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  "body": "\n## Statement\n\nLet $U \\subseteq \\R^n$ be open, let $T \u003e 0$, and let $0 \u003c \\lambda \\le \\Lambda$.\nLet $A = (a_{ij})$ be a measurable function on $U \\times (0, T]$ with values in the symmetric $n \\times n$ matrices such that $\\lambda \\abs{\\eta}^2 \\le \\sum_{i,j} a_{ij}(x, t)\\, \\eta_i \\eta_j \\le \\Lambda \\abs{\\eta}^2$ for all $\\eta \\in \\R^n$ and all $(x, t)$.\nLet $u \\ge 0$ be a [[weak solution|SJZX81A783]] of the [[parabolic equation|1CCTH8X2FT]]\n$$\\partial_t u = \\sum_{i,j=1}^{n} \\partial_{x_i}\\big(a_{ij}\\, \\partial_{x_j} u\\big) \\quad \\text{in } U \\times (0, T].$$\nLet $V$ be a connected open set with compact closure $\\overline{V} \\subset U$, and let $0 \u003c t_1 \u003c t_2 \\le T$.\nThen there is a constant $C$, depending only on $n$, $\\lambda$, $\\Lambda$, $U$, $V$, $t_1$ and $t_2$, such that\n$$\\sup_{x \\in V} u(x, t_1) \\le C \\inf_{x \\in V} u(x, t_2).$$\n\n## Remarks\n\nThis is Moser's parabolic Harnack inequality of 1964; the solution is Hölder continuous by the De Giorgi–Nash–Moser theory, so the pointwise values make sense.\nMoser's proof bounds $\\sup u$ and $\\inf u$ on small cylinders by integral means of powers of $u$, using iterated energy estimates.\nAn estimate for $\\log u$ then links the two bounds.\nA chain of overlapping cylinders then gives the statement on $V$, which is why $V$ must be connected.\nThe time lag $t_1 \u003c t_2$ is necessary.\nFor the [[heat equation|VP2HN86VAG]] $u_t = \\Delta u$ on $\\R^n$, the solutions $u_y(x, t) = (4\\pi t)^{-n/2} e^{-\\abs{x - y}^2/(4t)}$ with $y \\in \\R^n$ satisfy $u_y(x_1, t)/u_y(x_2, t) = e^{(\\abs{x_2 - y}^2 - \\abs{x_1 - y}^2)/(4t)}$, which is unbounded over $y$ for fixed distinct $x_1, x_2$.\nSo no bound $\\sup_V u(\\cdot, t) \\le C \\inf_V u(\\cdot, t)$ at a single time can hold for all nonnegative solutions.\nThe hypothesis $u \\ge 0$ is also necessary, as for [[Harnack's inequality|YNH832D50W]] for harmonic functions.\n\n## Explanations\n\n### L1 human\n\nA nonnegative temperature distribution cannot be very hot at one place now and almost zero at a nearby place a little later.\nHeat that is present now spreads, so it raises the temperature nearby at later times.\nThe inequality makes this precise: the largest value at an earlier time is controlled by the smallest value at a later time.\nLooking at the same time is not enough, because a very concentrated source far away can make two nearby points have very different temperatures.\n\n### L2 human\n\nThe parabolic Harnack inequality is the time-dependent analogue of [[Harnack's inequality|YNH832D50W]] for nonnegative harmonic functions.\nIt holds for equations with merely measurable coefficients, which makes it a central tool in the regularity theory of nonlinear parabolic equations.\nFrom it one derives Hölder continuity of solutions and a strong maximum principle.\nFor the heat equation on Riemannian manifolds with nonnegative Ricci curvature, Li and Yau proved a sharp differential form, which integrates to a Harnack inequality with explicit constants.\n",
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