  In this paper, we consider the Cauchy global 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 $s>0$. We here prove that there exists some
positive constant $\varepsilon_0$ 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 origin and in the negative Sobolev space $\dot
H^{-\varepsilon_0}(\R^d)$ including $L^p(\R^d)$ with certain $p<2$ as subspace.
Furthermore, unconditional uniqueness, and $L^2$-estimate both as time $t\to0$
and $t\to +\infty$ were considered.
