Global attractor for a class of nonlinear lattices

We consider a class of nonlinear lattices with nonlinear dampingequation(0.1)u¨n(t)+(−1)pΔpun(t)+αun(t)+h(un(t))+g(n,u˙n(t))=fn, where n∈Zn∈Z, t∈R+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].