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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1610.05722","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2016-10-18T17:54:51Z","cross_cats_sorted":[],"title_canon_sha256":"2210a0679b9ad1abe22d0220212a8ca21e889316da7d49a630c47fafdf5fd304","abstract_canon_sha256":"404ace9da8098a2401d4bfd9984152f823e818465053bc04631498da49e6f232"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:46:59.401489Z","signature_b64":"RP0JeICrl4/w+sj2L2a5gTyoGT9J6mUAIxinM5X07B67+MAAy/+TXXwtRpHyGYGOTnAOkgOmJIgksj7ZJjwaBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a861346ea45f885ede5b564c6e3076a58d322fec5eefbb515522c144b3e1fc91","last_reissued_at":"2026-05-18T00:46:59.400924Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:46:59.400924Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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