  In this paper, we consider the global Cauchy problem for the $L^2$-critical
semilinear heat equations $ \partial_t h=\Delta h\pm |h|^{\frac4d}h, $ with
$h(0,x)=h_0$, where $h$ is an unknown real function defined on $
\R^+\times\R^d$. In most of the studies on this subject, the initial data $h_0$
belongs to Lebesgue spaces $L^p(\R^d)$ for some $p\ge 2$ or to subcritical
Sobolev space $H^{s}(\R^d)$ with $s>0$. {\it First,} we prove that there exists
some positive constant $\gamma_0$ depending on $d$, such that the Cauchy
problem is locally and globally well-posed for any initial data $h_0$ which is
radial, supported away from the origin and in the negative Sobolev space $\dot
H^{-\gamma_0}(\R^d)$. In particular, it leads to local and global existences of
the solutions to Cauchy problem considered above for the initial data in a
proper subspace of $L^p(\R^d)$ with some $p<2$. {\it Secondly,} the sharp
asymptotic behavior of the solutions ( i.e. $L^2$-decay estimates ) as $t\to
+\infty$ are obtained with arbitrary large initial data $h_0\in \dot
H^{-\gamma_0}(\R^d)$ in the defocusing case and in the focusing case with
suitably small initial data $h_0$.
