open access

https://doi.org/10.14943/83498

https://hdl.handle.net/2115/69102

Nonexistence of stable turing patterns with smooth limiting interfacial configurations in higher dimensional spaces

英語

1996-10-01

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

352

1-21

When the thickness of the interface ( denoted by c) 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 small c, which makes a sharp contrast with one-dimensional case as in [5]. 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 order c1/3 as well as the rescaled reduced equation which 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 system is not of gradient type.

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