Layers and spikes in non-homogeneous bistable reaction-diffusion equations

We study epsilon(2)(u) double over dot = f(u, x) = Au(1-u) (phi-u), where A = A(u, x) > 0, phi = phi(x) is an element of (0, 1), and epsilon > 0 is sufficiently small, on an interval [0, L] with boundary conditions (u) over dot = 0 at x = 0, L. All solutions with an e 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 - phi(-)) (u - phi) (u - phi(+)) with phi(-)(x) < phi(x) < phi(+)(x) and also to an infinite interval.