Special Solutions to a Nonlinear Coarsening Model with Local Interactions

Constantin Eichenberg

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 $$\partial _t x = - \frac{\beta }{|\beta |} \Delta x^\beta $$ ∂ t x = - β | β | Δ x β , with $$\beta $$ β in the fast diffusion regime $$(-\infty ,0) \cup (0,1]$$ ( - ∞ , 0 ) ∪ ( 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 $$t^\frac{1}{1-\beta }$$ t 1 1 - β if $$\beta \ne 1$$ β ≠ 1 and exponential if $$\beta = 1$$ β = 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 $$\ell _+^\infty (\mathbb {Z})$$ ℓ + ∞ ( Z ) .