The Cauchy problem for a quasilinear system of equations with coupling in the linearization

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