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Blow-up phenomena and global existence for the periodic two-component Dullin-Gottwald-Holm system

Jingjing Liu1* and Dequan Zhang2

Author affiliations

1 Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, 450002, China

2 Faculty of Science, Guilin University of Aerospace Industry, Guilin, 541004, China

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Citation and License

Boundary Value Problems 2013, 2013:158  doi:10.1186/1687-2770-2013-158


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/158


Received:24 April 2013
Accepted:16 June 2013
Published:1 July 2013

© 2013 Liu and Zhang; 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

This paper is concerned with blow-up phenomena and global existence for the periodic two-component Dullin-Gottwald-Holm system. We first obtain several blow-up results and the blow-up rate of strong solutions to the system. We then present a global existence result for strong solutions to the system.

MSC: 35G25, 35L05.

Keywords:
periodic two-component Dullin-Gottwald-Holm system; blow-up; blow-up rate; global existence

1 Introduction

In this paper, we consider the following periodic two-component Dullin-Gottwald-Holm (DGH) system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M3">View MathML</a> and γ are constants.

System (1.1) has been recently derived by Zhu et al. in [1] by following Ivanov’s approach [2]. It was shown in [1] that the DGH system is completely integrable and can be written as a compatibility condition of two linear systems

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M4">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M5">View MathML</a>

where ξ is a spectral parameter. Moreover, this system has the following two Hamiltonians:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M6">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M7">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M9">View MathML</a>, (1.1) becomes the DGH equation [3]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M10">View MathML</a>

where A and α are two positive constants, modeling unidirectional propagation of surface waves on a shallow layer of water which is at rest at infinity, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M11">View MathML</a> stands for fluid velocity. It is completely integrable with a bi-Hamiltonian and a Lax pair. Moreover, its traveling wave solutions include both the KdV solitons and the CH peakons as limiting cases [3]. The Cauchy problem of the DGH equation has been extensively studied, cf.[4-13].

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M13">View MathML</a>, system (1.1) becomes the two-component Camassa-Holm system [2]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M14">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M15">View MathML</a> is in connection with the free surface elevation from scalar density (or equilibrium), and the parameter A characterizes a linear underlying shear flow. System (1.2) describes water waves in the shallow water regime with nonzero constant vorticity, where the nonzero vorticity case indicates the presence of an underlying current. A large amount of literature was devoted to the Cauchy problem (1.2); see [14-22].

The Cauchy problem (1.1) has been discussed in [1]. Therein Zhu and Xu established the local well-posedness to system (1.1), derived the precise blow-up scenario and investigated the wave breaking for it. The aim of this paper is to further study the blow-up phenomena for strong solutions to (1.1) and to present a global existence result.

Our paper is organized as follows. In Section 2, we briefly state some needed results including the local well posedness of system (1.1), the precise blow-up scenario and some useful lemmas to study blow-up phenomena and global existence. In Section 3, we give several new blow-up results and the precise blow-up rate. In Section 4, we present a new global existence result of strong solutions to (1.1).

Notation Given a Banach space Z, we denote its norm by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M16">View MathML</a>. Since all space of functions is over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M17">View MathML</a>, for simplicity, we drop <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M17">View MathML</a> in our notations if there is no ambiguity.

2 Preliminaries

In this section, we will briefly give some needed results in order to pursue our goal.

With <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M2">View MathML</a>, we can rewrite system (1.1) as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M20">View MathML</a>

(2.1)

Note that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M22">View MathML</a> is the kernel of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M23">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M24">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M26">View MathML</a>. Here we denote by ∗ the convolution. Using this identity, we can rewrite system (2.1) as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M27">View MathML</a>

(2.2)

The local well-posedness of the Cauchy problem (2.1) can be obtained by applying Kato’s theorem. As a result, we have the following well-posedness result.

Lemma 2.1[1]

Given the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M29">View MathML</a>, there exists a maximal<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M30">View MathML</a>and a unique solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M31">View MathML</a>

of (2.1). Moreover, the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M32">View MathML</a>depends continuously on the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M33">View MathML</a>and the maximal time of existence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M34">View MathML</a>is independent ofs.

Consider now the following initial value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M35">View MathML</a>

(2.3)

whereudenotes the first component of the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M32">View MathML</a>to (2.1).

