Global existences and asymptotic behavior for semilinear heat equation

In this paper, we consider the global Cauchy 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 $s>0$. {\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 $L^p({\mathbb{R}}^d)$ with some $p<2$. {\it Secondly,} the sharp asymptotic behavior of the solutions ( i.e. $L^2$-decay estimates ) as $t\to +\infty$areobtainedwitharbitrarylargeinitialdata$h_0\in \dot H^{-\gamma_0}(\R^d)$ in the defocusing case and in the focusing case with suitably small initial data $h_0$.