Omega_{R})}^{2}+\bigl\Vert \Phi^{\frac{1}{2}} \rho^{R} u_{x}^{R}\bigr\Vert _{L^{q}(\Omega_{R})}^{2}\bigr)\,\mathrm{d}t \\& \quad \leq C \int_{0}^{T_{0}}\bigl\Vert \Phi^{-\frac{1}{2}}u^{R} \bigr\Vert _{L^{\infty}(\Omega _{R})}^{2} \bigl\Vert \Phi\rho_{x}^{R} \bigr\Vert _{L^{q}(\Omega_{R})}^{2}\,\mathrm{d}t+C \\& \quad \leq C. \end{aligned}$$

(4.15)

By using (3.3), we have

$$\begin{aligned}& \int_{0}^{T_{0}}\bigl\Vert \bigl( \bigl(u_{x}^{R}\bigr)^{2}+\mu_{0} \bigr)^{\frac {p-2}{2}}u_{x}^{R}\bigr\Vert _{L^{2}(\Omega_{R})} \,\mathrm{d}t \\& \quad \leq C \int_{0}^{T_{0}}\bigl\Vert u_{x}^{R} \bigr\Vert _{L^{\infty}(\Omega_{R})}^{p-2} \bigl\Vert u_{x}^{R} \bigr\Vert _{L^{2}(\Omega_{R})}\,\mathrm{d}t+ C \int_{0}^{T_{0}}\mu_{0}^{\frac{p-2}{2}} \bigl\Vert u_{x}^{R}\bigr\Vert _{L^{2}(\Omega_{R})}\, \mathrm{d}t \\& \quad \leq C \int_{0}^{T_{0}} \bigl(\bigl\Vert u_{x}^{R}\bigr\Vert _{L^{p}(\Omega_{R})}^{1-\theta }\bigl\Vert u_{xx}^{R}\bigr\Vert _{L^{2}(\Omega_{R})}^{\theta} \bigr)^{p-2} \bigl\Vert u_{x}^{R}\bigr\Vert _{L^{2}(\Omega_{R})}\,\mathrm{d}t+C \\& \quad \leq C \int_{0}^{T_{0}} \bigl(\bigl\Vert u_{x}^{R}\bigr\Vert _{L^{p}(\Omega_{R})}^{1-\theta }t^{-\frac{\theta}{2}}t^{\frac{\theta}{2}} \bigl\Vert u_{xx}^{R}\bigr\Vert _{L^{2}(\Omega_{R})}^{\theta} \bigr)^{p-2} \bigl\Vert u_{x}^{R}\bigr\Vert _{L^{2}(\Omega_{R})}\,\mathrm{d}t+C \\& \quad \leq C \int_{0}^{T_{0}}t^{-\frac{p-2}{p+2}}\,\mathrm{d}t+C \\& \quad \leq C, \end{aligned}$$

(4.16)

where $\theta=\frac{2}{p+2}$.

With all these estimates (4.14)-(4.16), (3.3), (3.4), (3.24), and (3.34) at hand, we find that the sequence $(\rho^{R},u^{R})$ converges, up to the extraction of subsequences, to some limit $(\rho^{R},u^{R})$ in the obvious weak sense, that is, as $R\rightarrow\infty$, we have

$$\begin{aligned}& u^{R}\rightarrow u,\quad \mbox{weakly}^{\ast}\mbox{ in }L^{\infty}\bigl(0,T_{0};L^{2}(\mathbb{R})\bigr), \end{aligned}$$

(4.17)

$$\begin{aligned}& \Phi^{\frac{1}{2}}\rho^{R}\rightarrow\Phi^{\frac{1}{2}}\rho, \quad \mbox{in } C\bigl(\overline{\Omega}_{N}\times[0,T_{0}] \bigr),\forall N>0, \end{aligned}$$

(4.18)

$$\begin{aligned}& \Phi\rho^{R}\rightharpoonup\Phi\rho,\quad \mbox{weakly}^{\ast}\mbox{ in }L^{\infty}\bigl(0,T_{0};H^{1}(\mathbb{R}) \cap W^{1,q}(\mathbb{R})\bigr), \end{aligned}$$

(4.19)

$$\begin{aligned}& \Phi\rho^{R}\rightharpoonup\Phi\rho,\quad \mbox{weakly}^{\ast}\mbox{ in }L^{\infty}\bigl(0,T_{0};L^{1}(\mathbb{R}) \bigr), \end{aligned}$$

(4.20)

$$\begin{aligned}& \sqrt{\rho^{R}}u^{R}\rightharpoonup\sqrt{\rho}u,\quad \mbox{weakly}^{\ast}\mbox{ in }L^{\infty}\bigl(0,T_{0};L^{2}( \mathbb{R})\bigr), \end{aligned}$$

(4.21)

$$\begin{aligned}& u_{x}^{R}\rightharpoonup u_{x},\quad \mbox{weakly}^{\ast}\mbox{ in }L^{\infty}\bigl(0,T_{0};L^{2}( \mathbb{R})\cap L^{p}(\mathbb{R})\bigr), \end{aligned}$$

(4.22)

$$\begin{aligned}& u_{xx}^{R}\rightharpoonup u_{xx},\quad \mbox{weakly in }L^{(q+1)/q}\bigl(0,T_{0};L^{q}( \mathbb{R})\bigr)\cap L^{2}\bigl(0,T_{0};L^{2}( \mathbb{R})\bigr), \end{aligned}$$

(4.23)

$$\begin{aligned}& t^{1/2}u_{xx}^{R}\rightharpoonup t^{1/2}u_{xx},\quad \mbox{weakly in }L^{2} \bigl(0,T_{0};L^{2}(\mathbb{R})\cap L^{q}( \mathbb{R})\bigr), \end{aligned}$$

(4.24)

$$\begin{aligned}& t^{1/2}u_{xx}^{R}\rightharpoonup t^{1/2}u_{xx},\quad \mbox{weakly}^{\ast}\mbox{ in }L^{\infty}\bigl(0,T_{0};L^{2}(\mathbb{R})\bigr), \end{aligned}$$

(4.25)

$$\begin{aligned}& \sqrt{t}\sqrt{\rho^{R}}u_{t}^{R}\rightharpoonup \sqrt{t}\sqrt{\rho}u_{t},\quad \mbox{weakly}^{\ast}\mbox{ in }L^{\infty}\bigl(0,T_{0};L^{2}(\mathbb{R})\bigr), \end{aligned}$$

(4.26)

$$\begin{aligned}& \sqrt{t} u_{tx}^{R}\rightharpoonup\sqrt{t}u_{tx}, \quad \mbox{weakly in }L^{2}\bigl(\mathbb {R}\times(0,T_{0}) \bigr), \end{aligned}$$

(4.27)

$$\begin{aligned}& {\inf_{0\leq t\leq T_{0}}} \int_{\Omega_{2N_{0}}} \rho(x,t)\,\mathrm{d}x\geq\frac{1}{4}. \end{aligned}$$

(4.28)

Next, for $1< L< R$ and $\xi_{L}\in C^{1}(\Omega_{R})$ such that

$$ \textstyle\begin{cases} 0\leq\xi_{L}(x)\leq1, \quad |\xi_{Lx}|\leq\frac{C}{L}, \\ \xi_{L}(x)=0,\quad \mbox{if }|x|\leq\frac{L}{2}, \\ \xi_{L}(x)=1,\quad \mbox{if }|x|>L, \end{cases} $$

(4.29)

multiplying (1.1) by $\xi_{L}$, and integrating over $\Omega_{R}$, it follows that

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} \int_{-R}^{R}\xi_{L}\rho^{R} \,\mathrm{d}x =& \int_{-R}^{R}\xi_{Lx} \rho^{R}u^{R}\,\mathrm{d}x \\ \leq&\frac{C}{L}\bigl\Vert \rho^{R}\bigr\Vert _{L^{2}(\Omega_{R})}\bigl\Vert u^{R}\bigr\Vert _{L^{2}(\Omega_{R})} \\ \leq&\frac{C}{L}. \end{aligned}$$

