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 { u t = Δ u − χ 1 ∇ ⋅ ( u ∇ w ) , v t = Δ v − χ 2 ∇ ⋅ ( v ∇ w ) , w t = Δ w − γ w + α 1 u + α 2 v in R 2 × [ 0 , ∞ ) , where γ ⩾ 0 , χ 1 , χ 2 and α 1 , α 2 are real numbers. We obtain the global existence of solutions if ‖ u 0 ‖ 1 , ‖ v 0 ‖ 1 and ‖ ∇ w 0 ‖ 2 are small, and the asymptotic behavior of the small-data solution as follows:• 0 , the solution is asymptotic to a self-similar solution for large time; gt; 0 , the solution behaves like a multiple of the heat kernel as t → ∞ .