Formation of singularities for wave equations including the nonlinear vibrating string

Under very general assumptions, the authors prove that smooth solutions of quasilinear wave equations with small‐amplitude periodic initial data always develop singularities in the second derivatives in finite time. One consequence of these results is the fact that all solutions of the classical nonlinear vibrating string equation satisfying either Dirichlet or Neumann boundary conditions and with sufficiently small nontriviai initial data necessarily develop singularities. In particular, there are no nontrivial smooth small‐amplitude time‐periodic solutions.