Global in-time rough large data solution to complex-valued semilinear damped evolution equations

We study the semilinear Cauchy problem for complex-valued damped evolution equations \begin{align*} \partial_t^2u+(-Δ)^σu+(-Δ)^δ\partial_tu=u^p,\ \ u(0,x)=u_0(x),\ \partial_tu(0,x)=u_1(x), \end{align*} with $\delta \in [0,\sigma ]$, $\sigma \in {\mathbb{R}}_{+}$ and $p\in {\mathbb{N}}_{+}\mathrm{\backslash }\{1\}$, where the initial data belong to the rough space $E_s^{\alpha }$ endowed with the norm \begin{align*} |f|{E^α_s}=\big|\langleξ\rangle^s,2^{α|ξ|}\widehat{f}(ξ)\big|{L^2}\ \ \mbox{with}\ \ α<0, \ s\in$\mathbb{R}$. \end{align*} Concerning $(u_0,u_1)\in E_{s+\bar{\kappa }}^{\alpha }\times E_s^{\alpha }$ when $s⩾\frac{n}{2}-\frac{2\kappa +\bar{\kappa }-2\delta }{p-1}-\bar{\kappa }$ with $\kappa =\min \{2\delta ,\sigma \}$ and $\bar{\kappa }=\max \{2\delta ,\sigma \}$ whose Fourier transforms are supported in a suitable subset of first octant, we prove a global in-time existence result without requiring the smallness of rough initial data.