Dynamics On Some Chemotaxis System With Density-depressed Motility And Flux Limitation

In this paper,we study the properties of solutions to some reaction-diffusion systems with densitydepressed motility and flux-limitation,including local existence,global existence and uniformly boundedness,large time behavior of solutions and finite time blow-up.This dissertation is divided into five part.Chapter 1 gives an overview to the chemotaxis models involving the biological background and our main results.Chapter 2,we deal with an initial-boundary value problem about the chemotaxis-growth system with density-depressed motility and logistic source:in a bounded domain Ω?Rn(n≥2)with no-flux boundary conditions.Here one of the two densitydependent motility functions γ(v)describes the strength of diffusion while the other Φ(v)=(α1)γ’(v)(α>0)denotes the chemotactic sensitivity,also satisfy Φ(v)=(α-1)γ’(v)(α>0).It is proved that for a class generic motility functions there exists a unique global bounded classical solution to the above system with some suitable small initial data and some large μ.Furthermore,it asserts that the obtained global solution stabilizes to the spatially uniform equilibrium(1,0).Compare to the study already have,we consider a system with a normal α>0,which makes it is impossible to use dual method for the first equation.So we establish a energy-inequality of weighted integral∫ΩukΦ(v)to estimate u in Lp and we get the uniformly boundedness of u through the estimate of‖▽v‖L∞(Ω)together with Moser iteration procedure.On this basis,we obtained large time behavior by setting up a Lyapunov functional.Chapter 3,we study the Neumann initial-boundary value problem for the fully parabolic chemotaxis system where Ω is a ball in Rn with n≥3,function γ(v)describes the strength of diffusion.In this paper we choose γ(v)=e-χv,where χ is a positive constant.Based on the conclusion already obtained in low dimension,we consider the finite-time blow up in high dimension through a Lyapunov function.It is proved that for any prescribed m>0 there exist radially symmetric positive initial data(u0,v0)∈C0(Ω×W1,∞(Ω))with ∫Ω u0=m such that the corresponding solution blows up in finite time.Chapter 4,we consider the boundedness of the parabolic-parabolic chemotaxis system with nonlinear diffusion and flux-limitation coefficient under homogeneous Neumann boundary conditions in a bounded convex domain Ω?Rn with n≥ 2,D ∈ C2([0,∞))satisfy D(u)≥lum,u>0 and D(0)>0,where l>0 and m>0 are constants.On the basis of the existing results on linear diffusion systems,we consider chemotactic systems with nonlinear diffusion and flux-limitation in this chapter.To begin with,we set up differential inequalities d/dt{∫Ωur+∫Ω|▽v|2q}(r,q>1)to estimate u and v in Lp,respectively.By Moser iteration procedure we can improve the regularity of the classical solution,then we proved that if p∈(1,((m+1)n)/(n-1)),u0∈C0(Ω)and v0∈C1(Ω),the classical solutions to the above system are uniformly-in-time bounded.