A NEW RESULT FOR BOUNDEDNESS AND STABILIZATION IN A TWO-SPECIES CHEMOTAXIS SYSTEM WITH TWO CHEMICALS

This paper deals with the following competitive two-species chemotaxis system with two chemicals {u(t) = Delta u - chi(1)del . (u del v) + mu(1)u(1 - u - a(1)w), x is an element of Omega, t > 0, 0 = Delta v - v + w, x is an element of Omega, t > 0, w(t) = Delta w - chi(2)del . (w del z) + mu(2)w(1 - w - a(2)u), x is an element of Omega, t > 0, 0 = Delta z - z + u, x is an element of Omega, t > 0 under homogeneous Neumann boundary conditions in a bounded domain Omega subset of R-n (n >= 1), where the parameters chi(i) > 0, (mu i )> 0 and a(i) > 0 (i = 1,2). It is proved that the corresponding initial-boundary value problem possesses a unique global bounded classical solution if one of the following cases holds: (i) q(1) <= a(1); (ii) q(2) <= a(2); (iii) q(1) > a(1) and q(2) > a(2) as well as (q(1) - a(1))(q(2) - a(2)) < 1, where q(1) := chi(1)/mu(1) and q(2) := chi(2)/mu(2) which partially improves the results of Zhang et al. [53] and Tu et al. [34]. Moreover, it is proved that when a(1), a(2) is an element of (0, 1) and mu(1) and mu(2) are suf-ficiently large, then any global bounded solution exponentially converges to (1-a(1)/1-a(1)a(2), 1-a(2)/1-a(1)a(2), 1-a(2)/1-a(1)a(2), 1-a(1)/1-a(1)a(2)) as t -> infinity; When a(1) > 1 > a(2) > 0 and mu(2) is sufficiently large, then any global bounded solution exponentially converges to (0, 1, 1, 0) as t -> infinity; When al = 1 > a(2) > 0 and mu(2) is sufficiently large, then any global bounded solution algebraically converges to (0, 1, 1,0) as t -> infinity. This result improves the conditions assumed in [34] for asymptotic behavior.