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

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: u_t = Δu + ε^-2f(u, v), v_t = ε^-1 Δv + g(u, v). This is a fundamental fact for stuyding the domain size of patterns in higher-space dimensions.