Boundedness In A Four-dimensional Attraction-repulsion Chemotaxis System With Logistic Source

In this paper we study the attraction-repulsion chemotaxis system with logistic source:ut = △u-χ▽·(u▽v)+ζ▽·(u▽w)＋f(u),0 = △v-βu+αu,0=△w-δw+γu,subject to homogeneous Neumann boundary conditions in a bounded and smooth domain Ω(?)R4,where χ,α,ζ,γ,β and δ are positive constants,and f:R→R is a smooth function satisfying f(s)≤a-bs3/2 for all s ≥ 0 with a≥ 0 and b>0.It is proved that when the repulsion cancels the attraction(i.e.χa=ζγ),for any nonnegative initial data u0∈C0(Ω),the solution is globally bounded.This result corresponds to the one in the classical two-dimensional Keller-Segel model with logistic source bearing quadric growth restrictions.In Chapter 1,we summarize the biological background and the development of the problem considered,and briefly introduce the contents of this paper.Chapter 2 is to give some preliminaries.Then in Chapter 3,we state and prove the main result.