Abstract and Applied Analysis

Volume 2013 (2013), Article ID 682061, 8 pages

http://dx.doi.org/10.1155/2013/682061

Research Article

General Decay for the Degenerate Equation with a Memory Condition at the Boundary

Su-Young Shin and Jum-Ran Kang

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

Received 21 December 2012; Accepted 5 March 2013

Academic Editor: Abdelaziz Rhandi

Copyright © 2013 Su-Young Shin and Jum-Ran Kang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We consider a degenerate equation 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.

1. Introduction

The main purpose of this paper is to investigate the asymptotic behavior of the solutions of the degenerate equation with a memory condition at the boundary

where is a bounded domain of with a smooth boundary and let us assume that , can be divided into two nonnull parts and and and for all which satisfies some appropriate conditions. is the unit outward normal to and the corresponding unit tangent vector. Here, the relaxation functions () are positive and nondecreasing, the function and

and the constant , , represents Poisson’s ratio.

From the physical point of view, we know that the memory effect described in integral equations (3) and (4) can be caused by the interaction with another viscoelastic element. In fact, the boundary conditions (3) and (4) mean that is composed of a material which is clamped in a rigid body in and is clamped in a body with viscoelastic properties in the complementary part of its boundary named . Problems related to (1)–(5) are interesting not only from the point of view of PDE general theory, but also due to its applications in mechanics.

The existence of global solutions and exponential decay to the degenerate equation with has been investigated by several authors. See Cavalcanti et al. [1] and Menezes et al. [2]. For instance, when is equal to 1, (1) describes the transverse deflection of beams. There exists a large body of literature regarding viscoelastic problems with the memory term acting in the domain or at the boundary (see [3–17]). Santos et al. [18] studied the asymptotic behavior of the solutions of a nonlinear wave equation of Kirchhoff type with boundary condition of memory type. Cavalcanti et al. [19] proved the uniform decay rates of solutions to a degenerate system with a memory condition at the boundary. Santos and Junior [20] investigated the stability of solutions for Kirchhoff plate equations with a boundary memory condition. Rivera et al. [21] showed the asymptotic behavior to a von Karman plate with boundary memory conditions. Park and Kang [22] studied the exponential decay for the Kirchhoff plate equations with nonlinear dissipation and boundary memory condition. They proved that the energy decays uniformly exponentially or algebraically with the same rate of decay as the relaxation functions. In the present work, we generalize the earlier decay results of the solution of (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 [23], Mustafa and Messaoudi [24], and Santos and Soufyane [25] proved the general decay for the wave equation, Timoshenko system, and von Karman plate system with viscoelastic boundary conditions, respectively.

The organization of this paper is as follows. In Section 2, we present some notations and material needed for our work and state the existence result to system (1)–(5). In Section 3, we prove the general decay of the solutions to the degenerate equation with a memory condition at the boundary.

2. Preliminaries

In this section, we introduce some notations and establish the existence of solutions of the problem (1)–(5).

Note that, because of condition (2), the solution of system (1)–(5) must belong to the following space:

Let us define the bilinear form as follows:

Since , we know that is equivalent to the norm on ; that is,

and here and in the sequel, we denote by and generic positive constants. Simple calculation, based on the integration by parts formula, yields

We assume that there exists such that

If we denote the compactness of by , condition (12) implies that there exists a small positive constant such that , for all .

Next, we will use (3) and (4) to estimate the values and on . Denoting by

the convolution product operator and differentiating (3) and (4), we arrive at the following Volterra equations:

Applying the Volterra inverse operator, we get

where the resolvent kernels satisfy

Denoting that and , we have

Therefore, we use (17) instead of the boundary conditions (3) and (4).

Let us denote that

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

Lemma 1. For , one has

The proof of this lemma follows by differentiating the term .

Lemma 2 (see [26]). Suppose that ,   and ; then, any solution of

satisfies and also

We formulate the following assumptions.

(A1) Let satisfy

Additionally, we suppose that is superlinear; that is,

for some with the following growth condition:

for some and such that .

(A2);   with , for all , and satisfy the following condition

The well-posedness of system (1)–(5) is given by the following theorem.

Theorem 3 (see [27]). Consider assumptions (A1)-(A2) and let be such that

If , , satisfying the compatibility condition

then there is only one solution of the system (1)–(5) satisfying

3. General Decay

In this section, we show that the solution of system (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) is twice differentiable function such that

and there exists a nonincreasing continuous function satisfying

The following identity will be used later.

Lemma 4 (see [26]). For every and for every , one has

Let us introduce the energy function

Now, we establish some inequalities for the strong solution of system (1)–(5).

Lemma 5. The energy functional satisfies, along the solution of (1)–(5), the estimate

Proof. Multiplying (1) by , integrating over , and using (10), we get

Substituting the boundary terms by (17) and using Lemma 1 and the Young inequality, our conclusion follows.

Let us consider the following binary operator:

Then applying the Holder inequality for we have

Let us define the functional

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

Lemma 6. Suppose that the initial data , satisfying the compatibility condition (27). Then, the solution of system (1)–(5) satisfies

Proof. Differentiating using (1) and Lemma 4, we get

Let us next examine the integrals over in (39). Since on , we have on and

since

Therefore, from (39) and (40), we have

Using the Young inequality, we get

where is a positive constant. Since the bilinear form is strictly coercive on , using the trace theory, we obtain

where is a constant depending on ,  ,   and . Substituting inequalities (43)–(45) into (42) and taking into account that on , as well as (23) and (25), we have

Since the boundary conditions (17) can be written as

our conclusion follows.

Let us introduce the Lyapunov functional

with . Now, we are in a position to show the main result of this paper.

Theorem 7. Let . Suppose that the resolvent kernels ,   satisfy the condition (H). Then, there exist constants such that, for some large enough, the solution of (1)–(5) satisfies

Otherwise,

for all , where

Proof. Applying inequality (36) with in Lemma 6 and from Lemma 5, we obtain

We take and so small such that

Since and then choosing large enough, we obtain

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

If ,  , then, using (30) and (33), we have

which gives

Using the fact that is a nonincreasing continuous function as and are nonincreasing, and so is differentiable, with , for a.e. , then we infer that

Since using (55),

we obtain, for some positive constant ,

Case  1. If on , then (60) reduces to

A simple integration over yields

By using (33) and (59), we then obtain for some positive constant

Thus, estimate (49) is proved.

Case  2. If on , then (60) gives

where

In this case, we introduce

A simple differentiation of , using (64), leads to

Again, a simple integration over yields

which implies, for all ,

By using (59), we deduce that

Consequently, by the boundedness of , (50) is established.

Remark 8. Note that the exponential and polynomial decay estimates are only particular cases of (49) and (50). More precisely, we have exponential decay for and and polynomial decay for and , where and are positive constants.

Example 9. As in [24], we give some examples to illustrate the energy decay rates given by (49).

(1)If ,  , then, for , where . For suitably chosen positive constants and , satisfies (H) and (49) gives

(2)If ,  , and ,  , then, for , where . Then

The aforementioned two examples are included in the following more general one.(3)For any nonincreasing functions ,  , which satisfy (H), are also nonincreasing differentiable functions, and , for some , and (49) gives

Acknowledgment

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no. 2012R1A1A3011630).

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