The Hamilton-Jacobi equation on metric spaces has been studied by several authors; following the approach of Gangbo and Swiech, we show that the final value problem for the Hamilton-Jacobi equation has a unique solution even if we add a homological term to the Hamiltonian. In metric measure spaces which satisfy the $RCD(K,infty)$ condition one can define a Laplacian which shares many properties with the ordinary Laplacian on $R^n$; in particular, it is possible to formulate a viscous Hamilton-Jacobi equation. We show that, if the homological term is sufficiently regular, the viscous Hamilton-Jacobi equation has a unique solution also in this case.
Bessi, U. (2020). Hamilton-Jacobi in metric spaces with a homological term. COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS, 45(7), 776-819 [10.1080/03605302.2020.1737943].