# Spectral theory and time asymptotics of size-structured two-phase population models

## TL;DR

This paper conducts a spectral analysis of size-structured two-phase population models, establishing conditions for exponential growth and extending results to infinite maximal size cases.

## Contribution

It provides a comprehensive spectral analysis framework for two-phase population models, including criteria for irreducibility, spectral gap, and exponential growth.

## Key findings

- Characterization of irreducibility of the semigroup.
- Conditions for the spectral gap and exponential growth.
- Extension of the theory to infinite maximal size.

## Abstract

This work provides a general spectral analysis of size-structured two-phase population models. Systematic functional analytic results are given. We deal first with the case of finite maximal size. We characterize the irreducibility of the corresponding $L^{1}$ semigroup in terms of properties of the different parameters of the system. We characterize also the spectral gap property of the semigroup. It turns out that the irreducibility of the semigroup implies the existence of the spectral gap. In particular, we provide a general criterion for asynchronous exponential growth. We show also how to deal with time asymptotics in case of lack of irreducibility. Finally, we extend the theory to the case of infinite maximal size.