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We firstly study the decaying property of global solutions for the equation when $0<\\alpha<\\alpha^\\star$ where $\\alpha^\\star = \\frac{4-2b}{d-2}$ for $d\\geq 3$. The proof makes use of an argument of Visciglia. We next use this decay to show the energy scattering for the equation in the case $\\alpha_\\star<\\alpha<\\alpha^\\star$, where $\\alpha_\\star = \\frac{4-2b}{d}$."},"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":"1710.05766","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-10-13T16:49:16Z","cross_cats_sorted":[],"title_canon_sha256":"66cd38715710f917234a37fab2ba4947c88cff4c8964007a97651694f21ae3b6","abstract_canon_sha256":"81a0b8abf217cb44766838a41e81cd4a26be1c6f1cdc622358f7b5855cb34cfb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:32:48.230310Z","signature_b64":"gCRoMfAN4Xi5D5TCMSJXNbg1t4dhtvjPiFqJ8mTCTYq+4fqsHm9RYW4eWJwbti/XVgj3rxvuh+KwFzDle3O/CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e04eb1af9237811e02e8ac4175670b041ba245e91e122e214a81e8bf03d9b12e","last_reissued_at":"2026-05-18T00:32:48.229552Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:32:48.229552Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Energy scattering for a class of the defocusing inhomogeneous nonlinear Schr\\\"odinger equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Van Duong Dinh","submitted_at":"2017-10-13T16:49:16Z","abstract_excerpt":"In this paper, we consider a class of the defocusing inhomogeneous nonlinear Schr\\\"odinger equation \\[ i\\partial_t u + \\Delta u - |x|^{-b} |u|^\\alpha u = 0, \\quad u(0)=u_0 \\in H^1, \\] with $b, \\alpha>0$. We firstly study the decaying property of global solutions for the equation when $0<\\alpha<\\alpha^\\star$ where $\\alpha^\\star = \\frac{4-2b}{d-2}$ for $d\\geq 3$. The proof makes use of an argument of Visciglia. We next use this decay to show the energy scattering for the equation in the case $\\alpha_\\star<\\alpha<\\alpha^\\star$, where $\\alpha_\\star = \\frac{4-2b}{d}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.05766","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":"1710.05766","created_at":"2026-05-18T00:32:48.229692+00:00"},{"alias_kind":"arxiv_version","alias_value":"1710.05766v1","created_at":"2026-05-18T00:32:48.229692+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.05766","created_at":"2026-05-18T00:32:48.229692+00:00"},{"alias_kind":"pith_short_12","alias_value":"4BHLDL4SG6AR","created_at":"2026-05-18T12:30:58.224056+00:00"},{"alias_kind":"pith_short_16","alias_value":"4BHLDL4SG6AR4AXI","created_at":"2026-05-18T12:30:58.224056+00:00"},{"alias_kind":"pith_short_8","alias_value":"4BHLDL4S","created_at":"2026-05-18T12:30:58.224056+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/4BHLDL4SG6AR4AXIVRAXKZYLAQ","json":"https://pith.science/pith/4BHLDL4SG6AR4AXIVRAXKZYLAQ.json","graph_json":"https://pith.science/api/pith-number/4BHLDL4SG6AR4AXIVRAXKZYLAQ/graph.json","events_json":"https://pith.science/api/pith-number/4BHLDL4SG6AR4AXIVRAXKZYLAQ/events.json","paper":"https://pith.science/paper/4BHLDL4S"},"agent_actions":{"view_html":"https://pith.science/pith/4BHLDL4SG6AR4AXIVRAXKZYLAQ","download_json":"https://pith.science/pith/4BHLDL4SG6AR4AXIVRAXKZYLAQ.json","view_paper":"https://pith.science/paper/4BHLDL4S","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1710.05766&json=true","fetch_graph":"https://pith.science/api/pith-number/4BHLDL4SG6AR4AXIVRAXKZYLAQ/graph.json","fetch_events":"https://pith.science/api/pith-number/4BHLDL4SG6AR4AXIVRAXKZYLAQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4BHLDL4SG6AR4AXIVRAXKZYLAQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4BHLDL4SG6AR4AXIVRAXKZYLAQ/action/storage_attestation","attest_author":"https://pith.science/pith/4BHLDL4SG6AR4AXIVRAXKZYLAQ/action/author_attestation","sign_citation":"https://pith.science/pith/4BHLDL4SG6AR4AXIVRAXKZYLAQ/action/citation_signature","submit_replication":"https://pith.science/pith/4BHLDL4SG6AR4AXIVRAXKZYLAQ/action/replication_record"}},"created_at":"2026-05-18T00:32:48.229692+00:00","updated_at":"2026-05-18T00:32:48.229692+00:00"}