(4.30)

By virtue of $\xi_{L}$, we have

$$\begin{aligned}& \xi_{L}(x) (j_{\frac{1}{R}}*\rho_{0}) (x)-j_{\frac{1}{R}}*(\xi_{\frac {L}{2}}\rho_{0}) (x) \\& \quad \leq \xi_{L}(x) \int_{-\infty}^{+\infty}j_{\frac{1}{R}}(x-y) \rho_{0}(y)\, \mathrm{d}y- \int_{-\infty}^{+\infty}j_{\frac{1}{R}}(x-y) ( \xi_{\frac {L}{2}}\rho_{0}) (y)\,\mathrm{d}y \\& \quad \leq \int_{-\infty}^{+\infty}j_{\frac{1}{R}}(x-y) \bigl( \xi_{L}(x)-\xi_{\frac {L}{2}}(y)\bigr)\rho_{0}(y) \,\mathrm{d}y \\& \quad \leq 0. \end{aligned}$$

(4.31)

Integrating the above inequality with respect to the time variable over $(0, t)$, we get

$$\begin{aligned} \int_{-R}^{R}\xi_{L}\rho^{R} \,\mathrm{d}x \leq& \int_{-R}^{R}\xi_{L} \rho_{0}^{R}(x)\,\mathrm{d}x+\frac{C}{L} \\ \leq& \int_{-\infty}^{+\infty}\xi_{L}(j_{\frac{1}{R}}* \rho_{0}) (x)\,\mathrm {d}x+\frac{1}{R} \int_{-R}^{R}\xi_{L}e^{-x^{2}}\, \mathrm{d}x+\frac{C}{L} \\ \leq& \int_{-\infty}^{+\infty}\bigl(j_{\frac{1}{R}}*( \xi_{\frac{L}{2}}\rho _{0})\bigr) (x)\,\mathrm{d}x+ \frac{1}{R} \int_{-R}^{R}\xi_{L}e^{-x^{2}}\, \mathrm {d}x+\frac{C}{L} \\ \leq& \int_{-\infty}^{+\infty}(\xi_{\frac{L}{2}} \rho_{0}) (x)\,\mathrm {d}x+\frac{1}{R} \int_{-R}^{R}\xi_{L}e^{-x^{2}}\, \mathrm{d}x+\frac{C}{L} \\ \leq& \int_{\frac{L}{4}}^{+\infty}\rho_{0}(x)\,\mathrm{d}x+ \int_{-\infty }^{-\frac{L}{4}}\rho_{0}(x)\,\mathrm{d}x+ \int_{-\infty}^{-\frac {L}{2}}e^{-x^{2}}\,\mathrm{d}x \\ &{}+ \int_{\frac{L}{2}}^{+\infty}e^{-x^{2}}\,\mathrm{d}x+ \frac{C}{L}. \end{aligned}$$

(4.32)

Multiplying (2.2)₁ by $\xi_{L}^{2}\rho_{0}^{R}$, and integrating over $\Omega_{R}$, it follows that

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} \int_{-R}^{R}\bigl(\xi_{L} \rho^{R}\bigr)^{2}\,\mathrm{d}x =&- \int_{-R}^{R}\rho^{R}u_{x}^{R} \xi_{L}^{2}\rho^{R}\,\mathrm{d}x+2 \int _{-R}^{R}\rho^{R}u^{R} \xi_{L}\xi_{Lx}\rho^{R}\,\mathrm{d}x \\ \leq&\bigl\Vert \xi_{L}\rho^{R}\bigr\Vert _{L^{\infty}(\Omega_{R})}\bigl\Vert u^{R}\bigr\Vert _{L^{2}(\Omega_{R})}\bigl\Vert \rho^{R}\bigr\Vert _{L^{2}(\Omega_{R})} \\ &{}+\frac{C}{L}\bigl\Vert \rho^{R}\bigr\Vert _{L^{\infty}(\Omega_{R})}\bigl\Vert u^{R}\bigr\Vert _{L^{2}(\Omega_{R})}\bigl\Vert \rho^{R}\bigr\Vert _{L^{2}(\Omega_{R})} \\ \leq&\bigl\Vert \xi_{L}\rho^{R}\bigr\Vert _{L^{1}(\Omega_{R})}^{\theta}\bigl\Vert \rho_{x}^{R} \bigr\Vert _{L^{2}(\Omega _{R})}^{1-\theta}\bigl\Vert u^{R}\bigr\Vert _{L^{2}(\Omega_{R})}\bigl\Vert \rho^{R}\bigr\Vert _{L^{2}(\Omega_{R})} \\ &{}+\frac{C}{L}\bigl\Vert \rho^{R}\bigr\Vert _{L^{\infty}(\Omega_{R})}\bigl\Vert u^{R}\bigr\Vert _{L^{2}(\Omega_{R})}\bigl\Vert \rho^{R}\bigr\Vert _{L^{2}(\Omega_{R})} \\ \leq&C\bigl\Vert \xi_{L}\rho^{R}\bigr\Vert _{L^{1}(\Omega_{R})}^{\theta}+\frac{C}{L}, \end{aligned}$$

(4.33)

where $\theta=\frac{2}{3}$.

Integrating the above inequality with respect to the time variable over $(0, t)$, we get

$$\begin{aligned}& \int_{-R}^{R}\bigl(\xi_{L} \rho^{R}\bigr)^{2}\,\mathrm{d}x \\ & \quad \leq \int_{-R}^{R}\bigl(\xi_{L} \rho_{0}^{R}\bigr)^{2}\,\mathrm{d}+C \int_{0}^{t}\bigl\Vert \xi _{L} \rho^{R}\bigr\Vert _{L^{1}(\Omega_{R})}^{\theta}\,\mathrm{d}t+ \frac{C}{L} \\ & \quad \leq \int_{-R}^{R}\biggl[\xi_{L}(j_{\frac{1}{R}}* \rho_{0})+\frac {1}{R}e^{-x^{2}}\biggr]^{2}\, \mathrm{d}x +C \int_{0}^{T_{1}}\bigl\Vert \xi_{L} \rho^{R}\bigr\Vert _{L^{1}(\Omega_{R})}^{\theta}\,\mathrm {d}t+ \frac{C}{L} \\ & \quad \leq \int_{-\infty}^{\infty}\bigl(j_{\frac{1}{R}}*( \xi_{\frac{L}{2}}\rho _{0})\bigr)^{2}\,\mathrm{d}x+ \frac{2}{R}\Vert j_{\frac{1}{R}}*\rho_{0}\Vert _{L^{\infty}(\Omega_{R})} \int_{-R}^{R}\xi_{L}e^{-x^{2}}\, \mathrm{d}x+\frac{1}{R^{2}} \int _{-R}^{R}\xi_{L} \bigl(e^{-x^{2}}\bigr)^{2}\,\mathrm{d}x \\ & \qquad {} +C \int_{0}^{t}\bigl\Vert \xi_{L} \rho^{R}\bigr\Vert _{L^{1}(\Omega_{R})}^{\theta}\,\mathrm {d}t+ \frac{C}{L} \\ & \quad \leq \int_{-\infty}^{\infty}(\xi_{\frac{L}{2}} \rho_{0})^{2}\,\mathrm {d}x+\frac{2}{R}\Vert j_{\frac{1}{R}}*\rho_{0}\Vert _{L^{\infty}(\Omega_{R})} \int _{-R}^{R}\xi_{L}e^{-x^{2}} \,\mathrm{d}x+\frac{1}{R^{2}} \int_{-R}^{R}\xi _{L} \bigl(e^{-x^{2}}\bigr)^{2}\,\mathrm{d}x \\ & \qquad {} +C \int_{0}^{t}\bigl\Vert \xi_{L} \rho^{R}\bigr\Vert _{L^{1}(\Omega_{R})}^{\theta}\,\mathrm {d}t+ \frac{C}{L} \\ & \quad \leq \int_{\frac{L}{4}}^{\infty}\rho_{0}^{2}\, \mathrm{d}x+ \int_{-\infty }^{-\frac{L}{4}}\rho_{0}^{2}\, \mathrm{d}x+\Vert j_{\frac{1}{R}}*\rho_{0}\Vert _{L^{\infty}(\Omega_{R})} \biggl( \int_{-\infty}^{-\frac{L}{2}}e^{-x^{2}}\,\mathrm {d}x+ \int_{\frac{L}{2}}^{+\infty}e^{-x^{2}}\,\mathrm{d}x\biggr) \\ & \qquad {} + \int_{-\infty}^{-\frac{L}{2}}\bigl(e^{-x^{2}} \bigr)^{2}\,\mathrm{d}x+ \int_{\frac {L}{2}}^{+\infty}\bigl(e^{-x^{2}} \bigr)^{2}\,\mathrm{d}x+C \int_{0}^{t}\bigl\Vert \xi_{L}\rho ^{R}\bigr\Vert _{L^{1}(\Omega_{R})}^{\theta}\,\mathrm{d}t+ \frac{C}{L}, \end{aligned}$$

(4.34)

where $\theta=\frac{2}{3}$.

