### Abstract

The local and global well-posedness for the Cauchy problem for a class of nonlinear Klein-Gordon equations is studied in the Sobolev space H^{s} = H^{s}(R^{n}) with s ≥ n/2. The global well-posedness of the problem is proved under the following assumptions: (1) Concerning the nonlinearity f, f(u) behaves as a power u^{1+4}/^{n} near zero. At infinity f(u) has an exponential growth rate such as exp(κ|u|^{ν}) with κ > 0 and 0<ν≤2 if s = n/2, and has an arbitrary growth rate if s > n/2. (2) Concerning the Cauchy data (ϕ,Ψ) e H^{s} = H^{s} ⊕ H^{S-1}, ‖(ϕ,Ψ); H^{1/2}‖ is relatively small with respect to ‖(ϕ,Ψ); H^{s∗}‖, where s∗ is a number with s∗ = n/2 if s = n/2, n/2 < s∗ ≤ s if s > n/2, and the smallness of ‖(ϕ,Ψ); H^{n/2}‖ is also needed when s = n/2 and ν = 2.

Original language | English |
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Pages (from-to) | 255-293 |

Number of pages | 39 |

Journal | Publications of the Research Institute for Mathematical Sciences |

Volume | 37 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2001 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)