Diffusion in Media with Membranes and Some Nonlocal Parabolic Problems

We establish the classical solvability of a certain conjugation problem for one-dimensional (with respect to a spatial variable) Kolmogorov backward equation with discontinuous coefficients and some versions of the general nonlocal Feller–Wentzell boundary condition given on nonsmooth boundaries of the considered curvilinear domains. In addition, we prove, that the two-parameter Feller semigroup defined by the solution of this problem describes some inhomogeneous diffusion process with moving membranes on the given region of the real line. We also show the relationship between the constructed process and the generalized diffusion in a sense of M. I. Portenko.