Decay rate for systems of m-nonlinear wave equations with new viscoelastic structures

Abstract: The article discusses the effect of weak and strong damping terms on decay rate for systems of nonlinear mwave equations with new viscoelastic structures. The factors that allowed system (1.1) to coexist for a long time are the strong nonlinearities in the sources. We showed, under a novel condition on the kernel function in (2.4), a new scenario for energy decay in (3.7) by using an appropriate energy estimates. This result extend the results in [18, 27] for system of m-equations inspired from the paper [1].

Various non-linear sources have been combined as follows, we combine all two consecutive equations together and of course the last equation with the first one, which get the whole system closely linked by the strong nonlinear sources. The functions f j (u 1 , u 2 , . . . , u m ) ∈ (R m , R) are given for In order to use Poincare's inequality which is a key in calculus for the PDEs, we will study the problem (1.1) in the presence of a density function θ to find a generalized formula for Poincare's inequality that can be used in unbounded domain R n . The function Θ(x) > 0 for all x ∈ R n is a density and (Θ) −1 (x) = 1/Θ(x) ≡ θ(x) such that θ ∈ L τ (R n ) with τ = 2n 2n − rn + 2r for 2 ≤ r ≤ 2n n − 2 . (1.4) We define a new space related to the nature of our system, taking into account the boundless of space R n . The function spaces H is defined as the closure of C ∞ 0 (R n ), as in [20], we have is the norm of the weighted space L r θ (R n ). The following references in connection to our system for a single equation [6] and [7]. The work [6] was the pioneer in the literature for the single equation, source of inspiration of several works, while the work [7] is a recent generalization of [6] by introducing less dissipative effects (See [8,9,19,24,26]). With regard to the study of this type of systems without viscoelasticity, with the existence of both weak damping u t and strong damping ∆u t , under condition (3.2), we mention the work recently published in one equation in [14] where Ω is a bounded domain of R n , n ≥ 1 with a smooth boundary ∂Ω. The aim goal was mainly on the local existence of weak solution by using contraction mapping principle and of course the authors showed the global existence, decay rate and infinite time blow up of the solution with certain conditions on initial energy. In the case of non-bounded domain R n , we mention the paper recently published by T. Miyasita and Kh. Zennir in [18], where they considered equation as follows The authors succeeded in highlighting the existence of unique local solution and they continued to expand it to be global in time. The rate of the decay for solution was the main result, for more results related to decay rate of solution of this type of problems, please see [15,23,25,28].
Regarding the study of the coupled system of two nonlinear wave equations, we mention the work done by Baowei Feng et al. which was considered in [12], a coupled system for viscoelastic wave equations with nonlinear sources in bounded domain with smooth boundary as follows Under appropriate hypotheses, they established a general decay result by multiplication techniques to extend some existing results for a single equation to the case of a coupled system. There are several results in this direction, notably the generalization made by Shun in a complicate nonlinear case with degenerate damping term in [21]. The IBVP for a system of nonlinear viscoelastic wave equations in a bounded domain was considered in the problem where Ω is a bounded domain with a smooth boundary. Given certain conditions on the kernel functions, degenerate damping and nonlinear source terms, they got a decay rate of the energy function for some initial data. In n−equations, paper in [1] considered a system where the absence of global solutions with positive initial energy was investigated. Next, a nonexistence of global solutions for system of three semilinear hyperbolic equations was introduced in [3]. A coupled system of semilinear hyperbolic equations was investigated by many authors and many results were obtained with the nonlinearities in the form

Preliminaries
We introduce the Sobolev embedding and generalized Poincaré inequalities.
In the 1950s and 1970s, the linear theory of viscoelasticity was extensively developed and now, it becomes widely used to represent this term using several improvements to the nature of decreasing the kernel function. We assume that the kernel functions j ∈ C 1 (R + , R + ) satisfying We mean by R + the set {τ | τ ≥ 0}. Noting by and µ 0 (t) = min We assume that there is a function χ ∈ C 1 (R + , R + ) which is linear or is strictly convex C 2 function on (0, ε 0 ), ε 0 ≤ j (0), with χ(0) = χ (0) = 0 and a positive nonincreasing differentiable function ξ : [0, ∞) → [0, ∞), such that the novel properties satisfied for any ≥ 0. We note that, if χ is a strictly increasing convex C 2 − function on (0, τ] with χ(0) = χ (0) = 0, then χ has an extensionχ, which is strictly increasing and strictly convex C 2 −function on (0, ∞). For example,χ can be given bȳ Hölder and Young's inequalities give We need to define positive constants λ 0 and E 0 by The mainly aim of the present paper is to obtain a novel decay rate of solution from the convexity property of the function χ given in Theorem 3.4. We denote, as in [18], an eigenpair for any i ∈ N. Then 0 < λ 1 ≤ λ 2 ≤ · · · ≤ λ i ≤ · · · ↑ +∞, holds and {e i } is a complete orthonormal system in H.

