Upper semicontinuity of uniform attractors fornon-autonomous lattice systems under singular perturbations

Abstract

In this paper, we study the upper semicontinuity ofthe uniform attractors for the second order nonautonomous lattice systemsdriven almost periodic force with respect tothe coefficient of the second order derivative term under Hausdorffsemidistance. First, we present the existence of uniform attractors for thesecond order nonautonomous lattice systems and the corresponding first order nonautonomous lattice systems under certain 条件, respectively. Then we consider the relationship between these two uniform attractors when the coefficient of the second derivation term tendsto zero. The method is to construct a new set in the phase space of thesecond order lattice systems such that the uniform attractorof the first order lattice system is naturally embedded intothis set as the first component, and prove that the uniform attractors forthe second order nonautonomous lattice system can enter any neighborhood of thisset when the coefficient of the second derivation term is small enough. Finally,we discuss the upper semicontinuity of the uniform attractors of the secondorder nonautonomous lattice systems at any positive coefficient of the secondderivative term.