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Energy decay and blow-up of solution for a Kirchhoff equation with dynamic boundary condition

Hongwei Zhang*, Changshun Hou and Qingying Hu

Author Affiliations

Department of Mathematics, Henan University of Technology, Lianhua Street, Zhengzhou, 450001, P.R. China

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Boundary Value Problems 2013, 2013:166  doi:10.1186/1687-2770-2013-166


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/166


Received:16 November 2012
Accepted:27 June 2013
Published:12 July 2013

© 2013 Zhang 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

The energy decay and blow-up of a solution for a Kirchhoff equation with dynamic boundary condition are considered. With the help of Nakao’s inequality and a stable set, the energy decay of the solution is given. By the convexity inequality lemma and an unstable set, the sufficient condition of blow-up of the solution with negative and small positive initial energy are obtained, respectively.

1 Introduction

The aim of this article is to study the energy decay and blow-up of a solution of the following Kirchhoff-type equation with nonlinear dynamic boundary condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M1">View MathML</a>

(1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M2">View MathML</a>

(2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M3">View MathML</a>

(3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M4">View MathML</a>

(4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M5">View MathML</a>

(5)

here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M9">View MathML</a> are positive constants and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M10">View MathML</a>.

This problem is based on the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M11">View MathML</a>

(6)

which was proposed by Woinowsky-Krieger [1,2] as a model for a vibrating beam with hinged ends, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M12">View MathML</a> is the lateral displacement at the space coordinate x and time t. Equation (6) was studied by many authors such as Dickey [3], Ball Rivera [4], Tucsnak [5], Kouemou Patchen [6], Aassila [7], Oliveira and Lima [8]; Wu and Tsai [9] considered the following beam equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M13">View MathML</a>

(7)

They obtained the existence and uniqueness, as well as decay estimates, of global solutions and blow-up of solutions for the initial boundary value problem of equation (7) through various approaches and assumptive conditions. Feireisl [10] and Fitzgibbon et al.[11] showed the existence of a global attractor and an inertial manifold of equation (6) with damping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M14">View MathML</a>. Ma [12] studied the existence and decay rates for the solution of equation (6) with nonlinear boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M15">View MathML</a>

(8)

Pazoto and Menzala [13] were concerned with equation (6) with rotational inertia term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M16">View MathML</a> and nonlinear boundary condition (8). Santos et al.[14] established the existence and exponential decay of the Kirchhoff systems with nonlocal boundary condition. Guedda and Labani [15] gave the sufficient condition of the blow-up of the solution to equation (7) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M17">View MathML</a> and dynamic boundary condition. As the related problem, we mention the following:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M18">View MathML</a>

we refer the reader to [16-19].

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M19">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M20">View MathML</a>, problem (1)-(5) comes from the reference [20-22]. In this case, the model describes the weakly damped vibrations of an extensible beam whose ends are a fixed distance apart if one end is hinged while a load is attached to the other end [21]. One can find many references on problem (1)-(5) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M21">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M19">View MathML</a>, for example, Littman and Markus [23], Andrews et al.[24], Conrad and Morgul [25], Rao [26].

Dalsen [21,22] showed the exponential stability of problem (1)-(5) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M23">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M19">View MathML</a>. Park et al.[27] discussed the existence of the solution of the Kirchhoff equation with dynamic boundary conditions and boundary differential inclusion. Doronin and Larkin [28] and Gerbi and Said-Houari [29] were concerned with the wave equation with dynamic boundary conditions. Recently, Autuori and Pucci [30] studied the global nonexistence of solutions of the p-Kirchhoff system with dynamic boundary condition.

In this paper, we use the idea of references [31] to get the energy decay and blow-up of the solution for problem (1)-(5). We construct a stable set and an unstable set, which is similar to [32]. By the help of Nakao’s inequality, combining it with the stable set, we get the decay estimate. We find that the set of initial data such that the solution of problem (1)-(5) is decay, is smaller than the potential well in [32]. The blow-up properties of the solution of problem (1)-(5) with small positive initial energy and negative initial energy are obtained by using the convexity lemma [33]. These results are different from the results in [29,30].

