Upper semicontinuity of pullback attractors for lattice nonclassical diffusion delay equations under singular perturbations

We consider the following lattice nonclassical diffusion equation with delays <v(over dot)>(i)(t) + lambda(0)v(i)(t) + (-1)(p) Lambda(p) v(i)(t) + epsilon(-1)(p) Lambda(p) <v(over dot)>(i)(t) = f(i)(v(i)(t - rho(t))) + g(i)(t), i is an element of Z, where epsilon is an element of (0, 1], lambda(0) is a positive constant with lambda(0) < 1, p is any positive integer and Delta is the discrete one-dimensional Laplace operator. Under suitable conditions on f and g we prove the existence of pullback attractors for the multi-valued process associated with the epsilon-small perturbed systems for which the uniqueness of solutions need not hold. Moreover, we compare the dynamics of the original systems and the epsilon-small perturbed systems, and show that their attractors are "close'' in the sense of Hausdorff semidistance. (C) 2014 Elsevier Inc. All rights reserved.