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General decay for Kirchhoff plates with a boundary condition of memory type

Jum-Ran Kang

Author affiliations

Department of Mathematics, Dong-A University, Saha-Ku, Busan, 604-714, Korea

Citation and License

Boundary Value Problems 2012, 2012:129  doi:10.1186/1687-2770-2012-129


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/129


Received:30 July 2012
Accepted:25 October 2012
Published:7 November 2012

© 2012 Kang; 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 consider Kirchhoff plates with a memory condition at the boundary. For a wider class of relaxation functions, we establish a more general decay result, from which the usual exponential and polynomial decay rates are only special cases.

MSC: 35B40, 74K20, 35L70.

Keywords:
Kirchhoff plates; general decay rate; memory term; relaxation function

1 Introduction

We consider the following Kirchhoff plates with a memory condition at the boundary:

(1.1)

(1.2)

(1.3)

(1.4)

(1.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M6">View MathML</a> and Ω is an open bounded set of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M7">View MathML</a> with a regular boundary Γ. We divide the boundary into two parts:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M8">View MathML</a>

Let us denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M9">View MathML</a> the external unit normal to Γ, and let us denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M10">View MathML</a> the unit tangent positively oriented on Γ. We are denoting by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M12">View MathML</a> the following differential operators:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M13">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M14">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M15">View MathML</a> are given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M16">View MathML</a>

and the constant μ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M17">View MathML</a>, represents Poisson’s ratio.

In (1.1), u denotes the position of the plate. The integral equations (1.3) and (1.4) describe the memory effects which can be caused, for example, by the interaction with another viscoelastic element. The relaxation functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M18">View MathML</a> are positive and nondecreasing. This system models the small transversal vibrations of a thin plate whose Poisson coefficient is equal to μ. We assume that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M19">View MathML</a> such that

(1.6)

(1.7)

If we denote the compactness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M22">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M23">View MathML</a>, the condition (1.7) implies that there exists a small positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M24">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M26">View MathML</a>.

The uniform stabilization of Kirchhoff plates with linear or nonlinear boundary feedback was investigated by several authors; see, for example, [1-3] among others. The uniform decay for plates with memory was studied in [4-6] and the references therein. There exists a large body of literature regarding viscoelastic problems with the memory term acting in the domain or at the boundary (see [7-12]). Rivera and Racke [13] investigated the decay results for magneto-thermo-elastic system. Santos et al.[14] studied the asymptotic behavior of the solutions of a nonlinear wave equation of Kirchhoff type with a boundary condition of memory type. Cavalcanti and Guesmia [15] proved the general decay rates of solutions to a nonlinear wave equation with a boundary condition of memory type. Park and Kang [16] studied the exponential decay for the multi-valued hyperbolic differential inclusion with a boundary condition of memory type. Kafini [17] showed the decay results for viscoelastic diffusion equations in the absence of instantaneous elasticity. They proved that the energy decays uniformly exponentially or algebraically at the same rate as the relaxation functions. In the present work, we generalize the earlier decay results of the solution of (1.1)-(1.5). More precisely, we show that the energy decays at the rate similar to the relaxation functions, which are not necessarily decaying like polynomial or exponential functions. In fact, our result allows a larger class of relaxation functions. Recently, Messaoudi and Soufyane [18], Santos and Soufyane [19], and Mustafa and Messaoudi [20] proved the general decay for the wave equation, von Karman plate system, and Timoshenko system with viscoelastic boundary conditions, respectively.

The paper is organized as follows. In Section 2 we present some notations and material needed for our work. In Section 3 we prove the general decay of the solutions to the Kirchhoff plates with a memory condition at the boundary.

2 Preliminaries

In this section, we present some material needed in the proof of our main result. We use the standard Lebesgue and Sobolev spaces with their usual scalar products and norms. Define the following space:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M27">View MathML</a>

First, we shall use Eqs. (1.3) and (1.4) to estimate the values <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M12">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M22">View MathML</a>. Denoting by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M31">View MathML</a>

the convolution product operator and differentiating Eqs. (1.3) and (1.4), we arrive at the following Volterra equations:

Applying the Volterra inverse operator, we get

where the resolvent kernels of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M34">View MathML</a> satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M35">View MathML</a>

Denoting by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M36">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M37">View MathML</a>, we obtain

(2.1)

(2.2)

Therefore, we use (2.1) and (2.2) instead of the boundary conditions (1.3) and (1.4).

Let us define the bilinear form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M40">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M41">View MathML</a>

(2.3)

We state the following lemma which will be useful in what follows.

Lemma 2.1 ([2])

Letuandvbe functions in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M42">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M43">View MathML</a>

(2.4)

Let us denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M44">View MathML</a>

The following lemma states an important property of the convolution operator.

Lemma 2.2For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M45">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M46">View MathML</a>

The proof of this lemma follows by differentiating the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M47">View MathML</a>.

We formulate the following assumption:

(A1) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M6">View MathML</a> satisfy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M49">View MathML</a> in Ω for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M50">View MathML</a>.

Let us introduce the energy function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M51">View MathML</a>

In these conditions, we are able to prove the existence of a strong solution.

Theorem 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M52">View MathML</a>be such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M53">View MathML</a>

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M54">View MathML</a>satisfy the compatibility condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M55">View MathML</a>

(2.5)

then there is only one solutionuof the system (1.1)-(1.5) satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M56">View MathML</a>

Proof See Park and Kang [16]. □

3 General decay

In this section, we show that the solution of the system (1.1)-(1.5) may have a general decay not necessarily of exponential or polynomial type. For this we consider that the resolvent kernels satisfy the following hypothesis:

(H) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M57">View MathML</a> is a twice differentiable function such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M58">View MathML</a>

and there exists a nonincreasing continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M59">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M60">View MathML</a>

(3.1)

The following identity will be used later.

