Free boundary problems for a class of nonlinear parabolic equations : An approach by the theory of subdifferential operators

Nobuyuki KENMOCHI

doi:10.2969/jmsj/03410001

Nobuyuki KENMOCHI

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JOURNAL FREE ACCESS

1982 Volume 34 Issue 1 Pages 1-13

DOI https://doi.org/10.2969/jmsj/03410001

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Published: 1982 Received: March 03, 1980 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: TITLE Details: Wrong : Free boundary problems for a class of nonlinear parabolic equations
Right : Free boundary problems for a class of nonlinear parabolic equations : An approach by the theory of subdifferential operators

Date of correction: October 20, 2006 Reason for correction: - Correction: SUBTITLE Details: Wrong : An approach by the theory of subdifferential operators

Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) Ph. Bénilan, Equations d'évolution dans un espace de Banach quelconque et applications, Thèse, Publications Mathématiques d'Orsay, 25, Univ. Paris-Sud, 1972.
2) H. Brézis, On some degenerate non-linear parabolic equations, Proc. Symp. Pure Math., Amer. Math. Soc., 18 (1970), 28-38.
3) H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, Mathematical Studies, 5, North-Holland, 1973.
4) J.R. Cannon and C.D. Hill, Existence, uniqueness, stability and monotone dependence in a Stefan problem for the heat equation, J. Math. Mech., 17 (1967), 1-20.
5) A. Damlamian, Some results on the multi-phase Stefan problem, Comm. Partial Differential Equations, 2 (1977), 1017-1044.
6) J. Douglas, Jr., A uniqueness theorem for the solution of a Stefan problem, Proc. Amer. Math. Soc., 8 (1957), 402-408.
7) G. Duvaut, Résolution d'un problème de Stefan (Fusion d'un bloc de glace à zéro degré), C.R. Acad. Sci. Paris, 276 (1973), 1461-1463.
8) A. Fasano and M. Primicerio, General free-boundary problems for the heat equation I, J. Math. Anal. Appl., 57 (1977), 694-723.
9) A. Friedman, Partial differential equations of parabolic type, Prentice-Hall, Englewood Cliffs, N.J., 1964.
10) H. Kawarada, Stefan-type free boundary problems for heat equations, Publ. Res. Inst. Math. Sci., Kyoto University, 9 (1974), 517-533.
11) N. Kenmochi, Some nonlinear parabolic variational inequalities, Israel J. Math., 22 (1975), 304-331.
12) N. Kenmochi, On the quasi-linear heat equation with time-dependent obstacles, Nonlinear Anal., 5 (1981), 71-80.
13) N. Kenmochi, Solvability of nonlinear parabolic equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ., 30 (1981), to appar.
14) Y. Konishi, Some examples of nonlinear semi-groups in Banach lattices, J. Fac. Sci. Univ. Tokyo Sect. IA, 18 (1972), 537-543.
15) W. T. Kyner, An existence and uniqueness theorem for a nonlinear Stefan problem, J. Math. Mech., 8 (1959), 483-498.
16) O.A. Ladyzenskaja, V.A. Solonnikov and N.N. Ural'ceva, Linear and quasi-linear equations of parabolic type, Translations of Mathematical Monographs, 23, Amer. Math. Soc., Providence R.I., 1968.
17) J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod Gauthier-Villars, Paris, 1969.
18) L.I. Rubinstein, The Stefan problem, Translations of Mathematical Monographs, 27, Amer. Math. Soc., Providence R. I., 1971.
19) M. Yamaguchi and T. Nogi, Stefan problem (in Japanese), Sangyo-Tosho, Tokyo, 1977.

Right : [1] Ph. Bénilan, Equations d'évolution dans un espace de Banach quelconque et applications, Thèse, Publications Mathématiques d'Orsay, 25, Univ. Paris-Sud, 1972.
[2] H. Brézis, On some degenerate non-linear parabolic equations, Proc. Symp. Pure Math., Amer. Math. Soc., 18 (1970), 28-38.
[3] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, Mathematical Studies, 5, North-Holland, 1973.
[4] J. R. Cannon and C. D. Hill, Existence, uniqueness, stability and monotone dependence in a Stefan problem for the heat equation, J. Math. Mech., 17 (1967), 1-20.
[5] A. Damlamian, Some results on the multi-phase Stefan problem, Comm. Partial Differential Equations, 2 (1977), 1017-1044.
[6] J. Douglas, Jr., A uniqueness theorem for the solution of a Stefan problem, Proc. Amer. Math. Soc., 8 (1957), 402-408.
[7] G. Duvaut, Résolution d'un problème de Stefan (Fusion d'un bloc de glace à zéro degré), C. R. Acad. Sci. Paris, 276 (1973), 1461-1463.
[8] A. Fasano and M. Primicerio, General free-boundary problems for the heat equation I, J. Math. Anal. Appl., 57 (1977), 694-723.
[9] A. Friedman, Partial differential equations of parabolic type, Prentice-Hall, Englewood Cliffs, N. J., 1964.
[10] H. Kawarada, Stefan-type free boundary problems for heat equations, Publ. Res. Inst. Math. Sci., Kyoto University, 9 (1974), 517-533.
[11] N. Kenmochi, Some nonlinear parabolic variational inequalities, Israel J. Math., 22 (1975), 304-331.
[12] N. Kenmochi, On the quasi-linear heat equation with time-dependent obstacles, Nonlinear Anal., 5 (1981), 71-80.
[13] N. Kenmochi, Solvability of nonlinear parabolic equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ., 30 (1981), to appar.
[14] Y. Konishi, Some examples of nonlinear semi-groups in Banach lattices, J. Fac. Sci. Univ. Tokyo Sect. IA, 18 (1972), 537-543.
[15] W. T. Kyner, An existence and uniqueness theorem for a nonlinear Stefan problem, J. Math. Mech., 8 (1959), 483-498.
[16] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasi-linear equations of parabolic type, Translations of Mathematical Monographs, 23, Amer. Math. Soc., Providence R. I., 1968.
[17] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod Gauthier-Villars, Paris, 1969.
[18] L. I. Rubinstein, The Stefan problem, Translations of Mathematical Monographs, 27, Amer. Math. Soc., Providence R. I., 1971.
[19] M. Yamaguchi and T. Nogi, Stefan problem (in Japanese), Sangyo-Tosho, Tokyo, 1977.

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