2 Assumptions and preliminaries

In this section, we give some preliminaries which are used throughout this work.

We use the standard space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M25">View MathML</a> and the Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M27">View MathML</a> with their usual scalar products and norms. Especially, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M28">View MathML</a> denotes the norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M25">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M30">View MathML</a> the norms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M31">View MathML</a>.

We denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M32">View MathML</a>.

Lemma 2.1

(1) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M33">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M34">View MathML</a>

(9)

(2) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M35">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M36">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M37">View MathML</a>

(10)

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M38">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M39">View MathML</a>

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M40">View MathML</a>, we get the first inequality of (9). Integrating the above inequality over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M41">View MathML</a>, we get the second part of (9). From the following equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M42">View MathML</a>

and the Cauchy inequality, we can get the result of (10) with the help of (9). □

Lemma 2.2[34]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M43">View MathML</a>be a non-increasing and nonnegative function on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M44">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M45">View MathML</a>

(11)

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M46">View MathML</a>

whereC, ωare positive constants depending on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M47">View MathML</a>and other known qualities.

Lemma 2.3[33]

Suppose that a positive, twice-differentiable function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M48">View MathML</a>satisfies on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M49">View MathML</a>the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M50">View MathML</a>

(12)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M51">View MathML</a>, then there is a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M52">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M53">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M54">View MathML</a>.

A solution u of problem (1)-(5) means that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M55">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M56">View MathML</a>

and it satisfies the following identity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M57">View MathML</a>

(13)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M58">View MathML</a>.

In this paper, we always assume that a local solution exists for problem (1)-(5). In order to study the energy decay or the blow-up phenomenon of the solution of problem (1)-(5), we define the energy of the solution u of problem (1)-(5) by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M59">View MathML</a>

(14)

The initial energy is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M60">View MathML</a>

(15)

Then, after some simple computation, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M61">View MathML</a>

(16)

That is to say, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M62">View MathML</a> is a non-increasing function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M44">View MathML</a>. Moreover, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M64">View MathML</a>

(17)

We denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M65">View MathML</a>

(18)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M66">View MathML</a>

(19)

We can now define a stable set and an unstable set [31]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M67">View MathML</a>

3 Energy decay of the solution

In order to get the energy decay of the solution, we introduce the following set:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M68">View MathML</a>

(20)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M69">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M70">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M71">View MathML</a>.

Adapting the idea of Vitillaro [35], we have the following lemma.

Lemma 3.1Suppose thatuis the solution of (1)-(5), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M72">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M73">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M74">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M75">View MathML</a>, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M49">View MathML</a>.

Lemma 3.2Under the condition of Lemma 3.1 and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M8">View MathML</a>, then, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M49">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M79">View MathML</a>

(21)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M80">View MathML</a>

(22)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M81">View MathML</a>

(23)

Proof By (14) and (18), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M82">View MathML</a>

(24)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M83">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M84">View MathML</a> has the maximum at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M69">View MathML</a> and the maximum value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M86">View MathML</a>. We see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M84">View MathML</a> is increasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M88">View MathML</a>, decreasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M89">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M90">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M91">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M92">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M93">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M92">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M49">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M96">View MathML</a>.

By (24), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M97">View MathML</a>

then (21) holds since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M98">View MathML</a>. Furthermore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M99">View MathML</a>

(25)

So (22) holds. Similar to (25), the above equality becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M100">View MathML</a>

so (23) holds. □

Theorem 3.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M102">View MathML</a>, andube the solution of problem (1)-(5), then there exist two positive constantslandθindependent oftsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M103">View MathML</a>

Proof From (16), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M104">View MathML</a>

(26)

Now, for the above estimate and the mean value theorem, we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M105">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M106">View MathML</a> such that, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M107">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M108">View MathML</a>

