Free Boundary Value Problem for the One-Dimensional Compressible Navier-Stokes Equations with a Nonconstant Exterior Pressure

Abstract

We consider the free boundary value problem (FBVP) for one-dimensional isentropic compressible Navier-Stokes (CNS) equations with density-dependent viscosity coefficient in the case that across the free surface stress tensor is balanced by a nonconstant exterior pressure. Under certain assumptions imposed on the initial data and exterior pressure, we prove that there exists a unique global strong solution which is strictly positive from blow for any finite time and decays pointwise to zero at an algebraic time-rate.

1. Introduction

We will investigate the free boundary value problem for one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient for regular initial data in the case that across the free surface stress tensor is balanced by a nonconstant exterior pressure in the present paper. In general, the one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient read as 𝜌_𝑡+𝜌𝑢_𝑥=0,𝜌𝑢_𝑡+𝜌𝑢²+𝑃𝜌_𝑥=𝜇𝜌𝑢_𝑥𝑥,(𝑥,𝑡)∈𝑅×[0,𝑇], (1) where 𝜌>0, 𝑢, and 𝑃(𝜌)=𝜌^𝛾(𝛾>1) stand for the flow density, velocity, and pressure, respectively, and the viscosity coefficient is 𝜇(𝜌)=𝜌^𝛼 with 𝛼>0. We note here that as 𝛾=2 and 𝛼=1 in (1), the system corresponds to the viscous Saint-Venant system for shallow water.

There is huge literature on the studies of the compressible Navier-Stokes equations with density-dependent viscosity coefficients. For example, the mathematical derivations are achieved in the simulation of flow surface in shallow region [1, 2]. Bresch and Desjardins have investigated the existence of solutions to the 2D shallow water equations in [3, 4]. The global existence of classical solutions is proven by Mellet and Vasseur [5]. The qualitative patterns of behavior of global solutions and dynamical asymptotics of vacuum states are also shown, such as the finite time vanishing of finite vacuum or asymptotical formation of vacuum in large time, the dynamical behavior of vacuum boundary, the large time convergence to rarefaction wave with vacuum, and the stability of shock profile with large shock strength; refer to [6–11] and references therein.

Recently, there is much significant progress achieved on the free boundary value problems; for instance, the well-posedness of solutions to the free boundary value problem with initial finite mass and the flow density being connected with the infinite vacuum either continuously or via jump discontinuity is considered by many authors; refer to [12–24] and references therein. In addition, the free boundary value problems for multidimensional compressible viscous Navier-Stokes equations with constant viscosity coefficients for either barotropic or heat-conducive fluids are investigated by many authors, such as in the case that across the free surface stress tensor is balanced by a constant exterior pressure and/or the surface tension; classical solutions with strictly positive densities in the fluid regions to FBVP for CNS (1) with constant viscosity coefficients are shown locally in time for either barotropic flows [25–27] or heat-conductive flows [28–30]. In the case that across the free surface the stress tensor is balanced by exterior pressure [27], surface tension [31], or both exterior pressure and surface tension [32], respectively, as the initial data is assumed to be near to a nonvacuum equilibrium state, the global existence of classical solutions with small amplitude and positive densities in fluid region to the FBVP for CNS (1) with constant viscosity coefficients is proved. Global existence of classical solutions to FBVP for compressible viscous and heat-conductive fluids is also obtained with the stress tensor balanced by the exterior pressure and/or surface tension across the free surface; refer to [33, 34] and references therein.

In the present paper, we focus on the free boundary value problem for one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient and a nonconstant exterior pressure, and the existence, regularities, and dynamical behavior of global strong solution will be addressed, and so forth. As 𝛾>1, 0<𝛼≤1, we show that the free boundary value problem with regular initial data admits a unique global strong solution which is strictly positive from blow for any finite time and decays pointwise to zero at an algebraic time-rate (refer to Theorem 1 for details).

The rest of the paper is arranged as follows. In Section 2, the main results about the existence and dynamical behavior of global strong solution to FBVP for compressible Navier-Stokes equations are stated. Then, some important a priori estimates will be given in Section 3 and the theorem is proven in Section 4.

2. Main Results

We will investigate the global existence and dynamics of the free boundary value problem for (1) with the following initial data and boundary conditions: 𝜌,𝑢(𝑥,0)=𝜌₀,𝑢₀,𝑥∈𝑎₀,𝑏₀,𝜌^𝛾−𝜌^𝛼𝑢_𝑥(𝑎(𝑡),𝑡)=𝑝_𝑒(𝑡),𝜌^𝛾−𝜌^𝛼𝑢_𝑥(𝑏(𝑡),𝑡)=𝑝_𝑒(𝑡),𝑡>0, (2) where 𝑥=𝑎(𝑡) and 𝑥=𝑏(𝑡) are the free boundaries defined by 𝑑𝑎(𝑡)(𝑎(𝑡))(0)𝑑𝑡=𝑢,𝑡,𝑎=𝑎₀,𝑑𝑏(𝑡)(𝑏(𝑡))(0)𝑑𝑡=𝑢,𝑡,𝑏=𝑏₀,𝑡>0, (3) and the function 𝑝_𝑒(𝑡)>0 is the nonconstant exterior pressure.

Without the loss of generality, the total initial mass is renormalized to be one; that is, _𝑎(𝑡)^𝑏(𝑡)𝜌(𝑥,𝑡)𝑑𝑥=^𝑏0_𝑎0𝜌₀(𝑥)𝑑𝑥∶=1. (4) And we consider that the initial data satisfies inf_{[𝑎0,𝑏0]}𝜌₀≥𝜌>0,𝜌₀,𝑢₀∈𝑊^1,∞𝑎₀,𝑏₀,𝜌^𝛾₀−𝜌^𝛼₀𝑢_0𝑥𝑎₀=𝑝_𝑒0,𝜌^𝛾₀−𝜌^𝛼₀𝑢_0𝑥𝑏₀=𝑝_𝑒0,𝜌^𝛾₀𝑏₀≥𝜌^𝛾₀𝑎₀≥𝑝_𝑒0, (5) where 𝜌 is a positive constant and 𝑝_𝑒0∶=𝑝_𝑒(0); note that the compatibility conditions between initial data and boundary conditions hold. Then, we have the global existence and time-asymptotical behavior of strong solution as follows.

