Special solutions to a non-linear coarsening model with local interactions

TL;DR

This paper investigates special solutions to a non-linear lattice coarsening model driven by a backward fast diffusion equation, establishing their existence and analyzing long-term behavior through time-reversal techniques.

Contribution

It introduces a novel analysis of the backward fast diffusion equation on a lattice, proving the existence of coarsening solutions and deriving their long-time properties.

Findings

Existence of coarsening solutions for the model.

Long-time positivity and equilibrium estimates.

Validation of the scaling law for coarsening rate.

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.