Lemma 2.2[1]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M37">View MathML</a>be the solution of (2.1) with the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M29">View MathML</a>. Then Eq. (2.3) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M40">View MathML</a>. Moreover, the map<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M41">View MathML</a>is an increasing diffeomorphism ofwith

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M42">View MathML</a>

Lemma 2.3[1]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M37">View MathML</a>be the solution of (2.1) with the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M29">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M46">View MathML</a>be the maximal time of existence. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M47">View MathML</a>

Moreover, if there exists an<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M48">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M49">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M50">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M51">View MathML</a>.

Next, we give two useful conservation laws of strong solutions to (2.1).

Lemma 2.4[1]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M37">View MathML</a>be the solution of (2.1) with the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M29">View MathML</a>, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M46">View MathML</a>be the maximal time of existence. Then, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M56">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M57">View MathML</a>

Lemma 2.5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M37">View MathML</a>be the solution of (2.1) with the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M29">View MathML</a>, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M46">View MathML</a>be the maximal time of existence. Then, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M56">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M63">View MathML</a>

Proof By the first equation in (2.1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M64">View MathML</a>

This completes the proof of the lemma. □

Then we state the following precise blow-up mechanism of (2.1).

Lemma 2.6[1]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M37">View MathML</a>be the solution of (2.1) with the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M29">View MathML</a>, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M46">View MathML</a>be the maximal time of existence. Then the solution blows up in finite time if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M69">View MathML</a>

Lemma 2.7[23]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M70">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M71">View MathML</a>. Then, for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M72">View MathML</a>, there exists at least one point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M73">View MathML</a>with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M74">View MathML</a>

and the functionmis almost everywhere differentiable on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M75">View MathML</a>with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M76">View MathML</a>

Lemma 2.8[24]

(i) For every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M77">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M78">View MathML</a>

where the constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M79">View MathML</a>is sharp.

(ii) For every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M80">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M81">View MathML</a>

with the best possible constantclying within the range<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M82">View MathML</a>. Moreover, the best constantcis<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M79">View MathML</a>.

Lemma 2.9[25]

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M84">View MathML</a>is such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M85">View MathML</a>, then, for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M86">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M87">View MathML</a>

Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M88">View MathML</a>

Lemma 2.10[26]

Assume that a differentiable function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M89">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M90">View MathML</a>

(2.4)

whereC, Kare positive constants. If the initial datum<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M91">View MathML</a>, then the solution to (2.4) goes to −∞ beforettends to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M92">View MathML</a>.

3 Blow-up phenomena

In this section, we discuss the blow-up phenomena of system (2.1). Firstly, we prove that there exist strong solutions to (2.1) which do not exist globally in time.

Theorem 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M29">View MathML</a>, andTbe the maximal time of the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M32">View MathML</a>to (2.1) with the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M96">View MathML</a>. If there is some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M97">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M49">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M99">View MathML</a>

then the corresponding solution to (2.1) blows up in finite time.

Proof Applying Lemma 2.1 and a simple density argument, we only need to show that the above theorem holds for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M29">View MathML</a>. Here we assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M101">View MathML</a> to prove the above theorem.

Define now

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M102">View MathML</a>

By Lemma 2.7, we let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M103">View MathML</a> be a point where this infimum is attained. It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M104">View MathML</a>

Differentiating the first equation in (2.2) with respect to x and using the identity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M105">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M106">View MathML</a>

(3.1)

Since the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M41">View MathML</a> given by (2.3) is an increasing diffeomorphism of ℝ, there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M108">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M109">View MathML</a>. In particular, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M110">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M111">View MathML</a>, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M112">View MathML</a>. It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M113">View MathML</a>. By Lemma 2.3 and the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M49">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M115">View MathML</a>

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M116">View MathML</a>.

Valuating (3.1) at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M117">View MathML</a> and using Lemma 2.7, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M118">View MathML</a>

(3.2)

here we used the relations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M119">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M120">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M121">View MathML</a>. By Lemma 2.4 and Lemma 2.8, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M122">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M123">View MathML</a>

It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M124">View MathML</a>

(3.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M125">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M126">View MathML</a>, Lemma 2.10 implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M127">View MathML</a>

Applying Lemma 2.6, the solution blows up in finite time. □

Theorem 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M29">View MathML</a>, andTbe the maximal time of the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M32">View MathML</a>to (2.1) with the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M96">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M132">View MathML</a>. If there is some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M97">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M49">View MathML</a>and for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M86">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M136">View MathML</a>

then the corresponding solution to (2.1) blows up in finite time.

