### 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 Ḣ^{s} = Ḣ^{s}(ℝ^{n}) of fractional order s > n/2 under the following assumptions: (1) Concerning the Cauchy data (φ,ψ) ∈ Ḣ ≡ Ḣ^{s} ⊕ Ḣ^{s-1}, ∥(φ,ψ); Ḣ^{1/2}∥ is relatively small with respect to ∥(φ,ψ); Ḣ^{σ}∥ for any fixed σ with n/2 < σ ≤ s. (2) Concerning the nonlinearity f, f(u) behaves as a power u^{1+4/(n-1)} near zero and has an arbitrary growth rate at infinity.

Original language | English |
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Pages (from-to) | 613-632 |

Number of pages | 20 |

Journal | Houston Journal of Mathematics |

Volume | 27 |

Issue number | 3 |

Publication status | Published - 2001 Dec 1 |

Externally published | Yes |

### Keywords

- Besov spaces
- Sobolev spaces
- Wave equations

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Nakamura, M., & Ozawa, T. (2001). Small solutions to nonlinear wave equations in the Sobolev spaces.

*Houston Journal of Mathematics*,*27*(3), 613-632.