LONG-TIME BEHAVIOR OF A CLASS OF VISCOELASTIC PLATE EQUATIONS

. This paper is concerned with the initial-boundary value problem for a class of viscoelastic plate equations on an arbitrary dimensional bounded domain. Under certain assumptions on the memory kernel and the source term, the global well-posedness of solutions and the existence of global attractors are obtained.


Introduction
In this paper, we study the following initial-boundary value problem for nonlinear viscoelastic plate equations (2) u(x, t) = u 0 (x, t), u t (x, 0) = u 1 (x), x ∈ Ω, t ≤ 0, where Ω is a bounded domain of R N with a smooth boundary ∂Ω, the memory kernel g and the external forces f , h will be specified later. Problem 1-3 can be used to describe the vibrations of viscoelastic materials possessing a capacity of storage and dissipation of mechanical energy, see [17] for the details. And u(x, t) represents the displacement at time t of a particle having position x in a given reference configuration with the prescribed past history u 0 : Ω × (−∞, 0] → R. In view of the main results of [5], we see that the viscoelastic term (namely the memory term) produces the effect of strong dissipation, which prevails the effect of weak damping term on the decay of solutions in time.
There have been many works on the long-time behavior of viscoelastic plate equations, we refer the readers to [1,2,4,6,18,20,21,25] and the references therein. As for viscoelastic plate equations with past history, Pata [23] studied where A is a self-adjoint and strictly positive linear operator, α and µ are positive constants. Based on certain assumptions on g, he analyzed the exponential stability of the related semigroup. Guesmia and Messaoudi [13] investigated Under some assumptions on A and g, they established a general decay result which depends on the behavior of g. Jorge Silva and Ma [15] considered where h ∈ L 2 (Ω), and Ω is a bounded domain of R N (N ≥ 1) with a smooth boundary ∂Ω. Under some assumptions on f and g, they obtained the global wellposedness and regularity of weak solutions. Moreover, they proved the exponential decay of energy. Recently, Conti and Geredeli [9] studied where h ∈ L 2 (Ω), Ω is a bounded domain of R 3 with a smooth boundary ∂Ω. Under some assumptions on f 1 , f 2 and g, they obtained the existence and regularity of global attractors. In the works mentioned above, authors introduced a variable which reflects the relative displacement history so that the corresponding problem could be turned into an autonomous system. This scheme is so-called the past history approach [12] which suggests to consider some past history variables as additional components of the phase space corresponding to the equation under study.
In the present paper, in order to study the long-time behaviour of solutions of problem 1-3, we employ the past history approach and the operator technique so that Eq. 1 can be transformed into an abstract system in the history phase space. And thus the operator technique combined with the energy estimates becomes a crucial tool for the proof of the existence of global attractors. This paper is organized as follows. In Section 2 some notations and assumptions on f and g are displayed. Moreover, 1-3 is transformed into a generalized problem, and the main results of this paper are stated. In Section 3 the global well-posedness of regular solutions is obtained. And the global well-posedness of weak solutions is established by the density arguments [6]. In Section 4 the existence of global attractors is derived by means of the existence of an absorbing set and the semigroup decomposition [14,16,26].
(·, ·) denotes either the L 2 -inner product or a duality pairing between a space and its dual space. Moreover, |Ω| stands for the Lebesgue measure of Ω, C i , i = 1, 2, 3, · · · denote some different positive constants, and C i , i = 1, 2, 3 represent the positive constants for inequalities As in [3,10,27], we give the following assumptions on f in order to state the main results of this paper.
2.2. Reformulation of the problem. As in [2,3,8,10,27], we define the operator where the dense domain It is easy to verify that A is self-adjoint and strictly positive. Thus A γ is also selfadjoint and strictly positive for any γ > 0. Denote V 4γ := D(A γ ) and V −γ := V γ . Then, for any γ ∈ R, V γ and L g,γ are Hilbert spaces equipped with inner products and norms (u, w) Vγ = (A Thus . In this way, problem 1-3 can be seen as Now we are in a position to define the auxiliary function Thus the viscoelastic dissipation in 7 can be rewritten as Therefore, problem 1-3 is transformed into the following system for any w 1 ∈ V 2 , w 2 ∈ L g,2 and a.e. t ∈ (0, T ]. Thus, in order to deal with problem 1-3, we study the modified problem 8, 9. In fact, for a solution (u, u t , v t ) of problem 8, 9, we have In view of [17, Chapter 2, Section 4], we see that g(t) := −G (t), where G(t) is the viscoelastic flexural rigidity. From we deduce that Substituting this into 10, we obtain which shows that (u, u t ) is a solution of problem 1-3.

