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On the free boundary value problem for one-dimensional compressible Navier-Stokes equations with constant exterior pressure

Ruxu Lian12* and Jian Liu3

Author Affiliations

1 College of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou, 450011, P.R. China

2 Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, 100029, P.R. China

3 College of Teacher Education, Quzhou University, Quzhou, 324000, P.R. China

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Boundary Value Problems 2014, 2014:93  doi:10.1186/1687-2770-2014-93


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/93


Received:8 January 2014
Accepted:2 April 2014
Published:6 May 2014

© 2014 Lian and Liu; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Abstract

In this paper, we consider the free boundary value problem (FBVP) for one-dimensional isentropic compressible Navier-Stokes equations (CNS) with density-dependent viscosity coefficient and constant exterior pressure. Under certain assumptions imposed on the initial data, the global existence and uniqueness of a strong solution to FBVP for CNS are established, in particular, the strong solution tends pointwise to a non-vacuum equilibrium state at an exponential time-rate as the time tends to infinity.

Keywords:
Navier-Stokes equations; free boundary value problem; density-dependent viscosity coefficient; exterior pressure; strong solution

1 Introduction

In the present paper, we consider the free boundary value problem to one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient for regular initial data in the case that across the free surface the stress tensor is balanced by constant exterior pressure. In general, one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M2">View MathML</a>, u and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M3">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M4">View MathML</a>) stand for the flow density, velocity and pressure, respectively, and the viscosity coefficient is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M5">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M6">View MathML</a>. Note here that the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M8">View MathML</a> in (1.1) corresponds to the viscous Saint-Venant system for shallow water.

Recently, there have been much significant progress achieved on the compressible Navier-Stokes equations with density-dependent viscosity coefficients. For instance, the mathematical derivations are obtained in the simulation of flow surface in shallow region [1,2]. The existence of solutions for the 2D shallow water equations is investigated by Bresch and Desjardins [3,4]. 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 [5-15] and the references therein. The global existence of classical solutions is shown by Mellet and Vasseur [16]. The qualitative behaviors of global solutions and dynamical asymptotics of vacuum states are also made, such as the finite time vanishing of finite vacuum or the asymptotical formation of vacuum in large time, the dynamical behaviors of vacuum boundary, the large time convergence to rarefaction wave with vacuum, and the stability of shock profile with large shock strength; refer to [17-22] and the references therein.

In addition, some important progress has been made about free boundary value problems for multi-dimensional compressible viscous Navier-Stokes equations with constant viscosity coefficients for either barotropic or heat-conducive fluids by many authors; for example, 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 (1.1) with constant viscosity coefficients are proved locally in time for either barotropic flows [23-25] or heat-conductive flows [26-28]. In the case that across the free surface the stress tensor is balanced by exterior pressure [25], surface tension [29], or both exterior pressure and surface tension [30], respectively, as the initial data is assumed to be near to a non-vacuum equilibrium state, the global existence of classical solutions with small amplitude and positive densities in fluid region to the FBVP for CNS (1.1) with constant viscosity coefficients is obtained. The global existence of classical solutions to FBVP for compressible viscous and heat-conductive fluids is also established with the stress tensor balanced by the exterior pressure and/or surface tension across the free surface; refer to [31,32] and the references therein.

In this paper, we consider the free boundary value problem (FBVP) for one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient and constant exterior pressure, and we focus on the existence, regularities and dynamical behaviors of a global strong solution, etc. As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M4">View MathML</a> we show that the free boundary value problem with regular initial data admits a unique global strong solution which tends pointwise to a non-vacuum equilibrium state at an exponential time-rate as the time tends to infinity (refer to Theorem 2.1 for details).

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

2 Main results

We are interested in the global existence and dynamics of the free boundary value problem for (1.1) with the following initial data and boundary conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M10">View MathML</a>

(2.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M12">View MathML</a> are the free boundaries defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M13">View MathML</a>

(2.2)

and the positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M14">View MathML</a> is the exterior pressure.

Without loss of generality, the total initial mass is renormalized to be one, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M15">View MathML</a>

(2.3)

Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M16">View MathML</a>

(2.4)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M17">View MathML</a>

(2.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M18">View MathML</a>.

Firstly, we consider the initial data satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M19">View MathML</a>

(2.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M21">View MathML</a> are positive constants, δ is bounded, and we also consider the other initial data satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M22">View MathML</a>

(2.7)

note that the compatibility conditions between the initial data and boundary conditions hold. Then we have the global existence and time-asymptotical behavior of a strong solution as follows.

Theorem 2.1 (FBVP)

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M4">View MathML</a>. Assume that the initial data satisfies (2.6) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M24">View MathML</a>, and satisfies (2.6) together with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M25">View MathML</a>

(2.8)

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M26">View MathML</a>or the initial data satisfies (2.7) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M24">View MathML</a>, and satisfies (2.7) together with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M28">View MathML</a>

(2.9)

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M29">View MathML</a>, whereνis a positive constant. Then there exists a unique global strong solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M30">View MathML</a>to FBVP (1.1) and (2.1) satisfying, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M31">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M32">View MathML</a>

(2.10)

with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M33">View MathML</a>being a constant independent of time.

