Asymptotic behavior in time of small solutions to nonlinear wave equations in an exterior domain

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Abstract

We study the initial boundary value problem for the nonlinear wave equations { ∂2tu - (∂2r + n-1/r∂r) u = F(u, D̄u, D̄2u), t ∈ ℝ, R < r < ∞, (*) { u(0,r) = ε0u0(r), ∂tu(0,r) = ε0u1(r), R < r < ∞, { u(t, R) = 0, t ∈ ℝ, where D̄ = (∂t, ∂r), F : ℝ6 → ℝ and ε0 is sufficiently small. We assume that F(w) = F(w1, . . . , w6) = F(u, ∂ru, ∂tu, ∂2ru, ∂rtu, ∂2tu) is a polynomial with respect to arguments satisfying one of the following conditions formula presented where λαj ∈ ℝ for j = 1, . . . , 6. In this paper we show global existence and asymptotic behavior in time of solutions of (*).

Original languageEnglish
Pages (from-to)423-456
Number of pages34
JournalCommunications in Partial Differential Equations
Volume25
Issue number3-4
DOIs
Publication statusPublished - 2000 Jan 1
Externally publishedYes

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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