Stability result for a class of weakly dissipative second-order systems with infinite memory

: In this paper we consider the following abstract class of weakly dissipative second-order systems with infinite memory, u ′′ ( t ) + Au ( t ) − (cid:90) ∞ 0 g ( s ) A α u ( t − s ) ds = 0, t > 0, and establish a general stability result with a very general assumption on the behavior of g at infinity; that is g ′ ( t ) ≤ − ξ ( t ) G ( g ( t )) , t ≥ 0. where ξ and G are two functions satisfying some specific conditions. Our result generalizes and improves many earlier results in the literature. Moreover, we obtain our result with imposing a weaker restrictive assumption on the boundedness of initial data used in many earlier papers in the literature such as the one in [1–5]. The proof is based on the energy method together with convexity arguments.


Introduction
V iscoelastic materials exhibit an instantaneous elasticity effect and creep characteristics at the same time. The importance of the viscoelastic properties of materials has been realized because of the rapid developments in rubber and plastics industry. The modeling of the dynamics of physical phenomena such as heat flow in conductors with memory, hereditary polarization in dielectrics, population dynamics, viscolasticity can be described by an abstract integro-differential equation of the form where ′ represents a derivative with respect to time t, A : D(A) ⊂ H −→ H is a positive definite self-adjoint operator on H, g is the relaxation function (convolution kernel), α ∈ [0, 1], u 0 , u 1 are given history function and initial data respectively. The study of viscoelastic problems has attracted the attention of many authors and several decay and blow up results have been established. We start with the pioneer work [6,7] where Dafermos considered a one-dimensional viscoelastic problem and established various existence results and then proved, for smooth monotone decreasing relaxation functions, that the solutions go to zero as t goes to infinity. After that, many results dealing with the existence, uniqueness, regularity and asymptotic behavior of many systems of the form (1) have been studied; see, for example, [1,[8][9][10][11]. In the case of finite memory, that is, u 0 (t) = 0 for t < 0, see [12][13][14][15][16][17][18]. In particular, Rivera et al., [15] considered the interpolating cases α ∈ (0, 1) and a relaxation function g which decays exponentially to zero at infinity, that is, −c 0 g(s) ≤ g ′ (s) ≤ −c 1 g(s) ∀ s ∈ R + .
They showed that the energy decays polynomially at the rate of 1 t . Recently, Hassan and Messaoudi [19] considered    u ′′ (t) + Au(t) − t 0 g(t − s)A α u(s)ds = 0, t > 0, and established a new general decay rate result for which the relaxation function g satisfies condition For case of infinite memory, see [20][21][22][23][24][25]. In particular, Guesmia [1] considered and introduced a new ingenuous approach for proving a more general decay result based on the properties of convex functions and the use of the generalized Young inequality. He used a larger class of infinite history kernels satisfies the following condition such that where G : R + → R + is an increasing strictly convex function. Al-Mahdi and Al-Gharabli [2] considered the following viscoelastic problem and they established decay results with using a relaxation function g, satisfying the condition Very recently, Guesmia [26] considered two models of wave equations with infinite memory and established an explicit and general decay rate results where the relaxation function satisfying the condition (4). Motivated by the above works, we intend to study the following class of viscoelastic equations of the form where A : D(A) ⊂ H −→ H is a positive definite self-adjoint operator on H such that the embedding D(A β ) → D(A σ ) is compact for any β > σ ≥ 0 and α ∈ (0, 1).

Our main objectives
We intend to establish a two fold objective: 1. improve many earlier works such as the ones in [11,15,19] from finite memory to infinite memory; 2. prove a general decay estimate for the solution of Problem (10) with a wider class of relaxation functions than the ones considered in [1][2][3][4][5] by getting a better decay rate with imposing a weaker assumption on the boundedness of initial data than the one considered in the earlier papers such as the one in [1][2][3][4][5].
The paper is organized as follows: We present some assumptions and remarks in §3. We state and prove some technical lemmas in §4. The main result, its proof and some examples are presented in §5.

