Degenerate Kirchhoff-type hyperbolic problems involving the fractional Laplacian

In this paper we are concerned with a wave problem of Kirchhoff type driven by a nonlocal integro-differential operator. Under some appropriate assumptions, we obtain the global existence, vacuum isolating and blow up of solutions for the above problems by combining the Galerkin method with potential wells theory. Finally, we investigate the existence of global solutions for the above equations with critical initial conditions. The significant feature and difficulty of the equations are that the coefficient M can vanish at zero.