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The global solution and blow-up phenomena to a modified Novikov equation

Shaoyong Lai*, Haibo Yan and Nan Li

Author Affiliations

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

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Boundary Value Problems 2014, 2014:16  doi:10.1186/1687-2770-2014-16

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/16


Received:14 August 2013
Accepted:23 December 2013
Published:15 January 2014

© 2014 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

A modified Novikov equation with symmetric coefficients is investigated. Provided that the initial value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M1">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M2">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M3">View MathML</a> does not change sign and the solution u itself belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M4">View MathML</a>, the existence and uniqueness of the global strong solutions to the equation are established in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M5">View MathML</a>. A blow-up result to the development of singularities in finite time for the equation is acquired.

MSC: 35G25, 35L05.

Keywords:
global existence; strong solutions; blow-up result

1 Introduction

Many scholars have paid attention to the integrable equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M6">View MathML</a>

(1)

which was derived by Novikov [1]. Well-posedness of the Novikov equation in the Sobolev spaces on the torus was first done by Tiglay in [2], and was completed on both the line and the circle by Himonas and Holliman in [3]. Its Hölder continuity properties were studied in Himonas and Holmes [4]. The periodic and the non-periodic Cauchy problem for Eq. (1) and continuity results for the data-to-solution map in the Sobolev spaces are discussed in Grayshan [5]. A matrix Lax pair for Eq. (1) is acquired in [6] and is shown to be related to a negative flow in the Sawada-Kotera hierarchy. The scattering theory is applied to find non-smooth explicit soliton solutions with multiple peaks for Eq. (1) in [7]. Sufficient conditions on the initial data to guarantee the formation of singularities in finite time for Eq. (1) are given in Jiang and Ni [8]. This multiple peak property is common with the Camassa-Holm and Degasperis-Procesi equations [9-11]. Mi and Mu [12] established many dynamic results for a modified Novikov equation with peak solution. It is shown in Ni and Zhou [13] that the Novikov equation associated with initial value has locally well-posedness in a Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M7">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/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 [13]. Using the Littlewood-Paley decomposition and nonhomogeneous Besov spaces, Yan et al.[14] proved the global existence and blow-up phenomena for the weakly dissipative Novikov equation. For other methods to handle the Novikov equation and the related partial differential equations, the reader is referred to [15-22] and the references therein.

Observing the coefficients of the Novikov equation (1), we see that the coefficient of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M9">View MathML</a> is equal to the coefficient of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M10">View MathML</a> plus the coefficient of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M11">View MathML</a>. That is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M12">View MathML</a>

Indeed, this relationship among the coefficients plays important roles in the study of the essential dynamical properties of the Novikov model [1,2,11-13]. This motivates us to study the following equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M13">View MathML</a>

(2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M14">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M15">View MathML</a> are arbitrary constants. Clearly, letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M16">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M17">View MathML</a>, Eq. (2) becomes the Novikov equation (1). The essential difference between Eq. (2) and the Novikov equation (1) is that Eq. (2) does not conform with the following conservation law:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M18">View MathML</a>

which results in the bounds of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M19">View MathML</a> for Eq. (1).

Making use of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M2">View MathML</a>, the assumption that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M3">View MathML</a> does not change sign, and the assumption that the solution of Eq. (2) satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M23">View MathML</a>, we prove the global existence theorem of Eq. (2) in the Sobolev space,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M24">View MathML</a>

The objective of this work is to investigate Eq. (2). Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M14">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M15">View MathML</a> are arbitrary constants, we cannot obtain the boundedness of the solution u for Eq. (2) although the initial data satisfy the sign condition. To overcome this, assuming that the solution itself satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M23">View MathML</a> and the initial data satisfy the sign condition, we adopt the methods used in Rodriguez-Blanco [16] to derive that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M28">View MathML</a> possesses bounds for any time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M29">View MathML</a>. This leads us to establish the well-posedness of the global strong solutions to Eq. (2). Parts of the main results in [17,18] are extended. In addition, we acquire a blow-up result to the development of singularities in finite time, which includes the blow-up result in [12].

