On some classes of inverse parabolic problems of recovering the thermophysical parameters

S. G. Pyatkov   ; O. A. Soldatov

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 15 (2023) no. 3, pp. 23-33

Résumé

In the article we examine the question of regular solvability in Sobolev spaces of parabolic inverse coefficient problems. A solution is sought in the class of regular solutions that has all derivatives occurring in the equation summable to some power. The overdetermination conditions are the values of a solution at some collection of points lying inside the domain. The proof is based on a priori estimates and the fixed point theorem.

Détail

Keywords: inverse problem, initial-boundary value problem, uniqueness.
Mots-clés : parabolic equation, existence

Bibliographie

[1] Permyakov P.P., Identification of Parameters of the Mathematical Model of Heat and Moisture Transfer in Frozen Soils, Nauka Publ, Novosibirsk, 1989, 83 pp. (in Russ.)

[2] Prilepko A.I., Orlovsky D.G., Vasin I.A., Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, Inc, New York, 1999, 744 pp. | DOI | MR

[3] Belov Ya.Ya., Inverse Problems for Parabolic Equations, VSP, Utrecht, 2002, 211 pp. | MR

[4] Isakov V., Inverse Problems for Partial Differential Equations, Springer, Cham, 2018, 406 pp. | DOI | MR

[5] Kabanikhin S.I., Inverse and Ill-posed Problems: Theory and Applications, De Gruyter, Berlin–Boston, 2011, 459 pp. | DOI | MR

[6] Klibanov M.V., Li J., Inverse Problems and Carleman Estimates: Global Uniqueness, Global Convergence and Experimental Data, De Gruyter, Berlin–Boston, 2021, 344 pp. | DOI | MR | Zbl

[7] Hussein M.S., Huntul M.J., “Simultaneous Identification of Thermal Conductivity and Heat Source in the Heat Equation”, Iraqi Journal of Science, 62:6 (2021), 1968–1978

[8] Hussein M.S., Lesnic D., Ivanchov M.I., “Simultaneous Determination of Time-Dependent Coefficients in the Heat Equation”, Computers and Mathematics with Applications, 67:5 (2014), 1065–1091 | DOI | MR | Zbl

[9] Ivanchov M., Inverse Problems for Equations of Parabolic Type, Math. Studies. Monograph Series, 10, WNTL Publishers, Lviv, 2003 | MR

[10] Ivanchov N.I., Pabyrivska N.V., “On Determination of Two Time-Dependent Coefficients in a Parabolic Equation”, Siberian Mathematical Journal, 43 (2002), 323–329 | DOI | MR | Zbl

[11] Iskenderov A.D., “Multi-Dimensional Inverse Problems for Linear and Quasilinear Parabolic Equations”, Sov. Math. Dokl., 16 (1975), 1564–1568 | MR | Zbl

[12] Iskenderov A.D., Akhundov A.Ya., “Inverse Problem for a Linear System of Parabolic Equations”, Doklady Mathematics, 79:1 (2009), 73–75 | DOI | MR | Zbl

[13] Frolenkov I.V., Romanenko G.V., “On the Solution of an Inverse Problem for a Multidimensional Parabolic Equation”, Sibirskii Zhurnal Industrial'noi Matematiki, 15:2(50) (2012), 139–146 (in Russ.) | Zbl

[14] Pyatkov S.G., Samkov M.L., “On some Classes of Coefficient Inverse Problems for Parabolic Systems of Equations”, Siberian Advances in Mathematics, 22:4 (2012), 287–302 | DOI | MR | Zbl

[15] Pyatkov S.G., Tsybikov B.N., “On Some Classes of Inverse Problems for Parabolic and Elliptic Equations”, J. Evol. Equat., 11:1 (2011), 155–186 | DOI | MR | Zbl

[16] Pyatkov S.G., “On Some Classes of Inverse Problems for Parabolic Equations”, J. Inv. Ill-Posed problems, 18:8 (2011), 917–934 | DOI | MR

[17] Pyatkov S.G., “On Some Classes of Inverse Problems with Overdetermination Data on Spatial Manifolds”, Siberian Mathematical Journal, 57:5 (2016), 870–880 | DOI | MR | Zbl

[18] Kozhanov A.I., “The Heat Transfer Equation with an Unknown Heat Capacity Coefficient”, Journal of Applied and Industrial Mathematics, 14:1 (2020), 104–114 | DOI | MR | Zbl

[19] Kozhanov A.I., “Parabolic Equations with an Unknown Coeffcients Depending on Time”, Comput. Math. Math. Phys., 45:12 (2005), 2085–2101 | MR | Zbl

[20] Kamynin V.L., “Unique Solvability of the Inverse Problem of Determination of the Leading Coefficient in a Parabolic Equation”, Differential Equations, 47:1 (2011), 91–101 | DOI | MR | Zbl

[21] Pyatkov S.G., Rotko V.V., “Inverse Problems with Pointwise Overdetermination for some Quasilinear Parabolic Systems”, Siberian Advances in Mathematics, 30:2 (2020), 124–142 | DOI | DOI | MR | MR | Zbl

[22] Pyatkov S.G., Rotko V.V., “On some Parabolic Inverse Problems with the Pointwise Overdetermination”, AIP Conference Proceedings, 1907 (2017), 020008 | DOI

[23] S.G. Pyatkov, “Identification of Thermophysical Parameters in Mathematical Models of Heat and Mass Transfer”, Journal of Computational and Engineering Mathematics, 9:2 (2022), 52–66 | DOI | Zbl

[24] R. Denk, M. Hieber, J. Prüss, “Optimal $L^p$-$L^q$-Estimates for Parabolic Boundary Value Problems with Inhomogeneous Data”, Mathematische Zeitschrift, 257:1 (2007), 93–224 | DOI | MR

[25] O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Uraltseva, Ladyzhenskaya, O.A. Linear and Quasi-linear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, 1968, 648 pp. | DOI | MR

[26] H. Amann, “Compact Embeddings of Vector-Valued Sobolev and Besov Spaces”, Glasnik matematicki, 35:1 (2000), 161–177 | MR | Zbl

[27] V.P. Mikhailov, Partial Differential Equations, Mir, M., 1978, 396 pp. | MR

[28] S.M. Nikol'skii, Approximation of Functions of Several Variables and Imbedding Theorems, Grundl. Math. Wissensch., 205, Springer-Verlag, New York, 1975, 418 pp. | MR | Zbl

[29] Amann H., Linear and Quasilinear Parabolic Problems, v. I, Monographs in Mathematics, 89, Abstract Linear Theory, Birkhäuser Verlag, Basel, 1995, 338 pp. | DOI | MR | Zbl

[30] Triebel H., Interpolation Theory, Function Spaces, Differential Operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978, 528 pp. | DOI | MR