Open Access Research

General decay for a system of nonlinear viscoelastic wave equations with weak damping

Baowei Feng1*, Yuming Qin2 and Ming Zhang1

Author affiliations

1 College of Information Science and Technology, Donghua University, Shanghai, 201620, P.R. China

2 Department of Applied Mathematics, Donghua University, Shanghai, 201620, P.R. China

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Citation and License

Boundary Value Problems 2012, 2012:146  doi:10.1186/1687-2770-2012-146


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/146


Received:19 August 2012
Accepted:26 November 2012
Published:13 December 2012

© 2012 Feng 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

In this paper, we are concerned with a system of nonlinear viscoelastic wave equations with initial and Dirichlet boundary conditions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M1">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M2">View MathML</a>). Under suitable assumptions, we establish a general decay result by multiplier techniques, which extends some existing results for a single equation to the case of a coupled system.

MSC: 35L05, 35L55, 35L70.

Keywords:
viscoelastic system; general decay; weak damping

1 Introduction

In this paper, we are concerned with a coupled system of nonlinear viscoelastic wave equations with weak damping

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M3">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M4">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M2">View MathML</a>) is a bounded domain with smooth boundary Ω, u and v represent the transverse displacements of waves. The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M6">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M7">View MathML</a> denote the kernel of a memory, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M9">View MathML</a> are the nonlinearities.

In recent years, many mathematicians have paid their attention to the energy decay and dynamic systems of the nonlinear wave equations, hyperbolic systems and viscoelastic equations.

Firstly, we recall some results concerning single viscoelastic wave equation. Kafini and Tatar [1] considered the following Cauchy problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M10">View MathML</a>

(1.2)

They established the polynomial decay of the first-order energy of solutions for compactly supported initial data and for a not necessarily decreasing relaxation function. Later Tatar [2] studied the problem (1.2) with the Dirichlet boundary condition and showed that the decay of solutions was an arbitrary decay not necessarily at exponential or polynomial rate. Cavalcanti et al.[3] studied the following equation with Dirichlet boundary condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M11">View MathML</a>

The authors established a global existence result for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M12">View MathML</a> and an exponential decay of energy for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M13">View MathML</a>. They studied the interaction within the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M14">View MathML</a> and the memory term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M15">View MathML</a>. Later on, several other results were published based on [4-6]. For more results on a single viscoelastic equation, we can refer to [7-14].

For a coupled system, Agre and Rammaha [15] investigated the following system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M16">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M4">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M2">View MathML</a>) is a bounded domain with smooth boundary. They considered the following assumptions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M19">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M20">View MathML</a>):

(A1) Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M21">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M23">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M25">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M26">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M27">View MathML</a>.

(A2) There exist two positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M29">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M31">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M32">View MathML</a>

Under the assumptions (A1)-(A2), they established the global existence of weak solutions and the global existence of small weak solutions with initial and Dirichlet boundary conditions. Moreover, they also obtained the blow up of weak solutions. Mustafa [16] studied the following system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M33">View MathML</a>

(1.3)

in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M34">View MathML</a> with initial and Dirichlet boundary conditions, proved the existence and uniqueness to the system by using the classical Faedo-Galerkin method and established a stability result by multiplier techniques. But the author considered the following different assumptions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M19">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M20">View MathML</a>) from (A1)-(A2):

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M37">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M38">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M20">View MathML</a>) are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M40">View MathML</a> functions and there exists a function F such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M41">View MathML</a>

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M42">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M43">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M44">View MathML</a>, where the constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M45">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M47">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M48">View MathML</a>.

Han and Wang [17] considered the following coupled nonlinear viscoelastic wave equations with weak damping:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M49">View MathML</a>

(1.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M4">View MathML</a> is a bounded domain with smooth boundary Ω. Under the assumptions (A1)-(A2) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M19">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M20">View MathML</a>), the initial data and the parameters in the equations, they established the local existence, global existence uniqueness and finite time blow up properties. When the weak damping terms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M54">View MathML</a> were replaced by the strong damping terms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M55">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M56">View MathML</a>, Liang and Gao [18] showed that under certain assumption on initial data in the stable set, the decay rate of the solution energy is exponential when they take

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M22">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M59">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M61">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M26">View MathML</a>. Moreover, they obtained that the solutions with positive initial energy blow up in a finite time for certain initial data in the unstable set. For more results on coupled viscoelastic equations, we can refer to [19-21].

If we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M63">View MathML</a> in (1.4), the system will be transformed into (1.1). To the best of our knowledge, there is no result on general energy decay for the viscoelastic problem (1.1). Motivated by [16,17], in this paper, we shall establish the general energy decay for the problem (1.1) by multiplier techniques, which extends some existing results for a single equation to the case of a coupled system. The rest of our paper is organized as follows. In Section 2, we give some preparations for our consideration and our main result. The statement and the proof of our main result will be given in Section 3.

For the reader’s convenience, we denote the norm and the scalar product in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M64">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M65">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M66">View MathML</a>, respectively. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M67">View MathML</a> denotes a general constant, which may be different in different estimates.

2 Preliminaries and main result

To state our main result, in addition to (A1)-(A2), we need the following assumption.

(A3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M20">View MathML</a>, are differentiable functions such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M70">View MathML</a>

and there exist nonincreasing functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M71">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M72">View MathML</a>

Now, we define the energy functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M73">View MathML</a>

(2.1)

and the functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M74">View MathML</a>

(2.2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M75">View MathML</a>

The existence of a global solution to the system (1.1) is established in [17] as follows.

