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{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*} $

where $ M:[0, \infty)\rightarrow (0, \infty) $ is a nondecreasing and continuous function, $ [u]_{\alpha, 2} $ is the Gagliardo-seminorm of $ u $, $ (-\Delta)^\alpha $ and $ (-\Delta)^s $ are the fractional Laplace operators, $ g:\mathbb{R}^+\rightarrow \mathbb{R}^+ $ is a positive nonincreasing function and $ \lambda $ 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.