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Global existence and nonexistence of solutions for quasilinear parabolic equation

Xianghui Xu1, Yong-Hoon Lee1 and Zhong Bo Fang2*

Author Affiliations

1 Department of Mathematics, Pusan National University, Busan, 609-735, Republic of Korea

2 School of Mathematical Sciences, Ocean University of China, Qingdao, 266100, P.R. China

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Boundary Value Problems 2014, 2014:33  doi:10.1186/1687-2770-2014-33

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/33


Received:30 October 2013
Accepted:22 January 2014
Published:7 February 2014

© 2014 Xu 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 work is concerned with the global existence and nonexistence of solutions for a quasilinear parabolic equation with null Dirichlet boundary condition. Based on the Galerkin approximation technique and the theory of a family of potential wells, we obtain the invariant sets and vacuum isolating of global solutions including critical case, and we also give global nonexistence.

MSC: 35A01, 35B06, 35B08.

Keywords:
family of potential wells; global existence; nonexistence; vacuum isolating; critical value

1 Introduction

Our main interest lies in the following quasilinear p-Laplacian parabolic equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M1">View MathML</a>

(1.1)

subject to homogeneous Dirichlet boundary and initial conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M2">View MathML</a>

(1.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M3">View MathML</a>

(1.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M5">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M6">View MathML</a>) is a bounded domain with smooth boundary, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M7">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M8">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M9">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M10">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M11">View MathML</a>. For simplicity, we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M12">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M13">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M14">View MathML</a>.

Many natural phenomena have been formulated as the nonlinear diffusive equation (1.1) such as the model of non-Newton flux in the mechanics of a fluid, the model of a population, biological species and filtration; we refer to [1,2] and the references therein. In the non-Newtonian theory, the quantity p is a characteristic of the medium. Media with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M15">View MathML</a> are called dilatant fluids, while media with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M16">View MathML</a> are called pseudoplastics. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M17">View MathML</a>, they are Newtonian fluids.

There have been a lot of results on the global existence and nonexistence of solutions for nonlinear evolution equations and many effective methods have been developed such as the compactness method, the semi-group method, continuation of local solutions, the upper-lower solution method, and the concavity method; see [3-6]. Since the potential well was introduced by Sattinger [7] in order to prove the global existence of solutions for nonlinear hyperbolic equations which do not necessarily have positive definite energy, it has become an important method to study the global existence and nonexistence of solutions for various nonlinear evolution equations. It is an effective method to get a positive definite energy through one term, controlled by the other term in the principal part of the equations [7-19]. For instance, Levine [8] investigated the initial boundary value problem of the linear heat conduction equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M18">View MathML</a>

with nonlinear boundary condition, and he obtained the global existence and nonexistence of weak solutions. Payne and Sattinger [9] investigated the initial boundary value problem of the semilinear hyperbolic equation with fully nonlinear term

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M19">View MathML</a>

and proved the weak solution blows up in finite time. They also extended the results to the initial boundary value problem of the corresponding parabolic equation. Tsutsumi [10] studied the homogeneous Dirichlet initial boundary value problem of the nonlinear parabolic equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M20">View MathML</a>

(1.4)

and he obtained the sufficient conditions of the existence of global weak solutions and the solutions blow up in finite time for the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M21">View MathML</a>. Later, Liu [11] proved the global existence of solutions of the homogeneous Dirichlet initial boundary value problem for (1.4) with critical initial conditions. Pang and Zhang [12] investigated the initial boundary value problem of the quasilinear parabolic equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M22">View MathML</a>

and they obtained the sufficient conditions as regards global existence and nonexistence of solution by using a potential well method.

However, the potential wells used in these works were defined by the same method as Sattinger [7] and their results were similar. Until Liu [20] firstly introduced the theory of a family of potential wells, described the structure of potential wells and the estimates of the depth of potential wells. And he firstly found the phenomenon of vacuum isolating of solutions for nonlinear evolution equations. The study of applications about a family of potential wells has attracted more and more attention [20-25]. For instance, Liu and Zhao [21] not only proved the global existence and nonexistence of solutions, but they also obtained the vacuum isolating of solutions of the initial boundary value problem for semilinear hyperbolic equations and parabolic equations.

