  We consider a class of mass transfer models on a one-dimensional lattice with
nearest-neighbour interactions. The evolution is given by the discrete backward
fast diffusion equation, with exponent $\beta$ in the regime $(-\infty,0) \cup
(0,1]$. Sites with mass zero are deleted from the system, which leads to a
coarsening of the mass distribution. The rate of coarsening suggested by
scaling is $t^\frac{1}{1-\beta}$ if $\beta \neq 1$ and exponential if $\beta =
1$. We prove that such solutions actually exist by an analysis of the
time-reversed evolution. In particular we establish positivity estimates and
long-time equililibrium properties for discrete parabolic equations with
bounded initial data.
