Abstract
Consider an inhomogeneous equation of the mixed parabolic-hyperbolic type in a rectangular parallelepiped, where the inhomogeneity is the product of two factors, one being a function depending only on the spatial variables and the other being a function depending only on time. For this equation, we study the inverse problem of finding the factor depending on the spatial variables. A criterion for the uniqueness of the solution is established. The solution is constructed as the sum of a series in an orthogonal function system. When justifying the convergence of the series, one encounters the problem of small denominators depending on two positive integer arguments. We obtain estimates guaranteeing the separation of the denominators from zero with an indication of the asymptotics. These estimates permit justifying the convergence of the series in the class of regular solutions. The stability of the solution under perturbations of the boundary functions is established.
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REFERENCES
Kapustin, N.Yu., The Tricomi problem for a parabolic-hyperbolic equation with a degenerate hyperbolic. Part I, Differ. Uravn., 1987, vol. 23, no. 1, pp. 72–78.
Kapustin, N.Yu., The Tricomi problem for a parabolic-hyperbolic equation with a degenerate hyperbolic. Part II, Differ. Uravn., 1988, vol. 24, no. 8, pp. 1379–1386.
Kapustin, N.Yu. and Moiseev, E.I., On the spectral problem from the theory of the parabolic-hyperbolic heat equation, Dokl. Akad. Nauk SSSR, 1997, vol. 352, no. 4, p. 451.
Sabitov, K.B., Tricomi problem for a mixed parabolic-hyperbolic equation in a rectangular domain, Math. Notes, 2009, vol. 86, no. 2, pp. 249–254.
Sabitov, K.B., Pryamye i obratnye zadachi dlya uravnenii smeshannogo parabolo-giperbolicheskogo tipa (Direct and Inverse Problems for Equations of Mixed Parabolic-Hyperbolic Type), Moscow: Nauka, 2016.
Sabitov, K.B., Initial boundary and inverse problems for the inhomogeneous equation of a mixed parabolic-hyperbolic equation, Math. Notes, 2017, vol. 102, no. 3, pp. 378–395.
Sidorov, S.N., Nonlocal problem for a degenerating parabolic-hyperbolic equation, Dokl. Ross. Akad. Nauk, 2012, vol. 14, no. 3, pp. 34–44.
Sabitov, K.B. and Sidorov, S.N., On a nonlocal problem for a degenerating parabolic-hyperbolic equation, Differ. Equations, 2014, vol. 50, no. 3, pp. 352–361.
Sidorov, S.N., Nonlocal problems for an equation of mixed parabolic-hyperbolic type with power degeneration, Russ. Math., 2015, vol. 59, no. 12, pp. 46–55.
Sabitov, K.B. and Sidorov, S.N., Initial–boundary-value problem for inhomogeneous degenerate equations of mixed parabolic-hyperbolic type, Itogi Nauki i Tekh. Ser.: Sovrem. Mat. Pril., 2017, vol. 137, pp. 26–60.
Sabitov, K.B. and Sidorov, S.N., Initial–boundary-value problem for inhomogeneous degenerate equations of mixed parabolic-hyperbolic type, J. Math. Sci., 2019, vol. 236, no. 6, pp. 603–640.
Lavrent’ev, M.M., Reznitskaya, K.G., and Yakhno, V.G., Odnomernye obratnye zadachi matematicheskoi fiziki (One-dimensional Inverse Problems of Mathematical Physics), Novosibirsk: Vychisl. Tsentr Sib. Otd. Akad. Nauk SSSR, 1982.
Romanov, V.G., Obratnye zadachi matematicheskoi fiziki (Inverse Problems of Mathematical Physics), Moscow: Akad. Nauk SSSR, 1984.
Denisov, A.M., Vvedenie v teoriyu obratnykh zadach (Introduction to the Theory of Inverse Problems), Moscow: Izd. Mosk. Gos. Univ., 1994.
Prilepko, A.I., Orlovsky, D.G., and Vasin, I.A., Methods for Solving Inverse Problems in Mathematical Physics, New York–Basel: Marcel Dekker, 1999.
Bukhgeim, A.L., Introduction to the Theory of Inverse Problems, Utrecht: VSP, 2000.
Ivanchov, M., Inverse Problems for Equations of Parabolic Type, Lviv, 2003.
Isakov, V., Inverse Problems for Partial Differential Equations, New York: Springer, 2006.
Kabanikhin, S.I., Obratnye i nekorrektnye zadachi (Inverse and Ill-Posed Problems), Novosibirsk: Izd. Sib. Otd. Akad. Nauk SSSR, 2009.
Sabitov, K.B. and Safin, E.M., The inverse problem for a mixed-type parabolic-hyperbolic equation in a rectangular domain, Russ. Math., 2010, vol. 56, no. 4, pp. 48–54.
Sabitov, K.B. and Safin, E.M., The inverse problem for an equation of mixed parabolic-hyperbolic type, Math. Notes, 2010, vol. 87, no. 6, pp. 880–889.
Sabitov, K.B. and Sidorov, S.N., Inverse problem for degenerate parabolic-hyperbolic equation with nonlocal boundary condition, Russ. Math., 2015, vol. 59, no. 1, pp. 39–50.
Sidorov, S.N., Nonlocal inverse problem for determining the right-hand sides of a degenerate equation of mixed parabolic-hyperbolic type, Nauchn. Vedomosti Belgorod. Gos. Univ. Mat. Fiz., 2014, no. 19 (190), issue 36, pp. 45–57.
Sidorov, S.N., Inverse problems for an equation of a mixed parabolic-hyperbolic type with a degenerate parabolic part, Sib. Elektron. Mat. Izv., 2019, vol. 16, pp. 144–157.
Sidorov, S.N., Inverse problems for a degenerate mixed parabolic-hyperbolic equation on finding time-depending factors in right hand sides, Ufa Math. J., 2019, vol. 11, no. 1, pp. 75–89.
Sabitov, K.B. and Sidorov, S.N., Initial–boundary value problem for a three-dimensional equation of the parabolic-hyperbolic type, Differ. Equations, 2021, vol. 57, no. 8, pp. 1042–1052.
Il’in, V.A., Sadovnichii, V.A., and Sendov, Bl.Kh., Matematicheskii analiz. T. 2 (Mathematical Analysis. Vol. 2), Moscow: Izd. Mosk. Gos. Univ., 1987.
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This work was supported by the Russian Foundation for Basic Research, project no. 19-31-60016.
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Translated by V. Potapchouck
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Sidorov, S.N. Inverse Problem of Finding a Factor Depending on the Spatial Variables on the Right-Hand Side in a Parabolic-Hyperbolic Equation. Diff Equat 57, 1585–1597 (2021). https://doi.org/10.1134/S0012266121120053
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DOI: https://doi.org/10.1134/S0012266121120053