Abstract and Applied Analysis

Volume 2013 (2013), Article ID 420803, 12 pages

http://dx.doi.org/10.1155/2013/420803

Research Article

Stability for the Kirchhoff Plates Equations with Viscoelastic Boundary Conditions in Noncylindrical Domains

Jum-Ran Kang

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

Received 18 September 2012; Accepted 15 May 2013

Academic Editor: Zhengde Dai

Copyright © 2013 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 study Kirchhoff plates equations with viscoelastic boundary conditions in a noncylindrical domain. This work is devoted to proving the global existence, uniqueness of solutions, and decay of the energy of solutions for Kirchhoff plates equations in a non-cylindrical domain.

1. Introduction

Let be an open bounded domain of containing the origin and having boundary. Let be a continuously differentiable function. Consider the family of subdomains of given by

whose boundaries are denoted by , and let be the noncylindrical domain of given by

with boundary

In this paper, we consider the following Kirchhoff plates equations with viscoelastic boundary conditions:

where is the unit normal at directed towards the exterior of . We divide the boundary into two parts:

We are denoting by and the following differential operators:

where and are given by

and the constant , represents Poisson’s ratio. From the physics point of view, system (4) describes the small transversal vibrations of a thin plate with a moving boundary device. The integral equations (6) and (7) describe the memory effects which can be caused, for example, by the interaction with another viscoelastic element. The relaxation functions are positive and nondecreasing.

The uniform stabilization of plates equations with linear or nonlinear boundary feedback in cylindrical domain was investigated by several authors; see for example [1–3] among others. The uniform decay for viscoelastic plates with memory was studied by [4, 5] and the references therein. Santos et al. [6] studied the asymptotic behavior of the solutions of a nonlinear wave equation of Kirchhoff type with boundary condition of memory type. Santos and Junior [7] investigated the stability of solutions for Kirchhoff plate equations with boundary memory condition. Park and Kang [8] 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. But the existence of solutions and decay of energy for the Kirchhoff plate equations with viscoelastic boundary conditions in noncylindrical domain are not studied yet. In a moving domain, the transverse deflection of the thin plate which changes its configuration at each instant of time increases its deformation and hence increases its tension. Moreover, the horizontal movement of the boundary yields nonlinear terms involving derivatives in the space variables. To control these nonlinearities, we add in the boundary a memory type. This term will play an important role in the dissipative nature of the problem.

In [9–17], the authors considered the global existence and the uniform decay of solution in noncylindrical domains. Dal Passo and Ughi [15] investigated a certain class of parabolic equations in noncylindrical domains. Benabidallah and Ferreira [9] proved the existence of solutions for the nonlinear beam equation in noncylindrical domains. Santos et al. [17] studied the global solvability and asymptotic behavior for the nonlinear coupled system of viscoelastic waves with memory in noncylindrical domains. Park and Kang [14] investigated the global existence and stability for von Karman equations with memory in noncylindrical domains. Motivated by these results, we prove the exponential decay of the energy to the Kirchhoff plate equations with viscoelastic boundary conditions in noncylindrical domains.

This paper is organized as follows. In Section 2, we recall notations and hypotheses. In Section 3, we prove the existence and uniqueness of solutions by employing Faedo-Galerkin’s method. In Section 4, we establish the exponential decay rate of the solution.

2. Notations and Hypotheses

We begin this section introducing notations and some hypotheses. Throughout this paper we use standard functional spaces and denote that are norm and norm. We define the inner product

Also, let us assume that there exists such that

The method used to prove the result of existence and uniqueness is based on the transformation of our problem into another initial boundary value problem defined over a cylindrical domain whose sections are not time dependent. This is done using a suitable change of variable. Then we show the existence and uniqueness for this new problem. Our existence result on noncylindrical domains will follow by using the inverse transformation. That is, by using the diffeomorphism

and defined by

For each function we denote by the function

the initial boundary value problem (4)–(8) becomes

where

The above method was introduced by Dal Passo and Ughi [15] for studying a certain class of parabolic equations in noncylindrical domains. This idea was used in [11, 13, 14, 16, 17].

We will use (19) and (20) to estimate the values and on . Denoting by

the convolution product operator and differentiating (19) and (20) we arrive at the following Volterra equations:

Applying Volterra’s inverse operator, we get

where the resolvent kernels of satisfy

Denoting by and , we obtain

Therefore, we use (27) and (28) instead of the boundary conditions (19) and (20).

Let us define the bilinear form as follows:

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

where and are generic positive constants.

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 .

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

Lemma 2 (see [7]). Let and be functions in . Then one has

Lemma 3 (see [18]). Suppose that , , and ; then, any solution of

satisfies and also

To show the existence of solution, we will use the following hypotheses:

where , , and is a positive imbedding constant such that , for all .

3. Existence and Regularity

In this section we will study the existence and regularity of solutions for system (4)–(8).

