New scenario of decay rate for system of three nonlinear wave equations with visco-elasticities

Abstract: A system of three semilinear wave equations with strong external forces in Rn is considered. We use weighted phase spaces, where the problem is well defined, to compensate the lack of Poincare’s inequality. Using the Faedo-Galerkin method and some energy estimates, we prove the existence of global solution. By imposing a new appropriate conditions, which are not used in the literature, with the help of some special estimates and generalized Poincaré’s inequality, we obtain an unusual decay rate for the energy function. It is a generalization of similar results in [16,29,31]. The work is relevant in the sense that the problem is more complex than what can be found in the literature.

To enrich our topic, it is necessary to review previous works regarding the nonlinear coupled system of wave equations, from a qualitative and quantitative study. Let us begin with the single wave equation treated in [13], where the aim goal was mainely on the system where Ω is a bounded domain of R n , n ≥ 1 with a smooth boundary ∂Ω. The author firstly constructed a local existence of weak solution by using contraction mapping principle and of course showed the global existence, decay rate and infinite time blow up of the solution with condition on initial energy.
Next, a nonexistence of global solutions for system of three semi-linear hyperbolic equations was introduced in [3]. A coupled system for semi-linear hyperbolic equations was investigated by many authors and a different results were obtained with the nonlinearities in the form f 1 = |u| q−1 |v| q+1 u, f 2 = |v| q−1 |u| q+1 v. (Please, see [2,5,9,14,24,29]).
In the case of non-bounded domain R n , we mention the paper recently published by T. Miyasita and Kh. Zennir in [16], where the considered equation as follows The authors showed the existence of unique local solution and they continued to extend it to be global in time. The rate of the decay for solution was the main result by considering the relaxation function is strictly convex, for more results related to decay rate of solution of this type of problems, please see [6,17,25,26,30,31].
Regarding the study of the coupled system of two nonlinear wave equations, it is worth recalling some of the work recently published. Baowei Feng et al. considered in [10], a coupled system for viscoelastic wave equations with nonlinear sources in bounded domain ((x, t) ∈ Ω × (0, ∞)) with smooth boundary as follows Here, the authors concerned with a system in R n (n = 1, 2, 3). Under appropriate hypotheses, they established a general decay result by multiplication techniques to extends some existing results for a single equation to the case of a coupled system.
It is worth noting here that there are several studies in this field and we particularly refer to the generalization that Shun et al. made in studying a complicate non-linear case with degenerate damping term in [22]. 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.
The lack of existence (Blow up) is considered one of the most important qualitative studies that must be spoken of, given its importance in terms of application in various applied sciences. Concerning the nonexistence of solution for a more degenerate case for coupled system of wave equations with different damping, we mention the papers [19-21, 23, 27].
In m-equations, paper in [1] considered a system where the absence of global solutions with positive initial energy was investigated.
We introduce a very useful Sobolev embedding and generalized Poincaré inequalities.
Lemma 1.1. [16] Let ϑ satisfy (1.2). For positive constants C τ > 0 and C P > 0 depending only on ϑ and n, we have v 2n We assume that the kernel functions 1 we mean by R + the set {τ | τ ≥ 0}. Noting by and We assume that there is a function χ ∈ C 1 (R + , R + ) such that for any ξ ≥ 0. Hölder and Young's inequalities give and and Thanks to Minkowski's inequality to give Then there exist η > 0 such that We need to define positive constants λ 0 and E 0 by The mainely 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 3.1.

Local and global existence
The next Theorem is concerned on the local solution (in time [0, T ]).
We prove the existence of global solution in time. Let us introduce the potential energy J : for any w ∈ L 2 (R n ), j = 1, 2, 3. The modified energy is defined by The next, Lemma will play an important role in the sequel.
We sketch here the outline of the proof for local solution by a standard procedure(See [4,11,31]).
For any (u, v, w) ∈ X 3 T , we can obtain weak solution of the related system We reduces problem (2.5) to Cauchy problem for system of ODE by using the Faedo-Galerkin approximation. We then find a solution map : (u, v, w) → (z, y, ζ) from X 3 T to X 3 T . We are now ready show that is a contraction mapping in an appropriate subset of X 3 T for a small T > 0. Hence has a fixed point (u, v, w) = (u, v, w), which gives a unique solution in X 3 T . We will show the global solution. By using conditions on functions 1 , 2 , 3 , we have Noting that E 0 = G(λ 0 ), given in (1.19). Then Proof. Since 0 ≤ E(0) < E 0 = G(λ 0 ), there exist ξ 1 and ξ 2 such that G(ξ 1 ) = G(ξ 2 ) = E(0) with 0 < ξ 1 < λ 0 < ξ 2 .

Decay estimates
The decay rate for solution is given in the next Theorem there exists t 0 > 0 depending only on 1 , 2 , 3 , λ 1 and χ (0) such that