Large time behavior of the full attraction–repulsion Keller–Segel system in the whole space

In this paper, we study the Cauchy problem of the attraction–repulsion Keller–Segel chemotaxis model {ut=Δu−∇⋅(χu∇v)+∇⋅(ξu∇w),x∈Rn,t>0,vt=Δv+αu−βv,x∈Rn,t>0,wt=Δw+γu−δw,x∈Rn,t>0,u(x,0)=u0(x),v(x,0)=v0(x),w(x,0)=w0(x),x∈Rn. Here all parameters χ,ξ,α,β,γχ,ξ,α,β,γ and δδ are positive. When repulsion cancels attraction (i.e.,  ξγ=χαξγ=χα), the existence of global classical solution is established with large initial data in two or three spatial dimensions. Moreover, we show that the classical solution decays to zero as t→∞t→∞ and behaves like the heat kernel.