the porous-medium equation
Pour water into dry sand, or let gas seep through porous rock: the fluid spreads, but unlike heat it does NOT instantly reach every point. There is a sharp wet front advancing at a finite speed, with bone-dry ground just beyond it. Ordinary diffusion (the heat equation) cannot do this — it spreads instantly everywhere. The porous-medium equation is the nonlinear repair that captures finite-speed, front-led spreading: a diffusion whose rate depends on the density itself.
The equation is u_t = Laplacian(u^m) with m > 1 (often m = 2), where u >= 0 is the density of the fluid or gas. Rewrite it as u_t = div( m u^(m-1) grad u ): the effective diffusivity is m u^(m-1), which VANISHES where u = 0. That is the crucial twist — diffusion shuts off in empty regions, so the substance cannot leak into where there is none until the front physically reaches it. This 'degeneracy' (the diffusion coefficient hitting zero) is what produces a genuine free boundary moving at finite speed, in stark contrast to the linear heat equation's infinite propagation speed and instant smoothing. There is even an explicit self-similar solution, the Barenblatt profile, shaped like an inverted parabola raised to a power, supported on an expanding ball of radius that grows like a fixed power of t — the canonical spreading drop. The p-Laplacian is its close cousin, with the nonlinearity in grad u rather than in u itself.
Why does it matter? It is the model for gas through porous rock (its original derivation), groundwater flow, and population spread with crowding — anywhere diffusion should be slow when sparse and the substance should advance with a definite edge. Pedagogically it is the cleanest example that NONLINEAR diffusion behaves qualitatively unlike linear diffusion: finite propagation speed, a free boundary, and only limited smoothing (solutions can have a corner at the front). It is a parabolic equation, but its degeneracy gives it some of the finite-speed personality you would expect from a hyperbolic one.
Drop a finite blob of gas (say m = 2) onto empty space. The Barenblatt solution spreads it as an expanding paraboloid-shaped puddle: its support is a ball whose radius grows like t^(1/(d(m-1)+2)), and outside that radius u is exactly zero. There is a clean wet/dry boundary moving at finite speed — nothing like the heat equation, where any blob is instantly (if faintly) felt everywhere.
Density-dependent diffusion: a wet front advancing at finite speed.
The defining contrast with the heat equation is finite versus infinite propagation speed. Because the diffusivity m u^(m-1) vanishes where u = 0, the porous-medium equation has a genuine free boundary and does NOT smooth instantly — the very opposite of the linear heat equation.