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Existence and Uniqueness of the Classical Solution of the First Boundary Value Problem for Parabolic Systems on the Plane

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Abstract

We consider the first boundary value problem for uniformly parabolic systems of the second order with one spatial variable in bounded and semibounded domains with nonsmooth lateral boundaries. The coefficients of the system satisfy the Hölder condition and do not depend on the time variable. For continuous initial and boundary functions, the existence and uniqueness of the classical solution of this problem is established.

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REFERENCES

  1. Solonnikov, V.A., On boundary value problems for linear parabolic systems of differential equations of general form, Proc. Steklov Inst. Math., 1965, vol. 83, pp. 1–184.

    MathSciNet  Google Scholar 

  2. Baderko, E.A. and Cherepova, M.F., Simple layer potential and the first boundary value problem for a parabolic system on the plane, Differ. Equations, 2016, vol. 52, no. 2, pp. 197–209.

    Article  MathSciNet  MATH  Google Scholar 

  3. Baderko, E.A. and Cherepova, M.F., Dirichlet problem for parabolic systems with Dini continuous coefficients on the plane, Dokl. Math., 2017, vol. 96, no. 2, pp. 423–426.

    Article  MathSciNet  MATH  Google Scholar 

  4. Baderko, E.A. and Cherepova, M.F., Dirichlet problem for parabolic systems with Dini continuous coefficients, Appl. Anal., 2021, vol. 100, no. 13, pp. 2900–2910.

    Article  MathSciNet  MATH  Google Scholar 

  5. Maz’ya, V.G. and Kresin, G.I., On the maximum principle for strongly elliptic and parabolic second order systems with constant coefficients, Math. USSR-Sb., 1986, vol. 53, no. 2, pp. 457–479.

    Article  MATH  Google Scholar 

  6. Baderko, E.A. and Cherepova, M.F., Uniqueness of solutions to initial boundary value problems for parabolic systems in plane bounded domains with nonsmooth lateral boundaries, Dokl. Math., 2020, vol. 102, no. 2, pp. 357–359.

    Article  MathSciNet  MATH  Google Scholar 

  7. Baderko, E.A. and Cherepova, M.F., Uniqueness of solutions of the first and second initial–boundary value problems for parabolic systems in bounded domains on the plane, Differ. Equations, 2021, vol. 57, no. 8, pp. 1010–1019.

    Article  MathSciNet  MATH  Google Scholar 

  8. Baderko, E.A. and Sakharov, S.I., Uniqueness of the solution of the first initial–boundary value problem for a parabolic system with differentiable coefficients in a half-strip with nonsmooth lateral boundary, Differ. Equations, 2021, vol. 57, no. 5, pp. 605–614.

    Article  MathSciNet  MATH  Google Scholar 

  9. Baderko, E.A. and Sakharov, S.I., Uniqueness of solutions to initial–boundary value problems for parabolic systems with Dini-continuous coefficients in a semibounded domain on the plane, Comput. Math. Math. Phys., 2023, vol. 63, no. 4, pp. 553–563.

    Article  MathSciNet  MATH  Google Scholar 

  10. Friedman, A., Partial Differential Equations of Parabolic Type, Englewood Cliffs: Prentice Hall, 1964. Translated under the title: Uravneniya s chastnymi proizvodnymi parabolicheskogo tipa, Moscow: Mir, 1968.

  11. Konenkov, A.N., Classical solutions of the first boundary value problem for parabolic systems on the plane, Dokl. Math., 2022, vol. 105, no. 2, pp. 109–111.

    Article  MathSciNet  MATH  Google Scholar 

  12. Eidel’man, S.D., Parabolicheskie sistemy (Parabolic Systems), Moscow: Nauka, 1964.

    Google Scholar 

  13. Tveritinov, V.A., Gladkost’ potentsiala prostogo sloya dlya parabolicheskoi sistemy vtorogo poryadka (Smoothness of the simple layer potential for a second-order parabolic system), Available from VINITI Akad. Nauk SSSR, 1988, Moscow, no. 6850-В88.

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  1. Yesenin Ryazan State University, Ryazan, 390000, Russia

    A. N. Konenkov

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  1. A. N. Konenkov
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Correspondence to A. N. Konenkov.

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Translated by V. Potapchouck

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Konenkov, A.N. Existence and Uniqueness of the Classical Solution of the First Boundary Value Problem for Parabolic Systems on the Plane. Diff Equat 59, 904–913 (2023). https://doi.org/10.1134/S0012266123070042

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  • DOI: https://doi.org/10.1134/S0012266123070042

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