Linear Inverse Problems for a Class of Degenerate Equations of Sobolev Type

Considering degenerate equations of Sobolev type with principal part an elliptic parabolic operator, we study solvability of linear inverse problems with final and integral overdetermination and prove existence of regular solutions.

Keywords

linear inverse problems, final overdetermination, integral overdetermination, degenerate equations of Sobolev type, regular solutions, existence.

References

1. Sobolev S.L. On a Boundary Value Problem of Mathematical Physics [Ob odnoy kraevoy zadache matematicheskoy fiziki]. Izv. AN SSSR. Ser. Matem., 1954, vol. 18, no. 2, pp. 3 - 50.
2. Demidenko G.V., Uspenskii S.V. Partial Differential Equations and Systems Not Solvable with Respect to the Highest-Order Derivative. New York; Basel, Marcel Dekker, 2003.
3. Kozhanov A.I. Composite Type Equations and Inverse Problems. Utrecht, VSP, 1999.
4. Egorov I.E., Pyatkov S.G., Popov S.V. Neklassicheskie differentsial'no-operatornye uravneniya [Nonclassical Differential-operator Equations]. Novosibirsk, Nauka, 2000.
5. Pyatkov S.G. Operator Theory. Nonclassical Problems. Utrecht; Boston; Koln; Tokyo, VSP, 2002.
6. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroup of Operators. Utrecht, VSP, 2003.
7. Hayashi N., Kaikina E.I., Naumkin P.I., Shishmarev I.A. Asymptotic for Dissipative Nonlinear Equations. Springer, 2006.
8. Sveshnikov A.G., Al'shin A.B., Korpusov M.O., Pletner Yu.D. Lineynye i nelineynye uravneniya sobolevskogo tipa [Linear and Nonlinear Equations of Sobolev Type]. Moscow, Fizmatlit, 2007.
9. Korpusov M.O. Razrushenie v neklassicheskikh nelokal'nykh uravneniyakh [Blow-up of Nonclassical Non-local Equations]. Moscow, Librokom, 2011.
10. Kozhanov A.I. Nonlinear Loaded Equations and Inverse Problems [Nelineynye nagruzhennye uravneniya i obratnye zadachi]. Computational Mathematics and Mathematical Physics, 2004, vol. 44, no. 4, pp. 694 - 716.
11. Kozhanov A.I. The Solvability of Inverse Problems of Reconstruction Coefficients in the Equations of Composite Type [O razreshimosti obratnykh zadach vosstanovleniya koeffitsientov v uravneniyakh sostavnogo tipa]. Vestn. NGU. Seriya Matematika, mekhanika, informatika, 2008, vol. 8, issue 2, pp. 81 - 99.
12. Kozhanov A.I. On the Solvability of the Inverse Problem of Finding the Leading Coefficient of Equation of Composite Type [O razreshimosti obratnoy zadachi nakhozhdeniya starshego koeffitsienta v uravnenii sostavnogo tipa]. Vestnik Yuzhno-Ural'skogo universiteta. Seria 'Matematicheskoe modelirovanie i programmirovanie', 2008, no. 15 (115), issue 1, pp. 27 - 36.
13. Kozhanov A.I. On the Solvability of the Inverse Problem for Some Sobolev-type Equations [O razreshimosti koeffitsientnykh obratnykh zadach dlya nekotorykh uravneniy sobolevskogo tipa]. Nauch. vedomosti Belgorod. gos. un-ta. Matematika. Fizika, 2010, no. 5, issue 18, pp. 88 - 98.
14. Ablabekov B.S. Inverse Problems for Equations Benjamin - Bona - Mahoki [Obratnye zadachi dlya uravneniya Bendzhamena-Bona-Makhoki]. Informatsionnye tekhnologii i obratnye zadachi ratsional'nogo prirodoispol'zovaniya, Khanty-Mansiysk, Yugorskiy NII informatsionnykh tekhnologiy, 2005, pp. 6 - 9.
15. Fedorov V.E., Urazaeva A.V. An Inverse Problem for Linear Sobolev Type Equations. J. Inverse Ill-Posed Probl, 2004, vol. 12, no. 4, pp. 387 - 395.
16. Fedorov V.G., Ivanova N.D. A Nonlinear Inverse Problem for the Oskolkov System [Nelineynaya obratnaya zadacha dlya sistemy Oskolkova]. Teoriya i chislennye metody resheniya obratnykh i nekorrektnykh zadach (shkola-konferentsiya): tez. dokl. Novosibirsk, 2011, pp. 72.
17. Nazushev A.M. Uravneniya matematicheskoy biologii [The Equations of Mathematical Biology]. Moscow, Vyssh. shk., 1995.
18. Dzhenaliev M.T. K teorii lineynykh kraevykh zadach dlya nagruzhennykh differentsial'nykh uravneniy [To the Theory of Linear Boundary Value Problems for Loaded Differential Equations]. Almaty, Izd. In-ta Teor. i priklad. matematiki, 1995.
19. Dzhenaliev M.T., Ramazanov M.I. Nagruzhennye uravneniya kak vozmushcheniya differentsial'nykh uravneniy [Loaded Equation as Perturbations of Differential Equations]. Almaty, FYLYM, 2010.
20. Trenogin V.A. Funktsional'nyy analiz [Functional Analysis]. Moscow, Nauka, 1980.
21. Yakubov S.Ya. Lineynye differentsial'no-operatornye uravneniya i ikh prilozheniya [Linear Differential-operator Equations and Their Applications ]. Baku, Elm, 1985.
22. Ladyzhenskaya O.A., Ural'tseva N.N. Lineynye i kvazilineynye uravneniya ellipticheskogo tipa [Linear and Quasilinear Equations of Elliptic Type]. Moscow, Nauka, 1973.