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 = pa rtial derivative Delta u/partial derivative n = 0 on partial derivative Ome ga, 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 everywher e except for a finite number of points, which we call nuclei, in Omega, whe re the states tend to a different constant value. For any N we find such so lutions with N peaks located at certain geometrically identified points. Th e proof is based on a dynamical systems viewpoint where the stationary solu tions being sought are equilibrium points on a finite-dimensional invariant manifold of multi-peaked states. In addition to the existence of these sol utions we also discuss their strong instability, justifying the name nuclei for the points of concentration. (C) 2000 Academic Press.