On spectral gaps of growth-fragmentation semigroups in higher moment spaces
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Publication:6388157
DOI10.3934/KRM.2021050zbMath1508.47095arXiv2201.04832MaRDI QIDQ6388157
Mustapha Mokhtar-Kharroubi, Jacek Banasiak
Publication date: 13 January 2022
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 $L^{1}(mathbb{R} _{+}; left( 1+x
ight) ^{alpha }dx)$ for unbounded total fragmentation rates and continuous growth rates $r(.)$ such that $int_{0}^{+infty } frac{1}{r( au )}d au =+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.
One-parameter semigroups and linear evolution equations (47D06) Perturbation theory of linear operators (47A55) Positive linear operators and order-bounded operators (47B65) Integro-differential operators (47G20)
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