{ "id": "math/0412241", "version": "v1", "published": "2004-12-13T15:42:34.000Z", "updated": "2004-12-13T15:42:34.000Z", "title": "Uniqueness/nonuniqueness for nonnegative solutions of the Cauchy problem for $u_t=Δu-u^p$ in a punctured space", "authors": [ "Ross G. Pinsky" ], "comment": "12 pages", "categories": [ "math.AP" ], "abstract": "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.", "revisions": [ { "version": "v1", "updated": "2004-12-13T15:42:34.000Z" } ], "analyses": { "subjects": [ "35K15", "35K57" ], "keywords": [ "cauchy problem", "punctured space", "nonnegative solutions", "uniqueness/nonuniqueness", "nontrivial solution" ], "note": { "typesetting": "TeX", "pages": 12, "language": "en", "license": "arXiv", "status": "editable", "adsabs": "2004math.....12241P" } } }