Abstract
In this paper, we consider discrete growth–decay–fragmentation equations 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 is independent on the initial conditions.
Highlights
Coagulation and fragmentation models that describe the processes of objects forming larger clusters or, splitting into smaller fragments have received a lot of attention over several decades due to their importance in chemical engineering and other fields of science and technology, see, e.g., [21,42]
One of the most efficient approaches to modelling dynamics of such processes is through the kinetic equation which describes the evolution of the distribution of interacting clusters with respect to their size/mass
It turned out to be advantageous to allow clusters to be composed of particles of any size x > 0. This leads to the continuous integro-differential equation that was derived by Müller in the pure coagulation case [32] and extended to a coagulation–fragmentation version in [29]
Summary
Coagulation and fragmentation models that describe the processes of objects forming larger clusters or, splitting into smaller fragments have received a lot of attention over several decades due to their importance in chemical engineering and other fields of science and technology, see, e.g., [21,42]. In “Appendix,” we provide a description of an alternative method, in which we split the equations in a seemingly more natural way, considering the processes of growth and decay separately from the independent process of fragmentation and using the Trotter–Kato representation formula to prove the existence of the solution semigroup. We find that this approach produces weaker results due to worse properties of the growth–decay semigroup.
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