### Abstract

We first consider the wave equation in an exterior domain Ω in R^{N} with two separated boundary parts Γ_{0}, Γ_{1}. On Γ_{0}, the Dirichlet condition u |_{Γ0} = 0 is imposed, while on Γ_{1}, Neumann type nonlinear boundary dissipation ∂ u / ∂ ν = - g (u_{t}) is assumed. Further, a 'half-linear' localized dissipation is attached on Ω. For such a situation we derive a precise rate of decay of the energy E (t) for solutions of the initial boundary value problem. We impose no geometrical condition on the shape of the boundary ∂ Ω = Γ_{0} ∪ Γ_{1}. Secondly, when a T periodic forcing term works we prove the existence of a T periodic solution on R under an additional growth assumption on ρ (x, v) and g (v).

Original language | English |
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Pages (from-to) | 301-323 |

Number of pages | 23 |

Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 66 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2007 Jan 15 |

Externally published | Yes |

### Keywords

- Energy decay
- Exterior domain
- Nonlinear dissipation
- Periodic solution
- Wave equation

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics