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The existence of global weak solutions for a weakly dissipative Camassa-Holm equation in H 1 ( R )

Shaoyong Lai1*, Nan Li1 and Yonghong Wu2

Author Affiliations

1 Department of Mathematics, Southwestern University of Finance and Economics, Chengdu, 610074, China

2 Department of Mathematics and Statistics, Curtin University, Perth, WA, 6845, Australia

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Boundary Value Problems 2013, 2013:26  doi:10.1186/1687-2770-2013-26

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/26


Received:15 November 2012
Accepted:27 January 2013
Published:12 February 2013

© 2013 Lai et al.; licensee Springer

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

Abstract

The existence of global weak solutions to the Cauchy problem for a weakly dissipative Camassa-Holm equation is established in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M2">View MathML</a> under the assumption that the initial value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M3">View MathML</a> only belongs to the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M1">View MathML</a>. The limit of viscous approximations, a one-sided super bound estimate and a space-time higher-norm estimate for the equation are established to prove the existence of the global weak solution.

MSC: 35G25, 35L05.

Keywords:
global weak solution; Camassa-Holm type equation; existence

1 Introduction

In this work, we investigate the Cauchy problem for the nonlinear model

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M5">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M8">View MathML</a> is a polynomial with order n, N and m are nonnegative integers. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M11">View MathML</a>, Eq. (1) is the standard Camassa-Holm equation [1-3]. In fact, the nonlinear term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M12">View MathML</a> can be regarded as a weakly dissipative term for the Camassa-Holm model (see [4,5]). Here we coin (1) a weakly dissipative Camassa-Holm equation.

To link with previous works, we review several works on global weak solutions for the Camassa-Holm and Degasperis-Procesi equations. The existence and uniqueness results for global weak solutions of the standard Camassa-Holm equation have been proved by Constantin and Escher [6], Constantin and Molinet [7] and Danchin [8,9] under the sign condition imposing on the initial value. Xin and Zhang [10] established the global existence of a weak solution for the Camassa-Holm equation in the energy space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M1">View MathML</a> without imposing the sign conditions on the initial value, and the uniqueness of the weak solution was obtained under certain conditions on the solution [11]. Under the sign condition for the initial value, Yin and Lai [12] proved the existence and uniqueness results of a global weak solution for a nonlinear shallow water equation, which includes the famous Camassa-Holm and Degasperis-Procesi equations as special cases. Lai and Wu [13] obtained the existence of a local weak solution for Eq. (1) in the lower-order Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M14">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M15">View MathML</a>. For other meaningful methods to handle the problems relating to dynamic properties of the Camassa-Holm equation and other partial differential equations, the reader is referred to [14-19]. Coclite et al.[20] used the analysis presented in [10,11] and investigated global weak solutions for a generalized hyperelastic-rod wave equation (or a generalized Camassa-Holm equation), namely, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M11">View MathML</a> in Eq. (1). The existence of a strongly continuous semigroup of global weak solutions for the generalized hyperelastic-rod equation with any initial value in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M1">View MathML</a> was established in [20]. Up to now, the existence result of the global weak solution for the weakly dissipative Camassa-Holm equation (1) has not been found in the literature. This constitutes the motivation of this work.

The objective of this work is to study the existence of global weak solutions for the Eq. (1) in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M19">View MathML</a> under the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M20">View MathML</a>. The key elements in our analysis include some new a priori one-sided upper bound and space-time higher-norm estimates on the first-order derivatives of the solution. Also, the limit of viscous approximations for the equation is used to establish the existence of the global weak solution. Here we should mention that the approaches used in this work come from Xin and Zhang [10] and Coclite et al.[20].

The rest of this paper is as follows. The main result is given in Section 2. In Section 3, we present a viscous problem of Eq. (1) and give a corresponding well-posedness result. An upper bound, a higher integrability estimate and some basic compactness properties for the viscous approximations are also established in Section 3. Strong compactness of the derivative of the viscous approximations is obtained in Section 4, where the main result for the existence of Eq. (1) is proved.

2 Main result

Consider the Cauchy problem for Eq. (1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M21">View MathML</a>

(2)

which is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M22">View MathML</a>

(3)

where the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M23">View MathML</a>. For a fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M24">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M25">View MathML</a>

In fact, as proved in [13], problem (2) satisfies the following conservation law:

(4)

Now we introduce the definition of a weak solution to Cauchy problem (2) or (3).

Definition 1 A continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M27">View MathML</a> is said to be a global weak solution to Cauchy problem (3) if

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M28">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M29">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M30">View MathML</a> satisfies (3) in the sense of distributions and takes on the initial value pointwise.

The main result of this paper is stated as follows.

Theorem 1 Assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M20">View MathML</a>. Then Cauchy problem (2) or (3) has a global weak solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M32">View MathML</a> in the sense of Definition 1. Furthermore, the weak solution satisfies the following properties.

