### 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 |
---|---|

Pages (from-to) | 613-632 |

Number of pages | 20 |

Journal | Houston Journal of Mathematics |

Volume | 27 |

Issue number | 3 |

Publication status | Published - 2001 |

Externally published | Yes |

### Fingerprint

### Keywords

- Besov spaces
- Sobolev spaces
- Wave equations

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Houston Journal of Mathematics*,

*27*(3), 613-632.

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

Research output: Contribution to journal › Article

*Houston Journal of Mathematics*, vol. 27, no. 3, pp. 613-632.

}

TY - JOUR

T1 - Small solutions to nonlinear wave equations in the Sobolev spaces

AU - Nakamura, M.

AU - Ozawa, Tohru

PY - 2001

Y1 - 2001

N2 - 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.

AB - 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.

KW - Besov spaces

KW - Sobolev spaces

KW - Wave equations

UR - http://www.scopus.com/inward/record.url?scp=0035652960&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035652960&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0035652960

VL - 27

SP - 613

EP - 632

JO - Houston Journal of Mathematics

JF - Houston Journal of Mathematics

SN - 0362-1588

IS - 3

ER -