Global Classical Solutions, Stability of Constant Equilibria, and Spreading Speeds in Attraction–Repulsion Chemotaxis Systems with Logistic Source on R N

In this paper, we consider the following chemotaxis systems of parabolic–elliptic–elliptic type on RN, ut=Δu-χ1∇(u∇v1)+χ2∇(u∇v2)+u(a-bu),x∈RN,t>0,0=(Δ-λ1I)v1+μ1u,x∈RN,t>0,0=(Δ-λ2I)v2+μ2u,x∈RN,t>0,u(·,0)=u0,x∈RN,where χi≥0,λi>0,μi>0 (i=1,2) and a>0,b>0 are constant real numbers, and N is a positive integer. First, under some conditions on the parameters χi,μi,λi,a,b and N, we prove the global existence and boundedness of classical solutions (u(x,t;u0),v1(x,t;u0),v2(x,t;u0)) for nonnegative, bounded, and uniformly continuous initial functions u0(x). Next, we explore the asymptotic stability of the constant equilibrium (ab,μ1λ1ab,μ2λ2ab) and prove under some further assumption on the parameters that, for every strictly positive initial u0(x), limt→∞‖u(·,t;u0)-ab‖∞+‖λ1v1(·,t;u0)-abμ1‖∞+‖λ2v2(·,t;u0)-abμ2‖∞=0.Finally, we investigate the spreading properties of the global solutions with compactly supported initial functions. We show that under some conditions on the parameters, there are two positive numbers 0