Criteria on global boundedness versus finite time blow-up to a two-species chemotaxis system with two chemicals

This paper considers the two-species chemotaxis system with two chemicals {u(t) = Delta u chi del . (u del v), x subset of Omega, t > 0, 0 = Delta v - alpha v + beta w, x is an element of Omega, t > 0, w(t) = Delta w - xi del . (w del z), x is an element of Omega, t > 0, 0 = Delta z - gamma z + delta u, x is an element of Omega, t > 0 in a smooth bounded domain Omega subset of R-2, subject to the non-flux boundary condition, and chi, xi, alpha, beta, gamma, delta > 0. We obtain a blow-up criterion that if m(1)m(2) -2 pi(m(1)/chi beta + m(2)/xi delta) > 0, then there exist finite time blow-up solutions to the system with m1 := integral(Omega)u(0)(x)d(x) and m(2) := integral(Omega)w(0)(x)dx. When chi = xi = beta = delta = 1, the blow-up criterion becomes m(1)m(2) - 2 pi(m(1) + m(2)) > 0, and the global boundedness of solutions is furthermore established with alpha = gamma = 1 under the condition that max{m(1), m(2)} < 4 pi. This improves the current results for finite time blow-up with min{m(1), m(2)} > 4 pi and global boundedness with max{m(1), m(2)} < 4/C-GN respectively in Tao and Winkler