Boundary Stabilization of a Nonlinear Viscoelastic Equation with Interior Time-Varying Delay and Nonlinear Dissipative Boundary Feedback

and Applied Analysis 3 existence resulted for γ ≥ 0 and the exponential decay of the energy for γ > 0. This result has been extended to a situation γ = 0 byMessaoudi and Tatar [24] and exponential decay and polynomial decay results have been shown in the absence as well as presence of a source term. Later on, inspired by the ideas of [25–27], Han andWang [22] investigated the general decay of solutions of energy for the nonlinear viscoelastic equation 󵄨 󵄨 󵄨 󵄨 u t 󵄨 󵄨 󵄨 󵄨 ρ u tt − Δu − Δu tt + ∫ t 0 g (t − s) Δu (s) ds + u t 󵄨 󵄨 󵄨 󵄨 u t 󵄨 󵄨 󵄨 󵄨 k = 0. (13) In recent years, the control of partial differential equation with time delay effects has become an active area of research; see, for instance, [28, 29] and the references therein. The presence of delay may be a source of instability. For instance, it was proved in [30–34] that an arbitrarily small delay may destabilize a system which is uniformly asymptotically stable in the absence of delay unless additional conditions or control terms have been used. In [32], Nicaise and Pignotti examined (1) with ρ = 0, g ≡ 0, μ 1 > 0, μ 2 > 0, and τ(t) = τ being a constant delay in the case of mixed homogeneous Dirichlet-Neumann boundary conditions, under a geometric condition on the Neumann part of the boundary. More precisely, they investigated the following system with linear frictional damping term and internal constant delay: u tt (x, t) − Δu (x, t) + μ 1 u t (x, t) + μ 2 u t (x, t − τ) = 0,


Introduction
In this paper, we consider the global existence and asymptotic behavior of a nonlinear viscoelastic equation with interior time-varying delay and nonlinear dissipative boundary feedback as follows: where Ω is a bounded domain of   ( ≥ 1) with a smooth boundary Ω of  2 ,  is a positive real constant, () > 0 represents the time-varying delay effect and the initial data  0 ,  1 ,  0 are given functions belonging to suitable spaces, ℎ() is a positive function that represents the kernel of the memory term, (  ) is nonlinear dissipative boundary feedback, and  0 , ℎ,  satisfy suitable assumptions (see in Section 2).This model appears in viscoelasticity (see [1,2]).In the case of velocity-dependent material density (i.e.,  = 0) as well as presence of  2 = 0 and in the absence of the memory effect (i.e.,  = 0), (1) reduces to the wave equation.There is large literature on the global existence and uniform stabilization of wave equations.We refer the readers to [3][4][5].It is worth mentioning that Zhang and Miao [3] considered the nonlinear wave equation with dissipative term and boundary damping   − Δ +  ()   +  () = 0, in Ω × [0, ∞) ,  = 0, on Γ 1 × [0, ∞) ,