Next, by the fixed $\kappa\in(L, R)$ and by using (4.34), we have

$$\begin{aligned}& \int_{L}^{\kappa}\rho^{2}\,\mathrm{d}x \\& \quad \leq \int_{\frac{L}{4}}^{\infty}\rho_{0}^{2}\, \mathrm{d}x+ \int_{-\infty }^{-\frac{L}{4}}\rho_{0}^{2}\, \mathrm{d}x+\Vert j_{\frac{1}{R}}*\rho_{0}\Vert _{L^{\infty}(\Omega_{R})} \biggl( \int_{-\infty}^{-\frac{L}{2}}e^{-x^{2}}\,\mathrm {d}x+ \int_{\frac{L}{2}}^{+\infty}e^{-x^{2}}\,\mathrm{d}x\biggr) \\& \qquad {}+ \int_{-\infty}^{-\frac{L}{2}}\bigl(e^{-x^{2}} \bigr)^{2}\,\mathrm{d}x+ \int_{\frac {L}{2}}^{+\infty}\bigl(e^{-x^{2}} \bigr)^{2}\,\mathrm{d}x+C \int_{0}^{t}\bigl\Vert \xi_{L}\rho ^{R}\bigr\Vert _{L^{1}(\Omega_{R})}^{\theta}\,\mathrm{d}t+ \frac{C}{L}, \end{aligned}$$

(4.35)

where $\theta=\frac{2}{3}$.

Taking the limit on κ for inequality (4.35) we obtain, as $\kappa\rightarrow+\infty$,

$$\begin{aligned}& \int_{L}^{+\infty}\rho^{2}\,\mathrm{d}x \\& \quad \leq \int_{\frac{L}{4}}^{\infty}\rho_{0}^{2}\, \mathrm{d}x+ \int_{-\infty }^{-\frac{L}{4}}\rho_{0}^{2}\, \mathrm{d}x+\Vert j_{\frac{1}{R}}*\rho_{0}\Vert _{L^{\infty}(\mathbb{R})} \biggl( \int_{-\infty}^{-\frac{L}{2}}e^{-x^{2}}\,\mathrm {d}x+ \int_{\frac{L}{2}}^{+\infty}e^{-x^{2}}\,\mathrm{d}x\biggr) \\& \qquad {}+ \int_{-\infty}^{-\frac{L}{2}}\bigl(e^{-x^{2}} \bigr)^{2}\,\mathrm{d}x+ \int_{\frac {L}{2}}^{+\infty}\bigl(e^{-x^{2}} \bigr)^{2}\,\mathrm{d}x+\frac{C}{L} \\& \qquad {}+C \int_{0}^{t}\biggl( \int_{\frac{L}{4}}^{+\infty}\rho_{0}(x)\,\mathrm{d}x+ \int _{-\infty}^{-\frac{L}{4}}\rho_{0}(x)\, \mathrm{d}x \\& \qquad {}+ \int_{-\infty}^{-\frac {L}{2}}e^{-x^{2}}\,\mathrm{d}x+ \int_{\frac{L}{2}}^{+\infty}e^{-x^{2}}\, \mathrm{d}x+ \frac{C}{L}\biggr)^{\theta}\,\mathrm{d}t, \end{aligned}$$

(4.36)

where $\theta=\frac{2}{3}$.

Thus, we deduce that

$$\begin{aligned}& \int_{L}^{R}\rho^{2}\,\mathrm{d}x \\& \quad \leq \int_{\frac{L}{4}}^{\infty}\rho_{0}^{2}\, \mathrm{d}x+ \int_{-\infty }^{-\frac{L}{4}}\rho_{0}^{2}\, \mathrm{d}x+\Vert j_{\frac{1}{R}}*\rho_{0}\Vert _{L^{\infty}(\mathbb{R})} \biggl( \int_{-\infty}^{-\frac{L}{2}}e^{-x^{2}}\,\mathrm {d}x+ \int_{\frac{L}{2}}^{+\infty}e^{-x^{2}}\,\mathrm{d}x\biggr) \\& \qquad {}+ \int_{-\infty}^{-\frac{L}{2}}\bigl(e^{-x^{2}} \bigr)^{2}\,\mathrm{d}x+ \int_{\frac {L}{2}}^{+\infty}\bigl(e^{-x^{2}} \bigr)^{2}\,\mathrm{d}x+\frac{C}{L} \\& \qquad {}+C \int_{0}^{t}\biggl( \int_{\frac{L}{4}}^{+\infty}\rho_{0}(x)\,\mathrm{d}x+ \int _{-\infty}^{-\frac{L}{4}}\rho_{0}(x)\, \mathrm{d}x \\& \qquad {}+ \int_{-\infty}^{-\frac {L}{2}}e^{-x^{2}}\,\mathrm{d}x+ \int_{\frac{L}{2}}^{+\infty}e^{-x^{2}}\, \mathrm{d}x+ \frac{C}{L}\biggr)^{\theta}\,\mathrm{d}t, \end{aligned}$$

(4.37)

where $\theta=\frac{2}{3}$. In the same way, we can also handle $\int _{-R}^{-L}\rho^{2}\,\mathrm{d}x$.

By the Sobolev embedding theorem, we have

$$ \int_{-L}^{L}\bigl(\rho^{R}-\rho \bigr)^{2}\,\mathrm{d}x\rightarrow0,\quad R\rightarrow +\infty. $$

(4.38)

By virtue of

$$\begin{aligned}& \int_{-R}^{R}\bigl(\rho^{R}-\rho \bigr)^{2}\,\mathrm{d}x \\& \quad \leq \int_{-L}^{L}\bigl(\rho^{R}-\rho \bigr)^{2}\,\mathrm{d}x+ \int_{L}^{R}\bigl(\rho^{R} \bigr)^{2}\, \mathrm{d}x+ \int_{L}^{R}\rho^{2}\,\mathrm{d}x+ \int_{-R}^{-L}\bigl(\rho^{R} \bigr)^{2}\, \mathrm{d}x+ \int_{-R}^{-L}\rho^{2}\,\mathrm{d}x \\& \quad \leq \int_{-L}^{L}\bigl(\rho^{R}-\rho \bigr)^{2}\,\mathrm{d}x+ \int_{-R}^{R}\bigl(\xi_{L}\rho ^{R}\bigr)^{2}\,\mathrm{d}x+ \int_{L}^{R}\rho^{2}\,\mathrm{d}x+ \int_{-R}^{-L}\rho ^{2}\,\mathrm{d}x, \end{aligned}$$

(4.39)

and by using (4.34), (4.37), and (4.38), letting $R\rightarrow\infty$, and then letting $L\rightarrow\infty$, we have

$$ \lim_{R\rightarrow\infty} \int_{-R}^{R}\bigl(\rho^{R}-\rho \bigr)^{2}\,\mathrm{d}x=0. $$

