Global attractor for a class of nonlinear lattices

We consider a class of nonlinear lattices with nonlinear damping(0.1) u ¨ n ( t ) + ( − 1 ) p Δ p u n ( t ) + α u n ( t ) + h ( u n ( t ) ) + g ( n , u ˙ n ( t ) ) = f n , where n ∈ Z , t ∈ R + , α is a real positive constant, p is any positive integer and Δ is the discrete one-dimensional Laplace operator. Under suitable conditions on h and g we prove the existence of a global attractor for the continuous semigroup associated with (0.1). Our proofs are based on a difference inequality due to M. Nakao [M. Nakao, Global attractors for nonlinear wave equations with nonlinear dissipative terms, J. Differential Equations 227 (2006) 204–229].