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

We consider the following lattice nonclassical diffusion equation with delaysv̇i(t)+λ0vi(t)+(-1)pΔpvi(t)+ε(-1)pΔpv̇i(t)=fi(vi(t-ρ(t)))+gi(t),i∈Z,where ε∈(0,1],λ0 is a positive constant with λ0<1,p is any positive integer and ▵▵ 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 ε-small perturbed systems for which the uniqueness of solutions need not hold. Moreover, we compare the dynamics of the original systems and the ε-small perturbed systems, and show that their attractors are “close” in the sense of Hausdorff semidistance.