In physics, phenomena involving a finite propagation speed are typically expressed through hyperbolic (second-order) partial differential equations, such as, for instance, the well-known wave equation
$$f_{tt}=c^2\nabla^2f$$
where $c$ is the wave propagation speed.
What I'm interested to know is, could we find a parabolic PDE that models this kind of finite velocity limits, or are we automatically being thrown back to second-order equations? Is there a general mathematical theorem at play here about the solutions of parabolic vs hyperbolic PDEs?
(This question is a follow-up to this one, which I posted some months ago in Physics SE, and I consider wasn't fully answered by that time).