  The Cauchy problem for a quasilinear system of hyperbolic-parabolic equations
is addressed with the method of linearization and fixed point. Coupling between
the hyperbolic and parabolic variables is allowed in the linearization and we
do not assume the Friedrich's symmetrizability of the system. This coupling
results in linear energy estimates that prevent the application of Banach's
contraction principle. A metric fixed point theorem is developed in order to
conclude the local existence and uniqueness of solutions. We show that the
boundedness in the high norm and contraction in the low norm can be
incorporated into the formulation of the fixed point by introducing the notion
of a closed extension of the solution map. We apply our results to the
Cattaneo-Christov system for viscous compressible fluid flow, a system of
equations whose inviscid part is not hyperbolic.