(27)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M109">View MathML</a>

(28)

Multiplying equation (1) by u and integrating over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M110">View MathML</a>, by the boundary conditions (2) and (3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M111">View MathML</a>

(29)

Now, we estimate the terms of the right-hand side of (29). By (10), (23), (27) and the Young inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M112">View MathML</a>

(30)

It follows from (9), (10), (23), (28) and the Young inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M113">View MathML</a>

(31)

From (26), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M114">View MathML</a>

(32)

From the Young inequality, (10), (23), (26) and from the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M62">View MathML</a> is non-increasing, we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M116">View MathML</a>

(33)

By the Young inequality, (9), (10), (23), (26) and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M62">View MathML</a> is non-increasing, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M118">View MathML</a>

(34)

Substituting (30)-(34) into (29), we get the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M119">View MathML</a>

(35)

On the other hand, it follows from (22) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M120">View MathML</a>

(36)

Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M121">View MathML</a>

(37)

Therefore, by (37), (35) and (26), we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M122">View MathML</a>

(38)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M62">View MathML</a> is non-increasing, we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M124">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M125">View MathML</a>

(39)

Then, using (26), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M126">View MathML</a>, and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M62">View MathML</a> is non-increasing, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M128">View MathML</a>

(40)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M62">View MathML</a> is non-increasing, combining this with (40), (38) and (26), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M130">View MathML</a>

(41)

Choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M131">View MathML</a> sufficiently small, (41) leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M132">View MathML</a>

(42)

then, applying Lemma 2.2, we obtain the energy decay. □

4 Blow-up property

In this section, we show that the solution of problem (1)-(5) blows up in finite time if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M133">View MathML</a>.

Lemma 4.1Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M135">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M136">View MathML</a>

(43)

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M62">View MathML</a> is non-increasing, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M138">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M139">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M49">View MathML</a>. Similar to the proof of Lemma 3.2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M141">View MathML</a>

(44)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M142">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M143">View MathML</a> has the maximum at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M144">View MathML</a> and the maximum value is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M145">View MathML</a>. It is easy to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M143">View MathML</a> is increasing for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M147">View MathML</a>, decreasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M148">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M149">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M150">View MathML</a>. Therefore, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M133">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M152">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M153">View MathML</a>. By (44), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M154">View MathML</a>, which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M155">View MathML</a>. We claim that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M156">View MathML</a>

(45)

Otherwise, we suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M157">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M158">View MathML</a> and by the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M159">View MathML</a>, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M160">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M161">View MathML</a>.

Again the use of (44) leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M162">View MathML</a>

(46)

This is impossible since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M163">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M49">View MathML</a>. Hence, (45) holds. Furthermore, (43) is established since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M152">View MathML</a>. □

Theorem 4.2Suppose thatuis the local solution of problem (1)-(5), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M166">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M167">View MathML</a>, then the solutionublows up at some finite time.

Proof Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M168">View MathML</a>

(47)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M160">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M170">View MathML</a>, β are positive constants which will be fixed later (see Levine [33]). Then one finds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M171">View MathML</a>

(48)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M172">View MathML</a>

(49)

By (17) and (14), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M173">View MathML</a>

(50)

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M174">View MathML</a> and noticing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M8">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M176">View MathML</a>

(51)

By the Hölder inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M177">View MathML</a>

(52)

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M178">View MathML</a>

Then, by (48), (51), (52) and the Cauchy-Schwarz inequality, we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M179">View MathML</a>

(53)

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M160">View MathML</a> sufficiently large such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M181">View MathML</a>

(54)

Noticing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M182">View MathML</a>, by Lemma 2.3, we get the result. □

Theorem 4.3Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M12">View MathML</a>is the local solution of problem (1)-(5), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M166">View MathML</a>, and that either of the following conditions is satisfied:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M185">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M186">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M138">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M188">View MathML</a> (or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M135">View MathML</a>);

then the solutionublows up at some finite time.

