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

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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

**Energy decay and periodic solution for the wave equation in an exterior domain with half-linear and nonlinear boundary dissipations.** / Nakao, Mitsuhiro; Bae, Jeong Ja.

Research output: Contribution to journal › Article

*Nonlinear Analysis, Theory, Methods and Applications*, vol. 66, no. 2, pp. 301-323. https://doi.org/10.1016/j.na.2005.11.026

}

TY - JOUR

T1 - Energy decay and periodic solution for the wave equation in an exterior domain with half-linear and nonlinear boundary dissipations

AU - Nakao, Mitsuhiro

AU - Bae, Jeong Ja

PY - 2007/1/15

Y1 - 2007/1/15

N2 - We first consider the wave equation in an exterior domain Ω in RN with two separated boundary parts Γ0, Γ1. On Γ0, the Dirichlet condition u |Γ0 = 0 is imposed, while on Γ1, Neumann type nonlinear boundary dissipation ∂ u / ∂ ν = - g (ut) 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 = Γ0 ∪ Γ1. 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).

AB - We first consider the wave equation in an exterior domain Ω in RN with two separated boundary parts Γ0, Γ1. On Γ0, the Dirichlet condition u |Γ0 = 0 is imposed, while on Γ1, Neumann type nonlinear boundary dissipation ∂ u / ∂ ν = - g (ut) 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 = Γ0 ∪ Γ1. 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).

KW - Energy decay

KW - Exterior domain

KW - Nonlinear dissipation

KW - Periodic solution

KW - Wave equation

UR - http://www.scopus.com/inward/record.url?scp=33751028195&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33751028195&partnerID=8YFLogxK

U2 - 10.1016/j.na.2005.11.026

DO - 10.1016/j.na.2005.11.026

M3 - Article

AN - SCOPUS:33751028195

VL - 66

SP - 301

EP - 323

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

IS - 2

ER -