Metastable Bubble Solutions for the Allen-Cahn Equation with Mass Conservation

In a multi-dimensional domain, the slow motion behavior of internal layer solutions with spherical interfaces, referred to as bubble solutions, is analyzed for the nonlocal Allen-Cahn equation with mass conservation. This problem represents the simplest model for the phase separation of a binary mixture in the presence of a mass constraint. The bubble is shown to drift exponentially slowly across the domain, without change of shape, towards the closest point on the boundary of the domain. An explicit ordinary diierential equation for the motion of the center of the bubble is derived by extending, to a multi-dimensional setting, the asymptotic projection method developed previously by the author to treat metastable problems in one spatial dimension. An asymptotic formula for the time of collapse of the bubble against the boundary of the domain is derived in terms of the principal radii of curvature of the boundary at the initial contact point. An analogy between slow bubble motion and the classical exit problem for diiusion in a potential well is given.