††footnotetext: *Corresponding author.††footnotetext: _E-mail address:_
<EMAIL_ADDRESS>(H.-L. Li<EMAIL_ADDRESS>(S.
Zhao),††footnotetext<EMAIL_ADDRESS>(H.-W. Zuo).

#  Existence and Nonlinear Stability of Steady-States to Outflow Problem for
the Full Two-Phase Flow

Hai-Liang Li a,b, Shuang Zhaoa,b∗, Han-Wen Zuoa,b
a School of Mathematical Sciences, Capital Normal University, Beijing 100048,
P.R. China.
b Academy for Multidisciplinary Studies, Capital Normal University, Beijing
100048, P.R. China.

###### Abstract

The outflow problem for the viscous full two-phase flow model in a half line
is investigated in the present paper. The existence, uniqueness and nonlinear
stability of the steady-state are shown respectively corresponding to the
supersonic, sonic or subsonic state at far field. This is different from the
outflow problem for the isentropic Navier-Stokes equations, where there is no
steady-state for the subsonic state. Furthermore, we obtain either exponential
time decay rates for the supersonic state or algebraic time decay rates for
supersonic and sonic states in weighted Sobolev spaces.

Key words. Full two-phase flow, outflow problem, stationary solution,
nonlinear stability.

##  1 Introduction

Two-phase flow models play important roles in applied scientific areas, for
instance, nuclear, engines, chemical engineering, medicine, oil-gas,
fluidization, waste water treatment, biomedical, liquid crystals, lubrication
[12, 24, 6, 1, 21], etc. In this paper, we consider the full two-phase flow
model which can be formally obtained from a Vlasov-Fokker-Planck equation
coupled with the compressible Navier-Stokes equations through the Chapman-
Enskog expansion [19].

We consider the initial-boundary value problem (IBVP) for the full two-phase
flow model as follows:

$\left\\{\begin{split}&\rho_{t}+(\rho u)_{x}=0,\\\ &(\rho u)_{t}+[\rho
u^{2}+p_{1}(\rho)]_{x}=\mu u_{xx}+n(v-u),\\\ &n_{t}+(nv)_{x}=0,\\\
&(nv)_{t}+[nv^{2}+p_{2}(n)]_{x}=(nv_{x})_{x}-n(v-u),~{}~{}~{}x>0,~{}t>0,\end{split}\right.$
(1.1)

where $\rho>0$ and $n>0$ stand for the densities, $u$ and $v$ are the
velocities of two fluids respectively, the constant $\mu>0$ is the viscosity
coefficient, and the pressure-density functions take forms

$p_{1}(\rho)=A_{1}\rho^{\gamma},\quad p_{2}(n)=A_{2}n^{\alpha}$ (1.2)

with $A_{1}>0,~{}A_{2}>0$, $\gamma\geq 1$ and $\alpha\geq 1$. The initial data
are given by

$(\rho,u,n,v)(0,x)=(\rho_{0},u_{0},n_{0},v_{0})(x),~{}~{}~{}~{}\underset{x\in\mathbb{R}_{+}}{\inf}\rho_{0}(x)>0,~{}~{}~{}~{}\underset{x\in\mathbb{R}_{+}}{\inf}n_{0}(x)>0,$
(1.3)
$\begin{split}&\lim_{x\to+\infty}(\rho_{0},u_{0},n_{0},v_{0})(x)=(\rho_{+},u_{+},n_{+},u_{+}),\quad\rho_{+}>0,\quad
n_{+}>0,\end{split}$ (1.4)

and the outflow boundary condition is imposed by

$(u,v)(t,0)=(u_{-},u_{-}),\quad u_{-}<0,$ (1.5)

where $\rho_{+}>0$, $n_{+}>0$, $u_{+}$ and $u_{-}<0$ are constants.

The condition $u_{-}<0$ means that the fluids flow out the region
$\mathbb{R}_{+}$ through the boundary $x=0$ with the velocity $u_{-}$, and
therefore the problem $({\ref{f}})$-$({\ref{outflow c}})$ is called the
outflow problem[22]. On the other hand, the similar problem with the case
$u_{-}>0$ is called the inflow problem[22], and the densities on the boundary
$(\rho,n)(t,0)=(\rho_{-},n_{-})$ are also imposed for the well-posedness of
the inflow problem.

It is an interesting issue to study the outflow/inflow problem. There are many
important progress made recently about the existence and nonlinear stability
of steady-states and basic waves to the outflow/inflow problem for one-phase
flow, such as compressible Navier-Stokes equations [11, 16, 27, 15, 28, 29, 3,
9, 23, 31, 8, 26, 17, 10, 32, 33, 34, 5, 18, 30]. For instance, for the inflow
problem of isentropic Navier-Stokes equations, Matsumura-Nishihara [23] showed
the existence and stability of steady-states for both subsonic and sonic cases
together with the stability of the superposition of steady-states and
rarefaction waves under small perturbation, while Fan-Liu-Wang-Zhao [5]
investigated that steady-states and rarefaction waves were nonlinear stable
under large perturbation, and Huang-Matsumura-Shi [9] obtained the stability
of superposition of steady-states and shock waves under small perturbation.
For the inflow problem of full compressible Navier-Stokes equations, Nakamura-
Nishibata[26] got the existence and stability of steady-states also for both
subsonic and sonic cases, and Qin-Wang [32, 31] and Hong-Wang [8] showed the
combination of steady-states and rarefaction waves. For the outflow problem of
isentropic Navier-Stokes equations, the existence and nonlinear stability of
steady-states for both supersonic and sonic cases under small perturbation
were proved in [16, 27, 28, 29], the convergence rates toward steady-states
for supersonic and sonic cases were investigated in [28, 29], the stability of
the superposition of steady-states and rarefaction waves were got in [17, 11].
For the outflow problem of full compressible Navier-Stokes equations,
Kawashima-Nakamura-Nishibata[15] established the existence and nonlinear
stability of steady-states under small perturbation for three cases:
supersonic, sonic and subsonic flow while there is no steady-state to the
outflow problem of isentropic Navier-Stokes equations for the subsonic case,
in addition they also gained the convergence rates toward the stationary
solutions for supersonic and sonic cases [15], and Qin[30], Wan-Wang-Zhao[33]
and Wan-Wang-Zou[34] obtained the stability of steady-states, rarefaction
waves and their combination under large perturbation.

It is of interest and challenges to investigate the outflow/inflow problem for
two-phase flow models due to the coupled motions of two phases. Although it is
rather complicated, some important progress has been made about the existence
and nonlinear stability of steady-states and basic waves to the outflow/inflow
problem for two-phase flow models [4, 35, 38, 7, 37, 20]. For instance, Yin-
Zhu[37] obtained the existence of steady-states similar to isentropic Navier-
Stokes equations[16], and the nonlinear stability and convergence rates of
steady-states for the supersonic case to the outflow problem of the drift-flux
model. For the outflow problem of the two-fluid Navier-Stokes-Poisson system,
the existence of stedy-states which is similar to that of isentropic Navier-
Stokes equations[16], and the nonlinear stability of rarefactions waves and
steady-states together with the superposition of steady-states and rarefaction
waves were proved in [4, 35]. We established existence and nonlinear stability
of steady-states to inflow problem of the model $(\ref{f})$ for supersonic,
sonic and subsonic cases in [20], which is a different phenomena compared with
the inflow problem for isentropic Navier-Stokes equations [23] and full
compressible Navier-Stokes equations[26], where there is no steady-state for
the supersonic case.

However, there is no result about the existence and nonlinear stability of
steady-states for the outflow problem $(\ref{f})$-$(\ref{outflow c})$. The
main purpose of this paper is to prove the existence and nonlinear stability
of steady-states for the supersonic, sonic and subsonic case, and obtain
either exponential time decay rates for the supersonic flow or algebraic time
decay rates for both supersonic and sonic flows. Contrary to the isentropic
Navier-Stokes equations, the steady-state to the IBVP
$(\ref{f1})$-$(\ref{outflow c})$ exists for the subsonic case.

The steady-state
$(\widetilde{\rho},\widetilde{u},\widetilde{n},\widetilde{v})(x)$ to the
outflow problem $(\ref{f})$-$(\ref{outflow c})$ satisfies the following system

$\left\\{\begin{split}&(\widetilde{\rho}\widetilde{u})_{x}=0,\\\
&[\widetilde{\rho}\widetilde{u}^{2}+p_{1}(\widetilde{\rho})]_{x}=(\mu\widetilde{u}_{x})_{x}+\widetilde{n}(\widetilde{v}-\widetilde{u}),\\\
&(\widetilde{n}\widetilde{v})_{x}=0,\\\
&[\widetilde{n}\widetilde{v}^{2}+p_{2}(\widetilde{n})]_{x}=(\widetilde{n}\widetilde{v}_{x})_{x}-\widetilde{n}(\widetilde{v}-\widetilde{u}),~{}~{}~{}x>0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\end{split}\right.$
(1.6)

with the boundary conditions and spatial far field conditions

$\begin{split}&(\widetilde{u},\widetilde{v})(0)=(u_{-},u_{-}),\quad\lim_{x\to\infty}(\widetilde{\rho},\widetilde{u},\widetilde{n},\widetilde{v})(x)=(\rho_{+},u_{+},n_{+},u_{+}),\quad\underset{x\in\mathbb{R}_{+}}{\inf}\widetilde{\rho}(x)>0,\quad\underset{x\in\mathbb{R}_{+}}{\inf}\widetilde{n}(x)>0.\end{split}$
(1.7)

Integrating $(\ref{stationary f})_{1}$ and $(\ref{stationary f})_{3}$ over
$(x,+\infty)$, we have

$\begin{split}&\widetilde{u}=\frac{\rho_{+}}{\widetilde{\rho}}u_{+},\quad\widetilde{v}=\frac{n_{+}}{\widetilde{n}}u_{+},\end{split}$
(1.8)

which implies that the following relationship

$u_{+}=\frac{\widetilde{n}(0)}{n_{+}}u_{-}=\frac{\widetilde{\rho}(0)}{\rho_{+}}u_{-}<0$
(1.9)

is the necessary property of the steady-state to the boundary value problem
(BVP) $(\ref{stationary f})$-$(\ref{stationary boundary c})$.

Define the Mach number $M_{+}$ and the sound speed $c_{+}$ as below

$M_{+}:=\frac{|u_{+}|}{c_{+}},\quad
c_{+}:=(\frac{A_{1}\gamma\rho_{+}^{\gamma}+A_{2}\alpha
n_{+}^{\alpha}}{\rho_{+}+n_{+}})^{\frac{1}{2}}.$ (1.10)

Then, we have the following results about the existence and uniqueness of the
steady-state.

######  Theorem 1.1.

Let $\delta:=|u_{-}-u_{+}|>0$ and $u_{+}<0$ hold. Then there exists a set
$\Omega_{-}\subset\mathbb{R}_{-}$ such that if $u_{-}\in\Omega_{-}$ and
$\delta$ sufficiently small, there exists a unique strong solution
$(\widetilde{\rho},\widetilde{u},\widetilde{n},\widetilde{v})(x)$ to the BVP
$(\ref{stationary f})$-$(\ref{stationary boundary c})$ which satisfies either
for the supersonic or subsonic case $M_{+}\neq 1$ that

$~{}~{}~{}~{}~{}~{}|\partial_{x}^{k}(\widetilde{\rho}-\rho_{+},\widetilde{u}-u_{+},\widetilde{n}-n_{+},\widetilde{v}-u_{+})|\leq
C_{1}\delta e^{-c_{0}x},~{}~{}~{}k=0,1,2,3,$ (1.11)

or for the sonic case $M_{+}=1$ that

$|\partial_{x}^{k}(\widetilde{\rho}-\rho_{+},\widetilde{u}-u_{+},\widetilde{n}-n_{+},\widetilde{v}-u_{+})|\leq
C_{2}\frac{\delta^{k+1}}{(1+\delta x)^{k+1}},~{}~{}~{}k=0,1,2,3,$ (1.12)

and

$(\widetilde{u}_{x},\widetilde{v}_{x})=(a\sigma^{2}(x),a\sigma^{2}(x))+O(|\sigma(x)|^{3}),$
(1.13)

where $\sigma(x)$ is a smooth function satisfying
$\sigma_{x}=-a\sigma^{2}+O(|\sigma|^{3})$ and

$c_{1}\frac{\delta}{1+\delta x}\leq\sigma(x)\leq C_{3}\frac{\delta}{1+\delta
x},~{}~{}~{}~{}|\partial^{k}_{x}\sigma(x)|\leq
C_{3}\frac{\delta^{k+1}}{(1+\delta x)^{k+1}},~{}~{}~{}k=0,1,2,3,$ (1.14)

and $C_{i}>0$, $i=1,2,3$, $c_{0}>0$, $c_{1}>0$, and $a>0$ are positive
constants.

######  Remark 1.1.

Due to the drag force term, the existence of steady-states to the IBVP
$(\ref{f})$-$(\ref{outflow c})$ is obtained even for the subsonic case
$M_{+}<1$, which is different from that of the isentropic Navier-Stokes
equations [16]. Moreover, the existence of steady-states to the IBVP
$(\ref{f})$-$(\ref{outflow c})$ is similar to that of the full compressible
Navier-Stokes equations [15].

Then, we have the nonlinear stability of the steady-state to the IBVP
$(\ref{f})$-$(\ref{outflow c})$ for supersonic, sonic and subsonic cases.

######  Theorem 1.2.

Let the same conditions in Theorem 1.1 hold and assume that it holds

$|p^{\prime}_{1}({\rho_{+}})-p^{\prime}_{2}({n_{+}})|\leq\sqrt{2}|u_{+}|\min\\{(1+\frac{\rho_{+}}{n_{+}})[(\gamma-1)p^{\prime}_{1}(\rho_{+})]^{\frac{1}{2}},(1+\frac{n_{+}}{\rho_{+}})[(\alpha-1)p^{\prime}_{2}(n_{+})]^{\frac{1}{2}}\\}$
(1.15)

for the sonic case $M_{+}=1.$ Then, there exists a small positive constant
$\varepsilon_{0}>0$ such that if

$\|(\rho_{0}-\widetilde{\rho},u_{0}-\widetilde{u},n_{0}-\widetilde{n},v_{0}-\widetilde{v})\|_{H^{1}}+\delta\leq\varepsilon_{0},$
(1.16)

the IBVP $(\ref{f})$-$(\ref{outflow c})$ has a unique global solution
$(\rho,u,n,v)(t,x)$ satisfying

$\left\\{\begin{split}&(\rho-\widetilde{\rho},u-\widetilde{u},n-\widetilde{n},v-\widetilde{v})\in
C([0,+\infty);H^{1}),\\\ &(\rho-\widetilde{\rho},n-\widetilde{n})_{x}\in
L^{2}([0,+\infty);L^{2}),\\\ &(u-\widetilde{u},v-\widetilde{v})_{x}\in
L^{2}([0,+\infty);H^{1}),\end{split}\right.$

and

$\lim_{t\to+\infty}\sup_{x\in\mathbb{R_{+}}}|(\rho-\widetilde{\rho},u-\widetilde{u},n-\widetilde{n},v-\widetilde{v})(t,x)|=0.$
(1.17)

In addition, we have the time convergence rates of the global solution to the
IBVP $(\ref{f})$-$(\ref{outflow c})$ for both supersonic and sonic cases.

######  Theorem 1.3.

Assume that the same conditions in Theorem 1.1 hold. Then, the following
results hold.

  * (i)

For $M_{+}>1$ and $\lambda>0$, if the initial data satisfy

$(1+x)^{\frac{\lambda}{2}}(\rho_{0}-\widetilde{\rho},u_{0}-\widetilde{u},n_{0}-\widetilde{u},n-\widetilde{n})\in
L^{2}(\mathbb{R_{+}})$

and

$\|(\rho_{0}-\widetilde{\rho},u_{0}-\widetilde{u},n_{0}-\widetilde{n},v_{0}-\widetilde{v})\|_{H^{1}}+\delta\leq\varepsilon_{0},$
(1.18)

for a small positive constant $\varepsilon_{0}>0$, then the solution
$(\rho,u,n,v)(t,x)$ to the IBVP $(\ref{f})$-$(\ref{outflow c})$ satisfies

$\displaystyle~{}~{}~{}~{}\|(\rho-\widetilde{\rho},u-\widetilde{u},n-\widetilde{n},v-\widetilde{v})(t)\|_{L^{\infty}}\leq
C_{4}\delta_{0}(1+t)^{-\frac{\lambda}{2}},$ (1.19)

where $C_{4}>0$ and
$\delta_{0}:=\|(\rho_{0}-\widetilde{\rho},u_{0}-\widetilde{u},n_{0}-\widetilde{n},v_{0}-\widetilde{v})\|_{H^{1}}+\|(1+x)^{\frac{\lambda}{2}}(\rho_{0}-\widetilde{\rho},u_{0}-\widetilde{u},n_{0}-\widetilde{n},v_{0}-\widetilde{v})\|_{L^{2}}$
are constants independent of time.

  * (ii)

For $M_{+}=1$,
$1\leq\lambda<\lambda^{*},\lambda^{*}:=2+\sqrt{8+\frac{1}{1+b^{2}}}$ and
$b:=\frac{\rho_{+}(u_{+}^{2}-p^{\prime}_{1}(\rho_{+}))}{|u_{+}|\sqrt{(\mu+n_{+})n_{+}}}$,
if for arbitrary $\nu\in(0,\lambda]$, there exists a small positive constant
$\varepsilon_{0}>0$ such that

$\|\sigma^{-\frac{\lambda}{2}}(\rho_{0}-\widetilde{\rho},u_{0}-\widetilde{u},n_{0}-\widetilde{n},v_{0}-\widetilde{v})\|_{H^{1}}+\delta^{\frac{1}{2}}\leq\varepsilon_{0},$
(1.20)

then the IBVP $(\ref{f})$-$(\ref{outflow c})$ has a unique global solution
$(\rho,u,n,v)(t,x)$ satisfying

$\left\\{\begin{aligned}
&\sigma^{-\frac{\nu}{2}}(\rho-\widetilde{\rho},u-\widetilde{u},n-\widetilde{n},v-\widetilde{v})\in
C([0,+\infty):H^{1}),\\\
&\sigma^{-\frac{\nu}{2}}(\rho-\widetilde{\rho},n-\widetilde{n})_{x}\in
L^{2}([0,+\infty);L^{2}),\\\
&\sigma^{-\frac{\nu}{2}}(u-\widetilde{u},v-\widetilde{v})_{x}\in
L^{2}([0,+\infty);H^{1}),\end{aligned}\right.$ (1.21)

and

$\|\sigma^{-\frac{\nu}{2}}(\rho-\widetilde{\rho},u-\widetilde{u},n-\widetilde{n},v-\widetilde{v})(t)\|_{H^{1}}\leq
C_{5}\delta_{1}(1+t)^{-\frac{\lambda-\nu}{4}},$ (1.22)

where $\sigma(x)$ is defined by $(\ref{sig})$ satisfying $(\ref{sg0})$,
$C_{5}>0$ is a positive constant independent of time, and
$\delta_{1}:=\|\sigma^{-\frac{\lambda}{2}}(\rho_{0}-\widetilde{\rho},u_{0}-\widetilde{u},n_{0}-\widetilde{n},v_{0}-\widetilde{v})\|_{H^{1}}$
is a constant.

######  Remark 1.2.

For $M_{+}>1$, the exponential time convergence rates of the global solution
to the IBVP $(\ref{f})$-$(\ref{outflow c})$ can be established. Indeed, assume
that $M_{+}>1$, $u_{+}<0$ and a certain positive constant $\lambda>0$ hold.
For a certain positive constant $\kappa\in(0,\lambda]$, there exists a small
positive constant $\varepsilon_{0}>0$ such that if
$e^{\frac{\lambda}{2}x}(n_{0}-\widetilde{n},\rho_{0}-\widetilde{\rho},u_{0}-\widetilde{u},n_{0}-\widetilde{n},v_{0}-\widetilde{v})\in
L^{2}(\mathbb{R_{+}})$ and

$\|(\rho_{0}-\widetilde{\rho},u_{0}-\widetilde{u},n_{0}-\widetilde{n},v_{0}-\widetilde{v}))\|_{H^{1}}+\delta\leq\varepsilon_{0},$

then the solution $(\rho,u,n,v)(t,x)$ to the IBVP $(\ref{f})$-$(\ref{outflow
c})$ satisfies

$\|(\rho-\widetilde{\rho},u-\widetilde{u},n-\widetilde{n},v-\widetilde{v})(t)\|_{H^{1}}\leq
C_{6}\delta_{2}e^{-\frac{\kappa_{1}}{2}t},$ (1.23)

where $C_{6}>0$ and $\kappa_{1}\ll\kappa$ are positive constants independent
of time, and
$\delta_{2}:=\|(\rho_{0}-\widetilde{\rho},u_{0}-\widetilde{u},n_{0}-\widetilde{n},v_{0}-\widetilde{v})\|_{H^{1}}+\|e^{\frac{\lambda}{2}x}(\rho_{0}-\widetilde{\rho},u_{0}-\widetilde{u},n_{0}-\widetilde{n},v_{0}-\widetilde{v})\|_{L^{2}}$
is a constant.

The proof of $(\ref{M_{+}>1 exp d})$ can be obtained by similar arguments as
for $(\ref{M_{+}>1 algebra d})$. The details are omitted.

######  Remark 1.3.

In Theorem 1.3, we remove the restriction $(\ref{p small})$, and obtain the
nonlinear stability of steady-states and time decay rates of the solution to
the IBVP $(\ref{f})$-$(\ref{outflow c})$ for $M_{+}=1$ with the weighted
energy method. Moreover, if $p^{\prime}_{1}(\rho_{+})=p^{\prime}_{2}(n_{+})$,
time decay rates of the solution to the IBVP $(\ref{f})$-$(\ref{outflow c})$
for $M_{+}=1$ are the same as that of isentropic Navier-Stokes equations [28].

