Journal of Hyperbolic Diﬀerential Equations Vol. 15, No. 1 (2018) 149–174 c  World Scientiﬁc Publishing Company DOI: 10.1142/S0219891618500066 New structural conditions on decay property with regularity-loss for symmetric hyperbolic systems with non-symmetric relaxation Yoshihiro Ueda Faculty of Maritime Sciences, Kobe University Kobe 658-0022, Japan ueda@maritime.kobe-u.ac.jp Renjun Duan Department of Mathematics The Chinese University of Hong Kong Shatin, Hong Kong rjduan@math.cuhk.edu.hk Shuichi Kawashima Faculty of Mathematics, Kyushu University Fukuoka 819-0395, Japan kawashim@math.kyushu-u.ac.jp Received 2 September 2017 Accepted 26 September 2017 Published 21 March 2018 Communicated by P. G. LeFloch Abstract. This paper is concerned with the weak dissipative structure for linear symmetric hyperbolic systems with relaxation. The authors of this paper had already analyzed the new dissipative structure called the regularity-loss type in [Y. Ueda, R. Duan and S. Kawashima, Decay structure for symmetric hyperbolic systems with non-symmetric relaxation and its application, Arch. Ration. Mech. Anal. 205 (2012) 239–266]. Compared with the dissipative structure of the standard type in [T. Umeda, S. Kawashima and Y. Shizuta, On the devay of solutions to the linearized equations of electro-magneto-ﬂuid dynamics, Japan J. Appl. Math. 1 (1984) 435–457; Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J. 14 (1985) 249–275], the regularity- loss type possesses a weaker structure in the high-frequency region in the Fourier space. Furthermore, there are some physical models which have more complicated structure, which we discussed in [Y. Ueda, R. Duan and S. Kawashima, Decay structure of two hyperbolic relaxation models with regularity loss, Kyoto J. Math. 57(2) (2017) 235–292]. Under this situation, we introduce new concepts and extend our previous results devel- oped in [Y. Ueda, R. Duan and S. Kawashima, Decay structure for symmetric hyperbolic 149 J. Hyper. Differential Equations 2018.15:149-174. Downloaded from www.worldscientific.com by 104.227.203.200 on 07/15/19. Re-use and distribution is strictly not permitted, except for Open Access articles.