Global Existence and Decay of Solutions for Coupled NondegenerateKirchhoff Systemwith aTimeVaryingDelay Term

This paper deals with the global existence of solutions in a bounded domain for nonlinear viscoelastic Kirchhoff system with a time varying delay by using the energy and Faedo–Galerkin method with respect to the delay term weight condition in the feedback and the delay speed. Furthermore, by using some convex functions properties, we prove a uniform stability estimate.


Introduction
1.1. Model. Consider the following viscoelastic Kirchhoff system: where Ω is a bounded domain in IR n , n ϵ IN * , with a smooth boundary zΩ, l > 0, α, μ 1 and μ 2 are positive real numbers, h 1 and h 2 are positive functions which decay exponentially, τ(t) > 0 is a time varying delay, and the initial data (u 0 , v 0 , u 1 , v 1 , f 0 , g 0 ) are in a suitable function space.
M(r) � a + br c is a C 1 -function for r ≥ 0, with a, b > 0, and c ≥ 1. Time delay is often present in applications and practical problems. In last few years, the control of PDEs with time delay effects has become an active area of research (see, for example, [1][2][3][4] and the references therein). In [5], the authors showed that a small delay in a boundary control could turn a well-behaved hyperbolic system into a wild one and therefore delay becomes a source of instability. However, sometimes it can also improve the performance of the system.
By using the Faedo-Galerkin method, Wu in [6] proved the result of local existence and established the decay result by suitable Lyapunov functionals according to appropriate conditions on μ 1 , μ 2 and on the kernel h. Daewook [7] studied the following viscoelastic Kirchhoff equation with nonlinear source term and varying time delay: in Ω ×]0, +∞[, (2) which is a description of axially moving viscoelastic materials. According to the smallness condition taking into account of Kirchhoff coefficient and the relaxation function and by summing 0 ≤ m ≤ (2/(n − 2)) if n > 2 or 0 ≤ m if n ≤ 2, he got the uniform decay rate of the Kirchhoff type energy.
Very recently, in [1], we have proved the global existence and energy decay of solutions of the following viscoelastic nondegenerate Kirchhoff equation: with respect to some proposed assumptions. Under assumption setting on g 1 , g 2 , σ, and τ, the authors have obtained the global existence of solution and the decay rate of energy.
Recently, Mezouar and Boulaaras [1] have studied viscoelastic nondegenerate Kirchhoff equation with varying delay term in the internal feedback.
In the present paper, we extent our recent published paper in [1] for a coupled system (3). e famous technique using the presence of delay in PDE's problem is to set a new variable defined by velocity depending on delay, which will give us 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 Faedo-Galerkin procedure, and under a choice of a suitable Lyapunov functional, we establish an exponential decay result. e outline of the paper is as follows. In Section 2, some hypotheses related to problem are given and we state our main result. en, in Section 3, the global existence of weak solutions is proven. Finally, in Section 4, we give the uniform energy decay.
To state and prove our result, we need some assumptions.
and suppose that there exist positive constants ζ i satisfying for i � 1, 2.
Theorem 2 (decay rates of energy). Assume that Assumptions 1-3 hold. en, for every t 0 > 0 there exist positive constants K and c ′ such that the energy defined by (12) possesses the following decay: (16)
en, there exists a constant C s � C s (Ω, q) such that e following lemma states an important property of the convolution operator.
Proof. Multiplying the first equation in (7) by u t , integrating over Ω, and using integration by parts, we get d dt Consequently, by applying Lemma 2, equation (21) becomes d dt Similarly by multiplying the second equation in (7) by v t , integrating over Ω, and using integration by parts, we get d dt Multiply the third equation in (7) by ξΔz 1 and integrate the result over Ω × (0, 1) to obtain Consequently, Similarly, we get (26) From Assumption 3, we get Using (13), this completes the proof.

Global Existence (Proof of Theorem 1)
roughout this section we assume ). We will use the Faedo-Galerkin method to prove the existence of global solutions. Let T > 0 be fixed and let w k , k ∈ IN be a basis of H 2 (Ω) ∩ H 1 0 (Ω) and V k be the space generated by w k . Now, we define, for 1 ≤ j ≤ k, the sequence ϕ j (x, ρ) as follows: en, we may extend ϕ j (x, 0) by ϕ j (x, ρ) over L 2 (Ω × (0, 1)) such that (ϕ j ) j forms a basis of L 2 (Ω, H 1 (0, 1)) and denote Z k the space generated by ϕ k . We construct ap- 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, from the generalized Hölder inequality, we obtain Since Assumption 2 holds, according to 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 (31) make sense. e standard theory of ODE guarantees that systems (31)-(38) have a unique solution in [0, t k ), with 0 < t k < T. In the next step, we obtain a priori estimates for the solution of systems (31)-(38), so that it can be extended outside [0, t k ) to obtain one solution defined for all t > 0, using a standard compactness argument for the limiting procedure.

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 3 with employing Gronwall's lemma, we can find a positive constant C 1 independent of k such that where Noting Assumption 1 and estimate (40) yields that

Second Estimate.
Multiplying the first equation (respectively, the second equation) in (31) by a jk tt (respectively, by b jk tt ) and summing over j from 1 to k, it follows that Differentiating (36) with respect to t, we get 6 Complexity Multiplying the first equation by c jk t (respectively, the second equation by d jk t ) and summing over j from 1 to k, it follows that and then we have Integrating over (0, 1) with respect to ρ, we obtain Summing (43) and (47) and as M(r) ≥ a, we get Complexity By Young's inequality, the right hand side of (48) can be estimated as follows: Using Cauchy-Schwarz inequality and Sobolev-Poincare inequality, we obtain Similarly, By using (49)-(53) in (48), we deduce By using Assumption 3 and taking the first estimate (40) into account, we infer 8 Complexity where C 2 is a positive constant which depends on η, a, C 1 , C s , α.
For a suitable η > 0 such that 1 − (η(μ 2 i + 2)t + nC 2 s /2)) > 0 for i � 1, 2 and using Gronwall's lemma, we obtain the second estimate We observe from estimates (40) and (57) 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 (40) and Lemma 1, we deduce T.

Uniform Decay of the Energy (Proof of Theorem 2)
In this section we study the solution's asymptotic behavior of system (3).
To prove our main result, we construct a Lyapunov functional equivalent to E. For this, we define some functionals which allow us to obtain the desired estimate. (u, v, z 1 , z 2 ) be a solution of problem (7). en, the functional
We use Young's inequality with the conjugate exponents p � (l + 2)/(l + 1) and q � l + 2, and the second term in the right hand side can be estimated as Complexity 13 We have by Hölder's inequality