{"paper":{"title":"Optimal global second-order regularity and improved integrability for parabolic equations with variable growth","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Rakesh Arora, Sergey Shmarev","submitted_at":"2023-05-18T11:12:15Z","abstract_excerpt":"We consider the homogeneous Dirichlet problem for the parabolic equation\n  \\[ u_t- \\operatorname{div} \\left(|\\nabla u|^{p(x,t)-2} \\nabla u\\right)= f(x,t) + F(x,t, u, \\nabla u) \\] in the cylinder $Q_T:=\\Omega\\times (0,T)$, where $\\Omega\\subset \\mathbb{R}^N$, $N\\geq 2$, is a $C^{2}$-smooth or convex bounded domain. It is assumed that $p\\in C^{0,1}(\\overline{Q}_T)$ is a given function, and that the nonlinear source $F(x,t,s, \\xi)$ has a proper power growth with respect to $s$ and $\\xi$. It is shown that if $p(x,t)>\\frac{2(N+1)}{N+2}$, $f\\in L^2(Q_T)$, $|\\nabla u_0|^{p(x,0)}\\in L^1(\\Omega)$, then "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2305.10877","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2305.10877/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}