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Ann.}, DOI 10.1007/s00208-015-1219-7 (2015)] has established an optimal result for $m=2$ in dimension $N\\leq 2$, and partial results in higher dimensions in the range $p< N/(N-2)$. By nontrivial modifications of the techniques of Gidas and Spruck and of Bidaut-V\\'eron, we partially improve the results of Quittner in dimensions $N\\geq 3$."},"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":"1507.07192","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-07-26T12:11:03Z","cross_cats_sorted":[],"title_canon_sha256":"933f232c941bfd978d587c410c42d251735d55c694ccef7ec0f12036668da51d","abstract_canon_sha256":"5de8c2a0d871d7b3d9104a3743143127859f95fee68b36135c5bd0ad78e8baf3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:16.964603Z","signature_b64":"tmGl5r8gQINysWj+FtxcNhBI7k/Edbc/1rF+V7M6D/Bwr8Vvi29bG1ABaZTm5Of96462DI7q5HJh/EALRCfhBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"84f80365e21238d78da366844e27c0fb0f31d3cda5aa5b45e4033289cc6643e8","last_reissued_at":"2026-05-18T01:36:16.964134Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:16.964134Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Liouville-type theorem for the $3$-dimensional parabolic Gross-Pitaevskii and related systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Philippe Souplet, Quoc Hung Phan","submitted_at":"2015-07-26T12:11:03Z","abstract_excerpt":"We prove a Liouville-type theorem for semilinear parabolic systems of the form $${\\partial_t u_i}-\\Delta u_i =\\sum_{j=1}^{m}\\beta_{ij} u_i^ru_j^{r+1}, \\quad i=1,2,...,m$$ in the whole space ${\\mathbb R}^N\\times {\\mathbb R}$. 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