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+(-Δ)^σu+(-Δ)^δ\partial_tu=u^p,\ \ u(0,x)=u_0(x),\ \partial_tu(0,x)=u_1(x), \end{align*} with $\hat{I}´\in [0,\ddot{I}ƒ]$, $\ddot{I}ƒ\in {\mathbb{R}}_{+}$ and $p\in {\mathbb{N}}_{+}\backslash \{1\}$, where the initial data belong to the rough space $E^{\hat{I}}{\pm }_s$ 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 $(u_0,u_1)\in E^{\hat{I}}{\pm }_{s+\bar{\hat{I}}º}\times E^{\hat{I}}{\pm }_s$ when $s⩾\frac{n}{2}-\frac{2\hat{I}º+\bar{\hat{I}}º-2\hat{I}´}{p-1}-\bar{\hat{I}}º$ with $\hat{I}º=\min \{2\hat{I}´,\ddot{I}ƒ\}$ and $\bar{\hat{I}}º=\max \{2\hat{I}´,\ddot{I}ƒ\}$ 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.