Long-time behavior for impulsive generalized semiflows

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.

Automated summary

We propose new criteria for the existence of global attractors for problems with state-dependent impulses that are more general than those previously known. Our results are applicable to both nonunique and unique solutions, and we provide collective versions of the criteria that demonstrate the upper-semicontinuity of global attractors under perturbation. The theory is illustrated through examples of ODEs and PDEs.