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  "body": "\n## Statement\n\nLet $n \\ge 1$ and $T \u003e 0$.\nLet $u \\colon \\R^n \\times [0, T) \\to \\R$ be continuous, with $\\partial_t u$ and the second partial derivatives in $x$ continuous on $\\R^n \\times (0, T)$.\nAssume that $u$ solves the [[heat equation|VP2HN86VAG]] $\\partial_t u = \\Delta u$ on $\\R^n \\times (0, T)$ and that $u \\ge 0$.\nLet $g(x) = u(x, 0)$.\nThen for all $x \\in \\R^n$ and $0 \u003c t \u003c T$,\n$$u(x, t) = \\int_{\\R^n} \\Phi(x - y, t) \\, g(y) \\, dy,$$\nwhere $\\Phi$ is the [[heat kernel|KD404MF042]] of $\\R^n$.\n\n## Remarks\n\nIn particular, a nonnegative solution with initial value $0$ is identically $0$, so nonnegative solutions of the Cauchy problem are unique.\nNo growth condition at infinity is needed, which is the point of the theorem.\nThe sign condition cannot be dropped: [[Tychonoff's example|RN4J24CD83]] is a nonzero solution with initial value $0$, and it changes sign.\nOne half of the proof is a comparison on large balls with the maximum principle, which gives $u \\ge \\int \\Phi(x - y, t) g(y) \\, dy$.\nThe main work is to show that the difference, a nonnegative solution with initial value $0$, vanishes.\nDavid Widder proved the theorem for $n = 1$; the version stated holds in every dimension.\n\n## Explanations\n\n### L1 human\n\nTemperatures measured from absolute zero are never negative.\nWidder's theorem says that for such physical temperatures on all of space, the initial temperature determines the future completely.\nThe future is then given by the usual formula, which spreads the initial heat with Gaussian bells.\n\n### L2 human\n\nWithout a sign condition, uniqueness for the Cauchy problem needs a growth bound such as $\\abs{u} \\le A e^{a \\abs{x}^2}$.\nWidder's theorem replaces this bound by positivity, which is often easier to check in applications, for example for probability densities.\nIt is related to the parabolic Harnack inequality for nonnegative solutions.\n",
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