Global existence for damped $\sigma$-evolution equations with nonlocal nonlinearity

In this research, we would like to study the global (in time) existence of small data solutions to the following damped $\sigma$-evolution equations with nonlocal (in space) nonlinearity: \begin{equation*} \partial_{t}^{2}u+(-\Delta)^{\sigma}u+\partial_{t}u+(-\Delta)^{\sigma}\partial_{t}u=I_{\alpha}(|u|^{p}), \ \ t>0, \ \ x\in \mathbb{R}^{n}, \end{equation*} where $\sigma\geq1$, $p>1$ and $I_{\alpha}$ is the Riesz potential of power nonlinearity $|u|^{p}$ for any $\alpha\in (0,n)$. More precisely, by using the $(L^{m}\cap L^{2})-L^{2}$ and $L^{2}-L^{2}$ linear estimates, where $m\in[1,2]$, we show the new influence of the parameter $\alpha$ on the admissible ranges of the exponent $p$.