Local existence for a partially hyperbolic-parabolic system of quasilinear equations through a non-contractive fixed point argument

The Cauchy problem for a quasilinear system of second order hyperbolic-parabolic equations is addressed with the method of linearization and fixed point. Coupling between the hyperbolic and parabolic variables is allowed and we do not assume full symmetrizability of the system. This coupling results in weaker linear energy estimates and Banach's fixed point theorem cannot be applied. A non-contractive fixed point theorem is developed in order to conclude the local existence and uniqueness. We apply our results to the Cattaneo-Christov system for viscous compressible fluid flow, a system of equations whose non-viscous part is not hyperbolic.

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