初始超平面上系数递增和退化的kolmogorov型方程解的表示

H. Pasichnyk, S. Ivasyshen
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引用次数: 1

摘要

研究了系数在最低导数为|x|→∞且t = 0时退化的非齐次Kolmogorov型超抛物方程。证明了该方程解的积分表示定理。用泊松积分和由柯西问题的基本解产生的体积势来表示。所考虑的解,作为x的函数,可以在|x|→∞时无限增加,并且可以在t→0时以某种方式表现,这取决于方程在t = 0时退化的类型。注意,在很强退化的情况下,解,作为x的函数,是有界的。这些结果可用于确定所考虑的方程在t = 0时的弱退化情况下具有经典初始条件的正确可解性,在强退化情况下具有权初始条件或不具有初始条件。
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REPRESENTATION OF SOLUTIONS OF KOLMOGOROV TYPE EQUATIONS WITH INCREASING COEFFICIENTS AND DEGENERATIONS ON THE INITIAL HYPERPLANE
The nonhomogeneous model Kolmogorov type ultraparabolic equation with infinitely increasing coefficients 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 infinitely 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.
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