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  "body": "\n## Summary\n\nA parabolic partial differential equation is an evolution equation of the form $\\partial_t u + L u = f$, where $L$ is an elliptic operator in the space variables.\nThe model case is the [[heat equation|VP2HN86VAG]].\nParabolic equations describe diffusion: they smooth out the initial data, propagate information at infinite speed, and are well posed forward in time but not backward.\nSee the [[definition|WJGY7XG3QR]].\n\n## Explanations\n\n### L1 human\n\nDrop some ink in still water: it spreads out, its sharp edges blur, and after a while the colour is nearly uniform.\nA parabolic equation describes such a process.\nThe [[heat equation|VP2HN86VAG]] $\\partial_t u = \\partial_x^2 u$ says that the temperature rises where the graph of $u$ curves upward, that is, where the point is colder than its neighbours.\nSo hot spots cool down and cold spots warm up.\nRunning the process backward in time, from a blurred state to a sharp one, is unstable.\n",
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