We consider a problem without initial condition with free boundary for a parabolic equation with power nonlinearity and prove the uniqueness and existence theorems. The problem is reduced to a Stefan-type problem with initial condition. The equivalence of the problems and bilateral a priori estimates for the required functions are established. The behavior of the free boundary is investigated.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. I. Barenblatt and A. Yu. Ishlinskii, “On the impact of a viscoelastic rod upon a rigid obstacle,” Prikl. Mat. Mekh.,26, No. 3, 497–502 (1962).
S. N. Kruzhkov, “Some problems with unknown boundary for the heat-conduction equation,” Prikl. Mat. Mekh.,31, No. 6, 1009–1020 (1967).
S. N. Kruzhkov and S. Yakubov, “On the solvability of one class of problems with unknown boundary for the heat-conduction equation and the behavior of solutions as time infinitely increases,” Dinam. Sploshn. Sred., Issue 36, 46–70 (1978).
A. Fasano and M. Primicerio, “Viscoplastic impact of a rod on a wall,” Boll. Unione Mat. Ital. Ser. 4,7, No. 3, 531–555 (1975).
J. Takhirov and R. Turaev, “The free boundary problem without initial condition,” J. Math. Sci.,187, No. 1, 86–100 (2012).
M. L. Storm, “Heat conduction in simple metals,” J. Appl. Phys.,22, No. 7, 940–951 (1951).
J. M. Hill and V. G. Hart, “The Stefan problem in nonlinear heat conduction,” J. Appl. Math. Phys.,37, 206–229 (1986).
A. C. Briozzo and M. Natale, “One-dimensional nonlinear Stefan problem in Storm’s materials,” Mathematics,2, 1–11 (2014).
S. De Lillo and M. C. Salvatori, “A two-phase free boundary problem for the nonlinear heat equation,” J. Nonlin. Math. Phys.,11, No. 1, 134–140 (2004).
A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs (1964).
A. M. Meirmanov, Stefan Problem [in Russian], Novosibirsk, Nauka (1986).
L. I. Rubinshtein, Stefan Problem [in Russian], Zvaizgne, Riga (1967).
I. I. Danilyuk, “On the Stefan problem,” Usp. Mat. Nauk, 40, Issue 5(245), 133–185 (1985).
B. V. Bazaliy and A. Friedman, “A free boundary problem for an elliptic-parabolic system: Application to a model of tumor growth,” Comm. Partial Different. Equat.,28, 517–560 (2003).
V. A. Florin, “Consolidation of the ground medium and filtration in the case of variable porosity with regard for the influence of bound water,” Izv. Akad. Nauk SSSR, No. 11, 1625–1649 (1951).
S. De Lillo, G. Lupo, and M. Sommacal, “Half-line solution of a nonlinear heat-conduction problem” Teor. Mat. Fiz.,152, No. 1, 58–65 (2007).
S. N. Kruzhkov, “Nonlinear parabolic equations with two independent variables,” Tr. Mosk. Mat. Obshch.,16, 329–346 (1967).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Neliniini Kolyvannya, Vol. 21, No. 4, pp. 554–566, October–December, 2018.
Rights and permissions
About this article
Cite this article
Takhirov, Z.O. Florin-Type Problem for the Parabolic Equation with Power Nonlinearity. J Math Sci 246, 429–444 (2020). https://doi.org/10.1007/s10958-020-04749-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-020-04749-6