FACI-Faculdade Ideal

In this paper, we investigate a mathematical model for a nonlinear coupled system of Kirchho type of beam equations with nonlocal boundary conditions. We establish existence, regularity and uniqueness of strong solutions. Furthermore, we prove the uniform rate of exponential decay. The uniform rate of polynomial decay is considered. 2000 Mathematical Subject Classications : 11D61, 35J25

We observe that in problem (1.1)-(1.9)u and v represents the transverse displacement, the relaxation functions g i , i = 1, . . ., 4, are positive and non increasing belonging to W 1,2 (0, ∞) and the function f ∈ C 1 (R) satisfies f (s)s ≥ 0 ∀s ∈ R. (1.10) Additionally, we suppose that f is superlinear, that is for some δ > 0 with the following growth conditions for some C > 0 and ρ ≥ 1 .We shall assume that the function which was proposed by Woinowsky-Krieger [15] as a model for vibrating beams with hinged ends.
The equation (1.14) was studied by Dickey [3] by using Fourier sines series, and also by Bernstein [2] and Ball [1] by introducing terms to account for the effects of internal and external damping.
A larger class of nonlinear beam equations was studied by Rivera [9], which considered the question regarding viscoelastic effects and regularizing properties.Tucsnak [14] considered the beam equation with clamped boundary.He obtained the exponential decay of the energy when a damping of the type a(x)u t is effective near the boundary.In the same direction, Kouemou Patcheu [5] obtained the exponential decay of the energy for (1.15) when a nonlinear damping g(u t ) was effective in Ω.Also see Pazoto-Menzala [10], To Fu Ma [7] and the references therein.As far as we know there is no result concerning the asymptotic stability of solutions for the system (1.1)-(1.9)where the coupledness is nonlinear and boundary conditions are of memory type.So with the intention to fill this gap we consider here this problem.
Remark: The results obtained in the paper are not valid when M (s) = s, being like this the case M(s)=s is an important open problem.
As we have said before we study the asymptotic behavior of the solutions of system (1.1)-(1.9).We show that the energy decays to zero with the same rate of decay as g i .That is, when the relaxation functions g i decays exponentially then the energy decays exponentially.But if g i decays polynomially then the energy will also decay polynomially with the same rate.This means that the memory effect produces strong dissipation capable of making a uniform rate of decay for the energy.The method used here is based on the construction of a suitable functional for some positive constants C 1 , C 2 , γ and α.The notation we use in this paper are standard and can be found in Lion's book [6].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.The organization of this paper is as follows: In section 2 we prove a basic result on existence, regularity and uniqueness of strong solutions for the system(1.1)-(1.9).We use the Galerkin approximation, Aubin-Lions theorem, energy method introduced by Lions [6] and technical ideas to show existence, regularity and uniqueness of strong solutions for the problem (1.1)-(1.9).Finally in the sections 3 and 4 we study the stability of solutions for the system (1.1)-(1.9).We show that the dissipation is strong enough to produce exponential decay of solution, provided the relaxation function also decays exponentially.When the relaxation function decays polynomially, we show that the solution decays polynomially and with the same rate.We use the technique of the multipliers introduced by Komornik [4], Lions [6] and Rivera [8] coupled with some technical lemmas and some technical ideas.

. EXISTENCE AND REGULARITY OF GLOBAL SOLU-TIONS
In this section we shall study the existence and regularity of solutions for the system (1.1)-(1.9).
First, we shall use equations (1.4)-(1.7) to estimate the terms Applying the Volterra's inverse operator, we get where the resolvent kernels satisfy EJQTDE, 2005 No. 6, p. 5 Denoting by η 1 = 1 g 1 (0) and η 2 = 1 g 2 (0) we obtain Note that taking initial data such that u(L, Lemma 2.1 If h is a positive continuous function, then k also is a positive continuous function.
Moreover, 1.If there exist positive constants c 0 and γ with c 0 < γ such that then, the function k satisfies for all 0 < < γ − c 0 .

