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

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.


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 [1,6,12,21,24], 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].
The condition u − < 0 means that the fluids flow out the region R + through the boundary x = 0 with the velocity u − , and therefore the problem (1.1)-(1.5) 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 (ρ, n)(t, 0) = (ρ − , 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 [3, 5, 8-11, 15-18, 23, 26-34]. 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 [31,32] 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 [11,17]. 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 [4,7,20,35,37,38]. 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 stedystates 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 (1.1) 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 (1.1)- (1.5). 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 (3.3)-(1.5) exists for the subsonic case.
Remark 1.1. Due to the drag force term, the existence of steady-states to the IBVP (1.1)-(1.5) 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 (1.1)-(1.5) 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 (1.1)-(1.5) for supersonic, sonic and subsonic cases.
Theorem 1.2. Let the same conditions in Theorem 1.1 hold and assume that it holds  (i) For M + > 1 and λ > 0, if the initial data satisfy and where C 4 > 0 and δ 0 : are constants independent of time.
We explain main strategies to prove Theorems 1.1-1.3. The system (1.1)-(1.5) 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 (1.1) 2 and (1.1) 4 into conservation forms due to the influence of drag force, which implies that the steady-state satisfies the system (1.6)-(1.7) 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 w := u x , get the estimate | u x (0)| ≤ C|u − − u + | in Lemma 2.1, and then rewrite the system (1.6) into the 3 × 3 system (2.9) of autonomous ordinary differential equations. Since the condition is not necessary, we need subtle analysis to obtain the sign of Reλ i , i = 1, 2, 3, where λ i , i = 1, 2, 3 are three eigenvalues of the linearized 3 × 3 system of (2.9). It should be noticed that the linearized system of (2.9) 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 (φ, ψ,φ,ψ) := (ρ − ρ, u − u, n − n, v − 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 ( ρ u 2 + p 1 ( ρ)) x and ( n v 2 + p 2 ( n)) x decay slower than u xx for p ′ 1 (ρ + ) = p ′ 2 (n + ) owing to (1.6) 2 , (1.6) 4 and (1.12), which implies the term can not be controlled directly as in [16]. With the help of u x ≥ 0 and v x ≥ 0, we turn to deal with the terms the leading terms of which can be rewritten as two positive semidefinite 2 variable quadratic forms (3.21) under the condition (1.15).
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 (1.11) and the dissipation on the relaxation friction termψ − ψ, and decompose ψ as ψ =ψ + (ψ −ψ), as motivated by Li-Wang-Wang [19]. Due to the algebraic decay (1.12) 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 (1.12)-(1.14) and the dissipation on the drag forceψ − ψ, decompose ψ as ψ =ψ + (ψ −ψ), 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  26) to gain the 3 variable quadratic formλ 1ρ 2 +λ 2n 2 withλ 1 ,λ 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 (1.1)-(1.5) for M + = 1 for the initial perturbation satisfying and the function σ ≥ 0 satisfying (1.14). This is an interesting phenomena describing the influences of two fluids on each other somehow and it should be emphasized that λ * = 5 is the same as that of the isentropic Navier-Stokes equations [28] for u 2 Notation. We denote by · L p the norm of the usual Lebesgue space L p = L p (R + ), 1 ≤ p ≤ ∞. And if p = 2, we write · L p (R+) = · . H s (R + ) stands for the standard s-th Sobolev space over R + equipped with its norm For a scalar function W (x) > 0, the weighted L 2 (R + ) and H 1 (R + ) spaces are defined as follows: For a scalar function W a,ν : The rest of this paper will be organized as follows. We prove the existence and uniqueness of steadystates 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.

Existence of Steady-State
We prove Theorem 1.1 on the existence and uniqueness of steady-states to the BVP (1.6)-(1.7) with u + < 0 and δ sufficiently small as follows. In order to apply the center manifold theory [2], it is necessary to get the bounds of u x (0) or v x (0).

