Global boundedness in a two-species attraction-repulsion chemotaxis system with two chemicals and nonlinear productions

This paper studies the attraction-repulsion chemotaxis system of two-species with two chemical sut=triangle u-chi 1 del<middle dot>(u del v) +f1(u),vt=triangle v-v+w gamma 1,wt=triangle w+chi 2 del<middle dot>(w del z) +f2(w),0 =triangle z-z+u gamma 2, subject to the homogeneous Neumann boundary conditions in a bounded domain ohm subset of RN(N >= 2) with smooth boundary, where theparameters chi i,gamma i>0 (i= 1,2), and the logistic sources fi(s)is an element of C2[0,infinity)satisfyfi(s)<=mu is(1-s theta i)with mu i,theta i>0,fi(0)>= 0(i= 1,2). The interactions among the diffusion, attraction, repulsion, logistic sources, and nonlinear productions in the system determine the behavior of solutions. It is obtained that the solutions would be globally bounded whenever the nonlinear productions are dominated by one of the following three mechanisms: (i) the diffusion with gamma 1<2N, or gamma 2<4Nwith gamma 2 <= 1; (ii) the logistic sources with min{theta 1,theta 2}>= max{gamma 1,gamma 2}, and (iii) the cooperation of diffusion and logistic sources with theta 1+ 1> gamma 2min{1 +N2,1 +N gamma 12 theta 2},or theta 2+ 1> gamma 1min{1 +N2,1 + max{N4,1}gamma 2 theta 1}.(c) 2023 Elsevier Ltd. All rights reserved