We explain main strategies to prove Theorems 1.1-1.3. The system
$(\ref{f})$-$(\ref{outflow c})$ can be viewed as two compressible isentropic
Navier-Stokes equations coupled with each other through the drag force
relaxation mechanisms. Different from the isentropic Navier-Stokes
equations[16], we can not reformulate two momentum equations $(\ref{f})_{2}$
and $(\ref{f})_{4}$ into conservation forms due to the influence of drag
force, which implies that the steady-state satisfies the system
$(\ref{stationary f})$-$(\ref{stationary boundary c})$ consisting of a first-
order and a second-order ordinary differential equations instead of only a
first-order ordinary differential equation in [16], and it is not
straightforward to apply the center manifold theory[2]. To overcome the
difficulty, we introduce a new variable $\widetilde{w}:=\widetilde{u}_{x}$,
get the estimate $|\widetilde{u}_{x}(0)|\leq C|u_{-}-u_{+}|$ in Lemma 2.1, and
then rewrite the system $(\ref{stationary f})$ into the $3\times 3$ system
$(\ref{sf bar})$ of autonomous ordinary differential equations. Since the
condition $p^{\prime}_{1}(\rho_{+})=p^{\prime}_{2}(n_{+})$ is not necessary,
we need subtle analysis to obtain the sign of ${\rm Re}\lambda_{i},i=1,2,3$,
where $\lambda_{i},i=1,2,3$ are three eigenvalues of the linearized $3\times
3$ system of $(\ref{sf bar})$. It should be noticed that the linearized system
of $(\ref{sf bar})$ has at least one eigenvalue with negative real part due to
the effect of drag force, so that we obtain the existence of steady-states for
the supersonic, sonic and subsonic case in Theorem 1.1, which is different
from the outflow problem of the isentropic Navier-Stokes system[16], where
there is no steady-state for the subsonic case.

We establish the uniform estimates of the perturbation
$(\phi,\psi,\bar{\phi},\bar{\psi}):=(\rho-\widetilde{\rho},u-\widetilde{u},n-\widetilde{n},v-\widetilde{v})$
to prove the nonlinear stability of steady-states for the supersonic
$M_{+}>1$, sonic case $M_{+}=1$ and subsonic case $M_{+}<1$. For $M_{+}=1$, it
is easy to check that
$(\widetilde{\rho}\widetilde{u}^{2}+p_{1}(\widetilde{\rho}))_{x}$ and
$(\widetilde{n}\widetilde{v}^{2}+p_{2}(\widetilde{n}))_{x}$ decay slower than
$\widetilde{u}_{xx}$ for $p^{\prime}_{1}(\rho_{+})\neq p^{\prime}_{2}(n_{+})$
owing to $(\ref{stationary f})_{2}$, $(\ref{stationary f})_{4}$ and
$(\ref{sigma})$, which implies the term

$\int
R_{2}dx:=\int\phi\psi\frac{(\widetilde{\rho}\widetilde{u}^{2}+p_{1}(\widetilde{\rho}))_{x}}{\widetilde{\rho}}+\bar{\phi}\bar{\psi}\frac{(\widetilde{n}\widetilde{v}^{2}+p_{2}(\widetilde{n}))_{x}}{\widetilde{n}}dx$
(1.24)

can not be controlled directly as in [16]. With the help of
$\widetilde{u}_{x}\geq 0$ and $\widetilde{v}_{x}\geq 0$, we turn to deal with
the terms

$\int
R_{2}dx+\int(\rho\psi^{2}+p_{1}(\rho)-p_{1}(\widetilde{\rho})-p^{\prime}_{1}(\widetilde{\rho})\phi)\widetilde{u}_{x}dx+(n\bar{\psi}^{2}+p_{2}(n)-p_{2}(\widetilde{n})-p^{\prime}_{2}(\widetilde{n}))\widetilde{v}_{x}dx,$
(1.25)

the leading terms of which can be rewritten as two positive semidefinite 2
variable quadratic forms $(\ref{r_1'})$ under the condition $(\ref{p small})$.

By the weighted energy method, we get the exponential or algebraic time decay
rates for the supersonic case $M_{+}>1$ if the initial perturbation belongs to
the exponential or algebraic weighted Sobolev space, and obtain algebraic time
decay rates for the sonic case $M_{+}=1$. To get the basic weighted energy
estimates, we use $(\ref{M_{+}>1 stationary solution d})$ and the dissipation
on the relaxation friction term $\bar{\psi}-\psi$, and decompose $\psi$ as
$\psi=\bar{\psi}+(\psi-\bar{\psi})$, as motivated by Li-Wang-Wang[19]. Due to
the algebraic decay $(\ref{sigma})$ of steady-states, convergence rates of
steady-states for the sonic case $M_{+}=1$ is worse than that of the
supersonic case $M_{+}>1$. It is necessary to use the delicate algebraic decay
$(\ref{sigma})$-$(\ref{sig})$ and the dissipation on the drag force
$\bar{\psi}-\psi$, decompose $\psi$ as $\psi=\bar{\psi}+(\psi-\bar{\psi})$,
and obtain more delicate estimates to get the convergence rates for $M_{+}=1$.
It should be noticed that we make full use of the dissipation on drag force
term and the viscous terms, and take a linear coordinate transformation

$\begin{pmatrix}\phi\\\ \bar{\phi}\\\
\bar{\psi}\end{pmatrix}=\boldsymbol{P}\begin{pmatrix}\hat{\rho}\\\ \hat{n}\\\
\hat{v}\end{pmatrix},\quad\quad\quad{\rm
with~{}a~{}invertible~{}matrix~{}}\boldsymbol{P},$ (1.26)

to gain the 3 variable quadratic form
$\hat{\lambda}_{1}\hat{\rho}^{2}+\hat{\lambda}_{2}\hat{n}^{2}$ with
$\hat{\lambda}_{1},\hat{\lambda}_{2}>0$, which plays an important role in
basic weighted energy estimates together with some crucial cancellations. In
fact, we obtain the algebraic time decay rates of the solution to IBVP
$(\ref{f})$-$(\ref{outflow c})$ for $M_{+}=1$ for the initial perturbation
satisfying
$\sigma^{-\frac{\lambda}{2}}(\rho_{0}-\widetilde{\rho},u_{0}-\widetilde{u},n_{0}-\widetilde{n},v_{0}-\widetilde{v})\in
L^{2}(\mathbb{R}_{+})$, with
$\lambda<\lambda^{*},\lambda^{*}:=2+\sqrt{8+\frac{1}{1+b^{2}}}$,
$b=\frac{\rho_{+}(u_{+}^{2}-p^{\prime}_{1}(\rho_{+}))}{|u_{+}|\sqrt{(\mu+n_{+})n_{+}}}$
and the function $\sigma\geq 0$ satisfying $(\ref{sig})$. This is an
interesting phenomena describing the influences of two fluids on each other
somehow and it should be emphasized that $\lambda^{*}=5$ is the same as that
of the isentropic Navier-Stokes equations [28] for
$u_{+}^{2}=p^{\prime}_{1}(\rho_{+})=p^{\prime}_{2}(n_{+})$.

Notation. We denote by $\|\cdot\|_{L^{p}}$ the norm of the usual Lebesgue
space $L^{p}=L^{p}({\mathbb{R}_{+}})$, $1\leq p\leq\infty$. And if $p=2$, we
write $\|\cdot\|_{L^{p}(\mathbb{R}_{+})}=\|\cdot\|$. $H^{s}(\mathbb{R}_{+})$
stands for the standard $s$-th Sobolev space over $\mathbb{R}_{+}$ equipped
with its norm

$\|f\|_{H^{s}(\mathbb{R}_{+})}=\|f\|_{s}:=(\sum_{i=0}^{s}\|\partial^{i}f\|^{2})^{\frac{1}{2}}.$

$C([0,T];H^{1}(\mathbb{R}_{+}))$ represents the space of continuous functions
on the interval $[0,T]$ with values in $H^{s}(\mathbb{R}_{+})$.
$L^{2}([0,T];\mathcal{B})$ denotes the space of $L^{2}$ functions on the
interval $[0,T]$ with values in Banach space $\mathcal{B}$. For a scalar
function $W(x)>0$, the weighted $L^{2}(\mathbb{R}_{+})$ and
$H^{1}(\mathbb{R}_{+})$ spaces are defined as follows:

$\displaystyle L^{2}_{W}(\mathbb{R}_{+}):=$
$\displaystyle\left\\{\begin{matrix}~{}f\in
L^{2}(\mathbb{R}_{+})~{}|\end{matrix}\right.~{}\|f\|_{L^{2}_{W}}:=(\int_{\mathbb{R}_{+}}W(x)f^{2}dx)^{\frac{1}{2}}<+\infty\left.\begin{matrix}~{}\end{matrix}\right\\},$
$\displaystyle H^{1}_{W}(\mathbb{R}_{+}):=$
$\displaystyle\left\\{\begin{matrix}~{}f\in
H^{1}(\mathbb{R}_{+})~{}|\end{matrix}\right.~{}\|f\|_{H^{1}_{W}}:=(\sum_{i=0}^{1}\|\partial^{i}f\|^{2}_{L^{2}_{W}})^{\frac{1}{2}}<+\infty\left.\begin{matrix}~{}\end{matrix}\right\\}.$

For a scalar function $W_{a,\nu}:=(1+x)^{\nu}$ with $\nu\geq 0$, we denote
$\|f\|_{a,\nu}:=\|(1+x)^{\frac{\nu}{2}}f\|$.

The rest of this paper will be organized as follows. We prove the existence
and uniqueness of steady-states in Section 2, get the nonlinear stability of
steady-states in Section 3 for supersonic, sonic and subsonic cases, and
obtain convergence rates of steady-states for the supersonic flow in
Subsection 4.1 and the sonic flow in Subsection 4.2.

##  2 Existence of Steady-State

We prove Theorem 1.1 on the existence and uniqueness of steady-states to the
BVP $(\ref{stationary f})$-$(\ref{stationary boundary c})$ with $u_{+}<0$ and
$\delta$ sufficiently small as follows. In order to apply the center manifold
theory [2], it is necessary to get the bounds of $\widetilde{u}_{x}(0)$ or
$\widetilde{v}_{x}(0)$.

######  Lemma 2.1.

Assume that $u_{+}<0$ and $\delta=|u_{-}-u_{+}|$ hold with $\delta$
sufficiently small. Then the steady-state
$(\widetilde{\rho},\widetilde{u},\widetilde{n},\widetilde{v})$ to the BVP
$({\ref{stationary f}})$-$(\ref{stationary boundary c})$ satisfies

$|\widetilde{u}_{x}(0)|\leq C|u_{-}-u_{+}|,\quad|\widetilde{v}_{x}(0)|\leq
C|u_{-}-u_{+}|,$ (2.1)

where $C>0$ is a positive constant.

###### Proof.

Due to $\widetilde{\rho}=\frac{\rho_{+}u_{+}}{\widetilde{u}}$ and
$\widetilde{n}=\frac{n_{+}u_{+}}{\widetilde{v}}$, we have

$\left\\{\begin{aligned}
&(\rho_{+}u_{+}\widetilde{u}+A_{1}\rho^{\gamma}_{+}u^{\gamma}_{+}\widetilde{u}^{-\gamma})_{x}=(\mu\widetilde{u}_{x})_{x}+\frac{n_{+}u_{+}}{\widetilde{v}}(\widetilde{v}-\widetilde{u}),\\\
&(n_{+}u_{+}\widetilde{v}+A_{2}n^{\alpha}_{+}u^{\alpha}_{+}\widetilde{v}^{-\alpha})_{x}=(n_{+}u_{+}\frac{\widetilde{v}_{x}}{\widetilde{v}})_{x}-\frac{n_{+}u_{+}}{\widetilde{v}}(\widetilde{v}-\widetilde{u}).\end{aligned}\right.$
(2.2)

Adding $(\ref{sf})_{1}$ to $(\ref{sf})_{2}$ and integrating the resulted
equation over $(0,+\infty)$ lead to

$\mu\widetilde{u}_{x}(0)+\frac{n_{+}u_{+}}{u_{-}}\widetilde{v}_{x}(0)=\frac{1}{u_{+}}[(\rho_{+}+n_{+})u^{2}_{+}-(A_{1}\gamma\rho^{\gamma}_{+}+A_{2}\alpha
n^{\alpha}_{+})](u_{-}-u_{+})+O(|u_{-}-u_{+}|^{2}).$ (2.3)

With the help of $(\ref{wt u v})$, we multiply $(\ref{sf})_{1}$ by
$\widetilde{u}$, $(\ref{sf})_{2}$ by $\widetilde{v}$ respectively, then
integrate the summation of the resulted equations over $(0,+\infty)$ to gain

$\displaystyle\int_{0}^{+\infty}(\mu\widetilde{u}_{x}^{2}+n_{+}u_{+}\frac{\widetilde{v}^{2}_{x}}{\widetilde{v}})dx+\int_{0}^{+\infty}\frac{n_{+}u_{+}}{\widetilde{v}}(\widetilde{v}-\widetilde{u})^{2}dx$
(2.4) $\displaystyle=$ $\displaystyle-
u_{-}[\mu\widetilde{u}_{x}(0)+n_{+}u_{+}\frac{\widetilde{v}_{x}(0)}{u_{-}}]+[(\rho_{+}+n_{+})u_{+}^{2}-(A_{1}\gamma\rho^{\gamma}_{+}+A_{2}\alpha
n^{\alpha}_{+})](u_{-}-u_{+})+O(|u_{-}-u_{+}|^{2})$ $\displaystyle=$
$\displaystyle O(|u_{-}-u_{+}|^{2}).$

Multiplying $(\ref{sf})_{2}$ by $\frac{\widetilde{v}_{x}}{\widetilde{v}}$ and
then integrating the resulted equation over $(0,\infty)$ yield

$-n_{+}u_{+}\frac{\widetilde{v}^{2}_{x}(0)}{2u^{2}_{-}}+\int_{0}^{+\infty}A_{2}\alpha
n^{\alpha}_{+}\frac{u_{+}^{\alpha}}{\widetilde{v}^{\alpha+2}}\widetilde{v}^{2}_{x}dx=\int_{0}^{+\infty}\frac{n_{+}u_{+}}{\widetilde{v}}(\widetilde{v}-\widetilde{u})\frac{\widetilde{v}_{x}}{\widetilde{v}}dx+\int_{0}^{+\infty}n_{+}u_{+}\frac{\widetilde{v}^{2}_{x}}{\widetilde{v}}dx$
(2.5)

We estimate terms in the right hand side of $(\ref{sf'b3})$. With
$\inf\limits_{x\in\mathbb{R}_{+}}\widetilde{n}>0$, $\alpha\geq 1$ and
$(\ref{wt v-u})$, we have

$\displaystyle\int_{0}^{+\infty}\frac{n_{+}u_{+}}{\widetilde{v}}(\widetilde{v}-\widetilde{u})\frac{\widetilde{v}_{x}}{\widetilde{v}}dx+\int_{0}^{+\infty}n_{+}u_{+}\frac{\widetilde{v}^{2}_{x}}{\widetilde{v}}dx$
(2.6) $\displaystyle\leq$ $\displaystyle
C\int_{0}^{+\infty}\frac{n_{+}u_{+}}{\widetilde{v}}(\widetilde{v}-\widetilde{u})^{2}dx+\frac{1}{4}\int_{0}^{+\infty}A_{2}\alpha
n^{\alpha}_{+}\frac{u_{+}^{\alpha}}{\widetilde{v}^{\alpha+2}}\widetilde{v}^{2}_{x}dx$
$\displaystyle\leq$ $\displaystyle
C|u_{-}-u_{+}|^{2}+\frac{1}{4}\int_{0}^{+\infty}A_{2}\alpha
n^{\alpha}_{+}\frac{u_{+}^{\alpha}}{\widetilde{v}^{\alpha+2}}\widetilde{v}^{2}_{x}dx.$

Combining $(\ref{sf'b3})$ and $(\ref{sf''b2})$, we get

$|\widetilde{v}_{x}(0)|\leq C|u_{-}-u_{+}|,\quad|\widetilde{u}_{x}(0)|\leq
C|u_{-}-u_{+}|.$ (2.7)

∎

Then, we can prove Theorem 1.1 with the above lemma. For $(\ref{stationary
f})_{2}$ and $(\ref{stationary f})_{4}$, using
$\widetilde{\rho}=\frac{\rho_{+}u_{+}}{\widetilde{u}}$,
$\widetilde{n}=\frac{n_{+}u_{+}}{\widetilde{v}}$ and integrating the summation
of $(\ref{stationary f})_{2}$ and $(\ref{stationary f})_{4}$ over
$(x,+\infty)$, we have

$\left\\{\begin{split}&\widetilde{v}_{x}=\frac{\widetilde{v}}{n_{+}u_{+}}[\rho_{+}u_{+}(\widetilde{u}-u_{+})+A_{1}\rho^{\gamma}_{+}(\frac{u^{\gamma}_{+}}{\widetilde{u}^{\gamma}}-1)+n_{+}u_{+}(\widetilde{v}-u_{+})+A_{2}n^{\alpha}_{+}(\frac{u^{\alpha}_{+}}{\widetilde{v}^{\alpha}}-1)-\mu\widetilde{u}_{x}],\\\
&\widetilde{u}_{xx}=\frac{1}{\mu}[(\rho_{+}u_{+}-A_{1}\rho^{\gamma}_{+}u^{\gamma}_{+}\widetilde{u}^{-\gamma-1})\widetilde{u}_{x}-n_{+}u_{+}(1-\frac{\widetilde{u}}{\widetilde{v}})].\end{split}\right.$
(2.8)

Define $\widetilde{w}:=\widetilde{u}_{x}$ and
$\bar{U}:=(\bar{u},\bar{w},\bar{v})^{\rm
T}:=(\widetilde{u}-u_{+},\widetilde{w},\widetilde{v}-u_{+})^{\rm T}$. The
system $(\ref{2sf})$ can be reformulated into the autonomous system as follows

$\left\\{\begin{aligned}
&\bar{U}_{x}=\boldsymbol{J_{+}}\bar{U}+(0,\bar{g}_{2}(\bar{U}),\bar{g}_{3}(\bar{U}))^{\rm
T},\\\ &\bar{U}_{-}:=(\bar{u},\bar{w},\bar{v})^{\rm
T}(0)=(u_{-}-u_{+},\widetilde{u}_{x}(0),u_{-}-u_{+})^{\rm
T},~{}\lim_{x\to\infty}\bar{U}^{\rm T}=(0,0,0),\end{aligned}\right.$ (2.9)

where

$\displaystyle\boldsymbol{J_{+}}=$ $\displaystyle\begin{pmatrix}0&1&0\\\
\frac{n_{+}}{\mu}&\frac{\rho_{+}u_{+}^{2}-A_{1}\gamma\rho^{\gamma}_{+}}{\mu
u_{+}}&-\frac{n_{+}}{\mu}\\\
\frac{\rho_{+}u_{+}^{2}-A_{1}\gamma\rho^{\gamma}_{+}}{n_{+}u_{+}}&-\frac{\mu}{n_{+}}&\frac{n_{+}u_{+}^{2}-A_{2}\alpha
n^{\alpha}_{+}}{n_{+}u_{+}}\end{pmatrix},$ (2.10)
$\displaystyle\bar{g}_{2}(\bar{U})=$
$\displaystyle\frac{1}{2}(2\frac{n_{+}}{\mu}\frac{1}{u_{+}}\bar{v}^{2}-2\frac{n_{+}}{\mu}\frac{1}{u_{+}}\bar{u}\bar{v}+2\frac{A_{1}\gamma(\gamma+1)\rho^{\gamma}_{+}}{\mu
u^{2}_{+}}\bar{w}\bar{u})+O(|\bar{U}|^{3}),$ (2.11)
$\displaystyle\bar{g}_{3}(\bar{U})=$
$\displaystyle\frac{1}{2}[\frac{A_{1}\gamma(\gamma+1)\rho^{\gamma}_{+}}{n_{+}u^{2}_{+}}\bar{u}^{2}+2\frac{\rho_{+}u_{+}^{2}-A_{1}\gamma\rho^{\gamma}_{+}}{n_{+}u^{2}_{+}}\bar{v}\bar{u}+(2\frac{n_{+}u_{+}^{2}-A_{2}\alpha
n^{\alpha}_{+}}{n_{+}u^{2}_{+}}$
$\displaystyle+\frac{A_{2}\alpha(\alpha+1)n^{\alpha}_{+}}{n_{+}u^{2}_{+}})\bar{v}^{2}-2\frac{\mu}{n_{+}u_{+}}\bar{w}\bar{v}]+O(|\bar{U}|^{3}).$
(2.12)

Three eigenvalues $\lambda_{1},\lambda_{2},\lambda_{3}$ of matrix
$\boldsymbol{J_{+}}$ satisfy

$\left\\{\begin{aligned}
&\lambda_{1}\lambda_{2}\lambda_{3}=-\frac{(\rho_{+}+n_{+})u^{2}_{+}-(A_{1}\gamma\rho^{\gamma}_{+}+A_{2}\alpha
n^{\alpha}_{+})}{\mu u_{+}},\\\
&\lambda_{1}+\lambda_{2}+\lambda_{3}=\frac{\rho_{+}u^{2}_{+}-A_{1}\gamma\rho^{\gamma}_{+}}{\mu
u_{+}}+\frac{n_{+}u^{2}_{+}-A_{2}\alpha n^{\alpha}_{+}}{n_{+}u_{+}},\\\
&\lambda_{1}\lambda_{2}+\lambda_{1}\lambda_{3}+\lambda_{2}\lambda_{3}=\rho_{+}\frac{(u_{+}^{2}-A_{1}\gamma\rho_{+}^{\gamma-1})(u_{+}^{2}-A_{2}\alpha
n_{+}^{\alpha-1})}{\mu u^{2}_{+}}-1-\frac{n_{+}}{\mu}.\end{aligned}\right.$
(2.13)

If $M_{+}>1$, it is easy to obtain $\lambda_{1}\lambda_{2}\lambda_{3}>0$ and
$u_{+}^{2}>\min\\{A_{1}\gamma\rho_{+}^{\gamma-1},A_{2}\alpha
n_{+}^{\alpha-1}\\}$. Without loss of generality, we assume
$A_{1}\gamma\rho_{+}^{\gamma-1}\geq A_{2}\alpha n_{+}^{\alpha-1}$. Moreover,
we have

$\lambda_{1}+\lambda_{2}+\lambda_{3}<0{~{}\rm
for~{}}u_{+}^{2}>A_{1}\gamma\rho_{+}^{\gamma-1},\quad\lambda_{1}\lambda_{2}+\lambda_{1}\lambda_{3}+\lambda_{2}\lambda_{3}<0{~{}\rm
for~{}}A_{2}\alpha n_{+}^{\alpha-1}\leq u^{2}_{+}\leq
A_{1}\gamma\rho_{+}^{\gamma-1},$ (2.14)

which can imply ${\rm Re}\lambda_{1}<0,~{}{\rm Re}\lambda_{2}<0$ and
$\lambda_{3}>0$ for $M_{+}>1$. Using similar arguments, we have the following
results:

