Boundedness And Asymptotic Behavoir Of Solutions In Chemotaxis Systems

Chemotaxis and haptotaxis are two important mechanisms which describe the directed movement of cells toward the concentration of the chemical substance and the oriented movement of cells along a gradient of cellular adhesion,which is often fasciated by chemoattractants or enzymes bound in the extracellular matrix(ECM),respectively,and it plays an very important role in some situations including pattern formation,tumor invasion and embryo development.In mathematics,a variety of biological phenomenons can be modeled by chemotaxis models which are formed by nonlinear PDE systems.In this paper we mainly investigate global boundedness and asymptotic behavior of solutions to chemotaxis systems,including local existence,weak solution,global existence and large time behavior.In this paper,we divide the thesis into seven parts.Chapter 1 is mainly devoted to discussing the background and the development of the related topics about chemotaxis and haptotaxis.Moreover,we will introduce the main content of the present paper.In Chapter 2,we consider an attraction-repulsion parabolic-parabolic-parabolic chemotaxis system with logistic source.By using energy estimates,we first obtain if the logistic dampening effect is suitable strong,this system possesses a unique global bounded classical solution.Based on the boundedness property,we also obtian that the solution of the system approaches to the steady states under the same assumption.(The main results of this chapter are published in J.Math.Anal.Appl.2017(448):914-936.)In Chapter 3,we consider two-species chemotaxis system with two indirect signal production and logistic source.By making use of energy estimates,we obtain that this system possesses a unique global-bounded classical solution if the logistic growth is suitable large in the sense that the cells themselves decrease fast.Moreover,by constructing energy functional,we also obtain that the solution of the system exponentially stabilizes to the constant stationary solution if the logistic dampening effect is suitable strong.(The main results of this chapter are accepted in Z.Angew.Math.Phys.2017)In Chapter 4,we consider the chemotaxis-haptotaxis model of cancer invasion.Relying on a entropy-type inequality,we obtain the system possesses a unique classical solution which is global in time and bounded under an explicit condition.(The main results of this chapter are published in Computer and Mathematics with applications.)In Chapter 5,we study the quasilinear chemotaxis system modeling tumor invasion.If the diffusion exponent and tensor-valued function exponent satisfy some assumptions,we obtain for all reasonably regular initial data,a corresponding initial-boundary value problem for this system possesses a globally defined weak solution.Based on this boundedness property,we also obtain that any such solution approaches the spatially homogenous equilibrium in an appropriate sense.To the best of our knowledge,there are first results on boundedness and asymptotic behavior of the system.(The main results of this chapter are published in Mathematical Models and Methods in Applied Sciences,7(2018)1413-1451.)In Chapter 6,we consider a coupled chemotaxis-Stokes system with logistic source.If the diffusion exponent and tensor-valued function exponent satisfy m+2?>6/5,but the logistic source coefficient is arbitrarily small,we obtain the model possesses a uniform bounded global weak solutions.This extends a recent result by Liu el at.(J.Diff.Eqns,261(2016)967-999)which asserts global existence of weak solution under the constraints m+?>6/5 and m?1/3.(The main results of this chapter are published in Discrete Contin.Dyn.Syst.B)In Chapter 7,we summarize the main results of this paper.