Digital Object Identifier (DOI) 10.1007/s00205-009-0225-x Arch. Rational Mech. Anal. 195 (2010) 729–762 On Relaxation Hyperbolic Systems Violating the Shizuta–Kawashima Condition Corrado Mascia & Roberto Natalini Communicated by C. M. Dafermos Abstract In this paper, we start a general study on relaxation hyperbolic systems which violate the Shizuta–Kawashima ([SK]) coupling condition. This investigation is motivated by the fact that this condition is not satisfied by various physical sys- tems, and almost all the time in several space dimensions. First, we explore the role of entropy functionals around equilibrium solutions, which may not be constant, proposing a stability condition for such solutions. Then we find strictly dissipa- tive entropy functions for one dimensional 2 × 2 systems which violate the [SK] condition. Finally, we prove the existence of global smooth solutions for a class of systems such that condition [SK] does not hold, but which are linearly degenerated in the non-dissipative directions. 1. Introduction Prologue A hyperbolic system with relaxation is a particular type of hyperbolic sys- tem of balance laws, presenting a two-scale dynamic determined by the presence of a relaxation term with a characteristic time ε. For short times, the behavior is mainly determined by the interactions between hyperbolic propagation and relax- ations effect, which drive the system toward a given equilibrium manifold; for large times, a relaxed structure, which under suitable assumptions is described by a reduced diffusive system of conservation laws, emerges determining the main features of the asymptotic behavior of the solution. The main challenge in the math- ematical analysis of this kind of systems is to understand the interaction between hyperbolic convective/transport effects and zero order dissipative terms. Many physical models fit into this framework, the prototype of them being the compressible Euler system for isentropic flows with damping, see [20, 33, 37],