Some Problems Of A Class Of Chemotaxis Models

This thesis deals with some problems of a class of chemotaxis models.Chemotaxis model was first introduced by Keller and Segel[32]in 1970,then,lots of modified chemotaxis models have been widely studied by many researchers.Firstly,we studied the nonlinear parabolic chemotaxis-haptotaxis invasion model problem where the initial-boundary-value satisfies some conditions.where D(u,?u)=|?u|p-2,? is a bounded domain with the smooth boundary in R3,R>2,?,??0,?>0,u,v and w represent the cancer cell density,the matrix degrading enzyme(MDE)concentration and the extracellular matrix(ECM)density.D(u,?u)is the cancer cell random motility,to be a constant,which leads to linear isotropic diffusion,and during the past years,many authors have paid much attention to the D=1,see[56],[62],[54]."However,from a physical point of view,the migration of the cancer cells through the ECM should rather be regarded as movement in a porous medium,and so we are led to consider the cell motility D as a nonlinear function of the cancer cell density"([60]).In Chapter 2,we discuss the case of D(u,?u)=|?u|p-2.Based on the method of the parabolic regularization and the derivation of a series of integral estimates,we obtain that the chemotaxis-haptotaxis model problem admits a global bounded weak solution.Cancer cell invasion is a complicated process.In addition to random movement,cancer cells move toward the concentration gradient of diffusible enzymes secreted by cancer cells by capturing the matrix molecules that are adhered to them,and move to a higher density of non-diffusible tissue.Perumpanani and Byrne[50]proposed a model of cell invasion where it is assumed that the production of enzymes is related to the density and tissue concentration of cancer cells.On the other hand,the production of enzymes does not necessarily depend on the cancer cell density in a linear manner,but may,in some nonlinear forms.This forms account for saturation effects of chemotactic signal production at large densities of cells as widely discussed in the biomath literature,see for[23,45].Hence,for the above reasons,we discussed the following problem in Chapter 3.where ? is a bounded domain with the smooth boundary in R3,p>2,?,??0,?>0,?>0,??0,f and g are prescribed nonnegative and C1-smooth functions and f satisfies f(s)?Ksp for all s>0.with some constant K and parameter ? ?(0,1).u represent the cancer cell density,v is the enzyme concentration and w stands for the tissue density.Using the method of the parabolic regularization and the derivation of a series of integral estimates,it is asserted that the chemotaxis system admits a global bounded weak solution.Finally,we consider a class of chemotaxis-Navier-Stokes problems with two com-peting colonies.(?)(where ? is a bounded domain in R2,?1,?2,?1,?2,?1,?2,?,?,?,?>0,n1(x,t)is the density of one bacterial,n2(x,t)is the density of another bacterial,c(x,t)represents oxygen concentration,u is fluid velocity field,P represents fluid pressure,and ? is the gravity potential energy generated by the bacteria's own weight.The chemotaxis biological phenomenon of a single bacterium can be expressed as aerobic bacteria such as Bacillus subtilis living in the fluid layer moving to the contact surface of liquid and air under the chemotaxis of oxygen and the liquid flowing under the action of bacterial gravity.For this biological phenomenon,Tuval et al[66]proposed model#12Considering various factors in nature,the model was later popularized and widely s-tudied.For example,when the fluid flow is very slow,i.e.,?=0,the model was studied in[74]and[38].Other examples are also list,such as the model with chemotactic sen-sitivity([6,74]),the model with nonlinear bacterial diffusion coefficient([28,76]),the model with bacterial death and birth([60,67]),the multicolony model with competi-tion([24,41]).We know that there is not much research on the time period solution of the fluid chemotaxis model,so in Chapter 4,we study the existence of the periodic solution of a class of two colonies chemotaxis model with considering birth mortality and competitive relationship.By Leray-Schauder's fixed point theorem,we prove that the problem admits a time periodic solution.