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In this paper we study time-inhomogeneous versions of one-dimensionalStochastic Differential Equations (SDE) involving the Local Time of the unknownprocess on curves. After proving existence and uniqueness for these SDE undermild assumptions, we explore their link with Parabolic Differential Equations(PDE) with transmission conditions. We study the regularity of solutions ofsuch PDE and ensure the validity of a Feynman-Kac representation formula. Theseresults are then used to characterize the solutions of these SDE astime-inhomogeneous Markov Feller processes.

Time inhomogeneous Stochastic Differential Equations involving the local time of the unknown process, and associated parabolic operators