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  "body": "\n## Statement\n\nLet $k \\ge 1$ and $N \\ge 1$ be integers, and write the variables in $\\R \\times \\R^n$ as $(t, x)$.\nConsider a system of $N$ [[partial differential equations|GC5BDZ6QBC]] for an unknown $u = (u_1, \\dots, u_N)$ of the following form.\n$$\\partial_t^k u = F\\left(t, x, \\left(\\partial_t^j \\partial_x^\\alpha u\\right)_{j \u003c k, \\ j + \\abs{\\alpha} \\le k}\\right).$$\nHere $\\alpha$ runs over [[multi-indices|0PJXBR6QA3]] in $\\N^n$.\nLet $x_0 \\in \\R^n$, and let $\\varphi_0, \\dots, \\varphi_{k-1}$ be $\\R^N$-valued [[real-analytic functions|081WKEDK6C]] on a neighbourhood of $x_0$.\nAssume that $F$ is real-analytic on a neighbourhood of the point $\\left(0, x_0, (\\partial_x^\\alpha \\varphi_j(x_0))_{j \u003c k, \\ j + \\abs{\\alpha} \\le k}\\right)$.\nThen there is a neighbourhood $W$ of $(0, x_0)$ in $\\R \\times \\R^n$ and a real-analytic function $u$ on $W$ that solves the system in $W$.\nMoreover $u$ satisfies the initial conditions $\\partial_t^j u(0, x) = \\varphi_j(x)$ for $j = 0, \\dots, k - 1$ and all $x$ with $(0, x) \\in W$.\nAny two real-analytic solutions with these properties agree on a neighbourhood of $(0, x_0)$.\n\n## Remarks\n\nThe equation and the initial conditions determine all derivatives of $u$ at $(0, x_0)$, so the Taylor series of a real-analytic solution is fixed; this gives uniqueness.\nFor existence, the proof shows that this Taylor series converges by the method of majorants: one compares with an explicit system whose analytic solution dominates the coefficients.\nThe form of the equation matters: the equation must be solved for $\\partial_t^k u$, and the right side may contain neither $\\partial_t^k u$ nor any derivative of total order greater than $k$.\nWithout this, the conclusion fails: for the [[heat equation|VP2HN86VAG]] $\\partial_t u = \\partial_x^2 u$ with $u(0, x) = 1/(1 - x)$, the formal power series solution is $\\sum_{j \\ge 0} \\frac{(2j)!}{j!} \\frac{t^j}{(1 - x)^{2j + 1}}$, which diverges for every $t \\ne 0$.\nAnalyticity of the data is also needed; for smooth data, existence can fail even for a linear equation, as Lewy's example of an operator without [[local solvability|P6E9RZNRBZ]] shows.\nUniqueness among all smooth solutions, not only analytic ones, holds for linear equations by [[Holmgren's uniqueness theorem|1VVKF0Y19B]].\n\n## Explanations\n\n### L1 human\n\nSuppose that an equation and its starting values are given by convergent power series.\nThe theorem says that then there is a solution near the starting surface, again given by a convergent power series.\nIt is the basic existence theorem for partial differential equations.\nIt needs strong hypotheses: everything must be analytic, and the equation must be solved for the highest time derivative.\n\n### L2 human\n\nThe theorem gives local existence in the analytic class for any noncharacteristic Cauchy problem, after a change of variables that makes the initial surface $t = 0$.\nIt does not say that the problem is well posed: [[Hadamard's example|SZ6MEVM89M]] shows that for the Laplace equation the analytic solution exists but depends discontinuously on the data.\nSo for evolution equations one looks for theories in Sobolev or $C^\\infty$ spaces, such as the energy method for hyperbolic equations.\nFor ordinary differential equations, with $n = 0$, the theorem reduces to the existence of analytic solutions to analytic initial value problems.\n",
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