On decay and blow-up of solutions for a singular nonlocal viscoelastic problem with a nonlinear source term

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.