Small solutions to nonlinear wave equations in the Sobolev spaces

M. Nakamura*, T. Ozawa

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


The local and global well-posedness for the Cauchy problem for a class of nonlinear wave equations is studied. The global well-posedness of the problem is proved in the homogeneous Sobolev space Ḣs = Ḣs(ℝn) of fractional order s > n/2 under the following assumptions: (1) Concerning the Cauchy data (φ,ψ) ∈ Ḣ ≡ Ḣs ⊕ Ḣs-1, ∥(φ,ψ); Ḣ1/2∥ is relatively small with respect to ∥(φ,ψ); Ḣσ∥ for any fixed σ with n/2 < σ ≤ s. (2) Concerning the nonlinearity f, f(u) behaves as a power u1+4/(n-1) near zero and has an arbitrary growth rate at infinity.

Original languageEnglish
Pages (from-to)613-632
Number of pages20
JournalHouston Journal of Mathematics
Issue number3
Publication statusPublished - 2001
Externally publishedYes


  • Besov spaces
  • Sobolev spaces
  • Wave equations

ASJC Scopus subject areas

  • General Mathematics


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