(4.40)

By using (4.22) and (4.25), we have

$$\begin{aligned}& \int_{0}^{T_{0}} \int_{-R}^{R} \bigl(\bigl(u_{x}^{R} \bigr)^{2}+\mu_{0}\bigr)^{\frac {p-2}{2}} \bigl(u_{x}^{R}\bigr)^{2}\,\mathrm{d}x\, \mathrm{d}t \\& \quad \leq C \int_{0}^{T_{0}}\bigl\Vert u_{x}^{R} \bigr\Vert _{L^{\infty}(\Omega_{R})}^{p-2} \int _{-R}^{R}\bigl(u_{x}^{R} \bigr)^{2}\,\mathrm{d}x\,\mathrm{d}t+ C \int_{0}^{T_{0}}\mu_{0}^{\frac{p-2}{2}} \int_{-R}^{R}\bigl(u_{x}^{R} \bigr)^{2}\,\mathrm {d}x\,\mathrm{d}t \\& \quad \leq C \int_{0}^{T_{0}} \bigl(\bigl\Vert u_{x}^{R}\bigr\Vert _{L^{p}(\Omega_{R})}^{1-\theta }\bigl\Vert u_{xx}^{R}\bigr\Vert _{L^{2}(\Omega_{R})}^{\theta} \bigr)^{p-2} \int_{-R}^{R}\bigl(u_{x}^{R} \bigr)^{2}\,\mathrm{d}x\,\mathrm{d}t+C \\& \quad \leq C \int_{0}^{T_{0}} \bigl(\bigl\Vert u_{x}^{R}\bigr\Vert _{L^{p}(\Omega_{R})}^{1-\theta }t^{-\frac{\theta}{2}}t^{\frac{\theta}{2}} \bigl\Vert u_{xx}^{R}\bigr\Vert _{L^{2}(\Omega_{R})}^{\theta} \bigr)^{p-2} \int_{-R}^{R}\bigl(u_{x}^{R} \bigr)^{2}\,\mathrm{d}x\,\mathrm{d}t+C \\& \quad \leq C \int_{0}^{T_{0}}t^{-\frac{p-2}{p+2}}\,\mathrm{d}t+C \\& \quad \leq C, \end{aligned}$$

(4.41)

where $\theta=\frac{2}{P+2}$. Multiplying (2.2)₂ by $\xi_{L}u^{R}$, and integrating over $\Omega_{R}$, it follows that

$$\begin{aligned}& \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \int_{-R}^{R}\xi_{L}\rho^{R} \bigl(u^{R}\bigr)^{2}\,\mathrm {d}x+ \int_{-R}^{R}\xi_{L} \bigl(u^{R}\bigr)^{2}\,\mathrm{d}x+ \int_{-R}^{R}\xi _{L}\bigl( \bigl(u_{x}^{R}\bigr)^{2}+\mu_{0} \bigr)^{\frac{p-2}{2}}\bigl(u_{x}^{R}\bigr)^{2}\, \mathrm{d}x \\& \quad \leq -\frac{1}{2} \int_{-R}^{R}\xi_{L} \rho^{R}u_{x}^{R}\bigl(u^{R} \bigr)^{2}\,\mathrm {d}x+\frac{1}{2} \int_{-R}^{R}\xi_{L} \rho_{x}^{R}\bigl(u^{R}\bigr)^{3}\, \mathrm{d}x \\& \qquad{}+ \int_{-R}^{R}\xi_{Lx}\bigl( \bigl(u_{x}^{R}\bigr)^{2}+\mu_{0} \bigr)^{\frac {p-2}{2}}u_{x}^{R}u^{R}\,\mathrm{d}x + \int_{-R}^{R}P_{x}\xi_{L}u^{R} \,\mathrm{d}x \\& \leq 2\bigl\Vert u^{R}\bigr\Vert _{L^{\infty}(\Omega_{R})}^{2} \bigl\Vert \xi_{L}\rho^{R}\bigr\Vert _{L^{2}(\Omega_{R})} \bigl\Vert u_{x}^{R}\bigr\Vert _{L^{2}(\Omega_{R})}+ \frac{C}{L}\bigl\Vert u^{R}\bigr\Vert _{L^{\infty}(\Omega_{R})}^{2} \bigl\Vert \rho ^{R}\bigr\Vert _{L^{2}(\Omega_{R})}\bigl\Vert u^{R}\bigr\Vert _{L^{2}(\Omega_{R})} \\& \qquad {}+\frac{C}{L}\bigl\Vert \bigl(\bigl(u_{x}^{R} \bigr)^{2}+\mu_{0} \bigr)^{\frac{p-2}{2}}u_{x}^{R} \bigr\Vert _{L^{2}(\Omega_{R})}\bigl\Vert u^{R}\bigr\Vert _{L^{2}(\Omega_{R})} \\& \qquad {}+\bigl\Vert \rho^{R}\bigr\Vert _{L^{\infty}(\Omega _{R})}^{\gamma-1}\bigl\Vert \xi_{L} \rho^{R}\bigr\Vert _{L^{2}(\Omega_{R})}\bigl\Vert u_{x}^{R} \bigr\Vert _{L^{2}(\Omega _{R})} \\& \quad \leq C\bigl\Vert \xi_{L}\rho^{R}\bigr\Vert _{L^{2}(\Omega_{R})}+\frac{C}{L}. \end{aligned}$$

(4.42)

Integrating the above inequality with respect to the time variable over $(0, t)$, we get

$$\begin{aligned}& \frac{1}{2} \int_{-R}^{R}\xi_{L}\rho^{R} \bigl(u^{R}\bigr)^{2}\,\mathrm{d}x+ \int _{0}^{t} \int_{-R}^{R}\xi_{L} \bigl(u^{R}\bigr)^{2}\,\mathrm{d}x\,\mathrm{d}t+ \int _{0}^{t} \int_{-R}^{R}\xi_{L}\bigl( \bigl(u_{x}^{R}\bigr)^{2}+\mu_{0} \bigr)^{\frac {p-2}{2}}\bigl(u_{x}^{R}\bigr)^{2}\, \mathrm{d}x\,\mathrm{d}t \\& \quad \leq \frac{1}{2} \int_{-R}^{R}\xi_{L} \rho_{0}^{R}\bigl(u_{0}^{R} \bigr)^{2}\,\mathrm{d}x+C \int _{0}^{t}\bigl\Vert \xi_{L} \rho^{R}\bigr\Vert _{L^{2}(\Omega_{R})}\,\mathrm{d}t+ \int_{0}^{t}\frac {C}{L}\,\mathrm{d}t \\& \quad \leq C \int_{-R}^{R}\xi_{L} \rho_{0}^{R}\,\mathrm{d}x+C \int_{0}^{t}\bigl\Vert \xi _{L} \rho^{R}\bigr\Vert _{L^{2}(\Omega_{R})}\,\mathrm{d}t+ \int_{0}^{t}\frac{C}{L}\, \mathrm{d}t. \end{aligned}$$

(4.43)

It is easy to see that the following inequality is established:

$$ \int_{-R}^{R}\xi_{L}\bigl( \bigl(u_{x}^{R}\bigr)^{2}+\mu_{0} \bigr)^{\frac{p-2}{2}}\bigl(u_{x}^{R}\bigr)^{2}\, \mathrm{d}x \geq\frac{1}{2} \int_{-R}^{R}\xi_{L}\bigl\vert u_{x}^{R}\bigr\vert ^{p}\,\mathrm{d}x+ \frac{\mu _{0}^{\frac{p-2}{2}}}{2} \int_{-R}^{R}\xi_{L}\bigl\vert u_{x}^{R}\bigr\vert ^{2}\,\mathrm{d}x. $$

(4.44)

