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

We consider a singularly perturbed bistable reaction diffusion equation in a one-dimensional spatially degenerate inhomogeneous media. Degeneracy arises due to the choice of spatial inhomogeneity from some well-known class of normal forms or universal unfoldings. By means of a bilinear double well potential, we explicitly demonstrate the similarities and discrepancies between the bifurcation phenomena of the reaction diffusion equation and the limiting problem. The former is described by the location of the transition layer while the latter by the zeros of the spatial inhomogeneity function. Our result is the first which considers simultaneously the effects of singular perturbation, spatial inhomogeneity and bifurcation phenomena. (Part II [9] of this series analyzes the pitch-fork bifurcation for a general smooth double well potential where precise asymptotics and spectral analysis are needed.)