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  "body": "\n## Summary\n\nThe porous medium equation is the nonlinear diffusion equation $u_t = \\Delta(u^m)$ with $m \u003e 1$.\nIt models the flow of a gas through a porous rock, among other things.\nThe diffusion coefficient $m u^{m-1}$ vanishes where $u = 0$, so the equation is degenerate there.\nAs a consequence, disturbances travel with finite speed and solutions can have compact support with a moving free boundary.\n\n## Explanations\n\n### L1 human\n\nIn the ordinary [[heat equation|VP2HN86VAG]], heat spreads at once to the whole space, however little of it there is.\nIn the porous medium equation the spreading rate grows with the density, so where there is almost nothing, almost nothing moves.\nA blob of gas therefore keeps a sharp edge that advances at finite speed.\nThe explicit Barenblatt solution shows this: it is a bump of compact support whose radius grows like a power of $t$.\n",
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