Geodesic
The heat transfer equation with an unknown heat capacity coefficient
Sibirskij žurnal industrialʹnoj matematiki, Tome 23 (2020) no. 1, pp. 93-106.

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Under study are the inverse problems of finding, together with a solution $u(x,t)$ of the differential equation $cu_t -\Delta u + a(x,t)u = f(x,t)$ describing the process of heat distribution, some real $c$ characterizing the heat capacity of the medium (under the assumption that the medium is homogeneous). Not only the initial condition is imposed on $u(x,t)$, but also the usual conditions of the first or second initial-boundary value problems as well as some special overdetermination conditions. We prove the theorems of existence of a solution $(u(x,t),c)$ such that $u(x,t)$ has all Sobolev generalized derivatives entered into the equation, while $c$ is a positive number.
Keywords: heat transfer equation, heat capacity coefficient, inverse problem, final-integral overdetermination conditions
Mots-clés : existence. 
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A. I. Kozhanov. The heat transfer equation with an unknown heat capacity coefficient. Sibirskij žurnal industrialʹnoj matematiki, Tome 23 (2020) no. 1, pp. 93-106. http://geodesic.mathdoc.fr/item/SJIM_2020_23_1_a8/

[1] A. I. Prilepko, D. G. Orlovsky, I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, N.Y., 1999 | MR

[2] V. Isakov, Inverse Problems for Partial Differential Equations, Springer-Verl, Heidelberg, 2009 | MR

[3] M. Ivanchov, Inverse Problems for Equations of Parabolic Type, Math. Studies. Monograph Series, 10, VNTL Publ., Lviv, 2003 | MR | Zbl

[4] Yu. E. Anikonov, Inverse Problems for Kinetic and Other Evolution Equations, VSP, Utrecht, 2001 | MR

[5] A. I. Kozhanov, “Nonlinear loaded equations and inverse problems”, Zh. Vychisl. Mat. Mat. Fiz., 44:4 (2004), 694–716 (in Russian) | MR | Zbl

[6] A. I. Kozhanov, “Parabolic equations with unknown coefficients”, Zh. Vychisl. Mat. Mat. Fiz., 57:6 (2017), 963–973 (in Russian)

[7] A. I. Kozhanov, “A Nonlinear loaded parabolic equation and a related inverse problem”, Math. Notes, 76:6 (2004), 784–795 | DOI | DOI | MR | Zbl

[8] A. I. Kozhanov, “Parabolic equations with an unknown absorption coefficient”, Dokl. Akad. Nauk, 401:6 (2006), 740–743 (in Russian)

[9] A. I. Kozhanov, R. R. Safiullova, “Linear Inverse Problems for Parabolic and Hyperbolic Equations”, J. Inverse Ill-Posed Probl., 18:1 (2010), 1–18 | DOI | MR | Zbl

[10] A. Lorenzi, “Recovering two constants in a linear parabolic equation”, Inverse Problems in Applied Sciences, J. Phys. Conf. Series, 73, 2007, 012014 | DOI

[11] A. Lorenzi, G. Mola, “Identification of a real constant in linear evolution equation in a Hilbert spaces”, Inverse Probl. Imaging, 5:3 (2011), 695–714 | DOI | MR | Zbl

[12] G. Mola, “Identification of the diffusion coefficient in linear evolution equation in a Hilbert spaces”, J. Abstr. Differ. Equ. Appl., 2:1 (2011), 18–28 | MR

[13] A. Lorenzi, G. Mola, “Recovering the reaction and the diffusion coefficients in a linear parabolic equations”, Inverse Problems, 28:7 (2012), 075006 | DOI | MR | Zbl

[14] A. Lorenzi, E. Paparoni, “Identifications of two unknown coefficients in an integro-differential hyperbolic equation”, J. Inverse Ill-Posed Probl., 1:2 (1993), 331–348 | MR | Zbl

[15] A. S. Lyubanova, “Identification of a constant coefficient in an elliptic equation”, Appl. Anal., 87:10–11 (2008), 1121–1128 | DOI | MR | Zbl

[16] A. I. Kozhanov, R. R. Safiullova, “Definition of parameters in a telegraph equation”, Ufimsk. Mat. Zh., 9:1 (2017), 63–74 (in Russian)

[17] V. S. Vladimirov, Equations of Mathematical Physics, Nauka, M., 1976 (in Russian) | MR

[18] S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, Nauka, M., 1988 (in Russian) | MR

[19] O. A. Ladyzhenskaya, N. N. Ural'tseva, Linear and Quasilinear Equations of Elliptic Type, Nauka, M., 1973 (in Russian) | MR

[20] S. Ya. Yakubov, Linear DifferentialOperator Equations and Their Applications, Publ. Elm, Baku, 1985 (in Russian)

[21] A. I. Kozhanov, Composite Type Equations and Inverse Problems, VSP, Utrecht, 1999 | MR | Zbl

[22] A. G. Sveshnikov, A. B. Al'shin, M. O. Korpusov, Yu. D. Pletner, Linear and Nonlinear Sobolev Type Equations, Fizmatlit, M., 2007 (in Russian)