Global Existence And Blow-up For Some Chemotaxis Models

The present thesis studies several chemotaxis models in biomathematics, which describe the motion of cells, who, besides random diffusion, bias their movement towards a chemically more favorable environment. Our models are of two types:one about multi-species and a stimulus, the other about one species and multi-chemicals. This thesis is devoted to studying the global existence and blow-up for these chemotaxis models.The dissertation is divided into five parts.Chapter 1 gives an overview to the chemotaxis models involving the study background and our main results.In Chapter 2, a fully parabolic chemotaxis system for two species is considered under homogeneous Neumann boundary conditions in a smooth bounded domain Ω(?)R2. We obtain the global boundedness and asymptotic behavior with small initial data condition in critical space. More precisely, it is proved that one can find a small ε0> 0 such that for any initial data (u0,v0,w0) satisfying ‖u0‖L1(Ω)<ε0 and ‖▽w0‖L2(n)<ε0, the solution of the problem above is global in time and bounded. In addition, (u, v, w) converges to the steady state (m1, m2, (?)) as tâ†'∞, where (?) and (?)In Chapter 3 we consider the fully parabolic chemotaxis system for two species with homogeneous Neumann boundary condition in the dimension N≥ 3, where Ω is a ball in and x1, X2,γ,α1,α1 are positive constants. We consider the more general case (?).It is proved that for any mi> 0, (i = 1,2), there exists radially symmetric initial data (u10, u20, v0) ∈ (C0(Ω))2 × W1,∞(Ω) with mi=fΩ ui0, (i= 1,2) such that the corresponding solution blows up in finite time in the sense limtâ†'T‖u1‖L∞(Ω)+‖u2‖L∞(Ω)=∞ for some 0<T<∞.In chapter 4, we deal with the attraction-repulsion chemotaxis system under homogeneous Neumann boundary conditions in a smooth bounded domain in R2. The parameters x ≥ 0, ξ> 0 and α> 0,β> 0,γ> 0,δ> 0. We study the finite-time blowup of nonradial solutions in the parameter values xα -ζγ> 0 and β(?)δ.In Chapter 5, it deals with an initial-boundary value problem for the coupled chemotaxis-haptotaxis system with nonlinear diffusion under homogeneous Neumann boundary conditions in a bounded smooth domain Ω (?) Rn, n= 2,3,4, where x,ξ and μ are given positive parameters. The diffusivity D(u) is assumed to satisfy D(u)≥δum-1 for all u> 0 with some 8> 0. It is proved that for sufficiently regular initial data global bounded solutions exist whenever m> 2 -2/n. For the case of non-degenerate diffusion (i.e. D(0)> 0) the solutions are classical; for the case of possibly degenerate diffusion (D>(0)≥0), the existence of bounded weak solutions is shown.