Archive for Rational Mechanics and Analysis manuscript No. (will be inserted by the editor)

Decay structure for symmetric hyperbolic systems with non-symmetric relaxation and its application Abstract This paper is concerned with the decay structure for linear sym-metric hyperbolic systems with relaxation. When the relaxation matrix is symmetric, the dissipative structure of the systems is completely character-ized by the Kawashima-Shizuta stability condition formulated in [21,17], 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. [5,8,9, 20,19]). 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 relax-ation