生物数学趋化现象及相关偏微分方程的若干问题

We plan to study the mathematical theories of several partial differential equations that describe the chemotaxis in biology. We consider three kinds of problems. The first one is the stability of traveling waves, we will consider four problems: first is the stability of traveling waves with large amplitude, vacuum state and arbitrary diffusion rate; second is the convergence rate to traveling waves; third is the stability of traveling pulse; and fourth is the stability of multidimensional traveling waves. ..The second kind of problem is the effect of growth terms on the pattern formation and their stability for chemotaxis models. It includes the existence and stability of multidimensional traveling waves, the profile and stability of multidimensional non-constant steady state and the pattern formation under the effect of volume-filling. We will consider the cases of logistic growth and the Allee effect. ..The third one is to study the effect of fluid dynamics on the chemotaxis model. There are four problems. The first one is the formation of spiky pattern and its stability; the second one is the formation of plume, i.e. to show the Rayleigh-Taylor instability; the third one is the optimal temporal decay rate; and the last one is the optimal spatiotemporal decay rate.

本项目拟研究描述生物学中趋化现象的几类偏微分方程组的数学理论。包括三类问题。第一类是研究行波的稳定性，我们拟研究四个具体问题：一是大波强、带真空、任意扩散系数下波前的稳定性问题；二是行波稳定的收敛速度估计；三是脉冲波的稳定性；四是高维行波的稳定性问题。. 第二类是考虑增长项影响下趋化模型的模式形成和稳定性问题。包括高维行波的存在性和稳定性，高维非常数稳态解的存在性、形状和稳定性，体积填充效应下高维模式形成问题，我们计划分别考虑logistic增长和Allee效应两种情形。. 第三类是研究流体力学对趋化模型的影响。具体包括四个问题。一是尖峰模式的存在性和稳定性；二是羽毛状模式的产生机制，即证明Rayleigh-Taylor不稳定性；三是解的最优时间衰减率；四是解的最优时空衰减率。