FUNDAMENTAL SOLUTION OF THE CAUCHY PROBLEM FOR PARABOLIC EQUATION OF THE SECOND ORDER WITH INCREASING COEFFICIENTS AND WITH BESSEL OPERATORS OF DIFFERENT ORDERS

Abstract

The theory of the Cauchy problem for uniformly parabolic equations of the second order with limited coefficients is sufficiently fully investigated, for example, in the works of S.D. Eidelman and S.D. Ivasyshen, in contrast to such equations with unlimited coefficients. One of the areas of research of Professor S.D. Ivasyshen and students of his scientific school are finding fundamental solutions and investigating the correctness of the Cauchy problem for classes of degenerate equations, which are generalizations of the classical Kolmogorov equation of diffusion with inertia and contain for the main variables differential expressions, parabolic according to I.G. Petrovskyi and according to S.D. Eidelman (S.D. Ivasyshen, L.M. Androsova, I.P. Medynskyi, O.G. Wozniak, V.S. Dron, V.V. Layuk, G.S. Pasichnyk and others). Parabolic Petrovskii equations with the Bessel operator were also studied (S.D. Ivasyshen, V.P. Lavrenchuk, T.M. Balabushenko, L.M. Melnychuk). The article considers a parabolic equation of the second order with increasing coefficients and Bessel operators. In this equation, the some of coefficients for the lower derivatives of one group of spatial variables $x\in \mathbb{R}^n $ are components of these variables, therefore, grow to infinity. In addition, the equation contains Bessel operators of different orders in another group of spatial variables $y\in \mathbb{R}^m_+ $, due to which the coefficients in the first derivatives of these variables are unbounded around the point y=0. The paper defines a modified Fourier-Bessel transform that takes into account different orders of Bessel operators on different variables. With the help of this transformation and the method of characteristics, the solution of the Cauchy problem of the specified equation is found in the form of the Poisson integral, and its kernel, which is the fundamental solution of the Cauchy problem, is written out in an explicit form. Some properties of the found fundamental solution, in particular, estimates of its derivatives, have been established. They will be used to establish the correctness of the Cauchy problem.