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Given $f \\in L^2(0,T, V')$, one knows that for each $u_0 \\in H$ there is a unique solution $u\\in H^1(0,T;V')\\cap L^2(0,T;V)$ of $$\\dot u(t) + \\A(t) u(t) = f(t), \\, \\, u(0) = u_0.$$ %\\begin{align*} %&\\dot u(t) + \\A(t)u(t)= f(t)\\ %& u(0)=u_0. %\\end{align*} This result by J. L. Lions is well-known. The aim of this article is to find a criterion for the invariance of a clos"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1303.1167","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-03-05T20:34:25Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"c2808f8e21a7f15722aa659a0edb55060768d47e14adec318a2046c59f9f2b6e","abstract_canon_sha256":"124e9984dcd9f54b9aea6e7e204a9103041bb5284eb122d374b183ac5077010d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:31:45.470228Z","signature_b64":"5tWcvVl5/u9WE4gCP3itQYvQ9MC0XSQFjMUl0NLX+kvgmNx5a0GjKc1zU6AULQDJFd5k0qu2Y8NcZEDlrhGtBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"624cc7d112d2a852e41232486edabb4af56d3eff7ce4aacfe2d31e26707e2808","last_reissued_at":"2026-05-18T03:31:45.469290Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:31:45.469290Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Invariance of Convex Sets for Non-autonomous Evolution Equations Governed by Forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Dominik Dier, El Maati Ouhabaz (IMB), Wolfgang Arendt","submitted_at":"2013-03-05T20:34:25Z","abstract_excerpt":"We consider a non-autonomous form $\\fra:[0,T]\\times V\\times V \\to \\C$ where $V$ is a Hilbert space which is densely and continuously embedded in another Hilbert space $H$. Denote by $\\A(t) \\in \\L(V,V')$ the associated operator. Given $f \\in L^2(0,T, V')$, one knows that for each $u_0 \\in H$ there is a unique solution $u\\in H^1(0,T;V')\\cap L^2(0,T;V)$ of $$\\dot u(t) + \\A(t) u(t) = f(t), \\, \\, u(0) = u_0.$$ %\\begin{align*} %&\\dot u(t) + \\A(t)u(t)= f(t)\\ %& u(0)=u_0. %\\end{align*} This result by J. L. Lions is well-known. 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