Global existence and asymptotic properties of the solution to a two-species chemotaxis system

This paper deals with the Cauchy problem for a two-species chemotactic Keller–Segel system{ut=Δu−χ1∇⋅(u∇w),vt=Δv−χ2∇⋅(v∇w),wt=Δw−γw+α1u+α2v in R2×[0,∞)R2×[0,∞), where γ⩾0γ⩾0, χ1χ1, χ2χ2 and α1α1, α2α2 are real numbers. We obtain the global existence of solutions if ‖u0‖1‖u0‖1, ‖v0‖1‖v0‖1 and ‖∇w0‖2‖∇w0‖2 are small, and the asymptotic behavior of the small-data solution as follows:
• If γ=0γ=0, the solution is asymptotic to a self-similar solution for large time;
• If γ>0γ>0, the solution behaves like a multiple of the heat kernel as t→∞t→∞.