{"paper":{"title":"The existence and nonexistence of global solutions for a semilinear heat equation on graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Yiting Wu, Yong Lin","submitted_at":"2017-02-12T14:26:32Z","abstract_excerpt":"Let $G=(V,E)$ be a finite or locally finite connected weighted graph, $\\Delta$ be the usual graph Laplacian. Using heat kernel estimate, we prove the existence and nonexistence of global solutions for the following semilinear heat equation on $G$ \\begin{equation*} \\left\\{ \\begin{array}{lc} u_t=\\Delta u + u^{1+\\alpha} &\\, \\text{in $(0,+\\infty)\\times V$,}\\\\ u(0,x)=a(x) &\\, \\text{in $V$.} \\end{array} \\right. \\end{equation*} We conclude that, for a graph satisfying curvature dimension condition $CDE'(n,0)$ and $V(x,r)\\simeq r^m$, if $0<m\\alpha<2$, then the non-negative solution $u$ is not global, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.03531","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":""},"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"}