Electronic Journal of Qualitative Theory of Differential Equations

In this paper we prove the exponential decay in the case n > 2, as time goes to infinity, of regular solutions for a nonlinear coupled system of beam equations of Kirchhoff type with memory and weak damping utt + � 2 u M(||ru|| 2 2 (t) + ||rv|| 2 2 (t) )�u


. INTRODUCTION
Let us consider the Hilbert space L 2 (Ω) endowed with the inner product and corresponding norm We also consider the Sobolev space H 1 (Ω) endowed with the scalar product (u, v) H 1 (Ω) = (u, v) + (∇u, ∇v).
Let ν(•, t) be the x-component of the unit normal ν(•, •), |ν| ≤ 1.Then by the relation (1.5), one has ν(σ, t) = η( σ γ(t) ). (1.6)In this paper we deal with nonlinear coupled system of beam equations of Kirchhoff type with memory over a non cylindrical domain.We show the existence and uniqueness of strong solutions to the initial boundary value problem (1.1)- (1.4).The method we use 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 timedependent.This is done using a suitable change of variable.Then we show the existence and uniqueness for this new problem.Our existence result on non cylindrical domain will follow using the inverse transformation.That is, using the diffeomorphism τ : Q → Q defined by Denoting by φ and ϕ the functions the initial boundary value problem (1.1)-(1.4)becomes ) where To show the existence of a strong solution we will use the following hypotheses: (1.17) Note that the assumption (1.15) means that Q is decreasing if n > 2 and increasing if n ≤ 2 in the sense that when t > t and n > 2 then the projection of Ω t on the subspace t = 0 contain the projection of Ω t on the same subspace and contrary in the case n ≤ 2. The above method EJQTDE, 2007 No. 9, p. 4 was introduced by Dal Passo and Ughi [21] to study certain class of parabolic equations in non cylindrical domain.We assume that h ∈ C 1 ( ) satisfies h(s)s ≥ 0, ∀s ∈ .
Additionally, we suppose that h is superlinear, that is with the following growth conditions where M (τ ) = τ 0 M (s)ds and where λ 1 is the first eigenvalue of the spectral Dirichlet problem We recall also the classical inequality Unlike the existing papers on stability for hyperbolic equations in non cylindrical domain, we do not use the penalty method introduced by J. L. Lions [16], but work directly in our non cylindrical domain Q.To see the dissipative properties of the system we have to construct a suitable functional whose derivative is negative and is equivalent to the first order energy.This functional is obtained using the multiplicative technique following Komornik [8] or Rivera [20].
We only obtained the exponential decay of solution for our problem for the case n > The uniform stabilization of plates equations with linear or nonlinear boundary feedback was investigated by several authors, see for example [7,9,10,11,13,22] among others.In a fixed domain, it is well-known, the relaxation function g decays to zero implies that the energy of the system also decays to zero, see [2,12,19,23].But in a moving domain the transverse deflection u(x, t) and v(x, t) of a beam which charges its configuration at each instant of time, increasing its deformation and hence increasing its tension.Moreover, the horizontal movement of the boundary yields nonlinear terms involving derivatives in the space variable.To control these nonlinearities, we add in the system a frictional damping, characterized by u t and v t .This term will play an important role in the dissipative nature of the problem.A quite complete discussion in the modelling of transverse deflection and transverse vibrations, respectively, of purely for the nonlinear beam equation and elastic membranes can be found in J. Ferreira et al. [6], J. Límaco et al. [17] and L. A. Medeiros et al. [18].This model was proposed by Woinowsky [24] for the case of cylindrical domain, without the terms −∆u and t 0 g 1 (t − s)∆u(s)ds but with the term See also Eisley [5] and Burgreen [1] for physics justification and background of the model.We use the standard notations which can be found in Lion's book [15,16].In the sequel by C (sometimes C 1 , C 2 , . ..) we denote various positive constants which do not depend on t or on the initial data.This paper is organized as follows.In section 2 we prove a basic result on the existence, regularity and uniqueness of regular solutions.We use Galerkin approximation, Aubin-Lions theorem, energy method introduced by Lions [16] and some technical ideas to show existence regularity and uniqueness of regular solution for the problem (1.1)-(1.4).Finally, in section 3, we establish a result on the exponential decay of the regular solution to the problem (1.1)-(1.4).We use the technique of the multipliers introduced by Komornik [8], Lions [16] and Rivera [20] coupled with some technical lemmas and some technical ideas.

