Global attractors for non-linear viscoelastic equation with strong damping

. In this paper, we consider the long-time dynamical behavior of the viscoelastic equations with strong damping and further prove the existence of global attractors for this system.


Introduction
In this paper, we discuss the long-time dynamical behavior for the nonlinear viscoelastic problem: together with initial conditions u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x) and boundary conditions u(x, t) = 0, x ∈ ∂Ω, where Ω is a bounded domain in R n , n 1, with a smooth boundary and ρ, γ > 0 are real numbers. Here, u(x, t) represent displacement and g is a positive decaying function representing the kernel of memory term that will be specified below. Problems relate to the equation are interesting not only from the point of view of PDE general theory, but also due to its applications in mechanics. For instance, when the material density, f (u t ), is equal to 1, Eq. (2) describes the extensional vibrations of thin rods, see [1] for the physical details.
When the material density f (u t ) is not constant, we are dealing with a thin rod which possesses a rigid surface and whose interior is somehow permissive to slight deformations such that the material density varies according to the velocity. On the other hand, it is important to observe that similar equations to the one given in (2) arise in the study of viscoelastic plates with memory, more precisely More recently, Cavalcanti et al. [2] considered this problem by assuming 0 < ρ 2/(n−2) when n 3, ρ > 0 when n = 1, 2 and g decays exponentially. They obtained the global existence result for γ 0 and the uniform exponential decay of energy for γ > 0. Later, the decay result has been extended by Messaoudi and Tatar in [3] to the case γ = 0. Han and Wang [4] considered the following viscoelastic equation: more recently, Ma [5] considered the attractors of this problem. When there is no dispersion term, the related problems have been extensively studied and several results about existence, decay and blow-up been obtained. For instance, Cavalcanti et al. [6] deal with the equation with same boundary and initial conditions as that of system (1). Assuming that a(x) is a nonnegative function that may vanish outside of a subset Ω 0 ⊂ Ω of positive measure and g decays exponentially, they proved an exponential decay result of energy of (3). This result was later extended by Berrimi and Messaoudi [7] to the nonlinear damping case By introducing a new functional, they weakened the conditions on a(x) and g and obtained the decay result. Motivated by the ideas of Messaoudi [8], the authors established the general uniform decay of the energy for this model.
The aim of this paper is to prove the existence of the global attractor for system (1).

Preliminaries
Assume that ρ satisfies For the kernel function g, we assume that it verifies: (A2) There exists a positive function ξ(t) verifying where k is a positive constant.

Proof of the main result
In this section we prove our main result. Eqs. (1) can be transformed into the system in Ω × (0, ∞). We shall consider the problem (5)-(7) in the following Hilbert space Recall that the global attractor of S(t) acting on H is a compact set A ⊂ H enjoying the following properties: (i) A is fully invariant for S(t), that is, S(t)A = A for every t 0; (ii) A is an attracting set, namely, for any bounded set R ⊂ H, where δ H denotes the Hausdorff semi-distance on H.
More details on the subject of global attractors can be found in the books [10,11]. Remark 1. Lemma 2 implies that the existence of a bounded absorbing set R * ⊂ H for the C 0 -semigroup S(t). Indeed, if R * is any absorbing ball of H, then for any bounded set R ⊂ H, it is immediate to see that there exists t(R) 0 such that it is clear that R 0 is still a bounded absorbing set which is also invariant for S(t), that is, In the sequel, we define the operator A as Af = −∆f . It is well known that A is a positive operator on L 2 with domain D(A) = H 2 ∩ H 1 0 . Moreover, we can define the powers A s of A for s ∈ R. The space V 2s = D(A s ) turns out to be a Hilbert space with the inner product where · stands for L 2 -inner product on L 2 .
www.mii.lt/NA In particular, For further convenience, for s ∈ R, introduce the Hilbert space Clearly, where R 0 is the invariant, bounded absorbing set of S(t) given by Remark 1, take the inner product in H 0 of (5)- (7) and Here, the boundary term of integration by parts is neglected since we are working with more regular functions, similar application, please refer to [12]. Next, let and where M and ε are positive constants to be determined later.
By assumption (A2), we find that ξ(t) is a positive non-increasing function, then ξ(t) ξ(0) = L for every t 0. Repeating the similar arguments to those of the proof of Theorem 2.1 of Cavalcanti et al. [2], choosing our constants very carefully and properly, we get Finally (see Lemma 5.5 in [13]), we have the compact embedding where η t is defined in (4). Denote the closure of B in L 2 g (R + , H 1 0 ) byB. With reference to (8) and (9) for t 0, let R(t) be the ball of V 5/2 × V 3/2 and introduce the set (9), G(t) is compact in H. Then, due to the compactness of G(t) for every fixed t 0 and every d > 3l −1 F (0)e −(ε/2)C2t , there exist finitely many balls of H of radius d such that Φ(t) belongs to the union of such balls for every Φ 0 ∈ R 0 . This implies that α H (S(t)R 0 ) 3l −1 F (0)e −(ε/2)C2t ∀t 0, where α H is the Kuratowski measure of non-compactness, defined by α H (R) = inf{d: R has a finite cover of balls of H of diameter less than d}.
Since the invariant, connected, bounded absorbing set R 0 fulfills (10), exploiting a classical result of the theory of attractors of semigroups (see, e.g., [14]), we conclude that the ω-limit set of R 0 , that is, is a connected and compact global attractor of S(t). Therefore we have proved the following result.