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Particular cases of this problem are the Korteweg-de Vries-Burgers equation for $\\Phi(k)=-k^2$, the derivative Korteweg-de Vries-Kuramoto-Sivashinsky equation for $\\Phi(k)=k^2-k^4$, and the Ostrovsky-Stepanyams-Tsimring equation for $\\Phi(k)=|k|-|k|^3$."},"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":"1105.2995","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-05-16T02:47:43Z","cross_cats_sorted":[],"title_canon_sha256":"1ab7a20cb6cbfb70547706c644e09bbb8a1b7696c9a6c21d79260067a065d250","abstract_canon_sha256":"ed02cf03fb57da53dd7f5a1054ce9dbdf75f57bb6e56d62f18f12ba290585488"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:30:12.724288Z","signature_b64":"rV9BIRtJtWrSB4c4D/k26od4qljrJDkgC3roBi7STSI/C3jTLTODS+xXOEiyswnErJ4s+SHISEXRctDFlFgUAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"47aa38a9c596b22c1f5115795a410a7417ecde622aa74453f38d196f94070c64","last_reissued_at":"2026-05-18T03:30:12.723548Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:30:12.723548Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Well-posedness for a Family of Perturbations of the KDV Equation in Periodic Sobolev Spaces of Negative Order","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ricardo Pastran, Xavier Carvajal","submitted_at":"2011-05-16T02:47:43Z","abstract_excerpt":"We establish local well-posedness in Sobolev spaces $H^s(\\mathbb{T})$, with $s\\geq -1/2$, for the initial value problem issues of the equation $$ u_t + u_{xxx}+\\eta Lu + uu_x=0;\\; x\\in \\mathbb{T},\\; t\\geq0, $$ where $\\eta >0$, $(Lu)^{\\wedge}(k)=-\\Phi(k)\\hat{u}(k)$, $k\\in \\mathbb{Z}$ and $\\Phi \\in \\mathbb{R}$ is bounded above. 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