{"paper":{"title":"On the Global Existence and Blowup Phenomena of Schr\\\"{o}dinger Equations with Multiple Nonlinearities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Xianfa Song","submitted_at":"2010-09-02T04:26:13Z","abstract_excerpt":"In this paper, we consider the global existence and blowup phenomena of the following Cauchy problem \\begin{align*} \\left\\{\\begin{array}{ll}&-i u_t=\\Delta u-V(x)u+f(x,|u|^2)u+(W\\star|u|^2)u, \\quad x\\in\\mathbb{R}^N, \\quad t>0,\n  &u(x,0)=u_0(x), \\quad x\\in\\mathbb{R}^N, \\end{array} \\right. \\end{align*} where $V(x)$ and $W(x)$ are real-valued potentials with $V(x)\\geq 0$ and $W$ is even, $f(x,|u|^2)$ is measurable in $x$ and continuous in $|u|^2$, and $u_0(x)$ is a complex-valued function of $x$. We obtain some sufficient conditions and establish two sharp thresholds for the blowup and global exis"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.0328","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"}