The study is devoted to degenerate parabolic equations with a block structure, which under certain conditions are a generalization of the well-known degenerate parabolic diffusion equation with inertia of A.M. Kolmogorov. In this work, special Hölder conditions are formulated with respect to spatial variables on the coefficients of such equations, under which the existence of a classical fundamental solution of the Cauchy problem is proved, estimates for it and its derivatives are obtained, and properties such as normality, convolution formula, uniqueness are proved. Also, the correct solvability of the Cauchy problem in special weight spaces and integral representations of classical solutions of homogeneous equations in the form of Poisson integrals of functions or generalized measures, which specify the initial condition, are obtained. The correctness classes of the Cauchy problem are described. The obtained results can be used in further studies of the Cauchy problem and boundary value problems for linear and quasilinear degenerate parabolic equations.

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