Existence and Decay of Solution to Coupled System of Viscoelastic Wave Equations with Strong Damping in R n

: In this paper, we establish a general decay rate properties of solutions for a coupled system of viscoelastic wave equations in R n under some assumptions on g 1 ,g 2 and linear forcing terms. We exploit a density function to introduce weighted spaces for solutions and using an appropriate perturbed energy method. The ques-tions of global existence in the nonlinear cases is also proved in Sobolev spaces using the well known Galerkin method.

t 0 g 2 (t − s)u 2 (s, x)ds + u ′ 2 = 0, (u 1 (0, x), u 2 (0, x)) = (u 10 (x), u 20 (x)) ∈ (D(R n )) 2 , (u ′ 1 (0, x), u ′ 2 (0, x)) = (u 11 (x), u 21 (x)) ∈ (L l ρ (R n )) 2 , where α = 0, x ∈ R n , t ∈ R + * where the space D(R n ) defined in (2.4) and l, n ≥ 2, φ(x) > 0, ∀x ∈ R n , (φ(x)) −1 = ρ(x) defined in (A2 ). This type of problems is usually encountered in viscoelasticity in various areas of mathematical physics, it was first considered by Dafermos in [6], where the general decay was discussed. The problems related to (1.1) attract a great deal of attention in the last decades and numerous results appeared on the existence 32 Keltoum Bouhali and Fateh Ellaggoune and long time behavior of solutions but their results are by now rather developed, especially in any space dimension when it comes to nonlinear problems. The term t 0 g i (t − s)∆u i (s)ds corresponds to the memories terms and the scalar functions g i (t) (so-called relaxation kernel) is assumed to satisfy (2.1)-(2.3). The energy of (u 1 , u 2 ) at time t is defined by For α small enough we use Lemma 2.3, we deduce that: and the following energy functional law holds ∇u ′ i 2 2 , ∀t ≥ 0. (1.4) which means that, our energy is uniformly bounded and decreasing along the trajectories.
The following notation will be used throughout this paper (g • Ψ)(t) = t 0 g(t − τ ) Ψ(t) − Ψ(τ ) 2 2 dτ , for any Ψ ∈ L ∞ (0, T ; L 2 (R n )) (1.5) In the present paper we consider the solutions in an appropriate spaces weighted by the density function ρ(x) in order to compensate the lack of Poincare's inequality which play a decisive role in the proof. To motivate our work, we start with some results related to viscoelastic plate equations with strong damping in [23]: supplemented with the following conditions: u(t, x) = ∆u = 0, on ∂Ω × R + , u(0, x) = u 0 , u t (0, t) = u 1 , on Ω. (1.6) In this paper, Liu and all extend the exponential rate result obtained in [1] to the general case and show that the rate of decay for the solution is similar to that of the memory term under the following assumption for the function g is Coupled System of Viscoelastic Wave Equations with Strong Damping in R n 33 Paper [8] is concerned with a class of plate equations with memory in a history space setting and perturbations of p−Laplacian type for x ∈ Ω × R + , and results on the well-posedness and asymptotic stability of the problem were proved. In many existing works on this field, the following conditions on the kernel is crucial in the proof of the stability. For a viscoelastic systems with oscillating kernels, we mention the work by Rivera and all [17], the authors proved that if the kernel satisfies g(0) > 0 and decays exponentially to zero, that is for p = 1 in (1.8), then the solution also decays exponentially to zero. On the other hand, if the kernel decays polynomially, i.e. (p > 1) in the inequality (1.8), then the solution also decays polynomially with the same rate of decay. Recently the problem related to (1.1) in a bounded domain Ω ⊂ R n , (n ≥ 1) with a smooth boundary ∂Ω and g is a positive nonincreasing function was considered as equation in [15], where they established an explicit and very general decay rate result for relaxation functions satisfying: for a positive function H ∈ C 1 (R + ) and H is linear or strictly increasing and strictly convex C 2 function on (0, r], 1 > r. For the literature, In R n , we quote essentially the results of [2], [3], [4], [9]- [13], [15]- [20] and the references therein. In [10], authors showed for one equation that, for compactly supported initial data and for an exponentially decaying relaxation function, the decay of the energy of solution of a linear Cauchy problem (1.1) without strong damping in the case l = 2, ρ(x) = 1, is polynomial. The finitespeed propagation is used to compensate the lack of Poincare's inequality. In the case l = 2, in [9], author looked into a linear Cauchy viscoelastic equation with density. His study included the exponential and polynomial rates, where he used the spaces weighted by density to compensate the lack of Poincare's inequality in the absence of strong damping. The same problem treated in [9], was considered in [11], where under suitable conditions on the initial data and the relaxation function, they prove a polynomial decay result of solutions. The conditions which used, on the relaxation function g and its derivative g ′ are different from the usual ones. Coupled systems in R n , we mention, for instance, the work of [Takashi Narazaki, 2009. Global solutions to the Cauchy problem for the weakly coupled system of damped wave equations. Discrete And Continuous Dynamical Systems, 592-601], where the following weakly coupled system of a damped wave equations has considered: (1.9) Authors have shown the sufficient condition under which the Cauchy problem (1.9) admits global solutions when n = 1, 2, 3 provided that the initial data are sufficiently small in an associate space. Moreover, they have also shown the asymptotic behavior of the solutions and to generalize the existence result in [22] to the case n = 1, 2, 3 and improve time decay estimates when n = 3.