By virtue of

$$\begin{aligned}& \int_{0}^{t} \int_{-R}^{R}\bigl(u^{R}-u \bigr)^{2}\,\mathrm{d}x\,\mathrm{d}t \\& \quad \leq \int_{0}^{t} \int_{-L}^{L}\bigl(u^{R}-u \bigr)^{2}\,\mathrm{d}x\,\mathrm{d}t+ \int _{0}^{t} \int_{L}^{R}\bigl(u^{R} \bigr)^{2}\,\mathrm{d}x\,\mathrm{d}t+ \int_{0}^{t} \int _{L}^{R}u^{2}\,\mathrm{d}x\, \mathrm{d}t \\& \qquad {}+ \int_{0}^{t} \int _{-R}^{-L}\bigl(u^{R} \bigr)^{2}\,\mathrm{d}x\,\mathrm{d}t+ \int_{0}^{t} \int_{-R}^{-L}u^{2}\,\mathrm{d}x\, \mathrm{d}t \\& \quad = \int_{0}^{t} \int_{-L}^{L}\bigl(u^{R}-u \bigr)^{2}\,\mathrm{d}x\,\mathrm{d}t+ \int _{0}^{t} \int_{-R}^{R}\bigl(\xi_{L}u^{R} \bigr)^{2}\,\mathrm{d}x\,\mathrm{d}t \\& \qquad {}+ \int _{0}^{t} \int_{L}^{R}u^{2}\,\mathrm{d}x\, \mathrm{d}t+ \int_{0}^{t} \int _{-R}^{-L}u^{2}\,\mathrm{d}x\, \mathrm{d}t, \end{aligned}$$

(4.45)

and by using (4.42) and by a similar approach to (4.40), letting $R\rightarrow\infty$, and then letting $L\rightarrow\infty$, we have

$$ \lim_{R\rightarrow\infty} \int_{0}^{t} \int_{-R}^{R}\bigl(u^{R}-u \bigr)^{2}\,\mathrm {d}x\,\mathrm{d}t=0. $$

(4.46)

Subtracting both sides of the inequality (2.2)₂ from $[(u_{x}^{2}+\mu_{0} )^{\frac{p-2}{2}}u_{x}]_{x}$, multiplying by $\xi _{L}(u_{x}^{R}-u_{x})$, and integrating over $\Omega_{R}$, it follows that

$$\begin{aligned}& \int_{0}^{t} \int_{-R}^{R}\xi_{L} \bigl(u_{x}^{R}-u_{x}\bigr)^{2}\, \mathrm{d}x\,\mathrm{d}t \\& \quad \leq \frac{C}{L} \int_{0}^{t}\bigl\Vert \bigl(u_{x}^{2}+ \mu_{0} \bigr)^{\frac {p-2}{2}}u_{x}\bigr\Vert _{L^{2}(\Omega_{R})}\bigl(\bigl\Vert u^{R}\bigr\Vert _{L^{2}(\Omega_{R})}+ \Vert u\Vert _{L^{2}(\Omega_{R})}\bigr)\,\mathrm{d}t \\& \qquad {}+ \int_{0}^{t} \int_{-R}^{R}\xi_{L}^{2} \bigl(u_{x}^{2}+\mu_{0} \bigr)^{\frac {p-2}{2}}u_{x} \bigl(u_{x}^{R}-u_{x}\bigr)\,\mathrm{d}x\, \mathrm{d}t+ \int_{0}^{t}\Vert f\Vert _{L^{2}(\Omega_{R})}\bigl\Vert u^{R}-u\bigr\Vert _{L^{2}(\Omega_{R})}\,\mathrm{d}t \\& \qquad {}+ \int_{0}^{t}\bigl\Vert \rho^{R}\bigr\Vert _{L^{\infty}(\Omega_{R})}^{\frac{1}{2}}\bigl\Vert \sqrt{\rho ^{R}}u_{t}^{R}\bigr\Vert _{L^{2}(\Omega_{R})}\bigl\Vert u^{R}-u\bigr\Vert _{L^{2}(\Omega_{R})}\,\mathrm{d}t \\& \qquad {}+ \int_{0}^{t}\bigl\Vert \rho^{R}\bigr\Vert _{L^{\infty}(\Omega_{R})}\bigl\Vert u^{R}\bigr\Vert _{L^{\infty}(\Omega _{R})} \bigl\Vert u_{x}^{R}\bigr\Vert _{L^{2}(\Omega_{R})}\bigl\Vert u^{R}-u\bigr\Vert _{L^{2}(\Omega_{R})}\,\mathrm{d}t \\& \qquad {}+\frac{C}{L} \int_{0}^{t}\bigl\Vert \rho^{R}\bigr\Vert _{L^{\infty}(\Omega_{R})}^{\gamma-1}\bigl\Vert \rho^{R}\bigr\Vert _{L^{2}(\Omega_{R})}\bigl(\bigl\Vert u^{R}\bigr\Vert _{L^{2}(\Omega_{R})}+\Vert u\Vert _{L^{2}(\Omega _{R})}\bigr)\,\mathrm{d}t \\& \qquad {}+ \int_{0}^{t}\bigl\Vert \rho^{R}\bigr\Vert _{L^{\infty}(\Omega_{R})}^{\gamma-1}\bigl\Vert \xi_{L} \rho^{R}\bigr\Vert _{L^{2}(\Omega_{R})}\bigl(\bigl\Vert u_{x}^{R}\bigr\Vert _{L^{2}(\Omega_{R})}+\Vert u_{x}\Vert _{L^{2}(\Omega_{R})}\bigr)\, \mathrm{d}t \\& \quad \leq C \int_{0}^{t}\bigl\Vert u^{R}-u\bigr\Vert _{L^{2}(\Omega_{R})}^{2}\,\mathrm{d}t+\frac {C}{L}+C\bigl\Vert \xi_{L}\rho^{R}\bigr\Vert _{L^{2}(\Omega_{R})} \\& \qquad {}+C \int_{0}^{t} \int_{-R}^{R}\xi _{L}\bigl( \bigl(u_{x}^{R}\bigr)^{2}+\mu_{0} \bigr)^{\frac{p-2}{2}}\bigl(u_{x}^{R}\bigr)^{2}\, \mathrm{d}x\, \mathrm{d}t \\& \qquad {}+ \int_{0}^{t} \int_{-R}^{R}\xi_{L}|u_{x}|^{p-2} \bigl(u_{x}^{R}\bigr)^{2}\,\mathrm{d}x\, \mathrm{d}t+C \int_{0}^{t} \int_{-R}^{R}\xi_{L} \bigl(u_{x}^{R}\bigr)^{2}\,\mathrm{d}x\, \mathrm{d}t \\& \quad \leq C \int_{0}^{t}\bigl\Vert u^{R}-u\bigr\Vert _{L^{2}(\Omega_{R})}^{2}\,\mathrm{d}t+\frac {C}{L}+C\bigl\Vert \xi_{L}\rho^{R}\bigr\Vert _{L^{2}(\Omega_{R})} \\& \qquad {}+C \int_{0}^{t} \int_{-R}^{R}\xi _{L}\bigl( \bigl(u_{x}^{R}\bigr)^{2}+\mu_{0} \bigr)^{\frac{p-2}{2}}\bigl(u_{x}^{R}\bigr)^{2}\, \mathrm{d}x\, \mathrm{d}t \\& \qquad {} + \int_{0}^{t}\biggl( \int_{-R}^{R}|u_{x}|^{p}\, \mathrm{d}x\biggr)^{\frac {p-2}{p}}\biggl( \int_{-R}^{R}\xi_{L} \bigl(u_{x}^{R}\bigr)^{p}\,\mathrm{d}x \biggr)^{\frac{2}{p}}\, \mathrm{d}t+C \int_{0}^{t} \int_{-R}^{R}\xi_{L} \bigl(u_{x}^{R}\bigr)^{2}\,\mathrm{d}x\, \mathrm{d}t, \end{aligned}$$

(4.47)

where in the second inequality we have used (4.44).

