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

Mitsuhiro Nakao

Research output: Contribution to journalArticle

66 Citations (Scopus)

Abstract

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 + ∥u12 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.

Original languageEnglish
Pages (from-to)781-797
Number of pages17
JournalMathematische Zeitschrift
Volume238
Issue number4
DOIs
Publication statusPublished - 2001 Dec
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)

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