ON THE UNIQUE SOLVABILITY OF THE CAUCHY PROBLEM IN THE CLASS

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Abstract

The Cauchy problem for Petrovskii second-order parabolic systems in a strip on the plane is considered. The coefficients of the system satisfy the double Dini condition. The unique solvability of the problem in the space of functions that are continuous and bounded together with their spatial derivatives of the first order in the closure of the strip is established and corresponding estimates are obtained. An integral representation of the solution is given.

About the authors

E. A Baderko

Lomonosov Moscow State University; Moscow Center for Fundamental and Applied Mathematics

Email: baderko.ea@yandex.ru

S. I Sakharov

Lomonosov Moscow State University; Moscow Center for Fundamental and Applied Mathematics

Email: ser341516@yandex.ru

References

  1. Solonnikov, V.A., O kraevyh zadachah dlya linejnyh parabolicheskih sistem differencial’nyh uravnenij obshchego vida, Trudy Matematicheskogo instituta imeni V.A. Steklova, 1965, vol. 83, pp. 3–163.
  2. Ladyzhenskaya, O.A., Solonnikov, V.A., and Ural’tseva, N.N., Linear and Quasilinear Equations of Parabolic Type, American Mathematical Soc., 1968.
  3. Cherepova, M.F., O gladkosti resheniya zadachi Koshi dlya parabolicheskoj sistemy, Vestnik MEI, 2009, no. 6, pp. 38–44.
  4. Arnese, G. Su alcune proprieta dell’integrale di Poisson relativo ad una equazione parabolica di ordine 2 a coefficienti non costanti / G. Arnese // Ann. di Mat. Pura ed Appl. — 1971. — V. 91, № 1. — P. 1-16.
  5. Kamynin, L.I., On solution of the fundamental boundary value problems for a one-dimensional parabolic equation of second order by the method of potentials, Sib. Math. J., 1974, vol. 15, no. 4, pp. 573–592.
  6. Cherepova, M.F. The Cauchy problem for a multi-dimensional parabolic equation with Dini-continuous coefficients / M.F. Cherepova, I.V. Zhenyakova // J. Math. Sci. — 2022. — V. 264, № 5. — P. 581-602.
  7. Konenkov, A.N., The Cauchy problem for the heat equation in Zygmund spaces, Differ. Equat., 2005, vol. 41, no. 6, pp. 860–872.
  8. Konenkov, A.N., The Cauchy problem for parabolic equations in Zygmund spaces, Differ. Equat., 2006, vol. 42, no. 6, pp. 867–873.
  9. Tveritinov, V.A., O vtoroj kraevoj zadache dlya parabolicheskoj sistemy s odnoj prostranstvennoj peremennoj, Differ. Uravn., 1989, vol. 25, no. 12, pp. 2178–2179.
  10. Tveritinov, V.A., Reshenie vtoroi kraevoi zadachi dlya parabolicheskoi sistemy s odnoi prostranstvennoi peremennoi metodom granichnykh integral’nykh uravnenii, Moscow, 1989, dep. VINITI no. 6906–V89.
  11. Cherepova, M.F. The Cauchy problem for a parabolic system with nonuniform Holder coefficients / M.F. Cherepova // J. Math. Sci. — 2013. — V. 191, № 2. — P. 296-313.
  12. Baderko, E.A. Uniqueness theorem for parabolic Cauchy problem / E.A. Baderko, M.F. Cherepova // Appl. Anal. — 2016. — V. 95, № 7. — P. 1570-1580.
  13. Baderko, E.A. and Cherepova, M.F., Uniqueness of a solution to the Cauchy problem for parabolic systems, Dokl. Math., 2016, vol. 93, no. 3, pp. 316–317.
  14. Baderko, E.A. and Cherepova, M.F., Uniqueness of the solution of the Cauchy problem for parabolic systems, Differ. Equat., 2019, vol. 55, no. 6, pp. 806–814.
  15. Baderko, E.A. and Fedorov, K.D., On the smoothness of the poisson potential for second-order parabolic systems on the plane, Differ. Equat., 2023, vol. 59, no. 12, pp. 1613–1626.
  16. Dzyadyk, V.K. and Shevchuk, I.A., Theory of Uniform Approximation of Functions by Polynomials, Berlin, de Gruyter, 2008.
  17. Petrovskij, I.G., O probleme Koshi dlya sistem linejnyh uravnenij s chastnymi proizvodnymi v oblasti neanaliticheskih funkcij, Byull. Mos. Gos. Univ. Sek. A, 1938, vol. 1, no. 7, pp. 1–72.
  18. Friedman, A., Partial Differential Equations of Parabolic Type, Englewood Cliffs: Prentice-Hall, 1964.
  19. Zeineddin, M., O potenciale prostogo sloya dlya parabolicheskoj sistemy v klassah Dini (On the simple layer potential for a parabolic system in Dini classes), Cand. Sci. (Phys.-Math.) Diss., Moscow, 1992.
  20. Zeineddin, M., Gladkost’ potentsiala prostogo sloya dlya parabolicheskoi sistemy vtorogo poryadka v klassakh Dini, 1992, dep. VINITI no. 1294–V92.
  21. Kruzhkov, S.N., Estimates for the highest derivatives of solutions of elliptic and parabolic equations with continuous coefficients, Math. Notes of the Academy of Sciences of the USSR, 1967, vol. 2, no. 5, pp. 824–830.
  22. Vladimirov, V.S., Generalized Functions in Mathematical Physics, Moscow: Mir Publishers, 1979.
  23. Eidel’man, S.D., Parabolic Systems, North-Holland, Wolters-Nordhoff, 1969.

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