Besov spaces and self-similar solutions for the wave-map equation

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Abstract

Self-similar solutions play a crucial role in the blow-up theory for the wave-map equation; they correspond to self-similar data at the time of the blow-up. However, solutions to this equation are generally considered for data in the standard finite energy spaces (in dimension d) Hd/2 × Hd/2-1. We build up in this article solutions of the covariant wave-map equation for data which are small and of infinite energy, or large and self-similar. This provides us with a general framework which includes in particular the blowing up solutions of Shatah [14] and Bizon [3]. As an application, we describe more precisely the blow-up phenomenon.

Original languageEnglish (US)
Pages (from-to)1571-1596
Number of pages26
JournalCommunications in Partial Differential Equations
Volume33
Issue number9
DOIs
StatePublished - Sep 2008

Keywords

  • Besov space
  • Blow-up
  • Self-similar
  • Wave-map

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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