Nonlinear stability of steady-states
The function space Y (0, T ) for T > 0 is denoted by Then the perturbation (φ, ψ,φ,ψ) satisfies the following system where

4)
The initial and boundary conditions to the system (3.3) satisfy is satisfied, then it holds for arbitrary t ∈ [0, T ] that With the help of (3.8), it is easy to verify the following Sobolev inequality where δ > 0, c 0 > 0 and C > 0 are positive constants.
In order to complete the proof of Proposition 3.1, we need to establish the high order estimates of (φ, ψ,φ,ψ).
Lemma 3.4. Under the same conditions in Proposition 3.1, then the solution (φ, ψ,φ,ψ) to the problem xφ 2 Adding (3.24)-(3.27) together, integrating the resulted equation over R + , we have where First, we estimate terms in the left side of (3.28). Under the condition (3.7), the terms in the left side is estimated as follows We turn to estimate terms in the right hand side of (3.28).
Cauchy-Schwartz inequality and Young inequality with 0 < η < 1, we obtain Finally, the substitution of (3.29)-(3.34) into (3.28) for δ, ε and η small enough leads to 3) 4 by −ψ xx , respectively, then adding them together and inte- where We estimate terms in the left side of (3.37). By the decomposition 1 the second term is estimated as follows: (3.38) We turn to estimate terms in the right side of (3.37). With the aid of (1.11), Sobolev inequality and Cauchy-Schwarz inequality, we have
With the help of Lemmas 3.3-3.5, we get (3.9) and complete the proof of Proposition 3.1.

Time convergence rates
4.1 Convergence rate of supersonic steady-state for any time T > 0. Then for arbitrary ν ∈ [0, λ], there exist positive constants ε > 0 and is satisfied, it holds for arbitrary t ∈ [0, T ] that with θ > 0.
First, we estimate terms on the left hand side of (4.4). Under the condition (3.7), the second term on the left hand side is estimated as We decompose ψ as ψ =ψ + (ψ −ψ) and use (1.11) to gain where the symmetric matrix M 3 is denoted by It is easy to verify that 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 ε, δ and η small enough that By Young inequality with 0 < η < 1, the forth and fifth terms on the left hand side are estimated as By (1.11) and (3.11), it follows from Sobolev inequality and Cauchy-Schwarz inequality that (4.11) Finally, with η, δ and ε suitably small, the substitution of (4.5)-(4.11) into (4.4) leads to Similar to Lemmas 3.4-3.5, we get the following high order weighted estimates of (φ, ψ,φ,ψ). The details are omitted.
Repeating the same procedure, we get the desired estimates (4.16) and (4.17) for any λ > 0.

Convergence rate of sonic steady-state
The function space (4.20) , and that the is satisfied, it holds for arbitrary t ∈ [0, T ] that with β > 0.
By the fact λ ≥ 1 and (4.21), it is easy to verify the following estimate: To deal with some nonlinear terms, we use the following inequality as in [15,27,36].
To gain faster decay rates, it is necessary to use the following Hardy type inequality.
In order to show Proposition 4.5, we need to obtain the high order weighted estimates of (φ, ψ,φ,ψ). where Owing to (1.12), (3.7), and (4.23), we obtain the estimates for terms on the left hand side as below where we take δ and ε small enough.
We turn to estimate terms on the right hand side of (4.46). With the help of (1.12), (4.23), Young inequality and Cauchy-Schwarz inequality, we gain Finally, the substitution of (4.47)-(4.56) into (4.46) for δ and ε small enough leads to that with ξ ≥ 0.
Proof. Multiplying (3.3) 2 by −σ −ν ψ xx , (3.3) 4 by −σ −νψ xx respectively with the function σ satisfying (1.14) and (2.28), then adding them together and integrating the resulted equation in x over R + lead to where First, we estimate terms in the left side of (4.59). The second term is estimated as follows: With the help of δ and ε small enough, the third term is estimated as follows: (4.61) We turn to estimate terms on the right hand side of (4.59). With the help of (1.12), (4.23) and Cauchy-Schwarz inequality, we obtain Finally, we substitute (4.60)-(4.64) into (4.59) to gain under the condition δ and ε small enough that Multiplying (4.65) by (1 + δτ ) ξ and integrating the resulted inequality in τ over [0, t], and using Lemmas 4.8-4.9 and the smallness of δ and ε, we obtain (4.58). 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 δ and ε suitably small that where C > 0 is a positive constant independent of T , ν and ξ. Applying similar induction arguments as in [3,14,25] to (4.66), we have for β > 0, which implies