Assumptions
In this section, we state some assumptions needed in the proof of our main decay result. The strictly decreasing differentiable relaxation (kernel) function g : [0, ∞) −→ (0, ∞) satisfies the following assumptions: (A.2) There exists a non-increasing differentiable function ξ : R + −→ (0, ∞) and a C 1 function G : [0, +∞) −→ [0, +∞) which is linear or it is strictly increasing and strictly convex C 2 function on (0, r], with where ξ is satisfying  2) in the present paper is larger than the ones satisfying (6) and (7) used in some earlier papers such as the one in [1]. In fact, the boundedness of the sup in (6) use in [1], can be interpreted as the inequality in ( A.2) in the present paper (with ξ = 1). The conditions (6) and (7) used in [1] ask also the boundedness of the integral. So, it is better to consider the relaxation functions satisfy ( A.1) − ( A.2) used in the present paper than the one used in [1].

Remark 4.
As is in Mustafa [14], if G is a strictly increasing and strictly convex C 2 function on (0, r], with G(0) = G ′ (0) = 0, then there is a strictly convex and strictly increasing C 2 function G : [0, +∞) −→ [0, +∞) which is an extension of G. For instance, we can define G, for any t > r, by We state the existence, regularity and uniqueness theorem whose proof is in [15].
The "modified" energy functionals associated to our problem are given by Remark 5. The positiveness of the energy functionals comes from inequalities (12) and (13).

Technical lemmas
In this section, we state and prove some Lemmas that are useful in the proof of Theorem 2. Through out this work we use c > 1 to represent a generic constant, which is independent of t and the initial data.
Then, for any 0 < δ < 1, the functional I 1 defined by satisfies, along the solution of (10), the estimate Proof. Differentiating I 1 and exploiting the differential equation in Problem (10), we get Next, we estimate the terms in the right-hand side of the above identity. Using the Cauchy-Schwarz, Young and Hölder inequalities, Lemma 2 and inequalities (11) and (12), it follows that, for any 0 < δ < 1, and u ′ (t), Plugging the above estimates in (24), we obtain the desired estimate.
satisfies, along the solution of (10), the estimate Proof. Differentiating I 2 , using the equation in (10), and repeating the above computations, we get for any t ≥ 0. (12) and the fact that p(t) ≤ p(0) = 1−l 0 , we obtain, for any t ≥ 0,

Corollary 1.
There exists 0 < q 0 < 1 such that, for all t ≥ 0, we have the following estimate: G is defined in Remark 4 and f (t) is defined in (31).
Proof. We introduce a functional η defined by ds, ∀ t ≥ 0, and observe, from inequality (12), that Use of (15), (17) and (38) yields Thanks to (31), we can pick 0 < q 0 < min 1, l 8m(1+ω 0 ) so that To prove (35), we define another functional µ by Also, the strict convexity of G and the fact that G(0) = 0 entail that G(sτ) ≤ sG(τ), for 0 ≤ s ≤ 1 and τ ∈ (0, r]. Combining this with the hypothesis (A.2), Jensen's inequality and (39), we obtain, for any t ≥ 0, where G is a C 2 extension of G which is strictly increasing and strictly convex on (0, ∞). For simplicity, in the rest of this paper, we use G instead of G. Then we have for any t ≥ 0,

The main result
In this section, we state and prove our decay result. We introduce the following functions: It is not difficult to show that the above functions are convex and increasing on (0, r]. Now we state our main result.

Theorem 2. Assume that hypotheses (A.1)-(A.3) hold and the initial data satisfy
Then, for all 0 ≤ s ≤ t and for strictly positive constant C, we have the following decay results where q is defined in (37), h = h 0 + h 1 where h 0 , h 1 are defined in (20) and (21) and the functions G 2 (s) and G 4 (s) are defined in (41).