On solvability of some classes of transmission problems in a cylindrical space domain

In the article we examine the questions of regular solvability in the Sobolev spaces of the transmission problems with transmission conditions of imperfect contact type for parabolic second order systems in cylindrical space domains. A solution has all generalized derivatives occurring in the system summable to some power $p\in (1,\infty)$. At the interface the limit values of the conormal derivatives are expressed through the limit values of a solution. The problem does not belong to the class of classical diffraction problems and arises when describing heat-and-mass transfer processes in layered media. The proof relies on a priori bounds and the method of continuation in a parameter.