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

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