Boundedness in the higher dimensional attraction-repulsion chemotaxis-growth system

This paper deals with an attractionrepulsion chemotaxis system with logistic source {ut=u(uv)+(uw)+f(u),(x,t)(0,),vt=v1v+1u,(x,t)(0,),wt=w2w+2u,(x,t)(0,), under homogeneous Neumann boundary conditions in a smooth bounded domain Rn(n3) with nonnegative initial data (u0,v0,w0)W1,()W2,()W2,(), where >0, >0, i>0, i>0(i=1,2) and f(u)auu2 with a0 and >0. Based on the maximal Sobolev regularity and semigroup technique, it is proved that the system admits a unique globally bounded classical solution provided that n3, 1=2 and there exists 0>0 such that 1+2<0. The main aim of this paper is to solve the higher-dimensional boundedness question addressed by Xie and Xiang in [IMA J. Appl. Math. 81 (2016) 165198].

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