TY - JOUR

T1 - Nonexistence of higher dimensional stable turing patterns in the singular limit

AU - Nishiura, Yasumasa

AU - Suzuki, Hiromasa

PY - 1998/9

Y1 - 1998/9

N2 - 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 ε1/3. 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.

AB - 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 ε1/3. 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.

KW - Interfacial pattern

KW - Matched asymptotic expansion

KW - Reaction-diffusion system

KW - Singular perturbation

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U2 - 10.1137/S0036141096313239

DO - 10.1137/S0036141096313239

M3 - Article

AN - SCOPUS:0032339678

VL - 29

SP - 1087

EP - 1105

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 5

ER -