{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:OFYFVIEPOLFOV4XBE4ACGIMPYP","short_pith_number":"pith:OFYFVIEP","schema_version":"1.0","canonical_sha256":"71705aa08f72caeaf2e1270023218fc3e883485866c984fed96e2f1948299c30","source":{"kind":"arxiv","id":"1712.08213","version":1},"attestation_state":"computed","paper":{"title":"The nonlinear heat equation involving highly singular initial values and new blowup and life span results","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Fred B. Weissler, Slim Tayachi","submitted_at":"2017-12-21T21:16:19Z","abstract_excerpt":"In this paper we prove local existence of solutions to the nonlinear heat equation $u_t = \\Delta u +a |u|^\\alpha u, \\; t\\in(0,T),\\; x=(x_1,\\,\\cdots,\\, x_N)\\in {\\mathbb R}^N,\\; a = \\pm 1,\\; \\alpha>0;$ with initial value $u(0)\\in L^1_{\\rm{loc}}\\left({\\mathbb R}^N\\setminus\\{0\\}\\right)$, anti-symmetric with respect to $x_1,\\; x_2,\\; \\cdots,\\; x_m$ and $|u(0)|\\leq C(-1)^m\\partial_{1}\\partial_{2}\\cdot \\cdot \\cdot \\partial_{m}(|x|^{-\\gamma})$ for $x_1>0,\\; \\cdots,\\; x_m>0,$ where $C>0$ is a constant, $m\\in \\{1,\\; 2,\\; \\cdots,\\; N\\},$ $0<\\gamma0;$ with initial value $u(0)\\in L^1_{\\rm{loc}}\\left({\\mathbb R}^N\\setminus\\{0\\}\\right)$, anti-symmetric with respect to $x_1,\\; x_2,\\; \\cdots,\\; x_m$ and $|u(0)|\\leq C(-1)^m\\partial_{1}\\partial_{2}\\cdot \\cdot \\cdot \\partial_{m}(|x|^{-\\gamma})$ for $x_1>0,\\; \\cdots,\\; x_m>0,$ where $C>0$ is a constant, $m\\in \\{1,\\; 2,\\; \\cdots,\\; N\\},$ $0<\\gamma