As far as we know, there are fewer papers on the global existence and nonexistence of weak solutions for nonlinear parabolic equations by using the theory of a family of potential wells. In particular, for our problem (1.1)-(1.3), the analysis of the structure and depth of the potential well, the invariant sets, the vacuum isolating of global solutions, and the question of the global existence of solutions with critical initial conditions are still open. It is difficult to obtain an a priori estimate of the approximate solution for the study of the existence of global solutions by using the general Galerkin approximation method, but the theory of the potential well often makes up for the defect. The combination of the two methods can be used to solve the existence of solutions effectively. Moreover, the study of the phenomenon of vacuum isolating will be helpful for us in studying the distribution of solutions in Sobolev space. But the depth of potential well d for the problem (1.1)-(1.3) is usually very small etc. Our goal is to improve the theory of a family of potential wells for studying the global existence and nonexistence of solutions for our problem (1.1)-(1.3), including the critical case, and we further generalize the results in [10-12,20].

The outline of the paper is as follows. In Section 2, we firstly give the definition of the weak solution for problem (1.1)-(1.3), and the definition and properties of a family of potential wells. Then we prove the global existence of solutions for problem (1.1)-(1.3) by using the Galerkin approximation technique and the theory of a family of potential wells in Section 3. The invariant sets of global solutions and vacuum isolating are obtained in Section 4. Then the sufficient condition of global nonexistence of solutions is given in Section 5. Finally, we give the result of global existence with critical initial conditions.

2 Preliminaries

Due to the degeneracy of (1.1), problem (1.1)-(1.3) has no classical solutions in general. We need to give the definition of the weak solution firstly.

Definition 1 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M23">View MathML</a> is called a weak solution of problem (1.1)-(1.3) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M24">View MathML</a> if it satisfies the following conditions:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M26">View MathML</a>;

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M27">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M29">View MathML</a>;

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M30">View MathML</a>,

where T is either infinity or the limit of the existence interval of solution.

In order to study the problem (1.1)-(1.3), we also consider the auxiliary equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M31">View MathML</a>

(1.1a)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M32">View MathML</a>.

Next, we define the functionals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M34">View MathML</a>, and the potential well W as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M35">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M37">View MathML</a>.

Furthermore, for problem (1.1)-(1.3) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M38">View MathML</a>, we define the auxiliary functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M39">View MathML</a> and function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M40">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M41">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M42">View MathML</a>.

In the paper, we always assume that p and q satisfy (H):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M43">View MathML</a>

Before giving our main results, we show some preliminary lemmas which are very important in the following proofs. As for the proofs of these several lemmas, we will not repeat them again (see [11,20]).

Lemma 1 ([11], Lemma 2.2)

For any given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M45">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M46">View MathML</a>possesses the following properties:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M47">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M48">View MathML</a>;

(2) There exists a unique<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M49">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M50">View MathML</a>;

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M51">View MathML</a>, i.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M52">View MathML</a>is increasing for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M53">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M54">View MathML</a>, i.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M52">View MathML</a>is decreasing for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M56">View MathML</a>;

(4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M57">View MathML</a>.

Lemma 2 ([20], Lemmas 2.1-2.3)

The following sufficient and necessary conditions always hold:

(1) Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M58">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M59">View MathML</a>if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M60">View MathML</a>.

(2) Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M58">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M62">View MathML</a>if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M63">View MathML</a>.

(3) Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M64">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M65">View MathML</a>if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M66">View MathML</a>.

Lemma 3 ([20], Lemma 2.4)

The function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M40">View MathML</a>possesses the following properties on the interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M68">View MathML</a>:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M69">View MathML</a>;

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M40">View MathML</a>takes the maximum<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M71">View MathML</a>at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M72">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M74">View MathML</a>;

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M40">View MathML</a>is increasing on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M76">View MathML</a>and decreasing on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M77">View MathML</a>;

(4) For any given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M78">View MathML</a>, the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M79">View MathML</a>has exactly two roots<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M80">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M81">View MathML</a>.

Lemma 4 ([20], Lemma 2.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M82">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M84">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M65">View MathML</a>.

Proposition 1<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M86">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M84">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M89">View MathML</a>.

