Layers and Spikes in Non-homogeneous Bistable Reaction-diffusion Equations

We study ε2ü = f(u, x) = Au (1−u) (φ−u), where A = A(u, x) > 0, φ = φ(x) ∈ (0, 1), and ε > 0 is sufficiently small, on an interval [0, L] with boundary conditions u̇ = 0 at x = 0, L. All solutions with an ε independent number of oscillations are analyzed. Existence of complicated patterns of layers and spikes is proved, and their Morse index is determined. It is observed that the results extend to f = A(u, x) (u − φ−) (u − φ) (u − φ+) with φ−(x) < φ(x) < φ+(x) and also to an infinite interval.