Function spaces and statements
In this section we introduce some notation and results needed for our work. We omit the space variable x of u(x, t), u ′ (x, t) and for simplicity reason denotes u(x, t) = u and u ′ (x, t) = u ′ , when no confusion arises. The constants c used throughout this paper are positive generic constants which may be different in various occurrences also the functions considered are all real-valued. Here u ′ = du(t)/dt and u ′′ = d 2 u(t)/dt 2 . We denote by B R the open ball of R n with center 0 and radius R. First we recall and make use the following assumptions on the functions ρ and g for i = 1, 2 as: (A1 ) We assume that the function g i : R + −→ R + (for i = 1, 2) is of class C 1 satisfying: and there exist nonincreasing continuous functions ξ 1 ,ξ 2 : [19]). We define the function spaces of our problem and its norm as follows: 4) and the spaces L 2 ρ (R n ) to be the closure of C ∞ 0 (R n ) functions with respect to the inner product: Coupled System of Viscoelastic Wave Equations with Strong Damping in R n 35 The space L 2 ρ (R n ) is a separable Hilbert space. So, we are able to construct the necessary evolution triple for the space setting of our problem, which is: where all the embedding are compact and dense. The following technical Lemma will play an important role in the sequel.
Corollary 2.4. If q = 2, the Lemma 2.3. yields To study the properties of the operator φ∆, we consider as in [13], the equation and L 2 ρ (R n ) are defined with respect to the inner product (2.5), we may consider equation (2.10) as operator equation: Let us note that the operator φ∆ is not symmetric in the standard Lebesgue space L 2 (R n ), because of the appearance of φ(x) (see [ [21], pages 185-187]). From (2.9) and (2.11) we have From (2.11) and (2.12) we conclude that ∆ 0 is a symmetric, strongly monotone operator on L 2 ρ (R n ). The energy scalar product is given by: and the energy space is the completion of D(∆ 0 ) with respect to (u, v) E . It is obvious that the energy space X E is the homogeneous Sobolev space D(R n ). The energy extension ∆ E , namely is defined to be the duality mapping of D(R n ). For every η ∈ D −1 (R n ) the equation (2.10), has a unique solution. Define D(∆ 1 ) to be the set of all solutions of the equation (2.10) for arbitrary η ∈ L 2 ρ (R n ). The operator extension ∆ 1 of ∆ 0 , [see [24], Theorem 19.C] is the restriction of the energy extension ∆ E to the set D(∆ 1 ). The operator ∆ 1 is self-adjoint and therefore graph-closed. Its domain is a Hilbert space with respect to the graph scalar product which is equivalent to the norm Coupled System of Viscoelastic Wave Equations with Strong Damping in R n 37 So, we have established the evolution quartet where all the embedding are dense and compact. A consequence of the compactness of the embedding in (2.13) is that the eigenvalue problem has a complete system of eigenfunctions {w n , µ n } with the following properties: It can be shown, as in [4], that every solution of (2.14) is such that uniformly with respect to x. Finally, we give the definition of weak solutions for the problem (1.1).
We are now ready to state and prove our existence results.

Well-posedness result for nonlinear case
This section is devoted to prove the existence and uniqueness of solutions to the system (1.1) taking account the nonlinear case in the terms responsible on the relation between tow equations, that is replacing αu 1 , αu 2 by f 1 (u 1 , u 2 ), f 2 (u 1 , u 2 ) introduced in the last section. First, we prove the existence of the unique solution of the restricted problem on B R , the main ingredient used here is the Galerkin approximations introduced in [14].
Proof: The existence is proved by using the Galerkin method, which consists in constructing approximations of the solution, then we obtain a priori estimates necessary to guarantee the convergence of these approximations. So, we take be the eigen-functions of the operator −∆.
and the projection of the initial data on the finite dimensional subspace V m is given by: We search approximate solutions (3.1) Based on standard existence theory for differential equations, one can conclude the existence of solution (u m 1 , u m 2 ) of (3.1) on a maximal time interval [0, t m ), for each m ∈ N.

