On spectral gaps of growth-fragmentation semigroups with mass loss or death

Mustapha Mokhtar-Kharroubi

doi:10.3934/cpaa.2022019

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2022, Volume 21, Issue 4: 1293-1327. Doi: 10.3934/cpaa.2022019

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On spectral gaps of growth-fragmentation semigroups with mass loss or death

Mustapha Mokhtar-Kharroubi

Laboratoire de Mathématiques, CNRS-UMR 6623, Université de Bourgogne Franche-Comté, 16 Route de Gray, 25030 Besançon, France

Received: June 2021

Revised: December 2021

Early access: January 2022

Published: April 2022

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Abstract

We give a general theory on well-posedness and time asymptotics for growth fragmentation equations in $ L^{1} $ spaces. We prove first generation of $ C_{0} $-semigroups governing them for unbounded total fragmentation rate and fragmentation kernel $ b(.,.) $ such that $ \int_{0}^{y}xb(x,y)dx = y-\eta (y)y $ ($ 0\leq \eta (y)\leq 1 $ expresses the mass loss) and continuous growth rate $ r(.) $ such that $ \int_{0}^{\infty }\frac{1}{r(\tau )}d\tau = +\infty . $This is done in the spaces of finite mass or finite mass and number of agregates. Generation relies on unbounded perturbation theory peculiar to positive semigroups in $ L^{1} $ spaces. Secondly, we show that the semigroup has a spectral gap and asynchronous exponential growth. The analysis relies on weak compactness tools and Frobenius theory of positive operators. A systematic functional analytic construction is provided.

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

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