Notes on the asymptotic properties of some class of unbounded strongly continuous semigroups

The abstract Cauchy problem in the Banach and Hilbert space setting is considered and the asymptotic behavior of individual orbits of corresponding $C_0$-semigroup is studied. The possibility to find uniformly stable dense subset of initial states in the case of unstable semigroups (so-called polynomial stability) is discussed. Also, the existence of the fastest growing orbit (so-called maximal asymptotics) for certain class of semigroups is studied.