Decay Structure for Symmetric Hyperbolic Systems with Non-Symmetric Relaxation and its Application

Yoshihiro Ueda, Renjun Duan, Shuichi Kawashima*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

50 Citations (Scopus)


This paper is concerned with the decay structure for linear symmetric hyperbolic systems with relaxation. When the relaxation matrix is symmetric, the dissipative structure of the systems is completely characterized by the Kawashima-Shizuta stability condition formulated in Umeda et al. (Jpn J Appl Math 1:435-457, 1984) and Shizuta and Kawashima (Hokkaido Math J 14:249-275, 1985) and we obtain the asymptotic stability result together with the explicit time-decay rate under that stability condition. However, some physical models which satisfy the stability condition have non-symmetric relaxation term (for example, the Timoshenko system and the Euler-Maxwell system). Moreover, it had been already known that the dissipative structure of such systems is weaker than the standard type and is of the regularity-loss type (see Duan in J Hyperbolic Differ Equ 8:375-413, 2011; Ide et al. in Math Models Meth Appl Sci 18:647-667, 2008; Ide and Kawashima in Math Models Meth Appl Sci 18:1001-1025, 2008; Ueda et al. in SIAM J Math Anal 2012; Ueda and Kawashima in Methods Appl Anal 2012). Therefore our purpose in this paper is to formulate a new structural condition which includes the Kawashima-Shizuta condition, and to analyze the weak dissipative structure for general systems with non-symmetric relaxation.

Original languageEnglish
Pages (from-to)239-266
Number of pages28
JournalArchive for Rational Mechanics and Analysis
Issue number1
Publication statusPublished - 2012 Jul
Externally publishedYes

ASJC Scopus subject areas

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering


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