Geodesic
Coefficient inverse problem for an equation of mixed parabolic-hyperbolic type with nonlocal conditions
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2024), pp. 34-44 Cet article a éte moissonné depuis la source Math-Net.Ru

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The direct and inverse problems for a model mixed parabolic-hyperbolic equation with a characteristic line of type change have been studied. In the direct problem, a nonlocal problem for this equation with interior boundary conditions in the hyperbolic part of the domain has been investigated. The unknown of the inverse problem is the variable coefficient of the lower-order term in the parabolic equation. To determine it, the inverse problem is studied under the assumption that an integral overdetermination condition is known for the solution defined in the parabolic part of the domain. A theorem of unique solvability for the posed problems in the sense of a classical solution has been proven.
Keywords: equation of mixed parabolic-hyperbolic type, direct problem, inverse problem, uniqueness, integral equation, contraction mapping principle.
Mots-clés : existence 
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D. K. Durdiev. Coefficient inverse problem for an equation of mixed parabolic-hyperbolic type with nonlocal conditions. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2024), pp. 34-44. http://geodesic.mathdoc.fr/item/IVM_2024_10_a3/

[1] Zolina L.A., “O kraevoi zadache dlya modelnogo uravneniya giperbolo-parabolicheskogo tipa”, ZhVM i MF, 6:6 (1966), 991–1001 | MR | Zbl

[2] Bzhikhatlov Kh.G., Nakhushev A.M., “Ob odnoi kraevoi zadache dlya uravneniya smeshannogo parabolo-giperbolicheskogo tipa”, DAN SSSR, 183:2 (1968), 261–264 | MR

[3] Dzhuraev T.D., Kraevye zadachi dlya uravnenii smeshannogo i smeshanno-sostavnogo tipov, Izd-vo Fan, Tashkent, 1979 | MR

[4] Dzhuraev T.D., Sopuev A., Mamazhanov A., Kraevye zadachi dlya uravnenii parabolo-giperbolicheskogo tipa, Izd-vo Fan, Tashkent, 1986 | MR

[5] Salakhitdinov M.S., Mirsaburov M., “O nekotorykh kraevykh zadachakh dlya giperbolicheskogo uravneniya, vyrozhdayuschegosya vnutri oblasti”, Differents. uravneniya, 17:1 (1981), 129–136 | MR | Zbl

[6] Efimova S.V., Repin O.A., “Zadacha s nelokalnymi usloviyami na kharakteristikakh dlya uravneniya vlagoperenosa”, Differents. uravneniya, 40:10 (2004), 1419–1422 | MR | Zbl

[7] Repin O.A., “O zadache s operatorami M. Saigo na kharakteristikakh dlya vyrozhdayuschegosya vnutri oblasti giperbolicheskogo uravneniya”, Vestn. Samarsk. gos. tekhn. un-ta. Ser. fiz.-matem. nauki, 43 (2006), 10–14 | DOI

[8] Khubiev K.U., “Ob odnoi nelokalnoi zadache dlya uravneniya smeshannogo giperbolo-parabolicheskogo tipa”, Matem. zametki SVFU, 24:3 (2017), 12–18 | Zbl

[9] Sabitov K.B., Pryamye i obratnye zadachi dlya uravnenii smeshannogo parabolo-giperbolicheskogo tipa, Nauka, M., 2016 | MR

[10] Sabitov K.B., Safin E.M., “Obratnaya zadacha dlya uravneniya smeshannogo parabolo-giperbolicheskogo tipa v pryamougolnoi oblasti”, Izv. vuzov. Matem., 2010, no. 4, 55–62 | Zbl

[11] Sabitov K.B., Safin E.M., “Obratnaya zadacha dlya uravneniya smeshannogo parabolo-giperbolicheskogo tipa”, Matem. zametki, 87:6 (2010), 907–918 | DOI

[12] Sabitov K.B., “Nachalno-granichnaya i obratnye zadachi dlya neodnorodnogo uravneniya smeshannogo parabolo-giperbolicheskogo uravneniya”, Matem. zametki, 102:3 (2017), 415–435 | DOI | MR | Zbl

[13] Sabitov K.B., Sidorov S.N., “Obratnaya zadacha dlya vyrozhdayuschegosya parabolo-giperbolicheskogo uravneniya s nelokalnym granichnym usloviem”, Izv. vuzov. Matem., 2015, no. 1, 46–59 | Zbl