Proof The result can easily be obtained by Lemma 4 and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M90">View MathML</a> is equivalent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M89">View MathML</a>. □

Now we can define a family of potential wells as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M92">View MathML</a>

Obviously, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M93">View MathML</a>.

Remark 1 From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M94">View MathML</a>, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M59">View MathML</a> implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M96">View MathML</a>.

In the following, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M97">View MathML</a>

Obviously, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M98">View MathML</a>.

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M99">View MathML</a>, hence for any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M100">View MathML</a>, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M101">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M102">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M59">View MathML</a>. This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M104">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M105">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M106">View MathML</a>.

Lemma 5 ([20], Theorem 2.7)

Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M107">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M108">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M109">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M110">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M105">View MathML</a>are defined as the above, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M112">View MathML</a>

Lemma 6 ([20], Lemma 2.10)

Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M113">View MathML</a>for some given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M115">View MathML</a>are the two roots of the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M116">View MathML</a>, then the sign of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M39">View MathML</a>is not changed for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118">View MathML</a>.

Lemma 7 ([11], Lemma 2.8)

Letpandqsatisfy (H), then the solutions given in Theorem 1 satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M119">View MathML</a>

3 Existence of global weak solutions

In this section, we obtain the global existence of solutions for problem (1.1)-(1.3) by combining the Galerkin approximation technique and the theory of a family of potential wells.

Theorem 1Assume thatpandqsatisfy (H), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M120">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M115">View MathML</a>are the two roots of the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M123">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M124">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M125">View MathML</a>, then problem (1.1a)-(1.3) admits a global weak solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M126">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M128">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M129">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M131">View MathML</a>. Furthermore, we have

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M132">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M133">View MathML</a>;

(2) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M134">View MathML</a>, then the solution is uniquely determined by the initial function;

(3) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M135">View MathML</a>a.e. in Ω, the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M136">View MathML</a>a.e. in Ω for any fixed<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M137">View MathML</a>, hence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M126">View MathML</a>is a solution of the problem (1.1)-(1.3).

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M139">View MathML</a> be a system of base functions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M140">View MathML</a>. Construct approximate solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M141">View MathML</a> in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M142">View MathML</a>

satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M143">View MathML</a>

Multiplying (1.1a) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M144">View MathML</a>, summing over s and integrating with respect to t, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M145">View MathML</a>

(3.1)

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M124">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M147">View MathML</a>. By Lemma 6, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M148">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M149">View MathML</a>. From this and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M150">View MathML</a>, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M151">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M125">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M151">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M100">View MathML</a>. For any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M157">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M158">View MathML</a> (if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M124">View MathML</a>) or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M160">View MathML</a> (if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M125">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M105">View MathML</a> is defined in Lemma 5), thereby <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M163">View MathML</a> for sufficiently large n.

Next, we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M164">View MathML</a> for sufficiently large n and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M137">View MathML</a>. Otherwise, there must be a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M166">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M167">View MathML</a>, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M168">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M169">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M170">View MathML</a>. From (3.1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M171">View MathML</a>

hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M170">View MathML</a> is impossible. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M168">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M174">View MathML</a>, then by Lemma 4, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M175">View MathML</a>, which is also impossible. Thus from (3.1) and Lemma 2, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M176">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M177">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M137">View MathML</a> and sufficiently large n. From these and the compactness method, we can prove that problem (1.1a)-(1.3) admits a global weak solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M126">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M128">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M129">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M131">View MathML</a>.

Furthermore, by Theorem 1 in [10] we can easily get the results (1)-(3), here we omit the proofs. □

Similarly, we can get the following conclusions directly.

Corollary 1Under the conditions of Theorem 1, we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M185">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M186">View MathML</a>.

Corollary 2If the assumption<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M187">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M125">View MathML</a>is replaced by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M124">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M125">View MathML</a>, i.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M191">View MathML</a>, then the conclusion of Theorem 1 also holds.

Corollary 3If the assumption<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M192">View MathML</a>is replaced by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M124">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M125">View MathML</a>, then problem (1.1)-(1.3) admits a global weak solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M126">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M128">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M198">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M131">View MathML</a>.

