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.