Inverse Problems for Determining Boundary Regimes for Some Equations of Sobolev Type

We study the solvability of the inverse problems of finding a solution to some Sobolev type equations along with the unknown coefficients of a special type defining the boundary modes (boundary data) in the first or the third initial-boundary value problems respectively. The presence of an unknown coefficient in such problems supposes that there is an additional condition - an overdetermination condition along with the initial and boundary conditions that are typical for the corresponding class of differential equations. In the present work this condition is represented by the integral overdetermination, when some integrals by the cross-section of the cylindrical domain by planes t=const are equal to zero. The goal of this research is to prove the existence of regular (with all needed generalized according to S.L. Sobolev derivatives) solutions of equation. Along with the specific results there are also some of their possible generalizations.

Keywords

Sobolev type equations; inverse problems; unknown boundary data; integral overdetermination; regular solutions; solvability.

References

1. Prilepko A.I., Orlovsky D.G., Vasin I.A. Methods for Solving Inverse Problems in Mathematical Physics. New York, Marcel Dekker, 1999.
2. Kabanikhin S.I. Inverse and Ill-posed Problems. Novosibirsk, Siberian publishing, 2009.
3. Alekseev C.V. Optimization in the Stationary Problems of Heat and Mass Transfer and Magnetic Hydrodynamics. Science World, 2010, pp. 142-143.
4. Kostin A., Prilepko A. On Some Problems of Restoration of a Boundary Condition for a Parabolic Equation. Differential Equations, 1996, vol. 32, no. 1, pp. 113-122.
5. Boruhov V.T., Korzyuk V.I. [Application Neoclassical Boundary Value Problems for the Restoration of Boundary Modes of Transport Processes]. Vestnik Belorusskogo universiteta. Ser. I., 1998, no. 3, pp. 54-57. (in Russian)
6. Boruhov V.T., Vabishchevich P.N., Korzyuk V.I. [The Reduction of a Class of Inverse Problems of Heat Conduction to Direct the Initial-Boundary Value Problems]. Inzhenerno-fizicheskiy zhurnal, 2000, vol. 73, no. 4, pp. 742-747. (in Russian)
7. Korotkiy A.I. Reconstruction of Boundary Regimes in the Inverse Problem of Thermal Convection of an Incompressible Fluid. Tr. IMM DVO AN, 2006, vol. 12, no. 2, pp. 88-97. (in Russian)
8. Kozhanov A.I. The Problem of Recovery of the Boundary Condition for a Heat Equation. Analiticheskiye metody analiza i differetsial'nyh uravneniy: materialy 6 mezhdunarodnoy konferentsyi pamyati prof. A.A. Kilbasa, Minsk, Izd. BGU, 2012, pp. 87-96.
9. Kozhanov A.I. [Linear Inverse Problems for Certain Classes of Non-Stationary Equation]. Trudy VI mezhdunarodnoy molodezhnoy shkoly-konferentsyi 'Teoriya i chislennyye metody resheniy obratnyh i nekorrektnyh zadach'. Sibirskiye Elektronnyye Matematicheskiye Izvestiya, 2015, vol. 12, pp. 264-275.
10. Ionkin N.I. [Solution of a Boundary-Value Problem in Heat Condition with a Nonclassical Boundary Condition]. Differential Equations, 1977, vol. 13, no. 2, pp. 204-211.
11. Pulkina L.S. [A Nonlocal Problem with Two Integral Conditions for Hyperbolic Equations on a Plane]. Neklassicheskiye uravneniya matematicheskoy fiziki [Non-Classical Equations of Mathematical Physics]. Novosibirsk, Institut matematiki SO RAN, 2007, pp. 232-236.
12. Kozhanov A.I. [On the Solvability of Some Boundary Value Problems with a Shift for Linear Hyperbolic Equations]. Matematicheskiy zhurnal [Mathematical Journal], Almaty, 2009, vol. 9, no. 3, pp. 78-92.
13. Kozhanov A.I. [On the Solvability of Boundary Value Problems with Non-Local and Integral Conditions for Parabolic Equations]. Nelineynye granichnyye zadachi. IPMM NAN Ukrainy, 2010, vol. 20, pp. 54-76.
14. Pulkina L.S. Zadachi s neklassicheskimi usloviyami dlya giperbolicheskih uravneniy. Samara, Izd. Samarskogo Universiteta, 2012.
15. Kozhanov A.I. Problems with Integral-Type Conditions for Some Classes of Nonstationary Equations. Doklady Mathematics, 2014, vol. 90, no. 1, pp. 440-443. DOI: 10.1134/S1064562414050044
16. Kozhanov A.I. On the Solvability of Spatially Nonlocal Problems with Conditions of Integral Form for Some Classes of Non stationary Equations. Differential Equations, 2015, vol. 5, no. 18, pp. 1043-1050. DOI: 10.1134/S001226611508008X
17. Barenblatt G.I., Zheltov Yu.P., Kochina I.N. Basic Concepts in the Theory of Seepage of Homogeneous Fluids in Fissyrized Rocks. Journal of Applied Mathematics and Mechanics, 1960, vol. 24, no. 4, pp. 1268-1303.
18. Demidenko G.V., Uspenskii S.V. Partial Differential Equations and Systems not Solved with Respect to the Highest Order Derivative. N.Y., Basel, Hong Kong, Marcel Dekker, Inc., 2003.