{"paper":{"title":"The role of potential, Morawetz estimate and spacetime bound for quasilinear Schr\\\"{o}dinger equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Xianfa Song","submitted_at":"2019-04-22T02:53:03Z","abstract_excerpt":"In this paper, we deal with the following Cauchy problem \\begin{equation*} \\left\\{ \\begin{array}{lll} iu_t = \\Delta u + 2uh'(|u|^2)\\Delta h(|u|^2) + V(x)u,\\ x\\in \\mathbb{R}^N,\\ t>0\\\\ u(x,0) = u_0(x), \\quad x \\in \\mathbb{R}^N. \\end{array}\\right. \\end{equation*} Here $h(s)$ and $V(x)$ are some real functions. We take the potential $V(x)\\in L^q(\\mathbb{R}^N)+L^{\\infty}(\\mathbb{R}^N)$ as criterion of the blowup and global existence of the solution to (1.1). In some cases, we can classify it in the following sense: If $V(x)\\in S(I)$, then the solution of (1.1) is always global existence for any $u_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.10342","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"}