GENERAL DECAY FOR A VISCOELASTIC KIRCHHOFF EQUATION WITH BALAKRISHNAN-TAYLOR DAMPING, DYNAMIC BOUNDARY CONDITIONS AND A TIME-VARYING DELAY TERM

. In this paper, we consider a viscoelastic Kirchhoﬀ equation with Balakrishnan-Taylor damping, dynamic boundary conditions and a time- varying delay term acting on the boundary. By using the Faedo-Galerkin approximation method, we ﬁrst prove the well-posedness of the solutions. By in- troducing suitable energy and perturbed Lyapunov functionals, we then prove the general decay results, from which the usual exponential and polynomial decay rates are only special cases. To achieve these results, we consider the following two cases according to the coeﬃcient α of the strong damping term: for the presence of the strong damping term ( α > 0), we use the strong damping term to control the time-varying delay term, under a restriction of the size between the time-varying delay term and the strong damping term; for the absence of the strong damping term ( α = 0), we use the viscoelasticity term to control the time-varying delay term, under a restriction of the size between the time-varying delay term and the kernel function.

In recent years, wave equation with dynamic boundary conditions have been studied by many authors (see [1], [4], [7], [19], [23] for more details). The papers of [15], [16], [18] and [20] studied problem (1) with M ≡ 1 and without delay term. For example, in [15], Gerbi and Said-Houair studied the problem x ∈ Ω, t > 0, and proved the local existence by using the Faedo-Galerkin approximations combined with a contraction mapping theorem and showed the exponential growth of the energy. Later in [16], they established the global existence and asymptotic stability of solutions starting in a stable set by combining the potential well method and the energy method. A blow-up result for the case m = 2 with initial data in the unstable set was also obtained. Recently, when the additional relaxation function g = 0 is involved in problem (2), they got the existence and exponential growth results in [18]. It is worth mentioning that a wave equation with delay term has become an active area of research, see for instance [5], [10], [29], [30], [36] and the references therein. The delay term may be a source of instability. For example, it was proved in [11,29,30] 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 [17], Gerbi and Said-Houari considered the damped wave equation with dynamic boundary conditions and a delay boundary term: If the weight µ 2 of the delay term is less than the weight µ 1 of the term without delay or if µ 2 ≥ µ 1 and α > (µ 2 − µ 1 )B 2 (with B a constant), they proved the global existence of the solutions and the exponential stability of the system. This results indicated that even when µ 2 is greater than µ 1 , the strong damping term still provides exponential stability for the system. However, when α = µ 1 = 0, how can we stabilize the system?
Recently, Ferhat and Hakem in [12] considered the following problem: By using the potential well method and introducing suitable Lyapunov function, they proved the global existence and established general decay estimates depending on the decay rate of the relaxation function g. This result indicated that for a system of nonlinear viscoelastic wave equations with frictional damping, strong damping and nonlinear delay term, the viscoelasticity term also plays an important role in the stability of the whole system. For results of same nature, we refer the reader to [8], [9], [13], [14], [22], [24], [25], [26], [34], [38] and [41]. However, when α = µ 1 = 0, can the viscoelasticity term showed in problem (4) still cause the similar stability property?
We recall that the stability of PDEs with time-varying delays was studied in [6,31,32,33]. In [33], Nicaise et al. considered the system described by x ∈ (0, π), t > 0, and proved the exponential stability result, under the condition where d is a constant such that Later, Nicaise et al. [32] extended the above result to general space dimension. For equation (1) in the present of Balakrishnan-Taylor damping, it is mainly used to solve the spillover problem. The related problems were considered by Balakrishnan and Taylor in [2], Bass and Zes in [3], Tatar and Zarai in ( [35], [39], [40]), Mu in [28] and Wu in [37].
Motivated by these results, in this paper, we intend to study the asymptotic behavior and related decay rates of problem (1), in which no linear damping term is involved (i.e. µ 1 = 0). For our purpose, we consider two cases according to the coefficient α of the strong damping term: the case α > 0 and the case α = 0. As we shall see below, if α > 0, the presence of the strong damping term α∆u t in (1) plays a decisive role in the stability of the whole system. Thanks to the energy method, by introducing appropriate Lyapunov functionals, we show in this article that the decay rates of the solution energy are similar to the relaxation function, which are not necessarily decaying like polynomial or exponential function. If α = 0, the main difficulty arises since there is no strong damping term to control the time-varying delay term in the estimate of the energy decay. To overcome this difficulty, our basic idea is to control the time-varying delay term by making use of the viscoelasticity term. And to achieve this goal, a restriction of the size between the parameter µ 2 and the kernel g and a new Lyapunov functional is needed. Although the Balakrishnan-Taylor damping is present, we give the remark that it does not change the main result of this paper.
The paper is organized as follows. In Section 2, we present some assumptions needed for our work and state the main result. In Section 3, we prove the wellposedness of the solution. In Section 4, we prove the general decay result in the case of α > 0. For the case of α = 0, the general decay result is proved in Section 5.