The rest of this paper is organized as follows. Section 2 states the main results of this work. Section 3 proves the global existence result. The proof of a blow-up result is given in Section 4.

2 Main results

We let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M30">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M31">View MathML</a>) be the space of all measurable functions h such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M32">View MathML</a>. We define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M33">View MathML</a> with the standard norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M34">View MathML</a>. For any real number s, we let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M35">View MathML</a> denote the Sobolev space with the norm defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M36">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M37">View MathML</a>. Here we note that the norms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M39">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M40">View MathML</a> depend on variable t.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M41">View MathML</a> and nonnegative number s, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M42">View MathML</a> denotes the Frechet space of all continuous <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M7">View MathML</a>-valued functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M44">View MathML</a>. We set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M45">View MathML</a>.

In order to study the existence of solutions for Eq. (2), we consider its Cauchy problem in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M46">View MathML</a>

(3)

which is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M47">View MathML</a>

(4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M14">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M15">View MathML</a> are arbitrary constants. Now we give the main results for problem (3).

Theorem 1Assume that the solution of problem (3) satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M50">View MathML</a>and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M51">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M2">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M53">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M54">View MathML</a> (or equivalently<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M55">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M54">View MathML</a>). Then problem (3) has a unique solution satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M57">View MathML</a>

Theorem 2Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M58">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M2">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M60">View MathML</a>, then every solution of problem (3) exists globally in time. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M61">View MathML</a>, then the solution blows up in finite time if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M62">View MathML</a>becomes unbounded from below in finite time. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M63">View MathML</a>, then the solution blows up in finite time if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M62">View MathML</a>becomes unbounded from above in finite time.

3 Global strong solutions

For proving the global existence for problem (3), we cite the local well-posedness result presented in [18].

Lemma 3.1 ([18])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M58">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M2">View MathML</a>. Then the Cauchy problem (3) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M67">View MathML</a>where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M41">View MathML</a>depends on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M69">View MathML</a>.

Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M1">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M2">View MathML</a>. Then there exists a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M72">View MathML</a>to problem (3) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M73">View MathML</a>

with the maximal existence time<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M41">View MathML</a>. First, we study the differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M75">View MathML</a>

(5)

Lemma 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M77">View MathML</a>and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M41">View MathML</a>be the maximal existence time of the solution to problem (3). Then problem (5) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M79">View MathML</a>. Moreover, the map<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M80">View MathML</a>is an increasing diffeomorphism ofRwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M81">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M82">View MathML</a>.

Proof From Lemma 3.1, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M83">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M84">View MathML</a>. Thus we conclude that both functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M72">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M86">View MathML</a> are bounded, Lipschitz in space and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M87">View MathML</a> in time. Using the existence and uniqueness theorem of ordinary differential equations derives that problem (5) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M88">View MathML</a>.

Differentiating Eq. (5) with respect to x yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M89">View MathML</a>

(6)

which leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M90">View MathML</a>

(7)

For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M91">View MathML</a>, using the Sobolev imbedding theorem yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M92">View MathML</a>

It is inferred that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M93">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M94">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M82">View MathML</a>. It completes the proof. □

Lemma 3.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M76">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M77">View MathML</a>, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M41">View MathML</a>be the maximal existence time of the problem (3). We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M99">View MathML</a>

(8)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M100">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M101">View MathML</a>.

Proof Using Eqs. (2) and (6)-(8), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M102">View MathML</a>

(9)

Using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M103">View MathML</a> and solving the above equation, we complete the proof of the lemma. □

Remark 1 From Lemma 3.3, we conclude that, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M104">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M105">View MathML</a>. Since the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M106">View MathML</a> preserves positivity, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M107">View MathML</a>. Similarly, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M55">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M109">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M110">View MathML</a>.

Lemma 3.4If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M2">View MathML</a>, such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M53">View MathML</a> (or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M55">View MathML</a>) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M115">View MathML</a>, then there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M116">View MathML</a>such that the solution of problem (3) satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M117">View MathML</a>.

