Abstract
Coarsening of solutions of the integro-differential equationformula herewhere Ω ⊂ ℝn, J(·) [ges ] 0, ε > 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 ε 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(·) [ges ] 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.
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