NEW GENERAL DECAY RATES OF SOLUTIONS FOR TWO VISCOELASTIC WAVE EQUATIONS WITH INFINITE MEMORY

We consider in this paper the problem of asymptotic behavior of solutions for two viscoelastic 
 wave equations with infinite memory. We show that the stability of the system holds for a much larger class of kernels 
 and get better decay rate than the ones known in the literature. More precisely, we consider infinite memory kernels 
 satisfying, where and are given functions. Under this very general assumption on the behavior of g at infinity and for 
 each viscoelastic wave equation, we provide a relation between the decay rate of the solutions and the growth of g at 
 infinity, which improves the decay rates obtained in [15, 16, 17, 19, 40]. Moreover, we drop the boundedness assumptions 
 on the history data considered in [15, 16, 17, 40].


Aissa Guesmia
To cite this version: Aissa

Introduction
In this paper, we consider the following two viscoelastic problems: u tt (x, t) − ∆u(x, t) + +∞ 0 g(s)∆u(x, t − s)ds = 0 in Ω × R * + , u(x, t) = 0 on ∂Ω × R * + , u(x, −t) = u 0 (x, t), u t (x, 0) = u 1 (x) in Ω × R + g(s)u(x, t − s)ds = 0 in Ω × R * + , u(x, t) = 0 on ∂Ω × R * + , u(x, −t) = u 0 (x, t), u t (x, 0) = u 1 (x) in Ω × R + , (1.2) where u denotes the transverse displacement of waves, ∆ is the Laplacian operator with respect to the space variable x, u t and u tt denote, respectively, the first and second derivatives with respect to the time variable t, g : R + → R * + is a given function representing the infinite memory kernel and satisfying some hypotheses, Ω is a bounded domain of R N , N ∈ N * := {1, 2, . . .}, with a smooth boundary ∂Ω, and u 0 and u 1 are fixed history and initial data in a suitable Hilbert space.

