Small-amplitude nonlinear waves on a black hole background

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Abstract

Let G(x) be a C0 function such that G(x) ≤ K x p for x ≤ c, for constants K, c > 0. We consider spherically symmetric solutions of □gφ = G(φ) where g is a Schwarzschild or more generally a Reissner-Nordström metric, and such that φ and ∇φ are compactly supported on a complete Cauchy surface. It is proven that for p > 4, such solutions do not blow up in the domain of outer communications, provided the initial data are small. Moreover, φ ≤ C(max{ν, 1})-1, where ν denotes an Eddington-Finkelstein advanced time coordinate.

Original languageEnglish (US)
Pages (from-to)1147-1172
Number of pages26
JournalJournal des Mathematiques Pures et Appliquees
Volume84
Issue number9
DOIs
StatePublished - Sep 1 2005

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Keywords

  • Black hole
  • Non-linear wave equation
  • Reissner-Nordström
  • Schwarzschild

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