  In this paper, we obtain necessary conditions and sufficient conditions on
the initial data for the local-in-time solvability of the Cauchy problem \[
\partial_t u +(-\Delta)^\frac{\theta}{2} u=|x|^{-\gamma} u^p ,\quad x\in{\bf
R}^N, t>0, \qquad u(0)=\mu \quad \mbox{in} \quad {\bf R}^N, \] where $N\ge 1$,
$0<\theta\le2$, $p>1$, $\gamma>0$ and $\mu$ is a nonnegative Radon measure on
${\bf R}^N$. Using these conditions, we attempt to identify the optimal
strength of the singularity of $\mu$ for the existence of solutions to this
problem.
