{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:VVZ4DNIPMGXQ676VBGRMUKVWAH","short_pith_number":"pith:VVZ4DNIP","schema_version":"1.0","canonical_sha256":"ad73c1b50f61af0f7fd509a2ca2ab601cd1f5fa80f7ecb6382b5fa47ea22b794","source":{"kind":"arxiv","id":"1104.0306","version":1},"attestation_state":"computed","paper":{"title":"A general fractional porous medium equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ana Rodr\\'iguez, Arturo de Pablo, Fernando Quir\\'os, Juan Luis V\\'azquez","submitted_at":"2011-04-02T09:16:17Z","abstract_excerpt":"We develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion, $$ \\{ll} \\dfrac{\\partial u}{\\partial t} + (-\\Delta)^{\\sigma/2} (|u|^{m-1}u)=0, & \\qquad x\\in\\mathbb{R}^N,\\; t>0,  [8pt] u(x,0) = f(x), & \\qquad x\\in\\mathbb{R}^N.%. $$ We consider data $f\\in L^1(\\mathbb{R}^N)$ and all exponents $0<\\sigma<2$ and $m>0$. Existence and uniqueness of a weak solution is established for $m> m_*=(N-\\sigma)_+ /N$, giving rise to an $L^1$-contraction semigroup. In addition, we obtain the main qualitative properties of these solutions. In the lower range $0<"},"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":"1104.0306","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-04-02T09:16:17Z","cross_cats_sorted":[],"title_canon_sha256":"d7e6ad49cb6f90ecd1951fd32e2614c732205e20705bc3e1f7f71f8921ac1cec","abstract_canon_sha256":"edce3479efdd9f3a8fd320184262dd0459d6d4b9f246cbbaecfd31685e680329"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:25:04.259966Z","signature_b64":"K2D3bDONankAyfdfZYfewx9E4xuge4ltkFLjuawLOXwBATSxcFV6RhvwoWr/QiNJxU2P2m9WElXKcKnzvikUDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ad73c1b50f61af0f7fd509a2ca2ab601cd1f5fa80f7ecb6382b5fa47ea22b794","last_reissued_at":"2026-05-18T04:25:04.259513Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:25:04.259513Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A general fractional porous medium equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ana Rodr\\'iguez, Arturo de Pablo, Fernando Quir\\'os, Juan Luis V\\'azquez","submitted_at":"2011-04-02T09:16:17Z","abstract_excerpt":"We develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion, $$ \\{ll} \\dfrac{\\partial u}{\\partial t} + (-\\Delta)^{\\sigma/2} (|u|^{m-1}u)=0, & \\qquad x\\in\\mathbb{R}^N,\\; t>0,  [8pt] u(x,0) = f(x), & \\qquad x\\in\\mathbb{R}^N.%. $$ We consider data $f\\in L^1(\\mathbb{R}^N)$ and all exponents $0<\\sigma<2$ and $m>0$. Existence and uniqueness of a weak solution is established for $m> m_*=(N-\\sigma)_+ /N$, giving rise to an $L^1$-contraction semigroup. In addition, we obtain the main qualitative properties of these solutions. In the lower range $0<"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.0306","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":"1104.0306","created_at":"2026-05-18T04:25:04.259593+00:00"},{"alias_kind":"arxiv_version","alias_value":"1104.0306v1","created_at":"2026-05-18T04:25:04.259593+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.0306","created_at":"2026-05-18T04:25:04.259593+00:00"},{"alias_kind":"pith_short_12","alias_value":"VVZ4DNIPMGXQ","created_at":"2026-05-18T12:26:44.992195+00:00"},{"alias_kind":"pith_short_16","alias_value":"VVZ4DNIPMGXQ676V","created_at":"2026-05-18T12:26:44.992195+00:00"},{"alias_kind":"pith_short_8","alias_value":"VVZ4DNIP","created_at":"2026-05-18T12:26:44.992195+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/VVZ4DNIPMGXQ676VBGRMUKVWAH","json":"https://pith.science/pith/VVZ4DNIPMGXQ676VBGRMUKVWAH.json","graph_json":"https://pith.science/api/pith-number/VVZ4DNIPMGXQ676VBGRMUKVWAH/graph.json","events_json":"https://pith.science/api/pith-number/VVZ4DNIPMGXQ676VBGRMUKVWAH/events.json","paper":"https://pith.science/paper/VVZ4DNIP"},"agent_actions":{"view_html":"https://pith.science/pith/VVZ4DNIPMGXQ676VBGRMUKVWAH","download_json":"https://pith.science/pith/VVZ4DNIPMGXQ676VBGRMUKVWAH.json","view_paper":"https://pith.science/paper/VVZ4DNIP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1104.0306&json=true","fetch_graph":"https://pith.science/api/pith-number/VVZ4DNIPMGXQ676VBGRMUKVWAH/graph.json","fetch_events":"https://pith.science/api/pith-number/VVZ4DNIPMGXQ676VBGRMUKVWAH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VVZ4DNIPMGXQ676VBGRMUKVWAH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VVZ4DNIPMGXQ676VBGRMUKVWAH/action/storage_attestation","attest_author":"https://pith.science/pith/VVZ4DNIPMGXQ676VBGRMUKVWAH/action/author_attestation","sign_citation":"https://pith.science/pith/VVZ4DNIPMGXQ676VBGRMUKVWAH/action/citation_signature","submit_replication":"https://pith.science/pith/VVZ4DNIPMGXQ676VBGRMUKVWAH/action/replication_record"}},"created_at":"2026-05-18T04:25:04.259593+00:00","updated_at":"2026-05-18T04:25:04.259593+00:00"}