  The purpose of this work is to analyze the blow-up of solutions of the
nonlinear parabolic equation \[ u_t-\Delta u=|x|^{\alpha}|u|^{p}+{\mathtt
a}(t)\textbf{w}(x) \ \quad\mbox{for }
  (t,x)\in(0,\infty)\times\mathbb{R}^{N}, \] where $p>1$, $\alpha\in\mathbb{R}$
and ${\mathtt a}$, $\textbf{w}$ are suitable given functions. We improve
earlier results by considering a wide class of functions ${\mathtt a}(t)$.