By virtue of

$$ \int_{0}^{t} \int_{-R}^{R}\bigl(u_{x}^{R}-u_{x} \bigr)^{2}\,\mathrm{d}x\,\mathrm{d}t \leq \int_{0}^{t} \int_{-L}^{L}\bigl(u_{x}^{R}-u_{x} \bigr)^{2}\,\mathrm{d}x\,\mathrm {d}t+ \int_{0}^{t} \int_{-R}^{R}\xi_{L} \bigl(u_{x}^{R}-u_{x}\bigr)^{2}\, \mathrm{d}x\,\mathrm{d}t, $$

(4.48)

and by (4.47) and by the lower semi-continuity of various norms, letting $R\rightarrow\infty$, and then letting $L\rightarrow \infty$, we have

$$ \lim_{R\rightarrow\infty} \int_{0}^{t} \int_{-R}^{R}\bigl(u_{x}^{R}-u_{x} \bigr)^{2}\, \mathrm{d}x\,\mathrm{d}t=0. $$

(4.49)

By virtue of the nonlinear term, we have

$$\begin{aligned} \bigl[\bigl(s^{2}+\mu_{0}\bigr)^{\frac{p-2}{2}}s \bigr]' =&\bigl(s^{2}+\mu_{0} \bigr)^{\frac{p-4}{2}} \bigl[(p-1)s^{2}+\mu_{0} \bigr] \\ \geq&\bigl(s^{2}+\mu_{0}\bigr)^{\frac{p-2}{2}} \\ \geq&(\mu_{0})^{\frac{p-2}{2}}. \end{aligned}$$

First, we begin with the case $u^{R}u\geq0$ for fixed $(t,x)$, we get

$$\begin{aligned}& \bigl[\bigl(\bigl(u_{x}^{R}\bigr)^{2}+ \mu_{0}\bigr)^{\frac{p-2}{2}}u_{x}^{R}- \bigl(u_{x}^{2}+\mu_{0}\bigr)^{\frac {p-2}{2}}u_{x} \bigr]\bigl(u_{x}^{R}-u_{x}\bigr) \\& \quad \geq \int_{0}^{1}\bigl\vert \theta u_{x}^{R}+(1-\theta)u_{x}\bigr\vert ^{p-2}\,\mathrm {d}\theta\bigl(u_{x}^{R}-u_{x} \bigr)^{2} \\& \quad \geq \int_{0}^{1}\bigl\vert \theta u_{x}^{R}\bigr\vert ^{p-2}\,\mathrm{d}\theta \bigl(u_{x}^{R}-u_{x}\bigr)^{2} + \int_{0}^{1}\bigl\vert (1-\theta)u_{x} \bigr\vert ^{p-2}\,\mathrm{d}\theta \bigl(u_{x}^{R}-u_{x} \bigr)^{2} \\& \quad \geq \frac{1}{p-1}\bigl(\bigl\vert u_{x}^{R} \bigr\vert ^{p-2}+\vert u_{x}\vert ^{p-2} \bigr) \bigl(u_{x}^{R}-u_{x} \bigr)^{2} \\& \quad \geq C\bigl(\bigl\vert u_{x}^{R}\bigr\vert + \vert u_{x}\vert \bigr)^{p-2}\bigl(u_{x}^{R}-u_{x} \bigr)^{2} \\& \quad \geq C\bigl\vert u_{x}^{R}-u_{x}\bigr\vert ^{p}. \end{aligned}$$

(4.50)

Next, we discuss the case $u^{R}u< 0$ for fixed $(t,x)$, we have

$$\begin{aligned}& \bigl[\bigl(\bigl(u_{x}^{R}\bigr)^{2}+ \mu_{0}\bigr)^{\frac{p-2}{2}}u_{x}^{R}- \bigl(u_{x}^{2}+\mu_{0}\bigr)^{\frac {p-2}{2}}u_{x} \bigr] \bigl(u_{x}^{R}-u_{x} \bigr) \\& \quad \geq \bigl(\bigl(u_{x}^{R}\bigr)^{2}+ \mu_{0}\bigr)^{\frac{p-2}{2}}\bigl(u_{x}^{R} \bigr)^{2}-\bigl(\bigl(u_{x}^{R} \bigr)^{2}+\mu _{0}\bigr)^{\frac{p-2}{2}}u_{x}^{R}u_{x} -\bigl(u_{x}^{2}+\mu_{0}\bigr)^{\frac{p-2}{2}}u_{x}u_{x}^{R} \\& \qquad {}+\bigl(u_{x}^{2}+\mu_{0} \bigr)^{\frac{p-2}{2}}u_{x}^{2} \\& \quad \geq \bigl\vert u_{x}^{R}\bigr\vert ^{p}+\vert u_{x}\vert ^{p} \\& \quad \geq 2^{1-p} \bigl(\bigl\vert u_{x}^{R} \bigr\vert +\vert u_{x}\vert \bigr)^{p} \\& \quad = 2^{1-p}\bigl\vert u_{x}^{R}-u_{x} \bigr\vert ^{p}. \end{aligned}$$

(4.51)

Consequently, we obtain after using (4.50) and (4.51)

$$\begin{aligned}& \int_{-R}^{R} \bigl[\bigl(\bigl(u_{x}^{R} \bigr)^{2}+\mu_{0}\bigr)^{\frac{p-2}{2}}u_{x}- \bigl(u_{x}^{2}+\mu _{0}\bigr)^{\frac{p-2}{2}}u_{x} \bigr] \bigl(u_{x}^{R}-u_{x} \bigr)\, \mathrm{d}x \\& \quad \geq 2^{1-p} \int_{-R}^{R}\bigl\vert u_{x}^{R}-u_{x} \bigr\vert ^{p}\,\mathrm{d}x. \end{aligned}$$

(4.52)

By (4.52) and (4.47), we know

$$\int_{0}^{t} \int_{-R}^{R}\bigl\vert u_{x}^{R}-u_{x} \bigr\vert ^{p}\,\mathrm{d}x\,\mathrm{d}t \leq \int_{0}^{t} \int_{-L}^{L}\bigl\vert u_{x}^{R}-u_{x} \bigr\vert ^{p}\,\mathrm{d}x\,\mathrm {d}t+ \int_{0}^{t} \int_{-R}^{R}\xi_{L}\bigl\vert u_{x}^{R}-u_{x}\bigr\vert ^{p}\, \mathrm{d}x\,\mathrm{d}t $$

and

$$ \lim_{R\rightarrow\infty} \int_{0}^{t} \int_{-R}^{R}\bigl\vert u_{x}^{R}-u_{x} \bigr\vert ^{p}\, \mathrm{d}x\,\mathrm{d}t=0. $$

(4.53)

Next, for any function $\Psi\in C_{0}^{\infty}(\mathbb{R}\times[0,T_{0}))$, we take Ψ as a test function in the initial-boundary value problem (2.2) with the initial data $(\rho_{0}^{R},w_{0}^{R})$. By standard arguments, letting $R\rightarrow\infty$, it follows from (4.17)-(4.28), (4.40), (4.46), (4.49), and (4.53) that $(\rho,u)$ is a strong solution of (1.1)-(1.2) on $\mathbb{R}\times(0,T_{0}]$ satisfying (1.6) and (1.7). We only give a proof of the nonlinear term as follows:

