Geodesic
Inverse coefficient problem for a quasilinear hyperbolic equation with final overdetermination
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 4, pp. 647-666 Cet article a éte moissonné depuis la source Math-Net.Ru

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The inverse problem of recovering a solution-dependent coefficient multiplying the lowest derivative in a hyperbolic equation is investigated. As overdetermination is required in the inverse problem, an additional condition is imposed on the solution to the equation with a fixed value of the timelike variable. Global uniqueness and local existence theorems are proved for the solution to the inverse problem. An iterative method is proposed for solving the inverse problem. 
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A. Yu. Shcheglov. Inverse coefficient problem for a quasilinear hyperbolic equation with final overdetermination. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 4, pp. 647-666. http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_4_a8/

[1] Romanov V. G., Nekotorye obratnye zadachi dlya uravnenii giperbolicheskogo tipa, Nauka, Novosibirsk, 1972 | MR

[2] Romanov V. G., Inverse problems of mathematical physics, VNU Science Press, Utrecht, 1987 | MR

[3] Malyshev I., On the unique restoration of the coefficient in the wave equation, Springer, New York, Berlin, 1989 | MR

[4] Denisov A. M., Vvedenie v teoriyu obratnykh zadach, Izd-vo MGU, M., 1994

[5] Prilepko A. I., Orlovsky D. G., Vasin I.A., Methods for solving inverse problems in mathematical physics, Marcel Dekker, Inc., New York, Basel, 1999 | MR

[6] Denisov A. M., Elements of the theory of inverse problems, VSP, Utrecht, 1999 | MR | Zbl

[7] Bidaibekov E. H., “Ob odnoi obratnoi zadache dlya kvazilineinogo uravneniya giperbolicheskogo tipa”, Matem. probl. geofiz., 4, Novosibirsk, 1973, 61–68

[8] Glushkova E. S., “Über die Eindeutigkeit einiger inverser Probleme fur die Telegraphengleichung”, Matem. probl. geofiz., 6, Novosibirsk, 1975, 130–144 | Zbl

[9] Glushkova E. S., “Eindeutigkeitssatz eines inversen Problems für eine quasilineare Gleichung hyperbolischen Type”, Nekorrektnye matem. zadachi i probl. geofiz., VTs SO AN SSSR, Novosibirsk, 1976, 35–45

[10] Glushkova E. S., “Ob odnoi obratnoi zadache”, Nekorrektnye matem. zadachi i probl. geofiz., VTs SO AN SSSR, Novosibirsk, 1979, 49–59

[11] Cannon J. R., Du Chateau P., “An inverse problem for an unknown source term in a wave equation”, SIAM J. Appl. Math., 43:3 (1983), 553–564 | DOI | MR | Zbl

[12] Cavaterra C., “An inverse problem for a semilinear wave equation”, Boll. Univ. Mat. Ital. (B). Ser. 7, 2:3 (1988), 695–711 | MR | Zbl

[13] Guo Bao Qi, Wang Jian Tao, “An inverse problem for a one-dimensional semilinear hyperbolic equation with an unknown source”, J. Harbin Inst. Tech., 4 (1989), 14–20 | MR | Zbl

[14] Shcheglov A. Yu., “Iterative method for recovery a nonlinear source in a hyperbolic equation with final overdetermination”, J. Inv. Ill-Posed Problems, 10:6 (2002), 629–641 | MR | Zbl

[15] Scheglov A. Yu., “Metod priblizhennogo resheniya obratnoi zadachi dlya polulineinogo uravneniya giperbolicheskogo tipa”, Zh. vychisl. matem. i matem. fiz., 43:1 (2003), 111–126 | MR

[16] Grasselli M., “An identification problem for a semilinear hyperbolic equation”, Boll. Univ. Mat. Ital. (B). Ser. 7, 2 (1988), 15–30 | MR

[17] Grasselli M., “A stability result for a nonlinear hyperbolic inverse problem”, Ric. Mat., 38:1 (1989), 293–311 | MR

[18] Grasselli M., “Local existence and uniqueness for a quasilinear hyperbolic inverse problem”, J. Appl. Analys., 32:1 (1989), 15–30 | DOI | MR | Zbl

[19] Denisov A. M., “Edinstvennost resheniya zadachi opredeleniya nelineinogo koeffitsienta sistemy uravnenii v chastnykh proizvodnykh v malom i tselom”, Sibirskii matem. zhurnal, 36:1 (1995), 60–71 | MR | Zbl

[20] Denisov A. M., “Determination of a nonlinear coefficient in a hyperbolic equation for the Goursat problem”, J. Inv. Ill-Posed Problems, 6:4 (1998), 327–334 | DOI | MR | Zbl

[21] Denisov A. M., “Suschestvovanie resheniya obratnoi zadachi dlya kvazilineinogo uravneniya giperbolicheskogo tipa”, Differents. ur-niya, 38:9 (2002), 1155–1164 | MR | Zbl

[22] Denisov A. M., “Global theorem of uniqueness of solution to inverse coefficient problem for a quasilinear hyperbolic equation”, Ill-Posed and Inverse Problems, VSP, Utrecht, 2002, 73–85 | MR

[23] Scheglov A. Yu., “Metod priblizhennogo resheniya v $C^2$ uravneniya giperbolicheskogo tipa s lipshitsevoi nelineinostyu”, Zh. vychisl. matem. i matem. fiz., 41:3 (2001), 420–435