---
tag: 7E9R9X72JP
kind: definition
name: Reaction–diffusion system
msc: [35K57, 35Q92, 35K58]
defines: TAAHWE917M
uses: [1CCTH8X2FT, VP2HN86VAG, 0E0B6RA8K0]
sources:
  - url: https://en.wikipedia.org/wiki/Reaction%E2%80%93diffusion_system
---

## Definition

Let $n, m \ge 1$, let $\Omega \subseteq \R^n$ be an open set, and let $T > 0$.
Let $D = \operatorname{diag}(d_1, \ldots, d_m)$ with diffusion coefficients $d_i > 0$, and let $R \colon \R^m \to \R^m$ be a smooth map, the reaction term.
The unknown is a map $u = (u_1, \ldots, u_m) \colon \Omega \times (0, T) \to \R^m$.
The *reaction–diffusion system* is the [[parabolic|1CCTH8X2FT]] system
$$\partial_t u = D \Delta u + R(u) \quad \text{in } \Omega \times (0, T).$$
Here the [[Laplacian|0E0B6RA8K0]] $\Delta$ acts on each component.
So $\partial_t u_i = d_i \Delta u_i + R_i(u_1, \ldots, u_m)$ for $i = 1, \ldots, m$.

## Remarks

For $R = 0$ and $m = 1$ it is the [[heat equation|VP2HN86VAG]], up to the constant $d_1$.
Example: $m = 1$, $d_1 = 1$ and $R(u) = u(1 - u)$ give the Fisher–KPP equation $\partial_t u = \Delta u + u(1 - u)$; the constant $u = 1$ is a solution, since both sides are $0$ there.
Non-example: Burgers' equation $\partial_t u = \nu \partial_x^2 u - u \, \partial_x u$ is not of this form, since its nonlinear term involves the derivative $\partial_x u$ and not only the value $u$.
Some authors allow $d_i = 0$ for some components, or a reaction term that depends on $x$ and $t$.
The system is usually completed by initial values and, when $\Omega \ne \R^n$, by boundary conditions such as zero normal derivative.