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 equ al in dimension to that of the domain Omega: any sphere of fixed radiu s with interface contained in the domain is an equilibrium (indistingu ishable from the point of view of the perimeter functional). We resolv e 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 so lution preserves its ''bubble'' like shape until it reaches the bounda ry. Moreover, we show that the ''bubble'' moves superslowly. We also o btain an equation that determines those special spheres that correspon d to equilibria at the W-C-H level. Our work establishes the phenomeno n of superslow motion in higher space dimensions in the class of singl e interface solutions.