𝜕𝑢 𝜕]
+  (  ) = 0, on Γ 0 × [0, ∞) ,  (, 0) =  0 () ,   (, 0) =  1 () , in Ω, and they proved the existence and uniform decay of strong and weak solutions by using the nonlinear semigroup method, the perturbed energy method, and the multiplier technique.Quite recently, Cavalcanti et al. [6] considered the following model: (, 0) =  0 () ,   (, 0) =  1 () , for  ∈ M, where M is a smooth oriented embedded compact surface without boundary in  3 and Δ M is the Laplace-Beltrami operator on manifold M; furthermore, they obtained explicit and optimal decay rates of the energy.Later on, Cavalcanti et al. [7] extended the result for n-dimensional compact Riemannian manifolds (M, ) with boundary in two ways: (i) by reducing arbitrarily the region where the dissipative effect lies (this gives us a totally sharp result with respect to the boundary measure and interior measure where the damping is effective) and (ii) by controlling the existence of subsets on the manifold that can be left without any dissipative mechanism, namely, a precise part of radially symmetric subsets.An analogous result holds for compact Riemannian manifolds without boundary.
In the case  = 0 and in the absence of delay (i.e.,  2 = 0), there is large literature on the existence and decay of nonlinear viscoelastic equation during the past decades.In [8], Cavalcanti et Under the condition that () ≥  0 > 0 on  ⊂ Ω, with  satisfying some geometry restrictions and − 1  () ≤   () ≤ − 2  () ,  ≥ 0, they proved an exponential decay result for the energy.Berrimi and Messaoudi [9] improved Cavalcanti's result by introducing a differential functional which allowed to weaken the conditions on both () and .In [10], Cavalcanti and Oquendo studied Under some geometric restrictions on  and assuming that they established an exponential stability for the relaxation function  decaying exponentially and ℎ linear and polynomial stability for  decaying polynomially and ℎ nonlinear.
It is worth mentioning that Zhang et al. [11] studied the following initial boundary value problem: Furthermore, they showed that the solutions of ( 9) decay uniformly in time, with rates depending on the rate of decay of the kernel .More precisely, the solution decays exponentially to zero provided that  decays exponentially to zero.When  decays polynomially, we show that the corresponding solution also decays polynomially to zero with the same rate of decay.For other related works, we refer the readers to [12][13][14][15][16][17][18][19][20][21] and the references therein.
On the other hand, concerning the study of the following nonlinear viscoelastic equation with memory, there are a substantial number of contributions: Recently, Han and Wang [22] investigated the following problem: By introducing a new functional and using potential well method, the authors established the global existence and uniform decay if the initial data are in a suitable stable set.Cavalcanti et al. [23] studied a related problem with strong damping as follows: By assuming 0 <  ≤ 2/(−2), if  ≥ 3 or  > 0 and if  = 1, 2 and () decays exponentially, they established that the global existence resulted for  ≥ 0 and the exponential decay of the energy for  > 0. This result has been extended to a situation  = 0 by Messaoudi and Tatar [24] and exponential decay and polynomial decay results have been shown in the absence as well as presence of a source term.Later on, inspired by the ideas of [25][26][27], Han and Wang [22] investigated the general decay of solutions of energy for the nonlinear viscoelastic equation In recent years, the control of partial differential equation with time delay effects has become an active area of research; see, for instance, [28,29] and the references therein.The presence of delay may be a source of instability.For instance, it was proved in [30][31][32][33][34] that an arbitrarily small delay may destabilize a system which is uniformly asymptotically stable in the absence of delay unless additional conditions or control terms have been used.In [32], Nicaise and Pignotti examined (1) with  = 0,  ≡ 0,  1 > 0, 2 > 0, and () =  being a constant delay in the case of mixed homogeneous Dirichlet-Neumann boundary conditions, under a geometric condition on the Neumann part of the boundary.More precisely, they investigated the following system with linear frictional damping term and internal constant delay: or with boundary constant delay In the presence of delay ( 2 > 0), Nicaise and Pignotti [32] examined systems ( 14) and ( 15) and proved under the assumptions  2 <  1 that the energy is exponentially stable.Otherwise, they constructed a sequence of delays for which the corresponding solution is instable.The main approach used there is an observability inequality together with a Carleman estimate.See also [35] for treatment to these problems in more general abstract form and [36] for analogous results in the case of boundary time-varying delay.
We also recall the result by Nicaise et al. [36], where the researchers proved the same result as in [32] for the one space dimension by applying the spectral analysis approach.
Recently, the stability of PDEs with time-varying delays was studied in [38][39][40][41][42][43][44].In [40], Nicaise and Pignotti investigated the stabilization problem by interior damping of the wave equation with internal time-varying delay feedback and established exponential stability estimates by introducing suitable Lyapunov functionals, under the condition | 2 | < √ 1 −  1 in which the positivity of the coefficient  1 is not necessary.In [41], Nicaise et al. showed the exponential stability of the heat and wave equations with time-varying boundary delay in 1-D, under the condition 0 ≤  2 < √ 1 −  1 , where  is a constant such that   () ≤  < 1.
The rest of the paper is organized as follows.In Section 2, we show some assumptions and state our main result.In Section 3, we present the proof of our main result.That is, we will prove the global existence by using Faedo-Galerkin method and establish the general decay result (including exponential decay and polynomial decay) by using the perturbed energy method.Finally, in Section 4, we give further remarks on this context.

