### Abstract

We derive a precise decay estimate of the solutions to the initial-boundary value problem for the wave equation with a dissipation: u_{tt} - Δu + a(cursive Greek chi)u_{t} = 0 in Ω × [0, ∞) with the boundary condition u|∂Ω = 0, where a(cursive Greek chi) is a nonnegative function on Ω̄ satisfying a(cursive Greek chi) > 0 a.e. cursive Greek chi ∈ ω and ∫ω1/a(cursive Greek chi)^{p}dcursive Greek chi < ∞ for some 0 < p < 1 for an open set ω ⊂ Ω̄ including a part of ∂Ω with a specific property. The result is applied to prove a global existence and decay of smooth solutions for a semilinear wave equation with such a weak dissipation.

Original language | English |
---|---|

Pages (from-to) | 25-42 |

Number of pages | 18 |

Journal | Israel Journal of Mathematics |

Volume | 95 |

Publication status | Published - 1996 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Israel Journal of Mathematics*,

*95*, 25-42.

**Decay of solutions of the wave equation with a local degenerate dissipation.** / Nakao, Mitsuhiro.

Research output: Contribution to journal › Article

*Israel Journal of Mathematics*, vol. 95, pp. 25-42.

}

TY - JOUR

T1 - Decay of solutions of the wave equation with a local degenerate dissipation

AU - Nakao, Mitsuhiro

PY - 1996

Y1 - 1996

N2 - We derive a precise decay estimate of the solutions to the initial-boundary value problem for the wave equation with a dissipation: utt - Δu + a(cursive Greek chi)ut = 0 in Ω × [0, ∞) with the boundary condition u|∂Ω = 0, where a(cursive Greek chi) is a nonnegative function on Ω̄ satisfying a(cursive Greek chi) > 0 a.e. cursive Greek chi ∈ ω and ∫ω1/a(cursive Greek chi)pdcursive Greek chi < ∞ for some 0 < p < 1 for an open set ω ⊂ Ω̄ including a part of ∂Ω with a specific property. The result is applied to prove a global existence and decay of smooth solutions for a semilinear wave equation with such a weak dissipation.

AB - We derive a precise decay estimate of the solutions to the initial-boundary value problem for the wave equation with a dissipation: utt - Δu + a(cursive Greek chi)ut = 0 in Ω × [0, ∞) with the boundary condition u|∂Ω = 0, where a(cursive Greek chi) is a nonnegative function on Ω̄ satisfying a(cursive Greek chi) > 0 a.e. cursive Greek chi ∈ ω and ∫ω1/a(cursive Greek chi)pdcursive Greek chi < ∞ for some 0 < p < 1 for an open set ω ⊂ Ω̄ including a part of ∂Ω with a specific property. The result is applied to prove a global existence and decay of smooth solutions for a semilinear wave equation with such a weak dissipation.

UR - http://www.scopus.com/inward/record.url?scp=0030446223&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030446223&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0030446223

VL - 95

SP - 25

EP - 42

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

ER -