infinite speed of propagation
Put a single drop of hot water at one point of an infinitely long cold rod and ask: how long until the far end feels something? For the heat equation the startling answer is zero. The instant you place the drop, the temperature at every point of the rod, however distant, is strictly greater than zero. The effect is unimaginably tiny far away — but it is not exactly zero. Diffusion has no speed limit; it touches everything immediately. This is infinite speed of propagation.
Precisely: the solution of the heat equation from a point (delta) source is the heat kernel, a Gaussian proportional to e^(-x^2 / (4 k t)) divided by a power of t. For any t > 0 this Gaussian is strictly positive at EVERY x, no matter how large — it decays incredibly fast but never reaches zero. So the influence of data at one point spreads instantly across all of space. Contrast the wave equation, whose disturbance from a point stays trapped inside a cone of finite slope c. The heat kernel's positivity everywhere, for every positive time, is the mathematical statement of infinite speed, and it sits inseparably alongside the smoothing property: the same Gaussian mixing that smooths also spreads.
Is this physically real? No, and it is important to be honest: infinite speed is a feature of the model, an idealization, not of nature. It comes from Fourier's law treating heat flux as instantaneously proportional to the temperature gradient. The heat equation is the continuum limit of a random walk (Brownian motion), and a random walker has a tiny but nonzero chance of being very far very soon — which is exactly the nonzero tail. Models that respect a finite speed (like the telegrapher's equation, a damped wave) exist, but the plain heat equation, with its infinite speed, is an excellent and enormously useful approximation almost everywhere it is used.
Solve u_t = k u_xx with initial data a single point of heat at the origin. The heat kernel gives u(x, t) > 0 for every x at every t > 0, including a point a million kilometres away after a microsecond — astronomically small, but strictly positive, never zero.
The heat kernel is positive everywhere for every positive time — instantaneous global reach.
Infinite speed is a modelling idealization, not literal physics — it is a side effect of Fourier's law, and the far-field response is exponentially negligible. Do not read it as energy actually outrunning the speed of light; the genuinely physical wave equation keeps a finite speed.