带有矩阵值灵敏度的趋化模型的快速信号扩散极限研究

Study on the Fast Signal Diffusion Limit in Chemotaxis Model with Matrix-Valued Sensitivity

摘要: 趋化性是最基本的细胞生理反应之一，它能帮助生物寻找营养物质、逃避有害物质。Keller和Segel为描述细菌趋化性致使细菌种群聚集的现象，首次提出了由两个抛物型偏微分方程组成、用来刻画盘基网柄菌趋化现象的趋化模型。本文研究带有矩阵值灵敏度的抛物–椭圆型趋化模型和抛物–抛物型趋化模型的初值问题，并且在细胞初始质量适当小的条件下通过时空估计证明相应的初值问题存在整体解。此外，本文还证明当细胞扩散函数与信号扩散函数的比例趋于零时，抛物–抛物型趋化模型的解会收敛到相应的抛物–椭圆型趋化模型的解。

Abstract: Chemotaxis is one of the most fundamental cellular physiological responses, which helps organisms to seek nutrients and avoid harmful substances. Keller and Segel first proposed a chemotaxis model composed of two parabolic partial differential equations to describe the phenomenon of aggregation of bacterial populations caused by bacterial chemotaxis, which was used to characterize the chemotactic behavior of Dictyostelium discoideum. This paper investigates the initial value problem of parabolic-elliptic and parabolic-parabolic chemotaxis models with matrix value sensitivity, and proves the existence of global solutions for the corresponding initial value problems through spatio-temporal estimation under the condition that the initial mass of the cell is appropriately small. In addition, this article also proves that when the ratio of cell diffusigon function to signal diffusion function approaches zero, the solution of the parabolic-parabolic chemotaxis model will converge to the corresponding solution of the parabolic-elliptic chemotaxis model.