Theorem 1 (FBVP). Let 𝛾>1 and 0<𝛼≤1. Assume that the initial data satisfies (5); 𝑝_𝑒(𝑡)=𝑜((1+𝑡)^𝜈); 𝜈>0 is a constant; 𝑝_𝑒(𝑡)∈𝐿¹([0,𝑇]) and 𝑝^′_𝑒(𝑡)∈𝐿²([0,𝑇]) uniformly for 𝑇>0. Then, there exists a unique global strong solution (𝜌,𝑢,𝑎,𝑏) to the FBVP (1) and (2) satisfying 𝑐≤𝜌∈𝐿^∞0,𝑇;𝐻¹([𝑎(𝑡),𝑏(𝑡)])∩𝐶⁰([0,𝑇]×[𝑎(𝑡),𝑏(𝑡)]),𝑢∈𝐿^∞0,𝑇;𝐻¹([𝑎(𝑡),𝑏(𝑡)])∩𝐿²0,𝑇;𝐻²𝑎(𝑡)(𝑡,𝑎(𝑡)(𝑡)([,𝑏)]),𝑏∈𝐻¹,𝜌([0,𝑇])_𝛾−𝜌^𝛼𝑢_𝑥∈𝐿^∞0,𝑇;𝐿²([𝑎(𝑡),𝑏(𝑡)]), (6) with 𝑐>0 being a constant independent of time.
If it further holds that 𝑢₀∈𝐻²([𝑎₀,𝑏₀]), then (𝜌,𝑢,𝑎,𝑏) satisfies 𝜌,𝑢∈𝐶⁰([0,𝑇]×[𝑎(𝑡),𝑏(𝑡)]),𝜌∈𝐿^∞0,𝑇;𝐻¹𝑎(𝑡)(𝑡,𝜌([,𝑏)])_𝑡∈𝐿^∞0,𝑇;𝐿²([𝑎(𝑡),𝑏(𝑡)]),𝑢∈𝐿^∞0,𝑇;𝐻²([𝑎(𝑡),𝑏(𝑡)])∩𝐿²0,𝑇;𝐻³𝑎(𝑡)(𝑡,𝑢([,𝑏)])_𝑡∈𝐿^∞0,𝑇;𝐿²([𝑎(𝑡),𝑏(𝑡)])∩𝐿²0,𝑇;𝐻¹𝑎(𝑡)(𝑡,𝑎(𝑡)(𝑡)([,𝑏)]),𝑏∈𝐻²,𝜌([0,𝑇])_𝛾−𝜌^𝛼𝑢_𝑥∈𝐶⁰([0,𝑇]×([𝑎(𝑡),𝑏(𝑡)])). (7) The domain expands outwards in time as 𝑝_𝑒(𝑡)⁻¹(𝑡)(𝑡)≥𝑐()≥𝑏−𝑎1+𝑡^{𝜆/(𝛾−1)}𝑐(),0<𝛼<1,1+𝑡^1/(𝛾−1),𝛼=1, (8) where 0<𝜆≤𝛾−1 denotes a positive constant, and the density decays pointwise to zero for any 𝑥∈[𝑎(𝑡),𝑏(𝑡)] and 𝑡>0 as 𝜌()≤𝑐()𝑥,𝑡1+𝑡^{−𝜂𝜆/(𝛾−1)}𝑐(),0<𝛼<1,1+𝑡^{−𝜂/(𝛾−1)},𝛼=1, (9) where 𝜂>0 is a positive constant.

Remark 2. Theorem 1 holds for one-dimensional Saint-Venant model for shallow water; that is, 𝛾=2, 𝛼=1.

Remark 3. Equation (8) implies that as time goes to infinity, both the lower bound rate and the upper bound rate go to infinity.

Remark 4. In fact, we can choose the nonconstant exterior pressure 𝑝_𝑒(𝑡) like these 𝑝_𝑒(𝑡)=𝐶𝑡^𝑎𝑒^−𝑏𝑡,𝑎∈𝑅⁺,𝑏∈𝑅⁺,𝑝_𝑒(𝑡)=𝐶(1+𝑡)^𝑎1+𝑒^𝑏𝑡−1,𝑎∈𝑅,𝑏∈𝑅⁺, (10) or the linear combinations of these functions, and so forth.

Remark 5. In particular, let 𝑝_𝑒(𝑡)=𝑒^−𝑡; from (8), we have 𝑏(𝑡)−𝑎(𝑡)≤𝑒^𝑡, (11) which implies that the upper bound rate of (𝑏(𝑡)−𝑎(𝑡)) expands at an exponential rate; however, we proved that the upper bound rate of (𝑏(𝑡)−𝑎(𝑡)) expands at an algebraic rate in [16] where we consider the free boundary value problem without the nonconstant exterior pressure.

3. The A Priori Estimates

Making use of the Lagrange coordinates, we can establish some a priori estimates. Define the Lagrange coordinates transform 𝜉=^𝑥_𝑎(𝑡)𝜌𝑦,𝑡𝑑𝑦,𝜏=𝑡. (12) Since the conservation of total mass holds, the boundaries 𝑥=𝑎(𝑡) and 𝑥=𝑏(𝑡) are transformed into 𝜉=0 and 𝜉=1, respectively, and the domain [𝑎(𝑡),𝑏(𝑡)] is transformed into [0,1]. The FBVP (1) and (2) is reformulated into 𝜌_𝜏+𝜌²𝑢_𝜉𝑢=0,_𝜏+𝜌^𝛾_𝜉=𝜌^1+𝛼𝑢_𝜉𝜉,𝜌₀,𝑢₀=𝜌₀,𝑢₀(𝜉)[],𝜌,𝜉∈0,1_𝛾−𝜌^1+𝛼𝑢_𝜉(0,𝜏)=𝑝_𝑒(𝜏),𝜌^𝛾−𝜌^1+𝛼𝑢_𝜉(1,𝜏)=𝑝_𝑒(𝜏),𝜏∈[0,𝑇], (13) where the initial data satisfies inf_[]0,1𝜌₀≥𝜌>0,𝜌₀,𝑢₀∈𝑊^1,∞,𝜌([0,1])_𝛾0−𝜌₀^1+𝛼𝑢_0𝑥(0)=𝑝_𝑒0,𝜌^𝛾₀−𝜌₀^1+𝛼𝑢_0𝑥(1)=𝑝_𝑒0,𝜌^𝛾₀(1)≥𝜌^𝛾₀(0)≥𝑝_𝑒0, (14) and the consistencies between initial data and boundary conditions hold.

Next, we will deduce the a priori estimates for the solution (𝜌,𝑢) to the FBVP (13).

Lemma 6. Let 𝑇>0. Under the assumptions of Theorem 1, it holds for any strong solution (𝜌,𝑢) to the FBVP (13) that 𝜌^𝛾(1,𝜏)≥𝜌^𝛾(0,𝜏)≥𝑝_𝑒(𝜏). (15)

