{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:ZXXYCW3PF34JIR2ZUWICSHJCKU","short_pith_number":"pith:ZXXYCW3P","schema_version":"1.0","canonical_sha256":"cdef815b6f2ef8944759a590291d225531f183720d6adc37f032298919983e23","source":{"kind":"arxiv","id":"1711.04081","version":2},"attestation_state":"computed","paper":{"title":"On the second order derivative estimates for degenerate parabolic equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ildoo Kim, Kyeong-Hun Kim","submitted_at":"2017-11-11T05:51:11Z","abstract_excerpt":"We study the parabolic equation \\begin{align} \\notag &u_t(t,x)=a^{ij}(t)u_{x^ix^j}(t,x)+f(t,x), \\quad (t,x) \\in [0,T] \\times \\mathbf{R}^d \\\\ &u(0,x)=u_0(x) \\label{main eqn} \\end{align} with the full degeneracy of the leading coefficients, that is, \\begin{align} (a^{ij}(t)) \\geq \\delta(t)I_{d\\times d} \\geq 0. \\end{align} It is well known that if $f$ and $u_0$ are not smooth enough, say $f\\in \\mathbb{L}_p(T):=L_p([0,T] ; L_p(\\mathbf{R}^d))$ and $u_0\\in L_p(\\mathbf{R}^d)$, then in general the solution is only in $C([0,T];L_p(\\mathbf{R}^d))$, and thus derivative estimates are not possible. 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