{"paper":{"title":"Remarks on the Cauchy problem for the one-dimensional quadratic (fractional) heat equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Luc Molinet (LMPT), Slim Tayachi","submitted_at":"2013-04-03T09:13:39Z","abstract_excerpt":"We prove that the Cauchy problem associated with the one dimensional quadratic (fractional) heat equation: $u_t=D_x^{2\\alpha} u \\mp u^2,\\; t\\in (0,T),\\; x\\in \\R$ or $ \\T $, with $ 0<\\alpha\\le 1 $ is well-posed in $ H^s $ for $ s\\ge \\max(-\\alpha,1/2-2\\alpha) $ except in the case $ \\alpha=1/2 $ where it is shown to be well-posed for $ s>-1/2 $ and ill-posed for $ s=-1/2 $. As a by-product we improve the known well-posedness results for the heat equation ($\\alpha=1$) by reaching the end-point Sobolev index $ s=-1 $. Finally, in the case $ 1/2<\\alpha\\le 1 $, we also prove optimal results in the Be"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.0880","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"}