{"paper":{"title":"Global solutions to a non-local diffusion equation with quadratic non-linearity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Joachim Krieger, Robert M. Strain","submitted_at":"2010-12-13T21:44:18Z","abstract_excerpt":"In this paper we prove the global in time well-posedness of the following non-local diffusion equation with $\\alpha \\in[0,2/3)$: $$ \\partial_t u = {(-\\triangle)^{-1}u} \\triangle u + \\alpha u^2, \\quad u(t=0) = u_0. $$ The initial condition $u_0$ is positive, radial, and non-increasing with $u_0\\in L^1\\cap L^{2+\\delta}(\\threed)$ for some small $\\delta >0$. There is no size restriction on $u_0$. This model problem appears of interest due to its structural similarity with Landau's equation from plasma physics, and moreover its radically different behavior from the semi-linear Heat equation: $u_t ="},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.2890","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"}