McMillan, B.B.2021-07-162021-07-162021Symmetry, Integrability and Geometry: Methods and Applications, 2021; 17:1-241815-06591815-0659http://hdl.handle.net/2440/131197I consider the existence and structure of conservation laws for the general class of evolutionary scalar second-order differential equations with parabolic symbol. First I calculate the linearized characteristic cohomology for such equations. This provides an auxiliary differential equation satisfied by the conservation laws of a given parabolic equation. This is used to show that conservation laws for any evolutionary parabolic equation depend on at most second derivatives of solutions. As a corollary, it is shown that the only evolutionary parabolic equations with at least one non-trivial conservation law are of Monge-Ampère type.enCopyright status unknownConservation laws; parabolic symbol PDEs; Monge{Ampere equations; characteristic cohomology of exterior di erential systemsJournal article100004251110.3842/SIGMA.2021.0470006582042000012-s2.0-85107205080578229