Decay Rates for a Coupled Viscoelastic Lamé System with Strong Damping

In [6] Beniani, Taouaf and Benaissa studied a coupled viscoelastic Lamé system with strong dampings and established a general decay result. In this paper, we continue to study the system. Assuming g′ i(t) ≤ −ξi(t)Hi(gi(t)), i = 1, 2, we establish an explicit and general decay result, which is optimal, to the system. This result improves earlier results in [6].

In [6], the authors proved the well-posedness of solutions to problem (1.1)- (1.4). Under the assumptions on g i (t) g i (t) ≤ −ξ i (t)g i (t), i = 1, 2, ∀ t > 0, they established the general decay rates of energy of the form In this paper, we continue to consider problem (1.1)- (1.4), and improve the energy decay results in [6] to establish explicit and general energy decay results for a wider class of relaxation function. For a single Lamé equation, Bchatnia and Daoulatli [4] considered a Lamé system with localized nonlinear damping u tt − ∆ e u + a(x)g(u t ) = f (x), and established a general decay result of energy. Beniani et al. [7] studied energy decay of a time-delayed Lamé system. With respect to Lamé system with viscoelastic term, Bchatnia and Guesmia [5] investigated the system with past history and obtained a more general energy decay. When λ + µ = 0, the Lamé system reduces to classical wave system. For the following wave equation Messaoudi [17,18], by taking F = 0 and F = |u| γ u, γ > 0, respectively and assuming g (t) ≤ −ξ(t)g(t), obtained general decay results. We also mention Han and Wang [10], Liu [14,15], Messaoudi and Mustafa [16], Mustafa [24] and Park and Park [28], where the authors get general decay of energy for problems related to (1.5) use this assumption on g. Lasiecka et al. [11] considered another general assumption on g: g (t) ≤ −H(g(t)), where H is strictly convex and increasing function and was first introduced by Alabau-Boussouira and Cannarsa [2]. After that, there are some stability results established by using this condition. See Cavalcanti et al. [8,9], Lasiecka et al. [13], Mustafa [23], Mustafa and Messaoudi [27] and Xiao and Liang [30]. Very recently, in [25,26], Mustafa considered two classes of single wave equation and proved general and explicit decay results of energy under a more general class of relaxation function satisfying g (t) ≤ −ξ(t)H(g(t)).
For coupled wave system, Han and Wang [10] studied a coupled wave system with nonlinear weak dampings and finite memories. They proved local and global existence and finite time blow-up of solutions. A general decay result was established by Said-Houari et al. [29], and was extended by Messaoudi et al. [19] to wave system with past histories. Messaoudi and Tatar [20] considered a coupled system only with viscoelastic terms, and proved exponential decay and polynomial decay results, which was improved by Mustafa [22]. Recently, Al-Gharabli and Kafini considered the system in [20] and established a more general decay result by using some properties of convex functions, see [1].
The main question which can be asked here is the following: Whether can we get general and explicit decay rates for coupled Lamé system (1.1)-(1.4) under the different more general assumptions of different relaxations? Motivated by [6] and [25,26], in this paper, we intend to consider (1.1)-(1.4) with g i (t) ≤ −ξ i (t)H i (g i (t)), i = 1, 2, which is more general than the one in [6]. We establish explicit and general decay of system (1.1)-(1.4). Hence we extend the results of a single wave equation in [25,26] to coupled wave equations. It must to be point out that the decay results established here are optimal exponential and polynomial rates for 1 ≤ q < 2 when H(s) = s q , which improved the previous known results for 1 ≤ q < 3 2 . In addition, the energy decay result established in [6] is a special case of our result when the function H(s) is linear. Since the decay result in the present work holds for λ + µ = 0, our result also improves the ones in [1,20,22] and so on. Here the proof rely mainly on the construction of a Lyapunov functional. We adopt the idea of Mustafa [25,26] and Messaoudi and Hassan [21] and some properties of convex functions developed by Lasiecka and Tataru [12] and Alabau-Boussouira and Cannarsa [2].
The rest of this paper is as follows. In Section 2, we give some assumptions and our main results. In Section 3, we establish the general decay result of the energy.

