Advertisement

Inverse Parabolic Problems

  • Victor Isakov
Chapter
  • 2k Downloads
Part of the Applied Mathematical Sciences book series (AMS, volume 127)

Abstract

In this chapter, we consider the second-order parabolic equation
$$\displaystyle{ a_{0}\partial _{t}u -\mathrm{div}(a\nabla u) + b \cdot \nabla u + cu = f\,\mathrm{in}\,Q = \varOmega \times (0,T), }$$
where Ω is a bounded domain the space \(\mathbb{R}^{n}\) with the C2-smooth boundary ∂ Ω.

Keywords

Lateral Boundary Data Single Boundary Measurement Parabolic Initial Boundary Value Problem Final Overdetermination Lateral Cauchy 
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.

References

  1. [Be3]
    Belishev, M.I. Recent progress in the boundary control method. Inverse Problems, 23 (2007), R1-R67.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [BI]
    Bouchouev, I., Isakov, V. Uniqueness, stability, and numerical methods for the inverse problem that arises in financial markets. Inverse Problems, 15 (1999), R1–R22.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [BIV]
    Bouchouev, I., Isakov, V., Valdivia, N. Recovery of volatility coefficient by linearization. Quantit. Finance, 2 (2002), 257–263.MathSciNetCrossRefGoogle Scholar
  4. [CoV]
    Colombo, F., Vespri, V. Generations of analytic semigroups in W k, p(Ω) and \(C^{k}(\mathrm{\bar{\varOmega }})\). Differential and Integral Equations 9 (1996), 421–436.MathSciNetGoogle Scholar
  5. [ElI1]
    Elayyan, A., Isakov, V. On Uniqueness of Recovery of the Discontinuous Conductivity Coefficient of a Parabolic Equation. SIAM J. Math. Analysis, 28 (1997), 49–59.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [Fr]
    Friedman, A. Partial Differential Equations of Parabolic Type. Prentice-Hall, 1964.zbMATHGoogle Scholar
  7. [Gu]
    Guidetti, D. Convergence to a Stationary State and Stability for Solutions of Quasilinear Parabolic Equations. Ann. Mat. Pure Appl., CLI (1988), 331–358.Google Scholar
  8. [IY1]
    Imanuvilov, O., Yamamoto, M. Lipschitz stability in inverse parabolic problems by Carleman estimate. Inverse Problems, 14 (1998), 1229–1247.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [Is3]
    Isakov, V. On uniqueness of recovery of a discontinuous conductivity coefficient. Comm. Pure Appl. Math. 41 (1988), 865–877.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [Is4]
    Isakov, V. Inverse Source Problems. Math. Surveys and Monographs Series, Vol. 34, AMS, Providence, R.I., 1990.Google Scholar
  11. [Is6]
    Isakov, V. Inverse Parabolic Problems with the Final Overdetermination. Comm. Pure Appl. Math., 54 (1991), 185–209.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [Is10]
    Isakov, V. On uniqueness in inverse problems for semi linear parabolic equations. Arch. Rat. Mech. Anal., 124 (1993), 1–13.CrossRefzbMATHGoogle Scholar
  13. [Is21]
    Isakov, V. Recovery of time dependent volatility coefficient by linearisation. Evol. Equat. Control Theory, 3 (2014), 119–134.CrossRefzbMATHGoogle Scholar
  14. [J1]
    John, F. Numerical Solution of the heat equation for preceding time. Ann. Mat. Pura Appl., 40 (1955), 129–142.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [J3]
    John, F. Collected papers, vol. 1. Birkhäuser-Verlag, Basel-Boston, 1985.Google Scholar
  16. [KKL]
    Katchalov, A., Kurylev, Y., Lassas, M. Inverse boundary spectral problems. Chapman and Hall-CRC, 2000.zbMATHGoogle Scholar
  17. [Kre]
    Krein, S.G. Linear Differential Equations in Banach Space. Transl. Math. Monogr., 29, AMS, 1971.Google Scholar
  18. [LSU]
    Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’tseva, N.N. Linear and quasilinear equations of parabolic type. Transl. Math. Monogr., 23, AMS, Providence, R.I., 1968.Google Scholar
  19. [PrOV]
    Prilepko, A.I., Orlovskii, D.G., Vasin, I.A. Methods for solving inverse problems in mathematical physics. Marcel Dekker, New York-Basel, 2000.Google Scholar
  20. [PrS]
    Prilepko, A.I., Solov’ev, V.V. Solvability theorems and the Rothe method in inverse problems for an equation of parabolic type, II. Diff. Equat, 23 (1987), 1341–1349.MathSciNetzbMATHGoogle Scholar
  21. [Ti]
    Tikhonov, A.N. Theorèmes d’unicité pour l’équation de la chaleur. Mat. Sborn., 42 (1935), 199–216.zbMATHGoogle Scholar
  22. [Ve]
    Vessella, S. Stability estimates in an Inverse Problem for a Three-Dimensional Heat Equation. SIAM J. Math. Anal., 28 (1997), 1354–1370.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [Yo]
    Yosida, K. Functional Analysis. Springer, 1980.zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Victor Isakov
    • 1
  1. 1.Department of Mathematics and StatisticsWichita State UniversityWichitaUSA

Personalised recommendations