UPPER SEMICONTINUITY OF PULLBACK ATTRACTORS FOR NON-AUTONOMOUS LATTICE SYSTEMS UNDER SINGULAR PERTURBATIONS

Consider the second order nonautonomous lattice systems with singular perturbations epsilon(u)Over dot(m) + (u)Over dot(m) + (Au)(m) + lambda(m)u(m) + f(m)(u(j)vertical bar j is an element of I-mq) = g(m)(t), m is an element of Z(k), epsilon > 0 (*) and the first order nonautonomous lattice systems (u)Over dot(m) + (Au)(m) + lambda(m)u(m) + f(m)(u(j)vertical bar j is an element of I-mq) = g(m)(t), m is an element of Z(k). (**) Under certain conditions, there are pullback attractors {A(epsilon)(t) C l(2) x l(2)}(t is an element of R) and {A(t) subset of l(2)}(t is an element of R) for systems (*) and (**), respectively. In this paper, we mainly consider the upper semicontinuity of attractors A(t) subset of l(2) x l(2), t is an element of R, with respect to the coefficient of second derivative term under Hausdorff semidistance. First, we study the relationship between A(t) and A(t) when epsilon -> 0(+). We construct a family of compact sets A(epsilon)(t) C l(2) x l(2), t is an element of R such that A(epsilon)(t) is naturally embedded into A(0)(t) as the first component, and prove that A(t) can enter any neighborhood of A(0)(t) when is small enough. Then for epsilon(0) > 0, we prove that A(epsilon)(t) can enter any neighborhood of A(epsilon 0)(t) when epsilon -> epsilon(0). Finally, we consider the existence and exponentially attraction of the singleton pullback attractors of systems (*)-(**).