Second mixed boundary value problem in half space for the generalized Kolmogorov equation

Abstract

This article investigates the second boundary value problem in a half-space for a diffusion-type equation with inertia, where inertia is determined by four groups of variables. Each of these groups covers a corresponding number of variables, and they are characterized by the degeneration of parabolic features. To confirm the existence of solutions to
this boundary value problem, the method of limit transition is used, and an asymptotic expansion is constructed, which helps to study the behavior of the equation. The explicit analytical form of the fundamental solution of the Cauchy problem plays a particularly important role, as well as the study of the properties of its derivatives for a degenerate parabolic equation. To reduce the problem to a more convenient mathematical formulation, the method of potentials is used, where the kernel of these potentials corresponds to the fundamental solution of the generalized Kolmogorov equation. Thanks to this approach, the second boundary value problem in a half-space was reduced to a singular integral equation. To find the solution, classes of
differential functions were used, which allowed us to  guarantee the compression of the corresponding integral operator under the conditions of a small parameter t. In addition to the existence, a proof of the uniqueness of the solution of the given boundary value problem was also provided. This result is based on the use of the maximum principle in spaces of bounded functions, which, in turn, provides a theoretical foundation for similar studies. Thus, the article makes a significant contribution to the  development of mathematical methods for the analysis and solution of problems with degenerate parabolic equations in multidimensional spaces.