Equilibria with many nuclei for the Cahn-Hilliard equation

Let f be a bistable nonlinearity such as u - u(3). We consider multi-peaked stationary solutions to the Cahn-Hilliard equation u(t) = - Delta(epsilon(2) Delta u + f(u)) in Omega, partial derivative u/partial derivative n = partial derivative Delta u/partial derivative n = 0 on partial derivative Omega, with the average value of u in the metastable region. By "multi-peaked" we mean states which, as epsilon --> 0, tend to a constant value everywhere except for a finite number of points, which we call nuclei, in Omega, where the states tend to a different constant value. For any N we find such solutions with N peaks located at certain geometrically identified points. The proof is based on a dynamical systems viewpoint where the stationary solutions being sought are equilibrium points on a finite-dimensional invariant manifold of multi-peaked states. In addition to the existence of these solutions we also discuss their strong instability, justifying the name nuclei for the points of concentration. (C) 2000 Academic Press.