{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:AIOEY7OMORNDSYTU4PNWOEU7XF","short_pith_number":"pith:AIOEY7OM","canonical_record":{"source":{"id":"1202.1037","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-02-06T02:34:10Z","cross_cats_sorted":[],"title_canon_sha256":"9ad3d9db85cf411fb28b57c6c0b0115b8160e6d12493909b81d4201be7c9c2ee","abstract_canon_sha256":"ea1734a1742ca59356d133101105d4759536373ed739dad39f0ce234ff9dba90"},"schema_version":"1.0"},"canonical_sha256":"021c4c7dcc745a396274e3db67129fb95fc523c008e65c8ed772f5d5046779b1","source":{"kind":"arxiv","id":"1202.1037","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.1037","created_at":"2026-05-18T02:49:54Z"},{"alias_kind":"arxiv_version","alias_value":"1202.1037v1","created_at":"2026-05-18T02:49:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.1037","created_at":"2026-05-18T02:49:54Z"},{"alias_kind":"pith_short_12","alias_value":"AIOEY7OMORND","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"AIOEY7OMORNDSYTU","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"AIOEY7OM","created_at":"2026-05-18T12:26:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:AIOEY7OMORNDSYTU4PNWOEU7XF","target":"record","payload":{"canonical_record":{"source":{"id":"1202.1037","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-02-06T02:34:10Z","cross_cats_sorted":[],"title_canon_sha256":"9ad3d9db85cf411fb28b57c6c0b0115b8160e6d12493909b81d4201be7c9c2ee","abstract_canon_sha256":"ea1734a1742ca59356d133101105d4759536373ed739dad39f0ce234ff9dba90"},"schema_version":"1.0"},"canonical_sha256":"021c4c7dcc745a396274e3db67129fb95fc523c008e65c8ed772f5d5046779b1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:49:54.312138Z","signature_b64":"P/mOVJyM7SY6CRuLoLdyKyzpJnAbjZd6jR0swR/wFSnaCe7RY1XVvnUbvd+4lOijlLLUG28HL7ndY6HCk0V5AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"021c4c7dcc745a396274e3db67129fb95fc523c008e65c8ed772f5d5046779b1","last_reissued_at":"2026-05-18T02:49:54.311681Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:49:54.311681Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1202.1037","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:49:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xtTfTz9njrDmkZgOU2PZVRJ5wu9RmTleAPNnsgXPwSMgn/nvUaVbHA10OJGfDUdTjk0gfsEkjl0EEZpCPp1uBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T03:24:57.030534Z"},"content_sha256":"29a129690d0614380d6c022d95c93a8b8e66dac1cd4e2f0a27735e11dda197b3","schema_version":"1.0","event_id":"sha256:29a129690d0614380d6c022d95c93a8b8e66dac1cd4e2f0a27735e11dda197b3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:AIOEY7OMORNDSYTU4PNWOEU7XF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Asymptotic expansions of the solutions of the Cauchy problem for nonlinear parabolic equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Kazuhiro Ishige, Tatsuki Kawakami","submitted_at":"2012-02-06T02:34:10Z","abstract_excerpt":"Let $u$ be a solution of the Cauchy problem for the nonlinear parabolic equation $$ \\partial_t u=\\Delta u+F(x,t,u,\\nabla u) \\quad in \\quad{\\bf R}^N\\times(0,\\infty), \\quad u(x,0)=\\varphi(x)\\quad in \\quad{\\bf R}^N, $$ and assume that the solution $u$ behaves like the Gauss kernel as $t\\to\\infty$. In this paper, under suitable assumptions of the reaction term $F$ and the initial function $\\varphi$, we establish the method of obtaining higher order asymptotic expansions of the solution $u$ as $t\\to\\infty$. This paper is a generalization of our previous paper, and our arguments are applicable to th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.1037","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:49:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"a4h5EEsf2yaY2W+MS/BuhacF2rTkzjr7HSKDdGD2KxvPJ0964zQwrzjr5vCW68HkFBYdEaaLpRBFUDxt/Cx9BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T03:24:57.030868Z"},"content_sha256":"bd1657df5ebf7e5ff10d57b0f888902f4e52e904049761561abbac577b3f8bf5","schema_version":"1.0","event_id":"sha256:bd1657df5ebf7e5ff10d57b0f888902f4e52e904049761561abbac577b3f8bf5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AIOEY7OMORNDSYTU4PNWOEU7XF/bundle.json","state_url":"https://pith.science