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We assume that $(x,s)\\in\\Omega\\times\\R^+\\to f(x,s)$ is a bounded below Caratheodory function, locally Lipschitz with respect to $s$ uniformly in "},"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":"1104.1691","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-04-09T11:31:53Z","cross_cats_sorted":[],"title_canon_sha256":"dd34315939c0fce6df68d4c43cc14dae789e185eeb443fd00ac3ad185aacf129","abstract_canon_sha256":"2a449c07a31cd5b948b18f0ee423f45401b73e852b6d968bf921ed3a8bfd5653"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:24:43.360791Z","signature_b64":"2E4OkO1+2EEanogzZ6BaIMW7PRbgFJ/05sLKmKQ33pViWBp/vJshXtobvueD6Ntyr4zpp1DFn8435koC46AUCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2b15f4683b7c66df1797a9e98f382a6178655b5282003f32e827b662203aec7d","last_reissued_at":"2026-05-18T04:24:43.360320Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:24:43.360320Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Singular Parabolic Equation: Existence, Stabilization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jacques Giacomoni, Kaushik Bal, Mehdi Badra","submitted_at":"2011-04-09T11:31:53Z","abstract_excerpt":"We investigate the following quasilinear parabolic and singular equation,\n  {equation} \\tag{{\\rm P$_t$}} \\{{aligned} & u_t-\\Delta_p u =\\frac{1}{u^\\delta}+f(x,u)\\;\\text{in}\\,(0,T)\\times\\Omega, & u =0\\,\\text{on} \\;(0,T)\\times\\partial\\Omega,\\quad u>0 \\text{in}\\, (0,T)\\times\\Omega, &u(0,x) =u_0(x)\\;\\text{in}\\Omega, {aligned}. {equation} %\nwhere $\\Omega$ is an open bounded domain with smooth boundary in $\\R^{\\rm N}$, $1 < p< \\infty$, $0<\\delta$ and $T>0$. 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