On the asymptotic behaviour of semigroups for flows in infinite networks

TL;DR

This paper investigates the long-term behavior of operator semigroups modeling transport in infinite networks, especially focusing on non-strongly continuous cases, and establishes conditions for asymptotic periodicity.

Contribution

It extends understanding of asymptotic behavior to non-strongly continuous semigroups, proving asymptotic periodicity in these cases and revisiting classical results.

Findings

Semigroups exhibit asymptotic periodicity in operator norm.

Asymptotic behavior characterized for non-strongly continuous semigroups.

Extensions to bounded measures also show asymptotic periodicity.

Abstract

We study transport processes on infinite networks. The solution of these processes can be modeled by an operator semigroup on a suitable 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 where the corresponding semigroup is not strongly continuous. Due to this lack of strong continuity, there are currently only few results on the long-term behaviour of these semigroups. In this paper, we discuss the asymptotic behaviour for a certain class of these transport processes. In particular, it is proved 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 containing a multiplication operator. Furthermore, we revisit known results on the asymptotic behaviour of transport processes on infinite networks and prove the asymptotic periodicity of their extensions to the space of bounded measures.