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Cauchy Problem for Nonuniformly Parabolic Equations with Power Singularities

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We study the Cauchy problem for nonuniformly \(\overrightarrow{2b}\)-parabolic equations with degenerations. The coefficients of parabolic equations may have power singularities of any order with respect to any variables on some set of points. By using a priori estimates and the Arzelà and Riesz theorems, we establish the existence and integral representation for the unique solution of the formulated Cauchy problem. Estimates for the solution of the Cauchy problem and its derivatives in Hölder spaces with power weight are found. The order of power weight is determined via the orders of power singularities and degenerations of the coefficients of \(\overrightarrow{2b}\)-parabolic equations.

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Authors and Affiliations

  1. Yu. Fedkovych Chernivtsi National University, Chernivtsi, Ukraine

    I. D. Pukal’s’kyi

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  1. I. D. Pukal’s’kyi
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Correspondence to I. D. Pukal’s’kyi.

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Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 64, No. 2, pp. 31–41, April–June, 2021.

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Pukal’s’kyi, I.D. Cauchy Problem for Nonuniformly Parabolic Equations with Power Singularities. J Math Sci 277, 33–46 (2023). https://doi.org/10.1007/s10958-023-06811-5

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