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

We study $\varepsilon^2\ddot{u}=f(u,x)=A\, u\, (1-u)\,(\phi-u)$, where $A=A(u,x)>0$, $\phi=\phi(x)\in(0,1)$, and $\varepsilon>0$ is sufficiently small, on an interval $[0,L]$ with boundary conditions $\dot{u}=0$ at $x=0,L$. All solutions with an $\varepsilon$ 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.