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 > 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.

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