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Here $G=G(x,t)$ is a generalization of the heat kernel. We are interested in the asymptotic expansions of the solution of $(P)$ behaving like a multiple of the integral kernel $G$ as $t\\to\\infty$."},"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":"1309.7118","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-09-27T05:29:55Z","cross_cats_sorted":[],"title_canon_sha256":"e0f247613eb34a6774c63c9bfa64fb9d35bc697ffd2744992f657d3e1f8d82e9","abstract_canon_sha256":"c68edf74c5e7e282be7e16c3f35b6ecc7ff72d8dc4ec4b881be1ada3eb717302"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:49:54.024714Z","signature_b64":"aoAiJy1SvQ82CEw0UxwJJojXp7WeEZQxqwJyJ4SZO1oIgCu4r5ZEaKBAvudMvh2gjSxDhhETOTwZCmHPM44bBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"deaecb388ea42e7b41bd19eea1be32d6adde175f12ae792bafb6d2917f01b235","last_reissued_at":"2026-05-18T02:49:54.024064Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:49:54.024064Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotics for a nonlinear integral equation with a generalized heat kernel","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Kanako Kobayashi, Kazuhiro Ishige, Tatsuki Kawakami","submitted_at":"2013-09-27T05:29:55Z","abstract_excerpt":"This paper is concerned with a nonlinear integral equation $$ (P)\\qquad u(x,t)=\\int_{{\\bf R}^N}G(x-y,t)\\varphi(y)dy+\\int_0^t\\int_{{\\bf R}^N}G(x-y,t-s)f(y,s:u)dyds, \\quad $$ where $N\\ge 1$, $\\varphi\\in L^\\infty({\\bf R}^N)\\cap L^1({\\bf R}^N,(1+|x|^K)dx)$ for some $K\\ge 0$. 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