Equilibria with Many Nuclei for the Cahn–Hilliard Equation

Let f be a bistable nonlinearity such as u−u3. We consider multi-peaked stationary solutions to the Cahn–Hilliard equation ut=−Δ( 2 Δu+f(u)) in Ω, ∂u/∂n=∂ Δu/∂n=0 on ∂Ω, with the average value of u in the metastable region. By “multi-peaked” we mean states which, as →0, tend to a constant value everywhere except for a finite number of points, which we call nuclei, in Ω, 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.