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Differential Equations

, Volume 51, Issue 5, pp 605–619 | Cite as

Inverse problem of finding the coefficient of u in a parabolic equation on the basis of a nonlocal observation condition

  • A. B. KostinEmail author
Partial Differential Equations

Abstract

We consider the problem of reconstructing the coefficient c(x) multiplying u(x, t) in a parabolic equation. To find it, in addition to initial and boundary conditions, we pose a nonlocal observation condition of the form \(\int_0^T {u(x,t)} d\mu (t) = \chi (x)\) where the function χ(x) and the measure dµ(t) are known. We obtain sufficient conditions for the uniqueness and solvability of this problem, which have the form of easy-to-verify inequalities. We present examples of inverse problems for which the assumptions of our theorems are necessarily satisfied and an example of a problem that has a nonunique solution.

Keywords

Inverse Problem Parabolic Equation Monotone Operator Direct Problem Sobolev Embedding Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Prilepko, A.I. and Tikhonov, I.V., Reconstruction of the Inhomogeneous Term in an Abstract EvolutionEquation, Izv. Ross. Akad. Nauk Ser. Mat., 1994, vol. 58, no. 2, pp. 167–188.MathSciNetGoogle Scholar
  2. 2.
    Prilepko, A.I. and Solov’ev, V.V., On the Solvability of Inverse Boundary Value Problems for theDetermination of the Coefficient Multiplying the Lower Derivative in a Parabolic Equation, Differ.Uravn., 1987, vol. 23, no. 1, pp. 136–143.MathSciNetGoogle Scholar
  3. 3.
    Isakov, V.M., Inverse Parabolic Problems with the Final Overdetermination, Comm. Pure Appl. Math.,1991, vol. 44, pp. 185–209.zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Isakov, V.M., Inverse Problems for Partial Differential Equations, New York, 1998.zbMATHCrossRefGoogle Scholar
  5. 5.
    Prilepko, A.I. and Kostin, A.B., Inverse Problems of Determining the Coefficient in a Parabolic Equation.I, Sibirsk. Mat. Zh., 1992, vol. 33, no. 3, pp. 146–155.zbMATHMathSciNetGoogle Scholar
  6. 6.
    Prilepko, A.I. and Kostin, A.B., Inverse Problems of Determining the Coefficient in a Parabolic Equation.II, Sibirsk. Mat. Zh., 1993, vol. 34, no. 5, pp. 147–162.MathSciNetGoogle Scholar
  7. 7.
    Kamynin, V.L. and Kostin, A.B., Two Inverse Problems of Determination of a Coefficient in a ParabolicEquation, Differ. Uravn., 2010, vol. 46, no. 3, pp. 372–383.MathSciNetGoogle Scholar
  8. 8.
    Kamynin, V.L., The Inverse Problem of Determining the Lower-Order Coefficient in Parabolic Equationswith Integral Observation, Mat. Zametki, 2013, vol. 94, no. 2, pp. 207–217.CrossRefGoogle Scholar
  9. 9.
    Ladyzhenskaya, O.A., Solonnikov, V.A., and Ural’tseva, N.N., Lineinye i kvazilineinye uravneniyaparabolicheskogo tipa (Linear and Quasi-Linear Equations of Parabolic Type), Moscow: Nauka, 1967.Google Scholar
  10. 10.
    Kruzhkov, S.N., Nelineinye uravneniya s chastnymi proizvodnymi (Nonlinear Partial Differential Equations),Moscow, 1969.Google Scholar
  11. 11.
    Natanson, I.P., Teoriya funktsii veshchestvennoi peremennoi (Theory of Functions of a Real Variable), Moscow: Nauka, 1974.Google Scholar
  12. 12.
    Kostin, A.B., The Inverse Problem of Reconstructing a Source in a Parabolic Equation from the NonlocalObservation Condition, Mat. Sb., 2013, vol. 204, no. 10, pp. 3–46.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Lieberman, G.M., Second Order Parabolic Differential Equations, Singapore, 2005.Google Scholar
  14. 14.
    Trudinger, N.S., Pointwise Estimates and Quasilinear Parabolic Equations, Comm. Pure Appl. Math.,1968, vol. 21, pp. 205–226.zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Prilepko, A.I. and Solov’ev, V.V., Solvability Theorems and the Rothe Method in Inverse Problems foran Equation of Parabolic Type, Differ. Uravn., 1987, vol. 23, no. 11, pp. 1971–1980.MathSciNetGoogle Scholar
  16. 16.
    Prilepko, A.I. and Kostin, A.B., Some Inverse Problems for Parabolic Equations with Final and IntegralObservation, Mat Sb., 1992, vol. 183, no. 4, pp. 49–68.Google Scholar
  17. 17.
    Solonnikov, V.A., Estimates in Lp of Solutions of Elliptic and Parabolic Systems, Tr. Mat. Inst.Steklova, 1967, vol. 102, pp. 137–160.MathSciNetGoogle Scholar
  18. 18.
    Lyusternik, L.A. and Sobolev, V.I., Kratkii kurs funktsional’nogo analiza (A Short Course of FunctionalAnalysis), Moscow: Vyssh. Shkola, 1982.Google Scholar
  19. 19.
    Ladyzhenskaya, O.A. and Ural’tseva, N.N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa(Linear and Quasilinear Equations of Elliptic Type), Moscow: Nauka, 1973.zbMATHGoogle Scholar
  20. 20.
    Gilbarg, D. and Trudinger, N.S., Elliptic Partial Differential Equations of Second Order, Berlin: Springer-Verlag, 1983. Translated under the title Ellipticheskie differentsial’nye uravneniya s chastnymiproizvodnymi vtorogo poryadka, Moscow: Nauka, 1989.zbMATHCrossRefGoogle Scholar
  21. 21.
    Prilepko, A.I. and Tikhonov, I.V., The Principle of Positivity of a Solution to a Linear Inverse Problemand Its Application to the Heat Conduction Coefficient Problem, Dokl. Akad. Nauk, 1999, vol. 364,no. 1, pp. 21–23.MathSciNetGoogle Scholar
  22. 22.
    Kamynin, L.I. and Khimchenko, B.N., The Strict Extremum Principle for a Weakly Parabolically ConnectedSecond-Order Operator, Zh. Vychisl. Mat. Mat. Fiz., 1981, vol. 21, no. 4, pp. 907–925.zbMATHMathSciNetGoogle Scholar
  23. 23.
    Watson, G., A Treatise on the Theory of Bessel Functions, Cambridge, 1945. Translated under the title Teoriya besselevykh funktsii, Moscow: Inostrannaya Literatura, 1949.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)MoscowRussia

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