Backward Parabolicity, Cross-Diffusion and Turing Instability
Backward Parabolicity, Cross-Diffusion and Turing Instability
复制标题
DOI：
10.1007/s00332-018-9480-z
复制
发表时间：
2018-06
影响因子：
3
通讯作者：
A. Moussa;B. Perthame;Delphine Salort
中科院分区：
文献类型：
--
作者：
A. Moussa;B. Perthame;Delphine Salort
我们发现，在向后抛物方程，或交叉扩散系统中观察到的不适定性，可以解释为一个相应的半线性抛物系统的极限图灵不稳定性。我们的分析是基于，现在已经建立，从半线性反应扩散系统与快速反应速率的非线性抛物和交叉扩散系统的推导。我们用两个一般的例子来说明我们的观察和反应扩散系统。对于这些例子，我们证明了交叉扩散系统的后向抛物性等价于快速反应速率的图灵不稳定性。在一维中，图灵模式是周期解，其频率随反应速率而增加。此外，在某些特定情况下，手头的方程结构涉及经典的熵/李雅普诺夫函数，这导致先验估计，允许在不存在图灵不稳定性的情况下严格达到快反应极限。
We show that the ill-posedness observed in backward parabolic equation, or cross-diffusion systems, can be interpreted as a limiting Turing instability for a corresponding semi-linear parabolic system. Our analysis is based on the, now well established, derivation of nonlinear parabolic and cross-diffusion systems from semi-linear reaction–diffusion systems with fast reaction rates. We illustrate our observation with two generic examples forandreaction–diffusion systems. For these examples, we prove that backward parabolicity in cross-diffusion systems is equivalent to Turing instability for fast reaction rates. In one dimension, the Turing patterns are periodic solutions which have frequencies which increase with the reaction rate. Furthermore, in some specific cases, the structure of the equations at hand involves classical entropy/Lyapunov functions which lead to a priori estimates allowing to pass rigorously to the fast reaction limit in the absence of Turing instabilities.