{"paper":{"title":"Classical solutions for a logarithmic fractional diffusion equation","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, Juan Luis V\\'azquez","submitted_at":"2012-05-10T10:22:58Z","abstract_excerpt":"We prove global existence and uniqueness of strong solutions to the logarithmic porous medium type equation with fractional diffusion $$ \\partial_tu+(-\\Delta)^{1/2}\\log(1+u)=0, $$ posed for $x\\in \\mathbb{R}$, with nonnegative initial data in some function space of $L \\logL$ type. The solutions are shown to become bounded and $C^\\infty$ smooth in $(x,t)$ for all positive times. We also reformulate this equation as a transport equation with nonlocal velocity and critical viscosity, a topic of current relevance. Interesting functional inequalities are involved."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.2223","kind":"arxiv","version":2},"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"}