Properties Of Solutions To A Kellel-segel Model With Logistic Source In R~2

In this thesis,we study the properties of solutions of Keller-Segel chemotactic model with logistic source in unbounded domain.Scholars are interested in the interaction of chemotaxis,diffusion,growth,and damping effects of cells or microorganisms on the properties of model solutions.Such models can be used to solve practical problems in physics,biology and other fields,such as embryonic development,tumor cell growth.In this thesis,we consider the following parabolic-elliptic chemotaxis system with logistic source in the whole space R2,where f(u)has two cases,the first case is f(u)=au-buθ,θ≥ 2,and the second case is f(u)=au-bu2/(|ln(u+1)|γ),γ∈(0,1).For the first case,we prove that solutions of the whole space model exist globally for arbitrary initial data u0 ∈ L1 ∩ L∞(R2),a≥0 and b>0.For the second case,we prove that solutions of the whole space model exist globally for arbitrary initial data u0∈ L1∩L∞(R2),a≥0 and b>5/4.It is different from the classical Keller-Segel model that the existence of logistic source makes the model lose the mass conservation property,instead,we give an upper bound on ‖u‖L1(R2)for subsequent estimation.This paper mainly explores logistic source and diffusion term to suppress the concentrated term.Specifically,use analysis skills and property of the second moment to prove the boundedness of the term of ‖ulnu‖L1(R2)for θ=2,explores logistic source and diffusion term to suppress the concentrated term to prove the boundedness of‖u+1)ln(u+1)‖L1(R2)for θ>2 and logarithmic source.Further,obtain global existence of bounded solutions by the Moser iteration.Compared with the classical Keller-Segel model,the existence of logistic source prevents blow-up that holds in the whole space.That indicates logistic source has broken the threshold between diffusion and aggregation in the model such that solutions exist globally for any initial data.