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  "body": "\n## Definition\n\nLet $(X, d)$ be a [[metric space|0VKD1TP7F3]] and let $(S(t))_{t \\ge 0}$ be a semigroup of continuous maps $S(t) \\colon X \\to X$, with $S(0) = \\id$ and $S(t + s) = S(t) S(s)$.\nFor subsets $A, B \\subseteq X$, let $\\operatorname{dist}(B, A) = \\sup_{b \\in B} \\inf_{a \\in A} d(a, b)$.\nA *global attractor* of the semigroup is a set $\\mathcal{A} \\subseteq X$ such that:\n\n1. $\\mathcal{A}$ is [[compact|00R0X1WGPX]];\n2. $\\mathcal{A}$ is [[invariant|WWZTW11664]]: $S(t) \\mathcal{A} = \\mathcal{A}$ for every $t \\ge 0$;\n3. $\\mathcal{A}$ attracts bounded sets: $\\operatorname{dist}(S(t) B, \\mathcal{A}) \\to 0$ as $t \\to \\infty$ for every bounded set $B \\subseteq X$.\n\n## Remarks\n\nA global attractor, when it exists, is unique: it is the largest compact invariant set and the smallest closed set that attracts bounded sets.\n\nExample: let $X = \\R$ and let $S(t)$ be the flow of $u' = u - u^3$.\nEvery solution with $u(0) \u003e 0$ tends to $1$, every solution with $u(0) \u003c 0$ tends to $-1$, and $0$ is fixed.\nThe set $\\mathcal{A} = [-1, 1]$ is compact, invariant since the flow maps $[-1, 1]$ onto itself, and attracts every bounded interval.\n\nNon-example: for the same flow, the set $\\set{-1, 1}$ is compact and invariant, but it does not attract the bounded set $\\set{0}$, since $S(t) 0 = 0$ for all $t$.\nSo clause 3 fails.\n\nA standard sufficient condition is that the semigroup has a bounded absorbing set and is asymptotically compact.\n",
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