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.

Using a combination of Kawashima- and Goodman-type energy estimates, we establish spectral stability of general small-amplitude relaxation shocks of symmetric dissipative systems. This extends previous results obtained by Plaza and Zumbrun [8] by singular perturbation techniques under an additional technical assumption, namely, that the background equation be noncharacteristic with respect to the shock.

The present paper deals with the following hyperbolic-elliptic coupled system, modelling dynamics of a gas in presence of radiation, {(-qxx + Rq + G center dot ux = 0,) (ut + f(u)x + Lqx = 0,) x is an element of R, t > 0, where u is an element of R-n, q is an element of R and R > 0, G, L is an element of R-n. The function f : R-n -> R-n is smooth and such that del f has n distinct real eigenvalues for any u. The problem of existence of admissible radiative shock wave is considered, i.e., existence of a solution of the form (u, q)(x, t) := (U, Q)(x - st), such that (U, Q)(+/-infinity) = (u(+/-), 0), and u(+/-) is an element of R-n, s is an element of R define a shock wave for the reduced hyperbolic system, obtained by formally putting L = 0. It is proved that, if u(-) is such that del lambda(k)(u(-)) center dot r(k)(u(-)) not equal 0 (where lambda(k) denotes the k-th eigenvalue of del f and r(k) a corresponding right eigenvector), and (l(k)(u(-)) center dot L) (G center dot r(k)(u(-))) > 0, then there exists a neighborhood u of u(-) such that for any u(+) is an element of u, s is an element of R such that the triple (u(-), u(+); s) defines a shock wave for the reduced hyperbolic system, there exists a (unique up to shift) admissible radiative shock wave for the complete hyperbolic-elliptic system. The proof is based on reducing the system case to the scalar case, hence the problem of existence for the scalar case with general strictly convex fluxes is considered, generalizing existing results for the Burgers' flux f(u) = u(2)/2. Additionally, we are able to prove that the profile (U, Q) gains smoothness when the size of the shock vertical bar u(+) - u(-)vertical bar is small enough, as previously proved for the Burgers' flux case. Finally, the general case of nonconvex fluxes is also treated, showing similar results of existence and regularity for the profiles.