Long-time behavior for impulsive generalized semiflows

Abstract

We propose new criteria, which are more general than those previously known in the literature, to guarantee the existence of global attractors for problems with state-dependent impulses. Our result on the existence of global attractors is proved for the case where the solutions on the underlying semiflow can possibly be nonunique, and it is even new for the case of solution uniqueness. Additionally, we provide collective versions of the proposed criteria, under which, as we prove, the global attractors exhibit upper-semicontinuity upon perturbation of the problem. The theory is illustrated through several examples of ODEs and PDEs.