Slow dynamics for the Cahn-Hilliard equation in higher space dimensions: The motion of bubbles

It is known that the Van der Waals-Cahn-Hilliard (W-C-H) dynamics can be approximated by a Quasi-static Stefan problem with surface tension. It turns out that the Stefan problem has a manifold of equilibria equal in dimension to that of the domain Omega: any sphere of fixed radius with interface contained in the domain is an equilibrium (indistinguishable from the point of view of the perimeter functional). We resolve this degeneracy by showing that at the W-C-H level this manifold is replaced by a quasi-invariant stable manifold, on which the typical solution preserves its "bubble" like shape until it reaches the boundary. Moreover, we show that the "bubble" moves superslowly. We also obtain an equation that determines those special spheres that correspond to equilibria at the W-C-H level. Our work establishes the phenomenon of superslow motion in higher space dimensions in the class of single interface solutions.