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This paper is concerned with the decay structure for linear symmetrichyperbolic systems with relaxation. When the relaxation matrix is symmetric,the dissipative structure of the systems is completely characterized by theKawashima-Shizuta stability condition formulated in \cite{UKS84,SK85}, and weobtain the asymptotic stability result together with the explicit time-decayrate under that stability condition. However, some physical models whichsatisfy the stability condition have non-symmetric relaxation term (cf.~theTimoshenko system and the Euler-Maxwell system). Moreover, it had been alreadyknown that the dissipative structure of such systems is weaker than thestandard type and is of the regularity-loss type(cf.~\cite{D,IHK08,IK08,USK,UK}). Therefore our purpose of this paper is toformulate a new structural condition which include the Kawashima-Shizutacondition, and to analyze the weak dissipative structure for general systemswith non-symmetric relaxation.

Decay structure for symmetric hyperbolic systems with non-symmetric relaxation and its application