Study On Biological Chemotaxis Model With Indirect Signal Consumption Mechanism

In this paper,the homogeneous Neumann initial boundary value problems of the following chemotactic models with PDE indirect signal consumption mechanism are studied:(?)where u,v and w represent cell/bacterial density,chemical signal concentration and indirect signal concentration respectively.?(?)R~n,n?1 is a bounded domain with smooth boundary(?)?,the parameter ?>0.The logical source term f(u)represents cell proliferation and death.Firstly,the case of f(u)?0 is studied in this paper.In this part,by using the semigroup theory,the constant variation method,the parabolic comparison principle and the regularization properties of inhomogeneous linear heat equation,it is proved that there is a unique global bounded classical solution for the initial boundary value problem when the space dimension n?3.This part of the work removes the small assumption about initial values in the existing literature.Secondly,when considering the proliferation and death of bacteria or cells,logical source term should also be added to the model.Therefore,this paper further analyzes the situation of the initial boundary value problem when f(u)?0 and assumes that f(u)=?(u-u~r),where ??0,r>1.In this part,we prove that for any ??0,when r>1(n?3)or(?),there exists a unique global bounded classical solution to the initial boundary value problem by using the semigroup theory,the constant variation method,the parabolic comparison principle and the regularization properties of inhomogeneous linear heat equation.Finally,we make a brief summary of the main results of this paper,and put forward some problems that can be further studied in the future.