  We investigate the kinetics of systems in which particles of one species
undergo binary fragmentation and pair annihilation. In the latter, nonlinear
process, fragments react at collision to produce an inert species, causing loss
of mass. We analyse these systems in the reaction-limited regime by solving a
continuous model within the mean-field approximation. The rate of
fragmentation, for a particle of mass $x$ to break into fragments of masses $y$
and $x-y$, has the form $x^{\lambda-1}$ ($\lambda>0$), and the annihilation
rate is constant and independent of the masses of the reactants. We find that
the asymptotic regime is characterized by the annihilation of small-mass
clusters. The results are compared with those for a model with linear mass-loss
(i.e.\ with a sink). We also study more complex models, in which the processes
of fragmentation and annihilation are controlled by mutually-reacting
catalysts. Both pair- and linear-annihilation are considered. Depending on the
specific model and initial densities of the catalysts, the time-decay of the
cluster-density can now be very unconventional and even non-universal. The
interplay between the intervening processes and the existence of a scaling
regime are determined by the asymptotic behaviour of the average-mass and of
the mass-density, which may either decay indefinitely or tend to a constant
value. We discuss further developments of this class of models and their
potential applications.
