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Regularity of global solution to atmospheric circulation equations with humidity effect

Hong Luo

Author affiliations

College of Mathematics and Software Science, Sichuan Normal University, Chengdu, Sichuan, 610066, China

Citation and License

Boundary Value Problems 2012, 2012:143  doi:10.1186/1687-2770-2012-143

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/143


Received:1 June 2012
Accepted:14 November 2012
Published:5 December 2012

© 2012 Luo; 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 cited.

Abstract

In this article, the regularity of the global solutions to atmospheric circulation equations with humidity effect is considered. Firstly, the formula of the global solutions is obtained by using the theory of linear operator semigroups. Secondly, the regularity of the global solutions to atmospheric circulation equations is presented by using mathematical induction and regularity estimates for the linear semigroups.

MSC: 35D35, 35K20, 35Q35.

Keywords:
global solution; regularity; atmospheric circulation equations; humidity effect

1 Introduction

This paper is concerned with the regularity of solutions to the following initial-boundary problem of atmospheric circulation equations involving unknown functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M1">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M2">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M3">View MathML</a> is a period of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M4">View MathML</a> field <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M5">View MathML</a>):

(1.1)

(1.2)

(1.3)

(1.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M13">View MathML</a> are constants, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M14">View MathML</a>, T, q, p denote velocity field, temperature, humidity and pressure respectively, Q, G are known functions, and σ is a constant matrix

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M15">View MathML</a>

The problems (1.1)-(1.4) are supplemented with the following Dirichlet boundary condition at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M16">View MathML</a> and the periodic condition for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M17">View MathML</a>:

(1.5)

(1.6)

and initial value conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M20">View MathML</a>

(1.7)

Partial differential equations (1.1)-(1.7) are presented in atmospheric circulation with humidity effect [1]. Atmospheric circulation is one of the main factors affecting the global climate so it is very necessary to understand and master its mysteries and laws. Atmospheric circulation is an important mechanism to complete the transports and balance of atmospheric heat and moisture and the conversion between various energies. Moreover, it is also the important result of these physical transports, balance and conversion. Thus, it is of necessity to study the characteristics, formation, preservation, change and effects of the atmospheric circulation and master its evolution law, which is not only the essential part of human’s understanding of nature, but also a helpful method of changing and improving the accuracy of weather forecasts, exploring global climate change, and making effective use of climate resources.

The atmosphere and ocean around the earth are rotating geophysical fluids, which are also two important components of the climate system. The phenomena of the atmosphere and ocean are extremely rich in their organization and complexity, and a lot of them cannot be produced by laboratory experiments. The atmosphere or the ocean or the couple atmosphere and ocean can be viewed as initial and boundary value problems [2-5], or an infinite dimensional dynamical system [6-8]. We deduce atmospheric circulation models which are able to show the features of atmospheric circulation and are easy to be studied from the very complex atmospheric circulation model based on the actual background and meteorological data, and we present global solutions of atmospheric circulation equations with the use of the T-weakly continuous operator [1]. In [9], the steady state solutions to atmospheric circulation equations with humidity effect are studied. A sufficient condition of the existence of the steady state solutions to atmospheric circulation equations is obtained, and the regularity of the steady state solutions is verified. In this article, we investigate the regularity of the solutions to atmospheric circulation equations (1.1)-(1.7).

The paper is organized as follows. In Section 2, we present preliminary results. In Section 3, we present the formula of the solution to the atmospheric circulation equations. In Section 4, we obtain the regularity of the solutions to equations (1.1)-(1.7).

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M21">View MathML</a> denotes the norm of the space X, and C, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M22">View MathML</a> are variable constants.

2 Preliminaries

We consider the divergence form of the linear elliptic equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M23">View MathML</a>

(2.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M27">View MathML</a> is uniformly elliptic, i.e., there exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M28">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M29">View MathML</a>

The problem (2.1) is supplemented with the following Dirichlet boundary condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M30">View MathML</a>

(2.2)

Lemma 2.1[10] (Theory of linear elliptic equations)

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M31">View MathML</a>be a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M32">View MathML</a>field, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M36">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M37">View MathML</a>is a solution of Eqs. (2.1), (2.2), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M38">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M39">View MathML</a>depends onn, p, λ, Ω and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M40">View MathML</a>-norm or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M41">View MathML</a>-norm of the coefficient functions.

We consider the Stokes equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M42">View MathML</a>

(2.3)

Lemma 2.2[11,12] (ADN theory of the Stokes equation)

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M43">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M45">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M46">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M47">View MathML</a>) is a solution of Eq. (2.3), then the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M48">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M49">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M39">View MathML</a>depends onμ, n, k, α, Ω.

