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  "body": "\n## Definition\n\nLet $U \\subseteq \\R^n$ be open, let $T \u003e 0$, and write the variables in $U \\times (0, T)$ as $(x, t)$.\nConsider a [[linear PDE|6DZETZ5T9G]] in $U \\times (0, T)$ of the following form.\n$$\\partial_t u - \\sum_{i,j=1}^{n} a_{ij}(x, t) \\, \\partial_i \\partial_j u + \\sum_{i=1}^{n} b_i(x, t) \\, \\partial_i u + c(x, t) \\, u = f(x, t).$$\nHere all coefficients are real, $a_{ij} = a_{ji}$, and $\\partial_i$ is the derivative in $x_i$.\nThe equation is a *second-order linear parabolic equation* if its spatial part is [[uniformly elliptic|B4TTNHSTRX]].\nThat is, there is $\\theta \u003e 0$ with\n$$\\sum_{i,j=1}^{n} a_{ij}(x, t) \\, \\xi_i \\xi_j \\ge \\theta \\abs{\\xi}^2 \\quad \\text{for all } (x, t) \\in U \\times (0, T), \\ \\xi \\in \\R^n.$$\n\n## Remarks\n\nExample: the [[heat equation|VP2HN86VAG]] $\\partial_t u - \\Delta u = 0$ is parabolic with $a_{ij} = \\delta_{ij}$ and $\\theta = 1$.\nNon-example: the backward heat equation $\\partial_t u + \\Delta u = 0$ has $a_{ij} = -\\delta_{ij}$, which fails the inequality for every $\\theta \u003e 0$, so it is not parabolic in this sense.\nView a parabolic equation as a second-order equation in the $n + 1$ variables $(x, t)$.\nIts coefficient matrix has one zero eigenvalue, in the $t$ direction, and $n$ eigenvalues of the same sign.\nThis matches the [[classification of second-order PDEs|5FC6BMM1NA]].\nHigher-order parabolic equations, such as $\\partial_t u + \\Delta^2 u = 0$, are defined with an elliptic spatial operator of higher order.\n\n## Explanations\n\n### L1 human\n\nA parabolic equation is a first-order time derivative plus a uniformly elliptic operator in space, with the signs arranged as in the heat equation.\n",
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