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  "body": "\n## Definition\n\nLet $\\Omega$ be an open set in $\\R^n$ and let $m \\ge 1$.\nA [[partial differential equation|GC5BDZ6QBC]] of [[order|E7JDFDPM1X]] $m$ in $\\Omega$ is *semilinear* if it has the following form.\n$$\\sum_{\\abs{\\alpha} = m} a_\\alpha(x) \\, \\partial^\\alpha u + a_0\\left(D^{m-1} u, \\dots, D u, u, x\\right) = 0.$$\nHere the coefficients $a_\\alpha$ of the highest-order derivatives depend on $x$ only, and $a_0$ is a given function, which may be nonlinear.\nHere $\\alpha$ runs over [[multi-indices|0PJXBR6QA3]], and $D^j u$ denotes the collection of all partial derivatives of order $j$.\n\n## Remarks\n\nExample: $\\Delta u = u^3$ in $\\R^3$, with $\\Delta = \\partial_1^2 + \\partial_2^2 + \\partial_3^2$, is semilinear of order $2$; its highest-order part $\\Delta u$ has constant coefficients, and $a_0(u) = -u^3$.\nNon-example: $\\partial_t u + u \\, \\partial_x u = 0$ is not semilinear, because the coefficient $u$ of the highest-order derivative $\\partial_x u$ depends on the unknown; it is [[quasilinear|7X9N9H3RK6]].\nEvery [[linear PDE|6DZETZ5T9G]] is semilinear, with $a_0$ affine in the lower-order derivatives.\n\n## Explanations\n\n### L1 human\n\nSemilinear means linear in the highest derivatives, with coefficients that depend only on the position, and arbitrary in the rest.\n",
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