Given
Therefore which implies our first assertion. Therefore which proves our second assertion.

Remark:
The finiteness of the constant c p can be found in [13,Lemma 7.4 The following lemma states an important property of the convolution operator.
The first order energy of system (1.1)-(1.9) is given by The well-posedness of system (1.1)-(1.9) is given by the following Theorem.
then there exists only one solution (u, v) of the coupled system (1.1)-(1.9)satisfying EJQTDE, 2005 No. 6, p. 8 Proof.Let us solve the variational problem associated with (1.1)-(1.9),which is given by: find for all w ∈ W .This is done with the Galerkin approximations.Let {w j } be a complete orthogonal system of W for which We search for the functions such that for any w ∈ W , it satisfies the approximate equations We note that (2.10)-(2.12)are in fact an m × n system of ODEs in the variable t, which is known to have a local solution (u m (t), v m (t)) in an interval [0, t m [.After the estimate below the approximate solution (u m (t), v m (t)) will be extended to the interval [0, T ], for any given T > 0.

Estimate 1
By substituting of (2.10) and (2.11) with w = u m t (t) and w = v m t (t), respectively, we see that EJQTDE, 2005 No. 6, p. 10 Summing the equations (2.15) and (2.16) and using the hypotheses on f and (1.13) we arrive at Using the Young and Poincare's inequalities we get where E(0; u m (t), v m (t)) is the energy in t = 0. Integrating from 0 to t < t m and using Gronwall's inequality we obtain where C 2 is a positive constant independent of m and t.Therefore, the approximate solution (u m (t), v m (t)) can be extended to the whole interval [0, T ].In particular, there exist Using the compatibility conditions (2.6) and (2.8) and Poincare inequality we obtain Using the hypothesis on f and Sobolev imbedding we arrive at and therefore there exists M 2 > 0 such that Similarly we get Substituting the boundary conditions (2.1) and (2.2) yields 1 2 where EJQTDE, 2005 No. 6, p. 15 We analyzed some terms of (2.22).Let us denote by Integrating by parts we have that Now, from the Mean Value Theorem and estimate (2.17), there exist a constant (2.26) Then using (2.24)-(2.26)and Young inequality, there exists a positive constant C 2 such that (2.27) Putting and using a similar arguments as above yields (2.28) Using Young and Poincaré inequalities and hypothesis on k i we, after some calculations (2.29) similarly we obtain Dividing the above inequality by ε 2 and letting ε → 0 gives and from estimate (2.18) and (2.19) we find a constant M 3 , depending only in T , such that With the estimate (2.17

. EXPONENTIAL DECAY
In this section we shall study the asymptotic behavior of the solutions of system (1.1)-(1.9) when the resolvent kernels k 1 and k 2 are exponentially decreasing, that is, there exist positive Note that this conditions implies that Our point of departure will be to establish some inequalities for the solution of system (1.1)-(1.9).
Let us consider the following binary operator Then applying the Hölder's inequality for 0 ≤ ω ≤ 1 we have Let us introduce the following functionals where θ is a small positive constant.The following Lemma plays an important role for the construction of the Lyapunov functional.
Lemma 3.2 For any strong solution (u, v) of the system (1.1)-(1.9)we get for some positive constant C.

EJQTDE, 2005 No. 6, p. 19
Proof.Differentiating the functional ψ with respect to the time and substituting the equations (1.1) and (1.2) we obtain We analyze some of the terms of the equality (3.3).Integrating by parts we get Since f is superlinear we have Using the hypothesis (1.13) we have Noting that the boundary conditions(2.1)-(2.4)were written as and taking into account that ρ 1 , ρ 2 , ρ 3 , ρ 4 and θ are small, our conclusion follows.
To show that the energy decays exponentially we shall need the following Lemma.Proof.See e. g. [12] Finally, we shall show the main result of this section.
Proof.We shall prove this result for strong solutions, that is, for solutions with initial data (u 0 , v 0 ) ∈ (H 4 (0, L) ∩ W ) 2 and (u 1 , v 1 ) ∈ W 2 satisfying the compatibility conditions.Our conclusion follows by standard density arguments.Using hypothesis (3.1) in Lemma 3.1 we get On the other hand applying inequality (3.2) with ω = 1/2 in Lemma 3.2 we obtain Let us introduce the following functional with N > 0. Taking N large, the previous inequalities imply that

