Multi-layered stationary solutions for a spatially inhomogeneous Allen-Cahn equation

We consider stationary solutions of a spatially inhomogeneous Allen-Cahn-type nonlinear diffusion equation in one space dimension. The equation involves a small parameter ε, and its nonlinearity has the form h(x)²f(u), where h(x) represents the spatial inhomogeneity and f(u) is derived from a double-well potential with equal well-depth. When ε is very small, stationary solutions develop transition layers. We first show that those transition layers can appear only near the local minimum and local maximum points of the coefficient h(x) and that at most a single layer can appear near each local minimum point of h(x). We then discuss the stability of layered stationary solutions and prove that the Morse index of a solution coincides with the total number of its layers that appear near the local maximum points of h(x). We also show the existence of stationary solutions having clustering layers at the local maximum points of h(x).