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

Abstract:

The author discusses the semilinear parabolic equation $u_t=\Delta u + f(u) + g(u)|\nabla u|^2$ with $u|_{\partial \Omega }=0, \ u(x,0)=\phi (x)$. 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 $\Delta \psi +f(\psi )+g(\psi )|\nabla \psi |^2=0$, with $\psi |_{\partial \Omega }=0$.