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More precisely we find that for given points $q_1, q_2,\\ldots, q_k$ and any sufficiently small $T>0$ there is an initial condition $u_0$ such that the solution $u(x,t)$ of the problem blows-up at exactly those $k$ points with rates type II, namely with absolute size $ \\sim (T-t)^{-\\alpha} $ for $\\alpha > \\frac 34 $. The blow-up profile around each point is "},"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":"1808.10637","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-08-31T08:55:18Z","cross_cats_sorted":[],"title_canon_sha256":"d45ff5a55d731f262f1ae67eec0659cf4f3835f2d643171fd7641364bc5667fd","abstract_canon_sha256":"75583b6d66f45053f0d4579b374dec702d706441dc01f2a5800820ac8dfd769b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:06:36.461065Z","signature_b64":"tCoyQT2jSoUJbfkLPAZzRnOE0KzLliQJCmqo49VsuW8dAmhLwU7YlIfrFAyi1iWnB+o1LZi56Xplg/0O1a8jAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3f7719ebc0137655554850cb70e6c31046e16e2d8826a7aac2168627d25ce766","last_reissued_at":"2026-05-18T00:06:36.460631Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:06:36.460631Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Type II blow-up in the 5-dimensional energy critical heat equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Juncheng Wei, Manuel del Pino, Monica Musso","submitted_at":"2018-08-31T08:55:18Z","abstract_excerpt":"We consider the Cauchy problem for the energy critical heat equation $$ u_t = \\Delta u + |u|^{\\frac 4{n-2}}u {{\\quad\\hbox{in } }} \\ {\\mathbb R}^n \\times (0, T), \\quad u(\\cdot,0) =u_0 {{\\quad\\hbox{in } }} {\\mathbb R}^n $$ in dimension $n=5$. More precisely we find that for given points $q_1, q_2,\\ldots, q_k$ and any sufficiently small $T>0$ there is an initial condition $u_0$ such that the solution $u(x,t)$ of the problem blows-up at exactly those $k$ points with rates type II, namely with absolute size $ \\sim (T-t)^{-\\alpha} $ for $\\alpha > \\frac 34 $. 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