Optimal conditions for initial data leading to non-existence of non-negative solutions to the Cauchy problem for the parabolic Hardy-Hénon equation \( \partial _tu=\Delta u^m+|x|^{\sigma }u^p, \quad (t,x)\in (0,\infty )\times \mathbb {R}^N, \) with \(m>0\) , \(\sigma >0\) and \(p>\max \{1,m\}\) , are identified. Assuming that the initial condition satisfies \( u_0\in L^{\infty }(\mathbb {R}^N), \quad \lim \limits _{|x|\rightarrow \infty }|x|^{\gamma }u_0(x)=L\in (0,\infty ), \quad u_0\ge 0, \) it is shown that non-existence of solutions occurs for \( \gamma <\frac{\sigma +2}{p-m} - \frac{2\max {\{p-p_G,0\}}}{(p-1)(p-m)} \) with \( p_G:=1+\frac{\sigma (1-m)}{2}. \) The above threshold for non-existence is optimal, in view of the existence of self-similar solutions for the limiting value of \(\gamma \) .
