Global existence and exponential decay of solutions for generalized coupled non-degenerate Kirchhoff system with a time varying delay term

The paper studies a system of nonlinear viscoelastic Kirchhoff system with a time varying delay and general coupling terms. We prove the global existence of solutions in a bounded domain using the energy and Faedo–Galerkin methods with respect to the condition on the parameters in the coupling terms together with the weight condition as regards the delay terms in the feedback and the delay speed. Furthermore, we construct some convex function properties, and we prove the uniform stability estimate.


Introduction
The Kirchhoff equation belongs to the famous wave equation's models describing the transverse vibration of a string fixed in its ends. It has been introduced in 1876 by Kirchhoff [8] and it is more general than the D' Alembert equation. In one dimensional space it takes the following form: where the function u(x, t) is the vertical displacement at the space coordinate x, varying in the segment [0, L] and over time t > 0, ρ is the mass density, h is the area of the cross section of the string, P 0 is the initial tension on the string, L is the length of the string and E is the Young modulus of the material. The nonlinear coefficient is obtained by the variation of the tension during the deformation of the string. When we do not have an initial tension (i.e. P 0 = 0), we call that a degenerate case as opposed to the non-degenerate case.
In this paper, we are interested in studying, in A = Ω × (0, ∞), the following coupled viscoelastic Kirchhoff system: +∞[, (u(x, 0), v(x, 0)) = (u 0 (x), v 0 (x)), (u t (x, 0), v t (x, 0)) = (u 1 (x), v 1 (x)) in Ω, , v t (x, tτ (0))) = (f 0 (x, tτ (0)), g 0 (x, tτ (0))) in Ω × ]0, τ (0)[, (1.2) in which Ω is an n dimensional bounded domain of R n and we have a smooth boundary Γ , l > 0, μ 1 and μ 2 are positive real constants, h 1 and h 2 are positive functions with exponential decay, and τ (t) is a positive time varying delay. In addition the initial condition (u 0 , v 0 , u 1 , v 1 , f 0 , g 0 ) will be specified in their function space later. M is a smooth function defined by M : R + − → R, + r − → M(r) = a + br γ , with a, b > 0, and γ ≥ 1. f 1 and f 2 are two functions taking a particular form that we will make precise later. The problem (1.2) is a description of axially moving viscoelastic strings composed of two different materials (like the wires of electricity) that are nonhomogeneous and which will be of influence on its moving, specially on the acceleration. From the mathematical point of view, this influence is represented by |w t | l w , where |w t | l is the material density, varying the velocity. A lot of work has been published with this term, for example see [11] and [14], where we find different results about the global existence and nonexistence of solutions and the decay of energy.
In recent years, the study of wave equations with delay has become an active area and with different forms of delay (constant [7], switching [5], varying in time [12], distributed [6]). The delay appears in modeling of a lot of domains, like the physical, chemical, biological and engineering domains. It is introduced when we have a time lag between an action on a system and a response of the system to this action. Furthermore, a delay can be small enough in feedback yet can destabilize a system [10], or improve the performance of the system [17].
In the absence of delay, Cavalcanti et al. [3] studied the following viscoelastic wave equations with strong damping: They used the Fadeo-Galerkin method to prove the global existence of a solution; also an explicit decay rate of the energy has been given provided m > 0.
In the other hand, in the same case and for l = 0, Raslan et al. [16] and El-Sayed et al. [4] have studied coupled equal width wave equations with strong damping, as they were looking for the new exact solution.
The problem treated in [2] has the following form: Under the assumptions set on g 1 , g 2 , σ and τ , the authors have gotten the global existence of a solution and the decay rate of the energy.
Recently, Mezouar and Boulaaras [13] have studied the viscoelastic non-degenerate Kirchhoff equation with varying delay term in the internal feedback.
In the present paper, we extend our recently published paper in [13] for a coupled system (1.2). The famous technique of using the presence of a delay in the PDE problem is to set a new variable defined by a velocity dependent on the delay, which will give us a new problem equivalent to our studied problem; but the last one is a coupled system without delay.
After this, we can prove the existence of global solutions in suitable Sobolev spaces by combining the energy method with the Fadeo-Galerkin procedure and under the choice of a suitable Lyapunov functional, we establish an exponential decay result.
The outline of the paper is as follows: In the second section, some hypotheses related to the problem are given and we state our main result. Then in the third section, the global existence of weak solutions is proven. Finally, in the fourth section, we give the uniform energy decay.

Preliminaries and assumptions
Similar to that [12], we present the new variables Then we have In the same way, we have Throughout this work and for simplifying our formulas, we will adopt the notation z i , u and v instead of z i (x, ρ, t), u(x, t) and v(x, t), except if that makes things inconvenient.
In order to demonstrate the main result in this paper, a few assumptions are needed.
in the case n > 2, γ < ∞ in the case n ≤ 2.
(A-2) As regards the relaxation functions h i : R + → R + we see that they are bounded C 1 functions such that We assume also that there exist some positive constants ζ i verifying (p + 1)(p + q), b 2 = (q + 1)(p + q) such that p and q are conjugate (i.e. 1 p + 1 q = 1), p, q < γ -1 2 and satisfy The energy related to the system solution of (1.5) is defined as follows: where ξ is a positive constant such that and
Then there exists a constant C s = C s (Ω, q) such that u q ≤ C s ∇u 2 for u ∈ H 1 0 (Ω).
We present the following lemma.

