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

Chaoqun  Huang; Nung Kwan  Yip; Chaoqun  Huang; Nung Kwan  Yip

doi:10.3934/nhm.2015.10.897

Networks and Heterogeneous Media

2015, Volume 10, Issue 4: 897-948. doi: 10.3934/nhm.2015.10.897

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

Chaoqun Huang ^{1
,},
Nung Kwan Yip ^{2
,}

1.
School of Mathematics, Hunan University, Changsha 410082
2.
Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907

Received: 01 September 2014 Revised: 01 August 2015
Primary: 35B25, 35B32, 35K57; Secondary: 34D15, 34E05, 34E10.

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,
- reduced (bifurcation) equation,
- degenerate spatial inhomogeneity,
- (imperfect) pitchfork bifurcation
Citation: Chaoqun Huang, Nung Kwan Yip. Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part II[J]. Networks and Heterogeneous Media, 2015, 10(4): 897-948. doi: 10.3934/nhm.2015.10.897

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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.

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