L. Kerr, W. Lamb and M. Langer,
Discrete fragmentation systems in weighted ℓ1 spaces,
J. Evol. Equ. 20 (2020), 1419–1451
Abstract:
We investigate an infinite, linear system of ordinary differential equations
that models the evolution of fragmenting clusters.
We assume that each cluster is composed of identical units (monomers) and
we allow mass to be lost, gained or conserved during each fragmentation event.
By formulating the initial-value problem for the system as an abstract Cauchy problem (ACP),
posed in an appropriate weighted ℓ1 space, and then applying perturbation results
from the theory of operator semigroups, we prove the existence and uniqueness
of physically relevant, classical solutions for a wide class of
initial cluster distributions.
Additionally, we establish that it is always possible to identify a
weighted ℓ1 space on which the fragmentation semigroup is analytic,
which immediately implies that the corresponding ACP is well-posed
for any initial distribution belonging to this particular space.
We also investigate the asymptotic behaviour of solutions, and show that,
under appropriate restrictions on the fragmentation coefficients, solutions
display the expected long-term behaviour of converging
to a purely monomeric steady state.
Moreover, when the fragmentation semigroup is analytic, solutions are shown
to decay to this steady state at an explicitly defined exponential rate.