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

Mustapha Mokhtar-Kharroubi; Jacek Banasiak

doi:10.3934/krm.2021050

Article Contents

2022, Volume 15, Issue 2: 147-185. Doi: 10.3934/krm.2021050

This issue Previous Article Next Article Kinetic equations for processes on co-evolving networks

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

Mustapha Mokhtar-Kharroubi^1, , and
Jacek Banasiak^2,

1.
Laboratoire de Mathématiques, CNRS-UMR 6623, Université de Bourgogne Franche-Comté, 16 Route de Gray, 25030 Besançon, France
2.
Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa, Institute of Mathematics, Łódź University of Technology, Łódź, Poland

^*Corresponding author: Mustapha Mokhtar-Kharroubi
^*Corresponding author: Mustapha Mokhtar-Kharroubi

Received: March 31, 2021

Revised: October 31, 2021

Early access: January 2022

Published: April 2022

Both authors were supported by DSI/NRF SARChI Grant 82770. The second author was also supported by the National Science Centre of Poland Grant 2017/25/B/ST1/00051

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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.

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Mathematics Subject Classification: Primary: 47D06, 47G20; Secondary: 47B65, 47A55.

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