4 Invariant property and vacuum isolating of global solutions

In this section, we discuss the invariance of some sets under the flow of (1.1)-(1.3) and vacuum isolating behavior of solutions for problem (1.1)-(1.3).

4.1 Invariant property of global solutions

Theorem 2Assume thatpandqsatisfy (H), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M200">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M201">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M115">View MathML</a>are the two roots of the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M79">View MathML</a>, then the following hold.

(1) All solutions of problem (1.1)-(1.3) with initial energy<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M204">View MathML</a>belong to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M205">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M149">View MathML</a>, provided that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M187">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M125">View MathML</a>.

(2) All solutions of problem (1.1)-(1.3) with initial energy<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M204">View MathML</a>belong to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M210">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118">View MathML</a>, provided that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M212">View MathML</a>.

Proof Firstly, we consider the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M213">View MathML</a>. Here we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M214">View MathML</a>.

(1) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M215">View MathML</a> be any solution of problem (1.1)-(1.3) with initial energy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M213">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M187">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M125">View MathML</a>, T be the existence time of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M215">View MathML</a>. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M220">View MathML</a>

by Theorem 1, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M151">View MathML</a>.

Next we prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M222">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M224">View MathML</a>. Otherwise, there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M225">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M226">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118">View MathML</a>, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M228">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M229">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M230">View MathML</a>. From Lemma 7

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M231">View MathML</a>

(4.1)

we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M230">View MathML</a> is impossible. On the other hand, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M228">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M234">View MathML</a>, then by Lemma 4, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M235">View MathML</a>, which contradicts (4.1).

(2) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M215">View MathML</a> be any solution of problem (1.1)-(1.3) with initial energy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M213">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M212">View MathML</a>, T be the existence time of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M215">View MathML</a>. Since the sign of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M240">View MathML</a> is not changed for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M242">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118">View MathML</a>. From this and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M244">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118">View MathML</a> we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M246">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118">View MathML</a>.

Next we prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M248">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M224">View MathML</a>. Otherwise, there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M225">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M252">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118">View MathML</a>, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M228">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M230">View MathML</a>. From (4.1) we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M230">View MathML</a> is impossible. On the other hand, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M257">View MathML</a> be the first time such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M228">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M259">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M260">View MathML</a>. From (4.1) and Lemma 2, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M261">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M260">View MathML</a>. Hence we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M263">View MathML</a>, thus by Lemma 4, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M235">View MathML</a>, which contradicts (4.1).

For the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M265">View MathML</a>, we can obtain the same results as the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M213">View MathML</a> by Lemma 4, we omit it here. □

Remark 2 Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M204">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M107">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M108">View MathML</a> are invariant under the flow of (1.1)-(1.3) for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118">View MathML</a>.

From the above Theorem 2 and Lemma 2, we can easily get the following conclusions.

Theorem 3Letpandqsatisfy (H), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M200">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M201">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M115">View MathML</a>are the two roots of the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M79">View MathML</a>, then the following hold.

(1) All solutions of problem (1.1)-(1.3) with initial energy<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M204">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M276">View MathML</a>belong to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M277">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118">View MathML</a>.

(2) All solutions of problem (1.1)-(1.3) with initial energy<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M204">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M280">View MathML</a>belong to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M281">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118">View MathML</a>.

Remark 3 Let p and q satisfy (H), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M200">View MathML</a>. Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M204">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M109">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M110">View MathML</a> are invariant under the flow of (1.1)-(1.3) for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118">View MathML</a>.

4.2 Vacuum isolating of global solutions

The result of Theorem 3 shows that for any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M288">View MathML</a>, there exists a corresponding vacuum region of solutions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M289">View MathML</a>

for the set of all solutions of problem (1.1)-(1.3) with initial energy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M290">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M204">View MathML</a>, there is no solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M292">View MathML</a> and all solutions are isolated by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M292">View MathML</a>. This phenomenon is called the phenomenon of vacuum isolating of solutions. Obviously, the vacuum region <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M292">View MathML</a> of solutions becomes bigger and bigger with decreasing of e. As the limit case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M295">View MathML</a>, we obtain the biggest vacuum region of solutions (for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M296">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M297">View MathML</a>

Theorem 4Letpandqsatisfy (H), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M200">View MathML</a>. All nontrivial solutions of problem (1.1)-(1.3) with initial energy<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M299">View MathML</a>lie outside of the ball<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M300">View MathML</a> (maybe in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M301">View MathML</a>).