Proof. From (13), we can find 1𝛼𝜌^𝛼(0,𝜏)−𝑝_𝑒^𝛼/𝛾(𝜏)_𝜏+𝜌^𝛾(0,𝜏)−𝑝_𝑒(𝜏)+1𝛾𝑝_𝑒^{𝛼/𝛾−1}(𝜏)𝑝^′(𝜏)1=0,𝛼𝜌^𝛼(1,𝜏)−𝑝_𝑒^𝛼/𝛾(𝜏)_𝜏+𝜌^𝛾(1,𝜏)−𝑝_𝑒(𝜏)+1𝛾𝑝_𝑒^{𝛼/𝛾−1}(𝜏)𝑝^′(𝜏)1=0,𝛼𝜌^𝛼(1,𝜏)−𝜌^𝛼(0,𝜏)_𝜏+𝜌^𝛾(1,𝜏)−𝜌^𝛾(0,𝜏)=0, (16) which imply that 𝜌^𝛼(0,𝜏)−𝑝_𝑒^𝛼/𝛾(𝜏)=𝜌^𝛼₀(0)−𝑝^𝛼/𝛾_𝑒0×exp−𝛼^𝜏₀𝜌^𝛾(0,𝑠)−𝑝_𝑒(𝑠)+1/𝛾𝑝_𝑒^{𝛼/𝛾−1}(𝑠)𝑝^′(𝑠)𝜌^𝛼(0,𝑠)−𝑝_𝑒^𝛼/𝛾(𝑠)𝑑𝑠≥0,𝜌^𝛼(1,𝜏)−𝑝_𝑒^𝛼/𝛾(𝜏)=𝜌^𝛼₀(1)−𝑝^𝛼/𝛾_𝑒0×exp−𝛼^𝜏₀𝜌^𝛾(1,𝑠)−𝑝_𝑒(𝑠)+1/𝛾𝑝_𝑒^{𝛼/𝛾−1}(𝑠)𝑝^′(𝑠)𝜌^𝛼(1,𝑠)−𝑝_𝑒^𝛼/𝛾(𝑠)𝑑𝑠≥0,𝜌^𝛼(1,𝜏)−𝜌^𝛼()=𝜌0,𝜏_𝛼0(1)−𝜌^𝛼₀(0)×exp−𝛼^𝜏₀𝜌^𝛾(1,𝑠)−𝜌^𝛾(0,𝑠)𝜌^𝛼(1,𝑠)−𝜌^𝛼(0,𝑠)𝑑𝑠≥0. (17)(18)(19)

Lemma 7. Let 𝑇>0. Under the assumptions of Theorem 1, it holds for any strong solution (𝜌,𝑢) to the FBVP (13) that ¹₀12𝑢²+1𝜌𝛾−1^𝛾−1𝑑𝜉+^𝜏₀¹₀𝜌^1+𝛼𝑢²_𝜉+𝑑𝜉𝑑𝑠_𝜏0𝑝_𝑒(𝑠)(𝑏(𝑠)−𝑎(𝑠))^′=𝑑𝑠₁₀12𝑢²₀+1𝜌𝛾−1^0𝛾−1𝑑𝜉,𝜏∈[0,𝑇], (20) where 𝑎(𝜏) satisfies 𝑎^′(𝜏)=𝑢(0,𝜏) and 𝑎(0)=𝑎₀ and 𝑏(𝜏) satisfies 𝑏^′(𝜏)=𝑢(1,𝜏) and 𝑏(0)=𝑏₀.

Proof. Taking the product of (13)₂ with 𝑢, integrating on [0,1], and using boundary conditions, we have 12𝑑𝑑𝜏¹₀𝑢²𝑑𝜉+¹₀𝜌^1+𝛼𝑢²_𝜉=𝑑𝜉₁₀𝑢_𝜉𝜌^𝛾−𝑝_𝑒(𝜏)𝑑𝜉=−¹₀𝜌^𝛾−2𝜌_𝜏𝑑𝜉−𝑝_𝑒(𝜏𝑢()(1)(1,𝜏−𝑢0,𝜏))=−𝑑𝛾−1𝑑𝜏¹₀𝜌^𝛾−1𝑑𝜉−𝑝_𝑒(𝜏)(𝑏(𝜏)−𝑎(𝜏))^′, (21) which leads to (20) after the integration with respect to 𝜏∈[0,𝑇], where we use the fact that, from (15), the domain (𝑏(𝜏)−𝑎(𝜏)) expands as the time grows up, and it holds that (𝑏(𝜏)−𝑎(𝜏))^′=𝑏^′(𝜏)−𝑎^′(𝜏)=𝑢(1,𝜏)−𝑢(0,𝜏)≥0. (22)

Lemma 8. Let 𝑇>0. Under the assumptions of Theorem 1, it holds for any strong solution (𝜌,𝑢) to the FBVP (13) that ¹₀121𝑢+𝛼𝜌^𝛼_𝜉²+1𝜌𝛾−1^𝛾−1𝑑𝜉+𝛾^𝜏₀¹₀𝜌^{𝛾+2𝛼−1}𝜌²_𝜉𝑑𝜉𝑑𝑠+𝜌^𝛾(1,𝜏)𝑏(𝜏)−𝜌^𝛾()𝑎(𝜏)+𝛾0,𝜏𝛼^𝜏₀𝜌^𝛾−𝛼(1,𝑠)𝜌^𝛾(1,𝑠)−𝑝_𝑒(𝑠)𝑏(𝑠)−𝛾𝑑𝑠𝛼^𝜏₀𝜌^𝛾−𝛼(0,𝑠)𝜌^𝛾(0,𝑠)−𝑝_𝑒(𝑠)𝑎(𝑠)=𝑑𝑠₁₀12𝑢₀+1𝛼𝜌^𝛼_0𝜉²+1𝜌𝛾−1^0𝛾−1𝑑𝜉+𝜌^𝛾₀(1)𝑏₀−𝜌^𝛾₀(0)𝑎₀,𝜏∈[0,𝑇]. (23)

Proof. Multiplying (13)₁ by 𝜌^𝛼−1 gives 𝜌^𝛼_𝜏𝛼+𝜌^1+𝛼𝑢_𝜉=0, (24) which leads to 𝜌^𝛼_𝜏𝜉𝛼+𝜌^1+𝛼𝑢_𝜉𝜉=0. (25) Summing (13)₁ and (25), we deduce 𝜌𝑢+^𝛼𝜉𝛼_𝜏+𝜌^𝛾_𝜉=0. (26) Multiplying (26) by (𝑢+(𝜌^𝛼)_𝜉/𝛼) and integrating the result over [0,1]×[0,𝜏], we obtain ¹₀121𝑢+𝛼𝜌^𝛼_𝜉²+1𝜌𝛾−1^𝛾−1𝑑𝜉+𝛾^𝜏₀¹₀𝜌^{𝛾+2𝛼−1}𝜌²_𝜉𝑑𝜉𝑑𝑠+^𝜏₀𝜌^𝛾(1,𝑠)𝑢(1,𝑠)−𝜌^𝛾(0,𝑠)𝑢(0,𝑠)𝑑𝑠=0, (27) where we have the fact that ^𝜏₀𝜌^𝛾(1,𝑠)𝑢(1,𝑠)−𝜌^𝛾()𝑢()=0,𝑠0,𝑠𝑑𝑠_𝜏0𝜌^𝛾(1,𝑠)𝑏^′(𝑠)𝑑𝑠−^𝜏₀𝜌^𝛾(0,𝑠)𝑎^′(𝑠)𝑑𝑠=𝜌^𝛾(1,𝜏)𝑏(𝜏)−𝜌^𝛾₀(1)𝑏₀−𝛾𝛼^𝜏₀𝑏(𝑠)𝜌^𝛾−𝛼(1,𝑠)𝜌^𝛼(1,𝑠)_𝑠𝑑𝑠−𝜌^𝛾(0,𝜏)𝑎(𝜏)+𝜌^𝛾₀(0)𝑎₀+𝛾𝛼^𝜏₀𝑎(𝑠)𝜌^𝛾−𝛼(0,𝑠)𝜌^𝛼(0,𝑠)_𝑠𝑑𝑠=𝜌^𝛾(1,𝜏)𝑏(𝜏)−𝜌^𝛾(0,𝜏)𝑎(𝜏)−𝜌^𝛾₀(1)𝑏₀+𝜌^𝛾₀(0)𝑎₀+𝛾𝛼^𝜏₀𝜌^𝛾−𝛼(1,𝑠)𝜌^𝛾(1,𝑠)−𝑝_𝑒(𝑠)𝑏(𝑠)−𝛾𝑑𝑠𝛼^𝜏₀𝜌^𝛾−𝛼(0,𝑠)𝜌^𝛾(0,𝑠)−𝑝_𝑒(𝑠)𝑎(𝑠)𝑑𝑠, (28) which together with (15) and 𝑏(𝜏)−𝑎(𝜏)=¹₀1𝜌(𝜁,𝜏)𝑑𝜁>0 (29) gives rise to (23).

