Math 2.0

Partial differential equations Definition

Reaction–diffusion system

Builds on Parabolic partial differential equationHeat equationLaplacian

Definition

Let n,m≥1n, m \ge 1, let Ω⊆ℝn\Omega \subseteq \R^n be an open set, and let T>0T > 0. Let D=diag⁡(d1,…,dm)D = \operatorname{diag}(d_1, \ldots, d_m) with diffusion coefficients di>0d_i > 0, and let R ⁣:ℝm→ℝmR \colon \R^m \to \R^m be a smooth map, the reaction term. The unknown is a map u=(u1,…,um) ⁣:Ω×(0,T)→ℝmu = (u_1, \ldots, u_m) \colon \Omega \times (0, T) \to \R^m. The reaction–diffusion system is the parabolic system ∂tu=DΔu+R(u)in Ω×(0,T).\partial_t u = D \Delta u + R(u) \quad \text{in } \Omega \times (0, T). Here the Laplacian Δ\Delta acts on each component. So ∂tui=diΔui+Ri(u1,…,um)\partial_t u_i = d_i \Delta u_i + R_i(u_1, \ldots, u_m) for i=1,…,mi = 1, \ldots, m.

Remarks

For R=0R = 0 and m=1m = 1 it is the heat equation, up to the constant d1d_1. Example: m=1m = 1, d1=1d_1 = 1 and R(u)=u(1−u)R(u) = u(1 - u) give the Fisher–KPP equation ∂tu=Δu+u(1−u)\partial_t u = \Delta u + u(1 - u); the constant u=1u = 1 is a solution, since both sides are 00 there. Non-example: Burgers' equation ∂tu=ν∂x2u−u ∂xu\partial_t u = \nu \partial_x^2 u - u \, \partial_x u is not of this form, since its nonlinear term involves the derivative ∂xu\partial_x u and not only the value uu. Some authors allow di=0d_i = 0 for some components, or a reaction term that depends on xx and tt. The system is usually completed by initial values and, when Ω≠ℝn\Omega \ne \R^n, by boundary conditions such as zero normal derivative.

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  1. Euclidean space object
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  8. Differentiable function of several variables concept
  9. Partial derivative concept
  10. Vector field concept
  11. Gradient concept
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  13. Laplacian concept
  14. Reaction–diffusion system definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Reaction%E2%80%93diffusion_system
About this page
Tag7E9R9X72JP
URIm2://7E9R9X72JP
Kinddefinition
WritingW0 stub
MSC35K57, 35Q92, 35K58
DefinesReaction–diffusion system
UsesParabolic partial differential equation, Heat equation, Laplacian
Changed2 November 2026

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