Let X be a linear space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M51">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M52">View MathML</a> be two separable reflexive Banach spaces, and H be a Hilbert space. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M51">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M52">View MathML</a>, and H are completion spaces of X under the respective norm. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M55">View MathML</a> are dense embedding. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M56">View MathML</a> is a continuous mapping. We consider the abstract equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M57">View MathML</a>

(2.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M59">View MathML</a> is unknown.

Definition 2.3 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M58">View MathML</a> be a given initial value. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M61">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M62">View MathML</a>) is called a global solution of Eq. (2.4) if u satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M63">View MathML</a>

Definition 2.4 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M64">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M65">View MathML</a> is called uniformly weak convergence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M66">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M67">View MathML</a> is bounded, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M68">View MathML</a>

(2.5)

Definition 2.5 A mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M69">View MathML</a> is called T-weakly continuous if for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M70">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M62">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M72">View MathML</a> uniformly weakly converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M73">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M74">View MathML</a>

Lemma 2.6[3]

Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M75">View MathML</a>isT-weakly continuous and satisfies:

(A1) there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M70">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M77">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M78">View MathML</a>), such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M79">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M80">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M81">View MathML</a>are constants, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M82">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M62">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M84">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M85">View MathML</a>is a seminorm of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M87">View MathML</a>,

(A2) there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M88">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M89">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M90">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M91">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M39">View MathML</a>depends only onT, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M94">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M95">View MathML</a>.

Then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M58">View MathML</a>, Eq. (2.4) has a global weak solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M97">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M98">View MathML</a> is Frechét differentiable, then the regular solution can be presented under some condition.

We introduce a space sequence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M99">View MathML</a>

where X, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M51">View MathML</a>, H are such as in Lemma 2.6, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M101">View MathML</a> is a Banach space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M102">View MathML</a> is a Hilbert space, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M103">View MathML</a> are compact including. There exist a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M104">View MathML</a> and a nonnegative function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M105">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M62">View MathML</a>) such that

(2.6)

(2.7)

Lemma 2.7In addition to the assumptions about the existence of a global solution in Lemma 2.6, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M75">View MathML</a>is Frechét differentiable and satisfies (2.6), (2.7), then Eq. (2.4) has a unique global solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M110">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M111">View MathML</a>.

Lemma 2.8[13]

LetLbe a generator of a strongly continuous semigroup<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M112">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M113">View MathML</a>is a weak solution to the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M114">View MathML</a>

(2.8)

and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M115">View MathML</a>, then the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M116">View MathML</a>can be read as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M117">View MathML</a>

Note that we used to assume that the linear operator L in (2.8) is a sectorial operator which generates an analytic semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M112">View MathML</a>. It is known that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M119">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M120">View MathML</a> generates the fractional power operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M121">View MathML</a> and fractional order spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M122">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M123">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M124">View MathML</a>. Without loss of generality, we assume that ℒ generates the fractional power operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M121">View MathML</a> and fractional order spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M122">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M127">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M128">View MathML</a> is the domain of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M129">View MathML</a>. By the semigroup theory of linear operators (Pazy [13]), we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M130">View MathML</a> is a compact inclusion for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M131">View MathML</a>.

Lemma 2.9[13-15] (Imbedding theorem of factional order spaces)

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M31">View MathML</a>be a Lipschitz field, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M133">View MathML</a>be a sectorial operator, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M134">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M135">View MathML</a>. Then for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M136">View MathML</a>, the fractional order spaces<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M137">View MathML</a>satisfy the following relations:

and the inequalities

For sectorial operators, we also have the following properties which can be found in [13].

Lemma 2.10Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M140">View MathML</a>be a sectorial operator which generates an analytic semigroup<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M141">View MathML</a>. If all eigenvaluesλofLsatisfy<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M142">View MathML</a>for some real number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M143">View MathML</a>, then for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M144">View MathML</a>, we have

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M145">View MathML</a>is bounded for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M123">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M147">View MathML</a>,

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M148">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M149">View MathML</a>,

(3) for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M147">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M151">View MathML</a>is bounded and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M152">View MathML</a>

where some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M153">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M154">View MathML</a>is a constant only depending onα,

(4) the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M122">View MathML</a>-norm can be defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M156">View MathML</a>

(2.9)

(5) ifis symmetric, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M157">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M158">View MathML</a>

3 Formula of global solutions

We introduce the spaces

Let

Then Eqs. (1.1)-(1.7) can be rewritten as an abstract equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M161">View MathML</a>

Theorem 3.1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M162">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M163">View MathML</a>, then the global solutionϕof Eqs. (1.1)-(1.7) can be read as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M164">View MathML</a>

(3.1)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M112">View MathML</a>is an analytic semigroup generated byL, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M166">View MathML</a>is a Leray projection.

