TY - JOUR

T1 - Stability and characteristic wavelength of planar interfaces in the large diffusion limit of the inhibitor

AU - Taniguchi, Masaharu

AU - Nishiura, Yasumasa

PY - 1996/1/1

Y1 - 1996/1/1

N2 - A characteristic wavelength and its parametric dependency are studied for planar interfaces of activator-inhibitor systems as well as their stability in two-dimensional space. When an unstable planar interface is slightly perturbed in a random way, it develops with a characteristic wavelength, that is, the fastest-growing one. A natural question is to ask under what conditions this characteristic wavelength remains finite and approaches a positive definite value as the width of interface, say ε, tends to zero. In this paper, we show that the fastest-growing wavelength has a positive limit value as ε tends to zero for the system: ut = Δu + ε-2f(u, v), vt = ε-1 Δv + g(u, v). This is a fundamental fact for stuyding the domain size of patterns in higher-space dimensions.

AB - A characteristic wavelength and its parametric dependency are studied for planar interfaces of activator-inhibitor systems as well as their stability in two-dimensional space. When an unstable planar interface is slightly perturbed in a random way, it develops with a characteristic wavelength, that is, the fastest-growing one. A natural question is to ask under what conditions this characteristic wavelength remains finite and approaches a positive definite value as the width of interface, say ε, tends to zero. In this paper, we show that the fastest-growing wavelength has a positive limit value as ε tends to zero for the system: ut = Δu + ε-2f(u, v), vt = ε-1 Δv + g(u, v). This is a fundamental fact for stuyding the domain size of patterns in higher-space dimensions.

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U2 - 10.1017/S0308210500030638

DO - 10.1017/S0308210500030638

M3 - Article

AN - SCOPUS:21344453089

VL - 126

SP - 117

EP - 145

JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

SN - 0308-2105

IS - 1

ER -