The Allen-Cahn equation with generic initial datum

Abstract

We consider the Allen-Cahn equation ${\partial }_{t}u-\Delta u=u-u^3$ with a rapidly mixing Gaussian field as initial condition. We show that provided that the amplitude of the initial condition is not too large, the equation generates fronts described by nodal sets of the Bargmann-Fock Gaussian field, which then evolve according to mean curvature flow.