{"paper":{"title":"Uniqueness/nonuniqueness for nonnegative solutions of the Cauchy problem for $u_t=\\Delta u-u^p$ in a punctured space","license":"","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ross G. Pinsky","submitted_at":"2004-12-13T15:42:34Z","abstract_excerpt":"Consider classical solutions to the following Cauchy problem in a punctured space: $ &u_t=\\Delta u -u^p  \\text{in} (R^n-\\{0\\})\\times(0,\\infty); & u(x,0)=g(x)\\ge0 \\text{in} R^n-\\{0\\}; &u\\ge0 \\text{in} (R^n-\\{0\\})\\times[0,\\infty). $ We prove that if $p\\ge\\frac n{n-2}$, then the solution to \\eqref{abstract} is unique for each $g$. On the other hand, if $p<\\frac n{n-2}$, then uniqueness does not hold when $g=0$; that is, there exists a nontrivial solution with vanishing initial data."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0412241","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"}