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Nonexistence of Higher Dimensional Stable Turing Patterns in the Singular Limit

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Title: Nonexistence of Higher Dimensional Stable Turing Patterns in the Singular Limit
Authors: Nishiura, Yasumasa Browse this author →KAKEN DB
Suzuki, Hiromasa Browse this author
Keywords: reaction-diffusion system
interfacial pattern
singular perturbation
matched asymptotic expansion
MSC=35B25
MSC=35B35
MSC=35K57
MSC=35R35
Issue Date: Sep-1998
Publisher: Society for Industrial and Applied Mathematics
Journal Title: SIAM Journal on Mathematical Analysis
Volume: 29
Issue: 5
Start Page: 1087
End Page: 1105
Publisher DOI: 10.1137/S0036141096313239
Abstract: When the thickness of the interface (denoted by ε) tends to zero, any stable stationary internal layered solutions to a class of reaction-diffusion 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 smallε, 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 shrinking of stable patterns is of orderε^[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.
Rights: Copyright © 1998 Society for Industrial and Applied Mathematics
Type: article
URI: http://hdl.handle.net/2115/39999
Appears in Collections:電子科学研究所 (Research Institute for Electronic Science) > 雑誌発表論文等 (Peer-reviewed Journal Articles, etc)

Submitter: 西浦 廉政

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