Global solvability and asymptotic behavior in a two-species chemotaxis system with two chemicals

This paper deals with a two-species chemotaxis system with Lotka-Volterra competitive kinetics {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, v(t) = 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, z(t) = Delta z - z + u, x is an element of Omega, t > 0, under homogeneous Neumann boundary condition in a smooth domain Omega subset of R-n(n >= 1), where the parameters chi(i), mu(i), and a(i) are positive constants for i = 1, 2. Under appropriate regularity assumptions on the initial data, we obtain that the system possesses a globally bounded weak solution. Furthermore, we establish the asymptotic stabilization of this weak solution by constructing suitable energy functions in both coexistence and extinction cases.