Energy method in the partial Fourier space and application to stability problems in the half space

Yoshihiro Ueda, Tohru Nakamura, Shuichi Kawashima

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

The energy method in the Fourier space is useful in deriving the decay estimates for problems in the whole space Rn. In this paper, we study half space problems in R+n=R+×Rn-1 and develop the energy method in the partial Fourier space obtained by taking the Fourier transform with respect to the tangential variable x'∈Rn-1. For the variable x1∈R+ in the normal direction, we use L2 space or weighted L2 space. We apply this energy method to the half space problem for damped wave equations with a nonlinear convection term and prove the asymptotic stability of planar stationary waves by showing a sharp convergence rate for t→∞. The result obtained in this paper is a refinement of the previous one in Ueda et al. (2008) [13].

Original languageEnglish
Pages (from-to)1169-1199
Number of pages31
JournalJournal of Differential Equations
Volume250
Issue number2
DOIs
Publication statusPublished - 2011 Jan 15

Keywords

  • Asymptotic stability
  • Damped wave equation
  • Energy method
  • Fourier transform
  • Planar stationary wave

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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