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The local strong and weak solutions to a generalized Novikov equation

Shaoyong Lai* and Meng Wu

Author Affiliations

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

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Boundary Value Problems 2013, 2013:134  doi:10.1186/1687-2770-2013-134

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/134


Received:8 December 2012
Accepted:3 May 2013
Published:20 May 2013

© 2013 Lai and Wu; 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

A nonlinear partial differential equation, which includes the Novikov equation as a special case, is investigated. The well-posedness of local strong solutions for the equation in the Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M1">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M2">View MathML</a> is established. Although the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M3">View MathML</a>-norm of the solutions to the nonlinear model does not remain constant, the existence of its local weak solutions in the lower order Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M1">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M5">View MathML</a> is established under the assumptions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M6">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M7">View MathML</a>.

MSC: 35Q35, 35Q51.

Keywords:
local strong solution; local weak solution; generalized Novikov equation

1 Introduction

Novikov [1] derived the integrable equation with cubic nonlinearities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M8">View MathML</a>

(1)

which has been investigated by many scholars. Grayshan [2] studied both the periodic and the non-periodic Cauchy problem for Eq. (1) and discussed continuity results for the data-to-solution map in the Sobolev spaces. A Galerkin-type approximation method was used in Himonas and Holliman’s paper [3] to establish the well-posedness of Eq. (1) in the Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M1">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M2">View MathML</a> on both the line and the circle. Hone et al.[4] applied the scattering theory to find non-smooth explicit soliton solutions with multiple peaks for Eq. (1). This multiple peak property is common with the Camassa-Holm and Degasperis-Procesi equations (see [5-7]). A matrix Lax pair for Eq. (1) was acquired in [8,9] and was shown to be related to a negative flow in the Sawada-Kotera hierarchy. Sufficient conditions on the initial data to guarantee the formation of singularities in finite time for Eq. (1) were given in Jiang and Li [10]. Mi and Mu [11] obtained many dynamic results for a modified Novikov equation with a peak solution. It is shown in Ni and Zhou [12] that the Novikov equation associated with the initial value is locally well-posed in Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M11">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M2">View MathML</a> by using the abstract Kato theorem. Two results about the persistence properties of the strong solution for Eq. (1) are established in [12]. Tiglay [13] proved the local well-posedness for the periodic Cauchy problem of the Novikov equation in Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M1">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M14">View MathML</a>. The orbit invariants are used to show the existence of a periodic global strong solution if the Sobolev index <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M15">View MathML</a> and a sign condition holds. For analytic initial data, the existence and uniqueness of analytic solutions for Eq. (1) are obtained in [13]. Using the Littlewood-Paley decomposition and nonhomogeneous Besov spaces, Yan et al.[14] proved that Eq. (1) is locally well-posed in the Besov space under certain assumptions. For other methods to handle the Novikov equation and the related partial differential equations, the reader is referred to [15-24] and the references therein.

We note that the coefficients of the terms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M17">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M18">View MathML</a> in the Novikov equation (1) are 4, 3 and 1, respectively. Namely, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M19">View MathML</a>. This guarantees that the conservation law of Eq. (1) holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M20">View MathML</a>

which takes a key role in obtaining various dynamic properties in the previous works.

Motivated by the desire to extend parts of local well-posedness results in [3,11,12], we study the following model:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M21">View MathML</a>

(2)

where m, a and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M22">View MathML</a> are arbitrary constants. Clearly, letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M25">View MathML</a>, Eq. (2) becomes the Novikov equation (1).

The objective of this paper is to investigate Eq. (2). Since m, a and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M22">View MathML</a> are arbitrary constants, we do not have the result that the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M3">View MathML</a> norm of the solution of Eq. (2) remains constant. We will apply the Kato theorem for abstract differential equations to prove the existence and uniqueness of local solutions for Eq. (2) subject to the initial value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M28">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M2">View MathML</a>). In addition, the existence of local weak solutions for Eq. (2) is established in the lower-order Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M1">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M31">View MathML</a> under the assumptions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M32">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M7">View MathML</a>.

The rest of this paper is organized as follows. The main results are given in Section 2. The proof of a local well-posedness result is established in Section 3, while the existence of local weak solutions is proved in Section 4.

2 Main results

Firstly, we state some notations.

The space of all infinitely differentiable functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M34">View MathML</a> with compact support in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M35">View MathML</a> is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M36">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M37">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M38">View MathML</a>) is the space of all measurable functions h such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M39">View MathML</a>. We define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M40">View MathML</a> with the standard norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M41">View MathML</a>. For any real number s, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M42">View MathML</a> denotes the Sobolev space with the norm defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M43">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M44">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M45">View MathML</a> and nonnegative number s, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M46">View MathML</a> denotes the Frechet space of all continuous <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M11">View MathML</a>-valued functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M48">View MathML</a>. We set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M49">View MathML</a>. For simplicity, throughout this article, we let c denote any positive constant which is independent of parameter ε.

Defining

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M50">View MathML</a>

and setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M51">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M52">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M53">View MathML</a>, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M54">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M56">View MathML</a> (see [17]).

