  We are concerned with the following time-fractional semilinear heat equation
in the $N$-dimensional whole space ${\bf R}^N$ with $N \geq 1$.
  \[
  {\rm (P)}_\alpha \qquad
  \partial_t^\alpha u -\Delta u = u^p,\quad t>0,\,\,\, x\in{\bf R}^N, \qquad
  u(0) = \mu \quad \mbox{in}\quad {\bf R}^N,
  \]
  where $\partial_t^\alpha$ denotes the Caputo derivative of order $\alpha \in
(0,1)$, $p>1$, and $\mu$ is a nonnegative Radon measure on ${\bf R}^N$. The
case $\alpha=1$ formally gives the Fujita-type equation (P)$_1$ \
$\partial_tu-\Delta u=u^p$. In particular, we mainly focus on the Fujita
critical case where $p=p_F:=1+2/N$. It is well known that the Fujita exponent
$p_F$ separates the ranges of $p$ for the global-in-time solvability of
(P)$_1$. In particular, (P)$_1$ with $p=p_F$ possesses no global-in-time
solutions, and does not locally-in-time solvable in its scale critical space
$L^1(\mathbf{R}^N)$. It is also known that the exponent $p_F$ plays the same
role for the global-in-time solvability for (P)$_\alpha$. However, the problem
(P)$_\alpha$ with $p=p_F$ is globally-in-time solvable, and exhibites
local-in-time solvability in its scale critical space $L^1(\mathbf{R}^N)$. The
purpose of this paper is to clarify the collapse of the global and
local-in-time solvability of (P)$_\alpha$ as $\alpha$ approaches $1-0$.
