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
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)].
N. N. Tarkhanov, The Cauchy Problem for Solutions of Elliptic Equations, in Math. Top. (Akademie Verlag, Berlin, 1995), Vol. 7.
M. M. Lavrent’ev, V. G. Romanov, and S. P. Shishatskii, Ill-Posed Problems of Mathematical Physics and Analysis (Nauka, Moscow, 1980) [in Russian].
M. Ikehata, “Two analytical formulae of the temperature inside a body by using partial lateral and initial data”, Inverse Problems 25 (035011) (2009).
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).
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).
S. Bochner, Vorlesungen über Fouriersche Integrale (Akademie Verlag, Leipzig, 1932).
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).
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)].
M. Ikehata, “Inverse conductivity problem in the infinite slab,” Inverse Problems 17 (3), 437–454 (2001).
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)].
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.
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.
K. O. Makhmudov, O. I. Makhmudov, and N. Tarkhanov, “Equations of Maxwell type,” J. Math. Anal. Appl. 378 (1), 64–75 (2011).
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).
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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