On the Fujita exponent for a nonlinear parabolic equation with a forcing term

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