We consider the Cauchy problem for Eq. (2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M57">View MathML</a>

(3)

which is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M58">View MathML</a>

(4)

Now, we give our main results for problem (3).

Theorem 1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M28">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M2">View MathML</a>. Then the Cauchy problem (3) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M61">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M45">View MathML</a>depends on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M63">View MathML</a>.

It follows from Theorem 1 that for each ε satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M52">View MathML</a>, the Cauchy problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M65">View MathML</a>

(5)

has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M66">View MathML</a>, in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M67">View MathML</a> may depend on ε. However, we shall show that under certain assumptions, there exist two constants c and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M45">View MathML</a>, both independent of ε, such that the solution of problem (5) satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M69">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M70">View MathML</a> and there exists a weak solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M71">View MathML</a> for problem (3). These results are summarized in the following two theorems.

Theorem 2If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M28">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M73">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M7">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M75">View MathML</a>be defined as in system (5). Then there exist two constantscand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M45">View MathML</a>, which are independent ofε, such that the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M77">View MathML</a>of problem (5) satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M69">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M70">View MathML</a>.

Theorem 3Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M80">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M31">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M82">View MathML</a>. Then there exists a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M45">View MathML</a>such that problem (3) has a weak solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M71">View MathML</a>in the sense of distribution and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M85">View MathML</a>.

3 Proof of Theorem 1

Consider the abstract quasi-linear evolution equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M86">View MathML</a>

(6)

Let X and Y be Hilbert spaces such that Y is continuously and densely embedded in X, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M87">View MathML</a> be a topological isomorphism. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M88">View MathML</a> be the space of all bounded linear operators from Y to X. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M89">View MathML</a>, we denote this space by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M90">View MathML</a>. We state the following conditions in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M91">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M92">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M93">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M94">View MathML</a> are constants depending only on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M95">View MathML</a>.

(I) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M96">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M97">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M98">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M99">View MathML</a> (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M100">View MathML</a> is quasi-m-accretive), uniformly on bounded sets in Y.

(II) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M101">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M102">View MathML</a> is bounded, uniformly on bounded sets in Y. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M103">View MathML</a>

(III) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M104">View MathML</a> extends to a map from X into X, is bounded on bounded sets in Y, and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M105">View MathML</a>

Kato theorem (see [25])

Assume that (I), (II) and (III) hold. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M106">View MathML</a>, there is a maximal<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M45">View MathML</a>depending only on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M108">View MathML</a>and a unique solutionvto problem (6) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M109">View MathML</a>

Moreover, the map<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M110">View MathML</a>is a continuous map fromYto the space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M111">View MathML</a>

We set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M112">View MathML</a> with constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M115">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M116">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M117">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M118">View MathML</a>. We know that Q is an isomorphism of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M11">View MathML</a> onto <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M120">View MathML</a>. In order to prove Theorem 1, we only need to check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M121">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M122">View MathML</a> satisfy assumptions (I)-(III).

Lemma 3.1The operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M123">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M124">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M2">View MathML</a>belongs to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M126">View MathML</a>.

Lemma 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M123">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M124">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M2">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M130">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M124">View MathML</a>. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M132">View MathML</a>

(7)

Lemma 3.3For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M134">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M135">View MathML</a>, it holds that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M136">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M137">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M138">View MathML</a>

(8)

The above three lemmas can be found in Ni and Zhou [12].

Lemma 3.4Letrandqbe real numbers such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M139">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M140">View MathML</a>

This lemma can be found in [25,26].

Lemma 3.5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M141">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M2">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M117">View MathML</a>. Thenfis bounded on bounded sets in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M11">View MathML</a>and satisfies

(9)

(10)

Proof Using the algebra property of the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M147">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M148">View MathML</a>, we get

(11)

which completes the proof of (9). Using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M150">View MathML</a> and the first inequality in Lemma 3.4, we have

(12)

and

(13)

Using (12) and (13) yields

(14)

which completes the proof of inequality (10). □

Proof of Theorem 1 Using the Kato theorem, Lemmas 3.1, 3.2, 3.3 and Lemma 3.5, we know that system (3) or problem (4) has a unique solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M154">View MathML</a>

 □

4 Proofs of Theorems 2 and 3

Using the first equation of system (3) derives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M155">View MathML</a>

from which we have the conservation law

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M156">View MathML</a>

(15)

Lemma 4.1 (Kato and Ponce [26])

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M157">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M158">View MathML</a>is an algebra. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M159">View MathML</a>

wherecis a constant depending only onr.

Lemma 4.2 (Kato and Ponce [26])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M160">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M161">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M162">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M163">View MathML</a>

Lemma 4.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M164">View MathML</a>and the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M165">View MathML</a>is a solution of problem (3) and the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M28">View MathML</a>. Then the following results hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M167">View MathML</a>

(16)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M168">View MathML</a>, there is a constant c only depending on m, a and b such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M169">View MathML</a>

(17)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M170">View MathML</a>, there is a constant c only depending on m, a and b such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M171">View MathML</a>

(18)

Proof Using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M172">View MathML</a>, the Gronwall inequality and (15) derives (16).

