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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\R^n$ be open and let $m \u003e 1$ be a real number.\nThe *porous medium equation* is the [[parabolic equation|1CCTH8X2FT]] for a function $u \\colon \\Omega \\times (0, T) \\to [0, \\infty)$:\n$$u_t = \\Delta (u^m),$$\nwhere $\\Delta$ is the [[Laplacian|0E0B6RA8K0]] in $x$.\n\n## Remarks\n\nExample: the Barenblatt solution\n$$U(x, t) = t^{-\\alpha} \\Big( C - k \\abs{x}^2 t^{-2\\alpha/n} \\Big)_+^{1/(m-1)}, \\qquad \\alpha = \\frac{n}{n(m - 1) + 2}, \\quad k = \\frac{\\alpha (m - 1)}{2 m n},$$\nwith a constant $C \u003e 0$ and $s_+ = \\max(s, 0)$, solves the equation on $\\R^n \\times (0, \\infty)$ in the weak sense.\nFor each $t$ its support is the ball $\\abs{x} \\le (C/k)^{1/2}\\, t^{\\alpha/n}$, so it spreads at finite speed.\nNon-example: for $m = 1$ the equation is the [[heat equation|VP2HN86VAG]], which is excluded; its solutions with nonnegative nonzero data are positive everywhere for $t \u003e 0$.\nWriting $\\Delta(u^m) = \\nabla \\cdot (m u^{m-1} \\nabla u)$ shows that the equation is degenerate parabolic where $u = 0$.\nThe case $0 \u003c m \u003c 1$ is called the fast diffusion equation.\n",
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