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 = Delta u - x1 del center dot (u del w), {vt = Delta v - x2 del center dot (v del w) {wt = Delta w - gamma w + alpha 1u + alpha 2v in R-2 x [0,infinity), where gamma >= 0, X1, X2 and alpha 1, alpha 2 are real numbers. We obtain the global existence of solutions if parallel to u0 parallel to 1, parallel to v0 parallel to 1 and parallel to del w0 parallel to 2 are small, and the asymptotic behavior of the small-data solution as follows: If gamma = 0, the solution is asymptotic to a self-similar solution for large time; If gamma > 0, the solution behaves like a multiple of the heat kernel as t -> infinity. (C) 2014 Elsevier Inc. All rights reserved.