LP estimates for the linear wave equation and global existence for semilinear wave equations in exterior domains

Mitsuhiro Nakao

Research output: Contribution to journalArticle

8 Citations (Scopus)

Abstract

We consider the initial-boundary value problem for the semilinear wave equation utt - Δu + a(x)ut = f(u) in Ω x [0, ∞), u(x, 0) = u0(x), ut(x, 0) = u1(x) and u|∂Ω = 0, where Ω is an exterior domain in RN, a(x)ut is a dissipative term which is effective only near the 'critical part' of the boundary. We first give some LP estimates for the linear equation by combining the results of the local energy decay and LP estimates for the Cauchy problem in the whole space. Next, on the basis of these estimates we prove global existence of small amplitude solutions for semilinear equations when Ω is odd dimensional domain. When N = 3 and f = |u|αu our result is applied if α > 2√3-1. We note that no geometrical condition on the boundary ∂Ω is imposed.

Original languageEnglish
Pages (from-to)11-31
Number of pages21
JournalMathematische Annalen
Volume320
Issue number1
DOIs
Publication statusPublished - 2001
Externally publishedYes

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Lp Estimates
Semilinear Wave Equation
Exterior Domain
Global Existence
Wave equation
Linear equation
Local Energy Decay
Decay Estimates
Semilinear Equations
Energy Estimates
Initial-boundary-value Problem
Cauchy Problem
Odd
Term
Estimate

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

LP estimates for the linear wave equation and global existence for semilinear wave equations in exterior domains. / Nakao, Mitsuhiro.

In: Mathematische Annalen, Vol. 320, No. 1, 2001, p. 11-31.

Research output: Contribution to journalArticle

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