{"paper":{"title":"Variational problems related to self-similar solutions of Hardy-Sobolev heat equation in RN","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Fei Fang, Zhong Tan","submitted_at":"2026-05-24T01:26:31Z","abstract_excerpt":"In this paper, we apply a self-similar transformation to convert the parabolic equation with a Sobolev-Hardy term\n  \\begin{align*} u_t-\\Delta u= \\frac{|u|^{q-2}u}{\\left|x\\right|^s} & \\text { in } \\mathbb{R}^N \\times(0, \\infty), \\end{align*} into the following elliptic equation \\begin{equation*} -\\Delta v-\\frac{1}{2} y \\cdot \\nabla v=\\alpha v+ \\frac{|v|^{q-2} v}{|y|^s}, \\end{equation*} where $2 < q \\leq 2^*(s)=\\frac{2 N-2 s}{N-2}, 0 \\leq s < 2, \\alpha=\\frac{2-s}{2q-4}$. For this equation, we establish the weighted Hardy inequality and Sobolev inequality. Furthermore, by virtue of the variationa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.24804","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/2605.24804/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"}