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  "body": "\n## Definition\n\nLet $n, m \\ge 1$, let $\\Omega \\subseteq \\R^n$ be an open set, and let $T \u003e 0$.\nLet $D = \\operatorname{diag}(d_1, \\ldots, d_m)$ with diffusion coefficients $d_i \u003e 0$, and let $R \\colon \\R^m \\to \\R^m$ be a smooth map, the reaction term.\nThe unknown is a map $u = (u_1, \\ldots, u_m) \\colon \\Omega \\times (0, T) \\to \\R^m$.\nThe *reaction–diffusion system* is the [[parabolic|1CCTH8X2FT]] system\n$$\\partial_t u = D \\Delta u + R(u) \\quad \\text{in } \\Omega \\times (0, T).$$\nHere the [[Laplacian|0E0B6RA8K0]] $\\Delta$ acts on each component.\nSo $\\partial_t u_i = d_i \\Delta u_i + R_i(u_1, \\ldots, u_m)$ for $i = 1, \\ldots, m$.\n\n## Remarks\n\nFor $R = 0$ and $m = 1$ it is the [[heat equation|VP2HN86VAG]], up to the constant $d_1$.\nExample: $m = 1$, $d_1 = 1$ 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.\nNon-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 $u$.\nSome authors allow $d_i = 0$ for some components, or a reaction term that depends on $x$ and $t$.\nThe system is usually completed by initial values and, when $\\Omega \\ne \\R^n$, by boundary conditions such as zero normal derivative.\n",
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