It is investigated the inverse problems for the degenerate parabolic equation. The minor coefficient of this equation is a linear polynomial with respect to space variable with two unknown time-dependent functions. The degeneration of the equation is caused by the monotone increasing function at the time derivative. It is  established conditions of existence and uniqueness of the classical solutions to the named problems in the case of weak degeneration.

[1] Bodnarchuk A., Huzyk N. Coefficient inverse problem for a parabolic equation with an arbitrary weak degeneration. Visnyk Lviv. Univ. Ser. Mech.-Math. 2011, 75, 28-42. (in Ukrainian)
[2] Pabyrivska N., Varenyk O. Determination of the younger coefficient in a parabolic equation. Visnyk Lviv. Univ. Ser. Mech.-Math. 2005, 64, 181-189. (in Ukrainian)
[3] Azizbayov Elvin. The nonlocal inverse problem of the identification of the lowest coefficient and the right-hand side in a second-order parabolic equation with integral conditions. Bound Value Probl 2019, 11, 1-19. doi: 10.1186/s13661-019-1126-z
[4] Hussein M., Lesnic D. and Ismailov M. An inverse problem of finding the time-dependent diffusion coefficient from an integral condition. Mathematical Methods in the Applied Sciences 2016, 39 (5), 963-980. https://doi.org/10.1002/mma.3482
[5] Hussein M., Lesnic D., Ivanchov M., Snitko H. Multiple time-dependent coefficient identification thermal problems with a free boundary. Applied Numerical Mathematics 2016, 99, 24- 50. https://doi.org/10.1016/j.apnum.2015.09.001
[6] Hussein M.S., Lesnic D., Kamynin V. and Kostin A. Direct and inverse source problems for degenerate parabolic equations. Journal of Inverse and Ill-posed Problems 2020. 28 (3), 425-448. DOI: 10.1515/jiip- 2019-0046
[7] Huzyk N. Inverse problem of determining the coefficients in a degenerate parabolic equation. Electronic Journal of Differential Equations 2014, 2014 (172), 1–11.
[8] Ivanchov M., Inverse problems for equations of parabolic type, VNTL Publishers, Lviv, 2003.
[9] Ivanchov M. and Saldina N. An inverse problem for strongly degenerate heat equation. J. Inv. Ill-Posed Problems 2006, 14 (5), 465-480. DOI: https://doi.org/10.1515/156939406778247598
[10] Ivanchov M. and Vlasov V. Inverse problem for a two-dimensional strongly degenerate heat equation. Electronic Journal of Differential Equations 2018, 2018 (77), 1–17.
[11] Kabanikhin S., Inverse and ill-posed problems: theory and applications, De Gruyter, 2012. DOI: https://doi.org/10.1515/9783110224016
[12] Kinash N.YE. An inverse problem for a 2d parabolic equation with nonlocaloverdetermination condition. Carpathian Math. Publ. 2016, 8 (1), 107–117. doi:10.15330/cmp.8.1.107-117
[13] Ladyzhenskaya O.A., Solonnikov V.A. and Uraltseva N.N., Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, Rhode Island, USA, 1968.
[14] Zhi-Xue Zhaoa, Mapundi K. Bandab and Bao-Zhu Guo. Simultaneous identification of diffusion coefficient, spacewise dependent source and initial value for one-dimensional heat equation. Math. Meth. Appl. Sci. 2017, 40, 3552–3565. DOI: 10.1002/mma.4245
[15] Zui-Cha Deng, Liu Yang. 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. https://doi.org/10.1016/j.cam.2011.04.006