{"paper":{"title":"A non-local inequality and global existence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.CA","math.MP"],"primary_cat":"math.AP","authors_text":"Joachim Krieger, Philip T. Gressman, Robert M. Strain","submitted_at":"2012-02-18T17:50:19Z","abstract_excerpt":"In this article we prove a collection of new non-linear and non-local integral inequalities. As an example for $u\\ge 0$ and $p\\in (0,\\infty)$ we obtain $$ \\int_{\\threed} dx ~ u^{p+1}(x) \\le (\\frac{p+1}{p})^2 \\int_{\\threed} dx ~ \\{(-\\triangle)^{-1} u(x) \\} \\nsm \\nabla u^{\\frac{p}{2}}(x)\\nsm^2. $$ We use these inequalities to deduce global existence of solutions to a non-local heat equation with a quadratic non-linearity for large radial monotonic positive initial conditions. Specifically, we improve \\cite{ksLM} to include all $\\alpha\\in (0, 74/75)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.4088","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":""},"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"}