{
  "aliases": [
    "maximal-interval-of-existence"
  ],
  "backlinks": [
    {
      "field": "uses",
      "target": "02903SSA48"
    },
    {
      "field": "uses",
      "target": "0BFD3A6CRH"
    },
    {
      "field": "uses",
      "target": "0E4GXX7SJP"
    }
  ],
  "body": "\n## Definition\n\nLet $D$ be an open subset of $\\mathbb{R} \\times \\mathbb{R}^n$ and let $f : D \\to \\mathbb{R}^n$ be continuous.\nLet $I$ be an open interval and let $\\gamma : I \\to \\mathbb{R}^n$ be a [[solution|04SW7WECS8]] of $x' = f(t, x)$ whose graph lies in $D$.\nThe solution $\\gamma$ is maximal if there is no solution $\\tilde{\\gamma} : \\tilde{I} \\to \\mathbb{R}^n$ on an open interval $\\tilde{I}$ with $I \\subsetneq \\tilde{I}$ and $\\tilde{\\gamma}(t) = \\gamma(t)$ for each $t \\in I$.\nThe interval $I$ of a maximal solution is its maximal interval of existence.\n\n## Remarks\n\nIf $f$ is locally Lipschitz in $x$, each [[initial value problem|075VBY4K4X]] $x(t_0) = x_0$ with $(t_0, x_0) \\in D$ has exactly one maximal solution. Each other solution of the problem is a restriction of it.\nThe maximal interval $(t_-, t_+)$ can depend on the initial value, and it can be smaller than $\\mathbb{R}$.\nIf $t_+ \u003c \\infty$, then $(t, \\gamma(t))$ leaves every compact subset of $D$ as $t \\to t_+$. For $D = \\mathbb{R} \\times \\mathbb{R}^n$, this means $\\|\\gamma(t)\\| \\to \\infty$. See [[blow-up in finite time|0E4GXX7SJP]].\n\n## Explanations\n\n### L1 human\n\nA maximal solution is a solution that we cannot continue any further.\nIts interval ends only where the solution runs away to infinity or to the edge of the domain of the field.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "aliases": [
      "maximal-interval-of-existence"
    ],
    "defines": "01J7FZ65J0",
    "kind": "definition",
    "msc": [
      "34A12"
    ],
    "name": "Maximal solution",
    "sources": [
      {
        "locator": "Global uniqueness and maximum domain of solution",
        "url": "https://en.wikipedia.org/wiki/Ordinary_differential_equation"
      }
    ],
    "tag": "019VSXSFMQ",
    "uses": [
      "04SW7WECS8",
      "075VBY4K4X",
      "025PKW8AMR"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/019VSXSFMQ/",
  "links": [
    {
      "field": "defines",
      "target": "01J7FZ65J0"
    },
    {
      "field": "uses",
      "target": "04SW7WECS8"
    },
    {
      "field": "uses",
      "target": "075VBY4K4X"
    },
    {
      "field": "uses",
      "target": "025PKW8AMR"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/019VSXSFMQ.md",
  "msc": [
    "34A12"
  ],
  "name": "Maximal solution",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/019/019VSXSFMQ.md",
  "status": null,
  "tag": "019VSXSFMQ",
  "uri": "m2://019VSXSFMQ",
  "url": "https://math2.tamnd.com/wiki/maximal-solution-definition/",
  "writing": {
    "tag": "019VSXSFMQ",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}