Abstract
We consider the Stefan problem for a parabolic equation with a small parameter as the coefficient of the derivative with respect to time. We justify the limit transition as the small parameter tends to zero, which enables us to prove the classical solvability of the Hele-Shaw problem with free boundary in the small with respect to time.
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I. I. Danilyuk, “On the Stefan problem,” Usp. Mat. Nauk, 40, No. 5, 133–185 (1985).
E. Di Benedetto and A. Friedman, “The ill-posed Hele-Shaw model and the Stefan problem for supercooled water,” Trans. Amer. Math. Soc., 282, No. 1, 183–203 (1984).
B. V. Bazalii, “Stefan problem for the Laplace equation with regard for the curvature of the free boundary,” Ukr. Mat. Zh., 49, No. 10. 1299–1315 (1997).
O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type [in Russian], Nauka, Moscow (1967).
V. B. Bazalii, “Stefan problem,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 1, 3–7 (1986).
V. B. Bazalii, “On estimates of a solution of one model problem of conjugation in the theory of problems with free boundary,” Diffents. Uravn., 33, No. 10, 1374–1381 (1997).
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50; No. 11, pp. 1452–1462, November, 1998.
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Bazalii, B.V. On one proof of the classical solvability of the Hele-Shaw problem with free boundary. Ukr Math J 50, 1659–1670 (1998). https://doi.org/10.1007/BF02524473
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DOI: https://doi.org/10.1007/BF02524473