{"paper":{"title":"Semilinear heat equations and parabolic variational inequalities on graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Yong Lin, Yuanyuan Xie","submitted_at":"2021-08-30T06:34:12Z","abstract_excerpt":"Let $G=(V,E)$ be a locally finite connected weighted graph, and $\\Omega$ be an unbounded subset of $V$. Using Rothe's method, we study the existence of solutions for the semilinear heat equation $\\partial_tu+|u|^{p-1}\\cdot u=\\Delta u~(p\\ge1)$ and the parabolic variational inequality \\begin{eqnarray*} \\int_{\\Omega^\\circ} \\partial_tu\\cdot(v-u)\\,d\\mu\\ge \\int_{\\Omega^\\circ}(\\Delta u+f)\\cdot(v-u)\\,d\\mu \\qquad\\mbox{for any }v\\in \\mathcal{H}, \\end{eqnarray*} where $\\mathcal{H}=\\{u\\in W^{1,2}(V):u=0\\mbox{ on }V\\backslash\\Omega^\\circ\\}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2108.13007","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2108.13007/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"}