Ukrainian Mathematical Bulletin

Volume 1, Issue 1, 2004 pp. 67-75.

On Well-Posedness of the Cauchy Problem for Degenerate Parabolic Equations of Kolmogorov Type

Authors: V.S.Dron', S.D.Ivasyshen
Author institution: Fed'kovych Chernivtsi National University

Summary: In this paper we consider a second-order equation that generalizes the well-known Kolmogorov diffusion equation with inertia [1]. This equation contains three groups of space variables with different numbers of variables. A fundamental solution of the Cauchy problem for an equation of this type was constructed and studied in [2--4]. In [5,6] the well-posedness of the Cauchy problem was established and theorems on the integral representation of a solution of the Cauchy problem for homogeneous degenerate parabolic equation of the Kolmogorov type (not only of the second order but also of higher orders) were proved under the condition that the coefficients of the equation do not depend on space variables. Similar results for an inhomogeneous second-order equation with coefficients depending on all variables are obtained. The last theorem is devoted to the well-posedness of the Cauchy problem for this equation in Holder spaces. Results of this type were obtained for nondegenerate equations in [7] and for the equation under consideration with coefficients independent of space variables in [8].

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