Lemma 9. Let 𝑇>0. Under the assumptions of Theorem 1, it holds that 𝜌(𝜉,𝜏)≤𝐶,(𝜉,𝜏)∈[0,1]×[0,𝑇],𝑇>0, (30) where 𝐶 is the positive constant independent of time.

Proof. Integrating (24) with respect to 𝜏 over [0,𝜏], we know 𝜌^𝛼(𝜉,𝜏)=𝜌^𝛼₀(𝜉)−𝛼^𝜏₀𝜌^1+𝛼𝑢_𝜉(𝜉,𝑠)𝑑𝑠; (31) then integrating (13)₂ over [0,𝜉]×[0,𝜏] and using the boundary conditions, we have ^𝜉₀𝑢(𝜁,𝜏)𝑑𝜁−^𝜉₀𝑢₀(𝜁)𝑑𝜁+^𝜏₀𝜌^𝛾(𝑠)𝑑𝑠−^𝜏₀𝑝_𝑒(𝑠)𝑑𝑠=^𝜏₀𝜌^1+𝛼𝑢_𝜉(𝜉,𝑠)𝑑𝑠. (32) It holds from (31) and (32) that 𝜌^𝛼(𝜉,𝜏)+𝛼^𝜏₀𝜌^𝛾(𝜉,𝑠)𝑑𝑠=𝜌^𝛼₀(𝜉)−𝛼^𝜉₀𝑢(𝜁,𝜏)𝑑𝜁+𝛼^𝜉₀𝑢₀(𝜁)𝑑𝜁+𝛼^𝜏₀𝑝_𝑒(𝑠)𝑑𝑠≤𝐶+𝐶¹₀𝑢²𝑑𝜉^1/2+𝛼^𝜏₀𝑝_𝑒(𝑠)𝑑𝑠≤𝐶, (33) where 𝐶 denotes the positive constant independent of time.

Lemma 10. Let 𝑇>0. Under the assumptions of Theorem 1, it holds for any strong solution (𝜌,𝑢) to the FBVP (13) that ¹₀𝑢^2𝑛𝑑𝜉+𝑛(2𝑛−1)^𝜏₀¹₀𝜌^1+𝛼𝑢^2𝑛−2𝑢²_𝜉𝑑𝜉𝑑𝑠+^𝜏₀𝑝_𝑒(𝑠)𝑏^′(𝑠)^2𝑛−1−𝑎^′(𝑠)^2𝑛−1(𝑇),[],𝑑𝑠≤𝐶𝜏∈0,𝑇 (34) for any positive integer 𝑛∈𝑁, and 𝐶(𝑇)>0 denotes a constant dependent on time.

Proof. Multiplying (13)₂ by 2𝑛𝑢^2𝑛−1 and integrating the result over [0,1]×[0,𝜏], we obtain ¹₀𝑢^2𝑛𝑑𝜉+2𝑛(2𝑛−1)^𝜏₀¹₀𝜌^1+𝛼𝑢^2𝑛−2𝑢²_𝜉𝑑𝜉𝑑𝑠+2𝑛^𝜏₀𝑝(𝑠)𝑢^2𝑛−1(1,𝑠)−𝑢^2𝑛−1()=0,𝑠𝑑𝑠₁₀𝑢₀^2𝑛𝑑𝜉+2𝑛(2𝑛−1)^𝜏₀¹₀𝜌^𝛾𝑢^2𝑛−2𝑢²_𝜉𝑑𝜉𝑑𝑠; (35) then it holds from Young’s inequality and (22) that ¹₀𝑢^2𝑛𝑑𝜉+𝑛(2𝑛−1)^𝜏₀¹₀𝜌^1+𝛼𝑢^2𝑛−2𝑢²_𝜉𝑑𝜉𝑑𝑠+2𝑛^𝜏₀𝑝(𝑠)𝑏^′(𝑠)^2𝑛−1−𝑎^′(𝑠)^2𝑛−1=𝑑𝑠₁₀𝑢₀^2𝑛𝑑𝜉+𝐶^𝜏₀¹₀𝑢^2𝑛𝑑𝜉𝑑𝑠, (36) which together with Gronwall’s inequality gives (34).

Lemma 11. Let 𝑇>0, for 𝑛∈𝑁, and 𝑛>(1+𝛼)/4(𝛾−𝛼). Under the assumptions of Theorem 1, it holds for any strong solution (𝜌,𝑢) to the FBVP (13) that ^𝜏₀𝜌^𝛾_𝜉^2𝑛_{𝐿∞([0,1])}𝑑𝑠≤𝐶(𝑇),𝜏∈[0,𝑇]. (37)

Proof. Integrating (26) with respect to 𝜏 over [0,𝜏], we have 𝜌^𝛼_𝜉=𝜌^𝛼_0𝜉+𝛼𝑢₀(𝜉)−𝛼𝑢(𝜉,𝜏)−𝛼^𝜏₀𝜌^𝛾_𝜉(𝜉,𝑠)𝑑𝑠, (38) which together with (20) gives ^𝜏₀(𝜌^𝛾)_𝜉^2𝑛_{𝐿∞([0,1])}=𝛾𝑑𝑠_2𝑛𝛼^2𝑛𝜏₀𝜌^{2𝑛(𝛾−𝛼)}𝜌^𝛼_𝜉^2𝑛_{𝐿∞([0,1])}(𝑇)(𝑇)𝑑𝑠≤𝐶+𝐶^𝜏₀𝜌^{2𝑛(𝛾−𝛼)}𝑢^2𝑛_{𝐿∞([0,1])}(𝑇)𝑑𝑠+𝐶^𝜏₀^𝑠₀𝜌^𝛾_𝜉^2𝑛_{𝐿∞([0,1])}(𝑇)(𝑇)𝑑𝑙𝑑𝑠≤𝐶+𝐶^𝜏₀^𝑠₀𝜌^𝛾_𝜉^2𝑛_{𝐿∞([0,1])}𝑑𝑙𝑑𝑠, (39) where we have used ^𝜏₀𝜌^{2𝑛(𝛾−𝛼)}𝑢^2𝑛_{𝐿∞([0,1])}𝑑𝑠≤^𝜏₀¹₀𝜌^{2𝑛(𝛾−𝛼)}𝑢^2𝑛𝑑𝜉𝑑𝑠+^𝜏₀¹₀𝜌^{2𝑛(𝛾−𝛼)}𝑢^2𝑛_𝜉(𝑇)𝑑𝜉𝑑𝑠≤𝐶+𝐶^𝜏₀¹₀𝜌^{2(2𝑛(𝛾−𝛼)−𝛼)}𝜌^2𝛼−2𝜌²_𝜉+𝑢^4𝑛+𝜌^{4𝑛(𝛾−𝛼)−(1+𝛼)}𝑢^2𝑛+𝜌^1+𝛼𝑢^2𝑛−2𝑢²_𝜉(𝑇),𝑑𝜉𝑑𝑠≤𝐶 (40) which can be deduced from (20), (23), (30), and (34). Making use of Gronwall’s inequality to (40), we obtain (37).

