{"paper":{"title":"The generalized porous medium equation on graphs: well-posedness, extinction, and mass conservation","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alberto G. Setti, Bobo Hua, Davide Bianchi, Rados{\\l}aw K. Wojciechowski","submitted_at":"2026-07-25T08:04:03Z","abstract_excerpt":"We study the Cauchy problem for the generalized porous medium equation on infinite weighted graphs. For a general nonlinearity, we establish Dirichlet comparison and weak maximum principles on finite subgraphs and, through an exhaustion argument, construct minimal and maximal pointwise solutions for arbitrary $\\ell^\\infty$ initial data, controlled by explicit, possibly time-dependent, barriers.\n  For the porous nonlinearity $\\phi(s)=s|s|^{m-1}$, assuming a $\\nu$-Sobolev inequality with $\\nu>2$, we derive quantitative energy estimates for $\\ell^1$-mild solutions. These yield finite-time extinct"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.23091","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/2607.23091/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"}