(a) There exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M33">View MathML</a> depending on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M34">View MathML</a> and the coefficients of Eq. (1) such that the following one-sided <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M35">View MathML</a> norm estimate on the first-order spatial derivative holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M36">View MathML</a>

(5)

(b) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M38">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M40">View MathML</a>. Then there exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M41">View MathML</a> depending only on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M42">View MathML</a>, γ, T, a, b and the coefficients of Eq. (1) such that the following space higher integrability estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M43">View MathML</a>

(6)

3 Viscous approximations

Defining

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M44">View MathML</a>

(7)

and setting the mollifier <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M45">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M46">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M47">View MathML</a>, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M48">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M49">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M50">View MathML</a> (see Lai and Wu [13]). In fact, choosing the mollifier properly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M51">View MathML</a>

(8)

The existence of a weak solution to Cauchy problem (3) will be established by proving the compactness of a sequence of smooth functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M52">View MathML</a> solving the following viscous problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M53">View MathML</a>

(9)

Now we start our analysis by establishing the following well-posedness result for problem (9).

Lemma 3.1Provided that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M54">View MathML</a>, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M55">View MathML</a>, there exists a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M56">View MathML</a>to Cauchy problem (9). Moreover, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M57">View MathML</a>, it holds that

(10)

or

(11)

Proof For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M60">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M54">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M62">View MathML</a>. From Theorem 2.3 in [21], we conclude that problem (9) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M63">View MathML</a> for an arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M64">View MathML</a>.

We know that the first equation in system (9) is equivalent to the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M65">View MathML</a>

(12)

from which we derive that

(13)

which completes the proof. □

From Lemma 3.1 and (8), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M67">View MathML</a>

(14)

Differentiating the first equation of problem (9) with respect to x and writing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M68">View MathML</a>, we obtain

(15)

Lemma 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M38">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M40">View MathML</a>. Then there exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M41">View MathML</a>depending only on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M34">View MathML</a>, γ, T, a, band the coefficients of Eq. (1), but independent of ε, such that the space higher integrability estimate holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M76">View MathML</a>

(16)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M77">View MathML</a>is the unique solution of problem (9).

The proof is similar to that of Proposition 3.2 presented in Xin and Zhang [10] (also see Coclite et al.[20]). Here we omit it.

Lemma 3.3There exists a positive constantCdepending only on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M78">View MathML</a>and the coefficients of Eq. (1) such that

(17)

(18)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M77">View MathML</a>is the unique solution of system (9).

Due to strong similarities with the proof of Lemma 5.1 presented in Coclite et al.[20], we do not prove Lemma 3.3 here.

Lemma 3.4Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M77">View MathML</a>is the unique solution of (9). For an arbitrary<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M38">View MathML</a>, there exists a positive constantCdepending only on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M42">View MathML</a>and the coefficients of Eq. (1) such that the following one-sided<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M35">View MathML</a>norm estimate on the first-order spatial derivative holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M86">View MathML</a>

(19)

Proof From (15) and Lemma 3.3, we know that there exists a positive constant C depending only on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M42">View MathML</a> and the coefficients of Eq. (1) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M88">View MathML</a>. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M89">View MathML</a>

(20)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M90">View MathML</a> be the solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M91">View MathML</a>

(21)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M92">View MathML</a> is the value of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M93">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M94">View MathML</a>. From the comparison principle for parabolic equations, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M95">View MathML</a>

(22)

Using (14) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M96">View MathML</a>, we derive that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M97">View MathML</a>

(23)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M98">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M99">View MathML</a>. Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M100">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M101">View MathML</a>

(24)

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M102">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M103">View MathML</a>. From the comparison principle for ordinary differential equations, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M104">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M57">View MathML</a>. Therefore, by this and (22), the estimate (19) is proved. □

Lemma 3.5For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M106">View MathML</a>, there exists a sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M107">View MathML</a>tending to zero and a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M108">View MathML</a>such that, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M109">View MathML</a>, it holds that

(25)

(26)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M77">View MathML</a>is the unique solution of (9).

Lemma 3.6There exists a sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M107">View MathML</a>tending to zero and a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M114">View MathML</a>such that for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M115">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M116">View MathML</a>

(27)

The proofs of Lemmas 3.5 and 3.6 are similar to those of Lemmas 5.2 and 5.3 in [20]. Here we omit their proofs.

Throughout this paper, we use overbars to denote weak limits (the space in which these weak limits are taken is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M117">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M118">View MathML</a>).

Lemma 3.7There exists a sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M107">View MathML</a>tending to zero and two functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M121">View MathML</a>such that

(28)

(29)

for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M124">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M118">View MathML</a>. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M126">View MathML</a>

(30)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M127">View MathML</a>

(31)

Proof (28) and (29) are a direct consequence of Lemmas 3.1 and 3.2. Inequality (30) is valid because of the weak convergence in (29). Finally, (31) is a consequence of the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M128">View MathML</a>, Lemma 3.5 and (28). □

In the following, for notational convenience, we replace the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M129">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M130">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M131">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M132">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M133">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M134">View MathML</a>, respectively.

