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Struwe to investigate the Moser-Trudinger functional $E(v)=\\int_{\\Omega} (e^{v^2}-1)dx, v\\in H^1_0(\\Omega).$ We prove that if $u$ blows-up as $t\\to\\infty$ and if $E(u(t,\\cdot))$ remains bounded, then for a sequence $t_k\\to\\infty$ we have $u(t_k,\\cdot)\\rightharpoonup 0$ in $H^1_0$ and $\\|u(t_k,\\cdot)\\|_{H^1_0}^2\\to 4\\pi L$ for an integer $L\\ge 1$.\n  We couple these results with a topologic"},"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":"1108.5576","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-08-29T14:38:53Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"69ba2aec70502fffd72cc180c8a4947994d2be5d607058611d62f50e64fc6daa","abstract_canon_sha256":"438573b2bbfb251c0624e796154d98a0851274e773fc8161456a59a5f2651a8a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:12:54.295509Z","signature_b64":"Y9ovLpwK6kNc8t7IkmiU/nHNLiOosToV/zJ/0kOAHGIHzmWXLIhrFeZ3QsocEgXW7FtemoCQLJonWEbWOTIIBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9633bbd2f5efe23b34f2414e4106958dfa180dc0924118a4290ec837852b12b8","last_reissued_at":"2026-05-18T04:12:54.294904Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:12:54.294904Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Critical points of the Moser-Trudinger functional","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.FA","authors_text":"Andrea Malchiodi, Francesca De Marchis, Luca Martinazzi","submitted_at":"2011-08-29T14:38:53Z","abstract_excerpt":"On a smooth bounded 2-dimensional domain $\\Omega$ we study the heat flow $u_t=\\Delta u +\\lambda (t)ue^{u^2}$ ($\\lambda(t)$ is such that $d/dt ||u(t,\\cdot)||_{H^1_0}=0$) introduced by T. Lamm, F. Robert and M. Struwe to investigate the Moser-Trudinger functional $E(v)=\\int_{\\Omega} (e^{v^2}-1)dx, v\\in H^1_0(\\Omega).$ We prove that if $u$ blows-up as $t\\to\\infty$ and if $E(u(t,\\cdot))$ remains bounded, then for a sequence $t_k\\to\\infty$ we have $u(t_k,\\cdot)\\rightharpoonup 0$ in $H^1_0$ and $\\|u(t_k,\\cdot)\\|_{H^1_0}^2\\to 4\\pi L$ for an integer $L\\ge 1$.\n  We couple these results with a topologic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.5576","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1108.5576","created_at":"2026-05-18T04:12:54.295006+00:00"},{"alias_kind":"arxiv_version","alias_value":"1108.5576v2","created_at":"2026-05-18T04:12:54.295006+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.5576","created_at":"2026-05-18T04:12:54.295006+00:00"},{"alias_kind":"pith_short_12","alias_value":"SYZ3XUXV57RD","created_at":"2026-05-18T12:26:41.206345+00:00"},{"alias_kind":"pith_short_16","alias_value":"SYZ3XUXV57RDWNHS","created_at":"2026-05-18T12:26:41.206345+00:00"},{"alias_kind":"pith_short_8","alias_value":"SYZ3XUXV","created_at":"2026-05-18T12:26:41.206345+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/SYZ3XUXV57RDWNHSIFHECBUVRX","json":"https://pith.science/pith/SYZ3XUXV57RDWNHSIFHECBUVRX.json","graph_json":"https://pith.science/api/pith-number/SYZ3XUXV57RDWNHSIFHECBUVRX/graph.json","events_json":"https://pith.science/api/pith-number/SYZ3XUXV57RDWNHSIFHECBUVRX/events.json","paper":"https://pith.science/paper/SYZ3XUXV"},"agent_actions":{"view_html":"https://pith.science/pith/SYZ3XUXV57RDWNHSIFHECBUVRX","download_json":"https://pith.science/pith/SYZ3XUXV57RDWNHSIFHECBUVRX.json","view_paper":"https://pith.science/paper/SYZ3XUXV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1108.5576&json=true","fetch_graph":"https://pith.science/api/pith-number/SYZ3XUXV57RDWNHSIFHECBUVRX/graph.json","fetch_events":"https://pith.science/api/pith-number/SYZ3XUXV57RDWNHSIFHECBUVRX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SYZ3XUXV57RDWNHSIFHECBUVRX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SYZ3XUXV57RDWNHSIFHECBUVRX/action/storage_attestation","attest_author":"https://pith.science/pith/SYZ3XUXV57RDWNHSIFHECBUVRX/action/author_attestation","sign_citation":"https://pith.science/pith/SYZ3XUXV57RDWNHSIFHECBUVRX/action/citation_signature","submit_replication":"https://pith.science/pith/SYZ3XUXV57RDWNHSIFHECBUVRX/action/replication_record"}},"created_at":"2026-05-18T04:12:54.295006+00:00","updated_at":"2026-05-18T04:12:54.295006+00:00"}