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Persistence properties of solutions to the dissipative 2-component Degasperis-Procesi system

Sen Ming1, Han Yang1* and Ls Yong2

Author Affiliations

1 School of Mathematics, Southwest Jiaotong University, Chengdu, 610031, China

2 Department of Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, China

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Boundary Value Problems 2014, 2014:108  doi:10.1186/1687-2770-2014-108


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/108


Received:29 March 2014
Accepted:28 April 2014
Published:9 May 2014

© 2014 Ming et al.; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Abstract

The persistence properties of solutions to the dissipative 2-component Degasperis-Procesi system are investigated. We find that if the initial data with their derivatives of the system exponentially decay at infinity, then the corresponding solution also exponentially decays at infinity.

MSC: 35G25, 35L15, 35Q58.

Keywords:
2-component; Degasperis-Procesi system; dissipative; persistence properties

1 Introduction

We consider the following dissipative 2-component Degasperis-Procesi system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M1">View MathML</a>

(1.1)

where λ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M2">View MathML</a> are nonnegative constants, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M4">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M5">View MathML</a>).

In system (1.1), if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M6">View MathML</a>, we get the classical Degasperis-Procesi equation [1]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M7">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M8">View MathML</a> represents the fluid velocity at time t in x direction and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M9">View MathML</a>. The nonlinear convection term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M10">View MathML</a> causes the steepening of wave form. The nonlinear dispersion effect term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M11">View MathML</a> makes the wave form spread. The Degasperis-Procesi equation has been studied in many works [2-8]. Escher et al.[2] demonstrated that there exists a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M12">View MathML</a> to (1.2) with initial value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M13">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M5">View MathML</a>). Liu and Yin [3] obtained the global existence of solutions to (1.2). They derived several wave breaking mechanisms in Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M15">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M16">View MathML</a>. Yin [4] established the local well-posedness for the Degasperis-Procesi equation with initial value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M17">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M5">View MathML</a>) on the line. In [5], the author obtained the global existence of solutions to the Degasperis-Procesi equation on the circle. The precise blow-up scenario was also derived. The global existence of strong solutions and global weak solutions to the Degasperis-Procesi equation were shown in [6,7]. Guo et al.[9] studied the dissipative Degasperis-Procesi equation,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M19">View MathML</a>

(1.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M20">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M21">View MathML</a>) is the dissipative term. They obtained the global weak solutions to (1.3). Guo [10] established the local well-posedness for (1.3), and also obtained the global existence, persistence properties and propagation speed of solutions. Wu and Yin [8] obtained the local well-posedness for (1.3), and also studied the blow-up scenarios of solutions in periodic case.

On the other hand, many researchers have studied the integrable multi-component generalizations of the Degasperis-Procesi equation [11-16]. Yan and Yin [11] investigated the 2-component Degasperis-Procesi system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M22">View MathML</a>

(1.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M3">View MathML</a>. They established the local well-posedness for system (1.4) in Besov space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M24">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M25">View MathML</a>, and also derived the precise blow-up scenarios of strong solutions in Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M26">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M5">View MathML</a>. Zhou et al.[12] investigated the traveling wave solutions to the 2-component Degasperis-Procesi system. Manwai [16] studied the self-similar solutions to the 2-component Degasperis-Procesi system. Fu and Qu [13] obtained the persistence properties of solutions to the 2-component Degasperis-Procesi system in Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M28">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M29">View MathML</a>. For system (1.4), Jin and Guo [14] studied the blow-up mechanisms and persistence properties of strong solutions.

Recently, a large amount of literature has been devoted to the study of the 2-component Camassa-Holm system [17-28]. Hu [18] studied the dissipative periodic 2-component Camassa-Holm system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M30">View MathML</a>

(1.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M21">View MathML</a>. The author not only established the local well-posedness for system (1.5) in Besov space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M32">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M33">View MathML</a>, but she also presented global existence results and the exact blow-up scenarios of strong solutions in Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M34">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M16">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M36">View MathML</a> in system (1.5), Jin and Guo [19] considered the persistence properties of solutions to the modified 2-component Camassa-Holm system. Zhu [20] considered the persistence property of solutions to the coupled Camassa-Holm system, and also established the global existence and blow-up mechanisms of solutions. Guo [21,22] studied the persistence properties and unique continuation of solutions to the 2-component Camassa-Holm system in the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M36">View MathML</a>. It was shown in [29] that the dissipative Camassa-Holm, Degasperis-Procesi, Hunter-Saxton and Novikov equations could be reduced to the non-dissipative versions by means of an exponentially time-dependent scaling. One may refer to [30-34] and the references therein for more details in this direction.

Motivated by the work in [13,20,35], we study the dissipative 2-component Degasperis-Procesi system (1.1). We note that the persistence properties of solutions to system (1.1) have not been discussed yet. The aim of this paper is to investigate the persistence properties of solutions in Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M38">View MathML</a>. The main idea of this work comes from [35].