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, liquid crystals, lubrication, biomedical flows [8,13,22], 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 [20].
We are concerned with the initial-boundary value problem (IBVP) for the full two-phase flow model as follows: where ρ > 0 and n > 0 stand for the densities of two fluids, and u, v are the velocities of two fluids. The constant µ > 0 is the viscosity coefficient and the pressure-density functions take the form with A 1 > 0, A 2 > 0, γ ≥ 1 and α ≥ 1. The initial data are given by 4) and the outflow boundary condition is imposed where ρ + > 0, n + > 0, u + and u − < 0 are constants.
There are important progress made recently about the global existence and time decay rates of solutions for the two-phase flow models [3, 5-8, 14, 29, 31, 32, 34-36, 39]. For instance, the global existence of three-dimensional strong solutions to Cauchy problem for Euler-NS model was proved in [3]. Later, time decay rates of the three-dimensional strong solution to Cauchy problem for Euler-NS model were obtained in [31]. The global existence and time decay rates of three-dimensional strong solutions to the initial boundary problem for Euler-NS model was proved in [14]. The global well-posedness of strong solutions to Cauchy problem in Besov space or Sobolev space was obtained in [8,32,39] for drift-flux models. The existence of multi-dimensional global weak solutions for drift-flux models to both Cauchy problem [6,34] and the initial boundary value problem [29] was proved. The global weak solution to the free boundary value problem in 1D was shown in [5,7,35,36] for drift-flux models.
There are important progress made recently about the existence and nonlinear stability of steadystates for the inflow/outflow problem of the compressible Navier-Stokes system [2, 4, 9-12, 15, 18, 19, 21, 23, 25-28, 30, 33, 38, 40]. Among them, the existence and nonlinear stability of steady-states to the inflow problem of Navier-Stokes equations in a half line was obtained in [10,11,21,25,30] for both subsonic and sonic cases. Concerning with the outflow problem of Navier-Stokes equations, the nonlinear stability of the steady-states for both supersonic and sonic cases in a half line was proved in [12,19,26], and the stability of the multidimensional planar stationary solution for both supersonic and sonic cases was obtained in [15].
The existence and time decay rates of the solution to the outflow problem for Navier-Stokes equations in a half line were shown in [27,28] for supersonic and sonic cases. The existence and nonlinear stability of steady-states to the outflow problem for full compressible Navier-Stokes equations [2,18] were shown for supersonic, sonic and subsonic cases, and the time decay rates for the supersonic case or sonic case were obtained.
However, there are also important results about the outflow/inflow problem for two-phase flow models except for [4,9,33,38,40]. Among them, the existence of steady-states for supersonic and sonic cases, and the nonlinear stability of steady-states for the supersonic case to the outflow problem were gained in [38] concerning with the drift-flux model. For the two-fluid Navier-Stokes-Poisson system, the existence and nonlinear stability of steady-states to the outflow problem for supersonic and sonic cases in [4,33,40] were proved. The existence and nonlinear stability of steady-states to inflow problem of the two-fluid Navier-Stokes-Poisson system for both subsonic and sonic cases were shown in [9].
However, there is no result about the nonlinear time stability and time decay rates of steady-states for the outflow problem (1.1)-(1.5). The main purpose of this paper is to prove the nonlinear time stability of steady-states for supersonic, sonic and subsonic cases, and obtain either the exponential time decay rate for the supersonic flow or the algebraic time decay rate for both supersonic and sonic flows.
