Formation of singularities for wave equations including the nonlinear vibrating string

Sergiu Klainerman, Andrew Majda

Research output: Contribution to journalArticle

108 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)241-263
Number of pages23
JournalCommunications on Pure and Applied Mathematics
Volume33
Issue number3
DOIs
StatePublished - May 1980
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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