$\left\\{\begin{split}&{\rm if}~{}M_{+}>1,{\rm then}~{}{\rm
Re}\lambda_{1}<0,{\rm Re}\lambda_{2}<0,\lambda_{3}>0,\\\ &{\rm
if}~{}M_{+}<1,{\rm then}~{}{\rm Re}\lambda_{1}>0,{\rm
Re}\lambda_{2}>0,\lambda_{3}<0,\\\ &{\rm if}~{}M_{+}=1,{\rm
then}~{}\lambda_{1}>0,\lambda_{2}<0,\lambda_{3}=0.\end{split}\right.$ (2.15)

Then, applying the center manifold theory [2], it is not difficult to show the
supersonic or subsonic case $M_{+}\neq 1$ in Theorem 1.1 if $\delta$ is small.
Finally, we prove the sonic case $M_{+}=1$ in Theorem 1.1 which implies
$\lambda_{1}>0,\lambda_{2}<0,\lambda_{3}=0$. The eigenvectors
$r_{1},r_{2},r_{3}$ of $\lambda_{1},\lambda_{2},\lambda_{3}$ are obtained
respectively as follows

$r_{1}=\begin{pmatrix}1\\\ \lambda_{1}\\\
-\frac{\mu}{n_{+}}(\lambda_{1}^{2}-\frac{\rho_{+}u^{2}_{+}-A_{1}\gamma\rho^{\gamma}_{+}}{\mu
u_{+}}\lambda_{1})+1\end{pmatrix},r_{2}=\begin{pmatrix}1\\\ \lambda_{2}\\\
-\frac{\mu}{n_{+}}(\lambda_{2}^{2}-\frac{\rho_{+}u^{2}_{+}-A_{1}\gamma\rho^{\gamma}_{+}}{\mu
u_{+}}\lambda_{2})+1\end{pmatrix},r_{3}=\begin{pmatrix}1\\\ 0\\\
1\end{pmatrix}.$ (2.16)

Define the matrix $\boldsymbol{P_{1}}:=[r_{1},r_{2},r_{3}]$ and take a linear
transformation $Z:=(z_{1},z_{2},z_{3})^{\rm
T}=\boldsymbol{P_{1}}^{-1}\bar{U}$. With $(\ref{g2})$ and $(\ref{g})$, the
system $(\ref{sf bar})$ can be reformulated as follows

$\left\\{\begin{aligned} &\frac{d}{dx}\begin{pmatrix}z_{1}\\\ z_{2}\\\
z_{3}\end{pmatrix}=\begin{pmatrix}\lambda_{1}&*&0\\\ 0&\lambda_{2}&0\\\
0&0&\lambda_{3}\end{pmatrix}\begin{pmatrix}z_{1}\\\ z_{2}\\\
z_{3}\end{pmatrix}+\begin{pmatrix}g_{1}(z_{1},z_{2},z_{3})\\\
g_{2}(z_{1},z_{2},z_{3})\\\ g_{3}(z_{1},z_{2},z_{3})\end{pmatrix},\\\
&(z_{1},z_{2},z_{3})(0)=(z_{1-},z_{2-},z_{3-})=(\boldsymbol{P_{1}}^{-1}\bar{U}_{-})^{\rm
T},\quad\lim_{x\to\infty}(z_{1},z_{2},z_{3})=(0,0,0),\end{aligned}\right.$
(2.17)

where nonlinear functions $g_{i}(i=1,2,3)$ are denoted by

$\begin{pmatrix}g_{1}(z_{1},z_{2},z_{3})\\\ g_{2}(z_{1},z_{2},z_{3})\\\
g_{3}(z_{1},z_{2},z_{3})\end{pmatrix}=\boldsymbol{P_{1}}^{-1}\begin{pmatrix}0\\\
\bar{g}_{2}(\bar{u},\bar{w},\bar{v})\\\
\bar{g}_{3}(\bar{u},\bar{w},\bar{v})\end{pmatrix}.$ (2.18)

With the help of the manifold theory [2], there exist a local center manifold
$W^{c}(0,0,0)$ and a local stable manifold $W_{3}^{s}(0,0,0)$

$W^{c}(0,0,0)=\\{(z_{1},z_{2},z_{3})~{}|~{}z_{1}=f^{c}_{1}(z_{3}),z_{2}=f^{c}_{2}(z_{3}),|z_{3}|~{}{\rm
sufficient~{}small}\\},$ (2.19)
$W_{3}^{s}(0,0,0)=\\{(z_{1},z_{2},z_{3})~{}|~{}z_{1}=f^{s}_{1}(z_{2}),z_{3}=f^{s}_{2}(z_{2}),|z_{2}|~{}{\rm
sufficient~{}small}\\},$ (2.20)

where $f^{c}_{i},f^{s}_{i},i=1,2$ are smooth functions and
$f^{c}_{i}(0)=0,~{}Df^{c}_{i}(0)=0,~{}f^{s}_{i}(0)=0,~{}Df^{s}_{i}(0)=0,~{}i=1,2$.
Using $\bar{U}=PZ$, $(\ref{g3})$-$(\ref{g})$, and $(\ref{bg})$, we gain

$\bar{g}_{3}(z_{3})=az^{2}_{3}+O(|z_{1}|^{2}+|z_{2}|^{2}+|z_{3}|^{3}+|z_{1}z_{3}|+|z_{2}z_{3}|),$
(2.21)

where

$a=\frac{A_{1}\gamma(\gamma+1)\rho^{\gamma}_{+}+A_{2}\alpha(\alpha+1)n^{\alpha}_{+}}{2u_{+}^{2}(1+b^{2})(\mu+n_{+})},\quad
b:=\frac{\rho_{+}(u_{+}^{2}-p^{\prime}_{1}(\rho_{+}))}{|u_{+}|\sqrt{(\mu+n_{+})n_{+}}}.$
(2.22)

Therefore, the system $(\ref{sf z})$ can be reformulated as follows

$\left\\{\begin{aligned} &z_{1x}=\lambda_{1}z_{1}+O(|Z|^{2}),\\\
&z_{2x}=\lambda_{2}z_{2}+O(|Z|^{2}),\\\
&z_{3x}=az_{3}^{2}+O(|z_{1}|^{2}+|z_{2}|^{2}+|z_{3}|^{3}+|z_{1}z_{3}|+|z_{2}z_{3}|),\\\
&(z_{1},z_{2},z_{3})(0):=(z_{1-},z_{2-},z_{3-})=(\boldsymbol{P_{1}}^{-1}\bar{U}_{-})^{\rm
T},~{}\lim_{x\to\infty}(z_{1},z_{2},z_{3})=(0,0,0).\end{aligned}\right.$
(2.23)

Let $\sigma_{1}(x)$ be a solution to $({\ref{z}})_{1}$ restricted on the local
center manifold satisfying the equation

$\sigma_{1x}=a\sigma_{1}^{2}+O(\sigma_{1}^{3}),\quad\sigma_{1}(x)\to 0~{}{\rm
as}~{}x\to+\infty.$ (2.24)

which implies that there exists the monotonically increasing solution
$\sigma_{1}(x)<0$ to $(\ref{sg})$ if $\sigma_{1}(0)<0$ holds and
$|\sigma_{1}(0)|$ is sufficiently small. Therefore, if the initial data
$(z_{1-},z_{2-},z_{3-})$ belongs to the region
$\mathcal{M}\subset\mathbb{R}^{3}$ associated to the local stable manifold and
the local center manifold, then we have

$\left\\{\begin{aligned} &z_{i}=O(\sigma_{1}^{2})+O(\delta
e^{-cx}),~{}i=1,2,\\\ &z_{3}=\sigma_{1}+O(\delta
e^{-cx}),\end{aligned}\right.$ (2.25)

with $z_{3-}<0$, the smallness of $|(z_{1-},z_{2-},z_{3-})|$ and

$c\frac{\delta}{1+\delta x}\leq|\sigma_{1}|\leq C\frac{\delta}{1+\delta
x},\quad|\partial^{k}\sigma_{1}|\leq C\frac{\delta^{k+1}}{(1+\delta
x)^{k+1}},\quad C>0,~{}~{}k=0,1,2,3.$ (2.26)

Due to $\sigma_{1}(x)\leq 0$, we define

$\sigma(x):=-\sigma_{1},$ (2.27)

which satisfies

$\sigma_{x}=-a\sigma^{2}+O(|\sigma|^{3}),\quad\sigma\to 0~{}{\rm
as}~{}x\to+\infty$ (2.28)

It is easy to get

$|\partial_{x}^{k}(\widetilde{\rho}-\rho_{+},\widetilde{u}-u_{+},\widetilde{n}-n_{+},\widetilde{v}-u_{+})|\leq
C\frac{\delta^{k+1}}{(1+\delta x)^{k+1}},\quad C>0,~{}~{}k=0,1,2,3,$ (2.29)

and

$(\widetilde{u}-u_{+},\widetilde{v}-u_{+})=(-\sigma(x),-\sigma(x))+O(|\sigma(x)|^{2}),\quad(\widetilde{u}_{x},\widetilde{v}_{x})=(a\sigma^{2}(x),a\sigma^{2}(x))+O(|\sigma|^{3}),$
(2.30)

with the help of $\bar{U}=\boldsymbol{P}Z$ and $(\ref{ev})$.

##  3 Nonlinear stability of steady-states

The function space $Y(0,T)$ for $T>0$ is denoted by

$\displaystyle Y(0,T):=\\{~{}(\phi,\psi,\bar{\phi},\bar{\psi})~{}|{}$
$\displaystyle(\phi,\psi,\bar{\phi},\bar{\psi})\in
C([0,T];H^{1}(\mathbb{R}_{+})),$ (3.1)
$\displaystyle(\phi_{x},\bar{\phi}_{x})\in
L^{2}([0,T];L^{2}(\mathbb{R}_{+})),~{}(\psi_{x},\bar{\psi}_{x})\in
L^{2}([0,T];H^{1}(\mathbb{R}_{+}))~{}\\}.$

Let

$\phi=\rho-\widetilde{\rho},\quad\psi=u-\widetilde{u},\quad\bar{\phi}=n-\widetilde{n},\quad\bar{\psi}=v-\widetilde{v}.$
(3.2)

Then the perturbation $(\phi,\psi,\bar{\phi},\bar{\psi})$ satisfies the
following system

$\left\\{\begin{aligned}
&\phi_{t}+u\phi_{x}+\rho\psi_{x}=-(\psi\widetilde{\rho}_{x}+\phi\widetilde{u}_{x}),\\\
&\psi_{t}+u\psi_{x}+\frac{p^{\prime}_{1}(\rho)}{\rho}\phi_{x}-\frac{\mu\psi_{xx}}{\rho}-\frac{n(\bar{\psi}-\psi)}{\rho}=F_{1},\\\
&\bar{\phi}_{t}+v\bar{\phi}_{x}+n\bar{\psi}_{x}=-(\bar{\psi}\widetilde{n}_{x}+\bar{\phi}\widetilde{v}_{x}),\\\
&\bar{\psi}_{t}+v\bar{\psi}_{x}+\frac{p^{\prime}_{2}(n)}{n}\bar{\phi}_{x}-\frac{(n\bar{\psi}_{x})_{x}}{n}+(\bar{\psi}-\psi)=F_{2},\end{aligned}\right.$
(3.3)

where

$F_{1}=-[-\mu(\frac{1}{\rho}-\frac{1}{\widetilde{\rho}})\widetilde{u}_{xx}+\psi\widetilde{u}_{x}+(\frac{p^{\prime}_{1}(\rho)}{\rho}-\frac{p^{\prime}_{1}(\widetilde{\rho})}{\widetilde{\rho}})\widetilde{\rho}_{x}-(\frac{n}{\rho}-\frac{\widetilde{n}}{\widetilde{\rho}})(\widetilde{v}-\widetilde{u})],$
(3.4)
$F_{2}=-[-(\frac{1}{n}-\frac{1}{\widetilde{n}})(\widetilde{n}\widetilde{v}_{x})_{x}-\frac{(\bar{\phi}\widetilde{v}_{x})_{x}}{n}+\bar{\psi}\widetilde{v}_{x}+(\frac{p^{\prime}_{2}(n)}{n}-\frac{p^{\prime}_{2}(\widetilde{n})}{\widetilde{n}})\widetilde{n}_{x}].~{}~{}~{}~{}~{}~{}~{}$
(3.5)

The initial and boundary conditions to the system $(\ref{f1})$ satisfy

$(\phi,\psi,\bar{\phi},\bar{\psi})(0,x):=(\phi_{0},\psi_{0},\bar{\phi}_{0},\bar{\psi}_{0})=(\rho_{0}-\widetilde{\rho},u_{0}-\widetilde{u},n_{0}-\widetilde{n},v_{0}-\widetilde{v}),$
(3.6)
$\lim_{x\to\infty}(\phi_{0},\psi_{0},\bar{\phi}_{0},\bar{\psi}_{0})=(0,0,0,0),\quad(\psi,\bar{\psi})(t,0)=(0,0).$
(3.7)

######  Proposition 3.1.

Assume that the same assumptions in Theorem 1.2 hold. Let
$(\phi,\psi,\bar{\phi},\bar{\psi})$ be the solution to the problem
$(\ref{f1})$-$(\ref{boundary d1})$ satisfying
$(\phi,\psi,\bar{\phi},\bar{\psi})\in Y(0,T)$ for any time $T>0$. Then there
exist positive constants $\varepsilon>0$ and $C>0$ independent of $T$ such
that if

$\sup_{0\leq t\leq
T}\|(\phi,\psi,\bar{\phi},\bar{\psi})(t)\|_{1}+\delta\leq\varepsilon$ (3.8)

is satisfied, then it holds for arbitrary $t\in[0,T]$ that

$\begin{split}&\|(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}_{1}+\int_{0}^{t}\|(\phi_{x},\psi_{x},\bar{\phi}_{x},\bar{\psi}_{x})\|^{2}d\tau+\int_{0}^{t}\|(\bar{\psi}-\psi,\psi_{xx},\bar{\psi}_{xx})\|^{2}d\tau\leq
C\|(\phi_{0},\psi_{0},\bar{\phi}_{0},\bar{\psi}_{0})\|_{1}^{2}.\end{split}$
(3.9)

With the help of $(\ref{priori e})$, it is easy to verify the following
Sobolev inequality

$~{}~{}~{}~{}~{}\|(\phi,\psi,\bar{\phi},\bar{\psi})(t)\|_{L^{\infty}}\leq\|(\phi,\psi,\bar{\phi},\bar{\psi})(t)\|_{H^{1}}\leq\sqrt{2}\varepsilon.$
(3.10)

######  Lemma 3.2 ([16] ).

For any function $\psi(t,\cdot)\in H^{1}(\mathbb{R}_{+})$, it holds

$\displaystyle\int_{0}^{\infty}\delta e^{-c_{0}x}|\psi|^{2}dx\leq$
$\displaystyle C\delta(|\psi(t,0)|^{2}+\|\psi_{x}(t)\|^{2}),$ (3.11)
$\displaystyle\int_{0}^{\infty}\frac{\delta^{j}}{(1+\delta
x)^{j}}|\psi|^{2}dx\leq$ $\displaystyle
C\delta^{j-2}(|\psi(t,0)|^{2}+\|\psi_{x}(t)\|^{2}),\quad~{}j>2,$ (3.12)

where $\delta>0$, $c_{0}>0$ and $C>0$ are positive constants.

With the Lemma 3.2, we can gain the basic $L^{2}$ energy estimates of
$(\phi,\psi,\bar{\phi},\bar{\psi})$.

######  Lemma 3.3.

Under the same conditions in Proposition 3.1, then the solution
$(\phi,\psi,\bar{\phi},\bar{\psi})$ to the problem
$(\ref{f1})$-$(\ref{boundary d1})$ satisfies for $t\in[0,T]$ that

$\displaystyle\|(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}+\int_{0}^{t}\|(\psi_{x},\bar{\psi}_{x},\bar{\psi}-\psi)\|^{2}d\tau+\int_{0}^{t}|\phi(t,0)|^{2}+|\bar{\phi}(t,0)|^{2}d\tau$
(3.13) $\displaystyle\leq$ $\displaystyle
C\|(\phi_{0},\psi_{0},\bar{\phi}_{0},\bar{\psi}_{0})\|^{2}+C(\delta+\varepsilon)\int_{0}^{t}\|(\phi_{x},\bar{\phi}_{x})\|^{2}d\tau.$

###### Proof.

Define

$\Phi_{1}(\rho,\widetilde{\rho})=\int_{\widetilde{\rho}}^{\rho}\frac{p_{1}(s)-p_{1}(\widetilde{\rho})}{s^{2}}ds,\quad\mathcal{E}_{1}=\rho(\frac{\psi^{2}}{2}+\Phi_{1}),$
(3.14)
$\Phi_{2}(n,\widetilde{n})=\int_{\widetilde{n}}^{n}\frac{p_{2}(s)-p_{2}(\widetilde{n})}{s^{2}}ds,\quad\mathcal{E}_{2}=n(\frac{\bar{\psi}^{2}}{2}+\Phi_{2}).$
(3.15)

Then, by $(\ref{f})$ and $(\ref{stationary f})$, the direct computations lead
to

$\begin{split}&\quad(\mathcal{E}_{1}+\mathcal{E}_{2})_{t}+(G_{1}+G_{2})_{x}+n(\bar{\psi}-\psi)^{2}+\mu\psi_{x}^{2}+n\bar{\psi}^{2}_{x}+R_{1}+R_{2}=-R_{3},\end{split}$
(3.16)

where

$\left\\{\begin{aligned}
&G_{1}:=u\mathcal{E}_{1}+v\mathcal{E}_{2}+(p_{1}(\rho)-p_{1}(\widetilde{\rho}))\psi+(p_{2}(n)-p_{2}(\widetilde{n}))\bar{\psi},\\\
&G_{2}:=-(\mu\psi\psi_{x}+n\bar{\psi}\bar{\psi}_{x}+\bar{\phi}\bar{\psi}\widetilde{v}_{x}),\\\
&R_{1}:=[\rho\psi^{2}+p_{1}(\rho)-p_{1}(\widetilde{\rho})-p^{\prime}_{1}(\widetilde{\rho})\phi]\widetilde{u}_{x}+[n\bar{\psi}^{2}+p_{2}(n)-p_{2}(\widetilde{n})-p^{\prime}_{2}(\widetilde{n})\bar{\phi}]\widetilde{v}_{x},\\\
&R_{2}:=\phi\psi\frac{\widetilde{\rho}\widetilde{u}\widetilde{u}_{x}+(p_{1}(\widetilde{\rho}))_{x}}{\widetilde{\rho}}+\bar{\phi}\bar{\psi}\frac{\widetilde{n}\widetilde{v}\widetilde{v}_{x}+(p_{2}(\widetilde{n}))_{x}}{\widetilde{n}},\\\
&R_{3}:=\bar{\phi}(\bar{\psi}-\psi)(\widetilde{v}-\widetilde{u})+\bar{\phi}\bar{\psi}_{x}\widetilde{v}_{x}.\end{aligned}\right.$
(3.17)

Integrating $(\ref{f_0})$ in $x$ over $\mathbb{R}_{+}$ leads to

$\frac{d}{dt}\int\mathcal{E}_{1}+\mathcal{E}_{2}dx-G_{1}(t,0)+\int
n(\bar{\psi}-\psi)^{2}+\mu\psi_{x}^{2}+n\bar{\psi}^{2}_{x}dx+\int R_{1}dx+\int
R_{2}dx=-\int R_{3}dx.$ (3.18)

Under the condition $(\ref{boundary d1})$, we get

$-G_{1}(t,0)=-u_{-}[\Phi_{1}(\rho(t,0),\widetilde{\rho}(0))+\Phi_{2}(n(t,0),\widetilde{n}(0))]\geq
c(\phi^{2}(t,0)+\bar{\phi}^{2}(t,0)).$ (3.19)

For the supersonic or subsonic case $M_{+}\neq 1$, with the help of
$(\ref{M_{+}>1 stationary solution d})$, $(\ref{infty 1})$ and $(\ref{d'1})$,
we have

$\int_{0}^{\infty}|R_{1}|+|R_{2}|+|R_{3}|dx\leq
C\delta\|(\phi_{x},\psi_{x},\bar{\phi}_{x},\bar{\psi}_{x},\bar{\psi}-\psi)\|^{2}+C\delta(\phi^{2}(t,0)+\bar{\phi}^{2}(t,0)).$
(3.20)

For the sonic case $M_{+}=1$ and the restriction
$|p^{\prime}_{1}({\rho_{+}})-p^{\prime}_{2}({n_{+}})|\leq\sqrt{2}|u_{+}|\min\\{(1+\frac{\rho_{+}}{n_{+}})[(\gamma-1)p^{\prime}_{1}(\rho_{+})]^{\frac{1}{2}},(1+\frac{n_{+}}{\rho_{+}})[(\alpha-1)p^{\prime}_{2}(n_{+})]^{\frac{1}{2}}\\}$,
using $(\ref{sigma})$-$(\ref{sig})$, $(\ref{infty 1})$, and $(\ref{d'2})$, we
get

$\displaystyle\int R_{1}+R_{2}+R_{3}dx$ (3.21) $\displaystyle\geq$
$\displaystyle\int(\psi,\phi)\boldsymbol{M_{1}}(\psi,\phi)^{\rm
T}\widetilde{u}_{x}+(\bar{\psi},\bar{\phi})\boldsymbol{M_{2}}(\bar{\psi},\bar{\phi})^{\rm
T}\widetilde{v}_{x}dx-C\int\frac{\delta^{3}}{(1+\delta
x)^{3}}(\phi^{2}+\psi^{2}+\bar{\phi}^{2}+\bar{\psi}^{2})dx$
$\displaystyle-C\delta\int|\bar{\psi}-\psi|^{2}+\psi_{x}^{2}dx-C\delta^{\frac{1}{2}}\|(\phi,\psi,\bar{\phi},\bar{\psi})\|[\phi^{2}(t,0)+\bar{\phi}^{2}(t,0)+\|(\phi_{x},\psi_{x},\bar{\phi}_{x},\bar{\phi}_{x})\|^{2}]$
$\displaystyle\geq$
$\displaystyle-C(\delta+\varepsilon)\|(\phi_{x},\psi_{x},\bar{\phi}_{x},\bar{\psi}_{x},\bar{\psi}-\psi)\|^{2}-C\delta(\phi^{2}(t,0)+\bar{\phi}^{2}(t,0)),$

where $\boldsymbol{M_{1}}$, $\boldsymbol{M_{2}}$ are positive definite or non-
negative definite matrices defined by

$\begin{split}\boldsymbol{M_{1}}=\begin{pmatrix}&\rho_{+}&\frac{u_{+}^{2}-A_{1}\gamma\rho_{+}^{\gamma-1}}{2u_{+}}\\\
&\frac{u_{+}^{2}-A_{1}\gamma\rho_{+}^{\gamma-1}}{2u_{+}}&\frac{A_{1}\gamma(\gamma-1)\rho_{+}^{\gamma-2}}{2}\end{pmatrix},\quad\boldsymbol{M_{2}}=\begin{pmatrix}&n_{+}&\frac{u_{+}^{2}-A_{2}\alpha
n_{+}^{\alpha-1}}{2u_{+}}\\\ &\frac{u_{+}^{2}-A_{2}\alpha
n_{+}^{\alpha-1}}{2u_{+}}&\frac{A_{2}\alpha(\alpha-1)n_{+}^{\alpha-2}}{2}\end{pmatrix}.\end{split}$
(3.22)

Finally, with the help of $(\ref{f_})$-$(\ref{r_1'})$, we get $(\ref{e0})$.
Hence, the proof of Lemma 3.3 is completed. ∎

In order to complete the proof of Proposition 3.1, we need to establish the
high order estimates of $(\phi,\psi,\bar{\phi},\bar{\psi})$.

