Dron' V. The Cauchy problem for ultraparabolic equations of Kolmogorov type

Investigation object: the Cauchy problem for degenerate parabolic equations of Kolmogorov type. Investigation purpose: investigation of correct solvability. Investigation methods: potential method, maximum principle. Theoretical and practical results, novelty: Some properties of a fundamental solution of the Cauchy problem for degenerate parabolic second-order equations of Kolmogorov type with three group or space variables with depended of all variables coefficients, and potentials generated by it are proved. Modifications of a maximum principle for the equation with the real coefficients are proved. Theorems of uniqueness of solutions of the Cauchy problem in different classes of functions are proved. A correct resolvability and integral representations of solutions of the Cauchy problem for the generate nonhomogeneous equation, and also an isomorphism between weight Lp-spaces of the initial datas and classical solutions of the Cauchy problem are established. In a case of coefficients independed on space variables, theorems on a correct resolvability of the Cauchy problem in weight H(lder spaces are proved. Degree of application: it is planned. Sphere (area) of application: the differential equations theory, the probability process theory.