GLOBAL EXISTENCE AND ENERGY DECAY OF SOLUTIONS FOR A WAVE EQUATION WITH NON-CONSTANT DELAY AND NONLINEAR WEIGHTS

We consider the wave equation with a weak internal damping with non-constant delay and nonlinear weights given by utt(x, t)− uxx(x, t) + μ1(t)ut(x, t) + μ2(t)ut(x, t− τ(t)) = 0 in a bounded domain. Under proper conditions on nonlinear weights μ1(t), μ2(t) and non-constant delay τ(t), we prove global existence and estimative the decay rate for the energy.


Introduction
This paper is concerned with the initial boundary value problem (1) on Ω, u t (x, t − τ (0)) = f 0 (x, t − τ (0)) in Ω×]0, τ (0)[, where Ω =]0, L[, 0 < τ (t) are a non-constant time delay, µ 1 (t), µ 2 (t) are nonconstant weights and the initial data (u 0 , u 1 , f 0 ) belong to a suitable function space. This problem has been first proposed and studied in Nicaise and Pignotti [22] in case of constant coefficients µ 1 , µ 2 and constant time delay. Under suitable assumptions, the authors proved the exponential stability of the solution by introducing suitable energies and by using some observability inequalities. Some instability results are also given for the case of the some assumptions is not satisfied.
With a weight depending on time, µ 1 (t), µ 2 (t) and constant time delay, this problem was studied in [2], where the existence of solution was made by Faedo-Galerkin method and a decay rate estimate for the energy was given by using the multiplier method.
W. Liu in [19] studied the weak viscoelastic equation with an internal time varying delay term. By introducing suitable energy and Lyapunov functionals, he establishes a general decay rate estimate for the energy under suitable assumptions.
F. Tahamtani and A. Peyravi [29] investigated the nonlinear viscoelastic wave equation with source term. Using the Potential well theory they showed that the solutions blow up in finite time under some restrictions on initial data and for arbitrary initial energy.
Global existence and asymptotic behavior of solutions to the viscoelastic wave equation with a constant delay term was considered by M. Remil and A. Hakem in [28].
Global existence and asymptotic stability for a coupled viscoelastic wave equation with time-varying delay was studied in [3] by combining the energy method with the Faedo-Galerkin's procedure.
The stabilization problem by interior damping of the wave equation with boundary or internal time-varying delay was studied in [23] by introducing suitable Lyapunov functionals.
Energy decay of solutions for the wave equation with a time varying delay term in the weakly nonlinear internal feedbacks was considered in [11].
Time delay is the property of a physical system by which the response to an applied force is delayed in its effect, and the central question is that delays source can destabilize a system that is asymptotically stable in the absence of delays, see [7]. In fact, an arbitrarily small delay may destabilize a system that is uniformly asymptotically stable in the absence of delay unless additional control terms have been used, see for example [8,12,31] By energy method in [24] was studied the stabilization of the wave equation with boundary or internal distributed delay. By semigroup approach in [27] was proved the well-posedness and exponential stability for a wave equation with frictional damping and nonlocal time-delayed condition. Transmission problem with distributed delay was studied in [18] where was established the exponential stability of the solution by introducing a suitable Lyapunov functional.
Here we consider a wave equation with non-constant delay and nonlinear weights, thus, the present paper is a generalization of the previous ones. The remaining part of this paper is organized as follows. In the section 2 we introduce some notations and prove the dissipative property of the full energy of the system. In the section 3, for an approach combining semigroup theory (see [21] and [4]) with the energy estimate method we prove the existence and uniqueness of solution. In section 4 we present the result of exponential stability.
(H2) µ 2 : R + → R is a function of class C 1 (R + ),which is not necessarily positives or monotones, such that for some 0 < β < √ 1 − d and M 2 > 0. We now state a lemma needed later. [16]). Let E : R + → R + be a non increasing function and assume that there are two constants σ > −1 and ω > 0 such that Then As in [23], we assume that (5) τ (t) ∈ W 2,+∞ ([0, T ]), for T > 0 and there exist positive constants τ 0 , τ 1 and d satisfying We introduce the new variable Then and problem (1) takes the form We define the energy of the solution of problem (9) by (10) is a non-increasing function of class C 1 (R + ) andξ be a positive constant such that Our first result states that the energy is a non-increasing function.
Lemma 2.3. Let (u, z) be a solution to the problem (9). Then, the energy functional defined by (10) satisfies Proof. Multiplying the first equation (9) by u t (x, t), integrating on Ω and using integration by parts, we get Now multiplying the second equation (9) by ξ(t)z(x, ρ, t) and integrate on Ω×]0, 1[, to obtain From (10), (14) and (15) we obtain Due to Young's inequality, we have Inserting (17) into (16), we obtain Lemma 2.4. Let (u, z) be a solution to the problem (9). Then the energy functional defined by (10) satisfies and from (H1), we obtain and the lemma is proved.

