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

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)

### Cite this

**The Cauchy Problem for Nonlinear Klein-Gordon Equations in the Sobolev Spaces.** / Nakamura, Makoto; Ozawa, Tohru.

Research output: Contribution to journal › Article

*Publications of the Research Institute for Mathematical Sciences*, vol. 37, no. 3, pp. 255-293. https://doi.org/10.2977/prims/1145477225

}

TY - JOUR

T1 - The Cauchy Problem for Nonlinear Klein-Gordon Equations in the Sobolev Spaces

AU - Nakamura, Makoto

AU - Ozawa, Tohru

PY - 2001

Y1 - 2001

N2 - The local and global well-posedness for the Cauchy problem for a class of nonlinear Klein-Gordon equations is studied in the Sobolev space Hs = Hs(Rn) 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 u1+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 Hs = Hs ⊕ HS-1, ‖(ϕ,Ψ); H1/2‖ is relatively small with respect to ‖(ϕ,Ψ); Hs∗‖, 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); Hn/2‖ is also needed when s = n/2 and ν = 2.

AB - The local and global well-posedness for the Cauchy problem for a class of nonlinear Klein-Gordon equations is studied in the Sobolev space Hs = Hs(Rn) 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 u1+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 Hs = Hs ⊕ HS-1, ‖(ϕ,Ψ); H1/2‖ is relatively small with respect to ‖(ϕ,Ψ); Hs∗‖, 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); Hn/2‖ is also needed when s = n/2 and ν = 2.

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U2 - 10.2977/prims/1145477225

DO - 10.2977/prims/1145477225

M3 - Article

VL - 37

SP - 255

EP - 293

JO - Publications of the Research Institute for Mathematical Sciences

JF - Publications of the Research Institute for Mathematical Sciences

SN - 0034-5318

IS - 3

ER -