Now we rewrite system (1.1) as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M39">View MathML</a>

(1.6)

where the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M40">View MathML</a>.

The main results are presented as follows.

Theorem 1.1Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M41">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M4">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M5">View MathML</a>. Then the Cauchy problem (1.1) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M44">View MathML</a>.

Theorem 1.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M41">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M4">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M5">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M48">View MathML</a>is the corresponding solution to system (1.1). If there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M49">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M50">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M51">View MathML</a>

uniformly on the interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M52">View MathML</a>.

Theorem 1.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M41">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M4">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M5">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M48">View MathML</a>is the corresponding solution to system (1.1). Assume the constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M57">View MathML</a>.

(1) For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M58">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M59">View MathML</a>

and there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M60">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M61">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M62">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M63">View MathML</a>

(2) For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M64">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M65">View MathML</a>

and there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M60">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M61">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M62">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M69">View MathML</a>

Theorem 1.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M70">View MathML</a>in system (1.1). Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M41">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M4">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M5">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M74">View MathML</a>is the corresponding solution to system (1.1). If there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M57">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M76">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M77">View MathML</a>

uniformly on the interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M52">View MathML</a>.

Theorem 1.5Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M79">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M4">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M81">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M48">View MathML</a>is the corresponding solution to system (1.1). If there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M49">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M84">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M85">View MathML</a>

Theorem 1.6Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M41">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M4">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M81">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M48">View MathML</a>is the corresponding solution to system (1.1). If there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M90">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M91">View MathML</a>

and there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M92">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M93">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M94">View MathML</a>

The remainder of this paper is organized as follows. In Section 2, the proofs of Theorems 1.1 and 1.2 are presented. Section 3 is devoted to the proofs of Theorems 1.3 and 1.4. The proofs of Theorems 1.5 and 1.6 are given in Section 4.

Notation We denote the norm of Lebesgue space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M95">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M96">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M97">View MathML</a>, the norm in Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M99">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M100">View MathML</a> and the norm in Besov space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M101">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M99">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M103">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M104">View MathML</a>, we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M105">View MathML</a>

2 Proofs of Theorems 1.1 and 1.2

We write the definition of Besov space. One may check [36-39] for more details.

Proposition 2.1[39]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M99">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M107">View MathML</a>. The nonhomogeneous Besov space is defined by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M108">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M109">View MathML</a>

2.1 Proof of Theorem 1.1

Using the Littlewood-Paley theory and estimates for solutions to the transport equation, one may follow similar arguments as in [11] to establish the local well-posedness for system (1.1) with some modification. Here we omit the detailed proof. For system (1.1) with initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M110">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M111">View MathML</a>), we see that the corresponding solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M112">View MathML</a>. Thus we complete the proof of Theorem 1.1.

2.2 Proof of Theorem 1.2

We denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M113">View MathML</a>

Multiplying the second equation in (1.6) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M114">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M115">View MathML</a> and integrating the resultant equation with respect to x yield

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M116">View MathML</a>

(2.1)

We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M117">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M118">View MathML</a>

(2.2)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M5">View MathML</a>, using the Sobolev embedding theorem, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M120">View MathML</a>. Applying the Gronwall inequality to (2.2) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M121">View MathML</a>

Noting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M122">View MathML</a> gives rise to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M123">View MathML</a>

and, taking the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M124">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M125">View MathML</a>

Multiplying the first equation in system (1.6) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M126">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M115">View MathML</a> and integrating the resultant equation with respect to x yield

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M128">View MathML</a>

(2.3)

Using the Holder inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M129">View MathML</a>

(2.4)

which in combination with (2.3) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M130">View MathML</a>

Using the Gronwall inequality, one derives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M131">View MathML</a>

(2.5)

Taking the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M124">View MathML</a> in (2.5), one gets

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M133">View MathML</a>

(2.6)

Differentiating the first equation in (1.6) in the variable x yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M134">View MathML</a>

(2.7)

Multiplying (2.7) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M135">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M115">View MathML</a>, integrating the resultant equation with respect to x and using

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M137">View MathML</a>

(2.8)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M138">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M139">View MathML</a>

We obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M140">View MathML</a>

(2.9)

We introduce the weight function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M141">View MathML</a> which is independent on t

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M142">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M143">View MathML</a>. It follows <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M144">View MathML</a> a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M145">View MathML</a>. Multiplying the first equation in system (1.6) and (2.7) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M141">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M147">View MathML</a>

(2.10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M148">View MathML</a>

(2.11)

Multiplying (2.10) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M149">View MathML</a> and (2.11) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M150">View MathML</a>, respectively, and integrating the resultant equation with respect to x, we also note

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M151">View MathML</a>

As in the weightless case, we estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M152">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M153">View MathML</a> step by step as the previous estimates for u and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M154">View MathML</a>. Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M155">View MathML</a>

(2.12)

Multiplying the second equation in system (1.6) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M156">View MathML</a>, one deduces

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M157">View MathML</a>

(2.13)