Proposition[17]

Let (A1)-(A3) hold. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M77">View MathML</a>and that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M79">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M80">View MathML</a>is a computable constant and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M81">View MathML</a>. Then the problem (1.1) has a unique global solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M82">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M83">View MathML</a>

We are now ready to state our main result.

Theorem 2.1Let (A1)-(A3) hold. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M77">View MathML</a>and that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M79">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M80">View MathML</a>is a computable constant and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M81">View MathML</a>. Then there exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M90">View MathML</a>such that, fortlarge, the solution of (1.1) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M91">View MathML</a>

(2.3)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M92">View MathML</a>

(2.4)

3 Proof of Theorem 2.1

In this section, we carry out the proof of Theorem 2.1. Firstly, we will estimate several lemmas.

Lemma 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M94">View MathML</a>be the solution of (1.1). Then the following energy estimate holds for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M95">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M96">View MathML</a>

(3.1)

Proof Multiplying the first equation of (1.1) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M97">View MathML</a> and the second equation by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M98">View MathML</a>, respectively, integrating the results over Ω, performing integration by parts and noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M99">View MathML</a>, we can easily get (3.1). The proof is complete. □

Lemma 3.2Under the assumption (A3), the following hold:

(3.2)

(3.3)

Proof Using Hölder’s inequality, we get

On the other hand, we repeat the above proof with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M103">View MathML</a>, instead of g, we can get (3.3). The proof is now complete. □

Lemma 3.3Let (A1)-(A3) hold and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M94">View MathML</a>be the solution of (1.1). Then the functional<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M106">View MathML</a>defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M107">View MathML</a>

satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M108">View MathML</a>

(3.4)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M109">View MathML</a>.

Proof By (1.1), a direct differentiation gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M110">View MathML</a>

(3.5)

From the assumptions (A1)-(A2), we derive

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M112">View MathML</a>

(3.6)

By Young’s inequality and (3.2), we deduce for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M109">View MathML</a>

(3.7)

Similarly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M115">View MathML</a>

(3.8)

Using Young’s inequality and Poincaré’s inequality, we obtain for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M109">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M117">View MathML</a>

(3.9)

where λ is the first eigenvalue of −Δ with the Dirichlet boundary condition. Similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M118">View MathML</a>

which together with (3.5)-(3.9) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M119">View MathML</a>

(3.10)

Now, we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M109">View MathML</a> so small that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M121">View MathML</a>

which together with (3.10) gives (3.4). The proof is complete. □

Lemma 3.4Let (A1)-(A3) hold and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M94">View MathML</a>be the solution of (1.1). Then the functional<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M124">View MathML</a>defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M125">View MathML</a>

with

satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M127">View MathML</a>

(3.11)

Proof A direct differentiation for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M128">View MathML</a> yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M129">View MathML</a>

(3.12)

Using the first equation of (1.1) and integrating by parts, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M130">View MathML</a>

(3.13)

From Young’s inequality, Poincaré’s inequality and Lemma 3.2, we derive

(3.14)

(3.15)

(3.16)

(3.17)

Now, we estimate the first term on the right-hand side of (3.17). Using the assumptions (A1)-(A2) and Young’s inequality, we arrive at

(3.18)

where we used the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M136">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M137">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M26">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M139">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M140">View MathML</a> and the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M141">View MathML</a> proved in Lemma 5.1 in [17]. Combining (3.13)-(3.18), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M142">View MathML</a>

(3.19)

The same estimate to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M143">View MathML</a>, we can derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M144">View MathML</a>

which together with (3.19) gives (3.11). The proof is now complete. □

Proof of Theorem 2.1 For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M145">View MathML</a>, we define the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M146">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M147">View MathML</a>

and let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M148">View MathML</a>

for some fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M149">View MathML</a>.

Using Lemma 3.1 and Lemmas 3.3-3.4, a direct differentiation gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M150">View MathML</a>

(3.20)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M81">View MathML</a>.

Now, we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M152">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M153">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M154">View MathML</a> large enough so that

(3.21)

(3.22)

(3.23)

Inserting (3.21)-(3.23) into (3.20), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M158">View MathML</a>

(3.24)

Therefore, for two positive constants ω and C, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M159">View MathML</a>

(3.25)

On the other hand, we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M153">View MathML</a> even larger so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M161">View MathML</a> is equivalent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M162">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M163">View MathML</a>

(3.26)

Multiplying (3.25) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M164">View MathML</a> and using (A3), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M165">View MathML</a>

(3.27)

By virtue of (A3) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M166">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M167">View MathML</a>

(3.28)

Using (3.26), we can easily get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M168">View MathML</a>

(3.29)

which together with (3.28) yields, for some positive constant η,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M169">View MathML</a>

(3.30)

Integrating (3.30) over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M170">View MathML</a>, we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/146/mathml/M171">View MathML</a>

which together with (3.29) and the boundedness of E and ξ yields (2.3). The proof is now complete. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Acknowledgements

Baowei Feng was supported by the Doctoral Innovational Fund of Donghua University with contract number BC201138, and Yuming Qin was supported by NNSF of China with contract numbers 11031003 and 11271066 and the grant of Shanghai Education Commission (No. 13ZZ048).

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