Assume that (A1)-(A3) hold. Then the problem (1.2) admits a weak solution such that
Proof As in the previous assumptions in [2] for the initial conditions ) and the basic functions, we introduce the approximate solutions (u k , v k , z k 1 , z k 2 ), k = 1, 2, 3, . . . , in the form where a jk , b jk , c jk and d jk (j = 1, 2, . . . , k) are determined by the following ordinary differential equations: Noting that l 2(l+1) + 1 2(l+1) + 1 2 = 1, by applying the generalized Hölder inequality, we find Since (A1) holds, according to the Sobolev embedding the nonlinear terms (|u k t | l u k tt , w j ) and (|v k t | l v k tt , w j ) in (2.1) make sense (see [2]). A. First estimate. Since the sequences u k 0 , v k 0 , u k 1 , v k 1 , z k 1 (ρ, 0) and z k 2 (ρ, 0) converge and from Lemma 1.3 with employing Gronwall's lemma, we find C 1 > 0 independent of k such that Noting (A1) and the estimate (2.9) yields B. The second estimate. By multiplying the first side of equation (respectively, the second equation) in (2.1) by a jk tt (respectively, by b jk tt ), by summing j from 1 to k, then (2.13) Differentiating (2.6) with respect to t, we get Multiplying the first equation by c jk t (respectively the second equation by d jk t ), summing over j from 1 to k, we have Integrating over (0, 1) with respect to ρ, we obtain (2.14) Summing (2.13), (2.14) and as M(r) ≥ a, we get We estimate the right hand side of (2.15) as follows: From the integration by parts, we have Using the inequality ab ≤ 1 2 a 2 + 1 2 b 2 and Sobolev-Poincaré inequalities, we obtain On the other hand, by recalling (A-4) and Lemma 1.1 and using Young's inequality, we get Also by Young's inequality, we get (2.20) We have By using (A3) and taking the first estimate (2.9) into account, we infer where C 2 is a positive constant that depends on η, α, a, C s , |Ω|, b 1 , b 2 , p, q, C 1 for i = 1, 2. Integrating (2.24) over (0,t) we obtain For a suitable η > 0 such that 1 -(η(μ 2 i + 2) + (1+b i )C 2 s 2 ) > 0 for i = 1, 2, we obtain the second estimate t 0 ∇u k tt (s) 2 + ∇v k tt (s) 2 ds We observe from the estimate (2.9) and (2.25) that there exist subsequences (u m ) of (u k ) and (v m ) of (v k ) such that In the following, we will treat the nonlinear term. From the first estimate (2.9) and Lemma 1.1, we deduce where C 4 depends only on C s , C 1 , T, l.
On the other hand, from the Aubin-Lions theorem (see Lions [9]), we deduce that there exists a subsequence of (u m ), still denoted by (u m ), such that we have As we add and subtract |v k | q+1 |u| p u to the previous formula, we obtain We use the following elementary inequalities: for some constant C, ∀k ≥ 1 and ∀a, b ∈ R. Hence (2.38) becomes The typical term in the above formula can be estimated as follows. Noting that l 2p + 1 2q + 1 2 = 1, by applying the generalized Hölder inequality, we find Recalling (A4), Lemma 1.1 and (2.9), we get (2.41) Hence (2.39) yields As (u m ), (v m ) are Cauchy sequences in L ∞ (0, T, H 1 0 (Ω)) (we prove it as in [1]) then we deduce (2.36). Similarly we get the convergence (2.37).

Exponential decay rate
In order to make precise the asymptotic behavior of our solutions, we introduce some functionality to determine a suitable Lyapunov functional equivalent to E. Theorem 3.1 Assume that (A1)-(A3) hold. Then for every t 0 > 0 there exist positive constants K and c such that the energy defined by (1.6) obeys the following decay: (3.1)

Lemma 3.3 Along a solution of the problem (1.5) the functional
and Proof (i) Applying Young's inequality, Sobolev-Poincaré's inequality and L l+2 → L 2 , we find (ii) Taking a direct derivation of (3.2) and replacing |u t | l u tt , |v t | l v tt from the first and seconde equations of (1.5) give As M(r) ≥ a and making use of Young's inequality we obtain By use of Young's inequality, the third term in the right side is estimated as follows: Similarly and from (A4) Thus, (3.6) is valid.  (3.9) and

Lemma 3.4 Along a solution of the problem (1.5) the functional
where δ > 0 and c s is the Sobolev embedding constant.
Proof We have We use Young's inequality with the conjugate exponents p = l+2 l+1 and q = l + 2, then the second term in the right hand side can be estimated as We get by using Hölder's inequality Combining (3.12) with (3.11) we obtain In the same way, we get (3.14) Similarly (3.15) Combining (3.13),(3.14) and (3.15), we deduce (i).
(ii) We use the Leibnitz formula and the first and second equations of (1.5) to find  Next we will estimate I 1 , . . . , I 6 .