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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, "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1702.03531","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-02-12T14:26:32Z","cross_cats_sorted":[],"title_canon_sha256":"3097bd7a6276249412bc35329559d39aaf776aaa5392589be1aba504de008ea7","abstract_canon_sha256":"5dfe4abf40bd60a7d359fa56210655832b8928389625b283365926b629e5af96"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:50:53.566771Z","signature_b64":"bcqUZ2hse8IVhqb+eOHsj1aIYheTo3O3eDyVBMsmluiRqgH1/DmtoX+e//v/Cw/Tn1GXThaSHggKsP1qxXQLCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"16ab1aee62c072949189db5739e8ac6d72bac527c8498ebb850aa566c01886e9","last_reissued_at":"2026-05-18T00:50:53.566270Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:50:53.566270Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1702.03531","created_at":"2026-05-18T00:50:53.566354+00:00"},{"alias_kind":"arxiv_version","alias_value":"1702.03531v1","created_at":"2026-05-18T00:50:53.566354+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.03531","created_at":"2026-05-18T00:50:53.566354+00:00"},{"alias_kind":"pith_short_12","alias_value":"C2VRV3TCYBZJ","created_at":"2026-05-18T12:31:08.081275+00:00"},{"alias_kind":"pith_short_16","alias_value":"C2VRV3TCYBZJJEMJ","created_at":"2026-05-18T12:31:08.081275+00:00"},{"alias_kind":"pith_short_8","alias_value":"C2VRV3TC","created_at":"2026-05-18T12:31:08.081275+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/C2VRV3TCYBZJJEMJ3NLTT2FMNV","json":"https://pith.science/pith/C2VRV3TCYBZJJEMJ3NLTT2FMNV.json","graph_json":"https://pith.science/api/pith-number/C2VRV3TCYBZJJEMJ3NLTT2FMNV/graph.json","events_json":"https://pith.science/api/pith-number/C2VRV3TCYBZJJEMJ3NLTT2FMNV/events.json","paper":"https://pith.science/paper/C2VRV3TC"},"agent_actions":{"view_html":"https://pith.science/pith/C2VRV3TCYBZJJEMJ3NLTT2FMNV","download_json":"https://pith.science/pith/C2VRV3TCYBZJJEMJ3NLTT2FMNV.json","view_paper":"https://pith.science/paper/C2VRV3TC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1702.03531&json=true","fetch_graph":"https://pith.science/api/pith-number/C2VRV3TCYBZJJEMJ3NLTT2FMNV/graph.json","fetch_events":"https://pith.science/api/pith-number/C2VRV3TCYBZJJEMJ3NLTT2FMNV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/C2VRV3TCYBZJJEMJ3NLTT2FMNV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/C2VRV3TCYBZJJEMJ3NLTT2FMNV/action/storage_attestation","attest_author":"https://pith.science/pith/C2VRV3TCYBZJJEMJ3NLTT2FMNV/action/author_attestation","sign_citation":"https://pith.science/pith/C2VRV3TCYBZJJEMJ3NLTT2FMNV/action/citation_signature","submit_replication":"https://pith.science/pith/C2VRV3TCYBZJJEMJ3NLTT2FMNV/action/replication_record"}},"created_at":"2026-05-18T00:50:53.566354+00:00","updated_at":"2026-05-18T00:50:53.566354+00:00"}