{"paper":{"title":"On the Cauchy problem of fractional Schr\\\"{o}dinger equation with Hartree type nonlinearity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Gyeongha Hwang, Hichem Hajaiej, Tohru Ozawa, Yonggeun Cho","submitted_at":"2012-09-26T11:05:22Z","abstract_excerpt":"We study the Cauchy problem for the fractional Schr\\\"{o}dinger equation $$ i\\partial_tu = (m^2-\\Delta)^\\frac\\alpha2 u + F(u) in \\mathbb{R}^{1+n}, $$ where $ n \\ge 1$, $m \\ge 0$, $1 < \\alpha < 2$, and $F$ stands for the nonlinearity of Hartree type: $$F(u) = \\lambda (\\frac{\\psi(\\cdot)}{|\\cdot|^\\gamma} * |u|^2)u$$ with $\\lambda = \\pm1, 0 <\\gamma < n$, and $0 \\le \\psi \\in L^\\infty(\\mathbb R^n)$. We prove the existence and uniqueness of local and global solutions for certain $\\alpha$, $\\gamma$, $\\lambda$, $\\psi$. We also remark on finite time blowup of solutions when $\\lambda = -1$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.5899","kind":"arxiv","version":2},"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"}