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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).

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  • $\begingroup$ Can you clarify exactly what class of equations you're talking about? There are many first-order systems that I would say have finite propagation speed, e.g. the advection equation $\partial_t f + \partial_x f = 0$. Perhaps you mean hyperbolic vs parabolic rather than second vs first order? $\endgroup$ Commented Oct 22, 2016 at 22:41
  • $\begingroup$ @AnthonyCarapetis Yes, excuse me, I meant hyperbolic vs parabolic. I'm correcting the question. $\endgroup$
    – dahemar
    Commented Oct 22, 2016 at 22:47
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    $\begingroup$ Don't have time to write a good answer, but the quick version: no for linear, yes for nonlinear. See e.g. uv.es/mazon/trabajos/ARMA2.pdf for a nonlinear parabolic "relativistic heat equation". $\endgroup$ Commented Oct 22, 2016 at 22:51
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    $\begingroup$ Additionally to the answer of Anthony Carapetis, see, for instance, the article of Vazquez link.springer.com/chapter/10.1007/3-540-52595-5_96, and Diaz mat.ucm.es/~jidiaz/Publicaciones/ARTICULOS_PDF/A_004.pdf, concerning other nonlinear parabolic problems. $\endgroup$
    – Voliar
    Commented Oct 23, 2016 at 10:17

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Yes.

Famous ones include the parabolic $p$-Laplace equation and the porous medium equation. An example of a solution with finite speed of propagation is the Barenblatt solution. Note that both of the equations are non-linear.

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