{"paper":{"title":"Blowup for the nonlinear heat equation with small initial data in scale-invariant Besov norms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Fernando Cortez, Lorenzo Brandolese (ICJ)","submitted_at":"2019-02-17T18:15:56Z","abstract_excerpt":"We consider the Cauchy problem of the nonlinear heat equation  $u_t -\\Delta u= u^{b},\\ u(0,x)=u_0$, with $b\\geq 2$ and $b\\in \\mathbb{N}$. We prove that  initial data $u_0\\in \\mathcal{S}(\\mathbb{R}^{n})$ (the Schwartz class)arbitrarily small in the scale invariant Besov-norm$\\dot B^{-2/b}_{n(b-1) b/2,q}(\\mathbb{R}^{n})$, can produce solutions that blow up in finite time. The case $b=3$ answers a question raised by Yves Meyer.Our result also proves that the smallness assumption put in an earlier work by C. Miao, B.~Yuan and B. Zhang, for the global-in-time solvability, is essentially optimal."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.06302","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"}