{"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. In this"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.04081","kind":"arxiv","version":2},"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"}