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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $\\Omega$ be an open subset of $\\R^n \\times \\R$, with points written $(x, t)$, $x \\in \\R^n$, $t \\in \\R$.\nA *caloric function* on $\\Omega$ is a function $u \\colon \\Omega \\to \\R$ such that:\n\n1. $u$ is twice continuously differentiable in $x$ and once continuously differentiable in $t$ on $\\Omega$.\n2. $u$ satisfies the [[heat equation|VP2HN86VAG]]\n$$\\frac{\\partial u}{\\partial t}(x, t) = \\Delta_x u(x, t) \\quad \\text{for all } (x, t) \\in \\Omega,$$\nwhere $\\Delta_x = \\sum_{i=1}^n \\partial^2 / \\partial x_i^2$ is the [[Laplacian|0E0B6RA8K0]] in the space variables.\n\n## Remarks\n\nExample: on $\\R \\times \\R$, the function $u(x, t) = x^2 + 2t$ is caloric, since $\\partial_t u = 2$ and $\\partial_x^2 u = 2$.\nOn $\\R^n \\times (0, \\infty)$, the [[heat kernel|KD404MF042]] $\\Phi(x, t) = (4 \\pi t)^{-n/2} e^{-\\abs{x}^2 / (4t)}$ is caloric.\n\nNon-example: $u(x, t) = x^2 + t$ on $\\R \\times \\R$ is smooth but not caloric, because $\\partial_t u = 1$ while $\\partial_x^2 u = 2$.\n\nSome authors write the heat equation as $\\partial_t u = k \\Delta u$ with a constant $k \u003e 0$; a change of the time scale reduces this to $k = 1$.\nA function with $\\partial_t u \\le \\Delta u$ is called subcaloric, and one with $\\partial_t u \\ge \\Delta u$ is called supercaloric.\nCaloric functions are also called temperatures.\n",
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