Lemma 12. Let 𝑇>0. Under the assumptions of Theorem 1, it holds for any strong solution (𝜌,𝑢) to the FBVP (13) that 𝜌(𝜉,𝜏)≥𝐶(𝑇),(𝜉,𝜏)∈[0,1]×[0,𝑇]. (41)

Proof. Define that 𝑣()=1𝜉,𝜏𝜌().𝜉,𝜏 (42) By (13)₁, we have 𝑣_𝜏=𝑢_𝜉. (43) Multiplying (43) by 𝛽𝑣^𝛽−1, where 𝛽=2(1+𝛼), integrating the result over [0,1]×[0,𝜏], and using (37) and (38), we can find that ¹₀𝑣^𝛽𝑑𝜉+𝛽𝛽−1^𝜏₀¹₀𝑣^{𝛼+𝛽−1}𝑢²=𝑑𝜉𝑑𝑠₁₀𝑣^𝛽₀𝑑𝜉+𝛽^𝜏₀𝑣^𝛽−1𝑢𝑑𝑠|^𝜉=1_𝜉=0+𝛽𝛽−1𝛼^𝜏₀¹₀𝑣^{𝛼+𝛽−1}𝑢𝜌^𝛼_0𝜉𝑑𝜉𝑑𝑠+𝛽𝛽−1^𝜏₀¹₀𝑣^{𝛼+𝛽−1}𝑢𝑢₀×𝑑𝜉𝑑𝑠−𝛽𝛽−1_𝜏010𝑣^{𝛼+𝛽−1}𝑢^𝑠₀𝜌^𝛾_𝜉(𝑇)𝑑𝑙𝑑𝜉𝑑𝑠≤𝐶+𝛽^𝜏₀𝑣^𝛽−1()𝑢()−1,𝑠1,𝑠𝑑𝑠_𝜏0𝑣^𝛽−1()𝑢()+𝛽0,𝑠0,𝑠𝑑𝑠𝛽−12^𝜏₀¹₀𝑣^{𝛼+𝛽−1}𝑢²𝑑𝜉𝑑𝑠+𝐶(𝑇)^𝜏₀¹₀𝑣^𝛽𝑑𝜉𝑑𝑠. (44) Since it holds that 𝜌^𝛼(0,𝜏)=𝑝_𝑒^𝛼/𝛾(𝜏)+𝜌^𝛼₀(0)−𝑝^𝛼/𝛾_𝑒0×exp−𝛼^𝜏₀𝜌^𝛾(0,𝜏)−𝑝_𝑒(𝜏)+1/𝛾𝑝_𝑒^{𝛼/𝛾−1}𝑝^′(𝜏)𝜌^𝛼(0,𝜏)−𝑝_𝑒^𝛼/𝛾(𝜏)≥𝑝_𝑒^𝛼/𝛾(𝜏), (45) which implies 𝜌(0,𝜏)≥𝑝_𝑒^1/𝛾(𝜏)≥inf_{[]𝜏∈0,𝑇}𝑝_𝑒^1/𝛾(𝜏)∶=𝐶(𝑇), (46) we have from (20) and (23) that ^𝜏₀𝑣^𝛽−1𝑢()(𝑇)0,𝑠𝑑𝜉≤𝐶^𝜏₀¹₀𝑢²𝑑𝜉^1/2+¹₀𝜌^1+𝛼𝑢²_𝜉𝑑𝜉^1/2+¹₀𝜌^𝛼2_𝜉𝑑𝜉^1/21₀𝑢²𝑑𝜉^1/2(𝑇).𝑑𝑠≤𝐶 (47) Using the same method, we also have ^𝜏₀𝑣^𝛽−1𝑢(1,𝑠)𝑑𝜉≤𝐶(𝑇). (48) Substituting (47) and (48) into (44), we get ¹₀𝑣^𝛽𝛽𝑑𝜉+𝛽−12^𝜏₀¹₀𝑣^{𝛼+𝛽−1}𝑢²(𝑇)(𝑇)𝑑𝜉𝑑𝑠≤𝐶+𝐶^𝜏₀¹₀𝑣^𝛽𝑑𝜉𝑑𝑠, (49) which together with Gronwall’s inequality yields ¹₀𝑣^2(1+𝛼)𝑑𝜉≤𝐶(𝑇). (50) We have from (23), (46), and (50) that 𝑣(𝜉,𝜏)≤𝑣(0,𝜏)+¹₀𝑣_𝜉𝑑𝜉≤𝐶(𝑇)+𝐶¹₀𝑣^2(1+𝛼)𝑑𝜉^1/21₀𝜌^𝛼_𝜉²𝑑𝜉^1/2≤𝐶(𝑇), (51) which implies 𝜌(𝜉,𝜏)≥𝐶(𝑇). (52)

We also have the regularity estimates for the solution (𝜌,𝑢) to the FBVP (13) as follows.

Lemma 13. Let 𝑇>0. Under the assumptions of Theorem 1, it holds for any strong solution (𝜌,𝑢) to the FBVP (13) that 𝜌∈𝐿^∞0,𝑇;𝐻¹([0,1])∩𝐶⁰([0,𝑇]×[0,1]),𝑢∈𝐿^∞0,𝑇;𝐻¹([0,1])∩𝐿²0,𝑇;𝐻²,𝑎(𝜏)(𝜏)([0,1]),𝑏∈𝐻¹,𝜌([0,𝑇])_𝛾−𝜌^1+𝛼𝑢_𝜉∈𝐿^∞0,𝑇;𝐿²([0,1]). (53)
If it is also satisfied that 𝑢₀∈𝐻²([0,1]), (54) then the strong solution (𝜌,𝑢) has the regularities 𝜌,𝑢∈𝐶⁰([0,𝑇]×[0,1]),𝜌∈𝐿^∞0,𝑇;𝐻¹([0,1]),𝜌_𝜏∈𝐿^∞0,𝑇;𝐿²([0,1]),𝑢∈𝐿^∞0,𝑇;𝐻²([0,1])∩𝐿²0,𝑇;𝐻³,𝑢([0,1])_𝜏∈𝐿^∞0,𝑇;𝐿²([0,1])∩𝐿²0,𝑇;𝐻¹,𝑎(𝜏)(𝜏)([0,1]),𝑏∈𝐻²,𝜌([0,𝑇])_𝛾−𝜌^1+𝛼𝑢_𝜉∈𝐶⁰([0,𝑇]×([0,1])). (55)

