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Then we apply it to study the classical Keller-Segel system \\begin{equation} \\left\\{ \\begin{array}{llc} u_t=\\Delta u-\\nabla\\cdot(u \\nabla v), \\\\[6pt] \\displaystyle v_t=\\Delta v-v+u, \\end{array} \\right. \\end{equation} in a bounded domain $\\Omega\\subset\\mathbb{R}^N$ ($N\\ge 2$) with smooth boundary. It is known that for any $\\delta>0$, if $\\int_\\Omega u^{\\frac N2+\\delta}(\\cdot,t)$ is bounded, then the solution is global and bounded"},"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":"1707.09235","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-07-28T13:58:45Z","cross_cats_sorted":[],"title_canon_sha256":"1f7e57ce60b529bf4f5bf67154abd8f567db39d94bea1841fbbb843c37c4f8ea","abstract_canon_sha256":"4f74bd641159805196f5087390f2d8b47997d5d72cf4a35765c18fc12377c267"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:13:37.732807Z","signature_b64":"2H4ROtLpiy2w6hwDCQsidWN+9p1OvY2plD0T3LxfSrE9GZuGIXdt1x+i4OfXu/g16/a2px6jUQQlbiurcCp1Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"405cef558f90da3f562107756b904703d150944b633626b3d05f07c67aa0a9e6","last_reissued_at":"2026-05-18T00:13:37.732300Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:13:37.732300Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An interpolation inequality and its application in Keller-Segel model","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Xinru Cao","submitted_at":"2017-07-28T13:58:45Z","abstract_excerpt":"In this paper, we first prove an interpolation inequality of Ehrling-type, which is an improvement of a special case to the well known Gargliardo-Nirenberg inequality. Then we apply it to study the classical Keller-Segel system \\begin{equation} \\left\\{ \\begin{array}{llc} u_t=\\Delta u-\\nabla\\cdot(u \\nabla v), \\\\[6pt] \\displaystyle v_t=\\Delta v-v+u, \\end{array} \\right. \\end{equation} in a bounded domain $\\Omega\\subset\\mathbb{R}^N$ ($N\\ge 2$) with smooth boundary. 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