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

Abstract

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+(-Δ)^σu+\partial_{t}u+(-Δ)^σ\partial_{t}u=I_α(|u|^{p}), \ \ t>0, \ \ x\in \mathbb{R}^{n}, \end{equation*} where $\sigma \ge 1$, $p>1$ 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 $p$.