We consider one class of degenerate parabolic systems of equations of the type of diffusion equation with Kolmogorov inertia. For systems whose coefficients may depend only on the time variable, we construct a fundamental matrix of solutions of the Cauchy problem and obtain estimates for this matrix and all its derivatives.
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S. D. Éidel’man, S. D. Ivasyshen, and A. N. Kochubei, “Analytic methods in the theory of differential and pseudo-differential equations of parabolic type,” Operator Theory: Adv. Appl., 152 (2004).
S. Polidoro, “On a class of ultraparabolic operators of Kolmogorov–Fokker–Planck type,” Le Matematiche, 49, 53–105 (1994).
C. Cinti, A. Pascucci, and S. Polidoro, “Pointwise estimates for solutions to a class of non-homogeneous Kolmogorov equations,” Math. Ann., 340, No. 2, 237–264 (2008).
H. P. Malyts’ka, “Construction of a fundamental solution of the Cauchy problem for a diffusion equation with variable inertia,” Mat. Met. Fiz.-Mekh. Polya, 42, No. 3, 56–60 (1999).
H. P. Malyts’ka, “On the structure of a fundamental solution of the Cauchy problem for elliptic–parabolic equations that generalize diffusion equations with inertia,” Visn. Nats. Univ. “L’viv. Politekhnika,” No. 411, 221–228 (2000).
H. P. Malyts’ka, “On a fundamental solution of the Cauchy problem for a Kolmogorov-type parabolic equation of arbitrary order that is degenerate with respect to an arbitrary number of groups of variables,” Mat. Met. Fiz.-Mekh. Polya, 47, No. 4, 131–138 (2004).
R. Courant, Partial Differential Equations [Russian translation], Mir, Moscow (1964).
S. D. Éidel’man, Parabolic Systems [in Russian], Nauka, Moscow (1964).
S. D. Ivasyshen and S. D. Éidel’man, “\( \overrightarrow {2b} \)-Parabolic systems,” in: Proceedings of the Seminar on Functional Analysis [in Russian], Issue 1, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1968), pp. 3–175, 271–273.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1650–1663, December, 2008.
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Malyts’ka, H.P. Systems of equations of Kolmogorov type. Ukr Math J 60, 1937–1954 (2008). https://doi.org/10.1007/s11253-009-0182-4
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DOI: https://doi.org/10.1007/s11253-009-0182-4