{"paper":{"title":"Well-posedness of the Cauchy problem for the fractional power dissipative equations","license":"","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Baoquan Yuan, Bo Zhang, Changxing Miao","submitted_at":"2006-07-19T09:34:18Z","abstract_excerpt":"This paper studies the Cauchy problem for the nonlinear fractional power dissipative equation $u_t+(-\\triangle)^\\alpha u= F(u)$ for initial data in the Lebesgue space $L^r(\\mr^n)$ with $\\ds r\\ge r_d\\triangleq{nb}/({2\\alpha-d})$ or the homogeneous Besov space $\\ds\\dot{B}^{-\\sigma}_{p,\\infty}(\\mr^n)$ with $\\ds\\sigma=(2\\alpha-d)/b-n/p$ and $1\\le p\\le \\infty$, where $\\alpha>0$, $F(u)=f(u)$ or $Q(D)f(u)$ with $Q(D)$ being a homogeneous pseudo-differential operator of order $d\\in[0,2\\alpha)$ and $f(u)$ is a function of $u$ which behaves like $|u|^bu$ with $b>0$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0607456","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0607456/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}