The thesis consist of an introduction,
six chapters, conclusions, references and the appendix. The introduction consists of the relevance of research topic,
purpose, objectives, subject, object and research methods. The introduction substantiates the relevance of research topic. The goal, subject, object and methods
of the research are listed there. Scientific novelty, the practical significance of
the results, the relation to scientific topic and applicant's contribution are also
indicated in the introduction.
Section 1 Basic concepts and literature review are helpful. Section 1.1 defines four classes of degenerate parabolic equations, which are studied in the dissertation. Class K1 consists of ultraparabolic equations of the Kolmogorov type. Class K2 includes equations of the Kolmogorov type of arbitrary order. Equation of class K3 --- is an equation of the type of equations of class K1, in which there are additional degenerations at the initial time. The classes of equations K1, K2, and K3 are natural generalizations in different directions of the known diffusion equation with inertia of Kolmogorov. Class K4 includes Eidelman's parabolic systems of vector-order equations and degeneracy on the initial hyperplane. A feature of equations from this class is the inequality of spatial variables and the presence of degeneracy in the initial hyperplane. The conditions for the coefficients of the equations are given. These conditions are analyzed for the coefficients of degenerate parabolic equations of the Kolmogorov type and compared with traditional conditions.
In Section 1.2, for equations from the above classes, the definition of the classical fundamental solution of the Cauchy problem (FSCP) and the weaker Lie-FSCP is given. The classical method of Levy construction and research of FSCP, and also its modifications which are used in case of degenerate parabolic equations is described in detail in subsection 1.3. In particular, the step-by-step Levy method is described, which is used in the dissertation to construct classical FSCP for equations from classes K1 --- K3. Section 1.4 provides an overview of the literature, which studied the equations of the above classes, used the Levy method, investigated and applied the properties of FSCP.
The second chapter is called Supporting Information. Section 2.1 provides definitions and properties of estimating functions and some integrals containing estimating functions. Each class of equations has its own estimating function. The application of the Levy method involves solving integral equations of the Volterra type with a quasi-regular kernel. For each class of equations, the lemmas on the existence and estimation of the corresponding resolvent in terms of estimating functions are proved. Corresponding lemmas on the existence and estimates of solutions of such integral equations are given in Section 2.2. Similarly, information on the properties of integrals of the type derived from bulk potentials is given in Section 2.3. These properties are described in terms of belonging of the integral to a special weight space, depending on which space the density belongs to. To apply the Levy stepwise method, theorems on the properties and estimates of FSCP for auxiliary equations are necessary, ie equations whose coefficients depend only on the time variable and parameters from classes K1, K2 and K3 are given in Section 2.4.
The step-by-step Levy method consists of three stages, according to the number of groups of spatial variables. At each stage, the parametrix is taken FSCP, which was built at the previous stage. Thus it is necessary to select properties of density of volume potential so that the constructed FSCP had the necessary properties not only on time and spatial variables, but also on parameters. For class K1 it is stated in Sections 3.1, 3.2, 3.3. In Section 3.4, the obtained estimates of classical FSCP and its derivatives are used to establish accurate estimates of increments of derivatives from FSCP for the constructed classical FSCP. Also in this section the existence and estimation of Lee-FSCP for the equation from the specified class is proved. Similar results for equations from classes K2 and K3 are given in Sections 4 and 5, respectively. The most accurate results are obtained for ultraparabolic equations of the Kolmogorov type, ie equations from class K1 and K3. For equations of class K2, the estimates of the classical FSCP are less accurate. This is due to the structure and properties of the evaluation function, which has the form of a series.