[14] Sidorov S.N., “Obratnye zadachi dlya vyrozhdayuschegosya smeshannogo parabolo-giperbolicheskogo uravneniya po nakhozhdeniyu somnozhitelei pravykh chastei, zavisyaschikh ot vremeni”, Ufimsk. matem. zhurn., 11:1 (2019), 75–89 | MR | Zbl

[15] Sabitov K.B. Sidorov S.N., “On a nonlocal problem for a degenerating parabolic-hyperbolic equation”, Diff. Equat., 50:3 (2014), 352–361 | DOI | MR | Zbl

[16] Sabitov K.B., Sidorov S.N., “Nachalno-granichnaya zadacha dlya neodnorodnykh vyrozhdayuschikhsya uravnenii smeshannogo parabolo-giperbolicheskogo tipa”, Itogi nauki i tekhn. Sovrem. matem. i ee pril. Temat. obz., 137, 2017, 26–60

[17] Prilepko A.I., Kostin A.V., Solovev V.V., “Obratnye zadachi nakhozhdeniya istochnika i koeffitsientov dlya ellipticheskikh i parabolicheskikh uravnenii v prostranstvakh Geldera i Soboleva”, Sib. zhurn. chist. i prikl. matem., 17:3 (2017), 67–85 | Zbl

[18] Ivanchov N.I., “Ob obratnoi zadache odnovremennogo opredeleniya koeffitsientov teploprovodnosti i teploemkosti”, Sib. matem. zhurn., 35:3 (1994), 612–621 | MR | Zbl

[19] Durdiev D.K., Durdiev D.D., “The Fourier spectral method for determining a heat capacity coefficient in a parabolic equation”, Turkish J. Math., 46:8 (2022), 3223–3233 | DOI | MR | Zbl

[20] Denisov A.M., Vvedenie v teoriyu obratnykh zadach, Izd-vo Mosk. un-ta, M., 1994

[21] Prilepko A.I., Orlovsky D.G., Vasin I.A., Methods for Solving Inverse Problems in Mathematical Physics, Textbooks Pure Appl. Math., 231, Marcel Dekker, New York, 1999 | MR

[22] Durdiev D.K., Zhumaev Z.Z., “Memory kernel reconstruction problems in the integrodifferential equation of rigid heat conductor”, Math. Methods Appl. Sci., 45:14 (2022), 8374–8388 | DOI | MR | Zbl

[23] Durdiev D.K., Zhumaev Z.Z., “One-dimensional inverse problems of finding the kernel of integrodifferential heat equation in a bounded domain”, Ukrain. Math. J., 73:11 (2022), 1723–1740 | DOI | MR | Zbl

[24] Durdiev D.K., Zhumaev Zh.Zh., “Zadacha opredeleniya teplovoi pamyati provodyaschei sredy”, Differents. uravneniya, 56:6 (2020), 796–807 | DOI | Zbl

[25] Romanov V.G., Obratnye zadachi matematicheskoi fiziki, Nauka, M., 1984 | MR

[26] Kabanikhin S.I., Obratnye i nekorrektnye zadachi, Sib. nauchn. izd-vo, Novosibirsk, 2009

[27] Hasanogǧlu A. Hasanov, Romanov V.G., Introduction to Inverse Problems for Differential Equations, Springer Int. Publ., 2017 | MR | Zbl

[28] Durdiev D.K., Totieva Z.D., Kernel Determination Problems in Hyperbolic Integro-Differential Equations, Sci. Foundation Ser. Math. Sci., Springer Nature Singapore Pte Ltd, 2023 | MR | Zbl

[29] Durdiev D.K., “Ob opredelenii koeffitsienta uravneniya smeshannogo parabolo-giperbolicheskogo tipa s nekharakteristicheskoi liniei izmeneniya”, Differents. uravneniya, 58:12 (2022), 1633–1644 | MR | Zbl

[30] Nakhushev A.M., “O nekotorykh kraevykh zadachakh dlya giperbolicheskikh uravnenii i uravnenii smeshannogo tipa”, Differents. uravneniya, 5:1 (1969), 44–59 | Zbl

[31] Tikhonov A.N., Samarskii A.A., Uravneniya matematicheskoi fiziki, Nauka, M., 1977 | MR