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

The author discusses the semilinear parabolic equation $u_t=\Delta u + f(u) + g(u)\vert\nabla u\vert^2$ with $u\vert _{\partial \Omega}=0, u(x,0)=\phi(x)$. Under suitable assumptions on $f$ and $g$, he proves that, if $0 \leq \phi \leq \lambda \psi$ with $\lambda < 1$, then the solutions are global, while if $\phi \geq \lambda \psi$ with $\lambda > 1$, then the solutions blow up in a finite time, where $\psi$is a positive solution of $\Delta \psi+f(\psi)+g(\psi)\vert\nabla \psi\vert^2=0$, with $\psi\vert _{\partial \Omega}=0$.