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  "body": "\n## Counterexample\n\nThere is a function $u \\colon \\R \\times \\R \\to \\R$, smooth and not identically zero, such that\n$$\\partial_t u = \\partial_x^2 u \\ \\text{on } \\R \\times \\R \\quad \\text{and} \\quad u(x, t) = 0 \\ \\text{for all } t \\le 0.$$\nSo the Cauchy problem for the [[heat equation|VP2HN86VAG]] on $\\R \\times (0, \\infty)$ with initial value $0$ has a solution other than $u = 0$.\nOne construction: fix $\\alpha \u003e 1$, let $g(t) = \\exp(-t^{-\\alpha})$ for $t \u003e 0$ and $g(t) = 0$ for $t \\le 0$, and set\n$$u(x, t) = \\sum_{k=0}^{\\infty} g^{(k)}(t) \\, \\frac{x^{2k}}{(2k)!}.$$\nTerm by term, $\\partial_x^2$ of the $k$-th term is the $(k-1)$-th term with $g^{(k-1)}$ replaced by $g^{(k)}$, so formally $\\partial_x^2 u = \\sum_k g^{(k+1)}(t) x^{2k}/(2k)! = \\partial_t u$.\nThere are constants $\\theta, c \u003e 0$ with $\\abs{g^{(k)}(t)} \\le k! \\, (\\theta t)^{-k} \\exp(-c \\, t^{-\\alpha})$ for $t \u003e 0$.\nThese bounds show that the series and its differentiated series converge locally uniformly, so $u$ is a genuine solution.\nSince $g$ is not identically zero, neither is $u$.\n\n## Remarks\n\nThe solution grows very fast in $x$: on no strip $\\R \\times (0, T)$ is it bounded by $A e^{a x^2}$ for constants $A$ and $a$.\nTychonoff showed that uniqueness does hold in the class of solutions with $\\abs{u(x, t)} \\le A e^{a \\abs{x}^2}$.\nSo a growth condition at infinity is needed for uniqueness on the whole line; on a bounded interval with boundary values, the [[maximum principle|EMDVD4E83K]] gives uniqueness without one.\nNonnegative solutions are also unique, by Widder's theorem, so $u$ must change sign.\n\n## Explanations\n\n### L1 human\n\nOne would expect a rod that starts at temperature zero everywhere and receives no heat to stay at zero.\nOn an infinite rod this is false in a strange way: heat can arrive from infinity, through temperatures that grow enormously far away.\nThe example shows that the heat equation alone does not determine the future; one must also rule out such wild behaviour at infinity.\n",
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  "name": "Tychonoff's non-uniqueness example",
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