The porous-medium equation and finite propagation

On the whole space, consider uₜ=Δ(uᵐ) for nonnegative u and m>1. Where u is positive and smooth, it can be written in divergence form with diffusivity m uᵐ⁻¹. That coefficient vanishes at zero, so the equation is not uniformly parabolic across the edge of the support. Standard nonnegative weak solutions for bounded, integrable compactly supported initial data exist, are unique in the usual L¹-continuous energy class, and have finite propagation. Those statements belong to degenerate diffusion theory rather than following from a uniform heat-equation estimate.

A useful weak description tests uₜ=Δ(uᵐ) against compactly supported smooth functions and requires the relevant local energy integrability of uᵐ. The solution is smooth in suitable positive regions, but derivatives can lose regularity at the free boundary. Comparison and L¹ contraction organize the standard theory. Nonnegativity matters when interpreting a noninteger power and pressure variable; extending to signed data requires a different explicit convention. Changing to fast diffusion with exponent below one or to a nonlocal operator can alter propagation dramatically.

For one dimension and m=2, the displayed positive-part profile solves the equation for positive time. Its spatial radius grows like $t^{1/3}$ and its height falls like $t^{-1/3}$. The two exponents preserve mass. Inside its support, direct differentiation gives Bₜ=(B²)ₓₓ. At the interface, B vanishes and the derivative of B² matches the zero exterior flux, so the profile is a weak solution across the moving edge. B itself can have a gradient jump there, making global classical second derivatives inappropriate.

As time decreases to zero, this source-type profile approaches a point mass, not an ordinary bounded initial function. To produce an example with bounded compact initial data, shift the internal time: use B(x,1+s) with physical elapsed time s≥0. At s=0 the profile is already a bounded compact cap. Its later support remains finite and expands according to the same formula. This distinction avoids treating a singular source solution as though it had a regular initial trace in L∞.

Inside the positive phase, pressure satisfies vₜ=(m−1)vΔv+|∇v|². For sufficiently regular interfaces, the outward normal speed is related to the outward pressure gradient. This relation needs one-sided traces and interface regularity; it is not a license to differentiate an arbitrary weak free boundary pointwise. The explicit quadratic profile has enough structure to check the speed directly. General initial data can exhibit waiting times or complicated early interfaces, so one should not infer a universal smooth front law from one formula.

Finite propagation is a statement about support, not a claim that the profile is spatially constant or that its radius grows linearly. Scaling and conserved mass select a characteristic power law for the self-similar example. On bounded domains, boundary conditions can change mass conservation and long-time behavior. With forcing, mass can change even on the whole space. A valid comparison between linear and degenerate diffusion therefore keeps the exponent, sign class, domain, boundary conditions, and forcing fixed rather than transferring a Gaussian conclusion indiscriminately.