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Then the positive solution to problem (P) behaves like a positive solution to ODE $\\zeta'=\\zeta^\\alpha$ in $(0,\\infty)$ and it tends to $"},"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":"1809.04252","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-09-12T04:23:43Z","cross_cats_sorted":[],"title_canon_sha256":"33400d7aa78fae5d6223d19d0b42796b094951388414ba2c6df938758fc70520","abstract_canon_sha256":"36a715dc48c8218672cbeff6dbdf2b0a4c5c9057c8bf53c00327d2f2afd35fe2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:05:53.001745Z","signature_b64":"Tf1xLMz4ehCrO6IqvCk/2hID1SGAfPXaBRilDg/sIan1OgXbifJxT13WzJxZ22ygeYr1C+dLoj+s8UNRpLfaDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"423717ef9fcc8a54838a2b357f8bb2fcb8b2fc51ade7e5b698a62fe5184534d0","last_reissued_at":"2026-05-18T00:05:53.001023Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:05:53.001023Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Large time behavior of ODE type solutions to nonlinear diffusion equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Junyong Eom, Kazuhiro Ishige","submitted_at":"2018-09-12T04:23:43Z","abstract_excerpt":"Consider the Cauchy problem for a nonlinear diffusion equation \\begin{equation} \\tag{P} \\left\\{ \\begin{array}{ll} \\partial_t u=\\Delta u^m+u^\\alpha & \\quad\\mbox{in}\\quad{\\bf R}^N\\times(0,\\infty),\\\\ u(x,0)=\\lambda+\\varphi(x)>0 & \\quad\\mbox{in}\\quad{\\bf R}^N, \\end{array} \\right. \\end{equation} where $m>0$, $\\alpha\\in(-\\infty,1)$, $\\lambda>0$ and $\\varphi\\in BC({\\bf R}^N)\\,\\cap\\, L^r({\\bf R}^N)$ with $1\\le r<\\infty$ and $\\inf_{x\\in{\\bf R}^N}\\varphi(x)>-\\lambda$. Then the positive solution to problem (P) behaves like a positive solution to ODE $\\zeta'=\\zeta^\\alpha$ in $(0,\\infty)$ and it tends to $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.04252","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":"1809.04252","created_at":"2026-05-18T00:05:53.001129+00:00"},{"alias_kind":"arxiv_version","alias_value":"1809.04252v1","created_at":"2026-05-18T00:05:53.001129+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.04252","created_at":"2026-05-18T00:05:53.001129+00:00"},{"alias_kind":"pith_short_12","alias_value":"II3RP347ZSFF","created_at":"2026-05-18T12:32:31.084164+00:00"},{"alias_kind":"pith_short_16","alias_value":"II3RP347ZSFFJA4K","created_at":"2026-05-18T12:32:31.084164+00:00"},{"alias_kind":"pith_short_8","alias_value":"II3RP347","created_at":"2026-05-18T12:32:31.084164+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/II3RP347ZSFFJA4KFM2X7C5S7S","json":"https://pith.science/pith/II3RP347ZSFFJA4KFM2X7C5S7S.json","graph_json":"https://pith.science/api/pith-number/II3RP347ZSFFJA4KFM2X7C5S7S/graph.json","events_json":"https://pith.science/api/pith-number/II3RP347ZSFFJA4KFM2X7C5S7S/events.json","paper":"https://pith.science/paper/II3RP347"},"agent_actions":{"view_html":"https://pith.science/pith/II3RP347ZSFFJA4KFM2X7C5S7S","download_json":"https://pith.science/pith/II3RP347ZSFFJA4KFM2X7C5S7S.json","view_paper":"https://pith.science/paper/II3RP347","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1809.04252&json=true","fetch_graph":"https://pith.science/api/pith-number/II3RP347ZSFFJA4KFM2X7C5S7S/graph.json","fetch_events":"https://pith.science/api/pith-number/II3RP347ZSFFJA4KFM2X7C5S7S/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/II3RP347ZSFFJA4KFM2X7C5S7S/action/timestamp_anchor","attest_storage":"https://pith.science/pith/II3RP347ZSFFJA4KFM2X7C5S7S/action/storage_attestation","attest_author":"https://pith.science/pith/II3RP347ZSFFJA4KFM2X7C5S7S/action/author_attestation","sign_citation":"https://pith.science/pith/II3RP347ZSFFJA4KFM2X7C5S7S/action/citation_signature","submit_replication":"https://pith.science/pith/II3RP347ZSFFJA4KFM2X7C5S7S/action/replication_record"}},"created_at":"2026-05-18T00:05:53.001129+00:00","updated_at":"2026-05-18T00:05:53.001129+00:00"}