TABLE OF CONTENTS In this section we treat mixed systems in the strip 0 ? x ?1, t ?0, under initial and boundary conditions. For the uncoupled systems we assume a form as described in Section 7.2 (parabolic case) and Section 7.6 (hyperbolic case). Then we allow certain coupling terms in the differential equation and in the boundary conditions. The resulting systems are shown to be well-posed. We give an application to the linearized compressible Navier-Stokes equations.
Consider a parabolic system
in the strip 0 ? x ?1, t ?0, with boundary conditions
For each fixed t we assume the conditions formulated in Theorem 7.2.7 to be fulfilled (see also Lemma 7.2.1). We allow the matrices 
, etc. to depend smoothly on t but assume that the rank r j of 
is constant.
Consider further a hyperbolic system
in the same domain with boundary conditions
i.e., the ingoing characteristic variables are expressed at each boundary point in terms of the outgoing ones. (See Section 7.6.1 for notations.) We want to discuss the coupled system
with boundary conditions
and initial conditions
The coefficients B 11= B 11 (x, t), etc. and all inhomogeneous terms are assumed to be C ?-smooth. We first assume the existence of a solution and show
Lemma 7.7.1. Suppose that the boundary is not characteristic for the hyperbolic system; i.e., ?(0, t) and