######  Lemma 3.4.

Under the same conditions in Proposition 3.1, then the solution
$(\phi,\psi,\bar{\phi},\bar{\psi})$ to the problem
$(\ref{f1})$-$(\ref{boundary d1})$ satisfies for $t\in[0,T]$ that

$\begin{split}&\quad\|(\phi_{x},\bar{\phi}_{x})\|^{2}+\int_{0}^{t}\|(\phi_{x},\bar{\phi}_{x})\|^{2}d\tau+\int_{0}^{t}\phi_{x}^{2}(t,0)+\bar{\phi}_{x}^{2}(t,0)d\tau\\\
&\leq
C\|(\phi_{0},\psi_{0},\phi_{0x},\bar{\phi}_{0},\bar{\psi}_{0},\bar{\phi}_{0x})\|^{2}+C(\varepsilon+\delta)\int_{0}^{t}\|(\psi_{xx},\bar{\psi}_{xx})\|^{2}d\tau.\end{split}$
(3.23)

###### Proof.

Differentiating $(\ref{f1})_{1}$ in $x$, then multiplying the resulted
equation by $\mu\phi_{x}$, $(\ref{f1})_{2}$ by $\widetilde{\rho}^{2}\phi_{x}$
respectively, we gain

$\displaystyle(\mu\frac{\phi_{x}^{2}}{2})_{t}+(\mu
u\frac{\phi_{x}^{2}}{2})_{x}+\mu\widetilde{\rho}\phi_{x}\psi_{xx}$
$\displaystyle=$
$\displaystyle-\mu[\frac{3}{2}\psi_{x}\phi_{x}^{2}+\phi\phi_{x}\psi_{xx}+(\frac{1}{2}\phi_{x}\widetilde{u}_{x}+\psi_{x}\widetilde{\rho}_{x})\phi_{x}+(\phi\widetilde{u}_{x}+\psi\widetilde{\rho}_{x})_{x}\phi_{x}],$
(3.24)
$\displaystyle(\widetilde{\rho}^{2}\phi_{x}\psi)_{t}-(\widetilde{\rho}^{2}\phi_{t}\psi)_{x}+\widetilde{\rho}^{2}\frac{p^{\prime}_{1}(\rho)}{\rho}\phi_{x}^{2}-\mu\widetilde{\rho}\phi_{x}\psi_{xx}$
$\displaystyle=$
$\displaystyle-(\widetilde{\rho}^{2}\phi_{t}\psi_{x}+\widetilde{\rho}^{2}u\phi_{x}\psi_{x}+2\widetilde{\rho}\widetilde{\rho}_{x}\phi_{t}\psi)+\mu\widetilde{\rho}^{2}(\frac{1}{\rho}-\frac{1}{\widetilde{\rho}})\phi_{x}\psi_{xx}+\widetilde{\rho}^{2}\frac{n}{\rho}(\bar{\psi}-\psi)\phi_{x}+F_{1}\widetilde{\rho}^{2}\phi_{x}.$
(3.25)

Similarly, differentiating $(\ref{f1})_{3}$ in $x$, then multiplying the
resulted equation by $\bar{\phi}_{x}$, $(\ref{f1})_{4}$ by
$\widetilde{n}\bar{\phi}_{x}$ respectively lead to

$\displaystyle(\frac{\bar{\phi}_{x}^{2}}{2})_{t}+(v\frac{\bar{\phi}_{x}^{2}}{2})_{x}+\widetilde{n}\bar{\phi}_{x}\bar{\psi}_{xx}=-[\frac{3}{2}\bar{\psi}_{x}\bar{\phi}_{x}^{2}+\bar{\phi}\bar{\phi}_{x}\bar{\psi}_{xx}+(\frac{1}{2}\bar{\phi}_{x}\widetilde{v}_{x}+\bar{\psi}_{x}\widetilde{n}_{x})\bar{\phi}_{x}-(\bar{\phi}\widetilde{v}_{x}+\bar{\psi}\widetilde{n}_{x})_{x}\bar{\phi}_{x}],$
(3.26)
$\displaystyle(\widetilde{n}\bar{\phi}_{x}\bar{\psi})_{t}-(\widetilde{n}\bar{\phi}_{t}\bar{\psi})_{x}+\widetilde{n}\frac{p^{\prime}_{2}(n)}{n}\bar{\phi}_{x}^{2}-\widetilde{n}\bar{\phi}_{x}\bar{\psi}_{xx}$
$\displaystyle=$
$\displaystyle-(\widetilde{n}\bar{\phi}_{t}\bar{\psi}_{x}+\widetilde{n}v\bar{\phi}_{x}\bar{\psi}_{x}+\widetilde{n}_{x}\bar{\phi}_{t}\bar{\psi})+[\widetilde{n}\frac{(\bar{\phi}\bar{\psi}_{x})_{x}}{n}+\widetilde{n}(\frac{1}{n}-\frac{1}{\widetilde{n}})(\widetilde{n}\bar{\psi}_{x})_{x}-\widetilde{n}(\bar{\psi}-\psi)]\bar{\phi}_{x}$
(3.27)
$\displaystyle+\widetilde{n}_{x}\bar{\phi}_{x}\bar{\psi}_{xx}+F_{2}\widetilde{n}\bar{\phi}_{x}.$

Adding $(\ref{h_{x}})$-$(\ref{bs_{x}})$ together, integrating the resulted
equation over $\mathbb{R}_{+}$, we have

$\displaystyle\frac{d}{dt}\int(\mu\frac{\phi_{x}^{2}}{2}+\frac{\bar{\phi}_{x}^{2}}{2}+\widetilde{\rho}^{2}\phi_{x}\psi+\widetilde{n}\bar{\phi}_{x}\bar{\psi})dx+\int(\mu
u\frac{\phi_{x}^{2}}{2}+v\frac{\bar{\phi}_{x}^{2}}{2}-\widetilde{\rho}^{2}\phi_{t}\psi-\widetilde{n}\bar{\phi}_{t}\bar{\psi})_{x}dx$
(3.28)
$\displaystyle+\int(\widetilde{\rho}^{2}\frac{p^{\prime}_{1}(\rho)}{\rho}\phi_{x}^{2}+\widetilde{n}\frac{p^{\prime}_{2}(n)}{n}\bar{\phi}_{x}^{2})dx$
$\displaystyle=$ $\displaystyle\sum_{i=1}^{4}J_{i},$

where

$\displaystyle J_{1}=$
$\displaystyle-\int[\widetilde{\rho}^{2}(\phi_{t}+u\phi_{x})\psi_{x}+\widetilde{n}(\bar{\phi}_{t}+v\bar{\phi}_{x})\bar{\psi}_{x}+2\widetilde{\rho}\widetilde{\rho}_{x}\phi_{t}\psi+\widetilde{n}_{x}\bar{\phi}_{t}\bar{\psi}]dx,$
$\displaystyle J_{2}=$
$\displaystyle\int-\mu\phi\phi_{x}\psi_{xx}-\bar{\phi}\bar{\phi}_{x}\bar{\psi}_{xx}+\mu\widetilde{\rho}^{2}(\frac{1}{\rho}-\frac{1}{\widetilde{\rho}})\phi_{x}\psi_{xx}+\widetilde{n}(\frac{1}{n}-\frac{1}{\widetilde{n}})(\widetilde{n}\bar{\psi}_{x})_{x}\bar{\phi}_{x}+\widetilde{\rho}^{2}\frac{n}{\rho}(\bar{\psi}-\psi)\phi_{x}-\widetilde{n}(\bar{\psi}-\psi)\bar{\phi}_{x}dx,$
$\displaystyle J_{3}=$
$\displaystyle-\int\frac{3}{2}\mu\psi_{x}\frac{\phi_{x}^{2}}{2}+\frac{3}{2}\bar{\psi}_{x}\bar{\phi}_{x}^{2}dx+\int\widetilde{n}\frac{(\bar{\phi}\bar{\psi}_{x})_{x}}{n}\bar{\phi}_{x}dx,\quad$
$\displaystyle J_{4}=$
$\displaystyle-\int[\mu(\frac{1}{2}\phi_{x}\widetilde{u}_{x}+\psi_{x}\widetilde{\rho}_{x})\phi_{x}+\mu(\phi\widetilde{u}_{x}+\psi\widetilde{\rho}_{x})_{x}\phi_{x}-F_{1}\widetilde{\rho}^{2}\phi_{x}$
$\displaystyle+(\frac{1}{2}\bar{\phi}_{x}\widetilde{v}_{x}+\bar{\psi}_{x}\widetilde{n}_{x})\bar{\phi}_{x}-(\bar{\phi}\widetilde{v}_{x}+\bar{\psi}\widetilde{n}_{x})_{x}\bar{\phi}_{x}-\widetilde{n}_{x}\bar{\phi}_{x}\bar{\psi}_{xx}-F_{2}\widetilde{n}\bar{\phi}_{x}]dx.$

First, we estimate terms in the left side of $(\ref{1f})$. Under the condition
$(\ref{boundary d1})$, the terms in the left side is estimated as follows

$\int(\mu
u\frac{\phi_{x}^{2}}{2}+v\frac{\bar{\phi}_{x}^{2}}{2}-\widetilde{\rho}^{2}\phi_{t}\psi-\widetilde{n}\bar{\phi}_{t}\bar{\psi})_{x}dx=-u_{-}\frac{\mu\phi_{x}^{2}(0,t)+\bar{\phi}_{x}^{2}(0,t)}{2}\geq
0,$ (3.29)
$\int\widetilde{\rho}^{2}\frac{p^{\prime}_{1}(\rho)}{\rho}\phi_{x}^{2}+\widetilde{n}\frac{p^{\prime}_{2}(n)}{n}\bar{\phi}_{x}^{2}dx\geq\rho_{+}p^{\prime}_{1}(\rho_{+})\|\phi_{x}\|^{2}+p^{\prime}_{2}(n_{+})\|\bar{\phi}_{x}\|^{2}-C(\varepsilon+\delta)\|(\phi_{x},\bar{\phi}_{x})\|^{2}.$
(3.30)

We turn to estimate terms in the right hand side of $(\ref{1f})$. By
$(\ref{M_{+}>1 stationary solution d})$-$(\ref{sigma})$, $(\ref{f1})_{1}$,
$(\ref{f1})_{3}$, $(\ref{d'1})$-$(\ref{d'2})$, Cauchy-Schwartz inequality and
Young inequality with $0<\eta<1$, we obtain

$\displaystyle|J_{1}|\leq$ $\displaystyle
C\|(\psi_{x},\bar{\psi}_{x})\|^{2}+C\delta\|(\phi_{x},\bar{\phi}_{x})\|^{2}+C\delta(\phi^{2}(t,0)+\bar{\phi}^{2}(t,0)),$
(3.31) $\displaystyle|J_{2}|\leq$ $\displaystyle
C\|(\phi,\bar{\phi})\|_{L^{\infty}}\|(\phi_{x},\bar{\phi}_{x},\psi_{xx},\bar{\psi}_{xx})\|^{2}+C_{\eta}\|\bar{\psi}-\psi\|^{2}+\eta\|(\phi_{x},\bar{\phi}_{x})\|^{2}+C\delta\|(\bar{\phi}_{x},\bar{\psi}_{x})\|^{2}$
$\displaystyle\leq$ $\displaystyle
C(\varepsilon+\delta+\eta)\|(\phi_{x},\bar{\phi}_{x})\|^{2}+C\varepsilon\|(\psi_{xx},\bar{\psi}_{xx})\|^{2}+C_{\eta}\|\bar{\psi}-\psi\|^{2}+C\delta\|\bar{\psi}_{x}\|^{2},$
(3.32) $\displaystyle|J_{3}|\leq$ $\displaystyle
C\|(\psi_{x},\bar{\psi}_{x})\|_{L^{\infty}}\|(\phi_{x},\bar{\phi}_{x})\|^{2}+C\|\bar{\phi}\|_{L^{\infty}}\|(\bar{\phi}_{x},\bar{\psi}_{xx})\|^{2}$
$\displaystyle\leq$ $\displaystyle
C\varepsilon\|(\phi_{x},\psi_{x},\bar{\phi}_{x},\bar{\psi}_{x})\|^{2}+C\varepsilon\|(\psi_{xx},\bar{\psi}_{xx})\|^{2},$
(3.33) $\displaystyle|J_{4}|\leq$ $\displaystyle
C\delta\|(\phi_{x},\psi_{x},\bar{\phi}_{x},\bar{\psi}_{x})\|^{2}+C\delta\|\bar{\psi}_{xx}\|^{2}+C\delta(\phi^{2}(t,0)+\bar{\phi}^{2}(t,0)).$
(3.34)

Finally, the substitution of $(\ref{h(0)})$-$(\ref{I_{5}})$ into $(\ref{1f})$
for $\delta$, $\varepsilon$ and $\eta$ small enough leads to

$\displaystyle\frac{d}{dt}\int(\phi_{x}^{2}+\bar{\phi}_{x}^{2}+\widetilde{\rho}^{2}\phi_{x}\psi+\widetilde{n}\bar{\phi}_{x}\bar{\psi})dx+\|(\phi_{x},\bar{\phi}_{x})\|^{2}+\phi_{x}^{2}(t,0)+\bar{\phi}_{x}^{2}(t,0)$
(3.35) $\displaystyle\leq$ $\displaystyle
C\|(\psi_{x},\bar{\psi}_{x},\bar{\psi}_{x}-\psi_{x})\|^{2}+C(\delta+\varepsilon)\|(\psi_{xx},\bar{\psi}_{xx})\|^{2}+C\delta(\phi^{2}(t,0)+\bar{\phi}^{2}(t,0)).$

Integrating $(\ref{e1})$ in $\tau$ over $[0,t]$, and using Lemma 3.3 and Young
inequality, we have $(\ref{1'-order time e1})$. ∎

######  Lemma 3.5.

Under the same conditions in Proposition 3.1, then the solution
$(\phi,\psi,\bar{\phi},\bar{\psi})$ to the problem
$(\ref{f1})$-$(\ref{boundary d1})$ satisfies for $t\in[0,T]$ that

$\begin{split}&~{}~{}~{}~{}\|(\psi_{x},\bar{\psi}_{x})\|^{2}+\int_{0}^{t}\|(\psi_{xx},\bar{\psi}_{xx})\|^{2}d\tau\leq
C\|(\phi_{0},\psi_{0},\bar{\phi}_{0},\bar{\psi}_{0})\|^{2}_{1}.\end{split}$
(3.36)

###### Proof.

Multiplying $(\ref{f1})_{2}$ by $-\psi_{xx}$, $(\ref{f1})_{4}$ by
$-\bar{\psi}_{xx}$, respectively, then adding them together and integrating
the resulted equation in $x$ over $\mathbb{R}_{+}$ imply

$\begin{split}&\frac{d}{dt}\int\frac{\psi^{2}_{x}}{2}+\frac{\bar{\psi}_{x}^{2}}{2}dx+\int\mu\frac{1}{\rho}\psi^{2}_{xx}+\bar{\psi}_{xx}^{2}dx=\sum^{3}_{i=1}K_{i},\end{split}$
(3.37)

where

$\displaystyle K_{1}=$
$\displaystyle\int[u\psi_{x}\psi_{xx}-\frac{n}{\rho}(\bar{\psi}-\psi)\psi_{xx}+\frac{p^{\prime}_{1}(\rho)}{\rho}\phi_{x}\psi_{xx}+v\bar{\psi}_{x}\bar{\psi}_{xx}+(\bar{\psi}-\psi)\bar{\psi}_{xx}+\frac{p^{\prime}_{2}(n)}{n}\bar{\phi}_{x}\bar{\psi}_{xx}]dx,$
$\displaystyle K_{2}=$
$\displaystyle\int[\widetilde{u}_{x}\psi\psi_{xx}-\mu\widetilde{u}_{xx}(\frac{1}{\rho}-\frac{1}{\widetilde{\rho}})\psi_{xx}+(\frac{p^{\prime}_{1}(\rho)}{\rho}-\frac{p^{\prime}_{1}(\widetilde{\rho})}{\widetilde{\rho}})\psi_{xx}\widetilde{\rho}_{x}-(\frac{n}{\rho}-\frac{\widetilde{n}}{\widetilde{\rho}})(\widetilde{v}-\widetilde{u})\psi_{xx}+\widetilde{v}_{x}\bar{\psi}\bar{\psi}_{xx}$
$\displaystyle+(\widetilde{n}\widetilde{v}_{x})_{x}(\frac{1}{n}-\frac{1}{\widetilde{n}})\bar{\psi}_{xx}+(\frac{p^{\prime}_{2}(n)}{n}-\frac{p^{\prime}_{2}(\widetilde{n})}{\widetilde{n}})\bar{\psi}_{xx}\widetilde{n}_{x}-\frac{\widetilde{n}_{x}}{\widetilde{n}}\bar{\psi}_{x}\bar{\psi}_{xx}-\frac{(\bar{\phi}\widetilde{v}_{x})_{x}}{n}\bar{\psi}_{xx}]dx,$
$\displaystyle K_{3}=$
$\displaystyle-\int[\frac{(\bar{\phi}\bar{\psi}_{x})_{x}}{n}\bar{\psi}_{xx}+(\frac{1}{n}-\frac{1}{\widetilde{n}})(\widetilde{n}\bar{\psi}_{x})_{x}\bar{\psi}_{xx}]dx.$

We estimate terms in the left side of $(\ref{psi_{xx} space w1})$. By the
decomposition
$\frac{1}{\rho}=(\frac{1}{\rho}-\frac{1}{\widetilde{\rho}})+(\frac{1}{\widetilde{\rho}}-\frac{1}{\rho_{+}})+\frac{1}{\rho_{+}}$,
the second term is estimated as follows:

$\displaystyle\int\frac{\mu}{\rho}\psi_{xx}^{2}+\bar{\psi}_{xx}^{2}dx\geq$
$\displaystyle\frac{\mu}{\rho_{+}}\|\psi_{xx}\|^{2}+\|\bar{\psi}_{xx}\|^{2}-C(\|\phi\|_{L^{\infty}}+\delta)\|\psi_{xx}\|^{2}$
(3.38) $\displaystyle\geq$
$\displaystyle\frac{\mu}{\rho_{+}}\|\psi_{xx}\|^{2}+\|\bar{\psi}_{xx}\|^{2}-C(\varepsilon+\delta)\|\psi_{xx}\|^{2}.$

We turn to estimate terms in the right side of $(\ref{psi_{xx} space w1})$.
With the aid of $(\ref{M_{+}>1 stationary solution d})$, Sobolev inequality
and Cauchy-Schwarz inequality, we have

$\displaystyle|K_{1}|\leq$
$\displaystyle\frac{\mu}{16\rho_{+}}\|\psi_{xx}\|^{2}+\frac{1}{16}\|\bar{\psi}_{xx}\|^{2}+C\|(\phi_{x},\psi_{x},\bar{\phi}_{x},\bar{\psi}_{x},\bar{\psi}-\psi)\|^{2},$
(3.39) $\displaystyle|K_{2}|\leq$ $\displaystyle
C\delta\|(\phi_{x},\psi_{x},\bar{\phi}_{x},\bar{\psi}_{x})\|^{2}+C\delta\|(\psi_{xx},\bar{\psi}_{xx})\|^{2}+C\delta(\phi^{2}(t,0)+\bar{\phi}^{2}(t,0)),$
(3.40) $\displaystyle|K_{3}|\leq$ $\displaystyle
C\|\bar{\phi}\|_{L^{\infty}}\|(\bar{\psi}_{x},\bar{\psi}_{xx})\|^{2}+C\|\bar{\psi}_{x}\|_{L^{\infty}}\|\bar{\psi}_{xx}\|~{}\|\bar{\phi}_{x}\|\leq
C\varepsilon\|(\bar{\psi}_{x},\bar{\psi}_{xx})\|^{2}.$ (3.41)

Finally, taking $\delta$ and $\varepsilon$ small enough and substituting of
$(\ref{s_{xx} e})$-$(\ref{K_{3} e})$ into $(\ref{psi_{xx} space w1})$, we
obtain

$\frac{d}{dt}\int\psi_{x}^{2}+\bar{\psi}_{x}^{2}dx+\frac{\mu}{2\rho_{+}}\|\psi_{xx}\|^{2}+\frac{1}{2}\|\bar{\psi}_{xx}\|^{2}\leq
C\|(\phi_{x},\psi_{x},\bar{\phi}_{x},\bar{\psi}_{x},\bar{\psi}-\psi)\|^{2}+C\delta(\phi^{2}(t,0)+\bar{\phi}^{2}(t,0)).$
(3.42)

Integrating $(\ref{psi_{xx} e1'})$ in $\tau$ over $[0,t]$, and using Lemmas
3.3-3.4 and the smallness of $\delta$ and $\varepsilon$, we obtain the desired
estimate $(\ref{s 3})$. Therefore, we complete the proof of Lemma 3.5. ∎

With the help of Lemmas 3.3-3.5, we get $(\ref{e})$ and complete the proof of
Proposition 3.1.

