### Abstract

We study a decay property of solutions for the wave equation with a localized dissipation and a boundary dissipation in an exterior domain Ω with the boundary ∂Ω, = Γ_{0} ∪ Γ_{1}, Γ_{0} ∩ Γ_{1} = Ø. We impose the homogeneous Dirichlet condition on Γ_{0} and a dissipative Neumann condition on Γ_{1}. Further, we assume that a localized dissipation a(x)u_{t} is effective near infinity and in a neighborhood of a certain part of the boundary Γ_{0}. Under these assumptions we derive an energy decay like E(t) ≤ C(1 + t)^{-1} and some related estimates.

Original language | English |
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Pages (from-to) | 771-783 |

Number of pages | 13 |

Journal | Mathematische Nachrichten |

Volume | 278 |

Issue number | 7-8 |

DOIs | |

Publication status | Published - 2005 |

Externally published | Yes |

### Keywords

- Dissipation
- Energy decay
- Exterior problem
- Wave equation

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Bae, J. J., & Nakao, M. (2005). Energy decay for the wave equation with boundary and localized dissipations in exterior domains.

*Mathematische Nachrichten*,*278*(7-8), 771-783. https://doi.org/10.1002/mana.200310271