The discrete fragmentation equation: Semigroups, compactness and asynchronous exponential growth
Jacek Banasiak Wilson Lamb
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: fragmentation Semigroups of operators compactness long time asymptotics asynchronous exponential growth property.
Coagulation, fragmentation and growth processes in a size structured population
Jacek Banasiak Wilson Lamb
An equation describing the dynamical behaviour of phytoplankton cells is considered in which the effects of cell division and aggregration are incorporated by coupling the coagulation-fragmentation equation with the McKendrick-von Foerster renewal model of an age-structured population. Under appropriate conditions on the model parameters, the associated initial-boundary value problem is shown to be well posed in a physically relevant Banach space.
keywords: semilinear Cauchy problem coagulation algal population semigroups of operators size structured population models. fragmentation
Coagulation and fragmentation processes with evolving size and shape profiles: A semigroup approach
Wilson Lamb Adam McBride Louise Smith
We investigate a class of bivariate coagulation-fragmentation equations. These equations describe the evolution of a system of particles that are characterised not only by a discrete size variable but also by a shape variable which can be either discrete or continuous. Existence and uniqueness of strong solutions to the associated abstract Cauchy problems are established by using the theory of substochastic semigroups of operators.
keywords: semilinear Cauchy problems Semigroups of operators multicomponent coagulation-fragmentation processes.

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