# 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 backward parabolic equation \(\partial _t x = - \frac{\beta }{|\beta |} \Delta x^\beta \), with \(\beta \) in the fast diffusion 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 \(\beta \ne 1\) 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 equilibrium properties for discrete parabolic equations with initial data in \(\ell _+^\infty (\mathbb {Z})\).

## Keywords

Coarsening Infinite particle system Backward fast diffusion Discrete parabolic regularity## Mathematics Subject Classification

70F45 35K55 37L60## Notes

### Acknowledgements

The author would like to thank Barbara Niethammer and Juan J. L. Velázquez for inspiration, helpful discussions and proofreading. This work was supported by the German Research Foundation through the CRC 1060 *The Mathematics of Emergent Effects*.

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