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

Abstract

We study the semilinear Cauchy problem for complex-valued damped evolution equations \begin{align*} \partial_t^2u+(-\Delta)^{\sigma}u+(-\Delta)^{\delta}\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 $p\in\mathbb{N}_+\backslash\{1\}$, where the initial data belong to the rough space $E^{\alpha}_s$ endowed with the norm \begin{align*} \|f\|_{E^{\alpha}_s}=\big\|\langle\xi\rangle^s\,2^{\alpha|\xi|}\widehat{f}(\xi)\big\|_{L^2}\ \ \mbox{with}\ \ \alpha<0, \ s\in\mathbb{R}. \end{align*} Concerning $(u_0,u_1)\in E^{\alpha}_{s+\bar{\kappa}}\times E^{\alpha}_s$ when $s\geqslant\frac{n}{2}-\frac{2\kappa+\bar{\kappa}-2\delta}{p-1}-\bar{\kappa}$ 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.