{"paper":{"title":"Positive probability of explosion for stochastic heat equation with superlinear accretive reaction term and polynomially growing multiplicative noise","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Mild solutions to the stochastic heat equation explode with positive probability for beta in (1,3) with gamma in (beta/2, (beta+3)/4) or beta greater than 1 with gamma up to beta/2.","cross_cats":[],"primary_cat":"math.PR","authors_text":"Michael Salins, Yuyang Zhang","submitted_at":"2026-05-11T23:11:52Z","abstract_excerpt":"This paper studies the finite time explosion of the stochastic heat equation $\\frac{\\partial u}{\\partial t}(t,x)=\\frac{\\partial^2}{\\partial x^2} u(t,x)+(u(t,x))^{\\beta}+\\sigma(u(t,x))\\dot{W}(t,x)$. We consider an interval $D=[-\\pi,\\pi]$ under periodic boundary condition where $\\dot{W}(t,x)$ is a space-time white noise and $\\sigma(u)\\approx u^{\\gamma}$ near $\\infty$. Our results refine existing results by identifying behavior in a previously less understood regime, where we show that if $\\beta\\in(1,3),\\gamma\\in(\\frac{\\beta}{2},\\frac{\\beta+3}{4})$ or $\\beta>1,\\gamma\\in(0,\\frac{\\beta}{2}]$ then m"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"if β∈(1,3),γ∈(β/2,(β+3)/4) or β>1,γ∈(0,β/2] then mild solutions can explode with positive probability.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The assumption that σ(u) ≈ u^γ near infinity together with the existence of mild solutions up to the potential explosion time on the periodic interval.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Mild solutions explode with positive probability when β ∈ (1,3) and γ ∈ (β/2, (β+3)/4), or when β > 1 and γ ∈ (0, β/2].","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Mild solutions to the stochastic heat equation explode with positive probability for beta in (1,3) with gamma in (beta/2, (beta+3)/4) or beta greater than 1 with gamma up to beta/2.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"3faafaa6f96c01beaa393dffeb74575ad0d24bfc6dc119a76a0d3ff169f024b6"},"source":{"id":"2605.11319","kind":"arxiv","version":2},"verdict":{"id":"7fdc838f-507c-4392-8af4-4c0c527139d5","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-13T01:34:50.623194Z","strongest_claim":"if β∈(1,3),γ∈(β/2,(β+3)/4) or β>1,γ∈(0,β/2] then mild solutions can explode with positive probability.","one_line_summary":"Mild solutions explode with positive probability when β ∈ (1,3) and γ ∈ (β/2, (β+3)/4), or when β > 1 and γ ∈ (0, β/2].","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The assumption that σ(u) ≈ u^γ near infinity together with the existence of mild solutions up to the potential explosion time on the periodic interval.","pith_extraction_headline":"Mild solutions to the stochastic heat equation explode with positive probability for beta in (1,3) with gamma in (beta/2, (beta+3)/4) or beta greater than 1 with gamma up to beta/2."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.11319/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-20T04:42:00.650959Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T12:38:14.915870Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T10:01:17.025567Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T08:32:48.834034Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"cc0163872d6ff03b539e04f8fc03a5c616018d094f463a251ae13295b52d10be"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":1,"snapshot_sha256":"bdadb17b1ae65dfcd075607ea160b1219ca2eee09ea58978bd01e951ee6caf5d"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}