If it further holds that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M34">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M30">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M36">View MathML</a>

(2.11)

The solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M37">View MathML</a>tends to the non-vacuum equilibrium state exponentially

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M38">View MathML</a>

(2.12)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M39">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M40">View MathML</a>are positive constants independent of time.

Remark 2.1 Theorem 2.1 holds for the one-dimensional Saint-Venant model for shallow water, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M8">View MathML</a>.

Remark 2.2 The initial constraints (2.8) and (2.9) do not always require that the perturbation of the initial data around the equilibrium state <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M43">View MathML</a> is small. Indeed, it can be large provided that the state <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M44">View MathML</a> is large enough.

3 The a priori estimates

It is convenient to make use of the Lagrange coordinates in order to establish the a priori estimates. Take the Lagrange coordinates transform

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M45">View MathML</a>

(3.1)

Since the conservation of total mass holds, the boundaries <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M12">View MathML</a> are transformed into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M48">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M49">View MathML</a>, respectively, and the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M50">View MathML</a> is transformed into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M51">View MathML</a>. FBVP (1.1) and (2.1) is reformulated into

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M52">View MathML</a>

(3.2)

where the initial data satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M53">View MathML</a>

(3.3)

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M54">View MathML</a>

(3.4)

and the consistencies between the initial data and boundary conditions hold.

Next, we deduce the a priori estimates for the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M37">View MathML</a> to FBVP (3.2).

Lemma 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M31">View MathML</a>. Under the assumptions of Theorem 2.1, it holds for any strong solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M57">View MathML</a>to FBVP (3.2) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M58">View MathML</a>

(3.5)

Proof Taking the product of (3.2)2 with u, integrating on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M59">View MathML</a> and using boundary conditions, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M60">View MathML</a>

(3.6)

which leads to (3.5) after the integration with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M61">View MathML</a>. □

Lemma 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M31">View MathML</a>. Under the assumptions of Theorem 2.1, it holds for any strong solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M57">View MathML</a>to FBVP (3.2) with the initial data satisfying (3.3) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M64">View MathML</a>

(3.7)

As the initial data satisfies (3.4), it holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M65">View MathML</a>

(3.8)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M66">View MathML</a>satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M67">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M69">View MathML</a>satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M70">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M71">View MathML</a>.

Proof Multiplying (3.2)1 by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M72">View MathML</a> leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M73">View MathML</a>

(3.9)

which gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M74">View MathML</a>

(3.10)

Summing (3.2)1 and (3.10), we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M75">View MathML</a>

(3.11)

Multiplying (3.11) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M76">View MathML</a> and integrating the result over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M77">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M78">View MathML</a>

(3.12)

Next, we prove (3.7) in the case that the initial data satisfies (3.3) firstly, to obtain (3.7). We assume a priori that there are constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M79">View MathML</a> so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M80">View MathML</a>

(3.13)

By means of (3.2)1 and the boundary condition, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M81">View MathML</a>

(3.14)

which yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M82">View MathML</a>

(3.15)

From (3.13), we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M83">View MathML</a>

(3.16)

It holds from (3.5), (3.13), (3.15) and (3.16) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M84">View MathML</a>

(3.17)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M85">View MathML</a> is a positive constant independent of time, and we assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M86">View MathML</a>

(3.18)

Using the same method we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M87">View MathML</a>

(3.19)

From (3.12), (3.17) and (3.19), we have (3.7).

Then, we prove (3.8) as the initial data satisfies (3.4), where we have the fact

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M88">View MathML</a>

(3.20)

Applying equation (3.2)1 and the boundary condition, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M89">View MathML</a>

(3.21)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M90">View MathML</a>

(3.22)

which together with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M91">View MathML</a>

(3.23)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M92">View MathML</a>

(3.24)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M93">View MathML</a>

(3.25)

gives rise to (3.8). □

Lemma 3.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M31">View MathML</a>. Under the assumptions of Theorem 2.1, it holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M95">View MathML</a>

(3.26)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M96">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M97">View MathML</a>are positive constants independent of time.

Proof Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M98">View MathML</a>

(3.27)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M99">View MathML</a>

(3.28)

It is easy to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M100">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M101">View MathML</a>. In addition, it holds as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M102">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M103">View MathML</a>

(3.29)

and as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M104">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M105">View MathML</a>

(3.30)

As the initial data satisfies (3.3), it follows from (3.5) and (3.7) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M106">View MathML</a>

(3.31)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M107">View MathML</a> and ν are positive constants independent of time, and we assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M108">View MathML</a>

(3.32)

As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M24">View MathML</a> and as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M26">View MathML</a>, from the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M111">View MathML</a>

(3.33)

we can find that there are two positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M96">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M113">View MathML</a> independent of time and choose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M114">View MathML</a>

(3.34)

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M115">View MathML</a>

(3.35)

As the initial data satisfies (3.4), it follows from (3.5) and (3.8) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M116">View MathML</a>

(3.36)

where ν is a positive constant independent of time, and we use the fact that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M117">View MathML</a>

(3.37)

which together with Young’s inequality gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M118">View MathML</a>

(3.38)

where C is a positive constant independent of time. As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M24">View MathML</a> and as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M120">View MathML</a>, by means of the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M121">View MathML</a>

(3.39)

we can obtain two positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M96">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M97">View MathML</a> independent of time such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M124">View MathML</a>

(3.40)

The proof of this lemma is completed. □

We also have the regularity estimates for the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M37">View MathML</a> to FBVP (3.2) as follows.

