Global existence and blowup of solutions for a parabolic equation with a gradient term

The author discusses the semilinear parabolic equation u(t) = Deltau + f(u) + g(u)\delu\(2) with u\partial derivative Ohm = 0; u(x; 0) = phi (x). Under suitable assumptions on f and g, he proves that, if 0 less than or equal to phi less than or equal to lambda psi with lambda < 1, then the solutions are global, while if <phi> greater than or equal to lambda psi with lambda >1, then the solutions blow up in a finite time, where psi is a positive solution of Delta psi + f(psi) + g(psi)\del psi\(2) = 0, with psi\partial derivative Ohm = 0.