  In this paper, we are interested in a general equation that has finite speed
of propagation compatible with Einstein's theory of special relativity. This
equation without external force fields has been derived recently by means of
optimal transportation theory. We first provide an argument to incorporate the
external force fields. Then we are concerned with comparison and maximum
principles for this equation. We consider both stationary and evolutionary
problems. We show that the former satisfies a comparison principle and a strong
maximum principle. While the latter fulfils weaker ones. The key technique is a
transformation that matches well with the gradient flow structure of the
equation.