##  4 Time convergence rates

###  4.1  Convergence rate of supersonic steady-state

######  Proposition 4.1.

Assume that the same conditions in Theorem 1.3 for $M_{+}>1$ hold and let
$(\phi,\psi,\bar{\phi},\bar{\psi})$ be a solution to the IBVP
$(\ref{f1})$-$(\ref{boundary d1})$ satisfying
$(\phi,\psi,\bar{\phi},\bar{\psi})\in C([0,T];H^{1})$ and
${(1+x)^{\frac{\nu}{2}}}(\phi,\psi,\bar{\phi},\bar{\psi})\in C([0,T];L^{2})$
for any time $T>0$. Then for arbitrary $\nu\in[0,\lambda]$, there exist
positive constants $\varepsilon>0$ and $C>0$ independent of $T$ such that if

$\sup_{0\leq t\leq
T}\|(\phi,\psi,\bar{\phi},\bar{\psi})(t)\|_{1}+\delta\leq\varepsilon$ (4.1)

is satisfied, it holds for arbitrary $t\in[0,T]$ that

$\displaystyle(1+t)^{\lambda-\nu+\theta}(\|(\phi,\psi,\bar{\phi},\bar{\psi})\|_{1}+\|(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}_{a,\nu})+\nu\int_{0}^{t}(1+\tau)^{\lambda-\nu+\theta}\|(\phi,\bar{\phi},\psi,\bar{\psi})\|^{2}_{a,\nu-1}d\tau$
(4.2)
$\displaystyle+\int_{0}^{t}(1+\tau)^{\lambda-\nu+\theta}\|(\psi_{x},\bar{\psi}_{x},\bar{\psi}-\psi)\|^{2}_{a,\nu}d\tau+\int_{0}^{t}(1+\tau)^{\lambda-\nu+\theta}\|(\phi_{x},\psi_{xx},\bar{\phi}_{x},\bar{\psi}_{xx})\|^{2}d\tau$
$\displaystyle\leq$ $\displaystyle
C(1+t)^{\theta}(\|(\phi_{0},\psi_{0},\bar{\phi}_{0},\bar{\psi}_{0})\|^{2}_{1}+\|(\phi_{0},\psi_{0},\bar{\phi}_{0},\bar{\psi}_{0})\|^{2}_{a,\lambda}),$

with $\theta>0$.

Our first goal is to obtain the basic weighted energy estimates of
$(\phi,\psi,\bar{\phi},\bar{\psi})$.

######  Lemma 4.2.

Under the same conditions in Proposition 4.1, then the solution
$(\phi,\psi,\bar{\phi},\bar{\psi})$ to the IBVP $(\ref{f1})$-$(\ref{boundary
d1})$ satisfies for $t\in[0,T]$ that

$\displaystyle(1+t)^{\xi}\|(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}_{a,\nu}+\nu\int_{0}^{t}(1+\tau)^{\xi}\|(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}_{a,\nu-1}d\tau$
(4.3)
$\displaystyle+\int_{0}^{t}(1+\tau)^{\xi}\|(\psi_{x},\bar{\psi}_{x},\bar{\psi}-\psi)\|^{2}_{a,\nu}d\tau+\int_{0}^{t}(1+\tau)^{\xi}(\phi^{2}(t,0)+\bar{\phi}^{2}(t,0))d\tau$
$\displaystyle\leq$ $\displaystyle
C\|(\phi_{0},\psi_{0},\bar{\phi}_{0},\bar{\psi}_{0})\|^{2}_{a,\lambda}+C\delta\int_{0}^{t}(1+\tau)^{\xi}\|(\phi_{x},\bar{\phi}_{x})\|^{2}d\tau$
$\displaystyle+C\nu\int_{0}^{t}(1+\tau)^{\xi}\|(\psi_{x},\bar{\psi}_{x},\bar{\psi}-\psi)\|^{2}_{a,\nu-1}d\tau+C\xi\int_{0}^{t}(1+\tau)^{\xi-1}\|(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}_{a,\nu}d\tau.$

with $\xi\geq 0$.

###### Proof.

We multiply $(\ref{f_0})$ by $W_{a,\nu}$, where $W_{a,\nu}:=(1+x)^{\nu}$ is a
space weight function. We integrate the resulted equality over
$\mathbb{R}_{+}$ to obtain

$\displaystyle\frac{d}{dt}\int
W_{a,\nu}(\mathcal{E}_{1}+\mathcal{E}_{2})-(W_{a,\nu}G_{1})(t,0)-\int
W_{a,\nu-1}G_{1}dx-\int W_{a,\nu-1}G_{2}dx$ (4.4) $\displaystyle+\int
W_{a,\nu}[\mu\psi_{x}^{2}+n\bar{\psi}^{2}_{x}+n(\bar{\psi}-\psi)^{2}]dx$
$\displaystyle=$ $\displaystyle-\int W_{a,\nu}(R_{1}+R_{2}+R_{3})dx,$

where $\mathcal{E}_{i}$, $i=1,2$ are defined by $(\ref{mcE 1})$-$(\ref{mcE
2})$, and $G_{j}$ for $j=1,2$, $R_{k}$ for $k=1,2,3$ are defined by
$(\ref{G})$.
First, we estimate terms on the left hand side of (4.4). Under the condition
$(\ref{boundary d1})$, the second term on the left hand side is estimated as

$\begin{split}&~{}~{}~{}-(W_{a,\nu}G_{1})(t,0)=|u_{-}|[\Phi_{1}(\rho(t,0),\widetilde{\rho}(0))+\Phi_{2}(n(t,0),\widetilde{n}(0))]\geq
c(\phi^{2}(t,0)+\bar{\phi}^{2}(t,0)),~{}~{}~{}~{}~{}~{}~{}\end{split}$ (4.5)

We decompose $\psi$ as $\psi=\bar{\psi}+(\psi-\bar{\psi})$ and use
$(\ref{M_{+}>1 stationary solution d})$ to gain

$\displaystyle-\nu\int W_{a,\nu-1}G_{1}dx$ (4.6) $\displaystyle\geq$
$\displaystyle\nu\int
W_{a,\nu-1}[\frac{1}{2}(\phi,\bar{\phi},\bar{\psi})\boldsymbol{M_{3}}(\phi,\bar{\phi},\bar{\psi})^{\rm
T}-\rho_{+}u_{+}\bar{\psi}(\psi-\bar{\psi})-A_{1}\gamma\rho_{+}^{\gamma-1}\phi(\psi-\bar{\psi})]dx$
$\displaystyle-C(\delta+\varepsilon)\|(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}_{a,\nu-1},$

where the symmetric matrix $\boldsymbol{M_{3}}$ is denoted by

$\boldsymbol{M_{3}}=\begin{pmatrix}-A_{1}\gamma\rho_{+}^{\gamma-2}u_{+}&0&-A_{1}\gamma\rho_{+}^{\gamma-1}\\\
0&-A_{2}\alpha n_{+}^{\alpha-2}u_{+}&-A_{2}\alpha n_{+}^{\alpha-1}\\\
-A_{1}\gamma\rho_{+}^{\gamma-1}&-A_{2}\alpha
n_{+}^{\alpha-1}&-(\rho_{+}+n_{+})u_{+}\end{pmatrix}.$ (4.7)

It is easy to verify that $\boldsymbol{M_{3}}$ is a positive definite matrix
for $M_{+}>1$. Hence, the estimate of the third term on the left hand side is
obtained under the condition $\varepsilon$, $\delta$ and $\eta$ small enough
that

$\displaystyle-\nu\int W_{a,\nu-1}G_{1}dx$ (4.8) $\displaystyle\geq$
$\displaystyle
c\nu\|(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}_{a,\nu-1}-\eta\nu\|(\phi,\bar{\psi})\|^{2}_{a,\nu-1}-C_{\eta}\nu\|\bar{\psi}-\psi\|^{2}_{a,\nu-1}-C(\varepsilon+\delta)\|(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}_{a,\nu-1}$
$\displaystyle\geq$ $\displaystyle
c\nu\|(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}_{a,\nu-1}-C\nu\|\bar{\psi}-\psi\|^{2}_{a,\nu-1},$

By Young inequality with $0<\eta<1$, the forth and fifth terms on the left
hand side are estimated as

$\begin{split}&-\nu\int
W_{a,\nu-1}G_{2}dx\leq\nu\eta\|(\psi,\bar{\psi})\|^{2}_{a,\nu-1}+\nu
C_{\eta}\|(\psi_{x},\bar{\psi}_{x})\|^{2}_{a,\nu-1}+C\delta\|(\bar{\phi},\bar{\psi})\|^{2}_{a,\nu-1}.\end{split}$
(4.9) $\begin{split}&\quad\int
W_{a,\nu}[\mu\psi_{x}^{2}+n\bar{\psi}^{2}_{x}+n(\psi-\bar{\psi})^{2}]dx\geq
c\|(\psi_{x},\bar{\psi}_{x},\bar{\psi}-\psi)\|^{2}_{a,\nu}-C(\varepsilon+\delta)\|(\psi_{x},\bar{\psi}_{x},\psi-\bar{\psi})\|^{2}_{a,\nu}.\end{split}$
(4.10)

By $(\ref{M_{+}>1 stationary solution d})$ and $(\ref{d'1})$, it follows from
Sobolev inequality and Cauchy-Schwarz inequality that

$\displaystyle|\int W_{a,\nu}(R_{1}+R_{2}+R_{3})dx|\leq$ $\displaystyle
C\delta\int
e^{-\frac{c_{0}}{2}x}(\phi^{2}+\psi^{2}+\bar{\phi}^{2}+\bar{\psi}^{2}+|\bar{\psi}-\psi|^{2})dx$
(4.11) $\displaystyle\leq$ $\displaystyle
C\delta\|(\phi_{x},\psi_{x},\bar{\phi}_{x},\bar{\psi}_{x},\bar{\psi}-\psi)\|^{2}+C\delta(\phi^{2}(t,0)+\bar{\phi}^{2}(t,0)).$

Finally, with $\eta$, $\delta$ and $\varepsilon$ suitably small, the
substitution of $(\ref{second term e})$-$(\ref{mathcal{I}_{2} e})$ into
$(\ref{f_01})$ leads to

$\displaystyle\frac{d}{dt}\int
W_{a,\nu}(\mathcal{E}_{1}+\mathcal{E}_{2})dx+c\nu\|(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}_{a,\nu-1}+c\|(\psi_{x},\bar{\psi}_{x},\psi-\bar{\psi})\|^{2}_{a,\nu}+c(\phi^{2}(t,0)+\bar{\phi}^{2}(t,0))$
(4.12) $\displaystyle\leq$ $\displaystyle
C\delta\|(\phi_{x},\bar{\phi}_{x})\|^{2}+C\nu\|(\psi_{x},\bar{\psi}_{x},\psi-\bar{\psi})\|^{2}_{a,\nu-1}.~{}~{}~{}~{}~{}~{}~{}$

Multiplying (4.12) by $(1+\tau)^{\xi}$ and integrating the resulted equation
in $\tau$ over $[0,t]$, we gain the desired estimate $(\ref{L^{2} time e1})$.
∎

Similar to Lemmas 3.4-3.5, we get the following high order weighted estimates
of $(\phi,\psi,\bar{\phi},\bar{\psi})$. The details are omitted.

######  Lemma 4.3.

Under the same conditions in Proposition 4.1, then the solution
$(\phi,\psi,\bar{\phi},\bar{\psi})$ to the IBVP $(\ref{f1})$-$(\ref{boundary
d1})$ satisfies for $t\in[0,T]$ that

$\displaystyle(1+t)^{\xi}\|(\phi_{x},\bar{\phi}_{x})\|^{2}+\int_{0}^{t}(1+\tau)^{\xi}\|(\phi_{x},\bar{\phi}_{x})\|^{2}d\tau$
(4.13) $\displaystyle\leq$ $\displaystyle
C(\|(\phi_{0},\psi_{0},\bar{\phi}_{0},\bar{\psi}_{0})\|^{2}_{a,\lambda}+\|(\phi_{0x},\bar{\phi}_{0x})\|^{2})+C\nu\int_{0}^{t}(1+\tau)^{\xi}\|(\psi_{x},\bar{\psi}_{x},\psi-\bar{\psi})\|^{2}_{a,\nu-1}d\tau$
$\displaystyle+C\varepsilon\int_{0}^{t}(1+\tau)^{\xi}\|(\psi_{xx},\bar{\psi}_{xx})\|^{2}_{a,\nu}d\tau+C\xi\int_{0}^{t}(1+\tau)^{\xi-1}(\|(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}_{a,\nu}+\|(\phi_{x},\bar{\phi}_{x})\|^{2})d\tau,$

with $\xi\geq 0$.

######  Lemma 4.4.

Under the same conditions in Proposition 4.1 hold, then the solution
$(\phi,\psi,\bar{\phi},\bar{\psi})$ to the IBVP $(\ref{f1})$-$(\ref{boundary
d1})$ satisfies for $t\in[0,T]$ that

$\displaystyle(1+t)^{\xi}\|(\psi_{x},\bar{\psi}_{x})\|^{2}+\int_{0}^{t}(1+\tau)^{\xi}\|(\psi_{xx},\bar{\psi}_{xx})\|^{2}d\tau$
(4.14) $\displaystyle\leq$ $\displaystyle
C(\|(\phi_{0},\psi_{0},\bar{\phi}_{0},\bar{\psi}_{0})\|^{2}_{a,\lambda}+\|(\phi_{0x},\bar{\phi}_{0x},\psi_{0x},\bar{\psi}_{0x})\|^{2})+C\nu\int_{0}^{t}(1+\tau)^{\xi}\|(\psi_{x},\bar{\psi}_{x},\psi-\bar{\psi})\|^{2}_{a,\nu-1}d\tau$
$\displaystyle+C\xi\int_{0}^{t}(1+\tau)^{\xi-1}(\|(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}_{a,\nu}+\|(\phi_{x},\psi_{x},\bar{\phi}_{x},\bar{\psi}_{x})\|^{2})d\tau$

with $\xi\geq 0$.

Proof of Proposition 4.1 For $\nu\in[0,\lambda]$ and $\xi\geq 0$, it follows
from Lemmas 4.2-4.4 that

$\displaystyle(1+t)^{\xi}(\|(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}_{a,\nu}+\|(\phi_{x},\psi_{x},\bar{\psi}_{x},\bar{\phi}_{x})\|^{2})+\nu\int(1+t)^{\xi}\|(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}_{a,\nu-1}d\tau$
(4.15)
$\displaystyle+\int_{0}^{t}(1+\tau)^{\xi}\|(\psi_{x},\bar{\psi}_{x},\psi-\bar{\psi})\|^{2}_{a,\nu}d\tau+\int_{0}^{t}(1+\tau)^{\xi}\|(\phi_{x},\psi_{xx},\bar{\phi}_{x},\bar{\psi}_{xx})\|^{2}d\tau$
$\displaystyle\leq$ $\displaystyle
C(\|(\phi_{0},\psi_{0},\bar{\phi}_{0},\bar{\psi}_{0})\|^{2}_{a,\lambda}+\|(\phi_{0x},\psi_{0x},\bar{\phi}_{0x},\bar{\psi}_{0x})\|^{2})+C\nu\int_{0}^{t}(1+\tau)^{\xi}\|(\psi_{x},\bar{\psi}_{x},\bar{\psi}-\psi)\|^{2}_{a,\nu-1}d\tau$
$\displaystyle+C\xi\int_{0}^{t}(1+\tau)^{\xi-1}(\|(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}_{a,\nu}+\|(\phi_{x},\psi_{x},\bar{\psi}_{x},\bar{\phi}_{x})\|^{2}),$

where $C>0$ is a generic positive constant independent of $T,\nu,$ and $\xi$.
Hence, applying similar induction arguments as in [14, 25, 3] to (4.15), we
gain the desired estimate (4.2).

Indeed, for any $\lambda>0$ and $k=0,1,2,...[\lambda]$, we have

$\displaystyle(1+t)^{k}(\|(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}_{a,\lambda-k}+\|(\phi_{x},\psi_{x},\bar{\psi}_{x},\bar{\phi}_{x})\|^{2})+\nu\int(1+t)^{k}\|(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}_{a,\lambda-k-1}d\tau$
(4.16)
$\displaystyle+\int_{0}^{t}(1+\tau)^{k}\|(\psi_{x},\bar{\psi}_{x},\psi-\bar{\psi})\|^{2}_{a,\lambda-k}d\tau+\int_{0}^{t}(1+\tau)^{k}\|(\phi_{x},\psi_{xx},\bar{\phi}_{x},\bar{\psi}_{xx})\|^{2}d\tau$
$\displaystyle\leq$ $\displaystyle
C(\|(\phi_{0},\psi_{0},\bar{\phi}_{0},\bar{\psi}_{0})\|^{2}_{a,\lambda}+\|(\phi_{0x},\psi_{0x},\bar{\phi}_{0x},\bar{\psi}_{0x})\|^{2}),$

and

$\displaystyle(1+t)^{k}\|(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}_{1}+\int_{0}^{t}(1+\tau)^{k}\|(\phi_{x},\psi_{x},\psi_{xx},\bar{\psi}_{x},\bar{\phi}_{x},\bar{\psi}_{xx},\psi-\bar{\psi})\|^{2}d\tau$
(4.17) $\displaystyle\leq$ $\displaystyle
C(\|(\phi_{0},\psi_{0},\bar{\phi}_{0},\bar{\psi}_{0})\|^{2}_{a,\lambda}+\|(\phi_{0x},\psi_{0x},\bar{\phi}_{0x},\bar{\psi}_{0x})\|^{2}).$

To prove $(\ref{e1-2})$ and $(\ref{e1-3})$, we apply similar induction
arguments as in [14, 25, 3] to (4.15).

Step 1. Taking $\xi=0$, $\nu=\lambda$ in $(\ref{e1-1})$ and using $(\ref{e})$,
we have $(\ref{e1-2})$ and $(\ref{e1-3})$ for $k=0$. Therefore, $(\ref{e1-2})$
and $(\ref{e1-3})$ hold for $0<\lambda<1$.

Step 2. Taking $\xi=1$, $\nu=0$ in $(\ref{e1-1})$ and using $(\ref{e1-2})$
with $k=0$, we have $(\ref{e1-3})$ with $k=1$. Then, taking $\xi=1$,
$\nu=\lambda-1$ in $(\ref{e1-1})$ and using $(\ref{e1-3})$ with $k=1$ and
$(\ref{e1-2})$ with $k=0$, we obtain the desired estimate $(\ref{e1-2})$ with
$k=1$. Therefore, the proof is finished for $1\leq\lambda<2$.

Step 3. We repeat the same procedure as in Step 2. The estimate $(\ref{e1-1})$
(with $\xi=2$, $\nu=0$) together with $(\ref{e1-3})$ (with $k=1$) lead to
$(\ref{e1-3})$ (with $k=2$). Also, $(\ref{e1-1})$ (with $\xi=2$,
$\nu=\lambda-2$) together with $(\ref{e1-3})$ (with $k=2$) and $(\ref{e1-2})$
(with $k=1$) lead to $(\ref{e1-2})$ (with $k=2$), which proves the estimates
$(\ref{e1-2})$ and $(\ref{e1-3})$ for $2\leq\lambda<3$.

Repeating the same procedure, we get the desired estimates $(\ref{e1-2})$ and
$(\ref{e1-3})$ for any $\lambda>0$.

If $\lambda>0$ is integer, we obtain $(\ref{e__1})$ from $(\ref{e1-2})$
letting $k=\lambda$.

If $\lambda>0$ is not integer, we obtain $(\ref{e__1})$ as follows.
Taking $\nu=0$ in $(\ref{e1-1})$, we have

$\displaystyle(1+t)^{\xi}\|(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}_{1}+\int_{0}^{t}(1+\tau)^{\xi}\|(\phi_{x},\psi_{x},\psi_{xx},\bar{\phi}_{x},\bar{\psi}_{x},\bar{\psi}_{xx},\psi-\bar{\psi})\|^{2}d\tau$
(4.18) $\displaystyle\leq$ $\displaystyle
C(\|(\phi_{0},\psi_{0},\bar{\phi}_{0},\bar{\psi}_{0})\|^{2}_{a,\lambda}+\|(\phi_{0x},\psi_{0x},\bar{\phi}_{0x},\bar{\psi}_{0x})\|^{2})+C\xi\int_{0}^{t}(1+\tau)^{\xi-1}\|(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}_{1}d\tau.$

Using $(\ref{e1-2})$ with $k=[\lambda]$ and taking $s=1-(\lambda-[\lambda])$,
we have

$\displaystyle\int_{0}^{t}(1+\tau)^{\xi-1}\|(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}_{1}d\tau$
$\displaystyle\leq$
$\displaystyle\int_{0}^{t}(1+\tau)^{\xi-1-[\lambda]}\\{((1+t)^{[\lambda]}\|(\phi,\psi,\bar{\phi},\bar{\phi})\|^{2}_{a,\lambda-[\lambda]})^{s}((1+t)^{[\lambda]}\|(\phi,\psi,\bar{\phi},\bar{\phi})\|^{2}_{a,\lambda-[\lambda]-1})^{1-s}$
$\displaystyle+(1+t)^{[\lambda]}\|(\phi_{x},\psi_{x},\bar{\phi}_{x},\bar{\psi}_{x})\|^{2}\\}d\tau$
$\displaystyle\leq$ $\displaystyle
C(\|(\phi_{0},\psi_{0},\bar{\phi}_{0},\bar{\psi}_{0})\|^{2}_{a,\lambda}+\|(\phi_{0x},\psi_{0x},\bar{\phi}_{0x},\bar{\psi}_{0x})\|^{2})^{s}\int_{0}^{t}(1+\tau)^{\xi-1-[\lambda]}((1+t)^{[\lambda]}\|(\phi_{x},\psi_{x},\bar{\phi}_{x},\bar{\psi}_{x})\|^{2}$
$\displaystyle+(1+t)^{[\lambda]}\|(\phi,\psi,\bar{\phi},\bar{\phi})\|^{2}_{a,\lambda-[\lambda]-1})^{1-s}d\tau$
$\displaystyle\leq$ $\displaystyle
C(\|(\phi_{0},\psi_{0},\bar{\phi}_{0},\bar{\psi}_{0})\|^{2}_{a,\lambda}+\|(\phi_{0x},\psi_{0x},\bar{\phi}_{0x},\bar{\psi}_{0x})\|^{2})(\int_{0}^{t}(1+\tau)^{\frac{\xi-1-[\lambda]}{1+[\lambda]-\lambda}}d\tau)^{1+[\lambda]-\lambda}$
$\displaystyle\leq$ $\displaystyle
C(\|(\phi_{0},\psi_{0},\bar{\phi}_{0},\bar{\psi}_{0})\|^{2}_{a,\lambda}+\|(\phi_{0x},\psi_{0x},\bar{\phi}_{0x},\bar{\psi}_{0x})\|^{2})(1+t)^{\theta},$
(4.19)

where we take $\xi=\lambda+\theta(1+[\lambda]-\lambda)$ and $\theta>0$.