Preliminaries and main results.
In this section we present some assumptions and state the main results. For the relaxation function g, we assume the following: (G1) g: R + −→ R + is a nonincreasing differentiable function satisfying (G2) There exists a nonincreasing differentiable function ξ : R + −→ R + such that g (s) ≤ −ξ(s)g(s), ∀ s ∈ R + and +∞ 0 ξ(t)dt = ∞.
Let C p and C * p be the Poincaré's type constants defined as the smallest positive constants such that and where H 1 Γ0 = {u ∈ H 1 (Ω)|u |Γ 0 = 0}. We denote by (·, ·) the scalar product in L 2 (Ω), i.e., For the coefficient µ 2 of the time-varying delay term, we assume that where d is the positive constant of assumption (6) and a 0 is a small positive constant defined in (78) below. Now, we can state the following well-posedness result: Theorem 2.1. Suppose that (G1), (G2) and (7) hold. Then given u 0 ∈ H 1 Γ0 (Ω), u 1 ∈ L 2 (Ω) and f 0 ∈ L 2 (Ω × (0, 1)), there exist T > 0 and a unique weak solution . We define the energy of problem (1) as where ζ and λ are suitable positive constants and Then, we state the decay result as follows: (Ω) be given. Assume that g and ξ satisfy (G1) and (G2), and the assumption (7) holds. Then, for each t 0 > 0, there exist two positive constants K and k such that, for any solution of problem (1), the energy satisfies 3. Well-posedness of the problem. We now give, with a brief proof of the wellposedness of the problem, which can be established by using the Faedo-Galerkin approximation method (see [15,31,33] for the details). As in [29], we introduce the new variable Then, we have Then, problem (1) is equivalent to Proof of Theorem 2.1. We divide the proof into two steps: the construction of approximations and then thanks to certain energy estimates, we pass to the limit.
We define now the approximations: where (u n (t), z n (t)) are solutions to the finite dimensional Cauchy problem (written in normal form): and According to the standard theory of ordinary differential equations, the finite dimensional of problem (12), (13) has solution (g jn (t), h jn (t)) j=1,...,n defined on [0, t n ). The a priori estimates that follow imply that in fact t n = T .
The proof now can be completed arguing as in ( [27], Theorem 3.1).

4.
Decay of solutions for α > 0. As mentioned earlier, in this section, we prove the general decay result for problem (1) under the first inequality of assumption (7). For our purpose, we use the idea of Nicaise and Pignotti in [31]. We fix ζ such that which is possible by assumption (7). Moreover, the parameter λ is fixed satisfying whereτ is a positive constant such that Proposition 1. For any regular solution of problem (1), under the conditions of Theorem 2.2, the energy is non-increasing and for a suitable positive constant C, we have Proof. Differentiating (8) and by Cauchy-Schwarz's inequality, a trace estimate, Poincaré's inequality, we get Therefore, by (20) and (21) we can easily get (22). Now, we use the following modified functional, for positive constants N , ε 1 and ε 2 , we have where and It is easy to check that, by using Poincare's inequality, trace inequality, the functional L is equivalent to the energy E, that is, for ε 1 and ε 2 small enough, choosing N large enough, there exist two constants α 1 and α 2 such that Next, we estimate the derivative of L(t) according to the following lemmas.
Proof. By using the equation in (1), we get We now estimate the right-hand side of (29). For a positive constant δ 1 , we have the estimates as follows
Proof of Theorem 2.2 (case α > 0). Since g is continuous and g(0) > 0, then for any t 0 > 0, we have From (22), (28), (32) and (42), then from (24), we get At this point, we choose δ 2 < lg 0 l + 1 + 2(a − l) 2 such that Once δ 2 is fixed, we then pick Once δ 1 and δ 2 are fixed, the choice of ε 1 and ε 2 satisfying will make When δ 1 , δ 2 , ε 1 and ε 2 are fixed, we choose N > 0 large enough such that We have where C = N λζ 2 > 0. By (8), (G2) and (44), there exist two positive constants M and k 8 such that Multiplying (45) by ξ(t), we have Because ξ and g are nonincreasing, we get Inserting the last inequality in (46), we obtain Now, we define Since ξ(t) is a non-increasing positive function, we can easily get that H ∼ E. Thus (48) implies that for some k > 0. Then, by direct integration over (t 0 , t), we have Consequently, using the equivalent relations of H(t) and E(t) , we can conclude where k 9 is a positive constant and K = k 9 H(t 0 ). This completes the proof.

5.
Decay of solutions for α = 0. In this section, we prove the general decay result of problem (1) in the absence of the strong damping −∆u(t) (i.e. α = 0). For our purpose, we use the idea of Dai and Yang in [10]. Before proving the result, we need the following proposition and lemmas.
Proposition 2. For any regular solution of problem (1), under the conditions of Theorem 2.2, we have Proof. Differentiating (8) and by Cauchy-Schwarz's inequality, Poincaré's inequality, we get This completes the proof.
Remark 1. In Proposition 1, we proved that the energy functional is non-increasing.
However, since ζ 2 + |µ 2 | 2 u t (t) 2 2,Γ1 ≥ 0, E(t) may not be non-increasing here. Now, we define the Lyapunov functional, for positive constants ε 3 and ε 4 , we haveL where It is easy to check that, by using Poincare's inequality, trace inequality, the functionalL is equivalent to the energy E, that is, for ε 3 and ε 4 small enough, choosing N 1 large enough, there exist two constants β 1 and β 2 such that Next, we estimate the derivative ofL(t) according to the following lemmas.
for some positive constant δ 3 .
Lemma 5.2. Under the conditions of Theorem 2.2, the functional φ(t) defined in (26) satisfies for some positive constant δ 4 .
Remark 2. We note from the proof of Theorem 2.2 (Sections 3 and 4) that, in the case of σ = 0, the general decay result is still valid.