  In this paper, we prove the existence of an initial trace T u of any positive
solution u of the semilinear fractional diffusion equation (H) $\partial$ t u +
(--$\Delta$) $\alpha$ u + f (t, x, u) = 0 in R * + $\times$ R N , where N $\ge$
1 where the operator (--$\Delta$) $\alpha$ with $\alpha$ $\in$ (0, 1) is the
fractional Laplacian and f : R + $\times$ R N $\times$ R + $\rightarrow$ R is a
Caratheodory function satisfying f (t, x, u)u $\ge$ 0 for all (t, x, u) $\in$ R
+ $\times$ R N $\times$ R +. We define the regular set of the trace T u as an
open subset of R u $\subset$ R N carrying a nonnegative Radon measive $\nu$ u
such that lim t$\rightarrow$0 Ru u(t, x)$\zeta$(x)dx = Ru $\zeta$d$\nu$
$\forall$$\zeta$ $\in$ C 2 0 (R u), and the singular set S u = R N \ R u as the
set points a such that lim sup t$\rightarrow$0 B$\rho$(a) u(t, x)dx = $\infty$
$\forall$$\rho$ \&gt; 0. We study the reverse problem of constructing a
positive solution to (H) with a given initial trace (S, $\nu$) where S
$\subset$ R N is a closed set and $\nu$ is a positive Radon measure on R = R N
\ S and develop the case f (t, x, u) = t $\beta$ u p where $\beta$ \&gt; --1
and p \&gt; 1.
