Existence of solutions for a fractional semilinear parabolic equation with singular initial data

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 $\lambda \to +0$.