Special Solutions to a Nonlinear Coarsening Model with Local Interactions

Abstract

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.