arXiv:2611.08662v1 Announce Type: new Abstract: We consider the nonlinear Hardy-H\'enon parabolic equation $ u_t=\Delta u+|x|^l|u|^\alpha u,$ $t>0,$ $x\in\mathbb{R}^N$ or $x\in\mathbb{R}^N\setminus\{0\}$, with $N\ge1$, $l>-\min\{2,N\}$, $\alpha>0$, with nonnegative initial datum $u_0$. We first establish new criteria for finite-time blow-up. We then derive optimal upper bounds for the life-span $T_{\max}(\lambda\varphi)$ of solutions arising from the initial datum $u_0=\lambda\varphi$. These estimates establish a precise dependence of $T_{\max}(\lambda\varphi)$ on both the initial profile $\varphi$ and the parameter $\lambda$ in the small and large-amplitude regimes.
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