The following paper should be useful:
Hyperbolic conservation laws with stiff relaxation terms and entropy, G-Q Chen, C D Levermore, and T-P Liu, Comm. Pure Appl. Math., 47 (1994), 787-–830.
There's also a review of hyperbolic relaxation problems in:
R. Natalini, Recent results on hyperbolic relaxation problems, Analysis of systems of conservation laws (Aachen, 1997), 128--198, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., 99, Chapman & Hall/CRC, Boca Raton, FL, 1999.