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Networks and Heterogeneous Media

2013, Volume 8, Issue 4: 1009-1034. doi: 10.3934/nhm.2013.8.1009
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Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part I

  • 1.
    Mathematics Department, University of Pittsburgh, Pittsburgh, PA 15260
  • 2.
    Department of Mathematics, Purdue University, 150 N. University St., West Lafayette, IN 47907
  • Received: 01 February 2013 Revised: 01 September 2013

  • Primary: 35B25, 35B32, 35K57; Secondary: 34D15, 34E05, 34E10.

  • 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.)

    Citation: Chaoqun Huang, Nung Kwan Yip. Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part I[J]. Networks and Heterogeneous Media, 2013, 8(4): 1009-1034. doi: 10.3934/nhm.2013.8.1009

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  • 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.)


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  • This article has been cited by:

    1. Chaoqun Huang, Nung Kwan Yip, Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part II, 2015, 10, 1556-181X, 897, 10.3934/nhm.2015.10.897
    2. Arnd Scheel, Sergey Tikhomirov, 2017, Chapter 6, 978-3-319-64172-0, 88, 10.1007/978-3-319-64173-7_6
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