  We consider the Cauchy problem for the Hardy parabolic equation $\partial_t
u-\Delta u=|x|^{-\gamma}u^p$ with initial data $u_0$ singular at some point
$z$. Our main results show that, if $z\neq 0$, then the optimal strength of the
singularity of $u_0$ at $z$ for the solvability of the equation is the same as
that of the Fujita equation $\partial_t u-\Delta u=u^p$. Moreover, if $z=0$,
then the optimal singularity for the Hardy parabolic equation is weaker than
that of the Fujita equation. We also obtain analogous results for a fractional
case $\partial_t u+(-\Delta)^{\theta/2} u=|x|^{-\gamma}u^p$ with $0<\theta<2$.