Proof (i) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M185">View MathML</a>, similar to the proof of Theorem 4.2, we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M191">View MathML</a> in (51), then (53) holds. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M182">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M193">View MathML</a>, then the result holds by Lemma 2.3.

(ii) For the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M138">View MathML</a>, from (48), (49), (50) and (14), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M195">View MathML</a>

(55)

By Lemma 4.1,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M196">View MathML</a>

(56)

Combining (55) with (56), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M133">View MathML</a> and (17), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M198">View MathML</a>

(57)

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M199">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M200">View MathML</a>, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M202">View MathML</a>, then (57) can be rewritten

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/166/mathml/M203">View MathML</a>

(58)

The remainder of the proof is the same as the proof of Theorem 4.2. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Acknowledgements

We thank the referees for their valuable suggestions which helped us improve the paper so much. This work was supported by the National Natural Science Foundation of China (11171311) and the Key Science Foundation of Henan University of Technology (09XZD009).

References

  1. Woinowsky-Krieger, S: The effect of axial force on the vibration of hinged bars. J. Appl. Mech.. 17, 35–36 (1980)

  2. Ball, J: Stability theory for an extensible beam. J. Differ. Equ.. 14, 58–66 (1973)

  3. Dickey, KW: Infinite systems of nonlinear oscillation equations with linear damping. SIAM J. Appl. Math.. 7, 208–214 (1970)

  4. Munoz Rivera, JE: Global stabilization and regularizing properties on a class of nonlinear evolution equation. J. Differ. Equ.. 128, 103–124 (1996). Publisher Full Text OpenURL

  5. Tucsnak, M: Semi-internal stabilization for a nonlinear Euler-Bernoulli equation. Math. Methods Appl. Sci.. 9, 897–907 (1996)

  6. Patcheu, SK: On a global solution and asymptotic behavior for the generalized damped extensible beam equation. J. Differ. Equ.. 135(2), 123–138 (1997). PubMed Abstract OpenURL

  7. Aassila, M: Decay estimate for a quasi-linear wave equation of Kirchhoff type. Adv. Math. Sci. Appl.. 9(1), 371–381 (1999)

  8. Oliveira, ML, Lima, OA: Exponential decay of the solutions of the beams system. Nonlinear Anal.. 42, 1271–1291 (2000). Publisher Full Text OpenURL

  9. Wu, ST, Tsai, LY: Existence and nonexistence of global solutions for a nonlinear wave equation. Taiwan. J. Math.. 13B(6), 2069–2091 (2009)

  10. Feireisl, F: Exponential attractor for non-autonomous systems long-time behavior of vibrating beams. Math. Methods Appl. Sci.. 15, 287–297 (1992). Publisher Full Text OpenURL

  11. Fitzgibbon, WE, Parrott, M, You, YC: Global dynamics of coupled systems modelling non-planar beam motion. In: Ferreyra G, Goldstein GK, Neubrander F (eds.) Evolution Equation, pp. 187–189. Marcel Dekker, New York (1995)

  12. Ma, TF: Boundary stabilization for a nonlinear beam on elastic bearings. Math. Methods Appl. Sci.. 24, 583–594 (2001). Publisher Full Text OpenURL

  13. Pazoto, AF, Menzala, GP: Uniform rates of decay of a nonlinear beam with boundary dissipation. Report of LNCC/CNPq (Brazil), no 34/97, August 1997.

