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We treat in detail the equation $$u_t + \\{u^n(u_{xxx} + \\nu u^{m-n}u_x -A u^{M-n} u_x)\\}_x=0,$$ where $\\nu=\\pm 1,$ $n>0,$ $M>m,$ and $A \\ge 0.$ Global existence of weak nonnegative solutions is proven when $ m-n> -2$ and $A>0$ or $\\nu=-1,$ and when $-2< m-n<2,$ $A=0,$ $\\nu=1.$ From the weak solutions, we"},"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":"1010.0536","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-10-04T10:52:11Z","cross_cats_sorted":[],"title_canon_sha256":"d23b0a1861356c7f21a641c8f4137ed0fe04fd4e6151ee06e0d594b07216cde3","abstract_canon_sha256":"bb5a7e4f4d6591c3bb7099c81dba93b8e099841dd9deed0e4d0c1cb4bf58277b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:39:51.855235Z","signature_b64":"3n1AXLQWTFHz34m9FrOpClSV2hBXSIemeZRWGhOzEX3bmJu2z5O7uNPFp3bGUD0NiQmACiwTtS3pOUs8Eas2CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"08f8f92a82296c396e61dd4d855a60e16dc264c78609181ecedcfd0449521d9d","last_reissued_at":"2026-05-18T04:39:51.854586Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:39:51.854586Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The thin film equation with backwards second order diffusion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Amy Novick-Cohen, Andrey Shishkov","submitted_at":"2010-10-04T10:52:11Z","abstract_excerpt":"In this paper, we focus on the thin film equation with lower order \"backwards\" diffusion which can describe, for example, the evolution of thin viscous films in the presence of gravity and thermo-capillary effects, or the thin film equation with a \"porous media cutoff\" of van der Waals forces. 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