2)
Coupled System of Viscoelastic Wave Equations with Strong Damping in R n 39 This means, using (A1 ), that for some positive constant C independent of t and m, we have Then, integrating over (0, t) yields To estimate the terms on the right hand side of (3.6), we use (5.2)-(5.4), Young's inequality and (2.9) and take (3.3) into account to get Since 1 ≤ β ij , i, j = 1, 2. Now, we estimate First, we observe that and use (A2 ) and the generalized Hölde's inequality to infer Then, by (2.9), (3.3) and Young's inequality, we arrive at Since the other terms in (3.6) can be similarly treated and the norms of the initial data are uniformly bounded, we combine (3.6), (3.7), use (A1 ) and take δ small enough to end up with ). In the sequel, we will deal with the nonlinear term. By Aubin's Lemma (see [14]), we find, up to a subsequence, that Then, and therefore, from (5.5), (5.6), , then the use of (5.2)-(5.6) gives that ), for i = 1, 2. Combining the results obtained above, we can pass to the limit and conclude that (u 1 , u 2 ) is the solution of system (1.1) restricted un B R . ✷ In the next result, we will extend our solutions to R n . Proof: (a) Existence. Let R 0 > 0 such that supp(u 10 , u 20 ) ⊂ B R0 and supp(u 11 , u 21 ) ⊂ B R0 . Then, for R ≥ R 0 , R ∈ N, we consider the approximating problem (3.14) 42 Keltoum Bouhali and Fateh Ellaggoune By Lemma 3.1, problem (3.14) has a unique solution u R i such that We extend the solution of the problem (3.14) as The solution (u R 1 , u R 2 ) satisfies the estimates where the constant K is independent of R. The estimates (3.16) imply that Next using relations (3.16) and (3.17), the continuity of the embedding and the continuity of f i we may extract a subsequence of u R i , denoted by u Rm i , such that as R m → ∞ we get For fixed R = R m , let L m denote the operator of restriction It is clear that the restricted subsequence L m u Rm i satisfies the estimates obtained in Lemma 3.1. Therefore there exists a subsequence u Rm j i = u j i for which it can be shown by following the procedure of Lemma 3.1, that L m u j i converges weakly to solution u m i . We have Passing to the limit in (3.19) as j −→ ∞, we obtain that L m u i = u m i . The equalities (3.19) hold for any v ∈ C ∞ 0 ([0, T ] × R n ) since the radius R is arbitrarily chosen. Therefore u i is a solution of the problem (3.14). (b) Uniqueness. Let us assume that (u 11 , u 21 ), (u 12 , u 22 ) are two strong solutions of (1.1). Then, (z 1 , z 2 ) = (u 11 − u 12 , u 21 − u 22 ) satisfies, for all w ∈ D(R n ) (3.20) Substituting w = z ′ 1 in the first equation and w = z ′ 2 in the second equation, adding the resulting equations, integrating by parts and using (A1 ), yield Making use of (5.6) and following similar arguments that used to obtain (3.7), we Combining (3.20)-(3.21), integrating over (0, t) and using Gronwall's Lemma, then we deduce that which means that (u 11 , u 21 ) = (u 12 , u 22 ). This completes the proof. ✷ We can now state and prove the asymptotic behavior of the solution of (1.1).

Decay rate for linear cases
We show that our solution decays time asymptotically to zero and the rate of decay for the solution is similar to that of the memory terms, making some small perturbation in the associate energy, for this purpose, we introduce the functional The following Lemma will be useful in the proof of our next result.
for positive constants c.
Coupled System of Viscoelastic Wave Equations with Strong Damping in R n 45 Proof: From (4.1), integrate by parts over R n , we have Recalling that t 0 g i (s)ds ≤ ∞ 0 g i (s)ds = 1 − k i , using Young's inequality, Lemma 2.3 and Lemma 2.2, we obtain For α small enough and k = min{k 1 , k 2 }. ✷ Our main result reads as follows.
In order to prove this theorem, let us define for N 1 > 1, we need the next lemma, which means that there is equivace between the perturbed energy and energy functions.
holds for some positive constants β 1 and β 2 .
Proof: By (4.1) and (4.4), we have Thanks to Hölder's and Young's inequalities with exponents l l−1 , l, since 2n n+2 ≥ l ≥ 2, we have by using Lemma 2.3 Then, since l ≥ 2, we have by using (1.4) Consequently, (4.5) follows. ✷ At this point, we choose N 1 large and ε so small such that Multiplying (4.7) by ξ(t) gives The last term can be estimated, using (A1 ) as follows Thus, (4.7) becomes Using the fact that ξ is a nonincreasing continuous function as ξ 1 and ξ 2 are nonincreasing and so ξ is differentiable, with ξ ′ (t) ≤ 0 for a.e t, then Since, using (4.5) we obtain, for some positive constant ω Integration over (0, t) leads to, for some constant ω > 0 such that (4.14) Recalling (4.12), estimate (4.14) yields the desired result (4.3). This completes the proof of Theorem (4.2).

Concluding comments
1-One can easily obtain the same result in Theorem (4.2) in the nonlinear case 1) where our nonlinearity is given by the functions f 1 , f 2 satisfying the next assumptions: (hyp1 ) The functions f i : R 2 → R (for i=1,2) is of class C 1 and there exists a function F such that and for some constant d > 0 and 1 ≤ β ij ≤ n n−2 for i, j = 1, 2. (hyp2 ) There exists a positive constant k such that and for all (x, y), (r, s) ∈ R 2 and i = 1, 2. Noting that we follow the same steps in the linear cases with the same perturbed function and some calculations related with the presence of f 1 , f 2 .
For the reader we shall develop here the next important technical Lemma.

dxds.
Proof: It's not hard to see