Skip to main content
Log in

Some inverse problems for strong parabolic systems

  • Brief Communications
  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

The questions of well-posedness and approximate solution of inverse problems of finding unknown functions on the right-hand side of a system of parabolic equations are investigated. For the problems considered, theorems on the existence, uniqueness, and stability of a solution are proved and examples that show the exactness of the established theorems are given. Moreover, on the set of well-posedness, the rate of convergence of the method of successive approximations suggested for the approximate solution of the given problems is estimated.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. I. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type [in Russian], Moscow (1967).

  2. A. Friedman, Partial Differential Equations of Parabolic Type, Prentice Hall, Englewood Cliffs, NJ (1964).

    Google Scholar 

  3. A. D. Iskenderov, “On some inverse problems of the determination of the right-hand sides of differential equations,” Izv. Akad. Nauk Azer. SSR, Ser. Fiz.-Tekhn. Mat. Nauk, No. 2, 58–63 (1976).

  4. V. M. Isakov, “On one class of inverse problems for parabolic equations,” Dokl. Akad. Nauk SSSR, 263, No. 6, 1296–1299 (1982).

    MATH  MathSciNet  Google Scholar 

  5. J. R. Cannon and P. Duchateau, “An inverse problem for an unknown source in a heat equation,” J. Math. Anal. Appl., 75, 465–485 (1980).

    Article  MathSciNet  Google Scholar 

  6. A. I. Prilepko and A. D. Kostin, “On some inverse problems for parabolic equations with final and integral observation,” Mat. Sb., 183, No. 4, 49–68 (1992).

    Google Scholar 

  7. V. V. Solov’ev, “On the existence of a solution “on the whole” for the inverse problem of the determination of a source in a quasilinear equation of parabolic type,” Differents. Uravn., 32, No. 4, 536–544 (1996).

    Google Scholar 

  8. E. G. Savateev, “On the problem of the determination of the source function and the coefficient of a parabolic equation,” Dokl. Ross. Akad. Nauk, 344, No. 5, 597–598 (1995).

    MATH  MathSciNet  Google Scholar 

  9. O. Y. Shiyanenko, “On the uniqueness of a solution of one inverse problem for quasilinear heat equation,” Vestn. Mosk. Univ., Ser. 15, No. 3, 5–8 (1999).

  10. P. DuChateau and W. Rundell, “Unicity in an inverse problem for an unknown reaction term in a reaction-diffusion equation,” J. Different. Equat., 59, 155–164 (1985).

    Article  MathSciNet  Google Scholar 

  11. A. Y. Akhundov, “On determination of the right part in semilinear parabolic equation,” Proc. Inst. Math. Mech., 17, 3–9 (2002).

    MATH  MathSciNet  Google Scholar 

  12. A. Y. Akhundov, “A nonlinear parabolic inverse coefficient problem,” Trans. Issue Math. Mech., 22, No. 4, 19–24 (2002).

    MATH  MathSciNet  Google Scholar 

  13. M. M. Lavrent’ev, V. G. Romanov, and S. P. Shishatskii, Ill-Posed Problems in Mathematical Physics and Analysis [in Russian], Moscow (1980).

Download references

Author information

Authors and Affiliations

Authors

Additional information

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 1, pp. 115–124, January, 2006.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Akhundov, A.Y. Some inverse problems for strong parabolic systems. Ukr Math J 58, 127–138 (2006). https://doi.org/10.1007/s11253-006-0055-z

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-006-0055-z

Keywords

Navigation