$$\begin{aligned}& \int_{0}^{T_{0}} \int_{-R}^{R} \bigl[\bigl(\bigl(u_{x}^{R} \bigr)^{2}+\mu_{0}\bigr)^{\frac {p-2}{2}}u_{x}^{R}- \bigl(u_{x}^{2}+\mu_{0}\bigr)^{\frac{p-2}{2}}u_{x} \bigr]\Psi\, \mathrm{d}x\,\mathrm{d}t \\& \quad \leq \int_{0}^{T_{0}} \int_{-R}^{R}\biggl\vert \int_{0}^{1}\bigl[\mu_{0}+\bigl(\theta u_{x}^{R}+(1-\theta)u_{x}\bigr)^{2} \bigr]^{\frac{p-2}{2}}\,\mathrm{d}\theta \biggr\vert \bigl\vert u_{x}^{R}-u_{x}\bigr\vert \vert \Psi \vert \,\mathrm{d}x\,\mathrm{d}t \\& \quad \leq \int_{0}^{T_{0}} \int_{-R}^{R}\bigl\vert \bigl[ \mu_{0}+\bigl(\bigl\vert u_{x}^{R}\bigr\vert +\vert u_{x}\vert \bigr)^{2} \bigr]^{\frac {p-2}{2}}\bigr\vert \bigl\vert u_{x}^{R}-u_{x} \bigr\vert \vert \Psi \vert \,\mathrm{d}x\,\mathrm{d}t \\& \quad \leq C \int_{0}^{T_{0}} \int_{-R}^{R}\bigl(1+\mu_{0}+\bigl\vert u_{x}^{R}\bigr\vert ^{2}+\vert u_{x}\vert ^{2}\bigr)^{\frac {p-1}{2}}\bigl\vert u_{x}^{R}-u_{x}\bigr\vert \vert \Psi \vert \,\mathrm{d}x\,\mathrm{d}t \\& \quad \leq C \int_{0}^{T_{0}} \int_{-R}^{R}\bigl(1+\mu_{0}^{\frac {p-1}{2}}+ \bigl\vert u_{x}^{R}\bigr\vert ^{p-1}+\vert u_{x}\vert ^{p-1}\bigr)\bigl\vert u_{x}^{R}-u_{x} \bigr\vert \vert \Psi \vert \,\mathrm{d}x\, \mathrm{d}t \\& \quad \leq C \int_{0}^{T_{0}} \int_{-R}^{R}\bigl\vert u_{x}^{R}-u_{x} \bigr\vert \vert \Psi \vert \,\mathrm{d}x\,\mathrm {d}t+C \int_{0}^{T_{0}} \int_{-R}^{R}\bigl\vert u_{x}^{R} \bigr\vert ^{p-1}\bigl\vert u_{x}^{R}-u_{x} \bigr\vert \vert \Psi \vert \,\mathrm {d}x\,\mathrm{d}t \\& \qquad {} +C \int_{0}^{T_{0}} \int_{-R}^{R}\vert u_{x}\vert ^{p-1}\bigl\vert u_{x}^{R}-u_{x} \bigr\vert \vert \Psi \vert \,\mathrm{d}x\, \mathrm{d}t \\& \quad \leq C\biggl( \int_{0}^{T_{0}} \int_{-R}^{R}\bigl\vert u_{x}^{R}-u_{x} \bigr\vert ^{2}\,\mathrm{d}x\,\mathrm {d}t\biggr)^{\frac{1}{2}} \biggl( \int_{0}^{T_{0}} \int_{-R}^{R}\vert \Psi \vert ^{2}\, \mathrm{d}x\, \mathrm{d}t\biggr)^{\frac{1}{2}} \\& \qquad{} +C \int_{0}^{T_{0}}\bigl\Vert u_{x}^{R} \bigr\Vert _{L^{p}(\Omega_{R})}^{p-1}\bigl\Vert \bigl(u_{x}^{R}-u_{x} \bigr)\Psi\bigr\Vert _{L^{p}(\Omega_{R})}\,\mathrm{d}t \\& \qquad {}+C \int_{0}^{T_{0}}\Vert u_{x}\Vert _{L^{p}(\Omega_{R})}^{p-1}\bigl\Vert \bigl(u_{x}^{R}-u_{x} \bigr)\Psi\bigr\Vert _{L^{p}(\Omega_{R})}\,\mathrm{d}t \\& \quad \leq C\biggl( \int_{0}^{T_{0}} \int_{-R}^{R}\bigl\vert u_{x}^{R}-u_{x} \bigr\vert ^{2}\,\mathrm{d}x\,\mathrm {d}t\biggr)^{\frac{1}{2}} +C \int_{0}^{T_{0}}\Vert \Psi \Vert _{L^{\infty}(\Omega_{R})} \bigl\Vert \bigl(u_{x}^{R}-u_{x}\bigr)\bigr\Vert _{L^{p}(\Omega _{R})}\,\mathrm{d}t \\& \quad \leq C\biggl( \int_{0}^{T_{0}} \int_{-R}^{R}\bigl\vert u_{x}^{R}-u_{x} \bigr\vert ^{2}\,\mathrm{d}x\,\mathrm {d}t\biggr)^{\frac{1}{2}} +C \biggl( \int_{0}^{T_{0}} \int_{-R}^{R}\bigl\vert u_{x}^{R}-u_{x} \bigr\vert ^{p}\,\mathrm{d}x\,\mathrm {d}t\biggr)^{\frac{1}{p}}. \end{aligned}$$

(4.54)

The proof of the existence part of Theorem 1.2 is finished.

Finally, to finish the proof of the main theorem, we only need to prove the uniqueness of the solution of the problem (1.1). Let $(\rho,u)$ and $(\bar{\rho},\bar{u})$ be two solutions that satisfy the same initial condition. Denote

$$Z\triangleq\rho-\bar{\rho}, \qquad U\triangleq u-\bar{u}, \qquad \Theta \triangleq\pi (\rho)-\pi(\bar{\rho}). $$

Subtracting the momentum equations satisfied by $(\rho,u)$ and $(\bar {\rho},\bar{u})$ yields

$$\begin{aligned}& \rho U_{t}+\rho u\cdot U_{x}- \bigl\{ \bigl[ \bigl(u_{x}^{2}+\mu_{0}\bigr)^{\frac {p-2}{2}}u_{x} \bigr]_{x}- \bigl[\bigl(\bar{u}_{x}^{2}+ \mu_{0}\bigr)^{\frac{p-2}{2}}\bar {u}_{x} \bigr]_{x} \bigr\} \\& \quad =-\rho U\cdot\bar{u}_{x}-Z(\bar{u}_{t}+\bar{u}\cdot \bar{u}_{x}) -\Theta_{x}+f(u)-f(\bar{u}). \end{aligned}$$

(4.55)

By virtue of f, we have

$$ f_{s}=\frac{\partial f(t,x,s)}{\partial s}. $$

(4.56)

By (1.4), we know

$$\begin{aligned} \int_{-\infty}^{+\infty} \bigl[f(u)-f(\bar{u})\bigr]U\, \mathrm{d}x =& \int_{-\infty}^{+\infty} \int_{0}^{1}f_{s}\bigl(\theta u+(1- \theta)\bar {u}\bigr)\,\mathrm{d}\theta U^{2}\,\mathrm{d}x \\ \leq& -B \int_{-\infty}^{+\infty} U^{2}\,\mathrm{d}x. \end{aligned}$$

(4.57)

By virtue of

$$\begin{aligned} \bigl[\bigl(s^{2}+\mu_{0}\bigr)^{\frac{p-2}{2}}s \bigr]' =&\bigl(s^{2}+\mu_{0} \bigr)^{\frac{p-4}{2}} \bigl[(p-1)s^{2}+\mu_{0} \bigr] \\ \geq&\bigl(s^{2}+\mu_{0}\bigr)^{\frac{p-2}{2}} \\ \geq&(\mu_{0})^{\frac{p-2}{2}}, \end{aligned}$$

multiplying (4.55) by U, by using (1.4) and integrating by parts, leads to

$$\begin{aligned}& \frac{\mathrm{d}}{\mathrm{d}t} \int_{-\infty}^{+\infty}\rho|U|^{2}\,\mathrm{d}x+ \int _{-\infty}^{+\infty}|U_{x}|^{2}\, \mathrm{d}x+B \int_{-\infty}^{+\infty} U^{2}\,\mathrm{d}x \\& \quad \leq C \Vert \bar{u}_{x} \Vert _{L^{\infty}(\mathbb{R})} \int_{-\infty }^{+\infty}\rho|U|^{2}\,\mathrm{d}x+C \int_{-\infty}^{+\infty}|Z||U|\bigl(\vert \bar {u}_{t}\vert +\vert \bar{u}\vert |\bar{u}_{x}|\bigr) \,\mathrm{d}x \\& \qquad {} +C \Vert \Theta \Vert _{L^{2}(\mathbb{R})} \Vert U_{x} \Vert _{L^{2}(\mathbb{R})} \\& \quad \triangleq C \Vert \bar{u}_{x} \Vert _{L^{\infty}} \int_{-\infty}^{+\infty }\rho|U|^{2}\, \mathrm{d}x+A_{1}+A_{2}. \end{aligned}$$

