  In this paper we obtain necessary conditions and sufficient conditions on the
initial data for the solvability of the Cauchy problem $$ \partial_t
u+(-\Delta)^{\frac{\theta}{2}}u=u^p,\quad x\in{\bf R}^N,\,\,t>0, \qquad
u(0)=\mu\ge 0\quad\mbox{in}\quad{\bf R}^N, $$ where $N\ge 1$, $0<\theta\le 2$,
$p>1$ and $\mu$ is a Radon measure or a measurable function in ${\bf R}^N$. Our
conditions lead optimal estimates of the life span of the solution with $\mu$
behaving like $\lambda |x|^{-A}$ ($A>0$) at the space infinity, as $\lambda \to
+0$.
