  We investigate the $p-$Laplace heat equation $u_t-\Delta_p u=\zeta(t)f(u)$ on
a bounded smooth domain $\Omega\subset\mathbb{R}^N$. Using differential
inequalities arguments, we prove blow-up results under suitable conditions on
$\zeta, f$, and the initial data $u_0$. We also give an upper bound for the
blow-up time in each case.
