Global infinite energy solutions of the critical semilinear wave equation

Research output: Contribution to journalArticle

Abstract

We consider the critical semilinear wave equation (Equation Presented) set in ℝd, d ≥ 3, with 2* = 2d/d-2. Shatah and Struwe [22] proved that, for finite energy initial data (ie if (u0, u 1) ∈ Ḣ1 × L2), there exists a global solution such that (u, ∂tu) ∈ C(ℝ, Ḣ1 × L2). Planchon [17] showed that there also exists a global solution for certain infinite energy initial data, namely, if the norm of (u0, u1 ) in Ḃ2,∞ 1 × Ḃ2,∞0 is small enough. In this article, we build up global solutions of (NLW)2*-1 for arbitrarily big initial data of infinite energy, by using two methods which enable to interpolate between finite and infinite energy initial data: the method of Calderón, and the method of Bourgain. These two methods give complementary results.

Original languageEnglish (US)
Pages (from-to)463-497
Number of pages35
JournalRevista Matematica Iberoamericana
Volume24
Issue number2
DOIs
StatePublished - 2008

Keywords

  • Besov spaces
  • Critical wave equation
  • Global solutions
  • Infinite energy

ASJC Scopus subject areas

  • Mathematics(all)

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