Well-posedness and blow-up for an inhomogeneous semilinear parabolic equation

We consider the large-time behavior of sign-changing solutions of the inhomogeneous equation $u_t-\Delta u=|x|^\alpha |u|^{p}+\zeta(t)\,{\mathbf w}(x)$ in $(0,\infty)\times{\mathbb{R}}^N$, where $N\geq 3$, $p>1$, $\alpha>-2$, $\zeta, {\mathbf w}$ are continuous functions such that $\zeta(t)\sim t^\sigma$ as $t\to 0$, $\zeta(t)\sim t^m$ as $t\to\infty$ . We obtain local existence for $\sigma>-1$. We also show the following: $-$ If $m\leq 0$, $p 0$, then all solutions blow up in finite time; $-$ If $m> 0$, $p>1$ and $\int_{\mathbb{R}^N}{\mathbf w}(x)dx>0$, then all solutions blow up in finite time. The main novelty in this paper is that blow up depends on the behavior of $\zeta$ at infinity.