Volume 10, no. 2Pages 24 - 37

Nonlinear Inverse Problems with Integral Overdetermination for Nonstationary Differential Equations of High Order

A.I. Kozhanov, L.A. Telesheva
Nonlinear inverse coefficient problems for nonstationary higher order differential equations of pseudohyperbolic type are the object of research. More precisely, we study the problems of determining both the solution of the corresponding equation, an unknown coefficient at the solution or at the time derivative of the solution in the equation. A distinctive feature of these problems is the fact that the unknown coefficient is a function of time only. Integral overdetermination is used as an additional condition. We prove the existence theorems of regular solutions (those solutions that have all generalized derivatives in the sense of S.L. Sobolev). The technique of the proof relies on the transition from the original inverse problem to a new direct problem for an auxiliary integral-differential equation, and then on the proof of solvability of the latter and construction of some solution of the original inverse problem from a solution of the auxiliary problem.
Full text
pseudohyperbolic equations of higher order; inverse problem; regular solution; existence.
1. Prilepko A.I., Orlovsky D.G., Vasin I.A. Methods for Solving Inverse Problems in Mathematical Physics. N.-Y., Marcel Dekker, 1999.
2. Denisov A.M. Elements of the Theory of Inverse Problems. Utrecht, VSP, 1999. DOI: 10.1515/9783110943252
3. Kozhanov A.I. Composite Types Equations and Inverse Problems. Utrecht, VSP, 1999. DOI: 10.1515/9783110943276
4. Anikonov Yu.E. Inverse Problems for Kinetic and Other Evolution Equations. Utrecht, VSP, 2001. DOI: 10.1515/9783110940909
5. Lorenzi A. An Introduction to Mathematical Problems via Functional Analysis. Utrecht, VSP, 2001.
6. Romanov V.G. Investigation Methods for Inverse Problems. Utrecht, VSP, 2002. DOI: 10.1515/9783110943849
7. Belov Yu.Ya. Inverse Problems for Partial Differential Equations. Utrecht, VSP, 2002. DOI: 10.1515/9783110944631
8. Lavrentiev M.M. Inverse Problems of Mathematical Physics. Utrecht, VSP, 2003.
9. Ivanchov M. Inverse Problems for Equations of Parabolic Type. Lviv, WNTL Publishers, 2003.
10. Isakov V. Inverse Problems for Partial Differential Equations. N.-Y., Springer, 2006.
11. Kabanihin S.I. Obratnye i nekorrektnye zadachi [Inverse and Ill-Posed Problems]. Novosibirsk, Siberian publishing, 2009.
12. Cannon J.R., Lin Y. Determination of a Parameter p(t) in Some Quasilinear Parabolic Differential Equations. Inverse Problems, 1988, vol. 4, no. 1, pp. 35-45. DOI: 10.1088/0266-5611/4/1/006
13. Ivanchov M.I. Inverse Problem with Free Boundary for Heat Equation. Ukrainian Mathematical Journal, 2003, vol. 55, issue 7, pp. 1086-1098. DOI:10.1023/B:UKMA.0000010607.28568.a7
14. Slodicka M. Determination of a Solely Time-Dependent Source in a Semilinear Parabolic Problem by Means of Boundary Measurements. Journal of Computational and Applied Mathematics, 2015, vol. 289, pp. 433-440. DOI: 10.1016/j.cam.2014.10.004
15. Kozhanov, A.I. Parabolic Equations with an Unknown Time-Dependent Coefficient. Computational Mathematics and Mathematical Physics, 2005, vol. 45, no. 12, pp. 2085-2101.
16. Kozhanov A.I. [On the Solvability of the Inverse Problem of the Occurrence of the Senior Comfect in an Equation of the Compilation Type]. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2008, no. 15 (115), issue 1, pp. 27-36. (in Russian)
17. Kozhanov A.I. [Solvability of Inverse Problems for Recovery of Coefficients in Composite Type Equations]. Journal of Mathematical Sciences, 2008, vol. 8, no. 3, pp. 81-99. (in Russian)
18. Kozhanov A.I. [On the Solvability of Certain Nonlocal and Related Inverse Problems for Parabolic Equations]. Matematicheskie zametki Jakutskogo gosudarstvennogo universiteta, 2011, vol. 18, issue 2, pp. 64-78. (in Russian)
19. Telesheva L.A. [An Inverse Problem for Higher-Order Parabolic Equations: the Case an Unknown Time-Dependent Coefficient]. BSU Bulletin. Series: Mathematics, Informatics. 2010, no. 9, pp. 175-182. (in Russian)
20. Telesheva L.A. [On the Solvability of the Inverse Problem for a High-Order Parabolic Equation with an Unknown Coefficient for the Time Derivative]. Matematicheskie zametki Jakutskogo gosudarstvennogo universiteta, 2011, vol. 18, issue 2, pp. 180-201. (in Russian)
21. Ladyzhenskaya O.A., Ural'tseva N.N. Linear and Quasilinear Elliptic Equations, N.-Y., London, Academic Press, 1968.
22. Jakubov S.Ya. Linejnye differencial'no-operatornye uravnenija i ih prilozhenija [Linear Differential-Operator Equations and Their Applications]. Baku, Elm, 1995.
23. Trenogin B.P. Funkcional'nyj analiz [Functional Analysis]. Moscow, Nauka, 1980.
24. Demidovich B.P. Lekcii po matematicheskoj teorii ustojchivosti [Lectures on the Mathematical Theory of Stability]. Moscow, Fizmatlit, 2002.
25. Amandus N.E., Kozhanov A.I., Shvab I.V. Obyknovennye differencial'nye uravnenija. Ch. 1 [Ordinary Differential Equations. Vol. 1]. Novosibirsk, Novosibirsk State University, 2008.