Analogue of Bitsadze–Samarskii problem for loaded hyperbolic-parabolic equation with degeneration of order in the hyperbolicity domain

The paper considers a characteristically loaded equation of a mixed hyperbolicparabolic type with degeneration of order in the hyperbolicity part of the domain. In the hyperbolic part of the domain, we have a loaded one-velocity transport equation, known in mathematical biology as the Mac-Kendrick equation, in the parabolic part we have a loaded diffusion equation. In the paper is to study the uniqueness and existence of the solution of the nonlocal inner boundary value problem with Bitsadze–Samarskii type boundary conditions in the parabolic domain and the continuous conjugation conditions; the hyperbolic domain is exempt from the boundary conditions. The problem under investigation is reduced to a nonlocal problem for an ordinary second-order differential equation with respect to the trace of the unknown function in the line of the type changing. The existence and uniqueness theorem for the solution of the problem has been proved; the solution is written out explicitly in the hyperbolic part of the domain. In the parabolic part, the problem under study is reduced to the Volterra integral equation of the second kind, and the solution representation has been found.