  In this work, we are interested on the study of the Fujita exponent and the
meaning of the blow-up for
  the Fractional Cauchy problem with the Hardy potential, namely,
\begin{equation*} u_t+(-\Delta)^s u=\lambda\dfrac{u}{|x|^{2s}}+u^{p}\inn\ren,\\
u(x,0)=u_{0}(x)\inn\ren, \end{equation*} where $N> 2s$, $0<s<1$, $(-\Delta)^s$
is the fractional laplacian of order $2s$, $\l>0$, $u_0\ge 0$, and
$1<p<p_{+}(s,\lambda)$, where $p_{+}(\lambda, s)$ is the critical existence
power found in \cite{BMP} and \cite{AMPP}.
