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In the mass-critical case $\\alpha=\\alpha_\\star$, we prove that if $u_0$ has negative energy and satisfies either $xu_0 \\in L^2$ with $d\\geq 1$ or $u_0$ is radial with $d\\geq 2$, then the corresponding solution b"},"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":"1711.09088","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-11-22T21:51:19Z","cross_cats_sorted":[],"title_canon_sha256":"72e9657b6b7cd3a36d6d0440b31c894ca7a3c05f3362bc3a090f5d7a6f55a541","abstract_canon_sha256":"86028bc462a8fbc3334a37d9ead8b3ce230a0a299cf74850ed5a63522b4bb379"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:17:59.810648Z","signature_b64":"5wTWsJoSBCLd1XRTueRv8CeqIL8PRMH6pIsQXUvN//XMQpUQ9k6pbL5pchigklGvIvP8zjGT9LmsHdahXC55Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ec07fbefda32a8f14b42fd03fcc50f2711342456fbf24e01c4952f4a9ae2f470","last_reissued_at":"2026-05-18T00:17:59.809942Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:17:59.809942Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Blowup of $H^1$ solutions for a class of the focusing 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":"2017-11-22T21:51:19Z","abstract_excerpt":"In this paper, we consider a class of the focusing 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(\\mathbb{R}^d), \\] with $0<b<\\min\\{2,d\\}$ and $\\alpha_\\star\\leq \\alpha <\\alpha^\\star$ where $\\alpha_\\star =\\frac{4-2b}{d}$ and $\\alpha^\\star=\\frac{4-2b}{d-2}$ if $d\\geq 3$ and $\\alpha^\\star = \\infty$ if $d=1,2$. In the mass-critical case $\\alpha=\\alpha_\\star$, we prove that if $u_0$ has negative energy and satisfies either $xu_0 \\in L^2$ with $d\\geq 1$ or $u_0$ is radial with $d\\geq 2$, then the corresponding solution b"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.09088","kind":"arxiv","version":3},"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":"1711.09088","created_at":"2026-05-18T00:17:59.810045+00:00"},{"alias_kind":"arxiv_version","alias_value":"1711.09088v3","created_at":"2026-05-18T00:17:59.810045+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.09088","created_at":"2026-05-18T00:17:59.810045+00:00"},{"alias_kind":"pith_short_12","alias_value":"5QD7X362GKUP","created_at":"2026-05-18T12:31:00.734936+00:00"},{"alias_kind":"pith_short_16","alias_value":"5QD7X362GKUPCS2C","created_at":"2026-05-18T12:31:00.734936+00:00"},{"alias_kind":"pith_short_8","alias_value":"5QD7X362","created_at":"2026-05-18T12:31:00.734936+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/5QD7X362GKUPCS2C7UB7ZRIPE4","json":"https://pith.science/pith/5QD7X362GKUPCS2C7UB7ZRIPE4.json","graph_json":"https://pith.science/api/pith-number/5QD7X362GKUPCS2C7UB7ZRIPE4/graph.json","events_json":"https://pith.science/api/pith-number/5QD7X362GKUPCS2C7UB7ZRIPE4/events.json","paper":"https://pith.science/paper/5QD7X362"},"agent_actions":{"view_html":"https://pith.science/pith/5QD7X362GKUPCS2C7UB7ZRIPE4","download_json":"https://pith.science/pith/5QD7X362GKUPCS2C7UB7ZRIPE4.json","view_paper":"https://pith.science/paper/5QD7X362","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1711.09088&json=true","fetch_graph":"https://pith.science/api/pith-number/5QD7X362GKUPCS2C7UB7ZRIPE4/graph.json","fetch_events":"https://pith.science/api/pith-number/5QD7X362GKUPCS2C7UB7ZRIPE4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5QD7X362GKUPCS2C7UB7ZRIPE4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5QD7X362GKUPCS2C7UB7ZRIPE4/action/storage_attestation","attest_author":"https://pith.science/pith/5QD7X362GKUPCS2C7UB7ZRIPE4/action/author_attestation","sign_citation":"https://pith.science/pith/5QD7X362GKUPCS2C7UB7ZRIPE4/action/citation_signature","submit_replication":"https://pith.science/pith/5QD7X362GKUPCS2C7UB7ZRIPE4/action/replication_record"}},"created_at":"2026-05-18T00:17:59.810045+00:00","updated_at":"2026-05-18T00:17:59.810045+00:00"}