Define the generalized Mach number M + and the generalized sound speed c + as below (1.10) Then, we have the following results about the existence and uniqueness of the steady-state.  11) or for M + = 1 that and where σ(x) is a smooth function satisfying σ x = −aσ 2 + O(|σ| 3 ) and 14) and C i > 0, i = 1, 2, 3, c 0 > 0 c 1 > 0 and a > 0 are positive constants.
We explain the strategies to prove above main Theorems. The system (1.1)-(1.5) can be viewed as the two Navier-Stokes systems coupled by the drag force. On the one hand, compared with the single Navier-Stokes system, it is more difficult to prove the existence of the steady-states to IBVP (1.1)-(1.5).
Indeed, it is easy to check that there is no trivial steady-states which are constructed by steady-states to the Navier-Stokes system compared with [4,9,33,40]. In order to use the center manifold theory [1], we need to introduce the new variable w := u x , get the estimates | u x (0)| ≤ C|u − − u + | in Lemma 2.1, and reformulate the system (1.6) into the first-order autonomous system (2.9). On the other hand, the linearized reformulated system has at least one eigenvalue with negative real part due to the effect of the drag force. Thus, we obtain the existence of the steady-states for supersonic, sonic and subsonic cases in Theorem 1.1. The steady-states exist for the subsonic case, which is different from the single Navier-Stokes system [19].
To prove the nonlinear stability of the steady-states in Theorem 1.2, it is necessary to obtain the uniform estimates on the perturbations around steady-states (φ, ψ,φ,ψ). In fact, compared with the single Navier-Stokes system, it is more difficult to get the uniform estimates for sonic flow due to the drag force. Indeed, the bad term can be controlled by the good term With the weighted energy method, we get the exponential or algebraic time decay rates for the supersonic case. For sonic case, we obtain the existence and the algebraic time decay rates of the solution to IBVP (1.1)-(1.5) without the restriction |p 2 } by the weighted energy method. It is difficult to obtain the convergence rate for the supersonic case by the ideas used in compressible Navier-Stokes equations [28] except for p ′ 1 (ρ + ) = p ′ 2 (n + ). And for the sonic case, the ideas in compressible Navier-Stokes equations [28] can't work except for the special case n + = ρ + , µ = n + , p ′ 1 (ρ + ) = p ′ 2 (n + ) and γ = α. To get decay rates for both supersonic and sonic case, we make full use of the drag force term. Indeed, one important observation is decomposing ψ as ψ =ψ + (ψ −ψ) in basic energy estimates for both supersonic and sonic cases. For the sonic case, inspired by [27,28], we take the linear coordinate transformation  with a invertible matrix P . With the help of ψ =ψ + (ψ −ψ), the coordinate transformation and (1.13), we can get the good term for ν ∈ (0, 3] that a A 1 γ(γ + 1)ρ γ + + A 2 α(α + 1)n α+1 to control the bad term where the constants a, b are defined by (2.22) and the space weight function σ ≥ 0 satisfies (1.14) and (2.28). However, it is more difficult to get decay rates for p ′ 1 (ρ + ) = p ′ 2 (n + ) due to the term which is caused by the decomposition ψ =ψ + (ψ −ψ). To overcome the difficulty, it is necessary to control the bad term by using good terms for ν ∈ (0, 3] that where a > 0, b ≥ 0 are constants defined by (2.22), and the space weight function σ ≥ 0 satisfies (1.14) and (2.28).
Notation. We denote by · L p the norm of the usual Lebesgue space L p = L p (R + ), 1 ≤ p ≤ ∞. And if p = 2, we write · L p (R+) = · . H s (R + ) stands for the standard s-th Sobolev space over R + equipped with its norm For a scalar function W a,ν := (1 + x) ν with ν ≥ 0, we denote f a,ν : The rest of this paper will be organized as follows. We prove the existence and uniqueness of the steady-states in Section 2, get the nonlinear stability of the steady-states in Section 3 for supersonic, sonic and subsonic cases, and obtain the convergence rates of the steady-states for supersonic flow in