(4.58)

Then subtracting the mass equation for $(\rho,u)$ and $(\bar{\rho},\bar{u})$ gives

$$ Z_{t}+\bar{u}\cdot Z_{x}+Z \bar{u}_{x}+\rho U_{x}+U\cdot\rho_{x}=0. $$

(4.59)

Multiplying (4.59) by $2H\Phi$ and integrating by parts lead to

$$\begin{aligned}& \bigl(\bigl\Vert Z\Phi^{\frac{1}{2}}\bigr\Vert _{L^{2}(\mathbb{R})}^{2} \bigr)_{t} \\& \quad \leq C\bigl(\Vert \bar{u}_{x}\Vert _{L^{\infty}(\mathbb{R})}+\bigl\Vert \bar {u}\Phi^{-\frac{1}{2(1+\eta_{0})}}\bigr\Vert _{L^{\infty}(\mathbb{R})}\bigr)\bigl\Vert Z\Phi^{\frac{1}{2}}\bigr\Vert _{L^{2}(\mathbb{R})}^{2} \\& \qquad {}+C\bigl\Vert \rho\Phi^{\frac{1}{2}}\bigr\Vert _{L^{\infty}(\mathbb{R})} \Vert U_{x}\Vert _{L^{2}(\mathbb{R})}\bigl\Vert Z\Phi^{\frac{1}{2}} \bigr\Vert _{L^{2}(\mathbb {R})} \\& \qquad {}+C\bigl\Vert Z\Phi^{\frac{1}{2}}\bigr\Vert _{L^{2}(\mathbb{R})}\bigl\Vert U\Phi^{-\frac {1}{2}}\bigr\Vert _{L^{\frac{2q}{q-2}}(\mathbb{R})} \Vert \Phi \rho_{x}\Vert _{L^{q}(\mathbb{R})} \\& \quad \leq C\bigl(1+\Vert \bar{u}_{x}\Vert _{W^{1,q}(\mathbb{R})} \bigr)\bigl\Vert Z\Phi ^{\frac{1}{2}}\bigr\Vert _{L^{2}(\mathbb{R})}^{2} \\& \qquad {}+C\bigl\Vert Z\Phi^{\frac{1}{2}}\bigr\Vert _{L^{2}(\mathbb{R})}\bigl(\Vert U_{x}\Vert _{L^{2}(\mathbb{R})}+\Vert \sqrt{\rho} U\Vert _{L^{2}(\mathbb{R})}\bigr), \end{aligned}$$

where in the second inequality we have used Sobolev’s inequality, (1.7), (2.26), and (1.8). This combined with Gronwall’s inequality yields, for all $t \in(0,T^{*}]$,

$$ \bigl\Vert Z\Phi^{\frac{1}{2}}\bigr\Vert _{L^{2}(\mathbb{R})}\leq C \int_{0}^{t} \bigl(\Vert U_{x}\Vert _{L^{2}(\mathbb{R})}+\Vert \sqrt{\rho} U\Vert _{L^{2}(\mathbb{R})} \bigr)\, \mathrm{d}s. $$

(4.60)

As observed by Germain [21], putting (4.53) into (4.55) leads to

$$\begin{aligned} A_{1} \leq& C(\varepsilon) \bigl(1+t\Vert \bar{u}_{xt} \Vert _{L^{2}(\mathbb{R})}^{2}+t\Vert \bar{u}_{xx}\Vert _{L^{q}(\mathbb{R})}^{2} \bigr) \int \bigl(\Vert U_{x}\Vert _{L^{2}(\mathbb{R})}^{2}+ \Vert \sqrt{\rho} U\Vert _{L^{2}(\mathbb{R})}^{2} \bigr)\,\mathrm{d}s \\ &{}+ \bigl(\Vert \sqrt{\rho} U\Vert _{L^{2}(\mathbb{R})}^{2}+\Vert U_{x}\Vert _{L^{2}(\mathbb{R})}^{2} \bigr). \end{aligned}$$

(4.61)

Next, we will estimate $A_{2}$. In fact, one deduces from (2.2)₁ that

$$\Theta_{t}+u \Theta_{x}+U\pi(\bar{\rho})_{x}+ \gamma\Theta u_{x}+\gamma\pi(\bar {\rho}) U_{x}=0, $$

which gives

$$\begin{aligned}& \bigl(\Vert \Theta \Vert _{L^{2}(\mathbb{R})} \bigr)_{t} \\& \quad \leq C \bigl(1+\Vert u_{x}\Vert _{L^{\infty}(\mathbb{R})} \bigr) \Vert \Theta \Vert _{L^{2}(\mathbb{R})}+C\Vert U_{x}\Vert _{L^{2}(\mathbb {R})} \\& \qquad {}+C\bigl\Vert U\Phi^{-1}\bigr\Vert _{L^{\frac{2q}{q-2}}(\mathbb{R})}\bigl\Vert \pi (\bar{\rho})_{x}\Phi\bigr\Vert _{L^{q}(\mathbb{R})} \\& \quad \leq C \bigl(1+\Vert u_{x}\Vert _{L^{\infty}(\mathbb{R})} \bigr) \Vert \Theta \Vert _{L^{2}(\mathbb{R})}+C\Vert \sqrt{\rho} U\Vert _{L^{2}(\mathbb {R})}^{2}+C\Vert U_{x}\Vert _{L^{2}(\mathbb{R})}^{2}, \end{aligned}$$

(4.62)

which together with Gronwall’s inequality gives

$$\Vert \Theta \Vert _{L^{2}(\mathbb{R})}\leq C \int_{0}^{t} \bigl(\Vert U_{x}\Vert _{L^{2}(\mathbb{R})}+\Vert \sqrt{\rho} U\Vert _{L^{2}(\mathbb{R})} \bigr)\, \mathrm{d}s, $$

which shows

$$ A_{2}\leq\varepsilon \Vert U_{x}\Vert _{L^{2}(\mathbb{R})}^{2}+C(\varepsilon ) \int_{0}^{t} \bigl(\Vert U_{x}\Vert _{L^{2}(\mathbb{R})}^{2}+\Vert \sqrt{\rho} U\Vert _{L^{2}(\mathbb{R})}^{2} \bigr)\,\mathrm{d}s. $$

(4.63)

Denoting

$$H(t)\triangleq \int_{0}^{t}\Vert U\Vert _{L^{2}(\mathbb{R})}^{2} \,\mathrm {d}s+\Vert \sqrt{\rho} U\Vert _{L^{2}(\mathbb{R})}^{2}+ \int_{0}^{t} \bigl(\Vert \sqrt{\rho} U\Vert _{L^{2}(\mathbb{R})}^{2}+\Vert U_{x}\Vert _{L^{2}(\mathbb {R})}^{2} \bigr)\,\mathrm{d}s, $$

putting (4.61) and (4.63) into (4.58) and choosing ε suitably small lead to

$$H'(t)\leq C \bigl(1+\Vert \bar{u}_{x}\Vert _{L^{\infty}(\mathbb{R})}+t\Vert \bar{u}_{xx}\Vert _{L^{q}(\mathbb{R})}^{2}+t \Vert \bar{u}_{xt}\Vert _{L^{2}(\mathbb{R})}^{2} \bigr)G, $$

which together with Gronwall’s inequality and (1.8) yields $H(t) = 0$. Hence, $U(x,t) = 0$ for almost everywhere $(x,t)\in\mathbb {R}\times(0,T_{0})$. Then (4.60) implies that $Z(x,t) = 0$ for almost everywhere $(x,t)\in\mathbb{R}\times(0,T_{0})$. The proof of Theorem 1.2 is completed.