Multi-layered stationary solutions for a spatially inhomogeneous Allen-Cahn equation

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Abstract

We consider stationary solutions of a spatially inhomogeneous Allen-Cahn type nonlinear diffusion equation in one space dimension. The equation involves a small parameter ε, and its nonlinearity has the form h(x)2f(u), where h(x) represents the spatial inhomogeneity and f(u) is derived from a double-well potential with equal well-depth. When ε is very small, stationary solutions develop transition layers that can possibly cluster in the spatial region. We first show that those transition layers can appear only near the local minimum and local maximum points of the coefficient h(x) and that at most a single layer can appear near each local minimum point of h(x). We then discuss the stability of layered stationary solutions and prove that the Morse index of a solution coincides with the total number of its layers that appear near the local maximum points of h(x). We also show the existence of a stationary solution that has layers at any given set of local minimum and local maximum points of h(x) with the multiplicity of layers being arbitrary at the local maximum points.

Original languageEnglish
Pages (from-to)825-836
Number of pages12
JournalNonlinear Analysis, Theory, Methods and Applications
Volume47
Issue number2
DOIs
Publication statusPublished - 2001 Aug
Event3rd World Congres of Nonlinear Analysts - Catania, Sicily, Italy
Duration: 2000 Jul 192000 Jul 26

Keywords

  • Clustering layers
  • Double-well potential
  • Equal well-depth
  • Morse index
  • Nondegeneracy
  • Sharp transition layers

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