Existence of Stationary Solution
We prove Theorem 1.1 on the the existence and uniqueness of the stationary solution to (1.6)-(1.7) with u + < 0 and δ sufficiently small as follows. First, to apply the center manifold theory [1], it is necessary to get the bounds of u x (0) or v x (0). Lemma 2.1. Assume that u + < 0 and δ = |u − − u + | hold with δ sufficiently small. Let ( ρ, u, n, v) be the solution to the BVP (1.6)-(1.7). Then we have where C > 0 is a positive constant.
With (2.11) and (2.12), the system (2.9) can be reformulated as follows where nonlinear functions g i (i = 1, 2, 3) are denoted by  With the help of the manifold theory [1], there exist a local center manifold W c (0, 0, 0) and a local stable (2.22) Therefore, the system (2.17) can be reformulated as follows with z 3− < 0, the smallness of |(z 1− , z 2− , z 3− )| and (2.26) which satisfies It is easy to get and with the help ofŪ = P Z and (2.16).

Nonlinear stability of steady-states
The function space Y (0, T ) for T > 0 is denoted by Then the perturbation (φ, ψ,φ,ψ) satisfies the following system where

4)
The initial and boundary conditions to the system (3.3) satisfy is satisfied, it holds for arbitrary t ∈ [0, T ] that Under the condition (3.8), it is easy to verify the following Sobolev inequality 19] ). The function ψ(·, t) ∈ H 1 (R + ) satisfies for the constants δ > 0 and c 0 > 0 that where C > 0 is a positive constant independent of time.
In order to complete the proof of Proposition 3.1, we need to obtain the estimates of (φ x , ψ x ,φ x ,ψ x ).

(3.36)
Integrating (3.36) over [0, t], we can obtain where we have used (3.13) and Young inequality. Hence, we complete the proof of Lemma 3.4.

3.5.
With the help of Lemmas 3.3-3.5, we get (3.9) and complete the proof of Proposition 3.1.

Time convergence rates
4.1 Convergence rate of supersonic steady-state Then for ν ∈ [0, λ], there exist positive constants ε > 0 and C > 0 independent of T such that if is satisfied, it holds for arbitrary t ∈ [0, T ] that with θ > 0.
Proof. We multiply (3.16) by W a,ν , where W a,ν := (1 + x) ν is a space weight function. We integrate the resulted equality over R + to obtain where G i for i = 1, 2, R j for j = 1, 2, 3 and E k , k = 1, 2 are defined by (3.17).
First, we estimate terms on the left hand side of (4.4). Under the condition (3.7), the second term on the left hand side is estimated as We decompose ψ as ψ =ψ + (ψ −ψ) and use (1.11) to gain where the symmetric matrix M 3 is denoted by It is easy to verify that M 3 is a positive definite matrix under the condition M + > 1. Hence, the estimate of the third term on the left hand side is obtained under the condition ε, δ and η small enough that The forth and the fifth terms on the left hand side are estimated as below − ν W a,ν−1 G 2 dx ≤ νη (ψ,ψ) 2 a,ν−1 + νC η (ψ x ,ψ x ) 2 a,ν−1 + Cδ (φ,ψ) 2 a,ν−1 .
To prove Proposition 4.1, we need to obtain the following weighted energy estimates of (φ x , ψ x ,φ x ,ψ x ).
By similar arguments as showing Lemmas 3.4-3.5, we get the following Lemmas. The details are omitted.
Under the condition λ ≥ 1 and (4.18), it is easy to gain the following estimate: (4.20) To deal with some nonlinear terms, we use the following inequality as in [18,26,37].
To gain faster decay rates, it is necessary to use the following Hardy type inequality which is proved in [16]. for ψ satisfying ψ(t, 0) = 0 and wψ ∈ H 1 (R + ), with the function w := ζ 2 ζx .