## 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 |

## ASJC Scopus subject areas

- Mathematics(all)