Attractors and dimension of dissipative lattice systems

In this paper, by using the argument in [Q.F. Ma, S.H. Wang, C.K. Zhong, Necessary and sufficient conditions for the existence of global attractor for semigroup and application, Indiana Univ. Math. J., 51(6) (2002), 1541–1559.], we prove that the condition given in [S. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations 200 (2004) 342–368.] for the existence of a global attractor for the semigroup associated with general lattice systems on a discrete Hilbert space is a sufficient and necessary condition. As an application, we consider the existence of a global attractor for a second-order lattice system in a discrete weighted space containing all bounded sequences. Finally, we show that the global attractor for first-order and partly dissipative lattice systems corresponding to (partly dissipative) reaction–diffusion equations and second-order dissipative lattice systems corresponding to the strongly damped wave equations have finite fractal dimension if the derivative of the nonlinear term is small at the origin.