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Upper semicontinuity of attractors for lattice systems under singular perturbations. (English) Zbl 1189.34021

Summary: Consider the following first order lattice system

u ˙ m +(2u m -u m-1 -u m+1 )+λ m u m +f m (u m )=g m ,m,

which is perturbed by the ε-small two order term

εu ¨ m +u ˙ m +(2u m -u m-1 -u m+1 +λ m u m +f m (u m )=g m ,m·

Under certain conditions on f m , λ m and g m , the original systems and the ε-small perturbed systems have global attractors 𝒜 in 2 and 𝒜 ε in 2 × 2 , respectively, and 𝒜 can be naturally embedded into a compact set 𝒜 0 in 2 × 2 . We prove the upper semicontinuity of 𝒜 0 with respect to the attractors 𝒜 ε at zero by showing that for any neighborhood 𝒪(𝒜 0 ) of 𝒜 0 , 𝒜 ε enters 𝒪(𝒜 0 ) if ε is small enough.

34A33Lattice differential equations
34E15Asymptotic singular perturbations, general theory (ODE)