finite versus infinite propagation speed
If you shout, a friend across the field hears it a moment later — sound takes time to arrive, and someone far enough away hears nothing yet. But if you warm one end of an idealized rod, the mathematics of the heat equation says the far end gets very slightly warmer instantly. These two behaviours — a signal with a speed limit versus an effect felt everywhere at once — are the two opposite worlds of PDE propagation, and which one you live in is decided by the equation's type.
Finite propagation speed is the hyperbolic (wave-like) behaviour. The wave equation u_tt = c^2 u_xx moves disturbances at exactly the speed c: a bump that starts inside an interval cannot affect any point until enough time has passed for a signal at speed c to reach it. This gives a genuine domain of dependence (the value at (x, t) depends only on initial data within distance c*t) and a range of influence (a disturbance at a point only ever touches the cone reachable at speed c). Infinite propagation speed is the parabolic (diffusion) behaviour. The heat-kernel solution of u_t = k u_xx is a Gaussian that is strictly positive for EVERY x the instant t > 0, so a temperature spike at the origin raises the temperature everywhere immediately — there is no domain of dependence smaller than the whole line.
Why care, and an honest caveat. The distinction tells you what is physically reasonable to model with each equation and shapes the proofs: finite speed underlies the wave equation's well-posed Cauchy problem and energy estimates, while infinite speed comes bundled with the heat equation's instant smoothing and its irreversibility. But infinite speed is an IDEALIZATION — the classical heat equation predicts effects faster than light, which is unphysical; more refined models (the telegrapher / hyperbolic-heat equation) restore a finite speed. And not all of physics is so clean: nonlinear degenerate diffusions like the porous-medium equation actually have finite-speed fronts, defying the linear-diffusion intuition.
Light a match in a sealed room. By the wave picture, the flash of light reaches the far wall only after a tiny but definite travel time (finite speed). By the heat picture, the warmth of the flame — mathematically — raises the temperature at the far wall the very instant it is lit, if only by an unmeasurably tiny amount (infinite speed). Same room, two equations, two propagation laws.
Hyperbolic = a speed limit and a domain of dependence; parabolic = instant, everywhere — the type fixes which.
Infinite propagation speed is a feature of the LINEAR heat equation, not a universal law of diffusion: it is physically unrealistic (faster than light), and nonlinear degenerate diffusions such as the porous-medium equation genuinely propagate at finite speed.