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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 420803, 12 pages
Stability for the Kirchhoff Plates Equations with Viscoelastic Boundary Conditions in Noncylindrical Domains
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.
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.
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.  studied the asymptotic behavior of the solutions of a nonlinear wave equation of Kirchhoff type with boundary condition of memory type. Santos and Junior  investigated the stability of solutions for Kirchhoff plate equations with boundary memory condition. Park and Kang  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  investigated a certain class of parabolic equations in noncylindrical domains. Benabidallah and Ferreira  proved the existence of solutions for the nonlinear beam equation in noncylindrical domains. Santos et al.  studied the global solvability and asymptotic behavior for the nonlinear coupled system of viscoelastic waves with memory in noncylindrical domains. Park and Kang  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  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 ). Let and be functions in . Then one has
Lemma 3 (see ). 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
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.
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
We will use the following lemma.
Lemma 7 (see ). For every and for every , one has
The following lemma plays an important role for the construction of the Lyapunov functional.
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 ). 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