Skip to main content
SpringerLink
Log in

Problems with free boundaries for nonlinear parabolic equations

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

We establish necessary conditions for the existence of effects of space localization and stabilization in time that are qualitatively new for evolutionary equations. We suggest constructive methods for the solution of the corresponding one-dimensional problems with free boundaries that appear in ecology and medicine.

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

Access this article

Log in via an institution

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. Yu. A. Mitropol'skii and A. A. Berezovskii, “Stefan problems with limiting stationary state in special electrometallurgy, cryosurgery, and the physics of the sea,” Mat. Fiz. Nelin. Mekh., 7, 50–60 (1987).

    MathSciNet  Google Scholar 

  2. A. A. Berezovskii, “Two-dimensional models of cryodestruction of tissue,” in: Mathematical Simulation of Physical Processes [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1989), pp. 14–38.

    Google Scholar 

  3. N. A. Berezovskii and S. M. Dokuchaeva, “The Stefan problems in the problem of pollution and self-purification of the environment by a point source,” in: Nonlinear Boundary-Value Problems of Mathematical Physics and Their Applications [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1995), pp. 18–21.

    Google Scholar 

  4. M. A. Berezovskii, “Two-dimensional problems with free boundaries for weakly nonlinear parabolic equations,” in: Nonlinear Boundary-Value Problems of Mathematical Physics and Their Applications [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1996), pp. 27–30.

    Google Scholar 

  5. S. L. Sobolev, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  6. A. Friedman, Variational Principles and Free-Boundary Problems, Wiley, New York (1982).

    MATH  Google Scholar 

  7. M. A. Berezovskii, “One-dimensional problems with free boundaries in problems of ecology,” Ukr. Mat. Zh., 49, No. 7, 993–997 (1997).

    Article  MathSciNet  Google Scholar 

  8. Yu. A. Mitropol'skii, A. A. Berezovskii, and S. A. Berezovskii, Problems with Free Boundaries for Nonlinear Evolution Equations in Metallurgy, Medicine, and Ecology [in Russian], Preprint 94.20, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1994).

    Google Scholar 

  9. L. K. Martinson, “Investigation of a mathematical model of the process of nonlinear heat conductance in media with bulk absorption,” in: Mathematical Simulation [in Russian], Nauka, Moscow (1986), pp. 278–303.

    Google Scholar 

  10. A. S. Kalashnikov, “Some problems of the qualitative theory of nonlinear degenerating parabolic equations of the second order,” Usp. Mat. Nauk., 42, Issue 2 (254), 135–303 (1987).

    MathSciNet  Google Scholar 

  11. N. N. Bogolyubov and Yu. A. Mitropol'skii, Asymptotic Methods in the Theory of Nonlinear Oscillations [in Russian] Nauka, Moscow (1974).

    MATH  Google Scholar 

  12. Yu. A. Mitropol'skii, A. A. Berezovskii, and N. P. Konovalova, “Equivalent linearization of systems with distributed parameters,” Ukr. Mat. Zh., 38, No. 4, 464–471 (1986).

    MathSciNet  Google Scholar 

  13. Ph. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II, McGraw-Hill, New York (1953).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev

    Yu. A. Mitropol’skii & A. A. Berezovskii

  2. Kiev University, Kiev

    M. A. Berezovskii

Authors
  1. Yu. A. Mitropol’skii
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. A. Berezovskii
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. A. Berezovskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 10, pp. 1360–1372, October, 1997.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mitropol’skii, Y.A., Berezovskii, A.A. & Berezovskii, M.A. Problems with free boundaries for nonlinear parabolic equations. Ukr Math J 49, 1533–1547 (1997). https://doi.org/10.1007/BF02487439

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02487439

Keywords

Search

Navigation