Global existence and blow-up of solutions for a semilinear parabolic system

We discuss the initial-boundary value problem (u(i))(t) = Delta u(i) + f(i)(u(1),...,u(m)), with u(i)/partial derivative(Omega) = 0 and u(i)(x, 0) = phi(i)(x), i = 1,.., m, in a bounded domain Omega is an element of R-n, with n greater than or equal to 1 and m greater than or equal to 1. Under suitable assumptions on the nonlinear terms f(i) we will prove that, if 0 less than or equal to phi(i) < lambda psi(i) with lambda < 1, then the solutions are global, while if phi(i) > lambda psi(i) with lambda > i, then the solutions must blow up in finite time, where the psi(i) are positive solutions of Delta psi(i) + f(i)(psi(1),..., psi(m)) = 0 with psi i/partial derivative(Omega) = 0.