Open AccessREPRESENTATION OF SOLUTIONS OF KOLMOGOROV TYPE EQUATIONS WITH INCREASING COEFFICIENTS AND DEGENERATIONS ON THE INITIAL HYPERPLANE
Author(s) -
H. S. Pasichnyk,
S. D. Іvasyshen
Publication year - 2021
Publication title -
bukovinsʹkij matematičnij žurnal
Language(s) - English
Resource type - Journals
ISSN - 2309-4001
DOI - 10.31861/bmj2021.01.16
Subject(s) - mathematics , bounded function , type (biology) , representation (politics) , initial value problem , hyperplane , mathematical analysis , degeneration (medical) , pure mathematics , combinatorics , law , biology , medicine , ecology , pathology , politics , political science
The nonhomogeneous model Kolmogorov type ultraparabolic equation with innitely increasing coecients at the lowest derivatives as |x| → ∞ and degenerations for t = 0 is considered in the paper. Theorems on the integral representation of solutions of the equation are proved. The representation is written with the use of Poisson integral and the volume potential generated by the fundamental solution of the Cauchy problem. The considered solutions, as functions of x, could innitely increase as |x| → ∞, and could behave in a certain way as t → 0, depending on the type of the degeneration of the equation at t = 0. Note that in the case of very strong degeneration, the solutions, as functions of x, are bounded. These results could be used to establish the correct solvability of the considered equation with the classical initial condition in the case of weak degeneration of the equation at t = 0, weight initial condition or without the initial condition if the degeneration is strong.