Statement of main results.
The main results of this paper are stated as follows.
Then it is easy to see from Theorem 2.2 that {S(t)} t≥0 is a C 0 -semigroup generated by problem 8, 9. 3. Proof of Theorem 2.2 , which depends continuously on initial data.
Proof. Let {ω j } ∞ j=1 be an orthogonal basis of V 2 and an orthonormal basis of V 0 given by eigenfunctions of A. As in [11,15] is an orthonormal basis of L g,2 . We construct the approximate solutions of problem 8, 9 u n (t) = The approximate problem 11, 12 can be reduced to an ordinary differential system in the variables ξ jn (t) and ζ jn (t). In terms of standard theory for ODEs, there exists a solution (u n (t), u nt (t), v t n ) on some interval [0, T n ) with T n ≤ T . The following estimates will allow us to extend the local solutions to [0, T ] with any T > 0.
Replacing ω j in 11 1 with u nt and e j in 11 2 with v t n , summing for j and adding the two results, we obtain Since lim

≥0.
Hence, by integrating 13 with respect to t from 0 to t, we get It follows from 4 in (A 1 ) that, for any η > 0, there exists a constant C η > 0 such that Ω F (u n ) dx ≤ η u n 2 + C η |Ω|.
By virtue of Cauchy's inequality with > 0, we get Consequently, taking sufficiently small η and such that we deduce from 14 that Hence, from 15, 16 and 12, it follows that Replacing ω j in 11 1 with A 1 2 u nt and e j in 11 2 with A 1 2 v t n , summing for j and adding the two results, we obtain we conclude from 17 that Therefore, there exist u, v t and subsequences of {u n }, {v t n }, still represented by the same notations and we shall not repeat, such that, as n → ∞, We now claim that for any t ∈ [0, T ] and fixed j, as n → ∞. Indeed, for any w ∈ V 2 , we have If p > 2, then when N ≤ 4, .

Hence
If p = 2, then it is clear that 23 remains valid. Therefore, Thus assertion 22 follows from 21.
For fixed j, Hence lim n→∞ (v t nτ , e j ) g,2 = (v t τ , e j ) g,2 . Consequently, for fixed j, integrating 11 with respect to t and taking n → ∞, we get Moreover, it is easy to see from 12 that 3 . Therefore, (u, u t , v t ) is a solution of problem 8, 9.
According to Theorem 3.1, for any m ∈ N + , problem 8, 9 admits a unique regular solution (u m , u mt , v t m ) satisfying Then, by the arguments similar to the proof of 26, we get Thus (u, u t , v t ) is a global weak solution of problem 8, 9. Suppose that (u, u t , v t ) and (ū,ū t ,v t ) are two solutions of problem 8, 9 with initial data (u 0 , u 1 , v 0 ), (ū 0 ,ū 1 ,v 0 ), respectively. Then there exist such that Therefore, in terms of 28-32, the conclusions of Theorem 2.2 are derived immediately.

Proof of Theorem 2.3
In this section, for the sake of convenience, we denote Proof. Let U = u t + εu, t ∈ [0, ∞), where ε is a positive constant to be determined later. Then 8 1 becomes Multiplying 33 by U in V 0 and 8 2 by v t in L g,2 , integrating over Ω and adding the two results, we obtain Hence Applying Cauchy's inequality with 1 > 0, we get It follows from 5 that, for any η > 0, there exists a constant C η > 0 such that Moreover, Consequently, by taking sufficiently small 1 , η and 2 such that Since we conclude from 36 and the arguments similar to the proof of 16 that It follows from 34 and 35 that which yields This, together with 37, gives Hence S(t) possesses an absorbing set with the radius R > C14 + ε C17 ε C16 ( h 2 + |Ω|).