Statement of main results
The local solution (in time [0, T ]) is given in next Theorem. We will show now the global solution in time established in Theorem 3.3. Let us introduce the potential energy J : H m → R defined by The modified energy is defined by for any w ∈ L 2 (R n ), j = 1, 2, . . . , m.  u 21 ), . . . , (u m0 , u m1 ) ∈ H ×L 2 θ (R n ), problem (1.1) admits a unique global solution (u 1 , u 2 , . . . , u m ) such that The decay rate for solution is given in the next Theorem.
there exists t 0 > 0 depending only on j , a, ω, λ 1 and χ (0) such that holds for all t ≥ t 0 .
In particular, by the positivity of µ in (2.2), we have, as in [17], for a single wave equation.
Proof. For 0 ≤ t 1 < t 2 ≤ T , we have We define an inner product as and the associated norm is given by ∀v, w ∈ H. By (3.2), we get The following Lemma yields.

Proof of existence results
We give here the outline of the proof for local solution by a standard procedure (See [15,28]).
Proof. (Of Theorem 3.1.) Let (u 10 , u 11 ), (u 20 , u 21 ), . . . , (u m0 , u m1 ) ∈ H × L 2 θ (R n ). The presence of the nonlinear terms in the right hand side of our problem (1.1) gives us negative values of the energy. For this purpose, for any fixed (u 1 , u 2 , . . . , u m ) ∈ X m T , we can obtain first, a weak solution of the related system The Faedo-Galerkin's method consist to construct approximations of solutions (z 1n , z 2n , . . . , z mn , ) for (4.1), then we obtain a prior estimates necessary to guarantee the convergence of approximations. In the last step we pass to the limit of the approximations by using the compactness of some embedding in the Sobolev spaces. The uniqueness is obtain by letting two solutions for (4.1) and then, after ordinary calculations, we find that the solutions are equal. Some details regarding the transition to ODE systems are given, for this end let {e i } be the Galerkin basis and let W jn = span{e j1 , e j2 , ...., e jn }, j = 1, . . . , m.
Given initial data u j0 ∈ H, u j1 ∈ L 2 θ (R n ), we define the approximations which satisfy the following approximate problem with initial conditions z jn (x, 0) = u n j0 (x), z jnt (x, 0) = u n j1 (x), (4.4) which satisfies u n j0 → u j0 , strongly in H u jn 1 → u j1 , strongly in L 2 θ (R n ). (4.5) Taking e ji = g ji in (4.3) yields the following Cauchy problem for a ordinary differential equation with unknown g n ji .
g n jitt (t) + ag n jit (t) + λ i g n ji (t) + ωg n jit (t) By using the Caratheodory Theorem for standard ordinary differential equations theory, the problem from X m T to X m T . We are now ready to show that is a contraction mapping in an appropriate subset of X m T for a small T > 0. Hence has a fixed point which gives a unique solution in X m T .
We are ready to prove the decay rate. Let Then, we take t 0 > 0 such that Then, by definition of I(t), we have and Lemma 3.5, we have for all t 1 , t 2 ≥ 0  which completes the proof.

Conclusions
The paper deals with a kind of m−nonlinear wave equations with viscoelastic structures. We considered the local existence, global existence and exponential decay rate of solution. We discussed the effects of weak and strong damping terms on decay rate. The methods are standard for local existence and we extended the local solution to a global one by appropriate energy estimates. At last, We obtained a novel decay rate of solution from the convexity property of the function which extends the results in [Math. Meth. Appl. Sci., 43(3), 1138 (2020); Mathematics 8(2), 203 (2020)]. The treatment of Cauchy problem for a family of effectively damped single wave models with a nonlinear memory on the righthand side, that is for x ∈ R n u tt + (1 + t) r u t − ∆ (u + ωu t ) = t 0 (t − s) −γ u(s, .) p ds (6.1) where ω > 0, p > 1, r ∈ (−1, 1) and γ ∈ (0, 1), remains as an open problem, which will be our next work, based on [4,5,10,11].