  In this paper, we investigate the use of so called "duality lemmas" to study
the system of discrete coagulation-fragmentation equations with diffusion. When
the fragmentation is strong enough with respect to the coagulation, we show
that we have creation and propagation of superlinear moments. In particular
this implies that strong enough fragmentation can prevent gelation even for
superlinear coagulation, a statement which was only known up to now in the
homogeneous setting. We also use this control of superlinear moments to extend
a recent result about the regularity of the solutions in the pure coagulation
case, to strong fragmentation models.