Assumptions and main results
In the following, the constant δ > 0 is the embedding constant for u ∈ H 1 0 (Ω). We write · instead of · 2 . The constant c > 0 denotes a generic constant.
We assume for i = 1, 2, (A1) g i (t) : R + → R + are C 1 functions, which are increasing, satisfying (2.1) (A2) There exist two C 1 functions H i : R + → R + which are linear or are strictly increasing and strictly convex functions of class Remark 1. It follows from (A1) that lim t→+∞ g i (t) = 0. We know that there exists some t 1 ≥ 0 large enough such that For completeness, we give the existence of global solutions proved in [7].
The total energy of system (1.1)-(1.4) is defined by We give the following stability result.
where k 1 , k 2 are positive constants.
Assume (A1) and (A2) hold, then we have where k,k and k 1 are positive constants.
We end this section by giving two examples to illustrate explicit formulas for the decay rates of the energy. One can find in [25,26].
, then the functions H 1 and H 2 satisfy (A2) on the interval (0, r] for any 0 < r < 1. Then we can get E(t) ≤ c 1 e −c2t q .
We infer from (2.4) 1 that As c 2 ≤ 1, this is slower rate than g i (t). On the other hand, By (2.4) 2 , we get for large t This is the same rate as g i (t).

Proof of Theorem 2
In this section, we will prove Theorem 2.

Technical lemmas
The energy E(t) satisfies that for any t ≥ 0, As in [25,26], for any 0 < ζ < 1, we define satisfies for any t ≥ 0, Proof. From (1.1) we infer that Hölder's inequality gives us By Young's inequality and (3.4), we deduce that The same arguments as in [25,26], we can get the following three lemmas.
The same arguments as in Lemma 3, we can get the following lemma.
Lemma 4. The functional θ 2 (t) defined by Now we define the functional F (t) where N and N 1 are positive constants. It is easy to get that for N large, there exist β 1 > 0 and β 2 > 0 such that Lemma 5. It holds that for any t ≥ 0, Proof. Combining (3.1)-(3.2), and noting g i = ζg i − h i (i = 1, 2), we can infer that for any t > 0, where we used Poincaré's inequalities δ u t 2 ≤ ∇u t 2 and δ v t 2 ≤ ∇v t 2 . First of all we choose N 1 large so that Then for any s ∈ [0, ∞), we get By using the fact Thus there exist some ζ 0 (0 < ζ 0 < 1) such that if ζ < ζ 0 then At last, for any fixed N 1 , we choose N large enough and choose ζ satisfying Then we have

Proof of Theorem 2
Taking into account (3.9), we can get that there exist some constant m > 0, Case 1. The function H(t) is linear. We multiply (3.10) by ξ(t) and use (2.1) and (3.1) to get Define E(t) = ξ(t)F (t)+cE(t). We know that E(t) is equivalent to E(t). Noting that ξ(t) is nonincreasing, then we obtain from (3.11) that for any t ≥ 0, which gives us Case 2. The function H(t) is nonlinear. Define G(t) = F (t) + θ 1 (t) + θ 2 (t). It follows from (3.7), (3.8) and (3.9) that there exist a positive constant b such that for any t ≥ 0, By (3.12), we can choose a constant 0 < q < 1 so that We assume that I i (t) > 0 for all t ≥ 0, or else (3.10) implies an exponential decay. We define λ 1 (t) and λ 2 (t) by It is obvious that λ i (t) ≤ −cE (t), i = 1, 2. Noting that H i (t) is strictly convex on (0, r] and H i (0) = 0, we have provided 0 ≤ ν ≤ 1 and x ∈ (0, r]. By using (2.2), (3.13) and Jensen's inequality, we can obtain (3.14) Here H 1 is an extension of H 1 , which is strictly convex and strictly increasing C 2 function on (0, ∞). We have from (3.14) that Similarly, we have We infer from (3.10) that for any t ≥ 0, For ε 0 < r, we define the function K 1 (t) which is equivalent to E(t). Since E (t) ≤ 0, H i > 0 and H i > 0, we obtain from (3.15) that Now we denote the conjugate function of the convex function H i by H * i , see, for instance, Arnold [3]. Then which satisfies Young's inequality, ξi(t) , and using H * i (s)≤s(H i ) −1 (s) and (3.16), we conclude . Define the functional K 2 (t) by K 2 (t) = ξ(t)K 1 (t) + cE(t).
We know that there exist two positive constants β 3 and β 4 such that Making a appropriate choice of ε 0 , we infer from (3.18) that for some constant k > 0, where H 3 (t) = tH 0 (ε 0 t).