{"paper":{"title":"A Fujita-type blowup result and low energy scattering for a nonlinear Schr\\\"o\\-din\\-ger equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Fl\\'avio Dickstein, Fred B. Weissler, Sim\\~ao Correia, Thierry Cazenave","submitted_at":"2015-05-31T21:46:18Z","abstract_excerpt":"In this paper we consider the nonlinear Schr\\\"o\\-din\\-ger equation $i u_t +\\Delta u +\\kappa |u|^\\alpha u=0$. We prove that if $\\alpha <\\frac {2} {N}$ and $\\Im \\kappa <0$, then every nontrivial $H^1$-solution blows up in finite or infinite time. In the case $\\alpha >\\frac {2} {N}$ and $\\kappa \\in {\\mathbb C}$, we improve the existing low energy scattering results in dimensions $N\\ge 7$. More precisely, we prove that if $ \\frac {8} {N + \\sqrt{ N^2 +16N }} < \\alpha \\le \\frac {4} {N} $, then small data give rise to global, scattering solutions in $H^1$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.00294","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"}