  In this paper we consider a singular nonlocal viscoelastic problem with a
nonlinear source term and a possible damping term. We proved that if the
initial data enter into the stable set, the solution exists globally and decays
to zero with a more general rate, and if the initial data enter into the
unstable set, the solution with non-positive initial energy as well as positive
initial energy blows up in finite time. These are achieved by using the
potential well theory, the modified convexity method and the perturbed energy
method.
