Upper semicontinuity of attractors for lattice systems under singular perturbations

Consider the following first order lattice system u̇m+(2um−um−1−um+1)+λmum+fm(um)=gm,m∈Z, which is perturbed by the ϵϵ-small two order term ϵüm+u̇m+(2um−um−1−um+1)+λmum+fm(um)=gm,m∈Z. Under certain conditions on fmfm, λmλm and gmgm, the original systems and the ϵϵ-small perturbed systems have global attractors AA in ℓ2ℓ2 and AϵAϵ in ℓ2×ℓ2ℓ2×ℓ2, respectively, and AA can be naturally embedded into a compact set A0A0 in ℓ2×ℓ2ℓ2×ℓ2. We prove the upper semicontinuity of A0A0 with respect to the attractors AϵAϵ at zero by showing that for any neighborhood O(A0)O(A0) of A0A0, AϵAϵ enters O(A0)O(A0) if ϵϵ is small enough.