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Global existence of solutions for 1-D nonlinear wave equation of sixth order at high initial energy level

Jihong Shen2, Yanbing Yang1 and Runzhang Xu12*

Author Affiliations

1 College of Automation, Harbin Engineering University, Harbin, 150001, People’s Republic of China

2 College of Science, Harbin Engineering University, Harbin, 150001, People’s Republic of China

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Boundary Value Problems 2014, 2014:31  doi:10.1186/1687-2770-2014-31

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/31


Received:25 September 2013
Accepted:8 January 2014
Published:4 February 2014

© 2014 Shen 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

This paper considers the Cauchy problem of solutions for a class of sixth order 1-D nonlinear wave equations at high initial energy level. By introducing a new stable set we derive the result that certain solutions with arbitrarily positive initial energy exist globally.

Keywords:
Cauchy problem; sixth order wave equation; global existence; arbitrarily positive initial energy; potential well

1 Introduction

In this paper, we consider the Cauchy problem for the following 1-D nonlinear wave equation of sixth order:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M1">View MathML</a>

(1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M2">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M5">View MathML</a> are constants, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M6">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M7">View MathML</a> are given initial data, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M8">View MathML</a> is a given constant satisfying certain conditions to be specified later.

When Rosenau [1] was concerned with the problem of how to describe the dynamics of a dense lattice, he discovered Equation (1.1) by a continuum method. Meanwhile one-dimensional homogeneous lattice wave propagation phenomena can also be described by Equation (1.1). Since then the well-posedness of Equation (1.1) have been considered by many authors, we refer the reader to [2-4] and the references therein.

Recently, the authors in [5] first considered the Cauchy problem for Equation (1.1). By the contraction mapping principle, they proved the existence and the uniqueness of the local solution for the Cauchy problem of Equation (1.1). By means of the potential well method, they discussed the existence and nonexistence of global solutions to this problem at the sub-critical and critical initial energy level <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M9">View MathML</a>. So it is natural for us to ask what the weak solution for problem (1.1)-(1.2) behaves at sup-critical initial energy level <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M10">View MathML</a>. In this paper we intend to extend the existence of global solutions in [5] with arbitrarily positive initial energy. By using the potential well method [6-9] and introducing a new stable set we show that if the initial data satisfy some conditions, then the corresponding local weak solution with arbitrarily positive initial energy exists globally.

2 Some assumptions and preliminary lemmas

In this section we give some assumptions and preliminary results to state the main results of this paper. Throughout the present paper, just for simplicity, we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M12">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M13">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M14">View MathML</a>, respectively, with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M16">View MathML</a> and the inner product <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M17">View MathML</a>.

For the Cauchy problem (1.1), (1.2) we introduce the energy functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M18">View MathML</a>

(2.1)

and the Nehari functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M19">View MathML</a>

(2.2)

Moreover we define a new stable set, which will be used to obtain the existence of a global solution with arbitrarily positive initial energy,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M20">View MathML</a>

(2.3)

We show the following local existence theorem, which has been given in [5].

Theorem 2.1[5]

Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M23">View MathML</a>. Then problem (1.1)-(1.2) admits a unique local solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M24">View MathML</a>defined on a maximal time interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M25">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M26">View MathML</a>. Moreover if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M27">View MathML</a>

then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M28">View MathML</a>.

3 The main result and proof

In order to help the readers quickly get the main idea of the present paper, we show the main theorem in the beginning of this section.

Theorem 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M30">View MathML</a>be given and let (3.1) hold. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M32">View MathML</a>and the initial data satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M33">View MathML</a>

(3.1)

Then the existence time of a global solution for problem (1.1)-(1.2) is infinite.

In what follows, we show a preliminary lemma about the monotonicity of the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M34">View MathML</a>, which will be used to prove the invariance of the new stable set under the flow of problem (1.1)-(1.2).

Lemma 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M36">View MathML</a>be given and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M24">View MathML</a>be the solution of Equation (1.1) with initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M38">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M10">View MathML</a>and the initial data satisfy Equation (3.1), then the map<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M40">View MathML</a>is strictly decreasing as long as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M41">View MathML</a>.

