{"paper":{"title":"Nonlocal filtration equations with rough kernels","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ana Rodr\\'iguez, Arturo de Pablo, Fernando Quir\\'os","submitted_at":"2015-09-30T12:15:15Z","abstract_excerpt":"We study the nonlinear and nonlocal Cauchy problem \\[ \\partial_{t}u+\\mathcal{L}\\varphi(u)=0 \\quad\\text{in }\\mathbb{R}^{N}\\times\\mathbb{R}_+,\\qquad u(\\cdot,0)=u_0, \\] where $\\mathcal{L}$ is a L\\'evy-type nonlocal operator with a kernel having a singularity at the origin as that of the fractional Laplacian. The nonlinearity $\\varphi$ is nondecreasing and continuous, and the initial datum $u_0$ is assumed to be in $L^1(\\mathbb{R}^N)$. We prove existence and uniqueness of weak solutions. For a wide class of nonlinearities, including the porous media case, $\\varphi(u)=|u|^{m-1}u$, $m>1$, these solu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.09143","kind":"arxiv","version":3},"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"}