---
tag: CM7D3ES438
kind: definition
name: Keller–Segel model
msc: [35Q92, 92C17, 35K57]
defines: K04XRMH820
uses: [1CCTH8X2FT, 0E0B6RA8K0, 0ABK9V9Y64, 0MWV6ZTK6S]
sources:
  - url: https://en.wikipedia.org/wiki/Chemotaxis
---

## Definition

Let $\Omega \subseteq \R^n$ be open and let $\chi > 0$, $\tau \ge 0$, $\alpha \ge 0$ be constants.
The *Keller–Segel model* is the system for a cell density $u(x, t) \ge 0$ and a chemical concentration $v(x, t)$:
$$\partial_t u = \Delta u - \chi\, \nabla \cdot (u \nabla v), \qquad \tau\, \partial_t v = \Delta v - \alpha v + u.$$
Here $\Delta$ is the [[Laplacian|0E0B6RA8K0]], $\nabla$ the [[gradient|0ABK9V9Y64]] and $\nabla \cdot$ the [[divergence|0MWV6ZTK6S]] in $x$.
On a bounded $\Omega$ one usually imposes no-flux conditions $\partial_\nu u = \partial_\nu v = 0$ on $\partial\Omega$.
For $\tau > 0$ the system is [[parabolic|1CCTH8X2FT]]; the case $\tau = 0$ is the parabolic-elliptic Keller–Segel model.

## Remarks

Example: on a bounded $\Omega$ with $\alpha > 0$, the constants $u \equiv \alpha c$, $v \equiv c$ with $c > 0$ form a stationary solution, since all derivatives vanish and $-\alpha c + \alpha c = 0$.
Non-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 $\chi > 0$.
The total mass $\int_\Omega u\, dx$ is conserved under the no-flux conditions.
Consider the parabolic-elliptic model on $\R^2$ with $\tau = \alpha = 0$ and $\chi = 1$.
Solutions with mass less than $8\pi$ and suitable integrability of the initial data exist for all time.
Solutions with mass greater than $8\pi$ and finite second moment blow up in finite time.