Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

On a coagulation and fragmentation equation with mass loss

Jacek Banasiaka1 and Wilson Lamba2

a1 School of Mathematical Sciences, University of KwaZulu-Natal, Durban 4041, South Africa (

a2 Department of Mathematics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, UK (


A nonlinear integro-differential equation that models a coagulation and multiple fragmentation process in which continuous and discrete fragmentation mass loss can occur is examined using the theory of strongly continuous semigroups of operators. Under the assumptions that the coagulation kernel is constant, the fragmentation-rate function is linearly bounded, and the continuous mass-loss-rate function is locally Lipschitz, global existence and uniqueness of solutions that lose mass in accordance with the model are established. In the case when no coagulation is present and the fragmentation process is binary with constant fragmentation kernel and constant continuous mass loss, an explicit formula is given for the associated substochastic semigroup.

(Received August 29 2005)

(Accepted January 11 2006)