{"paper":{"title":"Refined regularity of the blow-up set linked to refined asymptotic behavior for the semilinear heat equation","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Hatem Zaag, Tej-Eddine Ghoul, Van Tien Nguyen","submitted_at":"2016-10-18T17:54:51Z","abstract_excerpt":"We consider $u(x,t)$, a solution of $\\partial_tu = \\Delta u + |u|^{p-1}u$ which blows up at some time $T > 0$, where $u:\\mathbb{R}^N \\times[0,T) \\to \\mathbb{R}$, $p > 1$ and $(N-2)p < N+2$. Define $S \\subset \\mathbb{R}^N$ to be the blow-up set of $u$, that is the set of all blow-up points. Under suitable nondegeneracy conditions, we show that if $S$ contains a $(N-\\ell)$-dimensional continuum for some $\\ell \\in \\{1,\\dots, N-1\\}$, then $S$ is in fact a $\\mathcal{C}^2$ manifold. The crucial step is to derive a refined asymptotic behavior of $u$ near blow-up. In order to obtain such a refined beh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.05722","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"}