{"paper":{"title":"Global rough solution for $L^2$-critical semilinear heat equation in the negative Sobolev space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Avy Soffer, Xiaohua Yao, Yifei Wu","submitted_at":"2019-03-20T02:14:04Z","abstract_excerpt":"In this paper, we consider the Cauchy global problem for the $L^2$-critical semilinear heat equations $\\partial_t h=\\Delta h\\pm |h|^{\\frac4d}h, $ with $h(0,x)=h_0$, where $h$ is an unknown real function defined on $ \\R^+\\times\\R^d$. In most of the studies on this subject, the initial data $h_0$ belongs to Lebesgue spaces $L^p(\\R^d)$ for some $p\\ge 2$ or to subcritical Sobolev space $H^{s}(\\R^d)$ with $s>0$. We here prove that there exists some positive constant $\\varepsilon_0$ depending on $d$, such that the Cauchy problem is locally and globally well-posed for any initial data $h_0$ which is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.08316","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"}