Nonexistence of Higher Dimensional Stable Turing Patterns in the Singular Limit
Yasumasa Nishiura, Hiromasa Suzuki · SIAM Journal on Mathematical Analysis · 1998
When the thickness of the interface (denoted by $\eps$) 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 $\eps$, 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 order $\eps^{1/3}$. 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.