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Using the strategy in \\cite{Cazenave2, Cazenave3} and taking some elementary techniques which differ from the pseudoconformal conservation law, we obtain some scatteri"},"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":"1104.2684","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-04-14T07:32:31Z","cross_cats_sorted":[],"title_canon_sha256":"f9acf3b9da6e7e9013b5542cd0bfd1f9b4b01769ac9194bcc39eab323c1c6921","abstract_canon_sha256":"18d4f057952c8ba7eb6bca68a9ce2b197d1d36dc9aa72d4b2c98fde3f453e2c2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:24:27.065588Z","signature_b64":"k6MYCFLCZ485rgKI4D0lqp6Nwhs5DkYQHZ2xolP8Vyv35Gs9xKjWXy2PZ9IYAjk3q8Lzf8XIDcW6BwmDxjuVDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5c999ea7890669fffabc314a734d055e5b44e7c6b220b3b25d3a47f1089e4af4","last_reissued_at":"2026-05-18T04:24:27.064794Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:24:27.064794Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some Results on the Scattering Theory for a Schr\\\"{o}dinger Equation with Combined Power-Type Nonlinearities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Xianfa Song","submitted_at":"2011-04-14T07:32:31Z","abstract_excerpt":"In this paper, we consider the Cauchy problem {align*} \\{{array}{ll}&i u_t+\\Delta u=\\lambda_1|u|^{p_1}u+\\lambda_2|u|^{p_2}u, \\quad t\\in\\mathbb{R}, \\quad x\\in\\mathbb{R}^N &u(0,x)=\\phi(x)\\in \\Sigma, \\quad x\\in\\mathbb{R}^N, {array}. {align*} where $N\\geq 3$, $0<p_1<p_2\\leq\\frac{4}{N-2}$, $\\lambda_1\\in\\mathbb{R}\\setminus\\{0\\}$ and $\\lambda_2\\in\\mathbb{R}$ are constants, $\\Sigma=\\{f\\in H^1(\\mathbb{R}^N); |x|f\\in L^2(\\mathbb{R}^N)\\}$. Using the strategy in \\cite{Cazenave2, Cazenave3} and taking some elementary techniques which differ from the pseudoconformal conservation law, we obtain some scatteri"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.2684","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":"1104.2684","created_at":"2026-05-18T04:24:27.064921+00:00"},{"alias_kind":"arxiv_version","alias_value":"1104.2684v1","created_at":"2026-05-18T04:24:27.064921+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.2684","created_at":"2026-05-18T04:24:27.064921+00:00"},{"alias_kind":"pith_short_12","alias_value":"LSMZ5J4JAZU7","created_at":"2026-05-18T12:26:34.985390+00:00"},{"alias_kind":"pith_short_16","alias_value":"LSMZ5J4JAZU776V4","created_at":"2026-05-18T12:26:34.985390+00:00"},{"alias_kind":"pith_short_8","alias_value":"LSMZ5J4J","created_at":"2026-05-18T12:26:34.985390+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/LSMZ5J4JAZU776V4GFFHGTIFLZ","json":"https://pith.science/pith/LSMZ5J4JAZU776V4GFFHGTIFLZ.json","graph_json":"https://pith.science/api/pith-number/LSMZ5J4JAZU776V4GFFHGTIFLZ/graph.json","events_json":"https://pith.science/api/pith-number/LSMZ5J4JAZU776V4GFFHGTIFLZ/events.json","paper":"https://pith.science/paper/LSMZ5J4J"},"agent_actions":{"view_html":"https://pith.science/pith/LSMZ5J4JAZU776V4GFFHGTIFLZ","download_json":"https://pith.science/pith/LSMZ5J4JAZU776V4GFFHGTIFLZ.json","view_paper":"https://pith.science/paper/LSMZ5J4J","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1104.2684&json=true","fetch_graph":"https://pith.science/api/pith-number/LSMZ5J4JAZU776V4GFFHGTIFLZ/graph.json","fetch_events":"https://pith.science/api/pith-number/LSMZ5J4JAZU776V4GFFHGTIFLZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LSMZ5J4JAZU776V4GFFHGTIFLZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LSMZ5J4JAZU776V4GFFHGTIFLZ/action/storage_attestation","attest_author":"https://pith.science/pith/LSMZ5J4JAZU776V4GFFHGTIFLZ/action/author_attestation","sign_citation":"https://pith.science/pith/LSMZ5J4JAZU776V4GFFHGTIFLZ/action/citation_signature","submit_replication":"https://pith.science/pith/LSMZ5J4JAZU776V4GFFHGTIFLZ/action/replication_record"}},"created_at":"2026-05-18T04:24:27.064921+00:00","updated_at":"2026-05-18T04:24:27.064921+00:00"}