Skip to main content
Log in

Problem with nonlocal condition on the free boundary

  • Published:
Ukrainian Mathematical Journal Aims and scope

We investigate the one-phase Florin problem for a parabolic equation with nonlocal condition. Theorems on the existence and uniqueness of a solution are proved, and a priori estimates for the solution are obtained.

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

Similar content being viewed by others

References

  1. L. I. Rubinshtein, Stefan Problem [in Russian], Zvaizgne, Riga (1967).

    Google Scholar 

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

    MATH  Google Scholar 

  3. A. M. Meirmanov, Stefan Problem [in Russian], Novosibirsk, Nauka (1986).

    Google Scholar 

  4. I. I. Danilyuk, “On the Stefan problem,” Usp. Mat. Nauk, 40, Issue 5(245), 133–185 (1985).

    MathSciNet  Google Scholar 

  5. B. V. Bazalii and S. P. Degtyarev, “On the classic solvability of a multidimensional problem for the convective motion of a viscous incompressible fluid,” Mat. Sb., 132(174), No. 1, 3–19 (1987).

    Google Scholar 

  6. V. A. Florin, “Compaction of a ground medium and filtration in the case of variable porosity with regard for the influence of bound water,” Izv. Akad. Nauk SSSR, Otdel. Tekhn. Nauk, 11, No. 11, 1625–1649 (1951).

    Google Scholar 

  7. G. I. Barenblatt and A. Yu. Ishlinskii, “On the impact of a viscoelastic rod against a rigid barrier,” Prikl. Mat. Mekh., 26, Issue 3, 497–502 (1962).

    Google Scholar 

  8. S. N. Kruzhkov, “On some problems with unknown boundaries for the heat conduction equation,” Prikl. Mat. Mekh., 31, Issue 6, 1009–1020 (1967).

    MathSciNet  Google Scholar 

  9. N. Cohen and S. J. Rubinov, “Some mathematical topics in biology,” in: Proceedings of the Symposium on Systems Theory, Polytechnic, New York (1965), pp. 321–337.

    Google Scholar 

  10. A. Fasano and M. Primicerio, “New results on some classical parabolic free-boundary problems,” Quart. Appl. Math., 38, No. 4, 439–460 (1981).

    MathSciNet  MATH  Google Scholar 

  11. A. M. Nakhushev, Problems with Shift for Partial Differential Equations [in Russian], Nauka, Moscow (2006).

    Google Scholar 

  12. J. R. Cannon and Hong-Ming Yin, “Non-classical parabolic equations,” J. Different. Equat., 79, 266–288 (1989).

    MathSciNet  MATH  Google Scholar 

  13. M. I. Ivanchov and H. A. Snitko, “Determination of time-dependent coefficients of a parabolic equation in a domain with free boundary,” Nelin. Gran. Zad., 20, 28–44 (2011).

    Google Scholar 

  14. I. E. Barans’ka and M. I. Ivanchov, “Inverse problem for the two-dimensional heat conduction equation in a domain with free boundaries,” Ukr. Mat. Visn., 4, No. 4, 457–484 (2007).

    MathSciNet  Google Scholar 

  15. S. De Lillo and M. Salvatori, “A two-phase free boundary problem for the nonlinear heat equation,” J. Nonlin. Math. Phys., 1, No. 1, 134–140 (2004).

    Article  Google Scholar 

  16. T. D. Dzhuraev and Zh. O. Takhirov, “Nonlinear Florin problem for a quasilinear parabolic equation,” Dokl. Akad. Nauk Resp. Uzb., 1, 3–7 (1998).

    Google Scholar 

  17. Zh. O. Takhirov, “Problem with internal bounded nonlocal conditions for a quasilinear parabolic equation,” Uzb. Mat. Zh., 6, 61–64 (1998).

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 1, pp. 71–80, January, 2012.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Takhirov, Z.O., Turaev, R.N. Problem with nonlocal condition on the free boundary. Ukr Math J 64, 78–88 (2012). https://doi.org/10.1007/s11253-012-0630-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-012-0630-4

Keywords

Navigation