Global existence and nonexistence for nonlinear wave equations with damping and source terms

We consider an initial-boundary value problem for a nonlinear wave equation in one space dimension. The nonlinearity features the damping term and a source term of the form , with . We show that whenever , then local weak solutions are global. On the other hand, we prove that whenever and the initial energy is negative, then local weak solutions cannot be global, regardless of the size of the initial data.