Proof. Multiplying (13)₂ by 𝜌^−(1+𝛼)𝑢_𝜏, integrating the result over [0,1], and making use of the boundary conditions, after a direct computation and recombination, we find 𝑑𝑑𝜏¹₀12𝑢²_𝜉−𝜌^𝛾−𝑝_𝑒(𝜏)𝜌^−(1+𝛼)𝑢_𝜉𝑑𝜉+¹₀𝜌^−(1+𝛼)𝑢²_𝜏=(𝑑𝜉𝛾−1+𝛼)¹₀𝜌^𝛾−𝛼𝑢²_𝜉+()𝑑𝜉1+𝛼¹₀𝑝_𝑒(𝜏)𝜌^−𝛼𝑢²_𝜉𝑑𝜉+¹₀𝑝^′_𝑒(𝜏)𝜌^−(1+𝛼)𝑢_𝜉−()𝑑𝜉1+𝛼¹₀𝜌^𝛾−𝑝_𝑒(𝜏)𝜌^−(2+𝛼)𝜌_𝜉𝑢_𝜏+()𝑑𝜉1+𝛼¹₀𝜌⁻¹𝜌_𝜉𝑢_𝜉𝑢_𝜏𝑑𝜉. (56) Integrating (56) over [0,𝜏], from (20), (23), (30), (41), 𝑝_𝑒(𝜏)∈𝐿¹([0,𝑇]), and 𝑝^′_𝑒(𝜏)∈𝐿²([0,𝑇]), it is easily verified that ¹₀𝑢²_𝜉𝑑𝜉+^𝜏₀¹₀𝑢²_𝑠(𝑇)(𝑇)𝑑𝜉𝑑𝑠≤𝐶+𝐶^𝜏₀¹₀𝑢²_𝜉𝑑𝜉𝑑𝑠+𝐶(𝑇)^𝜏₀¹₀𝜌²_𝜉(𝑇)𝑑𝜉𝑑𝑠+𝐶^𝜏₀¹₀𝜌²_𝜉𝑢²_𝜉(𝑇)(𝑇)𝑑𝜉𝑑𝑠≤𝐶+𝐶^𝜏₀¹₀𝜌²_𝜉𝑢²_𝜉𝑑𝜉𝑑𝑠, (57) where 𝐶(𝑇) denotes a constant dependent on time. From (13)₂, (20), (23), (30), and (41), it holds that 𝐶(𝑇)^𝜏₀¹₀𝜌²_𝜉𝑢²_𝜉≤𝐶(𝑇)𝑑𝜉𝑑𝑠2^𝜏₀¹₀𝜌²_𝜉𝑢²_𝜉1𝑑𝜉𝑑𝑠+4^𝜏₀¹₀𝑢²_𝑠(𝑇)𝑑𝜉𝑑𝑠+𝐶^𝜏₀¹₀𝜌²_𝜉𝑑𝜉𝑑𝑠+𝐶(𝑇)^𝜏₀¹₀𝑢²_𝜉𝑑𝜉𝑑𝑠. (58) Using (58), we can obtain that ¹₀𝑢²_𝜉𝑑𝜉+^𝜏₀¹₀𝑢²_𝑠𝑑𝜉𝑑𝑠≤𝐶(𝑇), (59) which together with (13)₂ implies ¹₀𝑢²_𝜉𝑑𝜉+^𝜏₀¹₀𝑢²_𝑠𝑑𝜉𝑑𝑠+^𝜏₀¹₀𝑢²_𝜉𝜉𝑑𝜉𝑑𝑠≤𝐶(𝑇). (60)
Differentiating (13)₂ with respect to 𝜏, we get 𝑢_𝜏𝜏+𝜌^𝛾−𝑝_𝑒(𝜏)_𝜉𝜏=𝜌^1+𝛼𝑢_𝜉𝜉𝜏. (61) Taking product between (61) and 𝑢_𝜏, integrating the results over [0,1], and using the boundary conditions (13)_4,5, we have 12𝑑𝑑𝜏¹₀𝑢²_𝜏=𝑑𝜉₁₀𝜌^𝛾−𝑝_𝑒(𝜏)_𝜏𝑢_𝜉𝜏𝑑𝜉−¹₀𝜌^1+𝛼𝑢_𝜉𝜏𝑢_𝜉𝜏𝑑𝜉. (62) The terms on the right-hand side of (62) can be bounded, respectively, as described below: ¹₀𝜌^𝛾−𝑝_𝑒(𝜏)_𝜏𝑢_𝜉𝜏𝑑𝜉=−¹₀𝛾𝜌^𝛾+1𝑢_𝜉𝑢_𝜉𝜏𝑑𝜉−¹₀𝑝^′_𝑒(𝜏)𝑢_𝜉𝜏𝛾𝑑𝜉≤−2𝑑𝑑𝜏¹₀𝜌^𝛾+1𝑢²_𝜉𝑑𝜉+𝐶¹₀𝜌^1+𝛼𝑢²_𝜉+𝜌^{2𝛾−𝛼+3}𝑢⁴_𝜉−𝑑𝜉₁₀𝑝^′_𝑒(𝜏)𝑢_𝜉𝜏−𝑑𝜉,₁₀𝜌^1+𝛼𝑢_𝜉𝜏𝑢_𝜉𝜏𝑑𝜉=−¹₀(1+𝛼)𝜌^𝛼𝜌_𝜏𝑢_𝜉+𝜌^1+𝛼𝑢_𝜉𝜏𝑢_𝜉𝜏1𝑑𝜉≤−2¹₀𝜌^1+𝛼𝑢²_𝜉𝜏𝑑𝜉+𝐶¹₀𝜌^3+𝛼𝑢⁴_𝜉𝑑𝜉. (63) Summing (62)-(63) together and making use of (30) and (60), we have 12𝑑𝑑𝜏¹₀𝑢²_𝜏𝛾𝑑𝜉+2𝑑𝑑𝜏¹₀𝜌^𝛾+1𝑢²_𝜉1𝑑𝜉+2¹₀𝜌^1+𝛼𝑢²_𝜉𝜏(𝑇)𝑑𝜉≤𝐶¹₀𝑢²_𝜉𝑑𝜉+𝐶(𝑇)𝑢_𝜉²_{𝐿∞([0,1])}¹₀𝑢²_𝜉+1𝑑𝜉4¹₀𝜌^1+𝛼𝑢²_𝜉𝜏𝑑𝜉+𝐶(𝑇)^𝜏₀𝑝^′_𝑒(𝑠)²(𝑇)(𝑇𝑢𝑑𝑠≤𝐶+𝐶)_𝜉²_{𝐿∞([0,1])}¹₀𝑢²_𝜉+1𝑑𝜉4¹₀𝜌^1+𝛼𝑢²_𝜉𝜏𝑑𝜉. (64) Integrating (64) over [0,𝜏], it holds from (13)₂, (20), and (60) that ¹₀𝑢²_𝜏𝑑𝜉+¹₀𝑢²_𝜉𝑑𝜉+^𝜏₀¹₀𝑢²_𝜉𝑠(𝑇)(𝑇)𝑑𝜉𝑑𝑠≤𝐶+𝐶sup_{[]𝜏∈0,𝑇}𝑢_𝜉²_𝐿∞≤𝐶(𝑇)+𝐶(𝑇)sup_{𝜏∈[0,𝑇]}¹₀𝑢²_𝜉𝑑𝜉^1/21₀𝑢²_𝜉𝜉𝑑𝜉^1/2(𝑇)+1≤𝐶2¹₀𝑢²_𝜏𝑑𝜉, (65) which gives ¹₀𝑢²_𝜏𝑑𝜉+¹₀𝑢²_𝜉𝑑𝜉+^𝜏₀¹₀𝑢²_𝜉𝜏𝑑𝜉𝑑𝑠≤𝐶(𝑇), (66) which implies (𝜌^𝛾−𝜌^1+𝛼𝑢_𝜉)∈𝐿^∞(0,𝑇;𝐻¹([0,1])), and it follows from the definition of 𝑎^′(𝜏)=𝑢(0,𝜏) and 𝑏^′(𝜏)=𝑢(1,𝜏) that 𝑎(𝜏), 𝑏(𝜏)∈𝐻²([0,𝑇]). The proof of this lemma is completed.

