Global rough solution for $L^2$-critical semilinear heat equation in the negative Sobolev space

In this paper, we consider the Cauchy global problem for the $L^2$-critical semilinear heat equations ${\mathrm{\partial }}_{t}h=\mathrm{\Delta }h\pm \mathrm{\mid }h{\mathrm{\mid }}^{\frac{4}{d}}h,$ with $h(0,x)=h_0$, where $h$ is an unknown real function defined on ${\mathbb{R}}^{+}\times {\mathbb{R}}^d$. In most of the studies on this subject, the initial data $h_0$ belongs to Lebesgue spaces $L^p({\mathbb{R}}^d)$ for some $p\ge 2$ or to subcritical Sobolev space $H^s({\mathbb{R}}^d)$ with $s>0$. We here prove that there exists some positive constant ${\epsilon }_0$ 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$L^p(\R^d)$withcertain$p<2$ as subspace. Furthermore, unconditional uniqueness, and $L^2$-estimate both as time $t\to 0$ and $t\to +\mathrm{\infty }$ were considered.