{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:7VRRBVGVKD5Q7IG6HVBSP3HMDH","short_pith_number":"pith:7VRRBVGV","schema_version":"1.0","canonical_sha256":"fd6310d4d550fb0fa0de3d4327ecec19e16f7ba83495425a66ba924349d48e93","source":{"kind":"arxiv","id":"1410.3400","version":2},"attestation_state":"computed","paper":{"title":"A Landesman-Lazer type result for periodic parabolic problems on $\\mathbb{R}^N$ at resonance","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Aleksander Cwiszewski, Renata Lukasiak","submitted_at":"2014-10-13T17:28:09Z","abstract_excerpt":"We are concerned with $T$-periodic solutions of nonautonomous parabolic problem of the form $u_t = \\Delta u + V(x) u + f(t,x,u)$, $t >0$, $x \\in \\mathbb{R}^N$, with $V \\in L^\\infty (\\mathbb{R}^N)+L^p(\\mathbb{R}^N)$, $p \\geq N$ and $T$-periodic continuous perturbation $f:\\mathbb{R}^N\\times \\mathbb{R} \\to \\mathbb{R}$. The so-called resonant case is considered, i.e. when ${\\cal N}:=\\mathrm{Ker} (\\Delta + V) \\neq \\{0\\}$ and $f$ is bounded by a square-integrable function. We derive a formula for the fixed point index of the associated translation along trajectories operator in terms of the Brouwer "},"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":"1410.3400","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-10-13T17:28:09Z","cross_cats_sorted":[],"title_canon_sha256":"c398b6810ecc03e0197a1e0d3ca78b1817d100a7be0c27766ddd513e26706de6","abstract_canon_sha256":"a2343a316fa98683da819a336cd47d0d7030e695c02dc0ba8b0aeded927b30e8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:35:46.870006Z","signature_b64":"jyus+xzzHN1p4It+1FJrFzjBxUjwCVl4GTHWL9MbKA11ysrDAazKdTbWpgFzQXkkoSZNbJO0aXATefgYjBQfAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fd6310d4d550fb0fa0de3d4327ecec19e16f7ba83495425a66ba924349d48e93","last_reissued_at":"2026-05-18T02:35:46.869609Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:35:46.869609Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Landesman-Lazer type result for periodic parabolic problems on $\\mathbb{R}^N$ at resonance","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Aleksander Cwiszewski, Renata Lukasiak","submitted_at":"2014-10-13T17:28:09Z","abstract_excerpt":"We are concerned with $T$-periodic solutions of nonautonomous parabolic problem of the form $u_t = \\Delta u + V(x) u + f(t,x,u)$, $t >0$, $x \\in \\mathbb{R}^N$, with $V \\in L^\\infty (\\mathbb{R}^N)+L^p(\\mathbb{R}^N)$, $p \\geq N$ and $T$-periodic continuous perturbation $f:\\mathbb{R}^N\\times \\mathbb{R} \\to \\mathbb{R}$. The so-called resonant case is considered, i.e. when ${\\cal N}:=\\mathrm{Ker} (\\Delta + V) \\neq \\{0\\}$ and $f$ is bounded by a square-integrable function. We derive a formula for the fixed point index of the associated translation along trajectories operator in terms of the Brouwer "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.3400","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":"1410.3400","created_at":"2026-05-18T02:35:46.869674+00:00"},{"alias_kind":"arxiv_version","alias_value":"1410.3400v2","created_at":"2026-05-18T02:35:46.869674+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.3400","created_at":"2026-05-18T02:35:46.869674+00:00"},{"alias_kind":"pith_short_12","alias_value":"7VRRBVGVKD5Q","created_at":"2026-05-18T12:28:19.803747+00:00"},{"alias_kind":"pith_short_16","alias_value":"7VRRBVGVKD5Q7IG6","created_at":"2026-05-18T12:28:19.803747+00:00"},{"alias_kind":"pith_short_8","alias_value":"7VRRBVGV","created_at":"2026-05-18T12:28:19.803747+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/7VRRBVGVKD5Q7IG6HVBSP3HMDH","json":"https://pith.science/pith/7VRRBVGVKD5Q7IG6HVBSP3HMDH.json","graph_json":"https://pith.science/api/pith-number/7VRRBVGVKD5Q7IG6HVBSP3HMDH/graph.json","events_json":"https://pith.science/api/pith-number/7VRRBVGVKD5Q7IG6HVBSP3HMDH/events.json","paper":"https://pith.science/paper/7VRRBVGV"},"agent_actions":{"view_html":"https://pith.science/pith/7VRRBVGVKD5Q7IG6HVBSP3HMDH","download_json":"https://pith.science/pith/7VRRBVGVKD5Q7IG6HVBSP3HMDH.json","view_paper":"https://pith.science/paper/7VRRBVGV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1410.3400&json=true","fetch_graph":"https://pith.science/api/pith-number/7VRRBVGVKD5Q7IG6HVBSP3HMDH/graph.json","fetch_events":"https://pith.science/api/pith-number/7VRRBVGVKD5Q7IG6HVBSP3HMDH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7VRRBVGVKD5Q7IG6HVBSP3HMDH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7VRRBVGVKD5Q7IG6HVBSP3HMDH/action/storage_attestation","attest_author":"https://pith.science/pith/7VRRBVGVKD5Q7IG6HVBSP3HMDH/action/author_attestation","sign_citation":"https://pith.science/pith/7VRRBVGVKD5Q7IG6HVBSP3HMDH/action/citation_signature","submit_replication":"https://pith.science/pith/7VRRBVGVKD5Q7IG6HVBSP3HMDH/action/replication_record"}},"created_at":"2026-05-18T02:35:46.869674+00:00","updated_at":"2026-05-18T02:35:46.869674+00:00"}