### Abstract

We derive the total energy decay E(t) ≤ I_{0}(1 + t)^{-1} and L^{2} boundedness ∥u(t)∥2 ≤ CI_{o} for the solutions to the initial boundary value problem for the wave equation in an exterior domain Ω: u_{tt} - Δu + a(x)u_{t} = 0 in Ω × (0, ∞) with u(x, 0) = u_{0}(x), u_{t}(x, 0) = u_{1}(x) and u|∂Ω = 0, where I_{0} = ∥u_{0}∥H_{1} + ∥u_{1}∥_{2} and a(x) is a nonnegative function which is positive near some part of the boundary ∂Ω and near infinity. We apply these estimates to prove the global existence of decaying solutions for semilinear wave equations with nonlinearity f(u) like |u|^{α}u, α > 0. We note that no geometrical condition is imposed on the boundary ∂Ω.

Original language | English |
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Pages (from-to) | 781-797 |

Number of pages | 17 |

Journal | Mathematische Zeitschrift |

Volume | 238 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2001 Dec |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Energy decay for the linear and semilinear wave equations in exterior domains with some localized dissipations.** / Nakao, Mitsuhiro.

Research output: Contribution to journal › Article

*Mathematische Zeitschrift*, vol. 238, no. 4, pp. 781-797. https://doi.org/10.1007/s002090100275

}

TY - JOUR

T1 - Energy decay for the linear and semilinear wave equations in exterior domains with some localized dissipations

AU - Nakao, Mitsuhiro

PY - 2001/12

Y1 - 2001/12

N2 - We derive the total energy decay E(t) ≤ I0(1 + t)-1 and L2 boundedness ∥u(t)∥2 ≤ CIo for the solutions to the initial boundary value problem for the wave equation in an exterior domain Ω: utt - Δu + a(x)ut = 0 in Ω × (0, ∞) with u(x, 0) = u0(x), ut(x, 0) = u1(x) and u|∂Ω = 0, where I0 = ∥u0∥H1 + ∥u1∥2 and a(x) is a nonnegative function which is positive near some part of the boundary ∂Ω and near infinity. We apply these estimates to prove the global existence of decaying solutions for semilinear wave equations with nonlinearity f(u) like |u|αu, α > 0. We note that no geometrical condition is imposed on the boundary ∂Ω.

AB - We derive the total energy decay E(t) ≤ I0(1 + t)-1 and L2 boundedness ∥u(t)∥2 ≤ CIo for the solutions to the initial boundary value problem for the wave equation in an exterior domain Ω: utt - Δu + a(x)ut = 0 in Ω × (0, ∞) with u(x, 0) = u0(x), ut(x, 0) = u1(x) and u|∂Ω = 0, where I0 = ∥u0∥H1 + ∥u1∥2 and a(x) is a nonnegative function which is positive near some part of the boundary ∂Ω and near infinity. We apply these estimates to prove the global existence of decaying solutions for semilinear wave equations with nonlinearity f(u) like |u|αu, α > 0. We note that no geometrical condition is imposed on the boundary ∂Ω.

UR - http://www.scopus.com/inward/record.url?scp=0035733190&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035733190&partnerID=8YFLogxK

U2 - 10.1007/s002090100275

DO - 10.1007/s002090100275

M3 - Article

AN - SCOPUS:0035733190

VL - 238

SP - 781

EP - 797

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

IS - 4

ER -