{"paper":{"title":"On the focusing mass-critical nonlinear fourth-order Schr\\\"odinger equation below the energy space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Van Duong Dinh","submitted_at":"2017-06-27T17:55:00Z","abstract_excerpt":"In this paper, we consider the focusing mass-critical nonlinear fourth-order Schr\\\"odinger equation. We prove that blowup solutions to this equation with initial data in $H^\\gamma(\\mathbb{R}^d), 5\\leq d \\leq 7, \\frac{56-3d+\\sqrt{137d^2+1712d+3136}}{2(2d+32)} <\\gamma<2$ concentrate at least the mass of the ground state at the blowup time. This extends the work in \\cite{ZhuYangZhang11} where Zhu-Yang-Zhang studied the formation of singularity for the equation with rough initial data in $\\mathbb{R}^4$. We also prove that the equation is globally well-posed with initial data $u_0 \\in H^\\gamma(\\mat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.09304","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"}