Lemma 3.1 ([2])

For every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M61">View MathML</a>and for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M62">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M63">View MathML</a>

Our point of departure will be to establish some inequalities for the strong solution of the system (1.1)-(1.5).

Lemma 3.2The energy functionalEsatisfies, along the solution of (1.1)-(1.5), the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M64">View MathML</a>

(3.2)

Proof Multiplying Eq. (1.1) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M65">View MathML</a>, integrating over Ω, and using Lemma 2.1, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M66">View MathML</a>

Substituting the boundary terms by (2.1) and (2.2) and using Lemma 2.2 and the Young inequality, our conclusion follows. □

Let us consider the following binary operator:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M67">View MathML</a>

Then applying the Holder inequality for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M68">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M69">View MathML</a>

(3.3)

Let us define the functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M70">View MathML</a>

The following lemma plays an important role in the construction of the desired functional.

Lemma 3.3Suppose that the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M71">View MathML</a>and satisfies the compatibility condition (2.5). Then the solution of the system (1.1)-(1.5) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M72">View MathML</a>

Proof Differentiating ψ, using Eq. (1.1), and taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M73">View MathML</a> in Lemma 3.1, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M74">View MathML</a>

(3.4)

Let us next examine the integrals over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M75">View MathML</a> in (3.4). Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M76">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M75">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M78">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M75">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M80">View MathML</a>

(3.5)

since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M81">View MathML</a>

Therefore, from (3.4) and (3.5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M82">View MathML</a>

(3.6)

Using the Young inequality, we have

(3.7)

(3.8)

where ϵ is a positive constant. Since the bilinear form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M85">View MathML</a> is strictly coercive on W, using the trace theory, we obtain

(3.9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M87">View MathML</a> is a constant depending on Ω and μ. Further, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M88">View MathML</a>

(3.10)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M89">View MathML</a> with some constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M90">View MathML</a>. Substituting the inequalities (3.7)-(3.10) into (3.6) and taking into account the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M91">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M75">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M93">View MathML</a>

Since the boundary conditions (2.1) and (2.2) can be written as

our conclusion follows. □

Let us introduce the Lyapunov functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M95">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M96">View MathML</a>. Now we are in a position to show the main result of this paper.

Theorem 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M97">View MathML</a>. Suppose the resolvent kernels<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M98">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M99">View MathML</a>satisfy the conditions (H) and (3.1). Then there exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M100">View MathML</a>such that for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M101">View MathML</a>large enough, the solution of (1.1)-(1.5) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M102">View MathML</a>

(3.11)

Otherwise,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M103">View MathML</a>

(3.12)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M104">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M105">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M106">View MathML</a>

Proof Applying the inequality (3.3) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M107">View MathML</a> in Lemma 3.3 and from Lemma 3.2, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M108">View MathML</a>

(3.13)

Then, choosing N large enough and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M109">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M110">View MathML</a>

On the other hand, we can choose N even larger so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M111">View MathML</a>

(3.14)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M112">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M113">View MathML</a>, then using (3.1) and (3.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M114">View MathML</a>

which gives

Using the fact that ξ is a nonincreasing continuous function as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M116">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M117">View MathML</a> are nonincreasing, and so ξ is differentiable, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M118">View MathML</a>, for a.e. t, then we infer that

Since using (3.14),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M120">View MathML</a>

(3.15)

we obtain for some positive constant ω,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M121">View MathML</a>

(3.16)

Case 1: If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M122">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M22">View MathML</a>, then (3.16) reduces to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M124">View MathML</a>

A simple integration over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M125">View MathML</a> yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M126">View MathML</a>

By using (3.2) and (3.15), we then obtain for some positive constant C

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M127">View MathML</a>

Thus, the estimate (3.11) is proved.

Case 2: If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M128">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M22">View MathML</a>, then (3.16) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M130">View MathML</a>

(3.17)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M131">View MathML</a>

In this case, we introduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M132">View MathML</a>

(3.18)

A simple differentiation of G, using (3.17), leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M133">View MathML</a>

Again, a simple integration over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M125">View MathML</a> yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M135">View MathML</a>

which implies, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M104">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M137">View MathML</a>

By using (3.15), we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M138">View MathML</a>

Consequently, by the boundedness of ξ, (3.12) is established. □

Remark 3.1 Estimates (3.11) and (3.12) are also true for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M139">View MathML</a> by virtue of continuity and boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M140">View MathML</a> and ξ.

Remark 3.2 Note that the exponential and polynomial decay estimates are only particular cases of (3.11) and (3.12). More precisely, we have exponential decay for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M141">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M142">View MathML</a> and polynomial decay for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M143">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M142">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M145">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M146">View MathML</a> are positive constants.

Example 3.1 As in [20], we give some examples to illustrate the energy decay rates given by (3.11).

(1) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M147">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M148">View MathML</a>, then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M149">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M150">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M151">View MathML</a>. For suitably chosen positive constants a and b, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M152">View MathML</a> satisfies (H) and (3.11) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M153">View MathML</a>

(2) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M154">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M155">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M156">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M148">View MathML</a>, then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M158">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M159">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M160">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M161">View MathML</a>

The above two examples are included in the following more general one.

(3) For any nonincreasing functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M162">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M158">View MathML</a>, which satisfy (H), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M164">View MathML</a> are also nonincreasing differentiable functions, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M165">View MathML</a>, for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M166">View MathML</a>, (3.11) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/129/mathml/M167">View MathML</a>

Competing interests

The author declares that she has no competing interests.

Author’s contributions

The work was realized by the author.

Acknowledgements

The author thanks the anonymous referee for a careful review. This work was supported by the Dong-A University research fund.

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