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  "body": "\n## Statement\n\nLet $\\Omega \\subseteq \\R^n$ be a bounded open set and let $T \u003e 0$.\nLet $\\Omega_T = \\Omega \\times (0, T]$, and let $\\Gamma_T = \\overline{\\Omega_T} \\setminus \\Omega_T$ be its parabolic boundary, which is the bottom $\\overline{\\Omega} \\times \\set{0}$ together with the sides $\\partial \\Omega \\times [0, T]$.\nLet $u \\colon \\overline{\\Omega_T} \\to \\R$ be continuous, with $\\partial_t u$ and the second partial derivatives in $x$ continuous on $\\Omega_T$, and assume that\n$$\\partial_t u - \\Delta u \\le 0 \\quad \\text{in } \\Omega_T.$$\nThen\n$$\\max_{\\overline{\\Omega_T}} u = \\max_{\\Gamma_T} u.$$\n\n## Remarks\n\nFirst assume the strict inequality $\\partial_t u - \\Delta u \u003c 0$.\nAt a maximum point $(x_0, t_0)$ in $\\Omega_T$ one would have $\\Delta u(x_0, t_0) \\le 0$ and $\\partial_t u(x_0, t_0) \\ge 0$, the latter also when $t_0 = T$, which is a contradiction.\nIn general apply this to $v = u + \\varepsilon \\abs{x}^2$, for which $\\partial_t v - \\Delta v \\le -2n\\varepsilon \u003c 0$, and let $\\varepsilon \\to 0$.\nApplied to $u$ and $-u$, it shows that a solution of the [[heat equation|VP2HN86VAG]] on $\\Omega_T$ is bounded by its values on the parabolic boundary.\nSo the initial-boundary value problem has at most one solution.\nThe strong form says more: if $\\Omega$ is connected and the maximum is reached at a point $(x_0, t_0)$ of $\\Omega_T$, then $u$ is constant on $\\overline{\\Omega} \\times [0, t_0]$.\nBoundedness of $\\Omega$ is needed.\nOn $\\R \\times (0, T]$, Tychonoff's example is a nonzero solution of the heat equation with zero initial values, so its maximum is not controlled by the initial values.\nWithout the side boundary, uniqueness only holds under growth conditions, such as $\\abs{u(x, t)} \\le A e^{a \\abs{x}^2}$.\n\n## Explanations\n\n### L1 human\n\nIn a body with no heat sources, the hottest moment and place must lie at the start or on the edge.\nHeat cannot pile up in the interior by itself; it can only spread out and even out.\nThis gives a simple way to compare and bound temperatures without solving the equation.\n\n### L2 human\n\nThe principle is the main tool for uniqueness and comparison results for parabolic equations.\nIt extends to operators $\\partial_t u - \\sum_{i,j} a_{ij} \\partial_i \\partial_j u - \\sum_i b_i \\partial_i u$ with $(a_{ij})$ positive definite.\nIt is the time-dependent analogue of the maximum principle for harmonic and subharmonic functions.\nIts strong form follows from a mean value property on heat balls or from a parabolic Harnack inequality.\n",
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