Abstract
In this paper we study the global existence and the relaxation limit for a 3×3 hyperbolic system of conservation laws with globally Lipschitz relaxation term. In particular, the convergence for solutions in Sobolev spaces toward the solutions to the equilibrium system, which is a 2×2 incompletely parabolic system, is proved. This is the first example of a semilinear relaxation approximation for an incompletely parabolic system. Thanks to the scaling we use, such convergence can also be viewed as the passage from the viscosity of the memory type to the viscosity of the rate type in the study of viscoelastic materials. Finally, a result of convergence for travelling waves is also provided.