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

Optimal singularities of initial data for solvability of the Hardy parabolic equation