Complex valued semi-linear heat equations in super-critical spaces $E^s_\sigma$

We consider the Cauchy problem for the complex valued semi-linear heat equation $$\partial_t u - \Delta u - u^m =0, \ \ u (0,x) = u_0(x),$$ where $m\geq 2$ is an integer and the initial data belong to super-critical spaces $E^s_\sigma$ for which the norms are defined by $$\|f\|_{E^s_\sigma} = \|\langle \xi\rangle^\sigma 2^{s|\xi|}\hat{f}(\xi)\|_{L^2}, \ \ \sigma \in \mathbb{R}, \ s<0.$$ If $s<0$, then any Sobolev space $H^{r}$ is a subspace of $E^s_\sigma$, i.e., $\cup_{r \in \mathbb{R}} H^r \subset E^s_\sigma$. We obtain the global existence and uniqueness of the solutions if the initial data belong to $E^s_\sigma$ ($s<0, \ \sigma \geq d/2-2/(m-1)$) and their Fourier transforms are supported in the first octant, the smallness conditions on the initial data in $E^s_\sigma$ are not required for the global solutions. Moreover, we show that the error between the solution $u$ and the iteration solution $u^{(j)}$ is $C^j/(j\,!)^2$. Similar results also hold if the nonlinearity $u^m$ is replaced by an exponential function $e^u-1$.