Finite memory
The problems related to viscoelasticity (see [8]) have attracted a great deal of attention during the last four decades and many results of existence and longtime behavior have been established. Many advances in the studies of well posedness and stability were made since the works of Dafermos [10,11].
Hrusa [24] considered a one-dimensional nonlinear viscoelastic wave equation and proved several global existence results for large data and an exponential decay result for strong solutions when the kernel is of the form g(s) = e −s . In [12], Dassios and Zafiropoulos considered a viscoelastic problem in R 3 and proved a polynomial decay result for exponentially decaying kernels. After that, Rivera [38] considered equations for linear isotropic homogeneous viscoelastic solids of integral type which occupy bounded domains or the whole space R N . In the bounded-domain case and for exponentially decaying kernels and regular solutions, he showed that the sum of the first and the second energy decays exponentially. Whereas, the decay is polynomial when the body occupies the whole space R N , even if the kernel is of an exponential decay.
For quasilinear problems, Cavalcanti et al. [4] studied, in a bounded domain, the following equation: for ρ > 0. A global existence result for γ ≥ 0, as well as an exponential decay result for γ > 0, have been established. This latter result was then extended to a situation, where γ = 0, by Messaoudi and Tatar [31,32], and exponential and polynomial decay results have been established in the absence, as well as in the presence, of a source term. In all the above mentioned works, the rates of decay of kernels were either of exponential or polynomial type. In [6], Cavalcanti et al. considered a semilinear viscoelastic wave equation with local frictional damping, where the kernel g satisfies, for two positive constants ξ 1 and ξ 2 , They established an exponential decay result under some restrictions on the control zone. Berrimi and Messaoudi [3] established the result of [6] under weaker conditions on both damping and kernel, for a problem where a source term is competing with the damping term. Cavalcanti and Oquendo [7] considered, in Ω × R * + , the following problem: and established, for a(x) + b(x) ≥ ρ > 0, an exponential stability result for g decaying exponentially and h linear, and a polynomial stability result for g decaying polynomially and h nonlinear. Li et al. [27] treated (1.5) with b(x) = 0 and f (u) = −|u| γ u, γ > 0. They showed the global existence and uniqueness of global solution of problem (1.5) and established uniform decay rate of the energy under suitable conditions on the initial data and g. For more general decaying kernels than the one defined by (1.4), Messaoudi [28,29] considered for q ≥ 2, b ∈ {0, 1} and g satisfying, for a nonincreasing function ξ : He established a more general decay result, from which the usual exponential and polynomial decay rates are only special cases. Said-Houari et al. [39] studied the well-posedness and stability of coupled two semilinear viscoelastic wave equations, where the kernels satisfy (1.7). They established the same stability estimate as in [28,29]. Using the same assumption (1.7), a similar stability estimate was proved in [20] and established a more general decay result, which leads to the optimal decay rate of solutions when g converges to zero at ifinity faster than s −2 . Optimality means that the solution converges to zero as fast as g. Mustafa and Messaoudi [34] considered (1.6) with b = 0 and kernels satisfying for some positive convex function G : R + → R + . They used some properties of convex functions together with the generalized Young inequality and established a general decay result depending on g and G.
The assumption (1.9) was also considered in [22] for an abstract viscoelastic wave equation, where the obtained decay rate depends on the solution of an ODE. The results of [34] were generalized in [33] to kernels satisfying g (t) ≤ −ξ(t)G(g(t)), ∀t ∈ R + (1.10) by getting a more general decay result depending on g, ξ and G. This general decay result extendeds the range of the polynomial decay rate optimality from p ∈ [1, 3 2 ) to p ∈ [1, 2) in (1.8), which allows kernels having a decay at infinity of the form g(s) = s −q , for q > 1. The arguments of [33] were recently used in [23] to prove the stability of the following abstract equation: where α ∈]0, 1[ and A is a given operator satisfying some hypotheses. For the case of memories acting on the boundary of domain, we refer the readers to [5,14] and the references therein.