Proof As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M167">View MathML</a> is a weak solution to Eqs. (1.1)-(1.7) [1], from the Hölder inequality and the Sobolev imbedding theorem, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M168">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M169">View MathML</a>. Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M170">View MathML</a>

(3.2)

From the Hölder inequality and the Sobolev imbedding theorem, we see

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M171">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M172">View MathML</a>. Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M173">View MathML</a>

(3.3)

Similarly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M174">View MathML</a>

(3.4)

According to the ADN theory and the theory of linear elliptic equations, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M175">View MathML</a>

is a sectorial operator and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M176">View MathML</a>

Therefore, L generates the analytic semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M112">View MathML</a>.

It follows from (3.2), (3.3), and (3.4) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M178">View MathML</a>

Applying Lemma 2.8 yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M179">View MathML</a>

 □

Remark 3.2 The analytic semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M112">View MathML</a> generated by L can be read as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M181">View MathML</a>

Remark 3.3 The semigroup generated by Eqs. (1.1)-(1.7) can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M182">View MathML</a>

4 Regularity of global solution

Theorem 4.1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M183">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M163">View MathML</a>, then Eqs. (1.1)-(1.7) have a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M185">View MathML</a>, and

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M62">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M188">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M189">View MathML</a>. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M190">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M191">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M192">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M193">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M194">View MathML</a>. Then (2.7) holds.

We prove (2.6).

By the interpolation inequality [16], we see

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M196">View MathML</a>

(4.1)

By the imbedding theorem of factional order spaces, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M197">View MathML</a>

(4.2)

Then it follows from (4.1) and 4.2) that

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M199">View MathML</a> is a weak solution to Eqs. (1.1)-(1.7), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M200">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M62">View MathML</a>. Then (2.6) and (2.7) hold. From Lemma 2.7, we deduce that the solution ϕ is unique and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M202">View MathML</a>

(4.3)

Multiplying (1.1) by u and integrating over Ω, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M203">View MathML</a>

Using the Young inequality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M204">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M205">View MathML</a> is a real constant satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M206">View MathML</a>. Then there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M39">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M208">View MathML</a>

Thanks to (4.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M209">View MathML</a>

(4.4)

We consider the Stokes equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M210">View MathML</a>

(4.5)

From (4.3), (4.4), and the Sobolev imbedding theorem, we find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M211">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M212">View MathML</a>. By the ADN theorem, Eq. (4.5) has a solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M213">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M214">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M215">View MathML</a>. Using the ADN theorem, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M216">View MathML</a>

(4.6)

Multiplying (1.2) by T and integrating over Ω, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M217">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M205">View MathML</a> is a constant. Then there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M39">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M220">View MathML</a>

Using (4.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M221">View MathML</a>

(4.7)

We consider the elliptic equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M222">View MathML</a>

(4.8)

It follows from (4.3), (4.7), and the Sobolev imbedding theorem that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M223">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M224">View MathML</a>. Using the theory of linear elliptic equations, Eq. (4.8) has a solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M225">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M226">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M227">View MathML</a>. Using the theory of linear elliptic equations, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M228">View MathML</a>

(4.9)

Multiplying (1.3) by q and integrating over Ω, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M229">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M205">View MathML</a> is a constant. Then there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M39">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M232">View MathML</a>

Using (4.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M233">View MathML</a>

(4.10)

We consider the elliptic equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M234">View MathML</a>

(4.11)

Using the arguments similar to those for (4.8), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M235">View MathML</a>

(4.12)

It follows from (4.6), (4.9), and (4.12) that

 □

Theorem 4.2If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M237">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M238">View MathML</a>, then Eqs. (1.1)-(1.7) have a higher regular solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M239">View MathML</a>and

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M241">View MathML</a>.

Proof We prove the theorem using mathematical induction.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M242">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M243">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M244">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M183">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M163">View MathML</a>. Using Theorem 4.1, we find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M247">View MathML</a>.

Thanks to the Sobolev imbedding theorem, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M248">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M249">View MathML</a>. We obtain

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M251">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M252">View MathML</a>.

We have from the formula (3.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M253">View MathML</a>

Then there exists α satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M88">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M255">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M256">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M257">View MathML</a> in Eq. (4.5). By the ADN theory, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M258">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M259">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M260">View MathML</a>.

