Large Time Behavior Of Solutions Of Keller-Segel Related Models

Since Engelmann first observed chemotaxis of organisms in 1881,this phenomenon has aroused great interest among scholars.In order to accurately describe the chemotaxis behavior of cells or organisms mathematically,Keller and Segel proposed the landmark Keller-Segel model in 1970.Since then,biologists and mathematicians have modeled and explored organisms in nature more extensively,and have proposed numerous relevant chemotaxis models.This dissertation focuses on the mountain pine beetle model,urban crime model and tumor cell model,which are related to the Keller-Segel model,and mainly studies the global existence,boundedness and large time behavior of the solutions to these three types of models.The specific research contents are as follows:1.This work considers the critical mass of the Cauchy problem for the chemotaxis model of mountain pine beetle with indirect signal production mechanism.Firstly,the global existence of the classical solutions can be shown by using the modified entropy and the Hardy-Littlewood-Sobolev inequality.Then,using the Gagliardo-Nirenberg inequality with the optimal constant,it is proved that for any mortality rate δ>0,the solutions are uniformly bounded when the initial mass of the flying mountain pine beetle is less than 47rδ.Furthermore,it is shown that by constructing the sub-solution and sup-solution in the radially symmetric setting,there is a threshold value 87rδ separating two different behaviors:all global solutions are uniformly bounded when the initial mass is below 8πδ,while there exist unbounded solutions when the initial mass exceeds 8πδ.This extends the results of Tao-Winkler(J.Eur.Math.Soc.,2017)in the plane unit disk to the two-dimensional whole space.2.This work investigates the relaxation of the urban crime model with porous media diffusion Δum in a bounded convex domain Ω(?)R2 with smooth boundary.By establishing an energy functional,the boundedness of the weak solutions of the urban crime model with nonlinear diffusion is obtained when m>3/2 and the chemotaxis coefficientχ>0,or when 1<m≤3/2 and 0<χ<(?).This work extends the relevant results of Rodriguez-Winkler(Math.Models Methods App.Sci.,2020),which were obtained in the case of m>3/2,to the case of arbitrary m>1 under the additional assumption of χ.3.This work is devoted to studying the influence of initial cell volume fraction on the migration behavior of tumor cells driven by interstitial flow.Firstly,the Lions-Magenes transformation is employed to convert the no-flux boundary condition into a homogeneous Neumann boundary condition.Then,by using the energy estimation method,two different migration behaviors of tumor cells are demonstrated under the conditions of low and high initial cell volume fractions,respectively.This extends the results of Evje-Winkler(J.Nonlinear Sci.,2020)in one dimensional case to arbitrary d≥ 2 dimensional case.