Global solution
For the semigroup setup we U = (u, u t , z) T and rewrite (9) as where the operator A(t) is defined by We introduce the phase space . Notice that the domain of the operator A(t) is independent of the time t, i.e., H is a Hilbert space provided with the inner product Using this time-dependent inner product and the next theorem we will get a result of existence and uniqueness. Our goal is then to check the above assumptions for problem (18). First, we prove D(A(0)) is dense in H. The proof is the same as the one Lemma 2.2 of [25], we give it for the sake of completeness.
Let (f, g, h) T be orthogonal to all elements of DA(0), namely for all (u, v, z) T ∈ D(A(0)).
We consequently Secondly, we notice that where Φ = (u, v, z) T and c is a positive constant and · is the norm associated the inner product (22). For all t, s ∈ [0, T ], we have |t−s| ≤ 0 for some c > 0. To do this , we have Hence ξ is a non increasing function and ξ > 0, we get Using (5) and τ is bounded, we deduce that which proves (24) and therefore (iii) follows. Now we calculate A(t)U, U t for a fixed t. Take U = (u, v, z) T ∈ D(A(t)). Then Integrating by parts, we obtain

So we get
Therefore, from (16) and (17), we deduce Then, we have where From the (13), we obtain which means that the operatorÃ = A(t) − κ(t)I is dissipative.
The existence and uniqueness are obtained by the following result. for problem (18).
Proof. A general theory for equations of type (18) has been developed using semigroup theory [14], [15] and [26]. The simplest way to prove existence and uniqueness results in to show that the triplet {(A, H, Y )}, with A = {A(t)/t ∈ [0, T ]}, for some fixed T > 0 and Y = A(0), forms a CD-systems (or constant domain system, see [14] and [15]). More precisely, the following theorem gives the existence and uniqueness results and is proved in Theorem 1.9 of [14] (see also Theorem 2.13 of [15] or [1]).

Asymptotic behavior
In this section we shall investigate the asymptotic behavior of problem (1). The stability result will be obtained using Lemma 2.2. Proof. From now on, we denote by c various positive constants which may be different at different occurrences.
Notice that u tt u = (u t u) t − u 2 t , using integration by parts and the boundary conditions we know that Similarly, we multiply the second equation of (9) by E q ξ(t)e −2ρτ (t) z(x, ρ, t) and then integrate over Ω × (0, 1) × (S, T ) to see that Using integration by parts and the boundary conditions we know that Since µ 1 is a non-increasing function of class C 1 (R), its derivatives is non-positive, which implies that ξ (t) ≤ 0. This result this Moreover, as then, from (40), (41) and (42), we have that where γ 0 = 2 min{1, e −2τ1 }.
Using the Young and Sobolev-Poincaré inequalities and Lemma 2.3, we find that From Lemma 2.4, we deduce that Now, we get that