Hardy type inequality and application to the stability of degenerate stationary waves

Shuichi Kawashima, Kazuhiro Kurata

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This paper is concerned with the asymptotic stability of degenerate stationary waves for viscous conservation laws in the half space. It is proved that the solution converges to the corresponding degenerate stationary wave at the rate t- α / 4 as t → ∞, provided that the initial perturbation is in the weighted space Lα2 = L2 (R+ ; (1 + x)α) for α < αc (q) : = 3 + 2 / q, where q is the degeneracy exponent. This restriction on α is best possible in the sense that the corresponding linearized operator cannot be dissipative in Lα2 for α > αc (q). Our stability analysis is based on the space-time weighted energy method combined with a Hardy type inequality with the best possible constant.

Original languageEnglish
Pages (from-to)1-19
Number of pages19
JournalJournal of Functional Analysis
Issue number1
Publication statusPublished - 2009 Jul 1
Externally publishedYes



  • Asymptotic stability
  • Degenerate stationary waves
  • Hardy inequality
  • Viscous conservation laws

ASJC Scopus subject areas

  • Analysis

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