Lemma 3.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M31">View MathML</a>. Under the assumptions of Theorem 2.1, it holds for any strong solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M37">View MathML</a>to FBVP (3.2) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M128">View MathML</a>

(3.41)

If it is also satisfied that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M129">View MathML</a>

(3.42)

then the strong solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M37">View MathML</a>has the regularities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M131">View MathML</a>

(3.43)

Proof Multiplying (3.2)2 by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M132">View MathML</a>, integrating the result over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M51">View MathML</a> and making use of the boundary conditions, after a direct computation and recombination, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M134">View MathML</a>

(3.44)

Integrating equation (3.44) over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M135">View MathML</a>, from (3.5), (3.7), (3.8) and (3.26), it is easily verified that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M136">View MathML</a>

(3.45)

where C denotes a constant independent of time. From (3.2)2, (3.5), (3.7), (3.8) and (3.26), we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M137">View MathML</a>

(3.46)

Using (3.46), we can obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M138">View MathML</a>

(3.47)

which together with (3.2)2 implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M139">View MathML</a>

(3.48)

Differentiating (3.2)2 with respect to τ, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M140">View MathML</a>

(3.49)

Taking product between (3.49) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M141">View MathML</a>, integrating the results over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M51">View MathML</a> and using the boundary conditions (3.2)4, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M143">View MathML</a>

(3.50)

The terms on the right-hand side of (3.50) can be bounded respectively as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M144">View MathML</a>

(3.51)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M145">View MathML</a>

(3.52)

Summing (3.50)-(3.52) together and making use of (3.26) and (3.48), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M146">View MathML</a>

(3.53)

Integrating equation (3.53) over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M135">View MathML</a>, we get from (3.2)2, (3.5) and (3.26) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M148">View MathML</a>

(3.54)

which gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M149">View MathML</a>

(3.55)

which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M150">View MathML</a>, and it follows from the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M67">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M70">View MathML</a> that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M153">View MathML</a>. The proof of this lemma is completed. □

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

Lemma 3.5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M31">View MathML</a>. Under the assumptions of Theorem 2.1, it holds for any strong solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M37">View MathML</a>to FBVP (3.2) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M156">View MathML</a>

(3.56)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M39">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M40">View MathML</a>denote two positive constants independent of time.

Proof Applying (3.6) and (3.12) with modification, we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M159">View MathML</a>

(3.57)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M160">View MathML</a>

(3.58)

We have from (3.5), (3.7), (3.26), the Gagliardo-Nirenberg-Sobolev inequality and Young’s inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M161">View MathML</a>

(3.59)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M162">View MathML</a>

(3.60)

and define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M163">View MathML</a>

(3.61)

As the initial data satisfies (3.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M164">View MathML</a>

(3.62)

where C and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M165">View MathML</a> are positive constants independent of time. By (3.57)-(3.62), a complicated computation gives rise to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M166">View MathML</a>

(3.63)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M167">View MathML</a> is a positive constant independent of time. From (3.63), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M168">View MathML</a>

(3.64)

As the initial data satisfies (3.4), from (3.22)-(3.24), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M169">View MathML</a>

(3.65)

which implies that the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M170">View MathML</a> expands as t grows up, so that we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M171">View MathML</a>

(3.66)

Then it holds from (3.65) and (3.66) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M172">View MathML</a>

(3.67)

Using (3.58), after a complicated computation, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M173">View MathML</a>

(3.68)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M174">View MathML</a> is a positive constant independent of time, and we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M175">View MathML</a>

(3.69)

where C is a positive constant independent of time.

By the fact

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M176">View MathML</a>

(3.70)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M33">View MathML</a> is a constant independent of time, and the Gagliardo-Nirenberg-Sobolev inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/93/mathml/M178">View MathML</a>

(3.71)

we can deduce (3.56). □

4 Proof of the main results

Proof The global existence of a unique strong solution to FBVP (1.1) and (2.1) can be established in terms of the short time existence carried out as in [7], the uniform a priori estimates and the analysis of regularities, which indeed follow from Lemmas 3.1-3.4. We omit the details. The large time behaviors follow from Lemma 3.5 directly. The proof of Theorem 2.1 is completed. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed to each part of this work equally. All authors read and approved the final manuscript.

Acknowledgements

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, Innovation Scientists and Technicians Troop Construction Projects of Henan Province. The research of Jian Liu is supported by NNSFC No. 11326140, the Doctoral Starting up Foundation of QuZhou University No. BSYJ201314.

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