Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part II

In this paper, we study the connection between the bifurcation of diffusetransition layers and that of the underlying limit interfacial problemin a degenerate spatially inhomogeneous medium. In dimension one,we prove the existence of bifurcation of diffuse interfaces in a pitchfork spatial inhomogeneityfor a partial differential equation with bistable type nonlinearity.Bifurcation point is characterized quantitatively as well. The main conclusionis that the bifurcation diagram of the diffuse transition layers inheritsmostly from that of the zeros of the spatial inhomogeneity. However, explicitexamples are given for which the bifurcation of these two are different interms of (im)perfection. This is a continuation of [8] whichmakes use of bilinear nonlinearity allowing the use of explicit solutionformula. In the current work, we extend the results to a generalsmooth nonlinear function. We perform detail analysis ofthe principal eigenvalue and eigenfunction of some singularly perturbed eigenvalue problems andtheir interaction with the background inhomogeneity. This is the firstresult that takes into account simultaneously the interaction betweensingular perturbation, spatial inhomogeneity and bifurcation.