Small solutions to nonlinear wave equations in the Sobolev spaces

M. Nakamura, Tohru Ozawa

Research output: Contribution to journalArticle

Abstract

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
Volume27
Issue number3
Publication statusPublished - 2001
Externally publishedYes

Fingerprint

Small Solutions
Global Well-posedness
Nonlinear Wave Equation
Sobolev Spaces
Homogeneous Space
Fractional Order
Cauchy
Cauchy Problem
Infinity
Nonlinearity
Zero
Arbitrary
Class

Keywords

  • Besov spaces
  • Sobolev spaces
  • Wave equations

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Small solutions to nonlinear wave equations in the Sobolev spaces. / Nakamura, M.; Ozawa, Tohru.

In: Houston Journal of Mathematics, Vol. 27, No. 3, 2001, p. 613-632.

Research output: Contribution to journalArticle

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