## Abstract

When the thickness of the interface (denoted by ε) tends to zero, any stable stationary internal layered solutions to a class of reaction-diffus on systems cannot have a smooth limiting interfacial configuration. This means that if the limiting configuration of the interface has a smooth limit, it must become unstable for small ε, which makes a, sharp contrast with the one-dimensional case. This suggests that stable layered patterns must become very fine and complicated in this singular limit. In fact we can formally derive that the rate of s irinking of stable patterns is of order ε^{1/3}. Using this scaling, the resulting rescaled reduced equation determines the morphology of magnified patterns. A variational characterization of the critical eigenvalue combined with the matched asymptotic expansion method is a key ingredient for the proof, although the original linearized system is not of self-adjoint type.

Original language | English |
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Pages (from-to) | 1087-1105 |

Number of pages | 19 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 29 |

Issue number | 5 |

DOIs | |

Publication status | Published - 1998 Sep |

## Keywords

- Interfacial pattern
- Matched asymptotic expansion
- Reaction-diffusion system
- Singular perturbation

## ASJC Scopus subject areas

- Analysis
- Computational Mathematics
- Applied Mathematics