###  4.2 Convergence rate of sonic steady-state

The function space $Y_{W}(0,T)$ for $T>0$ is denoted by

$\displaystyle Y_{W}(0,T):=\\{~{}(\phi,\psi,\bar{\phi},\bar{\psi})~{}|{}$
$\displaystyle(\phi,\psi,\bar{\phi},\bar{\psi})\in
C([0,T];H_{W}^{1}(\mathbb{R}_{+})),$ (4.20)
$\displaystyle(\phi_{x},\bar{\phi}_{x})\in
L^{2}([0,T];L_{W}^{2}(\mathbb{R}_{+})),~{}(\psi_{x},\bar{\psi}_{x})\in
L^{2}([0,T];H_{W}^{1}(\mathbb{R}_{+}))~{}\\}.$

######  Proposition 4.5.

Assume that $1\leq\lambda<\lambda^{*}$ with
$\lambda^{*}:=2+\sqrt{8+\frac{1}{1+b^{2}}}$,
$b:=\frac{\rho_{+}(u_{+}^{2}-p^{\prime}_{1}(\rho_{+}))}{|u_{+}|\sqrt{(\mu+n_{+})n_{+}}}$,
and that the same conditions in Theorem 1.3 hold for $M_{+}=1$. Let
$(\phi,\psi,\bar{\phi},\bar{\psi})$ be a solution to the IBVP
$(\ref{f1})$-$(\ref{boundary d1})$ satisfying
$(\phi,\psi,\bar{\phi},\bar{\psi})\in Y_{\sigma^{-\lambda}}(0,T)$ for any time
$T>0$. Then for arbitrary $\nu\in(0,\lambda]$, there exist positive constants
$\varepsilon>0$ and $C>0$ independent of $T$ such that if

$\sup_{0\leq t\leq
T}\|\sigma^{-\frac{\lambda}{2}}(\phi,\psi,\bar{\phi},\bar{\psi})(t)\|_{1}+\delta^{\frac{1}{2}}\leq\varepsilon$
(4.21)

is satisfied, it holds for arbitrary $t\in[0,T]$ that

$\displaystyle(1+\delta
t)^{\frac{\lambda-\nu}{2}+\beta}\|\sigma^{-\frac{\nu}{2}}(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}_{1}+\int^{t}_{0}(1+\delta\tau)^{\frac{\lambda-\nu}{2}+\beta}\|\sigma^{-\frac{\nu-2}{2}}(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}d\tau$
(4.22)
$\displaystyle+\int^{t}_{0}(1+\delta\tau)^{\frac{\lambda-\nu}{2}+\beta}\|\sigma^{-\frac{\nu}{2}}(\phi_{x},\psi_{x},\bar{\phi}_{x},\bar{\psi}_{x})\|^{2}d\tau+\int^{t}_{0}(1+\delta\tau)^{\frac{\lambda-\nu}{2}+\beta}\|\sigma^{-\frac{\nu}{2}}(\psi_{xx},\bar{\psi}_{xx},\bar{\psi}-\psi)\|^{2}d\tau$
$\displaystyle\leq$ $\displaystyle C(1+\delta
t)^{\beta}\|\sigma^{-\frac{\lambda}{2}}(\phi_{0},\psi_{0},\phi_{0x},\psi_{0x},\bar{\phi}_{0},\bar{\psi}_{0},\bar{\phi}_{0x},\bar{\psi}_{0x})\|^{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

with $\beta>0$.

By the fact $\lambda\geq 1$ and $(\ref{M_{+}=1 prior a})$, it is easy to
verify the following estimate:

$\|\sigma^{-\frac{1}{2}}(\phi,\psi,\bar{\phi},\bar{\psi})\|_{L^{\infty}}\leq\|\sigma^{-\frac{\lambda}{2}}(\phi,\psi,\bar{\phi},\bar{\psi})\|_{1}\leq\sqrt{2}\varepsilon.$
(4.23)

To deal with some nonlinear terms, we use the following inequality as in [27,
15, 36].

######  Lemma 4.6 ([36] ).

Let $\nu\geq 1$. Then a function $\sigma^{-\frac{\nu}{2}}(x)\phi(t,x)\in
H^{1}(\mathbb{R}_{+})$ satisfies

$\int\sigma^{-\frac{\nu-1}{2}}|\phi|^{3}dx\leq
C\|\sigma^{-\frac{1}{2}}\phi\|~{}(\sigma(0)\phi^{2}(t,0)+\|\sigma^{-\frac{\nu}{2}}\phi_{x}\|^{2}+\|\sigma^{-\frac{\nu-2}{2}}\phi\|^{2}),$
(4.24)

where the function $\sigma(x)\geq 0$ is defined by $(\ref{sg0})$ with
$\sigma(0)$ small enough.

To gain faster decay rates, it is necessary to use the following Hardy type
inequality.

######  Lemma 4.7 ([13] ).

Let $\zeta\in C^{1}[0,\infty)$ satisfies $\zeta>0$, $\zeta_{x}>0$ and
$\zeta(x)\rightarrow\infty$ for $x\rightarrow\infty$. Then we have

$\int_{\mathbb{R}_{+}}\psi^{2}\zeta_{x}dx\leq
4\int_{\mathbb{R}_{+}}\psi^{2}_{x}\frac{\zeta^{2}}{\zeta_{x}}dx$ (4.25)

for $\psi$ satisfying $\psi(t,0)=0$ and $\sqrt{w}\psi\in
H^{1}(\mathbb{R}_{+})$, with the function $w:=\frac{\zeta^{2}}{\zeta_{x}}$.

With the aid of Lemmas 4.6-4.7, we obtain the weighted $L^{2}$ estimate of
$(\phi,\psi,\bar{\phi},\bar{\psi})$.

######  Lemma 4.8.

Under the same conditions in Proposition 4.5, then the solution
$(\phi,\psi,\bar{\phi},\bar{\psi})$ to the problem
$(\ref{f1})$-$(\ref{boundary d1})$ satisfies for $t\in[0,T]$ that

$\displaystyle(1+\delta\tau)^{\xi}\|\sigma^{-\frac{\nu}{2}}(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}+\int^{t}_{0}(1+\delta\tau)^{\xi}\|\sigma^{-\frac{\nu-2}{2}}(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}d\tau$
(4.26)
$\displaystyle+\int^{t}_{0}(1+\delta\tau)^{\xi}\|\sigma^{-\frac{\nu}{2}}(\psi_{x},\bar{\psi}_{x},\bar{\psi}-\psi)\|^{2}d\tau+\int^{t}_{0}(1+\delta\tau)^{\xi}\frac{1}{\delta^{\nu}}(\phi^{2}(t,0)+\bar{\phi}^{2}(t,0))d\tau$
$\displaystyle\leq$ $\displaystyle
C\|\sigma^{-\frac{\lambda}{2}}(\phi_{0},\psi_{0},\bar{\phi}_{0},\bar{\psi}_{0})\|^{2}+C\delta\int^{t}_{0}(1+\delta\tau)^{\xi}\|\sigma^{-\frac{\nu}{2}}(\phi_{x},\bar{\phi}_{x})\|^{2}d\tau$
$\displaystyle+C\delta\xi\int^{t}_{0}(1+\delta\tau)^{\xi-1}\|\sigma^{-\frac{\nu}{2}}(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}d\tau,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

with $\xi\geq 0$.

###### Proof.

We multiply $(\ref{f_0})$ by the space weight function $\sigma^{-\nu}$, where
the space weight function $\sigma\geq 0$ satisfies $(\ref{sig})$ and
$(\ref{sg0})$. Then, we integrate the resulted equation over $\mathbb{R}_{+}$
to get

$\displaystyle\frac{d}{dt}\int\sigma^{-\nu}(\mathcal{E}_{1}+\mathcal{E}_{2})dx-(\sigma^{-\nu}G_{1})(t,0)-a\nu\int\sigma^{-(\nu-1)}G_{1}dx-a\nu\int\sigma^{-(\nu-1)}G_{2}dx$
(4.27)
$\displaystyle+\int\sigma^{-\nu}n(\bar{\psi}-\psi)^{2}dx+\int\sigma^{-\nu}(\mu\psi_{x}^{2}+n\bar{\psi}^{2}_{x})dx+\int\sigma^{-\nu}R_{1}dx$
$\displaystyle=$
$\displaystyle-\int\sigma^{-\nu}R_{2}dx-\int\sigma^{-\nu}R_{3}dx,$

where $\mathcal{E}_{i}$, $i=1,2$ are defined by $(\ref{mcE 1})$-$(\ref{mcE
2})$, and $G_{j}$ for $j=1,2$, $R_{k}$ for $k=1,2,3$ are defined by
$(\ref{G})$. First, we estimate terms on the left hand side of $(\ref{f_02})$.
Under the condition $(\ref{boundary d1})$, the second term on the left hand
side is estimated as

$\begin{split}&~{}~{}~{}-(\sigma^{-\nu}G_{1})(t,0)\geq\frac{c}{\delta^{\nu}}(\phi^{2}(t,0)+\bar{\phi}^{2}(t,0))\geq
0.~{}~{}~{}~{}~{}~{}~{}\end{split}$ (4.28)

For the third term on the left hand side, using $(\ref{nonlinear f})$ and
$\psi=\bar{\psi}+(\psi-\bar{\psi})$ yields

$\displaystyle-a\nu\int\sigma^{-(\nu-1)}G_{1}dx$ (4.29) $\displaystyle\geq$
$\displaystyle
a\nu\int\sigma^{-(\nu-1)}[\frac{1}{2}(\phi,\bar{\phi},\bar{\psi})\boldsymbol{M_{4}}(\phi,\bar{\phi},\bar{\psi})^{\rm
T}-\rho_{+}u_{+}\bar{\psi}(\psi-\bar{\psi})-A_{1}\gamma\rho^{\gamma-1}_{+}\phi(\psi-\bar{\psi})]dx$
$\displaystyle+a\nu\int\sigma^{-(\nu-1)}[-(A_{1}\gamma\widetilde{\rho}^{\gamma-2}\widetilde{u}-A_{1}\gamma\widetilde{\rho}_{+}^{\gamma-2}u_{+})\frac{\phi^{2}}{2}-(A_{2}\alpha\widetilde{n}^{\alpha-2}\widetilde{v}-A_{2}\alpha
n_{+}^{\alpha-2}u_{+})\frac{\bar{\phi}^{2}}{2}$
$\displaystyle-(A_{1}\gamma\widetilde{\rho}^{\gamma-1}-A_{1}\gamma\rho_{+}^{\gamma-1})\phi\psi-(A_{2}\alpha\widetilde{n}^{\alpha-1}-A_{2}\alpha
n_{+}^{\alpha-1})\bar{\phi}\bar{\psi}]dx-C\frac{\varepsilon}{\delta^{\nu}}(\phi^{2}(t,0)+\bar{\phi}^{2}(t,0))$
$\displaystyle-C\varepsilon(\|\sigma^{-(\frac{\nu-2}{2})}(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}+\|\sigma^{-\frac{\nu}{2}}(\phi_{x},\psi_{x},\bar{\phi}_{x},\bar{\psi}_{x})\|^{2}),$

where the symmetric matrix $\boldsymbol{M_{4}}$ is defined as

$\begin{split}&\boldsymbol{M_{4}}=\begin{pmatrix}-A_{1}\gamma\rho_{+}^{\gamma-2}u_{+}&0&-A_{1}\gamma\rho_{+}^{\gamma-1}\\\
0&-A_{2}\alpha n_{+}^{\alpha-2}u_{+}&-A_{2}\alpha n_{+}^{\alpha-1}\\\
-A_{1}\gamma\rho_{+}^{\gamma-1}&-A_{2}\alpha
n_{+}^{\alpha-1}&-(\rho_{+}+n_{+})u_{+}\end{pmatrix}.\end{split}$ (4.30)

Owing to $M_{+}=1$, it is easy to check that three eigenvalues of the matrix
$\boldsymbol{M_{4}}$ satisfy: $\hat{\lambda}_{1}>0$, $\hat{\lambda}_{2}>0$,
$\hat{\lambda}_{3}=0$. Take the coordinate transformation

$\begin{split}\begin{pmatrix}\phi\\\ \bar{\phi}\\\
\bar{\psi}\end{pmatrix}=\boldsymbol{P}\begin{pmatrix}\hat{\rho}\\\ \hat{n}\\\
\hat{v}\end{pmatrix},\end{split}$ (4.31)

where the matrix $\boldsymbol{P}$ is denoted by

$\displaystyle\boldsymbol{P}=\begin{pmatrix}r_{11}&r_{21}&-\frac{\rho_{+}}{u_{+}}\\\
r_{12}&r_{22}&-\frac{n_{+}}{u_{+}}\\\ r_{13}&r_{23}&1\end{pmatrix}{\rm
with~{}constants~{}}r_{ij}~{}{\rm for}~{}1\leq i\leq 2,~{}1\leq j\leq 3,$
(4.32)

such that

$\begin{split}(\phi,\bar{\phi},\bar{\psi})~{}\boldsymbol{M_{4}}~{}(\phi,\bar{\phi},\bar{\psi})^{\rm
T}=(\hat{\rho},\hat{n},\hat{v})\begin{pmatrix}\hat{\lambda}_{1}&0&0\\\
0&\hat{\lambda}_{2}&0\\\ 0&0&0\end{pmatrix}(\hat{\rho},\hat{n},\hat{v})^{\rm
T}=\hat{\lambda}_{1}\hat{\rho}^{2}+\hat{\lambda}_{2}\hat{n}^{2},\end{split}$
(4.33)

By $(\ref{sigma 1})$, $(\ref{infty e2})$, $(\ref{bound})$, and $(\ref{bar rho
u})$-$(\ref{P 1})$, the third term is estimated as

$\displaystyle-a\nu\int\sigma^{-(\nu-1)}G_{1}dx$ (4.34) $\displaystyle\geq$
$\displaystyle
a\nu\int\sigma^{-(\nu-1)}(\frac{\hat{\lambda}_{1}}{2}\hat{\rho}^{2}+\frac{\hat{\lambda}_{2}}{2}\hat{n}^{2})dx+a\nu\int\sigma^{-(\nu-2)}\frac{A_{1}\gamma(\gamma+1)\rho_{+}^{\gamma}+A_{2}\alpha(\alpha+1)n_{+}^{\alpha}}{2|u_{+}|^{2}}\hat{v}^{2}dx$
$\displaystyle+a\nu\int\sigma^{-(\nu-1)}\frac{\rho_{+}(u_{+}^{2}-A_{1}\gamma\rho_{+}^{\gamma-1})}{|u_{+}|}\hat{v}(\psi-\bar{\psi})dx-C\delta^{\frac{1}{2}}\|\sigma^{-\frac{\nu-1}{2}}(\hat{\rho},\hat{n})\|^{2}-C\delta^{\frac{1}{2}}\|\sigma^{-\frac{\nu-2}{2}}\hat{v}\|^{2}$
$\displaystyle-C\delta^{\frac{1}{2}}\|\sigma^{-\frac{\nu}{2}}(\psi-\bar{\psi})\|^{2}-C(\varepsilon+\delta)\|\sigma^{-\frac{\nu-2}{2}}(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}-C\varepsilon\|\sigma^{-\frac{\nu}{2}}(\phi_{x},\psi_{x},\bar{\phi}_{x},\bar{\psi}_{x})\|^{2}$
$\displaystyle-C\varepsilon\frac{1}{\delta^{\nu}}(\phi^{2}(t,0)+\bar{\phi}^{2}(t,0)),$

where we have used the following facts

$\displaystyle-(A_{1}\gamma\widetilde{\rho}^{\gamma-1}-A_{1}\gamma\rho_{+}^{\gamma-1})\geq\frac{A_{1}\gamma(\gamma-1)\rho_{+}^{\gamma-1}}{|u_{+}|}\sigma-C\sigma^{2},$
$\displaystyle-(A_{2}\alpha\widetilde{n}^{\alpha-1}-A_{2}\alpha
n_{+}^{\alpha-1})\geq\frac{A_{2}\alpha(\alpha-1)n_{+}^{\alpha-1}}{|u_{+}|}\sigma-C\sigma^{2},$
$\displaystyle-(A_{1}\gamma\widetilde{\rho}^{\gamma-2}\widetilde{u}-A_{1}\gamma\rho_{+}^{\gamma-2}u_{+})\geq
A_{1}\gamma(3-\gamma)\rho_{+}^{\gamma-2}\sigma-C\sigma^{2},$
$\displaystyle-(A_{2}\alpha\widetilde{n}^{\alpha-2}\widetilde{v}-A_{2}\alpha
n_{+}^{\alpha-2}u_{+})\geq
A_{2}\alpha(3-\alpha)n_{+}^{\alpha-2}\sigma-C\sigma^{2}.$

With the help of $(\ref{sigma})$-$(\ref{sig})$, $(\ref{boundary d1})$,
$(\ref{M_{+}=1 prior a})$, $(\ref{infty e2})$-$(\ref{nonlinear f})$,
$(\ref{bar rho u})$-$(\ref{P 1})$, and $\psi=\bar{\psi}+(\psi-\bar{\psi})$, it
holds that

$\displaystyle-a\nu\int\sigma^{-(\nu-1)}G_{2}(t,x)dx$ (4.35)
$\displaystyle\geq$
$\displaystyle-a^{2}\frac{\mu+n_{+}}{2}\nu(\nu-1)\|\sigma^{-\frac{\nu-2}{2}}\bar{\psi}\|^{2}-C(\delta+\varepsilon)\|\sigma^{-\frac{\nu-2}{2}}\bar{\psi}\|^{2}-C\varepsilon\|\sigma^{-\frac{\nu}{2}}\bar{\psi}_{x}\|^{2}-C\delta\|\sigma^{-\frac{\nu}{2}}(\bar{\psi}-\psi)\|^{2}$
$\displaystyle\geq$
$\displaystyle-a^{2}\frac{\mu+n_{+}}{2}\nu(\nu-1)\|\sigma^{-\frac{\nu-2}{2}}\hat{v}\|^{2}-C\delta^{\frac{1}{2}}(\|\sigma^{-\frac{\nu-1}{2}}(\hat{\rho},\hat{n})\|^{2}+\|\sigma^{-\frac{\nu-2}{2}}\hat{v}\|^{2})-C(\delta+\varepsilon)\|\sigma^{-\frac{\nu-2}{2}}\bar{\psi}\|^{2}$
$\displaystyle-C\varepsilon\|\sigma^{-\frac{\nu}{2}}\bar{\psi}_{x}\|^{2}-C\delta\|\sigma^{-\frac{\nu}{2}}(\bar{\psi}-\psi)\|^{2},$
$\int\sigma^{-\nu}n(\bar{\psi}-\psi)^{2}dx\geq[n_{+}-C(\delta+\varepsilon)]\|\sigma^{-\frac{\nu}{2}}(\bar{\psi}-\psi)\|^{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(4.36) $\displaystyle\int\sigma^{-\nu}R_{1}dx$ (4.37) $\displaystyle\geq$
$\displaystyle
a\int\sigma^{-(\nu-2)}[\frac{A_{1}\gamma(\gamma-1)\rho_{+}^{\gamma-2}}{2}\phi^{2}+\rho_{+}\psi^{2}+\frac{A_{2}\alpha(\alpha-1)n_{+}^{\alpha-2}}{2}\bar{\phi}^{2}+n_{+}\bar{\psi}^{2}]dx$
$\displaystyle-C(\delta+\varepsilon)\|\sigma^{-\frac{\nu-2}{2}}(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}\quad\quad$
$\displaystyle\geq$ $\displaystyle
a\frac{A_{1}\gamma(\gamma+1)\rho_{+}^{\gamma}+A_{2}\alpha(\alpha+1)n_{+}^{\alpha}}{2|u_{+}|^{2}}\|\sigma^{-\frac{\nu-2}{2}}\hat{v}\|^{2}-C\delta(\|\sigma^{-\frac{\nu-1}{2}}(\hat{\rho},\hat{n})\|^{2}+\|\sigma^{-\frac{\nu-2}{2}}\hat{v}\|^{2})$
$\displaystyle-C(\delta+\varepsilon)\|\sigma^{-\frac{\nu-2}{2}}(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}.$

For $\nu\in(0,3]$, with the help of $(\ref{sigma 1})$ and $(\ref{a})$, we add
$(\ref{G1-3})$-$(\ref{fifth term})$ together to have