Theorem 5Letpandqsatisfy (H), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M200">View MathML</a>. All nontrivial solutions of problem (1.1)-(1.3) with initial energy<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M303">View MathML</a>satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M304">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M305">View MathML</a>

Remark 4 The proofs of Theorems 4-5 are similar to Theorems 4.7-4.8 in [20], we omit them.

5 Nonexistence of global solutions

In this section, we given the sufficient condition of global nonexistence of solutions.

Theorem 6Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M126">View MathML</a>is a local solution of problem (1.1)-(1.3) on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M309">View MathML</a>, then no solution of (1.1)-(1.3) can exist on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M310">View MathML</a>when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M303">View MathML</a>.

Proof Assume for contradiction that there is a solution of (1.1)-(1.3) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M310">View MathML</a>.

Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M313">View MathML</a>

hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M314">View MathML</a>

(5.1)

Multiplying (1.1) by u and integrating over Ω, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M315">View MathML</a>

(5.2)

As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M316">View MathML</a>, we see from (5.1) and (5.2) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M317">View MathML</a>

(5.3)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M318">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M319">View MathML</a>. By the embedding theorem and (5.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M320">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M318">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M322">View MathML</a>

Hence we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M323">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M324">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M325">View MathML</a>

By setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M326">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M327">View MathML</a>

This is impossible, since the left hand side is finite and the right hand side goes to ∞ as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M328">View MathML</a>. □

6 Existence of global solution with critical initial conditions

In this section, we prove the result of global existence with critical initial conditions.

Theorem 7Assume thatpandqsatisfy (H), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M200">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M330">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M331">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M332">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M333">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M334">View MathML</a>are the two roots of the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M330">View MathML</a>, then problem (1.1)-(1.3) admits a global solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M126">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M128">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M339">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M131">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M342">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M343">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M344">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M345">View MathML</a> . Consider the initial condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M346">View MathML</a>

with the corresponding problem (1.1)-(1.3) and suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M347">View MathML</a> are two roots of the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M348">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M349">View MathML</a>, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M350">View MathML</a>, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M351">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M352">View MathML</a>. From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M353">View MathML</a> and Lemma 1, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M354">View MathML</a>

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M94">View MathML</a> implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M356">View MathML</a>. As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M357">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M358">View MathML</a> is increasing with δ, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M359">View MathML</a>.

Thus, by Theorem 1, the problem admits a global solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M141">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M361">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M362">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M363">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M364">View MathML</a>, satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M365">View MathML</a>

(6.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M366">View MathML</a>

(6.2)

i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M367">View MathML</a>

From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M368">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M369">View MathML</a>

(6.3)

Hence there exist u, ξ, and subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M370">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M371">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M372">View MathML</a>

By using the monotone operator method, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M373">View MathML</a>.

In (6.1), letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M374">View MathML</a> we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M375">View MathML</a>

On the other hand, letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M374">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M377">View MathML</a> we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M30">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M140">View MathML</a>. Also <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M380">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M381">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M382">View MathML</a>.

Furthermore, from (6.2) and Lemma 7, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M58">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M131">View MathML</a>. On the other hand, from (6.3) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M385">View MathML</a>

By Lemma 2, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M386">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M387">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M118">View MathML</a>. □

Remark 5 The invariant sets and vacuum of solutions for problem (1.1)-(1.3) with critical initial conditions also occur.

Remark 6 Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M389">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M17">View MathML</a>, Theorem 7 is still satisfied and generalizes the results of [11]. Similarly, the invariant sets and vacuum isolating of solutions also occur.

Remark 7 In fact, all the results in our paper also hold for the homogeneous Dirichlet initial boundary value problem for the more general equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/33/mathml/M391">View MathML</a>

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

Acknowledgements

The second and third authors were supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2012005767) and the National Science Foundation of Shandong Province of China (ZR2012AM018) and the Fundamental Research Funds for the Central Universities (No. 201362032), respectively. The authors would like to express their sincere gratitude to the anonymous reviewers for their insightful and constructive comments.

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