  We study the asymptotic properties of the steady state mass distribution for
a class of collision kernels in an aggregation-shattering model in the limit of
small shattering probabilities. It is shown that the exponents characterizing
the large and small mass asymptotic behavior of the mass distribution depend on
whether the collision kernel is local (the aggregation mass flux is essentially
generated by collisions between particles of similar masses), or non-local
(collision between particles of widely different masses give the main
contribution to the mass flux). We show that the non-local regime is further
divided into two sub-regimes corresponding to weak and strong non-locality. We
also observe that at the boundaries between the local and non-local regimes,
the mass distribution acquires logarithmic corrections to scaling and calculate
these corrections. Exact solutions for special kernels and numerical
simulations are used to validate some non-rigorous steps used in the analysis.
Our results show that for local kernels, the scaling solutions carry a constant
flux of mass due to aggregation, whereas for the non-local case there is a
correction to the constant flux exponent. Our results suggest that for general
scale-invariant kernels, the universality classes of mass distributions are
labeled by two parameters: the homogeneity degree of the kernel and one further
number measuring the degree of the non-locality of the kernel.
