Inverse  Stefan problem for a parabolic equation with weak power degeneration

N. M. Huzyk

In a free boundary domain there were established conditions of existence and uniqueness of the classical solution to the inverse problem for determination a time-dependent major coefficient in a parabolic equation with weak power degeneration near the derivative with respect to time. The Stefan condition and heat flux are given as overdetermination conditions.

1. B.F. Jones, The determination of a coefficient in a parabolic differential equation. I. Existence and uniqueness, J. Math. Mech., 11 (1962), №5, 907–918.

2. J.R. Cannon, Recovering a time-dependent coefficient in a parabolic differential equation, J. Math. Anal. Appl., 160 (1991), 572–582.

3. H. Azari, C. Li, Y. Nie, S. Shang, Determination of an unknown coefficient in a parabolic inverse problem, Dynamics of Continuous, Discrete and Impulsive Systems. Series A: Math. Analysis, 11 (2004), 665–674.

4. I.B. Bereznytska, A.Y. Drebot, M.I. Ivanchov, Yu.M. Makar, Inverse problem for a heat equation with integral overdetermination, Visnyk Lviv Univ., Ser. Mech-Math., 48 (1997), 71–79. (in Ukrainian)

5. M.I. Ivanchov, Some inverse problems for a heat equation with non-local boundary conditions, Ukr. Math. Journ., 45 (1993), №8, 1066–1071. (in Ukrainian)

6. M.I. Ivanchov, Reduction a free boundary problem for a parabolic equation to inverse problem, Nonlinear bondary problems, 12 (2002) 73–83. (in Ukrainian)

7. I. Barans’ka, Inverse problem with free boundary for a parabolic equation, Mat. Stud., 27 (2007), №1, 85–94. (in Ukrainian)

8. I. Barans’ka, M. Ivanchov, An inverse problem for two-dimentional heat equation in a free boundary domain, Ukr. Math. Bull., 4 (2007), №4, 457–484. (in Ukrainian)

9. H. Snitko, Inverse problem for a parabolic equation with free boundary, Mat. Met. Fiz.-Mekh. Polya, 50 (2007), №4, 7–18. (in Ukrainian)

10. N. Saldina, Inverse problem for a degenerate parabolic equation, Visnyk Lviv Univ., Ser. Mech-Math., 64 (2005), 245–257. (in Ukrainian)

11. M. Ivanchov, N. Saldina, An inverse problem for strongly degenerate heat equation, J. Inv. Ill-Posed Problems, 14 (2006), №5, 465–480.

12. N. Hryntsiv, Solvability of inverse problem for a degenerate parabolic equation in a free boundary domain, Scientific Bulletin of Chernivtsi University, 314-315 (2006), 40–49. (in Ukrainian)

13. M. Ivanchov, N. Hryntsiv, Inverse problem for a weakly degenerate parabolic equation in a domain with free boundary, J. of Math. Sciences., 167 (2010), №1, 16–29.

14. M. Ivanchov, Inverse problems for equations of parabolic type, VNTL Publishers, Lviv, 2003.

	Inverse Stefan problem for a parabolic equation with weak power degeneration(in Ukrainian)
Author	N. M. Huzyk hryntsiv@ukr.net Lviv National University
Abstract	In a free boundary domain there were established conditions of existence and uniqueness of the classical solution to the inverse problem for determination a time-dependent major coefficient in a parabolic equation with weak power degeneration near the derivative with respect to time. The Stefan condition and heat flux are given as overdetermination conditions.
Keywords	coefficient inverse problem; parabolic equation; weak power degeneration; Stefan condition
Reference	1. B.F. Jones, The determination of a coefficient in a parabolic differential equation. I. Existence and uniqueness, J. Math. Mech., 11 (1962), №5, 907–918. 2. J.R. Cannon, Recovering a time-dependent coefficient in a parabolic differential equation, J. Math. Anal. Appl., 160 (1991), 572–582. 3. H. Azari, C. Li, Y. Nie, S. Shang, Determination of an unknown coefficient in a parabolic inverse problem, Dynamics of Continuous, Discrete and Impulsive Systems. Series A: Math. Analysis, 11 (2004), 665–674. 4. I.B. Bereznytska, A.Y. Drebot, M.I. Ivanchov, Yu.M. Makar, Inverse problem for a heat equation with integral overdetermination, Visnyk Lviv Univ., Ser. Mech-Math., 48 (1997), 71–79. (in Ukrainian) 5. M.I. Ivanchov, Some inverse problems for a heat equation with non-local boundary conditions, Ukr. Math. Journ., 45 (1993), №8, 1066–1071. (in Ukrainian) 6. M.I. Ivanchov, Reduction a free boundary problem for a parabolic equation to inverse problem, Nonlinear bondary problems, 12 (2002) 73–83. (in Ukrainian) 7. I. Barans’ka, Inverse problem with free boundary for a parabolic equation, Mat. Stud., 27 (2007), №1, 85–94. (in Ukrainian) 8. I. Barans’ka, M. Ivanchov, An inverse problem for two-dimentional heat equation in a free boundary domain, Ukr. Math. Bull., 4 (2007), №4, 457–484. (in Ukrainian) 9. H. Snitko, Inverse problem for a parabolic equation with free boundary, Mat. Met. Fiz.-Mekh. Polya, 50 (2007), №4, 7–18. (in Ukrainian) 10. N. Saldina, Inverse problem for a degenerate parabolic equation, Visnyk Lviv Univ., Ser. Mech-Math., 64 (2005), 245–257. (in Ukrainian) 11. M. Ivanchov, N. Saldina, An inverse problem for strongly degenerate heat equation, J. Inv. Ill-Posed Problems, 14 (2006), №5, 465–480. 12. N. Hryntsiv, Solvability of inverse problem for a degenerate parabolic equation in a free boundary domain, Scientific Bulletin of Chernivtsi University, 314-315 (2006), 40–49. (in Ukrainian) 13. M. Ivanchov, N. Hryntsiv, Inverse problem for a weakly degenerate parabolic equation in a domain with free boundary, J. of Math. Sciences., 167 (2010), №1, 16–29. 14. M. Ivanchov, Inverse problems for equations of parabolic type, VNTL Publishers, Lviv, 2003.
Pages	182-192
Volume	40
Issue	2
Year	2013
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Inverse Stefan problem for a parabolic equation with weak power degeneration(in Ukrainian)