Nonexistence of higher dimensional stable Turing patterns in the singular limit
Nonexistence of higher dimensional stable Turing patterns in the singular limit
复制标题
在奇异极限下不存在高维稳定图灵模式
DOI：
10.1137/s0036141096313239
复制
发表时间：
1998
影响因子：
2
通讯作者：
Hiromasa Suzuki
中科院分区：
文献类型：
--
作者：
Y. Nishiura;Hiromasa Suzuki
当界面厚度（用 $\eps$ 表示）趋于零时，一类反应扩散系统的任何稳定的静态内部分层解都不能具有平滑的极限界面构型。这意味着，如果界面的极限配置具有平滑极限，则对于较小的$\eps$，它必定变得不稳定，这与一维情况形成鲜明对比。这表明稳定的分层图案在这个奇异极限下必须变得非常精细和复杂。事实上，我们可以正式推导出稳定模式的收缩率是 $\eps^{1/3}$ 量级。使用这种缩放，所得到的重新缩放的简化方程确定了放大图案的形态。尽管原始线性化系统不是自伴类型，但临界特征值的变分表征与匹配的渐近展开方法相结合是证明的关键要素。
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.
DOI：
--
发表时间：
--
期刊：
影响因子：
--
作者：
通讯作者：
--