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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\R^n$ be open and let $\\chi \u003e 0$, $\\tau \\ge 0$, $\\alpha \\ge 0$ be constants.\nThe *Keller–Segel model* is the system for a cell density $u(x, t) \\ge 0$ and a chemical concentration $v(x, t)$:\n$$\\partial_t u = \\Delta u - \\chi\\, \\nabla \\cdot (u \\nabla v), \\qquad \\tau\\, \\partial_t v = \\Delta v - \\alpha v + u.$$\nHere $\\Delta$ is the [[Laplacian|0E0B6RA8K0]], $\\nabla$ the [[gradient|0ABK9V9Y64]] and $\\nabla \\cdot$ the [[divergence|0MWV6ZTK6S]] in $x$.\nOn a bounded $\\Omega$ one usually imposes no-flux conditions $\\partial_\\nu u = \\partial_\\nu v = 0$ on $\\partial\\Omega$.\nFor $\\tau \u003e 0$ the system is [[parabolic|1CCTH8X2FT]]; the case $\\tau = 0$ is the parabolic-elliptic Keller–Segel model.\n\n## Remarks\n\nExample: on a bounded $\\Omega$ with $\\alpha \u003e 0$, the constants $u \\equiv \\alpha c$, $v \\equiv c$ with $c \u003e 0$ form a stationary solution, since all derivatives vanish and $-\\alpha c + \\alpha c = 0$.\nNon-example: with $\\chi = 0$ the cells do not respond to the chemical, so the system decouples into a heat equation for $u$; it is excluded by $\\chi \u003e 0$.\nThe total mass $\\int_\\Omega u\\, dx$ is conserved under the no-flux conditions.\nConsider the parabolic-elliptic model on $\\R^2$ with $\\tau = \\alpha = 0$ and $\\chi = 1$.\nSolutions with mass less than $8\\pi$ and suitable integrability of the initial data exist for all time.\nSolutions with mass greater than $8\\pi$ and finite second moment blow up in finite time.\n",
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