Global Existence Of Solutions In One-Dimensional Chemotactic Model With Mixed Boundary Conditions

Chemotaxis phenomenon is ubiquitous in ecology and biology,which describes the directed motion of cells in response to a chemical stimulus in the environment.If cells move towards the direction of an increasing concentration of a chemical signal,this di-rected motion is referred to as positive chemotaxis,while if cells are away from the direc-tion of an increasing concentration of a chemical signal,it is called negative chemotaxis.Chemotaxis plays a pivotal role in the survival and development of cells.In order to better understand the formation mechanism and evolution process of chemotaxis,many scholars have established various mathematical models and made lots of analysis.In par-ticular,Keller and Segel deduced a class of Partial Differential Equations that model the chemotactic process.This paper mainly investigates a Keller-Segel chemotaxis model with logarithmic sensitivity in one-dimensional bounded domain:where u(x,t)and v(x,t)represent the cell density and the chemical signal concentration,respectively.The parameter D denotes cell diffusion rate(D>0),ε describes chemical substances diffusion rate(ε≥0),μ describes chemical decay rate(μ>0)and χ stands for chemotaxis coefficient of measuring chemotactic sensitivity.In this paper,we concern the case of χ<0,namely,a repulsive chemotaxis model.According to the energy methods,we obtain the global existence and long-time behavior of solutions.By a Cole-Hopf type transformation,the singular Keller-Segel repulsive chemotaxis model is converted into a non-singular systemThen we establish energy estimates by utilizing a Lyapunov functional,the standard en-ergy methods and the relationship of time and space derivation to obtain a priori estimates of solutions for the transformed system.Finaly,under the mixed boundary conditions,we prove the global existence and exponential decay of solutions of the initial-boundary value problem.