The discrete fragmentation equation: Semigroups, compactness and asynchronous exponential growth

Jacek Banasiak; Wilson Lamb

doi:10.3934/krm.2012.5.223

Article Contents

2012, Volume 5, Issue 2: 223-236. Doi: 10.3934/krm.2012.5.223

This issue Previous Article Next Article Boltzmann equation and hydrodynamics at the Burnett level

The discrete fragmentation equation: Semigroups, compactness and asynchronous exponential growth

Jacek Banasiak^1, and
Wilson Lamb^2,

1.
School of Mathematical Sciences, UKZN, Durban
2.
Department of Mathematics and Statistics, University of Strathclyde, Glasgow

Received: August 2011

Revised: November 2011

Published: June 2012

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Abstract

In this paper we present a class of fragmentation semigroups which are compact in a scale of spaces defined in terms of finite higher moments. We use this compactness result to analyse the long time behaviour of such semigroups and, in particular, to prove that they have the asynchronous growth property. We note that, despite compactness, this growth property is not automatic as the fragmentation semigroups are not irreducible.

Keywords:

Semigroups of operators,
fragmentation,
long time asymptotics,
compactness,
asynchronous exponential growth property.

Mathematics Subject Classification: Primary: 34G10; Secondary: 35B40, 35P05, 47D06, 45K05, 80A30.

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