Relaxation limit and initial-layers for a class of hyperbolic-parabolic systems

We consider a class of hyperbolic-parabolic systems with small diffusion terms and stiff sources. The existence of solutions to the Cauchy problem with ill-prepared initial data is established by using composite expansions including initial-layer correctors and a convergence-stability lemma. New multitime expansions are introduced and lead to second-order error estimates between the composite expansions and the solution. Reduced equilibrium systems of second-order accuracy are also investigated as well as initial-layers of Chapman-Enskog expansions.