$\displaystyle-a\nu\int\sigma^{-(\nu-1)}G_{1}dx-a\nu\int\sigma^{-(\nu-1)}G_{2}dx+\int\sigma^{-\nu}n(\bar{\psi}-\psi)^{2}dx+\int\sigma^{-\nu}R_{1}dx$
$\displaystyle\geq$ $\displaystyle
c\|\sigma^{-\frac{\nu-1}{2}}(\hat{\rho},\hat{n})\|^{2}+\frac{1}{4}\\{a\frac{A_{1}\gamma(\gamma+1)\rho_{+}^{\gamma}+A_{2}\alpha(\alpha+1)n_{+}^{\alpha}}{2|u_{+}|^{2}}[1+\nu-\frac{\nu(\nu-1)}{2(1+b^{2})}]\|\sigma^{-\frac{\nu-2}{2}}\hat{v}\|^{2}$
$\displaystyle+n_{+}\|\sigma^{-\frac{\nu}{2}}(\psi-\bar{\psi})\|^{2}\\}+\int\sigma^{-\nu}(\psi-\bar{\psi},\hat{v})\boldsymbol{M_{5}}(\psi-\bar{\psi},\hat{v})^{\rm
T}dx-C\delta^{\frac{1}{2}}\|\sigma^{-\frac{\nu-1}{2}}(\hat{\rho},\hat{n})\|^{2}$
$\displaystyle-C\delta^{\frac{1}{2}}\|\sigma^{-\frac{\nu-2}{2}}\hat{v}\|^{2}-C(\varepsilon+\delta^{\frac{1}{2}})\|\sigma^{-\frac{\nu}{2}}(\psi-\bar{\psi})\|^{2}-C(\varepsilon+\delta)\|\sigma^{-\frac{\nu}{2}}(\phi_{x},\psi_{x},\bar{\phi}_{x},\bar{\psi}_{x})\|^{2}$
$\displaystyle-C\frac{\varepsilon}{\delta^{\nu}}(\phi^{2}(t,0)+\bar{\phi}^{2}(t,0))-C(\varepsilon+\delta)\|\sigma^{-\frac{\nu-2}{2}}(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}$
$\begin{split}&\geq
c\|\sigma^{-\frac{\nu-2}{2}}(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}+c\|\sigma^{-\frac{\nu-1}{2}}(\hat{\rho},\hat{n})\|^{2}+c\|\sigma^{-\frac{\nu}{2}}(\bar{\psi}-\psi)\|^{2}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\\
&\quad-C(\delta+\varepsilon)\|\sigma^{-\frac{\nu}{2}}(\phi_{x},\psi_{x},\bar{\phi}_{x},\bar{\psi}_{x})\|^{2}-C\varepsilon\frac{\phi^{2}(t,0)+\bar{\phi}^{2}(t,0)}{\delta^{\nu}},\end{split}$
(4.38)

where the positive definite matrix $\boldsymbol{M_{5}}$ is defined by

$\boldsymbol{M_{5}}=\begin{pmatrix}\frac{3}{4}n_{+}&\frac{\sqrt{(\mu+n_{+})n_{+}}}{2}ab\nu\sigma\\\
\frac{\sqrt{(\mu+n_{+})n_{+}}}{2}ab\nu\sigma&\quad\frac{3}{4}a\frac{A_{1}\gamma(\gamma+1)\rho_{+}^{\gamma}+A_{2}\alpha(\alpha+1)n_{+}^{\alpha}}{2|u_{+}|^{2}}[(1+\nu)-\frac{\nu(\nu-1)}{2(1+b^{2})}]\sigma^{2}\end{pmatrix}.$
(4.39)

Then, we consider the case $\nu\in[3,2+\sqrt{8+\frac{1}{1+b^{2}}})$ using the
Lemma 4.7 with $\zeta=\sigma^{-(\nu-1)}$. Therefore, with the aid of
$(\ref{a})$, the sixth term is estimated as below:

$\displaystyle\int\sigma^{-\nu}(\mu\psi^{2}_{x}+n\bar{\psi}_{x}^{2})dx$ (4.40)
$\displaystyle\geq$ $\displaystyle
a^{2}(\mu+n_{+})\frac{(\nu-1)^{2}}{4}\|\sigma^{-\frac{\nu-2}{2}}\hat{v}\|^{2}-C\delta^{\frac{1}{2}}(\|\sigma^{-\frac{\nu-2}{2}}\hat{v}\|^{2}+\|\sigma^{-\frac{\nu-1}{2}}(\hat{\rho},\hat{n})\|^{2})$
$\displaystyle-C(\delta+\varepsilon)\|\sigma^{-\frac{\nu}{2}}(\bar{\psi}-\psi,\bar{\psi}_{x})\|^{2}.$

For $\nu\in(3,\lambda]$, adding $(\ref{lower bdd})$ to $(\ref{six term})$,
taking $k=\nu B[4(1+b^{2})\nu+4b^{2}+5-\nu^{2}]^{-\frac{1}{2}}\in(0,1)$, and
using $(\ref{a})$ and
$c|(\phi,\bar{\phi},\bar{\psi})|\leq|(\hat{\rho},\hat{n},\hat{v})|\leq
C|(\phi,\bar{\phi},\bar{\psi})|$, we have

$\displaystyle-a\nu\int\sigma^{-(\nu-1)}G_{1}dx-a\nu\int\sigma^{-(\nu-1)}G_{2}dx+\int\sigma^{-\nu}n(\bar{\psi}-\psi)^{2}dx+\int\sigma^{-\nu}R_{1}dx$
(4.41)
$\displaystyle+\int\sigma^{-\nu}(\mu\psi_{x}^{2}+n\bar{\psi}_{x}^{2})dx$
$\displaystyle\geq$ $\displaystyle
c\|\sigma^{-\frac{\nu-1}{2}}(\hat{\rho},\hat{n})\|^{2}+\int\sigma^{-\nu}(\psi-\bar{\psi},\hat{v})\boldsymbol{M_{6}}(\psi-\bar{\psi},\hat{v})^{\rm
T}dx+(1-k)\\{n_{+}\|\sigma^{-\frac{\nu}{2}}(\psi-\bar{\psi})\|^{2}$
$\displaystyle+a\frac{A_{1}\gamma(\gamma+1)\rho_{+}^{\gamma}+A_{2}\alpha(\alpha+1)n_{+}^{\alpha}}{2|u_{+}|^{2}}[1+\nu-\frac{\nu(\nu-1)}{2(1+b^{2})}+\frac{(\nu-1)^{2}}{4(1+b^{2})}]\|\sigma^{-\frac{\nu-2}{2}}\hat{v}\|^{2}\\}$
$\displaystyle-C\delta^{\frac{1}{2}}(\|\sigma^{-\frac{\nu-1}{2}}(\hat{\rho},\hat{n})\|^{2}+\|\sigma^{-\frac{\nu-2}{2}}\hat{v}\|^{2})-C(\varepsilon+\delta)\|\sigma^{-\frac{\nu}{2}}(\phi_{x},\psi_{x},\bar{\phi}_{x},\bar{\psi}_{x})\|^{2}$
$\displaystyle-C(\varepsilon+\delta^{\frac{1}{2}})\|\sigma^{-\frac{\nu}{2}}(\psi-\bar{\psi})\|^{2}-C\frac{\varepsilon}{\delta^{\nu}}(\phi^{2}(t,0)+\bar{\phi}^{2}(t,0))-C(\varepsilon+\delta)\|\sigma^{-\frac{\nu-2}{2}}(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}$
$\displaystyle\geq$ $\displaystyle
c\|\sigma^{-\frac{\nu-2}{2}}(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}+c\|\sigma^{-\frac{\nu-1}{2}}(\hat{\rho},\hat{n})\|^{2}+c\|\sigma^{-\frac{\nu}{2}}(\bar{\psi}-\psi,\psi_{x},\bar{\psi}_{x})\|^{2}$
$\displaystyle-C(\delta+\varepsilon)\|\sigma^{-\frac{\nu}{2}}(\phi_{x},\bar{\phi}_{x})\|^{2}-C\frac{\varepsilon}{\delta^{\nu}}(\phi^{2}(t,0)+\bar{\phi}^{2}(t,0)),$

where $\delta^{\frac{1}{2}}$ and $\varepsilon$ are small enough, and the
positive definite matrix $\boldsymbol{M_{6}}$ is defined as

$\boldsymbol{M_{6}}=\begin{pmatrix}kn_{+}&\frac{\sqrt{(\mu+n_{+})n_{+}}}{2}ab\nu\sigma\\\
\frac{\sqrt{(\mu+n_{+})n_{+}}}{2}ab\nu\sigma&\quad
ka\frac{A_{1}\gamma(\gamma+1)\rho_{+}^{\gamma}+A_{2}\alpha(\alpha+1)n_{+}^{\alpha}}{2|u_{+}|^{2}}[1+\nu-\frac{\nu(\nu-1)}{2(1+b^{2})}+\frac{(\nu-1)^{2}}{4(1+b^{2})}]\sigma^{2}\end{pmatrix}.$
(4.42)

By $(\ref{sigma})$-$(\ref{sigma 1})$, $(\ref{bar rho u})$-$(\ref{P 1})$,
Cauchy-Schwarz inequality and $M_{+}=1$, we estimate terms on the right hand
side as

$\displaystyle|\int\sigma^{-\nu}R_{2}dx+\int\sigma^{-\nu}R_{3}dx|$ (4.43)
$\displaystyle\leq$ $\displaystyle
C\delta^{\frac{1}{2}}\|\sigma^{-\frac{\nu-1}{2}}(\hat{\rho},\hat{n})\|^{2}+C\delta^{\frac{1}{2}}\|\sigma^{-\frac{\nu-2}{2}}(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}+C\delta\|\sigma^{-\frac{\nu}{2}}(\bar{\psi}_{x},\bar{\psi}-\psi)\|^{2}.$

Finally, taking $\delta^{\frac{1}{2}}$ and $\varepsilon$ small enough, and
combining $(\ref{second term e2})$-$(\ref{R e})$, we obtain

$\displaystyle\frac{d}{dt}\int\sigma^{-\nu}(\mathcal{E}_{1}+\mathcal{E}_{2})dx+c\|\sigma^{-\frac{\nu-2}{2}}(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}+c\|\sigma^{-\frac{\nu}{2}}(\psi_{x},\bar{\psi}_{x},\bar{\psi}-\psi)\|^{2}+\frac{c}{\delta^{\nu}}[\phi^{2}(t,0)+\bar{\phi}^{2}(t,0)]$
$\displaystyle\leq$ $\displaystyle
C(\delta+\varepsilon)\|\sigma^{-\frac{\nu}{2}}(\phi_{x},\bar{\phi}_{x})\|^{2}.$
(4.44)

Multiplying $(\ref{r1})$ by $(1+\delta\tau)^{\xi}$ and integrating the
resulted equation in $\tau$ over $[0,t]$, we obtain $(\ref{L^{2} e2})$. The
proof of Lemma 4.8 is completed. ∎

In order to show Proposition 4.5, we need to obtain the high order weighted
estimates of $(\phi,\psi,\bar{\phi},\bar{\psi})$.

######  Lemma 4.9.

Under the same conditions in Proposition 4.5, then the solution
$(\phi,\psi,\bar{\phi},\bar{\psi})$ to the IBVP $(\ref{f1})$-$(\ref{boundary
d1})$ satisfies for $t\in[0,T]$ that

$\displaystyle(1+\delta
t)^{\xi}\|\sigma^{-\frac{\nu}{2}}(\phi_{x},\bar{\phi}_{x})\|^{2}+\int_{0}^{t}(1+\delta\tau)^{\xi}\|\sigma^{-\frac{\nu}{2}}(\phi_{x},\bar{\phi}_{x})\|^{2}d\tau$
(4.45) $\displaystyle\leq$ $\displaystyle
C(\|\sigma^{-\frac{\lambda}{2}}(\phi_{0},\psi_{0},\bar{\phi}_{0},\bar{\psi}_{0})\|^{2}+\|\sigma^{-\frac{\lambda}{2}}(\phi_{0x},\bar{\phi}_{0x})\|^{2})+C(\varepsilon+\delta)\int(1+\delta\tau)^{\xi}\|\sigma^{-\frac{\nu}{2}}(\psi_{xx},\bar{\psi}_{xx})\|^{2}d\tau$
$\displaystyle+\delta\xi\int_{0}^{t}(1+\delta\tau)^{\xi-1}\|\sigma^{-\frac{\nu}{2}}(\phi,\psi,\phi_{x},\bar{\phi},\bar{\psi},\bar{\phi}_{x})\|^{2}d\tau,$

with $\xi\geq 0$.

###### Proof.

Adding $(\ref{h_{x}})$-$(\ref{bs_{x}})$ together, and multiplying the resulted
equation by $\sigma^{-\nu}$ with the weight function $\sigma\geq 0$ satisfying
$(\ref{sig})$ and $(\ref{sg0})$, we integrate the resulted equation in $x$
over $\mathbb{R}_{+}$ to obtain

$\displaystyle\frac{d}{dt}\int\sigma^{-\nu}(\mu\frac{\phi_{x}^{2}}{2}+\frac{\bar{\phi}_{x}^{2}}{2}+\widetilde{\rho}^{2}\phi_{x}\psi+\widetilde{n}\bar{\phi}_{x}\bar{\psi})dx-[\sigma^{-\nu}(\mu
u\frac{\phi_{x}^{2}}{2}+v\frac{\bar{\phi}_{x}^{2}}{2}-\widetilde{\rho}^{2}\phi_{t}\psi-\widetilde{n}\bar{\phi}_{t}\bar{\psi})](t,0)$
(4.46) $\displaystyle-a\nu\int\sigma^{-(\nu-1)}[\mu
u\frac{\phi_{x}^{2}}{2}+v\frac{\bar{\phi}_{x}^{2}}{2}-\widetilde{\rho}^{2}\phi_{t}\psi-\widetilde{n}\bar{\phi}_{t}\bar{\psi}]dx+\int\sigma^{-\nu}(\widetilde{\rho}^{2}\frac{p^{\prime}_{1}(\rho)}{\rho}\phi_{x}^{2}+\widetilde{n}\frac{p^{\prime}_{2}(n)}{n}\bar{\phi}_{x}^{2})dx$
$\displaystyle=$ $\displaystyle\sum_{i=1}^{6}\mathcal{J}_{i},$

where

$\displaystyle\mathcal{J}_{1}=$
$\displaystyle-\int\sigma^{-\nu}[\widetilde{\rho}^{2}(\phi_{t}+u\phi_{x})\psi_{x}+\widetilde{n}(\bar{\phi}_{t}+v\bar{\phi}_{x})\bar{\psi}_{x}+2\widetilde{\rho}\widetilde{\rho}_{x}\phi_{t}\psi+\widetilde{n}_{x}\bar{\phi}_{t}\bar{\psi}]dx,$
$\displaystyle\mathcal{J}_{2}=$
$\displaystyle\int\sigma^{-\nu}[-\mu\phi\phi_{x}\psi_{xx}-\bar{\phi}\bar{\phi}_{x}\bar{\psi}_{xx}+\mu\widetilde{\rho}^{2}(\frac{1}{\rho}-\frac{1}{\widetilde{\rho}})\phi_{x}\psi_{xx}+\widetilde{n}(\frac{1}{n}-\frac{1}{\widetilde{n}})(\widetilde{n}\bar{\psi}_{x})_{x}\bar{\phi}_{x}]dx,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\mathcal{J}_{3}=$
$\displaystyle\int\sigma^{-\nu}[\widetilde{\rho}^{2}\frac{n}{\rho}(\bar{\psi}-\psi)\phi_{x}-\widetilde{n}(\bar{\psi}-\psi)\bar{\phi}_{x}]dx,$
$\displaystyle\mathcal{J}_{4}=$
$\displaystyle-\int\sigma^{-\nu}(\frac{3}{2}\mu\psi_{x}\frac{\phi_{x}^{2}}{2}+\frac{3}{2}\bar{\psi}_{x}\bar{\phi}_{x}^{2})dx+\int\sigma^{-\nu}\widetilde{n}\frac{(\bar{\phi}\bar{\psi}_{x})_{x}}{n}\bar{\phi}_{x}dx,\quad\mathcal{J}_{5}=\int\sigma^{-\nu}(F_{1}\widetilde{\rho}^{2}\phi_{x}+F_{2}\widetilde{n}\bar{\phi}_{x})dx,$
$\displaystyle\mathcal{J}_{6}=$
$\displaystyle-\int\sigma^{-\nu}[\mu(\frac{1}{2}\phi_{x}\widetilde{u}_{x}+\psi_{x}\widetilde{\rho}_{x})\phi_{x}+\mu(\phi\widetilde{u}_{x}+\psi\widetilde{\rho}_{x})_{x}\phi_{x}$
$\displaystyle-\widetilde{n}_{x}\bar{\phi}_{x}\bar{\psi}_{xx}+(\frac{1}{2}\bar{\phi}_{x}\widetilde{v}_{x}+\bar{\psi}_{x}\widetilde{n}_{x})\bar{\phi}_{x}-(\bar{\phi}\widetilde{v}_{x}+\bar{\psi}\widetilde{n}_{x})_{x}\bar{\phi}_{x}]dx.$

Owing to $(\ref{sigma})$, $(\ref{boundary d1})$, and $(\ref{infty e2})$, we
obtain the estimates for terms on the left hand side as below

$\displaystyle-$ $\displaystyle[\sigma^{-\nu}(\mu
u\frac{\phi_{x}^{2}}{2}+v\frac{\bar{\phi}_{x}^{2}}{2}-\widetilde{\rho}\phi_{x}\psi-\widetilde{n}\bar{\phi}_{x}\bar{\psi})](t,0)\geq\frac{c}{\delta^{\nu}}(\phi^{2}_{x}(t,0)+\bar{\phi}^{2}_{x}(t,0))\geq
0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ (4.47)
$\displaystyle-a\nu\int\sigma^{-(\nu-1)}(\mu
u\frac{\phi_{x}^{2}}{2}+v\frac{\bar{\phi}_{x}^{2}}{2})dx$ $\displaystyle\geq$
$\displaystyle\frac{a\nu|u_{+}|}{2}\int\sigma^{-(\nu-1)}(\mu\phi_{x}^{2}+\bar{\phi}_{x}^{2})dx-C(\varepsilon+\delta)\|\sigma^{-\frac{\nu-1}{2}}(\phi_{x},\bar{\phi}_{x})\|^{2},$
(4.48)
$\displaystyle-a\nu\int\sigma^{-(\nu-1)}(-\widetilde{\rho}^{2}\phi_{t}\psi-\widetilde{n}\bar{\phi}_{t}\bar{\psi})dx$
$\displaystyle\geq$
$\displaystyle-C\varepsilon\|\sigma^{-\frac{\nu}{2}}(\phi_{x},\bar{\phi}_{x})\|^{2}-C\|\sigma^{-\frac{\nu-2}{2}}(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}-C\|\sigma^{-\frac{\nu}{2}}(\psi_{x},\bar{\psi}_{x})\|^{2},$
(4.49) $\displaystyle\int$
$\displaystyle\sigma^{-\nu}(\widetilde{\rho}^{2}\frac{p^{\prime}_{1}(\rho)}{\rho}\phi_{x}^{2}+\widetilde{n}\frac{p^{\prime}_{2}(n)}{n}\bar{\phi}_{x}^{2})dx\geq\frac{A_{1}\gamma\rho_{+}^{\gamma}}{2}\|\sigma^{-\frac{\nu}{2}}\phi_{x}\|^{2}+\frac{A_{2}\alpha
n_{+}^{\alpha-1}}{2}\|\sigma^{-\frac{\nu}{2}}\bar{\phi}_{x}\|^{2},~{}$ (4.50)

where we take $\delta$ and $\varepsilon$ small enough.
We turn to estimate terms on the right hand side of $(\ref{1f-1})$. With the
help of $(\ref{sigma})$, $(\ref{infty e2})$, Young inequality and Cauchy-
Schwarz inequality, we gain

$\displaystyle|\mathcal{J}_{1}|\leq$ $\displaystyle
C\delta\|\sigma^{-\frac{\nu}{2}}(\phi_{x},\bar{\phi}_{x})\|^{2}+C\|\sigma^{-\frac{\nu}{2}}(\psi_{x},\bar{\psi}_{x})\|^{2}+C\|\sigma^{-\frac{\nu-2}{2}}(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2},$
(4.51) $\displaystyle|\mathcal{J}_{2}|\leq$ $\displaystyle
C(\|(\phi,\bar{\phi})\|_{L^{\infty}}+\delta)\|\sigma^{-\frac{\nu}{2}}(\phi_{x},\bar{\phi}_{x},\psi_{xx},\bar{\psi}_{xx})\|^{2}+C\delta^{2}\|\sigma^{-\frac{\nu}{2}}\bar{\psi}_{x}\|^{2}$
$\displaystyle\leq$ $\displaystyle
C(\delta+\varepsilon)\|\sigma^{-\frac{\nu}{2}}(\phi_{x},\bar{\phi}_{x})\|^{2}+C(\delta+\varepsilon)\|\sigma^{-\frac{\nu}{2}}(\psi_{xx},\bar{\psi}_{xx})\|^{2}+C\delta\|\sigma^{-\frac{\nu}{2}}\bar{\psi}_{x}\|^{2},$
(4.52) $\displaystyle|\mathcal{J}_{3}|\leq$
$\displaystyle\frac{A_{1}\gamma\rho_{+}^{\gamma}}{8}\|\sigma^{-\frac{\nu}{2}}\phi_{x}\|^{2}+\frac{A_{2}\alpha
n_{+}^{\alpha-1}}{8}\|\sigma^{-\frac{\nu}{2}}\bar{\phi}_{x}\|^{2}+C\|\sigma^{-\frac{\nu}{2}}(\bar{\psi}-\psi)\|^{2},$
(4.53) $\displaystyle|\mathcal{J}_{4}|\leq$ $\displaystyle
C\varepsilon\|\sigma^{-\frac{\nu}{2}}(\phi_{x},\bar{\phi}_{x})\|^{2}+C\varepsilon\|\sigma^{-\frac{\nu}{2}}(\psi_{xx},\bar{\psi}_{xx})\|^{2}+C\varepsilon\|\sigma^{-\frac{\nu}{2}}(\psi_{x},\bar{\psi}_{x})\|^{2},$
(4.54) $\displaystyle|\mathcal{J}_{5}|\leq$ $\displaystyle
C\delta\|\sigma^{-\frac{\nu}{2}}(\phi_{x},\bar{\phi}_{x})\|^{2}+C\delta\|\sigma^{-\frac{\nu-2}{2}}(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(4.55) $\displaystyle|\mathcal{J}_{6}|\leq$ $\displaystyle
C\delta\|\sigma^{-\frac{\nu}{2}}(\phi_{x},\bar{\phi}_{x})\|^{2}+C\delta^{2}\|\sigma^{-\frac{\nu}{2}}\bar{\psi}_{xx}\|^{2}+C\delta\|\sigma^{-\frac{\nu}{2}}(\psi_{x},\bar{\psi}_{x})\|^{2}+C\delta\|\sigma^{-\frac{\nu-2}{2}}(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}.$
(4.56)

Finally, the substitution of $(\ref{h-l1})$-$(\ref{h-r6})$ into $(\ref{1f-1})$
for $\delta$ and $\varepsilon$ small enough leads to that

$\displaystyle\frac{d}{dt}\int\sigma^{-\nu}[\mu\frac{\phi_{x}^{2}}{2}+\frac{\bar{\phi}_{x}^{2}}{2}+\widetilde{\rho}^{2}\phi_{x}\psi+\widetilde{n}\bar{\phi}_{x}\bar{\psi}]dx+c\|\sigma^{-\frac{\nu-1}{2}}(\phi_{x},\bar{\phi}_{x})\|^{2}$
(4.57)
$\displaystyle+\frac{A_{1}\gamma\rho_{+}^{\gamma}}{4}\|\sigma^{-\frac{\nu}{2}}\phi_{x}\|^{2}+\frac{A_{2}\alpha
n_{+}^{\alpha-1}}{4}\|\sigma^{-\frac{\nu}{2}}\bar{\phi}_{x}\|^{2}$
$\displaystyle\leq$ $\displaystyle
C(\varepsilon+\delta)\|\sigma^{-\frac{\nu}{2}}(\psi_{xx},\bar{\psi}_{xx})\|^{2}+C\|\sigma^{-\frac{\nu-2}{2}}(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}+C\|\sigma^{-\frac{\nu}{2}}(\psi_{x},\bar{\psi}_{x},\bar{\psi}-\psi)\|^{2}.~{}~{}~{}~{}~{}~{}~{}~{}~{}$

Multiplying $(\ref{high order weighted e})$ by $(1+\delta\tau)^{\xi}$ and
integrating the resulted equation in $\tau$ over $[0,t]$, and using Cauchy-
Schwarz inequality and Lemma 4.8, we obtain the desired estimate
$(\ref{h-e-t})$. The proof of Lemma 4.9 is completed. ∎

######  Lemma 4.10.

Under the same conditions in Proposition 4.5, then the solution
$(\phi,\psi,\bar{\phi},\bar{\psi})$ to the IBVP $(\ref{f1})$-$(\ref{boundary
d1})$ satisfies for $t\in[0,T]$ that

$\displaystyle(1+\delta
t)^{\xi}\|\sigma^{-\frac{\nu}{2}}(\psi_{x},\bar{\psi}_{x})\|^{2}+\int^{t}_{0}(1+\delta\tau)^{\xi}\|\sigma^{-\frac{\nu}{2}}(\psi_{xx},\bar{\psi}_{xx})\|^{2}d\tau$
(4.58) $\displaystyle\leq$ $\displaystyle
C\|\sigma^{-\frac{\lambda}{2}}(\phi_{0},\psi_{0},\phi_{0x},\psi_{0x},\bar{\phi}_{0},\bar{\psi}_{0},\bar{\phi}_{0x},\bar{\psi}_{0x})\|^{2}$
$\displaystyle+C\delta\xi\int^{t}_{0}(1+\delta\tau)^{\xi-1}\|\sigma^{-\frac{\nu}{2}}(\phi,\psi,\phi_{x},\psi_{x},\bar{\phi},\bar{\psi},\bar{\phi}_{x},\bar{\psi}_{x})\|^{2}d\tau,$

with $\xi\geq 0$.

