Diffusion in media with membranes and some nonlocal parabolic problems

Bohdan Kopytko; Mykhailo Osypchuk; Roman Shevchuk

doi:10.3842/umzh.v75i11.7379

Bohdan Kopytko Department of Mathematics, Faculty of Mechanical Engineering and Computer Science, Czestochowa University of Technology, Poland
Mykhailo Osypchuk Department of Mathematical and Functional Analysis, Faculty of Mathematics and Computer Science, Vasyl Stefanyk Precarpathian National University, Ivano-Frankivsk, Ukraine
Roman Shevchuk Department of Mathematics, Institute of Applied Mathematics and Fundamental Sciences, Lviv Polytechnic National University, Ukraine

DOI: https://doi.org/10.3842/umzh.v75i11.7379

Keywords: parabolic potential, Wentzell boundary condition, Feller semigroup, method of successive approximations

Abstract

UDC 519.21

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 variants of the general nonlocal Feller–Wentzell boundary condition given on nonsmooth boundaries of 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 the sense of M. I. Portenko.

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