Using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M173">View MathML</a> and the Parseval equality gives rise to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M174">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M168">View MathML</a>, applying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M176">View MathML</a> to both sides of the first equation of system (3) and integrating with respect to x by parts, we have the identity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M177">View MathML</a>

(19)

We will estimate the terms on the right-hand side of (19) separately. For the first term, by using the Cauchy-Schwarz inequality and Lemmas 4.1 and 4.2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M178">View MathML</a>

(20)

Using the above estimate to the second term yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M179">View MathML</a>

(21)

Using the Cauchy-Schwarz inequality and Lemma 4.1, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M180">View MathML</a>

(22)

For the last term in (19), using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M181">View MathML</a> results in

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M182">View MathML</a>

(23)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M183">View MathML</a>, it follows from (22) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M184">View MathML</a>

(24)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M185">View MathML</a>, applying Lemma 4.1 derives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M186">View MathML</a>

(25)

It follows from (20)-(25) that there exists a constant c such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M187">View MathML</a>

(26)

Integrating both sides of the above inequality with respect to t results in inequality (17).

To estimate the norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M188">View MathML</a>, we apply the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M189">View MathML</a> to both sides of the first equation of system (3) to obtain the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M190">View MathML</a>

(27)

Applying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M191">View MathML</a> to both sides of Eq. (27) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M170">View MathML</a> gives rise to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M193">View MathML</a>

(28)

For the right-hand of Eq. (28), we have

(29)

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M195">View MathML</a>

(30)

using Lemma 4.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M196">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M197">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M198">View MathML</a>

(31)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M199">View MathML</a>

(32)

Using the Cauchy-Schwarz inequality and Lemma 4.1 yields

(33)

Applying (29)-(33) into (28) yields the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M201">View MathML</a>

(34)

for a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M202">View MathML</a>. This completes the proof of Lemma 4.3. □

Lemma 4.4 ([17])

For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M32">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M205">View MathML</a>, the following estimates hold for anyεwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M52">View MathML</a>

(35)

(36)

(37)

(38)

wherecis a constant independent ofε.

Proof of Theorem 2 Using notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M211">View MathML</a> and differentiating both sides of the first equation of problem (5) or Eq. (27) with respect to x give rise to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M212">View MathML</a>

(39)

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M213">View MathML</a> be an integer and multiplying the above equation by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M214">View MathML</a> and then integrating the resulting equation with respect to x yield the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M215">View MathML</a>

(40)

Applying the Hölder’s inequality yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M216">View MathML</a>

(41)

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M217">View MathML</a>

(42)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M218">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M219">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M220">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M221">View MathML</a>, integrating both sides of the inequality (42) with respect to t and taking the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M220">View MathML</a> result in the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M223">View MathML</a>

(43)

where c only depends on m, a, b.

Using the algebraic property of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M147">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M148">View MathML</a> and the inequality (16) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M226">View MathML</a>

(44)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M227">View MathML</a>

(45)

where c is a constant independent of ε. From (45), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M228">View MathML</a>

(46)

It follows from (43) and (46) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M229">View MathML</a>

(47)

It follows from the contraction mapping principle that there is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M45">View MathML</a> such that the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M231">View MathML</a>

has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M232">View MathML</a>. Using the theorem presented on p.51 in Li and Olver [18] yields that there are constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M45">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M202">View MathML</a>, which are independent of ε, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M235">View MathML</a> for arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M236">View MathML</a>, which leads to the conclusion of Theorem 2. □

Using Theorem 2, (17), (18) and (44), notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M237">View MathML</a> and Gronwall’s inequality results in the inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M238">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M239">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M240">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M241">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M70">View MathML</a>. It follows from Aubin’s compactness theorem that there is a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M243">View MathML</a>, denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M244">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M244">View MathML</a> and their temporal derivatives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M246">View MathML</a> are weakly convergent to a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M165">View MathML</a> and its derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M188">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M249">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M250">View MathML</a>, respectively. Moreover, for any real number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M251">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M244">View MathML</a> is convergent to the function u strongly in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M253">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M240">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M246">View MathML</a> converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M188">View MathML</a> strongly in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M257">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M258">View MathML</a>.

Proof of Theorem 3 From Theorem 2, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M259">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M260">View MathML</a>) is bounded in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M261">View MathML</a>. Thus, the sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M244">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M259">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M264">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M265">View MathML</a> are weakly convergent to u, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M266">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M267">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M268">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M269">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M270">View MathML</a>, separately. Hence, u satisfies the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M271">View MathML</a>

(48)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M272">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M273">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M274">View MathML</a> is a separable Banach space and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M259">View MathML</a> is a bounded sequence in the dual space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M276">View MathML</a> of X, there exists a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M259">View MathML</a>, still denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M259">View MathML</a>, weakly star convergent to a function v in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M279">View MathML</a>. As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M259">View MathML</a> weakly converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M266">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M282">View MathML</a>, it results that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M283">View MathML</a> almost everywhere. Thus, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M85">View MathML</a>. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The article is a joint work of two 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 are very helpful to revise 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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