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

Abstract

In this paper, we study the connection between the bifurcation of diffuse transition layers and that of the underlying limit interfacial problem in a degenerate spatially inhomogeneous medium. In dimension one, we prove the existence of bifurcation of diffuse interfaces in a pitchfork spatial inhomogeneity for a partial differential equation with bistable type nonlinearity. Bifurcation point is characterized quantitatively as well. The main conclusion is that the bifurcation diagram of the diffuse transition layers inherits mostly from that of the zeros of the spatial inhomogeneity. However, explicit examples are given for which the bifurcation of these two are different in terms of (im)perfection. This is a continuation of [8] which makes use of bilinear nonlinearity allowing the use of explicit solution formula. In the current work, we extend the results to a general smooth nonlinear function. We perform detail analysis of the principal eigenvalue and eigenfunction of some singularly perturbed eigenvalue problems and their interaction with the background inhomogeneity. This is the first result that takes into account simultaneously the interaction between singular perturbation, spatial inhomogeneity and bifurcation.

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

“…With the use of a bilinear nonlinearity function and the associated explicit solution formula, we can give a fairly complete description. In the sequel [9], we will extend some of the results here to more general nonlinear functions. The technique is based on more PDE techniques and Liaponov-Schmidt reduction.…”

“…Because of this, we can obtain many precise quantitative statements. These can give us good indications for the more general Problem [G] which will be analyzed in the sequel [9] where detail spectral analysis and asymptotic expansion will be used.…”

“…In this paper and its sequel [9], we concentrate on bistable nonlinearity function f of the following form:…”

“…For the balanced case, G(1) = G(−1), the location is determined by balancing the mean curvature of the interface and the spatial inhomogeneity, leading to some prescribed mean curvature problem. In the current paper (and its sequel Part II [9]), we concentrate on the former unbalanced case.…”

Depinning asymptotics in ergodic media

“…Suitably defined averages of the speed can now vanish for open sets of the parameter a since zeros can be robust. Although this result appears intuitive, and although a formal expansion in ε gives such a result to leading order, we are not aware of a result that rigorously establishes such a description; see however [13,14] for results on depinning bifurcations in this context. Rapidly varying media, homogenization, and exponential asymptotics.…”