Initial boundary value problem for semilinear hyperbolic equations with weakly damping term on conical manifolds with critical and supcritical initial data

This paper investigates the qualitative properties of the solution to the IBVP for a class of wave equations that possess singular potentials and weak damping on manifolds with conical singularities.It is shown that the global existence and finite time blow up of the solution for the critical initial energy levels are obtained in the potential well theory.Moreover, the finite time blow up is obtained at the arbitrarily positive initial energy E(0) > 0.