. POLYNOMIAL RATE OF DECAY
Here our attention will be focused on the uniform rate of decay when the resolvent kernels k 1 , k 2 , k 3 and k 4 decay polynomially like (1 + t) −p .In this case we will show that the solution also decays polynomially with the same rate.Therefore, we will assume that the resolvent kernels k i satisfy for some p > 1 and some positive constants b 1 and b 2 .The following lemmas will play an important role in the sequel.
Lemma 4.1 Let (u, v) be a solution of system (1.1)-(1.9)and let us denote by (φ 1 , φ 3 ) = (u x (L, t), u(L, t)) and (ψ 2 , ψ 4 ) = (v x (L, t), v(L, t)).Then, for p > 1, 0 < r < 1 and t ≥ 0, we have while for r = 0 we get Proof.See e. g. [11] Lemma 4.2 Let f ≥ 0 be a differentiable function satisfying for some positive constants c 1 , c 2 , α and β such that Then there exists a constant c > 0 such that Proof.See e. g. [12] Theorem 4.1 Let us take (u 0 , v 0 ) ∈ W and (u 1 , v 1 ) ∈ L 2 (0, L).If the resolvent kernels k i satisfy the conditions (4.1), then there exists a positive constant C such that Proof.We shall prove this result for strong solutions, that is, for solutions with initial data (u 0 , v 0 ) ∈ (H 4 (0, L) ∩ W ) 2 and (u 1 , v 1 ) ∈ W 2 satisfying the compatibility conditions.Our conclusion will follow by standard density arguments.We use some estimates of the previous section which are independent of the behavior of the resolvent kernels k 1 , k 2 , k 3 , k 4 .Using hypothesis (4.1) in Lemma 3.1 yields Applying inequality (3.2) with ω = p+2 2(p+1) and using hypothesis (4.1) we obtain the following estimates 1+ Let us fix 0 < r < 1 such that 1 p+1 < r < p p+1 .From (4.1) we have that   Since (1 − r)(p + 1) > 1 we get, for t ≥ 0, the following bounds Using the above estimates in Lemma 4.1 with r = 0 we get Remark: The techniques in this paper may be used to study the problem (1.1)-(1.9)with moving boundary.This is a very important open problem.In this case, we define others appropriate functionals to prove the exponential and polynomial decay rates of the energy of weak solutions for the problem (1.1)-(1.9).Result concerning the above system in domains with moving boundary will appear in a forthcoming paper.
.13) EJQTDE, 2005 No. 6, p. 2 where M (λ) = λ 0 M (s)ds.Note that because of condition (1.3) the solution of the system (1.1)-(1.9)must belong to the following space W := {v ∈ H 2 (0, 1) : v(0) = v x (0) = 0}.This problem is based on the equation u tt + u xxxx − (α + β L 0 |u x (s, t)| 2 ds)u xx = 0 (1.14) Denoting by (g * ϕ)(t) = t 0 g(t − s)ϕ(s)ds, EJQTDE, 2005 No. 6, p. 4 the convolution product operator and differentiating the equations (1.4) and (1.7) we arrive at the following Volterra equations ) and (2.34) we can use Lions-Aubin lemma to get the necessary compactness in order to pass (2.10)-(2.11) to the limit.That concludes the proof of the existence of global solutions in [0, T ].To prove the uniqueness of solutions of the problem (1.1)-(1.9)we use the method of the energy introduced by Lions[6], coupled with Gronwall's inequality and the hypothesis introduced in the paper about the functions M , f , k i and the obtained estimate.