Remarks on blow-up phenomena in p-Laplacian heat equation with inhomogeneous nonlinearity

We investigate the $p-$Laplace heat equation $u_t-{\mathrm{\Delta }}_{p}u=\zeta (t)f(u)$ on a bounded smooth domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$. 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.