Existence and blow-up of solutions for fractional wave equations of Kirchhoff type with viscoelasticity

In this paper, we deal with the initial boundary value problem of the following fractional wave equation of Kirchhoff type\begin{document}$ \begin{align*} u_{tt}+M([u]_{\alpha, 2}^2)(-\Delta)^{\alpha}u+(-\Delta)^{s}u_{t} = \int_{0}^{t}g(t-\tau)(-\Delta)^{\alpha}u(\tau)d\tau+\lambda|u|^{q -2}u, \end{align*} $\end{document}where \begin{document}$ M:[0, \infty)\rightarrow (0, \infty) $\end{document} is a nondecreasing and continuous function, \begin{document}$ [u]_{\alpha, 2} $\end{document} is the Gagliardo-seminorm of \begin{document}$ u $\end{document} , \begin{document}$ (-\Delta)^\alpha $\end{document} and \begin{document}$ (-\Delta)^s $\end{document} are the fractional Laplace operators, \begin{document}$ g:\mathbb{R}^+\rightarrow \mathbb{R}^+ $\end{document} is a positive nonincreasing function and \begin{document}$ \lambda $\end{document} is a parameter. First, the local and global existence of solutions are obtained by using the Galerkin method. Then the global nonexistence of solutions is discussed via blow-up analysis. Our results generalize and improve the existing results in the literature.