Infinite memory
Giorgi et al. [13] considered the following semilinear hyperbolic equation with infinite memory in a bounded domain Ω ⊂ R 3 : under the condition K ≤ 0 and proved the existence of global attractors for the solutions. Conti and Pata [9] added a linear frictional damping to (1.11) and assumed that the kernel is a convex decreasing smooth function. They proved the existence of a regular global attractor. Pata [35] discussed the decay properties of the semigroup generated by the following abstract equation: where A is a strictly positive self-adjoint linear operator, α > 0, β ≥ 0 and µ is a decreasing function satisfying specific conditions. Subsequently, they established necessary as well as the sufficient conditions for the exponential stability. In [1], Appleby et al. studied (1.12) with α = 1 and β = 0 and established an exponential decay result for strong solutions in a Hilbert space. In [15], Guesmia considered the abstract equation in a Hilbert space H, where A and B are positive self-adjoint linear operators such that A ∼ B or A is stronger than B (in some sense). Under the boundedness conditions on the first initial data and for the larger class of kernels satisfying +∞ 0 g(s) G −1 (−g (s)) ds + sup s∈R + g(s) G −1 (−g (s)) < +∞, (1.15) where G : R + → R + is an increasing strictly convex function, he proved two general decay results corresponding to the cases A ∼ B and A is stronger than B. Using this approach, Guesmia and Messaoudi [18] later looked into u tt −∆u+ t 0 g 1 (t−s)div(a 1 (x)∇u(s))ds+ +∞ 0 g 2 (s)div(a 2 (x)∇u(t−s))ds = 0 under the first condition in (1.14) (with B = −∆) and suitable conditions on a 1 and a 2 and for a wide class of kernels g 1 and g 2 satisfying (1.7) and (1.15), respectively. They established a general decay result including, in particular, the usual cases of exponential and polynomial decay. The decay rate of solutions obtained in [15] and [18] is weaker than the one of g at infinity when g does not converge exponentially to zero. The authors of [17] considered (1.13) with A = B and under (1.7) and the first condition in (1.14). They proved a general decay result of the solution depending only on g and ξ by adopting, for infinite memory, the method introduced in [28] with some modifications imposed by the nature of their problem. In the particular cases where g converges to zero at infinity faster than any polynomial, the obtained decay rate of solutions in [17] is equal to the one of g; and so it improves the one presented in [15]. A similar stability result was proved in [21] for an abstract thermoelastic system by considering (1.7) and applying the arguments of [17]. Recently, Youkana [40] considered (1.13) in which g and u 0 satisfy (1.8) and (1.14). He obtained a better decay rate than the one of [15] and [17] when g has a decay at infinity of the form s −q , for q > 2.
In [16], the problem of indirect stability of two coupled abstract equations with one infinite memory was considered. More precisely, the author of [16] studied in a Hilbert space H, where A,Ã, B andB are positive self-adjoint linear operators such that A ∼ B or A is stronger than B, and g satisfies (1.15). Under the following weaker condition on u 0 than (1.14): where j = 0 if A ∼ B, and j = 1 if A is stronger than B, it was proved a general decay estimte depending also on the smoothness of initial data. For classical solutions, the estimate of [16] coincides with the one of [15] when A is stronger than B. The stability of the same system (1.16) was the subject of the paper [26], where the authors proved that the decay rate 1 t of the energy is guaranteed by the following weak condition on the decreasingness behavior of g: |{s ∈ R + : g(s) > 0 and g (s) = 0}| = 0, (1.18) (here | · | means the Lebesgue measure). The same result of [26] was proved in [25] for (1.16) with a finite memory (instead of the infinite one) and an additional semilinear term f (u) on the first equation. More precisely, the authors of [25] considered (1.18) and obtained the decay rate 1 t of energy for the coupled system as well as for the corresponding first single equation.
Without assuming any boundedness condition on u 0 and for g satisfying the authors of [19] proved the well posedness and stability of (1.13) with a distributed delay; that is where A ∼ B and f : R + → R is dominated by g (in some sense). The stability results of [19] lead to the same decay rate of [17] when g converges to zero at infinity faster than t −2 . However, when the decay rate of g is at most of the form t −2 , no stability result is obtained in [19].
In the present work, we study the asymptotic behavior of solutions of (1.1) and (1.2) as particular models of (1.13) corresponding to the cases, respectively, A ∼ B and A is stronger than B. Under the general assumption (1.10) instead of (1.15), (1.7), (1.19) and (1.8) considered in [15,17,19,40], respectively, we prove two general decay estimates of solutions, which improve the decay rates obtained in [15,16,17,19,40] in case when g has at most a polynomial decay at infinity (see examples in Section 4 ). Moreover, our class of admissible initial data is larger than the one considered in [15,16,17,40] because we do not assume any boundedness condition on u 0 .
The rest of this paper is organized as follows. In Section 2, we present some assumptions and material needed for our work and give the well posedness results of our two systems. Some technical lemmas are presented and proved in Section 3. Finally, we state and prove our main decay results and provide some examples in Section 4.