The well posedness of system (17)–(21) is given by the following theorem.

Theorem 4. Let be such that

The function satisfies that

If satisfy the compatibility condition

then there exists only one solution for system (17)–(21) satisfying

Proof. The main idea is to use the Galerkin method. To do this let us denote by the operator

It is well known that is a positive self-adjoint operator in the Hilbert space for which there exist sequences and of eigenfunctions and eigenvalues of such that the set of linear combinations of is dense in and as . Let us define

Note that for any , we have strong in and strong in .

Let us denote by the space generated by . Standard results on ordinary differential equations guarantee that there exists only one local solution

of the approximate system

By standard methods for differential equations, we prove the existence of solutions to the approximate equation (46) on some interval . Then, this solution can be extended to the whole interval , for all , by using the following first estimate.

The First Estimate. Multiplying (46) by , summing up the product result , and making some calculations using Lemma 1, we get

Now we will estimate terms of the right-hand side of (48). From the hypotheses on and Green’s formula, we get

Young’s inequality yields

Replacing the above calculations in (48) and using our assumptions and (30), we have

From our choice of and and integrating (51) over with , we obtain

We observe that, from (30) and (38),

for all . Hence, by Gronwall’s lemma we get

where is a positive constant which is independent of and .

The Second Estimate. First of all, we are going to estimate in -norm. Letting in (46), multiplying the result by , and using the compatibility condition (41), we have

Now, differentiating (46) with respect to , we obtain

Multiplying (56) by , summing up the product result in , and using Lemma 1, we have

Now we will estimate terms of the right-hand side of (57). From the hypotheses on and Green’s formula, we get

We know that

By using Hölder’s inequality and our assumption , we note that

and, hence, by applying Young’s inequality, we obtain

By the same argument of (63), we can obtain the similar estimate

Applying (58)–(64) to (57) and using the first estimate (54) and our assumptions and , we have

From (55) and our choice of and and integrating (65) over with , we obtain

Using the same arguments as for (53), we get

for all . Therefore, by Gronwall’s lemma, we obtain

where is a positive constant which is independent of and .

According to (54) and (68), we get

From (69) to (71), there exists a subsequence of , which we still denote by , such that

Letting in (46) and using (72)–(74), we obtain

for any . From Lemma 3 we obtain that . The uniqueness of solutions follows by using standard arguments.

Theorem 5. Under the hypotheses of Theorem 4, let , . Then there exists a unique solution of the problem (4)–(8) satisfying

Proof. This idea was used in [11, 13, 14, 16, 17]. To show the existence in noncylindrical domains, we return to our original problem in the noncylindrical domains by using the change variable given in (14) by , . Let be the solution obtained from Theorem 4 and defined by (16); then belongs to the class

Denoting by

then from (15) it is easy to see that satisfies (4)–(8) in the sense of . If , are two solutions obtained through the diffeomorphism given by (14), then , so . Thus the proof of Theorem 5 is completed.

4. Exponential Decay

In this section, we show that the solution of system (4)–(8) decays exponentially. First of all, we introduce the useful lemma for a noncylindrical domain.

Lemma 6 (see [11, 12]). Let be the smooth function defined in . Then

where is the -component of the unit normal exterior .

By the same argument of (27) and (28), it can be written as

We use (80) and (81) instead of the boundary conditions (6) and (7).

We will use the following lemma.

Lemma 7 (see [4]). For every and for every , one has

Now, we define the energy of problem (4)–(8) by

We observe that is a positive function. Using Lemmas 6 and 1, we have

Let us consider the following functional:

The following lemma plays an important role for the construction of the Lyapunov functional.

Lemma 8. Let one suppose that the initial data and satisfies the compatibility condition (41). Then the solution of system (4)–(8) satisfies

Proof. Differentiating and using (4) and Lemmas 6 and 7, we get

Let us next examine the integrals over in (87). Since on , we have

and hence

Therefore, from (87)–(90) we have

Noting that on follows from (91), we have the conclusion of the lemma.

Let us introduce the Lyapunov functional

with . Using Young’s inequality and choosing sufficiently large, we see that

for and are positive constants. We will show later that the functional satisfies the inequality of the following result.

Lemma 9 (see [7]). Let be a real positive function of class . If there exist positive constants , , and such that

then there exist positive constants and such that

Finally, we will show the main result of this section.

Theorem 10. Assume that there exist positive constants and such that

If then there exist constants such that

Proof. From (84) and Lemma 8 we have

Since the boundary conditions (80) and (81) can be written as

by using Young’s inequality we obtain

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

where is a constant depending on and . Substituting inequalities (100)–(102) into (98) we have

First, choose sufficiently small such that

Then, choosing large enough, we have

where and . From (93), (96), and (105), we obtain

By Lemma 9, there exist positive constants and such that

Using (93), we conclude that

for some positive constants and .

Acknowledgment

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A3011630).

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