###### Proof.

Multiplying $(\ref{f1})_{2}$ by $-\sigma^{-\nu}\psi_{xx}$, $(\ref{f1})_{4}$ by
$-\sigma^{-\nu}\bar{\psi}_{xx}$ respectively with the function $\sigma$
satisfying $(\ref{sig})$ and $(\ref{sg0})$, then adding them together and
integrating the resulted equation in $x$ over $\mathbb{R}_{+}$ lead to

$\displaystyle\frac{d}{dt}\int\sigma^{-\nu}(\frac{\psi_{x}^{2}}{2}+\frac{\bar{\psi}_{x}^{2}}{2})dx-a\nu\int\sigma^{-(\nu-1)}(\psi_{t}\psi_{x}+\bar{\psi}_{t}\bar{\psi}_{x})dx+\int\sigma^{-\nu}(\frac{\mu}{\rho}\psi_{xx}^{2}+\bar{\psi}_{xx}^{2})dx=\sum_{i=1}^{3}\mathcal{K}_{i},$
(4.59)

where

$\displaystyle\mathcal{K}_{1}=$
$\displaystyle\int\sigma^{-\nu}[u\psi_{x}\psi_{xx}-\frac{n}{\rho}(\bar{\psi}-\psi)\psi_{xx}+\frac{p^{\prime}_{1}(\rho)}{\rho}\phi_{x}\psi_{xx}+v\bar{\psi}_{x}\bar{\psi}_{xx}+(\bar{\psi}-\psi)\bar{\psi}_{xx}+\frac{p_{2}{{}^{\prime}}(n)}{n}\bar{\phi}_{x}\bar{\psi}_{xx}]dx,$
$\displaystyle\mathcal{K}_{2}=$
$\displaystyle\int\sigma^{-\nu}[\widetilde{u}_{x}\psi\psi_{xx}-\mu\widetilde{u}_{xx}(\frac{1}{\rho}-\frac{1}{\widetilde{\rho}})\psi_{xx}+(\frac{p^{\prime}_{1}(\rho)}{\rho}-\frac{p^{\prime}_{1}(\widetilde{\rho})}{\widetilde{\rho}})\psi_{xx}\widetilde{\rho}_{x}-(\frac{n}{\rho}-\frac{\widetilde{n}}{\widetilde{\rho}})(\widetilde{v}-\widetilde{u})\psi_{xx}+\widetilde{v}_{x}\bar{\psi}\bar{\psi}_{xx}$
$\displaystyle+(\widetilde{n}\widetilde{v}_{x})_{x}(\frac{1}{n}-\frac{1}{\widetilde{n}})\bar{\psi}_{xx}+(\frac{p^{\prime}_{2}(n)}{n}-\frac{p^{\prime}_{2}(\widetilde{n})}{\widetilde{n}})\bar{\psi}_{xx}\widetilde{n}_{x}-\frac{\widetilde{n}_{x}}{\widetilde{n}}\bar{\psi}_{x}\bar{\psi}_{xx}-\frac{(\bar{\phi}\widetilde{v}_{x})_{x}}{n}\bar{\psi}_{xx}]dx,$
$\displaystyle\mathcal{K}_{3}=$
$\displaystyle-\int\sigma^{-\nu}[\frac{(\bar{\phi}\bar{\psi}_{x})_{x}}{n}\bar{\psi}_{xx}+(\frac{1}{n}-\frac{1}{\widetilde{n}})(\widetilde{n}\bar{\psi}_{x})_{x}\bar{\psi}_{xx}]dx.$

First, we estimate terms in the left side of $(\ref{psi_{xx} space w 1})$. The
second term is estimated as follows:

$\displaystyle
a\nu\int\sigma^{-(\nu-1)}(\psi_{t}\psi_{x}+\bar{\psi}_{t}\bar{\psi}_{x})dx$
(4.60) $\displaystyle\leq$ $\displaystyle
C\delta\|\sigma^{-\frac{\nu}{2}}(\psi_{xx},\bar{\psi}_{xx})\|^{2}+C\delta\|\sigma^{-\frac{\nu}{2}}(\phi_{x},\psi_{x},\bar{\phi}_{x},\bar{\psi}_{x},\bar{\psi}-\psi)\|^{2}+C\delta\|\sigma^{-\frac{\nu-2}{2}}(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2},$

where we have used $(\ref{sigma})$, $(\ref{f1})_{2}$, $(\ref{f1})_{4}$,
$(\ref{infty e2})$ and Cauchy-Schwarz inequality.
With the help of $\delta$ and $\varepsilon$ small enough, the third term is
estimated as follows:

$\displaystyle\int\sigma^{-\nu}(\frac{\mu}{\rho}\psi_{xx}^{2}+\bar{\psi}_{xx}^{2})dx\geq$
$\displaystyle[\frac{\mu}{\rho_{+}}-C(\varepsilon+\delta)]\|\sigma^{-\frac{\nu}{2}}\psi_{xx}\|^{2}+\|\sigma^{-\frac{\nu}{2}}\bar{\psi}_{xx}\|^{2}$
(4.61) $\displaystyle\geq$
$\displaystyle\frac{\mu}{2\rho_{+}}\|\sigma^{-\frac{\nu}{2}}\psi_{xx}\|^{2}+\|\sigma^{-\frac{\nu}{2}}\bar{\psi}_{xx}\|^{2}.$

We turn to estimate terms on the right hand side of $(\ref{psi_{xx} space w
1})$. With the help of $(\ref{sigma})$, $(\ref{infty e2})$ and Cauchy-Schwarz
inequality, we obtain

$\displaystyle|\mathcal{K}_{1}|\leq\frac{\mu}{8\rho_{+}}\|\sigma^{-\frac{\nu}{2}}\psi_{xx}\|^{2}+\frac{1}{8}\|\sigma^{-\frac{\nu}{2}}\bar{\psi}_{xx}\|^{2}+C\|\sigma^{-\frac{\nu}{2}}(\phi_{x},\psi_{x},\bar{\phi}_{x},\bar{\psi}_{x},\bar{\psi}-\psi)\|^{2},$
(4.62) $\displaystyle|\mathcal{K}_{2}|\leq
C\delta\|\sigma^{-\frac{\nu}{2}}(\psi_{xx},\bar{\psi}_{xx})\|^{2}+C\delta\|\sigma^{-\frac{\nu-2}{2}}(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}+C\delta\|\sigma^{-\frac{\nu}{2}}(\bar{\phi}_{x},\bar{\psi}_{x})\|^{2},$
(4.63) $\displaystyle|\mathcal{K}_{3}|\leq
C(\varepsilon+\delta)\|\sigma^{-\frac{\nu}{2}}(\bar{\psi}_{x},\bar{\psi}_{xx})\|^{2}.$
(4.64)

Finally, we substitute $(\ref{psi_{xx} second term e})$-$(\ref{K_{4} e})$ into
$(\ref{psi_{xx} space w 1})$ to gain under the condition $\delta$ and
$\varepsilon$ small enough that

$\displaystyle\frac{d}{dt}\int\sigma^{-\nu}(\frac{\psi_{x}^{2}}{2}+\frac{\bar{\psi}_{x}^{2}}{2})dx+\frac{\mu}{4\rho_{+}}\|\sigma^{-\frac{\nu}{2}}\psi_{xx}\|^{2}+\frac{1}{4}\|\sigma^{-\frac{\nu}{2}}\bar{\psi}_{xx}\|^{2}$
(4.65) $\displaystyle\leq$ $\displaystyle
C\|\sigma^{-\frac{\nu}{2}}(\phi_{x},\psi_{x},\bar{\phi}_{x},\bar{\psi}_{x},\bar{\psi}-\psi)\|^{2}+C\delta\|\sigma^{-\frac{\nu-2}{2}}(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}$

Multiplying $(\ref{psi_{xx} space weighted e})$ by $(1+\delta\tau)^{\xi}$ and
integrating the resulted inequality in $\tau$ over $[0,t]$, and using Lemmas
4.8-4.9 and the smallness of $\delta$ and $\varepsilon$, we obtain
$(\ref{psi_{xx} time e 1})$. The proof of Lemma 4.10 is completed. ∎

Proof of Proposition 4.5 With the help of Lemmas 4.8-4.10, it holds for
$\delta$ and $\varepsilon$ suitably small that

$\displaystyle(1+\delta
t)^{\xi}\|\sigma^{-\frac{\nu}{2}}(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}_{1}+\int^{t}_{0}(1+\delta\tau)^{\xi}\|\sigma^{-\frac{\nu-2}{2}}(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}d\tau+\int_{0}^{t}(1+\delta\tau)^{\xi}\|\sigma^{-\frac{\nu}{2}}(\phi_{x},\psi_{x},\bar{\phi}_{x},\bar{\psi}_{x})\|^{2}$
$\displaystyle+C\int_{0}^{t}(1+\delta\tau)^{\xi}\|\sigma^{-\frac{\nu}{2}}(\psi_{xx},\bar{\psi}_{xx},\bar{\psi}-\psi)\|^{2}+C\frac{1}{\delta^{\nu}}\int_{0}^{t}(1+\delta\tau)^{\xi}(\phi^{2}(t,0)+\bar{\phi}^{2}(t,0))d\tau$
$\displaystyle\leq$ $\displaystyle
C\|\sigma^{-\frac{\lambda}{2}}(\phi_{0},\psi_{0},\bar{\phi}_{0},\bar{\psi}_{0})\|^{2}_{1}+C\delta\xi\int^{t}_{0}(1+\delta\tau)^{\xi-1}\|\sigma^{-\frac{\nu}{2}}(\phi,\psi,\bar{\phi},\bar{\psi})\|^{2}_{1}d\tau,$
(4.66)

where $C>0$ is a positive constant independent of $T$, $\nu$ and $\xi$.
Applying similar induction arguments as in [14, 25, 3] to $(\ref{high order
space time e})$, we have

$\displaystyle(1+\delta
t)^{\frac{\lambda-\nu}{2}+\beta}\|\sigma^{-\frac{\nu}{2}}(\phi,\psi,\phi_{x},\psi_{x},\bar{\phi},\bar{\psi},\bar{\phi}_{x},\bar{\psi}_{x})\|^{2}+\int^{t}_{0}(1+\delta\tau)^{\frac{\lambda-\nu}{2}+\beta}\|\sigma^{-\frac{\nu-2}{2}}(\phi,\bar{\phi},\psi,\bar{\psi})\|^{2}$
(4.67)
$\displaystyle+\int^{t}_{0}(1+\delta\tau)^{\frac{\lambda-\nu}{2}+\beta}\|\sigma^{-\frac{\nu}{2}}(\phi_{x},\psi_{x},\psi_{xx},\bar{\phi}_{x},\bar{\psi}_{x},\bar{\psi}_{xx},\bar{\psi}-\psi)\|^{2}$
$\displaystyle\leq$ $\displaystyle C(1+\delta
t)^{\beta}\|\sigma^{-\frac{\lambda}{2}}(\phi_{0},\psi_{0},\phi_{0x},\psi_{0x},\bar{\phi}_{0},\bar{\psi}_{0},\bar{\phi}_{0x},\bar{\psi}_{0x})\|^{2}$

for $\beta>0$, which implies

$\|\sigma^{-\frac{\nu}{2}}(\phi,\psi,\phi_{x},\psi_{x},\bar{\phi},\bar{\psi},\bar{\phi}_{x},\bar{\psi}_{x})(t)\|^{2}\leq
C(1+\delta
t)^{-\frac{\lambda-\nu}{4}}\|\sigma^{-\frac{\lambda}{2}}(\phi_{0},\psi_{0},\phi_{0x},\psi_{0x},\bar{\phi}_{0},\bar{\psi}_{0},\bar{\phi}_{0x},\bar{\psi}_{0x})\|^{2}.$
(4.68)

Acknowledgments

The research of the paper is supported by the National Natural Science
Foundation of China (Nos. 11931010, 11871047, 11671384), by the key research
project of Academy for Multidisciplinary Studies, Capital Normal University,
and by the Capacity Building for Sci-Tech Innovation-Fundamental Scientific
Research Funds (No. 007/20530290068).

## References

  * [1] F. Bubba, B. Perthame, C. Pouchol, M. Schmidtchen, Hele-Shaw limit for a system of two reaction-(cross-)diffusion equations for living tissues. Arch. Ration. Mech. Anal. 236 (2020), no. 2, 735-766.
  * [2] J. Carr, Applications of Center Manifold Theory, Springer-Verlag, 1981.
  * [3] Y.Z. Chen, H. Hong, X,D. Shi, Convergence rate of stationary solutions to outflow problem for full Navier-Stokes equations. Appl. Anal. 98 (2019), no. 7, 1267-1288.
  * [4] R.J. Duan, X.F. Yang, Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier- Stokes-Poisson equations, Commun. Pure Appl. Anal. 12 (2013) 985-1014.
  * [5] L.L. Fan, H.X. Liu, T. Wang, H.J. Zhao, Inflow problem for the one-dimensional compressible Navier-Stokes equations under large initial perturbation. J. Differential Equations 257 (2014), no. 10, 3521-3553.
  * [6] C.C. Hao and H.L. Li, Well-posedness for a multidimensional viscous liquid-gas two-phase flow model, SIAM J. Math. Anal. 44 (2012), 1304-1332.
  * [7] H. Hong, X.D. Shi, T. Wang, Stability of stationary solutions to the inflow problem for the two-fluid non-isentropic Navier-Stokes-Poisson system. J. Differential Equations 265 (2018), no. 4, 1129-1155.
  * [8] H. Hong, T. Wang, Stability of stationary solutions to the inflow problem for full compressible Navier-Stokes equations with a large initial perturbation. SIAM J. Math. Anal. 49 (2017), no. 3, 2138-2166.
  * [9] F.M. Huang, A. Matsumura, X.D. Shi, Viscous shock wave and boundary layer solution to an inflow problem for compressible viscous gas. Comm. Math. Phys. 239 (2003), no. 1-2, 261-285.
  * [10] F.M. Huang, J. Li, X.D. Shi, Asymptotic behavior of solutions to the full compressible Navier-Stokes equations in the half space. Commun. Math. Sci. 8 (2010), no. 3, 639-654.
  * [11] F.M. Huang, X.H. Qin, Stability of boundary layer and rarefaction wave to an outflow problem for compressible Navier-Stokes equations under large perturbation. J. Differential Equations 246 (2009), no. 10, 4077-4096.
  * [12] M. Ishii and T. Hibiki, Thermo-fluid Dynamics of Two-Phase Flow. New York: Springer-Verlag, 2006.
  * [13] S. Kawashima, K. Kurata, Hardy type inequality and application to the stability of degenerate stationary waves. J. Funct. Anal. 257 (2009), no. 1, 1-19.
  * [14] S. Kawashima, A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys. 101 (1985), no. 1, 97-127.
  * [15] S. Kawashima, T. Nakamura, S. Nishibata, P.C. Zhu, Stationary waves to viscous heat-conductive gases in half-space: existence, stability and convergence rate. Math. Models Methods Appl. Sci. 20 (2010), no. 12, 2201-2235.
  * [16] S. Kawashima, S. Nishibata and P. Zhu, Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space, Comm. Math. Phys. 240 (2003), 483-500.
  * [17] S. Kawashima, P.C. Zhu, Asymptotic stability of nonlinear wave for the compressible Navier-Stokes equations in the half space. J. Differential Equations 244 (2008), no. 12, 3151-3179.
  * [18] S. Kawashima, P.C. Zhu, Asymptotic stability of rarefaction wave for the Navier-Stokes equations for a compressible fluid in the half space. Arch. Ration. Mech. Anal. 194 (2009), no. 1, 105-132.
  * [19] H.L. Li, T. Wang, Y. Wang, Wave phenomena to the three-dimensional fluid-particle model, preprint, 2020.
  * [20] H.L. Li, S. Zhao, Existence and nonlinear stability of stationary solutions to the full two-phase flow model in a half line. Appl. Math. Lett. 116 (2021), 107039.
  * [21] T. Lorenzi, A. Lorz, B. Perthame, On interfaces between cell populations with different mobilities. Kinet. Relat. Models 10 (2017), no. 1, 299-311.
  * [22] A. Matsumura, Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas, Methods Appl. Anal. 8(2001), 645-666.
  * [23] A. Matsumura, K. Nishihara, Large-time behaviors of solutions to an inflow problem in the half space for a one-dimensional system of compressible viscous gas. Comm. Math. Phys. 222 (2001), no. 3, 449-474.
  * [24] A. Mellet, A. Vasseur, Asymptotic analysis for a Vlasov-Fokker-Planck/compressible Navier- Stokes system of equations, Comm. Math. Phys. 281 (2008), no. 3, 573-596.
  * [25] M. Nishikawa, Convergence rate to the traveling wave for viscous conservation laws, Funkcial. Ekvac. 41 (1998), no. 1, 107-132.
  * [26] T. Nakamura, S. Nishibata, Stationary wave associated with an inflow problem in the half line for viscous heat-conductive gas. J. Hyperbolic Differ. Equ. 8 (2011), no. 4, 651-670.
  * [27] T. Nakamura, S. Nishibata, Existence and asymptotic stability of stationary waves for symmetric hyperbolic-parabolic systems in half-line. Math. Models Methods Appl. Sci. 27 (2017), no. 11, 2071-2110.
  * [28] T. Nakamura, S. Nishibata, N. Usami, Convergence rate of solutions towards the stationary solutions to symmetric hyperbolic-parabolic systems in half space. Kinet. Relat. Models 11 (2018), no. 4, 757-793.
  * [29] T. Nakamura, S. Nishibata and T. Yuge, Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line, J. Differential Equations 241 (2007), 94-111.
  * [30] X.H. Qin, Large-time behaviour of solutions to the outflow problem of full compressible Navier-Stokes equations. Nonlinearity 24 (2011), no. 5, 1369-1394.
  * [31] X.H. Qin, Y. Wang, Stability of wave patterns to the inflow problem of full compressible Navier-Stokes equations. SIAM J. Math. Anal. 41 (2009), no. 5, 2057-2087.
  * [32] X.H. Qin, Y. Wang, Large-time behavior of solutions to the inflow problem of full compressible Navier-Stokes equations. SIAM J. Math. Anal. 43 (2011), no. 1, 341-366.
  * [33] L. Wan, T. Wang, H.J. Zhao, Asymptotic stability of wave patterns to compressible viscous and heat-conducting gases in the half-space. J. Differential Equations 261 (2016), no. 11, 5949-5991.
  * [34] L. Wan, T. Wang, Q.Y. Zou, Stability of stationary solutions to the outflow problem for full compressible Navier-Stokes equations with large initial perturbation. Nonlinearity 29 (2016), no. 4, 1329-1354.
  * [35] H.Y. Yin, J.S. Zhang, C.J. Zhu, Stability of the superposition of boundary layer and rarefaction wave for outflow problem on the two-fluid Navier-Stokes-Poisson system, Nonlinear Anal. Real World Appl. 31 (2016) 492-512.
  * [36] H.Y. Yin, Converge rates towards stationary solutions for the outflow problem of planar magnetohydrodynamics on a half line. Proc. Roy. Soc. Edinburgh Sect. A 149 (2019), no. 5, 1291-1322.
  * [37] H.Y. Yin, C.J. Zhu, Convergence rate of solutions toward stationary solutions to a viscous liquid-gas two-phase flow model in a half line, Commun. Pure Appl. Anal. 14 (2015), no. 5, 2021-2042.
  * [38] F. Zhou, Y.P. Li, Convergence rate of solutions toward stationary solutions to the bipolar Navier-Stokes-Poisson equations in a half line, Bound. Value Probl. 2013 (2013) 1.