### Abstract

We consider the initial-boundary value problem for the standard quasilinear wave equation: u_{tt} - div{σ(|∇u| ^{2})∇u} + a(x)u_{t} = 0 in Ω × [0, ∞) u(x, 0) = u_{0}(x) and u_{t}(x, 0) = u_{1}(x) and u|_{∂Ω} = 0 where Ω is an exterior domain in R ^{N}, σ(v) is a function like σ(v) = 1/√1 + v and a(x) is a nonnegative function. Under two types of hypotheses on a(x) we prove existence theorems of global small amplitude solutions. We note that a(x)u _{t} is required to be effective only in localized area and no geometrical condition is imposed on the boundary ∂Ω.

Original language | English |
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Pages (from-to) | 765-795 |

Number of pages | 31 |

Journal | Journal of the Mathematical Society of Japan |

Volume | 55 |

Issue number | 3 |

Publication status | Published - 2003 Jul |

Externally published | Yes |

### Fingerprint

### Keywords

- Decay
- Exterior domain
- Global solution
- Localized dissipation
- Quasilinear wave equation

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**On global smooth solutions to the initial-boundary value problem for quasilinear wave equations in exterior domains.** / Nakao, Mitsuhiro.

Research output: Contribution to journal › Article

*Journal of the Mathematical Society of Japan*, vol. 55, no. 3, pp. 765-795.

}

TY - JOUR

T1 - On global smooth solutions to the initial-boundary value problem for quasilinear wave equations in exterior domains

AU - Nakao, Mitsuhiro

PY - 2003/7

Y1 - 2003/7

N2 - We consider the initial-boundary value problem for the standard quasilinear wave equation: utt - div{σ(|∇u| 2)∇u} + a(x)ut = 0 in Ω × [0, ∞) u(x, 0) = u0(x) and ut(x, 0) = u1(x) and u|∂Ω = 0 where Ω is an exterior domain in R N, σ(v) is a function like σ(v) = 1/√1 + v and a(x) is a nonnegative function. Under two types of hypotheses on a(x) we prove existence theorems of global small amplitude solutions. We note that a(x)u t is required to be effective only in localized area and no geometrical condition is imposed on the boundary ∂Ω.

AB - We consider the initial-boundary value problem for the standard quasilinear wave equation: utt - div{σ(|∇u| 2)∇u} + a(x)ut = 0 in Ω × [0, ∞) u(x, 0) = u0(x) and ut(x, 0) = u1(x) and u|∂Ω = 0 where Ω is an exterior domain in R N, σ(v) is a function like σ(v) = 1/√1 + v and a(x) is a nonnegative function. Under two types of hypotheses on a(x) we prove existence theorems of global small amplitude solutions. We note that a(x)u t is required to be effective only in localized area and no geometrical condition is imposed on the boundary ∂Ω.

KW - Decay

KW - Exterior domain

KW - Global solution

KW - Localized dissipation

KW - Quasilinear wave equation

UR - http://www.scopus.com/inward/record.url?scp=0142055263&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0142055263&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0142055263

VL - 55

SP - 765

EP - 795

JO - Journal of the Mathematical Society of Japan

JF - Journal of the Mathematical Society of Japan

SN - 0025-5645

IS - 3

ER -