### Abstract

The local and global well-posedness for the Cauchy problem for a class of nonlinear Schrödinger equations is studied. The global well-posedness of the problem is proved in the Sobolev space H^{s} = H^{s}(R^{n}) of fractional order s > n/2 under the following assumptions. (1) Concerning the Cauchy data φ ∈ H^{s}: ∥φ; L^{2}∥ is relatively small with respect to ∥φ; H^{σ}∥ for any fixed σ with n/2 < σ ≤ s. (2) Concerning the nonlinearity f: f(u) behaves as a conformal power u^{1+4/n} near zero and has an arbitrary growth rate at infinity.

Original language | English |
---|---|

Pages (from-to) | 305-329 |

Number of pages | 25 |

Journal | Journal d'Analyse Mathematique |

Volume | 81 |

Publication status | Published - 2000 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)
- Analysis

### Cite this

*Journal d'Analyse Mathematique*,

*81*, 305-329.

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

Research output: Contribution to journal › Article

*Journal d'Analyse Mathematique*, vol. 81, pp. 305-329.

}

TY - JOUR

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

AU - Nakamura, M.

AU - Ozawa, Tohru

PY - 2000

Y1 - 2000

N2 - The local and global well-posedness for the Cauchy problem for a class of nonlinear Schrödinger equations is studied. The global well-posedness of the problem is proved in the Sobolev space Hs = Hs(Rn) of fractional order s > n/2 under the following assumptions. (1) Concerning the Cauchy data φ ∈ Hs: ∥φ; L2∥ is relatively small with respect to ∥φ; Hσ∥ for any fixed σ with n/2 < σ ≤ s. (2) Concerning the nonlinearity f: f(u) behaves as a conformal power u1+4/n 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 Schrödinger equations is studied. The global well-posedness of the problem is proved in the Sobolev space Hs = Hs(Rn) of fractional order s > n/2 under the following assumptions. (1) Concerning the Cauchy data φ ∈ Hs: ∥φ; L2∥ is relatively small with respect to ∥φ; Hσ∥ for any fixed σ with n/2 < σ ≤ s. (2) Concerning the nonlinearity f: f(u) behaves as a conformal power u1+4/n near zero and has an arbitrary growth rate at infinity.

UR - http://www.scopus.com/inward/record.url?scp=0001119867&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001119867&partnerID=8YFLogxK

M3 - Article

VL - 81

SP - 305

EP - 329

JO - Journal d'Analyse Mathematique

JF - Journal d'Analyse Mathematique

SN - 0021-7670

ER -