Existence Theory For Chemotaxis Systems Of Oxygen Consumption

This thesis studies two types of chemotaxis models in biomathematics,which describes the biased movement of cells in response to the concentration gradient of a diffusible chemical signal.The two models are specifically an oxygen consumption chemotaxis-fluid model with p-Laplacian diffusion and an oxygen consumption chemotaxis model with bounded chemotactic sensitivity.This thesis is devoted to studying the global existence of these chemotaxis models.The dissertation is divided into four parts.Chapter 1 provides a literature review of chemotaxis models,covers the research background of our problem,and gives the main results.In Chapter 2,we investigate an incompressible chemotaxis-Navier-Stokes system with slow p-Laplacian diffusionunder homogeneous boundary conditions of Neumann type for n and c,and of Dirichlet type for u in a bounded domain Ω(?)R3 with smooth boundary.Here,Φ∈W2,∞(Ω),0<χ∈C2([0,∞))and 0 ≤f∈C1([0,∞))with f(0)=0.It is proved that if p>32/15 and under appropriate structural assumptions on f and x,for all sufficiently smooth initial data(n0,c0,u0)the corresponding initial-boundary value problem possesses at least one global weak solution.In Chapter 3,a chemotaxis-Stokes system with slow p-Laplacian diffusion is considered under homogeneous boundary conditions of Neumann type for n and c,and of Dirichlet type for u,where Ω is a smooth bounded domain in R3 and Φ ∈ W2,∞(Ω)is a given function.For all sufficiently smooth initial data(n0,c0,w0),it is proved that global bounded weak solutions exist for the corresponding initial-boundary value problem whenever p>23/11.In Chapter 4,a chemotaxis model with bounded chemotactic sensitivity and signal absorp-tion(?)is considered under homogeneous Neumann boundary conditions in the ball Ω=BR(0)(?)Rn,where R>0 and n≥ 2.Here S is a scalar function with S(s,t)∈ C2([0,∞)×[0,∞))for s,t E[0,∞).Moreover,for some positive constant K,|S(s,t)|<K for all ∈[0,∞).For all appropriately regular and radially symmetric initial data(u0,uv)fulfilling u0≥ 0 and vo>0,this chapter shows that there is a globally defined pair(u,v)of radially symmetric functions which are continuous in(Ω\{0})×[0,∞)and smooth in(Ω\{0})×(0,∞),and which solve the corresponding initial-boundary value problem for(*)with(u(·,0),v(·,0))=(u0,v0)in an appropriate generalized sense.Moreover,in the two-dimensional setting,it is shown that these solutions are global mass-preserving in the flavor of the identity(?)and any nontrivial of these globally defined solutions eventually becomes smooth and satisfies#12 uniformally with respect to x ∈ Ω.