infinite propagation speed

Light a match at one end of a very long, cold metal bar. Intuitively, the far end stays cold for a while. But the heat equation says something startling: the temperature at the far end rises immediately — by an utterly tiny, undetectable amount, but strictly above zero, the very instant after you light the match. This is infinite propagation speed: in the heat equation, a disturbance anywhere is felt everywhere instantly.

The reason is baked into the heat kernel. The solution of the Cauchy problem is u(x,t) = integral of G(x - y, t) f(y) dy with the Gaussian G(x,t) = (1/sqrt(4 pi k t)) e^(-x^2/(4 k t)). For any t > 0, no matter how small, and for any x, no matter how far, G(x - y, t) is strictly positive. So if the initial heat f is positive anywhere, the integral is positive everywhere at every later instant. There is no wavefront, no finite distance the heat has reached so far and beyond which it is exactly zero. Compare the wave equation u_tt = c^2 u_xx, which has finite propagation speed c: there a signal travels at a definite speed and the region not yet reached is exactly undisturbed. The heat equation has no such speed limit.

This is genuinely unphysical — nothing in nature truly moves infinitely fast — and it is the price of Fourier's law assuming the flux responds instantly to the gradient. Yet the model is superb in practice because the amount of heat that races ahead is exponentially tiny: the Gaussian tail e^(-x^2/(4 k t)) is fantastically small at large x, so the 'instant' effect at the far end is far below any measurable level until a sensible diffusive time x^2/k has passed. The lesson is about the type of the equation: parabolic equations have infinite propagation speed, hyperbolic ones do not. If you need a true finite speed, you must change the model (e.g. the hyperbolic telegrapher's equation).