Global existence of solutions to the Cauchy problem of a two dimensional attraction–repulsion chemotaxis system

In this paper, we are interested in the following attraction–repulsion chemotaxis system: ut−Δu=−∇⋅(χu∇v)+∇⋅(ξu∇w),x∈R2,t>0,vt+βv−Δv=αu,x∈R2,t>0,δw−Δw=γu,x∈R2,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈R2,where the parameters χ,ξ,α,β,γ,δ are positive and the initial data u0,v0 are nonnegative and u0⁄≡0. By a modified free energy functional, we establish the global existence of classical solutions when the repulsion dominates over or cancels attraction (i.e. ξγ≥χα) or the attraction dominates (i.e. ξγ