Journal of Mathematical Sciences

, Volume 189, Issue 2, pp 274–283 | Cite as

Local solvability of a one-phase problem with free boundary

  • N. Yu. SelivanovaEmail author
  • M. V. Shamolin


A certain one-phase problem with free boundary is studied. The local (in time) solvability of this problem is proved; moreover, the general method elaborated is applied in a more concrete case. For this purpose, a new change of variables and the parametrization of the boundary are introduced, and the problem studied is reduced to a problem in a constant domain.


Free Boundary Hyperbolic System Variable Domain Stefan Problem Concrete Case 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    I. I. Danilyuk, “On Stefan problem,” Usp. Mat. Nauk , 40, No. 5(245), 133–185 (1985).MathSciNetGoogle Scholar
  2. 2.
    N. A. Eltysheva, “On qualitative properties of solutions of certain hyperbolic systems on the plane,” Mat. Sb., 132, No. 2, 186–209 (1988).Google Scholar
  3. 3.
    S. K. Godunov, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1979).Google Scholar
  4. 4.
    E. M. Kartashov, “Analytical methods for solving boundary-value problems of nonstationary heat conduction in domains with moving boundaries,” Izv. Ross. Akad. Nauk, Ser. Energ., No. 5, 3–34 (1999).Google Scholar
  5. 5.
    M. M. Lavrent’ev (Jr) and N. A. Lyul’ko, “Enlarging smoothness of solutions of certain hyperbolic problems,” Sib. Mat. Sh., 38, No. 1, 109–124 (1997).MathSciNetzbMATHGoogle Scholar
  6. 6.
    M. Slemrod, “Dynamics of measure-valued solutions to a backward-forward parabolic equation,”J. Dyn. Differ. Equ., 2, 1–28 (1991).MathSciNetCrossRefGoogle Scholar
  7. 7.
    S. L. Sobolev, “Locally nonequilibrium models of transport processes,” Usp. Fiz. Nauk, 167, No. 10, 1095–1106 (1997).CrossRefGoogle Scholar
  8. 8.
    A. D. Solomon, V. Alexiades, D. G. Wilson, and S. Drake, “On the formulation of hyperbolic Stefan problem,” Quart. Appl. Math., 43, No. 3, 295–304 (1985).MathSciNetzbMATHGoogle Scholar
  9. 9.
    Zh. O. Takhirov, “Two-phases problem with unknown boundaries for a first-order hyperbolic system of equations,” Uzb. Mat. Zh., No. 6, 48–56 (1991).Google Scholar
  10. 10.
    A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1977).Google Scholar
  11. 11.
    A. N. Tikhonov, A. V. Goncharskii, V. V. Stepanov, and A. G. Yagola, Numerical Methods for Solving Ill-Posed Problems [in Russian], Nauka, Moscow (1990).Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.All-Russian Institute for Scientific and Technical InformationRussian Academy of SciencesMoscowRussia
  2. 2.M. V. Lomonosov Moscow State UniversityMoscowRussia

Personalised recommendations