Multiplying (2.13) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M158">View MathML</a>, integrating the resultant equation with respect to x and using

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M159">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M160">View MathML</a>

Applying the Gronwall inequality and the Sobolev embedding theorem yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M161">View MathML</a>

Taking the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M124">View MathML</a>, one obtains

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M163">View MathML</a>

(2.14)

There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M164">View MathML</a> which depends on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M165">View MathML</a>, such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M143">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M167">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M168">View MathML</a>

(2.15)

Using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M169">View MathML</a> for all f, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M170">View MathML</a>

(2.16)

Plugging (2.15), (2.16) into (2.12) and using (2.14), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M171">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M172">View MathML</a>

(2.17)

Using the Gronwall inequality, one deduces that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M143">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M174">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M175">View MathML</a>

Finally, taking the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M176">View MathML</a>, one obtains

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M177">View MathML</a>

(2.18)

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M178">View MathML</a>

uniformly on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M52">View MathML</a>. This completes the proof of Theorem 1.2.

3 Proofs of Theorems 1.3 and 1.4

3.1 Proof of Theorem 1.3

(1) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M58">View MathML</a>, integrating the first equation in (1.6) over the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M181">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M182">View MathML</a>

(3.1)

From the assumption in Theorem 1.3, one deduces

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M183">View MathML</a>

(3.2)

It follows from Theorem 1.2 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M184">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M57">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M186">View MathML</a>

For the right side in (3.1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M187">View MathML</a>

(3.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M188">View MathML</a>. From Theorem 1.2, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M189">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M190">View MathML</a>

(3.4)

Noting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M191">View MathML</a>, if there is at least one of the equalities <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M192">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M193">View MathML</a> is valid, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M194">View MathML</a>. Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M195">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M196">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M197">View MathML</a>

which combined with the above estimates yields a contradiction. We obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M198">View MathML</a>. Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M199">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M200">View MathML</a>.

(2) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M64">View MathML</a>, similar to the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M58">View MathML</a>, one deduces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M199">View MathML</a>. Inserting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M199">View MathML</a> into the second equation in (1.1), one derives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M205">View MathML</a>

(3.5)

From (3.5), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M206">View MathML</a>. This completes the proof of case (2) in Theorem 1.3.

3.2 Proof of Theorem 1.4

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M70">View MathML</a>, integrating the first equation in system (1.6) on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M208">View MathML</a>, one obtains

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M209">View MathML</a>

(3.6)

From the assumption in Theorem 1.4 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M210">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M62">View MathML</a> and Theorem 1.2, one deduces

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M212">View MathML</a>

(3.7)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M57">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M214">View MathML</a>

For the right side in (3.6), firstly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M215">View MathML</a>

(3.8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M216">View MathML</a>. Using Theorem 1.2, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M217">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M218">View MathML</a>

(3.9)

Noting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M219">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M220">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M221">View MathML</a>

Similarly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M222">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M223">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M62">View MathML</a>. From Theorem 1.2, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M225">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M62">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M227">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M62">View MathML</a>. This completes the proof of Theorem 1.4.

4 Proofs of Theorems 1.5 and 1.6

4.1 Proof of Theorem 1.5

From the proof of Theorem 1.2, here we need to differentiate the second equation in (1.6) with variable x, and one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M229">View MathML</a>

(4.1)

Multiplying (4.1) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M230">View MathML</a> and integrating the resultant equation with respect to x, we also note

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M231">View MathML</a>

Thus, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M232">View MathML</a>

(4.2)

Taking the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M124">View MathML</a> and applying the Gronwall inequality yield

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M234">View MathML</a>

(4.3)

In order to obtain the estimates for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M235">View MathML</a>, we multiply (4.1) with the weight function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M141">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M237">View MathML</a>

(4.4)

Multiplying (4.4) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M238">View MathML</a> and integrating the resultant equation with respect to x, we note

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M239">View MathML</a>

Hence, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M240">View MathML</a>

(4.5)

Taking the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M124">View MathML</a> and using the Gronwall inequality, one obtains

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M242">View MathML</a>

(4.6)

From (4.6) and (2.17), one deduces that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M243">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M244">View MathML</a>

(4.7)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M245">View MathML</a>

Applying the Gronwall inequality to (4.7), for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M246">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M174">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M248">View MathML</a>

(4.8)

Now taking the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M176">View MathML</a> in (4.8), one obtains

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/108/mathml/M250">View MathML</a>

Using the assumption in Theorem 1.5, we complete the proof.

4.2 Proof of Theorem 1.6

The proof of Theorem 1.6 is similar to the proof of Theorem 1.3, here we omit it.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed to each part of this study equally and approved the final version of the manuscript.

Acknowledgements

The authors would like to express sincere gratitude to the anonymous referees for a number of valuable comments and suggestions. This work was partially supported by National Natural Science Foundation of P.R. China (71003082) and Fundamental Research Funds for the Central Universities (SWJTU12CX061 and SWJTU09ZT36).

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