Research On The Properties Of Solutions To The Chemotaxis System With Indirect Signal Absorption

Chemotaxis is a very common biological phenomenon in the nature.In the past several decades,many studies have been devoted to chemotaxis,and have obtained many important results.However,at present,there are still few studies on the chemotaxis system with indirect signal absorption.Therefore,this paper mainly considers a class of chemotaxis systems with indirect signal absorption under homogeneous Neumann boundary conditions:where ?(?)Rn(n?2)is a bounded region with smooth boundary.The parameter?>0 and f(u)=?(u-ur)with ??0,r>1.The main contents of this paper are as follows:Chapter 1 mainly introduces the background of chemotaxis systems and their research status as well as main research contents of this article.The second chapter studies the influence of the general logistic source term f(u)=?(u-ur)on the properties of the solution to the system,where ?>0,r>1.The result of this chapter clarifies the internal relationship between the logistic source exponent r and the spatial dimension n and the boundedness of the solution,and to a certain extent shows that the indirect signal absorption system is more conducive to the boundedness of the solution.The third chapter considers the influence of the classical logistic source term f(u)=?(u-u2)on the properties of the solution to the system,where ?>0.The result shows that the coefficient ? of the logistic source has a great influence on the properties of the solutionChapter 4 considers the long-time behavior of the solution.By constructing an appropriate energy functional,we showed the convergence of the solution.Furthermore,by constructing a new energy functional,we proved the convergence rate under the suitable assumption.This is a novel property that distinguishes the indirect signal absorption system from other chemotaxis systems.Chapter 5 considers the boundedness of the solution when the logistic source f(u)=0,and clarifies the effect of initial value v0 on the properties of the solution.The result improves part of the results of existing literatures.Chapter 6 summarizes all the results and proposes further problems to be studied.