Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media. I. (English) Zbl 1304.35050
Summary: 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 [the authors, “Singular perturbation and bifurcation of diffused transition layers
in degenerate inhomogeneous media. II”, preprint (2013)] of this series analyzes the pitchfork bifurcation for a general smooth double well potential where precise asymptotics and spectral analysis are needed.)
MSC:
35B25 | Singular perturbations in context of PDEs |
35B32 | Bifurcations in context of PDEs |
35K57 | Reaction-diffusion equations |
34D15 | Singular perturbations of ordinary differential equations |
34E05 | Asymptotic expansions of solutions to ordinary differential equations |
34E10 | Perturbations, asymptotics of solutions to ordinary differential equations |