Properties And Limit Behavior Of Solutions To Parabolic-parabolic Keller-segel Equations

This thesis considers the properties of weak solutions to parabolic-parabolic Keller-Segel equations with linear diffusion in a two-dimensional space. The properties include regularity of weak solutions, hyper-contractivity, uniqueness and limit behavior. Specifi-cally, for any p > 1, under some initial conditions: ∫JR2 ρ0 dx < 8π, ∫R2 ln(1 + |x|2)ρ0 dx <∞, ∫R2 ρ0 In ρ0 dx < ∞, and c0 ∈ H1, ▽c0 ∈ LP, we first give the regularity of space and time derivative for weak solutions, further obtain hyper-contractivity of weak solutions,and then use hyper-contractivity and semigroup theory to deduce uniqueness of weak solutions. Finally, we use the Lions-Aubin lemma to prove the solution of the parabolic-parabolic Keller-Segel model converges to the solution of parabolic-elliptical Keller-Segel model as ε → 0.