{"paper":{"title":"Existence and global behaviour of solutions of a parabolic problem involving the fractional $p$-Laplacian in porous medium","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Guillaume Warnault (1) ((1) LMAP-UMR CNRS 5142), Jacques Giacomoni (1), Lo\\\"ic Constantin (1)","submitted_at":"2024-11-21T16:14:52Z","abstract_excerpt":"In this paper, we prove the existence and the uniqueness of a weak and mild solution of the following nonlinear parabolic problem involving the porous $p$-fractional Laplacian:\n  \\begin{equation*} \\begin{cases} \\partial_t u+(-\\Delta)^s_p(|u|^{m-1}u)=h(t,x,|u|^{m-1}u) & \\text{in} \\; (0,T)\\times \\Omega,\\\\ u=0 & \\text{in} \\; (0,T) \\times \\mathbb{R}^d\\backslash \\Omega, \\\\ u(0,\\cdot)=u_0 & \\text{in} \\; \\Omega . \\end{cases}\\ \\end{equation*} We also study further the the homogeneous case $h(u)=|u|^{q-1}u$ with $q>0$. In particular we investigate global time existence, uniqueness, global behaviour of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.14260","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/2411.14260/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"}