Stability theory for the linear symmetric hyperbolic system with general relaxation

In this talk, we study the dissipative structure for the linear symmetric hyperbolic system with general relaxation. If the relaxation matrix of the system has symmetric properties, Shizuta and Kawashima(1985) introduced the suitable stability condition, and Umeda, Kawashima and Shizuta(1984) analyzed the dissipative structure. On the other hand, Ueda, Duan and Kawashima(2012,2018) focused on the system with non-symmetric relaxation and got partial results. Furthermore, they argued the new dissipative structure called the regularity-loss type. In this situation, this talk aims to extend the stability theory introduced by Shizuta and Kawashima(1985) and Umeda, Kawashima and Shizuta(1984) to our general system. Furthermore, we will consider the optimality of the dissipative structure. If we have time, I would like to discuss some physical models for its application and new dissipative structures.