  We address the well-posedness of the Cauchy problem corresponding to the
relativistic fluid equations, when coupled with the heat-flux constitutive
relation arising within the relativistic Chapman-Enskog procedure. The
resulting system of equations is shown to be non hyperbolic, by considering
general perturbations over the whole set of equations written with respect to a
generic time direction. The obtained eigenvalues are not purely imaginary and
their real part grows without bound as the wave-number increases. Unlike
Eckart's theory, this instability is not present when the time direction is
aligned with the fluid's direction. However, since in general the fluid
velocity is not surface-forming, the instability can only be avoided in the
particular case where no rotation is present.
