The discrete unbounded coagulation-fragmentation equation with growth, decay and sedimentation

Abstract

In this paper we consider discrete growth-decay-fragmentation equa-tions that describe the size-distribution of clusters that can undergo splitting, growth and decay. The clusters can be for instance animal groups that can split but can also grow, or decrease in size due to birth or death of individuals in the group, or chemical particles where the growth and decay can be due to surface deposition or erosion. We prove that for a large class of such problems the solution semigroup is analytic and compact and thus has the asynchronous exponential growth property; that is, the long term behaviour of any solution is given by a scalar exponential function multiplied by a vector, called the stable population distribution, that are independent on the initial conditions.