The global existence and blow-up of the solutions for a fractional Kirchhoff hyperbolic equations with viscoelastic term and logarithmic term

This article investigates a type of hyperbolic equation of the fractional Kirchhoff with viscoelastic and logarithmic nonlinear terms subject to homogeneous Dirichlet-boundary:

$$\{\begin{array}{ll}{u}_{tt}+M([u{]}_s^2)(-\mathrm{\Delta }{)}^{s}u-{\int }_0^{t}g(t-\tau )(-\mathrm{\Delta }{)}^{s}u(\tau )d\tau +u_t=|u{|}^{h-2}u\ln |u|,& \text{in}\mathrm{\Omega }\times (0,\mathrm{\infty }),\\ u(\cdot ,0)=u_0,u_t(\cdot ,0)=u_1,& \text{in}\mathrm{\Omega },\\ u(\cdot ,t)=0,& \text{on}\mathrm{\partial }\mathrm{\Omega }\times (0,\mathrm{\infty }),\end{array}$$

where $[u{]}_s$ is the Gagliardo semi-norm of $u,$ $(-\mathrm{\Delta }{)}^s$ is the fractional Laplacian with $s\in (0,1)$, $2<2\gamma <h<2_s^{\ast }$, $u_0$ and $u_1$ are the initial functions, and $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain with a smooth boundary. First, the global existence of solutions is established by combining the Galerkin method with the potential well theory. Subsequently, the finite-time blow-up of solutions is derived via the concavity method and a series of peculiar inequalities.