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(P1) consists of periodically extended delta-distributions, (P2) of periodic step potentials and (P3) contains certain periodic potentials $V,q\\in H^r_{\\per}(\\R)$ for $r\\in [1,3/2)$. Among other assumptions we suppose that $|f(x,s)|\\leq c(1+ |s|^p)$ for some $c>0$ and $p>1$. In each class we can find suitable potentials that give rise to a critical exponent $p^\\ast$ such that for $p\\in (1,p^\\ast)$ both in the \"+\" and the \"-\" case we can us"},"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":"1709.08443","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-09-25T12:13:20Z","cross_cats_sorted":[],"title_canon_sha256":"6b15fba099be9c7e1183c54e0c8ad0594213718b050f38e9b39090947b7de17e","abstract_canon_sha256":"bb31cda5320545bdbf8ae98d547244f38f17e409faf5ff6098ed24e13f22a2b2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:19:35.388102Z","signature_b64":"WlXklkGtMpWiJAqDvrE+LnUm+9PnTNJVN9EDz2SvYbk4duHt2PjyxOpoc7xNO2efZlgCdRcEXtCG4zA1kH7XAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ad95959ddbeda0989dce33827531169e63ea1e42aaa96321c2fa486b2fb9db08","last_reissued_at":"2026-05-18T00:19:35.387376Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:19:35.387376Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Real-valued, time-periodic localized weak solutions for a semilinear wave equation with periodic potentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Andreas Hirsch, Wolfgang Reichel","submitted_at":"2017-09-25T12:13:20Z","abstract_excerpt":"We consider the semilinear wave equation $V(x) u_{tt} -u_{xx}+q(x)u = \\pm f(x,u)$ for three different classes (P1), (P2), (P3) of periodic potentials $V,q$. (P1) consists of periodically extended delta-distributions, (P2) of periodic step potentials and (P3) contains certain periodic potentials $V,q\\in H^r_{\\per}(\\R)$ for $r\\in [1,3/2)$. Among other assumptions we suppose that $|f(x,s)|\\leq c(1+ |s|^p)$ for some $c>0$ and $p>1$. In each class we can find suitable potentials that give rise to a critical exponent $p^\\ast$ such that for $p\\in (1,p^\\ast)$ both in the \"+\" and the \"-\" case we can us"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.08443","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":"1709.08443","created_at":"2026-05-18T00:19:35.387485+00:00"},{"alias_kind":"arxiv_version","alias_value":"1709.08443v2","created_at":"2026-05-18T00:19:35.387485+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.08443","created_at":"2026-05-18T00:19:35.387485+00:00"},{"alias_kind":"pith_short_12","alias_value":"VWKZLHO35WQJ","created_at":"2026-05-18T12:31:49.984773+00:00"},{"alias_kind":"pith_short_16","alias_value":"VWKZLHO35WQJRHOO","created_at":"2026-05-18T12:31:49.984773+00:00"},{"alias_kind":"pith_short_8","alias_value":"VWKZLHO3","created_at":"2026-05-18T12:31:49.984773+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/VWKZLHO35WQJRHOOGOBHKMIWTZ","json":"https://pith.science/pith/VWKZLHO35WQJRHOOGOBHKMIWTZ.json","graph_json":"https://pith.science/api/pith-number/VWKZLHO35WQJRHOOGOBHKMIWTZ/graph.json","events_json":"https://pith.science/api/pith-number/VWKZLHO35WQJRHOOGOBHKMIWTZ/events.json","paper":"https://pith.science/paper/VWKZLHO3"},"agent_actions":{"view_html":"https://pith.science/pith/VWKZLHO35WQJRHOOGOBHKMIWTZ","download_json":"https://pith.science/pith/VWKZLHO35WQJRHOOGOBHKMIWTZ.json","view_paper":"https://pith.science/paper/VWKZLHO3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1709.08443&json=true","fetch_graph":"https://pith.science/api/pith-number/VWKZLHO35WQJRHOOGOBHKMIWTZ/graph.json","fetch_events":"https://pith.science/api/pith-number/VWKZLHO35WQJRHOOGOBHKMIWTZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VWKZLHO35WQJRHOOGOBHKMIWTZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VWKZLHO35WQJRHOOGOBHKMIWTZ/action/storage_attestation","attest_author":"https://pith.science/pith/VWKZLHO35WQJRHOOGOBHKMIWTZ/action/author_attestation","sign_citation":"https://pith.science/pith/VWKZLHO35WQJRHOOGOBHKMIWTZ/action/citation_signature","submit_replication":"https://pith.science/pith/VWKZLHO35WQJRHOOGOBHKMIWTZ/action/replication_record"}},"created_at":"2026-05-18T00:19:35.387485+00:00","updated_at":"2026-05-18T00:19:35.387485+00:00"}