Proof By Lemma 2.5, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M137">View MathML</a>. Using Lemma 2.4 and Lemma 2.9, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M138">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M139">View MathML</a>

Following a similar proof in Theorem 3.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M140">View MathML</a>

(3.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M141">View MathML</a>. Following the same argument as in Theorem 3.1, we deduce that the solution blows up in finite time. □

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M142">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M143">View MathML</a> in Theorem 3.2, we have the following result.

Corollary 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M29">View MathML</a>, andTbe the maximal time of the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M32">View MathML</a>to (2.1) with the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M96">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M148">View MathML</a>. If there is some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M149">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M49">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M151">View MathML</a>

then the corresponding solution to (2.1) blows up in finite time.

Remark 3.1 Note that system (2.1) is variational under the transformation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M152">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M153">View MathML</a> even <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M13">View MathML</a>. Thus, we cannot get a blow-up result according to the parity of the initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M155">View MathML</a> as we usually do.

Next, we will give more insight into the blow-up mechanism for the wave-breaking solution to system (2.1), that is, the blow-up rate for strong solutions to (2.1).

Theorem 3.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M32">View MathML</a>be the solution to system (2.1) with the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M29">View MathML</a>, satisfying the assumption of Theorem 3.1, andTbe the maximal time of the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M32">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M160">View MathML</a>

Proof As mentioned earlier, here we only need to show that the above theorem holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M101">View MathML</a>.

Define now

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M162">View MathML</a>

By the proof of Theorem 3.1, there exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M163">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M164">View MathML</a>

(3.5)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M165">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M166">View MathML</a> by Theorem 3.1, there is some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M167">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M168">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M169">View MathML</a>. Since m is locally Lipschitz, it is then inferred from (3.5) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M170">View MathML</a>

(3.6)

A combination of (3.5) and (3.6) enables us to infer

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M171">View MathML</a>

(3.7)

Since m is locally Lipschitz on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M172">View MathML</a> and (3.6) holds, it is easy to check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M173">View MathML</a> is locally Lipschitz on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M174">View MathML</a>. Differentiating the relation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M175">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M176">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M177">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M173">View MathML</a> absolutely continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M174">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M180">View MathML</a>. Integrating (3.7) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M181">View MathML</a> to obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M182">View MathML</a>

that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M183">View MathML</a>

By the arbitrariness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M165">View MathML</a> the statement of Theorem 3.3 follows. □

4 Global existence

In this section, we will present a global existence result.

Theorem 4.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M29">View MathML</a>, andTbe the maximal time of the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M32">View MathML</a>to (2.1) with the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M96">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M189">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M190">View MathML</a>, then the corresponding solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M32">View MathML</a>exists globally in time.

Proof Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M192">View MathML</a>

By Lemma 2.7, we let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M103">View MathML</a> be a point where this infimum is attained. It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M194">View MathML</a>

Since the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M41">View MathML</a> given by (2.3) is an increasing diffeomorphism of ℝ, there exists an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M108">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M109">View MathML</a>.

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M198">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M199">View MathML</a>. Valuating (3.1) at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M117">View MathML</a> and using Lemma 2.7, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M201">View MathML</a>

(4.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M202">View MathML</a>. By Lemma 2.4, Lemma 2.8 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M203">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M204">View MathML</a>

By Lemmas 2.2-2.3, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M205">View MathML</a> has the same sign with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M206">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M207">View MathML</a>. Moreover, there is a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M208">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M209">View MathML</a> because of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M210">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M190">View MathML</a>. Next, we consider the following Lyapunov positive function:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M212">View MathML</a>

(4.2)

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M213">View MathML</a> in (4.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M214">View MathML</a>

Differentiating (4.2) with respect to t and using (4.1), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M215">View MathML</a>

By Gronwall’s inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M216">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M51">View MathML</a>. On the other hand,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M218">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M219">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M220">View MathML</a>. It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/158/mathml/M221">View MathML</a>

This completes the proof by using Lemma 2.6. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

This paper is the result of joint work of all authors who contributed equally to the final version of this paper. All authors read and approved the final manuscript.

Acknowledgements

The authors would like to thank the editors and the referees for their valuable suggestions to improve the quality of this paper. This research is partially supported by the Doctoral Research Foundation of Zhengzhou University of Light Industry.

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