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We supply this equation by the initial condition $u(x,0)=\\phi$, and we allow $\\phi$ to be either bounded or unbounded in the origin but smaller than stationary singular solutions. We discuss local existence and long time behaviour for the solutions $u(t,x;\\phi)$ for a wide class of non-homogeneous non-linearities $f$. We show that in the supercritical case, Ground States with slow decay lie"},"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":"1503.02432","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-03-09T11:33:03Z","cross_cats_sorted":[],"title_canon_sha256":"690bbfa49c43d1945393494d82938d4bf619e9687a619475a3447ba814298827","abstract_canon_sha256":"1651a99581d8bea4416b3638f348d0fd801f420c6cc2a252ccfdae40b09c76c6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:25:19.591347Z","signature_b64":"tavsXbwNpvzzNIkZDgmqXQyjCspxKbCGgBnd1nYQu/gPWI7utys0aFm2AnuDYwizRpvoJyLYOYOO1EvupkkXDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"88c43f3510dba0f114fd7a1c2b4bf05dae3252d33eebe6c3cf9a31714ae81cb0","last_reissued_at":"2026-05-18T02:25:19.591010Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:25:19.591010Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On a non-homogeneous and non-linear heat equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Luca Bisconti, Matteo Franca","submitted_at":"2015-03-09T11:33:03Z","abstract_excerpt":"We consider the Cauchy-problem for a parabolic equation of the following type:\n  \\begin{equation*}\n  \\frac{\\partial u}{\\partial t}= \\Delta u+ f(u,|x|),\n  \\end{equation*} where $f=f(u,|x|)$ is supercritical. We supply this equation by the initial condition $u(x,0)=\\phi$, and we allow $\\phi$ to be either bounded or unbounded in the origin but smaller than stationary singular solutions. We discuss local existence and long time behaviour for the solutions $u(t,x;\\phi)$ for a wide class of non-homogeneous non-linearities $f$. 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