Coarsening in an integro-differential model of phase transitions

Coarsening of solutions of the integro-differential equation u_t = e ?_O J(|x - y|)(u(y) - u(x)) dy - f(u), x e O, where O ? Rⁿ, J(·) = 0, e > 0 and f(u) = u³ - u (or similar bistable nonlinear term), is examined, and compared with results for the Allen-Cahn partial differential equation. Both equations are used as models of solid phase transitions. In particular, it is shown that when e is small enough, solutions of this integro-differential equation do not coarsen, in contrast to those of the Allen-Cahn equation. The special case J(·) = 1 is explored in detail, giving insight into the behaviour in the more general case J(·) = 0. Also, a numerical approximation method is outlined and used on tests in both one- and two-space dimensions to verify and illustrate the main result.