Global asymptotic limit of solutions of the Cahn-Hilliard equation

We study the asymptotic limit, as epsilon SE arrow 0, of solutions of the Cahn-Hilliard equation u(t)(epsilon) = Delta(-epsilon Delta u(epsilon) + epsilon(-1) f(u(epsilon))) under the assumption that the initial energy integral(Omega) [epsilon/2\del u(epsilon)(.,0)\(2) + 1/epsilon F(u(epsilon)(.,0))] is bounded independent of epsilon. Here f = F', and F is a smooth function taking its global minimum 0 only at u = +/-1. We show that there is a subsequence of {u(epsilon)}(0<epsilon less than or equal to 1) converging to a weak solution of an appropriately defined limit Cahn-Hilliard problem. We also show that, in the case of radial symmetry, all the interfaces of the limit have multiplicity one for almost all time t > 0, regardless of initial energy distributions.