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Differential Equations

, Volume 54, Issue 5, pp 633–647 | Cite as

Asymptotic Behavior of Solutions of Inverse Problems for Degenerate Parabolic Equations

  • V. L. Kamynin
Partial Differential Equations
  • 11 Downloads

Abstract

We obtain theorems on the proximity as t → +∞ between the solution of the inverse problem for a second-order degenerate parabolic equation with one spatial variable and the solution of the inverse problem for a second-order degenerate ordinary differential equation under an additional integral observation condition. The conditions imposed on the input data admit oscillations of the functions on the right-hand side in the parabolic equation under study.

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References

  1. 1.
    Vasin, I.A. and Kamynin, V.L., On the asymptotic behavior of solutions of inverse problems for parabolic equations, Sib. Math. J., 1997, vol. 38, no. 4, pp. 647–662.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Vasin, I.A. and Kamynin, V.L., Asymptotic behavior of the solutions of inverse problems for parabolic equations with irregular coefficients, Sb. Math., 1997, vol. 188, no. 3, pp. 371–387.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Riganti, R. and Savateev, E., Solution of an inverse problem for the nonlinear heat equation, Commun. Partial Differ. Equations, 1994, vol. 19, no. 9–10, pp. 1611–1628.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Güvenilir, A.F. and Kalantarov, V.K., The asymptotic behavior of solutions to an inverse problem for differential operator equations, Math. Comput. Modelling, 2003, vol. 37, no. 9–10, pp. 907–914.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gözükizil, O.F. and Yaman, M., Long-time behavior of the solutions to inverse problems for parabolic equations, J. Appl. Math. Comput., 2007, vol. 184, no. 2, pp. 669–673.CrossRefzbMATHGoogle Scholar
  6. 6.
    Guidetti, D., Asymptotic expansion of solutions to an inverse problem of parabolic type with nonhomogeneous boundary conditions, Proc. Roy. Soc. Edinburgh. Sect. A, 2011, vol. 141, no. 4, pp. 777–817.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Guidetti, D., Convergence to a stationary state of solutions to inverse problems of parabolic type, Discrete Contin. Dyn. Syst. Ser. S, 2012, vol. 6, no. 3, pp. 711–722.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kamynin, V.L., On the solvability of the inverse problem for a degenerate parabolic equation with integral observation, Theses Intern. Sci. Seminar in Inverse and Ill-Posed Problems (Moscow, November 19–21, 2015), Moscow, 2015, pp. 90–91.Google Scholar
  9. 9.
    Kamynin, V.L., Inverse problem of determining the right-hand side in a degenerate parabolic equation with unbounded coefficients, Comput. Math. Math. Phys., 2017, vol. 57, no. 5, pp. 833–842.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kamynin, V.L., On the solvability of the inverse problem for determining the right-hand side of a degenerate parabolic equation with integral observation, Math. Notes, 2015, vol. 98, no. 5, pp. 765–777.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ladyzhenskaya, O.A., Solonnikov, V.A., and Ural’tseva, N.N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa (Linear and Quasilinear Equations of Parabolic Type), Moscow: Nauka, 1967.zbMATHGoogle Scholar
  12. 12.
    Lyusternik, L.A. and Sobolev, V.I., Kratkii kurs funktsional’nogo analiza (A Short Course of Functional Analysis), Moscow: Vysshaya Shkola, 1982.zbMATHGoogle Scholar
  13. 13.
    Kolmogorov, A.N. and Fomin, S.V., Elementy teorii funktsii i funktsional’nogo analiza (Elements of Function Theory and Functional Analysis), Moscow: Nauka, 1976.zbMATHGoogle Scholar
  14. 14.
    Kamynin, V.L., Asymptotic behavior of solutions to quasilinear parabolic equations in a bounded domain, Sib. Math. J., 1994, vol. 35, no. 2, pp. 305–323.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.National Research Nuclear University “MEPhI,”MoscowRussia

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