  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 \cite{UKS84,SK85}, 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 (cf.~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
(cf.~\cite{D,IHK08,IK08,USK,UK}). Therefore our purpose of this paper is to
formulate a new structural condition which include the Kawashima-Shizuta
condition, and to analyze the weak dissipative structure for general systems
with non-symmetric relaxation.
