{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:B4JHS7AAVQJYEZBYLIOTPIK7HO","short_pith_number":"pith:B4JHS7AA","canonical_record":{"source":{"id":"1501.05112","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-01-21T10:17:23Z","cross_cats_sorted":[],"title_canon_sha256":"438fd0633155ce4f58e24e3a55ddc8e4565af5ca2a87f07145f2f110666601e4","abstract_canon_sha256":"39eb05e215f354f0d8757a8d1adef1c4e2d16ecf4ff245b8ea49a3c1a7b0aa69"},"schema_version":"1.0"},"canonical_sha256":"0f12797c00ac138264385a1d37a15f3bb52fd641b2bab6596bedbaa3b20b52eb","source":{"kind":"arxiv","id":"1501.05112","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.05112","created_at":"2026-05-18T01:17:37Z"},{"alias_kind":"arxiv_version","alias_value":"1501.05112v1","created_at":"2026-05-18T01:17:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.05112","created_at":"2026-05-18T01:17:37Z"},{"alias_kind":"pith_short_12","alias_value":"B4JHS7AAVQJY","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_16","alias_value":"B4JHS7AAVQJYEZBY","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_8","alias_value":"B4JHS7AA","created_at":"2026-05-18T12:29:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:B4JHS7AAVQJYEZBYLIOTPIK7HO","target":"record","payload":{"canonical_record":{"source":{"id":"1501.05112","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-01-21T10:17:23Z","cross_cats_sorted":[],"title_canon_sha256":"438fd0633155ce4f58e24e3a55ddc8e4565af5ca2a87f07145f2f110666601e4","abstract_canon_sha256":"39eb05e215f354f0d8757a8d1adef1c4e2d16ecf4ff245b8ea49a3c1a7b0aa69"},"schema_version":"1.0"},"canonical_sha256":"0f12797c00ac138264385a1d37a15f3bb52fd641b2bab6596bedbaa3b20b52eb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:17:37.677445Z","signature_b64":"WaFgcLRFhcPVUOAnGNFrD/uctVyEOyV0PfuEBQCXNvuIP0Apcg+NKQqHVsH3p4Abtu7qp1G5GWKsjNiisPU0CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0f12797c00ac138264385a1d37a15f3bb52fd641b2bab6596bedbaa3b20b52eb","last_reissued_at":"2026-05-18T01:17:37.676792Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:17:37.676792Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1501.05112","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":"2026-05-18T01:17:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GOYvqa7QNB3VaFh/ryiOoz1QzvsR7aQ55ZF1fb7xhsp4AvK9q7dBD3UXXj2/c+MQhNyveKMXLZoqTJnOgIn0Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T11:50:41.179670Z"},"content_sha256":"3cacc1e31549277a51899ef9a729ca635455fb3a4dcf9c7735fc80623e268778","schema_version":"1.0","event_id":"sha256:3cacc1e31549277a51899ef9a729ca635455fb3a4dcf9c7735fc80623e268778"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:B4JHS7AAVQJYEZBYLIOTPIK7HO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Improved conditions for single-point blow-up in reaction-diffusion systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Nejib Mahmoudi, Philippe Souplet, Slim Tayachi","submitted_at":"2015-01-21T10:17:23Z","abstract_excerpt":"We study positive blowing-up solutions of the system: $$u_{t}-\\delta\\Delta u=v^p,\\,\\,\\, v_{t}-\\Delta v=u^{q},$$ as well as of some more general systems. For any $p,\\,q>1$, we prove single-point blow-up for any radially decreasing, positive and classical solution in a ball. This improves on previously known results in 3 directions:\n  (i) no type I blow-up assumption is made (and it is known that this property may fail);\n  (ii) no equidiffusivity is assumed, i.e. any $\\delta>0$ is allowed;\n  (iii) a large class of nonlinearities $F(u,v)$, $G(u,v)$ can be handled, which need not follow a precise "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.05112","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":"2026-05-18T01:17:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JAXL3Cgck5czKTvgg7RFn0/kW4V0xQdTVwNYe2Ueil9z1g24Ra3FJXuODJonH1xQogT4ULTQoI1vRFsBr1omBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T11:50:41.180031Z"},"content_sha256":"366239332f550a1a75e6139b13ec569a8b3f695f949e72d1ed10e1da64ab1516","schema_version":"1.0","event_id":"sha256:366239332f550a1a75e6139b13ec569a8b3f695f949e72d1ed10e1da64ab1516"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/B4JHS7AAVQJYEZBYLIOTPIK7HO/bundle.json","state_url":"https://pith.science/pith/B4JHS7AAVQJYEZBYLIOTPIK7HO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/B4JHS7AAVQJYEZBYLIOTPIK7HO/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":"2026-05-27T11:50:41Z","links":{"resolver":"https://pith.science/pith/B4JHS7AAVQJYEZBYLIOTPIK7HO","bundle":"https://pith.science/pith/B4JHS7AAVQJYEZBYLIOTPIK7HO/bundle.json","state":"https://pith.science/pith/B4JHS7AAVQJYEZBYLIOTPIK7HO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/B4JHS7AAVQJYEZBYLIOTPIK7HO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:B4JHS7AAVQJYEZBYLIOTPIK7HO","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":"39eb05e215f354f0d8757a8d1adef1c4e2d16ecf4ff245b8ea49a3c1a7b0aa69","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-01-21T10:17:23Z","title_canon_sha256":"438fd0633155ce4f58e24e3a55ddc8e4565af5ca2a87f07145f2f110666601e4"},"schema_version":"1.0","source":{"id":"1501.05112","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.05112","created_at":"2026-05-18T01:17:37Z"},{"alias_kind":"arxiv_version","alias_value":"1501.05112v1","created_at":"2026-05-18T01:17:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.05112","created_at":"2026-05-18T01:17:37Z"},{"alias_kind":"pith_short_12","alias_value":"B4JHS7AAVQJY","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_16","alias_value":"B4JHS7AAVQJYEZBY","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_8","alias_value":"B4JHS7AA","created_at":"2026-05-18T12:29:14Z"}],"graph_snapshots":[{"event_id":"sha256:366239332f550a1a75e6139b13ec569a8b3f695f949e72d1ed10e1da64ab1516","target":"graph","created_at":"2026-05-18T01:17:37Z","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":"We study positive blowing-up solutions of the system: $$u_{t}-\\delta\\Delta u=v^p,\\,\\,\\, v_{t}-\\Delta v=u^{q},$$ as well as of some more general systems. For any $p,\\,q>1$, we prove single-point blow-up for any radially decreasing, positive and classical solution in a ball. This improves on previously known results in 3 directions:\n  (i) no type I blow-up assumption is made (and it is known that this property may fail);\n  (ii) no equidiffusivity is assumed, i.e. any $\\delta>0$ is allowed;\n  (iii) a large class of nonlinearities $F(u,v)$, $G(u,v)$ can be handled, which need not follow a precise ","authors_text":"Nejib Mahmoudi, Philippe Souplet, Slim Tayachi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-01-21T10:17:23Z","title":"Improved conditions for single-point blow-up in reaction-diffusion systems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.05112","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":"sha256:3cacc1e31549277a51899ef9a729ca635455fb3a4dcf9c7735fc80623e268778","target":"record","created_at":"2026-05-18T01:17:37Z","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":"39eb05e215f354f0d8757a8d1adef1c4e2d16ecf4ff245b8ea49a3c1a7b0aa69","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-01-21T10:17:23Z","title_canon_sha256":"438fd0633155ce4f58e24e3a55ddc8e4565af5ca2a87f07145f2f110666601e4"},"schema_version":"1.0","source":{"id":"1501.05112","kind":"arxiv","version":1}},"canonical_sha256":"0f12797c00ac138264385a1d37a15f3bb52fd641b2bab6596bedbaa3b20b52eb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0f12797c00ac138264385a1d37a15f3bb52fd641b2bab6596bedbaa3b20b52eb","first_computed_at":"2026-05-18T01:17:37.676792Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:17:37.676792Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WaFgcLRFhcPVUOAnGNFrD/uctVyEOyV0PfuEBQCXNvuIP0Apcg+NKQqHVsH3p4Abtu7qp1G5GWKsjNiisPU0CA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:17:37.677445Z","signed_message":"canonical_sha256_bytes"},"source_id":"1501.05112","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3cacc1e31549277a51899ef9a729ca635455fb3a4dcf9c7735fc80623e268778","sha256:366239332f550a1a75e6139b13ec569a8b3f695f949e72d1ed10e1da64ab1516"],"state_sha256":"2a25cd2e6497c33cb46c70a2e8e5cdbb44bb9936b74e9c7d886d2c396d8e82ea"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vUAI9rvpn8+jHrL8U+AiGLgYU+VdHhTxrYdLx/3rorBLp3FFbIt1eid7qK71UhcE33SrpQ6KV+t/AcgVIRl0BQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T11:50:41.182350Z","bundle_sha256":"a5ddca6b01d7c246b5358fda11d34b393183c13724df49d87506a274bb972a94"}}