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

and Applied Analysis 3 We will use (19) and (20) to estimate the values B 1 and B 2 on Γ 1 . Denoting by


Introduction
Let Ω be an open bounded domain of R 2 containing the origin and having  2 boundary.Let  : [0, ∞[ → R be a continuously differentiable function.Consider the family of subdomains {Ω  } 0≤<∞ of R 2 given by Ω  =  (Ω) ,  :  ∈ Ω →  =  () , whose boundaries are denoted by Γ  , and let Q be the noncylindrical domain of R 3 given by with boundary In this paper, we consider the following Kirchhoff plates equations with viscoelastic boundary conditions:
We are denoting by B 1 and B 2 the following differential operators: where  1 and  2 are given by and the constant , 0 <  < 1/2, 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  1 ,  2 ∈  1 (0, ∞) 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][2][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][10][11][12][13][14][15][16][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.

Notations and Hypotheses
We begin this section introducing notations and some hypotheses.Throughout this paper we use standard functional spaces and denote that ‖ ⋅ ‖  , ‖ ⋅ ‖ , are   (Ω) norm and   (Ω  ) norm.We define the inner product Also, let us assume that there exists  0 ∈ R 2 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  −1 :  → Q defined by For each function  we denote by V 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 B 1 and B 2 on Γ 1 .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 −   /  (0) satisfy Denoting by  1 = 1/ 1 (0) and  2 = 1/ 2 (0), we obtain Therefore, we use ( 27) and (28) instead of the boundary conditions ( 19) and (20).Let us define the bilinear form (⋅, ⋅) as follows: where  0 and  0 are generic positive constants.Let us denote that The following lemma states an important property of the convolution operator.
Lemma 1.For , V ∈  1 ([0, ∞) : R) one has The proof of this lemma follows by differentiating the term ◻V.
We state the following lemma which will be useful in what follows.

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.
The function  satisfies that 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  2 (Ω) for which there exist sequences {  } ∈N and {  } ∈N of eigenfunctions and eigenvalues of  such that the set of linear combinations of Note that for any Let us denote by   the space generated by  1 ,  2 , . . .,   .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 [0,   ).Then, this solution can be extended to the whole interval [0, ], for all  > 0, by using the following first estimate.
The First Estimate.Multiplying (46) by    (), summing up the product result  = 1, 2, . . ., , 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   , −   ,    ≥ 0 and (30), we have From our choice of V 0 and V 1 and integrating (51) over (0, ) with  ∈ (0,   ), we obtain We observe that, from (30) and (38), for all  ≥ 0. Hence, by Gronwall's lemma we get where  5 is a positive constant which is independent of  and .
The Second Estimate.First of all, we are going to estimate V   (0) in  2 (Ω)-norm.Letting  → 0 + in (46), multiplying the result by    (0), 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   1 ≤ 0, we note that and, hence, by applying Young's inequality, we obtain By the same argument of (63), we can obtain the similar estimate ) Γ. (64) Applying ( 58)-( 64) to (57) and using the first estimate (54) and our assumptions   , −   ,    ≥ 0 and Abstract and Applied Analysis From (55) and our choice of V 0 and V 1 and integrating (65) over (0, ) with  ∈ (0,   ), we obtain Using the same arguments as for (53), we get for all  ≥ 0. Therefore, by Gronwall's lemma, we obtain where  11 is a positive constant which is independent of  and .

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 (, ) = (, ), (, ) ∈ Q.
Let V 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  ∞ (0, ∞;  2 (Ω  )).If  1 ,  2 are two solutions obtained through the diffeomorphism  given by ( 14), then Thus the proof of Theorem 5 is completed.

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.
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  ∈ Let us consider the following functional: The following lemma plays an important role for the construction of the Lyapunov functional.