Global Existence And Finite-time Blow-up Of Solutions To Some Chemotaxis Models

Chemotaxis is an important sensory phenomenon of cells or organisms,through which chemical signals are converted into motile behavior.In this thesis,we study the global existence and finite time blow-up of solutions for several kinds of chemotaxis models,including the attraction-repulsion model and chemotaxis-haptotaxis model.The dissertation is divided into five parts:In Chapter 1,we give an overview to the chemotaxis model,attraction-repulsion model and chemotaxis-haptotaxis model involving the study background and our main results.In Chapter 2,we investigate an attraction-repulsion model with nonlinear signal production:under homogeneous boundary conditions of Neumann type,where μ1(t)=1/|Ω|∫Ωf1(u)dx,μ2(t)=1/|Ω|∫Ωf2(u)dx,Ω=BR(0)(?)Rn(n≥)2.It is proved that,if f1(u)≥k1uγ1,f2(u)≤k2uγ2,γ1>γ2 and γ1>2/n,the corresponding solution blows up in finite time;and if f1(u)≤K1uγ1,f2(u)=K2uγ2 and γ1<2/n,the model possesses a global bounded classical solution.In Chapter 3,we consider a haptotaxis model describing cancer cells invasion and metastatic spread under homogeneous boundary conditions of Neumann type,where Ω(?)R3 is a bounded domain with smooth boundary.It is proved that,for suitable smooth initial data(u0,v0,m0,w0),the corresponding Neumann initial-boundary value problem possesses a global generalized solution.In Chapter 4,we investigate a chemotaxis-haptotaxis model with signal-dependent nonlinear diffusion under homogeneous boundary conditions of Neumann type,where Ω (?) Rn(n≥1)is a bounded domain with smooth boundary.It is proved that,for any τ≥0,the local classical solution exists.Especially,in the case of τ=0,for any μ>χ,the model admits a unique global bounded classical solution.In Chapter 5,we investigate a two-dimensional chemotaxis-haptotaxis model It is proved that,for any suitable smooth initial data(u0,u0,w0),this model admits a unique global strong solution if ‖u0‖L1<8π/χ.