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Assume that $h(x', x_N)>0$ when $x'\\neq 0$ but $h(x',x_N)\\to 0$ as $|x'|\\to 0$. For this class of equations we obtain sharp necessary and sufficient conditions in order that singularities on the boundary do not propagate in the interior."},"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":"1201.5325","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-01-25T17:07:27Z","cross_cats_sorted":[],"title_canon_sha256":"1b475508dc828a992d75b8ed6845a1159f9c23e665a9fe019444f14a4beac73e","abstract_canon_sha256":"294ffc9c0d979c0bf63426de94dac4480f0ce14681b4449c3710d943ae128291"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:58:41.997381Z","signature_b64":"bG3kCAi+E5DdBb7JlujunvMxA06loXp5Rqb39vNxT/VxStmxEXPIl3KtryxsQycd3piY8r4sF4xO2nK/IMXeCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8c35c11da13a125cd264a5ca15ba9c9d6f0cfe0d57917a255ee2ac129eb5fab8","last_reissued_at":"2026-05-18T01:58:41.996699Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:58:41.996699Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fading absorption in non-linear elliptic equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Andrey Shishkov, Moshe Marcus","submitted_at":"2012-01-25T17:07:27Z","abstract_excerpt":"We study the equation $-\\Delta u+h(x)|u|^{q-1}u=0$, $q>1$, in $R^N_+=R^{N-1}\\ti R_+$ where $h\\in C(\\bar{R^N_+})$, $h\\geq 0$. Let $(x_1,..., x_N)$ be a coordinate system such that $R^N_+=[x_N>0]$ and denote a point $x\\in \\RN$ by $(x',x_N)$. Assume that $h(x', x_N)>0$ when $x'\\neq 0$ but $h(x',x_N)\\to 0$ as $|x'|\\to 0$. For this class of equations we obtain sharp necessary and sufficient conditions in order that singularities on the boundary do not propagate in the interior."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.5325","kind":"arxiv","version":1},"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":"1201.5325","created_at":"2026-05-18T01:58:41.996792+00:00"},{"alias_kind":"arxiv_version","alias_value":"1201.5325v1","created_at":"2026-05-18T01:58:41.996792+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.5325","created_at":"2026-05-18T01:58:41.996792+00:00"},{"alias_kind":"pith_short_12","alias_value":"RQ24CHNBHIJF","created_at":"2026-05-18T12:27:20.899486+00:00"},{"alias_kind":"pith_short_16","alias_value":"RQ24CHNBHIJFZUTE","created_at":"2026-05-18T12:27:20.899486+00:00"},{"alias_kind":"pith_short_8","alias_value":"RQ24CHNB","created_at":"2026-05-18T12:27:20.899486+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/RQ24CHNBHIJFZUTEUXFBLOU4TV","json":"https://pith.science/pith/RQ24CHNBHIJFZUTEUXFBLOU4TV.json","graph_json":"https://pith.science/api/pith-number/RQ24CHNBHIJFZUTEUXFBLOU4TV/graph.json","events_json":"https://pith.science/api/pith-number/RQ24CHNBHIJFZUTEUXFBLOU4TV/events.json","paper":"https://pith.science/paper/RQ24CHNB"},"agent_actions":{"view_html":"https://pith.science/pith/RQ24CHNBHIJFZUTEUXFBLOU4TV","download_json":"https://pith.science/pith/RQ24CHNBHIJFZUTEUXFBLOU4TV.json","view_paper":"https://pith.science/paper/RQ24CHNB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1201.5325&json=true","fetch_graph":"https://pith.science/api/pith-number/RQ24CHNBHIJFZUTEUXFBLOU4TV/graph.json","fetch_events":"https://pith.science/api/pith-number/RQ24CHNBHIJFZUTEUXFBLOU4TV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RQ24CHNBHIJFZUTEUXFBLOU4TV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RQ24CHNBHIJFZUTEUXFBLOU4TV/action/storage_attestation","attest_author":"https://pith.science/pith/RQ24CHNBHIJFZUTEUXFBLOU4TV/action/author_attestation","sign_citation":"https://pith.science/pith/RQ24CHNBHIJFZUTEUXFBLOU4TV/action/citation_signature","submit_replication":"https://pith.science/pith/RQ24CHNBHIJFZUTEUXFBLOU4TV/action/replication_record"}},"created_at":"2026-05-18T01:58:41.996792+00:00","updated_at":"2026-05-18T01:58:41.996792+00:00"}