Assumptions and well posedness
In this section, we present some materials needed for the proof of our results and state the well-posedness results of (1.1) and (1.2). We use the standard Lebesgue space L 2 (Ω) and Sobolev space H 1 0 (Ω) with their usual scalar products that generate the norms || · || 2 and ||∇ · || 2 , respectively. We assume the following hypotheses: (A2) There exists a function G : which is increasing and strictly convex, with s → sG (s) and s → s(G ) −1 (s) are convex on R + , and there exists a nonincreasing function ξ : Remark 1. As a simple example of functions g satisfying (A1) and (A2), we can take Now, we state the existence results to (1.1) and (1.2); the proof is given in [15]. Systems (1.1) and (1.2) can be formulated as the following abstract linear first order system: (η t is the relative history of u, and it was introduced first in [11]), L i g is the weighted space with respect to the measure g(s)ds defined by Id is the identity operator and A is the linear operator given by in case (1.2), and Under the assumption (A1), the space H is a Hilbert space, D(A) ⊂ H with dense embedding, and A is the infinitesimal generator of a linear contraction C 0 -semigroup on H (see [15]). Therefore, from the classical semigroup theory (see [36]), we get the following well-posedness results for (2.3) (see [15] with Consequently, by assuming that (A1) is satisfied, the above theorem implies that, for any (u 0 (·, 0), Now, we consider the assumptions (A1) and (A2) and take initial data (u 0 , u 1 ) such that for (1.1), and for (1.2). We introduce the "modified" energy associated to (1.1) and (1.2) where f = ∇u for (1.1), f = u for (1.2) and we use the notation Also, following the idea of [37], we introduce the second "modified" energy associated to (1.2) Direct differentiation (multiplying (1.1) 1 by u t , multiplying (1.2) 1 , first, by u t and, second, by ∆u t , integrating by parts and using boundary conditions; see [37]) leads to On the other hand, using (2.1) and (2.2), we get

Preliminaries lemmas
In this section, we assume that (A1), (A2), (2.4) and (2.5) are satisfied, and we establish several lemmas needed for the proof of our stability results, which will be presented and proved in Section 4. These lemmas allow us to deal with the finite and infinite parts of the infinite memory. For the finite part, we apply the approach developped in [30] and [33] for the stability of wave equations with finite memory.  Then there exist positive constants N, N 1 , N 2 , a 1 , a 2 such that the functional where δ 0 = 1 for (1.1), and δ 0 = 0 for (1.2), satisfies
Proof. This lemma can be obtained by a direct application of the arguments of [33] used for (1.1) with finite memory. For (1.2), we need only to use Poincaré's inequality (2.2) to estimate (gou)(t) by c 0 (go∇u)(t).

Stability results
Before presenting our stability results, we put On the other hand, for fixed positive constants c 1 and c 2 , we introduce the class of functions α : where h is defined in (3.6); that ish(t) = t 0 h(s)ds, and G * 2 (t) = sup s∈R+ {ts−G 2 (s)}, t ∈ R + is the convex conjugate of G 2 in the sense of Young (see [2]). Thanks to (A2), G * 2 is given by According to (A2), G 0 is convex increasing on R + and defines a bijection from R + to R + , G 1 is decreasing and defines a bijection from ]0, 1] to R + , G 2 is convex and increasing on R + , γ is of class C 1 on R + , and G * 2 is convex and increasing on R + . Moreover, h is a C 1 (R + ) because, in both cases (1.1) and (1.2), we have η 0 ∈ L 1 g ; so, for any t ∈ R + , +∞ 0 g(t+s) 1 + ||∇u 0 (s)|| (since η 0 ∈ L 1 g thanks to (2.4) and (2.5)) and, using the left assumption in (2.1), Finally, we remark thath is of class C 1 and increasing on R + .
Case (1.2). Because 1 h+1 is nonicreasing and F 1 is nonegative, then (4.18) leads to is nonincreasing and integrating the above inequality over [0, t], we get The proof of Theorem 2 is now completed.
If s → ||∇u 0 (s)|| 2 is bounded (r = 0 in (4.32)) as it was assumed in [15,17], then the decay rate given in (4.36) with r = 0 is better than the one (ln (t+2)) −p (for some 0 <p small enough) obtained in [17], and it is little better than the one (ln (t + 2)) −p (for any 0 <p < q − 1) given in [15]. So we improve the results of [15] by droping the boundedness condition (1.14) on u 0 and obtaining a better decay rate.
2. The aprroach presented in this paper can be also applied to (1.16) and (1.20) in order to improve the decay rate of solutions obtained in [16,19], and drop the boundedness condition (1.17) on u 0 .