The nonlocal conjugation problem for one-dimensional parabolic equation with discontinuous coefficients and associated Feller semigroup

By the boundary integral equations method we establish the classical solvability of the conjugation problem for one-dimensional linear parabolic equation of the second order (backward Kolmogorov equation) with nonlocal Feller-Wentzell conjugation condition. Using the solution of this problem, we construct the two-parameter Feller semigroup associated with the inhomogeneous diffusion process in bounded domain with moving membrane.