Using (28), we conclude that for any convex function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M135">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M136">View MathML</a> being bounded and Lipschitz continuous on R and for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M124">View MathML</a>, we get

(32)

(33)

Multiplying Eq. (15) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M140">View MathML</a> yields

(34)

Lemma 3.8For any convex<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M135">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M136">View MathML</a>being bounded and Lipschitz continuous on R, it holds that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M144">View MathML</a>

(35)

in the sense of distributions on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M145">View MathML</a>. Here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M146">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M147">View MathML</a>denote the weak limits of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M148">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M149">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M117">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M151">View MathML</a>, respectively.

Proof In (34), by the convexity of η, (14), Lemmas 3.5, 3.6 and 3.7, taking limit for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M152">View MathML</a> gives rise to the desired result. □

Remark 3.9 From (28) and (29), we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M153">View MathML</a>

(36)

almost everywhere in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M145">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M155">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M156">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M157">View MathML</a>. From Lemma 3.4 and (28), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M158">View MathML</a>

(37)

where C is a constant depending only on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M42">View MathML</a> and the coefficients of Eq. (1).

Lemma 3.10In the sense of distributions on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M145">View MathML</a>, it holds that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M161">View MathML</a>

(38)

Proof Using (15), Lemmas 3.5 and 3.6, (28), (29) and (31), the conclusion (38) holds by taking limit for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M162">View MathML</a> in (15). □

The next lemma contains a generalized formulation of (38).

Lemma 3.11For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M135">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M164">View MathML</a>, it holds that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M165">View MathML</a>

(39)

in the sense of distributions on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M145">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M167">View MathML</a> be a family of mollifiers defined on R. Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M168">View MathML</a>, where the ⋆ is the convolution with respect to x variable. Multiplying (38) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M169">View MathML</a> yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M170">View MathML</a>

(40)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M171">View MathML</a>

(41)

Using the boundedness of η, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M136">View MathML</a> and letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M173">View MathML</a> in the above two equations, we obtain (39). □

4 Strong convergence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M128">View MathML</a>

Now, we will prove the strong convergence result, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M175">View MathML</a>

(42)

which is one of key statements to derive that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M32">View MathML</a> is a global weak solution required in Theorem 1.

Lemma 4.1Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M54">View MathML</a>. It holds that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M178">View MathML</a>

(43)

Lemma 4.2If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M54">View MathML</a>, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M180">View MathML</a>, it holds that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M181">View MathML</a>

(44)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M182">View MathML</a>

(45)

and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M183">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M184">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M157">View MathML</a>.

Lemma 4.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M180">View MathML</a>. Then for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M157">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M188">View MathML</a>

(46)

The proofs of Lemmas 4.1, 4.2 and 4.3 can be found in [10] or [20].

Lemma 4.4Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M54">View MathML</a>. Then for almost all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M57">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M191">View MathML</a>

(47)

Lemma 4.5For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M180">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M54">View MathML</a>, it holds that

(48)

We do not provide the proofs of Lemmas 4.4 and 4.5 since they are similar to those of Lemmas 6.4 and 6.5 in Coclite et al.[20].

Lemma 4.6Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M54">View MathML</a>. Then it has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M197">View MathML</a>

(49)

Proof

Applying Lemmas 4.4 and 4.5 gives rise to

(50)

From Lemma 3.6, we know that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M199">View MathML</a>, depending only on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M78">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M201">View MathML</a>

(51)

By Remark 3.9 and Lemma 4.3, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M202">View MathML</a>

(52)

Thus, by the convexity of the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M203">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M204">View MathML</a>

(53)

Using (51) derives

(54)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M206">View MathML</a> is concave, choosing M large enough, we have

(55)

Then, from (50) and (55), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M208">View MathML</a>

(56)

By using the Gronwall inequality, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M57">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M210">View MathML</a>

By the Fatou lemma, Remark 3.9 and (30), letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M211">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M212">View MathML</a>

(57)

which completes the proof. □

Proof of the main result Using (8), (10) and Lemma 3.5, we know that the conditions (i) and (ii) in Definition 1 are satisfied. We have to verify (iii). Due to Lemma 4.2 and Lemma 4.6, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M213">View MathML</a>

(58)

From Lemma 3.5, (27) and (58), we know that u is a distributional solution to problem (3). In addition, inequalities (5) and (6) are deduced from Lemmas 3.2 and 3.4. The proof of Theorem 1 is completed. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The article is a joint work of three authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.

Acknowledgements

Thanks are given to referees whose comments and suggestions were very helpful for revising our paper. This work is supported by both the Fundamental Research Funds for the Central Universities (JBK120504) and the Applied and Basic Project of Sichuan Province (2012JY0020).

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