Skip to main content

A modified quasi-boundary value method for a class of abstract parabolic ill-posed problems

Abstract

We study a final value problem for first-order abstract differential equation with positive self-adjoint unbounded operator coefficient. This problem is ill-posed. Perturbing the final condition, we obtain an approximate nonlocal problem depending on a small parameter. We show that the approximate problems are well posed and that their solutions converge if and only if the original problem has a classical solution. We also obtain estimates of the solutions of the approximate problems and a convergence result of these solutions. Finally, we give explicit convergence rates.

[12345678910111213]

References

  1. Ababna M: Regularization by nonlocal conditions of the problem of the control of the initial condition for evolution operator-differential equations. Vestnik Belorusskogo Gosudarstvennogo Universiteta. Seriya 1. Fizika, Matematika, Informatika 1998, (2):60-63, 81.

    Google Scholar 

  2. Ames KA, Payne LE: Asymptotic behavior for two regularizations of the Cauchy problem for the backward heat equation. Mathematical Models & Methods in Applied Sciences 1998,8(1):187-202. 10.1142/S0218202598000093

    Article  MathSciNet  MATH  Google Scholar 

  3. Ames KA, Payne LE, Schaefer PW: Energy and pointwise bounds in some non-standard parabolic problems. Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 2004,134(1):1-9. 10.1017/S0308210500003036

    Article  MathSciNet  MATH  Google Scholar 

  4. Clark GW, Oppenheimer SF: Quasireversibility methods for non-well-posed problems. Electronic Journal of Differential Equations 1994,1994(8):1-9.

    MathSciNet  Google Scholar 

  5. Denche M, Bessila K: A modified quasi-boundary value method for ill-posed problems. Journal of Mathematical Analysis and Applications 2005,301(2):419-426. 10.1016/j.jmaa.2004.08.001

    Article  MathSciNet  MATH  Google Scholar 

  6. Ivanov VK, Mel'nikova IV, Filinkov AI: Operator-Differential Equations and Ill-Posed Problems. Fizmatlit "Nauka", Moscow; 1995:175.

    MATH  Google Scholar 

  7. Lattès R, Lions J-L: Méthode de Quasi-Réversibilité et Applications, Travaux et Recherches Mathématiques, no. 15. Dunod, Paris; 1967:xii+368.

    Google Scholar 

  8. Lavrentiev MM: Some Improperly Posed Problems of Mathematical Physics, Springer Tracts in Natural Philosophy. Volume 11. Springer, Berlin; 1967.

    Book  Google Scholar 

  9. Mel'nikova IV: Regularization of ill-posed differential problems. Sibirskiĭ Matematicheskiĭ Zhurnal 1992,33(2):125-134, 221. translated in Siberian Math. J. 33 (1992), no. 2, 289–298

    MathSciNet  Google Scholar 

  10. Miller K: Stabilized quasi-reversibility and other nearly-best-possible methods for non-well-posed problems. In Symposium on Non-Well-Posed Problems and Logarithmic Convexity (Heriot-Watt Univ., Edinburgh, 1972), Lecture Notes in Mathematics. Volume 316. Springer, Berlin; 1973:161-176. 10.1007/BFb0069627

    Chapter  Google Scholar 

  11. Payne LE: Some general remarks on improperly posed problems for partial differential equations. In Symposium on Non-Well-Posed Problems and Logarithmic Convexity (Heriot-Watt Univ., Edinburgh, 1972), Lecture Notes in Mathematics. Volume 316. Springer, Berlin; 1973:1-30. 10.1007/BFb0069621

    Chapter  Google Scholar 

  12. Showalter RE: The final value problem for evolution equations. Journal of Mathematical Analysis and Applications 1974,47(3):563-572. 10.1016/0022-247X(74)90008-0

    Article  MathSciNet  MATH  Google Scholar 

  13. Showalter RE: Cauchy problem for hyperparabolic partial differential equations. In Trends in the Theory and Practice of Nonlinear Analysis (Arlington, Tex, 1984), North-Holland Math. Stud.. Volume 110. North-Holland, Amsterdam; 1985:421-425.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to M Denche.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Denche, M., Djezzar, S. A modified quasi-boundary value method for a class of abstract parabolic ill-posed problems. Bound Value Probl 2006, 37524 (2006). https://doi.org/10.1155/BVP/2006/37524

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/BVP/2006/37524

Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Equation
  • Convergence Rate