{"paper":{"title":"Solutions with time-dependent singular sets for the heat equation with absorption","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Hikaru Yamamoto, Jin Takahashi","submitted_at":"2017-12-17T06:43:14Z","abstract_excerpt":"We consider the heat equation with a superlinear absorption term $\\partial_{t} u-\\Delta u= -u^{p}$ in $\\mathbb{R}^n$ and study the existence and nonexistence of nonnegative solutions with an $m$-dimensional time-dependent singular set, where $n-m\\geq 3$. First, we prove that if $p\\geq (n-m)/(n-m-2)$, then there is no singular solution. We next prove that, if $1<p<(n-m)/(n-m-2)$, then there are two types of singular solution. Moreover, we show the uniqueness of the solutions and specify the exact behavior of the solutions near the singular set."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.06065","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"}