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

Abstract

This paper studies the attraction–repulsion chemotaxis system of two-species with two chemicals ut=Δu−χ1∇⋅(u∇¯v)+f1(u), vt=Δv−v+wγ1, wt=Δw+χ2∇⋅(w∇z)+f2(w), 0=Δz−z+uγ2, subject to the homogeneous Neumann boundary conditions in a bounded domain Ω⊂RN (N≥2) with smooth boundary, where the parameters χi,γi>0(i=1,2), and the logistic sources fi(s)∈C2[0,∞) satisfy fi(s)≤μis(1−sθi) with μi,θi>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 γ1<2N, or γ2<4N with γ2≤1; (ii) the logistic sources with min{θ1,θ2}≥max{γ1,γ2}, and (iii) the cooperation of diffusion and logistic sources with θ1+1>γ2min{1+N2,1+Nγ12θ2}, or θ2+1>γ1min{1+N2,1+max{N4,1}γ2θ1}.