Notes on the Asymptotic Properties of Some
Class of Unbounded Strongly Continuous
Semigroups

G.M. Sklyar

Institute of Mathematics, University of Szczecin, Wielkopolska 15, Szczecin 70-451, Poland
E-mail: sklar@univ.szczecin.pl

P. Polak

Institute of Mathematics, University of Szczecin, Wielkopolska 15, Szczecin 70-451, Poland
E-mail: piotr.polak@usz.edu.pl

Received April 4, 2018.

Abstract

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.