Degenerate Kirchhoff-type wave problems involving the fractional Laplacian with nonlinear damping and source terms

In this paper, we consider a Kirchhoff-type wave problem in a bounded domain, with Lipschitz boundary, involving nonlinear damping and source terms as well as the fractional Laplacian. Under some natural assumptions, we obtain global existence, vacuum isolating, asymptotic behavior and blow up of solutions for the problem under consideration, by combining the Galerkin method with potential wells theory. The significant feature and difficulty of the problem are that the coefficient of the fractional Laplacian may vanish at zero.