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.

We examine an infinite system of ordinary differential equations that models a discrete fragmentation process in which mass loss can occur. The problem is treated as an abstract Cauchy problem, posed in an appropriate Banach space. Perturbation techniques from the theory of semigroups of operators are used to establish the existence and uniqueness of physically meaningful solutions under minimal restrictions on the fragmentation rates. In one particular case, an explicit formula for the associated semigroup is obtained and this enables additional properties, such as compactness of the resolvent and analyticity of the semigroup, to be deduced. Another explicit solution of this particular fragmentation problem, in which mass is apparently created from a zero-mass initial state, is also investigated, and the theory of Sobolev towers is used to prove that the solution actually emanates from an initial infinite cluster of unit mass.