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We first show that self-similar solutions of this equation are infinitely differentiable and prove sharp results on the behavior of self-similar profiles at $y = 0$ in the case $\\alpha < 0$. We also give some partial uniqueness results for self-similar profiles: in the case $\\alpha = 0$ we prove that two profiles with the same mass and moment of order $\\lambda$ are necessarily equal, while in the "},"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":"0803.1462","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2008-03-10T17:45:51Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"2b38f0f2772e072d075f31f2ca1d7ab325e9b9dca7315f863d1d7625589fd823","abstract_canon_sha256":"e650a0534e3b566ce42e5f0d7be8d7243da081f4c67118b03659feefd3640824"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:06:56.980212Z","signature_b64":"zJzfCUXyvjEXnAwnua6+K5fhu9iAfHOgKADA0jilTdL/RiKS2XV8hx8BF+YE6rwahUgfn2thveTt4byXJTpsBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4f8f55b1023f3d4884b394a74dea835920d1d2b77e618f5f787faae26ddcf96d","last_reissued_at":"2026-05-18T04:06:56.979675Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:06:56.979675Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Regularity, asymptotic behavior and partial uniqueness for Smoluchowski's coagulation equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Jos\\'e Alfredo Ca\\~nizo, St\\'ephane Mischler","submitted_at":"2008-03-10T17:45:51Z","abstract_excerpt":"We consider Smoluchowski's equation with a homogeneous kernel of the form $a(x,y) = x^\\alpha y ^\\beta + x^\\beta y^\\alpha$ with $-1 < \\alpha \\leq \\beta < 1$ and $\\lambda := \\alpha + \\beta \\in (-1,1)$. 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We also give some partial uniqueness results for self-similar profiles: in the case $\\alpha = 0$ we prove that two profiles with the same mass and moment of order $\\lambda$ are necessarily equal, while in the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0803.1462","kind":"arxiv","version":2},"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":"0803.1462","created_at":"2026-05-18T04:06:56.979756+00:00"},{"alias_kind":"arxiv_version","alias_value":"0803.1462v2","created_at":"2026-05-18T04:06:56.979756+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0803.1462","created_at":"2026-05-18T04:06:56.979756+00:00"},{"alias_kind":"pith_short_12","alias_value":"J6HVLMICH46U","created_at":"2026-05-18T12:25:57.157939+00:00"},{"alias_kind":"pith_short_16","alias_value":"J6HVLMICH46URBFT","created_at":"2026-05-18T12:25:57.157939+00:00"},{"alias_kind":"pith_short_8","alias_value":"J6HVLMIC","created_at":"2026-05-18T12:25:57.157939+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/J6HVLMICH46URBFTSSTU32UDLE","json":"https://pith.science/pith/J6HVLMICH46URBFTSSTU32UDLE.json","graph_json":"https://pith.science/api/pith-number/J6HVLMICH46URBFTSSTU32UDLE/graph.json","events_json":"https://pith.science/api/pith-number/J6HVLMICH46URBFTSSTU32UDLE/events.json","paper":"https://pith.science/paper/J6HVLMIC"},"agent_actions":{"view_html":"https://pith.science/pith/J6HVLMICH46URBFTSSTU32UDLE","download_json":"https://pith.science/pith/J6HVLMICH46URBFTSSTU32UDLE.json","view_paper":"https://pith.science/paper/J6HVLMIC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0803.1462&json=true","fetch_graph":"https://pith.science/api/pith-number/J6HVLMICH46URBFTSSTU32UDLE/graph.json","fetch_events":"https://pith.science/api/pith-number/J6HVLMICH46URBFTSSTU32UDLE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/J6HVLMICH46URBFTSSTU32UDLE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/J6HVLMICH46URBFTSSTU32UDLE/action/storage_attestation","attest_author":"https://pith.science/pith/J6HVLMICH46URBFTSSTU32UDLE/action/author_attestation","sign_citation":"https://pith.science/pith/J6HVLMICH46URBFTSSTU32UDLE/action/citation_signature","submit_replication":"https://pith.science/pith/J6HVLMICH46URBFTSSTU32UDLE/action/replication_record"}},"created_at":"2026-05-18T04:06:56.979756+00:00","updated_at":"2026-05-18T04:06:56.979756+00:00"}