{"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 ${\\mathbb R}^N\\times {\\mathbb R}$. Very recently, Quittner [{\\em Math. 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 $N\\geq 3$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.07192","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"}