Critial Mass In A Quasilinear Parabolic-elliptic Keller-Segel Model

Chemotaxis is the characteristic of organisms that tend to chemicals for better survival.Keller-Segel model is a classic chemotaxis model.In this paper,the homogeneous Neumann initial boundary value problem of a parabolic-elliptic Keller-Segel equations is studied where Q is a bounded domain with smooth boundary in Rn(n≥3),u(x,t)represents the population density of the cells and v(x,t)represents the concentration of chemicals.Here,φ,ψ are nonnegative functions and respectively describe the diffusion rate and chemotactic sensitivity and ψ(s)/φ(s)grows like sα as s → ∞,α∈R.In Chapter 1,we introduce background and main achievements of the Keller-Segel equations related to the research content,and describe main result of this thesis.In Chapter 2,we prove that when α=2/n.there exists a critical mass m*>0,and when∫Ωu0<m*,the solution of the above model is bounded as a whole.In Chapter 3,we prove that the energy functional of steady-state solution has a lower bound.In Chapter 4,under the assumption that Ω is radially symmetric,there exists m*>0,we can construct radially symmetric initial data u0(x)=u0(|x|)satisfying ∫Ωu0<m*such that the corresponding solution must blowup.