L. Smith, W. Lamb, M. Langer and A. McBride,
Discrete fragmentation with mass loss,
J. Evol. Equ. 12 (2012), 181–201
Abstract:
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.