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A nonstandard Cauchy problem for the heat equation

Mathematical Notes volume 102, pages 250–260 (2017)Cite this article

Abstract

We consider the Cauchy problem for the heat equation in a cylinder C T = X × (0, T) over a domain X in Rn, with data on a strip lying on the lateral surface. The strip is of the form S × (0, T), where S is an open subset of the boundary of X. The problem is ill-posed. Under natural restrictions on the configuration of S, we derive an explicit formula for solutions of this problem.

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References

  1. L. A. Aizenberg and N. N. Tarkhanov, “Conditionally stable problems and Carleman formulas,” Sibirsk. Mat. Zh. 31 (6), 9–15 (1990) [Sib. Math. J. 31 (6), 875–880 (1990)].

    MathSciNet  Google Scholar 

  2. N. N. Tarkhanov, The Cauchy Problem for Solutions of Elliptic Equations, in Math. Top. (Akademie Verlag, Berlin, 1995), Vol. 7.

    Google Scholar 

  3. M. M. Lavrent’ev, V. G. Romanov, and S. P. Shishatskii, Ill-Posed Problems of Mathematical Physics and Analysis (Nauka, Moscow, 1980) [in Russian].

    Google Scholar 

  4. M. Ikehata, “Two analytical formulae of the temperature inside a body by using partial lateral and initial data”, Inverse Problems 25 (035011) (2009).

    Google Scholar 

  5. R. E. Puzyrev and A. A. Shlapunov, “On an ill-posed problem for the heat equation,” Zh. SFU. Ser. Matem. i Fiz. 5 (3), 337–348 (2012).

    Google Scholar 

  6. M. S. Agranovich and M. I. Vishik, “Elliptic problems with a parameter and parabolic problems of general type,” Uspekhi Mat. Nauk 19 (3 (117)), 53–161 (1964).

    MATH  MathSciNet  Google Scholar 

  7. S. Bochner, Vorlesungen über Fouriersche Integrale (Akademie Verlag, Leipzig, 1932).

    MATH  Google Scholar 

  8. I. M. Gel’fand and G. E. Shilov, “The Fourier transform of rapidly increasing functions and uniqueness of the Cauchy problem,” Uspekhi Mat. Nauk 8 (6 (58)), 3–54 (1953).

    MathSciNet  Google Scholar 

  9. Sh. Yarmukhamedov, “The Cauchy problem for the Laplace equation,” Dokl. Akad. Nauk SSSR 235 (2), 281–283 (1977) [Soviet Math. Dokl. 18 (4), 939–942 (1977)].

    MATH  MathSciNet  Google Scholar 

  10. M. Ikehata, “Inverse conductivity problem in the infinite slab,” Inverse Problems 17 (3), 437–454 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  11. Sh. Yarmukhamedov, “The Carleman function and the Cauchy problem for the Laplace equation,” Sibirsk. Mat. Zh. 45 (3), 702–719 (2004) [Sib. Math. J. 45 (3), 580–595 (2004)].

    MATH  MathSciNet  Google Scholar 

  12. Sh. Yarmukhamedov and I. Yarmukhamedov, “The Cauchy problem for the Helmholtz equation,” in Ill-Posed and Non-Classical Problems of Mathematical Physics and Analysis (VSP, Utrecht, 2003), pp. 143–172.

    Google Scholar 

  13. O. Makmudov, I. Niyozov, and N. Tarkhanov, “The Cauchy problem of couple-stress elasticity,” in Complex Analysis and Dynamical Systems. III, Contemp. Math. (Amer. Math. Soc., Providence, RI, 2008), Vol. 455, pp. 297–310.

    Google Scholar 

  14. K. O. Makhmudov, O. I. Makhmudov, and N. Tarkhanov, “Equations of Maxwell type,” J. Math. Anal. Appl. 378 (1), 64–75 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  15. E. M. Landis and O. A. Oleinik, “Generalized analyticity and related properties of solutions of elliptic and parabolic equations,” Uspekhi Mat. Nauk 29 (2 (176)), 190–206 (1974).

    Google Scholar 

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Authors and Affiliations

  1. Alisher-Navoi Samarkand State University, Samarkand, Uzbekistan

    K. O. Makhmudov & O. I. Makhmudov

  2. Institut für Mathematik, Universität Potsdam, Potsdam, Germany

    N. Tarkhanov

Authors
  1. K. O. Makhmudov
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  2. O. I. Makhmudov
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  3. N. Tarkhanov
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Correspondence to K. O. Makhmudov.

Additional information

Original Russian Text © K. O. Makhmudov, O. I. Makhmudov, N. Tarkhanov, 2017, published in Matematicheskie Zametki, 2017, Vol. 102, No. 2, pp. 270–283.

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Makhmudov, K.O., Makhmudov, O.I. & Tarkhanov, N. A nonstandard Cauchy problem for the heat equation. Math Notes 102, 250–260 (2017). https://doi.org/10.1134/S0001434617070264

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  • DOI: https://doi.org/10.1134/S0001434617070264