Asymptotic almost periodicity and motions of semigroups of operators

AbstractIn this survey we present an overview of recent work in resolving certain questions that arise in a qualitative study of solutions to the abstract Cauchy problem. Focusing on motions of C0-semigroups of operators and their asymptotic behavior, we address (1) the characterization of asymptotically almost periodic motions and (2) the characterization of those asymptotically almost periodic functions for which this property carries over to their integral. Moreover, we take up the problem raised by Nemytskii and Stepanov as to when the ω-limit set of a motion of a dynamical system is a minimal set of almost periodic motions