Relaxation limit for a symmetrically hyperbolic system

In this paper, we apply the invariant region theory to get an a prioriL∞ estimate of the relaxation approximated solutions to the Cauchy problem of a symmetrically hyperbolic system with stiff relaxation and dominant diffusion, and then obtain that the relaxation approximated solutions converge almost everywhere to the equilibrium state of the symmetrical system with the aid of the compactness framework about the scalar equation.