Proof We will prove this lemma to assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M118">View MathML</a> which results in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M119">View MathML</a> from Lemma 3.1. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M53">View MathML</a>, from Lemma 3.3, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M121">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M107">View MathML</a> does not change sign. From the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M115">View MathML</a> one derives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M124">View MathML</a>

(10)

where c is a positive constant. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M125">View MathML</a>

(11)

On the other hand, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M126">View MathML</a>

(12)

which results in

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M127">View MathML</a>

(13)

We conclude from Eqs. (11) and (13) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M117">View MathML</a>. To complete the proof, we use a simple density argument [16]. Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M129">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M130">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M131">View MathML</a>. Applying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M132">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M133">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M117">View MathML</a>. □

Using the first equation of system (3) one derives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M135">View MathML</a>

from which we have the conservation law

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M136">View MathML</a>

(14)

Lemma 3.5 (Kato and Ponce [23])

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M137">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M138">View MathML</a>is an algebra. Moreover

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M139">View MathML</a>

wherecis a constant depending only onr.

Lemma 3.6 (Kato and Ponce [23])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M140">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M141">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M142">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M143">View MathML</a>

Lemma 3.7Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M144">View MathML</a>and the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M72">View MathML</a>is a solution of problem (3) and the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M58">View MathML</a>. Then the following results hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M147">View MathML</a>

(15)

For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M148">View MathML</a>, there is a constantconly depending onaandbsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M149">View MathML</a>

(16)

Proof Using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M150">View MathML</a>, the Gronwall inequality and Eq. (14), one derives Eq. (15).

Using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M151">View MathML</a> and the Parseval equality gives rise to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M152">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M148">View MathML</a>, applying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M154">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/2014/1/16/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M155">View MathML</a>

(17)

We will estimate the terms on the right-hand side of Eq. (17) separately. For the first term, by using the Cauchy-Schwartz inequality and Lemmas 3.5 and 3.6, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M156">View MathML</a>

(18)

Using the above estimate to the second term yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M157">View MathML</a>

(19)

Using the Cauchy-Schwartz inequality and Lemma 3.5, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M158">View MathML</a>

(20)

For the last term in Eq. (17), using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M159">View MathML</a> results in

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M160">View MathML</a>

(21)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M161">View MathML</a>, it follows from Eq. (20) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M162">View MathML</a>

(22)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M163">View MathML</a>, applying Lemma 3.5 derives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M164">View MathML</a>

(23)

It follows from Eqs. (18)-(23) that there exists a constant c such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M165">View MathML</a>

(24)

Integrating both sides of the above inequality with respect to t results in inequality (16). □

Proof of Theorem 1 Using Eq. (16) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M166">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M167">View MathML</a>

(25)

Applying the Gronwall inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M168">View MathML</a>

(26)

Using Eq. (15) and Lemma 3.4, we complete the proof of Theorem 1. □

Remark 2 In fact, using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M169">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M2">View MathML</a>, Eqs. (15) and (26), we derive that the solution of Eq. (2) in space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M171">View MathML</a> blows up in finite time if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M172">View MathML</a>.

4 Proof of Theorem 2

Multiplying Eq. (2) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M173">View MathML</a> and integrating by parts, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M174">View MathML</a>

(27)

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M60">View MathML</a>, from Eq. (27), we derive <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M176">View MathML</a> is bounded. From Lemma 3.7 and Remark 2, we see that problem (3) has a global solution in the space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M177">View MathML</a>

Assume that the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M178">View MathML</a> of problem (3) blows up in finite time in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M171">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M2">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M181">View MathML</a>, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M62">View MathML</a> is bounded from below on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M183">View MathML</a>, i.e., there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M184">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M185">View MathML</a>

From Eq. (27), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M186">View MathML</a>

(28)

from which we derive that the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M187">View MathML</a> norm of the solution to problem (3) does not blow up in finite time. From Remark 2, we know that this is impossible. Therefore, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M188">View MathML</a>.

Similar to the above, we know that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M189">View MathML</a>, the solution of problem (3) blows up if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/16/mathml/M190">View MathML</a>.

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

This work is supported by both the Fundamental Research Funds for the Central Universities (JBK130401, JBK120504) and the Applied and Basic Project of Sichuan Province (2012JY0020).

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