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If $M(u)^{\\frac{1-s_{c}}{s_{c}}}E(u)<M(Q)^{\\frac{1-s_{c}}{s_{c}}}E(Q)$ and $\\|u_{0}\\|_{2}^{\\frac{1-s_{c}}{s_{c}}}\\|\\nabla u_{0}\\|_{2}>\\|Q\\|_{2}^{\\frac{1-s_{c}}{s_{c}}}\\|\\nabla Q\\|_{2},$ then either $u(t)$~blows up in finite forward time, or $u(t)$ exists globally for positive time and there exists a time sequence $t_{n}\\rightarrow+"},"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":"1101.2271","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-01-12T04:44:48Z","cross_cats_sorted":[],"title_canon_sha256":"1fea6c1d15256c6dc04a33032f37b91b675da56cfd37d4f6816e5dbb795b1302","abstract_canon_sha256":"febb57ba6a0e5969b77730465dcfe21653d9e819376e46a6c52e42a4756a0b19"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:31:20.981933Z","signature_b64":"nQqyM9CFn0onvXDSm73lL6eXXn3sGxYWimlZ/R0RGMz0yniR7bU+Oj899Nycw4HcCv6OkgjJQhYKoPgCfCKcDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"46bc33bb27f949f009a3630deb0f074a172f49bc54cdcedadd0ef532896eb420","last_reissued_at":"2026-05-18T04:31:20.981452Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:31:20.981452Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Nonscattering solutions to the $L^{2}$-supercritical NLS Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Qing Guo","submitted_at":"2011-01-12T04:44:48Z","abstract_excerpt":"We investigate the nonlinear Schr\\\"{o}dinger equation $iu_{t}+\\Delta u+|u|^{p-1}u=0$ with $1+\\frac{4}{N}<p<1+\\frac{4}{N-2}$ (when $N=1, 2$, $1+\\frac{4}{N}<p<\\infty$) in energy space $H^1$ and study the divergent property of infinite-variance and nonradial solutions. 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