In a domain with known boundaries, an inverse problem for a parabolic equation with strong degeneration is investigated. The degeneration of the equation is caused by a power function of time with the highest derivative of an unknown function. It is known that the lowest coefficient of the equation is a polynomial of the first degree in a spatial variable with two unknown coefficients of time. Boundary conditions of the second kind and the values ​​of thermal moments are given as redetermination conditions. The conditions for the existence and uniqueness of the classical solution of the specified problem are established.

[1] Azari H., Li C., Nie Y. and Shang S. Determination of an unknown coefficient in a parabolic inverse problem. Dynamics of Continuos, Discrete and Impulsive Systems. Series A: Math. Analysis 2004, 11, 665-674. doi: 124328203.
[2] Brodyak O., Huzyk N. Coefficient inverse problems for parabolic equation with general weak degeneartion. Bukovinian Math.Journal 2021, 9 (1), 91-106. doi: 10.31861/bmj2021.01.08. (in Ukrainian)
[3] Cannarsa Piermarco, Doubova Anna and Yamamoto Masahiro Inverse problem of reconstruction of degenerate diffusion coefficient in a parabolic equation. Inverse problems 2021, 37 (12), 1-39. doi: arXiv:2106.06832.
[4] Cannon J.R. and Peres-Esteva S. Determination of the coefficient of u_x in a linear parabolic equation. Inverse Problems 1994, 10 (3), 521-531. doi: 10.1088/0266-5611/10/3/002.
[5] Cannon J.R. and Rundell W. Recovering a time-dependent coefficient in a parabolic differential equation. J. Math. Anal. Appl. 1991, 160, 572-582. doi: https://core.ac.uk/reader/82169960.
[6] Deng Zui-Cha, Yang Liu. An inverse problem of identifying the coefficient of first order in a degenerate parabolic equation. J. of Computational and Applied Math 2011, 235, 4407-4417. doi: arXiv:1309.7421.
[7] Hazanee A. and Lesnic D. Determination of a time-dependent coefficient in the bioheat eqaution. International Journal of Mechanical Sciences 2014, 88, 259–266. doi: 10.1016/j.ijmecsci.2014.05.017.
[8] Huntul M.J. and Lesnic D. Determination of Time-Dependent Coefficients for a Weakly Degenerate Heat Equation. Computer Modeling in Engineering and Sciences 2020, 123 (2), 475-494. https://doi.org/10.32604/cmes.2020.08791.
[9] Hussein M.S. and Lesnic D. Identification of the time-dependent conductivity of an inhomogeneous diffusive material. Applied Mathematics and Computation 2015, 269, 35–58. doi: 10.1016/j.amc.2015.07.039.
[10] Huzyk N. Inverse problem of determining the coefficients in a degenerate parabolic equation. Electronic Journal of Differential Equations 2014, 172, 1-11. doi: 10.7153/dea-2021-13-14.
[11] Huzyk N. Inverse free boundary problems for a generally degenerate parabolic equation. Journal of Inverse and Ill-Posed Problems 2015, 23, 103-119. doi: 10.1515/jiip-2011-0016.
[12] Huzyk N. Determination of the lower coefficient in a parabolic equation with strong degeneration. Ukrainian Mathematical Journal 2016, 68 (7), 1049-1061. doi:10.1007/s11253-016-1276-4 (Translation from Ukrains’kyi Matematychnyi Zhurnal 2006, 68 (7), 922-932, https://umj.imath.kiev.ua/index.php/umj/article/view/1886. (in Ukrainian))
[13] Huzyk N. Coefficient inverse problem for the degenerate parabolic equation. Differential Equations and Applications 2021, 13 (3), 243-255. doi: 10.7153/dea-2021-13-14.
[14] Huzyk N.M., Pukach P.Y., Vovk M.I. Coefficient inverse problem for the strongly degenerate parabolic equation. Carpathian Math. Publ. 2023, 15 (1), 52-65. doi:10.15330/cmp.15.1.52-65.
[15] Ivanchov M. Inverse problems for equations of parabolic type. VNTL Publishers, Lviv, 2003.
[16] Ivanchov M.I., Saldina N.V. Inverse problem for a parabolic equation with strong power degeneration. Ukr Math J 2006, 58 (11), 1685–1703. doi: 10.1007/s11253-006-0162-x (Translation from Ukrains’kyi Matematychnyi Zhurnal 2006, 58 (11), 1487–1500. (in Ukrainian))
[17] Jones B.F. The determination of a coefficient in a parabolic differential equation. I. Existence and uniqueness. J. Math. Mech. 1962, 11 (5), 907-918. doi: http://www.jstor.org/stable/24900744.
[18] Lesnic D., Yousefi S.A., and Ivanchov M.Determination of a timedependent diffusivity from nonlocal conditions. Journal of Applied Mathematics and Computing 2013, 41 (1-2), 301–320. doi: 10.1007/s12190-012-0606-4.
[19] Lowe Bruce D. and Rundell William The determination of a coefficient in a parabolic equation from input sources. Journal of Applied Mathematics 1994, 52 (1), 31-50. doi: 10.1093/imamat/52.1.31.
[20] Nguyen Duc Phuong, Dumitru Baleanu, Tran Thanh Phong, Le Dinh Long. Recovering the source term for parabolic equation with nonlocal integral condition. Mathematical Methods in Applied Sciences 2021, 44 (11), 9026-9041. doi: 10.1002/mma.7331.
[21] Pabyrivska N. and Pabyrivskyy V. On the determination of an unknown source in the parabolic equation. Math modeling and computing 2017, 4 (2), 171-176. doi: 10.23939/mmc2017.02.171
[22] Ranran Li and Zhiyuan Li. Identifying unknown source in degenerate parabolic equation from final observation. Inverse Problems in Science and Engineering 2021, 29 (7), 1012-1031. doi: 10.1080/17415977.2020.1817005.
[23] Saldina N. Inverse problem for parabolic equation with weak degeneration. Маthеmatical Меthоds And Physicoмеchanical Fields 2006, 49 (3), 7-17. (in Ukrainian)
[24] Trong D.D. and Ang D.D.Coefficient identification for a parabolic equation. Inverse Problems 1994, 10 (3), 733-752. doi: 10.1088/0266-5611/10/3/015.