Some Blowup Results for a Nonlinear Parabolic Equation with a Gradient Term

Abstract

Under some conditions, a blowup result is proved for the solution u of: \[\begin{gathered} u_t = \Delta u - \left| {\nabla u} \right|^q + \left| {u} \right|^{p - 1} u,\quad t > 0,\quad x \in \Omega \hfill \\ u(t,x) = 0,\quad t > 0,\quad x \in \Gamma , \hfill \\ u(0,x) = \varphi (x),\quad x \in \Omega . \hfill \\ \end{gathered} \] The associated elliptic problem is also studied.