  14. Santos, ML, Rocha, MPC, Pereira, DC: Solvability for a nonlinear coupled system of Kirchhoff type for the beam equations with nonlocal boundary conditions. Electron. J. Qual. Theory Differ. Equ.. 6, 1–28 (2005)

  15. Guedda, M, Labani, H: Nonexistence of global solutions to a class of nonlinear wave equations with dynamic boundary conditions. Bull. Belg. Math. Sci.. 9, 39–46 (2002)

  16. Autuori, G, Pucci, P, Salvaton, MC: Asymptotic stability for nonlinear Kirchhoff systems. Nonlinear Anal., Real World Appl.. 10, 889–909 (2009). Publisher Full Text OpenURL

  17. Tawiguchi, T: Existence and asymptotic behavior of solutions to weakly damped wave equations of Kirchhoff type with nonlinear damping and source terms. J. Math. Anal. Appl.. 36, 566–578 (2010)

  18. Nakao, M: An attractor for a nonlinear dissipative wave equation of Kirchhoff type. J. Math. Anal. Appl.. 353, 652–659 (2009). Publisher Full Text OpenURL

  19. Li, FC: Global existence and blow-up of solutions for higher-order Kirchhoff-type equation with nonlinear dissipation. Appl. Math. Lett.. 17, 1409–1414 (2004). Publisher Full Text OpenURL

  20. Grobbelaar-van Dalsen, M: On the initial-boundary-value problem for the extensible beam with attached load. Math. Methods Appl. Sci.. 19, 943–957 (1996). Publisher Full Text OpenURL

  21. Grobbelaar-van Dalsen, M, Van der Merwe, A: Boundary stabilization for the extensible beam with attached load. Math. Models Methods Appl. Sci.. 9, 379–394 (1999). Publisher Full Text OpenURL

  22. Grobbelaar-van Dalsen, M: On the solvability of the boundary-value problem for the elastic beam with attached load. Math. Models Methods Appl. Sci.. 4, 89–105 (1994). Publisher Full Text OpenURL

  23. Littman, W, Markus, L: Stabilization of a hybrid system of elasticity by feedback boundary damping. Ann. Math. Pures Appl.. 152, 281–330 (1988). Publisher Full Text OpenURL

  24. Andrews, KT, Kuttler, KL, Shillor, M: Second order evolution equation with dynamic boundary conditions. J. Math. Anal. Appl.. 197, 781–795 (1996). Publisher Full Text OpenURL

  25. Conrad, F, Morgul, O: On the stabilization of a flexible beam with a tip mass. SIAM J. Control Optim.. 36, 1962–1966 (1998). Publisher Full Text OpenURL

  26. Rao, BP: Uniform stabilization of a hybrid system of elasticity. SIAM J. Control Optim.. 33, 440–454 (1995). Publisher Full Text OpenURL

  27. Park, JY, Park, SH: Solution for a hyperbolic system with boundary differential inclusion and nonlinear second-order boundary damping. Electron. J. Differ. Equ.. 80, 1–7 (2003)

  28. Doronin, GG, Larkin, NA: Global solvability for the quasi-linear damped wave equation with nonlinear second-order boundary condition. Nonlinear Anal.. 8, 1119–1134 (2002)

  29. Gerbi, S, Said-Houari, B: Local existence and exponential growth for a semi-linear damped wave equation with dynamical boundary conditions. Adv. Differ. Equ.. 13, 1051–1060 (2008)

  30. Autuori, G, Pucci, P: Kirchhoff system with dynamic boundary conditions. Nonlinear Anal.. 73, 1952–1965 (2010). Publisher Full Text OpenURL

  31. Todorova, G: Stable and unstable sets for the Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms. J. Math. Anal. Appl.. 239, 213–226 (1999). Publisher Full Text OpenURL

  32. Payne, L, Sattinger, D: Saddle points and instability on nonlinear hyperbolic equations. Isr. J. Math.. 22, 273–303 (1973)

  33. Levine, HA: Some additional remarks on the nonexistence of global solutions to nonlinear wave equations. SIAM J. Math. Anal.. 5, 138–146 (1974). Publisher Full Text OpenURL

  34. Nakao, M, Ono, K: Global existence to the Cauchy problem of the semi-linear evolution equations with a nonlinear dissipation. Funkc. Ekvacioj. 38, 417–431 (1995)

  35. Vitillaro, E: A potential well method for the wave equation with nonlinear source and boundary damping terms. Glasg. Math. J.. 44, 375–395 (2002). Publisher Full Text OpenURL