Asymptotic Limits Of A Chemotaxis Model With Logarithmic Sensitivity

Chemotaxis is the directional movement of living organisms(such as plants,bacteria)stimulated by the external environment(such as light,nutrition).Keller-Segel models are the basic models to describe biological chemotaxis,which are widely used in the fields of medicine,physiology and microbiology.In terms of mathematical structure,the model is a system of cross diffusion equations,according to the Weber-Fechner law,the sensitivity function in the model has logarithmic structure,so it is very difficult to study its mathematical theory.In this paper,we mainly study the asymptotic limit of the steady-state solution of the one-dimensional Keller-Segel model when the chemotaxis coefficient tends to infinity.Based on the asymptotic expansion technique,by overcoming the difficulties caused by cross diffusion and nonlinearity,we obtain the boundary layer equations of any order in the singular limit,The results of this paper clearly explain the phenomenon of microbial aggregation and help to study the stability of the spike solution of chemotaxis model.