Finally, we will give the large time behavior of the strong solution as follows.

Lemma 14. Let 𝑇>0. Under the assumptions of Theorem 1, it holds for 𝛼∈(0,1] and time 𝑡 large enough that 𝑝_𝑒(𝑡)⁻¹(𝑡)(𝑡)≥𝑐()≥𝑏−𝑎1+𝑡^{𝜆/(𝛾−1)}𝑐(),0<𝛼<1,1+𝑡^1/(𝛾−1),𝛼=1, (67) where 0<𝜆≤𝛾−1 denotes a positive constant, and the density decays pointwise to zero for any 𝑥∈[𝑎(𝑡),𝑏(𝑡)] and 𝑡>0 as 𝜌()≤𝑐()𝑥,𝑡1+𝑡^{−𝜂𝜆/(𝛾−1)}𝑐(),0<𝛼<1,1+𝑡^{−𝜂/(𝛾−1)},𝛼=1, (68) where 𝜂>0 is a positive constant.

Proof. From (13) we can find that 𝑑𝑑𝜏¹₀𝑢(𝜉,𝜏)𝑑𝜉=0, (69) and, without loss of generality, we can renormalize ∫¹₀𝑢₀(𝜉)𝑑𝜉 to be zero; then, we denote 1𝑤=𝑢−1+𝜏^𝜉₀1𝜌1𝑑𝜁+1+𝜏¹₀^𝜉₀1𝜌𝑑𝜁𝑑𝜉. (70) Applying (69), we can obtain 𝑤_𝜉=𝑢_𝜉−1()𝜌=11+𝜏𝜌_𝜏−1()𝜌,1+𝜏𝑤_𝜏+𝑤1+𝜏=𝑢_𝜏; (71)(72) then the system (13) becomes 𝜌_𝜏+𝜌²𝑤_𝜉+𝜌𝑤1+𝜏=0,_𝜏+𝑤+𝜌1+𝜏^𝛾𝜉=𝜌^1+𝛼𝑤_𝜉+𝜌^𝛼1+𝜏_𝜉,𝜌₀,𝑤₀=𝜌₀,𝑢₀−11+𝜏^𝜉₀1𝜌₀1𝑑𝜁+1+𝜏¹₀^𝜉₀1𝜌₀(𝜉),[],𝜌𝑑𝜁𝑑𝜉𝜉∈0,1_𝛾−𝜌^1+𝛼𝑤_𝜉+1()𝜌()1+𝜏0,𝜏=𝑝_𝑒(𝜏),𝜌^𝛾−𝜌^1+𝛼𝑤_𝜉+1()𝜌()1+𝜏1,𝜏=𝑝_𝑒(𝜏),𝜏∈[0,𝑇]. (73) Multiplying (73)₂ by 𝑤 and integrating the result over [0,1], after a straightforward calculation, we have 12𝑑𝑑𝜏¹₀𝑤²1𝑑𝜉+1+𝜏¹₀𝑤²=𝑑𝜉₁₀𝜌^1+𝛼𝑤_𝜉𝜉1𝑤𝑑𝜉+1+𝜏¹₀𝜌^𝛼_𝜉−𝑤𝑑𝜉₁₀𝜌^𝛾−𝑝_𝑒(𝜏)_𝜉𝑤𝑑𝜉=−¹₀𝜌^1+𝛼𝑤²_𝜉1𝑑𝜉−1+𝜏¹₀𝜌^𝛼𝑤_𝜉+𝑑𝜉₁₀𝜌^𝛾−𝑝_𝑒(𝜏)𝑤_𝜉𝑑𝜉, (74) which implies 12𝑑𝑑𝜏¹₀𝑤²1𝑑𝜉+1+𝜏¹₀𝑤²𝑑𝜉+¹₀𝜌^1+𝛼𝑤²_𝜉1𝑑𝜉=−1+𝜏¹₀𝜌^𝛼𝑤_𝜉𝑑𝜉+¹₀𝜌^𝛾𝑤_𝜉𝑑𝜉−¹₀𝑝_𝑒(𝜏)𝑤_𝜉𝑑𝜉, (75) and as 0<𝛼<1 it holds from (71) that −11+𝜏¹₀𝜌^𝛼𝑤_𝜉1𝑑𝜉=−1+𝜏¹₀𝜌^𝛼1𝜌_𝜏−1()𝜌=11+𝜏𝑑𝜉(𝛼−1)(1+𝜏)¹₀𝜌^𝛼−1_𝜏+1𝑑𝜉(1+𝜏)²¹⁰𝜌^𝛼−1𝑑𝜉,₁₀𝜌^𝛾𝑤_𝜉𝑑𝜉=¹₀𝜌^𝛾1𝜌_𝜏−1()𝜌=11+𝜏𝑑𝜉1−𝛾¹₀𝜌^𝛾−1_𝜏1𝑑𝜉−1+𝜏¹₀𝜌^𝛾−1−𝑑𝜉,₁₀𝑝_𝑒(𝜏)𝑤_𝜉𝑑𝜉=−¹₀𝑝_𝑒(𝜏1)𝜌_𝜏−1(1+𝜏)𝜌𝑑𝜉=−¹₀𝑝_𝑒(𝜏)𝜌_𝜏𝑑𝜉+¹₀𝑝^′_𝑒(𝜏)𝜌+𝑑𝜉₁₀𝑝_𝑒(𝜏)(1+𝜏)𝜌𝑑𝜉, (76) which together with (75) leads to 𝑑𝑑𝜏¹₀12𝑤²1𝑑𝜉+𝑑𝛾−1𝑑𝜏¹₀𝜌^𝛾−1𝑑𝑑𝜉+𝑑𝜏¹₀𝑝_𝑒(𝜏)𝜌+1𝑑𝜉1+𝜏¹₀𝑤²𝑑𝜉+¹₀𝜌^1+𝛼𝑤²_𝜉1𝑑𝜉+1+𝜏¹₀𝜌^𝛾−1=1𝑑𝜉(𝛼−1)(1+𝜏)¹₀𝜌^𝛼−1_𝜏1𝑑𝜉+(1+𝜏)²¹⁰𝜌^𝛼−1+𝑑𝜉₁₀𝑝^′_𝑒(𝜏)𝜌𝑑𝜉+¹₀𝑝_𝑒(𝜏)(1+𝜏)𝜌𝑑𝜉; (77) multiplying (77) by (1+𝑡)^𝜆 for some 0<𝜆<1 to be determined later, we have 𝑑𝑑𝜏¹₀(1+𝜏)^𝜆2𝑤²+(1+𝜏)^𝜆𝜌𝛾−1^𝛾−1+(1+𝜏)^𝜆−1𝜌1−𝛼^𝛼−1+(1+𝜏)^𝜆𝑝_𝑒(𝜏)𝜌+𝜆𝑑𝜉1−2(1+𝜏)^𝜆−11₀𝑤²𝑑𝜉+(1+𝜏)^𝜆1₀𝜌^1+𝛼𝑤²_𝜉+𝑑𝜉𝛾−1−𝜆()𝛾−11+𝜏^𝛾−11₀𝜌^𝛾−1+𝑑𝜉𝛼−𝜆()1−𝛼1+𝜏^𝜆−21₀𝜌^𝛼−1+𝑑𝜉₁₀(1+𝜏)^𝜆−1𝜌−(1+𝜏)𝑝^′_𝑒(𝜏)−(1+𝜆)𝑝_𝑒(𝜏)𝑑𝜉=0, (78) and we will prove the fact that as time 𝜏 is large enough, it holds that −(1+𝜏)𝑝^′_𝑒(𝜏)≥(1+𝜆)𝑝_𝑒(𝜏), (79) which needs 𝑑𝑑𝜏ln𝑝_𝑒⁻¹(𝜏)≥1+𝜆1+𝜏>0; (80) that is, 𝑝_𝑒(𝜏)≤𝐶(1+𝜏)^1+𝜆, (81) where 𝐶>0 is a positive constant independent of time and we can find that (81) is true as we assume 𝑝_𝑒(𝜏)=𝑜((1+𝜏)^𝜈), where 𝜈 is some positive constant. Then, as 0<𝜆≤min{𝛾−1,𝛼}, integrating (78) over [0,𝜏], we have ¹₀𝜌^𝛾−1𝑑𝜉≤(1+𝜏)^−𝜆. (82)
