{"paper":{"title":"Abstract kinetic equations with positive collision operators","license":"","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.SP","authors_text":"Canada), I. M. Karabash (the University of Calgary","submitted_at":"2007-08-18T23:14:45Z","abstract_excerpt":"We consider \"forward-backward\" parabolic equations in the abstract form $Jd \\psi / d x + L \\psi = 0$, $ 0< x < \\tau \\leq \\infty$, where $J$ and $L$ are operators in a Hilbert space $H$ such that $J=J^*=J^{-1}$, $L=L^* \\geq 0$, and $\\ker L = 0$. The following theorem is proved: if the operator $B=JL$ is similar to a self-adjoint operator, then associated half-range boundary problems have unique solutions. We apply this theorem to corresponding nonhomogeneous equations, to the time-independent Fokker-Plank equation $ \\mu \\frac {\\partial \\psi}{\\partial x} (x,\\mu) = b(\\mu) \\frac {\\partial^2 \\psi}{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0708.2510","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":""},"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"}