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

This paper deals with the two-species chemotaxis system with two chemicals {u(t) = d(1)Delta u - del . (uX(1)(v) del v) + mu(1)u(1-u - alpha 1w), x is an element of Omega, t > 0, v(t) = d(2)Delta v - alpha v +f(1)(w), x is an element of Omega, t > 0, w(t) = d(3)Delta w - del. (wx(2)(z) del z) +mu(2)w(1 - w - alpha 2u), x is an element of Omega, t > 0, under homogeneous Neumann boundary conditions in a bounded domain Q C R-n (n >= 1), where the parameters d(1), d(2), d(3,) d(4) > 0, mu(1), mu(2) > 0, a(1), a(2) > 0 and alpha, beta > 0. The chemotactic function x(i) (i = 1,2) and the signal production function f(i) (i = 1, 2) are smooth. If n = 2, it is shown that this system possesses a unique global bounded classical solution provided that vertical bar x'i vertical bar (i = 1, 2) are bounded. If n < 3, this system possesses a unique global bounded classical solution provided that iti (i = 1, 2) are sufficiently large. Specifically, we first obtain an explicit formula itio > 0 such that this system has no blow-up whenever iti > itio. Moreover, by constructing suitable energy functions, it is shown that: " If a(1), a(2) is an element of (0, 1) and mu(1) and mu(2) are sufficiently large, then any global bounded solution exponentially converges to (1-a(1)/1-a(1)a(2)), f1 (1-a(2)/1-a(1)a(2))/a, 1-a(2)/1-a(1)a(2), f2(1-a(1)/1-a(1)a(2/)beta) as t -> infinity; If a(1) > 1 > a(2) > 0 and mu(2) is sufficiently large, then any global bounded solution exponentially converges to (0, f(1) (1)/a, 1, 0) as t -> infinity; If a(1)= 1 > a(2) > 0 and mu(2) is sufficiently large, then any global bounded solution algebraically converges to (0, f(1)(1)/a, 1,0) as t -> infinity.