  We derive two different generalized heat-transport equations: The most
general one, of the first order in time and second order in space, encompasses
some well known heat equations and describes the hyperbolic regime in the
absence of nonlocal effects. Another, less general, of the second order in time
and fourth order in space, is able to describe hyperbolic heat conduction also
in the presence of nonlocal effects.
  We investigate the thermodynamic compatibility of both models by applying
some generalizations of the classical Liu and Coleman-Noll procedures. In both
cases, constitutive equations for the entropy and for the entropy flux are
obtained.
  For the second model, we consider a heat-transport equation which includes
nonlocal terms and study the resulting set of balance laws, proving that the
corresponding thermal perturbations propagate with finite speed.
