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For this equation, we establish the weighted Hardy inequality and Sobolev inequality. Furthermore, by virtue of the variationa"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.24804","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-05-24T01:26:31Z","cross_cats_sorted":[],"title_canon_sha256":"0301dba578b55515cb01689b4a41c1cca5c79ad8512d82b79e69f600c7566ab2","abstract_canon_sha256":"6e1439c3a3ed7b5d7b787a76c8aef923be3cb3588d5711b63a5c5700567da08a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-26T01:03:59.112660Z","signature_b64":"lLMHQeXImf5qV8ZmSYrRK/DtpMGBwvjSP2bsF4vrYF8JOa18itrwt9jCNTDwDhG8aUSom2Iz8winl/v6jjRSCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4a317d94734b311e4e02ae1bd2ab88049bcecf977557edd3ecdfe3780577cde7","last_reissued_at":"2026-05-26T01:03:59.112067Z","signature_status":"signed_v1","first_computed_at":"2026-05-26T01:03:59.112067Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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