As 𝛼=1 and 0<𝜆≤𝛾−1, it holds from (75) that 𝑑𝑑𝜏¹₀(1+𝜏)^𝜆2𝑤²+(1+𝜏)^𝜆𝜌𝛾−1^𝛾−1+(1+𝜏)^𝜆𝑝_𝑒(𝜏)𝜌+𝜆𝑑𝜉1−2(1+𝜏)^𝜆−11₀𝑤²+()𝑑𝜉1+𝜏^𝜆1₀𝜌^1+𝛼𝑤²_𝜉+𝑑𝜉𝛾−1−𝜆()𝛾−11+𝜏^𝛾−11₀𝜌^𝛾−1+𝑑𝜉₁₀(1+𝜏)^𝜆−1𝜌−(1+𝜏)𝑝^′_𝑒(𝜏)−(1+𝜆)𝑝_𝑒(𝜏)=𝑑𝑑𝜉()𝑑𝜏1+𝜏^𝜆−11₀+()ln𝜌𝑑𝜉1−𝜆)(1+𝜏^𝜆−21₀ln𝜌𝑑𝜉+(1+𝜏)^𝜆−2; (83) integrating (83) over [0,𝜏] and using ¹₀ln𝜌𝑑𝜉≤¹₀𝜌𝑑𝜉≤𝐶, (84) we have ¹₀𝜌^𝛾−1𝑑𝜉≤(1+𝜏)⁻¹. (85)
From (82) and (85), we know _𝑎(𝑡)^𝑏(𝑡)𝜌^𝛾𝑐()𝑑𝑥≤1+𝑡^−𝜆𝑐(),0<𝛼<1,1+𝑡⁻¹,𝛼=1, (86) and it holds from the conservation of the mass that 1=^𝑏0_𝑎0𝜌₀=𝑑𝑥_{𝑎(𝑡)𝑏(𝑡)}≤𝜌𝑑𝑥_{𝑎(𝑡)𝑏(𝑡)}𝜌^𝛾𝑑𝑥^1/𝛾(𝑏(𝑡)−𝑎(𝑡))^{(𝛾−1)/𝛾}, (87) which gives 𝑏(𝑡)(𝑡)≥𝑐()−𝑎1+𝑡^1/(𝛾−1)𝑐(),𝛼=1,1+𝑡^{𝜆/(𝛾−1)},0<𝛼<1. (88) Using (15), (17), and (23), we know 𝑏(𝑡)−𝑎(𝑡)≤𝐶𝜌^−𝛾(𝑎(𝑡),𝑡)≤𝐶𝑝_𝑒⁻¹(𝑡), (89) and we can choose the positive constant 𝜈 such that as time is large enough, it holds that 𝑝_𝑒(𝑡)=𝑜(1+𝑡)^𝜈≤𝑐(1+𝑡)^{−1/(𝛾−1)}𝑐(),𝛼=1,1+𝑡^{−𝜆/(𝛾−1)},0<𝛼<1. (90)
Finally, we will prove the decay rate of the density, and it holds from the Gagliardo-Nirenberg-Sobolev inequality, (23), and (30) that 𝜌^{(𝛾+2𝛼−1)/2}(𝑥,𝑡)=𝜌^{(𝛾+2𝛼−1)/2}()𝜌𝜉,𝜏≤𝐶^{(𝛾+2𝛼−1)/2}(𝜉,𝜏)_𝐿^1/2_1([0,1])𝜌^{(𝛾+2𝛼−1)/2}_𝜉(𝜉,𝜏)_𝐿^1/2_1([0,1])≤𝐶¹₀𝜌^𝛾−1𝑑𝜉^1/41₀𝜌^𝛼2_𝜉𝑑𝜉^1/4≤𝐶_𝑎(𝑡)^𝑏(𝑡)𝜌^𝛾𝑑𝑥^1/4, (91) which together with (86) gives (68).

Remark 15. We note that Lemmas 7–14 are proved by means of the methods used in [9, 13, 16, 24], and Lemma 6 is new.

4. Proof of the Main Results

Proof. The global existence of unique strong solution to the FBVP (1) and (2) can be established in terms of the short time existence carried out as in [15], the uniform a priori estimates, and the analysis of regularities, which indeed follow from Lemmas 6–13. We omit the details. The large time behavior follows from Lemma 14 directly. The proof of Theorem 1 is completed.

Conflict of Interests

The authors declare that they have no conflict of interests.

Authors’ Contribution

Both authors contributed to each part of this work equally.

Acknowledgments

The authors thank the referee for the helpful comments and suggestions on the paper. The research of Ruxu Lian is supported by NNSFC no. 11101145, China Postdoctoral Science Foundation no. 2012M520360, Doctoral Foundation of North China University of Water Sources and Electric Power no. 201032, and Innovation Scientists and Technicians Troop Construction Projects of Henan Province.