Asymptotic expansions for hyperbolic–parabolic systems with a small parameter

In this paper we consider initial‐boundary value problems for systems with a small parameter ϵ. The problems are mixed hyperbolic–parabolic when ϵ > 0 and hyperbolic when ϵ = 0. Often the solution can be expanded asymptotically in ϵ and to first approximation it consists of the solution of the corresponding hyperbolic problem and a boundary layer part. We prove sufficient conditions for the expansion to exist and give estimates of the remainder. We also examine how the boundary conditions should be choosen to avoid O (1) boundary layers.