Some Assumptions and Main Results
In this section, before proceeding to our analysis, we present some assumptions and state the main result.We use the standard Hilbert space  2 (Ω) and the Sobolev space  1 0 (Ω) with their usual scalar products and norms.Throughout this paper,   is used to denote a generic positive constant from line to line.
For the relaxation function ℎ, we assume that (G2) there exists a nonincreasing differentiable function () such that We assume that  satisfies For the time-varying delay, we assume that there exist positive constant  0 ,  such that Furthermore, we assume that the delay satisfies that and that  1 ,  2 satisfy Remark 1.We show an example of functions satisfying (G2) as follows: for ,  > 0 to be chosen properly; see [2].
Now, we are in a position to state our main results.

Proof of the Main Result
In this section, we will divide our proof into two steps.In Step 1, we prove the global existence of weak solutions by using Faedo-Galerkin method benefited from the ideas of [2,3,37].In Step 2, we establish the general decay of energy by introducing the new energy functional and using the perturbed energy method inspired by the contributions; see, for instance, [2-4, 11, 39].
To facilitate further our analysis, we need some notations and technical Lemmas 4 and 6.Let us first introduce some notations with these notations; we have the following lemma given in [2,11].
Step 2 (general decay of the energy).First, we introduce the new energy functional () and the perturbed energy   (); then we apply the perturbed energy method to establish general decay of the energy.More precisely, the method used is based on the construction of suitable Lyapunov functionals () and   () satisfying for some positive constants  1 ,  2 , .More details are present in [3, pp 1017] or [2,4,16].Now, we introduce the new energy functional as follows: where ,  are suitable positive constants.
Next, we will fix  such that Remark 7. In fact, the existence of such a constant  is guaranteed by the assumption (23).
Therefore, we have the following lemma.
Next, we introduce the following functionals: Set where  and  are suitable positive constants to be determined later.
Remark 9. Indeed, we easily see that, for  small enough while  large enough, there exist two positive constants  0 ,  1 , such that Concerning the estimates of Φ(), Ψ(), we have the following lemmas.
Now, we are ready to finalize our proof of general decay of the energy.Since ℎ is positive, we have It follows from ( 65), ( 72), (74), and (78) that If we choose some constants in the inequality (84), such that then we conclude that Hence, we have two cases to consider the general decay results as follows.
Case 2 (1 <  < 3/2).Due to (G2), we easily see that From the sketch of proof of Lemma 6, we observe that Thus, for  > 1, using (63) and (93), we get A simple integration of (96) over (0, ) yields As a consequence of (97), we obtain So, by using Lemma 6, we have which implies that Consequently, from (86) and (100), we have Thus, our main result is completed.
Remark 12.Our novel contribution is to show that our work improves earlier result in [37] in which only the exponential decay was investigated.More precisely, Kirane and Said-Houari [37] considered the exponential decay of problem (1) with a constant delay (i.e., () = ) and velocity-independent material density (i.e.,  = 0).

Further Remarks
In this section, we address some interesting problems of nonlinear viscoelastic equation with time-varying delay effects and velocity-dependent material density.Here, we mention some of them.
(1) An interesting problem is to show the well-posedness and stabilization of the nonlinear viscoelastic equation with boundary feedback with respect to timevarying delay effects.What will happen if the controller with time-varying delay effects is in the equation instead of on the boundary?More precisely, in our forthcoming work, we will investigate the wellposedness and general decay properties of the solutions for the following nonlinear viscoelastic equation with velocity-dependent material density: where Ω is bounded domain of   and  ≥ 1 with a smooth boundary Γ and let Γ 0 , Γ 1 be a partition of Γ such that Γ 0 ∩ Γ 1 = 0, Γ 0 ̸ = 0, Γ 1 ̸ = 0, ] = (] 1 , ] 2 ⋅ ⋅ ⋅ ]  ) denotes the unit outward normal to Γ.
(2) Another interesting problem is to give a positive answer of the open problem given by Kirane and Said-Houari [37].That is, the linear damping term  1   in the first equation of ( 16) plays a decisive role in their proofs.Thus, the problem of whether the stability properties they have proved are preserved when  1 = 0 is open.In order to overcome the above difficulty, our main idea is to contrast the effects of the time-varying delay by using the dissipative nonlinear where  2 is constant and (  ) is the dissipative nonlinear boundary feedback.