{"paper":{"title":"Almost Global Solutions of Kirchhoff Equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Duohui Xiang, Jianjun Liu","submitted_at":"2025-05-02T13:15:35Z","abstract_excerpt":"This paper is concerned with the original Kirchhoff equation $$\\left\\{\\begin{aligned}\n  & \\pa_{tt}u-\\Big(1+\\int_{0}^{\\pi}|\\pa_xu|^2 dx\\Big)\\pa_{xx}u=0,\n  \\\\&u(t,0)=u(t,\\pi)=0. \\end{aligned}\\right.$$ We obtain almost global existence and stability of solutions for almost any small initial data of size $\\varepsilon$. In Sobolev spaces, the time of existence and stability is of order $\\varepsilon^{-r}$ for arbitrary positive integer $r$. In Gevrey and analytic spaces, the time is of order $e^{\\frac{|\\ln\\varepsilon|^2}{c\\ln|\\ln\\varepsilon|}}$ with some positive constant $c$. To achieve these, we b"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.01248","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/2505.01248/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"}