{"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 $. The blow-up profile around each point is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.10637","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"}