Global existence and blow-up for a nonlinear reaction-diffusion system

We consider the nonlinear reaction-diffusion system u(t) - Delta u = u(m1)v(n1), v(t) - Delta v = u(m2)v(n2), subject to Dirichlet boundary conditions and m(2) > m(1) - 1, n(1) > n(2) - 1. We prove that if m(1) less than or equal to 1, n(2) less than or equal to 1, and m(2)n(1) less than or equal to (1 - m(1))(1 - n(2)) all nonnegative solutions are global, while if m(1) > 1, or n(2) > 1, or m(2)n(1) > (1 - m(1))(1 - n(2)) both global existence and finite time blow-up coexist. (C) 1997 Academic Press.