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 \u\(m-1) u(t) and a source term of the form \u\(p-1) u, with m, p > 1. We show that whenever m greater than or equal to p, then local weak solutions are global. On the other hand, we prove that whenever p > m and the initial energy is negative, then local weak solutions cannot be global, regardless of the size of the initial data.