. EXISTENCE AND REGULARITY OF GLOBAL SOLUTIONS
In this section we shall study the existence and regularity of solutions for the system (1.1)- (1.4).
For this we assume that the kernels g i : R + → R + is in C 1 (0, ∞), and satisfy where EJQTDE, 2007 No. 9, p. 6 To simplify our analysis, we define the binary operator With this notation we have the following lemma. and The proof of this lemma follows by differentiating the terms g2 ∇Φ(t) γ(t) and g2 ∆Φ(t) γ(t) .The wellposedness of system (1.10)-(1.13) is given by the following theorem.
Proof.Let us denote by B the operator It is well know that B is a positive self adjoint operator in the Hilbert space L 2 (Ω) for which there exist sequences {w n } n∈N and {λ n } n∈N of eigenfunctions and eigenvalues of B such that the set of linear combinations of {w n } n∈N is dense in D(B) and EJQTDE, 2007 No. 9, p. 7 Note that for any {(φ 0 , φ 1 ), (ϕ Let us denote by V m the space generated by w 1 , . . ., w m .Standard results on ordinary differential equations imply the existence of a local solution (φ m , ϕ m ) of the form Ω ∆φ m w j dy (j = 1, . . ., m), The extension of the solution to the whole interval [0, ∞) is a consequence of the first estimate which we are going to prove below. A From (1.16)-(1.17)and (1.19)-(1.20) it follows that where . Taking into account (1.16), (1.17), (2.1) and the last equality we obtain 1 2 Integrating the inequality (2.6), using Gronwall's Lemma and taking account (1.17) we get

.7)
A priori estimate II Now, if we multiply the equations (2.2) by λ j g jm and (2.3) by λ j f jm and summing up in EJQTDE, 2007 No. 9, p. 9 j = 1, . . ., m we get Summing the last two equalities and using the lemma 2.1 we obtain 1 2 where From (1.16)-(1.17)and (1.19)-(1.20),we have Using similar arguments as (2.7), the hypothesis for the function h and observing the above inequality we obtain Let us take p n = 2n n−2 .From the growth condition of the function h and from the Sobolev imbedding we obtain .

.18)
A priori estimate IV Multiplying the equations (2.2)-(2.3)by g jm (t) and f jm (t), respectively, summing up the product result in j = 1, 2, . . ., m and using the hypothesis on h we deduce 1 2 Choosing k > 2 α , we obtain From (2.21), using the Gronwall's Lemma we obtain the following estimate, taking into account (2.5) and (1.17) From estimates (2.7), (2.8), (2.18) and (2.22) it's follows that (φ m , ϕ m ) converge strong to where C is a positive constant independent of m and t, so that and Using similar arguments as above we conclude that EJQTDE, 2007 No. 9, p. 14 Letting m → ∞ in the equations (2.2)-( 2.3) we conclude that (φ, ϕ) satisfies (1.10)- (1.11) in To prove the uniqueness of solutions of problem (1.10), (1.11), (1.12) and (1.13) we use the method of the energy introduced by Lions [16], coupled with Gronwall's inequality and the hypotheses introduced in the paper about the functions g i , h,M and the obtained estimates.
To show the existence in non cylindrical domain, we return to our original problem in the non cylindrical domain by using the change of variable given in (1.8) by (y, t) = τ (x, t), (x, t) ∈ Q.
Theorem 2.2 Let us take

. EXPONENTIAL DECAY
In this section we show that the solution of system (1.1)-(1.4)decays exponentially.To this end we will assume that the memory g i satisfies: for all t ≥ 0, with positive constant C 1 .Additionally, we assume that the function γ(•) satisfies the conditions where d = diam(Ω).The condition (3.4) (see also (1.5)) imply that our domain is "time like" in the sense that |ν| < |ν| where ν and ν denote the t-component and x-component of the outer unit normal of ˆ .
Remark: It is important to observe that to prove the main Theorem of this section, that is, Theorem 3.1 as well the Lemmas 3.4 and 3.5 we use the following substantial hypothesis: To facilitate our calculations we introduce the following notation First of all we will prove the following three lemmas that will be used in the sequel.
where ν is the x-component of the unit normal exterior ν.
The proof is now complete.
Let us introduce the functional of energy Proof.Multiplying the equation (1.1) by u t , performing an integration by parts over Ω t and using the lemmas 3.
for k 0 and k 1 positive constants.Now we are in a position to show the main result of this paper.
Let Ω be an open bounded domain of R n containing the origin and having C 2 boundary.Let γ : [0, ∞[→ R be a continuously differentiable function.See hypothesis (1.15)-(1.17) on γ.Let EJQTDE, 2007 No. 9, p. 1 us consider the family of subdomains {Ω t } 0≤t<∞ of R n given by Ω t = T (Ω), T : y ∈ Ω → x = γ(t)y whose boundaries are denote by Γ t and Q the non cylindrical domain of R n+1

. 5 )
This because we worked directly in our domain with moving boundary, where the geometry of our domain influence directly in the problem, what generated several technical difficulties in limiting some terms in Lemma 3.5 and consequent to prove Theorem 3.1.