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

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 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**L ^{P} estimates for the linear wave equation and global existence for semilinear wave equations in exterior domains.** / Nakao, Mitsuhiro.

Research output: Contribution to journal › Article

^{P}estimates for the linear wave equation and global existence for semilinear wave equations in exterior domains',

*Mathematische Annalen*, vol. 320, no. 1, pp. 11-31. https://doi.org/10.1007/s002080100180

}

TY - JOUR

T1 - LP estimates for the linear wave equation and global existence for semilinear wave equations in exterior domains

AU - Nakao, Mitsuhiro

PY - 2001

Y1 - 2001

N2 - We consider the initial-boundary value problem for the semilinear wave equation utt - Δu + a(x)ut = f(u) in Ω x [0, ∞), u(x, 0) = u0(x), ut(x, 0) = u1(x) and u|∂Ω = 0, where Ω is an exterior domain in RN, a(x)ut is a dissipative term which is effective only near the 'critical part' of the boundary. We first give some LP estimates for the linear equation by combining the results of the local energy decay and LP 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|αu our result is applied if α > 2√3-1. We note that no geometrical condition on the boundary ∂Ω is imposed.

AB - We consider the initial-boundary value problem for the semilinear wave equation utt - Δu + a(x)ut = f(u) in Ω x [0, ∞), u(x, 0) = u0(x), ut(x, 0) = u1(x) and u|∂Ω = 0, where Ω is an exterior domain in RN, a(x)ut is a dissipative term which is effective only near the 'critical part' of the boundary. We first give some LP estimates for the linear equation by combining the results of the local energy decay and LP 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|αu our result is applied if α > 2√3-1. We note that no geometrical condition on the boundary ∂Ω is imposed.

UR - http://www.scopus.com/inward/record.url?scp=0035615371&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035615371&partnerID=8YFLogxK

U2 - 10.1007/s002080100180

DO - 10.1007/s002080100180

M3 - Article

AN - SCOPUS:0035615371

VL - 320

SP - 11

EP - 31

JO - Mathematische Annalen

JF - Mathematische Annalen

SN - 0025-5831

IS - 1

ER -