On the asymptotic behaviour of semigroups for flows in infinite networks

Abstract

We study transport processes on infinite networks which can be modeled by an operator semigroup on a Banach space. Classically, such semigroups are strongly continuous and therefore their asymptotic behaviour is quite well understood. However, recently new examples of transport processes emerged in which the corresponding semigroup is not strongly continuous. Due to this lack of strong continuity, there are currently no results on the long-term behaviour of these semigroups. In this paper, we close this gap for a certain class of transport processes. In particular, it is proven that the solution semigroups behave asymptotically periodic with respect to the operator norm as a consequence of a more general result on the long-term behaviour by positive semigroups that contain a multiplication operator. Furthermore, we revisit known results on the asymptotic behaviour of transport processes on infinite networks and prove the asymptotically periodicity of the extensions of those semigroups to the space of bounded measures.