On spectral gaps of growth-fragmentation semigroups in higher moment spaces

TL;DR

This paper develops a general functional analytic approach to establish spectral gaps and exponential growth in growth-fragmentation semigroups within specific moment spaces, accommodating unbounded rates and continuous growth functions.

Contribution

It introduces a new method using weak compactness and Frobenius theory to prove spectral gaps for growth-fragmentation semigroups in higher moment spaces, extending previous results.

Findings

Spectral gaps exist under certain moment space conditions.

The approach applies to unbounded fragmentation and growth rates.

Examples illustrate the theory's applicability.

Abstract

We present a general approach to proving the existence of spectral gaps and asynchronous exponential growth for growth-fragmentation semigroups in moment spaces $L^{1}(\mathbb{R}_{+};\ x^{\alpha }dx)$ and $L^{1}(\mathbb{R} _{+};\ \left( 1+x\right) ^{\alpha }dx)$ for unbounded total fragmentation rates and continuous growth rates $r(.)$\ such that $\int_{0}^{+\infty } \frac{1}{r(\tau )}d\tau =+\infty .\ $The analysis is based on weak compactness tools and Frobenius theory of positive operators and holds provided that $\alpha >\widehat{\alpha }$ for a suitable threshold $\widehat{ \alpha }\geq 1$ that depends on the moment space we consider. A systematic functional analytic construction is provided. Various examples of fragmentation kernels illustrating the theory are given and an open problem is mentioned.