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Complex valued semi-linear heat equations in super-critical spaces $E^s_σ$

We consider the Cauchy problem for the complex valued semi-linear heat equation $$ \partial_t u - Δu - u^m =0, \ \ u (0,x) = u_0(x), $$ where $m\geq 2$ is an integer and the initial data belong to super-critical spaces $E^s_σ$ for which the norms are defined by $$ \|f\|_{E^s_σ} = \|\langle ξ\rangle^σ2^{s|ξ|}\hat{f}(ξ)\|_{L^2}, \ \ σ\in \mathbb{R}, \ s<0. $$ If $s<0$, then any Sobolev space $H^{r}$ is a subspace of $E^s_σ$, i.e., $\cup_{r \in \mathbb{R}} H^r \subset E^s_σ$. We obtain the global existence and uniqueness of the solutions if the initial data belong to $E^s_σ$ ($s<0, \ σ\geq d/2-2/(m-1)$) and their Fourier transforms are supported in the first octant, the smallness conditions on the initial data in $E^s_σ$ 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 $e^u-1$.