一类具有非线性扩散的趋化模型的非线性不稳定性

Nonlinear Instability for a Nonlinear Diffusion Chemotaxis Model

本文考虑一类具有非线性扩散的趋化模型在d-维方体T^d=(0,π)~dd=1,2,3)上满足齐次Neumann边值条件时的不稳定正常数平衡解附近的非线性动力学性态.证明了对于任意给定的一般扰动δ,在以ln 1/δ为阶的时间段内,该扰动的非线性演化由相应的线性化模型的有限个固定的最快增长模式所控制.同时,每个初始扰动所产生的作用一定会与其它初始扰动所产生的作用截然不同,这就导致斑图的多样性.

This paper deals with nonlinear dynamics near an unstable constant equilibrium in a Neumann boundary value problem for a nonlinear diffusion chemotaxis model in a d-dimensional box Td = （0,π）d（d = 1,2,3）. It is proved that, given any general perturbation of magnitude 5, its nonlinear evolution is dominated by the corresponding linear dynamics along a finite number of fixed fastest growing modes, over a time period of the order In 1/5. Each initial perturbation certainly can behave drastically differently from another, which gives rise to the richness of patterns.