Asymptotic Behavior for Second Order Lattice Dynamical Systems

Recently, there are more and more authors to study the various properties of solutions for lattice dynamical systems, mainly are coupled map lattices and lattice ordinary differential equations, see [1-5] and the references therein. Lattice systems can be found in many fields of applications, for example, in chemical reaction theory, image processing and pattern recognition. Lattice systems have their own forms, in some cases, they arise in the spatially discretizations of partial differential equations. In this paper, we shall consider the asymptotic behavior of solutions for the following second order lattice dynamical system:


INTRODUCTION
Recently, there are more and more authors to study the various properties of solutions for lattice dynamical systems, mainly are coupled map lattices and lattice ordinary differential equations, see [1][2][3][4][5] and the references therein.Lattice systems can be found in many fields of applica- tions, for example, in chemical reaction theory, image processing and pattern recognition.Lattice systems have their own forms, in some cases, they arise in the spatially discretizations of partial differential equations.
In this paper, we shall consider the asymptotic behavior of solutions for the following second order lattice dynamical system: li -+-Oili (Ui-1 2Ui + Ui+l) + )Ui +f(ui) gi, iEZ (1) where a and A is a positive constants, gi is given, f(s) _,jm=o ajs 2+1 with a > 0, j-0, 1,... ,m, is a polynomial.By introducing a new weight inner product and norm in the space g2= {u (ui)iz 2 }, prove the existence of lui R, -i Z Ui < we a global attractor of system (1).The idea of using such a technique is due to Zhou [6] and Bates1, the later considered the existence of a global attractor for a first order lattice dynamical system.Equation (1) can be regarded as a discrete analogue of the following continuous damped semi-linear wave equation: utt + oeut Uxx + Au +f(u) g. (2) The global attractor and its dimension to Eq. (2) in bounded domain and unbounded domain have been studied in Hilbert spaces by Peter W. Bates, Kening Lu, Bixiang Wang, Attractors for lattice dynamical systems, Preprint, 1999.138 S. ZHOU many people, see [6][7][8][9][10][11][12] and the references therein.
This paper is organized as follows.In the second section, we present the existence and uniqueness of solutions for system (1).In Section 3, we prove the uniformly boundedness of solutions.In Section 3, we prove the existence of the global attractor.
IIF(qgl) F(2)lle < Z(ai, n)llq91 992i1, and thus, F() is locally Lipschitz from E to E. It is easy to see that the solutions of problem (3) is backward unique in time because if and a are replaced by -t and -a, the Eq. ( 3) is not changed.By the standard theory of ordinary differential equations, we obtain the existence and uniqueness of local solution o for problem (8).
From Lemma 3 below, it is obtained that the local solution (t) of ( 8) exists globally, that is, o CI(R, E), which implies that maps s(t) vo) e (u(t), v(t)) EE, t> 0 (11) generates a continuous semigroup {S(t)}t > o on E, where v(t) it(t) + eu(t). 4(e-o-) --e-a A Thus, the proof is completed.
LEMMA 3 If g g2, then there exists a bounded ball 0 OF(O, ro), centered at 0 with radius ro, such that for every bounded set B of E, there exists T(B) >_ 0 such that S(t)B C O, gt >_ T(B), where ro 2 (2/ccr)llg[I2.
Therefore, there exists a constant To _> 0 de- pending on O such that S(t)O c O, Vt >_ To.
4. GLOBAL ATTRACTOR Let H be a complete metric space and {S(t), > 0} be a continuous semigroup on H.

DEFINITION
A set X of H is called a global attractor for the semigroup {S(t), >_ 0} if (i) X is invariant set, i.e., S(t)X=X, gt >_ O. (ii) X is a compact set.(iii) X attracts any bounded set of H, i.e., for any bounded set BcH, d(S(t)B,X)-SUpx e s(0 infy e xd(x, y)-+O as too.
To obtain the existence of a global attractor for the semigroup {S(t)t >_ 0} associated with (8) on E. We need prove the asymptotic compactness of {s(t), > o}.