An initial-value problem modelling coagulation and fragmentation processes is studied. The results of earlier papers are extended to models where either one or both of the rates of coagulation and fragmentation depend on time. An abstract integral equation, involving the solution operator to the linear fragmentation part, is investigated via the contraction mapping principle. A unique global, nonnegative, mass-conserving solution to this abstract equation is shown to exist. The latter solution is used to generate a global, non-negative, mass-conserving solution to the original non-autonomous coagulation and multiple-fragmentation equation. (C) 1998 B. G. Teubner Stuttgart-John Wiley & Sons, Ltd.

Prior knowledge regarding the existence and uniqueness of nonnegative, mass-conserving solutions to a multiple-fragmentation equation is utilized to study a combined coagulation and fragmentation model. The coagulation and fragmentation equation is first recast as an abstract integral equation involving the solution operator associated with the fragmentation part. A contraction mapping argument is then used to prove the existence and uniqueness of a local solution. Detailed investigation of the related iteration scheme yields nonnegativity and mass conservation. The solution is shown to be global.

We investigate an initial-value problem modelling fragmentation processes where particles split into two or more pieces at a rate, γ, that not only depends on the sizes of the particles involved but also on time. The existence of non-negative, mass-conserving solutions is established by considering a truncated version of an associated non-autonomous abstract Cauchy problem. The latter has solutions of the form u(t)=Un(t,t0)f, t⩾t0, where f is the known data at some fixed time t0⩾0 and {Un(t,s)} is a uniformly continuous evolution system. A limit evolution system {U(t,s)} is shown to exist. Depending on the form of the known data f at time t0, the scalar-valued function u, obtained from the limit evolution system via u(x, t)=[U(t, t0)f](x) for a.e. x>0, t⩾t0, is a solution of either the original initial-value problem or an integral version of this problem. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.

An initial-value problem modelling fragmentation processes, where particles split into two or more pieces, is studied using the theory of linear semigroups. The existence and uniqueness of nonnegative, mass-conserving solutions are established.