FUNDAMENTAL SOLUTION TO CAUCHY PROBLEM FOR KOLMOGOROV-TYPE EQUATION WITH UNBOUNDED COEFFICIENTS THAT DO NOT DEPEND ON DEGENERATION VARIABLES

Abstract

The class of equations considered in the work is a generalization of the well-known equation diffusion with inertia of A.M. Kolmogorov. These equations are degenerate differential equations with partial derivatives of the parabolic type. They contain only lower derivatives in the sense of the theory of parabolic equations for some of the spatial variables. Such equations belong to the class of ultraparabolic or elliptic-parabolic equations. These equations and their various generalizations have been studied by many authors. Linear and nonlinear ultraparabolic equations arise in some problems of probability theory, mathematical modeling of options, Brownian motion theory, convective diffusion theory, binary electrolytes theory, modeling of diffusion processes with inertia and electron scattering, and other branches of science. In addition to the Kolmogorov structure, the equations under consideration have another peculiarity: the coefficients of the group of junior terms grow indefinitely. A degenerate second-order Kolmogorov equation with unbounded coefficients independent of the degeneracy variables is considered. The coefficients of the equation are differentiable, and their growth depends on some increasing and differentiable function. For such an equation, a fundamental solution of the Cauchy problem has been constructed and estimates of its derivatives with respect to the main group of variables are obtained. The obtained results can be used in studying the solvability of the Cauchy problem for the equation under consideration, as well as in constructing and investigating the fundamental solution of the Cauchy problem for an equation with coefficients depending on all groups of spatial variables and growing at |x| → ∞ .