Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part I
Chaoqun Huang, Nung Kwan Yip · Networks and Heterogeneous Media · 2013
We consider a singularly perturbed bistable reaction diffusion equationin a one-dimensional spatially degenerate inhomogeneous media.Degeneracy arises due to the choice of spatial inhomogeneityfrom some well-known class of normal forms or universal unfoldings.By means of a bilinear double well potential, we explicitly demonstratethe similarities and discrepancies between the bifurcation phenomenaof the reaction diffusion equation and the limiting problem.The former is described by the location of the transition layer whilethe latter by the zeros of the spatial inhomogeneity function.Our result is the first which considers simultaneously the effectsof singular perturbation, spatial inhomogeneity andbifurcation phenomena.(Part II [9] of this series analyzes the pitch-fork bifurcation for ageneral smooth double well potential where precise asymptotics and spectralanalysis are needed.)