  We study the fully nonlocal semilinear equation $\partial_t^\alpha
u+(-\Delta)^\beta u=|u|^{p-1}u$, $p\ge1$, where $\partial_t^\alpha$ stands for
the Caputo derivative of order $\alpha\in (0,1)$ and $(-\Delta)^\beta$,
$\beta\in(0,1]$, is the usual $\beta$ power of the Laplacian. We prescribe an
initial datum in $L^q(\mathbb{R}^N)$.
  We give conditions ensuring the existence and uniqueness of a solution living
in $L^q(\mathbb{R}^N)$ up to a maximal existence time $T$ that may be finite or
infinite. If~$T$ is finite, the $L^q$ norm of the solution becomes unbounded as
time approaches $T$, and $u$ is said to blow up in $L^q$. Otherwise, the
solution is global in time.
  For the case of nonnegative and nontrivial solutions, we give conditions on
the initial datum that ensure either blow-up or global existence. It turns out
that every nonnegative nontrivial solution in $L^q$ blows up in finite time if
$1<p<p_f:=1+\frac{2\beta}N$ whereas if $p\ge p_f$ there are both solutions that
blow up and global ones. The critical exponent $p_f$, which does not depend on
$\alpha$, coincides with the Fujita exponent for the case $\alpha=1$, in which
the time derivative is the standard (local) one. In contrast to the case
$\alpha=1$, when $\alpha\in(0,1)$ the critical exponent $p=p_f$ falls within
the situation in which global existence may occur. Our weakest condition for
global existence and our condition for blow-up are both related to the size of
the mean value of the initial datum in large balls.
