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In this work, we show that there is a strictly increasing function $f(s)$ such that $f^{-1}(\\vp(x))$ is convex for $0<\\lambda\\leq\\lambda^{\\ast}$, i.e., we prove t"},"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":"1305.1065","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-05-06T00:10:22Z","cross_cats_sorted":[],"title_canon_sha256":"9fb9b299dca41df9cd04f16e32a02d1b55690055b51b927d9ca406d8fbbed433","abstract_canon_sha256":"7e4e3c8eecdeff901561ae3a22a37090a724343ce8ef756949538d44aa3bfaa6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:17:44.192120Z","signature_b64":"0x4LvIVYVZyV9Koi88r0cSxRmNn+V7r3YMJ/f0RGnC0HU1YS5K8YDVcxwXQpf0M/+A4ZwGU0zD7mkoIvMRcmCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9b6ac59644a970d04171728c33f672516f533cf233f6db296d9d8c1f7b6d08da","last_reissued_at":"2026-05-18T03:17:44.191622Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:17:44.191622Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Geometric Properties of Gelfand's Problems with Parabolic Approach","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ki-Ahm Lee, SungHoon Kim","submitted_at":"2013-05-06T00:10:22Z","abstract_excerpt":"We consider the asymptotic profiles of the nonlinear parabolic flows $$(e^{u})_{t}= \\La u+\\lambda e^u$$ to show the geometric properties of the following elliptic nonlinear eigenvalue problems known as a Gelfand's problem: \\begin{equation*} \\begin{split} \\La \\vp &+ \\lambda e^{\\vp}=0, \\quad \\vp>0\\quad\\text{in $\\Omega$}\\\\ \\vp&=0\\quad\\text{on $\\Omega$} \\end{split}\n  \\end{equation*} posed in a strictly convex domain $\\Omega\\subset\\re^n$. 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