We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M261">View MathML</a>

which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M262">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M263">View MathML</a> in Eq. (4.5). Using the ADN theory, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M264">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M265">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M266">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M267">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M268">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M269">View MathML</a> from the formula (3.1).

Similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M270">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M271">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M272">View MathML</a>.

We have from the formula (3.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M273">View MathML</a>

Then there exists α satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M88">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M275">View MathML</a>

which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M276">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M277">View MathML</a> in Eq. (4.8). It follows from the linear elliptic equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M278">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M279">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M280">View MathML</a>.

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M281">View MathML</a>

which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M282">View MathML</a>. We obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M283">View MathML</a> in Eq. (4.8). Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M284">View MathML</a> from the theory of linear elliptic equations. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M285">View MathML</a>. From the formula (3.1), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M286">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M287">View MathML</a>.

Similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M288">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M289">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M290">View MathML</a>.

We have from the formula (3.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M291">View MathML</a>

Then there exists α satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M88">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M293">View MathML</a>

which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M294">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M295">View MathML</a> in Eq. (4.11). Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M296">View MathML</a> from the theory of linear elliptic equations. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M297">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M298">View MathML</a>.

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M299">View MathML</a>

which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M300">View MathML</a>. We see <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M301">View MathML</a> in Eq. (4.11). Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M302">View MathML</a> from the theory of linear elliptic equations. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M303">View MathML</a>. We have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M304">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M305">View MathML</a> from the formula (3.1).

It follows from Eq. (4.5) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M306">View MathML</a>

Clearly, the right-hand side of the above equality is continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M307">View MathML</a>. Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M308">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M309">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M310">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M311">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M312">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M313">View MathML</a>. From the hypothesis of mathematical induction, we see <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M314">View MathML</a>.

By the Sobolev imbedding theorem, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M315">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M249">View MathML</a>. Then it follows from the Sobolev imbedding theorem and the interpolation inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M317">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M318">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M319">View MathML</a>.

We have from the formula (3.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M320">View MathML</a>

Then there exists α satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M88">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M322">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M323">View MathML</a>. We see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M324">View MathML</a> in Eq. (4.5). Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M325">View MathML</a> from the ADN theory. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M326">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M327">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M328">View MathML</a>

which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M329">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M330">View MathML</a> in Eq. (4.5). Using the ADN theory, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M331">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M332">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M333">View MathML</a>

From the formula (3.1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M334">View MathML</a>

Similarly,

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M336">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M337">View MathML</a>.

We have from the formula (3.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M338">View MathML</a>

Then there exists α satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M88">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M340">View MathML</a>

which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M341">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M342">View MathML</a> in Eq. (4.8). It follows from the linear elliptic equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M343">View MathML</a> that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M344">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M345">View MathML</a>. We obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M346">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M347">View MathML</a>. We have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M348">View MathML</a> in Eq. (4.8). Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M349">View MathML</a> from the theory of linear elliptic equations. Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M350">View MathML</a>

From the formula (3.1), we induce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M351">View MathML</a>

Similarly,

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M353">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M354">View MathML</a>.

We have from the formula (3.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M355">View MathML</a>

Then there exists α satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M88">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M357">View MathML</a>

which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M358">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M359">View MathML</a> in Eq. (4.11). Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M360">View MathML</a> from the theory of linear elliptic equations. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M361">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M362">View MathML</a>. Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M363">View MathML</a>

which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M364">View MathML</a>. We find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M365">View MathML</a> in Eq. (4.11). Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M366">View MathML</a> from the theory of linear elliptic equations. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M367">View MathML</a>

From the formula (3.1), we see

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M368">View MathML</a>

It follows from Eq. (4.5) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M369">View MathML</a>

Clearly, the right-hand side of the above equality is continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M370">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M371">View MathML</a>

The proof is completed. □

Since the differentiability of time and of space can be transformed into each other, we obtain

Remark 4.3 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M237">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M238">View MathML</a>, then Eqs. (1.1)-(1.7) have a higher regular solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M239">View MathML</a>, and

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M241">View MathML</a>, where l, r, α, β are positive integers satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M377">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/143/mathml/M378">View MathML</a>.

Competing interests

The author declares that he has no competing interests.

Acknowledgements

The author is very grateful to the anonymous referees whose careful reading of the manuscript and valuable comments enhanced presentation of the manuscript. The project is supported by the National Natural Science Foundation of China (11271271), the NSF of Sichuan Science and Technology Department of China (2010JY0057) and the NSF of Sichuan Education Department of China (11ZA102).

References

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