/pith/AIOEY7OMORNDSYTU4PNWOEU7XF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AIOEY7OMORNDSYTU4PNWOEU7XF/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-27T03:24:57Z","links":{"resolver":"https://pith.science/pith/AIOEY7OMORNDSYTU4PNWOEU7XF","bundle":"https://pith.science/pith/AIOEY7OMORNDSYTU4PNWOEU7XF/bundle.json","state":"https://pith.science/pith/AIOEY7OMORNDSYTU4PNWOEU7XF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AIOEY7OMORNDSYTU4PNWOEU7XF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:AIOEY7OMORNDSYTU4PNWOEU7XF","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"ea1734a1742ca59356d133101105d4759536373ed739dad39f0ce234ff9dba90","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-02-06T02:34:10Z","title_canon_sha256":"9ad3d9db85cf411fb28b57c6c0b0115b8160e6d12493909b81d4201be7c9c2ee"},"schema_version":"1.0","source":{"id":"1202.1037","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.1037","created_at":"2026-05-18T02:49:54Z"},{"alias_kind":"arxiv_version","alias_value":"1202.1037v1","created_at":"2026-05-18T02:49:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.1037","created_at":"2026-05-18T02:49:54Z"},{"alias_kind":"pith_short_12","alias_value":"AIOEY7OMORND","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"AIOEY7OMORNDSYTU","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"AIOEY7OM","created_at":"2026-05-18T12:26:58Z"}],"graph_snapshots":[{"event_id":"sha256:bd1657df5ebf7e5ff10d57b0f888902f4e52e904049761561abbac577b3f8bf5","target":"graph","created_at":"2026-05-18T02:49:54Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $u$ be a solution of the Cauchy problem for the nonlinear parabolic equation $$ \\partial_t u=\\Delta u+F(x,t,u,\\nabla u) \\quad in \\quad{\\bf R}^N\\times(0,\\infty), \\quad u(x,0)=\\varphi(x)\\quad in \\quad{\\bf R}^N, $$ and assume that the solution $u$ behaves like the Gauss kernel as $t\\to\\infty$. In this paper, under suitable assumptions of the reaction term $F$ and the initial function $\\varphi$, we establish the method of obtaining higher order asymptotic expansions of the solution $u$ as $t\\to\\infty$. This paper is a generalization of our previous paper, and our arguments are applicable to th","authors_text":"Kazuhiro Ishige, Tatsuki Kawakami","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-02-06T02:34:10Z","title":"Asymptotic expansions of the solutions of the Cauchy problem for nonlinear parabolic equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.1037","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:29a129690d0614380d6c022d95c93a8b8e66dac1cd4e2f0a27735e11dda197b3","target":"record","created_at":"2026-05-18T02:49:54Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"ea1734a1742ca59356d133101105d4759536373ed739dad39f0ce234ff9dba90","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-02-06T02:34:10Z","title_canon_sha256":"9ad3d9db85cf411fb28b57c6c0b0115b8160e6d12493909b81d4201be7c9c2ee"},"schema_version":"1.0","source":{"id":"1202.1037","kind":"arxiv","version":1}},"canonical_sha256":"021c4c7dcc745a396274e3db67129fb95fc523c008e65c8ed772f5d5046779b1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"021c4c7dcc745a396274e3db67129fb95fc523c008e65c8ed772f5d5046779b1","first_computed_at":"2026-05-18T02:49:54.311681Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:49:54.311681Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"P/mOVJyM7SY6CRuLoLdyKyzpJnAbjZd6jR0swR/wFSnaCe7RY1XVvnUbvd+4lOijlLLUG28HL7ndY6HCk0V5AQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:49:54.312138Z","signed_message":"canonical_sha256_bytes"},"source_id":"1202.1037","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:29a129690d0614380d6c022d95c93a8b8e66dac1cd4e2f0a27735e11dda197b3","sha256:bd1657df5ebf7e5ff10d57b0f888902f4e52e904049761561abbac577b3f8bf5"],"state_sha256":"abffb2c1ae3b27ed7206f1900d0f9fb64f51202acec4606193fdf4b623a0ae0d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7eNfrDa03qxr1Cck+GdTxB16NfR9EPMlivaXFZXEmgAxmxXkdyA0rn4Ku1TeyYu0Fz6NRE7z0gggPvG2AzavBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T03:24:57.032709Z","bundle_sha256":"5a1a19e7314be0149cac1036773415a34ea26cf347ae1e672f73f74012268fd2"}}