### Abstract

We consider the initial-boundary value problem for the semilinear wave equation u_{tt} - Δu + a(x)u_{t} = f(u) in Ω x [0, ∞), u(x, 0) = u_{0}(x), u_{t}(x, 0) = u_{1}(x) and u|_{∂Ω} = 0, where Ω is an exterior domain in R^{N}, a(x)u_{t} is a dissipative term which is effective only near the 'critical part' of the boundary. We first give some L^{P} estimates for the linear equation by combining the results of the local energy decay and L^{P} estimates for the Cauchy problem in the whole space. Next, on the basis of these estimates we prove global existence of small amplitude solutions for semilinear equations when Ω is odd dimensional domain. When N = 3 and f = |u|^{α}u our result is applied if α > 2√3-1. We note that no geometrical condition on the boundary ∂Ω is imposed.

Original language | English |
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Pages (from-to) | 11-31 |

Number of pages | 21 |

Journal | Mathematische Annalen |

Volume | 320 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2001 |

Externally published | Yes |

### ASJC Scopus subject areas

- Mathematics(all)