Proof Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M42">View MathML</a>

(3.2)

then we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M43">View MathML</a>

(3.3)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M44">View MathML</a>

(3.4)

Note that by testing Equation (1.1) with u we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M45">View MathML</a>

Then, Equation (3.4) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M46">View MathML</a>

(3.5)

Furthermore, from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M47">View MathML</a> we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M48">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M49">View MathML</a>. Obviously from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M10">View MathML</a> and (3.1) we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M51">View MathML</a>

which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M52">View MathML</a>. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M53">View MathML</a>, namely, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M54">View MathML</a>. Therefore, we find that the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M40">View MathML</a> is strictly decreasing. □

Subsequently we show the invariance of the new stable set under the flow of problem (1.1)-(1.2), which plays a key role in proving existence of global solutions for problem (1.1)-(1.2) at high initial energy level <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M10">View MathML</a>.

Lemma 3.3 (Invariant set)

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M36">View MathML</a>be given and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M24">View MathML</a>be a weak solution of problem (1.1)-(1.2) with maximal existence time interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M61">View MathML</a>. Assume that the initial data satisfy (3.1). Then all solutions of problem (1.1)-(1.2) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M10">View MathML</a>belong to, provided<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M32">View MathML</a>.

Proof We prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M47">View MathML</a>. If it is false, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M66">View MathML</a> be the first time that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M67">View MathML</a>

(3.6)

namely,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M68">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M69">View MathML</a> be defined as (3.2) above. Hence by Lemma 3.2, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M69">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M71">View MathML</a> are strictly decreasing on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M72">View MathML</a>. And then by (3.1), for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M73">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M74">View MathML</a>

Therefore from the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M75">View MathML</a> in t we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M76">View MathML</a>

(3.7)

On the other hand, by (2.1) and (2.2) we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M77">View MathML</a>

Recalling (3.6) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M78">View MathML</a>

(3.8)

Then from the following equalities:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M79">View MathML</a>

and (3.8), we can derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M80">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M81">View MathML</a>. Or equivalently

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M82">View MathML</a>

(3.9)

which contradicts the first inequality of (3.7). This completes the proof. □

At this point we can prove the global existence for the solution of problem (1.1)-(1.2) with arbitrarily positive initial energy.

Proof of Theorem 3.1 From Theorem 2.1 there exists a unique local solution of problem (1.1)-(1.2) defined on a maximal time interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M25">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M84">View MathML</a> be the weak solution of problem (1.1)-(1.2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M32">View MathML</a> and (3.1). Then from Lemma 3.3 we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M41">View MathML</a>, namely,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M88">View MathML</a>

(3.10)

Therefore from (2.1)-(2.2) and (3.6), we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M89">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M90">View MathML</a>

Hence from Theorem 2.1 it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M28">View MathML</a> and the solution of problem (1.1)-(1.2) exists globally. This completes the proof of Theorem 3.1. □

Remarks

As we all know, the global existence and finite time blow-up results of solutions for problem (1.1)-(1.2) at the sub-critical and critical initial energy level <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M9">View MathML</a> were obtained in [5], where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M93">View MathML</a> is the total initial energy and d is the depth of the introduced potential well. The present paper derives some sufficient conditions on the initial data such that the solution exists globally at the sup-critical initial energy level <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/31/mathml/M10">View MathML</a>. However, to the best of our knowledge there is no result as regards the blow-up result of solutions with arbitrary positive initial energy for the Cauchy problem (1.1)-(1.2); obviously this is an open problem. Indeed we made a try to treat this issue at the same time, however, we did not derive the invariant of an unstable set as a result of the lack of a Poincaré inequality.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The work presented here was carried out in collaboration between all authors. RZ found the motivation of this paper and suggested the outline of the proofs. JH provided many good ideas for completing this paper. YB finished the proof of the main theorem. All authors have contributed to, read and approved the manuscript.

Acknowledgements

We thank the referees for their valuable suggestions which helped us improve the paper so much. This work was supported by the National Natural Science Foundation of China (11101102), China Postdoctoral Science Foundation (2013M540270), Postdoctoral Science Foundation OF Heilongjiang Province, the Support Plan for the Young College Academic Backbone of Heilongjiang Province (1252G020), Foundational Science Foundation of Harbin Engineering University and Fundamental Research Funds for the Central Universities.

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