Growth-fragmentation equations with McKendrick--von Foerster boundary condition

TL;DR

This paper investigates the mathematical properties and long-term behavior of growth-fragmentation equations with specific boundary conditions, providing new methods for analysis and explicit solutions for certain cases.

Contribution

It introduces three methods to prove well-posedness, analyzes spectral properties, and derives explicit solutions for growth-fragmentation equations with McKendrick--von Foerster boundary conditions.

Findings

Existence of a strongly continuous semigroup solution.

Conditions for the semigroup's irreducibility and exponential growth.

Explicit solution and Perron eigenpair for a special case.

Abstract

The paper concerns the well-posedness and long-term asymptotics of growth--fragmentation equation with unbounded fragmentation rates and McKendrick--von Foerster boundary conditions. We provide three different methods of proving that there is a strongly continuous semigroup solution to the problem and show that it is a compact perturbation of the corresponding semigroup with a homogeneous boundary condition. This allows for transferring the results on the spectral gap available for the later semigroup to the one considered in the paper. We also provide sufficient and necessary conditions for the irreducibility of the semigroup needed to prove that it has asynchronous exponential growth. We conclude the paper by deriving an explicit solution to a special class of growth--fragmentation problems with McKendrick--von Foerster boundary conditions and by finding its Perron eigenpair that determines its long-term behaviour.