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-??·(u?v)+??·(u?w)?f(u),0 = ?v-?u+?u,0=?w-?w+?u,subject to homogeneous Neumann boundary conditions in a bounded and smooth domain ?(?)R4,where ?,?,?,?,? and ? 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.?a=??),for any nonnegative initial data u0?C0(?),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.